THE ANALYSIS OF HOUSEHOLD SURVEYS THE ANALYSIS OF HOUSEHOLD SURVEYS A Microeconometric Approach to Development Policy Reissue Edition with a New Preface Angus Deaton WORLD BANK GROUP ©2018 International Bank for Reconstruction and Development/The World Bank 1818 H Street NW, Washington, DC 20433 Telephone: 202-473-1000; Internet: www.worldbank.org Some rights reserved 1 234 21 201918 This work is a product of the staff of The World Bank with external contributions. The findings, interpretations, and conclusions expressed in this work do not necessarily reflect the views of The World Bank, its Board of Executive Directors, or the governments they represent. The World Bank does not guarantee the accuracy of the data included in this work. The boundaries, colors, denominations, and other information shown on any map in this work do not imply any judgment on the part of The World Bank concerning the legal status of any territory or the endorsement or acceptance of such boundaries. 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Contents Preface ix Introduction 1 Purpose and intended audience 1 Policy and data: methodological issues 2 Structure and outline 4 Chapter 1: The design and content of household surveys 7 1.1 Survey design 9 Survey frames and coverage 10 Strata and dusters 12 Unequal selection probabilities, weights, and inflation factors 15 Sample design in theory and practice 17 Panel data 18 1.2 The content and quality of survey data 22 Individuals and households 23 Reporting periods 24 Measuring consumption 26 Measuring income 29 1.3 The Living Standards Surveys 32 A brief history 32 Design features ofLSMS surveys 34 l/l//?af have we learned? 35 1.4 Descriptive statistics from survey data 40 Finite populations and superpopulations 40 The sampling variance of the mean 43 Using weights and inflation factors 44 Sampling variation of probability-weighted estimators 49 References in this publication to "Taiwan," "Republic of China," and "Taiwan (China)" refer to the region/Taiwan, China "References to "Hong Kong" refer to the region, "Hong Kong SAR, China." v w THE ANALYSIS OF HOUSEHOLD SURVEYS Stratification 49 Two-stage sampling and dusters 51 A superpopulation approach to clustering 56 Illustrative calculations for Pakistan 57 The bootstrap 58 1.5 Guide to further reading 61 Chapter 2: Econometric issues for survey data 63 2.1 Survey design and regressions 66 Weighting in regressions 67 Recommendations for practice 71 2.2 The econometrics of clustered samples 73 77?e economics of clusters in developing countries 73 Estimating regressions from clustered samples 74 2.3 Heteroskedasticity and quantile regressions 78 Heteroskedasticity in regression analysis 79 Quantile regressions 80 Calculating quantile regressions 83 Heteroskedasticity and limited dependent variable models 85 Robust estimation of censored regression models 89 Radical approaches to censored regressions 91 2.4 Structure and regression in nonexperimental data 92 Simultaneity, feedback, and unobserved heterogeneity 93 Example 1. Prices and quantities in local markets 93 Example 2. Farm size and farm productivity 95 Example 3. The evaluation of projects 97 Example 4. Simultaneity and lags: nutrition and productivity 98 Measurement error 99 Selectivity issues 101 2.5 Panel data 105 Dealing with heterogeneity: difference- and within-estimation 106 Panel data and measurement error 108 Lagged dependent variables and exogeneity in panel data 110 2.6 Instrumental variables 111 Policy evaluation and natural experiments 112 Econometric issues for instrumental variables 115 2.7 Using a time-series of cross-sections 116 Cohort data: an example 117 Cohort data versus panel data 120 Panel data from successive cross sections 121 Decompositions by age, cohort, and year 123 CONTENTS vii 2.8 Two issues in statistical inference 127 Parameter transformations: the delta method 128 Sample size and hypothesis tests 129 2.9 Guide to further reading 131 Chapter 3: Welfare, poverty, and distribution 133 3.1 Living standards, inequality, and poverty 134 Social welfare 134 Inequality and social welfare 136 Measures of inequality 138 Poverty and social welfare 140 The construction of poverty lines 141 Measures of poverty 145 The choice of the individual welfare measure 148 Example 1. Inequality and poverty over time in Cote d'lvoire 151 Example 2: Inequality and poverty by race in South Africa 156 Exploring the welfare distribution: inequality 157 Lorenz curves and inequality in South Africa and Cote d'lvoire 160 Stochastic dominance 162 Exploring the welfare distribution: poverty 164 3.2 Nonparametric methods for estimating densities 169 Estimating univariate densities: histograms 170 Estimating univariate densities: kernel estimators 171 Estimating univariate densities: examples 175 Extensions and alternatives 176 Estimating bivariate densities: examples 180 3.3 Analyzing the distributional effects of policy 182 Rice prices and distribution in Thailand 182 The distributional effects of price changes: theory 183 Implementing the formulas: the production and consumption of rice 186 Nonparametric regression analysis 191 Nonparametric regressions for rice in Thailand 194 Bias in kernel regression: locally weighted regression 197 The distributional effects of the social pension in South Africa 200 3.4 Guide to further reading 202 Chapter 4: Nutrition, children, and intrahousehold allocation 204 4.1 The demand for food and nutrition 206 Welfare measures: economic or nutritional? 206 Nutrition and productivity 210 viii THE ANALYSIS OF HOUSEHOLD SURVEYS The expenditure elasticity of nutrition: background 211 Evidence from India and Pakistan 213 Regression functions and regression slopes for Maharashtra 216 Allowing for household structure 219 The effect of measurement errors 221 4.2 Intra-household allocation and gender bias 223 Gender bias in intrahousehold allocation 224 A theoretical digression 225 Adults, children, and gender 229 Empirical evidence from India 231 Boys versus girls in rural Maharashtra: methodology 234 Standard errors for outlay equivalent ratios 235 Boys versus girls in rural Maharashtra: results 236 Cote d'lvoire, Thailand, Bangladesh, and Taiwan (China) 238 4.3 Equivalence scales: theory and practice 241 Equivalence scales, welfare, and poverty 243 The relevance of household expenditure data 244 Cost-of-living indices, consumers'surplus, and utility theory 245 Calculating the welfare effect of price 246 Equivalence scales, the cost of children, and utility theory 247 The underidentification of equivalence scales 248 Engel's method 251 Rothbarth's method 255 Other models of equivalence scales 260 Economies of scale within the household 262 Utility theory and the identification of economies of scale 268 4.4 Guide to further reading 269 Chapter 5: Looking at price and tax reform 271 5.1 The theory of price and tax reform for developing countries 273 Tax reform 273 Generalizations using shadow prices 277 Evaluation ofnonbehavioral terms 278 Alternative approaches to measuring behavioral responses 279 5.2 The analysis of spatial price variation 283 Regional price data 283 Household price data 283 Unit values and the choice of quality 288 Measurement error in unit values 292 5.3 Modeling the choice of quality and quantity 293 A stripped-down model of demand and unit values 294 CONTENTS ix Modeling quality 296 Estimating the stripped-down model 299 An example from Cote d'lvoire 302 Functional form 303 Quality, quantity, and welfare: cross-price effects 306 Cross-price effects: estimation 311 Completing the system 314 5.4 Empirical results for India and Pakistan 315 Preparatory analysis 316 The first-stage estimates 316 Price responses: the second-stage estimates for Pakistan 317 Price estimates and taste variation, Maharashtra 320 5.5 Looking at price and tax reform 323 Shadow taxes and subsidies in Pakistan 324 Shadow taxes and subsidies in India 325 Adapting the price reform formulas 326 Equity and efficiency in price reform in Pakistan 328 Equity and efficiency in price-reform in India 330 5.6 Price reform: parametric and nonparametric analysis 332 5.7 Guide to further reading 334 Chapter 6: Saving and consumption smoothing 335 6.1 Life-cycle interpretations of saving 337 Age profiles of consumption 339 Consumption and saving by cohorts 342 Estimating a life-cycle model for Taiwan (China) 345 6.2 Short-term consumption smoothing and permanent income 350 Saving and weather variability 351 Saving as a predictor of income change? 354 6.3 Models of saving for poor households 357 The basic model ofintertemporal choice 357 Special cases: the permanent income and life-cycle models 359 Further analysis of the basic model: precautionary saving 361 Restrictions on borrowing 363 Borrowing restrictions and the empirical evidence 369 6.4 Social insurance and consumption 372 Consumption insurance in theory 375 Empirical evidence on consumption insurance 377 6.5 Saving, consumption, and inequality 383 Consumption, permanent income, and inequality 383 Inequality and age: empirical evidence 386 Aging and inequality 390 x THE ANALYSIS OF HOUSEHOLD SURVEYS 6.6 Household saving and policy: a tentative review 393 Motives, consequences, and policy 394 Saving and growth 395 Determinants of saving 397 6.7 Guide to further reading 399 Code appendix 401 Bibliography 439 Subject index 463 Author index 474 Preface It is a pleasure to write a new preface to The Analysis of Household Surveys on its twentieth birthday. It would be better still if this were a preface to a new edi- tion, and I hope that one day I shall write it. In the meantime, I hope that this Preface might be a guide to a new reader, by labeling those parts of the book that seem still relevant, as well as those that would be leading candidates for revision or updating. The origins of the book go back to the early days of the Living Standards Measurement Study (LSMS), which was set up in the World Bank around 1980. As its name suggests, the original idea was to promote household surveys that would enable the better measurement of poverty and of living standards around the world, something that was difficult to do with the data then avail- able. As time went on, and people came and went, the LSMS surveys evolved into multi-purpose tools that would permit not only measurement, but also analysis, permitting a better understanding of how people's lives work, what makes them tick, and why they are as well off or as poorly off as they are. Hence "the analysis" of household surveys, not just the measurement of income, or consumption, or wellbeing. In the original conception, there were to be a series of volumes on different specific topics, with this volume being more general, though with examples of topics that could be covered using the approach. Today, it is hard even to remember how relatively uncommon the analysis of individual household records was. Although there had been important very early studies, including the 1955 book on family budgets by Sigbert Prais and Hendrik Houthakker—which legend has it was the first economic analysis to use an electronic computer—and although micro analysis figured in some of the early textbooks, most economists were trained in econometric methods that explicitly or implicitly focused on aggregate time series. So there was a lot of important and survey-relevant material that remained uncovered. Students who had completed their econometrics courses and turned to household sur- vey data found much that puzzled them. Perhaps the most obvious gap in standard economics training was then (as it largely still is today) the topic of survey design, and how survey design should (if at all) be incorporated into analysis. Survey data come with "weights," related xi xii THE ANALYSIS OF HOUSEHOLD SURVEYS to survey design (for example, they might be the reciprocals of the probabilities of selection), and many generations of economists have had to wonder for themselves what should be done with them. Of course, this is standard material for survey design statisticians, but those who design surveys do not work in the same way economists do and sometimes have different ways of thinking and different objectives. So, when I started writing the book, I wanted to try to understand these issues better for myself, so I went back to the survey litera- ture, and wrote up what I found. Chapter 1 is still one of the few treatments of survey design from an economics perspective. One important change is that, at around the time I was writing, STATA introduced a full suite of survey-design- based econometric software, so it is no longer necessary for analysts to do what I had to do, and write my own code. Thinking about survey design forces the analyst to confront the difference between sample and population, and to think seriously about the population to which the analysis is supposed to apply. For survey samplers, the aim is often the estimation of some characteristic of a finite population, such as the median consumption of Indian families in 2015, something that could be known with certainty from a census, a complete listing of the population. Is this the right way of thinking about a regression analysis? Or should we consider some pos- sible super population, of which the current population is but one possible realization. These issues are important, and are rarely considered. As an exam- ple, even today papers in economics and in health routinely publish standard errors of means calculated from complete enumerations, such as mortality rates. In the same spirit, Chapter 2 was designed as a bridge from what is taught in an econometrics course and what applied economists will find when they confront microeconomic data. For several years, I used this chapter to teach a course at Princeton that helped prepare applied students in labor and develop- ment. When econometrics is taught by specialists, which has the huge advan- tage of providing an overarching statistical framework, it is often useful to work through some of the nitty-gritty issues of practice that are not conceptually interesting, but can make the difference between convincing and unconvinc- ing results. Sometimes this takes the form of warnings, that technical fixes rarely fix anything by themselves, and that techniques, such as panel data methods, which can work magic in ideal conditions, can be undermined by imperfections of various kinds, particularly measurement error. If I were rewriting Chapter 2 today, I would be even more skeptical. As I taught the material over the years, it became clear that many of the uses of instrumental variables and natural experiments that had seemed so compel- ling at first lost a good deal of their luster with time. One problem is the reliance of instrumental variables on exclusion restrictions. The orthogonality of instru- ments to the error term requires that they be uncorrelated with omitted vari- ables so that, when we are interested in the effect of x on y, and z is an instrument, then z can only affect y through its effect on x, and not through any other mechanism. This often seems plausible when an instrument is first PREFACE xiii proposed, but over time, other researchers, or other facts, can make the story much less plausible. There is no general rule here, and some of the studies using natural experiments and instruments have worn well, but that is more the exception than the rule. Twenty years later, I now find myself very much more skeptical about instru- ments in almost any situation. I should also note a mea culpa: the late Tony Atkinson, in his pre-publication review of the book, had noted that instrumen- tal variables hardly ever worked. I should have paid more attention to his views. Natural experiments are a form of instrumental variables. They often give a clean answer, eliminating effects that otherwise would cloud the analysis. Yet the "clean" answer is not always the answer that we want for policy or under- standing. This is one aspect of the familiar trade-off between internal and external validity; a natural experiment is like a laboratory experiment where many factors are held constant, but where we have little idea whether the effect will be replicated in settings that may be more relevant for policy. Of course, a good laboratory experiment tries to isolate some fundamental mechanism that will always be present, even when modified, but such experi- ments require more theory and background knowledge than is usually present in natural experiments in economics. Occasionally such experiments look more like anecdotes than analysis. If I were writing the book today, there would be a new chapter on random- ized controlled trials (RCTs).1 Two decades ago I, like most economists, thought that if only we could do RCTs, life would be straightforward; we could dis- cover the laws of behavior, understand poverty, and eliminate it. And indeed, one of the major developments in applied microeconomics in the last 20 years has been the widespread use of RCTs, particularly in economic development. As always, the practice has brought useful experience with its complement of successes and failures. RCTs often yield new insights and unexpected find- ings. Yet they also have more problems than we anticipated, both in theory and practice. They are not magic tools, any more than panel data or instrumental variables were magic tools. Indeed, once upon a time, economists thought of linear regression as a magic tool, and the history of econometrics teaching has been one in which the great enthusiasm of the early days gave way to a sadder catalog of regression diseases and diagnostics. The same is happening and will continue to happen with RCTs although for sure, and like regressions, they are likely to remain useful tools. But statistical inference with RCTs is much more difficult than it at first appears, causality can rarely be firmly established, the influence of omitted variables is not magically erased by randomization, and the lack of blinding—usually impossible in economics—can undermine estimation in the same way that failure of exclusion restrictions undermines 1. For readers interested in what such a chapter would look like, a good account can be found in my 2018 paper with Nancy Cartwright in Social Science and Medicine, Vol. 210, pp. 2-21: "Understanding and Misunderstanding Randomized Controlled Trials," https://doi.org/10.1016/j .socscimed.2017.12.005. xiv THE ANALYSIS OF HOUSEHOLD SURVEYS instrumental variable estimation. And, as is widely understood, but often for- gotten, the fact that some mechanism works in one place is no guarantee that it works anywhere else. RCTs cannot, by themselves, support a program of unconditional discovery of "what works." In the introduction to the book, I speculate about what can be done in the absence of experiments. While I am now more skeptical about what can be done with experiments, I continue to believe in what I wrote then. The trick is to use the data to tell us something that we didn't know before, and that can help us change our minds, or see things differently. Sometimes this can be done from simple descriptive statistics; that Indian per capita calorie consump- tion was falling during a period of rapid economic growth was an important finding in and of itself, and suggested a whole program of enquiry. That almost half of all children in India were severely malnourished was a finding that then Prime Minister Manmohan Singh described as "a national shame." Such straightforward descriptions can have huge effects on policy, as can correla- tions and regressions. Causal testing can also be supported in the same way through what philoso- phers call the"hypothetico deductive method." Given a hypothesis or an idea, the key is to develop and work it to the point where its implications can be transparently checked on the data. Sometimes, this will take an RCT, but if we are clever and diligent enough, it can often be tested by looking at old data in new ways, from a mean, a median, or a cross-tabulation. The analytical work goes into the drawing out of implications from the hypothesis, not into deriv- ing complex econometric methods or technical fixes. Chapters 3 through 6 are more substantive, though all use econometric methods to generate their findings. All of them could be updated with new results, and in some cases (such as the model of quality choice in Chapter 5), I would modify or be more skeptical about the underlying assumptions. But I believe that these chapters—as well as the STATA code that comes at the end—serve their original purpose of providing worked examples of the sort of analysis that survey data are suited to. The book has been widely used since 1997, it is frequently cited, and it has been downloaded from the Bank's website nearly 34,000 times. I am delighted that it will now be back in print for those who would like to have a "real" copy on their "real" desktops. Angus Deaton Princeton, December 2018 References in this publication to "Taiwan," "Republic of China," and "Taiwan (China)" refer to the region, "Taiwan, China "References to "Hong Kong" refer to the region, "Hong Kong SAR, China." Introduction The collection of household survey data in developing countries is hardly a new phenomenon. The National Sample Survey Organization in India has been col- lecting such data on a regular basis since the 1940s, and there are many other countries with long-running and well-established surveys. Until recently, however, the handling and processing of large microeconomic data sets was both cumber- some and expensive, so that survey data were not widely used beyond the pro- duction of the original survey reports. In the last ten or fifteen years, the availability of cheap and convenient microcomputers has changed both the collection and anal- ysis of household survey data. Calculations that could be done only on multi- million-dollar mainframes in 1980—and then with some difficulty—are now rou- tinely carried out on cheap laptop computers. These same machines can be carried into the field and used to record and edit data as they are provided by the res- pondents. As a result, survey data are becoming available in a more timely fashion, months rather than years after the end of the survey; freshly collected data are much more useful for policy exercises than are those that are many years old. At the same time, analysts have become more interested in exploring ways in which survey data can be used to inform and to improve the policy process. Such ex- plorations run from the tabulations and graphical presentation of levels of living to more basic research on household behavior. Purpose and intended audience This book is about the analysis of household survey data from developing countries and about how such data can be used to cast light on a range of policy issues. Much of the analysis works with household budget data, collected from income and expenditure surveys, though I shall occasionally address topics that require wider information. I shall use data from several different economies to illustrate the analysis, drawing examples of policy issues from economies as diverse as Cote d'lvoire, India, Pakistan, South Africa, Taiwan (China), and Thailand. I shall be concerned with methodology as well as substance, and one of the aims of the book is to bring together the relevant statistical and econometric methods that are useful for building the bridge between data and policy. The book is not intended as a manual for the analysis of survey data—it is hardly possible to reduce policy re- search to a formula—but it does provide a number of illustrations of what can be References in this publication to "Taiwan," "Republic of China," and "Taiwan (China)" refer to the region, "Taiwan, China "References to "Hong Kong" refer to the region, "Hong Kong SAR, China." 1 2 THE ANALYSIS OF HOUSEHOLD SURVEYS done, with fairly detailed explanations of how to do it. Nor can a "how-to" book provide a comprehensive review of all the development topics that have been ad- dressed with household survey data; that purpose has already been largely met by the microeconomic survey papers in the three volumes of the Handbook of Deve- lopment Economics. Instead, I have focused on topics on which I have worked my- self, in the hope that the lack of coverage will be compensated for by the detailed knowledge that can only come from having carried out the empirical research. The restriction to my own work also enables me to provide the relevant computer code for almost all of the empirical results and graphics, something that could hardly be combined with the broad coverage of a genuine survey. The Appendix gives code and programs using STATA; in my experience, this is the most convenient package for working with data from household surveys. The programs are not a package; users will have to substitute their own data sets and will need sufficient basic know- ledge of STATA to adapt the code. Nevertheless, the programs provide a template for generating results similar to those presented and discussed here. I have also tried to keep the programs simple, sometimes at the expense of efficiency or eleg- ance, so that it should not be too difficult to translate the logic into other packages. I hope that the material will be of interest to development practitioners, in the World Bank and elsewhere, as well as to a more academic audience of students of economic development. The material in the first two chapters is also designed to help readers interpret applied econometric work based on survey data. But the audi- ence that I most want to reach is that of researchers in developing countries. Statis- tical offices, research institutes, and universities in developing countries are now much less constrained by computation than they were only a few years ago, and the calculations described here can be done on personal computers using readily avail- able and relatively inexpensive software. I have also tried to keep the technical pre- sentation at a relatively modest level. I take for granted most of what would be familiar from a basic course in econometrics, but I devote a good deal of space to expositions of useful techniques—such as nonparametric density and regression estimation, or the bootstrap—that are neither widely taught in elementary econo- metrics courses nor described in standard texts. Nevertheless, there are points where there is an inevitable conflict between simplicity, on the one hand, and clar- ity and precision, on the other. When necessary I have "starred" those sections or subsections in which the content is either necessarily technical or is of interest only to those who wish to try to replicate the analysis. Occasional "technical notes," usually starred, are shorter digressions that can readily be skipped at a first reading. Policy and data: methodological issues Household surveys provide a rich source of data on economic behavior and its links to policy. They provide information at the level of the individual household about many variables that are either set or influenced by policy, such as prices, transfers, or the provision of schools and clinics. They also collect data on outcomes that we care about and that are affected by the policy variables, such as levels of nutrition, expenditure patterns, educational attainments, earnings, and health. Many impor- INTRODUCTION 3 tant research questions concern the link between the instruments of policy and the outcome variables: the rate of return to government-provided schooling, the effect- iveness of various types of clinics, the equity and efficiency effects of transfers and taxes, and the nutritional benefits of food subsidies. Because household surveys document these links, they are the obvious data bases for this sort of policy re- search, for evaluating the welfare benefits of public programs. Of course, associ- ations in the data establish neither causality nor the magnitude of the effects. The data from household surveys do not come from controlled experiments in which the effects of a "treatment" can be unambiguously and convincingly determined. In recent years, there has been a great deal of interest in social experiments, in- cluding the use of household survey data to evaluate the results of social experi- ments. Nevertheless, experiments are not always possible, and real experiments usually deviate from the ideal in ways that present their own difficulties of inter- pretation. In some cases, good luck, inspiration, and hard work throw up circum- stances or data that allow a clear evaluation of policy effects in the absence of con- trolled experiments; these "quasi" or "natural" experiments have been the source of important findings as well as of some controversy. Even without such solutions, it seems as if it ought to be possible to use standard survey data to say something about the policy effects in which we are interested. A good starting point is to recognize that this will not always be the case. Many policy questions are not readily answerable at all, often because they are not well or sharply enough posed, and even when an answer is available in principle, there is no reason to suppose that it can be inferred from the data that happen to be at hand. Only when this is appreciated is there much chance of progress, or even of a realistic evaluation of what can be accomplished by empirical analysis. Much of the empirical microeconomic literature in development uses economet- ric and statistical methodology to overcome the nonexperimental nature of data. A typical study would begin with a structural model of the process at hand, for example, of the effects on individual health of opening a new clinic. Integral to the model are statistical assumptions that bridge the gap between theory and data and so permit both the estimation of the parameters of the model and the subsequent interpretation of the data in terms of the theory. I have no difficulty with this approach in principle, but often find it hard to defend in practice. The statistical and economic assumptions are often arbitrary and frequently implausible. The econo- metric technique can be complex, so that transparency and easy replicability are lost. It becomes difficult to tell whether the results are genuine features of the data or are consequences of the supporting assumptions. In spite of these problems, I shall spend a good deal of space in Chapter 2 discussing the variety of econometric technique that is available for dealing with nonexperimental data. An understand- ing of these matters is necessary in order to interpret the literature, and it is impor- tant to know the circumstances in which technical fixes are useful. Most of the analysis in this book follows a different approach which recognizes that structural modeling is unlikely to give convincing and clean answers to the policy questions with which we are concerned. Rather than starting with the theory, I more often begin with the data and then try to find elementary procedures for 4 THE ANALYSIS OF HOUSEHOLD SURVEYS describing them in a way that illuminates some aspect of theory or policy. Rather than use the theory to summarize the data through a set of structural parameters, it is sometimes more useful to present features of the data, often through simple descriptive statistics, or through graphical presentations of densities or regression functions, and then to think about whether these features tell us anything useful about the process whereby they were generated. There is no simple prescription for this kind of work. It requires a good deal of thought to try to tease out implications from the theory that can be readily checked against the data. It also requires creative data presentation and processing, so as to create useful and interesting styl- ized facts. But in the end, I believe that we make more progress, not by pretending to estimate structural parameters, but by asking whether our theories and their policy implications are consistent with well-chosen stylized facts. Such facts also provide convenient summaries of the data that serve as a background to discussions of policy. I hope that the examples in this book will make the case that such an approach can be useful, even if its aims are relatively modest. Structure and outline Household budget surveys collect information on who buys what goods and ser- vices and how much they spend on them. Information on how poor people spend their money has been used to describe poverty and to build the case for social reform since the end of the eighteenth century, and household surveys remain the basis for documenting poverty in developing countries today. When surveys are carried out on a regular basis, they can be used to monitor the welfare of various groups in society and to keep track of who benefits and who loses from develop- ment. Large-scale national surveys allow a good deal of disaggregation and allow us to look beyond means to other features of distributions, distinguishing house- holds by occupational, regional, sectoral, and income groups. In most poor countries, a large fraction of government revenue is raised by in- direct taxes on goods and services, and many countries subsidize the prices of com- modities such as basic foodstuffs. Household expenditure surveys, by revealing who buys each good and how much they spend, tell us who pays taxes and who benefits from subsidies. They thus yield a reckoning of the gainers and losers from a proposed changes in taxes and subsidies. When data are collected on the use of services provided by the state, such as health and education, we also discover who benefits from government expenditures, so that survey data can be used to assess policy reform and the effectiveness of government taxation and expenditure. Data from household surveys are also a base for research, for testing theories about household behavior, and for discovering how people respond to changes in the economic environment in which they live. Some recent surveys, particularly the World Bank's Living Standards Measurement Surveys, have attempted to collect data on a wide range of household characteristics and activities, from fertility and physical measurement of weights and heights to all types of economic transactions. Such data allow us to examine all the activities of the household and to trace the behavioral links between economic events and individual welfare. INTRODUCTION 5 This book follows the progression of the previous three paragraphs, from data description through to behavioral analysis. Chapters 1 and 2 are preliminary to the main purpose and are concerned with the collection of household survey data, with survey design, and with its consequences for analysis. Chapter 1 is not meant to provide a guide to constructing surveys in developing countries, but rather to des- cribe those features of survey design that need to be understood in order to under- take appropriate analysis. Chapter 2 discusses the general econometric and statis- tical issues that arise when using survey data for estimation and inference; the tech- niques discussed here are used throughout the rest of the book, but I also attempt to be more general, covering methods that are useful in applications not explicitly considered. This is not a textbook of econometrics; these two chapters are designed for readers with a basic knowledge of econometrics who want some preparation for working with household survey data particularly, but not exclusively, from deve- loping countries. Chapter 3 makes the move toward substantive analysis and discusses the use of survey data to measure welfare, poverty, and distribution. I review the theoretical underpinnings of the various measures of social welfare, inequality, and poverty and show how they can be given empirical content from survey data, with illus- trations from the Ivorian and South African Living Standards Surveys. I highlight a number of techniques for data analysis that have proved useful in policy discus- sions, with particular emphasis on graphical methods for displaying large amounts of data. These methods can be used to investigate the distribution of income, in- equality, and poverty and to examine changes in the levels of living of various groups over time. The chapter also shows how it is possible to use the data to examine the distributional consequences of price changes directly, without having to construct econometric models. These methods are applied to an analysis of the effects of rice price policy on the distribution of real income in Thailand. Chapter 4 discusses the use of household budget data to explore patterns of household demand. I take up the traditional topic of Engel curve analysis in devel- oping countries, looking in particular at the demand for food and nutrition. For many people, nutritional issues are at the heart of poverty questions in developing countries, and Engel curve analysis from survey data allows us to measure the relationship between the elimination of hunger and malnutrition and more general economic development, as captured by increases in real disposable income. This chapter also addresses the closely related question of how goods are allocated with- in the household and the extent to which it is possible to use household data to cast light on the topic. One of the main issues of interest is how different members of the household are treated, especially whether boys are favored over girls. Analyses of the effects of household composition on demand patterns can perhaps shed some light on this, as well as on the old but vexed question of measuring the "costs" of children. In most surveys, larger households have more income and more expenditure, but they also have less income or expenditure on a per capita basis. Does this mean that large households are poorer on average or that small house- holds are poorer on average? The answer depends on whether there are economies of scale to large households—whether two people need twice as much as one—and 6 THE ANALYSIS OF HOUSEHOLD SURVEYS whether children, who are relatively plentiful in larger households, need less money to meet their needs than do adults. This chapter discusses the extent to which the survey data can be used to approach these questions. Chapter 5 is about price reform, its effects on equity and efficiency, and how to measure them. Because surveys provide direct information on how much is con- sumed of each taxed or subsidized good, it is straightforward to calculate the first- round effects of price changes, both on revenue and on the distribution of real income. What are much harder to assess are the behavioral responses to price changes, the degree to which the demand for the good is affected by the change in price, and the extent to which revenues and expenditures from taxes and subsidies on other goods are affected. The chapter discusses methods for estimating price re- sponses using the spatial price variation that is typically quite pronounced in devel- oping countries. These methods are sensitive enough to detect differences in price responses between goods and to establish important cross-price effects between goods, effects that are often large enough to substantially change the conclusions of a policy reform exercise. Reducing a subsidy on one staple food has very different consequences for revenue and for nutrition, depending on whether or not there is a closely substitutable food that is also subsidized or taxed. Chapter 6 is concerned with the role of household consumption and saving in economic development. Household saving is a major component and determinant of saving in most developing countries, and many economists see saving as the wellspring of economic growth, so that encouraging saving becomes a crucial com- ponent of a policy for growth. Others take the view that saving rates respond passively to economic growth, the roots of which must be sought elsewhere. Survey data can be used to explore these alternative views of the relationship between saving and growth, as well as to examine the role that saving plays in protecting living standards against fluctuations in income. The analysis of survey evidence on household saving, although fraught with difficulty, is beginning to change the way that we think about household saving in poor economies. I have benefited from the comments of many people who have given generously of their time to try to improve my exposition, to make substantive suggestions, and in a few cases, to persuade me of the error of my ways. In addition to the referees, I should like to thank—without implicating any of them—Martha Ains worth, Harold Alderman, Tony Atkinson, Dwayne Benjamin, TimBesley, Martin Browning, Kees Burger, Lisa Cameron, David Card, Anne Case, Ken Chay, John Dinardo, Jean Dreze, Eric Edmonds, Mark Gersovitz, Paul Glewwe, Margaret Grosh, Bo Honore, Susan Horton, Hanan Jacoby, Emmanuel Jimenez, Hendrik Juerges, Alan Krueger, Doug Miller, Juan Munoz, Meade Over, Anna Paulson, Menno Pradhan, Gillian Paull, James Powell, Martin Ravallion, Jeremy Rudd, Jim Smith, T. N. Srinivasan, David Stromberg, Alessandro Tarozzi, Duncan Thomas, and Galina Voronov. I owe special thanks to Julie Nelson, whose comments and corrections helped shape Chapter 5, and to Christina Paxson, who is the coauthor of much of the work reported here. Some of the work reported here was supported by grants from the National Institute of Aging and from the John D. and Catherine T. MacArthur Foundation. The book was written for the Policy Research Department of the World Bank. 1 The design and content of household surveys In his splendid essay on early studies of consumer behavior, Stigler (1954, p. 95) tells how the first collectors of family budgets, the Englishmen Reverend David Davies (1795) and Sir Frederick Morton Eden (1797), were "stimulated to this task by the distress of the working classes at this time." Davies used his results to draw attention to the living conditions of the poor, and to argue in favor of a minimum wage. The spread of working-class socialism in Europe in the late 1840s also spawned several compilations of household budgets, including the one of 200 Belgian households by Edouard Ducpetiaux in 1855 that was used two years later by Ernst Engel, not only as the basis for his law that the fraction of the budget devoted to food is larger for poorer families, but also to estimate the aggregate consumption, not of Belgium, but of Saxony! The use of budget data to expose poverty and living standards, to argue for policy reform, and to estimate national aggregates are all topics that are as relevant today as they were two centuries ago. The themes of the research were set very early in the history of the subject. The early investigators had to collect data where they could find it, and there was no attempt to construct representative samples of households. Indeed, the un- derstanding that population totals can be estimated from randomly selected samples and the statistical theory to support such estimation were developed only in the first quarter of this century. Around the turn of the century, Kiaer in Norway and Wright in the United States were among the first to use large-scale representative samples, but the supporting statistical theory was not fully worked out until the 1920s, with Bowley, Ronald Fisher, and Neyman making important contributions. The acceptance of sampling is well illustrated by the case of Rowntree, who was unpersuaded by the reliability of sampling when he undertook his survey of poverty in the city of York in 1936. Having collected a full census, he was later convinced by being able to reproduce most of the results from samples drawn from his data (see the supplementary chapter in Rowntree 1985). One of the first large- scale scientific surveys was carried out by Mahalanobis in Calcutta, who estimated the size of the jute crop in Bengal in 1941 to within 2.8 percent of an independent census at less than 8 percent of the cost—see Mahalanobis (1944, 1946) for the classic early accounts, and Seng (1951) and Casley and Lury (1981, ch. 1) for more history and citations to the early literature. References in this publication to "Taiwan," "Republic of China," and "Taiwan (China)" refer to the region, "Taiwan, China "References to "Hong Kong" refer to the region, "Hong Kong SAR, China." 7 8 THE ANALYSIS OF HOUSEHOLD SURVEYS Modern household surveys begin after World War II. Under the leadership of Mahalanobis at the Indian Statistical Institute in Calcutta, the Indian National Sample Survey (NSS) started the annual collection of household consumption data in 1950. Many other economies, both industrialized and developing, now have regular household consumption surveys, sometimes on an annual basis, as in India until 1973-74, or in Taiwan, The Republic of Korea, Britain, and the United States today, but more often less frequently, as for example in India after 1973-74 (quin- quennially), the United States prior to 1980, and many other countries. These sur- veys were often intended to provide data on poverty and income distribution, for example in the form of frequency distributions of households by levels of living— usually defined by per capita income or consumption—but this was by no means their only purpose. In many cases, the surveys were designed to produce aggregate data, to help complete the national accounts, to provide weights for consumer price indexes, or to provide the basis for projecting demand patterns in planning exer- cises. Once begun, it was typically difficult to change the mode of operation or to use the data for purposes different from those in the original design. The former would generate incompatibilities and inconsistencies in the data, while the latter required a computational capacity and willingness to release household-level data that were rarely in evidence. There are, of course, genuine confidentiality issues with household information, but these can be met by removing some information from the publicly available data, and hardly justify their treatment as state secrets. Recent years have seen a marked change in survey practice, in data collection, and in analysis. Although there are still laggard countries, many government statis- tical offices have become more open with their data and have given bona fide researchers and international organizations access to the individual household rec- ords. Reductions in the real cost of computation have led to more analysis, al- though it is only in the last few years that mass-storage devices and cheap memory have made it convenient to use microcomputers to analyze large data sets. Perhaps as important have been changes in the design of surveys, and there is now a much wider range of survey instruments in use than was the case a decade ago. Following a number of experimental and innovative surveys in the 1960s and 1970s—parti- cularly the Malaysian Family Life Survey in 1976-77—the World Bank's Living Standards Surveys first collected data in Peru and Cote d'lvoire in 1985 and in- corporated important innovations in data collection and in content. Originally de- signed to improve the World Bank's ability to monitor poverty and to make inter- national comparisons of living standards, poverty, and inequality, they evolved into vehicles for collecting comprehensive information on a wide range of household characteristics and activities. The rapid availability and ease of analysis of survey data has led to a productive feedback from analysis to design that was rare prior to 1980. In consequence, survey practice and questionnaire design are probably changing more rapidly now than ever before. This chapter and the next, which are preliminary to the analytical studies in the rest of the book, are concerned with the design and content of household surveys (this chapter) and with its implications for statistical and econometric analysis (the final section of this chapter and Chapter 2). In line with the substantive studies later THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 9 in the book, I give disproportionate attention to income and expenditure surveys, or to the income and expenditure sections of broader, integrated surveys such as the Living Standards Surveys. Even so, much of the discussion carries over to other types of household survey, for example to employment or fertility surveys, though I do not give explicit attention to those topics, The four sections of this chapter are concerned with the design of surveys, with the type of data that they collect, and with the effect of design on the calculation of descriptive statistics such as means. Section 1.1 discusses the practical and statis- tical issues concerned with choosing households for inclusion into a survey. Sec- tion 1.2 is concerned with the types of data that are usually collected, and their likely quality. Section 1.3 focusses on the particular features of the Living Stand- ards Surveys from which data are used in some of the later chapters. Many of the policy analyses that use household survey data were not contemplated when the surveys were designed, so that mechanical calculations that ignore the design of the survey can produce unpredictable results. For example, surveys that are designed to estimate means or population totals may be quite unsuitable for measuring dis- persion. In most surveys some types of households are overrepresented relative to their share in the population, while others are underrepresented, so that corrections have to be made to calculate genuinely representative totals. It is also wise to be sensitive to the possibility—in most cases the certainty—of measurement errors, to their effects on the calculations, and to strategies that can be used to protect inference in their presence. These issues are further complicated when, as in some of the Living Standards Surveys, households are observed on more than one occa- sion, and we are interested in analyzing changes in behavior over time. Section 1.4, which is more technical than the others, presents some of the most useful formulas for estimating means and their sampling variability taking into account the survey design. The discussion is useful both for Chapter 2, where I move from descriptive statistics to a more econometric approach, and for Chapter 3, where I deal with poverty measures, which are a particular kind of descriptive statistic. This section also contains a brief introduction to the bootstrap, a technique that is often useful for calculating standard errors and confidence intervals. 1.1 Survey design The simplest household survey would be one where there exists a reliable, up-to- date list of all households in the population, where the design assigns an equal probability to each household being selected from the list to participate in the survey and where, in the implementation stage, all households asked to participate actually do so. The sample would then be a simple random sample, with each household standing proxy for an equal number of households in the population. Such samples are easy to use and a few actual surveys approximate this simple structure. However, for a number of good and some not-so-good reasons, most surveys are a good deal more complex. I begin by discussing the list (or frame) from which households are selected and which defines the potential coverage of the survey, and then pass on to stratification and sampling issues. 10 THE ANALYSIS OF HOUSEHOLD SURVEYS Survey frames and coverage A typical household survey collects data on a national sample of households, ran- domly selected from a "frame" or national list of households. Sample sizes vary widely depending on the purpose of the survey, on the size of the population in the country being surveyed, and on the degree to which regional or other special sub- samples are required. Sample sizes of around 10,000 are frequently encountered, which would correspond to a sampling fraction of 1:500 in a population of 5 million households, or perhaps 25 million people. Since the accuracy of sample statistics increases less than proportionally with the sample size—usually in propor- tion to its square root—sampling fractions are typically smaller in larger popula- tions, a tendency that is reinforced by limits on the size of survey that can be mounted by many data collection agencies. Nevertheless, there are some very large surveys such as the current Indian NSS, where a full national sample contains around a quarter of a million households. The frame is often a census, which in principle provides a list of all households and household members, or at least of all dwellings. However, there are many countries where there is no up-to-date census, or no reliable recent census, so that other frames have to be constructed, usually from administrative records of some kind (see Casley and Lury 1981, ch. 6, who discuss some of the possibilities). Per- haps the most common method of selecting households from the frame uses a two- stage design. At the first stage, selection is from a list of "clusters" of households, with the households themselves selected at the second stage. In rural areas, the clusters are often villages but the choice will depend on the frame. Censuses have their own subunits that are suitable for first-stag'*, sampling. Once the clusters are chosen, households can be selected directly if an up-to-date list is available, and if the list is detailed enough to allow identification in the field. Otherwise, all house- holds in the selected clusters can be listed prior to the second stage. Since it is often possible to include some household information at the listing stage, the procedure allows the second-stage selection of individual households to be informed by prior knowledge, a possibility to which I shall return in the next subsection. Note finally that two-stage sampling is not inconsistent with each household in the population having an equal chance of selection into the sample. In particular, if clusters are randomly selected with probability proportional to the number of households they contain, and if the same number of households is selected from each cluster, we have a self-weighting design in which each household has the same chance of being included in the survey. The use of outdated or otherwise inaccurate frames is an important source of error in survey estimates. It should also be noted that in some countries—including the United States—censuses are politically sensitive so that various interest groups can be expected to try to interfere with the count. Even when the frame is accurate in itself, its coverage of the population will typically not be complete. Homeless people are automatically excluded from surveys that start from households, and in many countries people living in various institutional settings—the armed forces or workers* dormitories—will be excluded. THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 1 1 One example of the differences between a sample and population is provided by the data in Figure 1.1, which shows age-sex pyramids for Taiwan for selected years between 1976 and 1990. There are two pyramids for each year; those on the left-hand side were calculated from household survey data, and were previously reported in Deaton and Paxson (1994a), while those on the right-hand side are cal- culated from the official population data in Republic of China (1992). The survey data, which are described in detail in Republic of China (1989), come from a set of surveys that have been carried out on a regular basis since 1976, and that are carefully and professionally conducted. The 1976 sample has data on some 50,000 individuals, while the later years cover approximately 75,000 persons; the popu- lation of Taiwan grew from 16.1 million in 1975 to 20.4 million in 1990. The differences between the two sets of pyramids is partly due to sampling error—each year of age is shown in the graphs—but there are also a number of dif- ferences in coverage. The most obvious of these causes the notches in the sample pyramids for men aged 18 to 20. Taiwanese men serve in the military during those years and are typically not captured in the survey, and roughly two-thirds of the age group is missing. These notches tend to obscure what is one of the major common features of both population and sample, the baby boom of the early 1950s. In 1988 and 1990, there is some evidence that the survey is missing young women, although the feature is much less sharp than for men and is spread over a wider age range. The design feature in this case is that the survey does not include women attending college nor those living in factory dormitories away from home. As the Figure 1.1. Age and sex pyramids for survey data and population, Taiwan (China), selected years, 1976-90 Fraction of population in age-sex group Source: Author's calculations using survey data tapes, and Republic of China (1992). 12 THE ANALYSIS OF HOUSEHOLD SURVEYS population pyramids make clear, some of these women—together with the men in the same age group—are genuinely "missing" in the sense that the cohort of babies born around 1975 is substantially smaller than those immediately preceding or suc- ceeding it. A number of other distinctive features of these graphs are not design effects, the most notable being the excess of men over women that is greatest at around age 45 in 1976, and moves up the age distribution, one year per year, until it peaks near age 60 in 1990. These men are the survivors of Chiang Kai-shek's army who came to Taiwan after their defeat by the communists in 1949. The noncoverage of some of the population is typical of household surveys and clearly does not prevent us from using the data to make inferences. Nevertheless, it is always wise to be careful, since the missing people were not missed at random and will typically have different characteristics from the population as a whole. In the Taiwanese case, we should be careful not to infer anything about the behavior of young Taiwanese males. Another notable example comes from Britain, where the annual Family Expenditure Survey (FES) regularly underestimates aggregate alcohol consumption by nearly a half. Much of the error is attributed to coverage; there is high alcohol consumption among many who are excluded from the survey, primarily the military, but also innkeepers and publicans (see Kemsley and others 1980). To the effects of noncoverage by design can be added the effects of non- respondents, households that refuse to join the survey. Nonresponse is much less of a problem in developing countries than in (for example) the United States, where refusal to participate in surveys has been increasing over time. Although many surveys in developing countries report almost complete cooperation, there will al- ways be specific cases of difficulty, as when wealthy households are asked about incomes or assets, or when households are approached when they are preoccupied with other activities. Once again, some of the low alcohol reports in the British data reflect the relatively low response rates around Christmas, when alcohol consump- tion is highest (see Crooks 1989, pp. 39-44). It is sometimes possible to study sur- vey nonresponse patterns by tracing refusals in a contemporaneous census, using data from the census to assess the determinants of refusal in the survey (see Kemsley 1975 for such an exercise for the British FES using the 1971 census). Sometimes the survey itself will collect some information about nonrespondents, for example about housing. Groves (1989, ch. 4) discusses these and other tech- niques for assessing the consequences of nonresponse. Strata and clusters A two-stage sample design, first selecting clusters and then households, generates a sample in which sample households are not randomly distributed over space, but are geographically grouped. This arrangement has a number of advantages beyond the selection procedure. It is cost-effective for the survey team to travel from village to village, spending substantial time in each, instead of having to visit households that are widely dispersed from one another. Clustered samples also fa- cilitate repeat visits to collect information from respondents who may not have been present at the first visit, to monitor the progress of record keeping, or to ask THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 13 supplementary questions about previous responses that editing procedures have marked as suspect. That there are several households in each village also makes it worthwhile to collect village-level information, for example on schools, clinics, prices, or agroclimatic conditions such as rainfall or crop failures. (Though the clusters defined for statistical purposes will not always correspond to well-defined "communities.") For all these reasons, nearly all surveys in developing countries (and elsewhere, with telephone surveys the notable exception) use clustered samples. The purposes of the survey sometimes dictate that some groups be more in- tensively sampled than others, and more often that coverage be guaranteed for some groups. There may be an interest in investigating a "target" group that is of particular concern, and, if members of the group are relatively rare in the popu- lation as a whole, a simple random sample is unlikely to include enough group members to permit analysis. Instead, the sample is designed so that households with the relevant characteristic have a high probability of being selected. For example, the World Bank used a Living Standards-type survey in the Kagera region of Tan- zania to study the economic effects of AIDS. A random sample of the population would not produce very many households with an infected person, so that care was taken to find such households by confining the survey to areas where infection was known to be high and by including questions about sickness at the listing stage, so that households with a previous history of sickness could be oversampled. More commonly, the survey is required to generate statistics for population subgroups defined (for example) by geographical area, by ethnic affiliation, or by levels of living. Stratification by these groups effectively converts a sample from one population into a sample from many populations, a single survey into several surveys, and guarantees in advance that there will be enough observations to permit estimates for each of the groups. There are also statistical reasons for departing from simple random samples; quite apart from cost considerations, the precision of any given estimate can be en- hanced by choosing an appropriate design. The fundamental idea is that the survey- or typically knows a great deal about the population under study prior to the sur- vey, and the use of that prior information can improve the efficiency of statistical inference about quantities that are unknown. Stratification is the classic example. Suppose that we are interested in estimating average income, that we know that average rural incomes are lower than average urban incomes, and we know the proportions of the population in each sector. A stratified survey would be two identical surveys, one rural and one urban, each of which estimates average in- come. (It would not necessarily be the case that the sampling fractions would be the same in each stratum.) The average income for the country as a whole, which is the quantity in which we are interested, is calculated by weighting together the urban and rural means using the proportions of the population in each as weights—which is where the prior information comes in. The precision of this combined estimate is assessed (inversely) from its variance over replications of the survey. Because the two components of the survey are independent, the variance of the overall mean is the sum of the variances of the estimates from each strata. Hence, variance de- 14 THE ANALYSIS OF HOUSEHOLD SURVEYS pends only on vwWn-sector variance, and not on between-scctor variance. If instead of a stratified survey, we had collected a simple random sample, the vari- ance of the overall mean would still have depended on the within-sector variances, but there would have been an additional component coming from the fact that in different surveys, there would have been different fractions of the sample in rural and urban. If rural and urban means are different, this variability in the composition of the sample will contribute to the variability of the estimate of the overall mean. In consequence, stratification will have the largest effect in reducing variance when the stratum means are different from one another, and when there is relatively little variation within strata. The formulas that make this intuition precise are discussed in Section IA below. In household income and expenditure surveys, rural and urban strata are nearly always distinguished, and sometimes there is additional geographic stratification, by regions or provinces, or by large and small towns. Ethnicity is another possible candidate for stratification, as is income or its correlates if, as is often the case, some indication of household living standards is included in the frame or in the listing of households—landholdings and housing indicators are the most frequent examples. Stratification can be done explicitly, as discussed above, or "implicitly." The latter arises using "systematic" sampling in which a list of households is sampled by selecting a random starting point and then sampling every yth house- hold thereafter, with j set so as to give the desired sample size. Implicit stratifica- tion is introduced by choosing the order in which households appear on the list. An example is probably the best way to see how this works. In the 1993 South African Living Standards Survey, a list was made of clusters, in this case "census enumer- ator subdistricts" from the 1991 census. These clusters were split by statistical region and by urban and rural sectors—the explicit stratification—and then in order of percentage African—the implicit stratification. Given that the selection of clusters was randomized only by the random starting point, the implicit stratifica- tion guarantees the coverage of Africans and non-Africans, since it is impossible for a sample so selected not to contain clusters from high on the list, which are almost all African, and clusters low on the list, which are almost all non-African. While stratification will typically enhance the precision of sampling estimates, the clustering of the sample will usually reduce it. The reason is that households living in the same cluster are usually more similar to one another in behavior and characteristics than are households living in different clusters. This similarity is likely to be more pronounced in rural areas, where people living in the same village share the same agroclimatic conditions, face similar prices, and may belong to the same ethnic or tribal group. As a result, when we sample several households from the same cluster, we do not get as much information as we would from sampling several households from different clusters. In the (absurd) limit, if everyone in the same cluster were replicas or clones of one another, the effective sample size of the survey would not be the number of households, but the number of clusters. More generally, the precision of an estimate will depend on the correlation within the cluster of the quantity being measured; once again, the formulas are given in Section 1.4 below. THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 15 A useful concept in assessing how the sample design affects precision is Kish's (1965) "design effect," often referred to as deff. Deffis defined as the ratio of the variance of an estimate to the variance that it would have had under simple random sampling; some explicit examples are included in Section 1.4. Stratification tends to reduce dejff below one, while clustering tends to increase it above one. Estimates of the means of most variables in stratified clustered samples have deffs that are greater than one (Groves 1989, ch. 6), so that in survey design the practical conven- ience and cost considerations of clustering usually predominate over the search for variance-reduction. Unequal selection probabilities, weights, and inflation factors As we have seen, it is possible for a survey to be stratified and clustered, and for each household in the population to have an equal probability of inclusion in the sample. However, it is more common for probabilities of inclusion to differ, be- cause it costs more to sample some households than others, because differential probabilities of inclusion can enhance precision, and because some types of house- holds may be more likely to refuse to participate in the survey. Because noncoop- eration is rarely taken into account in design, even samples that are meant to have equal probabilities of selection often do not do so in practice. Variation in costs is common, for example between rural and urban households. In consequence, the cost of any given level of precision is minimized by a sample in which urban households are overrepresented and rural households underrepre- sented. The use of differential selection probabilities to enhance precision is per- haps less obvious, but the general principle is the same as for stratification, that prior information can be used to tell us where to focus measurement. To fix ideas, suppose again that we are estimating mean income. The estimate will be more precise if households that contribute a large amount to the mean—high-income households—are overrepresented relative to low-income households, who contri- bute little. This is "probability proportional to size," orp.p.s., sampling. Of course, we do not know household income, or we would not have to collect data, but we may have information on correlated variables, such as landholdings or household size. Overrepresentation of large households or large landholding households will typically lead to more precise estimates of mean income (see again Section 1.4 for formulas and justification). When selection probabilities differ across households, each household in the survey stands proxy for or represents a different number of households in the popu- lation. In consequence, when the sample is used to calculate estimates for the popu- lation, it is necessary to weight the sample data to ensure that each group of households is properly represented. Sample means will not be unbiased estimates of population means and we must calculate weighted averages so as to "undo" the sample design and obtain estimates to match the population. The rule here is to weight according to the reciprocals of sampling probabilities because households with low (high) probabilities of selection stand proxy for large (small) numbers of households in the population. These weights are often referred to as "raising" or 16 THE ANALYSIS OF HOUSEHOLD SURVEYS "inflation" factors because if we multiply each observation by its inflation factor we are estimating the total for all households represented by the sample household, and the sum of these products over all sample households is an estimate of the population total. Inflation factors are typically included in the data sets together with other variables. Table 1.1 shows means and standard deviations by race of the inflation factors for the 1993 South African survey. This is an interesting case because the original design was a self-weighting one, in which there would be no variation in inflation factors across households. However, when the survey was implemented there were substantial differences by race in refusal rates, and there were a few clusters that could not be visited because of political violence. As a result, and in order to allow the calculation of unbiased estimates of means, inflation factors had to be introduced after the completion of the fieldwork. The mean weight for the 8,848 households in the survey is 964, corresponding to a population of households of 8,530,808 (= 8,848 x 964). Because whites were more likely to refuse to participate in the survey, they attract a higher weight than the other groups. This South African case illustrates an important general point about survey weights. Differences in weights from one household to another can come from dif- ferent probabilities of selection by design, or from different probabilities by accid- ent, because the survey did not conform to the design, because of non-response, because households who cooperated in the past refused to do so again, or because some part of the survey was not in fact implemented. Whether by design or accid- ent, there are ex post differences in sampling probabilities for different households, and weights are needed in order to obtain accurate measures of population quan- tities. But the design weights are, by construction, the reciprocals of the sampling probabilities, and are thus controlled in a way that accidental weights are not. Weights that are added to the survey ex post do not have the same pedigree, and are often determined by judgement and modeling. In South Africa, the response rate among White households was lower, so the weights for White households were ad- justed upwards. But can we be sure that the response rate was truly determined by race, and not, for example, by some mixture of race, income, and location? Adop- tion of survey weights often involves the implicit acceptance of modeling decisions by survey staff, decisions that many investigators would prefer to keep to them- Table 1.1. Inflation factors and race, South Africa, 1993 Standard Households in Race Mean weight deviation sample Blacks 933 19 6,533 Coloreds 955 55 690 Asians 885 22 258 Whites 1,135 219 1,367 All 964 133 8,848 Source: Author's calculations using the South African Living Standards Survey, 1993. THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 17 selves. At the least, survey reports should document the construction of such weights, so that other researchers can make different decisions if they wish. Sample design in theory and practice The statistical arguments for stratification and differential sampling probabilities are typically less compelling in developing-country surveys than are the practical arguments. Optimal design for precision works well when the aim of the survey is the measurement of a single magnitude—average consumption, average income, or whatever. Once this objective is set, all the tools of the sample survey statistician can be brought to bear to design a survey that will deliver the best estimate at the lowest possible cost. Such single-purpose surveys do indeed occur from time to time and more frequently there is a main purpose, such as the estimation of weights for a consumer price index, or the measurement of poverty and inequality. Even in these cases, however, it is recognized that there are other uses for the data, and in general-purpose household surveys there is a range of possible applications, each of which would mandate a different design. Precision for one variable is impreci- sion for another, and it makes no sense to design a survey for each. In addition, optimizing for one purpose can make it difficult to use the survey for other pur- poses. A good example is the Consumer Expenditure Survey in the United States, where the main aim is the calculation of weights for the consumer price index. That object is relentlessly pursued, with some expenditures obtained by interviewing some households, some expenditures obtained by diary from other households, and each household is visited five times over fifteen months but with different kinds of data collected at each visit. All of this allows a relatively small sample to deliver good estimates of the average American spending pattern, but the complexity of the design makes it difficult—sometimes even impossible—to make calculations that would have been possible under simpler designs. Another problem with optimal schemes is that the selection of households according to efficiency criteria can compromise the usefulness of the data. For ex- ample, the use of public transport is efficiently estimated by interviewing travelers, and travelers are most easily and economically found by conducting "on-board" surveys on trains, on buses, or at stations. But if we are to study what determines the demand for travel, and who benefits from state subsidies to public transport, we need to know about nontravelers too, information that is better collected in standard household surveys. Indeed, if observations are selected into the sample according to characteristics that are correlated with the magnitude being studied—precisely the recipe inp.p.s. sampling—attempts to estimate models that explain that magni- tude are likely to be compromised by the selection of the sample. This "choice- based sampling" problem has been studied in the literature (see Manski and Ler- man 1977, Hausman and Wise 1977, 1981, and Cosslett 1993) and there exist tech- niques for overcoming the difficulties. But once again it is much easier to work with a simpler survey, and the results are likely to be more comprehensible and more convincing if they do not require complex corrections, especially when the corrections are supported by assumptions that are difficult or impossible to check. 18 THE ANALYSIS OF HOUSEHOLD SURVEYS There are also good practical reasons for straightforward designs. In their book on collecting data in developing countries, Casley and Lury (1981, p. 2) summarize their basic message in the words "keep it simple." As they point out: The sampling errors of any rational design involving at least a moderate sample size are likely to be substantially smaller than the nonsampling errors. Complications of design may create problems, resulting in larger nonsampling errors, which more than offset the theoretical benefits conferred. As we shall see, the econometric analysis will have to deal with a great many problems, among which nonsampling errors are not the least important. Correction for complex designs is an additional task that is better avoided whenever possible. Panel data The standard cross-sectional household survey is a one-time affair and is designed to obtain a snapshot of a representative group of households at a given moment in time. Although such surveys take time to collect (frequently a year) so that the "moment in time" varies from household to household, and although households are sometimes visited more than once, for example to gather information on income during different agricultural seasons, the aim of the survey is to gather information from each household about a given year's income, or about consump- tion in the month previous to the interview, or about the names, sexes, and ages of the members of the household on the day of the interview. By contrast, longitudinal or panel surveys track households over time, and collect multiple observations on the same household. For example, instead of gath- ering income for one year, a panel would collect data on income for a number of years, so that, using such data, it is possible to see how survey magnitudes change for individual households. Thus, the great attraction of panel data is that they can be used to study dynamics for individual households, including the dynamics of living standards. They can be used to address such issues as the persistence of poverty, and to see who benefits and who loses from general economic develop- ment, or who gains and loses from a specific shock or policy change, such as a devaluation, a structural adjustment package, or a reduction in the prices of com- modity exports. However, as we shall see in Chapter 2, panel data are not required to track outcomes or behavior for groups of individuals—that can be done very well with repeated cross-sectional surveys—but they are the only data that can tell us about dynamics at the individual level. Panel surveys are relatively rare in gener- al, and particularly so in developing countries. The panel that has attracted the greatest attention in the United States is the Michigan Panel Study of Income Dynamics (PSID), which has been following the members of about 4,800 original households since 1968. The most widely used panel data from a developing coun- try come from the Institute for Crop Research in the Semi-Arid Tropics (ICRISAT) in Hyderabad, India, which followed some 40 agricultural households in each of six villages in southwestern India for five or ten years between 1975 and 1985. THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 19 These long-standing surveys are not the only way in which panel data can be collected; an alternative is a rotating panel design in which some fraction of households is held over to be revisited, with the rest dropped and replaced by new households. Several of the Living Standards Surveys—to be discussed in Section 1.3 below—have adopted such a design. For example, in Cote d'lvoire, 1,600 households were selected into the 1985 survey, 800 of which were retained in 1986. To these original panelists 800 new households were added in 1986, and these were retained into 1987. By the pattern of rotation, no household is observed for more than two years, so that while we have two observations in successive years on each household (apart from half of the start-up households) we do not get the long-term observation of individual households that come from panel data. A third way of collecting data is to supplement cross-sectional data. Occasion- ally this can be done by merging administrative and survey data. More often, an earlier cross-sectional survey is used as the basis for revisiting households some years after the original survey. If records have been adequately preserved, this can be done even when there was no intention in the original survey of collecting panel data; indeed, it is good practice to design any household survey so as to maximize the probability of recontacting the original respondents. Such methods have been successful in a number of instances. Although the Peruvian Living Standards Sur- vey of 1985-86 was designed as a cross section, households living in Lima were revisited in 1990; of the 1,280 dwellings in the original survey, 1,052 were reinter- viewed (some dwellings no longer existed, or the occupants refused to cooperate) and, of these, 745 were occupied by the same family (Glewwe and Hall 1995). In 1988-89, RAND carried out a successful reinterview of nearly three-quarters of the individuals in the original 1976-77 Malaysian Family Life Survey (Haaga, Da- Vanzo, Peterson, and Peng 1994). Bevan, Collier, and Gunning (1989, Appendix) also appear to have been successful in relocating a high fraction of households in East Africa; in Kenya a 1982 survey reinterviewed nearly 90 percent of survey households first seen in 1977-78, while in Tanzania, 73 percent of the households in a 1976-77 survey were reinterviewed in 1983. In some cases, panel data can be constructed from a single interview by asking people to recall previous events. This works best for major events in people's lives, such as migration or the birth or death of a child; it is likely to be much more difficult to get an accurate recollection of earnings or expenditures in previous years. There is a substantial literature on the accuracy of recall data, and on the various biases that are induced by forgetting and selective memory (see Groves 1989, ch. 9.4). In the context of developing countries, Smith, Karoly, and Thomas (1992) and Smith and Thomas (1993) use their repeat of the Malaysian Family Life Survey to compare recollections about migrations in the first and second surveys. As well as their unique advantages, panel data have a number of specific prob- lems. One of the most serious is attrition, whereby for one reason or another, households are lost from the survey, so that as time goes on, fewer of the original households remain in the survey. The extent of attrition is affected by the design of the panel, whether or not the survey follows individuals who leave the original households or who move away from the original survey area. Another reason for 20 THE ANALYSIS OF HOUSEHOLD SURVEYS attrition is refusal; households that have participated once are sometimes unwilling to do so again. Refusal rates are typically lower in surveys in developing countries, and presumably attrition is too. In industrial countries with long-running surveys such as the PSID, there can be a substantial loss of panel members in the first few years until the panel "settles down." Becketti and others (1988) show that although 12 to 15 percent of the individuals in the PSID do not reappear after the first inter- view, the subsequent attrition rate is much lower so that, for example, of the indivi- duals in the first wave in 1968,61,6 percent were still present fourteen years later. Even when households are willing to cooperate, there may be difficulties in finding them at subsequent visits; individuals may move away, and the households may cease to exist if the head dies, or if children split off to form households of their own. Depending on whether the survey attempts to follow these migrants and "splits," as well as whether new births or immigrants are added to the sample, the process of household dissolution and formation can result in changes in the repre- sentativeness of the sample over time. (Or what appears to be a panel may not be, if enumerators substitute the new household for the old one without recognizing or recording the change,) There is therefore likely to be a tradeoff between, on the one hand, obtaining a representative sample, which is best done by drawing a new sample each year and, on the other hand, tracking individual dynamics, which re- quires that households be held over from year to year. Even so, Becketti and others (1988) found no serious problems of representativeness with the PSID when they compared the fourteen-year-old panel with the population of the United States. Although the main attractions of panel data are for analytical work, for the mea- surement of dynamics and for controlling for individual histories in assessing behavior, panel designs can also enhance the precision of estimates of aggregate or average quantities. The standard example is estimating changes. Suppose that we compare the case of two independent cross sections with a panel, in which the same households appear in the two time periods. From both designs, the change in average income, say, would be estimated by the difference in average incomes in the two periods. The variance of the estimate from the two cross sections would be the sum of the variances in the two periods because each cross-sectional sample is drawn independently. In the panel survey, by contrast, the same households appear in both periods, so that the variance of the difference is the sum of the variances of the individual means less twice the covariance between the two estimates of mean income. If there is a tendency for the same households to have high (or low) in- comes in both periods—which we should expect for incomes and will be true for many other quantities—the covariance will be positive and the variance of the esti- mated change will be less than the sum of the variances of the two means. The greatest precision will be obtained from a panel, a rotating panel, or inde- pendent cross sections depending on the degree of temporal autocorrelation in the quantity being estimated. The higher the autocorrelation, the larger the fraction of households that should be retained from one period to the next. The formulas are given in Hansen, Hurwitz, and Madow (1953, pp. 268-72) and are discussed in the context of developing countries by Ashenfelter, Deaton, and Solon (1986). Provid- ed precision is the main aim, a rotating panel is a good compromise; for example, THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 21 retaining only half the households from one period to the next will give a standard error for the change that is at most 30 percent larger than the standard error from the complete panel that is the optimal design, and will do better than this when the autocorrelation is low. Given that most surveys are multipurpose, and that there is a need to measure levels as well as changes, there is a good argument for consider- ing rotating panels. When using panel data to measure differences, it is important to be alive to the possibility of measurement (nonsampling) error and to its consequences for various kinds of analysis; indeed, the detection and control of measurement error will be one of the main refrains of this book. Suppose, to fix ideas, that household i in period t reports, not the true value x.t but *.* defined by (1.1) where eit is a mean zero measurement error with cross-sectional variance o>2, and where I assume for convenience that the error variance is the same in both periods. If the reporting error is uncorrelated with the truth, then the variance of measured x is the variance of the true x—the signal —plus the variance of the measurement error—the noise. If the observations are differenced over time, we have (1.2) A*; = A^ + Ae,.,: var(A^) = var(A*.,) + 2o> 2 (l-p) where p is the correlation between the errors in the two periods. There are several important consequences of (1.2). Note first that the presence of measurement error is likely to further enhance the advantages of panel data over independent cross sections for measuring changes in the means, at least if the same individuals tend to make the same reporting errors period after period. Second, the signal-to-noise ratio will be different for the changes in (1.2) than for the levels in (1.1). The prototypical example is where the underlying variable x changes only slowly over time, so that the variance of the true changes is smaller than the vari- ance of the levels. By contrast, the variance of the measurement error in changes will be double that in levels if p = 0, and will be increased by the differencing un- less p > 0.5 . There is no general result here, but there will be many cases in house- hold survey data where the variance of the measured changes will be dominated by measurement error, even when the measurement of the levels is relatively accurate. The data in Table 1.2 are taken from the 1985-86 panel of the Living Standards Survey of Cote d'lvoire and illustrate a number of these issues. The figures shown are summary statistics for consumption and income for 730 panel households who were in the survey in both years. Although the original design called for 800 panel- ists, not all households could be found in the second wave, nor yielded useful data. While there is no direct way of assessing the size of the measurement error, both magnitudes are hard to estimate, and for the reasons discussed in the next section, the individual data are likely to be very noisy. The upper part of the table shows that means and standard deviations of consumption and income are of similar size, so that with 730 households, the standard errors of the estimates of the mean levels 22 THE ANALYSIS OF HOUSEHOLD SURVEYS Table 1.2. Consumption and income for panel households, Cote d'lvoire, 1985-86 (thousands of CFAs per month) Standard Interquar- Mean deviation Median tile range Levels Consumption, 1985 1,561 1,513 1,132 1,344 Consumption, 1986 1,455 1,236 1,070 1,090 Income, 1985 1,238 1,464 780 1,137 Income, 1986 1,332 1,525 871 1,052 Differences 1986-85 Consumption -106 987 -17 679 Income 94 1,128 92 723 Notes: The figures shown are for total consumption and disposable income, both including imputed rental values of housing and durable goods. Data from 730 panel households. Source: Author's and World Bank calculations using the Cote d'lvoire Living Standards Surveys, 1985-86. are about 4 percent (1//(730) » 0.04) of the means. The changes in the bottom two rows have much smaller means than do the levels, and although the standard deviations are also smaller—the correlation coefficients between the years are 0.76 for consumption and 0.72 for income—the standard errors of the estimates of mean change are now relatively much larger. The correlation between the variables is also affected by the differencing. In the micro data, the levels of consumption and income are strongly correlated in both years, 0.81 in 1985 and 0.78 in 1986, but the correlation is only 0.46 for the first differences. There is, of course, nothing in these figures that proves that measurement error is in fact present, let alone that it is determining the outcomes. As we shall see in Chapter 6, there is no difficulty in accounting for the results in Table 1.2 even under the implausible assumption that consumption and income are perfectly measured. However, the phenomena that we observe here are typical of what happens when measurement error is present and important. 1.2 The content and quality of survey data One of the main reasons for collecting household survey data is the measurement and understanding of living standards. At the least, such measurement requires data on consumption, income, household size, and prices. For broader concepts of living standards, we also want information on health, nutrition, and life expectancy, and on levels of education, literacy, and housing. Moving from measurement to model- ing extends the scope a good deal wider. To understand consumption, we need to know about income and assets, and about their determinants, saving behavior, in- heritances, education, and the opportunities for working in the labor market, on a farm, or in small businesses of one kind or another. We also need information THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 23 about public goods, such as schools and hospitals, and on individuals* access to them. In the past, different types of data often have been collected by different kinds of surveys. Budget surveys collected data on consumption and its compo- nents; income and employment surveys have collected data on sources of income, on occupations, and on unemployment; fertility surveys have collected data on children ever born, contraceptive practices, and attitudes towards fertility; and nu- tritional surveys have collected data on how much food people consume, how it is prepared and eaten, and its calorie, protein, and nutrient content. The Living Stand- ards Surveys, described in Section 1.3, collect data on almost all of these topics from the same households in a single survey, sacrificing sample size for an inte- grated treatment of a relatively small number of households. This section is con- cerned with data quality issues that arise more or less independently whether we are dealing with multipurpose or more focussed surveys. Again, I make no attempt to be comprehensive; this section is not a list of what does or ought to appear in the ideal survey. My aim is rather to introduce a number of definitional and measure- ment issues that will recur throughout the book when using the data, and of which it is necessary to be aware in order to analyze them appropriately. Individuals and households The standard apparatus of welfare economics and welfare measurement concerns the well-being of individuals. Nevertheless, a good deal of our data have to be gathered from households, and while in some cases—earnings or hours worked— data are conceptually and practically available for both individuals and households, this is not the case for those measures such as consumption that are most immedi- ately relevant for assessing living standards. Some goods are consumed privately by each member of the household, but many others are shared, and even for food, the most important nonshared good in developing countries, information about each person's consumption usually cannot be inferred from the data on household purchases of food that are typically observed. Chapter 4 will take up these ques- tions in some detail, and review methods that have been proposed for inferring individual welfare from household-level data, as well as a broader avenue of re- search that uses household data to draw conclusions about allocations within the household. A prior question is the definition of the household in the data, why it is that some people are grouped together and others not. Broadly defined house- holds, containing servants and distant relatives, will be larger and more likely to have a membership that responds to changes in the economic environment. While this will be inconvenient for some purposes, and will certainly make it difficult to impute living standards, it may still be the unit that is relevant for decisions like migration or the allocation of work. There is no uniformity in definitions of the household across different surveys, although all are concerned with living together and eating together, and sometimes with the pooling of funds. A range of possibilities is reviewed by Casley and Lury (1981, pp. 186-88), who point out that different criteria are often in conflict, and emphasize that household arrangements are often not constant over time. Many of 24 THE ANALYSIS OF HOUSEHOLD SURVEYS the problems are associated with the complex structure of living arrangements in developing countries, and the fact that households are often production as well as consumption units so that a definition that is sensible for one may be inappropriate for the other. When men have several wives, each wife often runs what is effec- tively a separate household within a larger compound presided over by the hus- band. Even without polyandry, several generations or the families of siblings may live in a single compound, sometimes eating together and sometimes not, and with the group breaking up and reforming in response to economic conditions. In some countries, there are lineages to which groups of households belong, and the head of the lineage may have power to command labor, to order migration, to tax and reward individuals, and to control communal assets. Even so, members of the lineage will typically live in separate households, which will nevertheless not be the appropriate units for the analysis of at least some decisions. An example of the consequences of alternative definitions comes from Thai- land, where compound living arrangements are common. The National Statistical Office changed its survey practice between the 1975-76 and 1981 surveys, and now counts subunits as separate households. Between the two surveys, average household size fell from 5.5 persons to 4.5 persons, and at least some of the differ- ence can be attributed to the change in procedure. A decision to separate previously pooled households should not affect estimates of average consumption or income per head, but will increase measures of inequality, since the previous single esti- mate for the pooled household is replaced by multiple estimates for each of the sub- households, estimates that are not necessarily the same. Splitting households has the same effect on the distribution of income or consumption as an increase in dis- persion with no change in mean, and so must increase measures of inequality (see Kanbur and Haddad 1987 and Chapter 3 below). Reporting periods When households are asked to report their income or consumption, a choice has to be made about the reference or reporting period. As with optimal sample design, the ideal reporting period depends on the purpose of the survey. For example, if the object of the exercise is to estimate average consumption over a year, one extreme is to approach a sample of households on January 1 and ask each to recall expendi- tures for the last year. The other extreme is to divide the sample over the days of the year, and to ask each to report consumption for the previous day. The first method would yield a good picture of each household's consumption, but runs the risk of measurement error because people cannot recall many purchases long after they have been made. The second method is likely to be more economical, because the survey effort is spread over the year, and will give a good estimate of mean consumption over all households. However, unless each household is visited re- peatedly, the survey will yield estimates of individual expenditures that, while ac- curate on average, are only weakly related to the mean or normal expenditures that are appropriate measures of individual standards of living. Nor are short recall peri- ods immune to recall errors, such as "boundary" or "start-up" bias whereby respon- THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 25 dents report events that occurred just before the beginning of the reporting period, in an effort to be helpful and to ensure that the enumerator does not miss "relevant" events. Scott and Amenuvegbe (1990) cite a number of studies showing that reported rates of consumption diminish with the length of the recall period, and their own experiments with households from the Ghanaian Living Standards Survey showed that for 13 frequently purchased items, reported expenditures fell at an average of 2.9 percent for every day added to the recall period. They found no evidence of start-up bias; rather their results confirm that recall deteriorates with time, even over a matter of days. If these conclusions are more generally valid, recall periods of even two weeks—as is often the case—will result in downward-biased estimates of consumption. Even in the absence of reporting errors, so that sample means are unaffected by the choice of recall period, different designs applied to the same underlying popu- lation will give different estimates of inequality and of poverty. Many people receive no income on any given day, and many (albeit fewer) will spend nothing on any given day, but neither fact is a real indication that the individual is truly poor, nor that differences between individuals on a given day are indicative of the true extent of inequality. In practice, most surveys adopt sensible designs that tradeoff potential recall bias from long reporting periods against potential variance from short periods. For consumption, frequently purchased items like food have a recall period of between a week and a month, while larger or rarer items, like dur- able goods, are asked about on an annual recall basis. Even when the recall period is a day, households are revisited every day or two or, when practical, are asked to keep diaries for one or two weeks. Aggregate annual consumption can then be esti- mated for each household by multiplying monthly food expenditures by twelve and adding the durable and other items. Such a calculation will typically give a useful indication of household consumption, and the average over the households in the survey is likely to be a good estimate of average household consumption in the population. However, good estimates of means do not imply good estimates of dispersion. For any given household, expenditures will vary from one reporting period to an- other, so that even in a "survey" that repeatedly interviewed the same household once a week for a year and used the weekly reports to calculate 52 annual esti- mates, not all would be the same. The measured dispersion of annualized expendi- tures will contain both intrahousehold and interhousehold components, the former from the within-year dispersion for each household, and the latter from the genuine inequality across households in annual expenditure. Since it is the latter in which we are usually interested, the use of reporting periods shorter than a year will over- estimate dispersion. Some of the intrahousehold variation is seasonal, and seasonal patterns can be estimated from the data and used to make corrections. The same is not true for the random nonseasonal variation across weeks or months for each household. Scott (1992) makes calculations for this case and gives a plausible ex- ample in which the standard deviation of annual expenditures is overestimated by 36 percent from a survey that gathers consumption data on a monthly basis. 26 THE ANALYSIS OF HOUSEHOLD SURVEYS In several of the Living Standards Surveys (LSS), respondents were asked to report expenditures for more than one period. The standard LSS format calls for two visits, roughly two weeks apart, and the interviewer asks how much was spent on each food item "since my last visit." Respondents are also asked in how many months of each year they buy the item, and what they "normally" spend in each of those months. For some nonfoods, households report expenditures both "since the last visit" as well as uin the last year." Results of comparisons are reported in Grosh, Zhao, and Jeancard (1995) for the Living Standards Surveys of Ghana and Jamaica, and in Deaton and Edmonds (1996) for Cote d'lvoire, Pakistan, and Viet Nam. Estimates of mean expenditure tend to be larger for shorter reporting periods, which is consistent both with recall failure over time, which would bias down the long-period data, but also with boundary effects, which would bias up the short- period data. It is unclear which (or if either) measure is correct. For total expendi- tures, the ratio of means (short-to-long) are 1.04, 1.08, 1.10, and 1.01 for Cote d'lvoire, Ghana, Jamaica, and Viet Nam, respectively. There is also evidence, from Cote d'lvoire and Pakistan, but not from Viet Nam, that when the time between visits is longer, reported expenditures do not increase proportionately, so that the rate of expenditure is lower the longer the recall period; once again, this is consis- tent with progressive amnesia about purchases. Measures of dispersion are also somewhat higher for shorter reporting periods; in Cote d'lvoire for example, the standard deviations of monthly per capita consumption are 368 and 356 thousand CFAs for short and long periods, respectively. In general, "normal" expenditures on food are usually not very different from those reported "since the last visit," and while the discrepancies in both means and variances are larger for nonfoods, they account for a smaller share of the total budget. Given the other uncertainties associ- ated with defining and measuring consumption—see the next subsection—we should perhaps not be too concerned with the discrepancies that are attributable to differences in reporting periods, at least over the practical range. Of course, such a conclusion does not provide reassurance that the measures of inequality and poverty from the surveys correspond to the measures of inequality and poverty that we should like. Measuring consumption If our main concern is to measure living standards, we are often more interested in estimating total consumption that its components. However, some individual items of expenditure are of interest in their own right because their consumption is of direct interest—health care, education, food, especially nutrient-rich foods such as milk—or because the items are subsidized or taxed at differential rates, so that the pattern of demand has implications for public expenditures and revenues. We also need a separate accounting of public goods and their contribution to welfare, and we need to separate expenditures on nondurables from durables, since the latter do not contribute to living standards in the same way as the former. Forecasts of de- mand patterns are often useful, and are essential for the sort of planning exercises that were once a routine part of development policy. The rate at which consumption THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 2 7 switches from food to manufactures and services, and, in poorer economies, the rate at which a largely vegetarian and cereal-based diet is supplemented with meat, exert a powerful influence on the fraction of the population employed in agriculture and on the type and intensity of agricultural production. Even when the survey is concerned only with the measurement of living stan- dards, questions about total expenditure are unlikely to provoke accurate responses, and it is necessary to disaggregate to some extent in order to obtain satisfactory estimates. There are often other sources of information about components of con- sumption—crop surveys and trade data about cereals, for example—so that assess- ing the reliability of the survey typically requires disaggregation. Traditional household surveys in developing countries have surveyed consumption in great detail, and the Indian NSS, the Indonesian Survei Sosial Ekonomi Nasional (SUSE- NAS), and many other surveys collect information on around 200 separate food items alone, both in physical quantity and monetary units, together with several dozen more nonfood items. The World Bank's Living Standard Surveys have usu- ally been less detailed, on the grounds that the detail and the data on physical quan- tities are necessary only for calculating calorie and nutrient intake, but not to obtain accurate estimates of total consumption and living standards. There is mixed evidence on whether it is possible to obtain accurate estimates of total consumption from a small number of expenditure questions. A test survey in Indonesia (World Bank 1992, Appendix 4.2) subjected 8,000 households to both short and long questionnaires. In the former, the number of food items was reduced from 218 to 15, and the number of nonfood items from 102 to 8. Total measured food expenditures differed little between the questionnaires, either in mean or distribution, although the long questionnaire yielded about 15 percent more non- food expenditure. But these encouraging results have not been replicated else- where. A similar experiment in El Salvador with 72 versus 18 food and 25 versus 6 nonfood items gave ratios (long-to-short) of 1.27 for food and 1.40 overall (Jol- liffe and Scott 1995). A 1994 experiment in Jamaica produced similar results, with a long-to-short ratio of 1,26 for both food and nonfood (Statistical Institute and Planning Institute of Jamaica 1996, Appendix III). Although the shorter question- naires can sometimes lead to dramatic reductions in survey costs and times —in Indonesia from eighty minutes to ten—it seems that such savings come at a cost in terms of accuracy. The quality of consumption data has been subject to a good deal of debate. Minhas (1988) and Minhas and Kansal (1989) have compared various item totals from the Indian NSS consumption surveys with the independently obtained pro- duction-based totals of the amounts of various foods available for human consump- tion. While the results vary somewhat from food to food, and while it is important not to treat the production figures as necessarily correct, there is typically very close agreement between the two sets of estimates. The survey figures are, if any- thing, somewhat higher, for example by 4 percent for cereals in 1983. In Britain, the annual FES underestimates total consumption, although as we have already seen, some of the discrepancy is due to the downward bias in alcohol and tobacco expen- ditures, a part of which reflects the coverage of the survey. In the United States, the 28 THE ANALYSIS OF HOUSEHOLD SURVEYS Consumer Expenditure Survey also appears generally to underestimate consump- tion; again alcohol and tobacco are major offenders, underestimating the national accounts figures by a half and a third, respectively, but there are problems with other categories, and even food expenditures were some 15 percent lower than the estimates from the national income and product accounts (NIPA), with the discrep- ancy growing over time (see Gieseman 1987). There are peculiar sampling problems associated with variables whose distribu- tion in the population is extremely positively skewed. Assets—including land—are the most obvious example, but the same is true to a lesser extent of income, con- sumption, and many of its components. Consider the most extreme case, when one household owns all of the assets in the economy. Then surveys that do not include this household will yield an estimate of mean assets of zero, while those that do will yield an overestimate of the mean, by the ratio of population to sample size. Although the estimate of mean assets over all samples is unbiased, it will usually be zero. More generally, sample means will inherit some of the skewness of the distribution in the population, so that the modal survey estimate will be less than the population mean. There are two other issues that tend to compromise the quality of consumption data in developing countries. Both are associated with the fact that most agricul- tural households are producers as well as consumers, and both reflect the difficulty of disentangling production and consumption accounts for people who have no reason to make the distinction. The first problem, which will arise again in Chapter 4, is that wealthy households hire workers, both domestic servants and agricultural workers, and in many cases supply them with food, explicitly or implicitly as part of their wages. Food expenditures for wealthy households will therefore usually in- clude expenditures for items that are not consumed by the immediate family and for large agricultural households the discrepancy can be large. The second problem relates to consumption of home-produced items, typically food grown or raised on the farm or in kitchen gardens. Such items, often referred to as autoconsommation, are properly recorded as both income and consumption, but are often difficult to value, especially in economies—as in much of West Africa —where some markets are not well developed, and where home production and hunting may account for a large share—perhaps more than a half—of total con- sumption. In cases where prices are available, and where the items being consumed are similar to those that are sold nearby, imputation is not difficult, although there are often difficulties over the choice between buying and seeling prices. Where there are differences in quality, or where the item is rarely sold, some price must be imputed, and the choice is nearly always difficult. In one extreme example, a comprehensive survey in West Africa went so far as to collect data on the amount of water that people fetched from local rivers and ponds and used for cooking and washing. Regarding this as an item of autoconsomtnation, its price was calculated using an algorithm applied to all such items, which was to select the price for a similar traded item in the geographically nearest market. In this case, river water was effectively valued at the current price ofL'Eau Perrier in the nearest city, thus endowing rural households with immense (but alas illusory) riches. THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 29 Quite apart from being aware of the general measurement error that is likely to be introduced when imputations are responsible for a large fraction of total con- sumption, it is also important to recognize that such errors will be common to mea- sures of both income and consumption, since imputations are added to both. As a result, if we are interested in the relationship between consumption and income, the same measurement error will be present in both dependent and independent vari- ables, so that there is a spurious correlation between the two, something that needs to be taken into account in any analysis. Measuring income All of the difficulties of measuring consumption—imputations, recall bias, season- ality, long questionnaires—apply with greater force to the measurement of income, and a host of additional issues arise. Income is often a more sensitive topic than is consumption, especially since the latter is more obvious to friends and neighbors than the former. Accurate estimates of income also require knowledge of assets and their returns, a topic that is always likely to be difficult, and where respondents often have incentives to understate. Perhaps most important of all is the fact that for the large number of households that are involved in agriculture or in family business, personal and business incom- ings and outgoings are likely to be confused. Such households do not need the concept of income, so that respondents will not know what is required when asked about profits from farms or own enterprises. The only way to obtain such measures is by imposing an accounting framework on the data, and painstakingly con- structing estimates from myriad responses to questions about the specific com- ponents that contribute to the total Even in the industrialized countries, the mea- surement of self-employed income is notoriously inaccurate; for example, Coder (1991) shows that estimates of nonfarm self-employment income from the March round of the Current Population Survey (CPS) in the United States are 21 percent lower than independent estimates from fiscal sources, while the estimates for farm self-employment income are 66 percent lower. Yet the rafio of the CPS estimate to the tax estimate for wages and salaries is 99.4 percent. A farmer in a developing country (or in the United States for that matter) who buys seeds and food in the same market on the same day has no reason to know that, when computing income, it is only expenditure on the former that should be deducted from his receipts. A street trader selling soft drinks may report that his profits are zero, when the fact is that at the end of each day, after buying food and giving some money to his wife and children, he has just enough left to finance the next day's inventory. Those close to subsistence, whose outgoings are close to incomings, are quite likely to report that income is zero. To get better estimates, the survey must collect detailed data on all transactions, purchases of inputs, sales of outputs, and asset transactions, and do so for the whole range of economic activities for wage earners as well as the self-employed. This is an enormous task, especially in countries where households are large and complex and where people are involved in a wide range of income- generating activities. 30 THE ANALYSIS OF HOUSEHOLD SURVEYS The practical and conceptual difficulties of collecting good income data are severe enough to raise doubts about the value of trying; the costs are large and the data may not always be of great value once collected. Apart from some early experiments, the Indian NSS has not attempted to collect income data in their con- sumer expenditure surveys. Very few households refuse to cooperate with the con- sumption surveys, and, as we have seen, their estimates of aggregate consumption cross-check with independent estimates. The belief is that the attempt to collect in- come could compromise this success, and would lead, not only to poor income figures, but also to a deterioration in the quality of the estimates of consumption. That said, the World Bank's experience with the various Living Standards Surveys as well as that of RAND with the Malaysian and Indonesian Family Life Surveys has been that it is possible to collect data on income components—accurately or in- accurately—without any effect on response rates. On the conceptual issues, anyone who has made the calculations necessary to assemble a household income estimate from a detailed integrated survey, such as the Living Standards Survey, with the hundreds of lines of computer code, with the arbitrary imputations, and with allow- ances for depreciation and appreciation of capital goods and livestock, will inevit- ably develop a lively skepticism about the behavioral relevance of such totals, even if the calculation is useful as a rough measure of the flow of resources into the household. Survey-based estimates of income are often substantially less than survey-based estimates of consumption, even when national income estimates show that house- holds as a whole are saving substantial fractions of their incomes, and even in in- dustrialized countries where self-employment is less important and income easier to measure. Although there are often good reasons to doubt the absolute accuracy of the national income figures, the fact that surveys repeatedly show large fractions of poor people dissaving, and apparently doing so consistently, strongly suggests that the surveys underestimate saving. What little we know about the accuracy of the consumption estimates indicates that consumption is more likely to be under- estimated than overestimated, so that it seems likely that most survey estimates of income are too low. Some of the underestimation may come from positive skew- ness in the distribution of income, so that survey estimates of the mean will also be skewed with a mode below the mean (see p. 28 above.) But underestimation of individual incomes is almost certainly important too. While this conclusion is far from well documented as a general characteristic of surveys in developing count- ries, many statisticians find it plausible, given the conceptual and practical difficul- ties of measuring income. Discovering more about the discrepancies between national income and survey-based estimates of saving should be given high priority in future research. Most theories of saving relate to individual or family behavior, and yet much of the concern about saving, growth, and macroeconomic perfor- mance relates to national aggregates. If household data cannot be matched to national data, it is very difficult to make progress in understanding saving behavior. Table 1.3 presents data on income, expenditure, and saving from the Socio- economic Survey of the Whole Kingdom of Thailand in 1986. The left-hand panel groups the households in the survey according to a comprehensive definition of THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 31 Table 1.3. Household saving by income and expenditure deciles, Thailand, 1986 (baht per month) By income decile By expenditure decile Decile Income Saving Decile Income Saving 1 690 -683 1 950 49 2 1,096 -737 2 1,409 -6 3 1,396 -741 3 1,768 -23 4 1,724 -775 4 2,119 -73 5 2,102 -746 5 2,523 -145 6 2,589 -593 6 3,005 -221 7 3,231 -503 7 3,649 -103 8 4,205 -358 8 4,579 -379 9 5,900 82 9 6,119 -308 10 13,176 2,761 10 12,283 -1079 All 3,612 -229 All 3,841 -229 Notes: Figures are averages over all households in each decile; there are 10,918 households in the survey. In the left-hand panel, households are grouped by deciles of total income, in the right-hand panel, by deciles of total expenditure. Source: Author's calculations using the Social Survey of the Whole Kingdom of Thailand, 1986 (see Example 1.1 in the Code Appendix). household total income, and shows the average saving figures for households in each decile group. According to these estimates, households dissave in total, while the National Income Accounts for Thailand (U.N. 1991) estimate that household saving was 12.2 percent of household income in 1986. Moreover, low-income households dissave more than higher-income households, and the bottom nine dec- iles each show negative saving on average. Although such behavior is not univer- sal—see, for example, the results for Taiwan (China) in Chapter 6—it is common in surveys in developing countries (see, for example, Visaria and Pal 1980). Such evidence makes it easy to see why early observers of economic development inferred that saving was confined to rich households. However, the right-hand side of the table shows that such an explanation is not correct, at least not in any simple way. Households that are well-off in terms of income are also likely to be well-off in terms of consumption, so that if saving rates are higher for richer households, saving should also rise across consumption deciles. But the date show that the largest amount of dissaving is done by households in the top expenditure decile. These findings probably owe a good deal to measurement error. If income and consumption are independently measured, at least to some extent, households who overstate their incomes will also, on average, overstate their savings, while house- holds who overstate their consumption will correspondingly understate their sav- ing. The top deciles of income contain a large fraction of households with over- stated incomes and will thus show the highest saving rates, with the opposite effect for consumption. Of course, exactly the same story can be told with "transitory in- 32 THE ANALYSIS OF HOUSEHOLD SURVEYS come" and "transitory consumption" replacing "measurement error" in income and consumption, respectively, and it is this analogy that lies at the heart of Friedman's (1957) permanent-income theory of consumption. Indeed, since Friedman defined transitory and permanent income as if they were measurement error and the unob- served "true" income, respectively, an explanation in terms of measurement error can always be recast as a permanent-income story. Even so, the results in Table 1.3 are consistent with the importance of measurement error in income, as is the dis- crepancy between the national accounts and survey results. Paxson (1992) has argued that the presence of inflation also tends to overstate consumption relative to income given that surveys usually have different reference (reporting) periods for consumption and income. As we have seen, the reference period for consumption varies from item to item, but is often a week or two weeks for food—which may account for two-thirds or more of the budget—while the importance of seasonality in incomes means that reference periods for income are usually a year. Consumption is then denominated in more recent, higher prices than is income, imparting a downward bias to measures of saving. Since inflation in Thailand in 1986 was only 2 percent per annum, the appropriate correction makes little difference to the figures in Table 1.3, though Paxson shows that the correc- tions for Thailand in the previous 1980-81 survey, when inflation was 16 percent, increase saving by around 7 percent of income. 1.3 The Living Standards Surveys The Living Standards Measurement Study (LSMS) was begun in the World Bank in 1979 in the last months of the McNamara presidency. The original aim was to develop the World Bank's ability to monitor levels of living, poverty, and inequa- lity in developing countries, to allow more accurate statements about the number of people in poverty around the world, and to permit more useful comparisons between countries. In conjunction with host statistical offices, the project fielded its first surveys in Peru and in Cote d'lvoire in 1985-86. Since then, there have been several dozen LSMS or LSMS-related surveys. These surveys are different from the typical earlier survey in developing countries, and the experience with them has been influential in shaping current survey practice. It is therefore useful to devote some space to a description of their special features, and to attempt some assess- ment of what has been learned from the LSMS experience. A brief history In the late 1970s, it became clear that it was impossible for the World Bank—or for anyone else—to make well-supported statements about world poverty, especially statements that required internationally comparable data. There was no firm basis assessing such fundamental topics as the extent of poverty in the world, which countries were the poorest, or whether the inequality within and between nations was expanding or contracting. Even within countries, the simplest statements about distributional outcomes were difficult. One particularly important case was that of THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 33 Brazil, where there was dispute as to the extent that poor people had benefited from the "economic miracle" of the 1960s. According to an analysis by Fields (1977), the poor had done much better than the nonpoor, but Ahluwalia and others (1980) showed that neither this result, nor any other useful conclusion could be supported by the available evidence. While national income data were not above criticism, then as now, statistical offices worldwide were generating useful, credible, and comparable data on average economic performance, but there were no correspond- ing data on distribution. At the same time, writers in the "basic needs" literature, for example Streeten and others (1981), were arguing for a revaluation of the rela- tionship between economic growth and poverty. While there was no lack of econo- mists on either side of the issue, the data did not exist to settle what was largely a factual question. Even in India, the motherland of household surveys, evidence on poverty trends was controversial and hotly debated (see Bardhan 1971, Lai 1976, Ahluwalia 1978, and Griffin and Ghose 1979). Because data on consumer expendi- tures were collected only every five years, and because rural poverty in India is so sensitive to fluctuations in the harvest, it was impossible before the mid-1980s to separate trend from fluctuations, and be sure from the NSS data that poverty rates were indeed falling (see Ahluwalia 1985). The original aim of the LSMS project was to remedy this situation, by collecting —or at least by helping others to collect—comparable survey data across countries, and so allowing comparisons of poverty and inequality over time and space. In ret- rospect, it is unclear how such an objective could have been achieved except by the establishment of international standards for surveys that were comparable, for ex- ample, to the U.N.'s system of national accounts (SNA); even then, it would have been much more difficult to establish a common set of protocols for estimating dis- persion than for estimating means. However, by the time that the first full LSMS survey was ready for implementation in Cote d'lvoire in 1985, attention had shifted from measurement towards a more ambitious program of gathering data to be used to understand the processes determining welfare at the household level. Although either set of objectives would have required a multipurpose household income and expenditure survey of some kind, the new aims were best served by an intensive, integrated survey in which each household was asked about every aspect of its economic and domestic activity. While the expense of such a design necessitated a smaller sample size, as well as less detail on individual topics than had been the case in traditional single-purpose surveys, such as agricultural or consumption surveys, these were accepted as the costs of the opportunity to model household behavior as a whole. The shift in emphasis owed much to "Chicago" views of household and farm behavior, particularly Schultz's arguments that households respond rationally and purposively to prices and incentives and to the development by Becker of the "new household economics." These views first influenced survey practice in RAND's 1976-77 Malaysian Family Life Survey, the experience of which helped shape the first Living Standards Surveys. The latter have in turn influenced later RAND sur- veys, particularly the 1988-89 second Malaysian Family Life Survey and the 1993 Indonesian Family Life Survey. Behind these designs rests the belief that policy 34 THE ANALYSIS OF HOUSEHOLD SURVEYS advances should rest on an enhanced empirical understanding of how such house- holds respond to their economic and physical environments, and on the role of government policy in shaping those environments. Although such a perspective was different from the original one, the practical consequences were confined to the tradeoff between sample size and the amount of information from each indivi- dual. Nothing in the new design would prevent the data being used for its original purposes and indeed, in recent years, one of the main uses of LSMS surveys has once again been for the measurement of poverty. Another theme in the original design was an emphasis on collecting at least some panel data. As shown above, a panel design is often an efficient way to col- lect information on changes over time, which was one of the aims of the LSMS project. Panel data also seemed to be well suited to the documentation of the losses and gains from economic development, or from structural adjustment. In the late 1970s and early 1980s, there was also a great deal of interest in academic circles in the econometric possibilities associated with panel data, so that the collection of such data, which was rare in developing countries, was an exciting and promising new endeavor. The Ivorian Living Standards Survey collected data in 1985, 1986, 1987, and 1988, with an intended sample size of 1,600 households. There were three panels of 800 households each, which ran for two years each in 1985-86, 1986-87, and 1987-88; no household was retained for longer than two years, but in principle, all households would be interviewed at two visits a year apart, except for half of those in the first and last years. A larger (5,120-household) single-year survey was car- ried out in Peru in 1985-86, followed by a two-year panel survey with 3,200 households in Ghana in 1987-88 and 1988-89. Since then there have been LSMS or LSMS-related surveys in Mauritania (1988), Morocco (1990-91), Pakistan (1991), Venezuela (1991-93), Jamaica (1988 to date), Bolivia (1989 to date), the Kagera region of Tanzania (1991-93), Peru (1990, 1991, and 1994), Russia (1991-92), South Africa (1993), Viet Nam (1993), Kyrgyz Republic (1993), Nica- ragua (1993), Guyana (1993), Ecuador (1994), Romania (1994-95), Bulgaria (1995), and the Hebie and Liaoning provinces of China (1995). At the time of writ- ing, LSMS surveys are in the design or implementation stage for Brazil, Kazakhstan, Mongolia, Nepal, Paraguay, Tunisia, Turkmenistan, and Uzbekistan. These surveys are far from identical, although all have common design elements as described below. But the surveys have been adapted to different needs in different countries, so that, for example, the Jamaican survey is a compromise between a full LSMS survey and a previously existing and long-running labor force survey, while the survey in Kagera is concerned with monitoring economic responses to the AIDS epidemic. For further details, see Grosh and Glewwe (1995), from which the infor- mation in this paragraph is culled, or the LSMS homepage on the World Wide Web. Design features of LSMS surveys The design and implementation of the Ivorian survey is described in Ainsworth and Mufioz (1986) and their description remains a good account of a prototypical LSMS THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 35 survey. The implementation manual by Grosh and Munoz (1995) provides a fuller and more recent account, incorporating the lessons of past experience, and is essential reading for anyone designing an LSMS or LSMS-related survey. There are three separate questionnaires in the "standard" LSMS survey; a house- hold questionnaire, a community questionnaire, and a price questionnaire. The first is long by previous standards, and comprises (up to) seventeen sections or modules, some of which are technical, such as the section that identifies suitable respondents for subsequent modules, or the section that links individuals across years for panel households, Most modules are substantive and cover a long list of topics: house- hold composition, housing and its characteristics, education, health, economic acti- vities and time use, migration, agricultural and pastoral activities, nonfarm self- employment activities, food expenditures, durable-goods expenditures and inven- tories, fertility, other sources of income including remittances, saving, assets, and credit markets, and anthropometric measurement of household members. The com- munity questionnaire, which is sometimes used only in rural areas, gathers data from knowledgeable local people (such as chiefs, village headmen or elders, medical personnel, or teachers) about local demographics (population, ethnicity, religion, and migration), and about local economic and service infrastructure, such as transportation, marketing, extension services, primary and secondary schools, and health and hospital facilities. The price questionnaire, administered country wide, was collected by enumerators visiting local markets and observing prices, mostly of foods. The surveys are designed to produce high-quality data, and to deliver the results quickly; to this end, use is made of microcomputers both for the design of ques- tionnaires and for data entry and editing. The use of computer software to turn questions into a printed questionnaire, with appropriate pagination and skip pat- terns, not only cuts down on error, but also permits rapid redesign and reprinting after field tests. Many responses are preceded on the questionnaire, so that there is no coding by keyboard operators at the data entry stage. In each round, households are visited twice with an interval of two weeks, and roughly half the questionnaire administered at each visit. Between the two visits, the first data are entered into the computers and are automatically subject to editing and consistency checks by the software as they are entered. Such procedures not only minimize data entry errors, but permit the enumerators to correct some response errors during the second visit. They also mean that much less time is required to edit the data after the survey, so that instead of the several years that are common in some countries, the preliminary data and tabulations are available in three to six months after the completion of the fieldwork. What have we learned? It is too early for any final verdict on the contribution of LSMS surveys to data col- lection in developing countries in general, let alone to their ultimate aim, of im- proving policymaking and the understanding of economic behavior in developing countries. At the time of writing it is only eleven years since the first household 36 THE ANALYSIS OF HOUSEHOLD SURVEYS was interviewed. New data are continuously becoming available and new uses in policy and analysis are being developed. However, the LSMS surveys have taught us a great deal that we did not know in 1980, and the experience has certainly changed the way household data are—or at least ought to be—collected in develop- ing countries. The experiments with microcomputers have been successful. Computer-assisted questionnaire design works and the procedures for preceding, software-controlled data entry, and checking enhance data quality and speed delivery. Although local conditions typically preclude using computers during interviews, so that there are limits on the use of the computer-aided interview techniques that are rapidly devel- oping in the United States and elsewhere, it is feasible to install microcomputers at local survey headquarters, and these are used to enter and check the data within a few days of capture. Given the cost and robustness of microcomputers, there is every reason for their use to be universal in household survey practice around the world. There is no good reason—except for entrenched bureaucracy and vested interests—why survey results and tabulations should be delayed until years after fieldwork is completed. We also know now that long and complex household questionnaires are prac- tical. Because there are two visits separated by two weeks, and because different members of the household are interviewed for different parts of the questionnaire, no one person is subjected to an impossibly long interview. At the beginning of the LSMS project, opinions were sharply divided on the feasibility of long question- naires, with experienced surveyors arguing both for and against. In practice, and although each round of the LSMS survey was two to three hours long, depending on the number of household members, there were no refusals based on length, and no appearance of declining cooperation as the interviews progressed. Of course, at any given cost, there is a tradeoff between length of the questionnaires and the number of households that can be covered in the sample. The fact that LSMS surveys favor the former has meant that survey results are less useful than traditional, larger surveys for disaggregated measurement of living standards, by region, occupation, or other target group. As a result, and as interest has turned back to the use of sur- vey data to assess poverty, the later surveys have tended to be simpler and to have larger sample sizes, and there has been some reversion towards the original aims of measurement rather than analysis. While it is difficult to assess data quality without some sort of controlled experi- ment, the feedback from users has usually been positive, and there is no evidence to suggest that the timeliness of the data has been bought at the price of lower quality compared with data sets where the cleaning and editing process has taken a great deal longer. To be sure, there are difficulties with the LSMS data, but these appear to be the sort of problems that can and do arise in any survey, irrespective of design. The community and price questionnaires are elements that are more experimental in the LSMS design and there currently appears to be little hard evi- dence on their usefulness. The community questionnaire can be regarded as an efficient way of collecting information that could, in principle, have been collected from individual households. The data on the provision of services has been routine- THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 37 ly combined with the household data in various analyses, for example of the deter- minants of access to education and health facilities. One difficulty lies in the con- cept of a community. The simplest idea is of a village, whose inhabitants share common health, educational, and other facilities, and who buy and sell goods in the same markets. But communities may not conform to this model; people may be- long to different "communities" for different purposes, and in some parts of the world, there are no well-defined villages at all. Nor is there any guarantee that the primary sampling units or clusters in the survey necessarily correspond to units that have any unified social or administrative structure. In consequence, even when the community questionnaire yields data on schools, clinics, or transportation, we do not always have a clear delineation of the population served by those facilities, or on its relationship to the survey households in the cluster. Information about prices is not easy to collect. Enumerators are given a list of well-defined items, and are required to price at three different sites in local markets. For obvious reasons, an enumerator is not given money to make actual purchases, but instead approaches the seller, explains that he or she is conducting a survey (which has nothing to do with taxes or law enforcement), and asks the price of an item. There is no haggling, but the enumerator is supplied with scales and asks the seller's permission to weigh the potential purchase. While it is easy to see the problems that might accompany such a procedure, it is harder to devise alterna- tives. "Market price" is a concept that is a good deal more complex in an African market than in an American supermarket or an economics textbook. Different people pay different amounts, there are quantity discounts, and many foods are presented for sale in discrete bundles (half a dozen yams, or a bunch of carrots) whose weights or volumes may vary even at the same price. There are also wide geographical differences in the range of items in local markets, so that it is difficult to collect the sort of price data that will permit reliable calculation of the cost-of- living indexes required to compare real living standards across different areas of the country, and between rural and urban households. Even so, in those LSMS sur- veys (for example Pakistan and Viet Nam) where consumers reported both expen- ditures and physical quantities, the unit values from these reports are well corre- lated across space with the prices from the price questionnaire (see Deaton and Edmonds 1996). An alternative procedure is recommended by Grootaert (1993) who suggests collecting prices using the standard price surveys that usually exist to gather cost-of-living data, although such a solution sacrifices the exact match between survey households and the prices that they face. The experience of collecting panel data has been somewhat mixed. The original emphasis has been muted over time, and few of the recent surveys have been explicitly designed as panels. While the LSMS surveys should not be judged on their success (or lack of it) in collecting panel data, the experience has been useful, parti- cularly given the rarity of such data from developing countries. Some critics had suggested that it would be difficult to locate many households for the second annual visit and these worst fears were not realized. Even so, there is a good deal of migration of household members, and there are occasional refusals, so that actual panels are smaller than the design. In Cote d'lvoire, where the design called 38 THE ANALYSIS OF HOUSEHOLD SURVEYS for 800 panel households out of a total sample of 1,600, there were 793 "panel- designate" households from whom valid data were collected in 1985; 730 of them (92 percent) provided valid data in 1986. In the second panel in 1986-87, 693 (87 percent) of the 800 designated households provided data, and in 1987-88, there were 701 (88 percent) out of 800. Since attrition is generally at its worst in the first year, and since the LSMS surveys make no attempt to follow households that have moved, these fractions are impressively high, comparing well with the attrition rates in the PSHX Perhaps less satisfactory is the occasional difficulty of determining whether two so-called panel households in two years are in fact the same household. When panel households could not be located in the second year, replacement households were selected, and these were not always labeled as clearly as they should have been. The section of the questionnaire that links panel households and their members is a great help in this regard, and is a testament to the fact that the design- ers clearly anticipated the problems that arise with panel households, but there are cases where examination of this section, and the difficulty of matching individuals, raises questions about whether what is labeled as a panel household is not in fact a replacement. The underlying fear is that, in the absence of detailed supervision, enumerators may too readily substitute new households for the intended panelists, and that the fact that they have done so may not be easily detectable after the event. The "linking" section of the questionnaire is extremely important, and has to be a focus in any satisfactory panel survey. Linking is best done by having a list of the names of all household members, which can conflict with the usual promises of anonymity. There are also questions about the general value of the data obtained from two- year rolling panels as in the Ghanaian and Ivorian surveys. One issue is measure- ment error, particularly in income and, to a lesser extent, in consumption, Given that we want to measure changes in levels of living at the individual level, and given that in much of Africa economic progress has been slow at best, measured changes over a single year will be dominated by a combination of measurement error and the normal fluctuations of agricultural income, neither of which is of primary interest. It is probably also true that attrition is at its worst between the first and second year of a panel, so that a two-year design suffers a larger proportional loss per household year than would a longer panel. Longer panels also offer greater opportunities for assessing and controlling for measurement error (see, for ex- ample, Griliches and Hausman 1986 and Pischke 1995). What would be more useful is one of two things; first, estimates of change over much longer periods, five or ten years say, and second, estimates of change over periods when there have been major policy changes, such as those induced by structural adjustment programs, or clearly defined outside shocks that have affected living standards. Obtaining either of these can be a matter of good fortune, of having a survey in place at just the right time, but will be guaranteed only when surveys are organized on a continuing basis by a permanent local survey organiza- tion. As I have already noted, it is also sometimes possible to revisit households from an earlier survey even when there was no original intention of constructing THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 39 a panel. Such revisits can allow long observation periods without having to main- tain a permanent survey, and can be designed on an ad hoc basis to examine some event of interest (see Bevan, Collier, and Gunning 1989, who looked at the conse- quences of booms in coffee prices in Kenya and Tanzania in the mid-1980s, or Glewwe and Hall 1995, who looked at the effects of macroeconomic shocks in Peru). Although the Ivorian and Ghanaian surveys were originally intended to be permanent, data collection has ceased in both. The timing was particularly unfortu- nate in Cote d'lvoire because data collection ceased just before the decline in world prices forced the government to cut the procurement prices of cocoa and coffee, which are the major sources of agricultural income in much of the country. Studying the effects of this policy change, which terminated a thirty-year regime of approximately constant real prices, could have produced important insights about policy and welfare but the opportunity was lost. In general, it is unclear that, except by chance, there are major benefits to be expected from gathering short-term panel data in slowly growing or stagnant economies. The usefulness of the LSMS data for policymaking and research is another ques- tion on which a verdict is premature, although there is certainly a strong demand by the countries themselves for such surveys. For several of the LSMS countries, the surveys have replaced an almost total absence of information with at least some information, and the tabulations are routinely used in policy operations, project evaluation, and poverty assessment inside the World Bank and in the countries themselves. The surveys have also produced large amounts of microeconomic data that are potentially available for the sort of research described in this book. The LSMS group within the World Bank has also produced an impressive (and impress- ively long) list of working papers, most of which are concerned with policy-related empirical analysis of the data. However, there is less evidence that the data are being used to capacity in the countries from which they originally came. One of the aims of the LSMS project, from its first inception, was to provide an easily accessible data base for the analysis of policy on a daily basis. While no one supposed that cabinet ministers would sit in front of terminals displaying the LSMS data, there was hope that their assistants and advisors would, and that tabulations or graphics could be produced in hours, not months or years (or not at all) as is often the case. Questions about whether food subsidies are reaching their intended recipients, or who would be hurt by an increase in fertilizer prices, are examples of the sort of questions that arise regularly among policymakers and their advisors, and that can be quantified and clarified by survey data (see Chapter 3). Such distributional analyses are a regular feature of policymaking in the United States or Britain, for example, but for whatever reason—computing facilities, software, or personnel— similar analyses are rare in developing countries. Even so, there are signs of progress. Most notable is the case of Jamaica, where there was an unusually high degree of (powerful) local interest from the start, and where the LSMS data have been used in a wide range of domestic policy exercises, on poverty assessment, on the effectiveness of food stamps, and on health questions. The data are even used as part of the standard training in quantitative methods at the University of the West Indies (Grosh and Glewwe 1995). 40 THE ANALYSIS OF HOUSEHOLD SURVEYS In recent years there has also been a great deal of progress in the archiving of the data and in making them available to the research community, for example via the LSMS page on the World Wide Web. While there are real problems of owner- ship and confidentiality with all household survey data, arrangements can be—and in an increasing number of cases have been—worked out to give access to the indi- vidual household records (again see the LSMS homepage for up-to-date infor- mation). Finally, a note about survey costs. The Implementation Manual (Grosh and Munoz 1996, Table 8.1) presents total costs for eight actual surveys. These vary from $78 per household for a 2,000-household survey in Jamaica to over $700 per household for a 4,480-household survey in Brazil; in several other cases, the per household cost lies between $150 and $250. Much of the variation is explained by whether or not vehicles had to be purchased. The high estimates are inflated by these costs—only a fraction of which are directly attributable to the one-year sur- vey—but are understated by amount of technical assistance provided by the World Bank, which is not included in these estimates. 1.4 Descriptive statistics from survey data One of the first calls on survey data is to calculate descriptive statistics, often for a survey report containing a standard set of tables. In this section, I discuss some of the issues that arise in calculating these statistics, and in particular how we make sure that statistics describe the population rather than the particular sample that is available for analysis. To do so, we need to know how the sample was designed and to understand its relationship to the population of interest. Different sample designs require the data to be processed in different ways to estimate the same magnitude. I present some useful formulas for calculating the standard error of estimated means, again taking the sample design into account. These formulas are useful in themselves, as well as for the descriptive material that appears later in the book, particularly in Chapter 3 on the measurement of poverty. They also provide a starting point for the analytic and econometric material in Chapter 2. Common examples of descriptive statistics are measures of central tendency, such as means and medians, and of dispersion, such as variances and interpercentile ranges. Living standards are often measured by the means of income or consump- tion, inequality by their dispersion, and poverty by the fraction of the population whose income or consumption is below the poverty line. The quantities that are to be summarized are sometimes continuous, as with income or consumption, and sometimes discrete, as in poverty where the basic data are indicators of whether or not a household is in poverty. While the estimation of means and standard errors is familiar, it is worth recording the formulas that take account of differential weights (inflation factors) and that allow for stratification and clustering. Finite populations and superpopulations To make inferences using survey data we need a framework for thinking about how the data were generated, which means thinking about the population from which THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 41 the data came and about how data collection induces randomness into our sample. There are two different approaches. In the first, which is standard among survey statisticians, the population is a finite one—for example all households in Cote d'lvoire in calendar year 1985—and the sample households are randomly selected from that population just as, in the classic textbook example, balls are drawn from an urn. The survey data are random because replication of the survey would gene- rate different samples so that the variability of an estimate, such as the mean of household income, is assessed by thinking about how it would vary from one sample to another. The quantity of interest—in the example the average reported household income of Ivorian households in 1985—is a fixed number that could be measured with perfect accuracy from a census, which is when the sample coincides with the population. No assumptions are made about the distribution of income in the population; what is being estimated is simply the average income in the popula- tion in the survey year, not the parameter of a distribution. In the second approach, we are less interested in the actual population in the survey year, regarding it as only one of many possible populations that might have existed. The actual population is itself regarded as a sample from all possible such populations, the infinite superpopulation. The focus of attention is instead the statistical law or economic process that generated income in the superpopulation, in populations like the one under study, and the mean is of interest less for itself than as a parameter or a characteristic of that law or process. The finite-population or survey-statistical approach is typically associated with description, while the superpopulation approach is associated with modeling. The distinction is made by Groves (1989, esp. ch. 6) and will be useful in this book in which I shall be concerned with both. The distinction between description and modeling is often of no more than philosophical interest—there is no dispute about* the appropriate formula for a mean or standard deviation. But when we come to the econometric analysis in Chapter 2, where the tradition has been of modeling, there are sometimes sharp differences in the recommended calculation of apparently identical concepts. Recent trends in econometric practice have tended to emphasize description at the expense of modeling—an emphasis shared by this book—so that the traditional gulf between the practice of survey statisticians and econometricians is narrowing. In consequence, I shall sometimes follow one approach, and some- times the other, whichever seems more appropriate to the problem at hand. The example of the mean can be used to illustrate the two approaches and pro- vides an opportunity to record the basic formulas. Suppose that we have a simple random sample of n observations from a population of size Af, and that the quantity of interest is jc, with observations jc/t i running from 1 to n. The sample mean x is the obvious estimator of the population mean, where (1.3) x = n'lExr /=! The sample mean, (1.3), is a random variable that will vary from one sample to another from a given population and, in the superpopulation approach, from one population to another. 42 THE ANALYSIS OF HOUSEHOLD SURVEYS Suppose that we want to know the expectation of x over the different surveys. This seems like a complicated matter for the survey statistician, because there are a large number of different ways in which n objects can be selected from a popu- lation of W objects, and it is necessary to calculate the mean for each and its associ- ated probability of occurrence. A simple shortcut (see Cochrane, 1977, p. 28) is to rewrite (1.3) as N (1.4) x = n~lEaixi i=} where the sum now runs over the whole population, to N rather than n, and a. is a random variable that indicates whether i is in the sample, taking the value 1 if so, and 0 otherwise. The jc's on the right-hand side of (1.4) are no longer random vari- ables, but simply the fixed jc's in the (finite) population. Hence, when we take ex- pectations of (1.4), we need only take expectations of the a's. Since we have a simple random sample, each i has an equal probability of being included, and that probability is simply the ratio of n to N. The expectation of each a, is therefore 1 times the probability of its being 1, which is nIN, plus 0 times the probability of its being 0, which is (1 - nlN\ a total of n/N. We then have N N (1.5) E ( x ) = n-lE(n/N)x, = AT'Ejt. = X M /=i where X is the population average in which we are interested. A superpopulation approach makes more assumptions, but is in some ways more straightforward. It might postulate that, in the superpopulation, x is distribu- ted with mean JLI, a parameter that is the same for all households. We then have im- mediately, from (1.3), (1.6) E(x) = w - ' E n - n M also as desired. The finite-population approach is more general, in that it makes no assumptions about the homogeneity of the observations in the sample, but it is also more limited, in that it is specifically concerned with one population only, and makes no claim to generality beyond that population. The technical note at the end of this subsection shows that the variance of x is (1.7) V(*) = i^' where S 2 is given by (1.8) S2 =(N-1)-'E(* ( .-X) 2 1=1 and can be thought of as the population variance, and 1 -/ is the "finite-population correction" (fpc), (1.9) !-/ = (N-n)/N. THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 43 Except in the unusual situation where the sample is a large fraction of the popu- lation, thefpc is close enough to unity for the factor 1 -/ in (1.9) to be ignored. Indeed, in this book I shall typically assume that this is the case; sampling texts are more careful, and can be consulted for the more complex formulas where neces- sary. The superpopulation approach to the variance would postulate that each x. is independently and identically distributed with mean (i and variance a2, so that from (1.3), (1.10) V(x) = £(Jc-|i)2 = n~lo2. Since both o2 and S2 are estimated from the sample variance (1.11) * 2 - (n-ir'EOr,-*) 2 i-l and provided we ignore the fpc, there is no operational difference between the two approaches. Both use the same estimate of the mean, and both estimate its variance using the formula (1.12) v(jc) = n-ls2 = n -'("-I)" 1 £(*,-* >2* 1=1 Books on sampling often give separate treatment to the estimation of means and the estimation of proportions. However, a proportion is simply the mean of a binary (0,1) indicator that tells us whether the observation does or does not possess the attribute of interest. In consequence, the formulas above— as well as those in the next subsections—can also be used for estimating proportions and their sampling variability. To see how this works and to link up with the analysis of poverty in Chapter 3, suppose that x. is 1 if household i is in poverty, and is 0 otherwise. The estimate of the proportion of households in poverty is then the mean of x (1.13) p = n' 1 Ex, = njn M where nl is the number of households with x. = 1. If we have a simple random sample, the estimated variance of p is given by (1.12), which since xt can only take on the two values 0 or 1, takes the simple form (1.14) v(p) = iT'Cn-iy'lEd-^ E p2} = (n-\)'lp(\-p). i-l i»n,+l Formula (1.14) is useful because it is simple to remember and can be calculated on the back of an envelope. Even so, the same answer is given using the standard formulas and treating the ;t's as if they were continuous. ^Technical note: the sampling variance of the mean I follow the derivation in Cochrane (1977, p, 29) which starts from (1.4) and from its implication that 44 THE ANALYSIS OF HOUSEHOLD SURVEYS N N N (1.15) V(x) = n~2[ Ejcfvar(a.) + 2E Exix.cov(apaj)'\. i=l i=ij "<*->• v5*U"-) -fe-*) 46 THE ANALYSIS OF HOUSEHOLD SURVEYS (see the technical note on p. 49 below and Cochrane 1977, pp. 252-54). An un- biased estimate of (1.23) can be obtained from the sample by defining z. = w,xi and using the formula (1.24) HXlol)--^E(Zi-z)\ n~l 1=1 The sampling variance (1.23) can be used to give a formal answer to the ques- tion of why different probabilities can enhance efficiency, and to tell us what the optimal probabilities should be. In particular, it is a simple matter to show that (1.23) is minimized subject to the constraint that the IT'S add to 1 by selecting the IT'S to be proportional to the jt's. This is sampling with probability proportional to size (p.p.s.)\ larger values of x contribute more to the mean so that efficiency is en- hanced when larger values are overrepresented. Of course, if we knew the jc's, there would be no need to sample, so that in practice we can use only approximate p.p.s., in which the IT'S are set proportional to some other variable that is thought to be correlated with jc and that is known prior to sampling. We are often interested, not in the total of Jt, but its mean, X, which can be estimated from the ratio of the estimated total (1.21) to the estimated population, (1.19). This is the probability-weighted mean n n n (1.25) xw - Ew.*./ Ewi = Ev,*. 1=1 /=i «=i where the v. are the w. from (1.17) normalized to sum to unity, (1.26) v, = H > . / E H > fc=i A simple example of the effects of weighting is given in Table 1.4, which pre- sents the weighted and unweighted means and medians—see (1.30) below for the definition of a weighted median—for total household expenditure by race in the South African Living Standards Survey. These calculations use the weights sum- marized in Table LI and discussed on page 16 above. For the Blacks, Coloreds, Table 1.4. Household total expenditures, weighted and unweighted means, South Africa, 1993 (rand per month) Means Medians Race Weighted Unweighted Weighted Unweighted Blacks 1,053 1,045 806 803 Coloreds 1,783 1,790 1,527 1,547 Asians 3,202 3,185 2,533 2,533 Whites 4,610 4,621 4,085 4,083 All races 1,809 1,715 1,071 1,029 Source: Author's calculations using the South African Living Standards Survey, 1993. THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 47 and Asians, the weights do not vary much within each group, so that there is little difference between the weighted and unweighted estimates. For Whites, there is more within-group variance in weights, but it has little effect on either estimate be- cause there is little or no correlation between the weights and the level of income within the group. However, as we saw in Table 1.1, lower participation by Whites made the weights higher for Whites than for the other groups, so that when we calculate estimates for the whole country, their higher incomes result in weighted estimates that are larger than the unweighted estimates. This result illustrates the general point that, whenever there is an association between the sampling prob- abilities and the quantity being measured, unweighted estimates are biased. Because xw is the ratio of two random variables, it is not an unbiased estimator. However, because the variances of its numerator and denominator, (1.21) and (1.19), both converge to zero as n tends to infinity, it will converge to the popula- tion mean. The sampling variance of xw can be evaluated using standard approxi- mation techniques for ratio estimators; in the technical note on p. 49 below, I sketch the argument that leads to (1.27) V(xw) * N-2E^(xrX)2 1=1 which can be estimated from the sample data using (1.28) v(*J = _!-Evf2(*.-*J2. n - 1 1*1 Note that the estimated variance will usually need to be specially coded. In parti- cular and except for simple random sampling, (1.28) is not equal to the sample estimate of the population variance—equation (1.29) below—divided by the sample size (compare (1.11) and (1.12)). Formula (1.27) can also be used to find the probabilities that maximize the precision of the probability-weighted mean (1.25). Recalling that w. = ( mt.)~ l and choosing the TT'S to minimize (1.27) subject to the constraint that their sum be 1, it is easily shown that the optimal selection probabilities should be proportional to the absolute value of the deviation from the mean I jc. - XI It is information on the exceptional cases that adds most to the precision of the estimated mean. It is worth noting that the probability-weighted mean (1.25) is not the only pos- sible estimate. In particular, if the population size N is known, an estimate of the mean can be obtained by dividing Xtot by N. But there are a number of reasons why the weighted mean is frequently more useful. First, the population size is often not known, but is estimated from the survey itself, for example by randomly select- ing a set of villages, enumerating all households in each, and using the totals to estimate the number of households in the population. Second, when we come to use the survey data to calculate means or other statistics, some data are unusable, be- cause they are missing, because of transcription errors, or because they take on clearly implausible values. There is then little option but to average over the "good" observations, renormalizing the weights to sum to unity. Third, in many applications we are not interested in means per household, but in means per person. 48 THE ANALYSIS OF HOUSEHOLD SURVEYS We want to know the fraction of people in poverty, not the fraction of households in poverty. Or we may want to know the fraction of elderly, or women, or children who have some characteristic. In such cases, we weight the data, not by the number of households represented by the sample household, but by the number of people it represents. To get statistics about persons in the population, the quantity sur- veyed, say household per capita consumption, should be weighted not by the w. themselves, but by the w. multiplied by the number of people in household i. The total of these weights is (in expectation) the number of people in the population, not the number of households; as before, this population total may or may not be known in advance, but we often encounter cases where the relevant population total is estimated by summing weights from the survey estimate. Fourth and finally, we often want to calculate means for subgroups, and to do so in a way that is repre- sentative of the relevant subpopulations. Once again, xw is the relevant estimator unless we know the size of each subpopulation. The weights can be used to estimate other population statistics analogously to the mean in (1,25). For example, the population variance S2 is estimated from the sample by weighting the individual squared deviations from the mean so that (1.29) S2W = -2- E v(.(*. -xw )2 = _!- E W,(r, -*„)2 / E w. n - 1 1-1 n - 1 /=i /=i where the adjustment factor nl(n-l) is of no practical significance, but matches (1.29) to the standard unbiased estimator (1.11) in the case where w. = Nln, Ex- pression (1.29) would be used, for example, when using the variance of income or of the logarithm of income to measure inequality. Weights must also be used when ranking households, for example when calculating medians, quartiles, or other per- centiles in the population. In the sample, median household income (for example) is that level of income below which (and above which) lie half the sample obser- vations. When estimating the median for the population, we must find instead the level of income such that, when we take all sample households with lower income, the sum of their weights is half the total sum of weights. Formally, the median xw is defined by (1.30) ElCx^JcJv, = 0.5 where the v. are the normalized weights in (1.26) and the function \(e) is an indi- cator that takes the value 1 if the statement e is true and is 0 otherwise. Other per- centiles are calculated by replacing the 0.5 in (1.30) by the appropriate fraction. In practice, the easiest way to work is to sort the data in order of increasing jt(., and then to calculate a running sum of the normalized weights. The percentiles are then read off from this running sum. Example 1.1 in the Code Appendix gives the STATA program used to calculate the results in Table 1.3; this shows how to label households by their deciles of total household income and expenditure, and to sum- marize variables by those deciles. In principle, formulas can be derived for sampling variances for medians, vari- ances, and functions of these statistics. But the calculations are in some cases quite THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 49 complex, and the approximations and assumptions required are not always palat- able. As we shall see below, the bootstrap often offers a more convenient way to assess sampling variability in these cases. ^Technical note: sampling variation of probability-weighted estimates The probability-weighted mean is the ratio of two estimates, Xm and N. The vari- ance of such a ratio can be approximated by (1.31) var(J^) * AT2[var(4,) -2Xcov(X, o; tf) +X 2 var(tf)]. I illustrate only the derivation of the first term in square brackets; the other variance and the covariance are readily obtained in the same manner. From (1.21), (1.32) 4, = IjW,. The only random variables in (1.32) are the r's, so that AT N AT (1.33) var(X tor ) = E wfxf\3r(t.) + E E w / w.jc.jc.cov(r.r,). The /'s follow a multinomial distribution, so that the variance of t. is nrc^l - n.) and the covariance between t. and t, is -/iTC.it.. Substituting in (1.33) and rear- ranging gives (1.23) above. That (1.24) is unbiased for (1.23) is shown by inserting r's, and changing the summation from n to JV, and taking expectations (see also Cochrane). Substitution of (1.33) into (1.31) together with the comparable formulas for var(N) and cov(Xtot,N) gives (1.27). The sample estimate (1.28) is construc- ted by replacing the square of N in the denominator by the square of its estimate N, replacing X by its estimate xw, and replacing the population sum of squares by its sample equivalent, remembering that for every household in the sample, there are w. in the population. The scaling factor nl(n -1) is conventional and clearly has little effect if n is large (see also (1.29)). Stratification The effect of stratification is to break up a single survey into multiple independent surveys, one for each stratum. When we think of the different samples that might be drawn in replications of the survey, the strata will be held fixed while the parti- cular households selected from each will vary from sample to sample. Without stra- tification, the fraction of the sample in each stratum is left to chance. In conse- quence, estimates of population parameters vary across samples because each sample has different fractions of observations in each stratum so that, when the means differ across strata, their weighted average will also differ. As a result, strati- fication can reduce sampling variability whenever the means differ across strata. Suppose that there are S strata, labeled by sy that we know the total population Was well as the population in each stratum Ns, and that the mean for stratum s is Xs. The population mean, or "grand" mean, is then 50 THE ANALYSIS OF HOUSEHOLD SURVEYS S (1.34) X - L(NtIN)Xg which can be estimated from (1.35) x = E(Ns/N)xs where xs is the estimated mean for stratum s. Note that the calculation of these means will typically involve weights, as in (1.25) above. The population shares may or may not be the same as the sample shares; stratification is about breaking up the sample into subsamples, not about weighting. Nevertheless, it is often the case that the sampling fractions are different in different strata. In such cases, as I shall show below, it is possible to incorporate the stratum weights into the weights for each observation. Because the strata are independent, the variance of the estimate of the popula- tion mean (1.35) takes the simple form, ignoring thefpc, s (1.36) V(x) = E(Ns/N)2V(xs)t where V(xs) is the variance of the estimate of the stratum mean. If instead of the stratified survey, we had used a simple random sample, the numbers in each stra- tum, n v , would be random variables within the total sample size n. Hence, if we write the sample mean for the simple random sample design analogously to (1.35), s (1.37) xsrs = E(n r /M)jc f which looks very similar to (1.35), especially since the expectations of the sample ratios (nsln) are the population ratios (NSIN). However, in the unstratified design, the fractions of the sample in each stratum will vary from sample to sample, so that the variability of the estimate will not only have a component from the variability of the stratum means, as in the stratified sample, but also a component from the variability of the fractions in each stratum. With some algebra, it can be shown that the variances of the two estimates are linked by the approximation s 0 1Q\ rJO) Vf Y ^ » Vfr " \'*vrv/ V\A, } ) + ~ 11 £~f (N n "1 ^ IN \(Y y i V „''» /\**B - r •* ^ / • As would be expected, the variance is larger in the simple random sample than in the stratified sample, and will be the more so the larger is the heterogeneity across strata. When the strata means coincide with the grand mean, there is no increase in efficiency from stratification. In practice, estimation in stratified samples is usually done using simple adapta- tions of the formulas for the weighted estimates in the previous subsection. Sup- pose that we think of each stratum as a separate survey, and write the inflation fac- tors for households in stratum s as w.v, where, corresponding to (1.17), (1.39) w., = (n^r1 THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 57 where ns is the sample size from stratum s, and rc/v is the probability that i is drawn at each trial The within-stratum sum of these weights is an unbiased estimator of the stratum population, Ny, and the grand sum over all observations is an unbiased estimator of the total population. Hence, we can define a probability-weighted esti- mate of the grand mean corresponding to (1.35) (1.40) xw=X(Ns/N)xw X=\ where xsw is the probability- weighted mean (1.25) computed for stratum s. Note that in (1.40), unlike (1.35), the fractions of the population in each stratum are esti- mated, and are therefore random variables. As a result, (1.40) loses what might be expected from stratification, that without variation within strata, there is no vari- ance in an estimate from a stratified sample (see (1.36). That this is not the case is because there are differential weights within strata, so that different samples will give different weights to each stratum mean. Of course, if the ratios NS/N are avail- able, they can be used to replace the estimates in (1.40), with a gain in precision. If we substitute the appropriate sums of weights for Ns and N in (1.40), we get (1.41) xw = ( E E W f ) / ( E E w , , ) ,V=1 1*1 ' ' ,V=1 1 = 1 where jc.y is the observation from household i in stratum s. Note that (1.41) is simply the probability- weighted mean without any explicit allowance for the strati- fication; each observation is weighted by its inflation factor and the total divided by the total of the inflation factors for the survey. Like (1.25), it is also a ratio estimator; the mean is estimated by the ratio of the estimated total — in the numer- ator— to the estimated population size— in the denominator. In consequence, we can use the variance formula (1.31) to approximate the variance of (1.41) in terms of the variances and covariance of the totals which, in turn, are sums of stratum- specific terms because sampling is independent within each stratum. The algebra " is similar to that used to derive (1.23) and (1.27), and yields d.42) LE-LETT. v (J F) = _LE_L iI PJ.-AMM E*, \( -O--N.X\ -x\-L- 5 N 1 1 ' X \ ^ . v = , n v M -M «b ' *J As was the case for (1.23) and (1.24), a feasible sample-based estimator of (1.42) starts from defining z /v = x^wis, and using the formula (1.43) ?<*„) = E -\ E [(*,,-?,) -i>to-*,)]2 .¥=1 ns~ 1 1=1 where z v and u>v are the stratum means of the z's and the weights, respectively. Two-stage sampling and clusters Within strata, most household surveys collect their data in two stages, first samp- ling clusters, or primary sampling units (PSUS), and then selecting households from 52 THE ANALYSIS OF HOUSEHOLD SURVEYS within each cluster; this is the standard two-stage stratified design. Clustered sam- ples raise different statistical issues from stratified samples. When we imagine rep- licating a survey, which is how we think about sampling variability, the strata are held constant from sample to sample, but new clusters are drawn every time. For example, each potential survey might select an equal number of households from all of the provinces of the country (the strata), but would always select a new set of villages within provinces (the clusters). Probabilities of selection can differ at either or both stages of the survey, between clusters, or between households within clusters. The formulas for weighted and unweighted means are not affected by the two-stage design any more than they were affected by stratification. But the sampling variability of these estimates is affected by the design. Because house- holds within clusters are often similar to one another in their relevant character- istics, it is frequently the case that clustering will increase variability compared with simple random sampling. In this subsection, I introduce some notation, record the formulas for the means and their sampling variation, and explain how it is that clustering reduces precision, and the consequences of ignoring the clustering in calculating variability. This is perhaps the most important message of this sub- section, that it can be a serious mistake to treat a two-stage sample as if it were a simple random sample; the use of standard formulas can seriously overstate the precision of the estimates. I start by supposing that there is no stratification, or equivalently, that there is only a single stratum. Because separate strata can be thought of as separate surveys, a simple way to deal with a stratified sample is to work with one stratum at a time, and then to reassemble the survey as a whole from its components. I shall do so, and record the relevant formulas, at the end of the subsection. We need notation for the numbers of clusters and households in the sample and in the population. Suppose that there are N clusters in the population from which n are selected into the survey; this preserves the previous notation for the case where each observation is a cluster. I use the suffix c to denote a cluster or PSU, and mc and Mc to denote the number of sample and population households in cluster c. I shall use T for the total number of households in the population (1.44) T = £M C . c=l Suppose that sampling is with replacement, but with unequal probabilities at both stages. I use TIC to denote the probability of selection for cluster c in the first stage, and TT/C for the probability that i is selected at the second stage, conditional on c having been selected at the first. The unconditional probability that household i in cluster c is selected at a single two-stage draw is thereforeTC,CKC.Proceeding as before, we can define inflation factors for each stage of the survey. To differentiate these stage-specific inflation factors from the overall inflation factors, I use differ- ent notations for each. Define hc and h.c by (1.45) hc = (nig*1, hic = (m^)"1 THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 53 so that hc is the number of population clusters represented by cluster c, and hic is the number of cluster-c households represented by household i in cluster c. The overall inflation factor, the number of households in the population represented by household i, is the product of hc and /z /c , which corresponds to our previous survey weight w.c, (1.46) w.c = hchic = (Ti^m^r1. Note that the sum of these weights over cluster c, 0.47) wc = £ W f c - A r £* f c i=l i=l is an inflation factor that tells us how many population households are represented by the collectivity of sample households in cluster c. The probability-weighted mean is defined in the standard way; adapting (1.25) to recognize the clusters, n '"c n ZZwicx.c Zwcxm „ (1.48) n mc » r =i EEw. c Ew c = EV ^'^ ^ ^W i j_i /*=! where jtw is the probability-weighted mean for cluster c and the v's are the cluster weights (1.47) normalized to sum to 1. The evaluation of the variance of (1.48) is complicated by the randomness in the cluster means, as well as in the selection of clusters themselves. The algebra is simplified if we follow Cochrane (1977, pp. 275-76) and calculate expectations and variances in two stages, so that, for the mean xw (1.49) E(xw) = E}[E2(xw)] where the expectation E2 is taken with respect to the second-stage sampling, treat- ing the choice of clusters as fixed, and where E} is taken with respect to the choice of clusters. The corresponding variance formulas are (1.50) V(xw) = Vl[E2(xw)] *^[V 2 (5 W )]. The application of (1.49) and (1.50) is relatively straightforward using previous results, and after a good deal of algebra, we reach ft N Mc (1.51) V(xw) = n-lZnctfc(Xc-X)2 + T-2E Ew j c (X. -*c)2 C=l C=1>=1 where (|>c is the fraction of the population in cluster c, and Xc and X are the true means for the cluster and the population, respectively. (For comparison, note that (1.51) reduces to (1.27) when each cluster contains a single household and the weights satisfy n ~lnc = wc and 4>c = N ~l ). A consistent estimate of (1 .5 1) can be obtained from 54 THE ANALYSIS OF HOUSEHOLD SURVEYS n r* 2/-. _ vi (1-52) = -TEV(^-^)2 n-1 c=i which is identical to (1.28), with households replaced by clusters, and individual data points replaced by cluster means. It should be emphasized that, in spite of its formal similarity, the variance (1.52) is quite different from the corresponding formula when there is no clustering, and that the use of the incorrect formula can be seriously misleading. While it is some- times the case that estimated variances are not much altered by allowing for strati- fication or differential weights, clustering is ignored at one's peril. I illustrate for the simplest case, where there are M households in each of the N clusters, and at the first stage, clusters are selected by simple random sampling. Each cluster is then equally weighted, so that when we estimate the variance from (1,52) we get (1.53) If we were mistakenly to ignore the clustering and treat each observation as an in- dependent draw in a simple random sample of size m/z, we would use (1.12) to give ! (1.54) *„(*) = £ E (*,, -*) 2 = —S2. mn(mn- 1) c =i M mn If we substitute for the cluster means in (1.53) and rearrange, we get (1.55) v(x) - v v r v ( * ) [ l + ( m - l ) p ] where p is defined by n ^ EEE(Xjc-x)(Xkc--x) (1*56) p = c*\j-\k*j mn(m-l)s2 The quantity in (1.56) is a sample estimate of the intracluster correlation coef- ficient. Like any correlation coefficient, p measures the similarity of values, in this case within the clusters. When all the jc's are the same in the same cluster, p = 1, when they are unrelated, p = 0. In practice, for quantities like income and con- sumption in rural areas of developing countries, p is often substantially larger than zero and values of 0.3 to 0.4 are frequently encountered. Equation (1.55) shows how the magnitude of p affects the variability of sample estimates, at least in this simple case. When p = 0, the variance of the estimate from the clustered sample coincides with the variance of the estimate from the simple random sample. At the other extreme, when p = 1, the factor in square brackets in (1.55) is m, so that v(Jc) = s 2/n and the effective sample size is not the number of sample observations, mn, but the number of sample clusters, n. When the observations are the same within each cluster, sampling more than one from each does nothing to increase the precision of the estimate. In the next subsection, I shall give some practical examples of the way in which assumptions about sample THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 55 design affect calculations of standard errors, and of the potential for being misled by the wrong assumption. Two-stage samples often use a "self-weighting" design. At the first stage, clust- ers are selected with probability proportional to the number of households they contain while, at the second stage, an equal number of households is drawn from each cluster using simple random sampling. This has the effect of making the over- all, survey wide, inflation factors wic the same for all households. To see how this works, set Mc Mc i ft = = ft = m =m d.57) * ^ T * Mc < C =l so that, substituting into (1.46), we have (1.58) wfc - Tl(mn) which is the same as in a simple random sample. (Of course, this only applies to the weights and the computation of sampling variability must still allow for the two- stage design. A two-stage self-weighting sample is not the same thing as a simple random sample.) Self- weighting is simple and elegant. It also had practical utility when compu- tation was so difficult that the additional complexity of weights was best avoided if possible. However, self- weighting designs are rarely self- weighting in practice because adjustments are often made to the weights after the survey, for example to compensate for unanticipated nonresponse by some set of households, and weights have to be used in any case. An example is the South African Living Standards Survey, which had a self- weighting design, but which had to be weighted ex post (see Table 1.1). Since computation is hardly an issue today, it is unclear why the design remains so popular. In practice, it is necessary to combine the formulas for the clustered case with those that allow for multiple strata. This is conceptually straightforward, although the notation makes the formulas look forbidding. I denote the strata by the subscript s, and rewrite the mean for stratum s from (1.48) as a-59) ^-Ew^/Ew. c=l c=l where the only change is to add a suffix s to indicate the stratum. From (1.35), we can compute the grand mean over all the strata using (1.60) xw=iNsxM/N 5= 1 where the hats denote the usual estimates from the sums of the weights. Substitut- ing (1.59) into (1.60) gives the probability-weighted estimate of the grand mean in the familiar form of a ratio between the estimated total of X, and the estimated population size, 56 THE ANALYSIS OF HOUSEHOLD SURVEYS S «, S «, «c SEw E «*«r EEwto*ta X - ... 5=1 c*l _ s*l c=l 1=1 _ ^/or (1.61) W S «, 5 », «c ^ E £ Wcf E E E wto J=l C=l 5=1 C=l i»l which is simply the weighted mean using all the observations and all the weights in the survey. An estimate of the variance of (1,59) and (1.61) is obtained following the same general procedures as for the ratio estimator (1.41) in the stratified case, but making the adaptions for clustering for the variances within each stratum. The formulas are simplified if we define the cluster level variable wc (1.62) *« = J^ta^ta- If Zy is the mean of zcv over clusters in stratum s, the variance of (1.61) can be estimated from (compare (1.43)) (1.63) t(*J = ^z£ J^.£[C9 is the total weight in cluster c of stratum s, and ws is the stratum- s mean of wcj. Sample code for equation (1.63) is given in Example 1.2 of the Code Appendix; it is also available as a special case of the more general formulas available in Version 5.0 of STATA. A superpopulation approach to clustering It is also possible to take a superpopulation approach to clustering, and as was the case with simple random sampling, the results are in many ways simpler. They also provide a useful bridge to the discussion of clustering and regression in Section 1 of Chapter 2. Suppose that there are no weights and that (1.64) *te = »i + ac + e.c where JLI is the mean, occ is a cluster effect, and eic is a random variable with mean 0 and variance a] that is independently and identically distributed for all i and c. The cluster effects occ are also random with mean 0 and variance o£, are inde- pendently and identically distributed across clusters, and are independent of the e's. These independence assumptions are the counterpart of the independence of the two stages of the sampling in the finite-population approach and the presence of the a's allows cluster means to differ from the overall mean. As before, the obvious estimator of JLI is the sample mean jc, and straightforward calculation gives (1.65) E(x) = ji; V(Jc) = n-l<£ + (nmrl<£. The variance in (1.65) is the counterpart of (1.51) when both stages are by simple THE DESIGN AND CONTENT OF HOUSEHOLD SURVEYS 57 random sampling. It is also instructive to write it in the form (1.66) V(x) = - « • - « <• (•m' --1 ) - - ~« °2 nm nm where o2 = o^ + Q\ is the variance of jc.c and p, the ratio of o^ to o2 is the intra- cluster correlation coefficient (compare (1.55) above). An unbiased estimator of V(x) is given by (1.67) v(Jc) = /T1*,2 = n~l(n-irl E (xc~x)2 which corresponds to (1.52). In both cases the variance can be computed by con- sidering only the variation of the estimated cluster means, ignoring within cluster variability. Illustrative calculations for Pakistan For poverty and welfare calculations, we often use household per capita expendi- ture (PCE)—total expenditure on goods and services divided by household size—as a measure of living standards. I use measurements of PCE from the Pakistan Living Standards Survey—formally the Pakistan Integrated Household Survey, or PIHS— to illustrate the sort of design that is encountered in practice, as well as the conse- quences of the design for the calculation of statistics and their standard errors. The survey documentation will usually explain how the stratification was done and the primary sampling units selected. Identifiers for the stratum and cluster of each household are sometimes included as data in one of the household files, but are more usually incorporated into the household identifiers. This is the case for the PIHS, where the first three digits of the household code gives the stratum, the next three the cluster, and the last three the household within the cluster. Example 1.2 in the Code Appendix shows how the household identifiers are broken down to give stratum and cluster identifiers. In the PIHS there are 22 strata; the four pro- vinces—Punjab, Sindh, North-West Frontier (NWFP), and Baluchistan—which are further stratified by urban and rural and by income level. There are 280 PSUs, or clusters, between 2 and 37 in each of the strata, and there are between 13 and 32 households in each cluster. The probability weights are sufficiently correlated with PCE for the weighting to make a difference; the unweighted average of household PCE is 730 rupees a month, whereas the weighted mean is only 617 rupees a month. Table 1.5 shows these estimates for the country as a whole and for its four pro- vinces together with various calculated standard errors. The first two columns are doubly incorrect; the estimated means ignore the probability weights, and the standard errors ignore the sample design. The weights are negatively correlated with PCE—in this case, better-off households are oversampled—so that the un- weighted means are biased up, which makes the standard errors of little interest. The third column shows the (correct) weighted means, and the other columns show various possible standard errors, each calculated under different assumptions about 58 THE ANALYSIS OF HOUSEHOLD SURVEYS Table 1.5. Estimates of mean household per capita expenditure and calculated standard errors, Pakistan, 1991 (rupees per capita per month) Weight, Weight, strata, Province x sl/v = NJns% so that the weighted mean (1.25) is 5 E v V = 1 JM =< -v=l V —JC, _ 2^ = "* s = _ (2.5) *w = £ .v=i N Hence, provided that the sample means for each sector are unbiased for the cor- responding population means, so is the weighted mean for the overall population mean. Equation (2.5) also shows that it makes no difference whether we take a weighted mean of individual observations with inflation factors as weights, or whether we compute the sector means first, and then weight by population shares. Let us now move to the case where the parameters of interest are no longer population totals or means, but the parameters of a linear regression model. With- in each sector s = !,..,£, (2.6) y, = *,P, + ", and, in general, the parameter vectors P v differ across sectors. In such a case, we might decide, by analogy with the estimation of means, that the parameter of interest is the population-weighted average (2.7) P = W1 £ W V P,- 68 THE ANALYSIS OF HOUSEHOLD SURVEYS Consider the only slightly artificial example where the regressions are Engel curves for a subsidized food, such as rice, and we are interested in the effects of a general increase in income on the aggregate demand for rice, and thus on the total cost of the subsidy. If the marginal propensity to spend on rice varies from one sectors to another, then (2.7) gives the population average, which is the quantity that we need to know, Again by analogy with the estimation of means, we might proceed by estimat- ing a separate regression for each sector, and weighting them together using the population weights. Hence, (2.8) P = E-^IJ,, H, = (xXr'X/y,. .v=i N ' ' Such regressions are routinely calculated when the sectors are broad, such as in the urban versus rural example, and where there are good prior reasons for sup- posing that the parameters differ across sectors. Such a procedure is perhaps less attractive when there is little interest in the individual sectoral parameter esti- mates, or when there are many sectors with few households in each, so that the parameters for each are estimated imprecisely. But such cases arise in practice; some sample designs have hundreds of strata, chosen for statistical or adminis- trative rather than substantive reasons, and we may not be sure that the parameters are the same in each stratum. If so, the estimator (2.8) is worth consideration, and should not be rejected simply because there are few observations per stratum. If the strata are independent, the variance of $ is s(N}2 s ( N}2 (2.9) \ N ) ^" * = l { N ; where oj is the residual variance in stratum 5. Because the population fractions in (2.9) are squared, $ will be more precisely estimated than are the individual $ v . Instead of estimating parameters sector by sector, it is more common to esti- mate a regression from all the observations at once, either using the inflation factors to calculate a weighted least squares estimate, or ignoring them, and esti- mating by unweighted OLS. The latter can be written 5 V'( S \ (2.10) A'Y V Y A \ YA Y ; v ** s s\ ^ .vXv \ .vl / V '-1 / In general, the OLS estimator will not yield any parameters of interest. Suppose that, as the sample size grows, the moment matrices in each stratum tend to finite limits, so that we can write (2 (2 v m m plimw/'xX = A/,; pl\mnt'}X^ys = cf = A/ V P V v * * ' ' n .-« r n.-« where M and c are nonrandom and the former is positive definite. (Note that, as in Chapter 1, 1 am assuming sampling with replacement, so that it is possible to sample an infinite number from a finite population.) By (2.11), the probability limit of the OLS estimator (2.10) is ECONOMETRIC ISSUES FOR SURVEY DATA 69 ( s \-*s (2.12) plimfl = E(nsln)M \ E (n /n)c v Vi-i V 1-1 where I have assumed that, as the sample size grows, the proportions in each sector are held fixed. If all the Pv are the same, so that cs = M5P for all 5, then the OLS estimator will be consistent for the common p. However, even if the structure of the explanatory variables is the same in each of the sectors, so that Ms = M for all s and cs = MP V , equation (2.12) gives the sample-weighted average of the p^, which is inconsistent unless the sample is a simple random sample with equal probabilities of selection in all sectors. The inconsistency of the OLS estimator for the population parameters mirrors the inconsistency of the unweighted mean for the population mean. Consider then the regression counterpart of the weighted mean, in which each household* s contribution to the moment matrices is inflated using the weights, f 5 ", V1 S n, \ (2.13) I EEvvyU «—\ T^ / I I EEw^jc^ V-\ T-v I V .t=l i=l / \j=li=l ) where */v is the vector of explanatory variables for household i in sector s, and y /v is the corresponding value of the dependent variable. In this case, the weights are NJ n^ and vary only across sectors, so that the estimator can also be written as J (2.14) Pw 5^- IS«, N \iN'X'X\\i^X'y\ = (X'WX)~lX'Wy where X and y have their usual regression connotations—the Xs and ys matrices from each sector stacked vertically—and Wis an nxn matrix with the weights Nslns on the diagonal and zeros elsewhere. This is the weighted regression that is calculated by regression packages, including STATA. If we calculate the probability limits as before, we get instead of (2.12) ( S M \~l S ^ (2.15) plimft,- E-i M J E 7TM.vP.v V s't m ) j=l N so that, where we previously had sample shares as weights, we now have popu- lation shares. The weighted estimator thus has the (perhaps limited) advantage over the OLS estimator of being independent of sample design; the right-hand side of (2.15) contains only population magnitudes. Like the OLS estimator it is consis- tent if all the P v are identical, and unlike it, will also be consistent if the M v mat- rices are identical across sectors. We have already seen one such case; when there is only a constant in the regression, Mv = 1 for all s, and we are estimating the population mean, where weighting gives the right answer. But it is hard to think of other realistic examples in which the Af ¥ are common and the cs differ. In gen- eral, the weighted estimator will not be consistent for the weighted sum of the parameter vectors because 70 THE ANALYSIS OF HOUSEHOLD SURVEYS (2.16) E(Nx/N)M\ E(Nx/N)cs * E(Nx/N)M;lcs= E(Ns/N)$s = p. \ -V=l ' 7 A'=l ' ' .V=l ' ' ' lV = l In this case, which is probably the typical one, there is no straightforward analogy between the estimation of means and the estimation of regression parameters. The weighted estimator, like the OLS estimator, is inconsistent. As emphasized by Dumouchel and Duncan (1983), the weighted OLS estimator will be consistent for the parameters that would have been estimated using census data; as usual, the weighting makes the sample look like the population and re- moves the dependence of the estimates on the sample design, at least when sam- ples are large enough. However, the difference in parameter values across strata is a feature of the population, not of the sample design, so that running a regression on census data is no less problematic than running it on sample data. In neither case can we expect to recover parameters of interest. The issue is not sample design, but population heterogeneity. Of course, if the population is homoge- neous, so that the regression coefficients are identical in each stratum, both weighted and unweighted estimators will be consistent. In such a case, and in the absence of other problems, the unweighted OLS estimator is to be preferred since, by the Gauss-Markov theorem, least squares is more efficient than the weighted estimator. This is the classic econometric argument against the weighted estima- tor: when the sectors are homogeneous, OLS is more efficient, and when they are not, both estimators are inconsistent. In neither case is there an argument for weighting. Even so, it is possible to defend the weighted estimator. I present one argu- ment that is consistent with the modeling point of view, and one that is not. Sup- pose that there are many sectors, that we suspect heterogeneity, but the heteroge- neity is not systematically linked to the other variables. Consider again the proba- bility limit of the weighted estimator, (2.15), substitute CC=MS$J!J and write p v = p + (p,-p) to reach (s V's (2.17) plim^ = P + E (NJN)M \ £ (tf,W)M v (p v -p). \ J-l / A' = l The weighted estimate will therefore be consistent for p if (2.18) E(A7AOA/,(p f -p) = 0. ,v=l This will be the case if the variation in the parameters across sectors is random and is unrelated to the moment matrices Mx in each, and if the number of sectors is large enough for the weighted mean to be zero. The same kind of argument is much harder to make for the unweighted (OLS) estimator. The orthogonality con- dition (2.18) is a condition on the population, while the corresponding condition for the OLS estimator would have to hold for the sample, so that the estimator would (at best) be consistent for only some sampling schemes. Even then, its probability limit would not be P but the sample-weighted mean of the sector- specific p ? , a quantity that is unlikely to be of interest. ECONOMETRIC ISSUES FOR SURVEY DATA 71 Perhaps the strongest argument for weighted regression comes from those who regard regression as descriptive, not structural. The case has been put forcefully by Kish and Frankel (1974), who argue that regression should be thought of as a device for summarizing characteristics of the population, heterogeneity and all, so that samples ought to be weighted and regressions calculated according to (2.13) or (2.14). A weighted regression provides a consistent estimate of the population regression function—provided of course that the assumption about functional form (in this case that it is linear) is correct. The argument is effectively that the regression function itself is the object of interest. I shall argue in the next chapter that this is frequently the case, both for the light that the regression function sometimes sheds on policy, and when not, as a preliminary description of the data. Of course, if we are trying to estimate behavioral models, and if those mod- els are different in different parts of the population, the classic econometric argu- ment is correct, and weighting is at best useless. Recommendations for practice How then should we proceed? Should the weights be ignored, or should we use them in the regressions? What about standard errors? If regressions are primarily descriptive, exploring association by looking at the mean of one variable condi- tional on others, the answer is straightforward: use the weights and correct the standard errors for the design. For modelers who are concerned about heterogene- ity and its interaction with sample design, matters are somewhat more compli- cated. For descriptive purposes, the only issue that I have not dealt with is the com- putation of standard errors. In principle, the techniques of Section 1.4 can be used to give explicit formulas that take into account the effect of survey design on the variance-covariance matrices of parameter estimates. At the time of writing, such formulas are being incorporated into STATA. Alternatively, the bootstrap provides a computationally intensive but essentially mechanical way of calculating stand- ard errors, or at least for checking that the standard errors given by the conven- tional formulas are not misleading. As in Section 1.4, the bootstrap should be pro- grammed so as to reflect the sample design: different strata should be bootstrap- ped separately and, for two-stage samples, bootstrap draws should be made of clusters or primary sampling units (PSUs), not of the households within them. Because hypothetical replications of the survey throw up new households at each replication, with new values of x's as well as /s, the bootstrap should do the same. In this context, it makes no sense to condition on the original *'s, holding them fixed in repeated samples. Instead, each bootstrap sample will contain a resampling of households, with their associated Jt's, /s, and weights w's, and these are used to compute each bootstrap regression. In practice, the design feature that usually has the largest effect on standard errors is clustering, and the most serious problem with the conventional formulas is that they overstate precision by ignoring the dependence of observations within the same PSU. We have already seen this phenomenon for estimation of the mean 72 THE ANALYSIS OF HOUSEHOLD SURVEYS in Section 1.4, and it is sufficiently important that I shall return to it in Section 2.2 below. It is as much an issue for structural estimation as it is for the use of regres- sions as descriptive tools. The regression modeler has a number of different strategies for dealing with heterogeneity and design. At one extreme is what might be called the standard approach. Behavior is assumed to be homogeneous across (statistical or substan- tive) subunits, the data are pooled, and the weights ignored. The other extreme is to break up the sample into cells whenever behavior is thought likely to differ or where the sampling weights differ across groups. Separate regressions are then estimated for each cell and the results combined using population weights accord- ing to (2.8). When the distinctions between groups are of substantive interest—as will often be the case, since regions, sectors, or ethnic characteristics are often used for stratification—it makes sense to test for differences between them using covariance analysis, as described, for example, by Johnston (1972, pp. 192-207). When adopting the standard approach, it is also wise to adopt Dumouchel and Duncan's suggestion of calculating both weighted and unweighted estimators and comparing them. Under the null that the regressions are homogeneous across strata, both estimators are unbiased, so that the difference between them has an expectation of zero. By contrast, when heterogeneity and design effects are im- portant, the two expectations will differ. The difference between the weighted estimator (2.13) and the OLS estimator can be written as bw-toLS - (X'WX)-lX'Wy-&'X)-lX'y (2.19) = (X/WX)~lX/W(I-X(X/X)-lX/)y = (X'WXYWWMxy where Mx is the matrix /-AXX'X)"1*'. By (2.19) the difference between the two estimators is the vector of parameter estimates from a weighted regression of the unweighted OLS residuals on the jc's. Its variance-covariance matrix can readily be calculated in order to form a test statistic, but the easiest way to test whether (2.19) is zero is to run the "auxiliary" regression (2.20) y = Xb + WXg + v and to use an F-statistic to test g = 0 (see also Davidson and MacKinnon 1993. pp. 237-42, who discuss Hausman (1978) tests, of which this is a special case). In the case of many sectors, when we rely on the interpretation that the inter- sectoral heterogeneity is random variation in the parameters as in (2.17) above, note that the residuals of the regressions, whether weighted or unweighted, will be both heteroskedastic and dependent. Rewrite the regressions (2.6) as (2.21) y, -x,P+x/PrP> + «, a *,P + 5, where p is defined in (2.7) and where the compound residual £ v is defined by the second equality. If the intrasectoral variance-covariance matrix of the p v is Qo, ECONOMETRIC ISSUES FOR SURVEY DATA 73 the variances and covariances of the new residuals are zero between residuals in different sectors, while within each sector we have (2.22) £(£,0 = X,QPX.; * o\ where In is the n y x n v identity matrix. Hence, if the different sectors in (2.21) are combine'd, or "stacked," into a single regression, the variance-covariance matrix of the residuals will have a block diagonal structure, displaying both heteroske- dasticity and intercorrelation. In such circumstances, neither the weighted nor unweighted regressions will be efficient, and perhaps more seriously, the standard formulas for the estimated standard errors will be incorrect. In the next two sec- tions, we shall see how to detect and deal with these problems in a slightly differ- ent but mathematically identical context. 2.2 The econometrics of clustered samples In Chapter 1, we saw that most household surveys in developing countries use a two-stage design, in which clusters or PSUs are drawn first, followed by a selec- tion of households from within each PSU. In Section 1.4, I explored the conse- quences of clustered designs for the estimation of means and their standard errors. Here I discuss the use of clusters in empirical work more broadly. When the sur- vey data are gathered from rural areas in developing countries, the clustering is often of substantive interest in its own right. I begin with some of these positive aspects of clustered sampling, and then discuss its effects on inference in regres- sion analysis. The economics of clusters in developing countries In surveys of rural areas in developing countries, clusters are often villages, so that households in a single cluster live near one another, and are interviewed at much the same time during the period that the survey team is in the village. In many countries, these arrangements will produce household data where obser- vations from the same cluster are much more like one another than are observa- tions from different clusters. At the simplest, there may be neighborhood effects, so that local eccentricities are copied by those who live near one another and be- come more or less uniform within a village. Sample villages are often widely separated geographically, their inhabitants may belong to different ethnic and religious groups, they may have distinct occupational structures as well as differ- ent crops and cropping patterns. Where agriculture is important—as it is in most poor countries—there will usually be more homogeneity within villages than bet- ween them. This applies not only to the types of crops and livestock, but also to the effects of weather, pests, and natural hazards. If the rains fail for a particular village, everyone engaged in rainfed agriculture will suffer, as will those in occu- pations that depend on rainfed agriculture. If the harvest is good, prices will be low for everyone in the village, and although the effects will spread out to other 74 THE ANALYSIS OF HOUSEHOLD SURVEYS villages through the market, poor transport networks and high transport costs may limit the spread of low prices to other survey villages. Indeed, there is often only one market in each village, so that everyone in the village will be paying the same prices for what they buy, and will be facing the same prices for their wage labor, their produce, and their livestock. This fact alone is likely to induce a good deal of similarity between households within a given sample cluster. Cluster similarity has both costs and benefits, The cost is that inference is sim- plest when all the observations in the sample are independent, and that a positive correlation between observations not only makes calculations more complex, but also inflates variance above what it would have been in the independent case. In the extreme case, when all villagers are clones of one another, we need sample only one of them, and if the sample contains more than one person from each village, the effective sample size is the number of villages not the number of villagers. This argument applies just as much to regressions, and to other types of inference, as it does to the estimation of means. The benefit of cluster sampling comes from the fact that the clusters are vill- ages, and as such are often economically interesting in their own right. For many purposes it makes sense to examine what happens within each village in a differ- ent way from what happens between villages. In addition, cluster sampling gives us multiple observations from the same environment, so that we can sometimes control for unobservables in ways that would not otherwise be possible. One important example is the effects of prices, a topic to which I shall return in Chap- ter 5. Often, we do not observe prices directly, and since prices in each village will typically be correlated with other village variables such as incomes or agri- cultural production, it is impossible to estimate the effects of these observables uncontaminated by the effects of the unobservable prices. However, if we are prepared to maintain that prices have additive effects on the variable in which we are interested, differences between households within a village are unaffected by prices, and can be used to make inferences that are robust to the lack of price data. In this way the village structure of samples can be turned to advantage. Estimating regressions from clustered samples If the cluster design of the data is ignored, standard formulas for variances of estimated means are too small, a result which applies in essentially the same way to the formulas for the variance-covariance matrices of regression parameters esti- mated by OLS. At the very least then, we require some procedure for correcting the estimated standard errors of the least squares regression. There is also an efficiency issue; because the error terms in the regressions are correlated across observations, OLS regression is not efficient even within the class of linear estima- tors and it might be possible to do better with some other linear estimator. (Effi- ciency is also a potential issue for the sample mean, though I did not discuss it in Section 1.4.) The simplest example with which to begin is where the cluster design is bal- anced, so that there are m households in each cluster, and where the explanatory ECONOMETRIC ISSUES FOR SURVEY DATA 75 variables vary only between clusters, and not within them. This will be the case, for example, when we are studying the effects of prices on behavior and there is only one market in each village, or when the explanatory variables are govern- ment services, like schools or clinics, where access is the same for everyone in the same village. I follow the discussion in Section 1.4 on the superpopulation ap- proach to clustering and write the regression equation for household i in cluster c [compare (1.64)], (2-23) yk =* e Va e + ete =* e 'p+« t e so that the jc's are common to all households in the cluster, and the regression error term u.c is the sum of a cluster component occ and an individual component e/c. Both components have mean 0, and their covariance structure can be derived from the assumption that the cc's are uncorrelated across clusters, and the e's both within and across clusters. Hence, £(«*) = o2 = o*+o* / 7 1 (2.24) E(uicujc) = a2a = — -H—\ a2 - pa 2 , iVy lo. + o € J E(uicujc,) = 0, c*c!. Within the cluster, the errors are equicorrelated with intracluster correlation coef- ficient p, but between clusters, they are uncorrelated. This case has been analyzed by Kloek (1981), who shows that the special structure implies that the OLS estimator and the generalized least squares estimator are identical, so that OLS is fully efficient. Further, the true variance-covariance matrix of the OLS estimator—as well as of the generalized least squares (GLS) esti- mator—is given by (2.25) V(fo = a 2 ( X / X ) - 1 [ l + ( m - l ) p ] so that, just as in estimating the variance of the mean, the variance has to be scal- ed up by the design effect, a factor that varies from 1 to m, depending on the size of p. As before, ignoring the cluster design will lead to standard errors that are too small, and ?-values that are too large. There is also a (lesser) problem with estimating the regression standard error a2. If N is the sample size—the number of clusters n multiplied by m, the number of observations in each—and k is the number of regressors, the standard formula (N-k)~l e 'e is no longer unbiased for a2, although it remains consistent provided the cluster size remains fixed as the sam- ple size expands. Kloek shows that an unbiased estimator can be calculated from the design effectd = 1 + (m-l)p using the formula (2.26) a2 = efe(N-kdyl. 76 THE ANALYSIS OF HOUSEHOLD SURVEYS Moulton (1986, 1990) provides a number of examples of potential underesti- mation of standard errors in this case, some of which are dramatic. For example, in an individual wage equation for the U.S. with only state-level explanatory vari- ables, the design effect is more than 10; here a small but significant intrastate correlation coefficient, 0.028, is combined with very large cluster sizes, nearly 400 observations per state. In this case, ignoring the correction to (2.25) would understate standard errors by a factor of more than three. That this is likely to be the worst case is shown in papers by Scott and Holt (1982) and Pfefferman and Smith (1985). They show that when the explanatory variables differ within clusters, (2.25)—or when there are unequal numbers of observations in each cluster, (2.25) with the size of the largest cluster replacing m—provides an upper bound for the true variance-covariance matrix, and that in most cases, the bound is not tight. They also show that, although the OLS estima- tor is inefficient when the explanatory variables are not constant within clusters, the efficiency losses are typically small. These results are comforting because they provide a justification for using OLS, and a means of assessing the maximal extent to which the design effects are biasing standard errors. Even so, the biases might still be large enough to worry about, and to warrant correction. One obvious possibility is to estimate by OLS, use the residuals to estimate a2 from (2.26) — or even from the standard formula — as well as an estimate of the intracluster correlation coefficient n m m E E Eeice. (2.27) = c-iyi*j iCJC nm(m-\)d2 and then to estimate the variance-covariance matrix using (2.28) V(p) = d2(X/X)"lX/AX(X/X)-{ where A is a block-diagonal matrix with one block for each cluster, and where each block has a unit diagonal and a p in each off-diagonal position. An alterna- tive and more robust procedure is to use the OLS residuals from each cluster ec to form the cluster matrices Sc according to (2.29) 2C = ece'c and then to place these matrices on the diagonal of A in (2.28). This is equivalent to calculating the variance-covariance matrix using (2.30) V((3) = (X'X)-l(ixteee!Xc)(X'X)-1. c=l Provided that the cluster size remains fixed as the sample size becomes large— which is usually the case in practice—(2.30) will provide a consistent estimate of the variance-covariance matrix of the OLS estimator, and will do so even if the error variances differ across clusters, and even in the face of arbitrary correlation ECONOMETRIC ISSUES FOR SURVEY DATA 77 patterns within clusters (see White 1984, pp. 134-42.) In consequence, it can also be applied to the case of heterogeneity within strata discussed in the previous section; the strata are simply thought of as clusters, and the same analysis applied. As we shall see in Section 2.4 below, the same procedures can also be applied to the analysis of panel data where there are repeat observations on the same indivi- duals—the individuals play the role of the village, and successive observations play the role of the villagers (see also Arellano 1987). Note that the consistency of (2.30) does not suppose (or require) that the Sc matrices in (2.29) are consistent estimates of the cluster variance-covariance ma- trices; indeed it is clearly impossible to estimate these matrices consistently from a single realization of the cluster residuals. Nevertheless, (2.30) is consistent for the variance-covariance matrix of the parameters, and will presumably be more accurate in finite samples the more clusters there are, and the smaller is the cluster size relative to the number of clusters. Although (2.30) will typically require spe- cial coding or software, it is implemented in STATA as the option "group" in the "huber" or "hreg" command. Table 2.1 shows the effects of correcting the standard errors of "quality choice" regressions using data on the unit values—expenditures divided by quan- tities bought—of consumer purchases from the Pakistan Household Income and Expenditure Survey of 1984-85. The substantive issue here is that, because dif- ferent households buy different qualities of goods, even within categories such as rice and wheat, unit values vary systematically over households, with richer households reporting higher values. The OLS estimates of the expenditure elasticity of the unit values—what Prais and Houthakker (1955) christened "quality" elasticities—are given in the first column, and we see that there are quality elasticities of 0.13 for wheat and rice, while for the other two goods, which are relatively homogeneous and whose prices are supposedly controlled, the elasticities are small or even negative. Household size elasticities are the opposite sign to total expenditure elasticities, as would be the case (for example) if quality depended on household expenditure per head. Except for sugar, the size elasticities are all smaller in absolute value than the expenditure elasticities, so that, at constant per capita expenditure, unit values rise with household size, an effect that Prais and Houthakker attributed to economies of scale to household size. At the same level of per capita total ex- Table 2.1. Effects of cluster design on regression f-values, rural Pakistan, 1984-85 Expenditure t-value Size t-value Good elasticity Raw Robust elasticity Raw Robust Wheat 0.128 20.2 18.4 -0.070 -10.5 -9.0 Rice 0.129 12.2 8.7 -0.074 -6.9 -5.4 Sugar 0.005 3.1 1.5 -0.009 -5.2 -3.7 Edible oils -0.004 -3.0 -1.9 0.002 1.6 1.2 Note: Underlying regression has the logarithm of unit value as the dependent variable, and the logarithms of household total expenditure and of household size as independent variables. Source: Author's calculations using the Household Income and Expenditure Survey. 78 THE ANALYSIS OF HOUSEHOLD SURVEYS penditure, larger households are better-off than smaller households and, in con- sequence, buy better-quality foods. The robust f-values are smaller than the uncorrected values, although as suggested by the theoretical results, the ratios of the adjusted to unadjusted values are a good deal smaller than the (square roots of the) design effects. Even so, the reductions in the f-values for the estimated qua- lity elasticities for sugar and edible oils are substantial. Without correction, we would almost certainly (mistakenly) reject the hypothesis that the quality elastici- ties for these two goods are zero; after correction, the f-values come within the range of acceptance. 2.3 Heteroskedasticity and quantile regressions As we shall see in the next chapter, when we come to look at the distributions over households of the various components of living standards—income, con- sumption of various goods and their aggregate—it is rare to find variables that are normally distributed, even after standard transformations like taking logarithms or forming ratios. The large numbers of observations in many surveys permit us to look at the distributional assumptions that go into standard regression analysis, and even after transformation it is rarely possible to justify the textbook assump- tions that, conditional on the independent variables, the dependent variables are independently, identically, and normally distributed. The previous section dis- cussed how a cluster survey design is likely to lead to a violation of conditional independence. In this section, I turn to the "identically distributed" assumption, and consider the consequences of heteroskedasticity. Just as lack of independence appears to be the rule rather than the exception, so does heteroskedasticity seem to be almost always present in survey data. The first subsection looks at linear regression models, at the reasons for het- eroskedasticity, and at its consequences. I suggest that the computation of quan- tile regressions is useful, both in its own right, because quantile regression esti- mates will often have better properties than OLS, as a way of assessing the hetero- skedasticity in the conditional distribution of the variable of interest, and as a stepping stone to the nonparametric methods discussed in the next two chapters. As was the case for clustering, a consequence of heteroskedasticity in regression analysis is to invalidate the usual formulas for the calculation of standard errors, and as with clustering, there exists a straightforward correction procedure. Matters are much less simple when we move from regressions to models with limited dependent variables. In regression analysis, the estimation of scale param- eters can be separated from the estimation of location parameters, but the separa- tion breaks down in probits, logits, Tobits, and in sample selectivity models. I illustrate some of the difficulties using the Tobit model, and provide a simple but realistic example of censoring at zero where the application of maximum-likeli- hood Tobit techniques—something that is nowadays quite routine in the develop- ment literature—can lead to estimates that are no better than OLS. There are cur- rently no straightforward solutions to these difficulties, but I review some of the options and make some suggestions for practice. ECONOMETRIC ISSUES FOR SURVEY DATA 79 Heteroskedasticity in regression analysis It is a fact that regression functions estimated from survey data are typically not homoskedastic. Why this should be is of secondary importance; indeed it is just as reasonable to ask why it should be supposed that conditional expectations should be homoskedastic. Nevertheless, we have already seen in Section 2.1 above that even when individual behavior generates homoskedastic regression functions within strata or villages, but there is heterogeneity between villages, there will be heteroskedasticity in the overall regression function. Similar results apply to het- erogeneity at the individual level. If the response coefficients p differ by house- hold, and we treat them as random, we may write (2.31) E(y,\Xlfit) = P;*,.; V(y,|jc,,p,) = o2. Suppose that the P|t have mean P and variance-covariance matrix Q, then (2.31) generates the heteroskedastic regression model (2.32) E(y.\x.) = p7*,.; V ( y t \ x f ) = o2 + jt/QjC;. Models like (2.32) motivate the standard test procedures for heteroskedasticity such as the Breusch-Pagan (1979) test, or White's (1980) information matrix test (see also Chesher 1984 for the link with individual heterogeneity.) The Breusch- Pagan test is particularly straightforward to implement. The OLS residuals from the regression with suspected heteroskedasticity are first normalized by division by the estimated standard error of the equation. Their squares are then regressed on the variables thought to be generating the heteroskedasticity—if (2.32) is correct, these should include the original jc-variables, their squares, and cross-pro- ducts—and half the explained sum of squares tested against the x2 distribution with degrees of freedom equal to the number of variables in this supplementary regression. In the presence of heteroskedasticity, OLS is inefficient and the usual formulas for standard errors are incorrect. In cases where efficiency is not a prime concern, we may nevertheless want to use the OLS estimates, but to correct the standard errors. This can be done exactly as in (2.30) above, a formula that is robust to the presence of both heteroskedasticity and cluster effects. If there are no clusters, (2.30) can be applied by treating each household as its own cluster so that there are no cross-effects within clusters and the formula can be written (2.33) where x. is the column vector of explanatory variables for household i and e f is the squared residual from the OLS regression. This formula, which comes origin- ally from Bicker (1967) and Huber (1967), was introduced into econometrics by White (1980). Its performance in finite samples can be improved by a number of possible corrections; the simplest requires that ef in (2.33) be multiplied by 80 THE ANALYSIS OF HOUSEHOLD SURVEYS (n -k)~ln, where k is the number of regressors and n the sample size, see David- son and MacKinnon (1993,552-56.) In practice, the heteroskedasticity correction to the variance-covariance matrix (2.33) is usually quantitatively less important than the correction for intracluster correlations, (2.30). Quantile regressions The presence of heteroskedasticity can be conveniently analyzed and displayed by estimating quantile regressions following the original proposals by Koenker and Bassett (1978, 1982). To see how these work, it is convenient to start from the standard homoskedastic regression model. Figure 2.1 illustrates quantiles in the (standard) case where heteroskedasticity is absent. The regression line a + PJC is the expectation of y conditional on jc, and the three "humped" curves schematically illustrate the conditional densities of the errors given jc; in principle, these densities should rise perpendicularly from the page. For each value of jc, consider a process whereby we mark the percentiles of the conditional distribution, and then connect up the same percentiles for different values of x. If the distribution of errors is symmetrical, as shown in Figure 2.1, the conditional mean, or regression function, will be at the 50th percentile or median, so that joining up the conditional medians simply reproduces the regression. When the distribution of errors is also homoskedastic, the percentiles will always be at the same distance from the median, no matter what the value of x. Figure 2.1 shows the lines formed by joining the points corresponding to the 10th and 90th Figure 2.1. Schematic figure of a homoskedastic linear regression function Note: The solid line shows the regression function of y on x( assumed to be linear. The broken lines show the 10th and 90th percentiles of the distribution of y conditional on jr. ECONOMETRIC ISSUES FOR SURVEY DATA 81 percentiles of the conditional distributions. Because the regression is homoske- dastic, these are straight lines that are parallel to, and equidistant from, the regres- sion line. When regressions are heteroskedastic, or when the errors are asymmetric, marking and joining up percentiles will give quite different results. If the residuals are symmetric but heteroskedastic, the distance of each percentile from the regres- sion line will be different at different values of x. Joining up the percentiles for different values of x will not necessarily lead to straight lines or to any other sim- ple curve. However, we can still fit straight lines to the percentiles, and it is this that is accomplished by quantile regression. If the heteroskedasticity is linked to the value of jc, with the distribution of residuals becoming more or less dispersed as ;c becomes larger, then the quantile regressions for percentiles other than the median will no longer be parallel to the regression line, but will diverge from it (or converge to it) for larger values of x. Figure 2.2 illustrates using a food Engel curve for the rural data from the 1984-85 Household Income and Expenditure Survey of Pakistan. Previous experi- ence has shown that the budget share devoted to food can often be well approxi- mated as a linear function of the logarithm of household expenditure per capita, as first proposed by Working (1943). The points in the figure are a 10 percent ran- dom sample of the 9,119 households in the survey whose logarithm of per capita expenditure lies between 3 and 8; a small number of households at the extremes of the distribution are thereby excluded from the figure, but not from the calcula- Figure 2.2, Scatter diagram and quantile regressions for food share and total expenditure, Pakistan, 1984-85 i.o- 90th percentile 0.8- 0.6 0.4- 0.2- 0.0- 3 4 5 6 7 8 Logarithm of household expenditure per head Note: The scatter as shown is a ten percent random sample of the points used in the regressions. The regression lines shown were obtained using the "qreg" command in STATA and correspond to the 10th, 50th, and 90th percentiles. Source: Author's calculations based on Household Income and Expenditure Survey. 82 THE ANALYSIS OF HOUSEHOLD SURVEYS tions. The three lines in the figure are the quantile regressions corresponding to the 10th, 50th, and 90th percentiles of the distribution of the food share condi- tional on the logarithm of household expenditure per head; these were calculated using all 9,119 households. The procedures for estimating these regressions, calculated using the "qreg" command in STATA, are discussed in the technical note that follows, but the principle should be clear from the foregoing discussion. The slopes of the three lines differ; the median regression (50th percentile) has a slope of-0,094 (the OLS slope is -0.091), while the lower line has slope -0.121, and the upper -0.054, These differences and the widening spread between the lines as we move to the right show the increase in the conditional variance of the regression among better-off households; the 10th and 90th percentiles of the con- ditional distribution are much further apart among richer than poorer households. Those with more to spend in total devote a good deal less of their budgets to food, but there is also more dispersion of tastes among them. Quantile regressions are not only useful for discovering heteroskedasticity. By calculating regressions for different quantiles, it is possible to explore the shape of the conditional distribution, something that is often of interest in its own right, even when heteroskedasticity is not the immediate cause for concern. A very simple example is shown in Figure 2.3, which illustrates age profiles of earnings for black and white workers from the 1993 South African Living Standards Sur- vey. Earnings are monthly earnings in the "regular" sector, and the graphs use Figure 2.3. Quantile regressions of the logarithm of earnings on age by race, South Africa, 1993 (log of rand per month) Blacks Whites 9. 9. •a 90th pcrcenUIe 8. 7 - 7. 90th percentile 6 " I Oth percentile 50th percentik 5- lOthpcrcenule 20 30 40 50 60 70 20 30 40 50 60 70 Age Source: Author's calculations using the South African Living Standards Survey, 1993. ECONOMETRIC ISSUES FOR SURVEY DATA 83 only data for those workers who report such earnings. The two panels show the quantile regressions of the logarithm of earnings on age and age squared for the 10th, 50th, and 90th percentiles for Black and White workers separately. The use of a quadratic in age restricts the shapes of the profiles, but allows them to differ by race and by percentile, as in fact they do. The curves show not only the vast differences in earnings between Blacks and Whites—a difference in logarithms of 1 is a ratio difference of more than 2.7—but also that the shapes of the age pro- files are different Those whose earnings are at the top within their age group are the more highly-educated workers in more highly-skilled jobs, and because the human capital required for these jobs takes time to accumulate, the profile at the 90th percentile for whites has a stronger hump-shape than do the profiles for the 50th and 10th percentiles. There is no corresponding phenomenon for Blacks, presumably because, in South Africa in 1993, even the most able Blacks are res- tricted in their access to education and to high-skill jobs. These graphs and the underlying regressions do not tell us anything about the causal processes that generate the differences, but they present the data in an interesting way that can be suggestive of ideas for a deeper investigation (see Mwabu and Schultz 1995 for more formal analysis of earnings in South Africa, Mwabu and Schultz 1996 for a use of quantile regression in the same context, and Buchinsky 1994 for the use of quantile regressions to describe the wage structure in the U.S.) There are also arguments for preferring the parameters of the median regres- sion to those from the OLS regression. Even given the Gauss-Markov assumptions of homoskedasticity and independence, least squares is only efficient within the (restrictive) class of linear, unbiased estimators, although if the conditional distri- bution is normal, OLS will be minimum variance among the broader class of all unbiased estimators. When the distribution of residuals is not nonnal, there will usually exist nonlinear (and/or biased) estimators that are more efficient than OLS, and quantile regressions will sometimes be among them. In particular, the median regression is more resistant to outliers than is OLS, a major advantage in working with large-scale survey data. ^Technical note: calculating quantile regressions In the past, the applicability of quantile regression techniques has been limited, not because they are inherently unattractive, but by computational difficulties. These have now been resolved. Just as in calculating the median itself, median regression can be defined by minimizing the absolute sum of the errors rather than, as in least squares, by minimizing the sum of their squares. It is thus also known as the LAD estimator, for Least Absolute Deviations. Hence, the median regression coefficients can be obtained by minimizing (j) given by (2.34) $ = £lj>r*/pl = £ S * /P ''** = Z[*-l(y,**/P)]fy-*/P> 1= 1 where 0 is the c.d.f. of the standard normal distribution. The first two terms on the right-hand side of (2.42) are exactly those that would appear in the likelihood function of a standard normal regression, and would be the only terms to appear in the absence of censoring. The final term comes from the observations that are censored to zero; for each such observation we do not observe the exact value of the latent variable, only that it is zero or less, so that the contribution to the log likelihood is the logarithm of the probability of that event. Estimates of p and o are obtained by maximizing (2.42), a nonlinear problem whose solution is guar- anteed by the fact the log-likelihood function is convex in the parameters, and so has a unique maximum. This maximum-likelihood technique works well for the left-hand panel of the figure; in the 100 replications, the Tobit estimates of the slope averaged 1.009 with a standard deviation of 0.100. In this case, where the normality assumption is correct, and the disturbances homoskedastic, maximum likelihood overcomes the inconsistency of OLS. That all will not be as well in the presence of heteroskedasticity can be seen from the likelihood function (2.42) where the last term, which is the contribution to the likelihood of the censored observations, contains both the scale and loca- tion parameters. The standard noncensored likelihood function, which is (2.42) without the last term, has the property that the derivatives of the log-likelihood function with respect to the p's are independent of a, at least in expectation, and vice versa, something that is not true for (2.42). This gives a precise meaning to the notion that scale and location are independent in the regression model, but Figure 2.4. Tobit models with and without heteroskedasticity Without heteroskedasticity With heteroskedasticity OLS regression 0 20 40 60 80 100 0 20 40 60 80 100 Independent variable jc Note: See text for model definition and estimation procedures. Source: Author's calculations. 88 THE ANALYSIS OF HOUSEHOLD SURVEYS dependent in these models with limited dependent variables. As a result of the dependence, misspecification of scale will cause the P's that maximize (2.42) to be inconsistent for the true parameters, a result first noted by Hurd (1979), Nelson (1981), and Arabmazar and Schmidt (1981). The right-hand side of Figure 2.4 gives an illustration of the kind of problems that can occur with heteroskedasticity» Instead of being homoskedastic as in the left-hand panel, the u. are drawn from independent normal distributions with zero means and standard deviations o. given by (2.43) o. = 2o(l + 0.2^max(0,jc.-40)). According to this specification there is homoskedasticity to the left of the cutoff point (40), but heteroskedasicity to its right, and the conditional variance grows with the mean of the dependent variable beyond the cutoff. Although (2.43) does not pretend to be based on any actual data, it mimics reasonable models of behav- ior. Richer households have more scope for idiosyncracies of behavior than do the poor, and as we see in the right-hand panel, we now get zero observations among the rich as well as the poor, something that cannot occur in the homoskedastic model. This is what happens in practice; if we look at the demand for tobacco, alcohol, fee-paying schools or clinics, there are more nonconsumers among the poor, but there are also many better-off households who choose not to purchase. Not purchasing is partly a matter of income, and partly a matter of taste. The figure shows three lines. The dots-and-dashes line to the left is the OLS regression which is still biased downward; although the heteroskedasticity has generated more very high /s at high levels of x, the censoring at low values of jc keeps the OLS slope down. In the replications the OLS slope averaged 0.699 with a standard deviation of 0.100; there is more variability than before, but the bias is much the same. The second, middle (solid) line is the kinked line max(0,;c -40) which is (2,41) when all the a. are zero. (Note that this line is not the regression function, which is defined as the expectation of y conditional on x.) The third line, on the right of the picture, comes from maximizing the likelihood (2.42) under the (false) assumption that the w's are homoskedastic. Because the Tobit proce- dure allows it to deal with censoring at low values of jc, but provides it with no explanation for censoring at high values of x, the line is biased upward in order to pass through the center of the distribution on the right of the picture. The average MLE (Tobit) estimate of the slope in the replications was 1.345 with a standard error of 0.175, so that in the face of the heteroskedasticity, the Tobit procedure yields estimates that are as biased up as OLS is biased down. It is certainly possi- ble to construct examples where the Tobit estimators are better than least squares, even in the presence of heteroskedasticity. But there is nothing odd about the cur- rent example; heteroskedasticity will usually be present in practical applications, and there is no general guarantee that the attempt to deal with censoring by re- placing OLS with the Tobit MLE will give estimates that reduce the bias. This is not a defense of OLS, but a warning against the supposition that Tobit guarantees any improvement. ECONOMETRIC ISSUES FOR SURVEY DATA 89 In practice, the situation is worse than in this example. Even when there is no heteroskedasticity, the consistency of the Tobit estimates requires that the distri- bution of errors be normal, and biases can occur when it is not (see Goldberger 1983 and Arabmazar and Schmidt 1982). And since the distribution of the w's is almost always unknown, it is unclear how one might respecify the likelihood function in order to do better. Even so, censored data occur frequently in practice, and we need some method for estimating sensible models. There are two very dif- ferent approaches; the first is to look for estimation strategies that are robust against heteroskedasticity of the w's in (2.41) and that require only weak assump- tions about their distribution, while the second is more radical, and essentially abandons the Tobit approach altogether, I begin with the former. ^Robust estimation of censored regression models There are a number of different estimators that implement the first approach, yielding nonparametric Tobit estimators— nonparametric referring to the distribu- tion of the w's, not to the functional form of the latent variable which remains linear. None of these has yet passed into standard usage, and I review only one, PowelFs (1984) censored LAD estimator. It is relatively easily implemented and appears to work in practice. (An alternative is Powell's (1986) symmetrically trimmed least squares estimator.) One of the most useful properties of quantiles is that they are preserved under monotone transformations; for example, if we have a set of positive observations, and we take logarithms, the median of the logarithms will be the logarithm of the median of the untransformed data. Since max(0, z ) is monotone nondecreasing in z, we can take medians of (2.41) conditional on x. to get = (2.44) ^M) ™x[0>950(jc/P + M.ljc.)] = max(0,jc/P) where c + «fc *0 + P* c - Wc + ac + efc where yic is demand by household i in cluster c, xic is income or some other indi- vidual variable, pc is the common village price, and uic is the error term. As in previous modeling of clusters, I assume that uic is the sum of a village term ac and an idiosyncratic term e.c, both of which are mean-zero random variables. Suppose that aggregate supply for the village is zc per household, which comes from a weather-affected harvest but is unresponsive to price (or to income). Mar- ket clearing implies that (2.49) zc * yc = «„ + P*c - ypc + ac which determines price in terms of the village taste effect, supply, and average village income. Because markets have to clear at the village level, the price is higher in villages with a higher taste for the commodity. In consequence, the price on the right-hand side of (2.48) is correlated with the ccc component of the error term, and OLS estimates will be inconsistent. The inconsistency arises even if the village contains many households, each of which has a negligible effect on price. The bias can be large in this case. To make things simple, assume that P - 0, so that income does not appear in (2.48) nor average income in (2.49). According to the latter, price in village c is (2.50) Pc = Y~V 0 + a c -z c ). Write ^ for the OLS estimate of y obtained by regressing individual household demands on the price in the village in which the household lives. Provided that tastes are uncorrelated with harvests, it is straightforward to show that 2 Yoz hm = (2.51) P ? -: T- °« + °z The price response is biased downwards; in addition to the negative effect of price on demand, there is a positive effect from demand to price that comes from the effect on both of village-level tastes. The bias will only vanish when the vil- lage taste effects ac are absent, and will be large if the variance of tastes is large relative to the variance of the harvest. ECONOMETRIC ISSUES FOR SURVEY DATA 95 Example 2. Farm size and farm productivity Consider a model of the determinants of agricultural productivity, and in particu- lar the old question of whether larger or smaller farms are more productive; the observation of an inverse relationship between farm size and productivity goes back to Chayanov (1925), and has acquired the status of a stylized fact; see Sen (1962) for India and Berry and Cline (1979) for reviews. To examine the proposition, we might use survey data to regress output per hectare on farm size and on other variables not shown, viz. (2.52) where Q. is farm output, A. is farm size, and the common finding is that P < 0, so that small farms are "more productive*' than large farms. This might be inter- preted to mean that, compared with hired labor, family labor is of better quality, more safely entrusted with valuable animals or machinery, and needs less moni- toring (see Feder 1985; Otsuka, Chuma, and Hayami 1992; and Johnson and Ruttan 1994), or as an optimal response by small farmers to uncertainty (see Srinivasan 1972). It has also sometimes been interpreted as a sign of inefficiency, and of dualistic labor markets, because in the absence of smoothly operating labor markets farmers may be forced to work too much on their own farms, pushing their marginal productivity below the market wage (see particularly Sen 1966, 1975). However, if a relationship like (2.52) is estimated on a cross section of farms, and even if the amount of land is outside the control of the farmer, (2.52) is likely to suffer from what are effectively simultaneity problems. Such issues have the distinction of being among the very first topics studied in the early days of econometrics (see Marschak and Andrews 1944). Although it may be reasonable to suppose that the farmer treats his farm size as fixed when deciding what to plant and how hard*to work, this does not mean that A. is uncorrelated with «. in (2.52). Farm size may not be controlled by the farmer, but farms do not get to be the size they are at random. The mechanism determining farm size will differ from place to place and time to time, but it is unlikely to be independent of the quality of the land. "Desert" farms that are used for low-intensity animal grazing are typically larger than "garden" farms, where the land is rich and output per hectare is high. Such a correlation will be present whether farms are allocated by the market—low-quality land is cheaper per hect- are so that it is easier for an owner-occupier to buy a large farm—or by state- mandated land schemes—each farmer is given a plot large enough to make a living. In consequence, the right-hand side of (2.52) is at least partly determined by the left-hand side, and regression estimates of p will be biased downward. We can also give this simultaneity an omitted variable interpretation where land quality is the missing variable; if quality could be included in the regression instead of in the residual, the new residual could more plausibly be treated as orthogonal to farm size. At the same time, the coefficient p would more nearly measure the effect of land size, and not as in (2.52) the effect of land size contain- 96 THE ANALYSIS OF HOUSEHOLD SURVEYS inated by the (negative) projection of land quality on farm size. Indeed, when data are available on land quality— Bhalla and Roy (1988)—or when quality is con- trolled by IV methods—Benjamin (1993)—there is little or no evidence of a nega- tive relationship between farm size and productivity. The effect of an omitted variable is worth recording explicitly, since the for- mula is one of the most useful in the econometrician's toolbox, and is routinely used to assess results and to calculate the direction of bias caused by the omission. Suppose that the correct model is (2.53) y{ = « + P*{- + Y*/ + «, and that we have data on y and x, but not on z. In the current example, y is yield, and z is land quality. If we run the regression of y on jc, the probability limit of the OLS estimate of p is (2.54) plimp=P varjc In the example, it might be the case that p = 0, so that farm size has no effect on yields conditional on land quality. But y > 0, because better land has higher yields, and the probability limit of (5 will be negative because farm size and land quality are negatively correlated. The land quality problem arises in a similar form if we attempt to use equa- tions like (2.52) to measure the effects on output of extension services or "mod- ern" inputs such as chemical fertilizer. Several studies, Bevan, Collier and Gun- ning (1989) for Kenya and Tanzania, and Deaton and Benjamin (1988) for Cote d'lvoire, find that a regression of output on fertilizer input shows extremely high returns, estimates that, if correct, imply considerable inefficiency and scope for government intervention. Deaton and Benjamin use the 1985 Living Standards Survey of Cote d'lvoire to estimate the following regression between cocoa output, the structure of the orchard, and the use of fertilizer and insecticide, ln(Q ILM ) = 5.621 + 0.526 (LO ILM ) + 0.054 Insect (68.5) (4.3) (2.5) (2.55) (2.8) where Q is kilos of cocoa produced on the farm, LM and LO are the numbers of hectares of "mature" and "old" trees, respectively, and Insect and Pert are ex- penditures in thousands of Central African francs per cocoa hectare on insecticide and fertilizer, respectively. According to (2.55), an additional 1,000 francs spent on fertilizer will increase the logarithm of output per hectare by 0.158, which at a sample mean log yield of 5.64 implies an additional 48 kilos of cocoa at 400 francs per kilo, or an additional 19,200 francs. However, only slightly more than a half of the cocoa stands are fully mature, and the farmers pay the mettayeurs who harvest the crop between a half and a third of the total. But even after these ECONOMETRIC ISSUES FOR SURVEY DATA 97 adjustments, the farmer will be left with a return of 5,400 for an outlay of 1,000 francs. Insecticide is estimated to be somewhat less profitable, and the same cal- culation gives a return of only 1,800 for each 1,000 francs outlay. Yet only 1 in 14 farmers uses fertilizer, and 1 in 5 uses insecticide. On the surface, these results seem to indicate very large inefficiencies. How- ever, there are other interpretations. It is likely that highly productive farms are more likely to adopt fertilizer, particularly if the use of fertilizer is an indicator of farmer quality and the general willingness to adopt modern methods, high-yield- ing varieties, and so on. Credit for fertilizer purchases may only be available to better, or to better-off farmers. Suppose also that some farmers cannot use fertil- izer because of local climatic or soil conditions or because the type of trees in their stand, while others have the conditions to make good use of it. When we compare these different farms, we shall find what we have found, that farmers that use fertilizer are more productive, but there is no implication that more fertil- izer should be used. Expenditure on fertilizer in (2.55) may do no more than indicate that the orchard contains new hybrid varieties of cocoa trees, something on which the survey did not collect data. Example 3. The evaluation of projects Analysis of the effectiveness of government programs and projects has always been a central topic in development economics. Regression analysis seems like a helpful tool in this endeavor, because it enables us to link outcomes—incomes, consumption, employment, health, fertility—to the presence or extent of pro- grams designed to influence them. The econometric problems of such analyses are similar to those we encountered when linking farm outputs to farm inputs. In particular, it is usually impossible to maintain that the explanatory variables—in this case the programs—are uncorrelated with the regression residuals. Govern- ment programs are not typically run as experiments, in which some randomly selected groups are treated and others are left alone. A regression analysis may show that health outcomes are better in areas where the government has put clinics, but such an analysis takes no account of the pro- cess whereby sites are chosen. Clinics may be put where health outcomes were previously very poor, so that the cross-section regression will tend to underesti- mate their effects, or they may be allocated to relatively wealthy districts that are politically powerful, in which case regression analysis will tend to overstate their true impact. Rosenzweig and Wolpin (1986) found evidence of underestimation in the Philippines, where the positive effect of clinics on children's health did not show up in a cross section of children because clinics were allocated first to the areas where they were most needed. The clinics were being allocated in a desir- able way, and that fact caused regression analysis to fail to detect the benefits. In the next section, I shall follow Rosenzweig and Wolpin and show how panel data can sometimes be used to circumvent these difficulties. I shall return to the issue of project evaluation later in this section when I come to discuss selection bias, and again in Section 2.6 on IV estimation. 98 THE ANALYSIS OF HOUSEHOLD SURVEYS Example 4. Simultaneity and lags: nutrition and productivity It is important to realize that in cross-section data, simultaneity cannot usually be avoided by using lags to ensure that the right-hand side variables are prior in time to the left-hand side variables. If x precedes yt then it is reasonable to suppose that y cannot affect x directly. However, there is often a third variable that affects y today as well as x yesterday, and if this variable is omitted from the regression, today's y will contain information that is correlated with yesterday's x. The land quality issue in the previous example can be thought about this way; although farm size is determined before the farmer's input and effort decisions, and before they and the weather determine farm output, both output and inputs are affected by land quality, so that there remains a correlation between output and the prede- termined variables. As a final example, consider one of the more intractable cases of simultaneity, between nourishment and productivity. If poor people cannot work because they are malnourished, and they cannot eat because they do not earn enough, poor people are excluded from the labor market and there is persis- tent unemployment and destitution. The theory of this interaction was developed by Mirrlees (1975) and Stiglitz (1976), and it has been argued that such a mecha- nism helps account for destitution in India (Dasgupta 1993) and for the slow pace of premodern development in Europe (Fogel 1994), People who eat better may be more productive, because they have more en- ergy and work more efficiently, but people who work more efficiently also earn more, out of which they will spend more on food. Disentangling the effect of nutrition on wages from the Engel curve for food is difficult, and as emphasized by Bliss and Stern (1981), it is far from clear that the two effects can ever be - disentangled. One possibility, given suitable data, is to suppose that productivity depends on nutrition with a lag— sustained nutrition is needed for work—while consumption depends on current income. Hence, if yu is the productivity of indi- vidual i at time f, and cu is consumption of calories, we might write + c 56 cit = «i y a + Pi ir-i+Yi*ift + BH» it = 2 hyit^2z2it^u2il ( ' } where zl and z2 arc other variables needed to identify the system. Provided equa- tion (2.56) is correct and the two error terms are serially independent, both equa- tions can consistently be estimated by least squares in a cross section with infor- mation on lagged consumption. However, any form of serial dependence in the residuals uw will make OLS estimates of the first equation inconsistent. But there is a good reason to suppose that these residuals will be serially correlated, since permanent productivity differences across people that are not attributable to nutri- tion or the other variables will add a constant "individual" component to the error. Individuals who are more productive in one period are likely to be more produc- tive in the next, even when we have controlled for their nutrition and other ob- servable covariates. More productive individuals will have higher incomes and ECONOMETRIC ISSUES FOR SURVEY DATA 99 higher levels of nutrition, not only today but also yesterday, so that the lag in the equation no longer removes the correlation between the error term and the-right- hand-side variable. In a cross section, predetermined variables can rarely be legit- imately treated as exogenous. Measurement error Measurement error in survey data is a fact of life, and while it is not always pos- sible to counter its effects, it is always important to realize what those effects are likely to be, and to beware of inferences that are possibly attributable to, or con- taminated by, measurement error. The textbook case is the univariate regression model where both the explana- tory and dependent variables are subject to mean-zero errors of measurement. Hence, for the correctly measured variables y and x, we have the linear relation- ship (2.57) y,- = « + P*; + M, together with the measurement equations (2.58) *, = xi + e,. y. = y. + e2(. where the measurement error is assumed to be orthogonal to the true variables. Faute de mieux, y is regressed on Jc, and the OLS parameter estimate of p has the probability limit ft Pmj« (2.59) F r 2 " 0 m ** + °l where m^ is the variance of the unobservable, correctly measured x, and o] is the variance of the measurement error in x. Equation (2.59) is the "iron law of econo- metrics," that the OLS estimate of P is biased towards zero, or "attenuated." The degree of attenuation is the ratio of signal to combined signal and noise, X0, the reliability ratio. The presence of measurement error in the dependent variable does not bias the regression coefficients, because it simply adds to the variance of the equation as a whole. Of course, this measurement error, like the measurement error in jc, will decrease the precision with which the parameters are estimated. Attenuation bias is amplified by the addition of correctly measured explana- tory variables to the bivariate regression (2.57). Suppose we add a vector z to the right-hand side of (2.57), and assume that z is uncorrelated with the measurement error in x and with the original residuals. Then the probability limit of the OLS estimate of p, the coefficient of Jc, is now pA p where the new reliability ratio (2.60) A -* /V| 1 »"* ———«—^ o . /*/\ -^ Jl O 1-4 100 THE ANALYSIS OF HOUSEHOLD SURVEYS and R^ is the R 2 from the regression of x on z. The new explanatory variables z "soak up" some of the signal from the noisy regressor ;c, so that the reliability ratio for P is reduced, and the "iron law" more severely enforced. More generally, consider a multivariate regression where all regressors may be noisy and where the measurement error in the independent variables may be cor- related with the measurement error in the dependent variable. Suppose that the correctly measured variables satisfy (2.61) y = Xp + u. Then the OLS parameter estimates have probability limits given by (2.62) plimB - (M + Sl)-lMR + (M + nyly where M is the moment matrix of the true jc's, Q is the variance-covariance ma- trix of the measurement error in the Jc's, and y is the vector of covariances be- tween the measurement errors in the Jc's and the measurement error in y. The first term in (2.62) is the matrix generalization of the attenuation effect in the univariate regression—the vector of parameters is subject to a matrix rather than scalar shrinkage factor— while the second term captures any additional bias from a correlation between the measurement errors in dependent and independent vari- ables. The latter effects can be important; for example, if consumption is being regressed on income, and if there is a common and noisily measured imputation term in both—home-produced food, or the imputed value of owner-occupied housing—then there will be an additional source of bias beyond attenuation ef- fects. Even in the absence of this second term on the right-hand side of (2.62) and, in spite of the obvious generalization from scalar to matrix attenuation, the result does not yield any simple result on the direction of bias in any one coeffi- cient (unless, of course, Q is diagonal). One useful general lesson is to be specific about the structure of measurement error, and to use a richer and more appropriate specification than the standard one of mean-zero, independent noise. The analysis is rarely complex, is frequently worthwhile, and will not always lead to the standard attenuation result. One spe- cific example is worth a brief discussion. It arises frequently and is simple, but is nevertheless sometimes misunderstood. Consider the model (2.63) Jic - a + p* f c +yz c + Mte where / is an individual who lives in village c, yic is an outcome variable, xic and zc are individual and village-level explanatory variables. In a typical example, y might be a measure of educational attainment, x a set of family background vari- ables, and z a measure of educational provision or school quality in the village. The effect of health provision on health status might be another example. What often happens in practice is that the z- variables are obtained from administrative, not survey data, so that we do not have village-level data on z, but only broader ECONOMETRIC ISSUES FOR SURVEY DATA 101 measures, perhaps at a district or provincial level. These measures are error-rid- den proxies for the ideal measures, and it might seem that the iron law would apply. But this is not so. To see why, write z for the broad measure—p is for province—so that Z (2.64) P = npl Ez c cep where n is the number of villages in the province. Hence, instead of the measurement equation (2.58) where the observable is the unobservable plus an unrelated measurement error, we have (2.65) Z c = ZP +£ c and it is now the observable z that is orthogonal to the measurement error. Be- cause the measurement error in (2.65) is the deviation of the village-level z from its provincial mean, it is orthogonal to the observed z by construction. As a result, when we run the regression (2,63) with provincial data replacing village data, there is no correlation between the explanatory variables and the error term, and the OLS estimates are unbiased and consistent. Of course, the loss of the village-level information is not without cost. By (2.65), the averages are less vari- able than the individuals, so that the precision of the estimates will be reduced. And we must always be careful in these cases to correct standard errors for group effects as discussed in Section 2,2 above. But there is no errors-in- variables atten- uation bias. In Section 2.6 below, I review how, in favorable circumstances, IV techniques can be used to obtain consistent estimates of the parameters even in the presence of measurement error. Note, however, that if it is possible to obtain estimates of measurement error variances and covariances, o \ in (2.59) or Q and y in (2.62), then the biases can be corrected and consistent estimates obtained. by substituting the OLS estimate on the left-hand side of (2.62), replacing Q , y, and M on the right-hand side by their estimates, and solving for p. For (2.62), this leads to the estimator (2.66) b = (X / X-nQ)- 1 (Xy where n is the sample size, and the tildes denote variables measured with error. The estimator (2.66) is consistent if Q and y are known or are replaced by consis- tent estimates. This option will not always be available, but is sometimes possible, for example, when there are several mismeasured estimates of the same quantity, and we shall see practical examples in Section 5.3 and 5.4 below. Selectivity issues In Chapter 1 and the first sections of this chapter, I discussed the construction of samples, and the fact that the sample design frequently needs to be taken into account when estimating characteristics of the underlying population. This is 102 THE ANALYSIS OF HOUSEHOLD SURVEYS particularly important when the selection of the sample is related to the quantity under study; average travel time in a sample of travelers is likely to be quite unrepresentative of average travel time among the population as a whole: if wages influence the decision to work, average wages among workers—which are often the only wages observed—will be an upward-biased estimator of actual and po- tential wages. Sample selection also affects behavioral relationships. In one of the first and most famous examples, Gronau (1973) found that women's wages were higher when they had small children, a result whose inherent implausibility prompted the search for an alternative explanation, and which led to the selection story. Women with children have higher reservation wages, fewer of them work, and the wages of those who do are higher. As with the other cases in this section, the econometric problem is the induced correlation between the error terms and the regressors. In the Gronau example, the more valuable is a woman's time at home, the larger will have to be the unobserved component in her wages in order to induce her to work, so that among working women, there is a positive correla- tion between the number of children and the error term in the wage equation. A useful and quite general model of selectivity is given in Heckman (1990); according to this there are two different regressions or regimes, and the model switches between them according to a dichotomous "switch" that is itself ex- plained. The model is written: = (2.67) yw = *o/Po + "or' Vu *uPi+Mi/ together with the {1,0} variable di which satisfies (2.68) 0 ) where the indicator function 1(.) takes the value 1 when the statement it contains is true, and is zero otherwise. The observed variable yi is determined according to (2.69) y, = 4yyir The model is sometimes used in almost exactly this form; for example, the two equations in (2.67) could be wage equations in the formal and informal sectors respectively, while (2.68) models the decision about which sector to join (see, for example, van der Gaag, Stelcner and Vijverberg 1989 for a model of this sort applied to LSMS data from Peru and Cote d'lvoire). However, it also covers sev- eral special cases, many of them useful in their own right. If the right-hand side of the second equation in (2.67) were zero, as it would be if Pj =0 and the variance of ul were zero, we would have the censored regres- sion model or generalized Tobit. This further specializes to the Tobit model if the argument of (2.68) and the right-hand side of the first equation coincide, so that the switching behavior and the size of the response are controlled by the same factors. However, the generalized Tobit model is also useful; for example, it is often argued that the factors that determine whether or not people smoke tobacco ECONOMETRIC ISSUES FOR SURVEY DATA 103 are different from the factors that determine how much smokers smoke. In this case, (2.69) implies that for those values of y that are positive, the regression function is (2.70) E(yi\^zryi>0) = */p + X(z/y) where, since there is only one x and one p, I have dropped the zero suffix, and where the last term is defined by (2.71) X(z/Y) = E(uQi\u2i* -z/Y). (Compare this with the Tobit in (2.47) above.) This version of the model can also be used to think about the case where the data are truncated, rather than censored as in the Tobit and generalized Tobit. Censoring refers to the case where obser- vations that fall outside limits — in this case below zero— are replaced by the limit points, hence the term "censoring.'* With truncation, observations beyond the limit are discarded and do not appear in our data. Censoring is easier to deal with because, although we do not observe the underlying latent variable, individual ob- servations are either censored or not censored, and for both we observe the covariates x and z, so that it is possible to estimate the switching equation (2.68) as well as (2.70). With truncation, we know nothing about the truncated observa- tions, so that we cannot estimate the switching process, and we are restricted to (2.70). The missing information in the truncated regression makes it difficult to handle convincingly, and it should be avoided when possible. A second important special case of the general model is the "treatment" or "policy evaluation" case. In the standard version, the right-hand sides of the two switching regressions in (2.67) are taken to be identical apart from their constant terms, so that (2.69) takes the special form (2.72) y{ = a + 6rf. + jc/p + u. so that the parameter 0 is the effect on the outcome variable of whether or not the "treatment" is applied. If this were a controlled and randomized experiment, the randomization would guarantee that d. would be orthogonal to ur However, since u2 in (2.68) is correlated with the error terms in the regressions in (2.67), least squares will not yield consistent estimates of (2.72) because di is correlated with ur This model is the standard one for examining union wage differentials, for example, but it also applies to many important applications in development where d. indicates the presence of some policy or project. The siting of health clinics and schools are the perhaps the most obvious examples. As we have al- ready seen above, this version of the model can also be thought of in terms of simultaneity bias. There are various methods of estimating the general model and its variants. One possibility is to specify some distribution for the three sets of disturbances in (2.67) and (2.68), typically joint normality, and then to estimate by maximum likelihood. Given normality, the y-parameters in (2.68) can be estimated (up to 104 THE ANALYSIS OF HOUSEHOLD SURVEYS scale) by probit, and again given normality, the A-function in (2.71) has a specific form— the (inverse) Mills' ratio— and as Heckman (1976) showed in a famous paper, the results from the probit can be substituted into (2.70) in such a way that the remaining unknown parameters can be estimated by least squares. Since I shall refer to this again, it is worth briefly reviewing the mechanics. When MO and u2 are jointly normally distributed, the expectation of each conditional on the other is linear, so that we can write = a (2.73) KQ,. oPK-/a2) + 8; where st is orthogonal to u2i, aQ and a2 are the two standard deviations, and p is the correlation coefficient. (Note that pa0 /a2 = aQ2 /a2 is the large-sample regres- sion coefficient of UQ on M2, the ratio of the covariance to variance.) Given (2.73), we can rewrite (2.71) as (2.74) X ( z y ) - pa0£ - - > — = pa0 (v2 a2 a2J (z/y/a2) where q>(.) and O(.) are the density and distribution functions of the standard normal distribution, and where the final formula relies on the special properties of the normal distribution. The regression function (2.70) can then be written as , + P'<*«2-*,-. > + "a-*;. where the notation y. denotes the time mean for individual /. Equation (2.78) can be estimated as a pooled regression by OLS, although it should be noted (a) that there are n(T- 1) independent observations, not nT. Neither (2.77) nor (2.78) contains the individual fixed effects 0 y , so that these regressions are free of any correlation between the explanatory variables and the unobserved fixed effects, and the parameters can be estimated consistently by OLS. Of course, the fixed effect must indeed be fixed over time—which there is often little reason to sup- pose—and it must enter the equation additively and linearly. But given these assumptions, OLS estimation of the suitably transformed regression will yield con- sistent estimates in the presence of unobserved heterogeneity—or omitted vari- ables—even when that heterogeneity is correlated with one or more of the in- cluded right-hand side variables. In the example from the Philippines studied by Rosenzweig and Wolpin (1986), there are data on 274 children from 85 households in 20 barrios. The cross-section regression of child nutritional status (age-standardized height) on exposure to rural health units and family planning programs gives negative (and insignificant) coefficients on both. Because the children were observed in two years, 1975 and 1979, it is also possible to run (2.77), where changes in height are regressed on changes in exposure, in which regression both coefficients become positive. Such a result is plausible if the programs were indeed effective, but were allocated first to those who needed them the most. The benefit of eliminating unobserved heterogeneity does not come without cost, and a number of points should be noted. Note first that the regression (2.77) has exactly half as many observations as the regression (2.76), so that, in order to remove the inconsistency, precision has been sacrificed. More generally, with T periods, one is sacrificed to control for the fixed effects, so that the proportional 108 THE ANALYSIS OF HOUSEHOLD SURVEYS loss of efficiency is greatest when there are only two observations. Of course, it can be argued that there are limited attractions to the precise estimation of some- thing that we do not wish to know, but a consistent but imprecise estimate can be further from the truth than an inconsistent estimator. The tradeoff between bias and efficiency has to be made on a case-by-case basis. We must also beware of misinterpreting a decrease in efficiency as a change in parameter estimates be- tween the differenced and undifferenced equations. If the cross-section estimate shows that p is positive and significant, and if the differenced data yield an esti- mate that is insignificantly different from both zero and the cross-section esti- mate, it is not persuasive to claim that the cross-section result is an artifact of not "treating" the heterogeneity. Second, the differencing will not only sweep out the fixed effects, it will sweep out all fixed effects, including any regressor that does not change over the period of observation. In some cases, this removes the attrac- tion of the procedure, and will limit it in short panels. In the Ivorian cocoa farm- ing example in the previous section, most of the farmers who used fertilizer re- ported the same amount in both periods, so that, although the panel data allows us to control for farm fixed effects, it still does not allow us to estimate how much additional production comes from the application of additional fertilizer. Panel data and measurement error Perhaps the greatest difficulties for difference- and within-estimators occur in the presence of measurement error. Indeed, when regressors are measured with error, within- or difference-estimators will no longer be consistent in the presence of un- observed individual fixed effects, nor need their biases be less than that of the un- corrected OLS estimator. Consider the univariate versions of the regressions (2.76) and (2.77), and com- pare the probability limits of the OLS estimators in the two cases when, in addition to the fixed effects, there is white noise measurement error in x. Again, for sim- plicity, I compare the results from estimation on a single cross section with those from a two-period panel. The probability limit of the OLS estimator in the cross section (2.76) is given by (2.79) p l i m £ = —-—f" m +a ~ ' where cxQ is the covariance of the fixed effect and the true jc, a, is the variance of the measurement error, and I have assumed that the measurement errors and fixed effects are uncorrelated. The formula (2.79) is a combination of omitted variable bias, (2.54), and measurement error bias, (2.59). The probability limit of the difference-estimator in (2.77) is (2.80) where ,mA is the variance of the difference of the true jc, and OA is the variance of the difference of measurement error in jc. ECONOMETRIC ISSUES FOR SURVEY DATA 709 That the estimate in the levels suffers from two biases—attenuation bias and omitted variable bias—while the difference-estimate suffers from only attenuation bias is clearly no basis for preferring the latter! The relevant question is not the number of biases but whether the differencing reduces the variance in the signal relative to the variance of the noise so that the attenuation bias in the difference- estimator is more severe than the combined attenuation and omitted variable biases in the cross-section regression. We have seen one extreme case already; when the true x does not change between the two periods, the estimator will be dominated by the measurement error and will Converge to zero. Although the ext- reme case would often be apparent in advance, there are many cases where the cross-section variance is much larger than the variance in the changes over time, especially when the panel observations are not very far apart in time. Although measurement error may also be serially correlated, with the same individual mis- reporting in the same way at different times, there will be other cases where errors are uncorrelated over time, in which case the error difference will have twice the variance of the errors in levels. Consider again the two examples of farm productivity and nutritional wages, where individual fixed effects are arguably important. In the first case, ra^ is the cross-sectional variance of farm size, while mA is the cross-sectional variance of the change in farm size from one period to another, something that will usually be small or even zero. In the nutritional wage example, there is probably much great- er variation in eating habits between people than there is for the same person over time, so that once again, the potential for measurement error to do harm is much enhanced. One rather different case is worth recording since it is a rare example of direct evidence on measurement error. Bound and Krueger (1991) matched earnings data from the U.S. Current Population Survey with Social Security re- cords, and were thus able to calculate the measurement error in the former. They found that measurement error was serially correlated and negatively related to actual earnings. The reliability ratios—the ratios of signal variance to total vari- ance—which are also the multipliers of P in (2.79) and (2.80), fall from 0.82 in levels to 0.65 in differences for men, and from 0.92 to 0.81 for women. Since measurement error is omnipresent, and because of the relative ineffi- ciency of difference- and within-estimators, we must be careful never to assume that the use of panel data will automatically improve our inference, or to treat the estimate from panel data as a gold standard for judging other estimates. Neverthe- less, it is clear that there is more information in a panel than in a single cross section, and that this information can be used to improve inference. Much can be learned from comparing different estimates. If the difference-estimate has a dif- ferent sign from the cross-sectional estimate, inspection of (2.79) and (2.80) shows that the covariance between x and the heterogeneity must be nonzero; measurement error alone cannot change the signs. When there are several periods of panel data, the difference-estimator (2.77) and the within-estimator (2.78) are mathematically distinct, and in the presence of measurement error will have dif- ferent probability limits. Griliches and Hausman (1986) show how the compari- son of these two estimators can identify the variance of the measurement error 110 THE ANALYSIS OF HOUSEHOLD SURVEYS when the errors are independent over time—so that consistent estimators can be constructed using (2.66). When errors are correlated over time—as will be the case if households persistently make errors in the same direction—information on measurement error can be obtained by comparing parameters from regressions computed using alternative differences, one period apart, two periods apart, and soon. Lagged dependent variables and exogeneity in panel data Although it will not be of great concern for this book, I should also note that there are a number of specific difficulties that arise when panel data are used to esti- mate regressions containing lagged dependent variables. In ordinary linear regres- sions, serial correlation in the residuals makes OLS inconsistent in the presence of a lagged dependent variable. In panel data, the presence of unobserved individual heterogeneity will have the same effect; if farm output is affected by unobserved farm quality, so must be last period's output on the same farm, so that this pe- riod's residual will be correlated with the lagged dependent variable. Nor can the heterogeneity be dealt with by using the standard within- or difference-estimators. When there is a lagged dependent variable together with unobserved fixed effects, and we difference, the right-hand side of the equation will have the lagged differ- ence y.t_} - yit_l2, and although the fixed effects have been removed by the differ- encing, there is a differenced error term ui{ - uit_r which is correlated with the lagged difference because w.M is correlated with ;y.M. Similarly, the within- estimator is inconsistent because the deviation of lagged >>,M from its mean over time is correlated, with the deviation of uit from its mean, not because uit is cor- related with }>.M, but because the two means are correlated. These inconsistencies vanish as the number of time periods in the panel increases but, in practice, most panels are short. Nor are the problems confined to lagged-dependent variables. Even if all the right-hand side variables are uncorrelated with the contemporaneous regression error uir the deviations from their means can be correlated with the average over time, u.. For this not to be the case, we require that explanatory variables be uncorrelated with the errors at all lags and leads, a requirement that is much more stringent than the usual assumption in time-series work that a variable is predeter- mined. It is also a requirement that is unlikely to be met in several of the exam- ples I have been discussing. For example, farm yields may depend on farm size, on the weather, on farm inputs such as fertilizer and insecticide, and on (unob- served) quality. The inputs are chosen before the fanner knows output, but a good output in one year may make the farmer more willing, or more able, to use more inputs in a subsequent year. In such circumstances, the within-regression will eliminate the unobservable quality variable, but it will induce a correlation be- tween inputs and the error term, so that the within-estimator will be inconsistent. These problems are extremely difficult to deal with in a convincing and robust way, although there exist a number of techniques (see in particular Nickell 1981; Chamberlain 1984; Holtz-Eakin, Newey, and Rosen 1988; and particularly the ECONOMETRIC ISSUES FOR SURVEY DATA 111 series of papers, Arellano and Bond 1991, Arellano and Dover 1993, and Alonso- Borrego and Arellano 1996). But too much should not be expected from these methods; attempts to disentangle heterogeneity, on the one hand, and dynamics, on the other, have a long and difficult history in various branches of statistics and econometrics. 2.6 Instrumental variables In all of the cases discussed in Section 2.4, the regression function differs from the structural model because of correlation between the error terms and the ex- planatory variables. The reasons differ from case to case, but it is the correlation that produces the inconsistency in OLS estimation. The technique of IV is the standard prescription for correcting such cases, and for recovering the structural parameters. Provided it is possible to find instrumental variables that are corre- lated with the explanatory variables but uncorrelated with the error terms, then IV regression will yield consistent estimates. For reference, it is useful to record the formulas. If X is the nxk matrix of explanatory variables, and if W is an nxk matrix of instruments, then the IV estimator of p is given by (2.81) PJV = (W'XyWy. Since y = Xp +u and Wis orthogonal to u by assumption, (2.81) yields consistent estimators if the premultiplying matrix W'X is of full rank. If there are fewer instruments than explanatory variables—and some explanatory variables will often be suitable to serve as their own instruments—the IV estimate does not exist, and the model is underidentified. When there are exactly as many instruments as explanatory variables, the model is said to be exactly identified. In practice, it is desirable to have more instruments than strictly needed, because the additional in- struments can be used either to increase precision or to construct tests. In this overidentified case, suppose that Z is an nxkf matrix of potential instruments, with kf>k. Then all the instruments are used in the construction of the set Wby using two-stage least squares, so that at the first stage, each X is regressed on all the instruments Z, with the predicted values used to construct W. If we define the "projection" matrix Pz = Z(Z/Z)"1Z/, the iv estimator is written (2.82) Pn, = (X'Z(Z/Z)'lZfXrlX/Z(Z/Z)'lZ'y = (X*PzX)-lX'Pzy. Under standard assumptions, p^ is asymptotically normally distributed with mean p and a variance-covariance matrix that can be estimated by (2.83) V = (XPzXyl(XfPzDPzX)(XPzXrl. The choice of D depends on the treatment of the variance-covariance matrix of the residuals, and is handled as with OLS, replaced by d2 / under homoskedas- 112 THE ANALYSIS OF HOUSEHOLD SURVEYS ticity, or by a diagonal matrix of squared residuals if heteroskedasticity is sus- pected, or by the appropriate matrix of cluster residuals if the survey is clustered (see (2.30) above). (Note that the residuals must be calculated as y - X f y ^ , which is not the vector of residuals from the second stage of two-stage least squares. However, this is hardly ever an issue in practice, since econometric packages make the correction automatically.) When the model is overidentified, and kf>k, the (partial) validity of the instruments is usually assessed by computing an overidentification (OID) test statistic. The simplest—and most intuitive—way to calculate the statistic is to re- gress the IV residuals y - X$IV on the matrix of instruments Z and to multiply the resulting (uncentered) R2 statistic by the sample size n (see Davidson and Mac- Kinnon 1993, pp. 232-37). (The uncentered R2 is 1 minus the ratio of the sum of squared residuals to the sum of squared dependent variables.) Under the null hypothesis that the instruments are valid, this test statistic is distributed as a %2 statistic with k {-k degrees of freedom. This procedure tests whether, contrary to the hypothesis, the instruments play a direct role in determining y, not just an indirect role, through predicting the jc's. If the test fails, one or more of the instru- ments are invalid, and ought to be included in the explanation of y. Put differ- ently, the OID test tells us whether we would get (significantly) different answers if we used different instruments or different combinations of instruments in the regression. This interpretation also clarifies the limitations of the test. It is a test of overidentification, not of alt the instruments. If we have only k instruments and k regressors, the model is exactly identified, the residuals of the IV regression are orthogonal to the instruments by construction, so that the OID test is mechanically equal to zero, there is only one way of using the instruments, and no alternative estimates to compare. So the OID test, useful though it is, is only informative when there are more instruments than strictly necessary. Although estimation by IV is one of the most useful and most used tools of modern econometrics, it does not offer a routine solution for the problems diag- nosed in Section 2.4. Just as it is almost always possible to find reasons—meas- urement error, omitted heterogeneity, selection, or omitted variables—why the structural variables are correlated with the error terms, so is it almost always dif- ficult to find instruments that do not have these problems, while at the same time being related to the structural variables. It is easy to generate estimates that are different from the OLS estimates. What is much harder is to make the case that these estimates are necessarily to be preferred. Credible identification and estima- tion of structural equations almost always requires real creativity, and creativity cannot be produced to a formula. Policy evaluation and natural experiments One promising approach to the selection of instruments, especially for the treat- ment model, is to look for "natural experiments," cases where different sets of individuals are treated differently in a way that, if not random by design, was effectively so in practice. ECONOMETRIC ISSUES FOR SURVEY DATA 113 One of the best, and certainly earliest, examples is Snow's (1855) analysis of deaths in the London cholera epidemic of 1853-54, work that is cited by Freed- man (1991) as a leading example of convincing statistical work in the social sciences. The following is based on Freedman's account. Snow's hypothesis— which was not widely accepted at the time—was that cholera was waterborne. He discovered that households were supplied with water by two different water com- panies, the Lambeth water company, which in 1849 had moved its water intake to a point in the Thames above the main sewage discharge, and the Southwark and Vauxhall company, whose intake remained below the discharge. There was no sharp separation between houses supplied by the two companies, instead "the mixing of the supply is of the most intimate kind. The pipes of each Company go down all the streets, and into nearly all the courts and alleys,... The experiment, too, is on the grandest scale. No fewer than three hundred thousand people of both sexes, of every age and occupation, and of every rank and station, from gentlefolks down to the very poor, were divided into two groups without their choice, and in most cases, without their knowledge; one group supplied with water containing the sewage of London, and amongst it, whatever might have come from the cholera patients, the other group having water quite free from such impurity," Snow collected data on the addresses of cholera victims, and found that there were 8.5 times as many deaths per thousand among households sup- plied by the Southwark and Vauxhall company. Snow's analysis can be thought of in terms of instrumental variables. Cholera is not directly caused by the position of a water intake, but by contamination of drinking water. Had it then been possible to do so, an alternative analysis might have linked the probability of contracting cholera to a measure of water purity. But even if such an analysis had shown significant results, it would not have been very convincing. The people who drank impure water were also more likely to be poor, and to live in an environment contaminated in many ways, not least by the "poison miasmas" that were then thought to be the cause of cholera. In terms of the discussion of Section 2.4, the explanatory variable, water purity, is correlated with omitted variables or with omitted individual heterogeneity. The identity of the water supplier is an ideal IV for this analysis. It is correlated with the explana- tory variable (water purity) for well-understood reasons, and it is uncorrelated with other explanatory variables because of the "intimate" mixing of supplies and the fact that most people did not even know the identity of their supplier. There are a number of good examples of natural experiments in the economics literature. Card (1989) shows that the Mariel boatlift, where political events in Cuba led to the arrival of 125,000 Cubans in Miami between May and September 1980, had little apparent effect on wages in Miami, for either Cubans or non- Cubans. Card and Krueger (1994) study fast-food outlets on either side of the bor- der between New Jersey and Pennsylvania around the time of an increase in New Jersey's minimum wage, and find that employment rose in New Jersey relative to Pennsylvania. Another example comes from the studies by Angrist (1990) and Angrist and Krueger (1994) into earnings differences of American males by vet- eran status. The "treatment" variable is spending time in the military, and the out- 114 THE ANALYSIS OF HOUSEHOLD SURVEYS come is the effect on wages. The data present somewhat of a puzzle because veterans of World War II appear to enjoy a substantial wage premium over other workers, while veterans of the Vietnam War are typically paid less than other similar workers. The suspicion is that selectivity is important, the argument being that the majority of those who served in Vietnam had relatively low unobservable labor market skills, while in World War II, where the majority served, only those with relatively low skills were excluded from service. Angrist and Krueger (1994) point out that in the late years of World War II, the selection mechanism acted in such a way that those born early in the year had a (very slightly) higher chance of being selected than those born later in the year. They can then use birth dates as instruments, effectively averaging over all indivi- duals born in the same quarter, so that to preserve variation in the averages, Ang- rist and Krueger require a very large sample, in this case 300,000 individuals from the 1980 census. (Large sample sizes will often be required by "natural ex- periments'* since instruments that are convincingly uncorrelated with the residuals will often be only weakly correlated with the selection process.) In the IV esti- mates, the World War II premium is reversed, and earnings are lower for those cohorts who had a larger fraction of veterans. By contrast, Angrist (1990) finds that instrumenting earnings equations for Vietnam veterans using the draft lottery makes little difference to the negative earnings premium experienced by these workers, so that the two studies together suggest that time spent in the military lowers earnings compared with the earnings of those who did not serve. Impressive as these studies are, natural experiments are not always available when we need them, and some cases yield better instruments than others. Because "natural" experiments are not genuine, randomized experiments, the fact that the experiment is effectively (or quasi-) randomized has to be argued on a case-by- case basis, and the argument is not always as persuasive as in Snow's case. For example, government policies only rarely generate convincing experiments (see Besley and Case 1994). Although two otherwise similar countries (towns, or pro- vinces) may experience different policies, comparison of outcomes is always be- deviled by the concern that the differences are not random, but linked to some characteristic of the country (town or province) that caused the government to draw the distinction in the first place. However, it may be possible to follow Angrist and Krueger* s lead in looking, not at programs themselves, but at the details of their administration. The argu- ment is that in any program with limited resources or limited reach, where some units are treated and some not, the administration of the project is likely to lead, at some level, to choices that are close to random. In the World War II example, it is not the draft that is random, but the fact that local draft boards had to fill quotas, and that the bureaucrats who selected draftees did so partially by order of birth. In other cases, one could imagine people being selected because they are higher in the alphabet than others, or because an administrator used a list constructed for other purposes. While the broad design of the program is likely to be politically and economically motivated, and so cannot be treated as an experiment, natural or otherwise, the details are handled by bureaucrats who are simply trying to get the ECONOMETRIC ISSUES FOR SURVEY DATA 115 job done, and who make selections that are effectively random. This is a recipe for project evaluation that calls for intimate knowledge and examination of detail, but it is one that has some prospect of yielding convincing results. One feature of good natural experiments is their simplicity. Snow's study is a model in this regard. The argument is straightforward, and is easily explained to nonstatisticians or noneconometricians, to whom the concept of instrumental vari- ables could not be readily communicated. Simplicity not only aids communica- tion, but greatly adds to the persuasiveness of the results and increases the likeli- hood that the results will affect the policy debate. A case in point is the recent political firestorm in the United States over Card and Krueger's (1994) findings on the minimum wage. Econometric issues for instrumental variables IV estimators are invaluable tools for handling nonexperimental data. Even so, there are a number of difficulties of which it is necessary to be aware. As with other techniques for controlling for nonexperimental inconsistencies, there is a cost in terms of precision. The variance-covariance matrix (2.83) exceeds the corresponding OLS matrix by a positive definite matrix, so that, even when there is no inconsistency, the IV estimators—and all linear combinations of the IV esti- mates—will have larger standard errors than their OLS counterparts. Even when OLS is inconsistent, there is no guarantee that in individual cases, the IV estimates will be closer to the truth, and the larger the variance, the less likely it is that they will be so. It must also be emphasized that the distributional theory for IV estimates is asymptotic, and that asymptotic approximations may be a poor guide to finite sample performance. Formulas exist for the finite sample distributions of rv esti- mators (see, for example, Anderson and Sawa 1979) but these are typically not sufficiently transparent to provide practical guidance. Nevertheless, a certain amount is known, and this knowledge provides some warnings for practice. Finite sample distributions of rv estimators will typically be more dispersed with more mass in the tails than either OLS estimators or their own asymptotic distributions. Indeed, rv estimates possess moments only up to the degree of over- identification, so that when there is one instrument for one suspect structural vari- able, the rv estimate will be so dispersed that its mean does not exist (see David- son and MacKinnon 1993, 220-4, for further discussion and references). As a result, there will always be the possibility of obtaining extreme estimates, whose presence is not taken into account in the calculation of the asymptotic standard errors. Given sufficient overidentification so that the requisite moments exist— and note that this rules out some of the most difficult cases—Nagar (1959) and Buse (1992) show that in finite samples, rv estimates are biased towards the OLS estimators. This gives support to many students' intuition when first confronted with rv estimation, that it is a clever trick designed to reproduce the OLS estimate as closely as possible while guaranteeing consistency in a (conveniently hypo- thetical) large sample. In the extreme case, where there are as many instruments 116 THE ANALYSIS OF HOUSEHOLD SURVEYS as observations so that the first stage of two-stage least squares fits the data per- fectly, the iv and OLS estimates are identical. More generally, there is a tradeoff between having too many instruments, overfitting at the first stage, and being biased towards OLS, or having too few instruments, and risking dispersion and extreme estimates. Either way, the asymptotic standard errors on which we rou- tinely rely will not properly indicate the degree of bias or the dispersion. Nelson and Startz (1990a, 1990b) and Maddala and Jeong (1992) have ana- lyzed the case of a univariate regression where the options are OLS or iv estima- tion with a single instrument. Their results show that the central tendency of the finite-sample distribution of the IV estimator is biased away from the true value and towards the OLS value. Perhaps most seriously, the asymptotic distribution is a very poor approximation to the finite-sample distribution when the instrument is a poor one, in the sense that it is close to orthogonal to the explanatory variable. Additional evidence of poor performance comes from Bound, Jaeger, and Baker (1993), who show that the empirical results in Angrist and Krueger (1991), who used up to 180 instruments with 30,000 observations, can be closely reproduced with randomly generated instruments. Both sets of results show that poor instru- ments do not necessarily reveal themselves as large standard errors for the IV estimates. Instead it is easy to produce situations in which y is unrelated to *, and where z is a poor instrument for *, but where the IV estimate of the regression of y on x with z as instrument generates a parameter estimate whose "asymptotic t- value" shows an apparently significant effect. As a result, if IV results are to be credible, it is important to establish first that the instruments do indeed have pre- dictive power for the contaminated right-hand-side variables. This means display- ing the first-stage regressions—a practice that is far from routine—or at the least examining and presenting evidence on the explanatory power of the instruments. (Note that when calculating two-stage least squares, the exogenous x variables are also included on the right-hand-side with the instruments, and that it is the predic- tive power of the latter that must be established, for example, by using an F-test for those variables rather than the R2 for the regression as a whole.) In recent work, Staiger and Stock (1993) have proposed a new asymptotic theory for iv when the instruments are only weakly correlated with the regressors, and have produced evidence that their asymptotics provides a good approximation to the finite-sample distribution of IV estimators, even in difficult cases such as those examined by Nelson and Startz. These results may provide a better basis for IV inference in future work. 2.7 Using a time series of cross sections Although long-running panels are rare in both developed and developing count- ries, independent cross-sectional household surveys are frequently conducted on a regular basis, sometimes annually, and sometimes less frequently. In Chapter 1, I have already referred to and illustrated from the Surveys of Personal Income Distribution in Taiwan (China), which have been running annually since 1976, and I shall use these data further in this section. Although such surveys select ECONOMETRIC ISSUES FOR SURVEY DATA 117 different households in each survey, so that there is no possibility of following individuals over time, it is still possible to follow groups of people from one survey to another. Obvious examples are the group of the whole population, where we use the surveys to track aggregate data over time, or regional, sectoral, or occupational groups, where we might track the differing fortunes over time of farmers versus government servants, or where we might ask whether poverty is diminishing more rapidly in one region than in another. Perhaps somewhat less obvious is the use of survey data to follow cohorts of individuals over time, where cohorts are defined by date of birth. Provided the population is not much affected by immigration and emigration, and provided the cohort is not so old that its members are dying in significant numbers, we can use successive surveys to follow each cohort over time by looking at the members of the cohort who are randomly selected into each survey. For example, we can look at the average consumption of 30-year-olds in the 1976 survey, of 31-year-olds in the 1977 survey, and so on. These averages, because they relate to the same group of people, have many of the properties of panel data. Cohorts are frequently inter- esting in their own right, and questions about the gainers and losers from econo- mic development are often conveniently addressed by following such groups over time. Because there are many cohorts alive at one time, cohort data are more diverse and richer than are aggregate data, but their semiaggregated structure provides a link between the microeconomic household-level data and the macro- economic data from national accounts. The most important measures of living standards, income and consumption, have strong life-cycle age-related compon- ents, but the profiles themselves will move upward over time with economic growth as each generation becomes better-off than its predecessors. Tracking different cohorts through successive surveys allows us to disentangle the gene- rational from life-cycle components in income and consumption profiles. Cohort data: an example The left-hand top panel of Figure 2.5 shows the averages of real earnings for vari- ous cohorts in Taiwan (China) observed from 1976 through to 1990. The data were constructed according to the principles outlined above. For example, for the cohort born in 1941, who were 35 years old in 1976,1 used the 1976 survey to calculate the average earnings of all those aged 35, and the result is plotted as the first point in the third line from the left in the figure. The average earnings of 36- year-olds in the 1977 survey is calculated and forms the second point on the same segment. The rest of the line comes from the other surveys, tracking the cohort born in 1941 through the 15 surveys until they are last observed at age 49 in 1990. Table 2.2 shows that there were 699 members of the cohort in the 1976 survey, 624 in the 1977 survey, 879 in the 1978 survey (in which the sample size was increased), and so on until 691 in 1990. The figure illustrates the same pro- cess for seven cohorts, born in 1951, 1946, and so on backward at five-year inter- vals until the oldest, which was born in 1921, and the members of which were 69 years old when last seen in 1990. Although it is possible to make graphs for all Figure 2*5. Earnings by cohort and their decomposition, Taiwan (China), 1976-90 25-1 20 Earnings by -10 Cohort effects cohort I S. -30 I 1 -40 25 35 45 65 25 35 45 55 Age Cohort: age in 1976 118 2 -I 2 and is referred to by Atkinson as "equally distributed equivalent x" Equality is measured by the ratio OB/OC, or by x 7|ti, a quantity that will be unity if everyone has the same, or if the social welfare contours are straight lines perpen- dicular to the 45-degree line. This is the case where "a dollar is a dollar" whoever receives it so that there is no perceived inequality. Atkinson's measure of inequal- ity, defined by (3.3), is shown in the diagram as the ratio BC/OC. One of the advantages of the social welfare approach to inequality measure- ment, as embodied in (3.3), is that it precludes us from making the error of inter- preting measures of inequality by themselves as measures of welfare. It will some- times be the case that inequality will increase at the same time that social welfare is increasing. For example, if everyone gets better-off, but the rich get more than the poor, inequality will rise, but there has been a Pareto improvement, and most observers would see the new situation as an improvement on the original one. When inequality is seen as a component of social welfare, together with mean levels of living, we also defuse those critics who point out that a focus on inequal- ity misdirects attention away from the living standards of the poorest (see in particular Streeten et al 1981). Atkinson's formulation is entirely consistent with an approach that pays attention only to the needs of the poor or of the poorest groups, provided of course that we measure welfare through (3.3), and not through (negative) / alone. Just to reinforce the point, we might define a "basic-needs" social welfare function to be the average consumption of the poorest five percent of society, \ip say. This measure can be rewritten as \i (1 - 7), where 7 is the inequality measure 1 - \i p/\i. 138 THE ANALYSIS OF HOUSEHOLD SURVEYS Measures of inequality Given this basic framework, we can generate measures of inequality by specifying a social welfare function and solving for the inequality measure, or we can start from a standard statistical measure of inequality, and enquire into its consistency with the principle of transfers and with a social welfare function. The first ap- proach is exemplified by Atkinson's own inequality measure. This starts from the additive social welfare function 1 H Xl'£ (3.4a) W = lZ-^-, e * 1 NM 1-6 1 N (3.4V) InW = — £ In*,., e = 1. Afi-i The parameter e ;> 0 controls the degree of "inequality aversion" or the degree to which social welfare trades off mean living standards on the one hand for equality of the distribution on the other. In Figure 3.1, social welfare indifference curves are flatter when e is small, so that, for the same initial distribution 5, the point B moves closer to the origin as e increases. Atkinson's social welfare function, which will also prove useful in the tax reform analysis of Chapter 5, has the property that the ratio of marginal social utilities of two individuals is given by the reciprocal of the ratio of their jt's raised to the power of e: gw/g^ (3 5 '> awaT = W Hence, if e is zero so that there is no aversion to inequality, marginal utility is the same for everyone, and social welfare is simply ji, the mean of the jc's. If e is 2, for example, and i is twice as well-off as 7, then the marginal social utility of addi- tional jc to i is one-fourth the marginal social utility of additional x toj. As e tends to infinity, the marginal social utility of the poorest dominates over all other mar- ginal utilities, and policy is concerned only with the poorest. When social welfare is the welfare of the poorest, which is what (3.4) becomes as e tends to infinity, social preferences are sometimes said to be maximin (the object of policy is to maximize the minimum level of welfare) or Rawlsian, after Rawls (1972). Think- ing about relative marginal utilities according to (3.5) is sometimes a convenient way of operationalizing the extent to which one would want poor people to be favored by policies or projects. The inequality measure associated with (3.4) are, when e * 1, / - N \ i/d-0 (3.6a) = 1- -J-ECc/H)1"6 V N i=i / and, when e = 1, the multiplicative form <3.6b) / = 1=1 WELFARE, POVERTY, AND DISTRIBUTION 139 These expressions are obtained by raising social welfare to the power of l/( 1 -e), which makes the function homogeneous of the first degree, and then following through the procedures of the previous subsection. In line with the interpretation of e as an aversion or perception parameter, there is no (perceived) inequality when e is zero, no matter what the distribution of the jc's. Conversely, if e > 0 and one person has all but a small amount a, say, with a spread equally over the oth- ers, then / tends to one as the number of people becomes large. Values of e above 0 but below 2 appear to be useful, although in applications, it is often wise to look at results for a range of different values. We may also choose to start from the standard measures of inequality. Provided these satisfy the principle of transfers, they will be consistent with Atkinson's approach, and will each have an associated social welfare function that can be recovered by applying (3.3). Some statistical measures of inequality do not satisfy the principle of transfers. The interquartile ratio—the 75th percentile less the 25th percentile divided by the median—is one such. Transferring jc from a richer to a poorer person in the same quartile group will have no effect on inequality, and a transfer from someone at the bottom quartile to someone poorer will lower the bottom quartile and so will actually increase inequality. Less obviously, it is also possible to construct cases where a transfer from a better-off to a poorer person will increase the variance of logarithms. However, this can only happen when both people are far above the mean—which may not be relevant in some applications— and the other conveniences of the log variance may render it a competitive in- equality measure in spite of this deficiency. Other standard measures that do satisfy the principle of transfers are the Gini coefficient, the coefficient of variation, and Theil's "entropy" measure of inequal- ity. The Gini coefficient if often defined from the Lorenz curve (see below), but can also be defined directly. One definition is the ratio to the mean of half the average over all pairs of the absolute deviations between people; there are N(N~l)/2 distinct pairs in all, so that the Gini is = (37a) ( YY * > Note that when everyone has the same, ji, the Gini coefficient is zero, while if one person has N\i, and everyone else zero, there are N- 1 distinct nonzero absolute differences, each of which is N\i9 so that the Gini is 1. The double sum in (3.7a) can be expensive to calculate if N is large, and an equivalent but computationally more convenient form is 2 <3.7b) V ; rY -^i- EP,*, JV-1 N(N-l)\i i-i ' ' where p. is the rank of individual i in the jc-distribution, counting from the top so that the richest has rank 1. Using (3.7b), the Gini can straightforwardly and rapidly be calculated from microeconomic data after sorting the observations. I shall give examples below, together with discussion of how to incorporate sample weights, and how to calculate the individual-level Gini from household-level data. 140 THE ANALYSIS OF HOUSEHOLD SURVEYS Not surprisingly in view of (3.7b), the social welfare function associated with the Gini coefficient is one in which the jc's are weighted by the ranks of each individual in the distribution, with the weights larger for the poor. Since the Gini lies between zero and one, the value of social welfare in an economy with mean \JL and Gini coefficient y is n(l - y)» a measure advocated by Sen (1976a) who used it to rank of Indian states. The same measure has been generalized by Graaff (1977) to (i(l - Y)°, for o between 1 and 0; Graaff suggests that equity and effi- ciency are separate components of welfare, and that by varying o we can give different weights to each (see also Atkinson 1992 for examples). The coefficient of variation is the standard deviation divided by the mean, while Theil's entropy measure is given by (3.8) , = — / T f — 1 2^ In I — I *'!«(*') N i-i |i I \a) IT lies between 0, when all jt's are identical, and InW, when one person has every- thing. This and other measures are discussed at much greater length in a number of texts, for example, Cowell (1995) or Kakwani (1980). The choice between the various inequality measures is sometimes made on grounds of practical convenience, and sometimes on grounds of theoretical prefer- ence. On the former, it is frequently useful to be able to decompose inequality into "between" and "within" components, for example, between and within regions, sectors, or occupational groups. Variances can be so decomposed, as can Theil's entropy measure, while the Gini coefficient is not decomposable, or at least not without hard-to-interpret residual terms (see, for example, Pyatt 1976). It is also sometimes necessary to compute inequality measures for magnitudes—such as in- comes or wealth—that can be negative, which is possible with the Gini or the coef- ficient of variation, but not with the Theil measure, the variance of logarithms, or the Atkinson measure. Further theoretical refinements can also be used to narrow down the choice. For example, we might require that inequality be more sensitive to differences between the poor than among the rich (see Cowell 1995), or that inequality aversion be stronger the further we are away from an equal allocation (see Blackorby and Donaldson 1978). All of these restrictions have appeal, but none has acquired the universal assent that is accorded to the principle of transfers. Poverty and social welfare In developing countries, attention is often focussed less on social welfare and in- equality than on poverty. Indeed, poverty is frequently seen as the defining charac- teristic of underdevelopment, and its elimination as the main purpose of economic development. In such a context, it is natural for welfare economics to have a poverty focus. Even so, and although the poverty measurement literature has de- veloped in a somewhat different direction, the social welfare function approach of this section is quite general, and as we have already seen, can readily accommo- date a preference and measurement structure that is focusses attention exclusively towards the poor. WELFARE, POVERTY, AND DISTRIBUTION 141 The social welfare function (3.1) transforms the distribution of x's into a single number that can be interpreted as a summary welfare measure that takes into ac- count both the mean of the distribution and its dispersion. However, we are free to choose a function that gives little or no weight to the welfare of people who are well-off, so that social welfare becomes a measure of the welfare of the poor, in other words, a (negative) measure of poverty. In this sense, poverty measures are special cases of social welfare measures. However, in practical work, they serve rather different purposes. Poverty measures are designed to count the poor and to diagnose the extent and distribution of poverty, while social welfare functions are guides to policy. Just as the measurement of social welfare can be a inadequate guide to poverty, so are poverty measures likely to be an inadequate guide to policy. As far as measurement is concerned, what separates the social welfare from the poverty literatures is that, in the latter, there is a poverty line, below which people are defined as poor, and above which they are not poor. In the language of social welfare, this effectively assigns zero social welfare to marginal benefits that accrue to the nonpoor, whereas the inequality literature, while typically assigning greater weight to benefits that reach lower in the distribution, rarely goes as far as assign- ing zero weight to the nonpoor. While the simplicity of a poverty line concept has much to recommend it, and is perhaps necessary to focus attention on poverty, it is a crude device. Many writers have expressed grave doubts about the idea that there is some discontinuity in the distribution of welfare, with poverty on one side and lack of it on the other, and certainly there is no empirical indicator—income, consumption, calories, or the consumption of individual commodities—where there is any perceptible break in the distribution or in behavior that would provide an empirical basis for the construction of a poverty line. Even when there exists an acceptable, readily comprehensible, and uncontro- versial line, so that we know what we mean when we say that a percent of the population is poor, we should never minimize this measure as an object of policy. The poverty count is an obviously useful statistic, it is widely understood, and it is hard to imagine discussions of poverty without it. However, there are policies that reduce the number of people in poverty, but which just as clearly decrease social welfare, such as taxes on very poor people that are used to lift the just-poor out of poverty. Similarly, a Pareto-improving project is surely socially desirable even when it fails to reduce poverty, and it makes no sense to ignore policies that would improve the lot of those who are poor by many definitions, but whose incomes place them just above some arbitrary poverty line. The construction of poverty lines Without an empirical basis such as a discontinuity in some measure, the construc- tion of poverty lines always involves arbitrariness. In developed countries where most people do not consider themselves to be poor, a poverty line must be below the median, but different people will have different views about exactly how much money is needed to keep them out of poverty. Almost any figure that is reasonably 142 THE ANALYSIS OF HOUSEHOLD SURVEYS central within the distribution of these views will make an acceptable poverty line. The official poverty line in the United States evolved from work in the early 1960s by Orshansky (1963, 1965) who took the cost of the U.S. Department of Agricul- ture's "low-cost food plan" and multiplied it by three, which was the reciprocal of the average food share in the Agriculture Department's 1955 household survey of food consumption. While such a procedure might seem to be empirically well-grounded—and the perception that it is so has been important in the wide and continuing acceptance of the line—it is arbitrary to a considerable extent. The food plan itself was only one of several that were adapted by nutritional "experts" from the food consump- tion patterns of those in the lowest third of the income range in the 1955 survey, while the factor of three was based on food shares at the mean, not at the median, or at the 40th or the 25th percentile, for all of which a case could be mounted. In fact, Orshansky's line of a little over $3,000 for a nonfarm family of four was adopted, not because of its scientific foundations, but because her procedure yielded an answer that was acceptably close to another arbitrary figure that was already in informal use within the federal government (see Fisher 1992 for more on the history and development of the U.S. poverty line). In India, poverty lines and poverty counts have an even more venerable history stretching back to 1938 and the National Planning Committee of the Indian National Congress. The more recent history is detailed in Government of India (1993), from which the following account is drawn. In 1962, "a Working Group of eminent economists and social thinkers" recommended that people be counted as poor if they lived in a household whose per capita monthly expenditure was less than 20 rupees at 1960-61 prices in rural areas, or 25 rupees in urban areas. These "bare minimum" amounts excluded expenditure on health and education, both of which were "expected to be provided by the State according to the Constitution and in the light of its other commitments." The precise economic and statistical basis for these calculations is not known, although the cost of obtaining minimally adequate nutrition was clearly taken into account, and the difference between urban and rural lines made an allowance for higher prices in the former. Dandekar and Rath (1971a, 197 Ib) refined these poverty lines using a method that is still in widespread use. They started from an explicit calorie norm, 2,250 calories per person per day in both urban and rural areas. Using detailed food data from the National Sample Surveys (NSS), they calculated calorie consumption per capita as a function of total household expenditure per capita—the calorie Engel curve—and found that the norms were reached on average at 14.20 rupees per capita per month in rural areas, and 22.60 rupees per capita in urban areas, again at 1960-61 prices. These estimates were further refined by a "Task Force" of the Planning Commission in 1979, who revised the calorie norms to 2,400 in rural areas, and 2,100 in urban areas; the difference comes from the lower rates of physical activity in urban areas. The 28th round (1973-74) of the NSS was then used to estimate regression functions of calories on expenditure, and to convert these numbers to 49.09 rupees (rural) and 56.64 rupees (urban) at 1973-74 prices. These lines—updated for all-India price inflation—have been the basis for Indian WELFARE, POVERTY, AND DISTRIBUTION 143 poverty counts since 1979, although the "Expert Group" that reported in 1993 has recommended that allowance be made for interstate variation in price levels. In poor countries such as India, where food makes up a large share of the bud- get, and where the concern with poverty is closely associated with concerns about undernutrition, it makes more sense to use food and nutritional requirements to derive poverty lines than it does in the United States The "low-cost food plan" in the United States can be replaced by something closer to the minimum adequate diet for the country and type of occupation, and because food is closer to three- quarters than a third of the budget, the "multiplier" needed to allow for nonfood consumption is smaller, less important, and so inherently less controversial. Even so, the calorie-based procedure of setting a poverty line is subject to a number of serious difficulties. First, the minimum adequate calorie levels are themselves subject to uncertainty and controversy, and some would argue that resolving the arbitrariness about the poverty line with a calorie requirement simply replaces one arbitrary decision with another. Second, the concept of a behavioral Engel curve does not sit well with the notion that there is a subsistence level of calories. Suppose, for example, that a household is poor in that its expected calorie intake conditional on its income is inadequate, but has more than enough to buy the subsistence number of calories if it spent more of its budget on food. It seems that the subsistence number of calories is not really "required" in any absolute sense, or at least that the household is prepared to make tradeoffs between food and other goods, tradeoffs that are not taken into account in setting the line. Third, it is always dangerous to measure welfare using only a part of consumption, even when the part of consumption is as important as is food. When food is relatively cheap, people will consume more—even if only marginally so—and poverty lines will be higher where the relative price of food is higher, even though consumers may be compensated by lower prices elsewhere in the budget. Bidani and Ravallion (1994) have examined this phenomenon in Indonesia. They show that higher food prices in the cities, together with the lower caloric re- quirements of more sedentary urban jobs, imply that the urban calorie Engel curve is lower than the rural calorie Engel curve. At the same level of PCE, urban con- sumers consume less calories than do rural consumers. In consequence, a common nutritional standard requires a higher level of PCE in the cities. In the Indonesian case, this results in a poverty line so much higher in urban than rural areas that there appears to be more poverty in the former, even though real incomes and real levels of consumption are much higher in the cities. Once poverty lines are established they often remain fixed in real terms. In the United States, the current poverty line is simply Orshansky's 1961 poverty line updated for increases in the cost of living. In India, as detailed above, there have been revisions to methodology, but the lines have changed very little in real terms, and a number of studies, such as Bardhan (1973) and Ahluwalia (1978, 1985), have used poverty lines close to those proposed by Dandekar and Rath in 1971. This constancy reflects a view of poverty as an absolute; poverty is defined by the ability to purchase a given bundle of goods so that the poverty line should remain fixed in real terms. However, not everyone accepts this position, and it can be ar- 144 THE ANALYSIS OF HOUSEHOLD SURVEYS gued that poverty lines should move with the general standard of living, although perhaps not at the same rate. Some would argue that poverty is a purely relative phenomenon, defined by current social customs, and that the poor are simply those in the bottom percentiles of the distribution of welfare. An intermediate view comes from Sen's (1985, 1992) view of welfare in terms of the capability to function in society. If economic growth means that food is sold with an increased amount of packaging and service built in, if city center stores relocate to suburban areas that cannot be reached on foot, and if urban growth increases the cost and time to travel to work, then a fixed absolute poverty line makes no sense. There is also some relevant empirical evidence that comes from asking people whether they are poor and what the poverty line ought to be (see Mangahas 1979, 1982, 1985, who makes good use of such surveys to assess poverty in the Philippines). In the United States, Gallup polls have regularly asked respondents how much money they would need "to get along," and more occasion- ally what they think would be an adequate poverty line. In the 1960s, the mean responses about the latter were close to the official (Orshansky) line, but have since increased in real terms, although not always as fast as has average real dis- posable income (see Rainwater 1974 and Vaughan 1992). Ravallion (1993) has also examined the cross-country relationship between real gross domestic product (GDP) and poverty lines, and found that the elasticity is close to unity. While many people—including this author—are uncomfortable with an entirely relative con- cept of poverty, it is surely right that there should be some movement of the line in response to changes in mean levels of living. The conceptual and practical difficulties over the choice of a poverty line mean that all measures of poverty should be treated with skepticism. For policy evalu- ation, the social welfare function is all that is required to measure welfare, includ- ing an appropriate treatment of poverty. While it is possible—and in my view desirable—to give greater weight to the needs of the poorest, I see few advantages in trying to set a sharp line, below which people count and above which they do not. Poverty lines and poverty counts make good headlines, and are an inevitable part of the policy debate, but they should not be used in policy evaluation. Perhaps the best poverty line is an infinite one; everyone is poor, but some a good deal more so than others, and the poorer they are the greater weight they should get in measuring welfare and in policy evaluation. The concept of a poverty line is deeply embedded in the poverty literature, and measures of poverty are typically based on it. Even so, a good deal of the recent literature on poverty has followed Atkinson (1987) in recognizing that the poverty line is unlikely to be very precisely measured, and trying to explore situations in which poverty measures are robust to this uncertainty, I shall return to this ap- proach below once I have introduced some of the standard measures. Measures of poverty There are a number of good reviews of alternative poverty measures and their properties, see in particular Foster (1984) and Ravallion (1993), so that I can WELFARE, POVERTY, AND DISTRIBUTION 145 confine myself here to a brief discussion of the most important measures. The obvious starting point—and the measure most often quoted—is the headcount ratio, defined as the fraction of the population below the poverty line. If the line is denoted by z, and the welfare measure is jc, then the headcount ratio is (3.9) /V 1 = 1 where 1 (.) is an indicator function that is 1 if its argument is true and 0 otherwise. The sum of the indicators on the right-hand side of (3.9) is the number of people in poverty, so that PQ is simply the fraction of people in poverty. It is worth noting that with a change of sign, (3.9) could conceivably be re- garded as a social welfare function. It is the average value of a rather strange valu- ation function in which x counts as -1 when it is below the poverty line z, and as 0 when it is above z. This function is illustrated as the heavy line labeled PQ in Figure 3.2; it is nondecreasing in jc, so it has some of the characteristics of a utility function, but its discontinuity at the poverty line means that it is not concave. It is this lack of concavity that violates the principle of transfers, and makes it possible to increase social welfare by taking money from the very poor to lift some better- off poor out of poverty. Even if the poverty line were correctly set, and even if it were acceptable to view poverty as a discrete state, the headcount ratio would be at best a limited measure of poverty. In particular, it takes no account of the degree of poverty, and would, for example, be unaffected by a policy that made the poor even poorer. The Figure 3.2. Alternative poverty measures and social welfare Contribution to social welfare Individual welfare -1 146 THE ANALYSIS OF HOUSEHOLD SURVEYS headcount ratio gives the same measure of poverty whether all the poor are just below a generous poverty line, or whether they are just above an ungenerous level of subsistence. One way of doing better is to use the poverty gap measure (3.io) />, = j-rfi-^) !(*.**>. A^M\ zt According to (3.10), the contribution of individual i to aggregate poverty is larger the poorer is i. Pl can also be interpreted as a per capita measure of the total short- fall of individual welfare levels below the poverty line; it is the sum of all the shortfalls divided by the population and expressed as a ratio of the poverty line itself. Hence if, for example, Pl were 0.25, the total amount that the poor are below the poverty line is equal to the population multiplied by a quarter of the poverty line. It is tempting to think of Pl (or at least Ptz) as a measure of the per capita "cost" of eliminating poverty, but this is far from being so except in the impracti- cal case where lump-sum taxes and subsidies are possible. Even when tax and subsidy administration is efficient and is not corrupt, redistributive taxes have incentive effects that may render the elimination of poverty neither possible nor desirable given the actual range of feasible policies. This is clearly the case in an economy where everyone is poor, but applies much more widely. Once again, the appropriate way to think about tax systems for poverty alleviation is to go back to the social welfare function (3.1), to make sure that it incorporates the appropriate degree of weighting towards the poor, and to apply the general theory of tax design (see Newbery and Stern 1987 for a general discussion of such problems in the contexts of developing countries, and Chapter 5 below for some of the empi- rical issues). The poverty gap measure (3.10) has a number of advantages over the head- count ratio (3.9). In particular, the summand is now a continuous function of jt, so that there is no longer a discontinuity in the contribution of an individual to the poverty measure as that individual's x passes through the poverty line. When x is just below z, the contribution to poverty is very small, it is zero when x equals z, and remains at zero above z. Furthermore the function (1 -x/z) l(xzz) is convex in x—although not strictly so—so that the principle of transfers holds—at least in a weak form. As a result, the social welfare interpretation of the poverty gap mea- sure also makes more sense than that of the headcount ratio. The behavior of each individual's contribution to -Pl is illustrated in Figure 3.2 by the piecewise linear function rising from -1 to 0, a value which it retains above z- This function is increasing in x, and is (just) concave, so that while social welfare is not altered by transfers among the poor or among the nonpoor, it is no longer possible to increase social welfare by acting as an anti-Robin Hood, taking resources from the poor to give to the rich. The poverty gap measure will be increased by transfers from poor to nonpoor, or from poor to less poor who thereby become nonpoor. But transfers among the poor have no effect on the measure of poverty, and on this account we may wish to consider other poverty measures. Sen's (1976b) measure of poverty remedies WELFARE, POVERTY, AND DISTRIBUTION 147 the defect by incorporating the inequality among the poor. The definition is (3.11) Ps = P 0 ( l - ( l - Y ' ) ^ ) where \JLP is the mean of x among the poor, and yp is the Gini coefficient of in- equality among the poor, calculated by treating the poor as the whole population. Note that when there is no inequality among the poor, Ps reduces to the poverty gap measure Pr Conversely, when all but one of the poor has nothing, PS = PQ and the Sen measure coincides with the headcount ratio. More generally, the Sen measure is the average of the headcount and poverty gap measures weighted by the Gini coefficient of the poor, (3.12) Ps = P^p +P 1 (1-Y"). Because Sen's measure depends on the Gini coefficient, it shares two of its in- conveniences. First, the Gini—and thus the Sen index—is not differentiable. Al- though there is no economic reason to require differentiability, the inability to dif- ferentiate is sometimes a nuisance. More seriously, Sen's measure cannot be used to decompose poverty into contributions from different subgroups, something that is often informative when monitoring changes in poverty. If the aggregate poverty measure can be written as a weighted average of the poverty measures for the rural and urban sectors, or for households by age, or by occupation of the head, then changes over time can be similarly decomposed thus helping to identify groups that are particularly at risk, as well as sometimes pointing to the underlying mecha- nisms. While decomposability is hardly as fundamental a property as (say) the principle of transfers, it is extremely useful. Our final poverty measure, or set of measures, comes from Foster, Greer, and Thorbecke (1984). Their measures are direct generalizations of the poverty gap (3.10) and are written, for some positive parameter a, (3.13) Pa = AT1 £(l-yz)« !(*.«) M so that PQ and P^ are special cases corresponding to values for a of 0 and 1, res- pectively. The larger the value of a, the more does the measure penalize the pov- erty gaps. Most frequently used is a =2, which yields a poverty measure like the Sen index that is sensitive to distribution among the poor. The decomposability property of (3.13) follows immediately from its additive structure. In particular, if sectors are denoted by s, and there are 5 of them, we can write (3.1-4) Pa = AT1 £ £(l-*/2)0!(*,«) = E(ng/N)P; s-ljes s where ns is the number of people in sector s and P£ is the Foster, Greer, and Thorbecke index for poverty within the sector. Using (3.14), changes in aggregate poverty can be assigned to changes in sectoral poverty measures or to changes in the proportion of people in each sector. 148 THE ANALYSIS OF HOUSEHOLD SURVEYS Figure 3.2 shows how Pa measures up as a social welfare function. The third line, shown as dots and dashes, is the contribution of individual welfare to social welfare defined as -Pa in the case where a is greater than unity; as with Pv this line continues along the x-axis to the right of z. Because the function is strictly concave below e, it is sensitive to the degree of inequality among the poor. It is also continuously differentiate, even at the poverty line, so that the implicit marginal social utility declines continuously from its maximum among the poorest to zero at and above the poverty line. The choice of the individual welfare measure Apart from a brief reference in the context of choosing a poverty line, I have so far avoided discussion of exactly how welfare is to be measured, and what practical concept should replace the jt's in the various poverty and inequality formulas. Ideally, we should like a survey based measure that approaches as closely as pos- sible the individual welfare measures of economic theory. Particularly useful here is the concept of money metric utility—see Deaton and Muellbauer (1980a, ch. 7) for an overview—whereby the indifference curves of individual preference order- ings are labeled by the amount of money needed to reach them at some fixed set of prices. In order to avoid the specification of a parametric utility function, money metric utility can be approximated by real income or real expenditure, the two leading candidates for practical welfare measures. However, there are other possi- bilities, indicators of nutritional status being perhaps the most important, and even if we settle on income or expenditures, there are many other questions that have to be settled before going on to compute the measures. In this subsection, I discuss a few of the most important: the choice between consumption and income or other concepts, the choice between individual and household measures, the choice of time period, as well as some data issues, particularly the effects of measurement error and reporting periods. In the context of measuring welfare in developing countries, there is a very strong case in favor of using measures based on consumption not income. The standard argument—that by the permanent income hypothesis, consumption is a better measure of lifetime welfare than is current income—is much weaker than arguments based on practicality and data. It is unwise to condition a welfare mea- sure on the validity of a hypothesis whose empirical support is at best mixed. In particular and as we shall see in Chapter 6, there is very little evidence from devel- oping countries—or anywhere else—that lifetime consumption profiles are de- tached from lifetime income profiles as is required if consumption is to be superior to income as an indicator of lifetime welfare. Of course, there is no doubt that households smooth their consumption over shorter periods, certainly days, months, and seasons, and to some extent over runs of years. Income, especially agricultural income, can be extremely variable, and a farmer's income in any month is a poor indicator of living standards in that month. A better case can be made for annual income, but if farmers can even partially smooth out good years and bad, con- sumption will be the better measure. WELFARE, POVERTY, AND DISTRIBUTION 149 At the practical level, and as discussed in Section 1.2, the difficulties of measuring income are much more severe than those of measuring consumption, especially for rural households whose income comes largely from self-employ- ment in agriculture. Given also that annual income is required for a satisfactory estimate of living standards, an income-based measure requires multiple visits or the use of recall data, whereas a consumption measure can rely on consumption over the previous few weeks. Note that these arguments are likely reversed if we were dealing with, for example, the United States, where individual consumption surveys are much less developed than income surveys, where a much smaller fraction of the population is self-employed, where seasonality is much less of an issue, and where it is both feasible and economical to obtain accurate estimates of income for most people. The conversion of nominal measures of consumption to real measures requires a price index. In most cases, there will exist an adequate consumer price index or cost-of-living estimate that can be used to compare data collected in different time periods. In countries with rapid inflation, this may even have to be done within each survey year, since different households are interviewed at different times. What is often more difficult is the comparison of living costs across regions at a given time, for example, when we are trying to compare living standards or pov- erty rates across different regions. In some surveys—but not typically in the Living Standards Surveys—households are asked to report both quantities and expendi- tures on a range of goods, particularly foods, and these data can be used to calcu- late unit values. Although unit values are not the same as prices—an issue that will be discussed in some detail in Chapter 5—accurate price indexes for each region can nevertheless be obtained from the unit values by averaging within regions and calculating a Laspeyres index for each, that is by pricing out a fixed bundle of goods at the average unit values for each region. The Living Standards Surveys have usually collected price data, not from households, but from observations on prices in markets used by the households in the survey, and these data can also be used to construct regional price indexes. Although consumption and income are the standard measures of economic wel- fare, we will often want to supplement them with other measures of well-being, such as nutritional and health status, life expectancy, and education. While it is possible to consider methods for combining these indicators into a single measure, there is no adequate theory underlying such an aggregate so that weighting schemes are inevitably arbitrary, and it is more informative—as well as honest—to keep the different indicators separate. This is not to downplay the importance of these other indicators, nor to deny that public goods such as hospitals and schools contribute an important part of individual welfare. However, it is important not to confuse the components of economic welfare with their aggregate. We have al- ready seen how the definition of a poverty line in terms of calories can give mis- leading results when relative prices differ. The same argument applies to attempts to shortcut welfare measurement using indicators such as housing, or the owner- ship of durable goods. Immigrants to big cities often live in very poor-quality housing in order to have access to employment. In such cases, their poor housing 150 THE ANALYSIS OF HOUSEHOLD SURVEYS reflects the high price of housing in urban areas, but may tell us little about their living standards. Because surveys collect data at the level of the household, and not the indivi- dual, poverty and welfare measures must be based on consumption and income totals for the household, not for the individual. Although some surveys collect data on individual earnings, and even on individual income from assets, there is typi- cally a component of household income—a large component in the case of family farms—that is not readily attributable to individual household members. For con- sumption, the position is even worse. Data on purchases are inevitably purchases for the household as a whole, and although some items—such as food—are con- ceptually assignable to particular individuals, the cost of observing who eats what is too large for all but specialist nutritional surveys. Even then there are questions about contamination of behavior by the presence of the enumerator during food preparation and family meals. There are also public goods in most households— goods and services the consumption of which by one member of the household does not exclude consumption by others. The consumption of these goods cannot be assigned to specific individuals. As a result, we can either treat households as the units whose welfare is being measured, or we can use some rule to divide household total expenditure between its members, usually equally or in proportion to some measure of needs, and then treat each individual as the unit in the poverty and welfare calculations. Since it is hard to think of households as repositories for well-being, even in the best case where their membership does not change, an individual basis for measurement is conceptually clearer and is the recommendation carried throughout this book. One difficulty is that the assumption of equal division, or of each according to her or his needs, is bound to understate the true dispersion of consumption among indi- viduals, and thus understate inequality and poverty. As pointed out by Haddad and Kanbur (1990), who have also investigated the magnitude of the biases, the assum- ed equal distribution within the household could be reached from the unknown true one by a system of equalizing transfers, so that any welfare measure that respects the principle of transfers will be overstated (or understated if a poverty measure) using household data. It is also necessary to recognize that children do not have the same needs as adults. Assigning household PCE to each person gives too little to adults—especi- ally those who do heavy manual work—and too much to children. If there are economies of scale, PCE will understate individual welfare levels, even if all house- hold members are adults. Attempts to do better than PCE measures for individuals are discussed in Chapter 4, where I take up the question of allocation within the household, and the construction of "equivalence scales," numbers that are the theoretically appropriate deflators to move from household to individual welfare. However, I should point out in advance that the equivalence scale literature is still very far from providing satisfactory answers to these questions, and that the use of household PCE assigned to individuals is still best practice. Even so, it is wise to remain skeptical of estimates that appear to be purely statistical but rely heavily on arbitrary assignments, such as the number of children in poverty, or the average WELFARE, POVERTY, AND DISTRIBUTION 757 living standards of the elderly. The elderly rarely live by themselves in poor countries, and children do not do so anywhere, so that estimates of their welfare are determined as much by assumption as by measurement and should be treated as such. Measures of the fraction of children in poverty, or of women in poverty, are particularly fragile and international comparisons of such concepts cannot be treated seriously. The choice of time period, like all of these issues, is partly one of theory and partly one of practicality. In theory, we need to decide the reference period for welfare measurement, whether someone is poor if they go without adequate con- sumption or income for a week, a month, or a year. The reference period can be shorter for consumption than for income, and if we use income, the choice of reference period will depend on what mechanisms—credit markets, familial sup- port—are available to help people ride out fluctuations in income. In practice, long reference periods require either multiple visits or recall questions; the former are expensive and the latter risk measurement error. Note also that, if poverty and welfare measures are to be comparable across countries or over time, the reference periods must be the same. Because the dispersion of both consumption and income decrease the longer is the reference period, both the extent of inequality and pov- erty will be larger at short than at long reference periods. One of the most difficult practical issues in estimating poverty and inequality is to separate genuine dispersion from measurement error. If we start from any distribution of welfare and add measurement errors that have zero mean and are uncorrelated with the true values, the new distribution is a spread-out version of the original one, so that if our measures respect the principle of transfers, measured inequality will be higher and social welfare lower. Poverty measures that satisfy the principle of transfers will also be higher. For the headcount, which does not satisfy the principle, matters are more complicated and measurement error can bias the count in either direction. If the country is wealthy enough for the poverty line to be below the mode, the addition of mean-zero measurement error will cause the measured headcount to overstate the number in poverty, and vice versa. Similarly if we try to assess the persistence of poverty using panel data by seeing who re- mains in poverty and who escapes it, measurement error will exaggerate the extent of mobility, and make poverty seem less persistent than is truly the case. In most cases, we have little idea of the magnitude of measurement error, or how much of the variance of consumption or income changes is noise as opposed to signal. However, and bearing in mind the problems of estimating consumption and in- come in surveys in developing countries, it is always wise to consider the robust- ness of conclusions to the presence of substantial measurement error. Example L Inequality and poverty over time in Cote d'lvoire This subsection applies some of the foregoing concepts to Living Standards data from Cote d'lvoire for the four years 1985 through 1988, focussing on change over time, while the next subsection uses data from South Africa in 1993 to look at differences by race. The translation of the formulas into numbers is essentially 752 THE ANALYSIS OF HOUSEHOLD SURVEYS straightforward, but it is nevertheless useful to work through the exercise, partly to see how the theory can be used to help interpret the numbers, and partly to see some of the practical problems that are not dealt with by the theory. As always, a good place to start is with standard summary statistics, some of which are shown in Table 3.1. In fact there is a good deal of work prior to this, checking out the data, and looking for and investigating outliers, to which mea- sures of dispersion can be very sensitive. This sort of preliminary work is greatly aided by suitable methods of graphing the data; for example, the "oneway** com- mand in STATA plots a small vertical bar for each observation so that it is easy to see outliers in relation to the main body of the data. In line with the foregoing arguments, I have chosen consumption rather than income as the welfare measure, and I use the estimate of total monthly household expenditure calculated by the World Bank and provided on the diskettes of the Cote d*Ivoire LSS. I have also used the "corrective weights'* provided by the Bank. These were calculated after the surveys were complete, and are designed to correct some known deficiencies in the design that led to over- or undersampling. The most controversial component of the weight "corrects** the surveys* estimates of house- hold size. In the raw data, average household size declines over time at an implau- sible rate, something that could happen if large households were oversampled in the first survey, (see Coulombe and Demery 1993 and Demery and Grootaert 1993), or through progressive quality deterioration if enumerators became better at avoiding the very large households that can take many hours to interview (see Coulombe, McKay, and Pyatt 1993). The weights are based on the former hypoth- esis, and are controversial because they depend on its correctness; it is far from clear that they can be interpreted as inverse sampling probabilities. Unfortunately, inferences about per capita expenditures in Cote d*Ivoire are sensitive to these corrections (see again Coulombe, McKay, and Pyatt). The first column in the table shows the weighted average over the households in each survey of total annual expenditures in real terms. The units are thousands of Central African francs, with 1,000 CFAF worth between three and four U.S. dollars at purchasing power parity, and the prices are those of Abidjan in 1985.1 Table 3.1. Consumption measures, Cote d'lvoire, 1985-88 (thousands of CFAF per month) Total house- PCE hold Household Individual Standard Year expenditure Household size basis basis error 1985 1,548 6.51 292.2 237.8 8.75 1986 1,400 6.25 269.9 223.9 8.16 1987 1,345 6.20 264.3 217.0 9.13 1988 1,067 6.16 202.3 173.1 6.09 Note: There are 1,588 households in the 1985 sample, and 1,600 in 1986-88. Prices are those of Abidjan in 1985, and nominal amounts are converted to real terms using the regional price indexes for each year from Grootaert and Kanbur (1992). Standard errors are computed from the "svymean" command in STATA, with the regions as strata and the clusters as PSUs. Source: Author's calculations based on CILSS, 1985-88. WELFARE, POVERTY, AND DISTRIBUTION 153 have used Grootaert and Kanbur' s (1992) price indexes for the four years and five regions—Abidjan, Other Cities, East Forest, West Forest, and Savannah—to con- vert the nominal survey measures to constant prices. Mean household total expen- diture declines over the four years; indeed real per capita GDP declined in Cote d'lvoire over these years in line with the experience of many other African econo- mies during a period of falling world prices for their exports of primary commodi- ties, mainly cocoa and coffee in the case of Cote d'lvoire. The second column shows average household size; even after the correction there is some decline over time. The third column shows the (weighted) average over households of PCE which also shows a decline in every year, so that the decline in household size is not sufficient to offset the decline in the total. The average of household PCE, although often used as a welfare measure, is not what we want here; welfare resides in individuals, not households, and we need to recompute the averages on an individual basis. This is done in Table 3.1 by assign- ing the household levels of PCE to each individual and then averaging over the individuals. Of course, this is just a reweighting of the household data, so that if household h has nh members and PCE is xk, then the average of PCE on an indivi- dual basis is H H (3.15) x = Ewhnhxh/Ewhnh h=l h=l where H is the total number of households and wh is the household inflation factor or weight. Note that the weights and household size appear symmetrically in (3.15), so that is often convenient for calculation to redefine the weights by multi- plying by household size, or household size by multiplying by the weights. Since xhnh is simply total household expenditure, (3.15) implies that the individual average of PCE is the ratio of (weighted) average total household expenditure to (weighted) average household size, so that column 4 in Table 3.1 is the ratio of column 1 to column 2. Estimated standard errors for mean individual-level PCE are given in the final column; there are calculated using the "svymean" or "svyratio" commands in STATA, and take the regions as strata—although the CILSS was only implicitly strati- fied by region—and the clusters as PSUs. In the unlikely event that we are unconcerned with distributional issues, the averages of individual PCE would be adequate welfare indicators. But to move from mean consumption levels to social welfare, we must also calculate inequality measures, and these are shown in Table 3.2. Once again, these estimates are esti- mates of inequality between individuals, not between households. The conceptu- ally most straightforward way of going from one to the other is to "replicate" the observations in the data set, with one replication of the household PCE level for each individual in the household, and then to use the inequality formulas given above. In practice, this is both clumsy and inefficient compared with simple weighted estimates. Only in the case of the Gini coefficient is the calculation less than obvious. For example, to use household data to calculate the Atkinson in- equality measure on an individual basis, we calculate the level of social welfare according to the appropriately modified version of (3,4). Each sample household with nh members represents whnh individuals n the population, so that 154 THE ANALYSIS OF HOUSEHOLD SURVEYS I 1 Lj 1 H U/U-e) E^w*1****"8 h=l (3.16) x = H Ewhnh h^\ I (When e is 1, so that (3.16) is not defined, the calculation is replaced by exponent- iating the weighted averages of the logarithms.) The Atkinson inequality index is then calculated by dividing (3.16) by the (weighted) mean and subtracting the result from one. Calculation of the Gini coefficient from equation (3.7b) is only slightly more complicated by the need to convert household ranks to individual ranks. Since everyone in each household is assumed to have the same PCE, we can order them within the household in any way we choose. The first person in the best-off house- hold is then given rank 1, the first person in the second household rank 1 +nl, where nl is the number of people in the first household, and so on. As before, if there are nonzero survey weights, the simplest treatment is to pretend that there are, not nh, but whnh people in household h. Hence, and starting from pl = 1, the rank of the first person in household h +1 can be defined recursively by (3.17) PA+I = The average rank of all the persons in household h is therefore (3.18) so that the Gini coefficient for individual-level PCE is given from (3.7b) by (3.19) Y = - 1)] N-l with the best-off household coming first and the ranks calculated from (3.17). Since these inequality measures are independent of the scale of x, they will not be changed by deflation by the price index, provided that the price index does not vary from one household to another. Note too that, because the Gini coefficient is Table 3.2. Inequality measures between individuals. Cote d'lvoire, 1985-88 Gini coef- Coeff, vari- Atkinson measures Year flcient S.d. of logs ation e=0.5 e = 1.0 e = 2.0 1985 0.383 0.716 0.807 0.118 0.223 0.394 1986 0.358 0.627 0.786 0.103 0.190 0.382 1987 0.381 0.665 0.894 0.119 0.215 0.444 1988 0.345 0.615 0.745 0.091 0.180 0.357 Note: Measures of inequality use household PCE attributed to individuals and are calculated on an individual basis using corrective weights. S.d. is standard deviation and Coeff. variation the coefficient of variation. Source: Author's calculations based on CILSS, 1985-88. WELFARE, POVERTY, AND DISTRIBUTION 755 invariant to scaling of population size—doubling the number of people with each x has no effect on the index—the calculations are unaffected by any rescaling of the survey weights. The different measures are in broad agreement; all show that inequality is lowest in 1988, next lowest in 1986, and highest in 1985 and 1986. The Gini, the standard deviation of logs, and the Atkinson measure with e = 1 rank 1987 as more equal than 1985, while the reverse judgement is made by the coefficient of vari- ation, and the Atkinson measures with e set at either 0 or 2. The Atkinson mea- sures show more inequality the larger is the inequality aversion parameter e but, like the other measures, all rank 1988, 1986, and either 1985 or 1987 in the same way. Note that there is no simple relationship between the value of e and the rankings; the distribution in 1985 is more equal than that in 1987 when e is 0.5 or 2, but the reverse is true when e is L Recall that inequality should always be inter- preted jointly with the mean in order to assess changes in aggregate well-being. Even so, the decline in inequality from 1985 to 1988, although substantial by some of the measures, is not enough to have raised equality by the amount required to offset the decline in PCE and maintain the level of social welfare. Table 3.3 reports four poverty measures based on individual PCE, the head- count, the poverty gap ratio, the Foster-Greer-Thorbecke measure with a set at 2, and Sen's poverty measure. There are no new difficulties of calculation here; the headcount is simply the fraction of people who live in households whose house- hold PCE is less than equal to the poverty line, while Pv and P2 are calculated as weighted averages in the usual way. The Sen index is calculated according to (3.12) with the Gini for the poor calculated as described above but for the limited population of those in poverty. I have chosen a poverty line to match some arbit- rary but reasonable percentile in the base year and then held it fixed in real terms in the other years. In 1985, (just over) 30 percent of people were in households whose PCE was below 128,600 CFAF—an amount that is close to the useful (if arbitrary) guideline of one U.S. dollar per head per day—and I use this figure as the poverty line. Table 3.3 also shows bootstrapped standard errors for these poverty measures. These come from 100 bootstrap replications and take into account the clustered structure of the CILSS (see the Code Appendix for the STATA code for the bootstrap and for the inequality and poverty measures.) Table 3*3. Measures of individual poverty, Cote d'lvoire, 1985-88 (bootstrapped standard errors in brackets) Headcount Poverty-gap FGT index, Sen poverty Year ratio, PQ ratio, P{ index, Ps ^ 1985 0.300 (.030) 0.098 (.013) 0.045 (.007) 0.134 (.017) 1986 0.300 (.01 9) 0.082 (.007) 0.032 (.043) 0.1 12 (.009) 1987 0.348 (.025) 0.101 (.013) 0.043 (.008) 0.140 (.017) 1988 0.459 (.030) 0.142 (.016) 0.063 (.010) 0.196 (.021) Note: Measures of poverty are based on household PCE attributed to individuals, and are calculated on an individual basis. Source: Author's calculations based on CILSS, 1985-88. 156 THE ANALYSIS OF HOUSEHOLD SURVEYS These Ivorian estimates provide a good example of the fact that poverty does not always move in the same direction as average PCE. Although average PCE fell in every year from 1985 to 1988, the poverty measures decline from 1985 to 1986, only rising thereafter. The headcount ratio hardly changes from 1985 to 1986, but the decline is clear in the other measures. Although the mean of PCE fell between the two years, the fraction of people below the poverty line remained virtually unchanged and, as shown by the behavior of P{, the average PCE among the poor rose in spite of the fall in the overall average PCE. That people at the bottom of the distribution did better than the average presumably also contributed to the declines in inequality in Table 3.2. Note also the sensitivity of the poverty rankings of 1985 relative to 1987. The latter year has a large fraction of poor than the former, but once we take into account the average of PCE among the poor and its distribution between them, the increase in poverty is less, and as measured by P2, poverty declines. It should be emphasized once again that these results on poverty, and the poverty rankings of the years are in accord with the Atkinson social welfare ap- proach, provided enough weight is given to the poor. As argued in the theoretical section, there is little conceptual difference between social welfare measurement and poverty measurement, provided always that inequality indexes are interpreted jointly with means, and not on their own. The main difference between the two approaches is in the way the weighting is done, with everyone getting some weight in the social welfare function, although the poor get relatively more, while the non- poor get no weight in the poverty indices. Example 2. Inequality and poverty by race in South Africa The South African Living Standards Survey collected data at the end of 1993, shortly before the elections that established the government of President Nelson Mandela. The survey was intended to establish a picture of living standards and poverty on the eve of the elections, and to serve as a baseline against which future progress could be assessed. Because there have been no subsequent LSMS surveys in South Africa, these data cannot be used to track living standards over time, but they provide a snapshot of living standards by race at the end of the apartheid era. Table 3.4 brings together all of the statistics on the four racial groups. The top panel shows very large differences in levels of living, particularly between Blacks and Whites; the mean of individual PCE is almost seven times larger for Whites than for Blacks. The second panel shows that the ranking of groups by inequality is precisely the reverse of the ranking by average. Blacks are not only poorer than Whites, but their living standards are more unequally distributed. Indeed the Gini coefficient of 0.34 for Whites is low by any standards. As a result, even the seven- fold difference in mean PCE understates the difference in average welfare between Blacks and Whites. For example, if we measure social welfare as mean PCE multi- plied by one minus the Gini, Whites are 8.4 times better-off than Blacks. The dif- ferences between the groups also gives very high measures of inequality for the country as a whole so that, for example, the overall Gini coefficient is 0.59. WELFARE, POVERTY, AND DISTRIBUTION 757 Table 3.4. Welfare measures by race, South Africa, 1993 Measure Blacks Coloreds Indians Whites All Means Total household exp. 1,053 l',783 3,202 4,615 1,089 Household size 4.78 4.67 4.38 3.01 4.44 PCE (household) 325 483 828 1,793 615 PCE (individual) 220 378 732 1,531 406 Standard error 8.0 32.8 88.4 80.8 26.4 Inequality Gini 0.449 0.412 0.377 0.336 0.586 S. d. of logarithms 0.798 0.734 0.647 0.594 1.042 Coefficient of variation 1.062 0.994 0.924 0.733 1,563 Atkinson, 6=0.5 0.162 0.136 0.118 0.089 0.276 Atkinson, e = l,0 0.289 0.247 0.210 0.166 0.460 Atkinson, €=2.0 0.530 0.497 0.460 0.349 0.710 Poverty Headcount ratio P{} 0.317 0.080 0.0 0.0 0.250 Poverty gap ratio P{ 0.106 0.019 0.0 0.0 0.083 FGT/> 2 0.049 0.008 0.0 0.0 0.039 Sen measure Ps 0.144 0.029 0.0 0.0 0.113 Notes: Inequality and poverty are based on imputing household PCE to individuals. The poverty line is set at a PCE of 105 rand per month. Numbers of observations vary across statistics because of missing values; there are approximately 6,500 African, 700 Colored, 250 Indian, and 1,350 White households in the survey, Source: Author's calculations from South African Living Standards Survey, 1993. The poverty measures in the third panel use a poverty line of 105 rand per month per person. This is much lower than poverty lines commonly discussed in South Africa, but is approximately equal to $1 (U.S.) per person per day, the same line I used for Cote d'lvoire. More than 30 percent of Africans have less than even this very low cutoff; there are no Whites or Indians in poverty. As has often been noted, apartheid was a successful welfare state for the Whites, keeping incomes high, inequality low, and minimizing poverty. The majority of South Africa's citizens were less fortunate. Exploring the welfare distribution: inequality The welfare and poverty measures of the previous subsections are designed to aggregate the detailed distribution of welfare to a single number; provided that we know exactly the measure that we prefer, and provided that there is uncertainty neither about how to weight people at different income levels, nor about the level of the poverty line, then the summary measures are all we need. However, there is often substantial uncertainty about how much weight to give to the poor relative to the rich, and there is always a good deal of uncertainty about the poverty line. In consequence, it is often a good idea to explore the robustness of social welfare 158 THE ANALYSIS OF HOUSEHOLD SURVEYS measures to parameters about which we are unsure. At the same time, the reduc- tion of a distribution to a single number may be more aggregation than we want, and more fundamental insights into levels of living can often be obtained from graphical representations of either the whole distribution or of some part of it. I begin with inequality. The most familiar graphical tool for examining the distribution of income or consumption is the Lorenz curve, which is a plot of cumulative fraction of popula- tion—starting from the poorest—on the jt-axis against cumulative fraction of re- sources on the y-axis. Three (imaginary) distributions are sketched in Figure 3.3. If resources were equally distributed, with everyone receiving the same, the Lor- enz curve would be the 45-degree line, thus labeled the line of complete equality, whereas the case of complete inequality, with the richest person having everything, would generate a Lorenz curve running along the ;t-axis with a right angle at (100,0) to terminate at (100,100). Lorenz curves have positive slopes and positive second derivatives. At point/? on the horizontal axis, the slope of the curve is the ratio to mean x of the value of jc below which lie fraction p of the population; formally, if the distribution function of jc is F(JC) with mean |i, the slope of the Lorenz curve at/? is F ~l(p)/\Ji. This implies that the curves attain their maximum distance from the 45-degree line where the cumulative proportion of people is equal to the fraction of people with x below the mean (see Kakwani 1980, ch. 3, for this and further results). The Gini coefficient is also closely associated with the Lorenz curve. It is the area between the curve and the 45-degree line as a fraction of 0,5, which is the total area under the 45-degree line. The Lorenz curve is unaffected by multiplying Figure 3.3; Lorenz curves illustrating Lorenz dominance 100 Line of complete equality g u Distribution C Distribution B Cumulative percentage of people 100 WELFARE, POVERTY, AND DISTRIBUTION 759 everyone's allocation by a positive number, and so can tell us nothing about the mean of the distribution. Apart from this, all the information in the distribution is contained in the Lorenz curve so that, provided we know the mean, it is possible, for example, to recover the density or distribution function from the Lorenz curve. As first shown by Atkinson (1970), Lorenz curves play a very important role in characterizing the robustness of inequality measures. If two different Lorenz curves do not cross, as is the case for distributions B and either A or C in Figure 3.3, the lower curve can always be transformed into the upper curve by a series of equalizing transfers, by each of which welfare is transferred from a richer to a poorer individual. In consequence, when two Lorenz curves do not cross, the upper one represents an unambiguously more egalitarian distribution, one that will show a lower level of inequality using any measure of inequality that respects the principle of transfers. In Figure 3.3, the two distributions A and C cross one an- other (twice as shown), so that there is no unambiguous ranking of inequality without committing to more specific inequality measures. Distribution C is more equal among both the poorest and the richest, but is more unequal in the middle of the distribution than is distribution A. When one Lorenz curve is everywhere above another, we say that the distribution corresponding to the upper curve Lorenz dominates the distribution represented by the lower curve. Lorenz domination does not give a complete ordering of distributions; when Lorenz curves cross, neither distribution dominates the other. Because the Lorenz curves are unaffected by the mean of the distribution, they cannot be used to rank distributions in terms of social welfare, only in terms of inequality. This deficiency is easily repaired by looking at "generalized" Lorenz curves—a concept introduced by Shorrocks (1983). The horizontal axis for the generalized Lorenz curve is the same as that for the Lorenz curve, the cumulative fraction of the population, but the vertical axis, instead of showing the cumulative share of income, wealth, or consumption, shows the cumulative share multiplied by the mean, so that a Lorenz curve can be converted into a generalized Lorenz curve by multiplying by mean welfare. Clearly, for any single Lorenz curve, this is only a change of scale, and has no effect on its shape; generalized Lorenz curves are used for comparing different distributions with different means and thus with different aggregates. If the generalized Lorenz curve in one period lies above the generalized Lorenz curve in another period, it implies that for all p from 0 to 100, the poorest p percent of the population have more resources in total in the first period's distribution which will therefore be preferred by any equity respecting social welfare function. The poorest person has more, there is more in aggregate, and more generally, each quantile of the distribution is higher. Hence an equity respecting social welfare function will always prefer a distribution whose general- ized Lorenz curve lies above another. The generalized Lorenz curves corresponding to the three distributions in Figure 3.3 are shown in Figure 3.4, where I have assumed that A and B have the same mean, but that mean x in distribution C is higher. As drawn, the effect is to "lift" the distribution C clear of distribution A, so that C now dominates A by the generalized Lorenz criterion, although not by the Lorenz criterion. As a result, 160 THE ANALYSIS OF HOUSEHOLD SURVEYS Figure 3.4. Generalized Lorenz curves for Lorenz curves in Figure 3.3 Distribution C Distribution A Distribution B Cumulative percentage of people 100 distribution C will be preferred to A by any equality-preferring social welfare function. The generalized Lorenz curve of C now crosses that of B, so that the social welfare ranking of the two will depend on the precise social welfare func- tion used, on the tradeoff between more equality in B and the more mean in C. These examples should make it clear once again that inequality by itself is not a measure of welfare. If the mean of distribution C were further increased so that the generalized Lorenz curve for C were everywhere above that of B, we would have a situation where one distribution is preferred to another by all equity respecting social welfare functions, even though it is more unequal according to all measures of inequality that satisfy the transfer principle. Lorenz curves and inequality in South Africa and Cote d'lvoire Figure 3.5 shows three Lorenz curves for the individual PCE distributions in South Africa in 1993 for the whole population—the outer curve—for Blacks—the bro- ken line—and for Whites—the innermost line. These curves show, for example, that the poorest 20 (50) percent of South Africans receive only 3(13) percent of all of PCE, that the poorest 20 (50) percent of Blacks receive 5 (20) percent of all PCE received by Blacks, while the poorest 20 (50) percent of Whites receive 7.5 (28) percent of all PCE received by Whites. Also important to note is that the Lorenz curve for Blacks lies everywhere outside the Lorenz curve for Whites. As a result, the unanimous ranking in Table 3.4, where the distribution of PCE among Whites is shown as more equal than that among Blacks by all the measures, is not a special WELFARE, POVERTY, AND DISTRIBUTION 161 Figure 3.5. Lorenz curves for individual PCE by race, South Africa, 1993 0 20 40 60 Cumulative percentage of people Source: Author's calculations based on South African Living Standards Survey, 1993. feature of those particular measures but will be repeated for any other inequality measure that satisfies the principle of transfers. Put differently, provided we re- spect the principle of transfers, there can be no dispute that there is more inequality among Blacks than Whites. A second example comes from Cote d'lvoire. Figure 3.6 shows Lorenz curves for the four years of CILSS data, but in a slightly different way from usual. It is frequently the case with empirical Lorenz curves—as opposed to the theoretical curves in Figure 3.3—that different curves are very close to one another and are not easily told apart by inspection. That this was not the case for South Africa is because of that country's extreme differences in inequality between the races; Whites and Blacks are not only far apart in average living standards, but also in the dispersion of their living standards. Changes in inequality over time are likely to be less marked, and in Cote d'lvoire from 1985-88, the Lorenz curves do not move much. Differences are more easily seen if we plot, not the Lorenz curve itself, but the distance of the Lorenz curve from the 45-degree line, and these cur- ves are plotted in Figure 3,6. Because signs are changed, the higher curves are now those with the greatest inequality. The figure shows why the results in Table 3.2 come out as they do and they tell us what would happen if we were to work with alternative measures of inequality. The curve for 1988 lies entirely below the curve for 1986, and both lie below the curves for 1985 and 1987; these last cannot be ranked relative to one another, but cross at around the 70th percentile of the population. Below the 70th percentile, the 1985 curve is higher because the poorest part of the distribution was poorer in 162 THE ANALYSIS OF HOUSEHOLD SURVEYS Figure 3.6. Transformed Lorenz curves, Cote d'lvoire, 1985-88 30- 1987 ..••5^ "SN. 1985 I tob 0 20 40 60 80 100 Cumulative percentage of people Note\ The curves show the vertical distance of the Lorenz curve from the 45-degree line. Source: Author's calculations using CILSS 1985 than in 1987. Above the 70th percentile, which is a part of the distribution about which we may be less concerned, the curve for 1985 is lower because the share of PCE in the hands of the best-off people is less. As we saw in Table 3.2, all the measures agreed that 1988 was more equal than 1986, and that both were more equal than either 1985 or 1987, while disagreeing on the relative ranking of those two years. This is exactly what must happen given the shapes of the Lorenz curves. Table 3.1 shows that the average level of PCE is falling over time, so that a graph of generalized Lorenz curves over time will be one in which the standard Lorenz curves are progressively rotated clockwise from the origin. Because the changes in the mean are large relative to the changes in inequality, the generalized Lorenz curves move downward over time, with 1986 below 1985, 1987 below 1986, and 1988 below 1987. A small exception is that the generalized Lorenz curve for 1986 is slightly higher than that for 1985 at the very bottom of the distri- bution; the poorest people had absolutely more in 1986 than a year earlier, not just a larger share. This apart, social welfare fell in Cote d'lvoire from 1985 to 1988, and the conclusion does not depend on how we measure inequality nor on our degree of aversion to it. ^Stochastic dominance The mechanics of ranking welfare distributions are clarified by reference to the concept of stochastic dominance. While it is possible to read discussions of in- equality and of poverty without knowing anything about stochastic dominance, a WELFARE, POVERTY, AND DISTRIBUTION 163 deeper understanding can be obtained with the aid of the few straightforward definitions and results given in this subsection. For those who wish to skip this material, I shall label the stochastic dominance results when I come to them, so that it is possible to move directly to the next subsection without loss of continuity. Stochastic dominance is about ranking distributions, and the Lorenz dominance discussed above is only one of a set of definitions and results. All work by treating consumers as a continuum, so that instead of dealing with concepts such as the fraction of people whose consumption (say) is less than ;t, we think of x as being continuously distributed in the population with CDF F(x). We shall typically be concerned with the comparison of two such (welfare) distributions, whose CDFs I write as F{(x) and F2(x), and we want to know whether we can say that one is "better" than the other, in some sense to be defined. The first definition is first-order stochastic dominance. We say that distribution with CDF F{(x) first-order stochastically dominates distribution F2(x) if and only if, for all monotone nondecreasing functions a(;t) (3.20) faWdF^x) * fa(x)dF2(x) where the integral is taken over the whole range of ;c. The way to appreciate this definition is to think of oc(;t) as a valuation function, and monotonicity as meaning that more is better (or at least no worse.) According to (3.20), the average value of a is at least as large in distribution 1 as in distribution 2 no matter how we value x, so long as more is better. In consequence, distribution 1 is "better," in the sense that it has more of jc, and it stochastically dominates distribution 2. There is a useful result that provides an alternative characterization of first- order stochastic dominance. The condition (3.20) is equivalent to the condition that, for all x, (3.21) F2(x) fcfjOc) so that the CDF of distribution 2 is always at least as large as that of distribution 1. (The proof of the equivalence of (3.20) and (3.21) is straightforward, and is left as an exercise. But we can go from (3.20) to (3.21) by choosing ,(*) and D2(x), themselves the integrals of F^x) and F2(x)y whose relative rankings tells us about first-order dominance, so can third-order dominance be tested using the integrals of D} (x) and D2(x), Exploring the welfare distribution: poverty If robustness analysis is desirable for social welfare and inequality comparisons, it is even more so for the measurement of poverty if we are not to be hostage to an WELFARE, POVERTY, AND DISTRIBUTION 165 ill-defined and arbitrarily selected poverty line. At the least, we need to explore the sensitivity of the various poverty measures to the choice of z, although when we do so, it should not be a surprise that we are thereby led back to something closely akin to the social welfare approach. If we have literally no idea where the poverty line is, and it is even possible that everyone is poor, poverty measures have to give at least some weight to everyone, although as always, poorer individuals will get more weight than richer ones. But this is exactly what a standard social welfare function does, so that the social welfare approach will only differ from the poverty approach when we can set some limits on permissible poverty lines. Start from the headcount ratio, and consider what happens as we vary the poverty line z. Since PQ is the fraction of the population whose welfare level is below z, we have (3.23) PQ(z\F) =*<*) where the notation on the left-hand side emphasizes not only that the headcount is a function of the poverty line z, but also that it is a function (technically a func- tional) of the distribution F. If we have two different distributions F{ and F2, relating to different years, sectors, or countries, and we want to know which shows more poverty and the extent to which the comparison depends on the choice of poverty line z, then (3.23) tells us that if, for all poverty lines z (3.24) F,(z)>F2(z) the headcount will always be higher for the first distribution than the second. Hence, all we have to do to test the robustness of the headcount ratio is to plot the CDFs of the two distributions that we are interested in comparing, and if one lies above the other over the range of relevant poverty lines, then the choice of poverty line within that range will make no difference to the outcome. In the language of the previous subsection, the poverty ranking of two distri- butions according to the headcount ratio is robust to all possible choices of poverty line if, and only if, one distribution first-order stochastically dominates the other. In practice, we usually have some idea of the poverty line, or are at least able to rule out some as implausible, so that the more useful requirement is that (3.24) hold over some relevant range, which is a restricted form of stochastic dominance. Figures 3.7 and 3.8 show part of the cumulative distributions for individual PCE in South Africa and in Cote d'lvoire for 1985 through 1988. Since poverty lines at the very top of the distribution are usually implausible—even for a poor country— it is not necessary to show the complete range of PCE levels. In the South African case in Figure 3.7, the cutoff of 2,000 rand per capita per month excludes about 20 percent of Whites, but no one else. As was the case for the Lorenz curves, the extraordinary inequalities in South Africa produce an unusually clear picture. The four distribution functions are quite separate so that, no matter what poverty line we choose, there will be a higher fraction of people in poverty among Blacks than among Coloreds, a higher fraction among Coloreds than among Indians, and a higher fraction among Indians than among Whites. 166 THE ANALYSIS OF HOUSEHOLD SURVEYS Figure 3.7. Distribution functions of individual PCE by race, South Africa, 1993 1.0- Blacks I 0.8- I o. e 0.6 •s 0.4- 0.2- 0- 0 500 1,000 1,500 2,000 PCE: rand per capita per month Source: Author's calculations based on South African Living Standards Survey, 1993. The situation for Cote d'lvoire is less clear because several of the distribution functions cross. Here I have excluded people living in households with per capita monthly expenditure of more than 300,000 CFAF, which is two and a half times the poverty line used in constructing Table 3.3. Given the declines in PCE over time, it is no surprise that, over most of the range, the curves are higher in the later years so that, for most poverty lines, the fraction of poor people will be increasing from 1985 through to 1988. However, around the poverty line of 128,600 CFAF used in Table 3.3, the distribution functions for the first three years are very close and, at the lowest values of PCE, the curve for 1985 lies above that for 1986 and 1987. As we have already seen in making inequality comparisons, the poorest did better in 1985 than in 1986, even though average PCE fell. To examine the robustness of the other poverty measures, consider the "poverty deficit curve," defined as the area under the CDF up to some poverty line z t. (3.25) D(z;F) = fF(x)dx. U Why this measure is useful is revealed by integrating the right-hand side of (3.25) to give (3.26) = ZF(z)-ff(x)xdx = zF(z)(\ --^) = where, as before, u p is the mean welfare among the poor and P t (z;F) is the pov- erty-gap measure of poverty. Equation (3.26) establishes that we can use the WELFARE, POVERTY, AND DISTRIBUTION 167 Figure 3.8. Distribution functions of individual PCE, Cote d'lvoire, 1985-88 1.0- 0.8- 1988 <§ 0.6 ° 1987 . ,„ 1986 I 1 0.2- 0- 0 100 200 300 PCE: thousands of CFAF per capita per month Source: Author's calculations based on CILSS. poverty deficit curve to examine the robustness of the poverty-gap measure to different choices of the poverty line in exactly the same way that we used the CDF to examine the robustness of the headcount ratio. If the poverty deficit curve for one distribution lies above the poverty deficit curve of another, the first distribu- tion will always have more poverty according to the poverty-gap measure. Figure 3.9 shows the lower segments of the poverty deficit curves for the Ivor- ian data. These curves, which are marked in the same way as Figure 3.8, show that the poverty-gap ratio is higher in 1988 than in 1987 for a range of poverty lines, results that establish some robustness for the estimates in Table 3.3. The poverty deficit curve for 1987 is above that for 1986 and, except for low poverty lines, above that for 1985. Given previous results, the crossing of the 1985 and 1986 curves is to be expected; 1986 was better than 1985 at the bottom of the distribu- tion, but worse on average. It is possible to continue this type of robustness analysis beyond the headcount and poverty-gap ratios to the other poverty measures. However, it is better at this point to look at the pattern that is emerging, and once again to link the analysis of poverty back to the social welfare function. Note first that if, as happens in South Africa, or in Cote d'lvoire for 1988 and 1987, one of the distribution functions had been higher than another from 0 up to some plausible upper limit for the poverty line z+, say, then the same would automatically have been true for the poverty deficit curves. Formally, if for two distributions Fl and F2 (3.27) Ffo) * F2(x), Qzxzz+ 168 THE ANALYSIS OF HOUSEHOLD SURVEYS Figure 3.9. Poverty deficit curves, Cote d'lvoire, 1985-88 150- 1986 g 100- 1 € 1987 I ' " 1985 & 50J 1988 0- 0 100 200 300 PCE: thousands of CFAF per capita per month Source: Author's calculations based on CILSS. then i z (3.28) £>(r,F,) = fF}(x)dx * fF2(x)dx = D(z;F2), 0 * z * z + . 0 0 Hence if the distributions do not cross before the maximum possible poverty line, then not only are the headcount ratios robust to the choice of line, but so are the poverty-gap ratios. Indeed, if we were to push the analysis a stage further, and look at the area under the deficit curve, the resulting curves would not cross if the poverty deficit curves did not cross, so that measures like P2 would also be robust to choice of the line. Of course, these results only work one way; it is possible for the distribution functions to cross and for the poverty deficit curves not to do so, as indeed is the case for 1987 versus 1986 in C6te d'lvoire. But if we find that the distribution functions do not cross, we need look no further because all the poverty measures will be robust. If they do cross but the poverty deficit curves do not, then any measure that is sensitive to the depth of poverty will be robust, and so on. While these results that take us from one type of robustness to another are useful, they are not always exactly what we need. As emphasized by Atkinson (1987), we may have a lower as well as an upper limit for the poverty line, and it may turn out that the distribution functions do not cross between the limits, so that (3.27) holds for z ~ £ x z z+, say, so that the headcount ratio is robust to the choice of poverty line within the range of possibilities. Since the distribution functions may still cross below z", we no longer have the implication that (3.28) holds even over the restricted range. WELFARE, POVERTY, AND DISTRIBUTION 769 It is also worth considering again what happens when the poverty line can be anything, from zero to infinity. This is the case where the robustness of the head- count ratio is equivalent to the distribution functions never crossing, which is (unrestricted) first-order stochastic dominance. If the poverty deficit curves never cross, we have second-order stochastic dominance; by definition, Fl second-order stochastically dominates F2 if the expectation of all monotone increasing concave functions is larger under F{ than under Fr If the social welfare function is addi- tive over individual welfare levels, concavity is equivalent to diminishing marginal social utility, and thus to a preference for transfers from richer to the poorer peo- ple. Hence, noncrossing of the poverty deficit curves means that an additive equity-respecting social welfare function would prefer the distribution with the lower curve. Not surprisingly, the poverty deficit curve of one distribution is everywhere above that of another if and only if the generalized Lorenz curve of the former is everywhere below that of the former; generalized Lorenz curves and social welfare functions will rank distributions in the same way as one another and in the same way as poverty deficit curves when nothing is known about the pov- erty line. If the poverty deficit curves cross or, equivalently, if the generalized Lorenz curves cross, we might want to consider one more order of integration or of domi- nance. If we draw the curves formed by integrating under the poverty deficit curves, and if these do not cross, then one distribution dominates the other at the third order. This means that it would be preferred by any additive social welfare function where social marginal utility is positive, diminishing, and diminishing at a decreasing rate. This last condition, sometimes referred to as the principle of diminishing transfers (see Kolm 1976), means that, not only does the social wel- fare function increase when transfers are made from rich to poor, but that a transfer from someone earning 300 rand to someone earning 200 rand is to be preferred to a transfer from someone with 500 rand to someone with 300 rand. These third- order connections were first recognized by Atkinson, who reviews them in more detail in Atkinson (1992). Finally, it is worth emphasizing again the potential role of measurement error in household expenditures. The addition of random noise to a distribution spreads it out, and the contaminated distribution will be second-order stochastically domi- nated by the true distribution. If the contamination is similar across years, inter- temporal comparisons may be unaffected, but there is no reason to suppose that this is the case. Surveys often have start-up problems in their first year, and accu- racy can be expected to improve over time in a well-run survey, or to deteriorate if enumerators are not carefully supervised. 3.2 Nonparametric methods for estimating density functions All of the techniques discussed in the previous section relate to the distribution functions of welfare, to transformations like Lorenz curves or generalized Lorenz curves or to the integrals of areas beneath them. However, for many purposes, we are also interested in the density functions of income, consumption, or welfare. 170 THE ANALYSIS OF HOUSEHOLD SURVEYS Standard measures of central tendency and dispersion are often most easily visual- ized in terms of densities, and so it is useful to have techniques that provide esti- mates of densities. These techniques are the topic of this section which takes the form of a largely methodological digression between two more substantive discus- sions. It presents an introduction to the tools of nonparametric density estimation to be used in the next section, as well as in several subsequent chapters. These tools also help understand the nonparametric regressions that will be used in the next section. The distribution of welfare is only one of many examples where it is useful to calculate and display a density function. In the next section, where I con- sider the effects of pricing policies, I shall show how the joint density of the con- sumption or production of a commodity and levels of living can be used to des- cribe the differential effects of price changes on the well-being of rich and poor. Estimating univariate densities: histograms A good place to start is with the distributions of PCE by race in South Africa that were discussed in the previous section. For each of the four groups, Figure 3.10 shows the standard histograms that are used to approximate densities. The histo- grams are drawn for the logarithm of real PCE at the individual level, not for the level; the distribution of the latter is so positively skewed as to preclude the draw- ing of informative histograms. The logarithmic transformation yields a distribution that is more symmetric and much closer to normal. Indeed, the curves drawn on each histogram show the normal densities with mean and variance equal to the Figure 3.10. Histograms of log(PCE) by race, South Africa, 1993 Blacks Coloreds 0.15" 0.10- 0.05- 8, 5 Indians Whites g 0.15- 0.10- 0.05- 0- 8 10 10 Logarithm of PCE Source: Author's calculations based on South African Living Standards Survey, 1993. WELFARE, POVERTY, AND DISTRIBUTION 1 71 means and variances of the underlying distributions of the logarithm of PCE; these appear to provide a reasonably good approximation to the empirical densities, at least as represented by the histograms. The histograms and normal distributions of Figure 3.10 are all that we need for many purposes. They provide a visual impression of the position and spread of the data, and allow comparison with some convenient theoretical distribution, in this case the (log)normal. However, there are also a number of disadvantages of histo- grams. There is a degree of arbitrariness that comes from the choice of the number of "bins" and of their widths. Depending on the distribution of points, the choice of a bin boundary on one side or another of a cluster of points can change the shape of the histogram, so that the empirical distribution appears to be different. Perhaps more fundamental are the problems that arise from using histograms— which are tools for representing discrete distributions—to represent the continu- ously differentiable densities of variables that are inherently continuous. Such representations can obscure the genuine shape of the empirical distribution, and are inherently unsuited to providing information about the derivatives of density functions, quantities that are sometimes of interest in their own right, as we shall see in Chapters 4 and 5. It is worth asking why these difficulties did not arise when graphing CDFs, or the areas beneath them as in Figures 3.7, 3.8, and 3.9 above. The answer is that they did, but because the data are cumulated and there are a large number of data points, the discontinuities are less apparent. The empirical distribution functions in Figures 3.7 and 3.8 are calculated from the formula (3.29) F(x) = /i 1=1 which is simply the fraction of the sample whose jt's are less than or equal to the value x. F(x) is a step function that jumps whenever x is equal to a data point, and that is flat between data points. Because there are many data points, the steps are all very small relative to the scale of the figure, and there are many steps, so that the eye does not perceive the jagged shape of the graph, at least away from the upper tail of the distribution where the points thin out. These considerations apply even more strongly to poverty deficit curves, which are the integrals of distribution functions; they are no longer discontinuous at the data points, but only have slopes that are discontinuous step functions. But changes in slopes are even harder to see, and the pictures give the impression of smooth and ever accelerating slopes. When we move from a CDF to a density, we are moving in the opposite direction, differ- entiating rather than integrating, so that the discontinuities in the empirical distri- bution function present serious difficulties in estimating densities, difficulties that are magnified even further if we try to estimate the derivatives of densities. ^Estimating univariate densities: kernel estimators The problems with histograms have prompted statisticians to consider alternative ways of estimating density functions, and techniques for doing so are the main / 72 THE ANALYSIS OF HOUSEHOLD SURVEYS topic of this subsection. One familiar method is to fit a parametric density to the data; the two-parameter lognormal is the simplest, but there are many other possi- bilities with more parameters to permit a better fit (see, for example, Cramer 1969 or Kakwani 1980.) Here I look at nonparametric techniques which, like the histo- gram, allow a more direct inspection of the data, but which do not share the histo- gram's deficiencies. Readers interested in following the topic further are encour- aged to consult the splendid (and splendidly accessible) book by Silverman (1986), on whose treatment the following account is based. Perhaps the simplest way to get away from the "bins" of the histogram is to try to estimate the density at every point along the jt-axis. With a finite sample, there will only be empirical mass at a finite number of points, but we can get round this problem by using mass at nearby points as well as at the point itself. The essential idea is to estimate the density f ( x ) from the fraction of the sample that is "near" to jc. One way of doing this is to choose some interval or "band," and to count the number of points in the band around each jr. Think of this as sliding the band (or window) along the jc-axis, calculating the fraction of the sample per unit interval within it, and plotting the result as an estimate of the density at the mid-point of the band. If the "bandwidth" is A, say, the so-called naive estimator is (3.30) _ rtrt /=i \ 2 At each point jc, we pass through the sample, giving a score of 1 to each point within h!2 of jc, and zero otherwise; the density is estimated as the total score as a fraction of the sample size and divided by h to put it on a per unit basis. The choice of the bandwidth h is something to which I shall return. For the moment, the point to note is that the bandwidth ought to be smaller the larger is the sample size. If we only have a few points, we need large bands in order to get any points in each, even if the wide bandwidth means that we risk biasing the estimate by bringing into the count data that come from a different part of the distribution. However, as the sample size grows, we can shrink the bandwidth so that, in the limit when we have an infinite amount of data, the bandwidth will be zero, and we will know the true density at each point. In fact, we have to do a little more than this. What has to happen in order to get a consistent estimate of the density at each point is that the bandwidth become smaller at a rate that is less fast than the rate at which the sample size is increasing. As a result, not only does the shrinking band- width guarantee that bias will eventually be eliminated by concentrating only on the mass at the point of interest, but it also ensures that variance will go to zero as the number of points within each band increases and the average within the band becomes more precise. In this way, the increase in the sample size is shared be- tween more points per band, to increase the precision, and smaller bandwidths so as to ultimately eliminate bias. Of course, because some of the benefits of larger sample sizes have to be "diverted" from putting more observations into each band and devoted to shrinking the bandwidth, the rate of convergence to the density at each x is bound to be slower than the normal root-W convergence that is standard in sampling or regression analysis. WELFARE, POVERTY, AND DISTRIBUTION 173 While the naive estimator captures the essential idea of nonparametric density estimation using the "kernel" method—the kernel is the band or indicator function in (3.30)—it does not solve one of the problems with which we began. In parti- cular, there will be steps in f(x) every time a data point enters or exits the band. But this can be dealt with by a simple modification. Instead of giving all the points inside the band equal weight, we give more weight to those near to x and less to those far away, so that points have a weight of zero both just outside and just inside the band. We can do this quite generally by replacing the indicator function in (3.30) by a "kernel" function K ( . ) t so that (3.31) f(x) = J-E nh i=\ which is the "kernel estimate" of the density /(jc). There are many possible choices for the kernel function. Because it is a weight- ing function, it should be positive and integrate to unity over the band, it should be symmetric around zero, so that points below x get the same weight as those an equal distance above, and it should be decreasing in the absolute value of its argu- ment. The "rectangular" kernel in (3.30)—so called because all observations in the band get equal weight— satisfies all these criteria except the last. A better choice is a kernel function that uses quadratic weights. This is the Epanechnikov kernel n ( wl) K(z) = *'* = 0, whose weights have an inverted U-shape that decline to zero at the band's edges. Another obvious source of kernels is the class of symmetric density functions, the most popular of which is the Gaussian kernel (3.33) K(z) =(2*r°'5exp(-z2/2). The Gaussian kernel does not use a discrete band, within which observations have weight and outside of which they do not, but instead gives all observations some weight at each point in the estimated density. Of course, the normal density is very small beyond a few standard deviations from the mean, so that the Gaussian kernel will assign very little weight in the estimate of the density at x to observations that are further than (say) 3h from x. A third useful kernel is the quartic or "biweight" (334) = 0, lzl>l. The quartic kernel behaves similarly to the Epanechnikov kernel, declining to zero at the band's edges, but has the additional property that its derivative is continuous at the edge of the band, a property that is useful in a number of circumstances that we shall see as we proceed. Although the choice of kernel function will influence the shape of the estimated density, especially when there are few points and the bandwidth is large, the litera- 174 THE ANALYSIS OF HOUSEHOLD SURVEYS tare suggests that this choice is not a critical one, at least among sensible alterna- tives such as those listed in the previous paragraph. As a result, a kernel can be chosen on other grounds, such as computational convenience or the requirement that it be continuously differentiable at the boundary. More important is the choice of the bandwidth, and this is a practical problem that has to be faced in every application. As we have already seen, the bandwidth controls the trade-off between bias and variance; a large bandwidth will provide a smooth and not very variable estimate but risks bias by bringing in observations from other parts of the density, while a small bandwidth, although helping us pick up genuine features of the underlying density, risks producing an unnecessarily variable plot. Estimating densities by kernel methods is an exercise in "smoothing" the raw observations into an estimated density, and the bandwidth controls how much smoothing is done. Oversmoothed estimates are biased, and undersmoothed estimates too variable. A formal theory of the trade-off between bias and variance provides helpful insights and is a useful guide to bandwidth selection. In standard parametric infer- ence, optimal estimation is frequently based on minimizing mean-squared error between the estimated and true parameters. In the nonparametric case, we are at- tempting to estimate, not a parameter, but a function, and there will be a mean- squared error at each point on the estimated density. One natural procedure is to attempt to minimize the mean integrated squared error, defined as the expectation of the integral of the squared error over the whole density. Silverman (1986, pp. 38-40) shows how to approximate the mean integrated square error for a kernel estimate of a density, and shows that the (approximate) optimal bandwidth is r-1/5 (3.35) h' = Since the evaluation of (3.35) requires knowledge of the very density that we are trying to estimate, it is not directly useful, but is nevertheless informative. It con- firms that the bandwidth should shrink as the sample size increases, but that it should do so only very slowly, in (inverse) proportion to the fifth root of N. Note too the importance of the absolute size of the second derivative of the density. If there is a large amount of curvature, then estimates based on averaging in a band will be biased, so that the bandwidth ought to be small, and conversely on seg- ments of the density that are approximately linear. For most of the applications considered in this book, an adequate procedure is to consider a number of different bandwidths, to plot the associated density esti- mates, and to judge by eye whether the plots are undersmoothed or oversmoothed. This can perhaps be regarded as an informal version of an iterative procedure that computes a "pilot" estimate, calculates the optimal bandwidth from (3.35) assum- ing that the pilot is the truth, and then repeats. Applying the informal procedure usually leads to an easy separation of features in the density that are driven by ran- dom sampling from those that appear to be genuine characteristics of the under- lying law from which the sample is drawn. There should also be some preference for undersmoothing when using graphical methods; the eye can readily ignore WELFARE, POVERTY, AND DISTRIBUTION 7 75 variability that it judged to be spurious, but it cannot discern features that have been covered up by oversmoothing. It is very useful in these calculations to have a good bandwidth with which to start. If it is assumed that the density has some specific form, normal being the obvious possibility, then (3.35) yields an optimal bandwidth once a kernel is chosen. If the estimated density has the right shape, we have a good choice—in- deed the best choice—and otherwise we can make informal adjustments by hand. If both the kernel and the density are Gaussian, (3.35) gives an optimal bandwidth of 1.06oW ~ 1/5 , where o is the standard deviation of the density, which can be calculated prior to estimation. Silverman suggests that better results will typically be obtained by replacing o by a robust measure of spread, in which case the opti- mal bandwidth is (3.36) h* = 1.06 min(o, 0.75 IQR)N~lls where IQR is the interquartile range—the difference between the 75th and 25th percentiles. Similar expressions can easily be obtained from (3.35) using other kernels; for the Epanechnikov, the multiplying factor 1.06 should be replaced by 2.34, for the quartic by 2.42. Estimating univariate densities: examples The techniques of the previous subsection can be illustrated using the distributions of log (PCE)in Cote d'lvoire and South Africa, as well as similar distributions from Thailand that I shall refer to again in the next section. Figure 3.11 shows the esti- mated density function for PCE in South Africa, with the data pooled by race. The calculations were done using the "kdensity" command in STATA, weighted using the sampling weights multiplied by household size, and with STATA's default choice of kernel—the Epanechnikov. The weighting is done using the obvious modification of (3.31) (3.37) fw(x) =h'lEvh /»=i where vh are the normalized weights, the sampling weights normalized by their sum (see equation 1.25). The top left panel uses STATA's default bandwidth, which comes from a formula similar to (3.36), while the other panels show what happens if the bandwidth is scaled up or down. The South African expenditure distribution is a good illustration because the distributions for Africans and Whites are so far apart that the combined density has a distinct "bump," perhaps even a second mode, around the modal log (PCE) for Whites. With the default bandwidth, this can be clearly seen in the estimated den- sity. It is a good deal less clear when the bandwidth is doubled, and is altogether smoothed away when the bandwidth is doubled again. Choosing too large a band- width runs the risk of smoothing out genuine, and genuinely important, features of the data. The figure in the bottom right panel uses a bandwidth that would usually be regarded as too small. Certainly, the graph is not very smooth, and the rough- ness is almost certainly a feature of the estimation, and not of the underlying true 176 THE ANALYSIS OF HOUSEHOLD SURVEYS Optimal bandwidth Twice optimal Half optimal Logarithm of PCE Source: Author's calculations based on South African Living Standards Survey, 1993. density. Nevertheless, the mode for Whites is clear enough, and the eye is usually capable of smoothing away irrelevant roughness. In this sort of visual exercise, variance is less of a risk than bias, and some undersmoothing is often useful. Figure 3.12 shows the results of estimating kernel densities for the logarithm of PCE for the four years of data from Cote d'lvoire, and provides yet another repre- sentation of data that should by now be familiar. We see once again, although in a different form, the features displayed by the summary statistics in Table 3.1, and in the Lorenz, distribution, and deficit curves in Figures 3.6, 3.8, and 3.9. The densities generally move to the left over time, although 1986 moves to the right at the bottom, decreasing both poverty and inequality. ^Extensions and alternatives Although kernel estimators are the only nonparametric estimators of densities that I shall use in this book, there are several alternative techniques. Even within the kernel class, there are different ways of selecting the bandwidth, and there are several useful estimators based on other techniques. When the estimates are used for more than graphical presentation, for example as the input into further calcu- lations, it is essential to have a method of bandwidth selection that is objective and replicable across investigators. Apart from the use of rules based on specific distributions, such as (3.36), the most commonly used technique is cross-valida- tion (see again Silverman 1986, pp. 48-6, or Stoker 1991, pp. 123-24). The essen- tial idea here is to compute alternative estimates of the density based on different WELFARE, POVERTY, AND DISTRIBUTION 177 Figure 3.12. Estimated density functions for log(PCE), Cote d'lvoire, 1985-88 0.6- 1988 .§* 0.4- 2 4 6 8 Logarithm of PCE Source: Author's calculations based on QLSS. bandwidths, to use the result to get some idea of the relationship between the band width and mean-squared error, and thus to select a bandwidth that minimizes error. Such calculations are computer-intensive, and are unnecessary for the types of ap- plications considered in this book: when our main concern is the graphical inspec- tion of data, it is usually sufficient to try several bandwidths, and to select a suit- able one by inspection of the results. Kernel methods can also be generalized to allow different bandwidths at differ- ent points of the distribution. One nonkernel approach that does so automatically is the nearest neighbor method. Here the estimate of the density is obtained by cal- culating the distance from each point x to its fcth nearest neighbor; k is a number chosen by the investigator, equal perhaps to a fifth of the sample size, and which plays the same role in nearest neighbor estimation as does the bandwidth in kernel estimation. If dk(x) is the distance from jc to the Jtth nearest of its neighbors, whet- her to the left or the right, the estimate of the density at x is given by (3.38) /,(*) = (*-l)/2) = Vh(mk + nk,p) where i|f is the indirect utility function, p is a vector of prices of consumption goods, including the price of rice, TIA is profits (net income) from farm activity, and mh is income from nonfarm activities, such as wage-labor or transfers. Some- what schizophrenically in view of the analysis to come in Chapter 5, 1 am assum- ing that prices are the same for all farmers, thus ignoring regional variation in prices. However, the survey data do not report household-specific prices, and provided all prices move with the export price—as seems to be the case — individ- ual price variation will not affect the analysis. Although the utility function in (3.43) is a standard one— given that goods' prices are /?, it is the maximum utility that can be obtained from the efficient spending of all sources of income, whether from the farm or elsewhere— it rests on a number of important and by no means obviously correct assumptions. Most crucial is the "separation" property, that the only effect of the household having a farm is that it has farm income (see again Singh, Squire, and Strauss 1986 for further discussion). The validity of separation rests on the existence of efficient rural labor markets—a plausible assumption for Thailand — but also, and more 184 THE ANALYSIS OF HOUSEHOLD SURVEYS dubiously, on the supposition that household and hired labor are perfect substitutes on the family farm. When these assumptions hold, the shadow price of family labor is its market price, which is the wage rate, and working on the household's farm is no different from any other job. As a result, workers in the household can be modeled as if they were wage earners at a fixed wage, and the ownership of the farm brings a rental income, as would the ownership of any other real or financial asset. There are good reasons to doubt the perfect substitutability of household and hired-in labor—differential monitoring costs or transport costs are two obvious examples—but the validity of the separation property as an adequate approxima- tion is ultimately an empirical question. I am unaware of evidence from Thailand, but tests for Indonesia in Pitt and Rosenzweig (1986) and Benjamin (1992) have yielded reasonably favorable results. The validity of (3.43) also requires a different sort of separability, that goods and leisure are separable in preferences, so that the direct utility function takes the form u(l,v(q)) for leisure / and vector of goods q. Given this, the indirect utility function in (3.43) corresponds to the utility of the goods that enter the direct sub- utility function \)(q). It is straightforward to relax this assumption in the present context, but only at the price of complicating the algebra. The same can be said for my ignoring intertemporal aspects and assuming that saving is zero. Suppose now that there is a change in the *th price, in this case the price of rice. If we confine ourselves to small changes, the effects can be analyzed through the derivatives of the indirect utility function (3.43). In particular, using the chain rule (34 4) ^ = a^3^ + a^ a/?,. dxk ePi dpi since, for households that are both producers and consumers, the price of rice affects both farm profits and the cost of living. For households, such as urban households, that produce nothing, the first term on the right-hand side of (3,44) will be zero, since there are no farm profits, and similarly for farmers who do not produce rice, whose profits are unaffected by the price of rice. The two terms in (3.44) can be further elucidated using general results. The effects of prices on profits—often referred to as Hotelling's Lemma after Hotelling (1932)—are given by - h (3.45) — = yu where yhi is the production of good i by h. The effect on utility of an increase in price is given by Roy's (1942) theorem, 3 46 a*» = a** <- > ^ ~'"a* where qhi is the amount of good i consumed by household h, and the quantity d\|f h /dxh is the marginal utility of money to household /i, a nonoperational concept with which I shall dispense shortly. Producers benefit from a price change in pro- portion to the amount of their production and consumers lose in proportion to the amount of their consumption. For households that are both producers and con- WELFARE, POVERTY, AND DISTRIBUTION 185 sumers— farm-households —the gain or loss is proportional to the difference between production and consumption, which is yhi~qhi. It is often more convenient to work with proportional changes in prices, and if we substitute (3.45) and (3.46) into (3.44) and multiply by p. we obtain (3 47) xh I shall refer to the last term on the right-hand side of (3.47) as the "net benefit ratio." It measures the elasticity with respect to price of money-equivalent utility, or consumers' surplus for those who prefer that much-abused concept. For policy work, we are concerned with the second of these two terms, but not the first. When considering a policy-induced price change, we need to know the money-equivalent losses and gains for different individuals, so that we can calculate the distributional effects as well as anticipate the likely political repercussions. The marginal utility of money to each individual, quite apart from being unobservable, will be sub- sumed in policy discussions by decisionmakers' views about the value of giving money to each individual, whether based on levels of living, region, caste, or political preference. One simple way to summarize these effects is to write (3.48) where W is to be thought of as a social welfare function, and f ^ captures, not the private marginal utility of money, but the social marginal utility of money. This is not something observable, but summarizes the attitudes of the policymaker to- wards giving resources to individual ft, depending on that household's consump- tion level xh as well as on other relevant characteristics zh, such as perhaps region or ethnic characteristics. (For the remainder of this book, z will denote household characteristics, and not a poverty line.) We still need to allow for the fact that the rice that is produced—paddy —is not the same commodity as the rice that is consumed— milled rice. Suppose that one kilogram of paddy generates A< 1 kilograms of milled rice, and that p. is the price of the latter, so that the price of paddy is Xpr If we then rework the preceding analysis, the net benefit ratio is as before, but with Kyhi in place of yhi. But this is still the ratio of the value of net sales of rice (or paddy) to total expenditure, so that provided everything is measured in money terms, the benefit ratio is correctly computed by subtracting the value of consumption from the value of sales. There are a number of caveats that ought to be entered before going on to the empirical results. Note first that the use of the differential calculus, although con- venient, limits the analysis to the effects of "small" price changes and may not give the correct answer for actual (finite) price changes. Of course, the result that producers gain and consumers lose from a price increase is a general one, not a local approximation. However, the magnitude of the welfare effects of finite price changes will depend, not just on the amounts produced and consumed, but also on the second-order effects, which involve the amounts by which consumption and Table 3.5. Household PCE, rice production, and rice consumption in Thailand, 1981-82 Upper Lower Whole Upper Lower North North Kingdom North North East East Center South Bangkok Municipal areas (urban) PCE 1,516 1,349 1,362 1,171 1,172 1,497 1,361 1,680 Value of rice sold 20 26 134 4 12 33 18 4 Value of rice bought 243 326 261 298 276 247 234 216 Budget share of rice 5.59 8.80 7.03 8.18 7.68 5.72 6.06 4.05 Villages (rural) PCE 675 560 647 472 441 862 712 1,021 186 Value of rice sold 909 426 1,471 580 711 1,183 366 1,515 Value of rice bought 388 377 377 469 468 374 324 285 Budget share of rice 18.70 21.77 19,44 24.64 26.78 14.21 14.38 8.59 Note: All figures are averages over all households in the sample. Amounts in the first three rows are baht per month. Production values are one-twelfth of the annual value of crops and are averaged over all households whether or not they produce rice. The budget shares of rice are the percentages of the nondurable expenditures devoted to rice derived for each household and averaged over households. Glutinous rice and nonglutinous rice are taken together. Source: Deaton (1989a). WELFARE, POVERTY, AND DISTRIBUTION 187 production respond to price changes. Since my main interest here is in the distribu- tional effects of price changes, and in locating the benefits and costs of price changes in the distribution of living standards, these effects will change the con- clusions only to the extent that the elasticities of supply and demand differ system- atically between poor and rich. Although there is no reason to rule out such effects a priori, there is no reliable evidence on the topic; as I shall argue in Chapter 5, the measurement of the slopes of demand functions is difficult enough without trying to determine how the slopes vary with the level of living. A more serious deficiency in the analysis is its neglect of repercussions in the labor market. Changes in the price of the basic staple will affect both supply and demand for labor, and these effects can cause first-order modifications to the results. Once again, although it is possible to write down models for how these effects might operate—see, for example, Sah and Stiglitz (1992) who work through several cases—there is little point in doing so here without more hard information about the structure and functioning of the rural labor market in Thai- land. Implementing the formulas: the production and consumption of rice The empirical analysis starts from (3.48), and uses the 1981-82 Socioeconomic Survey of the Whole Kingdom of Thailand to provide the information. The general approach is to calculate net benefit ratios for each household, and to examine the distribution of these ratios in relation to living standards and regional variation. While it is evident without survey data that lower rice prices redistribute real resources from rural to urban areas, what is less obvious is how price changes redistribute real income between rich and poor people within the rural sector. The Thai socioeconomic surveys collect data on households by three levels of urbanization, municipal areas (4,159 households in the 1981-82 survey), sanitary districts (1,898 households), and villages (5,836 households). The first corresponds to the urban sector, while the last is the rural area with which I am primarily con- cerned. (Sanitary districts are semiurban conglomerates of villages, and I shall typically ignore them.) Table 3.5 shows averages of household per capita expendi- ture as well as information on rice for the country as a whole and for seven major regions. Households in Bangkok are nearly half of urban households and their per capita expenditure is a good deal higher than elsewhere. The rural households that live on the fringes of Bangkok—an excellent rice growing region—are much better-off than other rural households, although substantially poorer than urban Bangkok households. Over the other regions, the average levels of per capita expenditure in the rural areas are a half to a third of the corresponding urban esti- mates. Once again, I do not have satisfactory price indices for rural versus urban, but there is little doubt that the urban households have a substantially higher standard of living. In consequence, there is no equity argument in favor of the redistribution from rural to urban that is brought about by export taxes on rice. Not surprisingly, urban households produce little rice compared with rural households. Of the latter, those on the fringes of Bangkok, in the Center, and in the Figure 3.15. Living standards and rice consumption in Thailand, 1981-82 Univariate densities: all regions Joint density contours: villages Villages "8 0,3 1 02 Log(PCE):log(baht) Log(PCE):log(baht) Joint density netmap: municipal areas Joint density contours: municipal areas 00 Log(PCE): log (baht) Source: Author's calculations based on Socioeconomic Survey of Thailand, 1981-82. WELFARE, POVERTY, AND DISTRIBUTION 189 Lower North are the most productive; these are the regions in the alluvial river basin of central Thailand. In the North Eastern and Upper Northern regions, most production (and consumption) of rice is of the glutinous variety. Although there is little substitution in consumption between glutinous and nonglutinous rice, with distinct groups of people eating each, there is substitutability in production, and for the purpose of the current analysis I assume that the prices of the two kinds of rice move in parallel. Rice consumption is also higher in rural areas in spite of the higher levels of living among urban households. However, urban Thais spend sub- stantial amounts on precooked meals—often good value for single-person house- holds—and although the survey collects data on these expenditures, it cannot decompose these meals into their constituent commodities, so that urban consump- tion of rice is understated by reported purchases of rice. For this reason, and be- cause rural households are poorer, their budget shares for rice are very high, around a quarter of all expenditures in the North East, which is the poorest region. The four panels of Figure 3.15 provide a detailed graphical presentation of the data whose averages appear in Table 3.5. The top left-hand panel shows the esti- mated densities for the logarithm of PCE for the three regions, with the richer urban areas clearly separated from the rural areas with the sanitary districts in between. Note also that, unlike the traditional presumption that cities are more unequal, the Thai urban distribution shows the least dispersion of the three. Relative to a normal distribution, the densities have thick tails, particularly in the rural areas where there are a number of households with very high consumption levels. The other three panels show the relationship between the share of rice in household expendi- tures and the logarithm of per capita household expenditure. In the top right panel is the contour map of the bivariate distribution in the villages, while the bottom two panels show the netmap and contour map for the urban areas. All three dia- grams show that the share of rice in the budget is lower for better-off households, but also that there is considerable dispersion at all levels of living. In line with the averages in the table, budget shares of rice are much lower in the urban than rural areas, and the netmap shows clearly that the urban density has finite mass at zero, presumably corresponding to those who get their rice by buying prepared meals. While the relationship between rice purchases and living levels is of interest in its own right, and I shall return to these general questions in the next chapter, our main interest here is in net purchases of rice, or at least in the benefit ratio—net sales of rice as a ratio of total household expenditure. The bivariate density for the net benefit ratios and the logarithm of PCE is shown in Figure 3.16. The horizontal line corresponds to a net benefit of zero so that households who are self-sufficient in rice will be along that line at a position determined by their level of living. Households above the line—somewhat less than a half of all rural households—are net producers of rice (or at least were so in the survey year) while those below the line consume more than they grow. This majority of households includes non- farmers as well as those who are farmers but grow no rice. The figure shows that those who benefit from higher rice prices, and among them those who benefit the most, tend to be located in the middle of the distribu- tion of the logarithm of PCE. Among the best-off households, there are relatively 190 THE ANALYSIS OF HOUSEHOLD SURVEYS few with positive benefit ratios, and none with large benefit ratios. The best-off rural households in Thailand are either nonfarmers (government employees or teachers, for example) or specialize in crops other than rice, which is usually a small-scale family enterprise unsuited for large-scale commercial agriculture. At the other end of the distribution, among the poorest families, there are many more rice growers, and a larger fraction of households who obtain a substantial fraction of their consumption by sales of rice. But on average, the poorest households, although they would be net beneficiaries from high rice prices, do not have benefit ratios as large as those for the households in the middle of the distribution. In consequence, higher prices for rice would benefit average rural households at all levels of living, but the greatest proportional benefits would be to the rural middle class, not to either rich or poor. Although higher rice prices would not be a very efficient way of helping the poor in rural Thailand—and bear in mind that I am ignoring any effects that work through the labor market—it is clearly not the case, as with plantation crops in some countries, that higher prices benefit only the richest and largest farmers. Although Figure 3.16 contains essentially all the information that we need, the discussion in the previous paragraph rested, not so much on the detail of the den- sity, but on an averaging over net benefit levels for different PCE groups. This con- ditional averaging is the essence of nonparametric (and indeed of parametric) re- gression, to which the next subsection provides an introduction. Figure 3.16. Joint distribution of net benefit ratios and log(PCE), rural Thailand, 1981-82 1.2 I JS I 3,001 households make net purchases Log(PCE): log(baht) Source: Author's calculations based on Socioeconomic Survey of Thailand, 1981-82. WELFARE, POVERTY, AND DISTRIBUTION 191 Nonparametric regression analysis Economists often think of regression as linear regression, with one variable fitted by ordinary least squares on one or more explanatory variables. But recall from Chapter 2 that the definition of a regression is as a conditional expectation. For a variable y and a vector of covariates jt, the expectation ofy conditional on x is (3.49) E(y\x) = m(x) where the function m(x), which in general will not be linear, is the regression function of y on x. Since y differs from its expectation by a residual that is, by construction, uncorrelated with the latter, the statistical definition coincides with the standard linear regression model when the regression function is linear. How- ever, it should also be noted that, given a joint distribution for any set of variables, we can always calculate the regression function for any one variable conditional on the others, so that there can be no automatic association between regression and causality, or anything else that depends on an asymmetric treatment of the vari- ables. It is instructive to rehearse the standard links between a conditional expectation and the underlying distributions. In particular, we can write (3.50) E(y\x) = f y f c ( y \ x ) d y = fyfj(y>x)dy/fM(x) where the C, A/, and J subscripts denote conditional, marginal, and joint distribu- tions, respectively. Alternatively, the regression function can be written entirely in terms of the joint distribution, (3.51) m(*) = E(y\x) = fyfj(y,x)dylffj(y,x)dy. This equation suggests a nonparametric method for estimating the regression function; estimate the joint density using kernel (or other) methods, and use the results to calculate (3.51). In practice, there is a somewhat more direct approach through equation (3.50). Perhaps the obvious way to calculate a regression function is to use the sample information to calculate the average of all y-values corresponding to each x or vector of jt's. With an infinite sample, or with discrete explanatory variables, such an approach would be feasible. With finite samples and continuous x> we face the same problem as in density estimation, that there are no sample points exactly at each jt, and we adopt the same solution, which is to average over the points near x, nearness being defined with reference to a bandwidth that will shrink to zero as the sample size increases. As with density estimation, weighting is desirable so as to avoid discontinuities in the regression function as individual observations move into and out of the bands, and this can be dealt with by calculating kernel regres- sions that are closely analogous to kernel estimates of densities. Indeed, the con- cept is perhaps already more familiar with regressions; the common practice of 792 THE ANALYSIS OF HOUSEHOLD SURVEYS smoothing time series by calculating a moving average over a number of adjacent points is effectively a (rectangular) kernel regression. The kernel regression estimator can be written as follows: n n ( X~X\ l ( X-X\ (3.52) m« = Ey.K ^ / £* — /-l V A / ' 1-1 \ h ) which, using f(x) the kernel estimate of the density at x from (3.31) above, can be written as (3.53) m(x) = (nhy^y^l i / f(x). /-i \ h ) ' This equation can be thought of as an implementation of (3.50) with the kernels acting to smooth out the discrete sample points. Using (3.52), the estimate of the regression function can also be written as a weighted average of the /s, (3.54) m(jc) = E wi(x)yi where the weights are given from (3.52). According to (3.54), which makes clear the moving average analogy, the estimated regression function is a weighted aver- age of all the yt in the sample with the weights depending on how far away each corresponding jc. is from the point at which we are calculating the function. As when estimating densities, bandwidths can be chosen by trying different values and by inspecting the plots of the resulting estimates, or by some more automatic and computationally intensive technique. The same kernels are available for regression analysis as for density estimation (see the formulas on p. 173). These matters, as well as other topics relating to nonparametric regression, are discussed in Hardle (1991) who also (p. 101) gives formulas for asymptotic confi- dence bands around the regression (3.54): if ca is the (1-a) quantile of the /-distri- bution, the upper and lower (1-a) confidence bands for the regression are given by r^ ii/2 (3.55) ba(x) = m(x)±c\fK(z)2dz\ Q(x)[Nhf(x)]'lf2 where 5(;c) is the estimate of the local regression standard error, and is calculated from (3.56) Although these bands are useful for assessing the relative precision of different points along the estimated regression function, they should not be treated very seri- ously. The asymptotic results are obtained by ignoring a number of terms—such as the bias of the regression—which tend to zero only very slowly. A better proce- dure is to use the bootstrap, and I shall give some substantive examples in Chapter 4. Even so, the routine presentation of confidence bands together with regressions is not always advisable, since it tends to clutter the diagrams and obscure more important features. WELFARE, POVERTY, AND DISTRIBUTION 793 Nonparametric estimation of regression functions is a harder task than nonpara- metric estimation of densities. In particular, it is not possible to calculate a condi- tional expectation for values of jc where the density is zero; if x cannot occur, it makes no sense to condition y on its occurrence, and the attempt to calculate the regression will involve dividing by zero; see (3.53). In practice, there will be dif- ficulties whenever the estimated density is small or zero; while only the latter calls for division by zero, the former will make the regression function imprecise; see (3.55). Unlike linear regression, or regression with an assumed functional form, it is impossible to use nonparametric regression to calculate predictions for out-of- sample behavior. It is also necessary when calculating (3.52) to take care to calcu- late the regression only for values of x where the calculated density is reasonably large. This is not a problem for the density, since the estimated density is simply zero in places where there are no observations. In consequence, the regression can be calculated by evaluating the numerator and denominator of (3.53) for each value of x for which an estimate is desired, then calculating and presenting the ratio for only those values of jc where f ( x ) is above some critical value, set for ex- ample so as to exclude 5 percent of the sample observations. The main strength of nonparametric over parametric regression is the fact that it assumes no functional form for the relationship, allowing the data to choose, not only the parameter estimates, but the shape of the curve itself. The price of the flexibility is the much greater data requirements to implement the nonparametric methods, the difficulties of handling high-dimensional problems, and to a lesser extent, computational costs. When data are scarce, the best that can be done is to focus on a few key parameters, and to make inferences conditional on plausible functional forms. But in many problems using household survey data—as in the Thai rice pricing example—there is enough information for it to make sense to ask the data to determine functional form, something that will be particularly attractive when functional form is an important issue, and when the dimension of the prob- lem is low. There are also many other regression techniques that bridge the gap between linear regression on the one hand and nonparametric kernel regression on the other. Polynomial regressions are a familiar tool, and are capable of modeling a wide range of functional forms provided the degree of the approximating polyno- mial is increased as the sample size increases. Fourier series offer an alternative way of approximating functional forms, an alternative that has been explored by Gallant and his colleagues, for example Gallant (1981). To regress y on jc nonpara- metrically, run an OLS regression of y on a constant, jc, jc2, and a series of terms of the form sinO'*) and cosO'*)» for) running from 1 to /. For larger sample sizes, larger values of / are used, and increases in J correspond to reductions in band- width in kernel methods. In practice and with the usual sample sizes of a few thousand, setting / to 2 or 3 seems to work well. Cleveland (1979) has proposed a local regression method, LOWESS, that can be thought of as a series of linear regressions at different points appropriately stitched together; I shall return to these locally weighted regressions below. There is also substantial experience in using spline functions for regression analysis (see, for ex- 194 THE ANALYSIS OF HOUSEHOLD SURVEYS ample, Hardle 1991, pp. 56-65, and Engle et al 1986). There are also methods that allow some covariates to be treated nonparametrically, while others appear in the standard linear form (Robinson 1988 and Estes and Honore 1996). Indeed, there is a rapidly growing literature on semiparametric estimation, well reviewed by Stoker (1991). All of these techniques have their strengths and weaknesses, and some are more appropriate for some problems than others. For example, polyno- mials tend not to work very well with household survey data, because their shapes can be very sensitive to the position of a few outliers. Some methods, such as kernels, are more readily generalizable to higher dimensions than others, such as splines. There are also important differences in computational costs, as well as in data requirements, topics that are still being actively researched, and on which it is currently difficult to give adequate practical guidance. One important distinction between nonparametric and parametric econometrics is that the former lacks the menu of options that is available in the latter, for ex- ample for dealing with simultaneity, measurement error, selectivity, and so forth. In some cases, such as the selectivity models discussed in Chapter 2 where the models are identified by functional form assumptions, nonparametric alternatives are by definition impossible, which is only a dubious disadvantage. In other cases, such as simultaneity, it is because the techniques do not yet exist, although there will undoubtedly be further developments in this area. However, one problem for linear regression that is dealt with automatically by kernel estimation is when the dependent variable is discrete, as for example in the Thai case when we are inter- ested in whether a given household is or is not a rice farmer. In the binary case where the dependent variable y. takes the value 1 or 0, with the probability of the former given by some unknown function K(JC.), the expectation of y. conditional on xi is simply the probability n(jc / ) which is therefore also the regression func- tion. Hence, if we simply treat the 1's and O's as we would any other values of a dependent variable, and mechanically apply the regression formulas (3.52) or (3.53), the results will converge to the probability function TI(JC). Nonparametric regressions for rice in Thailand Nonparametric regression techniques can be used to complete the previous analy- sis, calculating the average benefit ratios at each level of living, and also to look behind the averages to the structures of rice farming that underlie them. Figure 3.17 starts with the latter and shows nonparametric regressions where the inde- pendent variable is—as usual—the logarithm of PCE, and the dependent variable is a dichotomous variable taking the value one if the household produces rice, and zero if not. Each panel shows this regression together with a similar regression for whether or not the household sells rice; the latter always lies below the former, since a household must be a producer to be a seller. The top left-hand panel shows the results for the whole rural sector, and indicates that the fraction of households growing rice diminishes with the level of living. Three-quarters of the poorest households are rice farmers, but less than one-quarter of the best-off households are so. However, conditional on being a rice farmer, the probability of being a WELFARE, POVERTY, AND DISTRIBUTION 195 Figure 3.17. Proportions of households producing and selling rice, all rural and selected regions, Thailand, 1981-82 All mral household* Upper North households Producer* Producers Sellers O 0.2 •§ 3 0 Lower North households Bangkok Fringe households Seller* Sellers 4,5 5.5 6,5 7.5 S3 4,5 5.5 ft.5 7.5 Log(PCE): log(baht) Source: Author's calculations based on Socioeconomic Survey of Thailand, 1981-82. seller increases with log(PCE); the lower line draws closer to the upper line as we move from left to right. While most poor households in rural Thailand grow rice, many do so to meet their own food needs and have no surplus to sell on the mar- ket. And while better-off rice farmers are much more likely to be selling rice, the richer the household the less likely it is to farm rice at all. The other three panels in the figure remind us that the top left panel is an aver- age, and that there is considerable regional variation. The Upper North is a rela- tively poor area; nearly all its poor households grow rice, but only a fifth sell any. The Lower North in the next panel is a good deal better-off, and a higher fraction of rice farmers sell rice at all levels of living. In the extreme case of the Bangkok Fringe in the bottom right panel, only between 30 and 40 percent of households grow rice, but they do so extremely productively and almost all participate in the market. These facts about rice production and rice sales help to interpret Figure 3.18, which shows the nonparametric regression of the net benefit ratio on the logarithm of household per capita expenditure. This regression is the conditional expectation corresponding to the joint density in Figure 3.16, and contains no new information. However, the regression provides the answer to the question of by how much the people at each level of living would benefit from an increase in the price of rice. Since the net benefit ratio expresses the benefit as a fraction of total household consumption, a flat line would show that all rural households benefit proportion- ately, so that the change would be neither regressive nor progressive. A positive 796 THE ANALYSIS OF HOUSEHOLD SURVEYS Figure 3*18. Net benefit ratios averaged by log(PCE), rural households, Thailand, 1981-82 0.3 1 0.2 o -a 0.0 4.5 5.5 6.5 7.5 8.5 Log(PCE): log(baht) Source: Author's calculations based on Socioeconomic Survey of Thailand, 1981-82. slope would indicate that the benefits are proportionately larger for those who are better-off, and vice versa for a negative slope. In fact, the graph shows none of those patterns; instead—and as we have already seen from Figure 3.16—it is the households in the middle of the distribution that benefit the most. The poor gain from a price increase, but not very much; although they grow rice, they sell rela- tively little on the market and many of them have to buy rice in addition to their own production, so that as a group they benefit only modestly from the price in- crease. Wealthy households also benefit modestly, but for precisely the opposite reasons. Wealthy rice farmers sell most of their crop on the market, but few wealthy households are rice farmers at all. In consequence, the regression function for the net benefit ratio has the inverted U-shape shown in Figure 3.18. If we put this analysis together with the data on the sectoral expenditure distri- butions in Figure 3.15, we can conclude that the survey data do not support the argument that an export tax on rice is desirable on distributional grounds. The tax redistributes real resources from relatively poor rural growers to better-off urban consumers, while within the rural sector, all income groups lose, though the largest losses are born by those in the middle of the welfare distribution. Of course, these conclusions do not tell us everything that we would need to know in deciding on tax policy in Thailand. I have said nothing about the distortions caused by the tax, nor about the desirability or otherwise of alternative instruments for raising government revenue. Even so, the techniques of this section give information on who gets (or produces) what, and provide in a readily accessible form part of what WELFARE, POVERTY, AND DISTRIBUTION 797 is required for making sensible decisions. Since the actual distributional conse- quences of policies are frequently misrepresented for political purposes, with mid- dle-class benefits disguised as benefits for the poor, such analyses can play a valuable role in policymaking. Bias in kernel regression: locally weighted regression Two potential sources of bias in kernel regression are illustrated in Figure 3.19. The graph shows two highly simplified examples. On the *-axis, there are three equally spaced data points, xr XT and xy Ignore the points XA and XB for the moment. There are two regression functions shown; m} which is a straight line, and m2 which is curved. For each of these I have marked the y-points correspond- ing to the *'s; yv yr and y3 on mr and >>,*( = y,), y^ and yj on mr Since nothing depends on their being a scatter of points, I am going to suppose that the ^-values lie exactly on the respective regression lines. Consider what happens when we try to estimate the two regression functions using kernel regression, and focus on the point jc2, so that, for regression mv the right answer is yr while for regression m2, it is y2*. Start with the concave regression function mr If the bandwidth is as shown, only the points x}, xr and jc3 will contribute, so that the estimate of the regression at jc2 will be a weighted average of yr y* and y* in which y^ gets the most weight, yl and y^ get equal weight, and the weights add up to 1. Because the regression function is concave, such an average must be less than y£, so that the estimate is biased downward. If the regression function had been convex, the bias would have been upward, and it is only in the linear case that there will be no bias. Figure 3.19. Sources of bias in kernel regressions 198 THE ANALYSIS OF HOUSEHOLD SURVEYS The bias will be gradually eliminated as the sample size gets larger because the bandwidth will get smaller, so that, in the limit, only the points at x2 contribute to the estimate of the conditional mean. In practice, there is little comfort in the fact that there is no bias in some hypothetical situation, and we must always be careful when the bandwidth is large and there is curvature. Preliminary data transforma- tions that induce approximate linearity are likely to be useful in reducing bias in kernel regression. Even when the regression function is linear, there is another potential source of bias. Turn now to the linear regression function mv and consider once again esti- mating the regression function at xr When we use the points jcp *2, and *3, everything works as it should, and the weighted average of yv y2, and y3 is yr But now introduce the two additional points XA and XB> so that there are now five points within the bandwidth, and they are no longer equally spaced. The kernel estimate of the regression at x2 is now the weighted average of the five corres- ponding y-values. It is still the case that y2 gets the most weight, and that yl and y3 get equal weights, but the y-values corresponding to XA and XB also get positive weight, so that the estimate is biased upward. More generally, there will be bias of this kind when, at the point of estimation, both the regression function and the density of x have nonzero derivatives. In practice, this bias is likely to be most serious at the "ends" of the estimated regression. For example, suppose that xl is the smallest value of x in the sample. When we try to estimate a kernel regression at JCP the average of nearby points can include only those to the right so that, if the regression function is positively (negatively) sloped, there will be an upward (downward) bias. There will be corresponding biases in the opposite direction for the largest x-values. These biases will diminish as we move from the extremes towards the center of the distribution but, if the bandwidth is large, they can seri- ously distort the estimated regression. As shown by Fan (1992), the biases associated with unequally spaced jc's can be eliminated by moving away from kernel regression. The idea, like its imple- mentation, is straightforward. Note first that, in the example of the linear regres- sion function in Figure 3.19, an OLS regression would not encounter the same difficulties as the kernel regression. Indeed, when the regression function is linear, OLS will be unbiased and consistent. The problem with OLS is that it cannot adapt to the shape of the regression function so that, no matter how large the sample, it can never estimate a nonlinear function such as wr But this can be dealt with by estimating a series of local regressions. Instead of averaging the y's around jc2, as in kernel regression, and instead of running a regression using all the data points as in OLS, we adopt the best of both procedures and run a regression using only the points "close" to xr As with kernel regression, we use a kernel to define "close," but instead of averaging, we run a weighted or GLS regression at jc2, where the weights are nonzero only within the band, and are larger the closer are the obser- vations to jc2. We repeat this procedure for each point at which we want to esti- mate the regression function. Fan's locally weighted regression smoother can be summarized as follows. As in the Thai example, suppose that we want to plot the estimated regression func- WELFARE, POVERTY, AND DISTRIBUTION 799 tion for a range of values of jt. As before, we divide up the range of A: into a grid of points, usually 50 or 100 depending on how much detail we need to show. For each point jc on the grid, calculate a series of weights for each data point using a kernel with a suitably chosen bandwidth. This can be done exactly as in kernel regression (3.52), so that we first define e (M7) 'W We then estimate the locally weighted regression parameter estimates (3.58) |J(jc) = [X/&(x)X]'lX/Q(x)y where 0(jt) is an n x n diagonal matrix with 0.(;c) in the ith position, and the n x 2 matrix X has ones in its first column and the vector of jt- values in its second. The predicted value of this regression at jc is then the estimated value of the regres- sion function at the grid point, i.e., (3.59) /fl(jc) = P,(jc) + |J 2 (jc)jc. The predicted value in (3.59) is calculated for each point on the grid, and the results plotted as in Figures 3.17 and 3.18. As Fan notes, the value of (^(jc) from (3.58) is a natural estimator of the slope of the regression function at jc and, as we shall see in the next chapter, plots of this function can also be informative. The local regression in (3.58) can be extended to incorporate quadratic as well as linear terms in jc, in which case the X matrix would have three columns, with ;c2 in the third. The nonlinearity will generally help alleviate the first problem discus- sed above and illustrated in Figure 3.19, of forcing a linear structure on the data, even locally. The estimated regression will also provide a local estimate of the second derivative of the regression function. Presumably, higher-order polynomi- als could also be considered, and will yield estimates of higher-order derivatives. The role of the bandwidth in (3.57) is the same for these locally weighted regressions as it is in kernel regressions. It controls the tradeoff between bias and variance and, for consistency of the estimates, it must tend to zero as the sample size tends to infinity and must do so slowly enough that the number of observa- tions in each local regression also tends to infinity. As with kernel regressions, standard errors are best calculated using the bootstrap, and I shall give examples in the next chapter. The "ksm" command in STATA implements Cleveland's (1979) LOWESS estimation, which is closely related to Fan's locally weighted smoother; the difference is that LOWESS uses a nearest-neighbor definition of closeness in place of the kernel, but this should make little difference to the operating charac- teristics of the procedure. However, as implemented in STATA, "ksm" estimates the regression function at every observation, which is likely to be prohibitively expen- sive for large data sets. STATA code for a direct implementation of Fan's smoother is provided in the Code Appendix for the South African example in the next subsection. 200 THE ANALYSIS OF HOUSEHOLD SURVEYS The distributional effects of the social pension in South Africa The following account is based on Case and Deaton (1996), which should be consulted for fuller details and documentation. The social security system in South Africa is unlike any other in the world. At the time of the Living Standards Survey of 1993, and well before the elections in the spring of 1994, the government paid a monthly "social" pension to age-qualified men and women whose (individual) income fell below a cutoff. In late 1993, the monthly payment was 370 rand, a little more than $100 at what was then the exchange rate. For comparison, 370 rand is around half of average household income in the survey, and is more than twice the median per capita household income of Blacks. That such comparatively enormous sums should be paid as pensions is a historical accident. In the apartheid era, most White workers were covered by privately funded occupational pension schemes, and the social pension was designed as a safety measure for those few Whites who reached retirement age without adequate coverage. During the transi- tional period, the social pension was gradually extended to non-Whites, first at lower rates, but ultimately uniformly subject to the means test. But because of the enormous disparity between the incomes of Whites and Blacks, a pension that is very small by White standards can be very large relative to the typical earnings of the majority population. We also have a situation where the means test rules out the vast majority of Whites while, at the same time, more than three-quarters of Black women over 59 and men over 64 are receiving payments. In 1993 (and it remains true at the time of writing) the social pension accounted for most of social welfare expenditure in South Africa. Not only is there concern about the cost of the scheme (around seven billion rand), but there are also ques- tions about whether a pension is the best way of spending the very limited social budget at the expense, for example, of payments to children or the unemployed. Nor is it clear that transferring cash to the elderly is an effective antipoverty strat- egy. In the United States, the elderly are somewhat less subject to poverty than the population as a whole, and because they usually live alone, or with other elderly, there is no automatic presumption that nonelderly benefit from pension payments. Much the same is true of White pensioners in South Africa but, by and large, they are not recipients of the social pension. In Black households, the elderly do not live alone, they live in households that are larger than average, and households with a pensioner actually have more children than households without a pensioner. Indeed, because of South African patterns of migrant labor, there are many "hol- low" households, with elderly and children, but without working-age adults. In such circumstances, transfers in the form of a pension may be well-targeted to- wards poverty reduction, and the distributional effects are certainly worth serious investigation. Figure 3.20 looks at the distributional incidence of pension income for White households and for Black households separately. The data from the Living Stan- dards Survey are used to give, for each household, actual pension receipts, which are reported as a component of individual income, and "potential" pension re- ceipts, defined as the number of age-qualified people—women aged 60 and over WELFARE, POVERTY, AND DISTRIBUTION 201 Figure 3.20. Regressions of actual and potential pension receipts, by race, South Africa, 1993 '400 $1 (U.S.) a day $1 (U.S.) a day per capita per capita §300 'K I |"200 Potential receipts I a 100 - Reported receipts Reported receipts 2 4 6 8 1 0 0 2 4 6 10 Log of per capita household income, excluding pensions Black households White households Note: Locally weighted regressions with quartic (biweight) kernels and bandwidths of 1.0 for African households and 1.5 for White households. The solid line is the regression function of household pension receipts conditional on the logarithm of household income (excluding pensions) per capita. The broken line is the corresponding regression for potential pension receipts, where potential receipts are 370 rand times the number of age-qualified people in the household. Source: Case and Deaton (1996). and men aged 65 and over—multiplied by the maximum monthly pension of 370 rand. Households would receive this potential amount if there were no means test—or if everyone qualified—and if everyone who was entitled to the pension actually received it in full. Locally weighted regressions are then used to calculate the conditional expectation of these actual and potential receipts as a function of household per capita income, excluding the pension. In the left-hand panel, for Black households, the two lines are very close to- gether, showing that the means test has little effect. At all levels of Black pre- pension income, most of those who are age-qualified receive the maximum pay- ment. More importantly, both lines slope down from left to right, showing that the pension payments are progressive; households at the bottom of the per capita income distribution receive between 300 and 400 rand per month, while those at the top are entitled to—and receive—very little. Note that this is automatic target- ing that owes nothing to the operation of the means test—potential receipts differ little from actual receipts—but works because the Black elderly live in households with low per capita income. The situation is very different for the White house- holds shown in the right-hand panel. Although there are a few Whites with non- pension per capita household income below the dollar-a-day poverty line—though as we have seen there are no Whites with PCE below this level—and these people receive social pensions, very few of better-off White households receive pensions. At typical levels of per capita income among Whites, the means test rules out receipt of the social pension so that potential and actual receipts are far apart. That 202 THE ANALYSIS OF HOUSEHOLD SURVEYS there is no similar discrepancy among Black households at the same level of per capita income suggests that the means test is not very consistently applied among Blacks—for which there is some independent if anecdotal evidence—but it should also be noted that there are very few Black households with incomes in this range. As with the case of rice prices in Thailand, the nonparametric regressions pro- vide a straightforward way of calculating and presenting the immediate distribu- tional incidence of policy. But that this is only a starting point should be obvious. In the Thai case, I emphasized that such calculations take no account of behavioral consequences of price changes—particularly in the labor market—and there are similar caveats in the South African case. In particular, I have taken no account of changes in private transfers in response to the social transfers, nor of possible effects on the migration patterns that make them possible (see Jensen 1996 for evidence). 3.4 Guide to further reading There are several good reviews of the material on inequality and distribution in Section 3.1. Atkinson (1970) remains the cornerstone of the inequality literature, and repays careful reading. Sen (1973), updated, with a review of the subsequent literature in Foster and Sen 1997) takes a somewhat different approach, starting from the statistical measures, and enquiring into their suitability for assessing eco- nomic inequality. This monograph should be read in conjunction with Sen (1992) which takes a much broader view of the meaning of inequality and its place in social arrangements more generally. Cowell (1995) is a useful review of alterna- tive measures of inequality and of the practical problems of implementing them. Wolfson (1994) shows that the the polarization of incomes (as in the vanishing middle) is quite different from expanding inequality. Kakwani (1980) is recom- mended. His book focusses on inequality and poverty in developing countries, and covers a great deal of useful material not discussed in here, such as families of dis- tributions for fitting to income distributions, and how to estimate inequality mea- sures from the grouped data that are published in survey reports. He also provides a thorough treatment of inequality measures and of the Lorenz curve. The book also contains a useful discussion of poverty methods, but was written too early to cover the more recent developments. Sundrum (1990) is also a good discussion of inequality in developing countries. Shorrocks (1983) should be consulted for generalized Lorenz curves; the paper also contains empirical calculations for sev- eral countries. The literature on poverty measurement has grown rapidly in the last decade and has come into much closer alignment with the social welfare approach to welfare and inequality. Foster (1984) is a survey of the various different mea- sures. Atkinson (1987, 1992) ties together the strands in the literature and his analysis and presentation strongly influenced the layout and ideas in this chapter. Ravallion's (1993) monograph is an excellent review that focusses on poverty in developing countries and together with Atkinson (1992) is strongly recommended as supplementary reading for this chapter. Anderson (1996) discusses possible methods for constructing statistical tests that can be used in conjunction with WELFARE, POVERTY, AND DISTRIBUTION 203 stochastic dominanance comparisons, and provides references to the earlier litera- ture. Nonparametric density and regression estimation is becoming more used in economics, but much of the econometric literature remains (unnecessarily) obscure and impenetrable, so that this useful material has found fewer applications than it merits. A bright spot in this literature is the book by Silverman (1986), which is a delight to read; it is focussed towards applications, with the level of technique determined by what is strictly necessary to explain the ideas, and no more. Al- though Silverman is concerned only with density estimation, and not with regres- sion analysis, the basic ideas are very similar, and this book remains the first stop for anyone interested in pursuing these topics. The book by Hardle (1991) is about nonparametric regression, and covers all of the important techniques, but it is not as transparent as Silverman, nor is it as immediately useful for someone who wants to use the techniques. Hardle and Linton (1994) is a useful summary review, and although not all topics are covered, it is a better introduction than Hardle's book. An excellent, up-to-date review of locally weighted regression is Hastie and Loader (1993). Semi-parametric techniques are a live research area in economet- rics. In many applications, especially those in high dimensions, a fully nonpara- metric approach requires too much data, so that there is a role for techniques that blend parametric and nonparametric methods. Stoker (1991) is a good entry point into this literature. For applications of nonparametric methods that are related to those given in this chapter, see Bierens and Pott-Buter (1990) and Budd (1993). 4 Nutrition, children, and intrahousehold allocation One of the traditional uses of household survey data has been the analysis of family budgets, starting with the descriptive work of the 18th and 19th centuries, and becoming more analytic and econometric in the 20th, with Prais and Houth- akker's (1955) classic monograph perhaps still the best known. That literature has investigated the distribution of the budget over goods, how that allocation chan- ges with levels of living—Engel curve analysis—and the relationship between the demographic structure of the household—the sexes and ages of its members—and the way in which it spends its resources. These studies have had a wide audience in the policy community. The early work came from social activists who hoped that documenting the living standards of the poor would generate a political de- mand to improve them. Engel curve analysis has been an important ingredient in understanding how the structure of economies changes in the process of economic growth (see in particular the classic work of Kuznets 1962, 1966). Much of the work on demographic structure has been motivated by attempts to derive "equi- valence scales," numbers that tell us how much a child costs relative to an adult, and that might allow us to correct crude measures of living standards such as in- come or total expenditure for differences in household composition. Such correc- tions have a major impact on measures of poverty and inequality, on the identi- fication of who is poor, and on the design of benefit programs. This chapter is concerned with these traditional topics in the context of surveys from developing countries. In the eyes of many people, including many development economists, poverty is closely related to whether or not people get enough to eat, so that documenting the living standards of the poor becomes a question of counting calories and a major task of household surveys is to assess nutritional adequacy. Household sur- vey data can also be used to examine how levels of nutrition change with the amount of money people have to spend. The topic is important in the debate over development strategy, between growth versus basic needs, and between less and more interventionist positions. If the elasticity of calories with respect to income is high, general economic development will eliminate hunger, while if the elasti- city is low, we are faced with a choice, between a strategy for economic growth with hunger remaining a problem for a long time, perhaps indefinitely, or a more References in this publication to "Taiwan," "Republic of China," and "Taiwan (China)" refer to the region, "Taiwan, China "References to "Hong Kong" refer to the region, "Hong Kong SAR, China." 204 NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 205 interventionist strategy that targets the nutrition of the poor while letting general economic development look after itself. This topic is addressed in Section 4.1 using survey data from India and Pakistan. I look at some of the theoretical as well as empirical issues, and argue that some of the questions in the debate can be approached using the nonparametric techniques discussed in Chapter 3. Section 4.2 is about the demographic composition of the household, its effects on demand patterns, and about the use of such information to make inferences about the allocation within the household. Household surveys nearly always col- lect data on household consumption (or purchases), not on individual consump- tion, and so cannot give us direct information about who gets what. In the devel- opment literature, much attention has focussed on gender issues, particularly al- though not exclusively among children, and on the question of whether girls are treated as well as boys. I review some of this work, as well as recent theoretical developments on how to use household data to make inferences about intrahouse- hold allocation. I implement one specific methodology that tries to detect whether girls get more or less than boys, and look at evidence from India and Pakistan as well as from a number of other countries. Section 4.3 turns from the relatively firm empirical territory of Sections 4.1 and 4.2 to the more controversial ground of equivalence scales. Although the con- struction of scales is of great importance for any enterprise that uses household survey data to draw conclusions about welfare, the state of knowledge and agree- ment in the area is not such as to allow incontrovertible conclusions or recom- mendations. Even so, it is important to understand clearly what the difficulties are in passing from the empirical evidence to the construction of scales, and to see the assumptions that underlie the methodologies that are used in practice. Clarifica- tion of assumptions is the main issue here; equivalence scales are not identified from empirical evidence alone although, once an identifying assumption has been made, the empirical evidence is relevant and useful. Nor is there any lack of prac- tical empirically based scales once identifying assumptions have been made. The problem with much of the literature, both in the construction and use of equiva- lence scales, is that identifying assumptions are often implicit, and that the effects of the assumptions on the results can be hard to see. As a result, it is difficult to know whether different investigators are actually measuring the same thing, and those who use the scales run the risk of implicitly incorporating an assumption that they would have no hesitation in rejecting were it made explicit. In most of this chapter, I adopt the standard convention of household budget analysis, that prices are the same for all households in the survey. The assumption of uniform prices is what has traditionally separated the fields of family budget analysis on the one hand from demand analysis on the other. The former investi- gates the nature of Engel curves and the effects of household composition, while the latter is mostly concerned with the measurement of price effects. Chapter 5 is about the effects of prices on demand in the context of tax and price reform, and while prices are typically not central to the questions of this chapter, there is no satisfactory justification for the uniform price assumption. Because transportation and distribution networks tend to develop along with economic growth, there is 206 THE ANALYSIS OF HOUSEHOLD SURVEYS much greater scope for spatial price variation in less developed than more deve- loped countries. In consequence, the uniform price assumption, while possibly defensible in the context of the United States or Great Britain, is certainly false in the countries analyzed in this book. I shall indicate places where I think that it is potentially hazardous to ignore price variation, but this is a poor substitute for the research that builds price variation into the analysis. That work is not straight- forward. Price data are not always available, and when they are, they frequently come in a form that requires the special treatment that is one of the main issues in the next chapter. 4.1 The demand for food and nutrition One attractive definition of poverty is that a person is poor when he or she does not have enough to eat, or in more explicitly economic terms, when they do not have enough money to buy the food that is required for basic subsistence. For the United States or other developed economies, where few people spend more than a third of their incomes on food, such a definition is clearly inadequate on its own, and must be supplemented by reference to commodities other than food. However, in countries such as India and Pakistan, where a substantial fraction of the population spend three-quarters or more of their budgets on food, a hunger- based definition of poverty makes sense. This section explores the relationship between measures of nutritional status, typically the number of calories consum- ed, and the standard economic measures of living standards, such as income or total expenditure. As usual, the analysis will be largely empirical, using data from India and Pakistan, but there are a number of theoretical issues that have to be given prior consideration. Welfare measures: economic or nutritional? If everyone spent all their income on food, and did so in the ways that are recom- mended by nutritionists, there would be no conflict between economic and nu- tritional views of living standards. However, people choose to buy goods other than food, some of which are obvious necessities like housing, shelter, and medi- cal care, but others less obviously so, like entertainment or tobacco, and they buy such goods even when food intake is below the best estimates of subsistence. Furthermore, food purchases themselves are rarely organized according to purely nutritional considerations. As has been known (before and) since the first applica- tions of linear programming—see Stigler (1945) and Dorfman, Samuelson, and Solow (1958, pp. 9-28) for an account—minimum nutritional requirements can usually be met for very small amounts of money, even by the standards of the very poorest. But minimum-cost diets are tedious and uninteresting, and they often bear no relation to what is actually eaten by poor people who presumably have interests beyond nutritional content. As a result, measures of welfare based on nutritional status will differ from the standard economic measures based on expenditures, income, or assets. There is, of course, no reason why we cannot NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 207 have multidimensional measures of welfare—someone can be wealthy but hung- ry, or well-fed but poor—but we can run into difficulties if we do not keep the differences clear, especially when the two views have different implications for the design of policy. The conflict between nutritional status and economic welfare is sharpest when we look at price changes, where it is possible for something that is desirable from the point of view of nutrition to be undesirable according to the standard eco- nomic criteria. In particular, economists tend to think that individuals with high substitution elasticities are in a good position to deal with price fluctuations, since they are well equipped both to avoid the consequences of price increases and to take advantage of price decreases. By contrast, nutritionists see high substitution elasticities as a cause for concern, at least among the poor, since nutritional status is thereby threatened by price increases. To clarify these issues, we need a simple formulation of welfare under the two alternative approaches. For the economist, welfare is defined with reference to a preference ordering or utility function, which for these purposes we can write as v ( q f , q n ) , where the two components are food and non-food respectively. We can think of q, and qn as vectors, but nothing is lost here if we consider only two goods, one food and one nonfood. Corresponding to this utility function, there is an indirect utility function, written ty(x,pf,pn), whose value is the maximum utility that can be reached by someone who has jc to spend and when the prices of food and nonfood are p, and pn, respectively. In practice, indirect utility would usually be approxi- mated by real total expenditure, which is x deflated by a price index formed from p, and pn. To consider the effect of price changes on welfare, it is convenient to follow the usual route of consumers' surplus and convert price changes into their money equivalents. For this, we use the cost or expenditure function c(u,pf1pn), which is defined as the minimum expenditure needed to reach the welfare level u at prices /?, and pn; see Deaton and Muellbauer (1980a, ch. 2) for a full discussion of the cost function and its properties. The partial derivatives of the cost function with respect to prices are the quantities consumed, while the matrix of second derivatives is the matrix of compensated price effects, the Slutsky matrix. The cost function is concave in the prices; holding utility constant, the response of cost to price is linear (and proportional to consumption) if consumption is held constant in face of the price increase, but will typically increase less rapidly be- cause it is possible to substitute away from the more expensive good. In particu- lar, if prices change by an amount A/?, the associated change in costs Ac satis- fies the inequality (4.1) Ac * q.bp = qfkpf + qnbpn. Equation (4.1) is illustrated for a single price change in Figure 4.1. The straight line through the origin is the case of no substitution, where the same is bought irrespective of price, and costs are proportional to price with slope given by the amount consumed. The other two cases show two different degrees of substitu- 208 THE ANALYSIS OF HOUSEHOLD SURVEYS Figure 4.1. Substitution and the costs of price change Cost change Ac Limited substitution \ High substitution Slope is q Price change No substitution tion; because there is substitution away from the good as it becomes more expen- sive, as well as toward it when it becomes cheaper, consumers with more ability to substitute are hurt less by price increases and benefit more by price decreases. Suppose now that only the price of food changes. A second-order Taylor ap- proximation to the change in the cost of living can be obtained from the cost function using the fact that its first derivative is the quantity consumed and the second the substitution (Slutsky) term: (4.2) Ac * <7y A/7,+ 0.5^ A/7 where sff is the compensated derivative of demand with respect to price. Because the substitution effects of price must be nonpositive, so that sff<. 0, the second term in (4.2) is always zero or negative; the larger the opportunities for substitu- tion, the more is the consumer able to offset the costs of the price increase. In Figure 4.1, sff is the curvature of each line at the origin, and thus (locally) deter- mines how much consumers benefit from substitution. Clearly, substitution is a good thing; the higher the substitution, the less vulnerable is the consumer to in- creases in price, and the more he or she benefits from decreases. Further, any policy or other change that increases substitution possibilities (while preserving consumption levels) will make people better-off and is thus to be encouraged. For the nutritionist, the costs of a price increase are measured in lost calories or other nutrients. If K is the calorie content per unit of food, and k is total calorie intake (k for kilocalories,) the change in calories induced by the price change can be approximated by NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 209 d2qf 2 2 (4.3) «K - 3 where the derivatives are uncompensated. If we compare (4.2) and (4.3), both depend on the responsiveness of food consumption to prices, the latter through the uncompensated first and second derivatives, and the former through the com- pensated price derivative. We can perhaps be permitted to assume that the second term in (4.3) is negligible—it is the price derivative of the price derivative and certainly there is little or no empirical evidence about such a quantity — in which case the distinction between the two equations becomes quite sharp. In (4.2), the cost of a price increase is smaller the larger (absolutely) is the (compensated) price response, while in (4.3) the nutritional cost of the price increase is larger the larger (absolutely) is the (uncompensated) price response. The difference between the compensated and uncompensated price effects will be large when the good whose price is changing is a large share of the budget, which is true for food as a whole, but in practice we are usually concerned with the price of a particular food where the difference will be small. The crux of the conflict between the two approaches can be seen by consider- ing an example. Suppose that milk is being subsidized, that the poor receive a good deal of their nutrients from milk, and that the government is considering reducing the subsidy for budgetary reasons. Suppose also, for the purpose of the argument, that, according to the best empirical evidence, there is a large price elasticity among the poor, so that if the price is increased, the poor will reduce their consumption of milk and its associated nutrients. The nutritional advice is therefore to maintain the subsidy, while the economic advice is likely to be the re- verse. By their high price elasticity, the poor have revealed that there are good substitutes for milk, in their own eyes if not in those of the nutritionists, so that the withdrawal of the subsidy is unlikely to hurt them much. For those who, like most economists, base welfare on people's purchasing power and leave the choice of individual goods up to people themselves, the large price elasticity is welcome since it is evidence that people are not vulnerable to increases in the price of milk. This conflict between commodity-specific and money approaches to welfare occurs in many other situations; health care, education, and even telephones, are cases where policymakers and their constituents sometimes believe that consump- tion of a specific good is valuable independently of the general living standards of the consumer, and independently of whether people appear to value the good themselves. At the heart of the matter is whether or not we accept people's own judgments of what is good for them. While it is easy to find examples of people making poor choices, it is also difficult to find convincing cases where policy- makers or other "experts" have done better on their behalf. One such case that is relevant for the current discussion is the history of food rationing in Britain dur- ing World War II; in spite of widespread shortages there was a marked improve- ment in general nutritional standards brought about by policies that simultane- ously limited consumption (by rationing) while redirecting it towards commodi- ties such as fresh milk that had not been widely consumed prior to the war and 210 THE ANALYSIS OF HOUSEHOLD SURVEYS whose supply was guaranteed during it (see Hammond 1951). The policy seems also to have narrowed long-standing health and mortality inequalities, at least temporarily (see Wilkinson 1986). In any case, my quarrel is not with those who wish to change tastes, or who wish to eliminate hunger even at the expense of more general economic well-being. Although many economists would disagree with such prescriptions, there is no logical flaw in the arguments. What is both incorrect and logically flawed is to try to follow both the com- modity-specific and economic approaches simultaneously. High substitution effects are either a good thing or a bad thing; they cannot be both. It is entirely legitimate to worry about the effects of food prices on nutrition or of user charges on consumption of hospital services or education by the poor—and such has been a major research topic in the World Bank in recent years (see Jimenez 1987 or Gertler and van der Gaag 1990)—but it is necessary to take a view about what high price elasticities mean. If our goal is to provide these services to the poor even when their behavior suggests that they do not value them, then that fact should be explicitly recognized and its implications—for education, for exam- ple—taken into account. Alternatively, if we accept that the decisions of the poor have a reasonable basis—perhaps because the services are of poor quality and worth very little—then it is necessary to think hard about the justification for the subsidy, and whether or not the funds could not be better employed elsewhere, perhaps in improving quality and delivery. Spending scarce resources to subsidize facilities that are not valued by the poor will not do much to reduce poverty, and the true beneficiaries of such policies are often to be found elsewhere. Nutrition and productivity An important issue is the direction of causation, whether the link runs, not only from income to nutrition, but also from nutrition to income. One possibility is that those who do heavy manual labor require more calories than those who do not— see for example Pitt, Rosenzweig, and Hassan (1990), who also consider the im- plications for intrahousehold allocation. This possibility can be dealt with by controlling for the appropriate occupational variables in the nutrition demand function. However, it is also possible that not getting enough to eat impairs pro- ductivity to the point where poverty and malnutrition become a trap; it is impos- sible to work because of malnourishment, and impossible to escape from mal- nourishment without working. Following Leibenstein (1957), the theory of nutritional wages has been work- ed out by Mirrlees (1975) and Stiglitz (1976) in some of the finest theoretical work in economic development. These models can account for the existence of unemployment; attempts by workers without jobs to underbid those with jobs will only succeed in reducing their productivity, so that the employer gains nothing by hiring them. By the same token, the theory can explain the existence of high wages in modern sector jobs. It is also consistent with unequal allocations within the family because equal shares may leave no one with enough energy to work the farm or to be productive enough to get a job. This model has been used to NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 277 account for destitution in India by Dasgupta (1993, pt. IV see also Dasgupta and Ray 1986, 1987) as well as by Fogel (1994), who sees the mechanism as the major impediment to historical growth in Europe. While there has been some empirical work that looks for such effects—most notably by Strauss (1986) for Sierra Leone—there are formidable difficulties in the way of constructing a convincing test. If we use data on self-employed work- ers, and find a relationship between income and nutrition, we need some means of knowing whether what we see is the common-or-garden consumption function, by which higher income generates more spending and more nutrition, or is instead what we are looking for, the hypothesized effect of nutrition on productivity, out- put, and income. In principle, such identification problems can be solved by the application of instrumental variables, but it is doubtful whether there are any vari- ables that can convincingly be included in one relationship and excluded from the other. These points were argued by Bliss and Stern (1981), who also identified the corresponding difficulties in looking for nutritional effects among employed workers. Since employers will only hire well-nourished workers—which is the source of the model's predictions about unemployment—nutritional effects will only be found among the employees of those employers who are unaware of the effects of nutrition on productivity! The Mirrlees-Stiglitz theory hinges on non- linearities in the effects of nutrition on productivity, and requires that productivity be a convex function of nutrition at low levels, becoming concave as nutritional status improves. These nonlinearities would have to provide the basis for identi- fication in the econometric analysis, which is likely to be both controversial and unconvincing. There is one final point on which the empirical evidence is directly relevant. For the nutritional wage theory to be a serious contender to explain the phenom- ena for which it was invented, the calories required to maintain productivity must be costly. If it is possible to obtain enough calories to do a day's work for a very small fraction of the daily wage, then low productivity rooted in malnutrition is an implausible explanation for unemployment. In the empirical analysis below, we shall use Indian data to make these calculations. The expenditure elasticity of nutrition: background The relationship between nutritional intake and total expenditure (or income) in poor countries is the link between economic development and the elimination of hunger. In the simplest of worlds, one might imagine that food is the "first neces- sity/' and that people whose incomes do not permit them to buy enough food would spend almost everything on food. Even admitting the existence of other necessities, such as clothing and shelter, it is still the case that the poorest house- holds in India (for example) spend more than three-quarters of their budget on food, and that this share does not fall by much until the top quartile of the PCE dis- tribution. For such people, the demand for food is elastic with respect to PCE, so that it might be supposed that the elasticity of nutrient intake would also be high, perhaps even close to unity as in the simplest case. 272 THE ANALYSIS OF HOUSEHOLD SURVEYS However, it has been recognized for many years that this view is too simple. Even if the expenditure elasticity of food were unity, the elasticity of calories need not be, since the composition of foods will change as income rises. This can happen through substitution between broad groups of goods, as when meats are substituted for cereals, or when "fine" cereals such as rice and wheat are substi- tuted for roots (e.g., cassava) or coarse cereals such as sorghum or millet, or it can happen within food groups, as people substitute quality, variety, and taste for quantity and calories. As a result, the nutrient elasticity will be lower than the food elasticity, perhaps by as much as a half. Indeed, Reutlinger and Selowsky (1976), in one of the most cited studies of malnutrition and development, as- sumed that calorie intake and total expenditure were linked by a semilogarithmic relationship—with the implication that the elasticity falls as calorie consumption rises—but that even among households just meeting their caloric need, the elastic- ity was between 0.15 and 0.30. The empirical evidence has produced a wide range of estimates of the elastic- ity, from close to unity to close to zero (see Bouis and Haddad 1992 and Strauss and Thomas 1995 for reviews of the literature) but, at least until recently, there would have been assent for the range suggested by Reutlinger and Selowsky, as well as for their assumption that the elasticity is higher among the poor than the rich. To paraphrase Alfred Marshall's dictum on the demand for food, the size of the elasticity is ultimately limited by the capacity of the human stomach, elastic though that may be. However, a number of recent studies have claimed that the true elasticity is very low, perhaps even zero, and that the earlier presumptions are based on studies that are flawed by inadequate data as well as by econometric and theoretical flaws. Bouis and Haddad (1992), using data from the Philippines, in- vestigate a wide range of reporting and econometric biases, and conclude that the true estimate, although significantly positive, is in the range of 0.08 to 0.14. Behr- man and Deolalikar (1987) use data from the ICRISAT villages in south India, and although their preferred point estimate is 0.37, it has an estimated standard error that is also 0.37, leading Behrman and Deolalikar to conclude that "for communi- ties like the one under study, increases in income will not result in substantial improvements in nutrient intakes." (p. 505, italics in the original). If we were to accept this extreme revisionist opinion, the conflict between basic needs and economic development is stark indeed. Even if structural adjust- ment or other policy reform were to succeed in accelerating (or in some countries just starting) economic growth, the poor will still not have enough to eat, and we will be left with the problems of endemic hunger and malnutrition, albeit in a richer world. Much of the applied economics of development policy would have to be rethought; the standard prescriptions of project evaluation, price reform, tax policy, and trade policy are all derived under the assumption that what we want to promote is people's general economic well-being, not their intake of nutrients, or if we are concerned with the latter, it is assumed that it will follow from the for- mer. One can ultimately imagine a pricing policy that is designed, not to target the poor, to avoid distortion, nor to raise revenue, but to induce people to eat the recommended foods. NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 213 Evidence from India and Pakistan The main results reported in this subsection are taken from Subramanian and Deaton (1996), which should be consulted for the details of the calculations. We use Indian NSS data for the state of Maharashtra in 1983 (38th round) to construct household calorie availabilities from the consumption data. Household surveys (including the NSS) typically collect data on consumption levels, not on nutrition, so that data on calorie intake have to be calculated ex post from the data on con- sumption of food. Because of substitution between foods as incomes rise, the multiplication of total food expenditures by an average "calorie per rupee" factor will systematically understate the calories of the poor and overstate those of the rich, so that the elasticity of nutrition with respect to income will be overstated. Since substitution can take place both between broad groups and within broad groups, it is good practice to evaluate calories using as much detail as possible. In the Maharashtran data, 5,630 rural households report consumption during the last 30 days on each of more than 300 items, of which 149 are food items for which there are data on both expenditures and quantities. The latter are converted to calories using the tables on the nutritive values of Indian foods in Gopalan et al. (1971). In Indian households, a substantial amount of food is not consumed by household members, but is given to employees and guests, and conversely, many members of poorer households receive a fraction of their calories in the form of meals provided by employers. Although the data do not provide direct estimates of the calories involved in these transactions, the 38th round of the NSS asked respondents to report the number of meals taken by family members, provided to employees and by employers, and served to guests at both ceremonial and other occasions. These figures can be used to estimate the average calorie content of each type of meal by regressing the total calorie content of the food bought by the household on the numbers of meals provided by the household in each of the categories. Doing so suggests that each person-meal at home provides 727 calo- ries, each meal to a servant 608 calories, while meals to guests have 1,550 calo- ries at ceremonies and 1,379 at less formal occasions. These numbers can be used to correct the total calories, to exclude food not consumed by family members, and to include food received by family members as employees. Table 4.1 shows how rural Maharashtran households obtained their calories in 1983, what foods they consumed and how much they spent on each. The top half of the table shows the allocations over broad groups of foods, while the bottom half looks at cereals alone. On the left side, I show the allocation of expenditure between the goods, in the middle the allocation of calories, and on the right, the prices that people paid for 1,000 calories obtained by purchases of each of the various foods or groups of foods. In each case, the table shows the mean for all rural households in the sample together with means for the bottom and top deciles of the PCE distribution. (Tabulations of means by decile is a special case of the method of fractile analysis introduced by Mahalanobis (1960), and can be thought of as a crude but often useful form of nonparametric regression, just as histo- grams are crude but useful nonparametric estimators of densities.) Table 4.1. Expenditure patterns, calorie consumption, and prices per calorie, rural Maharashtra, 1983 Expenditure shares (percent) Calorie shares (percent) Price per lfOOO calories (rupees) Bottom Top Bottom Top Bottom Top Food Mean 10% 10% Mean 10% 10% Mean 10% 10% Food groups Cereals 40.7 46.0 31.0 70.8 77.3 57.3 0.64 0.51 0.79 Pulses 8.9 10.2 7.8 6.6 6.2 7.2 1.51 1.44 1.60 Dairy 8.1 4.9 11.8 2.8 1.3 4.9 3.69 3.59 3.92 Oils & fats 9.0 9.2 9.2 5.9 4.8 7.6 1.74 1.67 1.81 Meat 5.1 3.4 6.4 0.7 0.4 1.0 11.7 11.0 12.2 Fruit & vegetables 10.5 8.5 12.0 3.5 2.3 5.4 3.90 3.83 3.85 Sugar 6.5 7.4 5.9 7.2 7.0 8.0 1.01 0.94 1.09 214 Other food 11.3 10.4 16.1 2.5 0.8 8.6 17.4 16.8 15.9 Cereals Rice 11.6 9.0 10.9 15.2 10.1 16.5 0.95 0.89 1.02 Wheat 5.6 3.8 7.9 8.5 4.7 14.4 0.79 0.73 0.82 Jowar 18.2 27.4 9.3 37.8 52.9 21.6 0.50 0.43 0.55 Bajra 3.0 2.7 1.3 6.6 4.9 3.2 0.48 0.48 0.50 Other coarse cereals 1.2 2.8 0.3 2.2 4.5 0.6 0.66 0.58 0.99 Cereal substitutes 1.1 0.5 1.3 0.6 0.2 0.8 2.23 2.22 2.22 Total food 67.4 73.4 54.1 2,120 1385 3382 1.14 0.88 1.50 Calories adjusted 2,098 1429 3167 Note: Mean refers to the mean of the whole sample and bottom (top) 10 percent to the mean of households in the bottom (top) decile of PCE. Shares of calories and of expenditures are calculated on an individual-household basis and are averaged over all appropriate households. Calorie prices are averages over consuming households. Source: Subramanian and Deaton (1996, table 1). NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 215 On average, rural Maharashtran households spent more than two-thirds (67.4 percent) of their budget on food; the corresponding figure is near three-quarters for the bottom decile (73.4 percent) and is still above a half (54,1 percent) among households in the top decile. From this they obtained 2,120 calories on average. More than 40 percent of expenditures were on cereals, with the poor buying most- ly coarse cereals, mainly jowar (sorghum vulgare) while better-off households spent more on wheat and rice. Cereals provide calories more cheaply than do other foods. On average each 1,000 calories from cereals cost only 64 paise in 1983, and a large share of total calories—more than three-quarters in the bottom decile—came from this source. The coarse cereals provided calories even more cheaply, jowar and bajra (millet) at 50 paise per 1,000 calories, and the focus of poor households' consumption on these goods means that poor households spent 88 paise per 1,000 calories compared with 1.50 rupees for households in the top decile and 1.14 rupees on average. For comparison, the wage rate in rural Maha- rashtra in 1983 was around 15 rupees per day, so it is hard to believe that the inadequate nutrition is a bar to employment or productive labor for the individu- als in this sample. Given that a substantial fraction of calories are required to maintain the body even at rest, and that an additional 2,000 calories can be pur- chased for one rupee, malnutrition alone is surely an insufficient force to keep people in destitution, There are two other important points in the table. First, the increase in the cost per calorie from poor to rich is a consequence of the substitution between the broad groups of foods, at least provided that the different types of cereals are distinguished from one another. There is not much evidence of higher prices being paid within categories, so that, for example, the best-off households pay about the same per kilo for their dairy products or their wheat as do households in the bottom decile. Secondly, the correction for meals given and taken, which is shown in the last row of the table, works as expected, increasing the calories of the poorest households, decreasing the calories of the better-off, and leaving the mean relatively unchanged. If the correction is not made, we run the risk of over- estimating the responsiveness of calories to income. Table 4.1 also gives a rough idea of the various elasticities, and how the in- crease of food expenditure with income is divided between additional calories on the one hand and additional expenditure per calorie on the other. Since the budget share of a good is identically equal to the price multiplied by quantity (per capita) divided by total expenditure (per capita), the logarithm of the budget share is the logarithm of price plus the logarithm of quantity less the logarithm of total expen- diture. The average of the logarithm of PCE among households in the top decile is 5.6, and that among those in the bottom decile is 3.8. If this is compared with the change in the logarithm of the budget share, from (the logarithm of) 73.4 to 54.1, we get a total elasticity of food expenditure per head of about 0.8, matching the slow decline of the food share with PCE. The corresponding rise in expenditure per calorie, from 0.88 to 1.50 rupees per 1,000, translates into an elasticity of around 0.3, so that the residual, which is the elasticity of calories with respect to PCE, is around 0.5. As we shall see, this estimate is on the high side, but the gene- 216 THE ANALYSIS OF HOUSEHOLD SURVEYS ral picture is not; the food expenditure elasticity is large with substantial fractions accounted for by each of its two components, the elasticity of calories and the elasticity of the price of calories. The general relationship between calories and PCE can also be illustrated using the bivariate density estimation techniques of Section 3.2. Figure 4.2 shows the estimated contours of the joint density of the logarithm of calories per head per day and the logarithm of monthly PCE. For comparative purposes, and using the same vertical scale (for calorie intake), Figure 4.3 shows the same plot for the Pakistani province of Sindh in 1984-85 based on data from the National Income and Expenditure Survey. I will not use the Pakistani data further in this section, but the general similarity between the two graphs is worthy of note. In both cases, the contours are approximately elliptical, as would happen if the joint density were in fact normal. The implicit regression functions are close to being both line- ar and homoskedastic, and the slope in each case is approximately one-half. Al- though both surveys come from the subcontinent, the dietary habits of the two regions are different, with rice the staple in the Sindh Province of Pakistan, while in Maharashtra the hierarchy of basic foods starts with jowar and progresses to rice, and eventually to wheat. Other similarities and differences between these two surveys will be further explored in Chapter 5 when I consider the response of food demand to prices. Regression functions and regression slopes for Maharashtra Given the results in Figure 4.2, it is hardly surprising that the associated regres- sion function is close to linear. Figure 4.4 shows the estimated regression using a locally weighted smoother with a quartic kernel and a bandwidth of 0.50, The graphs also show confidence bands for the regression line. These were obtained by bootstrapping the locally weighted regressions 50 times, so that, at each point along the 100-point grid used for the regressions, there are 50 bootstrapped repli- cations. These are used to calculate a standard error at each point, and then to plot lines that are two standard errors above and below the estimated regression func- tion. The graph shows two such pairs of lines; the inner pair ignores the clustered structure of the sample and the outer pair takes it into account in the bootstrap- ping. In this case, the cluster structure makes little difference, and the confidence bands are tight around the regression line. The graph suggests that the slope falls with PCE but is close to linear with slope 0.5. Given the economic plausibility of a slope that is higher among the poor, it is worth calculating and plotting a nonparametric estimate of the slope itself, allowing it to be different at different levels of PCE. Such estimates are a natural by-product of the locally weighted regressions, since each local regression delivers a slope parameter, and are shown in Figure 4.5, once again with two sets of estimated confidence bands. The graph shows an elasticity that declines gently with the level of PCE, from about 0.7 at the bottom to 0.3 at the top. These data do not show any evidence of a sharp change in the elasticity at some critical level of calorie intake as found by Strauss and Thomas (1995) for Brazil. NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 277 Figure 4.2. Calories per head and log(PCE), rural Maharashtra, 1983 9.2 8.4 8. 3 '5. s I 7.6 I W) 3 6.8 3.0 3.8 4.6 5.4 6.2 Log(PCE): Iog(rupees per capita per month) Source: Subramanian and Deaton (1996). Figure 4.3. Calories per head and log(PCE), rural Sindh, 1984-85 9.2 t I i 7 I '° « " 6.8 4.0 5.0 6.0 7.0 Log (PCE): log(rupees per month) Source: Author's calculations based on National Income and Expenditure Survey, 1984-85. 218 THE ANALYSIS OF HOUSEHOLD SURVEYS Figure 4.4. Estimated regression function for log of calories per head and log(PCE), rural Maharashtra, 1983 8.2- 8.0- 7.8- 7.6- s 7.2- 7.0- 3.4 3.8 4.2 4.6 5.0 5.4 5.8 6.2 LogofPCE Source: Subramanian and Deaton (1996). Figure 4.5. Elasticity of calories per head to PCE, Maharashtra, 1983 i.o- 0.8- !§ 0.4- 0.2- 0.0- 3.4 3.8 4,2 4.6 5.0 5.4 5.8 6.2 Log(PCE) Source: Subramanian and Deaton (1996). NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 279 Allowing for household structure One serious problem with the regressions and regression slopes in Figures 4.4 and 4.5 is that they portray only the bivariate relationship between calories per head and total household expenditure per head. While the latter is certainly the most important variable determining calorie consumption, it is by no means the only one, and even if we are less interested in other covariates than in the total budget, the calorie expenditure relationship will not be consistently estimated if the omit- ted variables are correlated with PCE. The most important omitted variables are those representing the size and composition of the household. While a relation- ship between calories per head and total expenditure per head is a natural starting point, and allows for household size in a rough way, it makes no concession to the fact that children consume fewer calories than do their parents, so that, controlling for PCE, households with more children will typically consume fewer calories per head. Even among all-adult households, we might expect the presence of econo- mies of scale to imply that larger households with the same PCE are better-off than smaller households (but see Section 4.3 below) so that food consumption per capita will be a function of both PCE and household size. The same will be true if there are economies of scale in food consumption itself, for example if wastage is reduced when more people share the same kitchen and take meals together. As always, nonparametric techniques are much better at handling bivariate than multivariate relationships, but it is possible to look at the calorie expenditure relationship nonparametrically for a range of household sizes. Figure 4.6 illus- trates this, reproducing the nonparametric regression functions between the loga- rithm of calories per head and the logarithm of PCE, but now separately for house- holds of different sizes. There are 10 curves, for household sizes 1 through 10, and each curve is based on a different number of households. The curve for five- person households uses 1,100 observations and is the most precisely estimated; the number of households in each category rises from 283 single-person house- holds to 1,100 with 5 persons, and falls to only 96 households with 10 persons. There are several important points to note. First, the curves are lower for larger households, so that, conditional on PCE, calorie consumption per capita falls with household size, an effect that is at least partly due to the fact that larger households have a larger proportion of children. Second, the elasticity of calories with respect to expenditure is not much affected by the size of the household. The different regression lines are approximately parallel to one another. Third, because per capita calorie consumption is nega- tively related to household size at constant PCE, and because in rural Maharashtra PCE is negatively related to household size—as it is in most household surveys— the omission of household size from the regression of per capita calorie consump- tion on PCE will have the effect of biasing upward the estimate of the calorie expenditure elasticity. As PCE rises, household size falls, which will have an inde- pendent positive effect on calories per head, and this effect will be attributed to—or projected on—PCE if household size is not included in the regression. A careful examination of Figures 4.6 and 4.4 shows that the slopes in the former are 220 THE ANALYSIS OF HOUSEHOLD SURVEYS Figure 4.6. Calorie-outlay regression functions by household size 8.4- 8.2 One-person households 8 Two-person households I '° I o S 7-8 Three-person households o ^^^ v 7.6- I ^ Eight-person households 7.2- 7.0- 3.4 3.8 4.2 4.6 5.0 5.4 5.8 6.2 Log(PCE) Source: Subramanian and Deaton (1996). indeed lower. In Figure 4.4, a difference in the logarithm of PCE from 3.8 to 5.8 is associated with a change from 7.2 to 8.0 in the logarithm of per capita calories, a difference of 0.8, compared with a corresponding difference of about 0.7 in Fig- ure 4.6. Controlling for household size will therefore tend to reduce the calorie expenditure elasticity, from about 0.40 to about 0.35, figures that are much more like the conventional than the revisionist wisdom. Since the nonparametric regressions suggest that the relationship between the logarithm of calories per head and the logarithm of PCE is close to linear and that household size has an effect that is additive without interactions with log(PCE), the nonparametric results can be closely approximated by a linear regression of the logarithm of calories per head on the logarithms of PCE and of household size. This regression can then be used as a basis for examining the role of other covari- ates, since the completely nonparametric approach cannot handle any further in- crease in dimension. In fact, apart from household size, the inclusion of other variables—even including dummy variables for each of the 563 villages in the survey—has little effect on the estimates of the elasticity, which remains close to 0.35. Several of the other variables are important—see Subramanian and Deaton (1996, Table 2) for details—but unlike household size, they are sufficiently or- thogonal to the logarithm of PCE to have little effect on the key coefficient, One objection to the use of household size and household composition vari- ables to "explain" calorie consumption is that these variables are themselves en- dogenous, so that their coefficients—as well as that on the total expenditure vari- NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 227 able—are not readily interpreted. This is partly an argument about the endoge- neity of fertility—families do not have children by random and unanticipated delivery (perhaps by storks!)—but more generally recognizes that family size is adjusted in many ways, by migration, by younger adults splitting off to form their own households, by marriage, and in some parts of the world, by large-scale fostering, All of these processes are (arguably) influenced by the economic posi- tion of the household. As usual, the fundamental problem is a possible correlation between the regression errors and the explanatory variables, so that the calorie expenditure elasticity will be biased if households who consume more calories than predicted are also those that are more fertile, more likely to absorb members than to shed them, and so forth, possibilities that can hardly be excluded on theo- retical grounds. In principle, the calorie expenditure relationship could be reestimated using instrumental variables, although in practice, our understanding of fertility or the adaptation of household size to economic change is far from sufficiently precise to provide instruments that are well-correlated with household size, let alone that are credibly excluded from the calorie expenditure relationship itself. Even so, it is useful to consider interpreting the relationships with and without household size as short-run versus long-run relationships, so that the reduced-form relation- ship, between calories and income (or total outlay), is taken as revealing the long- run influence of higher incomes on calorie consumption, taking into account those effects that operate indirectly through the adjustment of household size through fertility or other mechanisms. For some purposes, this is what we want to know; the effect of increasing per capita income on levels of nutrition once all the offsetting and reinforcing second-order effects have worked themselves out. If so, the appropriate elasticity would be at the upper end of the range, closer to 0.40 than 0.30. The effect of measurement errors The estimates of the calorie expenditure elasticity for Maharashtra—and the results are very similar for the 1984-85 Pakistani survey—are close to what I have characterized as the conventional wisdom, and are very much higher than the revisionist estimates of Behrman and Deolalikar (1987) and Bouis and Had- dad (1992). Bouis and Haddad (see also Bouis 1994) note that their low esti- mates—as well as those of Behrman and Deolalikar—are obtained using data obtained by the direct observation by nutritionists of calorie consumption, and not, as in the Indian and Pakistani data and in most other studies, by using con- sumption data that has been converted to calories using tables of conversion ra- tios. Bouis and Haddad provide evidence from both the Philippines and from Kenya, where the data allow implementation of both methodologies, that the esti- mates from the direct nutritional data are lower. (Behrman and Deolalikar's esti- mates, which also come from Maharashtran villages, are not lower, but are insig- nificantly different from zero; their point estimate of 0.37 is essentially the same as that discussed above.) There can be no automatic presumption that the direct 222 THE ANALYSIS OF HOUSEHOLD SURVEYS nutrition surveys provide the superior data, since the increased accuracy from observing what people actually eat could well be offset by the artificial and intru- sive nature of the survey. However, as Bouis and Haddad argue, there is one very good reason for sus- pecting that standard methods lead to upwardly biased estimates; because calories are counted by multiplying consumption quantities by positive factors (calorie conversion factors) and adding, and because total expenditure is obtained by mul- tiplying those same quantities by other positive factors (prices) and adding, measurement error in the quantities will be transmitted to both totals, thus leading to a spurious positive correlation between total expenditure and total calories. Although the mismeasurement of total expenditure will typically bias downward the estimate of the elasticity—the standard attenuation effect of measurement error in an explanatory variable—there is here the additional effect of the com- mon measurement error which will bias the elasticity upwards, and as shown in the paper by Bouis and Haddad, this effect will typically dominate. By contrast, the direct observation of calories, even if subject to measurement error, will yield estimates whose errors are independent of the measurement of total expenditure, so that the calorie elasticity, even if measured imprecisely because of the mea- surement errors, will not be biased in either direction. The standard procedure for dealing with measurement error is to reestimate using an instrument for total expenditure, the obvious choice being the sum of the nonimputed components of income. This quantity is measured independently of expenditure, is well-correlated with expenditure, and is plausibly excluded from the calorie equation, conditional on expenditure. Unfortunately, the NSS consump- tion surveys do not collect income data. However, Subramanian and Deaton show that, just as the OLS estimator is biased up, the (inconsistent) estimator obtained by using nonfood expenditure as an instrument is biased down. But this estimate of the elasticity is still much larger than zero, 0.3 as opposed to 0.4, and is quite pre- cisely estimated. This result is certainly consistent with the result that the calorie to PCE elasticity is biased upward by the standard method of imputing calories by converting quantities purchased, but it is not consistent with a true elasticity that is much less than 0.3, and certainly not with one that is 0.10 or even zero. If the difference between direct calorie measurement and imputed calorie measurement is correlated measurement error, we cannot sustain an argument that direct calorie estimation in Maharashtra would lead to very low estimates of the expenditure elasticity of calories. It remains possible that the calorie data from the NSS are compromised in other ways. Bouis and Haddad observe in their Filipino data that meals given to ser- vants by rich households tend to be understated as are meals received by servants and employees. If this is true in the Indian data, even our lowest estimates will be biased upward, although it is worth noting that the nonparametric estimates of regression functions and slopes at any given point depend only on data within the bandwidth around that point, so that estimates of the slope in the middle of the range of PCE, for example, are unaffected by measurement errors that affect only the richest and poorest. NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 225 4.2 Intrahousehold allocation and gender bias Welfare, living standards, and poverty are all characteristics of individuals, not households, and although households are often the primary income recipients and are nearly always the units for which we observe consumption, we cannot be indifferent to the allocations to individual members of the household. If women systematically get less than men, or if children and old people are systematically worse-off than other members of the household, social welfare will be overstated when we use measures that assume that everyone in the household is equally treated. When people are treated differently but we assume the opposite, the true distribution of welfare among family members can be obtained from the supposed equal distribution by a set of mean-preserving but disequalizing transfers. In consequence, the supposed distribution always understates true inequality and overstates true social welfare. Measures of poverty are also usually evaluated by attributing household per capita income or consumption to each member of the household, a practice that is used even when we are counting the number of wom- en, children, or old people in poverty. Without some idea of how resources are allocated within households, these measures are little more than guesses. Even in cases where welfare is the same for all household members, per capita consumption measures will generally not provide a correct ranking of the living standards of different households or of the members within them. Children will often require less than adults to obtain the same level of living, while old people may need more of some things—health services or warmth in cold countries— and less of others—food or work-related consumption. Members of large families may therefore be better-off than members of smaller families with the same level of resources per capita, although we have no means of making the correction unless we understand enough about intrahousehold allocation to know how much money is needed to attain equal welfare levels for different types of household members. There are two separate issues. The first is the positive question of whether and to what extent allocations within the household differ according to the gender and age of the recipient. The second is whether we can find a theoretical framework that will allow us to use the empirical evidence on the effects of household com- position on household demand patterns to say enough about individual welfare to improve our measures of social welfare and poverty. The issues also differ depending on whether we are dealing with gender is- sues, with children versus adults, or with the elderly. For issues of men versus women, or boys versus girls, we are most often concerned with whether differ- ences exist, with their magnitude, and under what circumstances allocations de- pend on gender. For children versus adults, there are at least two questions. The first is the welfare of children as separate entities from their parents. Since the allocation of resources within the household is controlled by adults, and since children are not voluntary members of the families to which they belong, there can be no general presupposition that their interests are fully taken into account. The second question concerning children is about "equivalence scales," which is 224 THE ANALYSIS OF HOUSEHOLD SURVEYS how much it costs to support children, whether and under what assumptions such calculations make sense, and how they can be used—if at all—to make compari- sons of welfare levels between members of households of different compositions. For the elderly, there are other issues that are only now coming into focus for developing countries. In much of Asia, the demographic transition that is either underway or recently completed is "aging" the population; reductions in mortality are leading to more elderly people, while reductions in mortality combined with reductions in fertility are increasing the ratio of the old to the young. There is much concern, although so far little hard knowledge, about the economic conse- quences of this change, about changes in living arrangements of the elderly, about support mechanisms both inside and outside the extended family, about the eco- nomic burdens for working-age people, and about the effects on saving and in- vestment (see Cowgill 1986, Deaton and Paxson 1992, and Chapter 6 for further discussion and some empirical evidence). In this section, I shall be concerned with the positive, empirical issues of how to use household survey data to look for patterns of discrimination between indi- viduals by gender, and particularly between boys and girls. In the next, I turn to the issue of equivalence scales. Gender bias in intrahousehold allocation There is a large literature documenting that, at least in some areas of the world, the allocation of household resources favors males over females. Much of this work, which is reviewed for example by Dreze and Sen (1989), Harriss (1990), Dasgupta (1993), and Strauss and Thomas (1995), is not concerned with the allo- cation of consumption within the family, but with the direct comparison by gen- der of nutrition, health, or education. Perhaps best known are the findings on excess infant mortality among girls, particularly (but not exclusively) in China, Northern India, Pakistan, and Bangladesh (see among others Government of India 1988, Sen 1989, Dreze and Sen 1989, pp. 50-54, Rosenzweig and Schultz 1982, and Coale 1991). The mechanisms underlying these differences are not fully un- derstood, but it is widely believed that female children are given less medical attention when they are sick, and it is certainly possible that they are provided with fewer resources in general. Other research has found more complex gender- related differences in nutrition; for example, Thomas (1994) used surveys from the United States, Ghana, and Brazil to document a positive association between a mothers education and her daughters' heights and between father's education and his sons' heights, an association that is attenuated or absent between mothers and sons and between fathers and daughters. At first sight, household survey data on consumption do not appear to be useful for sorting out who gets what; data on consumption are for the household as a whole, not for its individual members. Even so, the surveys tell us about household composition, the numbers of household members, and the age and sex of each, so that it is certainly possible to study the relationship between household composition and household consumption patterns. Indeed, if the consumption of NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 225 (say) food relative to nonfood depends on the ratio of males to females in the household, then we have established that allocations depend on gender, although we are very far from having understood what determines the differences, and in particular, whether they are the result of tastes or of discrimination. It is some- times the case that we directly observe the consumption of individual members of the household, at least for some goods, or that there are goods that, by their very nature, are only consumed by a subset of family members. An example of the former would be surveys where each member of the household is asked to record each expenditure, some of which are directly assignable to that person's con- sumption. More common is the second case, where there are goods (adult cloth- ing, women's clothing, alcohol, tobacco, or baby formula) that are exclusively used by some members or some groups and not others, such as men versus women or adults versus children. Given a suitable theoretical structure, these features of the data can be used as a lever to pry open the black box and to ob- serve the internal workings of the household. A theoretical digression There are many different theories of how resources are allocated within the household. The simplest, which has dominated empirical research until recently, is one in which households are treated as monolithic entities, endowed with pre- ferences as if the household were an individual. This can be thought of as a dicta- torial model, in which (presumably) the paterfamilias decides on behalf of every- one so that the consumption behavior of the household will look very much like the behavior of the individual consumer of the textbook. At the other extreme, the household can be thought of as a group of individuals who bargain with each other over resources. Such models provide a richer structure to behavior than does the dictatorial model. For example, the allocation between a husband and wife will depend on what each would get should the marriage be dissolved, so that such models predict differences in household consumption patterns according to the relative earnings of each partner, unlike the dictatorial model in which all re- sources are pooled. It also makes sense to suppose that people in the family care for each other, and get pleasure from each other's consumption as well as from their own, or get pleasure from each other's pleasure. The consequences of these different assumptions have been explored in the literature, most notably by Man- ser and Brown (1980), McElroy and Horney (1981), Becker (1981), McElroy (1990), and Lundberg and Pollak (1993). In a series of papers, Chiappori (1988, 1992), Bourguignon and Chiappori (1992), and Browning et al. (1994) have developed a methodology that is consis- tent with a number of these models and that permits empirical testing of their different predictions. Under appropriate, although not always uncontroversial as- sumptions, these methods allow recovery of the rules for resource allocation with- in the household. Their procedures are also useful for thinking about the empirical results later in this chapter and for discussing equivalence scales, so that it is useful to provide a brief introduction here. 226 THE ANALYSIS OF HOUSEHOLD SURVEYS The simplest case is where the household consists of two members A and B whose private consumption vectors are denoted qA and qB, respectively. Sup- pose also that there is a vector of public goods z, that are available to each, so that the two utility functions can be written \)A(qA,z) and \)B(qB,z). Chiappori and his coauthors remain agnostic about exactly how the allocative conflict between the two people is resolved, but instead focus on the assumption that the allocation is efficient, so that, given whatever each member gets, each individual's utility function is maximized subject to the effective budget of each. Given efficiency, the optimal choice for A can be written as the solution to the problem (4.4) max \>A(q,z) s.t. pA.q = QA(p,pz,y) where z is the optimal choice of the public good, p is the price vector for all goods, p A is the price of goods consumed by A, pz is the price vector of public goods, and QA(p>pz,y) is the sharing rule, the function that determines the total amount that A gets conditional on the prices of goods, including the prices of public goods, and total household resources y. The solution to (4.4) will be a set of demand functions (4.5) q,A = g^p.p^ylp^z}. If we hold fixed the allocation to A, QA(p,pz,y)> then because these demand functions are the result of the standard maximization problem (4.4), they have all the usual properties associated with well-behaved demand functions from the textbook. There is a precisely analogous maximization problem and set of de- mand functions for B, and the overall budget constraint implies that #'s sharing rule satisfies (4.6) 0V./w> = y-Pz*-pA'1, plus the para- meters for any other included variables. Since all the right hand side variables are the same in the regressions for all goods, we have the classical multivariate regression model as described, for example, in Goldberger (1964, pp. 201-12). For this model, for which equation-by-equation OLS coincides with full informa- tion maximum likelihood, the variances and covariances of the OLS estimates of the b's both within and across equations is given by the formula (4.16) £(£.-*.)(£,-*/ - (Dyflfftr 1 Table 4.3. Outlay-equivalent ratios for adult goods and possible adult goods, Maharashtra, 1983 Males: Age group Females: Age group 0-4 5-9 10-14 15-54 55 + 0-4 5-9 10-14 15-54 55 + Adult goods Tobacco and pan -0.42 -0.12 -0.13 0.57 0.87 -0.04 -0.01 -0.17 -0.01 0.03 Alcohol 0.02 0.11 -0.89 0.37 0.10 -0.31 -0.02 -0.76 -0.30 -0.24 236 Possible adult goods Men's clothing -0.39 -0.48 -0.14 -0.14 0.07 -0.56 -0.23 -0.45 -0.57 -0.52 Women's clothing -0.21 -0.37 -0.39 -0.54 -0.53 -0.31 -0.27 -0.40 -0.22 -0.20 Leather footwear -0.60 -0.77 -0.09 0.22 -0.03 -0.69 -0.12 -0.59 -0.64 -0.40 Amusements -0.25 -0.22 -0.46 0.97 -0.32 -0.46 -0.33 -0.35 -0.46 -0.44 Personal care 0.00 0.02 -0.08 0.12 -0.42 0.19 -0.14 0.16 0.26 -0.13 Source: Subramanian and Deaton (1991, Table 5). NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 237 where X is the matrix of explanatory variables and to .. is the covariance between the residuals in the ith andjth equations. This residual variance-covariance matrix is estimated from the OLS residuals from each equation, so that (4.17) G>u = n-}e!ej where e{ and e. are the vectors of estimated residuals from the ith andjth equa- tions, respectively. The covariance of any two outlay-equivalent ratios can then be obtained from (4.16) and (4.17) by the delta method, viz. (4J8) £(w) where hJ(u,p) is the Hicksian (or compensated) demand function for good i. However, we can use the indirect utility function (4.20) to rewrite (4.23) as = (4.24) qs = *,(«./>) hf$(x,p),p) = gfap) where g f a p ) is the familiar Marshallian (or uncompensated) demand function that gives demands as a function of total outlay and prices. These Marshallian demands are what we can hope to estimate from the data, and we need a proce- dure for evaluating the cost-of-living numbers from knowledge of these functions alone. To illustrate, suppose that we wish to use base-period utility u ° as the refer- ence. Since c(uQ,p°) - jc°, outlay in the base period, the only quantity that needs to be calculated for (4.21) and (4.22) is the counterfactual c(w°,/? 1 ), the mini- mum cost of obtaining base utility at the new prices. As a first stage, write this as PI = (4.25) c(« V) '(ii°,/> V / X[dc(u*,p)/dpk]dpk n° * which by (4.23) and (4.24) is PI + (4.26) c(«V) = *(«V) f Z[gk(c(u»,plp)]dpk. NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 247 The principle of the calculation can now be seen. We start from the initial price vector pQ. For this value, the terms in the integral are known from the demand function, so we can use (4.26) to calculate the new value of c(u °,/?), not for p *, but for some other price close to p° in the direction of pl. Given this new value, we can move to a new price, so that by dividing up the path from pQ to pl into many small steps, we can calculate (4.26) as accurately as we want, with more steps giving greater accuracy. The details of how to do this in practice need not concern us here; elegant algorithms have been proposed by Vartia (1983) and Hausman (1981) for the case where the parametric form of the Marshallian de- mands is known, while Hausman and Newey (1995) show how to perform the calculations when demands are estimated by the nonparametric kernel techniques discussed in Chapter 3. For the purposes of this chapter, what is required is the demonstration of the principle, that because the derivatives of the cost function are known or can be estimated, the cost-of-living indexes and consumers' surplus measures are known or can be estimated. Equivalence scales, the cost of children, and utility theory Consider now the extension of cost-of-living theory to the theory of the cost of children. At first sight the analogy seems close to perfect, with children, or more generally household characteristics, playing the role of prices. In particular, it is supposed that the cost function depends on demographic characteristics z, so that, for a utility-maximizing household, total expenditure is the minimum cost of reaching their utility level, and (4.19) is replaced by (4.27) c(w,p,z) = x. This equation is not without its problems, and in particular it does not make clear whose utility level is referred to. We are no longer dealing with the single agent of the standard cost-of-living theory, and there is more than one possible interpre- tation of (4.27). For example, we might be dealing with parents' welfare, with the cost function calculating the minimum expenditure required to maintain parents' utility in the face of social conventions about what needs to be spent on their off- spring. Alternatively, each person in the household might have a separate utility function, and the sharing rule between them is set so that all obtain the same wel- fare level, in which case (4.27) is the minimum cost of bringing each member to the utility level u. In the next section, I shall sometimes have to be more explicit, but for the moment I note only that (4.27) is consistent with a number of sensible models of intrahousehold allocation. The equivalence scale compares two households with compositions z° and z *, in exactly the same way that the cost-of-living index number compares two price levels. Hence, if u R and p R are the reference utility level and price vector, respectively, the equivalence scale is written, by analogy with (4.21), (4.28) m(z\z°iu R,pR) = c(u*,p*9zl) + c(u*,p*,zQ). 248 THE ANALYSIS OF HOUSEHOLD SURVEYS In the simplest case, z° would represent a family of two adults and z l a family with two adults and a newborn child, so that using the prechild utility level and current prices as base, the excess of (4.28) over unity would be the cost of the newborn as a ratio of total household expenditure. Alternatively, we can follow the consumers' surplus analogy, and measure not the equivalence scale— which is an index number— but the cost of children — which corresponds to a consumers' surplus measure. This last is the cost of the difference in household characteris- tics, cf. (4.22) (4.29) D(z\z*\u\PR) = c(u*,p*,zl)-c(u*,pR,zQ) which, in the simple example above, would be the amount of money necessary to restore the original welfare level. The underidentification of equivalence scales The analogy between the cost of living and the cost of children runs into diffi- culty when we consider how the theoretical concepts are to be implemented. As before, I assume that we know the demand functions, which is more than before, since we must know not only how demands respond to incomes and to prices, but also how they respond to changes in household composition, Of course, these demand functions have to be estimated, but the survey data contain the informa- tion to do so. But that done, and in contrast to the cost-of-living case, knowledge of the demand functions is not sufficient to identify the equivalence scale (4.28) nor the cost measure (4.29), a result due to Pollak and Wales (1979). I first sketch their argument, then interpret the result, and finally consider possible remedies, Suppose that we start from a consumer whose preferences are represented by a cost function augmented by compositional variables as in (4.27) and that we track through the relationship between the cost function and the observable de- mands as in (4.23) and (4.24). With the addition of the z-vector, (4.24) becomes = (4.30) ,z) where the Hicksian demand functions h. are the partial derivatives of the cost function as in (4.23). Now consider a new (and different) cost function that is constructed from the original one by means of (4.31) c(u,p,z) = cK(u,z),p,z] where 5(i*,z) is a function that is increasing in its first argument, utility, so that in the new cost function, as in the old, it costs more to be better-off. Since £ can be any increasing function, the new cost function and the old can be quite different, and are capable of embodying quite different attitudes of parents to their children, and quite different sharing rules between them. The new Hicksian demand functions are given by differentiating the new cost function, so that by (4.31) they are related to the old Hicksian demands by NUTRITION, CHILDREN, AND MTRAHOUSEHOLD ALLOCATION 249 (4.32) gt = hfu9p9z) = hf£(u9z),p,z] where u is the utility level associated with the new cost function and the original total outlay x. However, since (4.31) is equal to jc, it must be true that (4.33) 5(fi,z) = ty(x,P,z). By the definition of the new cost function, the expression £(w,z) plays the same role as did u in the original cost function, and in particular both are equal to the value of the original indirect utility function. As a result, when we convert from the Hicksian to the Marshallian demand functions by substituting for utility, we obtain, from (4,32) and (4.33), (4.34) qt = h.[ty(x,p,z),p,z] = gt(x,p,z) so that the new demand functions are identical to the old. The two cost functions, the new and the old, although representing different preferences and different costs of children (or other compositional variables) re- sult in the same observable behavior. As a result, if all we observe is the behavior, it is impossible to discover which cost function generated the data. But the costs of children depend on which is correct, so neither the scale nor the cost measure can be calculated from knowledge of the demand functions alone. This is the essence of Pollak and Wales' argument. Like all situations where there is a failure of identification, the result does not mean that measurement is impossible, only that additional information is required, and in the next subsections, where I discuss specific proposals for estimating equivalence scales, I shall be explicit about the source of that additional informa- tion. However, before going on to that analysis, there are several general issues of interpretation with which we have to deal. It is sometimes argued that it makes no sense even to try to measure the cost of children. While fertility control is not perfect, parents generally have children as a result of conscious choice, and are presumably made better-off by the birth of a child. By this argument, the costs of a child—the benefits of losing a child—are negative, perhaps infinitely so, so that to consider the additional expenditures associated with the child as a welfare cost to the parents makes no more sense than to claim that the cost of a car is a welfare loss to its purchaser. While this argument admits the reality of the costs, it would deny their relevance for welfare calculations, and views the identification problem as a symptom of the absurdity of trying to measure the happiness that a child brings to its parents by counting the cost of clothing and feeding them. If we insist on trying to define equivalence scales simply by making cost functions depend on household characteristics, the argument is entirely valid. If welfare in these equations is taken to be that of the parents, then the choice of the function £ affects the welfare that parents get from their children without affecting the household demand functions, so that it is perfectly legitimate to interpret Pollak and Wales' argument about identification 250 THE ANALYSIS OF HOUSEHOLD SURVEYS in terms of a story of endogenous fertility. Whatever additional information is used to make the identification, it must clarify exactly what costs and benefits are being considered, and somehow rule out the direct benefits that parents get from the existence of their children. Even if we were to suppose that having children is involuntary, and that all births are unanticipated events over which parents have no control, then the equi- valence scales would still not be identifiable from demand functions alone. The most obvious case is if children were simply smaller versions of their parents, with identical tastes, but needing less of everything. Suppose that for every unit of each good that the parent receives, the child gets an amount a - a + pln* + (Ti-p)ln3 + Ya(2/3) + Y c (l/3^ where yc is the coefficient for the ratio of the appropriate child category. The compensating level of expenditure x1 is obtained by setting (4.36) equal to (4.35) and solving for x. Hence, (4.37) '"feH'iH^- If T| = 0, so that the food share is independent of family size holding PCE constant, and if ya = y c , so that switching adults for children has no effect on food con- sumption, the ratio of x l to x° is simply the ratio of family sizes, here 3 to 2. But even if r| is zero, this can be expected to overstate the compensation required because Y f l >Y c , (adults eat more than children) and P is negative (Engel' s Law) so that the last term in (4.37) will be negative. Table 4.5 shows the coefficients from the food share regressions from the Indian and Pakistani data. The p-coefficients are both negative, as they must be for Engel' s Law to hold, and so are the coefficients on the logarithm of household size, so that in both data sets, the food share decreases with household size when PCE is held constant. This result, which is a good deal stronger in the Indian than in the Pakistani data, means that, if we were to accept Engel' s contention that the food share indicates welfare, larger households behave as if they are better-off than smaller households with the same PCE. (Whether this is a sensible interpretat- ion will be discussed below.) The coefficients on the demographic ratios show the NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 253 Table 4.5. Regression coefficients of food share in India and Pakistan Coefficient of Maharashtra, India Pakistan ln(jc/n) -0.1270 -0.1016 Inn -0.0355 -0.0133 Ratio of adults 15-54 0.0070 0.0128 Ratio of children 0-4 -0.0231 -0.0205 5-9 -0.0116 -0.0020 10-14 -0.0049 0.0037 Source: India, Subramanian and Deaton (1991); Pakistan, author's calculations based on Household Income and Expenditure Survey, 1984-85. expected pattern, increasing with the age of the group; small children consume less food than older children, who consume less than adults. The calculations corresponding to (4.37) are shown in Table 4.6. The numbers show the estimated costs of a family of two adults plus one additional person of various ages all calculated relative to the costs of a childless couple. For the In- dian data, a child aged between 0 and 4 years is equivalent to 0.24 of a couple, or 0.48 of an adult, a ratio that rises to 56 percent and 60 percent for children aged 5 to 9 and 10 to 14, respectively. The age groups for Pakistan are a little older, and the estimates a little higher; 56 percent of an adult for the youngest category, and 72 percent and 76 percent for the two older groups. However, these estimates must be compared with the last row of the table, which shows the equivalence scale when an additional adult is added to the household. The third adult is only 68 percent of each of the original pair in India and 84 percent in Pakistan, so that even in the latter, the method yields large economies of scale. But these econo- mies of scale also operate for the child equivalence scales in the first three rows, so that when we say that a child is 48 percent of an adult, the effect is as much the consequence of the apparent economies of scale that operate for adults and chil- dren alike, as it is a consequence of children costing less than adults. One way of purging the child scales of the effects of the economies of scale is to look at the first three rows as ratios of the fourth row so as to measure the cost of an addi- tional child relative to the cost of an additional adult. By this measure, children Table 4.6. Equivalence scales using Engel's method, India and Pakistan Age Maharashtra, India Pakistan 0-4 1.24 1.28 5-9 1.28 1.36 10-14 1.30 1.38 15-54 1.34 1.42 Note: The numbers shown are the estimated ratio of the costs of a couple with the child as shown to a couple without children. Source: Table 4.5. 254 THE ANALYSIS OF HOUSEHOLD SURVEYS are very expensive indeed, with even the youngest child costing more than three- quarters of an adult. Such results appear to be typical of the Engel method in developing countries; for example, Deaton and Muellbauer (1986) report even higher estimates from Sri Lanka and Indonesia. In their work as in the results reported here, the estimates are large because the coefficients of the demographic ratios do not differ very much between adults and children, so that the replacement of an adult by a child does not shrink the food share by much relative to the amount that the food share would be reduced by an increase in total outlay. By the Engel method and its identifying assumption, this finding translates into children being almost as ex- pensive as adults. Although we have no standard by which to judge these estimates other than crude observation and intuition, by these criteria they seem very high. However, we must be careful about the source of such intuition and in particular rule out other estimates obtained from different models. When another methodology is used, the identifying assumption is different from that in the Engel method, so that even though the results are called "child equivalence scales'* or "child costs," we are in effect measuring different things. A comparison between scales from different models is not the same thing as a comparison of, say, the expenditure elasticity of food from different models. Models of child costs, unlike models of demand, not only provide estimates, they also provide the definition of what is being estimated. As a result, the validity of the Engel estimates can only be tested by considering the basic assumptions, and trying to decide whether or not they make sense. It is to that task I now turn. Underlying the Engel methodology are two empirical regularities, and one assertion. The first regularity is Engel's Law itself, that the share of food in the budget declines as income or total outlay increases. The second regularity is that, with resources held constant, the food share increases with the number of chil- dren. The assertion, which was made by Engel himself, is that the food share is a good indicator of welfare. More precisely, if we rank households (inversely) according to their food shares, we have correctly ranked them in terms of well- being, and the procedure can be applied across households of different demo- graphic compositions. It is important to note that this is indeed an assertion, and not an implication of the two empirical regularities. The truth of Engel's Law certainly implies that among households with the same demographic composition, those with higher food shares are generally those with the lower levels of income, and other things constant, with lower levels of welfare, but this is no more than a restatement of Engel's Law itself. Because the presence of additional children tends to increase the household's food share, it is true that additional children affect the budget in the same direction as does a reduction in income, but that is very different from a demonstration that an increase in income sufficient to re- store the food share is the precise amount needed to compensate for the additional expenditures associated with the children. Because food is so important in the budgets of poor households, the assertion that the food share indicates welfare has a superficial plausibility; food is the first NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 255 necessity and, at least among the very poor, we can probably do a good job of assessing welfare by checking whether people have enough to eat. But the claim needs to be argued, and the primacy of food is not by itself enough to establish Engel's assertion. Even if our main concern is with food, and if we believe that food consumption is a rough but useful measure of welfare, why focus on the share of food in the budget in preference to more direct measures such as food consumption or nutrient intake? That the share of the budget on food does not correctly indicate welfare over households of different compositions has been convincingly argued by Nicholson (1976). His argument runs as follows. Consider the standard case of a child born to a previously childless couple, and suppose that we know the true compensa- tion, defined as the amount of money needed to provide for the child without cutting into the parents' consumption. If this compensation is paid, the parents are exactly as well-off as before, and will presumably consume much the same pat- tern of goods as before. However, the child's consumption patterns are different from those of its parents; in particular, they will be relatively biased towards food, which is one of the few goods consumed by small children. As a result, even when the correct compensation has been paid, the consumption pattern of the household will be tipped towards food in comparison with the pattern prior to the birth of the child. But according to Engel, the food share is an inverse indicator of welfare, so that the household is worse-off, and requires further compensation to reduce the food share to its original level. As a result, Engel compensation is overcompensation, and the estimates of child costs using the Engel methodology are too high. Nicholson's argument has its weaknesses, for example in not being based on an explicit model of allocation, and in not allowing for the substitution in adult consumption that is likely to come about from the presence of the child. But it is nevertheless convincing. All that is required for its general validity is that the compensated household's food share is increased by the presence of a child, something that is hard to dispute. Note again that this is not a question that can be settled with respect to the empirical evidence; we are discussing the plausibility of an identifying assumption, whether or not we would find compelling a claim for compensation that cited as evidence an increase in the food share after the addi- tion of a child to the household. The point of Nicholson's argument is to show that the "food share identifies welfare" assumption is unsupported, that it does not follow from the importance of food in the budget nor from the validity of Engel's Law, and that it is likely to lead to an overestimation of child costs and child equivalence scales. Because the argument is persuasive, it must be concluded that the identifying assumption of the Engel methodology is not an acceptable one. The method is unsound and should not be used. Rothbarth's method In Section 4.2 on sex bias, I used expenditure on adult goods to detect differential treatment of children by gender. This methodology is a simple extension of that 256 THE ANALYSIS OF HOUSEHOLD SURVEYS suggested by Rothbarth in 1943 for measuring the costs of children. Rothbarth's idea was that expenditures on adult goods could be used to indicate the welfare of the adults, so that, if additional children reduce such expenditures, it is because of the resources that have been redirected towards the children. By calculating how much of a reduction in income would have produced the same drop in expendi- tures on adult goods, Rothbarth calculated the decrease in income that was equi- valent to the additional children and used this as his measure of child costs. As I argued in the previous section, this method is effectively the same as the identi- fication of a sharing rule on the assumption that there exist goods that are exclu- sively consumed by one group in the household, in this case adults. In his paper, Rothbarth used a very broad selection of adult goods, including virtually all luxu- ries and saving, but the subsequent literature has used much narrower definitions of adult goods, often confined to alcohol, tobacco, and adult clothing. Although we are now using a different indicator of welfare—expenditure on adult goods rather than the food share—the procedure for calculating the Roth- barth measure is similar to that for calculating the Engel scale. Figure 4.8 corre- sponds to Figure 4.7, but instead of the food share we plot expenditure on adult goods against total outlay; the graph has a positive slope on the supposition that adults goods are normal goods. The larger household spends less on adult goods at the same level of total outlay, so that, if we pick the original ^° as the refer- ence outlay for the small household, the cost of children is once again xl - jc°. Engel curves of the form (4.14) are not quite so convenient for the Rothbarth calculations as for the Engel method, although it is possible to follow through the Figure 4.8. Rothbarth's method for measuring the cost of children •a Large household 8- w x} Outlay NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 257 calculations that are parallel to (4.35) through (4.37). The equation for the share of adult goods in the budget is used to derive the expenditure on adult goods for both the reference family and the larger family, and equating the two defines the compensating value xl. For this version of Working* s Engel curve, the equation has no explicit analytical solution but can be readily solved iteratively. However, we need not take the trouble here since most of the calculations have already been done in the previous section. Table 4.3 lists the outlay-equivalent ratios for boys and girls corresponding to various possible adult goods in the Maharashtran data. Up to a linear approxima- tion, these ratios (with the sign changed) tell us how much total outlay would have to be increased in order to restore the adult expenditure to its level prior to addition of one more child. For tobacco and pan, which is the most reliable of the adult goods, the six child-gender combinations generate outlay-equivalent ratios of between 0.42 and 0.01, so that if we take 0.20 (say) as representative, the re- sults suggest that an additional child costs rather less than half of an adult. For the Pakistani results in Table 4.4, the outlay-equivalent ratios are higher, a little less than a half for girls using men's footwear, and between a third and a half for boys using women's footwear, so that an additional child costs two-thirds or more of an adult. Deaton and Muellbauer (1986) used data from Sri Lanka and Indonesia together with the extreme assumption that all nonfoods are adult goods; they found that children cost between 20 and 40 percent of an adult. The specific estimates are of less interest than the general point that the Roth- barth scales are a good deal smaller than the Engel scales, in these examples about half as much. There is no general result here; the relationship between the Roth- barth and Engel scales will depend on which goods are selected as adult goods, and on the empirical responses of food and adult goods to outlay and to demo- graphic composition. However, in the special case where children consume only food, so that all nonfoods can be treated as adult goods, then Engel's Law implies that the Engel scale must be larger than the Rothbarth scale. The argument is from Deaton and Muellbauer (1986) and runs as follows. Suppose that, when a child is born, the parents are paid the Rothbarth compensation, so that expenditure on adult goods, here nonfood, is restored to its original level. But since positive compensation has been paid, total expenditure has risen with nonfood expenditure unchanged, so that the share of food in the budget has risen. According to Engel, this is a decrease in welfare, and insufficient compensation has been paid. The Engel scale is therefore larger than the Rothbarth scale. What then are the problems with the Rothbarth methodology, and is its identi- fying assumption, that expenditure on adult goods is an indicator of adult welfare, any more reasonable than Engel's assertion about the food share? There are cer- tainly a number of practical problems. As we saw in the previous section, it is not always easy to find convincing examples of adult goods, either because the sur- vey does not collect such data—men's and women's clothing rather than adult or child clothing—or because adult goods are consumed by very few households— alcohol in Muslim societies. And while it is hard enough to find goods that are consumed only by adults, it can be even harder to find goods that are consumed 258 THE ANALYSIS OF HOUSEHOLD SURVEYS only by adults and where it is plausible that children do not affect their consump- tion through substitution effects. Babies may not go to the movies, nor eat meals in restaurants, but their presence may alter their parents' consumption of movies and restaurant meals, even when they have been fully compensated for the costs of the children. When we have a group of possible adult goods, so that it is possible to test the restriction that all give the same estimate of child costs, the restrictions are some- times rejected, leaving the choice of adult goods controversial and arbitrary (see the discussion of the evidence from Thailand in Deaton 1989b or from Spain in Deaton, Ruiz-Castillo, and Thomas 1989). Another difficulty arises if adult goods as a group have an expenditure elasticity of zero or close to it, in which case it be- comes impossible to calculate the compensating changes in total expenditure. If this happens, the theory is simply not working, since the children do not affect adult expenditures in the same way as do changes in income. If additional chil- dren cause changes in expenditure on alcohol and tobacco, and if income has no such effects, then children must be exerting substitution effects and the goods cannot be used as adult goods. However, this difficulty seems to occur only in developed countries when alcohol and tobacco are the adult goods (see Cramer 1969); in work on developing countries that I have seen, adult expenditures— including expenditures on alcohol and tobacco—respond strongly to changes in total outlay. From a theoretical perspective, there are certainly problems with the Rothbarth identifying assumption, but they are less severe than those associated with that required for the Engel method, Nicholson's argument destroys the foundation of the Engel approach, but the arguments against the Rothbarth procedure are more about details, and with its failure to include some factors that might be important. And while we know that the Engel procedure biases upwards the estimates of the scales, it is much harder to sign any bias associated with Rothbarth's method. The most serious arguments are concerned with the possible substitution ef- fects of children, with the rearrangements in the budget that the children cause even when compensation has been paid. One model of substitution is that children exert price-like effects that cause substitution away from the goods in which they are most intensive. The idea here, first formalized by Barten (1964), is that goods that are consumed by both adults and children become more expensive to the adult than goods that are only consumed by adults. For example, on a visit to a restaurant, the father who prefers a soft drink and who would order it were he alone, finds that in the company of a child his soft drink is twice as expensive but that a beer costs the same, and so is encouraged to substitute towards the latter. If so, perfectly compensated adults will consume more adult goods in the presence of children than they would without, so that the Rothbarth compensation would be too small. Barten's analogy between children and price effects is both elegant and insightful, but it is hard to believe that this is the only way that substitution effects operate. Much of the budget reallocation in the presence of children is concerned with the need to allocate time differently, and it is just as likely that compensated adults will cut adult expenditures as increase them. NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 259 We must also question whether it is appropriate to measure adult welfare as indicated by their consumption of adult goods, and beyond that, whether the welfare of adults as indicated by expenditures on such goods can tell us anything useful about the welfare of other members of the household. Of course, the Roth- barth methodology does not claim that adults get no welfare from other goods, only that their welfare is a monotonic increasing function of adult goods' expen- ditures. But the demand function for adult goods will certainly also depend on the relative price of adult goods, and so we must be careful not to apply the method to compare welfare over situations where such price changes have taken place. More generally, and even in developing countries, the price of adult time is likely to be sensitive to the presence of children, and may affect purchases of adult goods, especially time-intensive adult goods. As far as the welfare of nonadults is con- cerned, it is certainly possible to imagine situations in which the welfare of the adults is monotonically related to the welfare of the children, as for example when the parents allocate resources so as to equalize welfare over all members of the family. If such an assumption makes sense in context, there is a justification for using Rothbarth scales to correct total household expenditure, and to attribute the results to each member of the household. In the next subsection, I briefly discuss a number of other methods for calcu- lating equivalence scales, but none of these is as readily implemented as are either the Engel or Rothbarth methods, and in none is the identifying assumption so transparent. As a result, the choice for practical policy applications is to do nothing—or at least to work with either total or PCE to use one of either Engel or Rothbarth. Since the Engel method is indefensible, and since the attribution of total household expenditure to each household member requires quite implausible economies of scale, the choice is between calculating equivalents according to Rothbarth, or using PCE as in Chapter 3. More realistically, since it would be cumbersome to calculate a set of equivalents on a case by case basis, a modified Rothbarth alternative would be to choose a set of scales in line with the results that are generally obtained by the method, for example a weight of 0.40 for young children aged from 0 to 4, and 0.50 for children aged from 5 to 14. These num- bers are obviously arbitrary to a degree, but there is no good evidence that can be marshaled against them, and they are broadly consistent with results from a pro- cedure that has much to commend it. In Chapter 3,1 followed standard practice, using PCE for welfare and poverty calculations, rather than these equivalence scales. However, the decision was more for convenience of presentation than from intellectual conviction. Indeed, welfare measures based on PCE certainly overstate the costs of children, and understate the welfare levels of members of large families relative to those of small families. Whether or not the procedure of counting children at about half of adults is considered to be an improvement depends on the weights we give to the various arguments and compromises that underlie these estimates. The measure- ment of welfare and poverty would rest on a much firmer footing if there existed a solid empirical and theoretical basis for the construction of equivalence scales. But given current knowledge, numbers like a half are as good as we have, and in 260 THE ANALYSIS OF HOUSEHOLD SURVEYS my view, it is better to use such estimates to construct welfare measures than to assume that everyone is equal as we do when we work with per capita measures, Other models of equivalence scales There are several other methods for estimating equivalence scales, and at least one of these, the method proposed by Prais and Houthakker (1955), has been used in the development literature, The basic idea behind this method, which traces back to earlier work by Sydenstricker and King (1921), is to write the household demand functions in the form (4.38) Pflifai = fi(x/mQ) where m. and w0 are commodity-specific and "general" scales that are functions of household composition, and are to be thought of as measuring the need for each good and for total outlay of different types of household. A household with children would have large commodity specific scales for child foods, children's clothing and education, and the overall scale would reflect each of these specific needs. Indeed, because the budget constraint must hold, the general scale in (4.38) can be defined as the solution to (4.39) Em./;U/m0) = x. This model is estimated by specifying functional forms for each m. in terms of the observable demographic characteristics, substituting into (4.38), and estimat- ing (4.38) and (4.39) as a set of nonlinear demand equations. This apparently straightforward procedure is incomplete, at least conceptually if not numerically, because the model is not identified, a result first shown by Muellbauer (1980) (see also Deaton and Muellbauer 1980a, pp. 202-05 for further discussion). To make things worse, researchers sometimes select functional forms for the demand functions—such as double logarithmic demands—that do not permit the budget constraint (4.39) to be satisfied exactly. This failure of the model can result in the empirical version of the model being econometrically identified, essentially because of a failure of approximation to the theoretically underidentified true model, and there are examples in the literature where authors have thus succeeded in estimating a model that is theoretically unidentified. Parameters obtained in such a way are clearly of no practical value. In order to obtain genuine identification, it is necessary to have prior informa- tion about at least one of the commodity scales, a result that should come as no surprise given the earlier discussion of identification in general. Note too the close relationship with Bourguignon and Chiappori's approach, where the identi- fication of the sharing rule is obtained by finding goods that are exclusive or assignable, and with Rothbarth's, where we specify exclusively adult goods. In the Prais-Houthakker method, an obvious way to proceed would be to identify the model by assuming that the commodity-specific scale for adult goods is unity. NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 267 Having identified the model in this or some other way, estimation can proceed and scales be obtained. However, it is not clear what is gained over the much simpler Rothbarth methodology. The calculation of the Prais-Houthakker scales requires estimation of a potentially large system of nonlinear equations, a task that is a good deal easier than it once was, but the effort requires a justification that is not apparent, at least to this author. I have already mentioned a second alternative, the model originally proposed by Barten (1964), and extended by Gorman (1976). Barten's procedure is even less suitable than Prais and Houthakker's for applied work, but it contains an important insight that I shall use again in the next subsection, so that it is worth a brief statement. Barten built the idea of commodity-specific scales into the frame- work of utility theory, writing the household utility function—best thought of as the utility of each member of the household—in the form (4.40) w = v^/mj,...,^//^) where the m's have essentially the same commodity-specific scale interpretation as in Prais and Houthakker's model. The maximization of (4.40) subject to the usual budget constraint p.q - jc, can be conveniently rewritten by defining the scaled quantities q* - qi/mi and scaled prices p* =pimr so that the consumer's problem can be rewritten as the maximization of \)(q *) subject to the budget constraint that p *.q * = jc. This problem immediately generates standard-form demand functions in which q* is a function of the prices p .*, so that (4.41) ?,/*f, = gfai*iPv — *r*nPn)' Note first that if all the mi were identical and equal to household size, (4.41) would give the sensible result that consumption per head is equal to the standard Marshallian demand function of outlay per head and prices. (Recall that demand functions are homogeneous of degree zero.) More generally, when the scaling factors differ from one commodity to another, the income reduction effects will still be present, working by scaling up each price by a greater or lesser amount. However, and this is the central insight of the Barten model, there are also substi- tution effects. Goods that are child-intensive are relatively expensive in a house- hold with many children, and family decisionmakers will substitute against them. While the recognition of the possibility of such substitution is an important contribution of the Barten model, the model is clearly incomplete as it stands. The fundamental identification of equivalence scales comes from the assumption that household composition acts entirely analogously to prices, so that, not surpris- ingly, the model cannot be estimated without data with price variation (see again Muellbauer 1980 and Deaton and Muellbauer 1980a, pp. 202-5). But the analogy with price, while conceivably a part of the story, is surely not all of it. For exam- ple, as pointed out by Deaton and Muellbauer (1980a, p. 200) and forcefully argued by Nelson (1993), there is no reason to suppose that a child-intensive good, such as milk, could not be highly price elastic in families without children; 262 THE ANALYSIS OF HOUSEHOLD SURVEYS for adults, beer or soda may be good substitutes for milk, though one might hesi- tate to make the same substitutions for children. Within the Barten model, if the substitution effect is large enough, this situation can lead to a reduction in milk demand when a child is added to the family, a result that compromises the empi- rical usefulness of the model as well as any claim that the model might have to represent the welfare of children as well as adults. It is possible to extend the model to improve the situation, for example by following Gorman's (1976) suggestion to make allowance for fixed costs of children, but the empirical imple- mentation of such models is cumbersome out of all proportion to their advantages over the simpler Rothbarth methodology. Economies of scale within the household It is an old and plausible idea that two can live more cheaply together than two can live separately, yet there has been relatively little serious empirical analysis of the phenomenon, and our understanding of the topic is even less complete than that of child costs. One good approach to the issue is through the recognition that there are public goods within the household, goods that can be shared by several people without affecting the enjoyment of each. In this last subsection, I discuss some of the literature—which is again dominated by Engel's method—and then sketch out an approach to economies of scale based on the existence of public goods that can be shared within the household. The public good approach reveals serious flaws in the Engel methodology, and suggests an alternative approach. Attempts to implement this alternative founder on an empirical paradox that is currently unresolved. I start by dismissing children, and thinking only about households containing n identical adults. The conceptual issues with which I am concerned are not affected by the presence of children, and if children are to be included, we can suppose that they have already been converted to adults using a child equivalence scale. Consider the direct utility function u(<7pp--->P m ) where ^(jc,/? p ...,/7 n ) is the indirect utility of a single individual with outlay jc. Because both the budget shares and indirect utility depend on family size only through the term x/n e , welfare is correctly indicated by the budget share of any good, and two households of different size are equally well-off when their pat- terns of budget shares are the same. There is an empirical restriction—that we get the same answer whichever budget share we use—but I ignore it for the moment. Instead consider one particular budget share, that of food. As before, I use the subscript/for food, and rewrite (4.44) as (4.46) wf = . = (a.'Eaklnpk) + p^.ln* = a* + p.ln*. We can now introduce household size and economies of scale by multiplying the cost function by an isoelastic function of n, so that (4.51) becomes NUTRITION, CHILDREN, AND INTRAHOUSEHOLD ALLOCATION 269 (4.54) where 6 is, as before, the parameter controlling economies of scale. If we derive the budget share equations as before, we get, in place of (4.53), cf. (4.47) above, w. = a* + p.ln(jc/n e ) = a* + p^ln* - p.01n* (4.55) = a* + pJn(jc/n) + p,(l-0)lnn which is the equation that I have been using to interpret the data. To see the identification problem, suppose that, instead of (4.54), we also make the economies of scale parameter 0, not a constant, but a function of the utility level w, in the form 0 + 0jlnw. This is plausible enough and certainly cannot be ruled out in advance; additional people may not affect costs proportion- ately, but have larger or smaller effects the better-off is the household. Substitu- tion gives our last cost function (4.56) c(u,p,n) = n°*l*'tHp)u*» = n'a(p)u*f}*9**ut. If we follow through the calculations for a third time, we simply come back to (4.55); the incorporation of the household size function into the elasticity has had no effect on behavior, and so we cannot tell (4.54) and (4.56) apart on the data. But the two cost functions have different implications for household welfare. The indirect utility function for (4.56) is (4.57) Inw = [P(p) + Ojlnn]- 1 ! In* - 01nn - lna(/?)] which contains the term 0 t lnn in the denominator. Welfare levels, unlike the de- mand functions, are affected by the presence of this term so that, once again, we cannot infer welfare from behavior. 4.4 Guide to further reading There is an enormous background literature on the topics covered in this chapter, and it is probably best approached through surveys. The recent book by Dasgupta (1993) is much concerned with gender discrimination (Chapter 1 1) and nutrition (Chapters 14 through 17). Dasgupta argues the case for the nutritional wage hypo- thesis as the root cause of destitution, but one does not have to accept all his argu- ments to benefit from his extensive reviews of the literatures in economics and in nutrition. Dreze and Sen (1989) is also concerned with undernutrition, its causes, and the design of public policy for its alleviation; it also provides extensive refer- ences to the literature. At the narrower and more technical level, both Bouis and Haddad (1992) and Strauss and Thomas (1995) provide reviews of earlier litera- ture as well as their own contributions to measurement. The reports of the Inter- national Food Policy Research Institute in Washington are frequently concerned with nutritional issues in a range of countries. On equivalence scales, the easiest 270 THE ANALYSIS OF HOUSEHOLD SURVEYS place to begin is with Chapter 8 of Deaton and Muellbauer (1980a), and to update with Deaton and Muellbauer (1986). Buhmann et al. (1988) provide a review of some of the equivalence scales that have been used in the academic literature as well as in government programs, and show how different scales affect poverty and inequality measurement for the (developed) countries in the Luxembourg Income Study. There is also a literature on "subjective" approaches to the con- struction of equivalence scales that parallels attempts to measure poverty lines by direct questioning (see, for example, Rainwater 1974; Danziger et al. 1984; and van Praag, Hagenaars, and van Weerden 1982). The issue of economies of scale that is sketched in the penultimate subsection is dealt with much more fully in Deaton and Paxson (1996). 5 Looking at price and tax reform In a world of laissez-faire, prices set in free competitive markets guarantee econo- mic efficiency. As has long been recognized, such efficiency is of a limited na- ture; its achievement requires strong assumptions and takes no account of the distribution of income, even to the extent of being consistent with some individu- als being unable to survive. Laissez-faire pricing also leaves little room for the government to collect revenue, whether to influence the distribution of resources or to finance its own activities. The lump sum taxes that would (by definition) leave behavior unaffected generate little revenue if they exist at all, and the lim- ited range of tax instruments on which governments in poor countries must rely are often quite distortionary. As a result, the best that can be hoped for is a price system that is efficient in the sense of minimizing distortion for any given amount of revenue. The design of such systems is the topic of optimal tax theory, starting from Mirrlees (1971) and Diamond and Mirrlees (1971), a brief account of which is given in the first section below. In recent years, public economics has been greatly influenced by what is now referred to as "political economy," a branch of economics and political science that uses the tools of economic analysis, particularly the typical economic as- sumptions of rationality and greed, to help explain the behavior of political actors. These models often seem relevant to poor countries, where the empirical reality is often in stark contrast to the assumptions of a previous generation of models in which the government was seen as a benevolent dictator acting in the public inter- est. As a result, there has been a good deal of suspicion—and in some cases rejection—of optimal tax models in which prices are set to maximize a social welfare function subject to budgetary and behavioral constraints. These criticisms are justified in that it is absurd to regard the optimal models as descriptions of the social and political equilibria that actually exist in developing countries. Even when the motivation of a given government is clear, existing taxes are likely to be a web of only partially consistent levies and subsidies, the fossilized traces of long-dead administrations of differing purposes and philosophies. Even so, it is important to think rationally about what system of prices would be desirable, at least to the extent of meeting the legitimate needs of government for revenue and redistribution while minimizing the costs of collection and distor- 277 272 THE ANALYSIS OF HOUSEHOLD SURVEYS tion. Indeed, international organizations such as the World Bank and the Interna- tional Monetary Fund are required to make proposals for reform, sometimes in opposition to what governments would choose left to themselves. Although opti- mal tax theory is unlikely to predict the tax structure in any given situation, it provides a systematic and indispensable framework for thinking about the stan- dard issues of distribution and economic efficiency that are the basis for any intelligent discussion of price and tax reform. My main concern in this chapter is not with the theory itself, but with its implementation, and with the use of household survey data to make the calcula- tions. In Chapter 3,1 showed how the surveys could be used to see who gets what and who produces what, so that we could calculate who benefits and who loses from price changes, and thus assess the distributional consequences of a change in pricing policy. That analysis is an important part of the analysis of price reform, but to go further, we must also know something about efficiency, which means finding out how behavior responds to the incentives provided by price changes. Traditionally, such evidence has been obtained by examining the historical experience, linking demand and production decisions to historical variations in price. What I shall mostly be concerned with here is a different approach that uses variation in prices across space to identify behavioral responses. This line of work has been developed in a series of papers over recent years (Deaton 1987a, 1988, 1990a and Deaton and Grimard 1992), and the account in this chapter is intended to be more complete and comprehensible than is possible within the confines of a journal article. It also aims to show how the results of the calculations can be used to look at price reform in practice, with examples from India and Pakistan using the data sets introduced in previous chapters. Section 5.1 presents a brief review of the theory that is required for practical implementation in poor countries. It also provides a first discussion of how the formulas might be used, and reviews the various procedures that have been adopt- ed in the literature. I argue that, without using the data from the surveys, it is almost impossible to derive policy recommendations that are firmly based on the empirical evidence. While it is certainly possible to make policy based on prior reasoning, on the results of computable general equilibrium (CGE) models, or on econometric evidence from historical data, such recommendations—sensible though they may be—are largely grounded in prior assumption, and are not much affected by empirical evidence. Section 5.2 is a first look at the evidence on spa- tial price variation from the Pakistani and Maharashtran data, with a particular focus on whether the unit value data that come from the surveys can be trusted for an econometric analysis of behavioral responses to price. This section also con- siders some of the difficulties that lie in the way of a straightforward use of spatial price variation to estimate demand responses. Section 5.3 presents a model of consumer behavior that is designed to accommodate the behavior we see in the surveys, particularly the fact that better-off households pay more for each unit of even quite narrowly defined commodities. I show how to derive a simple econo- metric model that is based on this theory. Section 5.4 works through the stages of the estimation using the Indian and Pakistani data. Section 5.5 uses the results to LOOKING AT PRICE AND TAX REFORM 273 examine price reform, and highlight the differences that come from using the data-based methodology. In particular, I emphasize interaction effects; because there are important substitution effects between goods, it is important not to con- sider price reform one good at a time. The final Section 5.6 considers a number of directions for future research, and in particular the possible use of nonparametric estimation. 5.1 The theory of price and tax reform for developing countries The theory of taxation for developing countries is extensively treated in the monograph by Newbery and Stern (1987), which is not only a collection of appli- cations from several countries, but which also contains introductory chapters that review the underlying theory. Interested readers are strongly advised to consult that source for a fuller account than given here; my purpose is to provide only those results that are required for practical analysis of reform, and to identify those quantities for which empirical evidence is required. The analysis will be concerned almost exclusively with indirect taxes (and subsidies). Although in- come taxes provide a large share of government revenue in developed countries, they are less important in poorer countries, for obvious administrative reasons. Most poor countries do indeed have income taxes, but it is difficult to collect revenue from the self-employed, particularly in agriculture, or from those em- ployed in small businesses or informal enterprises. As a result, most income taxes come from the relatively small formal sectors, or from the government paying income taxes to itself on the incomes of its own employees. A simple linear in- come tax can be effectively replicated (except in its treatment of saving) by an indirect tax system that taxes all goods at the same rate and that uses a poll tax or subsidy that is identical for all individuals. However, the practical usefulness of this identity is limited by the impossibility of taxing all consumer goods in most countries; in particular, it is not possible to drive a tax wedge between producer and consumer when the producer and consumer are one and the same, as is the case for farm households. And although many countries provide food subsidies for urban residents, arbitrage places limits on differences in prices between urban and rural areas. For these (and other) reasons, the tax instruments that are avail- able to governments in developing countries are typically much more limited than is the case in richer countries with their more elaborate distribution networks, and the fact must always be kept in mind when discussing reforms. Tax reform The theory of tax reform is concerned with small departures from an existing tax structure. This is in contrast to the theory of optimal taxes, which is concerned with characterizing the (constrained) optimum in which taxes and subsidies are set at their socially desirable level. The differences are sometimes blurred in both theory and practice; the formulas for tax reform often have obvious implications for the optimum, and any practical reform will involve a discrete change in price. 274 THE ANALYSIS OF HOUSEHOLD SURVEYS However, the empirical burdens of calculating tax reforms are much lighter than those of calculating optima. The former asks us to evaluate the current position, and to calculate the desirability of various directions of reform, for which we need information about supply and demand and behavioral responses calculated at the current position of the economy. The calculation of an optimal tax system, by contrast, will require us to know the same quantities at the hypothetical wel- fare optimum, a position that is most likely far from the current one, or from anything in the historical record. We begin by specifying the accounting apparatus for judging prices, as well as the rules that govern the behavioral responses. This is done in exactly the same way as in Chapter 3, by specifying a social welfare function and its determinants. As in Chapter 3, the social welfare function should be regarded as an accounting device that keeps track of who gets what, and that is used to consider alternative distributions of real resources among individuals or households. In Chapter 3, equation (3.1), social welfare was defined over the "living standards" jc of each member of the population, where x was later equated with real expenditure levels. Here, because we are interested in the effects of changes in prices, we need to be more explicit about the link between prices and welfare. To do so, rewrite social welfare as a function of the individual welfare levels w, which are in turn given by the values of the indirect utility functions that give the maximum attainable wel- fare in terms of prices and outlays. Hence, (5.1) W = V( W l ,w 2 , ...,«„) where N is the number of people in the economy and the welfare levels are given by (5.2) "/, = $(xh>P) for household A's indirect utility function ty(xh,p), where xh is total outlay, and p the vector of prices. The indirect utility function defines not only the way that welfare is affected by prices, but also encapsulates the behavioral responses of individuals; given the indirect utility function ty(xh,p), demand functions can be derived by Roy's identity, (3.46). I am using the subscript h to identify individuals or households, and I shall deal with the difference between them later. I also assume that everyone faces the same price, an assumption that will be relaxed below, but which simplifies the initial exposition. Note finally that I am avoiding dealing either with labor supply, or with intertemporal choice and saving. Because there is no income tax in this model, there is no need to model earnings, not because labor supply is unaffected by commodity taxes and subsidies, but because with an untaxed wage, the in- duced changes in labor supply have no affect on tax collections. A similar argu- ment applies to saving, and I interpret (5.2) as the single period utility associated with an additively intertemporal utility function of the kind to be discussed in Chapter 6. Even so, the current approach fails to recognize or account for differ- ences between the distribution of welfare over lifetimes and over single periods. LOOKING AT PRICE AND TAX REFORM 275 We can also begin with a simplistic model of price determination whereby the consumer price is the sum of a fixed (for example, world) price and the tax or subsidy, so that (5.3) Pi = p? + t. where t. is the tax (if positive) or subsidy (if negative) on good /. The simplest (and best) justification of (5.3) is in the case where the good is imported, there is a domestic tax (or import tariff), and the country is small enough so that changes in domestic demand have no effect on the world price. Another possible justifica- tion runs in terms of a production technology displaying constant returns to scale, with a single nonproduced factor, and no joint products, in which case the non- substitution theorem holds, and producer prices are unaffected by the pattern of demand (see, for example, the text by Mas-Colell, Whinston, and Green 1995, pp. 157-60). The assumptions required for this are overly strong, even for a modern industrialized economy, and make less sense in a poor one with a large agricul- tural sector. We shall see how to dispense with them in the next subsection. Government revenue is the last character in the cast. It is simply the sum over all goods and all households of tax payments less subsidy costs: M H (5.4) R = ££',-(0) = Ep,.ln( r I P / ( * - / > . Y ) where the 2M - 1 parameters (the p's must add to 1) are to be estimated. The sys- LOOKING AT PRICE AND TAX REFORM 281 tern (5.17) is readily estimated on short time-series data; indeed the p's are identi- fied from the Engel curves in a single cross section, as are the y's up to a con- stant. Once estimates have been obtained, (5.17) can be used to calculate the complete set of own- and cross-price responses, so that (5.9) or (5.12) can be evaluated without need for further assumptions. The convenience of the linear expenditure system has led to its extensive use in the applied policy literature, including work on tax reform in developing coun- tries (see Ahmad and Stern 1991, or Sadoulet and de Janvry 1995). The model is certainly superior to that based on the crude assumption that cross-price effects are zero; it is based on an explicit utility function and thus satisfies all theoretical requirements. However, the linear expenditure system still offers a relatively limited role for measurement, relying instead on the assumptions built into its functional form. Of course, that is the point. Without much data, there is little alternative but to use prior restrictions to aid estimation, and there is much to be said for using restrictions that are consistent with economic theory. But having done so, it is important to investigate which of our results come from the mea- surement, and which from the assumptions. In the case of the linear expenditure system, much is known about the effects of its supporting assumptions. As shown in Deaton (1974b), additive preferences imply that own-price elasticities are approximately proportional to total expendi- ture elasticities, so that the efficiency effects of the taxes, which work through the price elasticities, tend to be offset by the equity effects, which work through the total expenditure elasticities. Indeed, Atkinson (1977) showed that if all prefer- ences are given by the Stone-Geary form (5.16), and if the government has avail- able to it a lump sum polltax or subsidy that is the same for everyone in the econ- omy and that is optimally set, then the optimal solution is to tax all commodities at the same rate. This result can be generalized somewhat; Deaton (1979) shows that the result holds provided only that the Engel curves are linear and there is separability between goods and leisure. Furthermore, if to these requirements is added the assumption that preferences are additive—as in the linear expenditure system—then not only are uniform taxes optimal, but any move toward uniform taxes is welfare improving (see Deaton 1987b). It is even possible to allow for variation in tastes across individuals, provided that the optimal poll taxes or subsi- dies are set to take such variation into account (see Deaton and Stern 1986). In all these cases, the tax reform exercise does not require any econometric estimation at all! The answer is known before we start, and the analysis of the data adds nothing to the policy recommendations. If the linear expenditure system is plausible, then these results are very useful, since they tell us what sort of price reforms to recommend. Unfortunately, the model is much too restrictive to be credible, especially in the context of price re- form in poor countries. Demand functions from additive preferences do not per- mit inferior goods, and the substitution effects between commodities are of a simple and restrictive kind that do not permit complementarities between goods, nor any specific kinds of substitutability (see Deaton and Muellbauer 1980a, ch. 5, for a more precise discussion). When we are thinking about tax and subsidy 282 THE ANALYSIS OF HOUSEHOLD SURVEYS policy in poor countries, many of the commodities involved are foods, some of which may be close substitutes, such as different types of cereals, and some of which are complements, such as cooking oil and the foods with which it is used. In several countries, there are also important inferior goods, such as cassava in Indonesia, the consumption of which is largely confined to members of the poor- est households. Neither the linear expenditure system nor any other model of additive preferences can handle this sort of consumption pattern. A somewhat different approach to tax and price reform in developing coun- tries is to use CGE models. These (often quite elaborate) representations of the economy usually conform to the standard textbook representations of a general equilibrium system, with specifications for consumer preferences, for the technol- ogy of production, and for the process by which supply and demand are brought into equilibrium, usually by setting prices at their market clearing levels. The parameters of these models are not estimated using standard statistical methods; instead, the models are "calibrated" by choosing the parameters of preferences and technology so as to match the historical record (if only roughly) as well as to bring key responses into line with estimates in the literature. Many of these mod- els use the linear expenditure system, or one of its variants; the parameters of (5.16) could be calibrated from (5.17) by looking at a set of expenditure Engel curves, which would yield the P's and the intercepts Y/'Y-PI *e individual y parameters can be solved out given one more piece of information, for example the own-price elasticity of food, for which there exist many estimates in the litera- ture. Alternatively, parameters can be taken from the many estimates of the linear expenditure system for developing countries, such as those assembled in the World Bank study by Lluch, Powell, and Williams (1977). While such models can be useful as spreadsheets for exploring alternative policies and futures, the incorporation of the linear expenditure system has ex- actly the same consequences as it does in the more explicitly econometric frame- work. The results of tax reform exercises in CGE models are entirely determined by the assumptions that go into them. When a CGE model incorporates the linear expenditure system—or the even more restrictive Cobb-Douglas or constant elas- ticity of substitution preferences—a prescription for uniform taxes or for a move towards uniform taxes is a prescription that comes directly from the assumptions about functional form, and is unaffected by the values of the parameters attached to those forms. The finding cannot therefore be shown to be robust by calculating alternative outcomes with different parameter values; it is guaranteed to be robust provided the functional forms are not changed. It is also guaranteed not to be robust with respect to changes in those functional forms. Atkinson and Stiglitz (1976) and Deaton (1981) provide examples of cases where major changes in optimal tax structure, for example between regressive and nonregressive taxation, can be brought about by minor variations in separability assumptions, variations that would be hard to detect in an econometric analysis. If we are to make recommendations about tax reform that recognize the actual structure of demand in developing countries, there is no alternative to the estima- tion of a model of demand that is sufficiently general to allow for that structure. LOOKING AT PRICE AND TAX REFORM 283 5.2 The analysis of spatial price variation In the literature on demand analysis in developed countries, almost all of the iden- tifying price variation has come from price changes over time, with little attention paid to variation in prices over space (although see Lluch 1971 for an exception). The reason for this is obvious enough; in countries like the United States, where transport and distribution systems are highly developed, and where transport costs are relatively low, there is little price variation between localities at any given time. In developing countries, transport is often more difficult, markets are not always well integrated, and even the presence of potential arbitrage cannot equal- ize prices between different geographical locations. In consequence, when we are trying to identify behavioral responses in demand, we have in developing coun- tries a source of price variation that is not usually available in developed econo- mies. Regional price data Data on regional price differences are often available from the statistical offices responsible for constructing consumer price indexes; even when national esti- mates are the main focus of interest, standardized bundles of goods are priced in a number of (usually urban) locations around the country. These price data can be merged with the household survey data by associating with each household the prices in the nearest collection center at the time closest to the reporting period for that household. The combined data can then be used to estimate a demand system with individual households as the units of analysis. Alderman (1988) uses such a procedure to estimate a demand system for Pakistan. However, there are a number of difficulties. One is that with only a few sites where prices are collected, the data may give inaccurate estimates of the prices faced by at least some house- holds. In particular, urban prices may be a poor guide to rural prices, especially when there are price controls that are more effective in urban areas. The paucity of collection sites may also leave us in much the same situation as with the histor- ical record, with too many responses to estimate from too few data points. The situation is made worse by the fact that it is often desirable to allow for the effects of regional and seasonal taste variation in the pattern of demand by entering re- gional and seasonal dummies into the regression, so that the price effects on de- mand are only identified to the degree that there are multiple observations within regions or that regional prices do not move in parallel across seasons. Household price data The individual household responses often provide a useful source of price data. In many surveys, although not in all and in particular not in the original LSMS sur- veys, households are asked to report, not only their expenditures on each good, but also the physical amount that they bought, so that we get one observation in kilos and one in currency. The ratio of these two observations is a measurement 284 THE ANALYSIS OF HOUSEHOLD SURVEYS of price, or more accurately, of unit value. The attraction of such measures is the amount of data so provided. If the current Indian NSS were available on a nation- wide basis, we would have upwards of a quarter of a million observations on prices, a data set that would certainly permit the estimation of a much richer pattern of responses than could ever be obtained from a few dozen years of the historical record. Other surveys are not so large, but a typical comparison would still be between many thousands of observations from a survey and less than 50 from time series. As always, there is a price to be paid. Unit values are not the same thing as prices, and are affected by the choice of quality as well as by the actual prices that the consumer faces in the market. Not every household in each survey reports expenditures on every good, and no unit values can be obtained from those that do not. And when there are measurement errors in the data— which means always—there are obvious risks in dividing expenditure by quantity and using the result to "explain" quantity. These problems and others have to be dealt with if the analysis is to be convincing. Even so, it is worth beginning by looking at the unit values, and trying to see whether they give us useful information about prices. I do this using (once again) the Pakistani data from the 1984-85 Household Income and Expenditure Survey and the Maharashtran state sample from the 38th round of the Indian NSS. The results summarized here are reported in full in Deaton and Grimard (1992) for Pakistan, and in Deaton, Parikh, and Subramanian (1994) for Maharashtra. Table 5,1 for Pakistan and Figures 5.1 and 5.2 for Maharashtra provide differ- ent ways of looking at the spatial variation in the unit values. Table 5.1 reports two different sets of calculations for the Pakistani data. In the first, reported in the top panel, the logarithms of the unit values are regressed against a set of dummies for each of the four provinces in the survey—Punjab, Sindh, North West Frontier Province (NWFP), and Baluchistan (the omitted category)—and for each of the four quarters (omitting the fourth) of the calendar year during which the survey was conducted. These regressions are intended to capture broad regional and seasonal patterns. The bottom half of the table looks at variation from village to village within each province, and uses analysis of variance to decompose price variation into its within-village and between-village components. The unit values for all goods but the last come directly from the survey, al- though the wheat category includes bread and flour converted to wheat content. The unit value for the final category, other food, is computed from a set of weights—here the average budget share for each component over all households in the survey—that are used to construct a weighted geometric index from the unit values for each of the individual foods in the other food category. Although the same weights are applied to all households, they are rescaled in each case to account for goods not purchased to ensure that the weights add to one for the cal- culated unit value for each household. If a unit value were calculated for such a heterogeneous category by adding up expenditures and dividing by the sum of the physical weights, the result would be as much determined by the composition of demand between items in the category as by any variation in prices. The rescaling by fixed weights has some of the same effect since households that buy only ex- Table 5.1. Variation in log unit values of foods by provinces and villages, rural Pakistan, 1984-45 Provinces and quarters Wheat Rice Dairy products Meat Oils &fats Sugar Otherfoods Regression Punjab -7.2 -8.0 -68.7 -22.1 -2.0 -6.6 -9.7 (6.0) (4.5) (15.8) (16.4) (8.5) (21.9) (11.8) Sindh 7.3 ^3.1 -78.3 -13.8 -2.6 -2.9 -11.8 (5.2) (22.3) (15.5) (9.5) (10.4) (9.0) (13.3) NWFP 6.3 -2.8 -26.0 -31.1 -1.3 -3.7 -6.3 (4.7) (1.4) (5.4) (20.8) (5.0) (10.9) (6.9) Jul-Sep 1984 -2.1 0.4 2.0 -1.4 -0.2 0.3 0.9 (2.6) (0.4) (0.7) (1.6) (1.4) (1.7) (1.7) Oct-Dec 1984 -2.4 -1.0 -1.5 -3.4 -0.5 -0.2 ^.3 (3.0) (0.8) (0.5) (4.1) (2.9) (1.1) (8.4) 285 Jan-Mar 1985 0.0 -2.0 -3.0 -1.9 -0.3 0.3 -6.9 (0.0) (1.6) (1.1) (2.2) (1.8) (1.5) (13.3) Analysis of variance F R2 F R2 F R2 F R2 F R2 F R2 F R2 Punjab 4.02 0.33 9.14 0.59 3.07 0.39 7.85 0.45 13.7 0.61 7.2 0.42 12.8 0.53 Sindh 5.42 4.40 4.40 0.44 9.44 0.78 8.61 0.48 10.4 0.51 19.9 0.66 14.1 0.57 NWFP 6.06 7.12 7.12 0.57 10.63 0.67 4.46 0.30 13.5 0.54 13.9 0.59 10.8 0.48 Baluchistan 9.14 8.52 8.52 0.61 17.70 0.80 10.97 0.61 7.74 0.50 18.0 0.70 17.9 0.69 Note: The top panel shows coefficients from a regression of the logarithms of unit values on province and quarter dummies. The figures in brackets are absolute r-values. The bottom panel shows F- and /^-statistics for cluster (village) dummies within each of the five regions. Regressions and analyses of variance use data from only those households that report expenditures on the good in question. Source: Deaton and Grimard (1992, Table 3). 286 THE ANALYSIS OF HOUSEHOLD SURVEYS pensive items will have higher unit values, but there is no way of completely avoiding the difficulty, and the effect will have to be taken into account in the analysis. The top panel of Table 5.1 shows that interprovincial price differences are much larger than are seasonal differences, though the latter are statistically signi- ficant for most of the categories. Apart from wheat, which is more expensive in Sindh and in NWFP than in Baluchistan, all province means are lower than those in Baluchistan. For oils and fats and for sugar, there is a good deal of state price con- trol, and there is relatively little variation in price across the provinces, and the same holds to a lesser extent for wheat. Rice is very much cheaper in Sindh than elsewhere. The bottom panel reports, not parameter estimates, but the F-tests and #2- statistics from what can be thought of as a regression of the logarithm of unit value on dummies for each village, of which there are 440 in Punjab, 152 in Sindh, 110 in NWFP, and 55 in Baluchistan, although not all villages appear in each regression since the village is excluded if no household reports that type of expenditure. It is impossible (and unnecessary) to include seasonal dummies in these regressions because all the members of a village are interviewed within a few days of one another in the same calendar quarter. What we are looking for here is evidence that unit values are informative about prices, and since prices should not vary by much within villages over a short period of time, there should be significant F-statistics for the village effects, or put differently, the village dummies should explain a large share of the total variance in the logarithms of the unit values. While it is not clear how much is enough, all the F-statistics are sig- nificant at conventional values, and the village effects explain around a half of the total variance. Given that unit values are contaminated by measurement error, and given that there is variation within villages as richer households buy higher-qual- ity goods, the results in the table seem sufficient to justify the suppositions both that there is spatial price variation and that variation in unit values provides a (noisy) guide to it. Figures 5.1 and 5.2 show the behavior of the unit values for rice and forjowar in the Maharashtran data; similar graphs can be drawn for other foods, but these illustrate the point and the commodities are important in their own right. Instead of looking at the unit values at the village level, these diagrams display averages over households by district and quarter, so that each panel shows the seasonal pat- tern for that district in 1983, while a comparison of different panels reveals the differences in prices from one district to another. As for Pakistan, there is varia- tion in unit values, both across space and over the seasons of the year. Both dia- grams show pronounced seasonal patterns that are correlated—although far from perfectly so—between the districts. For rice, the logarithm of the unit value rose on average by 15 percent from the first to the third quarter, and by a further 2 percent between the third and fourth quarters. There is also substantial variation by district, both in levels and in the nature of the seasonal pattern. For example, in Dhule and Jalgaon, the average unit value of rice peaked in the second quarter, at which point the price of rice was nearly a third higher than in Chandrapur, for ex- LOOKING AT PRICE AND TAX REFORM 287 Figure 5.1. Log unit price of rice by district and subround, Maharashtra, 1983 Thane Kulaba Ratnagiri Dhule Jalgaon 1.4- 1.2 : » 1.0: 1.4l 1.2:' l.o: r 1.4; '•2:, 1.0 T l,4l' 2 »- i.o; , r ( 1.4; 1.2: 10 ' 1 2 3 4 1 2 3 4 1 2 3 4 Source: Deaton, Parikh, and Subramanian (1995, Fig. 1). Figure 5.2. Log unit price ofjowar by district and subround, Maharashtra, 1983 Ratnagiri Nasik Dhule Jalgaon 0.9 1, 0, 0.1 I 0.9 •/ 0, 0.1 I 0.9 { 0, 0.1 I 0.9 1; .Nagpur Bhandara °-5 o.i; • •• 0.9 f 0,1 0.1 I 2 3 4 1 2 3 4 1 2 3 4 Source: Deaton, Parikh, and Subramanian (1995, Table 2). 288 THE ANALYSIS OF HOUSEHOLD SURVEYS ample. Jowar prices also rise through the year, by 13 percent from the first to the third quarter, falling by 2 percent in the final quarter. Interdistrict variation in jowar prices is much larger than for rice. In the inland eastern districts, Buldana, Akola, Amravati, Yevatmal, Wardha, and Nagpur, which are also the main pro- ducing districts, prices are lower throughout the year, by a half or 50 paise per kilogram, than in the coastal districts where no jowar is grown. Of course, these figures come from only the 1983 survey, but the spatial and seasonal patterns appear to be typical of other years. For Maharashtra, it is also possible to compare the unit values from the survey with other price information. Government of India (1987) publishes monthly data on market prices in selected locations for a number of foodstuffs and crops. In cases where these prices can be matched by district, the two sets of numbers correspond very closely (see Deaton, Parikh, and Subramanian for details). Al- though the market price data are not sufficiently complete to serve as an alterna- tive for estimating price responses, they are nevertheless useful as a check on the unit values that are reported by individual households in the NSS survey data. While the examination of the unit value data cannot establish that they can be treated as if they were prices, the results here indicate that unit values are at the very least useful as an indicator of seasonal and spatial price variation. Unit values and the choice of quality One reason why unit values are not the same thing as prices is that unit values are affected by the choice of quality. A kilo of the best steak costs a great deal more than does a kilo of stewing beef, Roquefort costs more than cheddar, and even for relatively homogeneous commodities such as rice, there are many different grades and types. Unit values are computed by dividing expenditures by physical quan- tities, and high-quality items, or mixtures that have a relatively large share of high-quality items, will have higher unit prices. As a consequence, and in contrast to a market price, over which the consumer has no control, a unit value is chosen, at least to some extent. In particular, since better-off households tend to buy higher-quality goods, unit values will be positively related to incomes or total outlays. Because unit values are choice variables, there is a risk of simultaneity bias in any attempt to use them to "explain" the patterns of demand. Hence, in- stead of explaining the demand for rice by the price of rice, we are in effect re- gressing two aspects of the demand for rice on one another, with results that are unlikely to give what we want, which is the response of demand to price. The size of quality effects can be assessed by running regressions of the loga- rithm of unit value on the logarithm of total expenditure and the usual list of household demographics and other characteristics. Such a model is written K-\ (5.18) Inv = a1 + p'ln* + y'lnn + E L(n./n) J J + u} i where v is the unit value, x is total household expenditure, and the demographic terms are as in equation (4.14), the logarithm of household size and household LOOKING AT PRICE AND TAX REFORM 289 structure as represented by the ratios to household size of persons in various age and sex categories. Although we might expect income (or total expenditure) to be the most important variable in explaining the choice of quality and thus the unit value, there is no reason to rule out other household characteristics; some groups within the household may choose particular items within a food category, and religious dietary preferences will also affect quality. Regressions like (5.18) were first examined by Prais and Houthakker (1955), who used a 1937-39 survey of British households to estimate values of p1, a parameter that they referred to as the "quality elasticity," or more precisely as the expenditure elasticity of quality. For working-class households, they estimated quality elasticities ranging from 0.02 for onions, cocoa, and lard, to as high as 0.28 for sausages, 0.29 for fresh fish, and 0.33 for cake mixes (Prais and Houthakker 1972, Table 27, p. 124). They also emphasized local variations in the range of varieties; for example, the quality elasticity of cheese in Holland is 0.20 as compared with only 0.05 in Bri- tain—British working-class households in the 1930s ate only cheddar—and the quality elasticity of tea in Bombay was 0.25 as against 0.06 in Britain. Because unit values will vary not only with the choice of quality, but also with actual market prices, (5.18) should ideally be estimated with price data included on the right-hand side. This is impossible from the data that we have, but it is nevertheless possible to estimate the nonprice parameters of (5.18) consistently, provided that we are prepared to make the assumption that market prices do not vary within each village over the relevant reporting period. In rural areas in most developing countries, this is a reasonable assumption. Most villages have only a single market, and prices are likely to be much the same for a group of house- holds that are in the same location and surveyed at the same time. Hence, equa- tion (5.18) can be extended to include prices simply by adding dummy variables for each village. With large surveys, there may be several thousand villages, so the regression must usually be run, not by asking the computer to calculate it directly, but by first calculating village means for all variables, and then running a regression using as left- and right-hand-side variables the deviations from vil- lage means. By the Frisch-Waugh (1933) theorem, the regression of deviations from village means gives identical parameter estimates to those that would have been obtained from the regression containing the village dummies. Two practical points are worth noting. First, the removal of village means must be done on exactly the same data that are to be used in the regressions. This seems obvious enough, but in the case of (5.18), where different subsets of house- holds purchase different goods so that the samples with usable unit values are different for each good, the village means for variables such as Inx, which are available for all households, will have to be different for each good. One simple way of checking that the demeaning has been done correctly is to check that the estimated intercept in the demeaned regression is zero, at least up to the accuracy of the machine. Second, the standard errors given by the usual OLS formula will not be correct, although the error will typically be small, and is in any case easily corrected. If there are n households in the survey, grouped into C clusters, the hypothetical regression with all the dummies would have n-k-C-\ degrees of 290 THE ANALYSIS OF HOUSEHOLD SURVEYS freedom when there are k other regressors (such as those listed in (5.18) above). But the regression using the demeaned data includes all n observations, and has k regressors, so that when the equation standard error is estimated, the computer will divide the sum of squares of the residuals not by n-t-C-1, as it should, but by n~k. The correct standard errors can be calculated by scaling the calculated standard errors by the square root of the ratio of n-fc-C-1 to n~fe, which with (say) 10 households per cluster, amounts to inflating the standard errors by the square root of 0.9, or by 5,4 percent. (All of these issues can be avoided by using the "areg" command in STATA, which when used with the option "absorb (clus- ter)" will estimate the regression with implicit dummies for each cluster.) Tables 5.2 and 5.3 present parameter estimates of (5.18) for the two data sets using the methodology of the previous paragraph. Only the coefficients on the logarithms of total expenditure and household size are shown, although the regres- sions contain a full range of compositional and socioeconomic variables in addi- tion to the village dummies. Apart from the case of sugar in urban Pakistan, where the coefficient is zero, all the expenditure elasticities of quality are positive, and most are statistically significant. The heterogeneous meat commodity has the highest quality elasticities, and the elasticity of 0.242 in urban Pakistan is the highest in either table. The important cereal categories—rice, wheat, and coarse cereals—have modest quality elasticities, from 3 to 11 percent. Price control for sugar and oil in Pakistan is reflected in very low quality elasticities, although the estimate of 0.006 for sugar in the rural sector is significantly different from zero. The response of quality to household size is negative, except for the intriguing case of fruit in Maharashtra, where both rural and urban estimates are positive. As Table 5.2. Within-village regressions for unit values, Pakistan, 1984-85 Rural Urban Food Inx Inn In* Inn Wheat 0.095 -0.033 0.027 -0.028 (14.7) (4.3) (7.6) (6.7) Rice 0.107 -0.062 0.111 -0.075 (12.4) (6.3) (16.5) (9-7) Dairy products 0.139 -0.031 0.010 -0.023 (6.4) (1.3) (0.9) (1.9) Meat 0.154 -0.099 0.242 -0.156 (23.7) (13.5) (34.0) (19.0) Oils -1 , so that demand is price inelastic, the estimate of the price elasticity will be biased downward, with the spurious negative correlation dominating the attenuation bias. More generally, but still on the assumption that o12 = -a\ , the estimated price elasticity will be biased toward -1. In the next section, I propose a methodology that permits consistent estimation of the price elasticities in the presence of both measurement errors and quality contamination of unit values. 5.3 Modeling the choice of quality and quantity The purpose of this section is to construct an empirical model that can be used with the survey data, including the unit value information, to provide consistent estimates of price elasticities of demand. The basic ideas that underlie the model are straightforward, but the application involves a certain amount of detail, and the final model can appear to be complex if encountered all at once. To avoid unnecessary complexity, I proceed in stages, introducing a very simple version of the model first, and using it to explain the basic principles. I then introduce the various complications one at a time. For the reader only interested in the empiri- cal results, only the first few subsections need be read, leaving the more technical material to be consulted only by those who wish to use the methods. 294 THE ANALYSIS OF HOUSEHOLD SURVEYS A stripped-down model of demand and unit values In the simplest possible model of demand, the quantities consumed are related to total outlay (or income) and to price. To that basic model, we must add the special features of the current problem, notably that we observe not price but unit value, and that prices do not vary within villages (clusters) but only between them. One version of such a model has a demand function and a unit value function (5.23) lnqhc = 5E4 = p e''6-' The elasticity of unit value with respect to price, or the parameter ty in (5.24) is one plus the elasticity of quality to price, so that we have finally that (5.34) * = 1+ PVv According to (5.33), separability implies that quality shading in response to price is determined by the price, income, and quality elasticities of the commodity group. When prices rise holding relative prices constant, there is a reduction in demand for the group as a whole, and the size of the reduction is controlled by the price elasticity e . When less is bought, there is a quality effect whose magnitude depends on the elasticity of quality with respect to expenditure on the group, an elasticity that is the quality elasticity with respect to total expenditure divided by the expenditure elasticity of the group. As a result, there will be no quality shad- ing if either the price elasticity or the quality elasticity is zero. The parameter ijr, LOOKING AT PRICE AND TAX REFORM 299 the elasticity of unit value with respect to price in the unit value equation (5.24), is one if there is no quality shading, and will be less than one by an amount that is larger the larger are the price and quality elasticities. The separability assumption thus provides the basis for quantifying (and correcting) the bias that would arise if quality effects are ignored, and unit values treated as prices. Because (5.34) provides an additional relationship between the quantity and unit value elasticities e and v|j, it can be used to identify the two elasticities separately from estimates of the ratio ty~le , which as we have seen is all that can be identified from the quantity and unit value data. If we write (J) for the ratio of e to i|j, equation (5.34) implies that (5 35) ( } 4> = — = -~- * d+PVe,) so that, upon rearrangement, we have (5.36) e = * . " l-4>p'/e, As a result, if we know the ratio (J) and the quantity and quality elasticities ex and P°, we can use (5.36) to calculate the price elasticity e . Since all of the magni- tudes on the right-hand side of (5.36) can be estimated from the data, the separability-based model of quality shading provides the basis for identifying the price elasticity. Equation (5.36) can also be thought of as a correction formula, where the uncorrected (for the effects of quality) price elasticity is $, and where the correc- tion will be small if p1, the total expenditure elasticity of quality, is small. In the next subsection, I show how the data can be used to give consistent estimates of all of the identified parameters up to and including the ratio ((). Once this is done, a final correction is made by applying equation (5.36) to the estimated parameters to obtain an estimate of the price elasticity. Estimating the stripped-down model We are now in a position to estimate all of the parameters of the two-equation quantity and unit value model (5.23) and (5.24). There are two stages. The first, which we have already seen—uses within-village information to estimate the total expenditure elasticities, and the second, discussed here, uses between-village in- formation to estimate the price elasticities. Suppose that the first-stage estimates of ex and p1 have been obtained. Use these estimates to construct the two variables (5.37) fhc = In^-^Iiu^ (5.38) ylhc = \nvhc~$llnXfic. Note that these are not the residuals from the first-stage regressions; these regres- 300 THE ANALYSIS OF HOUSEHOLD SURVEYS sions contain village dummies, and in (5.37) and (5.38), it is only the effects of total expenditure that are being netted out, and the price information contained in the village dummies must be left in. The residuals from the first-stage regressions play a different role, which is to estimate the variances and covariances of the residuals in the quantity and unit value equations. Let e ° and el be the residuals from the quantity and unit value regressions, respectively, and write n-k-C-1 for the number of degrees of freedom in these regressions; recall n is the total number of observations, C the number of clusters, and k—which is one here—is the number of other right-hand-side variables. Then we can estimate the variance of uhc and the covariance of uhc and uhc using (5.39) a11 = el'el/(n-k-C-l) a01 = eG/el/(n-k-C~~l). We shall shortly see how these estimates are to be used. Since the second stage uses the between-village information, it begins from (5.37) and (5.38), the quantities and unit values purged of the effects of total ex- penditure, averaging them by cluster. Denote these averages yc and ylc, respectively, the absence of the household subscript h denoting village averages. Corresponding to these estimates are the underlying "true" values y® and y^ which we would have obtained had the first-stage parameters been known rather than estimated. From (5.23) and (5.24), these can be written as (5.40) y° = a° + zp\nnc+fc + u° (5.41) ylc = aUv^lnjr^wJ where, once again, the w's have been averaged over villages. For a large enough sample of villages, we would therefore have (5.42) covO^1) = e^m + a01/^ where m is the large-sample intervillage variance of log prices, and nc is the num- ber of households per village. The last term comes from noting that wc° and wc! in (5.40) and (5.41) are averages over the nc households in each cluster. (When I come to deal with the complications, I shall return to these formulas and to what happens when there are different numbers of households in different clusters, as well as when there are different numbers of observations on quantities and unit values.) Corresponding to (5.42), we can also write the variance of the corrected unit values as (5.43) var^1 - v|/2m + o n /w c . The comparison of (5.42) and (5.43) yields *P cov(y c V)-a 01 /n c (5.44)

= ; var(y c ! )-d n //i c where the terms on the right-hand side come from (5.37), (5.38), and (5.39). Equations (5.44) and (5.45) look somewhat unfamiliar at first sight, but they are readily related to standard estimators, especially errors-in-variables estima- tors. Note first that the bivariate OLS estimator of the regression of y on x is the ratio of the covariance of jc and y to the variance of x. In the absence of a01 and o11, the error variances and covariances from the first stage, (5.45) is simply the OLS regression of average village demand on the village average unit value, where both right-hand and left-hand variables have been purged of the effects of house- hold total expenditure. The a01 and o11 terms, which would vanish if each vil- lage were large enough, are designed to correct for the part of the between-village variances and covariances that comes from measurement and econometric error in the underlying first-stage equations. Indeed, (5.45) is a standard errors-in-vari- ables estimator in which the XfX matrix and Xfy vector are purged of the contri- butions of measurement error (see 2.66). That the estimator is relatively unfamil- iar in practice is because it is not feasible in the absence of information about the moments of the measurement error, information that is rarely available, at least in economics. In the current case, because of the two-stage structure of the problem, the information can be obtained from the residuals of the first-stage within-village regressions. That (5.45) reduces to OLS when the number of households in each village is infinite, gives a useful insight into the nature of the estimation procedure. It is de- signed to give a consistent estimate of the ratio (j) as the number of villages tends to infinity, holding fixed the number of households in each village. Although the first stage of estimation uses only the within-village information, it pools such data over many villages so that, provided the fixed effects are the only form of village heterogeneity, the first-stage parameters will converge to their true values as the number of villages increases, even if there are only a few households per village. But even if the first-stage parameters were known, so that the "hats" in (5.45) could be removed, the correction for the first-stage errors would still have to be made if there are not enough households in each village to average them away. Apart from this correction, (5.45) is simply a least squares estimate of the price elasticity using the variation in quantities and prices across the villages. In the case where y's and o's are known, the asymptotic variance of (j) is given by the delta method, or from Fuller (1987, p. 108) (5.46) v( we can write (5.48) ^ = «° + p°ln^ + f.zhc + 8lnn c +/c - «£ where whc is the share of the good in the budget of household h in cluster c. This equation is paired with the unit value equation (5.24), which need only be supple- mented with demographic terms to read (5.49) lnv fcc = a1 + p1 \nxhc + f,zhc + ^f\^c + u\c. Although the same right-hand-side variables appear in (5.48) as in the logarithmic model (5.23), their coefficients are no longer elasticities, so that the model needs reinterpretation and the quality correction formulas must also be reworked. I shall turn to this below, but first it is worth thinking briefly about what kind of demand function is represented by (5.48) and what, if any, is its relationship to standard demand theory. At a superficial level, (5.48) is closely related to standard demand models like the "almost ideal demand system" in which the budget share is a linear function of the logarithms of total expenditure and prices. However, the analogy is at best incomplete. First, this is a model where utility depends, not only on quantity, but also on quality, so that, at the very least it has to be shown that (5.48) is consistent with the appropriately augmented utility theory. I shall examine this question below, but even when the model is utility-consistent, the fact may be largely irrelevant in practice. Standard empirical demand systems, such as the almost ideal system, are derived under the assumption that all goods are purchased, so that the budget shares are strictly positive. While choice theory certainly permits "corner solutions" in which nothing is bought of some goods, the formal incor- poration of corners into tractable empirical demand functions remains a largely unsolved problem. For a one-good model, the presence of corners generates a Tobit model which, as we have seen in Chapter 2, is problematic enough in prac- tical situations. However, when several goods are considered at once, as will be the case in the real applications below, there are a large number of possible re- gimes depending on which combination of goods is being bought. As a result, equation (5.48) can at best be thought of as a linear approximation to the regres- sion function of the budget share conditional on the right-hand-side variables, averaging over zero and nonzero purchases alike. While this may be more or less satisfactory for the purpose at hand—the modeling of average behavior—the LOOKING AT PRICE AND TAX REFORM 305 averaging will remove any direct link between (5.48) and utility theory. In this respect, the model is on much the same footing as an aggregate demand system where the averaging over agents almost never permits an interpretation in terms of a representative agent, Consider now what (5.48) and (5.49) imply for the elasticities and how the estimation of the previous section must be modified to deal with the new func- tional form. In (5.48) the elasticity of the budget share with respect to total expen- diture is p°/w, but since the budget share is unit value times quantity divided by total expenditure, we have (5.50) ex + p1 = (p°/w) + 1 where p1 is, as before, the total expenditure elasticity of quality from the (un- changed) unit value equation (5.24) or (5.49). Similarly, for the price elasticity e we have (5.51) ep + i|r = (0/w). Because both total expenditure and price elasticities depend on the budget share, neither will be constant but will vary from household to household. The two-stage estimation technique works in the same general way as before. At the first stage, the budget shares and the logarithms of the unit values are each regressed on the household demographics and the logarithms of total expenditure. The "corrected" y® and y* are formed as before, although the former is now a corrected budget share rather than a log quantity. As before the residuals from the first-stage regressions are used to estimate the covariance and variance o01 and o11, and the results used to calculate $ in (5.45), although given the budget share equation (5.48), what is being estimated is now the ratio ty~lQ. Again as before, we use the simple quality theory to allow the separate recovery of both parame- ters, although the formulas are now different. The theoretical relationship between the quality, price and expenditure elasticities in (5.34) gives, on substitution from (5.50) and (5.51) and rearrangement , 1 PV-9) (5.52) * = 1-"^ i- P°+H> Given cj> = i|r~l 0 and (5.52), 0 can therefore be recovered from (5.53) 6 = S l+(w-)C where (5.54) c- £ ftO . ,,,/i _ ftl\ . If P1 is small, C will be small, and so will be the correction to 0 in (5.53); when the income elasticity of quality is small, there will be little shading of unit value in response to price. 306 THE ANALYSIS OF HOUSEHOLD SURVEYS With the parameters recovered, the elasticities can be calculated from (5.50) and (5.51). Equations (5.46) and (5.47) remain correct for the variance of <[> and the variances and covariances of the other parameters can be obtained using the delta method (see Deaton 1988 for the formulas). One other practical detail is worth noting. Since not every household reports a purchase, and since unit values are only reported for those who do, there will be unequal numbers of observations for the budget share equation (5.48) and the unit value equation (5.49). Suppose that in cluster c there are nv households of which rcc+ record purchases. At the first stage of the estimation, the imbalance in numbers makes no difference. At the second stage, however, when the corrected budget shares and unit values are averaged by cluster, there are different numbers in the budget share average than in the unit value average, and the numbers will differ from village to village. The difference is important when we compute the vari- ances and covariance of the averaged residuals wc° and wj. Since the former is averaged over nc observations, its variance is a lnc> while the variance for the latter is clearly o n /w c + . Straightforward algebra shows that the covariance is ool/nc. As a result, the nc in equations (5.42) through (5.46) must be replaced by T and T + , respectively, where (5.55) T'1 = limC-'Sn/1; (tT1 = lira (T'E («;)-'. In practice, these magnitudes are unknown, but estimates of T"1 and CO"1 can be calculated from the averages in (5.55) ignoring the limits. Rewriting the model in terms of budget shares, (5.48) and (5.49), takes us close to what we require for implementation. What remains is an exploration of the welfare foundations of the model, of how quality and quantity interact in util- ity. This analysis will also show how to embed the demand for each good in a system of demand equations. Although the general model is the one that should be used in practice, the simpler case in this subsection is useful for investigating practical econometric issues. In Deaton (1990a), I report a number of Monte Carlo experiments using (5.48) and (5.49). The experiments yield estimates that are correctly centered, in contrast to estimates that do not allow for village fixed effects, or that ignore measurement error. As usual, the price of consistency is an increase in variance. However, the experiments show that the estimator performs adequately even when there are as few as two observations in each village. In- creasing the number of villages or clusters is much more important than increas- ing the number of observations in each. The asymptotic variance formulas work- ed well and accurately predicted the variance across experiments. *Quality, quantity, and welfare: cross-price effects In this subsection, I show how quality and quantity can be interpreted within the standard model of utility maximization. At the same time, we shall see how to incorporate cross-price effects. For price and tax reform, where subsidies and taxes are frequently levied on closely related goods, some knowledge of cross- LOOKING AT PRICE AND TAX REFORM 307 price effects is essential, and as I argued in Section 5.1, it is one of the main ad- vantages of household survey data that they provide some hope of estimating these effects. The extension of the model from a single equation to a system of demand equations is conceptually straightforward, and consists of little more than adding additional price terms to the budget share equation (5.48). Even so, there are some theoretical issues that must also be taken care of, particularly the exten- sion of the model of quality under separability to the multigood case, so that we can handle the effects of changes in the price of one good on the quality choice of another. These theoretical issues are the topic of this subsection. In the economet- ric analysis, which is discussed in the next subsection, there is a good deal of what is essentially book-keeping, and the resulting algebra is a good deal more com- plex than that required so far. Readers who are willing to take this material on trust can safely omit the details and move on to Section 5.4 where the empirical results are presented. The theoretical basis for a multigood model with quality effects is a utility function that is separable in each of M commodity groups. Write this as (5.56) where each qG is a vector of goods, each element of which is perfectly homoge- neous. The group utility functions, which are sometimes referred to as subutility or felicity functions, have all the properties of the usual utility functions, and I shall denote the value of each by UG. Since overall utility is increasing in each of the subutilities, and since consumption in each group, qG, affects overall utility only in so far as it affects group subutility, the consumer will maximize each u G (<7 G ) subject to the amount spent on the group, an amount that I denote by XG - pG.qG. For this subutility maximization problem, define the cost function cG(uG,pG), the minimum amount of money needed to attain UG at prices pG. For a utility-maximizing consumer, money is spent efficiently, so that = C U (5.57) XG G( G>PG)- I can now set up the definitions of quality and quantity. Physical quantity is defined as before, equation (5.25) repeated here for convenience (5.58) QG = kG.qG. The utility from the consumption of group G is conveniently measured using "money-metric utility," the minimum cost need to reach UG at a reference price vector pG. Define quality £G implicitly by the identity (5.59) cG(uG,pG) = 5 G g G . According to (5.59), the utility from group G consumption can be expressed as the product of quality and quantity; in the special case where there is only one 308 THE ANALYSIS OF HOUSEHOLD SURVEYS way of attaining utility UG, which is to buy the quantity qG, (5.59) reduces to (5.28). We also define the price index TCG as the ratio of costs at actual prices to costs at the reference prices, ^("G^G) (5.60) *G = — r; C U G( G>PG) so that, as before, we have = (5.61) XG ^C^GG- Expenditure is the product of quantity, quality, and price. In general, the price indexes, K G , are functions of the utility levels UG\ this will not be so when prefer- ences are homothetic within the group, in which case the pattern of demand in the group is independent of the level of living, and there are no quality effects! In this formulation, group utility UG is a monotone increasing function of the product of quality and quantity, (5.59), so that we can write overall utility as (5.62) u = V*« 1 fi 1 ,...,5 G G c ,..-^G w ), which, given (5,61), is maximized subject to the budget constraint M (5.63) ZnG$GQG *x. This is a standard-form utility maximization problem, and will have standard de- mand functions as solutions. I write these = (5.64) 1GQG gG(x>*i>-~>*c- — *M) which will satisfy all of the standard properties of demand functions: homogene- ity, Slutsky symmetry, and negative definiteness of the Slutsky matrix. Provided we treat the product of quality and quantity as the object of preference, the stan- dard apparatus of demand analysis applies as usual. The demand functions (5.64) are somewhat more complicated than they look. The prices of the composite goods, TTG, defined by (5.60), are each a function of UG , or equivalently, of £ G <2 G » so that (5.64) is not an explicit set of demand func- tions, but a set of equations that has to hold when choices are made optimally. The dependence of the n's on the M'S vanishes under the conditions of the Hicks' composite commodity theorem which, in the current context, means that the rela- tive prices of the goods within each group are the same in each cluster. If the con- stant of proportionality in cluster c is nGc, so that the price vector in the cluster is pGc = 7iGcpG, (5.60) gives nGc = rcGc, which is independent of UQ. In practice, we will not usually want to assume the constancy of relative prices—in areas by water, fish will be cheap, while in arid areas, fish will be dear—and we need only the higher-level assumption that the dependence of TCG on UG is negligible, for which constant relative prices is a sufficient, but not necessary, condition. LOOKING AT PRICE AND TAX REFORM 309 For the empirical work, we draw from the literature and adopt a standard flex- ible functional form for the demand system (5.64). One suitable form is the al- most ideal demand system of Deaton and Muellbauer (1980b), whereby (5.64) takes the form M (5.65) wc = a° + Peln(*/70 + E O^lnT^ //-i where (the unsubscripted) TI is a linearly homogeneous price index formed from all the prices (see Deaton and Muellbauer for the theoretical definition). Here I follow their practical suggestion of using a Stone index for 7t whereby M (5.66) In 71 ~ E wH\niiH //=i and WH is the sample average of the budget share of //. If (5.66) is substituted into (5.65), we have M (5.67) WG = aG + P G ln* + E 8 OT lnn ff //=! M = c£ + p£ In* + E (Octf-Pc*!/) lnnH. //=! The unit values VG = £GnG are themselves functions of total expenditure and the prices, and for lack of any theoretical guidance on functional form, I use the generalization of (5.49), (5.68) lnv G = 4 + $Glnx+ E i^lni^ //=! where the original scalar quality shading elasticity i[; is replaced by a matrix of elasticities i[/G//. In the absence of quality shading, unit value would be equal to price, the Y matrix would be the identity matrix, and a1 and p1 would be zero. Because the budget shares add to unity, the vector oc° in (5.65) must sum to unity, while the vector P°, and each column of the 0-matrix, must add to zero. The system should also be homogeneous of degree zero in total expenditure and prices; a doubling of both should double unit values and leave the budget shares unchanged. This will occur if, and only if, the rows of QGH add to zero, or equiva- lently that, for all G, (5.69) £/A* + P°G = 0- Similarly, linear homogeneity of the unit value equation (5.68) requires that, for all groups G, (5.70) Effi|rGW + p^ = 1. The adding-up and homogeneity restrictions enable us to "complete" the demand system by adding another commodity defined as "all other goods," and to infer its own- and cross-price elasticities from the adding-up and homogeneity restrictions. 310 THE ANALYSIS OF HOUSEHOLD SURVEYS When using unit values to indicate prices, which is the methodological basis for this work, we will never obtain prices for a complete enumeration of consumers' expenditures, since there are many goods that do not have obvious quantity units, and for which only expenditure data are collected. For these goods, we can do no better than an "other goods" category. The Slutsky (or substitution) matrix of the demand system will be symmetric if, and only if, Q*GH - 6#G, or equivalently, (5.71) & GH + $S*H = Otfc + PffWo- It is common in demand analysis to use the symmetry restrictions to add to the precision of the parameter estimates. In the context of price reform it might be thought that such considerations would carry less weight, partly because there are so many observations in the survey data, and partly because the existence of corner solutions—zero purchases—for many households precludes a clean rela- tionship between the theory and its empirical application. Unless we build differ- ent models for the different "regimes" comprising different combinations of pur- chased goods, and then link these regimes within a general statistical model of selection, we cannot follow the parameters of the theory through into the empiri- cal specification. As a result, even though the utility restrictions hold for a single agent who buys all of the goods, they will not hold in the survey data. However, there are also some arguments for using the restrictions. First, al- though there are many observations, there is not always a great deal of price vari- ation. In the Pakistani data, there are at least partially successful controls on the prices of sugar and of edible oil, so that even with nearly 10,000 observations, it will be difficult to estimate price elasticities involving these goods. Indeed, the interest in price reform is often at its most intense in countries with a history of price control and we are frequently faced with using survey data with little price variation to try to predict what will happen when markets are liberalized. Second, not all of the restrictions from the theory have the same status, and we might be much more willing to impose absence of money illusion on the empirical demand functions than to impose Slutsky symmetry. Unfortunately, it is the latter that is likely to be the most helpful when we are trying to infer the price elasticities involving a controlled good. In such a situation, we need restrictions from some- where, and we might use the utility restrictions for want of better ones, just as symmetry and homogeneity are frequently imposed on aggregate data without rigorous justification. One task remains, to generalize the quality model so as to link the matrices 0 and T, so that they can be separately identified. The multigood treatment of this section permits a direct generalization of equation (5.34). From the definitions of quality and quantity, equations (5.58) and (5.59), we have (5.72) ln5G = lnc(u G ,/> G ) - \n(kG.qc). As before, note that qG =/ G (* G /TC G ,/? G ), so that (5.31) generalizes to LOOKING AT PRICE AND TAX REFORM 311 81ngc _ ain£G ( ain*G \ (5.73) H dlnxG[d\nnH J Equation (5.32) remains unchanged, so that (5.74) ^S. - $>*». ain7tH ea In consequence, we have ainv . G , , PC CCH (5.75) Wru ~ VCH = VarH + ain^ " ec where &GH is the Kronecker delta, equal to unity when G = H and zero otherwise. When G = H, (5.75) and (5.34) are identical. Otherwise, when the price of one commodity increases, the effects on quality of another are controlled by how much the price increase affects quantity, i.e. by the cross-price elasticity. ^Cross-price effects: estimation A good place to begin is with a final statement of the two equations for the budget share and for the logarithm of unit value. With demographics, fixed effects, and errors included, (5.67) and (5.68) become, for good G, household ht in cluster c M + (5.76) wcftc = a° + pJXc + Y G-«* + £ ^CH^HC =1 " M (5.77) lnvG,c = alG + $lclnxHc + YG-^ C + ^ *G//ln^//c + «ikc H*l and our task is to estimate the parameters. The first stage is exactly as before; the budget shares, the logarithms of the unit values, the z's, and the logarithms of the jc's are demeaned by their cluster means, and the two regressions run using the demeaned variables. Because all of the prices are constant within clusters, the demeaning removes all the prices as well as the fixed effects and so allows con- sistent estimation of the ^ *- eGfce//Ac c h€c & = (5.81) GH (n-C-kyl^^eGHceHhc c hzc (5.82) XG* GH = -c- (n~ = » -kYl^€GhceHhc c h€c where el and eQ are the residuals from the first-stage, within- village unit value and budget share regressions, respectively. In practice, it will often be too much to expect the data to deliver good estimates of all elements of these three matrices, in which case only the diagonal elements need be calculated, and the off-diagonals set to zero. The empirical results in the next section are constructed in this way. The between- village variance-covariance matrix of the (theoretical, not esti- mated) y$ is written Q, that of the y$ is 5, their covariance is /?, and the elements of these matrices are estimated from (5.78) and (5.79), = VGH cov(^,^c)> SGH = cov(^^c)f P-Q-^ - , A! *o N = COV *. As in that case, its probability limit in the presence of the quality effects is not 0, but the matrix generalization of the ratio of 0 to T, (5.86) plim£ = B = (YV©'. The transposition is because B has the estimates for each equation down the col- umns, whereas in 0 they are arranged along the rows. LOOKING AT PRICE AND TAX REFORM 373 As before, the 0 matrix is not identified without further information, here sup- plied by the separability theory of quality. The quality theory delivered equation (5.75) above, which in matrix notation is (5.87) T = I + D($l)D(e)-lE. E is the matrix of price elasticities, and the diagonalization operator D(.) con- verts its vector argument into a diagonal matrix. The matrix of price elasticities and the vector of total expenditure elasticities are linked to the model parameters by (5.88) E = -Y +D(w)-l@ (5.89) e = i-pl+p°/)(w)-1. Equations (5.87) through (5.89) can be combined with (5.86) so as to retrieve 0 and thus E from B, (5.90) 9 = B X 7 = B'[I-D({)B' + D({)D(w)rl (5.91) E = [D(w)- 1 fl / -/]'F = [ D ( w y l B f - I ] [ I - D ( £ ) B ' + D(£)D(w)rl where the elements of ( are defined by (5.92) CG = [(l-PcK + P^T'Pc. This completes the estimation stage; the first-stage parameters and residuals are used to make the covariance matrices in (5.80) through (5.83), the results are used to calculate the matrix B in (5.85), an estimate that is corrected using the first-stage estimates to give the 0 parameters or elasticity matrix using (5.90) or (5.91). Variance-covariance matrices for the estimated parameters and elasticities can be obtained by bootstrapping— which is the approach used for the results reported here —or analytical formulas can be derived that are valid in large samples. The algebra for the latter requires the introduction of a good deal of new notation, and is not given here. Intrepid readers can use the results in Deaton and Grimard (1992) and Deaton (1990a) as templates for the slightly different expressions here. The main value of such an exercise would be to provide asymptotic approxi- mations that could be used to improve the bootstrap calculations, as described in Section 1.4. However, I suspect that a straightforward application of the bootstrap is adequate, and it certainly delivers standard errors that are close to those from the asymptotic formulas. In practice, I have adopted a further shortcut, which is to bootstrap only the second stage of the estimation. The first-stage, within-cluster, estimates are calculated—with standard errors routinely provided by STATA — and (5.78) and (5.79) are used to calculate estimates of the cluster averages of the "purged" unit values and budget shares. This cluster-level data set is then treated as the base data from which bootstrap samples are drawn. This shortcut saves a 314 THE ANALYSIS OF HOUSEHOLD SURVEYS large amount of computer time because the first-stage estimation is much more time-consuming than the second stage. It is justified by inspection of the asymp- totic variance formulas, which show that it is the contribution from the second stage that dominates, and from practical experience which shows that the first- stage contribution to the variance is negligible. ^Technical note: completing the system This note, which is intended only for those who wish to pursue their own calcula- tions, and need to understand the code, lays out the calculations needed to com- plete the system by adding a single composite commodity, "nonfood." I also ex- plain how to impose symmetry, at least approximately. The calculations start from the MxM matrix 0 defined in (5.67) and calculated from (5.90). Write the corresponding (M+l)x(M+l) matrix for the complete system 0*, which differs from 0 by having an additional row and column. Using the homogeneity restric- tion (5.69), the final column of 0" is given by (5.93) 0 GMtl = -p G -£6 G f f . H=\ The final row of 0* is computed from the adding-up restriction, so that (5.94) 0^1G = -£0^ G . H=\ In the same way, the adding-up restrictions are used to extend the vectors oc°, P°, and w to a0*, P°x, and vP*. The vector of quality elasticities p l cannot be ex- tended in the same way; instead it is necessary to assume some plausible quality elasticity for the nonfood category. That done, (5.92) is used to calculate the extended vector £*. From (5.90) applied to the complete system, we have (5.95) 0* = B^Y* (5.96) vv) =0 where I have used the fact that \ec(ab')=b®a. The selection matrix L is re- quired here in order to prevent each symmetry restriction being applied twice. Without it, we would be trying to impose that element {1,2} should equal element {2,1 } as well as that element {2,1 } should equal element { 1,2}, a redundancy that will lead to singularities in the matrices to be inverted. If we write b for vecl?, (5.100) can be written in the familiar linear form (5.101) Rb = r with R and r defined from (5.100). We already have an unrestricted estimate from (5.85). Write A for the matrix S-ClN~l in (5.85), and define the restricted esti- mate of B, by (5.102) vecBR = vecB + (I®Arl[R(I®ArlR/]'l(r-RvecB). 5.4 Empirical results for India and Pakistan In this section, I take the technical apparatus for granted and return to my main purpose of deriving demand parameters for the analysis of price reform. I begin with a preliminary subsection on data issues and the selection of goods, and then turn to the first- and second-stage results. 316 THE ANALYSIS OF HOUSEHOLD SURVEYS Preparatory analysis The first choice in any demand analysis is the choice of goods to be included. Because we are interested in price and tax reform, the analysis must distinguish goods with different (actual or potential) tax rates. In some cases, there are goods of particular concern, for example those whose nutrient content is important, or goods that are substitutes or complements for the goods whose prices might be altered. There are also data limitations. We cannot estimate the demand for a good whose price is constant across the sample, and it will be difficult to obtain reliable estimates of goods that are purchased by only a few households. Larger demand systems are also harder to deal with than small ones; the more goods, the greater the computational problem, and the harder it is to report the results. It is of the greatest importance to subject the data to close scrutiny before undertaking the formal analysis. In addition to outliers or incorrectly coded data, there are special problems associated with the unit values which must be carefully checked for plausibility. We have already seen one way of doing so, by checking the reported unit values against other price data, by looking at seasonal and spatial patterns, and by an analysis of variance. Univariate analysis of the unit values is also important in order to expose difficulties that can arise if spatial variation in tastes causes spatial differences in unit values. The problem arises in its most acute form when the commodity group consists of a single good that comes in different forms. Consider, for example, the category "fish." The main components of the group will usually be "dried fish" and "fresh fish," with different unit val- ues per kilo; suppose that, allowing for transport and processing margins (the cost of drying the fish), a dried fish costs a little more than the same fish fresh, but weighs less because it contains less water. Suppose also that the country has a coastline but no lakes or rivers, and that the proportion of dried fish rises with the distance from the sea. Even if everyone consumes exactly the same number of fish, some fresh and some dried but with added water, there will be a negative spatial correlation between the weight of fish purchased and their unit values, a correlation that we are in danger of interpreting as the price elasticity of demand. The first-stage estimates The first-stage estimates of the unit value equations are presented in Tables 5.2 and 5.3, and have already been discussed in Section 5.2 above. Tables 5.5 and 5.6 present a selection of coefficients from the corresponding within-village regres- sions for the budget shares. The last column of each table lists the sample aver- ages of the budget shares for each good. Wheat, including bread and flour, is the basic staple in most of Pakistan, and accounts for 12.8 percent of the budget of rural households. In rural Maharashtra, wheat accounts for only 3.7 percent of the budget, and the most important single category is jo war, which accounts for 12.3 percent of the total, and is followed in importance by rice, with 8.2 percent. All of these basic foodstuffs attract negative P° coefficients, and thus have total ex- penditure elasticities that are less than unity. The total expenditure elasticity of LOOKING AT PRICE AND TAX REFORM 577 Table 5.5. Within-village regressions for budget shares, rural Pakistan, 1984-85 Food In x In n ex w Wheat -0.068 0.070 0.37 0.128 (53.4) (47.0) Rice -0.006 0.007 0.68 0.027 (7.3) (8.4) Dairy products 0.024 -0.007 1.05 0.127 (12.0) (3.1) Meat 0.009 -0.006 1.10 0.037 (13.1) (7.6) Oils and fats -0.024 0.013 0.42 0.041 (39.9) (17.7) Sugar -0.004 0.002 0.86 0.029 (7.9) (3.9) Other foods -0.040 0.020 0.59 0.122 (38.3) (16.1) Note: Absolute values of/-values are shown in brackets. Source: Deaton and Grimard (1992). wheat in Pakistan is 0.37, that ofjowar in Maharashtra is 0.29. The elasticity of rice, which is the secondary cereal in both surveys, is 0.68 in Pakistan and 0.67 in Maharashtra. Meat and dairy products have greater than unity elasticities in both countries, and edible oil and sugar are necessities in both. These results suggest two similar populations, but with different cereals playing the primary and sec- ondary roles in each. The relationship between the coefficients on total expenditure and household size conform to the by now familiar pattern where the estimate for In n has the opposite sign to that for In jt, but is smaller in absolute magnitude. For the basic cereals, wheat in Pakistan and coarse cereals and rice in Maharashtra, the coeffi- cients on household size are close to (minus) the coefficients on total expenditure, so that the per capita consumption is approximately determined by PCE independ- ently of household size. Price responses: the second-stage estimates for Pakistan The top panel of Table 5.7 presents the first set of own- and cross-price elastici- ties for rural Pakistan together with bootstrapped estimates of "standard errors." The numbers are arrayed so that the elasticity in row i and column j is the re- sponse of consumption of good i to the price of good j. The bootstrapped "stan- dard errors" are calculated by making 1,000 draws from the cluster (second-stage) data, recalculating the estimates for each, and then finding (half) the length of the interval that is symmetric around the bootstrapped mean, and that contains 68.3 percent of the bootstrapped estimates. If the distribution of estimates were nor- mal, this interval would be one standard deviation on either side of the mean, 318 THE ANALYSIS OF HOUSEHOLD SURVEYS Table 5.6. Within-village regressions for budget shares, rural Maharashtra, 1983 Food In x In n e x w Rice -0.023 0.020 0.67 0.082 (11.5) (9.6) Wheat 0.006 0.003 1.12 0.037 (4.2) (2.1) Jowar -0.082 0.084 0.29 0.123 (33.2) (7.5) Other cereals -0.015 0.017 0.46 0,035 (9.3) (9.6) Pulses and gram -0.018 0.011 0,67 0.060 (20,8) (11.8) Dairy products 0.012 -0,006 1.11 0.053 (7.6) (3.4) Oils and fats -0.011 0.008 0.79 0.060 (12,7) (8.3) Meat, eggs, and fish 0.008 0.000 1.05 0.034 (6.3) (0.1) Vegetables -0.018 0.007 0.59 0.047 (26.5) (10.3) Fruit and nuts 0.002 -0.001 0.98 0.023 (3.0) (0.9) Sugar and gur -0.013 0.008 0.65 0.043 (20.2) (11.9) Note: Absolute values of /-values are shown in brackets. Source: Deaton, Parikh, and Subramanian (1994, Tables 2, 5, and 6). which is why I refer to the numbers as "standard errors" but, in general, there is no reason to suppose that the distributions have finite moments. In the top panel of the table, I have completed the system, but have not imposed symmetry. Prov- ince and quarter effects in demands are allowed for by regressing the corrected cluster averages of budget shares and unit values on quarterly and province dum- mies and then using the residuals in the second-sjtage calculations. The rationale for this will be discussed in the next subsection. The estimates are relatively well determined, at least in comparison to similar estimates from time-series data. Apart from sugar—on which more below—all of the diagonal terms—the own-price elasticities—are negative, and that for rice is less than -1. Only a minority of the cross-price effects is significantly different from zero; several of these involve the two goods rice and meat. According to these estimates, increases in the price of rice increase the demand for all other goods except other foods, and there are statistically significant effects for oils and fats and for sugar. Similarly, an increase in the price of meat has significantly positive effects on the demand for rice and for oils and fats. The estimated elasti- cities in the sugar and oils and fats columns are poorly determined; this is because there is little variance of sugar and oil prices in the sample. Although the estimated Table 5.7. Unconstrained and symmetry-constrained estimates of price elasticities, rural Pakistan, 1984-85 Food Wheat Rice Dairy Meat Oils &fats Sugar Other foods Nonfoods Unconstrained estimates Wheat -0.61 (0.10) 0.16 (0.07) 0.02 (0.04) -0.06 (0.07) 0.30 (0,31) -0.14 (0,27) -0.09 (0.10) -0.03 (0.39) Rice -0.06 (0.45) -2.16 (0.25) -0.25 (0.12) 0.66 (0.32) -1.20 (1.00) -0.59 (1.10) -0.14 (0.37) 2.95 (1.58) Dairy 0.13 (0.14) 0.09 (0.07) -0.89 (0.04) -0.13 (0.09) 0.10 (0.39) 0.72 (0.34) -0.24 (0.14) -0.84 (0.46) Meat -0.58 (0.22) 0.18 (0.14) -0.05 (0.08) -0.57 (0.18) -1.57 (0.70) -0.75 (0.60) 0.82 (0.20) 1.29 (0.81) Oils & fats 0.09 (0.05) 0.12 (0.03) 0.01 (0.02) 0.09 (0.04) -0.80 (0.18) -0.12 (0.14) -0.01 (0.05) 0.46 (0.23) Sugar 0.02 (0.17) 0.34 (0.09) 0.07 (0.05) 0.15 (0.10) -0.16 (0.51) 0.11 (0.53) 0.05 (0.15) -1.36 (0.75) Other foods 0.22 (0.09) -0.12 (0.05) 0.02 (0.03) 0.08 (0.06) 0.24 (0.41) 0.09 (0.25) -0.51 (0.10) -0.62 (0.40) Nonfoods 0.09 (0.03) 0.02 (0.01) 0.00 (0.01) 0.03 (0.02) -0.06 (0.07) 0.08 (0.06) 0.03 (0.02) -0.60 (0.09) Symmetry-constrained estimates Wheat -0.63 (0.10) 0.11 (0.05) 0.03 (0.04) -0.08 (0.05) 0.08 (0.05) -0.01 (0.05) 0.10 (0.07) -0.04 (0.15) LOOKING AT PRICE AND TAX REFORM Rice 0.49 (0.27) -2.04 (0.22) -0.14 (0.11) 0.47 (0.17) 0.44 (0.13) 0.36 (0.12) -0.58 (0.18) 0.20 (0.38) Dairy -0.05 (0.03) -0.04 (0.02) -0.90 (0.04) -0.03 (0.02) -0.02 (0.01) 0.02 (0.01) -0.05 (0.03) 0.00 (0.05) Meat -0.40 (0.20) 0.33 (0.13) -0.10 (0.08) -0.54 (0.18) 0.14 (0.10) 0.04 (0.09) 0.40 (0.16) -1.11 (0.32) Oils & fats 0.10 (0.06) 0.11 (0.03) 0.01 (0.02) 0.06 (0.03) -0.81 (0.18) -0.10 (0.10) 0.02 (0.07) 0.44 (0.23) Sugar -0.08 (0.21) 0.29 (0.10) 0.11 (0.06) 0.06 (0.11) -0.34 (0.34) 0.09 (0.53) 0.12 (0.18) -1.04 (0.72) Other food 0.07 (0.07) -0.11 (0.10) 0.01 (0.03) 0.13 (0.04) 0.01 (0.06) 0.03 (0.04) -0.50 (0.09) -0.24 (0.12) Nonfoods 0.05 (0.02) 0.00 (0.01) 0.00 (0.01) 0.04 (0.01) -0.03 (0.03) 0.04 (0.03) 0.07 (0.02) -0.58 (0.06) Note: The rows show the commodity being affected and the columns the commodity whose price is changing. The figures in brackets are obtained from 1,000 replications of the bootstrap using the cluster-level data and are defined as half the length of the interval around the bootstrap mean that contains 0.638 (the fraction of a normal random variable within two standard deviations of the mean) of the bootstrap replications. Source: Author's calculations using Household Income and Expenditure Survey, 1984-85. 319 320 THE ANALYSIS OF HOUSEHOLD SURVEYS own-price elasticity of sugar is positive, it has a large standard error, and is not significantly different from zero. Indeed, the standard errors in these two columns are two to three times larger than those elsewhere in the table. Given the lack of information in the survey on sugar and oil prices, the sym- metry-constrained estimates deserve attention. Although the theoretical restriction cannot be rigorously defended, it offers us the choice between some answers and no answers, and will guarantee that the estimates satisfy unique substitution-com- plementarity patterns, ruling out the possibility that good a is a substitute for good b when good b is a complement for good a. The restricted estimates for the full system are given in the bottom panel of the table. In those cases where the previous estimates were well determined, which are the goods whose unit values show substantial variation in the data, there is little change in the estimates, although the restrictions bring some increase in estimated precision. However, there are large reductions in the standard errors in the sugar and oil columns, because symmetry allows us to use the effect of (say) rice on sugar, something that is well determined because of the variability in rice prices, to "fill in" the estimate of the effect of sugar on rice. Of course, symmetry cannot help with the own-price elasticities, nor can adding up and homogeneity, which are being used to complete the system, not to impose cross-equation restrictions. As a result, the own-price elasticities for both sugar and oil still have large stan- dard errors, and the estimate for the former remains (insignificantly) positive. The symmetry restriction also helps pin down some of the other estimates. The substitutability between rice and wheat—an effect that is of considerable impor- tance for price reform in an economy where both are subsidized—is similar in the top and bottom panels, but is more precise in the latter. The pattern of elasticities shows that meat, oils, and sugar are also substitutes for rice. The final row and column relates to the "ghost" category of nonfood, where we have no price data, but where the elasticities are determined by the theoretical restrictions. Appro- priately enough, the standard errors in the last column are large. The configuration of these elasticities is very different from what would be the case if we had used a demand model based on additive preferences, such as the linear expenditure system. Such models not only restrict the cross-price elasticities to be small, as is mostly the case here, but also enforces an approximate propor- tionality that has dramatic effects for tax policy. The estimates in Tables 5.5 and 5.7 show no such proportionality. Table 5.5 estimates that meat and dairy products are luxuries and rice a necessity, and yet Table 5.7 estimates that rice is more price elastic than is either dairy products or meat. Taxing rice is therefore doubly unat- tractive compared with taxing either of the other two goods. I shall turn to these issues in more detail in the next and final section. Price estimates and taste variation: Maharashtra Table 5.8 provides a first look at the price elasticities for Maharashtra, but is pri- marily designed to explore another issue, which is the extent to which we should LOOKING AT PRICE AND TAX REFORM 321 Table 5.8. The effects of alternative specification on own-price elasticities, rural Maharashtra, 1983 Neither Food Quarters & quarters nor regions regions Quarters only Regions only Rice -1.19 -2.38 -2.31 -1.10 Wheat -1.29 -1.17 -1,25 -1.22 Jowar -0.52 -0.80 -0.81 -0.55 Other cereals -3.31 -3.39 -3.65 -3.17 Pulses -0.53 -0.76 -0.77 -0.53 Dairy products -0.15 -0.61 -0.53 -0.22 Edible oils -0.40 -0.48 -0.42 -0.49 Meat -1.08 -1.84 -1.85 -1.05 Vegetables -0.73 -0.74 -0.82 -0.71 Fruit -1.09 -1.04 -1.02 -1.12 Sugar and gur -0.20 -0.29 -0.33 -0.18 All cereals -0.28 -0.38 -0.38 -0.22 All foods -0.27 -0.32 -0.30 -0.26 Source: Deaton, Parikh, and Subramanian (1994 Table 7). make separate allowance for regional and temporal taste variation that is unre lated to price, and whether we wish to require that demand respond to seasonal price differences in the same way as it responds to spatial price differences. Rice may usually be 10 percent cheaper at the beginning of the year than in the sum- mer, and it may generally be 10 percent cheaper in one district than another. But seasonal and regional differences in tastes may result in differences in demands not being the same in the two situations, even once we have controlled for in- comes, demographics, and other observable covariates. Of course, if we imposed no structure on tastes, and allow demands to vary arbitrarily from village to vil- lage, it would be impossible to estimate any parameters. My preferred procedure is to allow quarterly and regional dummies, and the (unconstrained) own-price elasticities from this specification are reported in the first column of Table 5.8, together with aggregate elasticities for cereals and the sum of the foods, calcu- lated on the assumption that prices change proportionally for all components of the aggregate. Columns two through four show the effects on these own-price elasticities of excluding both sets of dummies, of including only quarterly dummies, and of including only regional dummies. The important issue turns out to be whether or not regional dummies are included. The estimates in the first and last columns, where regions are included, are close to one another, as are the estimates in the second and third columns, where regions are excluded. In most but not all cases, the estimates in the central columns are absolutely larger than those in the outer columns. One interpretation runs in terms of the long-run effects of prices. Some interregional price differences are of very long standing, and it is plausible on Table 5.9. Symmetry-constrained estimates of price elasticities, rural Maharashtra, 1983 Other Dairy Other Food Rice Wheat Jowar cereals Pulses products Edible oils Meat Vegetables Fruit Sugar goods Rice -1.05 0.28 0.37 -0.46 -0.71 -0.33 -0.19 -0.30 -0.02 -0.14 0.24 1.13 Wheat 0.63 -1.32 -0.18 0.42 0.40 0.28 -0.05 -0.09 -0.03 0.02 -0.27 -0.94 Jowar 0.27 -0.02 -0.45 0,54 0.04 0.12 0.08 0.22 0.06 0.03 -0.26 -0.94 Other cereals -1.15 0.47 2.04 -3.29 -0.12 0.16 0.21 -0.33 0.14 0.07 0.21 1.02 Pulses -0.95 0.25 0.04 -0.07 -0.57 0.04 0.10 -0.03 0.22 0.11 0.29 -0.04 Dairy products -0.57 0.19 0.18 0.08 0.02 -0.13 0.17 -0.01 -0.09 -0.05 0.21 -1.18 Edible oils -0.27 -0.01 0.12 0.11 0.09 0.16 -0.28 0.06 0.03 0.04 0.32 -1.11 322 Meat -0.81 -0.10 0.73 -0.37 -0.10 -0.01 0.09 -1.12 0.16 -0.02 0.15 0.18 Vegetables -0.03 -0.00 0.12 0.10 0.31 -0.08 0.05 0.13 -0.66 0.05 -0.15 -0.44 Fruit -0.57 0.04 0.11 0.09 0.29 -0.12 0.09 -0.03 0.09 -1.09 -0.02 0.04 Sugar and gur 0.48 -0.22 -0.82 0.16 0.44 0.28 0.47 0.13 -0.17 -0.00 -0.33 -1.12 Other goods -0.18 0.06 0.26 -0.04 0.03 0.10 0.12 -0.01 0.05 0.00 0.09 -0.85 Note: Estimates shown in boldface are greater in absolute value than twice their bootstrapped standard errors. For the description of bootstrapped standard errors, see the note to Table 5.7. Source: Deaton, Parikh, and Subramanian (1994 Table 8). LOOKING AT PRICE AND TAX REFORM 323 general theoretical grounds that long-run price elasticities are absolutely larger than short-term elasticities. In consequence, regional dummies may capture some of the long-run price effects, and estimated price elasticities will be lower when regional dummies are included. When we consider price reform proposals, we are probably not greatly interested in effects that take many years, perhaps even centu- ries, to be established, so that the relevant estimates are the generally lower figures that come from inclusion of the dummies. Table 5.9 gives the final set of results, obtained by completing the system and by imposing the symmetry restriction; the patterns in this table are not very differ- ent from the unconstrained estimates which are therefore not shown here. With 12 goods, the inclusion of standard errors makes it very hard to read the table, so I have highlighted those estimates that are more than twice their (bootstrapped) standard errors. As in the Pakistani case, the credibility and usefulness of symme- try is enhanced by the fact that the important and well-determined elasticities in the table, particularly the own-price elasticities, are changed very little by the imposition of the restriction. The results show a number of important patterns, particularly of substitutability between the various foods. Rice, wheat, andjowar are substitutes, as arejowar and other coarse cereals. Sensibly enough, pulses and dairy products are comple- ments with rice, but pulses are substitutes for vegetables, fruit and nuts, and sugar and gur. Sugar is a substitute for other cereals, edible oils, and dairy products, as well as for pulses, but is complementary withjowar. The substitutability between edible oils and sugar is important because the former is taxed—implicitly through protection of domestic processing—and the latter subsidized, so that we need to look at both together when thinking about reform. It is instructive to compare these estimates with those for Pakistan. In Pakistan, the staple food is wheat, rather thanjowar, and as incomes rise, the movement is towards rice, as opposed to towards rice and then wheat in Maharashtra. As was the case for the total expenditure elasticities, and with allowance for the different roles of the different foods, the elasticities are similar. The own-price elasticity for wheat in rural Pakistan is estimated to be -0.63, while that for the "superior" rice is -2.04. In Table 5.9Jowar has a price elasticity of-0.45, while the estimates for rice and wheat are -1.05 and -1.32, respectively. In both cases rice and wheat are substitutes, in Pakistan the elasticity of rice with respect to the wheat price is 0.49, compared with 0,63 for the estimated elasticity of wheat with respect to the rice price in Maharashtra. 5.5 Looking at price and tax reform I am now in a position to return to the main purpose of the chapter, the analysis of price and tax reform. Spatial price variation has been investigated, a model pro- posed, and parameters estimated. It is now time to use the results to investigate the consequences of various kinds of price reform in India and Pakistan. A good deal of the work in such exercises is concerned with the description of the en- vironment, with how actual taxes and subsidies work, and with the derivation of 324 THE ANALYSIS OF HOUSEHOLD SURVEYS shadow prices. Given the length of this chapter, and the fact that I have nothing new to add to these topics, I content myself with a brief description of the back- ground in each country, and with a description of how the results of Section 5.4 can be combined with the theoretical formulas in Section 5.1.1 start with the des- criptions of each country, then discuss how to adapt the formulas of Section 5.1 to accommodate the empirical results, and conclude with the policy evaluation. Shadow taxes and subsidies in Pakistan The implementation of the tax reform formula (5.9) or (5.12) requires information on the consumption of the various commodities, which we have from the survey, the price derivatives, which have been estimated in the previous section, social in- come weights, which are discussed below, and shadow prices. Ahmad, Coady, and Stern (1988) have estimated a complete set of shadow prices for Pakistan based on an 86 x 86 input-output table for 1976, and Ahmad and Stern (1990, 1991) use these shadow prices in their tax reform experiments. However, these prices embody a large number of assumptions, about shadow prices for land, labor, and assets as well as, most crucially, judgments about which goods are tradeable (further divided into importables and exportables) and which goods are nontraded. These assumptions all have a role in determining the shadow prices, and while the procedure itself is unimpeachable, the use of their prices for the current exercise makes it very difficult to isolate the role played in the reform proposals by each of the different elements in the story. Instead, I shall work here with an illustrative set of shadow prices for the eight goods in the demand analy- sis. These prices are illustrative only in the sense that they do not claim to take into account all the ingredients that should ideally be included in shadow prices. However, they are based on the actual situation in Pakistan, and they are simple and transparent enough so that we can see how features of that situation enter into the analysis of price reform. The crucial commodities are wheat, rice, and sugar. Rice and wheat sell at domestic prices that are substantially lower than their world prices at the official exchange rate. According to the Pakistan Economic Survey (Government of Paki- stan, annual), the border price for both rice and wheat was approximately 40 percent above the domestic price in the mid-80s. Both are tradeable, and for both I work with accounting ratios (shadow prices divided by consumer prices) of 1.4. In the case of sugar, there is a heavily protected domestic refining industry, and the border price of refined sugar is some 60 percent of the domestic price. Oils and fats are also protected by a system of taxes and subsidies with the result that the border price is perhaps 95 percent of the domestic price. For the other four goods in the system, I work with accounting ratios of unity. Although there are tariffs and taxes on many nonfood items, which would call for accounting ratios less than unity, many other items are nontraded, so that their shadow prices de- pend on the shadow price or conversion factor for labor, which, given the pricing of wheat and rice, is likely greater than unity. The experiments are therefore conducted under the supposition that the ratio of shadow prices to consumer LOOKING AT PRICE AND TAX REFORM 325 prices for the eight goods (wheat, rice, dairy, meat, oils and fats, sugar, other food, nonfood) are given by the vector (1.4, 1.4, 1.0, 1.0, 0.95, 0.6, 1.0, 1.0). It is important to be precise about exactly which policy experiments are being considered, and which are not. The instruments that are being considered here are instruments that increase or decrease the consumer price of the good, as would be the case, for example, if there were a value-added tax, or if the good had a fixed world price and were imported over a tariff. Such instruments may be available in Pakistan for sugar or for oils. However, to the extent that the rice and wheat sub- sidies are maintained by export taxes, the experiments described here do not correspond to what would happen if the export tax were changed. An increase in an export tax increases consumer welfare at the same time as it increases govern- ment revenue, so that if only these two effects were considered, as is the case in (5.9) and (5.12), it would be desirable to increase the tax indefinitely. Of course, export taxes are limited by the supply responses of farmers, as well as by interna- tional demand when the country has some monopoly power, as is the case for Pakistan with basmati. An export tax is an instrument that alters not only con- sumer prices but also producer prices, and it cannot be fully analyzed without looking at the supply responses. The calculations here, which look only at the demand side, correspond to a different hypothetical experiment, in which pro- ducer and consumer prices are separated. This would suppose that the govern- ment procures the total output of rice and wheat at one set of prices, that it sells the commodities either to foreigners at world prices, or to domestic consumers at a third set of prices. It is the consequences of changing these domestic prices that can be examined using the apparatus described above. Of course, these are artifi- cial experiments which are not feasible in practice. Farmers who grow wheat or rice cannot be charged one price for consumption and paid another for procure- ment. However, more realistic policies will have the same effects as those de- scribed here, plus additional effects that work through the supply side. What I have to say here is only a part of the story, but it is a part that has to be under- stood if well-informed policy decisions are to be made. Shadow taxes and subsidies in India The Indian reform exercise is rendered somewhat artificial by the fact that the survey data come from Maharashtra, rather than all India. In consequence, I focus only on the major distortions in Indian agricultural policy, and once again make no attempt to derive a full set of shadow prices; for an example of the latter for 1979/80 based on the data used in the Technical Note to the Sixth Indian Plan (see Ahmad, Coady, and Stern 1986). Indian domestic prices of rice and of wheat are held below their world prices by the Public Distribution System (PDS) which procures and stockpiles cereals, and which sells them to consumers through "fair- price" shops. While purchases in fair-price shops are rationed, so that above the ration, the marginal price is the free-market price—which is presumably higher than it would be in the absence of PDS—we simplify by treating the PDS as if it straightforwardly subsidized rice and wheat. The other important distortion that 526 THE ANALYSIS OF HOUSEHOLD SURVEYS we incorporate is the effective taxation of edible oils. The government protects the domestic groundnut processing industry, and in consequence the prices of edible oils are above the world prices. Gulati, Hansen, and Pursell (1990) calcu- late that, on average from 1980 to 1987, the domestic price of rice was 67 percent of its world price, that of wheat was 80 percent of the world price, and that of groundnuts was 150 percent of the world price. These translate into tax factors, the (shadow) tax share in the domestic price T./ (1 +t.) of-0.50, -0.25, and 0.33 for rice, wheat and groundnut oil, so that the accounting ratios for the 12 goods (rice, wheat, jowar, other cereals, pulses, dairy products, edible oils, meat, vegeta- bles, fruit, sugar, other goods) are (1.5, 1.25, 1.00, 1.00, 1.00, 1.00, 0.67, 1.00, 1.00, 1,00, 1.00, LOO). While these are stylized figures, and ignore other price distortions in food prices, like the Pakistani figures, they are based on the reality of the 1980s, and will serve to illustrate the general points of the analysis. *Adapting the price reform formulas Before using the formulas of Section 5.5,1 need to match them to the empirical model, and more substantively, I need to adapt the theory to a world in which prices are not the same for everyone and where both quality and quantity adapts to price. To do the latter requires an assumption about how tax changes affect the different prices that people pay, and the simplest such assumption is that changes in taxes affect all prices proportionately within a commodity group. This makes most sense when taxes are ad valorem, and may fail in some cases, for example if a good is taxed through an import tariff, if spatial price variation reflects transport costs, and if transport costs depend on volume rather than value. Suppose that the (shadow) tax rate on good i is T., so that the price paid, TC., is ft.(l + 'r), where ft. is taken to be fixed as the tax rate changes. When utility de- pends on quality as well as quantity as in equation (5.62), the derivative of the cost of living with respect to price changes is the product of quality and quantity. A change in the tax rate AT. induces a price change ft.AT., so that if h buys £«7»2i/i' *e compensation required is ft,£.^2^AT, = xHwjhfa.(l +T.)' 1 where, as before, xh is total expenditure and wih is the budget share of good L I adopt the standard Atkinson social welfare function (5.13), so that (5.103) -iHC- The numerator of the X. ratio in (5.9) is therefore replaced by (5.104) |j~ = o^r^w^G,*= d+^r^^/^r^w^ A value of e of unity implies that additional income is twice as valuable to some- one with half the income, with higher values implying a greater focus on the poor. Note that, by (5.103), social welfare accounting is done for individuals, not households, but under the assumptions that each person in the household receives the same, and that PCE is an adequate measure of the welfare of each. LOOKING AT PRICE AND TAX REFORM 327 Tax revenue collected from household h is (5.105) Rh = Etjlt^fi^ = £(T t /l+T t )*,w M . fc=l *=1 The derivative of revenue with respect to a change in the rate can be related to the empirical model by looking first at the derivative of the budget shares with re- spect to the tax rate, 3vv. dw,. din 71. 0,. (5.106) —- Substituting into (5.105) and averaging over all households, we have (5.107) a.^Wi-i.zJ^] where x is the mean of xh and vv. is the ''plutocratic" average budget share, H H (5.108) >v. = ExHwth/ExH. /i=i /i=i Finally, if we define the "socially representative budget share" \vf by (5.109) wf = [ X(xhtn,)-exhWih] /Zxh h=\ h=l the marginal cost-benefit ratio of a tax increase A. can be written Wj /Wf W; /VV. (5.110) ,*_L.(3s-,).E_SA 1 +T; \ W. I k*i 1 +T. W. W, The numerator of (5.110) is a pure distributional measure for good i; it can be interpreted as the relative shares of the market-representative individual (the representative agent) and the socially representative individual, whose income is lower the higher is the inequality aversion parameter e. This measure is modified by the action of the terms in the denominator. The first of these (apart from 1) is the (shadow) tax factor multiplied by the elasticity of expenditure on good i with respect to its price, quality and quantity effects taken together. This term mea- sures the own-price distortionary effect of the tax. If it is large and negative, as would be the case for a heavily taxed and price-elastic good, the term will con- tribute to a large A, .-ratio and would indicate the costliness of raising further reve- nue by that route. The last term is the sum of the tax factors multiplied by the cross-price elasticities, and captures the effects on other goods of the change in the tax on good i, again with quantity and quality effects included. From a theo- retical point of view, this decomposition is trivial, but when we look at the results, it is useful to separate the own- and cross-price effects, because the former are likely to be more reliably measured than the latter, and because the latter are often 328 THE ANALYSIS OF HOUSEHOLD SURVEYS omitted or handled by assumption in more restrictive systems. (In the calculations below, the term w* is arbitrarily scaled to sum to unity across goods. Changes in social welfare are not measured in money units, and for any given value of e, we are concerned only with the relative values of the A,'s across different goods.) Equity and efficiency in price reform in Pakistan Table 5.10 shows the efficiency effects of raising taxes on each of the goods, distinguishing between the terms in the denominator of (5.110). The first column shows the tax factors t./(l +T.) calculated from the accounting ratios discussed above; these are the shadow taxes calculated from comparison of the world and domestic prices. Wheat and rice carry shadow subsidies, oils and sugar, shadow taxes. The second column shows the own-price elasticities of quantity times qual- ity; these are the own-good contributions to the tax distortion. Because the quality elasticities are small, the combined elasticities are approximately equal to the own-price elasticities (see equations (5.90) and (5.91) above), but they are con- ceptually distinct. Taxes are ad valorem, so that quality shading in response to tax increases depresses revenue just as do decreases in quantity. The third column shows the own-price distortions and corresponds to the middle term in the deno- minator of (5.110). Goods that bear no shadow taxes do not appear in this column since there are no distortionary effects of small tax increases at zero tax rates. The large effects are on wheat and rice, with the latter much larger because its own- price elasticity is much larger. By themselves, these own-price terms will generate small cost-benefit ratios, particularly for rice. Subsidies are being paid, they are distortionary, particularly for rice, and it is desirable to reduce them, at least from this limited perspective. The cross terms in the next column are typically smaller, the exception being oils and fats, where the negative term acts so as to decrease the attractiveness of raising those prices. By checking back through the matrix of price elasticities in the lower half of Table 5.8, we can see that the cross-price elasticity responsible is that between the price of sugar and the demand for rice. An increase in the price of edible oil causes the demand for rice to rise—the elasticity is 0.44—which is distortionary because rice subsidized, an effect which is enhanced by a smaller but still positive cross-price elasticity with wheat, which is also subsidized. The final column in the table presents the sum of both effects, plus 1 according to (5.110). According to these—and bearing in mind that nothing has yet been said about distributional issues—rice is the commodity whose price should be raised, and oils and fats the commodity whose price should be lowered. The wheat sub- sidy too is distortionary, and there is an efficiency case for raising the price. Equity effects are incorporated in Table 5.11 for a range of values of the distributional parameter e. The first panel corresponds to e =0 where there are no distributional concerns, and the cost-benefit ratios are simply the reciprocals of the last column in Table 5.10. Rice is a good candidate for a price increase, and edible oils for a price decrease. As we move through the table to the right, the distributional effects modify this picture to some extent. The first column in each LOOKING AT PRICE AND TAX REFORM 329 Table 5.10. Efficiency aspects of price reform in Pakistan T ,- Own Cross °'-l 1+T. effect effects Total Food ** Wheat -0,40 -0.64 0.25 -0.05 1.20 Rice -0.40 -2.08 0.83 -0.05 1.78 Dairy products 0.00 -1.01 0.00 0.01 1.01 Meat 0.00 -0.57 0.00 0.01 1.01 Oils and fats 0.05 -2.33 -0.12 -0.38 0,51 Sugar 0.40 0.13 0.05 -0.14 0.91 Other foods 0.00 -0.53 0.00 0.02 1.02 Nonfoods 0.00 -1.11 0.00 -0.02 0.98 Note: The columns correspond to the elements in the denominator of (5.110). Column 3 is the product of columns 1 and 2; column 4 is the last term in the denominator; and column 5 is 1 plus the sum of columns 3 and 4, Source: Author's calculations using Household Income and Expenditure Survey, 1984-85. case, which shows the relative budget shares of an increasingly poor individual relative to the market-representative individual, moves away from luxuries and towards necessities as the distributional parameter increases. Wheat, the basic staple, attracts a larger weight as we move down the income distribution, as do oils and fats. Other food, which contains pulses, also has a weight that increases with e. Wheat is a target for price increases on efficiency grounds, although not as much as is rice, but the equity effects tend to make it less attractive, and its A- value becomes relatively large as we move to the right in the table. However, for oils and fats, which were the main candidates for price decreases, the equity ef- fects only strengthen the conclusions. Oils and fats figure relatively heavily in the budgets of low-income consumers, and reducing their price is desirable for both equity and efficiency reasons. Rice remains the best candidate for consumer tax increases; it is not consumed by the very poor, and its subsidy is the most distor- tionary. Table 5.11. Equity effects and cost-benefit ratios for price increases, Pakistan €=0 6=0.5 €=1.0 € = 2.0 e e £ e Food w /vv A, kv /u A, ^ /H A, ^ /n A, Wheat 1.00 0.83 1.09 0.90 1.16 0.97 1.28 1.06 Rice 1.00 0.56 1.04 0.58 1.06 0.59 1.06 0.59 Dairy prod. 1.00 0.99 1.01 1.00 1.01 1.00 0.97 0.96 Meat 1.00 0.99 0.98 0.97 0.95 0.94 0.88 0.87 Oils and fats 1,00 1.97 1.07 2.12 1.13 2.24 1.22 2.41 Sugar 1.00 1.10 1.01 1.10 1.00 1.10 0.97 1.06 Other foods 1.00 0.98 1.04 1.01 1.06 1.04 1.10 1.08 Nonfoods 1.00 1.02 0.96 0.99 0.94 0.96 0.92 0.94 Note: The two columns for each value of e are the numerator of (5.110) and (5.110) itself. Source: Author's calculations using Household Income and Expendiutre Survey, 1984-85. 330 THE ANALYSIS OF HOUSEHOLD SURVEYS In conclusion, I emphasize once again that these policy implications do not apply directly to the tax instruments currently in use in Pakistan. The findings suggest that raising revenue from an increase in the price of rice to consumers would be desirable on grounds of efficiency and equity. But the main instrument for controlling the price of rice is the export tax, a decrease in which would cer- tainly increase the consumer price of rice, but it would also decrease rather than increase government revenue, and it would change supplies of rice, effects which we are in no position to consider here. Nevertheless, the fact that rice and wheat are substitutes in consumption, and that there are further substitution effects be- tween rice, wheat, and edible oils, would have to be taken into account in a re- form of any export tax, just as they are taken into account here. Indeed, perhaps the most useful lesson of this analysis is how important it is to measure the way in which price changes affect the pattern of demand. The Pakistani substitution patterns between rice, wheat, and edible oils are not consistent with additive pre- ferences, and so cannot be accommodated within a model like the linear expendi- ture system. In countries where some prices are far from opportunity cost, cross- price effects of tax changes will often be as important as the effects on the good itself, and these effects must be measured in a flexible way. Nor could additive preferences accommodate the pattern of total expenditure and own-price elastici- ties that characterize demand patterns in Pakistan. Oils and fats are a necessity, but have a high own-price elasticity, so that the ratio of the own-price to income elasticity is many times greater than for a rice, which has high own-price and income elasticities. Additive preferences require that this ratio be (approximately) the same for all goods. Yet it is this ratio that is the principal determinant of how the balance between equity and efficiency ought to be struck. Equity and efficiency in price reform in India Table 5.12 shows the calculated efficiency effects for the Indian case, and corres- ponds to Table 5.10 for Pakistan. As before, the first column shows the factors t./(l +Tf.) calculated from the accounting ratios, while the second column shows the own-price elasticities of quality and quantity together. The product of the first and second columns, which is shown as the third column, gives the contribution of the own-price effects to the measure of the distortion that would be caused by a marginal increase in price. These are nonzero only for the goods that bear taxes or subsidies, rice, wheat, and oils. Wheat is more price elastic than rice, but its subsidy is less, and the own-price distortion effect is also less. However, as the next column shows, the distortion caused by the wheat subsidy is alleviated by the cross-price effects, largely because a lower wheat price draws demand^away from rice, which is even more heavily subsidized. There are also important cross-price effects for other cereals, pulses, dairy products, meat, and fruit. Increases in the price of any of these goods decreases the demand for rice, and helps reduce the costs of the rice subsidy. On efficiency grounds alone, the prices of rice, other cereals, pulses, dairy products, and meat should be increased, and those ofjowar and nonfoods decreased. LOOKING AT PRICE AND TAX REFORM 331 Table 5.12, Efficiency aspects of price reform in India T, 8| Own Cross Food 1+T, */ effect effects 1 "otal Rice -0.50 -1.13 0.57 -0.15 .42 Wheat -0.25 -1.33 0.33 -0.32 .01 Jowar 0.00 -0.39 0.00 -0.12 ().88 Other cereals 0.00 -3.51 0.00 0.58 .58 Pulses 0.00 -0.60 0.00 0.55 ,55 Dairy products 0.00 -0.21 0.00 0.26 .26 Edible oils 0.33 -0.27 -0.09 0.16 .07 Meat 0.00 -1.13 0.00 0.43 .43 Vegetables 0.00 -0.66 0.00 0.04 .04 Fruit 0.00 -1.08 0.00 0.27 .27 Sugar and gur 0.00 -0.28 0.00 -0.03 (197 All other 0.00 -1.43 0.00 -0.20 (180 Note: The columns correspond to the elements of the denominator of (5.110). Column 3 is the product of columns 1 and 2; column 4 is the last term in the denominator; and column 5 is 1 plus the sum of columns 3 and 4. Source: Deaton, Parikh, and Subramanian, (1994 Table 9). Table 5.13 brings in the equity effects, and computes the cost-benefit ratios that trade off both equity and efficiency. In the first pair of columns, the Atkinson inequality aversion parameter is zero, so that all individuals are treated alike, no matter how much their household spends. In this case, the A-ratios are simply the reciprocals of the last column in Table 5.12, and we get the same ranking of rela- tive tax costs. As we move to the right, and the e parameter increases, the equity Table 5.13. Equity effects and cost-benefit ratios for price increases, India € =0 e = 0.5 € = 1.0 € = 2.0 Food vv€/w A H i> e /w A * € /H A V£/M A Rice 1 .00 0.71 11.02 0.72 1.03 0.73 1.02 0.72 Wheat 1 .00 0.99 ().98 0.97 0.96 0.95 0.91 0.90 Jowar .00 1.14 .11 1.27 1.21 1.38 1.39 1.58 Other cereals .00 0.63 .09 0.69 1.16 0.73 1.24 0.78 Pulses .00 0.65 .05 0.68 1.09 0.70 1.15 0.75 Dairy prod. .00 0.79 ().97 0.77 0.94 0.74 0.87 0.69 Edible oils .00 0.94 .02 0.96 1.04 0.98 1.07 1.01 Meat .00 0.70 .00 0.70 0.98 0.69 0.94 0.66 Vegetables .00 0.96 .04 1.00 1.06 1.02 1.09 1.05 Fruit .00 0.79 ().96 0.76 0.93 0.73 0.84 0.66 Sugar & gur .00 1.03 11.03 1.07 1.07 1.10 1.12 1.16 All other .00 1.25 ().96 1.20 0.92 1.15 0.88 1.10 Note: The two columns for each value of e are the numerator of (5.110) and (5.110) itself. Source: Deaton, Parikh, and Subramanian (1994 Table 10). 332 THE ANALYSIS OF HOUSEHOLD SURVEYS column gives larger values to the goods most heavily consumed by the poor, and relatively smaller values to those that are most heavily consumed by those who live in households that are better-off. For e = 0.5, jowar receives the highest weight, with other cereals, pulses, vegetables, sugar, rice, and edible oils all showing equity effects greater than unity. These are the goods most heavily con- sumed by the poor. As e increases further, the importance of these goods for distribution increases further. Of the two subsidized cereals, the equity case is stronger for rice than wheat—which is indeed why rice carries the larger subsi- dy—and the difference between them increases with the degree of inequality aversion. Neither cereal is consumed as heavily by the poor as are the coarse cereals, although there are limitations on the extent to which the government could intervene in that market. The cost-benefit ratios A bring together the equity and efficiency effects. Be- cause of the subsidy, and because it is less heavily consumed by the poor than coarse cereals, rice is always a candidate for a price increase. But when concern for the poor is relatively low, increases in the prices of pulses and of other cereals are less socially costly, because of the large cross-price elasticities with rice. Given any positive degree of equity preference, the most attractive candidate for a price decrease is jowar. When equity preference is large, the efficiency case for raising prices of pulses and other cereals is outweighed by equity criteria, and meat and fruit become the best candidates for tax increases. 5.6 Price reform: parametric and nonparametric analysis This has been a long chapter, and although the topic has been the same through- out, we have followed it from the theory through the data problems to the econo- metric implementation and policy evaluation. Although the methodology is inevi- tably complex in parts, it has been successfully applied in a number of countries, there is standardized software which is included in the Code Appendix, and the real difficulties lie, as always, not in the econometric sophistication, but in setting up the problem in a sensible way, picking the right goods for the analysis, and in the crucial preliminary analysis of the data, looking for outliers, interpreting and examining the unit value data, and making whatever adaptations are necessary for local conditions or institutions. It is rarely possible to do satisfactory policy analy- sis with "off-the-shelf econometrics. Even so, the procedures of this chapter should be helpful as a basis for analysis of tax reform and pricing issues. In this brief final section, I want to draw attention to areas where the methodology is weakest, and where there is likely to be the greatest payoff to further research. A major contrast between the analysis of this chapter and that in Chapters 3 and 4 is that the absence of nonparametric methods. Instead, I have used a para- metric model of demand to deliver the price responses, and this is in sharp con- trast to exercises like that in Chapter 3, where I looked at the distributional effects of rice pricing in Thailand without having to resort to a specific functional form. The reason for the difference is the difficulty in estimating price responses; even with thousands of observations, there is often limited price variation, so that there LOOKING AT PRICE AND TAX REFORM 333 is little hope of direct measurement without imposing prior structure. Although the model in this chapter also used a parametric form to specify the effects of total expenditure and sociodemographics, this is only necessary to allow isolation of the price responses, and no use of the results is made in the policy exercises in Section 5.5; the survey data themselves are used directly to calculate the average and socially representative budget shares in the cost-benefit formulas. An important topic for research is whether it is possible to do better, and bring a nonparametric element to the estimation of the price responses. The hope comes from the fact that the cost-benefit formula (5.9) does not require an estimate of the price derivative at every point in the sample, or even for different values of total expenditure or the socio-demographics. Instead, all that appears in (5.9) is the aggregate of each household's response, or equivalently, since we can divide top and bottom of (5.9) by the number of households, the average price response for all households. Among recent developments in econometrics have been meth- ods for estimating such average derivatives without requiring the specification of a functional form (see Hardle and Stoker 1989; Stoker 1991; and Powell, Stock, and Stoker 1989, for the basic techniques, and Deaton 1995, for a discussion in the context of pricing policy in developing countries). These "semiparametric" methods deliver only averages, not estimates for each household, but in large enough samples they do so almost as efficiently as does parametric regression. A preliminary investigation of average derivative techniques using the Paki- stani data is provided by Deaton and Ng (1996). The method has both advantages and disadvantages. Among the former is the removal of any need to write down a theoretically consistent model of demand. Even more importantly, the absence of a functional form provides a clean solution to the problems associated with some households buying the good, and some not. The average derivative methods esti- mate the derivative of the regression function, the expectation of demand condi- tional on prices, total expenditure, and sociodemographics, but unconditional on whether or not the household buys the good. This is exactly what the cost-benefit formulas require, and so the method avoids the conventional detour through the complexity of utility theory and switching regression models. There are equally serious disadvantages. First, there is no apparent way of allowing for either quality effects or measurement error. The former is perhaps not too serious, but there are many examples where the neglect of the latter could produce serious biases. Second, although we can do without a functional form, it remains necessary to specify which are the conditioning variables, and the omis- sion of relevant variables will lead to inconsistent estimates of price derivatives when the omitted variables are correlated with prices, in exactly the same way that OLS is biased by omitted variables. Third, although the average derivative method finesses the problem of zero observations in the unit values, it provides no assistance with the fact that unit value observations are missing for such households. Because the missing households are unlikely to be randomly select- ed—more households will buy nothing where prices are high than where they are low—there is a risk of selectivity bias, and that risk is present whether we use parametric or semiparametric methods. In the methods of this chapter, the effects 334 THE ANALYSIS OF HOUSEHOLD SURVEYS of selectivity are moderated by the averaging over clusters, so that a village is included as long as one of its households purchases the good. Such averaging is much less attractive once the parametric framework is abandoned, since although the average derivative estimation is asymptotically as efficient as ordinary least squares, it requires a good many more observations in practice. The final disadvantage, at least for the present, is the time required to compute the estimates. The formulas are not complex, and the estimators can be written in closed form, so that there is no need for iterative solution, but with realistic sam- ple sizes, the calculations are slow even with very fast machines. Although this problem will inevitably diminish over time, the methods are difficult to recom- mend now, partly because many researchers in developing countries have access to fast machines with only a considerable lag, and because the computational costs have limited the experience with the methods in real applications, so that we cannot be sure that there are not hidden difficulties that will only become appar- ent through use. 5.7 Guide to further reading The World Bank volume edited by Newbery and Stern (1987) contains the theo- retical background for this chapter, as well as a number of applied essays that discuss applications to specific countries. The introductory essays by Stern, Chap- ters 2 and 3, as well as Chapters 7 and 13 on taxation and development, and on agricultural taxation by Newbery are worth special attention. Newbery*s Chapter 18 on agricultural price reform in the Republic of Korea is a model for how such studies should be done. The book by Ahmad and Stern (1991) also reviews the theory and, like this chapter, applies it to tax reform in Pakistan. Sah and Stiglitz (1992) develop the theory of pricing in the context of a conflict between agri- cultural producers and urban consumers. The theory of price reform is closely re- lated to the theory of cost-benefit analysis, the former dealing with the effect on social welfare of small changes in prices, the latter with the effects of small changes in quantities—projects. Survey papers that emphasize this link are Dreze and Stern (1987, 1990) as well as the splendid review by Squire (1989). Quite different insights into pricing policy are obtained from the perspective of political economy; this literature is more concerned with why prices and taxes come to be what they are, and less with prescriptions based on equity and efficiency (see the World Bank volumes by Krueger, Schiff, and Valdes 1991-92, the early political analysis by Bates 1981, Grossman and Helpman 1994, and the immensely read- able monograph by Dixit 1996). 6 Saving and consumption smoothing There are many reasons to be interested in the saving behavior of households in developing countries. One is to understand the link between saving and growth, between saving and economic development itself. Although economists are still far from a generally accepted understanding of the process of development, sav- ing plays an important role in all the theoretical accounts and the data show a strong positive correlation between saving and growth, both over time in individ- ual countries, and in international comparisons across countries, If we believe that saving generates growth, and since there is a close link between household and national saving rates, both over time and over countries—see Figure 6.1 for evi- dence from a number of developing countries—then it is to the determinants of household saving that we must look if we are to understand economic growth. If instead we suppose that saving rates are a by-product of economic growth gener- ated in some other way—for example by government-directed investment pro- grams, or by the dynamic gains from trade—our understanding is not complete unless we can explain why it is that economic growth causes households to save a larger fraction of their incomes. The second reason for examining household saving is to understand how people deal with fluctuations in their incomes. The majority of people in poor countries are engaged in agriculture, and their livelihoods are often subject to great uncertainty, from weather and natural calamities, from sickness, and from fluctuations in the prices of their crops. Individuals close to subsistence need to free consumption from income, so that they are not driven to extremities simply because their incomes are temporarily low. These "smoothing" or "insurance" mechanisms can take a number of forms, of which saving is one. By laying aside some income in good times, people can accumulate balances for use in bad times, a procedure that would be available even to Robinson Crusoe, provided that he had some commodity that could be stored. In a society with many people, risks can be pooled, either through formal financial intermediaries or through more informal networks of personal credit or social insurance, at the local or national levels. One useful way of classifying saving behavior is by the length of the time period over which households can detach consumption from income. We would References in this publication to "Taiwan," "Republic of China," and "Taiwan (China)" refer to the region, "Taiwan, China." References to "Hong Kong" refer to the region,"Hong Kong SAR, China" 555 336 THE ANALYSIS OF HOUSEHOLD SURVEYS Figure 6.1. National and household saving, selected countries (percent) Botswana Colombia Jamaica 40- 30- 20- 10- 0- Korea, Republic of Peru Philippines 40 30 20- 10- o- T I I I i I I I 1 I T Sri Lanka Thailand 1980 1985 1990 40- 30- 20- National saving 10- ,---•""' 0- Household saving 1980 1985 1990 1980 1985 1990 Source: UN Statistical Yearbook, various years. normally expect even the poorest households to be able to consume on days on which they received no income, but it is more of an issue whether or not con- sumption is at risk from seasonal fluctuations in income, and there is a substantial literature in development economics on the risks associated with seasonal short- ages and fluctuations in prices. Another literature, often couched in terms of the permanent income hypothesis, looks at whether farmers set something aside in good years and so detach their annual consumption levels from their annual in- comes. Beyond this, people may smooth their consumption over their lifetimes, saving when they are young to make provision for their old age. Whether and how well individuals undertake these various kinds of saving has implications not only for welfare, but also for the behavior of the macroeconomy and for the rela- tionship between saving and growth. In particular, the life-cycle hypothesis, that households successfully undertake lifetime consumption smoothing, has the fa- mous implication that changes in the rate of economic growth will cause changes in saving rates in the same direction (Modigliani 1966, 1970, 1986). Paralleling the taxonomy of consumption smoothing over different time peri- ods is a taxonomy of consumption smoothing over different groups of people. At one extreme, every individual is an island, and must deal alone with income fluctuations and other threats to consumption by saving and dissaving. At the other extreme, we can imagine social insurance schemes, spanning large groups of agents, in which individual risk is diversified away by being spread over every- SAVING AND CONSUMPTION SMOOTHING 337 one, leaving consumption subject only to economy-wide risk. Intermediate cases are of the greatest practical interest. We usually suppose that consumption risk is pooled within families, and the anthropological literature gives many examples of risk pooling in agricultural villages. Social security exists in one form or another, but rarely provides more than partial protection. My concern in this book is with the analysis of household survey data, and the main task of this chapter is to present evidence from surveys that helps us under- stand the determinants of household saving. This is a difficult and speculative undertaking, and one that is a good deal more provisional than anything else in the book. First, for the reasons discussed in Chapter 1, the measurement of saving in household surveys is subject to large margins of error, so that especially where household saving rates are low, it may be almost impossible to obtain any useful measure of household saving. Second, there are several different theoretical ex- planations of household saving, and there is no general agreement on which (if any) is correct. We are not yet at the stage where the policy prescriptions are clear, and where the only remaining task for empirical analysis is to provide the magnitudes necessary to set the controls. Instead, new empirical evidence and theoretical insights are still coming in, and we remain at the stage where there coexist different models and different interpretations with quite different policy implications In the main sections of this chapter, I present some of the most important models of household consumption, and discuss methods and results for using them to interpret the survey data. As always, my aim is not to provide a compre- hensive treatment of the topic, nor to survey previous research in the area, but to provide template examples of empirical analyses that cast light on the topic in question. I consider in turn the life-cycle hypothesis (Section 6.1) and the shorter- term smoothing over years and seasons that is predicted by various versions of the permanent income hypothesis (Section 6.2). Section 6.3 develops the theory of intertemporal choice and uses the results to consider extensions beyond the per- manent income and life-cycle models, extensions where households have a pre- cautionary motive for saving, or where households have no access to credit mar- kets, but use assets as a buffer to help smooth their consumption. Section 6.4 re- views recent work on insurance mechanisms, on whether households are able to stabilize their consumption by pooling risks across individuals instead of by sav- ing and dissaving over time. Section 6.5 is about the relationship between saving, age, and inequality. Section 6.6 tries to bring all of the material together and to summarize its implications for a number of policy questions. In particular, I draw some tentative conclusions about the relationship between saving and growth. 6.1 Life-cycle interpretations of saving The life-cycle model, originally proposed by Modigliani and Brumberg (1954, 1979) sees individuals (or couples) smoothing their consumption over their life- times. In the simplest "stripped-down" model, people receive labor income (earn- ings) only until the retirement age, but hold their consumption constant over their 338 THE ANALYSIS OF HOUSEHOLD SURVEYS lives. As a result, they are net savers during their working years, and dissavers during retirement. The assets that are sold by the elderly to finance their con- sumption are accumulated by the young, and provided there is neither population nor income growth, the provision for old age will support a constant ratio of wealth to earnings in the economy as a whole, and there will be no net saving. With either population growth or per capita earnings growth, the total amount of saving by the young will be magnified relative to the total amount of dissaving by the elderly, who are fewer in number and have less lifetime wealth, so that in- creases in either type of growth will increase the saving rate. Simplified versions of the model predict that the saving ratio should increase by about 2 percentage points for each percentage point increase in the growth rate, a prediction that is consistent both with the cross-country relationship between saving and growth as well as with the declines in saving rates that have accompanied the productivity slowdown in the industrialized countries (Modigliani 1993). The positive causality from growth to saving is qualitatively robust to changes in assumptions about the shapes of lifetime profiles of consumption, earnings, and saving, provided only that the average age at which saving takes place is lower than the average age of dissaving. This may not be the case if the rate of econo- mic growth is expected to be very high, because the young may wish to dissave in anticipation of higher earnings in the future. Even so, it is plausible that these groups will be unable to borrow the resources needed to finance such spending, so that a presumption remains that growth will increase saving rates for the econ- omy as a whole. A fuller account of this summary can be found in many sources, for example in Modigliani's (1986) Nobel address; a review by the author is Deaton (1992c, Chapter 2). The validity of the life-cycle description of saving has been the subject of a great deal of debate and research, mostly in the context of developed economies. Although there is still room for differences of opinion, my own belief, argued more fully in Deaton (1992c), is that the life-cycle model overstates the degree to which consumption is in fact detached from income over the life cycle, and that aside from institutionalized employer or national pension schemes, relatively few households undertake the long-term saving and dissaving that is predicted by the model. There are also questions about applying the model to developing coun- tries. In the poorest economies, where family size is large, and life expectancy relatively low, the fraction of old people is small and very few of them live alone. In the LSMS for Cote d'lvoire in 1986, less than 1 (10) percent of females (males) aged 70 or over live alone or with their spouse. Even in Thailand, where the de- mographic transition is much further advanced, the corresponding figures for men and women in rural areas in the 1986 Socioeconomic Survey were 15 and 25 percent (see Deaton and Paxson 1992, Table 6.4). When the elderly live with their children, they can be provided for directly and personally; there is no need for the accumulation and decumulation of marketable assets that is done in developed economies through anonymous and impersonal financial markets. Nevertheless, there is a wide range of income levels, demographic structures and customary living arrangements among developing countries. For those coun- SAVING AND CONSUMPTION SMOOTHING 339 tries where economic development and the demographic transition have pro- ceeded the furthest, living arrangements are already changing, and provision for consumption by the elderly is moving into the forefront of public policy. These changes are at their most rapid in the fast-growing economies of East and South East Asia, countries which also have national and household saving rates that are very high by historical and international standards. The mechanisms that produce this conjunction are far from clear. Is the high saving a consequence of the rapid growth, or its cause? Are the high saving rates generated by demographic change and the fact that many current savers will have to fend for themselves in their old age? Does the political pressure for social security spring from the same concerns, and will its introduction cut saving rates and threaten economic growth? Indeed, will the aging of the population that is already underway mean lower saving rates as the fraction of dissavers increases? One key to many of these questions is whether or not the life-cycle hypothesis provides an adequate description of the behavior of saving. In this section, I look at life-cycle saving behavior in the data from Cote d'lvoire, Thailand, and Taiwan (China), starting with simple age profiles of con- sumption, and moving on to a more sophisticated analysis based on the cohort methods of Section 2.6 to interpret the data in terms of the life-cycle model. Age profiles of consumption Perhaps the most obvious way to examine the life-cycle behavior of consumption from survey data is to plot consumption and saving against age. But as soon as we try to do so, we encounter the difficulty of measuring income and thus saving, as well as the fact that much income data and almost all consumption data relate to households and not individuals. It is far from clear that the large households that exist in many rural areas around the world, and that often contain subunits of several families, can be adequately described in terms of the simple life cycle of the nuclear family. Even so, we have little choice but to classify households by the age of the head, to measure income as best we can, and to see whether the results make sense. Figure 6.2, taken from Deaton (1992c, p. 55), shows age profiles of consump- tion and income for 1,600 or so households in the Cote d'lvoire Living Standards Surveys for 1985 and 1986, as well as profiles for 3,589 urban and 5,012 rural households from the 1986 Thai Socioeconomic Survey, In order to keep the pro- files relatively smooth, I have used the averages for each age to calculate the five- year moving averages shown in the figures; five-year smoothing also eliminates problems of "age-heaping" when some people report their age rounded to the nearest five years. The graphs do not look much like the stylized life-cycle profiles of the text- book; there is little evidence of either "hump saving" among the young or of dis- saving among the elderly. Instead, and with the exception of urban Thailand, there is very little saving (or dissaving) at any age, with the consumption profile very close to the income profile. Even in urban Thailand, where there is positive 340 THE ANALYSIS OF HOUSEHOLD SURVEYS Figure 6.2. Age profiles of income and consumption, Cote d'lvoire, 1985-86, and Thailand, 1986 Consumption 2,850 2,850-t Income 2,350 CA I 1,350- SS u 850- Ivorian households, 1985 Ivorian households, 1986 350- 10,550 4,000- Income Income 3,500- 8,550- 3,000- ^ 6,550- 2,500- Consumption Consumption 4,550- 2,000- Thai urban households, 1986 Thai rural households, 1986 2,550- 1,500- 20 30 40 50 60 70 20 30 40 50 60 70 Age of household head Source: Author's calculations using CILSS 1985 and 1986, and Socioeconomic Survey of Thailand, 1986. saving, the consumption profile is similar to the income profile, and most saving takes place after age 40. There is also no sign of the predicted dissaving among older households. This close co-ordination between consumption and income has been encountered in several other cross-sectional data sets. For example, Carroll and Summers (1991) find a close correlation in the United States between con- sumption and income profiles for different educational and occupational groups, and conclude that consumption "tracks" income more closely than would be ex- pected from life-cycle models whose essence is that consumption and income need bear no relationship to one another. Similarly, Attanasio and Davis (1993) use cohort data from the United States to find a close relationship between five- year consumption and income changes, although the relationship is much weaker between one-year changes. More comprehensively, Paxson (1996) uses cohort data from the United States, Britain, Thailand, and Taiwan (China) to document consumption and income tracking. While it is possible that the age profiles of consumption are simply those that people like, and just happen to match the shape of income profiles, it stretches belief that the coincidence should happen for every educational and occupational group as well as for a wide range of countries, the preferences of each just happening to look like its income path, even though the income profiles differ from case to case. Even so, the evidence in Figure 6.2 has to be treated with a great deal of cau- tion and is subject to several interpretations. In spite of positive saving among SAVING AND CONSUMPTION SMOOTHING 341 urban Thai households, the household saving rate from the 1986 survey is much lower than that reported in the National Income and Product Accounts (NIPA). Although NIPA measures of saving should never be automatically credited over measures from household surveys—NIPA measures of household saving are fre- quently derived as residuals—the discrepancies serve to remind us of the general difficulty of measuring saving, whether by surveys or other means. In this partic- ular case, Paxson (1992) has pointed out that in the presence of inflation, the practice of comparing money income reported for the last year with twelve times the money consumption reported for the last month will depress estimates of saving. However, her corrections for 1986 are not large, and she reports negative total savings on average for all nonurban households; among the latter, total sav- ing is less than ten percent of their total income. These figures are not consistent with the national accounts data shown in Figure 6.1. Even in the absence of measurement error, there are a number of difficulties with age profiles of consumption such as those in Figure 6.2. First, these profiles are simply the cross sections for the households in the surveys, and there is no reason to suppose that the profiles represent the typical or expected experience for any individual household or group of households. We are not looking across ages for the same household or same cohort of households, but at the experience at dif- ferent ages of different groups of households, whose members were born at dif- ferent dates and have had quite different lifetime experiences of education, wealth, and earnings. Without controlling for these other variables, many of which are likely to affect the level and shape of the age profiles, we cannot isolate the pure effect of age on consumption and saving. In the examples in Figure 6.2, these problems are likely to be most serious for Thailand, where there has been more than 3 percent annual growth in per capita income over the past quarter- century. As a result, the 20-year-olds at the left of the graphs are four to five times better-off over their lifetimes than are the 70-year-olds on the right. Life- time resources are diminishing as we move from left to right in the diagrams, and since, according to the life-cycle hypothesis it is lifetime resources that determine the level of the consumption path, the cross-sectional consumption trajectories will be raised among the young and pulled down among the older households, so that the cross-sectional profiles will be rotated clockwise relative to the genuine lifetime profiles. The Thai panels are therefore consistent with an interpretation in which consumption grows steadily with age, and in which the hump shape comes from the lower lifetime wealth of the elderly. The second problem with these simple age profiles of consumption is that they take no account of variation in household size with the age of the head. Without a good procedure for counting adult equivalents, there is no ideal correction, but even a crude conversion to a per capita basis does a great deal to flatten out the profiles in Figure 6.2. The age profile of household size has much the same shape as the age profile of household income. If we were to discover the lifetime pro- files of individual consumption were strongly correlated with age profiles of indi- vidual income, there would be cause for concern about the individual's apparent lack of ability to protect consumption from swings in income, and in particular to 342 THE ANALYSIS OF HOUSEHOLD SURVEYS make adequate provision for old age when earnings capacity is low. However, it is less clear that we should be concerned when household consumption tracks household income, and when household numbers and composition are changing as children are born, grow up, and leave home, and as elderly parents and rela- tions move in. Indeed, since decisions about household division and augmentation are not made in a vacuum, household size may itself respond to the earnings opportunities available to the household, not only through fertility, but also by migration and the splitting and joining of family subgroups. But even if there were no consequences for individual welfare of household consumption tracking household income, the lack of "hump" saving and subsequent dissaving removes the supposition of a link between growth and saving in the aggregate economy. A third difficulty with the age profiles is selection as households age, a diffi- culty that is particularly serious when looking at saving or dissaving among the old. For example, the bottom left panel of Figure 6.2 shows positive saving at all ages, including households headed by people to age 70. But many households cease to exist as independent units even before the head reaches age 70, because of death, or because old people tend to live with their children. Although there is little evidence on the subject, it is easy to imagine that high-saving households are more likely to survive as independent units, in which case the profiles will show saving rates that are upward biased to a degree that increases with age. (See Shor- rocks 1975 for a similar analysis of the relationship between age and wealth hold- ings, where a correction for differential mortality by wealth converts upward- sloping age profiles of wealth into downward-sloping profiles, at least among the elderly.) Consumption and saving by cohorts Where there exists a time series of household surveys with information on con- sumption and income, it is possible to track the behavior of cohorts over time, and thus to avoid one of the major difficulties associated with a single cross section of consumption and income. Such data also have a direct interpretation in terms of the life-cycle hypothesis, with the age effects, which come from tastes, separated from the wealth effects, which come from the lifetime budget constraint, and which differ between cohorts by an amount that depends on how fast the econ- omy is growing. In this subsection, I show how to fit the life-cycle model to co- hort data using the results of the study of saving in Taiwan (China) by Deaton and Paxson (1994a); similar analyses for Britain, Thailand, and the United States are reported by Paxson (1996). The Taiwanese surveys, which have already been used in Chapters 1 and 2, begin in 1976, so that with data up to 1990, there are 15 successive surveys that can be used to track cohorts over time. The surveys collect data on the various components of both consumption and income over the past year, and although the annual savings estimates are far from identical to—and are usually lower than— those in the national accounts, both sources show that households save between a fifth and a quarter of their incomes. By the standards of most countries, this is an SAVING AND CONSUMPTION SMOOTHING 343 unusually close correspondence, perhaps because saving rates are so high in Tai- wan (China). Figures 6.3 and 6.4 present the consumption data in two different ways. Figure 6.3 shows the cross-sectional age profiles of (constant-price) consumption for every second year from 1976 through 1990. Each graph plots against head's age the average consumption (in 1986 prices) of all households with heads of that age. The growth of real consumption is raising these profiles through time, al- though the second oil shock had the effect of slowing growth, and the 1980 and 1982 consumption profiles are not far apart. As was the case for Thailand in Figure 6.2, the age profiles rise somewhat from age 20 through 50, and decline thereafter. Note the sampling effects in these graphs. The averages are for a single year of age in each year, and are not smoothed by combining adjacent years into moving averages. As a result, the profiles for the youngest and oldest heads are relatively imprecisely estimated. The rapid growth of consumption in Taiwan (China) means that these profiles bear no relation to the experience of any given cohort. Someone who is 25 years old in 1976 is 27 years old, not in 1976 but in 1978, so that to trace the average consumption of such people, the points should be connected, not within years as in Figure 6.3, but within cohorts, linking up the behavior, if not of the same individuals, at least of the randomly selected representatives of the same cohort over time. Figure 6.4 shows the cohort consumption experience for every fifth cohort, beginning with those born in 1966, and then moving back (increasing in age) five years at a time until we reach the cohort of those born in 1911. The first line seg- ment connects the average consumption of those who were 20 years old in 1976 to the average consumption of 21-year-olds in 1977, of 22-year-olds in 1978, until the last observation of the cohort in 1990, when they were 34 years old. The second line segment repeats the exercise for those who were five years older, and so on. Figures 6.3 and 6.4 use exactly the same data on average consumption by age of head, and if Figure 6.3 showed every year, and Figure 6.4 showed every cohort, the two figures would contain exactly the same points but connected differently. Figure 6.4, which shows a marked resemblance to the cohort earnings profiles in the first panel of Figure 2.5—a similarity to which I shall return—shows just how misleading are the cross-section age profiles when what we are interested in is the life-cycle profiles. Far from consumption first rising gently then declining with age, as in Figure 6.3, consumption rose with age for all cohorts, except pos- sibly the very oldest. For the youngest cohorts, consumption rose very rapidly with age, and the rate of growth was slower the older the cohort, so that it was young households who gained the most from consumption growth in this period. (Of course, old people may have gained too, since many presumably relocated into younger households.) The decline in consumption after age 45 in the cross- section age profiles in Figure 6.3 is entirely due to the fact that, at any given date, older people have lower consumption, and not to any tendency for consumption to decline with age for anyone. (This effect presumably also applies to the Thai figures in the lower half of Figure 6.2.) According to the life-cycle model, the 344 THE ANALYSIS OF HOUSEHOLD SURVEYS Figure 6.3. Cross-sectional consumption profiles, alternate years, Taiwan (China), 1976-90 400- 1990 1984 * 300- 200- 1978 1980 1982 1976 100- 20 30 40 50 60 70 Age of household head Source: Author's calculations from Surveys of Personal Income Distribution, Taiwan (China). Figure 6.4. Consumption profiles, selected cohorts, Taiwan (China) Born in 1946 400- Born in 1936 Born in 1956 X 300- I 200- 1 Born in 1951 100 Born in 1911 20 30 40 50 60 70 Age of household head Source: Author's calculations from Surveys of Income Distribution, Taiwan (China). SAVING AND CONSUMPTION SMOOTHING 345 lower consumption of older cohorts is attributed to their lower lifetime wealth, so that Figure 6.4 would be interpreted as showing a preference for consumption to rise with age, albeit more slowly at higher ages. This interpretation can be pre- cisely quantified using the cohort decomposition techniques of Section 2.6. Estimating a life-cycle model for Taiwan (China) Suppose that individual i is born in year b and is observed in year f. If the life- cycle hypothesis is correct and if uncertainty is not important, the individual's consumption level is proportional to lifetime resources with a factor of propor- tionality that depends on age, so that (6.1) cibt = gi(t-b)Wib where W is lifetime wealth—the sum of financial wealth and the discounted pres- ent value of expected future earnings—and t~bt the difference between today's date and the year of birth, is age. In this simple model with no uncertainty, life- time wealth does not vary over time, and we can think of the individual—or at least the household—having its lifetime resources set at birth (or at incorporation) and then choosing how to allocate consumption over time according to its prefer- ences as represented by the function g. In general, this function will also depend on the real rate of interest, with higher rates causing the age profile to tip counter- clockwise, but I ignore this dependence for the moment. If we first take logarithms of equation (6.1) and then average over all house- holds in the cohort born at time b and observed at /, we obtain (6.2) InCfc = ]ng(t-b)+]nWb so that the average of the logarithms of individual consumptions is the sum of two components, one of which depends only on age, and one of which depends only on cohort. As a result, we can estimate (6.2) using the methods of Section 2.6, regressing the average log of cohort consumption on a set of age and cohort dum- mies. The former recovers preferences about intertemporal choice and the latter the lifetime wealth levels of each cohort. There is no need to assume a functional form for preferences nor to measure lifetime wealth levels. Equation (6.2) is even consistent with some uncertainty; as long as the members of the cohort can esti- mate their future earnings correctly on average, there is no need to suppose that each can do so. While this is a better assumption than is complete certainty, it still rules out the sort of macroeconomic shocks that would provide "surprises" for all members of the cohort. We are effectively assuming that, at least by 1976, mem- bers of each of these Taiwanese cohorts neither systematically over- nor underes- timated the growth in consumption that was to come in the next decade and a half. Although there is no general theoretical framework to support such a construc- tion, we can treat income in exactly the same way as consumption in (6.2), taking 346 THE ANALYSIS OF HOUSEHOLD SURVEYS logarithms, averaging, and regressing on age and cohort dummies. The validity of the decomposition requires that incomes follow an unchanging lifetime profile, but that the position of the profile moves up from older to younger cohorts as economic growth raises earnings. The difference between the logarithm of in- come and the logarithm of consumption is a monotone increasing function of the saving-to-income ratio, and is approximately equal to the saving ratio when the saving ratio is small, so that the cohort-age decomposition of log consumption and log income automatically delivers a cohort-age decomposition of the saving ratio. The results of the calculations are shown in Figures 6.5 and 6.6, and come from two regressions. In the first, the average of the logarithm of consumption for each cohort in each year is regressed on age and cohort dummies, and on the average numbers of adults and children in each cohort, the last two being includ- ed to make allowance for the differential consumption requirements of adults and children; family composition can be thought of as an argument of the preference function g in (6.1). The second regression does the same for the averages of log income. Year effects are not included, although the results are very much the same when they are, provided they are restricted to average to zero and to be orthogonal to time (see Section 2.7). The cohort effects for both regressions, as well as the difference between the two sets of age effects, which I am interpreting as the cohort effects in the ratio of saving to income, are plotted as a function of cohort age (age of the household head in 1976) in Figure 6.5. Figure 6.6 shows the age effects as a function of the age of the household head; the three graphs are the age structure of log income in the absence of growth, the preferred age profile of log consumption, and the resulting age profile of the saving ratio. Each point on each graph corresponds to a coefficient from one or other regression; the curves are not constrained to take on any particular shape, so that the relatively smooth curves in the figures come directly from the data. What do Figures 6.5 and 6.6 tell us about the usefulness of the life-cycle model for explaining saving behavior among Taiwanese households? Figure 6.5 shows that the cohort effects in both log consumption and log income are declin- ing with age in 1976, so that the younger the household, the higher is its lifetime profile of both real income and real consumption. The cohort effects increase at about 5 percent per annum for income and at about 3.5 percent per annum for consumption, rates that are considerably lower than the rate of growth of per capita GDP. The difference between the cohort income and consumption growth rates means that the saving rate is growing from older to younger cohorts. A household headed by a 25-year-old in 1990 is saving 20 percent more of its in- come than was a 25-year-old household 35 years before, in 1955. According to the life-cycle model, consumption profiles can only grow less rapidly than lifetime wealth profiles if bequests are a luxury good, so that the share of lifetime consumption in lifetime resources, including inherited bequests, is falling over time. Put differently, pure accumulation is an important motive for saving, accumulation that is not simply designed to be run down in old age. This is entirely plausible in Taiwan (China), where many families are accumulating SAVING AND CONSUMPTION SMOOTHING 347 Figure 6.5. Cohort effects in consumption, income, and saving, Taiwan (China) o- Saving rate J03JJ3 jjoqoa pajBuiijsg -2- Log income -3 - 10 20 30 40 50 60 70 Cohort: age of household head in 1976 Source: Author's calculations from Surveys of Personal Income Distribution, Taiwan (China), and Deaton and Paxson (1994a). Figure 6.6. Age effects in consumption, income, and saving, Taiwan (China) 2 - Log income "8 i - Saving rate 0 - 20 30 40 50 60 70 Age of household head Source: Author's calculations from Surveys of Personal Income Distribution, Taiwan (China), and Deaton and Paxson (1994a). Figure 6.7. Averages of log consumption and log income, by year and cohort, Taiwan (China) siuootii pire uoijdiunsuoo jo stuqiucSoi jo sSejaAU uoqo3 £ Source: Author's calculations from Surveys of Income Distribution, Taiwan (China), and Deaton and Paxson (1994a). SAVING AND CONSUMPTION SMOOTHING 349 property—housing and small businesses—which they intend to leave to their children, but it remains unclear why this motive should have become so much stronger over time unless bequests are luxury goods. It is also possible that Taiwanese of working age, who are wealthier and have fewer children than did their parents, are saving more to support themselves in a retirement that ever fewer can expect to spend with their children. Such a change should not cause consumption to grow less rapidly than income in the long run, since there will be additional dissaving when these people retire. However, it is possible that, with few of these people retired to date, the analysis could mistake the phenomenon for an increase in the demand for bequests. The age profiles of consumption and saving in Figure 6.6 also present prob- lems for a life-cycle interpretation of the data. In particular, there is no evidence of dissaving among the old; indeed, the saving rate increases with age at least up to age 60. It is possible that age selectivity is responsible for this finding, and that the potential dissavers are those who have already exhausted their assets, are living with their children, or are dead. While it seems impossible to rule out such an explanation, the lack of saving among the elderly is a common finding in other countries (see for example Mirer 1979 for the United States, Borsch-Supan and Stahl 1991 for Germany, Attanasio and Weber 1992 for Britain—though see also Shorrocks 1975) and the selectivity explanation is no more plausible than a num- ber of other stories, for example that the elderly have little need for consumption at the margin, or that they feel a duty to pass on their assets to their heirs. The rapidly rising age profile of consumption, although very far from the flat profile that is often assumed in the textbooks, is not in itself inconsistent with the theory. There is no reason why people should not choose a consumption profile that increases with age, and the high real interest rates in Taiwan (China) certainly provide an incentive to postpone consumption until late in life. However, there must be at least a suspicion that the similarity between the income and consump- tion profiles in Figure 6.6 is not entirely coincidental, and that consumption might be determined by income. Such a suspicion is reinforced by looking at the same data in yet another way. Figure 6.7 plots the averages of log consumption and log income for each cohort starting with those who were 14 in 1976 (in the top left panel) and finishing with those who were 62 in 1976 (in the bottom right panel). For most of these cohorts, the income and consumption lines move apart over time, restating the fact that the saving ratio is rising, but there is a very marked similarity in the two lines for each cohort. Whatever is the time profile of income, so too is the time profile of consumption. According to the life-cycle model, such tracking could come about through a coincidence of the age profiles of preferenc- es and incomes; the desired life-cycle profile of consumption happens to match the actual life-cycle profile of income. Alternatively, the data are consistent with people saving a fraction of their incomes, a fraction that drifts upward over time. To say so is not much more than a description of the data, and it is far from clear why people should behave in this way. For people who have no interest in accumulation, and who would like to borrow but cannot, consumption would move closely with income because consumption is equal to income. But for these 350 THE ANALYSIS OF HOUSEHOLD SURVEYS Taiwanese households, who save a great deal and have consequently accumulated assets that are several multiples of their income, there is no obvious reason why consumption and income should be so closely tied. Nor is this phenomenon con- fined to Taiwan (China); Paxson (1996) finds similar close links between con- sumption and income for cohort data from Thailand, Britain, and the United States, and Carroll and Summers (1991) find the same across occupational and educational groups in the United States. In summary, the Taiwanese data permit a life-cycle interpretation but there are uncomfortable anomalies. The saving rate rises with age, which may be a selec- tion effect, and there is secular increase in saving rates across cohorts, which is perhaps attributable to bequest motives, or to changes in living arrangements. But neither phenomenon has a natural interpretation in the life-cycle model, and both need to be explained away. It is worth recalling from Section 2.7 that the identification of the cohort and age effects in Figures 6.5 and 6.6 rests on the assumption that there is no trend in year effects. In consequence, if we were to extend the model to include an upward time trend in the saving rate, it would be possible to eliminate both the downward trend in the cohort saving rates—Figure 6.5—and the upward trend in the age saving rates—Figure 6.6—after which both figures would be closer to the predic- tions of the theory. But this is just another way of admitting that the theory cannot explain the data. Aggregate saving rates rose in Taiwan (China) between 1976 and 1990, and the rise is not explained by any simple version of the life-cycle model, but by a residual time-trend. Life-cycle theory is supposed to explain aggregate saving in terms of changes in cohort size—demographics—and changes in cohort lifetime resources—wealth effects. Instead, we are left with the uninformative conclusion that, for some unknown and unmodeled reason, saving rates went up. Paxson (1996) encounters the same failures of the life-cycle model in trying to explain trends in the saving rates in Britain, the United States, and Thailand; indeed, in her calculations, the Taiwanese case is the most favorable for the life-cycle story! 6.2 Short-term consumption smoothing and permanent income The evidence in Section 6.1 shows that consumption moves parallel to income over the life cycle and indeed, that they are tied together over periods longer than a few years. Such a finding does not imply that consumption and income are closely related at shorter frequencies of days, months, seasons, or years. Such short-term or "high-frequency" smoothing reveals the extent to which households can maintain their consumption and living standards in the face of short-term fluctuations in their incomes, fluctuations that can be frequent and large for agri- cultural households. Short-term consumption smoothing is one way that house- holds insure their consumption in the presence of risky and variable incomes. The extent to which consumption is smoothed is important for a number of policy issues in developing countries. If agricultural households are somehow in- capable of suitable mtertemporal allocation, then there is a possible custodial role SAVING AND CONSUMPTION SMOOTHING 351 for the government to make the allocation on their behalf. Indeed, the supposed inability of poor households to save lies at the heart of the once-standard project evaluation guidelines (Dasgupta, Marglin, and Sen 1972; Little and Mirrlees 1974; and Squire and van der Tak 1975). The supposition that consumption is constrained by current income, even in the short run, is also frequently used as a justification for commodity-price stabilization schemes, or marketing boards for crops that pay farmers fixed prices for their crops and "protect'* them from the fluctuations in market prices. For example, in an early discussion, Bauer and Paish (1952) argued that marketing boards should make payments to farmers based on a moving average of world prices, on the grounds that "Small producers are unlikely to have the self-restraint and foresight to set aside in good times sufficient reserves to cushion the effects of worse ones, or, even if they have, may be debarred from doing so by social customs and obligations." (p. 766). The link with saving was made in a reply by Friedman (1954), who emphasized that the scheme forced farmers to save, and who doubted whether marketing boards were a suitable substitute for education and the development of credit markets. (Of course, the question of a custodial saving policy depends not only on the extent to which peasants can make good intertemporal decisions, but also on whether the government can do better.) Just as low-frequency smoothing is associated with the life-cycle hypothesis, so can high-frequency smoothing be associated with the at least some versions of Friedman's (1957) permanent income hypothesis. According to Friedman's the- ory, consumption is determined not by current income, but by permanent income, where permanent income can be defined over various possible time horizons. If the horizon is the lifetime, the permanent income model coincides with the life- cycle model, but we can also consider an infinite horizon, which would be appro- priate for a dynastic or long-lived extended family, and which has been much used in the macroeconomic literature for a "representative" agent, or a short hori- zon of just a few years. It is not straightforward to use household survey data to discover the extent to which consumption is smoothed over time. Looking at changes in income and consumption requires panel data, and as we have seen, such changes are typically very poorly measured in the few examples of panel data that we have. However, our understanding of these issues has been enhanced by a number of valuable studies based on household survey data, and in this section, I look at some of the methods that are available, and some of the results. Saving and weather variability A convenient way to state the permanent income model for empirical analysis is to write consumption as a function of permanent and transitory income, together with other variables, for example the demographic structure of the household. Write this in the form: (6.3) c^ = a + py^ + y^ + 0.zA, + uht 352 THE ANALYSIS OF HOUSEHOLD SURVEYS where the subscripts h and / refer to households and time periods, the superscripts P and T denote permanent and transitory, respectively, and total measured income is the sum of its transitory and permanent components. In a "strict" version of the permanent income hypothesis, the parameters p and y would be one and zero, re- spectively, but if we can find evidence that P is greater than y> that a greater fraction of what is seen as transitory income is saved, it would suggest that house- holds are acting so as to smooth their consumption relative to income. The estimation of (6.3) requires additional assumptions about what is transi- tory and what is permanent. Suppose that we substitute yp = y-yT so that (6.3) can be written as (6-4) '*, = « + Py», + e-z*, + y and the covariance of total and transitory income is positive, P will be biased down by the omission of transitory income, so that we would expect the estimate of p to be larger under iv than OLS if households are smoothing their consumption. Of course, lagged income might not be a valid instrument, either because there are dynamics in consumption patterns so that consumption is di- rectly affected by both current and lagged income, or because transitory income is correlated from one year to the next. If lagged income is negatively correlated with current transitory income, which would be true if unusually high incomes are followed by unusually low ones, as is the case for some tree crops, then the IV estimates using lagged income will be biased down just as the OLS estimates are biased up. If instead of lagged income, we use as instruments the head's educa- tion and the beginning of period values of assets—personal, business, and agricul- tural assets are separately distinguished—the estimate of p rises to 0.516 (t = 13), this time in accord with the permanent income model in which education and assets are correlated with permanent but not transitory income. However, we know how difficult it was to calculate income estimates for these data, and that the margin of error is large. So it is possible that the increase in the coefficient has nothing to do with permanent income, but comes from the fact that assets and education are uncorrelated with the measurement errors in income. Business and agricultural income are also by far the most difficult to measure, and income from earnings the easiest, so we could construct an instrumental variables estimate using current (1987) earnings to instrument total income, not on perma- nent income grounds, but simply to control the measurement error. This estimate is 0.410 (t = 17) and takes us most of the way from the OLS to the IV estimates using assets and education. Of course, it can also be argued that earnings are less variable than either agricultural or business income, so that these results too are consistent with a permanent income interpretation. Measurement error and transi- tory income are essentially indistinguishable in these models, and it is hopeless to try to test hypotheses that depend on the distinction between them. Paxson's (1992) study of rice farmers in Thailand circumvents these problems an provides convincing evidence that farmers respond differently to transitory and permanent income. Most rice farms in Thailand are not irrigated, so that crop sizes depend on the amount of rainfall, with more rainfall resulting in more in- come over the relevant range. Although average rainfall is predictably different from place to place, the deviation of each year's rainfall from its local mean is serially uncorrelated and thus unpredictable. Hence income associated with good rainfall is transitory, with close to no direct effect on future or permanent income. Paxson uses data on farmers' incomes from the Thai Socioeconomic Surveys of 1975-76, 1980-81, and 1986, and matches the geographical location of each household to 21 local rainfall stations, from which monthly rainfall data are avail- 354 THE ANALYSIS OF HOUSEHOLD SURVEYS able over 35 years. She then regresses measured income on the rainfall and other variables, which allows a transitory component of income to be isolated. Entered into a consumption regression, this component attracts a lower coefficient than does the remainder of income. These Thai rice farmers appear to recognize that it is wise to lay aside a substantial fraction of the rain-induced income. Paxson (1993) has also used the Thai data to investigate consumption smooth- ing over seasonal frequencies. One would perhaps expect even the most myopic of agricultural consumers to understand the implications of seasonal income fluctuations, and the need to make adequate provision in good seasons for the lean times to come. Nevertheless, there is a literature in economic development (see for example the papers in Sahn 1989), that argues that poor households are sometimes unable to make adequate provision for seasonal fluctuations. Paxson's idea is to compare the seasonal consumption patterns of groups whose seasonal income patterns are very different. If consumption is constrained by seasonal income patterns, the seasonal patterns of consumption will be different across the different groups, while if seasonal patterns of consumption are determined purely by seasonal variations in prices or tastes, they should plausibly be the same across groups. Paxson compares the seasonal consumption patterns of rural versus urban consumers, of farm versus nonfarm households, and of single-cropping rice farm- ers versus double-cropping rice farmers. Even though the Thai surveys do not track individuals over even a whole year, seasonals can be computed by averag- ing over different members of each group. The results show that consumption patterns of the different groups are very similar, even for those groups with large differences in their income patterns. At least for these data, consumption season- ality is a matter of choice, not of constraint. Saving as a predictor of income change? Another method for testing whether people make provision for the future is to test whether their saving predicts future changes in their incomes. That this is a conse- quence of the permanent income hypothesis was first stated and investigated by Campbell (1987) using American macroeconomic data, but the idea is even more appropriate for microeconomic analysis, particularly for farm households. Con- sider the case of a tree-crop farmer, say a cocoa or coffee smallholder in Cote d'lvoire. If trees are destroyed by bush fires or by infestation by pests, the farmer knows that his harvest will be low for the next year or two. Similarly, even within the harvest year, the state of a tree provides substantial information about its yield long before the harvest is collected, and even longer before the farmer receives payment. Farmers thus have a good deal of advance information about their fu- ture incomes, information that they should use when deciding how much to con- sume and how much to save in the present. A farmer who discovers that his trees have been destroyed in the previous night should cut consumption immediately rather than waiting for the misfortune to be turned into hard cash. When people look to the future in setting their behavior, there are important econometric implications. In particular, if people use information about the future SAVING AND CONSUMPTION SMOOTHING 355 that is not available to the econometrician, the existence of the information will show up in their current behavior, which can thus be used to predict the future, even though there is no causal link. If we know that someone has checked the weather forecast, and observe her carrying an umbrella, we can use the umbrella to "predict" the rain, even though the causality runs from rain to umbrellas, not the other way around. In the same way, the saving behavior of the Ivorian small- holders, who can read something of their future income in their trees, can tell us what they expect to happen to their incomes. Of course, if the survey data col- lected all the information relevant for predicting income, the health of the trees, rainfall to date, and myriad other factors, and if we conditioned our forecasts on all of these things, then saving would not add additional information. But in prac- tice, there is no hope that the econometrician could know as much as the farmer, so that, if farmers are deliberately smoothing their consumption, saving should predict future income. More specifically, bad news should engender a cut in con- sumption and good news an increase, so that saving should negatively predict changes in income. The prediction that saving predicts a fall in income can be tested if we have panel data with several observations on income, consumption, and saving, and this is the case for the two successive panels (1985-86 and 1986-87) of the Cote d'lvoire LSS. However, it is necessary to be careful when constructing a test. Given a long enough panel, we could simply test for each household whether saving in year t helped predict income in year / + 1 using, for example, a standard Granger causality test. However, panels are typically short—in the Ivorian case only two years—so that we have to use the cross-sectional dimension of the data, and test whether people who saved a great deal in one year were also those whose income was lowest in the next. But different individuals have different average income levels, and we do not want to contaminate the test with the cross-section relationship between saving and income levels. However, if each household's income varies around its own individual (constant) mean—a household fixed ef- fect—then the change in income will be purged of the fixed effect, and we can test whether saving in one year is negatively correlated with the change in income from that year to the next. Under the permanent income hypothesis, households with stationary income streams use saving to smooth consumption, not to accu- mulate, so that saving itself will be stationary and, over a long enough period of time, will average to zero. Hence, if all that households are doing is smoothing their consumption, and if income is a stationary process for each, the cross-sec- tion regression of income change on lagged saving is free of household-specific fixed effects and can be used to test the proposition that saving helps predict income change. Table 6.1 shows the results of regressing the change in income for the Ivorian panel households on their saving in the previous year. (The data used here and in subsequent tables in this chapter come from an earlier release than those used in Chapter 1, and are therefore not strictly comparable with them.) The left-hand side of the table shows the regressions for the 1985-86 panel households, and the right-hand side those for the 1986-87 panel. Each of the five regions is shown 556 THE ANALYSIS OF HOUSEHOLD SURVEYS Table 6.1. The effects of saving on the next period's income, Cote d'lvoire, 1985-86 and 1986-87 Region 1985-86 1986-87 Abidjan -0.439 (1.9) -0.022 (2.0) Other urban -0.627 (3.6) -0.465 (3.6) West Forest -0.694 (10.8) -0.781 (4.0) East Forest -0.868 (14.7) -0.761 (5.4) Savannah -0.625 (3.5) -0.792 (11-9) Note: The numbers are the regression coefficients (and absolute /-values, corrected for cluster effects) of year /'s saving in a regression where the dependent variable is the change in income from year r to / +1. Source: Author's calculations using CILSS data. separately. It is reasonable to expect that the degree of private information will vary from region to region, that it will probably be larger for farmers than wage earners, and larger for tree-crop farmers than for cultivators of perennials like cotton. The coefficients appear to confirm the hypothesis. All are negative and all except for Abidjan are significantly different from zero. The households who saved the most were those whose income fell the most in the next period. Unfortunately, this evidence is far from convincing. The culprit, as so often, is measurement error in income. Suppose, for example, that income in period t is overstated. As a result, saving in t will be overstated while the increase in income from t to t + I will be understated and there will be a spurious negative correlation between lagged saving and income change, a correlation that might account for the OLS results in the table. Of course, since saving is mismeasured and since saving is the explanatory variable, there will also be an attenuation bias, but if we suppose that there is no true relation between saving and future income, zero will be attenuated to zero, and the spurious negative correlation will generate a nega- tive probability limit for the OLS estimator. Formally, if the true relationship is (6.5) A;y]} where a> is the amount from total resources At+yt (or "cash on hand") that is withheld for future consumption, and which will be augmented in the next period by the stochastic real rate of interest. If we differentiate (6.9) with respect to real assets A,, we have (6.10) V,(A,) = x > ( c , ) E A,(c,),say. This equation is an immediate consequence of the envelope theorem, that when differentiating the right-hand side of (6.9), we can ignore the implicit dependence of CD on Af (see for example Dixit 1976, ch. 3). The expression Xt(ct) is intro- duced to denote the marginal utility of consumption, a quantity with which we shall be much concerned. The other condition that we need from (6.9) is the first- order condition associated with the optimal choice of 0). Provided that there are no restrictions on borrowing or lending, so that current consumption can be set independently of current resources, this is (6.11) ufo) = EJ(l+r, + XiGVi)]* If we use the definition of A,,(c,) on the left-hand side, and equation (6.10) to sub- stitute for the right-hand side, we obtain (6.12) M c /) = £ /[( I + 'M) A 'M( C M>]- Equation (6.12), often referred to as the Euler equation of intertemporal optimiza- tion, ensures that, up to a discount factor, money is expected to be equally valu- able at the margin in all periods, or more loosely, that the marginal rate of substitution between consumption in any two periods should reflect the relative SAVING AND CONSUMPTION SMOOTHING 359 opportunity cost of funds in the two periods. It provides a number of useful in- sights about saving behavior, as well as serving as a basis for empirical analysis. Special cases: the permanent income and life-cycle models Under special assumptions, the Euler equation (6.12) is consistent with the permanent income hypothesis and with the simplest form of the life-cycle model. In particular, suppose that each period's subutility function is identical up to a discounting factor, so that (6.13) u,(c,) = (l+6)-'i>(c,) for time-invariant function u(.). The parameter 5 controls the rate at which sub- utility tomorrow is discounted relative to subutility today, and is thus the rate of time preference. Suppose also that the rate of interest is a constant r, so that if A(c) is the derivative of u(c), (6.12) takes the special form (6.14) *(*,) = y;£ £ / A < c /+i>- If we make two further assumptions, (i) that the rate of time preference is equal to the rate of interest, and (ii) that the subutility functions are quadratic so that the marginal utility functions are linear in consumption, then (6.14) implies that (6.15) ct = Etct+l. This equation says that consumption is a "martingale," a stochastic process whose expected future value is its current value; indeed, by successive substitution in (6.15), current consumption is also the expected value of consumption in any future period. (6.15) also can be rewritten as (6.16) ct+l = ct + ut +l where ut+l is a stochastic shock or "innovation" whose expectation at time t is zero. Equations (6.15) and (6.16), which trace back to the seminal work of Hall (1978), are often referred to as the "random walk" property of consumption. However, a random walk is a special case of a martingale, when the variance of the innovations ut+l is constant over time, something that is not implied by (6.15). Note too that (6.15) is itself obtained from (6.14) under very special —and very implausible—assumptions, and that even (6.14) is a specialization of the Euler equation (6.12). Substantively, the transition from (6.12) to (6.15) rules out variations in real interest rates and imposes a quite unreasonable homogeneity on household structure, something that would have to be modified in practice. As we shall see in the next subsection, the specialization from (6.14) to (6.15) rules out a type of precautionary behavior that is plausible for poor people in developing 360 THE ANALYSIS OF HOUSEHOLD SURVEYS countries. And all of these special assumptions are added to the basic assumption underlying the Euler equation (6.12), that consumers have access to as much borrowing as they need when they need it, and that they pay the same rate for loans as they get on their savings. If we accept all the assumptions, (6.15) provides a formal justification for models in which planned consumption is constant. In a life-cycle interpretation, where the focus is less on uncertainty than on the age profiles of consumption, earnings, and savings, the martingale property becomes the constant consumption property of the simplest "stripped-down" life-cycle model. The permanent in- come hypothesis is essentially the same, though it tends to be stated, not that con- sumption is constant, but that consumption is equal to permanent income, defined as the annuity value of the sum of current assets and the discounted present value of expected future earnings. In the absence of uncertainty, permanent income will be constant, and so will be consumption. With stochastic earnings, permanent income is revised as expectations about income are revised, but since permanent income is itself an expected value, its expected change is zero, and it is itself a martingale. Although constant consumption, or constant expected consumption, is an attractive and simple baseline against which to compare more elaborate models of intertemporal allocation, its derivation requires very strong assumptions that we cannot reasonably expect to hold for poor households in developing countries. A richer version of the life-cycle model can be derived by recognizing the dependence of the marginal utility of consumption on factors other than con- sumption, such as age or household composition. Suppose that there is no uncer- tainty, but that (6.14) is extended to include household characteristics zt, some- times referred to as "taste shifters," (6.17) A(c,,z,) = ltlA(c,+1>Z,+1) = |^Ij\(c^,Vt) so that, since the last equality holds for all k, we have the result that the age pro- file of consumption is determined by household characteristics, or tastes, and by the relationship between the real rate of interest and the rate of time preference (see Ghez and Becker 1975 for one of the first derivations along such lines). If r > 6, so that the rewards for waiting are sufficient to overcome impatience, the marginal utility of consumption will be falling with age, and consumption will be rising. Conversely, if impatience overcomes the incentives to wait, marginal utility will be rising, and consumption falling. The level of consumption, as op- posed to its profile with age, is determined by the intertemporal budget constraint, which in the absence of uncertainty is summarized by lifetime resources. If the marginal utility functions in (6.17) are homogeneous—for example if they are of the standard isoelastic constant relative risk aversion type—then the ratio of con- sumption levels in different periods will be independent of the level of lifetime resources, so that changes in lifetime resources will simply move the profile up or down in parallel, without altering its shape. Such a model provides a formal basis for the life-cycle model that was applied to Taiwan (China) in Section 6.1 (see equation 6.1). SAVING AND CONSUMPTION SMOOTHING 361 Further analysis of the basic model: precautionary saving Although the permanent income and life-cycle models are perhaps the two lead- ing examples that are consistent with the general theory of intertemporal choice, they are by no means the only ones. Each simplifies in a different direction, the permanent income model downplaying the effects of age and family composition to focus on the way in which consumption responds to new information, and the life-cycle model abstracting from uncertainty in order to derive simple forms for the age profiles of consumption and saving. As we have seen in the first two sections, each of these simplifications is potentially useful for thinking about different aspects of consumption smoothing in developing countries, the perma- nent income model for thinking about fluctuations in seasonal or annual incomes, and the life-cycle model for examining relationships between saving and age. However, there are important cases that are covered by neither one, perhaps the most important of which is behavior when there is uncertainty, and when the marginal utility of consumption is not linear. The marginal utility of consumption function X(c) can usefully be thought of as determining a "shadow-price" for consumption; because there is diminishing marginal utility of consumption, this price is higher when consumption is low than when it is high. Additional consumption is more urgent in the lean season than in the fat season, and consumers will be prepared to give up more for it when they have little, for example by paying higher interest rates for loans to tide them over. It is also possible—perhaps even reasonable—to suppose that the marginal utility of consumption rises more rapidly when consumption is low than when consumption is high. For example, if there is a subsistence level of consumption, then the marginal value of consumption might well tend to infinity as consump- tion falls to that level. Based on such arguments, we might want to suppose that the marginal utility of consumption function is (strictly) convex. Consider the implications of equation (6.14) in the case where the marginal utility of consumption function is convex. Suppose that, for some unspecified reason, future consumption becomes more uncertain, in that its mean remains un- changed but its spread around the mean is larger. Because the marginal utility function is convex, the increase in spread will increase its expected future value, and so the current marginal utility of consumption must also increase to preserve the equality in (6.14). Since marginal utility is diminishing, this means that cur- rent consumption will fall and saving increase. Since (6.14) also implies that the marginal utility of current consumption is the appropriately discounted expecta- tion of the marginal utility of consumption in any future period, the same argu- ment shows that an increase in uncertainty about consumption in any future pe- riod will also increase current saving. When the marginal utility function is con- vex, increases in future uncertainty in the sense defined lead to an increase in current saving, something that can be thought of as precautionary saving. This argument is limited by the fact that it does not specify the cause of the mean-preserving spread in future consumption, and the proposition cannot be readily translated into statements about the effects of increased uncertainty in the 362 THE ANALYSIS OF HOUSEHOLD SURVEYS more fundamental underlying variables, such as earnings and interest rates. For example, it is not true without further assumptions that a farmer with convex marginal utility will increase current saving in response to a mean-preserving spread in next period's harvest (see for example the discussion and references in Dasgupta 1993, pp. 260-64). However, the convexity of marginal utility has proved to be a useful way of characterizing the precautionary motive, and seems particularly promising for analyzing the behavior of people for whom subsistence or worse is a real possibility. It should be noted that the convexity of the marginal utility of consumption— which characterizes how prudent or cautious people are—is not at all the same thing as the concavity of the utility function—which characterizes the degree of risk aversion. Declining marginal utility, as in the permanent income hypothesis, implies risk aversion, but it does not necessarily imply the existence of a precau- tionary motive. Risk aversion relates to the second derivative of the utility func- tion, and prudence to the third, a fact that has been used by Kimball (1990) to provide simple measures of prudence that parallel—but differ from—the standard measures of risk aversion. Of course, for specific utility functions, prudence and risk aversion may be related. For example, in the standard isoelastic utility func- tion with marginal utility c ~ p , where p is the coefficient of relative risk aversion, prudence is also controlled by the single parameter p, so that consumers who are more risk averse are automatically also more prudent. One reason that the permanent income and life-cycle models have dominated discussion is that they are easy to work with, and it is possible to provide closed- form solutions that characterize consumption and saving. This is not possible with precautionary motives, and apart from a few special cases of limited interest, there are no closed-form solutions to (6.14) when there is uncertainty and when A(c) is nonlinear. As a result, we understand less about precautionary savers than we do about permanent income or life-cycle savers. Even so, much recent prog- ress has been made using approximations by letting computers solve sample dynamic programs and then simulating behavior (see Skinner 1988; Zeldes 1989a; Carroll forthcoming, and Hubbard, Skinner, and Zeldes 1994, 1995). Several points are worth noting. When there is a precautionary motive for saving, intertemporal transfers of funds that are actuarially neutral are capable of affecting behavior. In the life- cycle or permanent income models, where consumption depends on expected lifetime resources or on permanent income, a fully funded but compulsory social security system that accumulates contributions with interest cannot affect con- sumption because it does not affect the discounted present value of lifetime re- sources. If the contributions leave the consumer short of liquidity, loans can be negotiated with the accumulated contributions as collateral. However, if consum- ers are prudent, and like to have precautionary balances on hand, the compulsory contributions will generate an addition to saving. Savings that are accumulated for retirement can perfectly well double as a reserve for emergencies, but if direct access to them is removed, or if they cannot be used as collateral for loans, they must be replaced. SAVING AND CONSUMPTION SMOOTHING 363 The obvious question is why such consumers do not borrow, and why the availability of loans does not remove the need for precautionary balances. The answer is that prudent consumers may be reluctant to borrow, even when loans are available, so that in the absence of precautionary balances, consumption may have to be cut when income is low. Although a loan may protect consumption now, it has to be repaid in the future, and if the consumer is unlucky enough, may have to be repaid when things are even worse than they are at present. In the limit, if utility is infinitely low at subsistence, and if income can fall as low as subsistence levels, consumers will never borrow (see Skinner 1988 and Carroll forthcoming). A plan that involves entering the last period of life with nonpositive assets involves a finite risk of subsistence, and thus cannot be optimal. But a consumer with non-positive assets in the second to last period also has a finite risk of exiting the period with non-positive assets, so such a plan cannot be opti- mal either, and so on back to the present. Such an argument raises more questions than it answers—about the support from friends or relatives that might be avail- able in extreme circumstances, and about incentives to provide, accept, or exploit such support—but it provides an illustration of why credit availability might be of limited value to consumers for whom precautionary motives are important. Even when consumers are not completely unwilling to borrow, the computer analyses show that prudent consumers smooth their consumption less than do per- manent income consumers. When income is low, at the beginning of life, or be- cause of a poor harvest, consumption is also lowered at least to some extent, be- cause to maintain consumption would run down precautionary balances or run up loans, either of which would expose the consumer to possible low consumption in a future in which low incomes might occur again. As a result, if incomes are suf- ficiently uncertain, if marginal utility functions sufficiently convex, and if the rate of time preference is sufficiently high to prevent people wanting to accumulate, the consumption of prudent consumers may track their incomes quite closely (see particularly Carroll forthcoming). Restrictions on borrowing The other major challenge to the permanent income and life-cycle accounts of consumption comes from models that explicitly impose a borrowing constraint. Although it is clear that most consumers in developing countries have access to credit on some terms, many people do not have access to the low borrowing rates of formal credit markets, and must rely when necessary on the very high rates charged by moneylenders. Such a situation could be modeled by assuming that borrowing rates are much higher than lending rates, but a simpler way to start is to make the borrowing rate infinite, i.e. to prohibit borrowing altogether. This extreme assumption is also an important baseline, because it allows us to investi- gate the extent to which it is possible for consumption to be smoothed in "autarky" when agents have no one to rely on but themselves, when there are no friends, relatives, or social systems to pool risks and provide insurance, and when there are no credit markets to provide loans when resources are low. 364 THE ANALYSIS OF HOUSEHOLD SURVEYS A consumer with a borrowing constraint can be modeled in the same way as in the basic case, maximizing (6.7) subject to (6.8) but with the additional constraint that assets are nonnegative, so that (6.18) At * 0. The asymmetry in (6.18) is important; consumers cannot borrow, but nothing pre- vents them from accumulating assets. This may seem obvious, but for many poor people around the world, even this may be difficult. Savings institutions may not exist, or may not be trusted, and there may be no safe place to store cash. As a re- sult, people may be forced to save by stockpiling grain, or by hoarding jewelry or gold, assets that may have low or even negative returns. For those people whose original plan does not call for negative assets, the re- striction will have no effect on their intertemporal allocation between t and / + 1 , so that the original Euler equation (6.12) will hold as before. For others, or for the same people in different circumstances, the Euler equation might represent behav- ior in the absence of borrowing, but its satisfaction calls for higher consumption than is available out of current income or assets, something that is impossible in the absence of borrowing. Indeed, the borrowing restriction (6.18) implies that consumption is also limited, and must satisfy (6.19) c, * A,*?/- By (6.19), the marginal utility of money (which is lower the higher is consump- tion) cannot be less than A (At +yt) , so that the effect of the borrowing restriction is to replace the original Euler equation (6.12) by (6.20) *-t(ct) = maxfA^+y^.^Kl+r^)^^)]). The solution of (6.20) appears to be even less readily accessible than the solution to the precautionary saving problem without the credit constraint but, in fact, the problem is rather better understood, and yields insights that seem useful for de- scribing what we see for at least some poor, agricultural households. Consider the special case of (6.20) that ignores taste shifters, and that assumes constant rates of interest and time preference; this is the modification of (6.14) that incorporates the borrowing constraint: (6.21) A(c,) = max[ ^(At+yt)^E l+d tX(ct+})}. \ ) What (6.21) implies for behavior depends on how income behaves or is expected to behave, as well as on the relationship between r and 6. If the consumer wishes to save rather than borrow, the constraint will not bind, and behavior will be the same as in the standard case. This will be so if people are relatively patient, with low enough rates of time preference, or if income is not expected to rise too much. Much can be learned from examining the special case where income yf is stationary, and where impatience outweighs the reward for waiting, so that 5 > r. SAVING AND CONSUMPTION SMOOTHING 365 Consider a farmer whose income is determined by the weather, who has no access to credit, who has no wish to accumulate, but who needs assets to provide a cushion in bad years. In the absence of uncertainty, his income would be the same every year. If the farmer could borrow, he would do so. According to the unconstrained solution (6.14) with 6>r, the marginal utility of consumption is rising through time, so that consumption is falling. This relatively impatient con- sumer wants consumption to be higher than income in the short run, and will pay off the resulting debts by having consumption lower than income in the future. If borrowing is not permitted, he will first run down any assets that he has, and then consume his income in every period thereafter. However, as soon as we introduce uncertainty, this may not be the best strategy. Setting consumption equal to in- come will introduce variations in consumption that can at least partially be avoid- ed, not by accumulating assets over the long run, which is never desirable for this impatient individual, but by putting something aside in good years that can be drawn on in the bad years that will sooner or later inevitably follow. The simplest case is when income is independently and identically distributed, so that each harvest tells the farmer nothing about harvests in the future. Provided his horizon is long enough, the optimal solution to (6.21) is for the farmer to determine his consumption in relation to Af +yt, his stock of "cash on hand" after each harvest (see Schechtman 1976 and Schechtman and Escudero 1977, for the original analysis, and Deaton 1990b, 1991, for a fuller exposition in the current context). The relationship between consumption and cash on hand has the prop- erty that when cash on hand is low, because the harvest has been poor, or because previous harvests were poor so that few assets were carried over from previous years, or both, everything should be consumed. This is the situation in which the farmer would borrow if he could, but since that option is not available, he does the next best thing, and consumes everything. And while it is true that the harvest may be poor again next year, the risk has to be taken if things are bad enough in the present; even when everything is spent, the marginal utility of consumption is higher now than it is expected to be in the future. When cash on hand is above the critical level, something should be put aside as a precautionary reserve for future consumption, and the more cash is available, the larger is the fraction that ought to be saved. This optimal behavior takes into account not only the inability to borrow today, and the tradeoff between consumption today and consumption tomorrow, but also the farmer's knowledge that he will not be able to borrow in the future. Indeed, the fact that future consumption is limited by future income will enhance the precautionary motive, since it makes the marginal utility of consumption higher than it would otherwise have been when income is low, and thus strength- ens the effect of uncertainty on raising the expected marginal utility of future consumption. The model can also be extended to handle the case where harvests are not serially independent. Once again tree crops provide a natural example. Suppose that once damage to trees has been inflicted, it lasts more than one sea- son, so that low income in one year usually means that income will also be low in the following year. The farmer must now take into account, not only his cash on 366 THE ANALYSIS OF HOUSEHOLD SURVEYS hand, but also his expectations about next year's harvest. He will then be less likely than before to spend everything in bad years, because he knows that next year is also likely to be bad. But in good years, he can spend more than he other- wise could have done, because a good harvest now also means that the trees are in good shape for next year. The formal mechanism for recovering the farmer's consumption rule from (6.21) is discussed in the references given above. It is not possible to derive an analytical solution, but given a specification for the utility function, for the sto- chastic process characterizing income, and for the values of the rates of real inter- est and time preference, the rule can be calculated numerically. But it turns out that most of the properties of the optimal rule can be captured by relatively simple rules of thumb according to which the farmer spends everything if cash on hand is below a critical value close to mean income, and then saves a fixed fraction of anything above that level (see Deaton 1992b, from which the following example is taken). Suppose that the farmer's income yt is an independent and identically distri- buted (i.i.d.) normal variable with mean 100 and standard deviation 10. Write xt for cash on hand At+y(t and consider the consumption rule of the form (6.22) c, = x, - 0.7(jr,- 100) 1 (*,> 100) where 1 (.) takes the values of 1 or 0 depending on whether its argument is true or not. By (6.22), consumption is equal to cash on hand whenever cash on hand is less than 100, which is the mean harvest, while 70 percent of any excess is saved. The choice of mean income for the critical value and of 70 percent for the saving parameter comes from roughly matching (6.22) to the optimal nonlinear rule (which is not discussed here), but the choice can also be justified directly in a way that will be described below. But first it is useful to examine what happens to consumption and assets for a consumer who behaves according to (6.22). This can be done by assuming that initial assets (at the end of period 0) are zero, so that jCj = y}, then randomly generating a time series for "income" from independ- ent drawings from N(100,102). Consumption in the first period is given by ap- plying (6.22) to jCj, and the next period's cash on hand using (6.8) to update x, (6.23) *,+i = (l + r)(*,-c,)+y, + 1 so that we get a simulated series for income, consumption, and assets. Results for one particular 100-period horizon are shown in Figure 6.8 where I have chosen a real interest rate of 5 percent. The top line of the graph shows the simulated income series, the middle line (the scale for which appears on the right) is consumption, and the bottom series assets. The most obvious feature of these simulations is how much smoother is consumption than income, a result that is achieved purely by the smoothing rule (6.22) and in the absence of any ability to borrow. In Figure 6.8, the standard deviation of consumption is 65 percent of the standard deviation of income; the average over a larger number of simulations is SAVING AND CONSUMPTION SMOOTHING 567 Figure 6.8. Simulation of income, consumption, and assets 140 Income 120 100 g BO 120 60 100 40 80 20 60 10 20 30 40 50 60 70 80 90 100 Period Source: Author's calculations as described in the text. 62 percent. This relative smoothness is made possible by the existence of assets that are usually available to buffer the effects of poor harvests. The bottom line on the graph shows that the amount of assets required to achieve the smoothing is quite small; the average value of assets in the figure is 7.3, less than 10 percent of average income or one standard deviation of income. The bottom line in the figure shows how assets are built up when income is good and drawn down when income is poor, and although there are long periods when assets are positive—for example in the long series of good harvests around period 30 or near period 70—there comes a time when assets are exhausted, a period when cash on hand is low enough to make it desirable to spend everything now rather than to hold something for a future that is likely to be better. This periodic—but random—exhaustion of assets is a general feature of these models; although assets will run out more or less frequently depending on the nature of harvest uncertainty and on the preferences of the individual, sooner or later they will always do so. Once assets are zero, and since the harvest process is i.i.d., there will eventually come a poor harvest without assets, and the consumer will have no choice but to cut consumption by the shortfall in income. There are sev- eral such episodes in Figure 6.8: in period 3 before the consumer has had time to build up any assets, and in period 54 where assets were already exhausted so that consumption had to bear the consequence of the poor harvest. As a result, this sort of buffering leaves consumers vulnerable to a succession of poor harvests, or to poor harvests that are close to one another. For example, there are two poor 368 THE ANALYSIS OF HOUSEHOLD SURVEYS harvests in periods 72 and 74 in the figure. In the first, consumption is protected by assets, but in the second it is not, partly because assets were low after the first shock. The figure also shows that the distribution of consumption is negatively skewed. It is always possible for the farmer to stop consumption becoming too high, by building up assets, but consumption can only be protected from falling when there are assets, which is usually but not invariably the case. Figure 6.8 illustrates the consequences of only one particular rule, and we might ask what are the effects of changing the parameters—the cutoff point and the fraction saved—and how we should choose between the various possible rules. Figure 6.9 shows what would have happened in Figure 6.8 if the farmer had been more conservative and had saved 85 percent of the excess of cash on hand above mean income. The income series in the top lines of the two figures are the same, so that we can isolate the effects of changes in the saving rule. As is to be expected from the more cautious behavior, consumption is now much smoother than before, although there are still occasional downward spikes; while the fre- quency of these spikes can be affected by the policy, they cannot be eliminated entirely and will occur from time to time however conservative the policy. The price of the additional smoothness is that assets are much higher on average, and for an impatient consumer whose rate of time preference is higher than the rate of interest, these assets are unwelcome because they represent missed consumption opportunities. The choice between the two figures therefore depends on the trade- off between security and impatience, something that can only be settled by appeal Figure 6.9, The effects of a conservative saving policy 140 120 100 120 180 3 S 60 100 40 80 60 20 0 10 20 30 40 50 60 70 80 90 100 Period Source: Author's calculations as described in the text. SAVING AND CONSUMPTION SMOOTHING 369 to a specific utility function. For the commonly used isoelastic single-period utility function c r ~ 3 and rate of time preference of 10 percent, the less conserva- tive rule in the top figure is superior. Such utility comparisons also offer a straightforward way of computing opti- mal parameters for a piecewise linear policy rule like (6.22). For any combination of cutoff and saving propensity, run a simulation like those described here, and calculate the average per period utility value over a long enough simulation to approximate expected utility. The parameters that maximize this value—found by grid search or some other method—are the optimal ones within this class of rules. In general, the propensity to save will be larger and the cutoff lower the more cautious is the consumer, and the more variable is income, both of which increase the value of a conservative saving policy. Borrowing restrictions and the empirical evidence It is far from straightforward to construct a direct test of the buffering model. We would require accurate data on consumption and income that tracked households over several years, and it is hard to see how to use one or two cross sections to substitute for the lack of such a panel. Nevertheless, it is worth emphasizing the consistency between the buffering model and the evidence on saving that I have already reviewed in this chapter. First, the evidence from the permanent income literature, while far from estab- lishing that the permanent income hypothesis is an accurate description of con- sumption behavior by poor households, has convincingly established that the pro- pensity to save out of transitory income is relatively high so that consumption is smoother than income. Such behavior is also predicted by the buffering model; consumption is smoothed over the random time horizon defined by the time be- tween "stockouts" of assets. Second, as we saw in Section 6.1 above, there is a good deal of evidence against the life-cycle proposition that people smooth con- sumption over their lifetimes, with the data showing a fairly close "tracking" bet- ween consumption and income. Once again, this is consistent with the buffering model which predicts that consumption is smoothed relative to income in the short run, but is tied to income in the longer run, because assets are not accumu- lated over long periods of time. However, I should note that the buffering model does not offer any clear explanation for why consumption tracks income in Tai- wan (China). Taiwanese households own very substantial assets, and are clearly not in a position where they would like to borrow but cannot. Households who follow the buffering rule accumulate no assets in the long run, and dissave as often as they save. The model is therefore consistent with data showing a large fraction of households with consumption greater than income in any given year, although because some income shocks are common to all house- holds, we would not expect to see half of households saving in each year, but more in one year and less in another. As was discussed in Chapter 1, household surveys from developing countries frequently do record dissaving by substantial fractions of households. There is no doubt that some of this is due to underestima- 370 THE ANALYSIS OF HOUSEHOLD SURVEYS tion of income relative to consumption, but the observation may have more truth to it than is often credited. Like the permanent income hypothesis, the buffering model implies that sav- ing will predict declines in income. Farmers who see that their incomes will be low next period can accumulate assets in advance, and those who see good times coming can run down their assets. It is only those people who receive advance notice of good fortune, but who have no assets, who will be prevented from bor- rowing to turn their anticipations into consumption now. This last has been used to construct formal tests for the presence of liquidity constraints, tests that can also be regarded as attempts to distinguish between the two Euler equations (6.14)—with no borrowing restrictions—and (6,21)—where borrowing is prohib- ited (see Hayashi 1985a, 1985b, and Zeldes 1989b). To see how these work, rewrite (6.21) in the form (6.24) A(c,) = ^£,Mc,t])^, where £, is positive if the credit restrictions are binding—which requires that assets are zero—and will be zero otherwise. If £, is positive, marginal utility of consumption is higher than the consumer would wish, and so consumption is lower; without assets and without access to borrowing, it is not possible to raise consumption above the amount of cash on hand. Suppose that we can isolate a group of consumers for whom we are sure that the borrowing constraints are not binding, for example those with assets, land- holdings, or regularly high incomes. For them, £, will be zero, so that if we select a specific utility function, we can estimate (6.24) ignoring the last term. Using the estimated parameters, we can then estimate £, for a group of possibly constrained consumers and see whether, as predicted by the theory, it tends to be positive. To take a concrete example, suppose that utility is isoelastic, so that the mar- ginal utility of consumption is c ~p, in which case (6.24) can be written for the unconstrained households as 1 +r -p -p (6.25) 778C'*' C ' = "< +1 * ut+l is a zero-mean shock that is the difference between the realization and its expectation and, as such, is orthogonal to any variable known to the consumer at time t or earlier. In particular, the mean of the left-hand side of (6.25) should be zero for a large enough group of unconstrained consumers, as should be the mean of the left-hand side multiplied by any variable known at time t or earlier. Sup- pose that we have data on / such variables, and that zr is the realization of thej'th in period t. Then the sample means, (6.26) ,.= w -.^ill c ,;p- c/ -pj should converge to zero as the number of unconstrained households becomes large. Choosing the parameters to make the s. as small as possible is the general- SAVING AND CONSUMPTION SMOOTHING 371 ized methods of moments (GMM) estimation method first applied in a closely related context by Hansen and Singleton (1982). Once the parameters have been estimated, the average (6,26) can be recalculated for the supposedly constrained households with z>t replaced by unity, and the result checked to see if it is posi- tive. This general framework has been applied to developed economies by Haya- shi (1985a, 1985b) and Zeldes (1989b) with results that supported the supposition that households are credit constrained. A closely related but perhaps somewhat simpler methodology is to note that the unconstrained Euler equation (6.25) can be thought of as determining the rate of growth of consumption between t and t+ 1. In the absence of credit restric- tions, consumption growth will reflect the different needs in the two periods, for example associated with changes in family size, as well as the rate of interest and the rate of time preference. Such a rate of growth equation can be estimated for unconstrained households, and the average residual from the equation calculated for possible constrained households. If some of these households are credit con- strained, and are being forced to postpone consumption from t to / +1, their con- sumption growth should be higher than predicted, and the average residual posi- tive. Put differently, if low income and low or zero assets are a sign of being credit constrained, consumption growth should be negatively related to initial assets and income in the presence of credit restrictions. This appears to be the case in Zeldes' (1989b) work for the United States, and a similar result has been discovered for the ICRISAT households in southwestern India by Morduch (1990), who also investigates the impact of the borrowing restrictions on these farmers' production decisions. That poor people in developing countries face limits on the amount that they can borrow is a plausible conclusion, and one that is supported by a good deal of less formal observation and description. However, there are some serious prob- lems with these tests. First, there are difficulties with "macro" shocks, unpredict- able changes in income that affect all households in common (see Hayashi 1985a, 1985b, whose tests are constructed to be immune to the problem). If there is an unexpectedly good harvest in a region as a whole, consumption growth will be unexpectedly high for everyone, the shocks in (6.25) will be correlated across individuals, and neither their average shock over households nor (6.26) will be zero, even though there are no borrowing restrictions. Simple macro shocks that are the same for everyone can readily be dealt with by allowing year dummies in the growth rate regressions, but there is no guarantee that macro shocks work in this way—for example high rainfall is good for some crops and bad for others. The basic problem here is that tests of the various models require time-series data on individual households, and that cross sections or short panels are not an ade- quate substitute. The second problem has been pointed out by Carroll (forthcoming). He con- structs a model of precautionary saving without borrowing restrictions, in which people are impatient but choose not to borrow (because they fear the possible consequences) and where assets buffer consumption in very much the same way as in the model with borrowing restrictions in the previous subsection. In such a 372 THE ANALYSIS OF HOUSEHOLD SURVEYS model, households with low initial income and assets are households with little protection against a poor harvest next year, and so must save more than house- holds with more income or assets. As a result, consumption growth will be larger for households with lower income and assets, not because of borrowing restric- tions, but because of precautionary saving. The third and final problem is possible lack of power of tests based on viola- tions of the unconstrained Euler equation (6.14). Consider once again Figures 6.8 and 6.9, which illustrate the behavior of consumers who cannot borrow at all, and whose rule-of-thumb behavior delivers a close approximation to the optimal behavior in such circumstances. These consumers have rates of time preference greater than the real rate of interest, and would have declining consumption tra- jectories based on initial borrowing if loans were available to them. As a result of their inability to borrow, their behavior is quite different, with each person's consumption following a stationary process. For these people, the borrowing re- strictions change consumption in a fundamental way, replacing a downward- trending series with a stationary one. In spite of this radical change in behavior, the liquidity constrained consumers in these figures usually hold assets, and so satisfy the unrestricted Euler equation in most periods. As a result, tests for bor- rowing constraints that look for zero assets and violation of Euler equations may have few observations to work with, even when consumers are never allowed to borrow. Once again, the tests are only likely to be powerful when we have long time series on individual households. Given the wealth of informal observation on rural credit markets in developing countries, almost no one would suppose that the typical household can lend and borrow unlimited amounts at a common real interest rate. Even so, we have made surprisingly little progress in detecting the effects on consumption and welfare of these less than perfect credit markets. Data quality is responsible for a good num- ber of the difficulties, but there is also room for imaginative new research that finds new ways of examining the issue. 6.4 Social insurance and consumption Saving is only one of the ways that people can protect their consumption against fluctuations in their incomes. An alternative is to rely on other people, to share risk with friends and kin, with neighbors, or with anonymous other participants through private or government insurance schemes, or through participation in financial markets. Imagine a group of rural cultivators who live in the same vil- lage, but whose plots are scattered around the village, are sown to a range of different crops, and are subject to different risks. Some farmers will be richer than others by virtue of their larger or more productive landholdings and from assets accumulated in the past, but suppose that all are risk averse, and would sacrifice some of their average income for greater income stability. Each has diminishing marginal utility of consumption, so that the welfare loss associated with a poor harvest is greater than the welfare gain associated with a good one. If there are enough farmers and their risks are uncorrelated so that their average income is SAVING AND CONSUMPTION SMOOTHING 373 nonstochastic, then before each farmer knows how good his or her harvest will be, they would agree to a pact whereby each agrees to pool the surpluses above (or shortfalls from) their individual means, so that each would be guaranteed their own mean income. When risks are correlated—as we would expect them to be in an agricultural village—it is not possible to guarantee a nonstochastic income for everyone, but it will still be advantageous to pool individual deviations from the average outcome, so that each farmer's consumption is smoothed as much poss- ible. Members of the village will also have incentives to seek partnerships with people outside the village. The problems of implementing such schemes are as obvious as their advan- tages. Although individuals have an incentive to join the pact ex ante, the lucky ones will want to renege ex post. They will also wish to conceal income if it is possible to do so. The schemes also reduce individual incentives for effort and to avoid risk—the good husbandry that minimizes losses from pests, diseases, and fires—so that complete consumption insurance may be neither desirable nor possible. As a result, the schemes are more likely to exist among groups of people among whom information is good, where income is difficult to hide, and where behavior can be monitored and influenced by the group. Families, kin groups, or villages are the obvious examples. And indeed, there are various local institutions that provide this sort of insurance to a greater or lesser extent (see the excellent review by Platteau 1991). The mutual insurance schemes that are common among certain occupations are one example; "tithes" paid to the church and used for poor relief are another. Some villages have extensive sharing systems. In traditional Indian villages each occupation or caste within the village receives a prespecified share of the harvest in exchange for labor or other services; the blacksmith, the fisherman, and the basket-maker is each entitled to some share of the harvest for goods or services rendered earlier in the year (see Srinivas 1976). Similar institu- tions in Thailand are described by Townsend (1993a, 1993b, 1995), by Platteau (1991) for fishing communities in Kerala, and by Udry (1990), who details how villagers in northern Nigeria have evolved schemes to punish participants who renege or otherwise do not abide by the rules. Risk sharing among individuals can also be accomplished anonymously through financial and insurance markets. Formal insurance is available for some well-defined and easily observed events, such as death, injury, flooding, or hail, but its scope is limited by the same moral hazard and information problems that limit the provision of local personalized insurance. Risky assets whose values fluctuate with events can also provide some coverage, but few poor agricultural households in developing countries hold financial assets. Forward sales of agri- cultural crops can be thought of in this way, as can crops in the ground if market price fluctuates inversely with output. While it is hard to believe that any of these mechanisms could provide com- plete consumption insurance for poor households, the very multiplicity of existing mechanisms makes it likely that there is at least partial insurance through finan- cial or social institutions, and that such risk sharing adds to the possibilities for autarkic consumption smoothing through intertemporal transfers of money or 374 THE ANALYSIS OF HOUSEHOLD SURVEYS goods. There has also been an increased interest in the possibility that local groups can deal with some of the externalities between people that arise in admin- istering schemes that are potentially beneficial (see for example Wade's 1988 discussion of water rights in India). There has therefore arisen a line of research that examines consumption behavior to find the extent of insurance, if any. Al- though it is also possible to examine the mechanisms, the insurance contracts, tithes, and transfers, their multiplicity makes it attractive to look directly at the magnitude that is supposed to be smoothed, namely consumption. In the United States, there is now overwhelming evidence against complete consumption insur- ance at the national level (see Cochrane 1991 and Attanasio and Davis 1993 as well as Attanasio and Weber's 1992 and Nelson's 1994 refutations of Mace's 1991 earlier, apparently positive, results). More interesting is the hypothesis that there is risk sharing within extended families, but Altonji, Hayashi, and Kotlikoff (1992) found no evidence for complete risk sharing between parents and their grown-up children in the U.S. Panel Study of Income Dynamics. But none of this evidence denies the existence of partial risk sharing, nor is it directly relevant to the extent of risk sharing in the very different environments of high-risk, low- income villages in developing countries. The evidence on transfers in poor countries is quite extensive and almost all of it is consistent with a risk-sharing and consumption-smoothing interpretation. For example, Lucas and Stark (1985) find that transfers in Botswana are responsive to drought, and Rosenzweig (1988) shows how transfers in the ICRISAT villages in India are sensitive to shortfalls in income. There is also some evidence that more than money is involved. Rosenzweig and Stark (1989) investigate the extent to which marriage decisions are influenced by the desirability of constructing a risk- sharing network, and Ainsworth (1992) documents the relationship between child-fostering and risk in Cote d'lvoire. There is an impressive series of papers by Cox and Jimenez (1991a, 1991b, 1992a, 1992b, 1992c, 1993) who use survey data—including several Living Stan- dards Surveys—to investigate the correlates of private transfers in Peru, Ghana, Cote d'lvoire, Colombia, and the Philippines. Although the details vary from country to country, there is a remarkably consistent picture overall. Transfers are typically from better-off households to poorer households, so that they equalize the distribution of income, and households are more likely to receive transfers when they are headed by a woman, or when their members are ill or unemployed. The data also show a remarkable inverse correlation between the age profile of consumption and the age profile of the fraction of households receiving transfers, evidence which suggests that transfers even play a role in smoothing consumption over the life cycle. In most of these studies, the transfers do not appear to be very large, except in the Philippines, where remittances from abroad are important, and where transfers averaged nearly one-fifth of income for urban recipients. In the other surveys, transfers of only a few percentage points of consumption are more common, although there is some evidence that standard surveys understate the size of transfers, perhaps by a large amount. Even so, the extent to which transfers stabilize consumption remains more at issue than the fact that they do so. SAVING AND CONSUMPTION SMOOTHING 375 In this section, I discuss first the theoretical framework that justifies the claims made at the beginning of this section, that insurance is socially desirable and can be brought about by an appropriate system of state-contingent assets. This theory also delivers predictions that can be tested on the consumption data. I then look at some of the empirical results. ^Consumption insurance in theory Start from the social problem for the agricultural village. Suppose that each household h has an intertemporal utility function (6.27) B'-EE^ufo,*) t s where s indicates states of nature, t time periods, and TIV, is the probability of state s in time period t. This is the expected utility function as in (6.7) above, but with a finite number of states of nature with explicit probabilities. If the farmers are to cooperate in a village pact to provide mutual insurance in an optimal fashion, they will have to decide who gets what share of the total—presumably by some pro- cess of initial bargaining that takes into account relative wealth and thus the utility of each member without the scheme—and then maximize an appropriately weighted sum of utilities subject to an overall constraint on total consumption in each state in each time period. The Lagrangean of this problem can be written (6.28) * = Eu'EEivta?) + EE5,,(C,,-£c,J) h t s s t h where [a* is the social weight for household h that controls the share obtained by that household, Cst is aggregate village consumption in state s and time t, c* is consumption of h in s and t, and £ v/ are the Lagrange multipliers associated with the resource constraints. Note that this formulation simplifies, not only by ruling out any transactions between the villagers and outsiders, but by also assuming away saving, borrowing, or buffering activities by each villager on his or her own account. The village council is assumed to control the consumption level of each member. Allowing for autarkic consumption smoothing as well as risk sharing makes the problem a great deal more complex. If the individual consumption levels are the controls, the first-order condition for (6.28) can be written in logarithms as (6.29) lnA*(c,*) = ln£ v , - ln>i* - In7iv, where, as before, X(c) is the marginal utility function of consumption. Since only one state is actually realized in period t, the s subscript in (6.29) can be dropped, and the equation will hold for each time period. In particular, we can take first differences, which will eliminate the individual fixed effect lnji /I to yield (6.30) AlnAfr,*) = Aln£,-Aln7i, = A In?; 376 THE ANALYSIS OF HOUSEHOLD SURVEYS so that, with complete consumption insurance, the rate of growth of each partici- pant' s marginal utility of consumption is the same. The implications of (6.30) for consumption depend on the shape of the mar- ginal utility functions, as well as on other factors that affect marginal utility and contribute to interpersonal differences in tastes. For example, suppose that house- hold utility functions take the form (6.31) u?(c,*) = Q", «,V,V) = (l-pr'eVW)1-" so that household utility is the utility of consumption per head multiplied by the number of people, and where I have further specialized to the isoelastic form. The quantity 6/ is an unobservable "taste shifter" that accounts for intertemporal and interpersonal variations in needs that are not captured by the household size. Given this specification, (6.30) becomes (6.32) Aln(c,*//i,*) - V t A l n ^ - A l n e ? ) = -p-'Aln^+e," so that, up to taste shifters that can be treated as mean-zero error terms, the growth of per capita consumption is the same for all households. More generally, we might suppose that the growth of consumption depends on any factors that alter needs from period to period. However, and this is the main empirical predic- tion of complete insurance, the growth of each household's consumption should not depend on the change in that household's resources once the change in aggre- gate resources has been taken into account. The individual harvests are all pooled in the insurance scheme, and while the aggregate village harvest affects each person's consumption outcome, there is no further influence of individual circum- stance. While I have presented the analysis in terms of a village insurance scheme, exactly the same result can be obtained by using financial markets, provided there is a complete set of assets, meaning that for each state and time period there exists an asset that will pay one unit if that state occurs in that time period, and zero otherwise. These securities are then freely traded among the participants and their prices competitively set, and will enable each person to achieve exactly the same outcomes as those described above (see Deaton 1992c, pp. 34-37, for an exposi- tion in this context). This market outcome is simply the decentralized version of the "central planner's" problem that here is handled by the village mutual insur- ance scheme. Empirical evidence on consumption insurance Empirical evidence on consumption insurance in developing countries comes from Townsend (1994), who looks at the ICRISAT villages in southwestern India, and from Townsend (1995), who looks at amphoes (roughly, counties) in Thai- land. The ICRISAT data track individual households over a number of years, and their consumption streams appear to move more closely together than do their SAVING AND CONSUMPTION SMOOTHING 577 income streams, something that is certainly consistent with (6.32), although also with standard autarkic intertemporal smoothing, with or without borrowing con- straints. However, Townsend finds that the growth of household consumption is positively influenced by household income flows, which should not happen if consumption is completely insured. There are no similar panel data for Thailand, but Townsend uses successive surveys to trace income and consumption changes at the amphoe level, and once again finds that the change in amphoe consumption less the average national consumption change is affected by the change in am- phoe income. Of course, the absence of consumption insurance over such wide areas does not imply that there is not perfect insurance at the village or lower level. The Ivorian Living Standards Survey data can be used to look at consumption insurance at the village level, and the following results are based on the (early release version of the) data from 1985, 1986, and 1987, which between them con- tain two two-year panels each intended to cover 800 households. Each survey has 100 sample clusters, 43 of which are in urban areas, so that there are 57 clusters corresponding to rural villages, each of which is designed to contain 16 house- holds. There are 31 village clusters that are common to the 1985 and 1986 sur- veys, and which comprise the first panel. In 1986 and 1987 there are 26 common panel clusters; these are not the same villages as made up the first panel. The structure of the sample is shown in Table 6.2. These figures are for the house- holds actually used in the analysis below; inevitably, some households have been excluded. The majority are households that produced no useable data but a few others were excluded because the data on income or consumption were implausi- ble either in levels or in changes. Consumption and income totals are the sums of a large number of components, more than a hundred for consumption, and several hundred for income, and for all but a very few cases it was possible to repair the estimates by identifying and replacing a component. There are three regions of the country shown in the table. The West and East Forest comprise the tropical south of the country. Both areas derive much of their incomes from cocoa and coffee production, while the northern Savannah depends on rainfed agriculture, with cotton and rice the main cash crops. Table 6.2. Structure of the samples, Cote d'lvoire, 1985-86 and 1986-87 (numbers of households and clusters) 1985-86 1986-87 Region Clusters Households Clusters Households West Forest 11 150 5 77 East Forest 12 181 11 165 Savannah 8 123 10 150 All rural 31 454 26 392 Note: One cluster in the West Forest contains only a single household. 378 THE ANALYSIS OF HOUSEHOLD SURVEYS Table 6.3 shows estimates of income and consumption for the panel house- holds in each of the years covered by the surveys. There are two sets of estimates for 1986 because that year appears in both of the panels, but the two estimates come from different households in different clusters. As always, and for the rea- sons discussed in Chapter 1, the consumption estimates are almost certainly a good deal more reliable than those for income. Even so, for the purposes of this analysis, it is less important that income be precisely measured than that it capture at least some of the idiosyncratic resource flows to the household. Income from sales of cocoa and coffee, which is an important component of agricultural in- come in the Forest, is almost certainly well measured; the crop is acquired by a parastatal marketing board at a predetermined price—a price confirmed by the survey data—and the quantities are carefully measured and credibly reported. Table 6.3. Income and consumption estimates, 1985,1986, and 1987 (thousands of CFAS, annual rates, household averages) 1985 1986(1) 1986(2) 1987 Region c y c y c y c y West Forest 1,188 1,098 903 790 1,026 1,225 893 1,151 East Forest 1,005 896 1,042 1,056 1,147 1,283 1,048 1,151 Savannah 671 630 828 788 770 742 659 716 All rural 975 891 938 896 977 1,061 867 1,011 Note: c is consumption and y is income. The two sets of numbers for 1986 are for the panel households held over from the first year (1) and for the new panel households that were carried forward into 1987 (2). Changes in incomes and consumption are shown in Table 6.4, together with F- tests for the significance of cluster effects. These F-tests, which can be obtained straightforwardly in STATA by regressing consumption and income changes on village dummies, or by the use of the "anova" command, measure the extent to which changes in income and consumption are more similar within villages than between villages. Statistically insignificant values of the tests indicate that changes in household income or consumption within the same village are no more similar than changes in income and consumption from households in different villages. Large F-values for income change would tend to indicate the presence of covariant risk, most plausibly coming from the similarity of weather for house- holds located in the same village. We would also expect large F-values for con- sumption changes in the presence of consumption insurance at the village but not national level. Village insurance schemes will induce a stronger comovement in consumption within villages than is observed between villages, and we would expect the F-statistics to be larger for consumption, which is smoothed over households, than for income, which is not. The statistics for income changes are generally low; common village compo- nents explain very little of the variation in individual income changes. In the 1985-86 panel, the F-statistic for the West Forest has an associated p-value of SAVING AND CONSUMPTION SMOOTHING 379 Table 6.4. Income and consumption changes, 1985-86 and 1986-87 (thousands of CFAS, household averages) 1985-86 1986-87 Income Ay F df{ 4T2 Ay F 4T, df2 West Forest -308 2.04 10 139 -74 1.93 4 72 East Forest 160 1.28 11 169 -64 2.05 10 154 Savannah 159 1.52 7 115 -26 1.17 9 144 All rural 5 2.30 30 423 -51 1.65 25 370 Consumption Ac F #1 #2 Ac F 4f, #2 West Forest -285 4.10 10 139 -133 4.04 4 72 East Forest 37 4.15 11 169 -99 2.16 10 154 Savannah 157 1.35 7 115 -111 0.70 9 114 All rural -37 4.62 30 423 -110 2.14 25 370 Source: Author's calculations using CILSS data. 0.03, and that for the rural areas as a whole one of 0.0002, albeit from a much larger sample size. Clearly, a few of the West Forest villages did worse than oth- ers in 1986, and it is those same income falls in one region that generate the F- statistic for all regions. But these effects, as well as those for 1986-87, are much smaller than might be expected if we think that covariant agricultural risk is im- portant. None of the F-statistics are close to the log(sample size) critical value of the Schwarz test. Nor is it easy to explain these results by appealing to measure- ment error. If, for example, the observed within-village variation in income changes is composed of equal parts of measurement error and "real" change, then it is easy to show that the observed F-statistics would be a little more than half what they ought to be. But doubling the figures in the top half of the table still does not generate very impressive values. The literature on agricultural risk in developing countries places great empha- sis on the covariation between income risks for different fanners in the same location, and the consequent desirability of risk-pooling schemes that link the village to different agricultural zones or to urban areas. There is little evidence of such covariation in these data, even though they span at least two quite distinct agricultural zones, the Forest and the Savannah. Cote d'lvoire may be an excep- tion to the general pattern, or these two years may be atypical in some ways. One possibility is that there is little covariance in normal years, but that there are infre- quent but important events, such as fires or droughts, that affect everyone to- gether, and that it is these rare events that shape risk-bearing strategy. Some risks can be an ever-present threat without occurring often. Even so, crops in C6te d'lvoire are certainly affected by weather conditions, and fires in the cocoa- and coffee-growing areas are frequent occurrences, so that the absence of significant village-level effects remains something of a puzzle. It is also possible that village effects do indeed exist, but are swamped by intravillage variation from one house- hold to another. 380 THE ANALYSIS OF HOUSEHOLD SURVEYS Inspection of the lower half of Table 6.4 reveals that the F-tests for village effects in consumption changes are typically much larger than those for income changes. Although there is little evidence of village effects in the Savannah, there are strong effects in the East and West Forest regions in 1985-86, and in the West Forest in 1986-87. Some factor is causing consumption levels within each village to move more closely together than do income levels. Risk pooling is certainly one possibility. Another is that consumption is measured more accurately than income, and that for correctly measured magnitudes, the F-statistics would be closer. Nor is it difficult to think of explanations for intravillage comovement in consumption that do not involve risk sharing; the effective real rate of interest may vary between but not within villages, or there may simply be village taste effects related to the geographical distribution of tribal groups in Cote d'lvoire. Whatever its cause, the comovement does indeed exist; consumption changes are more similar within villages than between them, and are more similar than are income changes. Table 6.4 shows a close relationship across regions and over time between consumption change and income change. Table 6.5 explores these connections more closely. The top panel of the Table gives estimates for 1985-86, and the bottom panel for 1986-87. The first row in each case shows that for each of the regions the ordinary least squares regression of the change on consumption on the change in income generates a significant positive coefficient. These values differ a good deal from region to region, and from the first panel to the second, but the coefficient is always positive and the f-values are large, ranging from 3.2 to 8.8. Of course, this finding is consistent with complete risk sharing; consumption changes are determined by the village average change in resources, which—by construction—is positively correlated with the individual changes. However, if consumption insurance is complete, the coefficients on income change should fall to zero when the change in village average income is introduced into the regres- sion, or when village dummies are included, since the average change is a linear combination of the village dummies. Indeed, given the difficulties in measuring incomes, and given that there may be village resources that are not counted in individual incomes, the test is more convincing if village dummies are used. The second rows for each of the panels show that the introduction of village dummies makes little difference to the original income coefficients. The dummies themselves are often jointly significant, but their presence does not affect the esti- mates of the income coefficients, something that effectively follows from the result in Table 6.3 that the village dummies do not predict of income change. The coefficients on individual income changes remain significantly positive as before, in contradiction to the complete insurance story which would require them to be zero. One possible explanation for the results could be an upward bias in the income coefficients that comes about as follows. For many of these rural households a significant share of income and consumption is accounted for by food that is pro- duced and consumed by the household and neither sold nor bought in the market. A value is imputed to this home-produced food and the figure is included in both SAVING AND CONSUMPTION SMOOTHING 381 Table 6.5. OLS and IV estimates of the effects of income on consumption West Forest East Forest Savannah All rural OLS 1985-86 No dummies 0.290 (6.2) 0.153 (3.2) 0.368 (5.8) 0.259 (8.8) Village dummies 0.265 (5.7) 0.155 (3.5) 0.373 (5.7) 0.223 (7.7) Own income 0.265 (5.3) 0.155 (3.2) 0.373 (5.6) 0.223 (7.1) Village income 0.199 (1.4) -0.031 (0.2) -0.050 (0.2) 0.252 (3.0) IVE 1985-86 No dummies 0.192 (3.9) -0,003 (0.1) 0.271 (4.0) 0.126 (4.0) Village dummies 0.171 (3.5) 0.029 (0.6) 0.270 (3.8) 0.107 (3.4) Own income 0.171 (3.2) 0.029 (0.5) 0.270 (3.7) 0.107 (3.1) Village income 0.161 (1.1) -0.417 (2.0) 0.020 (0.1) 0.144 (1.6) OLS 1986-87 No dummies 0.458 (8.8) 0.162 (5.3) 0.168 (4.0) 0.239 (10.4) Village dummies 0.424 (8.1) 0.173 (5.6) 0.164 (3.8) 0.235 (10.1) Own income 0.424 (7.9) 0.173 (5.3) 0.164 (3.8) 0.235 (9.7) Village income 0.350 (2.0) -0.094 (1.0) 0.061 (0.4) 0.039 (0.5) IVE 1986-87 No dummies 0.418 (7.8) 0.090 (2.8) 0.088 (2.0) 0.177 (7.4) Village dummies 0.388 (7.3) 0.105 (3.2) 0.087 (1.9) 0.177 (7.3) Own income 0.388 (7.1) 0.105 (3.1) 0.087 (1.9) 0.177 (7.0) Village income 0.353 (2.0) -0.127 (1.3) 0.015 (0.1) -0.002 (0.0) Note: Absolute values of r-values are shown in brackets The first row of each panel shows the coefficient on income change of a regression of consumption changes on income changes. The second row reports the same result when village dummies are included in the regression. The third and fourth rows show the estimates from a regression of consumption changes on individual household and village average changes in income. The iv regressions use the change in the value of cash income, individual and village average, as instruments for total income including imputations; the t-values on these instruments in the first-stage regressions are large, typically larger than 30. Because village dummies "sweep out" the village means, the coefficients—but not the standard errors—are identical in the second and third rows in each panel. income and consumption. In consequence, any errors in this imputation—and finding the correct price for the imputation is an obvious source of difficulty— will add a common error component to whatever other errors of measurement are present in the income and consumption totals. Positively correlated measurement errors do not necessarily generate an upward bias in the regressions in Table 6.5. For positive coefficients, the upward bias works in the opposite direction to the standard downward attenuation bias produced by the income errors alone, and the net effect cannot be signed in advance. To try to correct for these problems, I have calculated IV estimators using as an instrument income changes excluding changes in the imputed value of home-produced food and other income in kind. This quantity is a major component of the income change, and is therefore strongly positively correlated with it, but by excluding the imputations, the spuri- ous correlation is avoided. Note that I am not challenging the appropriateness of 382 THE ANALYSIS OF HOUSEHOLD SURVEYS including these imputations in both income and consumption. Rather the IV tech- nique is an attempt to correct for the consequences of measurement error given the imputation procedures. The IV estimates in the lower second and fourth panels show that these con- cerns have some foundation, especially for the East Forest and the Savannah, where the estimated marginal propensities to consume are a good deal lower. Indeed for the East Forest in 1985-86 and for the Savannah in 1986-87, there is no longer any significant relationship between consumption and income changes. However, the instrumentation makes no difference to the relationship between the estimates with and without the village effects. As was the case for OLS, the intro- duction of the village dummies does not shift the estimates, and for the West Forest in both panels, the East Forest in the second panel, and the Savannah in the first, there remains a strong positive relationship between consumption change and income change that is robust to the inclusion of the village dummies. These results seem to provide fairly firm evidence against the most extreme hypothesis, that there is complete consumption insurance within each of these villages in C6te d'lvoire. But given the moral hazard, incentive, and information difficulties associated with complete consumption insurance, it is hardly surpris- ing that there should exist at least some link between individual effort and its consumption reward. What would be more interesting would be evidence on whether there is partial insurance between households in the same village. One possibility is that the change in consumption for each household is partly deter- mined by its own income change, and partly determined by the change in average village income, so that (6.33) Ac," = a + pAy,* + YAy, + e? where A y f is the change in average village income. If there is no saving in the village as a whole, so that the average income change is equal to the average con- sumption change, then a = 0 and y + p = 1, but it is not necessary to impose such a restriction, especially given the measurement errors in income changes, and the fact that the average village income changes will be more accurately measured than the individual income changes. Our concern here is simply whether, condi- tional on individual income change, there is a role for the village average income change in explaining individual consumption changes. This test, which is the converse of the test for complete consumption insurance, is a test for whether there is any role for village-level insurance in an otherwise autarkic world. The results are shown in the third and fourth rows of each of the four panels of Table 6.5; the third row shows the coefficient of individual income, which is (mechanically) identical to the coefficient in the presence of dummy variables, and the fourth row shows the coefficient on the change in the village averages. As before, the second and fourth sets of results are estimated by IV; by analogy, I have used the change in average village cash income as an instrument for the change in average village income. In the East Forest and in the Savannah, there is no effect of the village averages on individual consumption. However, the con- sumption changes of households in the West Forest show some effect of average SAVING AND CONSUMPTION SMOOTHING 383 income, especially in the second panel from 1986-87. This effect is robust to instrumentation, and in this single case, the effect of average village income is almost as large as the effect of individual income. But given the failure of this effect to appear in the other case, as well as the possibility that the average in- come change is simply better measured than the individual income change, these estimates from the West Forest provide no more than a tantalizing suggestion that there is some risk sharing taking place at the village level. 6.5 Saving, consumption, and inequality This section describes research by Deaton and Paxson (1994b, 1995) on the re- lationship between saving, inequality, and population growth. If consumption insurance is not perfect, as the empirical evidence shows, then, at least under some circumstances, the process whereby a group of individuals save and dissave in order to smooth their consumption will lead to widening consumption dispari- ties within the group. Even if everyone gets the same luck on average, the accu- mulation of assets and ultimately the level of consumption will depend on their accumulated luck. A group of people flipping unbiased coins will each throw heads half the time if they throw for long enough, but there will be steadily in- creasing inequality in the total numbers of heads accumulated by each member of the group. In the same way, a group of farmers who are not co-insured, and who borrow and lend to smooth their consumption as in the permanent income model, will show gradually expanding inequality of assets and consumption over time. An implication of the analysis is that the inequality of income and consumption within any given cohort should be increasing with the age of the cohort. In conse- quence, the decline in the rate of population growth that comes with the demo- graphic transition, and which expands the numbers of older relative to younger people, will also tend to increase aggregate inequality, as population weights are shifted from relatively equal young cohorts to relatively unequal older cohorts. In this section, I explain the theory behind these mechanisms and use the Tai- wanese household survey data to show that the predictions are indeed supported by the evidence. I also explain how the aging of the population that accompanies a slowing of population growth acts so as to increase inequality, again with illust- rations from the Taiwanese case. Taiwan (China) is one of the leading cases of "growth with equity" in economic development, and has experienced rapid rates of economic growth while maintaining low measures of inequality. However, there has been some upward drift in measures of inequality over the last decade, and at least some of this can be attributed to the mechanisms described here. The increase in inequality, far from being a problem, is a concomitant of the essen- tially welcome decrease in rates of population growth. Consumption, permanent income, and inequality The simplest relationship between intertemporal choice and inequality is for the permanent income hypothesis with uncertainty. Recall from Section 6.3 that the 384 THE ANALYSIS OF HOUSEHOLD SURVEYS permanent income hypothesis extends the constant lifetime consumption rule to one in which consumption does not change except when the consumer receives new information, so that today's expectation of consumption in any future period is the same as today's consumption, equation (6.15) above. If we write this in the form (6.16), and add an i suffix to distinguish each individual, we have (6.34) c,,+1 = <* + n /M so that w. /+1 is the component of consumption in period N-l that comes from new information received by / after period /. Given what is known in period /, this innovation has a zero expectation. Take variances of both sides of (6.34) over the cross section of individuals that are present at both / and H-l, and assume that the cross-sectional covariance of cit and M. /+I is zero, so that (6.35) varc,+1 = varc, + where the variances are taken across the cross section of individuals, and o/+1 is the cross-sectional variance of consumption innovations. If we measure consump- tion inequality by the variance, inequality is increasing through time. The result can be made a good deal stronger if we are prepared to assume that the innova- tions are independent of the previous level of consumption. Given this assump- tion, (6.34) says that the distribution of consumption at H-l is the distribution of consumption at time t with the addition of mean-zero random variation, so that the Lorenz curve of consumption in / will lie everywhere within the Lorenz curve of consumption in any later period. In consequence, any measure of inequality that respects the principle of transfers will show increasing inequality over time (see Section 3.1). What is the intuition behind the result, and how robust is it to relaxation of the various assumptions necessary to derive it? A useful way of looking at (6.34) is to write consumption at time t as original consumption at time zero plus the sum of all the subsequent innovations: / (6.36) cit = c w + E w . T . T=l Each of the individual innovations M / T has zero mean, and their average will con- verge to zero given a long enough period of time. But consumption does not depend on the average, but on the sum, and the variance of the sum is increasing with time. While good luck and bad luck can be expected to average out over time, consumption depends on cumulated luck, and the cumulated histories of fortune for different individuals can be expected to diverge with the passage of time. Note the crucial role of uncertainty in driving the expanding inequality. Without uncertainty, the permanent income hypothesis predicts constant con- sumption streams over time, inequality is set by the distribution of lifetime re- sources, and once set, remains constant over time. In the presence of uncertainty, SAVING AND CONSUMPTION SMOOTHING 385 forward-looking permanent-income consumers adjust their consumption only when they get new information, so that provided everyone does not get the same news, the flow of new information will constantly act to widen the distribution of consumption. The consumption streams of any group of people will be "fanning out" over time. The reasoning requires no assumption about what is happening to earnings of different people, and it is perfectly possible for the permanent income hypothesis to hold for a group of people for whom the distribution of earnings is constant over time. For such a group, the expanding distribution of consumption is sup- ported by an expansion in the distribution of assets. The lucky consumers, those who have a positive run of innovations, will accumulate assets, and the income from those assets permits higher consumption. As a result, not only is the distri- bution of consumption becoming more dispersed with time, but so is the distribu- tion of income, provided that income is (properly) taken to include both earnings and asset income. Formally establishing this result takes some algebra, and the interested reader is referred to Deaton and Paxson (1994b). The fact that the inequality of consumption and income is increasing over time for any group of people does not necessarily imply that inequality is increasing for society as a whole. For any given cohort, consumption and income inequality will increase as the cohort ages, so that inequality is increasing over time for any given group. What happens to aggregate inequality will depend on whether old generations pass on their inequality to their children, so that each new generation has a higher level of inequality at the start of their lives, and overall inequality is increasing, or whether there is no inheritance of inequality, so that each gene- ration begins with the same starting level of inequality, and aggregate inequality is constant. The analysis here casts no light on these questions, and so is consis- tent with any pattern of aggregate inequality. The simple story of within-cohort inequality growth that comes from the permanent income hypothesis has to be modified in various ways once the as- sumptions are made more relaxed, and more realistic versions of the model are discussed in Deaton and Paxson (1994b). In summary: in the presence of covari- ant risk, consumption innovations will be correlated across individuals, and it is possible for there to be a nonzero cross-sectional covariance between the innova- tions and lagged consumption. However, there are plausible assumptions under which this covariance has to average to zero over a run of years, so that even if inequality does not increase year by year, it must increase on average. It is also necessary to recognize that consumption will change from year to year, not only when there is new information as in (6.34), but also when needs change, for ex- ample as people age, or as household size and composition changes. The "taste shifters" are certainly capable of destroying the relationship between inequality and cohort age. For example, the dispersion of household size tends to contract during late middle age of household heads, and there is likely to be an associated contraction in the dispersion of household consumption and income. Even so, the spread of inequality associated with good and poor fortune will still be present, and will be superimposed on whatever changes are wrought by changes in house- 386 THE ANALYSIS OF HOUSEHOLD SURVEYS hold size or tastes. Finally, if we move away from the permanent income hypoth- esis, and allow for precautionary motives in saving, the spread of inequality can be slowed down or even halted. Consumption becomes more unequal because people respond to the opportunities that fortune lays before them. Extreme precautionary behavior makes people reluctant to respond to opportunities, and a cohort of savers can be prudent enough to choke off the inequality-promoting effects of uncertainty. In spite of all the qualifications, there remains a presumption that intertem- poral choice in the presence of uncertainty will generate increases in inequality within a group of people. This presumption does not extend to all other models of consumption. In particular, consider the buffering model in Section 6.3 and the model of complete consumption insurance in Section 6.4. In the former, where consumers are relatively impatient but cannot borrow, the consumption of each household follows a stochastic process like those described in Figures 6.8 and 6.9. As the figures illustrate, and as can be formally demonstrated, the stochastic, pro- cesses for consumption are stationary so that if we look at a group of consumers, all of whom are buffering their consumption, the distribution of consumption among them will not be spreading out over time, and there will be no trend in inequality. Because all of these consumers are impatient, none are accumulating assets in the long term, and since borrowing is ruled out by assumption, none are accumulating debts, so that in all but the short run, consumption moves with earnings. In consequence, there will be growing consumption inequality if and only if the inequality of earnings is growing. The case of complete consumption insurance, although very different from autarkic buffering behavior by individual households, has the same implications for the development of inequality. The operation of the insurance scheme will cause the marginal utilities of individual consumption levels to move in lockstep, so that unless there are systematic divergences in tastes over time, there is no scope for an increase in the dispersion of consumption. Indeed, it is precisely to prevent the dispersion of consumption that the insurance scheme exists. It is possible for there to be increasing consumption inequality for reasons dif- ferent than those given above. In particular, if the distribution of earnings within a cohort is dispersing with the age of the cohort, and if households are liquidity constrained so that consumption is tied to income over all but the short run, then consumption inequality will be driven by the rising inequality in earnings and will also increase. That the inequality of earnings should rise with age is predicted by standard models of education and earnings in which different people have differ- ent amounts of education (see Mincer 1974 and Dooley and Gottschalk 1984). Even within a homogenous education group, earnings inequality could increase if different individuals experience different rates of return to human capital. Inequality and age: empirical evidence In order to test the prediction that inequality increases with time within any given group of people, we need either panel data, in which the members of the group SAVING AND CONSUMPTION SMOOTHING 557 are tracked over a period of time, or a time series of cross sections through which we can follow the randomly selected representatives of a number of cohorts as in Section 2.6.1 look first at the latter, using the repeated cross sections from Tai- wan (China), and then briefly consider evidence from the ICRISAT panel of house- holds in India. Figures 6.10 and 6.11 are reproduced from Deaton and Paxson (1994b) and show how consumption inequality evolves with the age of the household head in Taiwan (China) from 1976 through 1990. Figure 6.10 shows the variance of the logarithm of consumption for every fifth cohort, as measured by tracking the cohort through the surveys. For example, the top left graph in the figure shows the cohort born in 1961, whose members were 15 years old in the first survey year, 1976. Of course, there are no 15-year-old household heads, so that the first time we observe the cohort is in 1981, when they are 20, and when at least a few show up as heads. The subsequent points in the graph show the variance of loga- rithms of household consumption for households headed by 21-year-olds in 1982, 22-year-olds in 1983, through to 29-year-olds in 1990. The series of graphs show that there is little increase in inequality at young ages, but once household heads reach age 40, the inequality lines begin to turn upward, rising quite rapidly through the 40s and 50s, with dispersion ceasing or reversing after age 60. The final panel of the figure shows the variance of log consumption for all households in Taiwan (China) tracking through each of the 15 surveys. As documented else- where (Republic of China 1989), there has been a mild increase in consumption Figure 6.10. Variances of log household consumption, selected cohorts, Taiwan (China), 1976-90 0.8- Agel5inl976 Age 20 in 1976 Age 25 in 1976 Age 30 in 1976 0.4- 0.0- 0.8- Age 35 in 1976 Age40inl976 Age 45 in 1976 Age 50 in 1976 E .a Age 55 in 1976 Age 60 in 1976 •s Full sample 0.4. Age 65 in 1976 0.0- 75 80 85 90 75 80 85 90 75 80 85 90 75 80 85 90 Source: Deaton and Paxson (1994b). 388 THE ANALYSIS OF HOUSEHOLD SURVEYS Figure 6.11. Age effects in (log) consumption inequality, Taiwan (China) 0.6 _ > uopduinsuoo Soj jo GDUEUBA sqi ui siDOjga a3y Inequality plus two standard errors O O Inequality minus two standard errors 0.0 — C I I I I 20 40 60 80 Age Source: Deaton and Paxson (1994b). inequality over the period, but it is dwarfed by the changes within the middle- aged cohorts. Figure 6.11 graphs the age effects from a regression of the variance of logs on age and cohort dummies as described in Section 2.6, and extracts the relationship between inequality and age from the multiple graphs in the previous figure. In- equality does not change very much until age 40, and then increases until age 60, after which there is no further dispersion. This is what is to be expected from the theory if dispersion is being driven by innovations in earnings, and if people either retire or have only predictable changes in earnings after age 60. The broken lines in the graph are drawn at two estimated standard errors from the age in- equality lines, and show that the results are not being driven by the fact that the cohort inequality measures are based on sample data. Deaton and Paxson (1994b, 1997b) provide similar analyses for Britain, the United States, and Thailand, with similar results. The shapes of the age inequality profiles are somewhat different. Taiwan (China) is the only country of the three to show the convex profile displayed in Figure 6.11, but the increase in inequality from age 20 to age 70 is the same in the four economies. In all four countries, the inequality profiles of earnings and income also increase with age, so that the results on consumption inequality are consistent, not only with the permanent in- come story of the previous subsection, but also with an explanation in which the fundamental cause of expanding inequality lies in dispersion of earnings with age, with consumption tied to earnings through borrowing restrictions. SAVING AND CONSUMPTION SMOOTHING 389 A very different context for examining inequality is provided by the house- hold panel data from three of the ICRISAT villages in Maharashtra and Karnataka in southwestern India. These households are followed over time, so there is no need to create cohort data, and we can look directly at the individual consumption trajectories. Figure 6.12 presents the evidence. Each panel comes from a separate village, and shows the deviation of each household's consumption from the mean logarithm of consumption taken over the sample households in the relevant vil- lage. The trajectories are thus centered around zero, and add to zero in each agri- cultural year (June to May). In none of these villages is there any apparent ten- dency for the bundle of consumption trajectories to fan out over time. Nor do cor- rections to the consumption data for household size in the form of division by the number of adult equivalents yield different results. These households are appar- ently not setting their consumption according to the permanent income hypothesis or one of the other models of intertemporal choice that predicts increasing in- equality in consumption. Although complete consumption insurance is consistent with constant inequality, we already know from Townsend's (1994) work on these data that household consumption levels respond to changes in individual incomes, so that insurance is at best imperfect. Figure 6.12 is more obviously consistent with the buffering model, in which households are not insured, but accumulate assets only in the short run. Household incomes in these villages also show no increase in inequality over time, so that both income and consumption data are consistent with the autarkic behavior of impatient and liquidity-cons- Figure 6.12. Consumption trajectories for households in the Indian ICRISAT villages, 1976-91 4- Aurepalle Shirapur ireaui 3§ej|iA UIQJJ uopduinsuoD §oj jo UOHBIAOQ 76 77 78 79 80 81 4- Kanzara 2- 0- -2- 76 77 78 79 80 81 Source: Author's calculations from ICRISAT data. 390 THE ANALYSIS OF HOUSEHOLD SURVEYS trained consumers, who smooth their consumption as best they can without help from their neighbors. Aging and inequality In economies where inequality rises with age, as in Taiwan (China), the United States, and Britain, changes in the age structure of the population will affect over- all levels of inequality. In particular, declines in the rate of population growth will tip the structure of the population towards the elderly, so that the older, more unequal generations will become more heavily represented and the younger, more equal generations will be less heavily represented, a process that will tend to increase overall inequality. The age-sex pyramids for Taiwan (China) in Figure 1.2 become steadily more top-heavy over time as the population ages, and Taiwan (China) is only one of many developing countries in Asia where there have been similar declines in population growth rates, and similar increases in the average age of the population. If aging in these economies has an automatic tendency to increase inequality, it is important that the mechanism be understood and quanti- fied, so that the process is not misinterpreted, and inappropriate policy action taken to try to correct a non-problem. The relationship between population aging and inequality is not as straight- forward as this summary might suggest, but a number of results can be proved under the appropriate assumptions. One approach that requires strong assump- tions about economic growth and the inequality process but only very weak as- sumptions about population structure is based on the analysis of inequality in Chapter 3, and is adapted from a related analysis by Anand and Kanbur (1992). Suppose that F(x) is the cumulative distribution function of consumption (or income) in the population as a whole, and that F,(x) is the corresponding cumu- lative distribution function for age group i. These functions are then related by the identity T (6.37) F(x) = Ew,F.(jr) i=0 where T is the age of the oldest group and it. is the proportion of the population aged i. We might then consider social welfare and inequality in this society through the function r (6.38) W = fv(x)dF(x) = Enifi)(x)dFi(x) J /=o J where v(x) is a monotone increasing concave function, which can be thought of as an individual or social evaluation function for the welfare of each individual. Suppose that inequality is being generated according to the simplest version of the permanent income hypothesis described at the beginning of the previous sub- section, so that consumption inequality is fanning out with age, and that the distri- bution of consumption at any age is second-order stochastically dominated by the distribution of consumption at any earlier age. Suppose too that the distribution of SAVING AND CONSUMPTION SMOOTHING 391 consumption within the economy is stationary over time, so that the distributions of consumption for each age group in the cross section are the same as the distri- butions of consumption for each cohort as it ages. This means that the distribution of consumption among the old now is the same as the distribution of consumption will be among those who are young now when they are old, so that economic growth (or decline) is ruled out by assumption. Given these assumptions, any aging of the population will unambiguously increase inequality. To see the result, imagine a new age distribution ffc . that is the same as the old, but with a transfer from a younger to an older age group, so that for some j > k, and 5 > 0 it. = w r a, i=t (6.39) ft. = 71. + 8, i =j = §9fc %i ^i* « 7.*- From (6.38), we then have at once that (6.40) W - W = 5 [fv(x)dFk(x) - Ju(jc) 0 where the inequality follows from the fact that when kfv(x)dF.(x). The result (6.40) shows that, under the assumptions, any transfer of population from younger to older age groups will result in a distribution that is stochastically dominated by the original distribution. Since the assumptions guarantee that the mean of the distribution is unchanged, population aging shifts out the Lorenz curve, with an unambiguous increase in inequality according to any measure that respects the principle of transfers. Once we allow for economic growth, it is no longer possible to establish such general results on the relationship between aging and inequality. When consump- tion is growing from one generation to the next, older cohorts are poorer over their lifetimes than the younger cohorts, so that the lifetime experience of each cohort will be quite different from the cross-section relationship between con- sumption and age at any given moment of time. Aggregate inequality depends, not only on inequality within cohorts, as in the analysis above, but also on the distribution of consumption between cohorts. In consequence, even if we suppose that age profiles of consumption are unchanged in the face of changes in the age structure of the population, such changes will change not only inequality, but also means. Even so, it is possible to make some fairly general statements provided that we restrict ourselves to stable demographic equilibria associated with con- stant rates of population growth, and provided we work with a specific inequality measure, the variance of logs. 392 THE ANALYSIS OF HOUSEHOLD SURVEYS Deaton and Paxson (1995, 1997b) show that in demographic equilibrium, the variance of log consumption is a decreasing function of the rate of population growth provided that the lifetime consumption profile for each cohort displays steady growth, steady decline, or is flat. The most important practical case where the result fails is when the cross-sectional consumption to age profile is increasing with age, but is concave, so that the lowest levels of consumption are found among the youngest households, either because they have low income and face borrowing restrictions, or because they contain a large proportion of children. If these households show the lowest levels of consumption, and are the major con- tributors to the cross-cohort component of inequality, aging the population by redistributing people from these groups to older groups will decrease cross-cohort inequality, and the effect is potentially large enough to overcome the increase in the weight attached to the within-cohort inequality of the older groups. Under the life-cycle model, higher rates of economic growth make such effects less likely, because the higher the rate of growth, the higher are the lifetime wealth levels of younger relative to older consumers, which will tend to tip the cross-sectional consumption-to-age profile in a clockwise direction, raising the consumption of the young relative to that of the old. As we shall see below, there is not much evidence that this effect exists in the data; instead, the age profile of consumption is determined by the age profile of earnings. Table 6.6 provides some illustrative calculations based on Taiwanese para- meters. They are based on the validity of the life-cycle model, with the consump- tion-age effects taken from the calculations reported in Section 6.2 above. These age effects determine the age profile of consumption for any given cohort, and the assumed rates of economic growth determine the relative lifetime wealth levels for each cohort. Higher economic growth rotates clockwise the cross-sectional consumption-age profile, and thus changes the intercohort component of inequal- ity. The estimated age effects, shown in Figure 6.6, rise sharply with age, so that we are in the situation described in the previous paragraph where increases in the rate of population growth increase intercohort inequality when economic growth is low and the cross-section consumption profiles are steeply upward sloping, but decrease intercohort inequality when economic growth is high and the cross- sectional age profiles are flatter. In these calculations, the middle figure in each row is increasing to the right when economic growth is 0 or 2 percent, is non- monotonic at 4 percent, and is decreasing at 6 percent. Increased rates of popula- tion growth always decrease the intracohort component of inequality, a compo- nent that is unaffected by the rate of economic growth, and is shown in the first row of each panel of the table. In these particular calculations, the effects of pop- ulation growth on intercohort inequality is larger than the effect on intracohort inequality, and it is the former that dominates the overall result/However, the large intercohort effects come from the assumption that economic growth does in fact tip the cross-sectional consumption profile—something that probably does not happen in reality—and are thus likely to be overstated. Although it will take a good deal more research to refine these sort of calcu- lations, the table illustrates plausible magnitudes for the effects of economic and SAVING AND CONSUMPTION SMOOTHING 393 Table 6.6. The relationship between inequality and growth (Taiwan (China), hypothetical calculations: variances of logarithms) Percentage rate of population growth 0 2 4 Consumption growth = 0 Within-cohort inequality 0.289 0.262 0.240 Between-cohort inequality 0.261 0.309 0.347 Total inequality 0.550 0.572 0.588 Consumption growth = 2 percent Within-cohort inequality 0.289 0.262 0.240 Between-cohort inequality 0.082 0.112 0.142 Total inequality 0.371 0.374 0.382 Consumption growth = 4 percent Within-cohort inequality 0.289 0.262 0.240 Between-cohort inequality 0.037 0.035 0.042 Total inequality 0.326 0.297 0.283 Consumption growth = 6 percent Within-cohort inequality 0.289 0.262 0.240 Between-cohort inequality 0.126 0.080 0.049 Total inequality 0.416 0.342 0.289 Source: Deaton and Paxson (1995, Table 2). population growth on aggregate consumption inequality. Given that aggregate in- equality is usually rather insensitive to economic change, with inequality mea- sures changing little over long periods of time, the effects shown in the table are large. For example, if we assume the distributions of consumption are (at least ap- proximately) lognormal, the variances of logarithms can be converted into Gini coefficients (see Aitchison and Brown 1969, pp. 12-13 for the formula and its derivation). Table 6.6 then shows that, at a 6 percent annual rate of growth of consumption per head, the difference between 3 and 1 percent annual population growth rate is the difference between a Gini coefficient of 0.309 and 0.352, a difference that would typically be sufficient to cause concern if not correctly interpreted as a composition effect. 6.6 Household saving and policy: a tentative review This chapter, like the research it has reported, has inevitably been fragmentary; there is no coherent story in which the fragments can be set, and that tells us what is the role of saving in economic development and in the lives of people in deve- loping countries. In this final section, I try to pull together, not a complete ac- count, but some themes that have appeared in different places in the chapter, and to address the issues that are most important for policy. 394 THE ANALYSIS OF HOUSEHOLD SURVEYS Motives, consequences, and policy Some people save to accumulate wealth, for themselves or for their heirs. Others save so as to have something for a "rainy day," a rainy year, or a rainy retirement, and yet others save because they fear the consequences of facing an uncertain future with no assets. Saving may even reflect the lack of suitable consumption opportunities, or simply the inability of a traditionally poor people to adjust their habits quickly to their newfound prosperity. Where credit markets are poorly developed, people need to save in anticipation of large purchases of items such as houses or cars. Different motives generate different patterns of saving, among individuals and in the aggregate economy, and imply different responses of indi- vidual welfare and of aggregate saving to policies such as interest rates, direct saving incentives such as tax deductions, or the provision of state-sponsored social security. Empirical research to date has found at least some support for several different interpretations of saving behavior, but failures of measurement, and perhaps of imagination, have prevented economists from coming to the solid understanding that could serve as a basis for policy. Recent research has focussed on interpretations of saving that are associated with smoothing consumption, whereby individuals free their consumption from earnings in either (or both) the short run (seasons, years) or long run (lifetimes). The results reviewed in this chapter provide good evidence that households in poor countries are capable of purposeful intertemporal allocation of resources, at least sufficient to help protect their consumption against seasonal and year-to-year fluctuations in incomes. The evidence that individuals or households in develop- ing countries make effective provision for retirement by saving is weaker or nonexistent, nor is it even clear that it makes sense for them to try given tradi- tional social and family arrangements. I shall return in the next section to the implications of this failure of the life-cycle model, but the success of short-term smoothing is itself of major importance for thinking about policy. Many countries operate tax and pricing schemes, the ostensible purpose of which is to stabilize farmers' incomes; for example, many countries in Africa pay their farmers a price for cash crops that is both more stable and a good deal lower than world prices. Although there are undoubtedly cases where price stabilization is used as a screen for taxation, there is an arguable case for the state attempting to stabilize farmers' incomes, though it is far from clear that price stabilization is the best way to do so (see Newbery and Stiglitz 1981). The arguments for stabil- ization typically assume a close relationship between farm incomes and consump- tion. When incomes are low, and credit markets are absent or imperfect, con- sumption will also be low, while high incomes will generate conspicuous and wasteful consumption or investment, since uneducated farmers cannot be ex- pected to make adequate provision for the future (see for example the statement quoted on p. 351 above). As a result, farmers need to be protected against their own fecklessness by the stabilization board. Not only do the empirical results suggest that these concerns are exaggerated, but there is also a considerable body of historical and contemporary evidence that governments find it very difficult to SAVING AND CONSUMPTION SMOOTHING 395 deal with the fluctuations in government revenues that are generated by the stabi- lization and taxation schemes (see for example Hirschman 1977; Bauer 1984; Bevan, Collier, and Gunning 1989; Deaton and Miller 1995; and Collier and Gunning 1997 for discussion and evidence). Of course, none of this should be taken to mean that poor agricultural communities are capable of costlessly dealing with all fluctuations in their incomes, nor that outside help should be denied in time of need, only that policy should not be based on a presumption that rural households are incapable of making sensible intertemporal choices, nor on the presumption that the state can do better. Section 6.4 discussed some of the recent research concerned with consumption insurance. This work is relatively new and the results are both controversial and unclear. Some of the empirical work appears to show evidence of the operation of local insurance schemes, although there is increasing agreement that such schemes cannot and do not provide the complete pooling of resources that would maximize the degree of insurance cover. This is a topic that deserves a great deal more research, in more and in different contexts, and with attention to the links with other literatures on risk sharing, credit markets, and sharecropping contracts. We need to know, not only whether and to what extent consumption is insured, but what mechanisms are involved. The understanding of this topic is once again a prerequisite for sensible policy design, particularly as the process of develop- ment inevitably brings greater demands for formal and state-sponsored insurance schemes. Saving and growth What have we learned from the household data about the macroeconomics of saving, and particularly about one of the longest-standing questions in economic development, the relationship between saving and growth? Is it true, as the classi- cal economists believed, and as has become credible once again, that higher sav- ing is the precursor of economic growth, so that the key to raising growth rates is to design incentives to encourage saving, almost certainly in conjunction with educational expansion so that physical and human capital formation are expanded together? Or are high savings rates simply a symptom of rapid growth, whose wellsprings lie elsewhere? One of the great attractions of the life-cycle hypothesis is that it offers a coher- ent microeconomic theory of individual saving that is capable of delivering important macroeconomic conclusions through a process of explicit aggregation over households. Instead of simply invoking the anthropomorphic fallacy of aggregation, imputing human motives to the behavior of national averages, aggre- gate saving is explicitly linked to the rates of growth of income and of population. Faster growth, whether of population or of per capita income, expands the total scale of saving among the young relative to the scale of dissaving among the old and so raises the average saving rate. In a stationary population with constant per capita incomes, saving would be zero, even though each household saves at some point during its existence. On the surface, the empirical evidence supports this 396 THE ANALYSIS OF HOUSEHOLD SURVEYS account. There is a strong positive relationship between per capita growth and saving rates across the cross section of countries. Countries like Hong Kong, Indonesia, Korea, Taiwan (China), and Thailand, that have increased their growth rates, have also increased their saving rates, and in the industrialized countries where growth rates have been lower in the 80s than in the 70s, and lower in the 70s than in the 60s, there has been a corresponding and almost universal decline in national saving rates. Moreover, the magnitude of these effects is very close to that predicted by straightforward "back-of-the-envelope" calculations using the life-cycle model. Unfortunately, recent evidence has not been kind to this elegant and appar- ently satisfactory account. First, the simple cross-sectional age profiles of con- sumption tend to be similar to those illustrated in Figure 6.2 for Thailand and Cote d'lvoire; there is some evidence of hump saving in middle age, but it is limited in extent, and apparently unmatched by dissaving among the elderly, As discussed in Section 6.1 above, there are arguments that help reconcile these findings with the life-cycle model, but the extent of saving among the young and dissaving among the old (if it exists) is not sufficient to support an account in which growth is responsible for the increase in saving rates that appears in the data. Second, detailed analysis of the experience using repeated cross-sectional surveys does not support the life-cycle interpretation of secular changes in the saving ratio. Bosworth, Burtless, and Sabelhaus (1991) and Attanasio and Weber (1992) for the United States, Blundell and Banks (1992) for Britain, and particu- larly Paxson (1996) for the United States, Britain, Taiwan (China) and Thailand, show that changes in national saving rates are not aggregation effects associated with the relative wealth and size of different cohorts, but come from secular changes of saving rates within cohorts. Paxson's work, like the earlier work of Carroll and Summers (1991) also establishes that the cross-country correlation of saving with growth cannot be attributed to the life-cycle model. If the correlation between saving and growth does not come from life-cycle behavior, how can we explain it? The obvious possibility is that the causation runs in reverse, not from growth to saving, but from saving to growth. Such an explanation has always had its adherents among development economists, and Lewis (1954) was probably expressing no more than the then widely accepted view when he saw the problem of raising the saving rate as the central problem of economic development. The supposition that increases in saving rates could in- crease the steady-state rate of growth was (perhaps only) temporarily discredited by the Solow (1956) model, which showed that in a neoclassical framework, increases in the saving rate would increase the level of per capita income, but would raise the rate of growth only along the transition from one growth path to the other, not permanently. But as has long been known from the early work of Atkinson (1969), the transitions can last for a long period, and more recent empir- ical work by Mankiw, Romer, and Weil (1992) has shown that the Solow model is capable of giving a surprisingly good account of the data, with the ratio of investment to GDP an important determinant of growth. In a world of perfect international capital markets, there is no obvious reason why there should be a SAVING AND CONSUMPTION SMOOTHING 397 relationship between investment and saving ratios at the country level, but the correlation is nevertheless strong and positive—Feldstein and Horioka (1980)— even if weakening over time. Recent research on economic growth has also given a good deal of attention to models in which there are constant returns to scale to some broadly defined concept of capital—typically including human as well as physical capital—and such models clearly have the ability to generate permanent increases in growth rates in response to increases in the (broadly defined) saving ratio. Attributing differences in growth rates to differences in saving rates leaves aside the question of what determines international differences in saving rates. Treating saving rates as determined by tastes (the Confucian ethic) is hardly satis- factory and does a poor job (for example) of explaining why Korean and Taiwan- ese saving rates should only recently have become so high. Nevertheless, if the life-cycle explanation is ruled out, we are left with only partial and incomplete explanations. Determinants of saving One popular explanation of the rate of saving looks to interest rates, and sees the high real interest rates that accompany economic growth—in unofficial if not official markets—as promoting saving and providing the fuel for growth. In many countries, governments have attempted to keep interest rates low (once thought to be the recipe for high investment and growth) in order to keep down the costs of their own borrowing. The accompanying physical controls on rates—financial repression—are seen as discouraging saving, diverting what saving there is to the finance of government deficits, and stifling economic growth by limiting private investment. I have written elsewhere about the theoretical and empirical support for this argument (see Deaton 1990b but also Balassa 1990 and Fry 1988 for more sympathetic treatments). The arguments may be summarized briefly as follows. First, in spite of stubborn beliefs to the contrary, the economic theory of intertem- poral choice does not predict that higher interest rates raise saving, but is ambi- guous about the direction of the effect. Both the life-cycle hypothesis and the buffering model described in Section 6.3 are consistent with modest positive effects of interest rates on saving, but it is hard to make an argument for anything other than small effects. Second, the lack of interest rate effects on saving should not be confused with the existence of possibly large interest rate effects on portfo- lio allocation. Where households have been forced to buffer their consumption with expensive assets such as stocks of grain, or hard-to-protect assets like cash, the introduction of sound financial intermediation is likely to be welcome, and to generate a flow of "saving" into the institutions. But this is a portfolio reallocation effect, and does not represent a cut in consumption in response to the higher interest rate. Third, of the econometric work that shows strong interest rate effects, much is nonreplicable, and very little is credible. Most careful econometric work is consistent with the theoretical prediction, that interest rate effects are small or 398 THE ANALYSIS OF HOUSEHOLD SURVEYS nonexistent. Fourth, most households do not hold interest-bearing assets, in either rich or poor countries. Nonsmokers are not much affected by the price of tobacco, and people who hold no interest-bearing assets are not much affected by the price of assets. Even in the United States, where financial intermediation is as sophisti- cated as anywhere in the world, around a quarter of total household consumption is located in households where no one possesses any financial assets whatever, not even a bank or savings account. In the Ivorian LSS, there are no rural house- holds that admit to holding financial assets. It is certainly true that these people, like their counterparts in the United States, are not thereby absolved from making intertemporal choices, but the returns that they face, on loans from moneylenders or on their own physical investments, bear little or no relationship to the interest rates that are influenced by the government. A second avenue of research is to recognize that saving may be prompted more by motives of accumulation than by the need to transfer resources from youth to old age. Such an explanation is perhaps consistent with the evidence from Taiwan (China) in Section 6,1, where the elasticity of consumption with respect to lifetime resources is estimated to be substantially less than unity, so that, as people become better-off, they commit a larger fraction of their lifetime resources to accumulating bequests. It is also consistent with the recent upward reevaluation of the contribution of bequests to national wealth in the United States and other developed economies (see Kotlikoff and Summers 1981, Modigliani 1988, and Kotlikoff 1988). However, if it is true that bequests are a luxury good, saving rates will be positively correlated with the level of economic development, rather than with the rate of growth, which is what the data appear to show. A third promising research topic is the relationship between saving and hous- ing, an account that potentially combines life-cycle and accumulation motives. Saving for house purchase is an explanation that is frequently offered by high savers in Hong Kong, Korea, and Taiwan (China), countries where house prices are also extremely high relative to incomes, at least judged by similar ratios in the United States. High house prices are also cited as an explanation for the high saving ratios, though given limited land in these relatively small countries, it would make just as much sense to attribute the high land prices to the high saving ratios. Older people also emphasize the desirability of bequeathing housing to their heirs. As far as I am aware, no one has combined these sort of considerations into a coherent account of growth, land prices, and saving, let alone tested such a model against the data, but the topic deserves a good deal further investigation. A fourth explanation of saving is that consumption is subject to habits or "ratchet effects/* According to such models—which date back at least to Duesen- berry (1949)—it hurts more to reduce consumption than to increase it, so con- sumers respond to growth in their incomes only slowly, and attempt not to reduce their consumption when incomes fall. Alternatively, it might be supposed that tastes and technology of consumption respond slowly to growth, so that it takes time for Korean and Taiwanese entrepreneurs to construct the leisure and service industries that would facilitate the increase in consumption. According to such models, saving rates should be highest among those consumers who have recently SAVING AND CONSUMPTION SMOOTHING 399 experienced the most rapid rates of real income growth. As reported above, there is some support for this idea in the Taiwanese data, but the effect is much too small to explain much of Taiwanese saving rates, or to account for the cross- country relationship between saving and growth. The simple buffer-stock model of Section 6.3 also predicts a positive relation- ship between saving and growth, but as in the case of the habit model, the effect is too small. When households hold assets only for precautionary purposes, the amount will depend on the variance of earnings, rather than on its level or its ex- pected rate of growth. Suppose that in equilibrium, and for whatever reason, assets are a constant multiple of income. If income is growing at rate g, so must be assets, and since saving is the rate of increase of assets, we have AA s s y (6.42) g = ~— = 7 = -f- A A y A From the first and last equalities, we have immediately that (6.43) -=£-• y y The ratio of assets to income for buffering households will depend on how much smoothness people want, and how much they are prepared to pay for it. However, the ratio is only 0.075 in the simulation in Figure 6.8, and could hardly be more than a third, so that, according to (6.43), buffering cannot account for the cross- country relationship. For the equation to match the data requires an asset to in- come ratio of two to three, which is roughly in accord with actual national magni- tudes, which are also much the same as those predicted by stripped-down versions of the life-cycle model. All of which would suggest that the life-cycle model might do a good job of explaining the cross-country results. The difficulty, of course, is that the evidence is not consistent with the supposition that saving for retirement is such a large multiple of income. Disentangling these various explanations from one another and confronting them with the evidence is a major research task, and one on which we can expect progress in the near future. Progress to date has been hampered by data quality, especially in household surveys where saving rates are very low, and by the lim- ited information to be obtained from short aggregate time series, either for indi- vidual countries or from international comparisons. But there exist good house- hold surveys from many of the high-saving nations of East and South-East Asia, and as the information in these is sifted, our knowledge should greatly increase. 6.7 Guide to further reading There is an immense literature on saving in both developing and developed econ- omies, and only a few of the topics are discussed in this chapter. Gersovitz (1988) provides a comprehensive review of the literature on saving and development and deals with many of the topics that are not covered here. My own review for the 400 THE ANALYSIS OF HOUSEHOLD SURVEYS World Bank's first Annual Conference on Development Economics, Deaton (1990b), builds on Gersovitz's survey and also focusses on some of the same topics covered here. Besley's (1995a) review takes a broader perspective and is more up to date. Deaton (1992c) is a fuller treatment that contains some develop- ment material, but is more concerned with the American literature, much of which focusses on macroeconomic time-series issues (see also Browning and Lusardi 1996, for an update). Nevertheless, it provides an up-to-date treatment of the theory of intertemporal choice, and a review of the most important empirical results. There is much to be learned from going back to the classics, to Lewis (1954) and to Modigliani and Brumberg's (1954, 1979) original account of the life-cycle model. Modigliani's (1986) Nobel address is a particularly useful state- ment of the achievements of the life-cycle model, achievements that have been the basis for almost all subsequent work. Some recent work on social insurance and consumption smoothing, which also provides a further introduction to the literature, is contained in a symposium in the Journal of Economic Perspectives, Townsend (1995), Morduch (1995), and Besley (1995b). Code appendix This appendix provides the STATA code used to produce the results in the text. These are not intended to be self-contained; they cannot be run as they stand without modification and, at the very least, the addition of the user's own data set. The idea is not to supply a software package that will replicate the analyses in the text, but to provide the code to serve as a template for the user's own analysis. I have not provided code for straightforward cases such as tabulations or standard regressions. I have tried to add enough comments to the code to make it broadly comprehensible and to aid those who wish to translate it into languages other than STATA; in most cases, the translation should not be difficult. Apart from some later addition of comments, the code given here is the code that was actually used to produce the results in the text. I have taken a good deal of care to check the code, but I give no guarantee that it is error free, let alone that it will automatically work on different data sets and different machines. The code listed here is available on the following site: http://www.worldbank.org/householdsurveys A.I Chapter 1 There are three examples. The first is for the saving, expenditure, and income tabulation for Thailand in Table 1.3, the second for the alternative estimates of standard errors for mean PCE in Pakistan in Table 1.4, and the third for the bootstrap estimates of standard errors, also for Pakistan, discussed on pp. 57-58. Example 1.1. Saving, expenditure, and income in Thailand The program, which was used to calculate Table 1.3, takes data on income, expen- diture, and weights from a file thaicy. dta, creates a measure of saving, sorts by income and by expenditure, and creates deciles for each, taking into account the weights. Income, expenditure, and saving are then summarized by decile. version 4.0 #delimit ; References in this publication to "Taiwan," "Republic of China," and "Taiwan (China)" refer to the region, "Taiwan, China." References to "Hong Kong" refer to the region, "Hong Kong SAR, China." 401 402 THE ANALYSIS OF HOUSEHOLD SURVEYS cap log close; log using table!3, replace; drop _all; set more 1; * data with income, expenditure, and weights; use thaicy; * remove missing values; drop if incl==.; drop if expl--.; * getting the sum of the weights for normalization; qui summ weight; global bigw=_N*_result(3); *first sorting by household total income; sort incl; gen sumwt=sum(weight)/$bigw; gen dec=int(sumwt*10)+1; * puts the richest household in the top decile, not the llth; replace dec=10 if dec==ll; sort dec; *average saving by deciles of total household income; by dec: summ savl [aweight=weight]; *all; summ savl [aweight=weight]; *average income by deciles of total household income; by dec: summ incl [aweight=weight]; *all; summ incl [aweight=weight]; drop dec sumwt; *now repeating by household total expenditure; sort expl; gen sumwt=sum(weight)/$bigw; gen dec=int(sumwt*10)+1; replace dec=10 if dec==ll; sort dec; *average saving by deciles of total household expenditure; by dec: summ savl [aweight=weight]; summ savl [aweight=weight]; *average expenditure by deciles of total household expenditure; by dec: summ expl [aweight=weight]; summ expl [aweight=weight]; log close; Example 1.2. Estimating standard errors for average PCE in Pakistan The program takes data from a file consdata. dta, which contains information on total expenditure, household size, and survey design variables, that were taken from the expenditure files of the PIHS. It then computes weighted and unweighted means, together with various correct and incorrect variances and standard errors. This program should not be used in preference to the complex survey design procedures that became available in Version 5.0 of STATA. version 4.0 ttdelimit ; cap log close; log using table!5, replace; drop _all; set more 1; CODE APPENDIX 403 use consdata; *per capita household expenditure; gen pce=totmexp/hhsize; *stratum identifiers; gen strat=int(hhid/1000000); ^cluster identifiers; gen clust=int(hhid/1000); *province identifiers; gen prov=int(hhid/100000000); tab prov; keep pee strat clust weight prov; save temp, replace; ^defining the program to do the work; cap program drop surv; program def surv; dis("first the ordinary unweighted mean"); *ci provides a standard error of the estimate, col 2; ci pee; dis{"now the weighted mean"); summ pee [aweight=weight]; local xw=__result (3) ; *xw is the correct, weighted mean; qui summ weight; local nhat=_result(1)*_result(3); gen dx2=(pce-'xw1)A2; qui summ dx2 [aweight=weight]; *this is a weighted variance estimate; local varxl=_result{3)/_result{1) ; display("first incorrect variance"); dis "varxl'; display("standard error for col 4"); dis sqrt(N varxl'); dis("allowing weights but no stratification or clusters"); *this corresponds to (1.28); gen vi2dx2={weight/"nhat1)A2*dx2; qui summ vi2dx2; local varx2=_result(1)*_result(3); display("second incorrect variance"); dis Nvarx2'; display("standard error"); dis sqrt(x varx2'); dis("allowing weights and stratification, but no clusters" *this uses (1.28) and adds over strata using (1.36); egen svarx2=mean(vi2dx2), by(strat); egen ns=count(pee), by(strat); gen fac=ns/(ns-1); replace svarx2=svarx2*fac; qui summ svarx2; local varx3=_result(1)*_result(3); display{"third incorrect variance"); dis Vvarx3'; display("standard error"); dis sqrt(v varx3'); dis("now allowing for everything"); *this is evaluated according to (1.63); gen z=weight*pce; egen zcs=sum(z), by(clust); egen wcs-sum(weight), by(clust); sort clust; qui by clust: keep if __n= = l; egen zs=mean(zcs), by(strat); egen ws=mean(wcs), by(strat); gen dall-((zcs-zs)-Nxw'*(wcs-ws))A2; egen tm=sum(dall), by(strat); sort strat; qui by strat: keep if _n==l; qui replace tm=tm*fac; qui summ tm; 404 THE ANALYSIS OF HOUSEHOLD SURVEYS local varok=__result (1) *_result(3) / ( N nhat' *2) ; display{"correct variance"); dis "varok*; display("standard error"); dis sqrtTvarok1) ; end; surv; **Now doing the provinces; cap program drop cont; program def cont; local pno=l; while "pno4 <= 4 {; drop _all; use temp; keep if prov==Npno'; surv; local pno=Npno'+1; }; end; cont; log close; Example 1.3. Bootstrapping weighted means and medians The program uses the same data as in Example 1.2 to construct bootstrap samples with the same stratified and clustered structure as the original. In each bootstrap replication, weighted and unweighted means and medians are calculated and saved in a data set. The standard deviations over the bootstrap replications are used to in- dicate the sampling variability of the original estimates. version 4.0 ttdelimit ; cap log close; log using ex!3, replace; set more 1; drop _all; use consdata; gen clust=int(hhid/1000); gen strat=int(hhid/1000000) ; gen pce=totmexp/hhsize; keep clust strat pee weight; egen newstrat=group(strat); *egen numbers the strata 1,2, etc; drop strat; save tempbase, replace; **Bootstrapping for srs and weighted mean; **Using strata and clusters; cap program drop sbsmean; program def sbsmean; tempname sim; tempfile bsres samp ssamp; postfile %sim' x xw xmed xmedw using "bsres', replace; local ic=l; while "ic 1 <= 100 {; display "replication no. " % ic'; /* at each replication build up bs sample ** stratum by stratum: *isl is stratum counter */ local is=l; while "is* <- 22 {; CODE APPENDIX 405 use tempbase, clear; qui keep if newstrat==N is ' ; sort clust; qui save ^ssamp 1 , replace; keep clust; qui by clust: drop if _n -= 1; /* bs sampling of clusters */ bs ample _N; sort clust; qui by clust: gen dum=_N; qui by clust: drop if _n ~= 1; merge clust using xssamp'; qui drop if _merge ~= 3; qui expand dum; drop _merge dum; if s is'==l {; qui save "samp 1 , replace; }; else {; append using "samp 1 ; qui save * samp ' , replace ; } ; local is= N is'+l; }; N use samp ' , clear; /* the unweighted estimate */ qui summ pee, d; local md=_result (10) ; local mn=__result (3) ; /* the weighted estimate */ qui summ pee [aweight=weight] ,d; local mnw=_result (3) ; local mdw=_result (10) ; post "sim 1 x mn' xmnw( N md* "mdw1 local ic= x ic'+l; }; postclose v sim'; drop _all; use 'bsres(; summ; save ex!3x replace; end; sbsmean; log close; A.2 Chapter 2 There are two examples. The first concerns the heteroskedastic censored regression model and the second, the decomposition of data from a time series of cross- sections into age, cohort, and year effects. Example 2.1. Comparing OLS, Tobit, and Powell's censored LAD estimators The heteroskedastic censored regression model is discussed on pp. 85-90.1 give the code for the Monte Carlo experiment comparing Powell* s censored LAD estimator with OLS and MLE Tobit, the results of which are discussed on p. 90. version 4.0 ttdelimit ; set more 1; /* control program */ /* number of observations, either 100 or 1000 */ 406 THE ANALYSIS OF HOUSEHOLD SURVEYS global nobs=100; cap program drop doit; program define doit; local ct = 1; while "ct 1 < "I 1 {; dis ("Experiment number "ct' 0 ); runtest; append using temp; qui save temp, replace; local ct="ct'+l; }; end; cap program drop runtest; program define runtest; drop _all; qui set obs $nobs; gen x=_n/($nobs/100); gen u=invnorm(uniform()) ; qui replace u=u*20; gen ytrue=max(0,x-40); /* this is the heteroskedasticity */ qui replace u=u*(1+0.03*ytrue); gen ystar=-40+x+u; qui replace ystar=Q if ystar < 0; qui regress ystar x; /* ols regression */ global bols=_coef[x]; qui tobit ystar x, 11; /* mle tobit estimation */ mac def btob=_coef[x]; /* censored LAD estimation: 10 loops */ qui qr-eg ystar x, qu(0.5); predict xbl; local iloop=2; while viloop' <=10 {; qui qreg ystar x, qu(0.5), if xbl > 0; predict xb2; qui replace xbl=xb2; drop xb2; local iloop-Niloop'+l; }; /* powell estimate coefficient */ mac def bpow=__coef [x] ; drop _all; qui set obs 1; gen zols-$bols; gen ztob=$btob; gen zpow=$bpow; list; end; drop _all; set obs 1; gen zols=999; gen ztob=999; gen zpow=999; cap save temp, replace; doit 100; drop if zols==999; gen absols=abs(zols-1); gen abstob=abs(ztob-1); gen abspow=abs(zpow-1); gen powbest=0; replace powbest=l if (abspow <=abstob) & (abspow < absols) \ summ; CODE APPENDIX 407 Example 2.2. Decompositions into age, cohort, and year effects The second example presents the code for the decomposition of earnings in Taiwan (China) into age, cohort, and year effects; the methods are discussed on pp. 116-27, and the results are shown in Figure 2.5. version 4.0 ^delimit ; set more 1; drop _all; use finp; /* **finp is a data set that is constructed from the **original surveys from 1975-1990. For the individual- **level data, in each survey, the average earnings at **each year is calculated, and this is saved, so that **for each year from 1975 to 1990, finp contains age, **average earnings of individuals of that age, and **the survey year **/ keep age yr iearn; gen cohort=age-yr+65; drop i f age < 25; drop if age > 65; /* generating dummies for each age, cohort and year */ tab age, gen(aged); tab cohort, gen(cohd); tab yr, gen(yrd); drop agedl; drop cohdl; /**constructing variables so that the year effects **add to zero, and are orthogonal to a time trend */ replace yrdl5=yrd!5-14*yrd2+13*yrdl; replace yrdl4=yrd!4-13*yrd2+12*yrdl; replace yrdl3 =yrdl3-12 *yrd2 +11*yrdl; replace yrdl2 =yrdl2-11*yrd2 +10 *yrdl; replace yrdl1=yrdl1-10*yrd2 + 9*yrdl; replace yrdl0 =yrdl0 - 9*yrd2 + 8*yrdl; replace yrd9-yrd9-8*yrd2+7*yrdl; replace yrd8 =yrd8- 7*yrd2 + 6*yrdl; replace yrd7=yrd7-6*yrd2 + 5*yrdl; replace yrd6~yrd6-5*yrd2+4*yrdl; replace yrd5=yrd5-4*yrd2+3*yrdl; replace yrd4=yrd4-3 *yrd2 +2 *yrdl; replace yrd3 =yrd3 -2*yrd2 +1 *yrdl ,- drop yrd2; drop yrdl; regress iearn aged* cohd* yrd*; /* calculating & storing the year in variable yreff */ gen yreff=_b[yrd3] if yr==67; replace yreff=_b[yrd4] if yr==68 replace yreff=_b[yrd5] if yr==69 replace yreff=_b[yrd6] if yr==70 replace yreff=_b[yrd7] if yr==71 replace yreff=_b[yrd8] if yr==72 replace yreff=_b[yrd9] if yr==73; replace yreff=_b[yrdl0] if yr==74; replace yreff=_b[yrdl1] if yr==75; replace yreff=_b[yrd!2] if yr==76; replace yref f =__b [yrd!3] if yr==77; replace yreff=_b[yrd!4] if yr==78; replace yreff=_b[yrdlS] if yr==79; global tern = -2*_b [yrd3] ~3*_b [yrd4] -4*_b [yrd5] -5*__b [yrd6] -6*_b [yrd7] -7*_b[yrd8] -8*_b[yrd9] -9*__b [yrdlO] -10*_b [yrdll] -ll*_b [yrd!2] -12*_b[yrdl3]-13*_b[yrd!4]-14*_b[yrdlS]; 408 THE ANALYSIS OF HOUSEHOLD SURVEYS replace yreff=$tem if yr==66; global alphal= -$tem-_b[yrd3]-_b[yrd4]-_b[yrd5]-_b[yrd6]- JD[yrd7] -_b [yrd8] -_b [yrd9 ] ~_b [yrdlO] -__b [yrdll ] -_b[yrd!2] -JD [yrd!3 ] -_b [yrd!4 ] -_b [yrdl5 ] ; replace yreff=$alphal if yr==65; replace yr=yr+ll; /* calculating and storing cohort effects in coheff */ global nc=2; global ic=12; gen coheff=0 if cohort==ll; cap program drop doit; program def doit; while $ic < 66 {; global ix="cohd$nc"; replace coheff =_b[$ix] if cohort"$ic; global ic=$ic+l; global nc=$nc+l; }; end; doit; /* calculating and storing age effects */ mac def nc=2; mac def ic=26; gen ageff=0 if age==25; cap program drop doit; program def doit; while $ic < 66 {; mac def ix="aged$nc"; replace ageff=_b[$ix] if age==$ic; mac def ic=$ic+l; mac def nc=$nc+l; }; end; doit; * Look at the results; set more 0; graph yreff yr, c(l) s(p) xlabel ylabel sort; graph ageff age, c(l) s(.) xlabel ylabel sort; graph coheff cohort, c(l) s(.) xlabel ylabel sort; A.3 Chapters The examples provide code for the summary measures of living standards, in- equality, and poverty in Cote d'lvoire and South Africa shown in Tables 3.1-3.4 and Figures 3.5 and 3.6. Figures 3.7-3.12 were calculated using the functions provided by STATA and do not require special coding. The GAUSS code for Figures 3.13 and 3.14 is given as Example 3.7.1 have not provided the code for the Thai rice calculations; Figures 3.15 and 3.16 are calculated in the same way as Figures 3.13 and 3.14. The nonparametric regressions shown in Figures 3.17 and 3.18 are kernel regressions, a method that has been superceded by the locally weighted regressions shown in Figure 3.20 and the code for which is given in Example 3.8. Example 3.1. Consumption measures for Cote d'lvoire (Table 3.1) The code uses data on household expenditures, prices, and weights to calculate means for households and individuals, as well as their standard errors taking into account the survey design. CODE APPENDIX 409 version 4.0 /* NB the code uses the survey commands incoporated into the main code from version 5.0 of STATA */ #delimit ; drop __all; cap log close; log using table31.log, replace; use c:\cdil996\1985dta\hhexp85; matrix prices=(100,107.3,107.75,115.31\92.84,100.45,98.58,106.74\ 87.01,93.36,94.95,99.84X78.25,80.11,81.50,83.51\ 75.97,85.97,88.20,94.42); /* **These are the prices reported on p 49 of the CLISS documentation **Rows are regions, Abidjan/Other Cities, EForest, WForest, Savannah ** Columns are 1985, 1986, 1987, 1988 */ matrix prices=prices*.01; gen hhid=clust*100+nh; ren reg region; ren hhsizeS hhs; keep hhid region hhexp hhs dust; replace hhexp=hhexp/1000; replace hhexp=hhexp/prices[1,1] if region—I; replace hhexp=hhexp/prices[2,1] if region==2; replace hhexp=hhexp/prices[3,1] if region==3; replace hhexp=hhexp/prices[4,1] if region-=4; replace hhexp=hhexp/prices[5,1] if region==5; gen year=1985; sort hhid; save temp, replace; use c:\cdil996\1985dta\weight85; gen hhid=clust*100+nh; keep hhid allwaitn; sort hhid; merge hhid using temp; tab _jn; drop _m; gen pce=hhexp/hhs; sort hhid; save temp, replace; drop _all; use c:\cdil996\1986dta\hhexp86; gen hhid=clust*100+nh; ren reg region; ren hhsize6 hhs; keep hhid region hhexp hhs clust; replace hhexp=hhexp/1000; replace hhexp=hhexp/prices[1,2] if region==l; replace hhexp=hhexp/prices[2,2] if region==2; replace hhexp=hhexp/prices[3,2] if region==3; replace hhexp=hhexp/prices[4,2] if region==4; replace hhexp=hhexp/prices[5,2] if region=-5; gen year=1986; sort hhid; save tempa, replace; use c:\cdil996\1986dta\weight86; gen hhid=clust*100+nh; keep hhid allwaitn; sort hhid; merge hhid using tempa; tab _m; drop _m; gen pce=hhexp/hhs; append using temp; save temp, replace; use c:\cdil996\1987dta\hhexp87; gen hhid=clust*100+nh; ren reg region; ren hhsize7 hhs; 410 THE ANALYSIS OF HOUSEHOLD SURVEYS keep hhid region hhexp hhs clust; replace hhexp=hhexp/1000; replace hhexp=hhexp/prices[I/3] if region==l; replace hhexp=hhexp/prices(2,3] if region==2; replace hhexp=hhexp/prices[3,3j if region==3; replace hhexp=hhexp/prices[4,3] if region==4; replace hhexp=hhexp/prices[5,3] if region==5; gen year=1987; sort hhid; save tempa, replace; use c:\cdil996\1987dta\weight87; gen hhid=clust*100+nh; keep hhid allwaitn; sort hhid; merge hhid using tempa; tab _m; drop _m; gen pce=hhexp/hhs; append using temp; save temp, replace; use c:\cdil996\1988dta\hhexp88; gen hhid=clust*100+nh; ren reg region; ren hhsizeS hhs; keep hhid region hhexp hhs clust; replace hhexp=hhexp/1000; replace hhexp=hhexp/prices[1,4] if region==l; replace hhexp=hhexp/prices[2,4j if region==2; replace hhexp=hhexp/prices[3,4] if region==3; replace hhexp=hhexp/prices[4,4] if region==4; replace hhexp=hhexp/prices[5,4] if region==5; gen year=1988; sort hhid; save tempa, replace; use c:\cdil996\1988dta\weight88; gen hhid=clust*100+nh; keep hhid allwaitn; sort hhid; merge hhid using tempa; tab _m; drop _m; gen pce=hhexp/hhs; append using temp; save cdi85_88, replace; tab year; varset strata region; varset psu clust; varset pweight allwaitn; svymean hhexp if year==1985; svymean hhexp if year==1986; svymean hhexp if year==1987; svymean hhexp if year==1988; svymean hhs if year==1985; svymean hhs if year==1986; svymean hhs if year==1987; svymean hhs if year==1988; svymean pee if year==1985; svymean pee if year==1986; svymean pee if year==1987; svymean pee if year==1988; svyratio hhexp/hhs if year==1985; svyratio hhexp/hhs if year==1986; svyratio hhexp/hhs if year==1987; svyratio hhexp/hhs if year==1988; CODE APPENDIX 411 gen combwt=allwaitn*hhs; varset pweight combwt; svymean pee if year==1985; svymean pee if year==1986; svymean pee if year=-1987; svymean pee if year==1988; log close; Example 3.2. Inequality measures for Cote d'lvoire (Table 3.2) The code calculates each of the inequality measures in Table 3.2, with and without weights. version 4.0 #delimit ; drop _all; cap log close; log using table32, replace; /* defining program to calculate gini without weights */ cap program drop giniu; program def giniu; local yr=1985; while xyr' <- 1988 {; drop _all; use cdi85_88; keep if year==vyr'; gsort - pee; gen hrnk=sum(hhs); gen hhrnk=hrnk[_n-l]+1; replace hhrnk=l in 1; qui summ hhs; local bign=_result(1)*_result(3); qui summ pee [aweight=hhs]; local mn=_result(3); gen rnkx=pce*(hhrnk+0.5*(hhs-1)); qui summ rnkx [aweight=hhs]; x x local g=( => bign'+l)-(2/ 1 mn')*_result(3); local g ; g ' / Tbign -1) x disC'Gini coefficient yr' ^ ^ g 1 " ) ; local y r = v y r ' + l ; }; end; /* defining program to calculate gini with weights */ cap program drop giniw; program def giniw; local yr=1985; while Nyr' <= 1988 {; drop _all; use cdi85_88; keep if year=="yr'; gsort - pee; replace hhs=hhs*allwaitn; gen hrnk=sum(hhs); gen hhrnk=hrnk[_n-l]+1; replace hhrnk=l in 1; qui summ hhs; local bign=_result(1)*_result{3); qui summ pee [aweight-hhs]; local mn=_result(3); gen rnkx=pce*(hhrnk+0.5*(hhs-1)); qui summ rnkx [aweight=hhs]; local g=(vbign'+l)-(2/xmn')*_result(3); local g="g ( /(^bign 1 -1); 412 THE ANALYSIS OF HOUSEHOLD SURVEYS disC'Gini coefficient ^yr' = " g > M ) ; local yr=syr'+l; }; end; **Unweighted ginis; giniu; **Weighted ginis; giniw; /* program for Atkinson measures */ /* argument "I 1 is the inequality aversion */ cap program drop aba; program def aba; local yr=1985; while xyr' <= 1988 {; drop _all; use cdi85_88; keep if year=="yr'; replace hhs=hhs*allwaitn; qui summ pee [aweight=hhs]; local mn=_result(3); gen pcerat=pce/vmn'; if *!• ==1 {; gen tm=ln(pce/^mn'); qui summ tm [aweight=hhs]; local ineq=l-exp(__result(3) ) ;} ; else {; gen tm=(pce/'mn1)A%1'; qui summ tm [aweight=hhs]; local ineq=l-(_result(3))A(I/(1-^1'});}; dis("Nyr' Atkinson coefficient, eps=Nl( =%ineq'"); local yr=Nyr'+l; }; end; aba 0.5; aba 1; aba 2; /* program for standard deviation of logs */ cap program drop sdlogs; program define sdlogs; local yr=1985; while *yr' <= 1988 {; drop —all; use cdi85_88; keep if year==vyr'; replace hhs=hhs*allwaitn; gen lnpce=ln(pce); qui summ Inpce [aweight=hhs]; local sdln=sqrt(_result(4)); dis(""yr' s.d. of Inpce, = 'sdln'"); local yr=Nyr'+l; }; end; sdlogs; /* program for coefficient of variation */ cap program drop coefvar; program define coefvar; local yr=1985; while %yr' <= 1988 {; drop _all; use cdi85_88; x keep if year=- yr'; replace hhs=hhs*allwaitn; qui summ pee [aweight=hhs]; local cv=sqrt(_result(4))/_result(3); CODE APPENDIX 413 dis(M>yr' c.o.v. of pee, = ^cv 1 "}; local yr=vyr'+l; }; end; coefvar; log close; Example 3.3. Poverty measures for Cote d'lvoire (Table 3.3) The code calculates the four poverty measures, together with their bootstrapped standard errors. version 4.0 #delimit ; drop _all; cap log close; log using table33,replace; /* setting the poverty line */ global pline=128.6; /* program for various poverty measures */ /* this iterates over the four years */ cap program drop peas; program def peas; local yr=1985; while vyr' <= 1988 {; use cdi85_88; keep if year==*vyr'; gen wt=hhs*allwaitn; gen pind=pce <= $pline; qui summ pind [aweight=wt]; local pO=_result(3); gen df=(l-pce/$pline)*pind; qui summ df [aweight=wt]; local pl=_result(3); gen dfa=(l-pce/$pline)A2*pind; qui summ dfa [aweight=wt]; local p2=__result(3) ; drop if pee > $pline; giniw; local psen=xpO'*$g+xpl'*(l-$g); dis("PO=spO',Pl=Npl',P2="p2'^sen^'psen1"); drop _all; local yr=^yr'+l; }; end; /* we need this for the gini among the poor */ cap program drop giniw; program def giniw; gsort - pee; qui {; gen hrnk=sum(wt); gen hhrnk=hrnk[_n-l]+1; replace hhrnk=l in 1;}; qui summ wt; local bign=_result(l)*_result(3); qui summ pee [aweight=wt]; local mn=_result(3); gen rnkx=pce*(hhrnk+0.5*(wt-1)); qui summ rnkx [aweight-wt]; global g=rbign'+l) -(2/%mn' ) *_result(3) ; global g=$g/(^bign1-1); end; 414 THE ANALYSIS OF HOUSEHOLD SURVEYS peas; /* bootstrapping the measures*/ drop __all; set more 1; cap log close; log using t33boot, replace; /* program for creating clustered bootstrap sample */ /* and calculating poverty measures */ cap program drop bootit; program def bootit; use tempclus; bsample _N; sort clus; qui {; by clus: gen nreps=_N; by clus: keep if _n==l; merge clus using tempall; keep if __merge==3; expand nreps; drop nreps;}; weepeas; drop _all; end; /* control program */ cap program drop runit; cap program def runit; local yr=1985; while Nyr' <= 1988 {; use cdi85_88; keep if year==xyr'; gen wt=hhs*allwaitn; keep clus hhid pee wt; sort clus; save tempall, replace; keep clus; qui by clus: keep if _n==l; save tempclus, replace; drop _all; local mcrep=l; while "mcrep1 <= 100 {; dis ("Year x yr ( , Replication ymcrep'")i bootit; drop _all; qui set obs 1; gen year=xyr'; gen pO=$pO; gen pl=$pl; gen p2=$p2; genx psen=$psen; if mcrep' != 1 | "yr1 i- 1985 {; append using t33boot; }; qui save t33boot, replace; drop __all; N local mcrep= mcrep'-fl; >; local yr^yr'+l; }; end; /* calculates the various measures */ cap program drop weepeas; program def weepeas; CODE APPENDIX 415 global pline=128.6; qui {; gen pind=pce <= $pline; qui summ pind [aweight-wt]; global pO=_result(3); gen df=(l-pce/$pline)*pind; qui summ df [aweight=wt]; global pl=_result(3); A gen dfa=(1-pee/$pline) 2 *pind; qui summ dfa [aweight=wt]; global p2=_result(3); drop if pee > $pline; giniw; global psen=$pO*$g+$pl*(l-$g); drop _all; }; end; runit; drop _all; use t33boot; sort year; by year: summ; log close; Example 3.4. Welfare measures for South Africa (Table 3.4) The code is similar to that for Cote d'lvoire but is adapted to the South African data. version 4.0 ftdelimit ; drop _all; cap log close; log using sastats, replace; /* household roster file */ use c:\safrica\work\m6_hrost; /* keeping only members */ keep if mcode==l; drop if gender_n==l; drop if age==.; drop if age <- 0; keep hhid; sort hhid; qui by hhid: gen hhs=_N; qui by hhid: keep if _n==l; /* picking up the weights */ merge hhid using c:\safrica\work\strata2; tab _m; drop _m; keep hhid hhs race rsweight; sort hhid; /* picking up household expenditures */ merge hhid using c:\safrica\work\hhexptl; keep hhid hhs race rsweight totmexp; gen pce=totmexp/hhs; gen wt=rsweight*hhs; gen clus=int(hhid/1000); sort race; summ totmexp hhs pee [aweight=rsweight]; summ pee [aweight=wt]; by race: summ totmexp hhs pee [aweight=rsweight]; by race: summ pee [aweight=wt]; varset pweight wt; varset psu clus; svymean pee; svymean pee, subpop(race==l); 416 THE ANALYSIS OF HOUSEHOLD SURVEYS svymean pee, subpop(race=-2); svymean pee, subpop{race==3); svymean pee, subpop(race==4); varset pweight rsweight; svyratio totmexp/hhs; svyratio totmexp/hhs, subpop(race==l); svyratio totmexp/hhs, subpop{race==2); svyratio totmexp/hhs, subpop(race==3); svyratio totmexp/hhs, subpop(race==4); save sastats, replace? drop _all; cap program drop ginisa; program def ginisa; local rg=0; while N rg' <= 4 {; drop __all; use sastats; if "rg l > 0 (;keep if race=="rg';); giniw; disC'Gini coefficient race = v rg' =$g"); local rg=vrg'+l; }; end; cap program drop abasa; program def abasa; local rg=0; while x rg' <= 4 {; drop _all; use sastats; if "rg1 > 0 {;keep if race==xrg';}; qui summ pee [aweight=wt]; local mn=_result(3); gen pcerat=pce/vmn'; if v l ' ==1 {; gen tm=ln{pce/%mn'); qui summ tm [aweight=wt]; local ineq=l-exp(__result (3) ) ;}; else {; gen tm=(pce/ x mn l ) AX 1'; qui summ tm [aweight=wt]; local ineq=l-{_result(3))A(I/{!-'1'));}; dis("race N rg' Atkinson coefficient, eps=Nl' =vineq'"); local rg=vrg'+l; }; end; cap program drop sdlogssa; program define sdlogssa; local rg=0; while N rg* <= 4 {; drop __all; use sastats; if N rg* > 0 {;keep if race=="rg';}; gen lnpce=ln{pce); qui summ Inpce [aweight=wt]; local sdln=sqrt(_result(4)) ; disC'Race ^rg' s,d. of Inpce, = xsdln'"); local rg=Nrg'+l; }; end; cap program drop coefvar; program define coefvar; local rg=0; while N rg' <= 4 {; drop _all; use sastats; CODE APPENDIX 417 if N rg' > 0 {;keep if race=="rg' ; } ; qui summ pee [aweight=wt] ; local cv=sqrt (_result (4) ) /_result (3) ; dis("Race x % rg* c.o.v. of pee, = N cv'") local rg= rg'+l; ); end; cap program drop peassa; program def peassa; local rg=0; while x rg' <= 4 {; drop _all; use sastats; if x rg' > 0 {;keep if race==xrg';}; gen pind=pce <= $pline; qui summ pind [aweight=wt]; local pO=_result(3); gen df=(l-pce/$pline)*pind; qui summ df [aweight=wt]; local pl=_result(3); gen dfa=(l-pce/$pline)A2*pind; qui summ dfa [aweight^wt]; local p2=_result(3); drop if pee > $pline; if J* > 0 {; giniw;}; else {;global g=l;>; local psen=spO**$g+xpl'*(l-$g); dis (<(PO =N pO' ,Pl=xpl' ,P2 =xp2 ' ,Psen=xpsen' ") ; drop _all; local rg=xrg'+l; >;end; cap program drop giniw; program def giniw; gsort - pee; qui {; gen hrnk=sum(wt); gen hhrnk=hrnk[_n-l]+1; replace hhrnk=l in 1;}; qui summ wt; local bign=_result(1)*_result(3); qui summ pee [aweight=wt]; local mn=_result(3) ; gen rnkx=pce *(hhrnk+ 0.5 *{wt-1) ) ; qui summ rnkx [aweight=wt]• global g=(xbign<+l)-{2/xmn')*_result(3); global g=$g/("bign'-l); end; ginisa; abasa 0.5; abasa 1; abasa 2; sdlogssa; coefvar; global pline-105; peassa; log close; Example 3.5. Lorenz curves for South Africa (Figure 3.5) The code shows how to calculate the cumulative fractions of population and of PCE and uses the results to draw the standard Lorenz curves. 418 THE ANALYSIS OF HOUSEHOLD SURVEYS version 4.0 #delimit ; drop __all; cap log close; log using lorenz, replace; cap program drop lc; program def lc; drop _all; /* data saved from code in Ex3.4 above */ use sastats; drop if pce==.; if v l ' > 0 {; keep if race^ 1' ;}; sort pee; gen cumpop=sum{wt); gen cumpce=sum(pce); qui summ wt; replace cumpop-cumpop/{^result(1)*_result(3)); qui summ pee; replace cumpce=cumpce/ (_result (1) *_result(3) ) ; local gg=int(_N/100); keep if (("gg'*int(_n/"gg')-_n)==0)|_n==l|_n==_N; sort pee; graph cumpce cumpop cumpop, c(ll) s{..) xlabeKO,0.2,0.4,0.6,0.8,1.0) ylabel(0, 0.2,0.4,0.6,0.8,1.0) border; gen rcode=xl'; keep cumpce cumpop rcode; if v l ' > 0 {; append using temp; }; save temp, replace; end; lc 0; 1C 1; lc 4; drop _all; use temp; gen cumpceO=cumpce if rcode-=0; gen cumpcel=cumpce if rcode==l; gen cumpce4=cumpce if rcode==4; drop cumpce; graph cumpce* cumpop cumpop, c(llll) s( ) xlabeKO,0.2,0.4,0.6,0.8,1.0) ylabelfO, 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8 , 1 . 0 ) saving(lorenz, replace) border; log close; Example 3.6. Transformed Lorenz curves for Cote d'lvoire (Figure 3.6) This example adapts the code for the standard Lorenz curves so as to plot the distance between the Lorenz curve and the 45-degree line. version 4.0 #delimit ; drop _all; cap log close; log using lorenzic, replace; cap program drop Icic; program def Icic; drop _all; use cdi85_88; drop if pce==.; if *1( > 0 {; keep if year==N1';}; sort pee; CODE APPENDIX 419 gen wt=hhs*allwaitn; gen cumpop=sum(wt); gen cumpce=sum(pce); qui summ wt; replace cumpop=cumpop/(_result(1)*_result(3)); qui summ pee; replace cumpce=cumpce/(_result(1)*_result(3)); local gg=int(_N/100); keep if ((vgg'*int(_n/vgg')-_n)==0)|_n==l|_n==__N; sort pee; graph cumpce cumpop cumpop, c(ll) s(..) xlabel(0,0.2,0.4,0.6,0.8,1.0) ylabel(0, 0.2,0.4,0.6,0.8,1.0); gen ycode=x1'; keep cumpce cumpop ycode; if N l ' > 1985 {; append using temp; }; save temp, replace; end; Icic 1985; Icic 1986; Icic 1987; Icic 1988; drop _all; use temp; gen cumpce5=cumpce if ycode==1985; gen cumpce6=cumpce if ycode==1986; gen cumpce?=cumpce if ycode==1987; gen cumpce8^cumpce if ycode==1988; drop cumpce; graph cumpce* cumpop cumpop, c(lllll) s( ) xlabel(0,0.2(0.4,0.6f0.8,1.0) ylabel(0, 0.2,0.4,0.6,0.8,1.0) saving(lorenzic, replace); /* defining the complementary lorenz curves as graphed */ gen compce5=cumpop-cumpce5; gen compce6=cumpop-cumpce6; gen compce7=cumpop-cumpce7; gen compce8=cumpop-cumpce8; graph compce* cumpop, c(lllll) s( ) xlabel(0,0.2,0.4,0.6, 0.8,1.0) ylabel(0, 0.1,0.2,0.3,0.4) saving(complc,replace); log close; Example 3.7. Contours and netmaps (Figures 3.13 and 3.14) The code for the contours and netmaps is given in GAUSS because these facilities do not exist in STATA. I have also calculated the kernel estimates in GAUSS, though it would also have been possible to go to this stage in STATA. /* GAUSS program for contouring joint density of log pee for first two years of GDI data, results shown in Figures 3.13 and 3.14 */ /* the data pan8586 of log pee in the two years was created in STATA and converted to GAUSS format */ library pgraph; open fI="pan8586"; x=readr(fl,10000) ; fl=close(fl); x=packr(x); n=rows(x); v=(x-meanc(x)')'*(x-meanc(x)')/n; dt=det(v); v_l=invpd(v); 420 THE ANALYSIS OF HOUSEHOLD SURVEYS /* the inverse variance-covariance matrix of the data */ xmin-3; xmax=8; ymin=3; ymax= 8; grid=99; stx~(xmax-xmin)/(grid-1); sty=(ymax-ymin)/{grid-1); ans=zeros(grid,grid); ggx=seqa(xmin,stx,grid); ggy=seqa(ymin,sty,grid); /* setting the bandwidth */ h=l.0; /* doing the density estimation */ icsl; do until ic==grid+l; ic; jc=l; do until jc-=grid+l; locate 10,1; ic;;jc; xx=ggx[ic,1]|ggy[jc, 1] ; df=(x-xx')/h; t2=v_l[l,l]*df [ . , l ] A 2 + 2 * vA_ l [ l , 2 ] * d f [ . , 1] . *df [. , 2 ] +v^l[2,2]*df[./2] 2; anstic,jc]=sumc(selif((l-t2) , t2 .It 1)); jc=jc+l; endo ; ic=ic-Hl; endo ; ans=ans*2/(pi*n*hA2*sqrt(dt)); ans=missrv(ans,0) ; /* save for future use */ save ansplhl=ans; /* making Figure 3.13 */ graphset; begwind; xtics(3(8f1.0,2) ; ytics(3,8,1.0,2); _pdate=""; ^3lctrl=-l; _pstype=4; _psymsiz=l; _plevs{0,0.01,0.02,0.03,0.05,0.1,0.2,0.4,0.6,0.8,1.0,1.2,1.4}; makewinddO^S^O.O,!) ; contour(ggx',ggy,ans'); makewind(10,8,0,0,l) ; xy(x[.,l],x[.,2]); endwind; gkp^seqad^^O) ; graphset; xtics(3,8,1.0,2) ; ytics(3,8,1.0,2) ; ztics(0,0.4,0.1,1) ; _pdate=""; _pticout=l; begwind; window(1,2,0); surface(ggx[gkp,.] 'rggytgkp,.],ans[gkp,gkp]') ; nextwind; _pview={l/10/-3,4); surface(ggx[gkp,.]'/ggy[gkp,.]/ans[gkp,gkp]'); endwind; end; CODE APPENDIX 421 Example 3.8. Locally weighted regression (Figure 3.20) Figure 3.20 shows plots of expected actual and potential pension receipts against the logarithm of household per capita income. These are calculated using locally weighted regressions, the code for which is given below. /* this is for Africans, i.e., left panel of Figure 3.20 */ version 4.0 ttdelimit ; cap log close; log using fanpen, replace; set more 1; drop _all; use socpenh; keep if race==l; /* prepinc is total income less pension income */ gen prepinc=totminc-socpenl; drop if prepinc <= 0; gen ppens=n60f+n65m; gen lnpcy=log(prepinc/hhsizem); gen incpc=prepinc/hhsizem; global penamt=370; gen psocpen=ppens*$penamt; keep psocpen socpenl Inpcy; / psocpen is potential pension receipts socpenl is receipts of social pensions Inpcy is the log of nonpension income lowrex does the locally weighted regression argument 1 is the regressand (input) argument 2 is the regressor (input) argument 3 is the estimated regression function (output) argument 4 is the derivative of the regression function (output) argument 5 is the bandwidth (input) argument 6 is the grid over the regressor for evaluation (input) / cap program drop lowrex; program def lowrex; local ic=l; gen x 3' =.; gen " 4' =.; while x ic' <= $gsize {; dis x ic'; quietly {; local xx= ^ 6' pic 1 ]; gen z=abs(P 2'-"xx')/"5'); gen kz=(15/16)*(l-z*2)*2 if z <= 1; reg x l' *2' =kz if kz *-= .; replace ^ 4'=_b[^ 2'] in ^ ic'; * replace X3'=_b[_cons]+_b[x2']*xxx' in x ic'; drop z kz; }; local ic=xic'+l; }; end; global xmin=0; global xmax= 9; global gsize=50; global st=($xmax-$xmin)/($gsize-l); global h=1.0; gen xgrid=$xmin+(_n-l)*$st in l/$gsize; lowrex psocpen Inpcy smthp dsmth $h xgrid; 422 THE ANALYSIS OF HOUSEHOLD SURVEYS drop dsmth; lowrex socpenl Inpcy smtha dsmth $h xgrid; sort xgrid; keep in l/$gsize; graph smthp smtha xgrid, c ( l l ) s ( i i ) ylabel x l a b e l ( 0 , 2 , 4 , 6 , 8 , 1 0 ) x l i n e ( 4 . 6 5 4 ) saving(fanaf, replace); log close; A.4 Chapter 4 The programs below were used to generate Figures 4.4 and 4.5 in the text. The code for the contour maps in Figures 4.2 and 4.3 is a straightforward application of Example 3,7 above. The results in Section 4.2 are obtained from straightforward regression analysis, and the code is not given here. Example 4.1. Locally weighted regressions for calorie Engel curves There are three sections of code which, for logical transparency, are presented in the reverse order in which they should be run. The first section calculates the estimated regression function and its derivatives and then picks up the bootstrap replications from the second and third sections of the code to construct Figures 4.4 and 4.5. /* ** This is Section 1 of the code and calulates the main results */ version 4.0 # delimit ; drop _all; set more 1; capture log close; capture log using ex41a.log, replace; /* setting up the data */ use ..\feb95\working; gen lnpce=log{totexp/nmems); gen lncal=log{tcall/nmems); drop if Incal == .; drop if Inpce == . ; keep Inpce Incal; /* the locally weighted regression code */ cap program drop lowrex; program def lowrex; local ic=l; gen x^31-.; gen 4 ' = . ; while "ic 1 <= 100 {; dis "ic' ; quietly {; local xx=x 6' [ x ic ' ]; gen z=abs((x 2 l -'xx')/ x 5'); kz~(15/16)*(l-z*2)A2 if z <= 1; gen x reg l' Xx2' =kz if kz -= .; replace x4'=_b["2'} in x ic'; replace 3'-_b[_cons]+_b[x2']*'xx' in x ic'; drop z kz; }; local ic=xic'+l; };end; CODE APPENDIX 423 global xmin=3.5; global xmax=6.0; global st=($xmax-$xmin)799; global h=0.5; gen xgrid=$xmin+(_n~l)*$st in 1/100; lowrex Incal Inpce smth dsmth $h xgrid; set more 0; /* graphing the estimated regression function */ graph smth xgrid, c(l) s(i) xlabel ylabel; /* graphing the slope */ graph dsmth xgrid, c(l) s(i) xlabel ylabel; keep smth dsmth xgrid; sort xgrid; save temp, replace; /* bringing in the bootstrap results {w/o clustering) */ use ..\feb95\lowboot; drop if replic==0; egen sdsmth=sd(smth), by(xgrid); egen sddsmth=sd(dsmth), by(xgrid); sort xgrid; qui by xgrid: drop if _n ~- 1; keep xgrid sdsmth sddsmth; merge xgrid using temp; tab _merge; drop _merge; sort xgrid; /* bands for the smooth and its slope */ gen smthu=smth+2*sdsmth; gen smthd=smth-2*sdsmth; gen dsmthu=dsmth+2*sddsmth; gen dsmthd=dsmth-2*sddsmth; save temp, replace; /* repeating with clustered bootstrap */ use clusam; /* clustered version of lowboot */ drop if replic—0; egen sdsmthc=sd(smth), by(xgrid); egen sddsmthc=sd(dsmth), by(xgrid); sort xgrid; qui by xgrid: drop if _n -= 1; keep xgrid sdsmthc sddsmthc; merge xgrid using temp; tab _merge; drop _merge; sort xgrid; gen smthuc=smth+2*sdsmthc; gen smthdc=smth-2*sdsmthc; gen dsmthuc=dsmth+2*sddsmthc; gen dsmthdc=dsmth-2*sddsmthc; graph smth* xgrid, c(11111) pen(22222) s(iiiii) xlabel(3.4,3.8,4.2,4.6,5.0,5.4,5.8,6.2) ylabel(7,7.2,7.4,7.6,7.8,8.0,8.2)saving(1owregxl, replace); graph dsmth* xgrid, c(11111) pen(22222) s(iiiii) xlabel(3.4,3.8,4.2,4.6,5.0,5.4,5.8,6.2) ylabel(0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1) saving(Iowregx2, replace); log close; /* Section 2: Bootstraps the locally weighted regression ** Without any allowance for cluster design */ version 4.0 #delimit ; drop __all; set more 1; cap log close; log using ex41b, replace; use ..\feb95\working; gen lnpce=log(totexp/nmems); 424 THE ANALYSIS OF HOUSEHOLD SURVEYS gen lncal=log(tcall/nmems); drop if Incal == .; drop if Inpce == . ; keep Inpce Incal; save temp, replace; ren Incal smth; ren Inpce xgrid; gen dsmth=.; gen replic=0; save lowboot, replace; /* this program controls the bootstrap */ cap program drop lowrxit; program def lowrxit; local jc=l; while *jc« <= 50 {; dis v jc'; drop __all; use temp; bsample _N; local xmin=3.5; local xmax=6.0; local st=(xxmax'-xxmin')/99; local h=0.5; qui gen xgrid="xmin'+(_n~l)*Nst' in 1/100; qui lowregx Incal Inpce smth dsmth N h * xgrid; drop Incal Inpce; keep in 1/100; gen replic= N jc 1 ; append using lowboot; save lowboot, replace; local jc=xjc'+l; >; end; lowrxit; /* plotting bunches of regression functions and derivatives */ set more 0; gen lnc=smth if replic==0; replace smth=. if replic==0; sort replic xgrid; graph Inc smth xgrid, c(.L) s(oi) xlabel ylabel; graph smth xgrid if replic > 0, c(L) s(i) xlabel(3.5, 4, 4.5, 5, 5 . 5, 6) ylabel saving(lowbootl, replace); graph dsmth xgrid if replic > 0, c(L) s(i) xlabel(3.5,4,4.5,5,5.5,6) ylabel saving(Iowboot2, replace); log close; /* Section 3: Bootstrapping the locally weighted regression ** With allowance for cluster design */ version 4.0 #delimit; cap log close; log using ex41c/ replace; set more 1; drop _all; use ..\feb95\working; gen lnpce=log(totexp/nmems); gen lncal=log(tcall/nmems); drop if Incal == . ; drop if Inpce ~~ .; keep Inpce Incal hid; gen long cluster=int(hid/100); sort cluster; save tempwork, replace; CODE APPENDIX 425 keep cluster; qui by cluster: drop if _n ~= 1; save tempclus, replace; cap program drop csampl; program define csampl; use tempclus; bsample _N; sort cluster; qui by cluster: gen dum=_N; qui by cluster: drop if _n ~= 1; merge cluster using tempwork; qui drop if _merge ~= 3; drop _merge; qui expand dum; drop dum; sort cluster; end; use tempwork; ren Incal smth; ren Inpce xgrid; gen dsmth=.; gen replic=0; save clusam, replace; cap program drop loxit; program def loxit; local jc=l; while x jc' <= 50 {; dis v jc'; drop _all; csampl; local xmin=3.5; local xmax=6.0; local st=(vxmax'-vxmin')/99; local h=0.5; qui gen xgrid=xxmin'+(_n-l)*xst' in 1/100; qui lowregx Incal Inpce smth dsmth x h' xgrid; drop Incal Inpce; keep in 1/100; gen replic=Njc'; append using clusam; save clusam, replace; local jc=>jc'+l; };end; loxit; /* plotting bunches of regression functions and derivatives */ set more 0; gen lnc=smth if replic==0; replace smth=. if replic==0; sort replic xgrid; graph Inc smth xgrid, c ( . L ) s ( . i ) xlabel ylabel; graph smth xgrid if replic > 0, c ( L ) s { i ) x l a b e l ( 3 . 5 , 4 , 4 . 5 , 5 , 5 . 5 , 6 ) ylabel saving(clusaml, replace); graph dsmth xgrid if replic > 0, c ( L ) s ( i ) x l a b e l ( 3 . 5 , 4 , 4 . 5 , 5 , 5 . 5 , 6 ) ylabel saving{cIusam2, replace); log close; A.5 Chapters I provide the code for calculating the system of demand equations, including the own- and cross-price elasticities, for completing the system, and for calculating the 426 THE ANALYSIS OF HOUSEHOLD SURVEYS symmetry-constrained estimates. The code here is for the Maharashtran case; except for minor details—the number and names of the goods, and the definition of the other variables—the Pakistani code is the same. There are four separate programs: the first, all India. do, is for estimating the demand system. Append- ed to it is a program mkmats . do, that calculates the commutation and selection matrices required for the symmetry-constrained estimates, as well as procedures for making the "vec" of a matrix, and for reversing the operation. The code in boo tall. do bootstraps the procedure in order to obtain measures of sampling variability. Finally, the program policy. do calculates the efficiency and equity components of the cost-benefit ratios for price reform and was used to give the results in Tables 5.12 and 5.13. program allIndia.do program for doing eveything in one shot requires a data set indinr on log unit values budget shares, Inx, and Inn, as well as subrounds and regions and household characteristics version 4.0 #delimit ; drop _all; set matsize 400; set maxvar 1724 width 1832; set more 1; capture log close; log using allindia, replace; *The log unit values begin with luv, e.g. luvwhe; *The budget shares begin with w, e.g. wric; *lnexp is log of outlay, Inhhs log of household size; *regional and subround (seasonal) variables; *Demographics and other variables to taste; *Here they are rml-rm5 and rfl-rf4, age-sex ratios; *Caste, religion, and labor types; use indinr; *These are the commodity identifiers: used as three-letter prefixes; global goods "ric whe jow oce pul dai oil mea veg fru sug"; *Number of goods in the system; global ngds=ll; matrix define sig=J($ngds,1,0); matrix define ome=J($ngds, 1, 0) ; matrix define lam=J($ngds,1,0); matrix define wbar=J($ngds,1,0); matrix define bl=J{$ngds,1,0); matrix define bO=J{$ngds,1,0); *Average budget(shares; cap program drop mkwbar; program def mkwbar; local ig=l; while ""1'" -= "" {; qui summ w'l'; matrix wbar[x ig',1]=_result(3); local ig=vig'+l; mac shift); end; mkwbar $goods; *First stage regressions: within village; cap program drop stlreg; program def stlreg; local ig=l; CODE APPENDIX 427 while "x l' " -= ""{; *Cluster fixed-effect regression; areg luv v l' rm* rf* Inexp Ihhs schcaste hindu buddhist labtyp*, absorb(cluster); *Measurement error variance; matrix ome[* ig1,1]=$S_E_sse/$S_E_tdf; *Quality elasticity; matrix bl[Nig',1]=_coef[Inexp]; *These residuals still have cluster effects in; predict ruv"1', resid; *Purged y's for next stage; gen ylxl'=luvNl'-_coef[Inexp]*lnexp-_coef[Ihhs]*lhhs-_coef[rml]* rml-_coef [rm2] *rm2-_coef [rm3 ] *rm3-_coef [rm4] *rm4~__coef [rm5] * rm5-mcoef[rf1]*rfl-_coef[rf2]*rf2-_coef[rf3]*rf3-_coef[rf4]*rf4 -__coef[schcaste]*schcaste-_coef[hindu]*hindu-__coef[buddhist]* buddhist-_coef[labtypl]*labtypl-_coef[Iabtyp2]*labtyp2 -_coef[Iabtyp3]*labtyp3-_coef[Iabtyp4]*labtyp4; drop luvxl'; *Repeat for budget shares; areg w x l' rm* rf* Inexp Ihhs schcaste hindu buddhist labtyp*, absorb(cluster); predict rw'l1, resid; matrix sigpig1 ,1] =$S_E_sse/$S__E_tdf ; matrix bO[xig',1]-_coef[Inexp]; gen yONl'=wvl'-_coef[Inexp]*lnexp-_coef[Ihhs]*lhhs-_coef[rml]* rml-_coef [rm2] *rm2-__coef [rm3] *rm3-_coef [rm4] *rm4-_coef [rm5] * rm5-__coef [rf 1] *rf l-_coef [rf 2] *rf 2-_coef [rf 3 ] *rf 3-_coef [rf 4] *rf 4 -_coef[schcaste]*schcaste-_coef[hindu]*hindu-_coef[buddhist]* buddhist-_coef[labtypl]*labtypl-_coef[Iabtyp2]*labtyp2 -_coef[labtyp3]*labtyp3-_coef[Iabtyp4]*labtyp4; *This next regression is necessary to get covariance of residuals; qui areg ruvNl' rw N l' Inexp Ihhs rm* rf* schcaste hindu buddhist labtyp*, absorb(cluster); matrix lamp ig',l]=_coef[rwxl']*sig[xig',1]; s drop w l' rw x l' ruvxl'; local ig=xig'+l; mac shift); end; stlreg $goods; matrix list sig; matrix list ome; matrix list lam; matrix list bO; matrix list bl; drop Inexp rm* rf*; drop labtyp* schcaste hindu buddhist Ihhs; *Saving so far as a protection; save tempa, replace; drop _all; use tempa; *Averaging by cluster; *Counting numbers of obs in each cluster for n and n+; cap program drop clustit; program def clustit; local ig=l; while B X l i n -- ""{; egen yOcx ig'=mean(yOx1'), by(cluster); egen nOcNig'=count(yON1'), by(cluster); egen ylcxig1=mean(yl"1'), by(cluster); egen nlcNig1=count(ylN1'), by(cluster); drop yO"1' yl v 1 *; local ig=xig'+l; 428 THE ANALYSIS OF HOUSEHOLD SURVEYS mac shift }; end; clustit $goods; sort cluster; *keeping one obs per cluster; *NB subround and region are constant within cluster; qui by cluster: keep if _n==l; *Saving cluster-level information; *Use this for shortcut bootstrapping; save tempclus, replace; *Removing province and quarter effects; qui tab region, gen(regiond); qui tab subround, gen(quard); drop regiondS quard4; cap program drop purge; program define purge; local ig=l; while v ig' <- $ngds {; qui regress yOc N ig' quard* regiond*; predict tm, resid; replace yOc"ig'=tm; drop tm; qui regress ylc^ig1 quard* regiond*; predict tm, resid; replace ylcxig'=tm; drop tm; local ig=%ig'+l; }; end; purge; drop regiond* quard*; matrix define nO=J($ngds,1,0); matrix define nl=J($ngds,1,0); *Averaging (harmonically) numbers of obs over clusters; cap program drop mkns; program define mkns; local ig=l; while x ig' <= $ngds {; replace nOc * ig'=1/nOcx ig'; replace nlcxig'=l/nlc"ig'; qui summ nOcN ig'; matrix nO Tig 1 ,1J = (..result (3) )A (-1) ; qui summ nlc^ig'; matrix nlpig 1 ,1] = (^result (3)) A (-1) ; drop nOc'ig' nlc N ig'; local ig=>ig'+l; }; end; mkns ; *Making the intercluster variance and covariance matrices; *This is done in pairs because of the missing values; matrix s=J($ngds,$ngds,0); matrix r=J($ngds,$ngds,0}; cap program drop mkcov; program def mkcov; local ir=l; while "ir* <= $ngds {; local ic=l; while N ic' <= $ngds {; qui corr ylcxir' ylc^ic', cov; CODE APPENDIX 429 matrix s [ x ir( ,N ic ' ] =_resul t ( 4 ) ; qui corr ylcxir' yOc x ic', cov; matrix r [ N ir ' , v ic ' ] =__result ( 4 ) ; local ic-Nic'+l; j; local ir^ir'+l; }; end; mkcov; *We don't need the data any more; drop _all; matrix list s; matrix list r; *Making OLS estimates; matrix bols=syminv(s); matrix bols=bols*r; display{"Second-stage OLS estimates: B-matrix"); matrix list bols; display("Column 1 is coefficients from 1st regression, etc"); *Corrections for measurement error; cap program drop fixmat; program def fixmat; matrix def sf=s; matrix def rf=r; local ig=l; while v ig' <= $ngds {; matrix sf [ x ig' , x ig 1 ] =sf [ v ig' , x ig' ] -ome[xig( ,1] /nlpig' ,1] ; matrix rf[v ig * , x ig']=rf[v ig',*ig'j-lam[v ig',1]/nO[x ig',1]; local ig^ig'+l; >; end; fixmat; matrix invs=syminv(sf); matrix bhat=invs*rf; *Estimated B matrix without restrictions; matrix list bhat; *Housekeeping matrices, including elasticities; cap program drop mormat; program def mormat; matrix def xi=J($ngds,1,0); matrix def el=J($ngds, 1, 0) ; local ig=l; while "ig' <= $ngds {; matrix xi [ x ig',1] =bl t " igM] / (bO [ * ig',1] + (1-bl[x ig M]*wbar[^ ig M])}; pig1,1]=l-bl["ig1,l]+bO[xig',1]/wbar[xig',1]; matrix el x local ig= ig'+l; }; end; mormat; global ngl=$ngds+l; matrix iden=I($ngds); matrix idenl=I($ngl); matrix itm=J($ngds, 1,1) ; matrix itml=J($ngl, 1,1) ; matrix dxi=diag(xi); matrix dwbar-diag(wbar); 430 THE ANALYSIS OF HOUSEHOLD SURVEYS matrix idwbar=syminv(dwbar); display("Average budget shares"); matrix tm=wbar'; matrix list tm; display{"Expenditure elasticities"); matrix tm=el'; matrix list tm; display("Quality elasticities"); matrix tm=bl'; matrix list tm; *This all has to go in a program to use it again later; *Basically uses the b matrix to form price elasticity matrix; cap program drop mkels; program define mkels; matrix cmx=bhat' ; matrix cmx=dxi*cmx; matrix cmxl=dxi*dwbar; matrix cmx=iden-cmx; matrix cmx=cmx-(-cinxl ; matrix psi-inv(cmx); matrix theta=bhat'*psi; display{"Theta matrix"); matrix list theta; matrix ep=bhat'; matrix ep=idwbar*ep; matrix ep=ep-iden; matrix ep=ep*psi; display("Matrix of price elasticities"); matrix list ep; end; mkels; **Completing the system by filling out the matrices; cap program drop complet; program define complet; *First extending theta; matrix atm=theta*itm; matrix atm=-l*atm; matrix atm=atm-bO; matrix xtheta=theta,atm; matrix atm=xtheta'; matrix atm=atm*itm; matrix atm=atm'; matrix xtheta=xtheta\atm; *Extending the diagonal matrices; matrix wlast=wbar'*itm; matrix won=(l); matrix wlast=won-wlast; matrix xwbar=wbar\wlast; matrix dxwbar^diag(xwbar); matrix idxwbar=syminv(dxwbar); matrix bllast=(0.25); matrix xbl=bl\bllast; matrix b01ast=bO'*itm; matrix b01ast=~l*bOlast; matrix xbO=bO\b01ast; matrix xe=itml-xbl; matrix tm=idxwbar*xbO; matrix xe=xe+tm; matrix tm=xe'; display("extended outlay elasticities"); matrix list tm; matrix xxi=itml-xbl; matrix xxi=dxwbar*xxi; matrix xxi=xxi+xbO; CODE APPENDIX 431 matrix tm=diag(xbl); matrix tm=syminv(tm); matrix xxi=tm*xxi; matrix dxxi=diag(xxi); *Extending psi; matrix xpsi=dxxi*xtheta; matrix xpsi=xpsi+idenl; matrix atm=dxxi*dxwbar; matrix atm=atm+idenl; matrix atm=syminv{atm); matrix xpsi=atm*xpsi; matrix ixpsi=inv(xpsi); ^Extending bhat & elasticity matrix; matrix xbhatp=xtheta*ixpsi; matrix xep=idxwbar*xbhatp; matrix xep=xep-idenl; matrix xep=xep*xpsi; display("extended matrix of elasticities"); matrix list xep; end; complet; **Calculating symmetry-restricted estimators; **These are only approximately valid & assume no quality effects; run mkmats; vecmx bhat vbhat; ** R matrix for restrictions; Imx $ngds llx; commx $ngds k; global ng2=$ngds*$ngds; matrix bigi-I($ng2); matrix k=bigi-k; matrix r=llx*k; matrix drop k; matrix drop bigi; matrix drop llx; ** r vector for restrictions, called rh; matrix rh=bO#wbar; matrix rh=r*rh; matrix rh=-l*rh; **doing the constrained estimation; matrix iss=iden#invs; matrix rp=r'; matrix iss=iss*rp; matrix inn=r*iss; matrix inn=syminv(inn); matrix inn=iss*inn; matrix dis=r*vbhat; matrix dis=rh-dis; matrix dis=inn*dis; matrix vbtild=vbhat+dis; unvecmx vbtild btild; **the following matrix should be symmetric; matrix atm=bO'; matrix atm=wbar*atm; matrix atm-btild+atm; matrix list atm; **going back to get elasticities and complete sytem; matrix bhat=btild; mkels; complet; log close; program mkmats.do 432 THE ANALYSIS OF HOUSEHOLD SURVEYS *calculates two matrices, the commutation matrix and *the lower diagonal selection matrix that are needed *in the main calculations; these are valid only for *square matrices *also a routine for taking the vec of a matrix *and a matching unvec routine *for calculating the commutation matrix k *the matrix is defined by K*vec(A)=vec(A1) cap program drop commx program define commx local n2='l'A2 matrix "2'=J(Xn2',"n2',0) local i=l local ik=0 while % i' <= '!' { local j=l local ij= v i' while v j ' <= N l ' { local ir=*j'+*ik ( matrix N 2'[ v ir', x ij']=1 local ij =x ij' +s 1' local j="j'+1 } local i= s i'+l local ik= v ik'+ s l' } end **for vecing a matrix, i.e., stacking it into a column vector cap program drop vecmx program def vecmx local n=rowsof(vl') local n2= N n' A 2 matrix def N 2 ' =J ( V n2 M, 0) local j=l while N j' <= "n' { local i=l while v i ' <= "n( { local vcel=( N j'~1)* x n'+ % i' matrix X 2'pvcel',1] =x 1't'i' ,x j'] local i= v i'+l } local j= v j'+1 } end *program for calculating the matrix that extracts *from vec(A) the lower left triangle of the matrix A cap program drop Imx program define Imx local ng2=xl'"2 local nr=0.5**l'*ri'-l) matrix def N2'=J("nr',"ng2',0) local ia=2 local ij=l while v ij' <= xnr'{ local ik=0 local klim=*l'-*ia' while "ik 1 <= "klim1 { local ip-'ia'+( N ia l -2)* s l'+*ik' matrix N 2'[ s ij',*ip']=1 local ij=>ij'+l local ik=xik'+l } local ia= N ia'+l CODE APPENDIX 433 } end **program for unvecing the vec of a square matrix; cap program drop unvecmx program def unvecmx local n2=rowsof{x1') local n=sqrt{Nn2') matrix def "2'=J{ v n', x n' ( 0) local i=l while x i' <= N n' { local j=l while N j' <= x n' { local vcel=T j ' - D ^ n ' + U1 2'[xi',xj']='1'[xvcel«,1] matrix N v local j = j ' + 1 } local i = x i ' + l } end program bootall.do **for bootstrapping Indian demand estimates version 4.0 ^delimit ; capture log close; set more 1; drop _all; do allindia; log using bootall, replace; drop _all; set maxvar 300; vecmx xep vxep; set obs 1; gen reps=0; global nels=$ngl*$ngl; global nmc=1000; cap program drop vtodat; program define vtodat; local ic=l; while N ic' <= $nels {; gen e"ic'=vxep[Nic',1]; local ic= N ic'+l; >; end; vtodat; save bootall, replace; drop _all; cap program drop purge; program define purge; local ig=l; while s ig' <- $ngds {; qui regress yOc'ig' quard* regiond*; predict tm, resid; replace yOc'ig^tm; drop tm; qui regress ylcxig' quard* regiond*; predict tm, x resid; replace ylc ig'=tm; drop tm; local ig= v ig'+l; }; end; cap program drop mkns; program define mkns; 434 THE ANALYSIS OF HOUSEHOLD SURVEYS local ig=l; while "ig1 <= $ngds {; replace nOc *ig'=1/nOcxig'; replace nlcxig'=l/nlcvig'; qui summ nOc x ig'; matrix n O T i g 1 , 1 ] = (^result(3))A(-l) ; qui summ n l c ^ i g 1 ; matrix nlTig 1 ,1] = (^result (3) ) A ( - l ) ; drop n Q c ' i g ' n l c ' i g 1 ; local ig="ig'+l; }; end; cap program drop mkcov; program def mkcov; local ir=l; while v ir' <= $ngds {; local ic=l; while x ic' <= $ngds {; qui corr ylcxir' ylc'ic1, cov; matrix s[*ir',xic']-_result(4); qui corr ylc^ir' yOc'ic 1 , cov; matrix r[*ir', N ic']=_result(4); local ic=Nic'+l; >; local ir= N ir'+l; }; end; cap program drop fixmat; program def fixmat; matrix def sf=s; matrix def rf=r; local ig=l; while ^ig 1 <= $ngds {; matrix sf [ v i g * , N ig* ]=sf [ ^ig 1 ,N ig' J-omepig* ,1] /nl f v ig' , 1] ; matrix rf [ x ig' ,x ig l ]=rf Pig' , x ig' ]-lam[ "ig 1 ,1]/nO[Nig' , 1] \ local ig=5xig'+l; }; end; cap program drop mkels; program define mkels; matrix cmx=bhat'; matrix cmx=dxi*cmx; matrix cmxl=dxi*dwbar; matrix cmx=iden-cmx; matrix cmx=cmx+cmxl; matrix psi=inv(cmx); matrix theta=bhat'*psi; display("Theta matrix"); matrix list theta; matrix ep=bhat'; matrix ep=idwbar*ep; matrix ep=ep-iden; matrix ep=ep*psi; end; cap program drop complet; program define complet; *First extending theta; matrix atm=theta*itm; matrix atm=-l*atm; matrix atm=atm-bO; matrix xtheta=theta(atm; matrix atm=xtheta'; matrix atm=atm*itm; matrix atm=atm'; matrix xtheta=xtheta\atm; CODE APPENDIX 435 *Extending the diagonal matrices; matrix wlast=wbar'*itm; matrix won=(1) ; matrix wlast=won-wlast; matrix xwbar=wbar\wlast; matrix dxwbar=diag(xwbar); matrix idxwbar=syminv{dxwbar); matrix bllast=(0.25) ; matrix xbl=bl\bllast; matrix b01ast=bO'*itm; matrix b01ast=-l*b01ast; matrix xbO=bO\b01ast; matrix xe=itml-xbl; matrix tm=idxwbar*xbO; matrix xe=xe+tm; matrix tm=xe'; matrix xxi=itml-xbl; matrix xxi=dxwbar*xxi; matrix xxi=xxi+xbO; matrix tm=diag(xbl); matrix tm=syminv(tm); matrix xxi=tm*xxi; matrix dxxi=diag(xxi); *Extending psi; matrix xpsi=dxxi*xtheta; matrix xpsi=xpsi+idenl; matrix atm=dxxi*dxwbar; matrix atm=atm+idenl; matrix atm=syminv(atm); matrix xpsi=atm*xpsi; matrix ixpsi=inv(xpsi); *Extending bhat & elasticity matrix; matrix xbhatp=xtheta*ixpsi; matrix xep=idxwbar*xbhatp; matrix xep=xep-idenl; matrix xep=xep*xpsi; end; cap program drop bootindi; program define bootindi; local expno=l; while ^expno' <= $nmc {; display{"Experiment number xexpno'"); quietly {; use tempclus; bsample _N; qui tab region, gen(regiond); qui tab subround, gen(quard); drop regiond6 quard4; purge; drop regiond* quard*; matrix define nO=J($ngds,1,0); matrix define nl=J($ngds,1,0); *Averaging (harmonically) numbers of obs over clusters; mkns; *Making the intercluster variance and covariance matrices; *This is done in pairs because of the missing values; matrix s=J($ngds,$ngds,0); matrix r=J{$ngds,$ngds,0); mkcov; *We don't need the data any more; drop _all; *Making OLS estimates; matrix bols=syminv(s); matrix bols=bols*r; *Corrections for measurement error; fixmat; matrix invs=syminv(sf); matrix bhat=invs*rf; 436 THE ANALYSIS OF HOUSEHOLD SURVEYS global ngl=$ngds+l; matrix iden=I($ngds); matrix idenl=I($ngl); matrix itm=J($ngds,1,1); matrix itml=J($ngl,1,1) ; matrix dxi=diag{xi); matrix dwbar=diag(wbar); matrix idwbar=syminv{dwbar); mkels; **Completing the system by filling out the matrices; complet; **Calculating symmetry restricted estimators; vecmx bhat vbhat; ** R matrix for restrictions; Imx $ngds llx; commx $ngds k; global ng2=$ngds*$ngds; matrix bigi^I($ng2); matrix k=bigi-k; matrix r=llx*k; matrix drop k; matrix drop bigi; matrix drop llx; ** r vector for restrictions, called rh; matrix rh=bO#wbar; matrix rh=r*rh; matrix rh=-l*rh; **doing the constrained estimation; matrix iss=iden#invs; matrix rp=r'; matrix iss=iss*rp; matrix inn=r*iss; matrix inn=syminv{inn); matrix inn=iss*inn; matrix dis-r*vbhat; matrix dis=rh-dis; matrix dis=inn*dis; matrix vbtild=vbhat+dis; unvecmx vbtild btild; **going back to get elasticities and complete sytem; matrix bhat^btild; mkels; complet; vecmx xep vxep; set obs 1; gen reps-x expno'; vtodat; append using bootall; save bootall, replace; drop _all; local expno="expno'+1; }; } ; end; bootindi; use bootall; display{"Monte Carlo results"); summ; log close; program policy.do ** Does the policy analysis tables starting by running allindia ** and then processes to produce the two tables, Table 5.12 and 5.13 ** version 4.0 #delimit ; CODE APPENDIX 437 set more 1; drop _all; /* do allindia; */ capture log close; log using policy, replace; drop _all; global xgoods "ric whe jow oce pul dai oil mea veg fru sug oth"; matrix ap=(1.5\1.25\1.0\1.0\1.0\1.0\0.67\1.0\1.0\1.0\1.0\1.0); cap program drop mkws; program def mkws; use indinr; keep Inexp Ihhs w*; gen xh=exp(Inexp); gen nh=exp(Ihhs); gen woth=l-wwhe-wric-wjow~woce-wpul-wdai-wmea-woil-wsug-wveg-wfru; matrix wtild=J(12,1, 0) ; matrix weps=J(12,4,0) ; ^Plutocratic and soc rep budget shares; local ig=l; while » v l ' " ~= '•"{; qui summw v l' [aweight=xh]; matrix wtild[xig' ,l]=_result(3) ; matrix weps[vig1,1]=_result(3); gen tem=wx1'*(xh/nh)A(-0.5); qui summ tern [aweight=xh]; matrix weps [ v ig' , 2 ] =__resul t (3) ; drop tern; gen tem=wx1'*(xh/nh)A(-1); qui summ tern [aweight=xh]; matrix weps["ig',3]=_result(3); drop tem; v gen tem=w 1'*(xh/nh)A(-2); qui summ tem [aweight=xh]; matrix weps[xig',4]=_result(3); drop tem; local ig=xig'-«-l; mac shift); ^Scaling the matrices; matrix itmlt=itml'; matrix wsum=itmlt*weps; local ic-1; while xic'<=$ngl {; local id=l; while x id' <=4 {; local stmp=l/wsum[l,xid1]; matrix weps["ic',"id1]=xstmp'*weps[xic',xid']; local id="id'+l; }; local ic=xic'+l; }; end; mkws $xgoods; cap program drop equi; program def equi; local ic=l; while "ic 1 <= 4 {; matrix wtmp=weps[.,"ic']; matrix tmp=mxw*wtmp; matrix tmpl=dtot*tmp; display("equity effect"); matrix list tmp; display("Cost-benefit ratio"); matrix list tmpl; local ic=xic'+l; 438 THE ANALYSIS OF HOUSEHOLD SURVEYS end; cap program drop calpol; program def calpol; matrix tauotau=itml-ap; display{"Column one"); matrix list tauotau; matrix dth=vecdiag(xtheta); matrix dth=dth'; matrix mxw=diag(wtild); matrix mxw=syminv(mxw); matrix midt=mxw*dth; matrix col2=midt~itml; display("second column"); matrix list co!2; matrix dtau=diag{tauotau); matrix co!3=dtau*co!2; display("third column"); matrix list co!3; matrix co!4=xtheta';; matrix col4=co!4*tauotau; matrix co!4=mxw*col4; matrix tmp=dtau*midt; matrix col4=co!4-tmp; display{"fourth column"); matrix list co!4; matrix tot=itml+co!3; matrix tot=tot+co!4; display("totals"); matrix list tot; display("Going on to next Table"); matrix dtot=diag(tot); matrix dtot=syminv(dtot); equi ; end; calpol; log close; A.6 Chapter 6 None of the calculations in Chapter 6 require new or nonstandard coding. 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Subject index Accounting ratios, 324, 326, 328, 330 Bootstrap, 58-61; confidence intervals, Accumulation motives, 346, 349, 398 127, 216; further reading, 62; poverty Adult equivalents, 242 measures, 155; standard deviation, Adult goods, See Goods 404-05; standard errors, 71, 122, 192, Age dummies, 124-25 413-15; variance-covariance matrix, Age effects: code, 407-08; normalization, 313-14; weighted means and medians, 125-26; Taiwan (China), 346, 347, 350 404-05 Age profile, 123; consumption, 124, 339- Borrowing constraints, 363-72 42, 387-88; earnings, 118,127, 343 Botswana, 374 AIDS, 13, 34 Boundary effects, 24-25, 26, 169 Altruism, 226, 243 Brazil, 33, 34, 40, 216, 224 Apartheid, 156-57, 200 Breusch-Pagan test, 79 Assets, 12, 22, 24, 28-29, 35, 149-50, Britain: age inequality profiles, 388; con- 184, 206, 227, 241, 324, 337-38, 348, sumption and income, 340; Family Ex- 350, 352, 353, 357-58, 360, 363-76, penditure Survey (FES), 8, 12, 27; food 383-89, 394, 397-99 expenditure, 267; London cholera epi- Assignable goods, See Goods demic, 113; savings rate, 350; World Attenuation bias, 99-100,109 War II food rationing, 209-10 Attrition, 19-20, 37-39,120 Budget: constraint, 357, 360; conventional Autoconsommation, 28 analysis, 205; surveys, 23 Autocorrelation, 20 Bulgaria 34 Average: sample- weighted, 69 Average derivative estimator, 333-34 Calories: household structure and, 219-21; elasticities, 204, 212, 215-18; Engel Bandwidth, See Kernel curve, 142-43; productivity and, 211; Bangladesh, 239, 241 price change and, 208-09; regression Bayesian hypothesis testing, 131 functions, 216-18 Behavioral relationships, 102 Censored regression models, 89-92, 105 Bequests, 346, 348, 350, 398 Census: data, 70; political sensitivity, 10 Bias: attenuation, 99-100, 109; boundary Central tendency, 170 effects, 24-25, 26, 169; efficiency Children: cost of, 244-45, 247-49; gender trade-off, 108; gender, 223-41, 269; bias, 224, 231, 237-38; needs, 150, omitted variables, 109; variance trade- 223, 241; nutrition, 107; substitution off, 174 effect, 258, 261-62; weights, 242. See Bolivia, 34 also Equivalence scales References in this publication to "Taiwan," "Republic of China," and "Taiwan (China)" refer to the region, "Taiwan, China "References to "Hong Kong" refer to the region, "Hong Kong SAR, China." 463 464 THE ANALYSIS OF HOUSEHOLD SURVEYS China, 11, 34, 224. See also Taiwan policy implications, 350-51; short-term, (China) 350-57, 394. See also Life-cycle model; Clusters, 10, 155; advantages of, 12-13; Permanent income model; Saving balanced, 74-75; bootstrap, 60; econo- Contour plots, 180-81,419-20 metrics of, 73-78; effects of, 345-47, Convexity: of marginal utility, 361-63; of 378; Pakistan study, 57; precision and, nutrition to productivity relation, 211; 14, 52, 71-73; primary sampling units of regression functions, 197; of social (psus), 51-56, 71; in regressions, 74- welfare indifference curves, 135, 145 78; superpopulation and, 56-57; un- Correlation: autocorrelation, 20; errors, balanced, 76,306 92-105, 110; intracluster, 54, 76; not Cohorts, 117-20; code, 407-08; combined causation, 65; spurious, 29 surveys, 121; effects of, 125-26, Cost-benefit ratios: efficiency and equity, 345-47; inequality within, 121; in the 277, 331; evaluation, 278-79; of tax life-cycle model, 342-45; profiles, 127; changes, 276-78 versus panel data, 120-21 Cost functions, 207, 245, 249-50 Colombia, 374 Cost-of-living index, 245-46, See also Commodity prices, 18, 153, 351 Price index. Community, concept of, 37 Costs: children, 244, 247-49; computa- Composite commodity theorem, 308 tional, 179, 194; inequality, 136; sur- Computable general equilibrium (CGE) vey, 15,40 model, 272, 282 Cote d'lvoire: annual expenditure, 152- Computers, 1-2, 8, 35-36, 59, 181 53; consumption, 408-11; consumption Consumer surplus, 245-47 insurance, 377-83; Enquete Budget Consumption: age profiles, 124, 339-42, Consommation, 302; gender effects, 387-88; buffering model, 386, 389; by 238-39; household size, 153; income, age of household head, 341-42; cash on 41; inequality, 151-56, 161-62, 411- hand and, 365-66; code, 408-11; cohort 13; living arrangements, 338-39; effect, 345-47; with cumulated luck, Living Standards Survey, 8, 21-22, 26, 384-85; data quality debate, 27-28; 32-34, 96-97, 133, 151-56, 339, 352- farmers' rule, 366; habit effects, 53, 355-56, 377-83; Lorenz curve, 398-99; household composition and, 418-19; panel data, 37-39; PCE, 165- 224-45; income and, 345-50; inflation 66, 175-76, 180-81; poverty, 151-56, effect on data, 32; innovations, 384, 167-68,413-15; price elasticity of beef, 385; marginal propensity, 353; marginal 302-03; private transfers, 374 utility, 360, 361-63, 365; "martingale", Covariance analysis, for heterogeneity,72 359-60; measuring, 26-29; perman- Cross-price effects: estimation, 311-14; ent-income theory, 32; social insurance subsidies and 330; tax revenue and, and, 372-83; tracking with income, 277, 279-80, 327-28; in utility maxi- 339, 349-50, 369, 380, 392; transitory, mization model, 306-11 32; village effects, 380; welfare mea- Cross-price elasticity, 317-20, 328, surement, 148-49 425-38 Consumption insurance, 386; cluster effects, 378; Cote d'lvoire, 377-83; Data: content and quality, 22-32; cross- empirical evidence, 376-83; mutual, sectional, 19; features of, 4; from house- 373; partial, 382-83; in theory, 375-76 hold to individual, 23-24, 48, 150, Consumption smoothing: autarkic, 363, 219-21, 241; household expenditure, 375; buffering model, 364-72; fre- 244-45; nonexperimental, 3, 65, 92- quencies, 336, 350-51, 354; further 105, 115; plausibility, 316; price, 149, reading, 400; over individuals, 336-37; 279-80,283-88; quality, 27-28, 32, 62; SUBJECT INDEX 465 recall, 19, 24-25; truncated and cen- Dynamics, 18,20-21 sored, 103-05, See also Panel data; Samples; Surveys Earnings, 3, 4, 19, 23, 109, 113-14, 150, Decomposition: into age, cohort, and year 169, 200, 274, 337-38, 341-46, 353, effects, 118-26, 345-49; of inequality, 357-62, 385-86, 388, 392, 394, 399; 140; of prices, 284; of poverty, 147 intrahousehold allocation and, 225-29; Deff (design effect), 15 quantile regressions of, 82-83; in South Delta method, 122,128-29 Africa, 82-83; in Taiwan (China), Demand; almost ideal system, 304; code, 117-27; 425-38; prices and, 92, 93-94, 293- Econometric approach, 3, 64-65, 73-78, 315; Stone-Geary system, 280-82; with 131-32 symmetry restrictions, 310, 320, 323 Economic growth: effect on nutrition, Demand functions, 308; error terms in, 204-05; inequality and, 390-93; 294; Hicksian, 248-49; Marshallian, poverty and, 33; saving and, 335, 338- 246-47, 249, 261 89 Demand model: choice of goods, 316; Economies of scale: Dreze method, estimation of, 295-96, 299-302; with 264-67; Engel method, 264; equi- unit values, 294-96 valence scale and, 253-54; household, Demographic structure: of the household, 241-42, 243; public good approach, 204-05, 231-35, 241-69, 288, 290-91, 262-63, 264-67; utility theory of, 262- 294-96, 298, 304-05, 311, 321, 333, 69 338, 350-51; of the population, 35, 123, Ecuador, 34 224,338-39,383,391-92 Education, 3, 22, 26, 35, 37, 65, 83, 100, Density: bivariate, 180-81,189-90; furth- 106, 142, 149, 209-10, 244, 260, 340, er reading, 203; high dimensional, 179; 341,386, 395; gender bias in, 224, 231; nearest neighbor method, 177-78; non- as instrument for permanent income, parametric methods, 169-81; two-para- 350-53 meter lognormal, 172; univariate, 170- Efficiency; equity tradeoff, 108; from lais- 76 sez-faire, 271; policy effect, 135; in Descriptive statistics, 40-61, 63, 64, 71 price reform, 328-30; taxation and, 277; Design effect (deff), 15 welfare component, 140 Developing countries: tax collection prob- Efficiency: versus bias in OLS estimation, lems, 273; tax and price reform theory, 75-76; 273-315; villages, 73-74; welfare mea- Elasticities: calories, 204, 212, 215-18; surements, 148-49. See also named cross-price, 317-20, 328, 425-38; ex- countries penditure, 305-06; own-price, 425-38; Diets: minimum cost, 206. See also Calo- quality, 77, 289, 298-99, 305-06, 314- ries; Food; Nutrition 15; substitution, 207. See also Price Dimensionality, curse of, 179 elasticities Dispersion, 25-26, 170 El Salvador, 27 Dissaving, 338, 340, 349, 396 Engel curve, 204-05, 244; calorie, 142- Distribution: cumulative, 165-66; further 43, 422-25; expenditure, 282; food, reading, 202-03; ranking, 163, 165; rice 81-82, 98, 252; linear, 281; in Roth- prices and, 182-87; shape of, 82-83; barth calculation, 256; Working, 231, single number representation, 135; taxa- 233, 257, 304-05 tion and, 196. See also Inequality Engel's Law, 251,257 Dominance, stochastic, 162-64, 202-03, Envelope theorem, 358 242-43 Equity: measures, 137; preference, 135; in Durable goods, 22, 25-26, 35, 149, 187 price reform, 328-30; taxation and, 277, 466 THE ANALYSIS OF HOUSEHOLD SURVEYS Equity (continued) Foster-Greer-Thorbecke index, 147-48, 281; a welfare component, 140 155-56 Equivalence scales; assumptions, 205; Fourier series, 193 economies of scale and, 253-54; France, 227, 267 Engel's method, 251-55; further read- ing, 269-70; identification, 260; need GAUSS, 181,408,419-20 for, 204, 223-24, 243-44; policy appli- Gauss-Markov theorem, 70, 83 cations, 259; problems with, 150, 260- Gender bias, 205, 223-41, 269 62; Rothbarth's method, 255-60; under- Generalized least squares estimator (GLS), identification of, 248-51; utility theory 75 of, 247-48 Generalized methods of moments (GMM), Error-in-variables problem, 122, 301 370-71 Errors: boundary or start-up bias, 24-25; Ghana, 25, 26, 34, 38-39, 224, 374 conditional mean, 93; correlation, 92- Gini coefficient, 139, 140, 147, 153-56, 105, 110; differencing, 107-09; distri- 158,393 bution of, 89; inaccurate frames, 10-12; Goods: adult, 237, 238, 240, 256-58; mean integrated squared, 174; non- assignable, 228-29; exclusive, 228-29, sampling, 18, 21-22; recall, 24-25; 230 Type I and II, 130-3L See also Mea- Government: expenditure, 4; revenue, 327, surement error; Standard error 329-30,395. See also Policy; Subsidies Estimators: compared, 405-06; general- Granger causality, 355 ized least squares, 75; kernel, 171-75; Graphical techniques, 133 LAD, 83, 89-90, 405-06; ratio, 51; Guyana, 34 standard unbiased, 48; Tobit, 405-06; two-step, 90; weighted, 69-72; un- Habits, See Consumption weighted, 72. See also OLS estimators Headcount ratio, 145-46,151,155-56; ro- Euler equation, 358-60, 364, 370-72 bustness, 165, 168-69 Exclusive goods, See Goods Heckit (Heckman's probit), 90, 104 Expectations (mathematical), 42, 45, 53, Heterogeneity, 70-72, 77, 93, 106-08, 191 110 Experiments: natural, 3, 105, 112-15; Heteroskedasticity, 78,79-83, 85-89 policy, 92-93 Histograms, 133, 170-71 Homoskedastic regression model, 80-81 Farm-household model, 183-85 Hong Kong (China), 396, 398 Farms: fertilizers, 96-97, 108; insect- Hotelling's Lemma, 184 icides, 96-97; management quality, Household allocation: with adults and 106; productivity, 109; size determina- children, 229-31; efficient, 226-27; tion, 95; size-productivity relationship, gender bias, 223, 224-25, 231-35, 237- 95-97, 98 38; rules, 225-31, 245, 256, 260. See Fertility, 23, 221,250 also Equivalence scales Financial repression, 397 Household composition: by gender, 233- Finite population correction (/J?c), 42-44, 34; consumption patterns and, 224-25, 50,60 229; demand patterns and, 205; ranking, Fixed effects, 106-08, 121-22 242-43, See also Demographic struc- Food: beef, 302-03; demand for, 206-22; ture diets, 206; expenditure, 222, 267; Household expenditure: code, 401-02; rationing, 209-10; shares, 252-53; Cote d'lvoire, 165-66, 175, 176, 180- subsidies, 68; substitution, 323, 330. 81; data relevance, 244-45; elasticities, See also Rice 305-06; on food, 267; individual from SUBJECT INDEX 467 household data, 219-21, 241; Pakistan, Survey, 30, 33; SUSENAS, 27 402-04; South Africa, 165-66, 170-71, Inequality, 33; age effect, 386-90; among 175; Thailand, 30-31; as welfare mea- the poor, 148; aversion to, 138, 331; surement, 148. See also Per capital ex- code, 411-13; costs of, 136; Cote penditure (PCE) d'lvoire, 151-56; further reading, 202- Households: behavioral responses, 4-5, 03; international comparisons, 243-44; 34; definition, 23-24; dictatorial or measures, 136-37, 138-40, 157-60; pooling model, 226; farm-household and saving, 121, 383-93; South Africa, model, 183-85; food consumption by 156-57; within-cohort, 385-86. See size, 267; ranked, 48. See also Econo- also Distribution mies of scale Infant mortality, gender bias in, 224 Household structure, See Demographic Inflation: effects, 32, 341; factors, 16, 45, structure, Household composition 53, 55, 67-68, see also Weights Housing: indicators, 149; saving and, 398 Inheritance, 346, 349 Hypothesis tests, 129-31 Instrumental variables, 111-16 Instruments, 116, 123 Identically distributed assumption, 78 Insurance. See Consumption insurance Identification, 104-05; exact, 111; over Interest rates, and saving, 397 (OID), 112, 115 International Monetary Fund, 272 Impatience, 360, 364-68, 371, 386, 389 Interquartile ratio, 139 Income: accounting framework, 29; age Intertemporal choice model, 357-59,400 profiles, 339-40; agricultural, 38; Intracluster correlation coefficient, 54, 76 choice and, 88; code, 401-02; cohort Investment, and saving, 397 effect, 345-47; consumption and, 345- 50; effects, 230; lagged, 353; measure- Jamaica, 26, 27, 34, 39, 40 ment, 29-32; prediction, 354-57 370; saving and, 395-96; seasonally, 32; Kazakstan, 34 self-employed, 29; shocks, 371; single Kenya, 19, 96 location risk, 379; stabilization policy, Kernel, 191-92; advantages of, 178; band- 394-95; Thailand, 30-31; tracking with width, 174-75, 176, 192, 216; bias, consumption, 349-50, 369, 380, 392; 197-99; bivariate, 178-79; cross- transitory, 31-32, 352-53, 369; as validation, 176-77; Epanechnikov, 173, welfare measure, 148 175, 178-79, 180; estimators, 171-75; India: age inequality profiles, 389-90; Gaussian, 173, 179; quartic, 173, 175, economies of scale, 263-64, 267; equity 179, 216; rectangular, 173, 178, 192; and efficiency, 330-31; food demand, smoothing, 174-75 206; food share regression, 252-53; Korea, Republic of, 8, 396, 398 gender bias, 205, 231-35, 237-38; Kyrgyz Republic, 34 ICRISAT, 18, 212, 371, 374, 376-77, 389-90; Living Standards Survey, 27; Labor markets and farm productivity, 95 National Sample Survey, 8, 142; Labor supply: and saving, 358; and taxes, National Sample Survey Organization, 274 1, 27,30, 213-14; poverty line, 142-43; LAD estimator, 83, 89-90, 405-06 poverty trends, 33; price reform, Lagrange multipliers, 129, 375 330-31; Public Distribution System Lags, 98-99, 110-11 (PDS), 325; shadow taxes and subsidies, Land quality, 95-96, 98 325-26; taste variation, 320-23; unit Laspeyres index, 149 values 316. See also Maharashtra Least squares regression, See OLS esti- Indonesia, 106,143, 257, 396; Family Life mators 468 THE ANALYSIS OF HOUSEHOLD SURVEYS Life-cycle model, 337-50, 359-60, 362; 404-05; zero, 28 applied to developing countries, 338- Measurement error, 99-101; calorie ex- 39; attractions of, 395-96; with cohort penditure, 221-22; causing spurious data, 342-45; with economic growth, correlations, 29, 31; difference- and 392; simplified, 337-38; to explain within-estimation, 108-10; earnings cross-country effects, 399. See also and, 109; household expenditure, 169; Permanent income model income, 31, 38, 352-53, 356; in multi- Linear expenditure system, 280-82 variate regression, 100; in panel data, Linear probability model, and selection, 21-22, 108-10, 120; problems of 105 isolating, 151; savings, 337; unit values, Linear programming, and quantile regres- 292-93; variances and covariances, 101 sions 84 Median: compared to mean, 59; defined, Linear regression model, 67, 106-08, See 48; preferred to OLS, 83; unweighted, also OLS estimators 46-47; weighted, 46-47,404-05 Liquidity constraints test, 370-72 Migration, 19, 23-24, 35, 37, 117, 121, Living arrangements, 338-39 202,221,342 Living standards, 22, 134-69. See also Mills' ratio, 104-05 Social welfare function; Welfare Modeling, 41, 64-65, 67 Living Standards Measurement Study Money: marginal utility, 185; metric (LSMS), 32-40; data quality, 36-37; utility, 148; value of a man, 244 design features, 34-35; for policy mak- Mongolia, 34 ing, 39; for research, 39-40 Monte Carlo experiments, 306, 405-06 Living Standards Surveys, 4, 8, 23, 26, 27, Morocco, 34 30, 32-40, 62, 149. See also under indi- Multivariate regression measurement vidual countries error, 100 Local markets, 93-94, 289 Logit models, 85-86 Needs: age-related, 223; children's, 150, Lorenz curve: code, 417-19; of con- 223, 241 sumption, 384, 391; Cote d'lvoire, Nepal, 34 161-62; generalized, 159-60, 162, 164, Net benefit ratio, 185,187,189-90 169, 202; inequality measure, 133,139, Netmap, 181,419-20 158-60; South Africa, 160-61 Nicaragua, 34 Lorenz dominance, 159,163-64 Nonexperimental data, 115; causal infer- LOWESS, 193 ence from, 65; regression in, 92-105; treatment and control variables, 92-93 Maharashtra: code, 425-38; nutrition, Nonparametric methods, 133-34, 191-97, 213-18, 221-22, 231-38; price vari- 202 ation, 272, 285-88, 316, 320-23 Nonresponse, 12,16 Malaysia, 8, 19,30,33 Null hypothesis, 129-31 Malnutrition, 5, 210-15 Nutrition: child, 107; demand for, 206- Mauritania, 34 22; economic growth and, 204-05; Maximum-likelihood estimation (MLE), 85 effect of income change, 221; effect on Mean: approaches compared, 41-43; with wages, 98, 109; expenditure elasticity, bootstrap, 60; compared to median, 59; 211-12; further reading, 269; gender consumption and income, 21-22; of in- bias, 224; indicators, 148; PCE and, flation factor, 16; probability-weighted, 219-21; productivity and, 210-11; 46-48, 51, 53; sampling variance, 43- surveys, 23; wage theory, 210-11. See 44; stratification, 50; unweighted, 46- also Calories; Food 47, 52,60; weighted, 46-47, 52, 60, 67, SUBJECT INDEX 469 Observation periods, 24-26, 39 advantage of, 362; behavioral predic- OLS estimators, 88, 94, 356, 381-82; tion, 354-57; for empirical analysis, attenuated, 99-100; bivariate, 301; 351-52; with growth, 391; inequality, compared with other estimators, 115- 390-92; panel data, 355; problems of, 16, 353, 405-06; efficiency, 75-76,79; 352; a special case, 359-60; with food expenditure, 222; for linear reg- uncertainty, 383-86. See also Life-cycle ression, 198; median preferred to, 83; model net-outlay ratios, 235; pooled reg- Peru, 8, 19, 34, 374 ression, 107; unweighted, 68-69 Philippines, 106, 107, 212, 221-22, 374 Omitted variable bias, 109 Policy, 34, 259; effects, 3, 182-202; Outlay-equivalent ratios, 238; boys and evaluation, 97, 103, 105, 112-15, 134- girls, 257; contamination, 240; standard 35; experiments, 92-93; focus on poor, errors, 235-37 137, 138, 140-41; income stabilization, Overidentification (OID), 112, 115 394-95 Ownership indicators, 149-50 Population: aging, 390-93; compared to Own-price: effect, 327-28; elasticity, 280- sample, 11; finite, 40-44; growth, 395- 81,317-21,330,425-38 96; heterogeneity, 70-72; variance, 48, See also Superpopulation Pakistan, 425; demand for food, 206; Poverty: Atkinson index, 153-55; code, Economic Survey, 324; economies of 413-15; cost of elimination, 146; Cote scale, 263-64, 267; food share regres- d'lvoire, 151-56; definition, 206; diag- sion, 252-53; gender effects, 205, 239, nosis, 141; and economic growth, 33; 241; Household Income and Expendi- estimation, 43; Foster-Greer-Thorbecke ture Survey, 77-78, 81-82, 239, 283- index, 147-48, 155-56; further reading, 88; Living Standards Survey, 26, 34, 202-03; international comparisons, 57-58, 264; PCE, 402-04; prices, 272, 243-44; measures of, 144-48, 164-69, 283, 317-20; shadow taxes and sub- 413-15; Sen index, 146-47, 155-56; sidies, 324-25; Sindh Province, 216; social welfare approach, 140-41; South unit values, 316 Africa, 156-57; tracking, 4, See also Panel data, 18-22, 77; advantages of, 34, Inequality; Welfare 97, 105-11; collection experience, 37- Poverty deficit curve, 166, 168 39; exogeneity, 110-11; from succes- Poverty gap, 146, 147, 155-56 sive cross sections, 121-23; inequality Poverty line, 151; calorie basis, 142-43; changes, 386-87; lags, 110-11; mea- absolute or relative, 143-44; construc- surement error, 108-10; poverty persist- tion, 141-44; Cote d'lvoire, 167-68; a ence, 151; rotating design, 19-21; ver- crude device, 141; South Africa, 157, sus cohort data, 120-21 167-68 Paraguay, 34 Precautionary motives, 361-63, 365, Parameter transformation, 127-29 371-72,386 Pareto condition, 135 Precision: and clusters, 14, 52; design Pareto improvement, 137 effect (deff), 15; from panel design, 20; Pensions, 133, 182,200-202 optimal design, 17-18; stratification Per capita expenditure (PCE): code, 402- and,14 03; Cote d'lvoire, 165-67, 175-76, Preferences: additive, 280-81; time, 180; disadvantages of measure, 150, 363-65 153; India, 215-20; Indonesia, 143; Price change: behavioral response, 272, Pakistan, 57-58; South Africa, 165-67, 332-33; demand and, 92-94, 205-06; 170-71 policy induced, 185; profits and, 184; Permanent income model, 336, 370; quality and, 297-99; utility and, 184; 470 THE ANALYSIS OF HOUSEHOLD SURVEYS Price change (continued) long, 36; pre-coded responses, 35; welfare and, 133, 207-08, 246-47 price, 35, 36-37 Price elasticities, 305-06; Cote d'lvoire, 302-03; estimates, 293-315; long- and RAND, 19, 30, 33 short-term, 323; Pakistan and India Randomness, 41, 92 compared, 323 Random walk, 359 Price index; consumers' surplus and, 245- Ratio estimator, 51 47; indirect utility and, 207; in poverty Recall data, 19, 24-25 measures, 149, 151, 153-54; quality Reference period, 151 and, 308-09; regional, 37, 283-84; Refusal rates, 20 Stone, 309; weights for, 8, 17; Regression, 66-73; censored models, Price reform: with ad valorem taxes, 89-92, 105; code, 420-21, 422-25; 326-28; equity and efficiency in, 328- confidence bands, 192; defined, 63-64, 31; further reading, 334 191; descriptive, 71; "diseases", 65; Prices; accounting apparatus, 274; data, from clusters, 74-78; function, 4, 63- 279-80, 283-88; distribution and, 276; 66,71,79-80,88,91-93, 103-05, 111, laissez-faire, 271; market, 37; relation 134, 142, 191-99, 216-222, 231-32, with unit values, 286-88, 291-92; sea- 304, 333, 421-25; further reading, 203; sonal, 286; shadow, 277-78, 324; heteroskedasticity in, 79-80; IV, 111- spatial variation, 272, 279, 283-93; 12, 115, 353, 381-82; least squares, 74; stabilization, 394; vector, 297; in vill- linear, 64, 67, 91, 106-08, 198; locally ages, 74. See also Commodity prices; weighted, 198-99; multivariate, 100; Cross-price effects; Own-price; World nonparametric, 133-34, 191-97, 202; prices parametric, 193-94; polynomial, 193, Primary sampling units (PSUs). See Clus- 199; pooled, 107; quantile, 78, 80-85; ters residuals, 72-73; weights, 67-71, 191- Principle of diminishing transfers, 169 92,420-21,422-25 Principle of transfers, 135-36, 139,145 Reliability ratio, 99 Probability: of inclusion, 15-17; pro- Reporting periods, 24-26, 39 portional to size (p.p.s)> 15, 17, 46; of Rice: consumption, 187-90; nonparamet- selection, 16, 52; weighted estimates, 49 ric regression, 194-97; paddy and mill- Probit models, 85-86, 104-05 ed, 185; prices, 182-87; production, Production, 28 187-90, 194-95; sales, 194-95 Productivity, nutrition and, 210-11 Risks, 362, 372-73, 375 Profits, prices and, 184 Romania, 34 Project: administration, 114-15; evalu- Russia, 34 ation, 97, 106 Samples: choice-based, 17; compared to Quality: choice, 77; definition, 296, 297; population, 11; early use of, 7; effects, effects, 288-89; elasticity, 77, 289, 343; probability of inclusion, 15-17, 298-99, 305-06, 314-15; with price 46; random, 42, 67; replacement, 38, change, 297-99 44, 52, 59-60; selection, 101-05; Quality model, 296-99; estimation, 305; simple random, 9, 13-15, 41-47, 50, functional form, 303-06 52, 54-60, 67, 69; size, 10, 127, Quantile regression, 78; heteroskedasticity 129-31; theory and practice, 17-18; and, 80-83; technical note, 83-85. See two-stage, 51-56; variance, 43-44, 45, also LAD estimator 47, 49, 52. See also Clusters; Data; Questionnaires: community, 35, 36-37; Selection; Stratification; Surveys household, 35; linking sections, 38; Saving: buffering model, 364-72, 386, SUBJECT INDEX 471 389, 399; by age, 349; cohort effect, 46-47, 165-66, 170-71, 175; inequal- 345-47; custodial policy, 351, 362; ity, 156-57, 160-61; inflation factors, determinants of, 397-99; effect of 16; Living Standards Survey, 14, inflation, 341; explanations of, 337; 82-83, 182-202; Lorenz curve, 417-18; from transitory income, 369; further poverty, 156-57, 167-68; social pen- reading, 399-400; growth and, 335, sions, 133, 200-202; welfare measures, 395-96; housing and, 398; as income 415-17 predictor, 354-57, 370; and inequality, Spline function, 193 121, 383-93; instrument, 352, 356-57; Sri Lanka, 257 measurement error, 337; motives, 346, Standard deviation, 404-05 349, 361-63, 365, 371-72, 386, 394, Standard error: asymptotic, 115-16; boot- 398; in poor households, 357-72; ratios, strapped, 122, 192, 413-15; with 338-39, 397; Thailand, 30-31; uncert- clusters, 75-76; code, 402-04; com- ainty and, 361-63, 365; underestimated, putation of, 71; effects of correcting, 30; upward time trend, 350; weather 77; of mean change, 21-22; for out- variability and, 351-54. See also Con- lay-equivalent ratios, 235-37; regres- sumption smoothing; Life-cycle model; sion, 74, 75, 192 Permanent income model; Social insur- Start-up problems, 24-25, 26, 169 ance STATA, 2, 48, 56, 62, 69, 71, 77, 82, 84, Scale, 85, 87 86, 90, 152, 153, 175, 181, 198, 290, Scatter plots, 180-81 313,378,401-38 Schools, 2, 13, 23, 35, 37, 75, 88, 92, 100, Statistical perspective, 3, 40-61, 63, 64, 103, 149 71, 127-31 Schwarz test, 379 Stochastic dominance, 162-64, 202-03, Score test, 129 242-43. See also Lorenz dominance. Seasonality, 18, 25, 29, 32, 148-49, Stone-Geary demand system, 280-82 283-84, 286, 288, 302-03, 316, 321, Stone index, 309 336-37, 350, 354, 361, 365, 394 Stratification, 13-15, 49-51; bootstrap, Selection, 101-05; by efficiency criteria, 60; combined with clusters, 55-56; esti- 17; probability, 44-49, 67. See also mation, 50-51; explicit and implicit, 14; Samples mean, 50; Pakistan study, 57; precision Self-weighting, 10, 55 and, 14; sampling variation and, 49; Separability, 298 weights, 50-51 Shadow prices, 184, 277-78, 324-28 Stylized facts, 4, 95 Sharing rules, 225-31, 245, 256, 260 Subsidies, 4, 68, 209, 320, 324-26, 330 Simultaneity, 93, 98-99 Substitution: effect of children, 230, 258, Slutsky matrix, 207-08, 310 261-62; elasticities, 207; food, 323, 330 Slutsky symmetry, 310 Superpopulation, 40-44, 56-57 Social insurance, 372-83. See also Saving Surveys: aims of, 22, 32, 33, 39, 133; Social welfare function, 134-36; Atkin- budget, 23; costs, 15, 40; design, 9-22, son, 276, 278, 326; basic needs, 137; 66-73,101-02; fertility, 23; frames and concavity, 135-36; equity-respecting, coverage, 10-12; further reading, 61-62; 169; focus on poor, 137, 138, 140-41; historical use of, 7-8; income and em- indifference curves, 138; inequality, ployment, 23; longitudinal. See panel 137, 138-39; policy guides, 141; sym- data; non-coverage, 10-12; nutrition, metry or anonymity, 135; in tax model, 23; optimal, 17; price data, 283-88; 274-76. See also Living standards; range of data, 2-3; savings under- Welfare estimation, 30; single-purpose, 17; two- South Africa, 34; household expenditure, stage, 10, 315-23. See also Data; Sam- 472 THE ANALYSIS OF HOUSEHOLD SURVEYS Surveys (continued) Two-step estimator, 90 pies; Selection Uncertainty, 361-63, 365, 383-86 Taiwan (China), 8, 396; age, cohort and United States, 8, 398; age inequality year effects, 407-08; age inequality profiles, 388; Consumer Expenditure profiles, 387-88; age-sex pyramids, Survey, 27-28; consumption and in- 11-12, 390; consumption and income, come, 340; consumption insurance, 339; food expenditure, 267; gender 374; credit restrictions, 371; Cubans in effects, 240-41; "growth with equity," Miami, 113; Current Population Survey 383; inequality, 392-93; life-cycle (CPS), 29, 109; distributional analysis, model, 342-50; Surveys of Personal 39; food, 142-43, 206, 267; gender Income Distribution, 116-27 bias, 224; Michigan Panel Study of Tanzania, 13, 19,34,96 Income Dynamics (PSID), 18, 20, 374; Tariffs, 275, 324-26 minimum wages, 113, 115; poverty Taste shifters, 360, 376, 385 line, 142, 144; savings rate, 350, 396; Tax, 4; ad valorem, 326-28; avoidance, veteran wage premium, 113-14 277; distribution and, 196; efficient and Unit value function, 294-95 equitable, 277; export, 182, 187, 196, Unit values: elasticity, 298; estimates, 325, 330; food, 324-25; further reading, 316-17; household structure and, 291; 334; incentive effects, 146; optimal market prices and, 289-90; measure- models, 271-72, 273-74; optimal solu- ment errors, 292-93; quality effects, tion, 281; reform theory, 273-77; 288-92; regressions, 290-91; relation revenue, 91-92, 327, 329-30, 395; with prices, 149, 286-88, 291-92; shadow, 278, 324-26; small change, spatial variation, 284-88; weights, 285 275-76 U.N. system of national accounts (SNA), Technical fixes, 3, 65 33 Test statistic, 72, 112, 128, 129, 131 Utility function, 207; farm-household, Thailand, 396; age inequality profiles, 183-84; indirect, 274; intertemporal, 388; code, 401-02; export tax, 182, 357; marginal, 376; separation assump- 187,196; gender effects, 238-39; living tions, 183-84 arrangements, 24, 338-39; NIPA, 341; Utility maximization model, 306-11, regression for rice, 194-97; rice farm- 413-15 ers, 353-54; rice prices, 133, 182-87; Utility theory: with commodity-specific rice production and consumption, 187- scales, 261; in cost-of-living indices, 90; rural redistribution, 187; savings, 245-46; of economies of scale, 262- 341, 350, 401-02; Socioeconomic 69; in equivalence scales, 247-48 Survey, 30-31, 182-202, 339; urban Uzbekistan, 34 distribution, 189 Theirs entropy measure of inequality, Variance: bias tradeoff, 174; coefficients, 139,140 139, 140; sampling, 43-44, 45, 47, 49, Time preference rate, 363-65 52 Time series: data, 366, 372, 399; of cross- Variance-covariance matrix, 72-73, sections, 116-27, 342-50, 387 75-77, 80, 84, 100, 111, 115, 127-28, Tobit model, 92, 304; generalized, 90-91, 312 102-03; MLE, 88; simulation, 86-89, Venezuela, 34 405-06 Viet Nam, 26, 34 Tunisia, 34 Villages, 73-74, 93-94, 289, 380 Turkmenistan, 34 Two-stage design, 51-56 Wald test statistic, 128, 129 SUBJECT INDEX 473 Water, valuation of, 28 17; price effect, 207-08, See also Weighted estimator, 69-71 Poverty; Social welfare function Weighted least squares estimate, 68 Women's wages, 98, 102, 109, 113-14 Weights, 15-17; probabilities of selection World Bank; price reform and, 272; cor- and, 16; in regressions, 67-71; self- rective weights, 152; Living Standards weighting, 10, 55; stratification, 50-51; Surveys, 4, 8, 27, 30, 32, 39, 62 use of, 44-49, See also Inflation factor World prices, 133, 153, 182, 275, 278, Welfare: code, 415-17; consumption of 324-26,351,394 goods and, 259; food share and, 254- World Wide Web, 40,401 55; from household to individual, 23-24, 48, 150, 219-21, 241; measures Year effects, 125-26; code, 407-08 of, 134, 136-37, 148-51, 206-10, 415- Author index Ahluwalia, Montek, S. 33, 143 Bevan, David, 19,39,96,395 Ahmad, Asif, 239, 241 Bhalla, Surjut S., 96, 352 Ahmad, S. Ehtisham, 281, 324-25, 334 Bidani, Benu, 143 Ainsworth, Martha, 34, 62, 374 Bierens, Herman J., 203 Aitchison, J., 393 Blackorby, Charles, 140 Alderman, Harold, 283 Bliss, Christopher J., 98, 211 Alonso-Borrego, C£sar, 111 Blundell, Richard, 396 Altonji, Joseph, 374 Bond, Steven R,, 111 Amenuvegbe, Ben, 25 Borsch-Supan, Axel, 349 Anand, Sudhir, 390 Bosworth, Barry, 396 Anderson, Theodore W., 115, 202 Bouis, Howaarth E., 212, 221-22, 269 Andrews, William H., 95 Bound, John, 109, 116 Angrist, Joshua D., 113-14,116 Bourguignon, Francois, 225, 227-28, 242, Arabmazar, Abbas, 88-89 245, 260 Arellano, Manuel, 77, 111 Bover, Olympia, 111 Ashenfelter, Orley, 20 Bowley, Arthur L., 7 Atkinson, Anthony B., 134, 136-40, 144, Breusch, Trevor S., 79 156, 159, 168-69, 202, 242, 276, Brown, J. A. C, 393 281-82,331,396 Brown, Murray, 225 Attanasio, Orazio, 340, 349, 374, 396 Browning, Martin, J., 225, 229, 400 Brumberg, Richard, 337,400 Baker, Regina, 116 Buchinsky, Moshe, 89 Balassa, Bela, 397 Budd, John W., 203 Banks, James, 396 Buhmann, Brigitte, 244, 270 Bardhan, Pranab K., 33,143 Burtless, Gary, 396 Batten, Anton P., 258, 261-62, 265, 280 Buse, Adolf, 115 Bassett, Gilbert, 80, 84 Bates, Robert H., 334 Campbell, John Y., 354 Bauer, Peter, 351,395 Card, David E., 113, 115 Becker, Gary S., 33, 225, 360 Carroll, Christopher D., 340, 350, 362-63, Becketti, Sean, 20 371,396 Behrman, Jere R., 212, 221 Case, Anne C., 114, 182, 200-201 Benjamin, Dwayne, 96, 184 Casley, Dennis J., 7, 10, 18, 23, 61 Berry, Albert R., 95 Chamberlain, Gary, 104,110 Besley, Timothy, 114,400 Chayanov, Aleksander, V., 95 474 AUTHOR INDEX 475 Chesher, Andrew, 79 Dooley, Martin D., 386 Chiappori, Pierre-Andre, 225-26, 228, Dorfman, Robert, 206 231,245,260 Dreze, Jean, 224, 264, 269, 278, 334 Chow, Gregory C., 131 Ducpetiaux, Edouard, 7 Christensen, Laurits R., 250 Duesenberry, James S., 398 Chuma, Hiroyuka, 95 Dumouchel, W. H., 70, 72 Cleveland, William S., 193, 199 Duncan, Greg J., 70, 72 Cline, William R,, 95 Coady, David, 324-25 Eden, Sir Frederick M,, 7 Coale, Ansley, 224 Edmonds, Eric V., 26, 37 Cochrane, John H., 374 Efron, Bradley, 59, 62 Cochrane, Willam G., 42-43, 46, 54, 61 Bicker, F., 79 Coder, John, 29 Engel, Ernst, 7, 244, 251-55, 258, 262-64 Collier, Paul, 19, 39,96, 395 Engle, Robert R, 194 Cosslett, Stephen R., 17 Escudero, Vera, 365 Coulombe, Harold, 152 Espenshade, Thomas J., 230 Cowell, Frank A., 140, 202 Estes, Eugena, 194 Cowgill, Donald O., 224 Cox, Donald, 374 Fan, Jianqing, 198-99 Cramer, Jan S., 132, 172, 230, 258 Feder, Gershon, 95 Crooks, Edmund, 12 Feldstein, Martin, 397 Fields, Gary S., 33 Dalton, Hugh, 135-36 Fisher, Gordon M,, 142 Dandekar, V. M., 142-43 Fisher, Ronald A., 7 Danziger, Sheldon, 270 Fogel, Robert W., 98, 211 Das Gupta, Monica, 241 Foster, James, 144, 147, 202 Dasgupta, Partha, 98, 211, 224, 269, 351, Frankel, Martin R., 71 362 Freedman, David A., 65, 113 DaVanzo, Julie, 19 Friedman, Milton, 351-52 Davidson, Russell, 72, 80, 112, 115, 129, Frisch, Ragnar, 289 132 Fry, Maxwell J., 397 Davies, David, 7 Fuller, Wayne A., 123, 301-02 Davis, Steven, 340, 374 De Janvry, Alain, 281 Gallant, A. Ronald, 193 Deaton, Angus, 11, 20, 26, 37, 96, 123, Gersovitz, Mark, 399-400 125, 148, 182, 186, 200-201, 207, Gertler, Paul, 92, 210 213-14, 217-18, 220, 222, 224, Ghez, Gilbert R., 360 231-32, 235-38, 243, 247, 250, Ghose, A. K., 33 253-54, 257-58, 260-61, 264, 267, Gibbons, Donna M., 106 270, 272, 275, 280-82, 284-85, 287, Gieseman, Raymond, 28 290-91, 297, 302-03, 306, 309, 313, Glewwe, Paul, 19, 34-35, 39 317-18, 321-22, 331-3, 338-39, 342, Goldberger, Arthur S., 89, 235 347-48, 357, 365-66, 383, 385, 387- Gopalan, C, 213 88, 392-93, 395, 397, 400 Gorman, William M., 261-62 Demery, Lionel, 152 Gottschalk, Peter, 386 Deolalikar, Anil B., 212, 221 Graaff, Jan de-Villiers, 140 DiNardo, John, 131 Green, Jerry G., 275 Dixit, Avinash K., 334, 358 Greenhalgh, Susan, 240 Donaldson, David, 140 Greer, J., 147 476 THE ANALYSIS OF HOUSEHOLD SURVEYS Gregory, A. W., 129 Howes, Stephen, 61 Griffin, Keith, 33 Hubbard, R. Glenn, 362 Griliches, Zvi, 38, 109 Huber, Peter J,, 79 Grimard, Franque, 272, 284-85, 290, 313, Hurd, Michael, 88 317 Hurwitz, W. N., 20, 61 Gronau, Rueben, 102, 105 Grootaert, Christiaan, 37, 62,152-53 Jaeger, David A., 116 Grosh, Margaret E., 26, 34-35, 39-40, 62, Jeancard, Henri-Pierre, 26 65 Jensen, Robert T., 202 Grossman, Gene M, 334 Jeong, Jinook, 62, 116 Grossman, Jean Baldwin, 92 Jimenez, Emmanuel, 210, 374 Groves, Robert M., 12, 15, 19,41,61, 132 Johnson, Nancy L., 95 Gulati, Ashok, 326 Johnston, John, 72, 131 Gunning, Jan Willem, 19, 39, 96, 395 Jolliffe, Dean, 27 Jorgenson, Dale W,, 250 Haaga, John, 19 Haddad, Lawrence J., 24, 150, 212, 221- Kakwani, NanakC, 140, 172, 202 22, 269 Kanbur, S. M. Ravi, 24, 150, 152-53, 390 Hagenaars, Aldi J., 270 Kansal, S. M., 27 Hall, Gillette, 19, 39 Karoly, Lynn A., 19 Hall, Peter, 61 Kemsley, W, R R, 12 Hall, Robert E., 125,359 Kennan, John, 93 Hammond, R. J., 210 Kimball, Miles S., 362 Hansen, James, 326 King, W. L, 260 Hansen, Lars Peter, 371 Kish, Leslie, 15,61,71 Hansen, M.R, 20, 61 Kloek, Teun, 75 Hardle, Wolfgang, 179, 192, 194, 203, Koenker, Roger, 80, 84 333 Kolm, Serge Christophe, 169 Harris, Christopher J., 266 Konus, A, A., 245-46 Harriss, Barbara, 224 Kotlikoff, Laurence J., 374, 398 Hassan, N. Nazmul, 210 Krueger, Alan B., 109, 113-16 Hastie, Trevor, 203 Krueger, Anne O., 334 Hausman, Jerry A., 17, 38, 72, 109, 247 Kuznets, Simon, 204, 243 Hay ami, Yujiro, 95 Hayashi, Fumio, 370-71, 374 Lanjouw, Peter, 61, 264, 267 Heckman, James J,, 86, 90, 102, 104 Lau, Laurence J., 250 Helpman, Elhanan, 334 Lazear, Edward P., 230 Henderson, A. M, 230 Learner, Edward E,, 130-31 Hicks, John, 308 Leibenstein, Harvey, 210 Hirschman, Albert (X 395 Lemenshow, Stanley, 61 Hoch, Irving, 106 Lerman, Steven, 17 Holt, D., 62, 76, 132 Levy, PaulS,, 61 Holtz-Eakin, Doug, 110 Lewis, W. Arthur, 396, 400 Honore, Bo, 194 Lillard, Lee A., 125 Horioka, Charles, 397 Lindert, Peter H,, 244 Horney, Mary Jean, 225 Linton, Oliver, 203 Hotelling, Harold, 184 Lipton, Michael, 65 Houthakker, Hendrik S., 77, 204, 260-61, Little, Ian M. D., 351 268, 289, 298 Lluch, Constantino, 282-83 AUTHOR INDEX 477 Loader, Clive, 203 Newman, John, 92 Lucas, Robert E.B., 374 Neyman, Jerzy, 7 Lundberg, Shelley, 225 Ng, Serena, 333 Lury,D.A.,7,10, 18,23,61 Nicholson, J. Leonard, 230, 255, 258 Lusardi, Annamaria, 400 Nickell, Stephen, 110 Mace, Barbara J., 374 Orshansky, Mollie, 142-44 McElroy, Marjorie, 225 Otsuka, Keijiro, 95 McKay, Andrew, 152 MacKinnon, James G., 72, 80, 112, 115, Pagan, Adrian R., 79 129,132 Paish, Frank, 351 Maddala,G. S.,62, 116 Pal, Shyamalendu, 31 Madow,W.G.,20,61 Parikh, Kirit, 285, 287, 318, 321-22, Magnus, JanR., 315 331-32 Mahalanobis,P.C.,7-8,213 Parrish, William L., 240 Mangahas, Mahar, 144 Paxson, Christina H., 11, 32, 125, 224, Mankiw, N. Greg, 396 243, 264, 267, 270, 338, 340-42, 348, Manser, Marilyn, 225 350, 353-54, 383, 385, 387-88, 392- Manski, Charles R, 17,104 93, 396 Marglin, Stephen A., 351 Peng, Tey Nai, 19 Marschak, Jacob, 95 Peterson, Christine, 19 Marshall, Alfred, 212 Pfefferman, D., 76 Mas-Colell, Andreu, 275 Pindyck, Robert, S. 131 Michael, Robert T., 230 Pischke, Jorn-Steffen, 38 Miller, Ronald L, 395 Pitt, Mark M., 106,184,210 Mincer, Jacob A., 386 Platteau, Jean-Phillipe, 373 Minhas, B. S., 27 Pollak, Robert A., 225, 248-50 Mirer, Thad W., 349 Pott-Buter, Hettie A., 203 Mirrlees, James A., 98, 211, 271, 351 Powell, Alan A., 282 Modigliani, Franco, 336-38, 398, 400 Powell, James L., 89-91, 105, 333, 405- Morduch, Jonathan, 239, 241, 371,400 06 Moulton, Brent R., 76 Prais, Sigbert J., 77, 204, 260-61, 268, Muellbauer, John N. J., 148, 207, 231, 289, 298 250, 254, 257, 260-61, 270, 281, 297, Pudney, Stephen, 62 309 Pursell, Gary, 326 Mundlak, Yair, 106 Pyatt, Graham, 140, 152 Munoz, Juan, 34-35, 40, 62 Murthy, M. N., 61 Rainwater, Lee, 144, 270 Musgrove, Philip, 352 Rao, J, N. K., 60 Mwabu, Germano, 83 Rath, N., 142-43 Ravallion, Martin, 65, 143^4, 202, 264, Nagar,A.L., 115 267 Nelson, Charles, 116 Rawlings, Laura, 92 Nelson, Forrest D., 88 Rawls,John, 138 Nelson, Julie A., 261, 265 Ray, Debraj, 211 Neudecker, Heinz, 315 Reutlinger, Shlomo, 212 Newbery, David M. G., 146, 273, 334, Robinson, Peter M., 194 394 Romer, David, 396 Newey, Whitney K., 90, 105, 110, 247 Rosen, Harvey, 110 478 THE ANALYSIS OF HOUSEHOLD SURVEYS Rosenzweig, Mark R., 97, 106-07, 184, Staiger, Douglas, 116 210, 224, 374 Stark, Oded, 374 Rothbarth, Erwin, 230-31, 255-61, 264- Startz, Richard, 116 65 Stelcner, Morton, 102 Rowntree, B. Seebohm, 7 Stern, Nicholas H., 98, 146, 211, 273, 275, Roy, Prannoy, 96 278,281,324-25,334 Roy, Rene, 184,274,276 Stigler, George J., 7, 206 Rubinfeld, Daniel L., 131 Stiglitz, Joseph E., 98, 187, 210, 282, 334, Rudd, Jeremy B., 240 394 Ruiz-Castillo, Javier, 235, 258 Stock, James H,, 116,333 Ruttan, Vernon W., 95 Stoker, Thomas M., 176, 179, 194, 203, 333 Sabelhaus, John, 396 Strauss, John, 183, 210, 212, 216, 269 Sadoulet, Elisabeth, 281 Streeten, Paul, 33, 137 Sah,Raaj, 187,334 Subramanian, Shankar, 213-14, 217-18, Sahn, David, 354 220, 222, 231-32, 236-39, 253, 285, Samuelson, Paul A., 206 287,318,321-22,331-32 Sawa, Takemitsu, 115 Summers, Lawrence J., 340, 350, 396, 398 Schechtman, Jack, 365 Sundrum, R. M., 202 Schiff, Maurice, 334 Sydenstricker, E., 260 Schmidt, Peter, 88-89 Schultz, T, Paul, 33, 83, 224 Thomas, Duncan, 19, 212, 216, 224, 235, Schwarz, Gideon, 131 258, 269 Scott, A. J,, 76 Thorbecke, Eric, 147 Scott, Christopher, 25 Tibshirani, Robert J., 59, 62 Scott, Kinnon, 27 Tobin, James, 86 Selowsky, Marcelo, 212 Townsend, Robert M., 373, 376, 389, 400 Sen, Amartya K., 95, 140, 144, 146, 202, Trairatvorakul, Prasarn, 182 224,269,351 Seng, You Poh, 7 Udry, Christopher, 373 Setboongsarng, S., 182 Shorrocks, Anthony R, 159, 202, 342, 349 Valdes, Alberto, 334 Siamwalla, Ammar, 182 Van der Gaag, Jacques, 102, 210 Silverman, Bernard W., 172, 174, 176, Van der Tak, Hermann, 351 203 Van Praag, Bernard M. S., 270 Singh, Inderjit, 183 Van Weerden, H., 270 Singleton, Kenneth J., 371 Vartia, Yrjo (X, 247 Skinner, C. J. D., 62, 132 Vaughan, Denton R., 144 Skinner, Jonathan, 362-63 Veall,M.R., 129 Slesnick, Daniel T., 250 Vijverberg, Wim, 102 Smith, James P., 19 Visaria, Pravin, 31 Smith, T. M.F.,62,76, 132 Snow, John, 113, 115 Wade, Robert, 374 Solon, Gary, 20 Wales, Terence J., 248-50 Solow, Robert M., 206, 396 Walker, James R., 105 Som, Ranjan K., 61 Waugh, F. V., 289 Squire, Lyn, 183,334,351 Weber, Guglielmo, 349, 374, 396 Srinivasan, T. N., 95 Weil, David, 396 Stahl, Konrad, 349 Weiss, Yoram, 125 AUTHOR INDEX 479 Whinston, Michael D., 275 Wolter, Kirk M., 61-62 White, Halbert, 77, 79 Working, Holbrook, 81, 231, 233, 304 Wilkinson, Richard G,, 210 Wu,C.F.J.,60 Williams, Ross A., 282 Willis, Robert J., 240 Yule, George Udny, 65 Wise, David A., 17 Wolfson, Michael C, 202 Zeldes, Stephen P., 362, 370-71 Wolpin, Kenneth I., 97,106-07, 352 Zhao, Qing-hua, 26