LS.fllS LSM - 61 Living Standards JAN.1990 Measurement Study Working Paper No. 61 Large Sample Distribution of Several Inequality Measures With Application to Cote d'Ivoire Nanak Kakwani LSMS Working Papers No. I Living Standards Surveys in Developing Countries No. 2 Poverty and Living Standards in Asia: An Overview of the Main Results and Lessons of Selected Household Surveys No. 3 Measuring Levels of Living in Latin America: An Overview of Main Problems No. 4 Towards More Effective Measurement of Levels of Living, and Review of Work of the United Nations Statistical Office (UNSO) Related to Statistics of Levels of Living No. 5 Conducting Surveys in Developing Countries: Practical Problems and Experience in Brazil, Malaysia, and the Philippines No. 6 Household Survey Experience in Africa No. 7 Measurement of Welfare: Theory and Practical Guidelines No. 8 Employment Data for the Measurement of Living Standards No. 9 Income and Expenditure Surveys in Developing Countries: Sample Design and Execution No. 10 Reflections on the LSMS Group Meeting No. 11 Three Essays on a Sri Lanka Household Survey No. 12 The ECIEL Study of Household Income and Consumption in Urban Latin America: An Analytiral History No. 13 Nutrition and Health Status Indicators: Suggestions for Surveys of the Standard of Living in Developing Countries No. 14 Child Schooling and the Measurement of Living Standards No. 15 Measuring Health as a Component of Living Standards No. 16 Procedures for Collecting and Analyzing Mortality Data in LSMS No. 17 The Labor Market and Social Accounting: A Framework of Data Presentation No. 18 Time Use Data and the Living Standards Measurement Study No. 19 The Conceptual Basis of Measures of Household Welfare and Their Implied Survey Data Requirements No. 20 Statistical Experimentation for Household Surveys: Two Case Studies of Hong Kong No. 21 The Collection of Price Data for the Measurement of Living Standards No. 22 Household Expenditure Surveys: Some Methodological Issues No. 23 Collecting Panel Data in Developing Countries: Does It Make Sense? No. 24 Measuring and Analyzing Levels of Living in Developing Countries: An Annotated Questionnaire No. 25 The Demand for Urban Housing in the Ivory Coast No. 26 The CBte d'Ivoire Living Standards Survey: Design and Implementation No. 27 The Role of Employment and Earnings in Analyzing Levels of Living: A General Methodology with Applications to Malaysia and Thailand No. 28 Analysis of Household Expenditures No. 29 The Distribution of Welfare in Cite d'lvoire in 1985 No. 30 Quality, Quantity, and Spatial Variation of Price: Estimating Price Elasticities from Cross-Sectional Data No. 31 Financing the Health Sector in Peru No. 32 Informal Sector, Labor Markets, and Returns to Education in Peru No. 33 Wage Determinants in C6te d'Ivoire No. 34 Guidelines for Adapting the LSMS Living Standards Questionnaires to Local Conditions No. 35 The Demand for Medical Care in Developing Countries: Quantity Rationing in Rural C6te d'Ivoire (List continues on the inside back cover) Large Sample Distribution of Several Inequality Measures With Application to C8te d'Ivoire The Living Standards Measurement Study The Living Standards Measurement Study (LSMS) was established by the World Bank in 1980 to explore ways of improving the type and quality of house- hold data collected by statistical offices in developing countries. Its goal is to foster increased use of household data as a basis for policy decisiornaking. Specifically, the LSMS is working to develop new methods to monitor progress in raising levels of living, to identify the consequences for households of past and proposed gov- ernment policies, and to improve communications between survey statisticians, an- alysts, and policymakers. The LSMS Working Paper series was started to disseminate intermediate prod- ucts from the LSMS. Publications in the series indude critical surveys covering dif- ferent aspects of the LSMS data collection program and reports on improved methodologies for using Living Standards Survey (LSS) data. More recent publica- tions recommend specific survey, questionnaire, and data processing designs, and demonstrate the breadth of policy analysis that can be carried out using LSS data. LSMS Worldng Paper Number 61 Large Sample Distribution of Several Inequality Measures With Application to Cote d'Ivoire Nanak Kakwani The World Bank Washington, D.C. Copyright 0 1990 The International Bank for Reconstruction and Development/THE WORLD BANK 1818 H Street, N.W. Washington, D.C. 20433, U.S.A. All rights reserved Manufactured in the United States of America First printing January 1990 This is a working paper published informally by the World Bank. To present the results of research with the least possible delay, the typescript has not been prepared in accordance with the procedures appropriate to formal printed texts, and the World Bank accepts no responsibility for errors. The findings, interpretations, and conclusions expressed in this paper are entirely those of the author(s) and should not be attributed in any manner to the World Bank, to its affiliated organizations, or to members of its Board of Executive Directors or the countries they represent. Any maps that accompany the text have been prepared solely for the convenience of readers; the designations and presentation of matexial in them do not imply the expression of any opinion whatsoever on the part of the World Bank, its affiliates, or its Board or member countries concerning the legal status of any country, territory, city, or area or of the authorities thereof or concerning the delimitation of its boundaries or its national affiliation. The material in this publication is copyrighted. Requests for permission to reproduce portions of it should be sent to Director, Publications Department, at the address shown in the copyright notice above. The World Bank encourages dissemination of its work and will normally give permission promptly and, when the reproduction is for noncommercial purposes, without asking a fee. Permission to photocopy portions for classroom use is not required, though notification of such use having been made will be appreciated. The complete backlist of publications from the World Bank is shown in the annual Index of Publications, which contains an alphabetical title list and indexes of subjects, authors, and countries and regions; it is of value principally to libraries and institutional purchasers. The latest edition is available free of charge from the Publications Sales Unit, Department F, The World Bank, 1818 H Street, N.W., Washington, D.C. 20433, U.S.A., or from Publications, The World Bank, 66, avenue d'Iena, 75116 Paris, France. Nanak Kakwani is a professor and head of the Department of Econometics at the University of New South Wales, Australia, and a consultant to the Welfare and Human Resources Division of the World Bank's Population and Human Resources Department. Library of Congress Cataloging-in-Publication Data Kakwani, Nanak. Large sample distribution of several inequality measures with application to Cote d'Ivoire / Nanak Kakwani. p. cm. - (LSMS working paper, ISSN 0253-4517; no.61) Includes bibliographical references. ISBN 0-8213-14254 1. Income distribution-Ivory Coast-Statistical methods. I. Title. II. Series. HC1025291514 1989 339.2'096668-dc2O 89-70646 CIP ABSTRACT Several measures have been devised to analyze income or consumption distribution inequality. The paper presents the large sample distributions of several inequality measures that are used to test if the observed differences in sample estimates of their values are statistically significant. The results presented in the paper are applied to analyze income inequality in C6te d'Ivoire from the data of the Living Standards Survey, 1985. - vi - ACKNOWLEDGEMENT I would like to thank Paul Glewwe and J. Dutta for their comments on an earlier draft. Kalpana Mehra provided me with expert computational assistance for which I am grateful to her. I am indebted to Maria Felix for typing and to Brenda Rosa for editing the manuscript. - vii - TABLE OF CONTEfTS 1. Introduction. ....... ............. .o...... ............. l 2. Review of Inequality Measures..... ... ....a............. **a ... . ..... . r 2 3. Confidence Interval and Hypothesis Testing .............a*... ........6 4. Hypothesis Testing Based on the Lorenz Curve, Gini Index and Relative Mean Deviation ........................ .... Lorenz Curve* ... Cini Index ......o.. ............ ... .9 Relative Mean Deviation........... ... ...... .... 12 5. Asymptotic Distributions of a General Class of InequalityMesrs1 6. Specific Inequality Measures.*.. ....15 Coefficient of Variation.. 1.......... ........ The Class of Decomposable Measures. 15 Atkinson's Inequality Measures.**............... .......oooo. 17 7. Methodology: Application to C6te d'Ivoire.ooo l....o.ooe ... oo.....19 8. Conclusions*** .. ooo..o..s ...*oo31 REFERENCES .................... .......... o..33 LIST OF TABLES TABLE 1: Inequality Measures and Standard Errors: Cote d'Ivoire, 1985 ... .*.... o***..oooo*ooo2l TABLE 2: Inequality Comparisons by Sex of Household Head: Cote d'Ivoire, 1985 .....23.*.... ........ TABLE 3: Inequality Comparisons by Nationality of Household Head: Cote d'Ivoire, 1985 ... TABLE 4: Mean Consumption and its Standard Error by Region: C8te d'Ivoire, 1985 .......9.oooo26 TABLE 5: Statistics for Testing Significance of Differences in Mean Income Between Regions: C6te d'Ivoire, 1985..ooooooooo26 - viii - LIST OF TABLES (Continued) TABLE 6: Inequality Within Regions: Cote d'Ivoire, 1985................29 TABLE 7: Testing for Significance of Inequality Differences Between Regions: C6te d'Ivoire, 1985 ...... ..............30 - 1 - 1. INTRODUCTION Several inequality measures have been proposed in the literature to answer a wide range of questions: How large is income inequality and which way is it moving? How does income inequality in developing countries compare with that in developed countries? Is there any trade off between growth and equity? To answer these and other related questions, it is essential to measure the income inequality and to draw valid conclusions on the size of the inequality. Because inequality measures are estimated on the basis of sample observations, we need to test whether the observed differences in their values are statistically significant, that is, whether the differences are due to sampling errors or due to some other factors affecting income inequality. The paper presents the large sample distributions of several inequality measures which would be used to provide distribution free confidence interval and statistical inference for income inequality. The results are derived in a simple manner, presented so that they are easily applicable to empirical work and are applied to analyze income inequality in Cote d'Ivoire. - 2 - 2. REVIEW OF INEQUALITY MEASURES Suppose income x of an individual is a random variable with the. probability distribution function F(x). Let L(p) be the Lorenz function which measures the income share of the bottom lOOp percent of the population. If the society is concerned only with the welfare of the bottom l00p percent of the population, L(p) will be a suitable measure of income equality. This measure has been popular with World Bank economists who assume p = 0.40, that is, the income share of the bottom 40 percent of the population (Ahluwalia 1974). An inequality measure based on the income share of the bottom lOOp percent population may be defined as *(p) = p - L(p) (2.1) which takes value zero if the income share of the bottom lOOp percent of the equation is equal to lOOp percent. +(p) will be zero for all p in the range O S p < 1 if, and only if, every individual in the society receives exactly the same income. Among all the inequality measures, the Gini index is the most widely used to analyze size distributions of income and wealth. It is defined as G =A2 where f1 = f |x-yI f(x)f(y)dxdy, -3- f(x) being the probability density function and p the mean income. This measure is also equal to one minus twice the area under the Lorenz curve.l/ The relative mean deviation is another inequality measure most widely mentioned in the literature. It is defined as R = 2 J' ix-pi f(x)dx which is also equal to *(p ) where pp = F(p). The main drawback is that it is completely insensitive to transfers of income among individuals on one side of the mean income. The coefficient of variation as a measure of dispersion, suggested by Pearson, is defined as _ = a where a is the standard deviation of the distribution. This measure has been criticized on the grounds that it is equally sensitive to income transfers at all income levels (Sen 1973). Theil (1967) proposed two inequality measures that are based on the notion of entropy in information theory. These measures have gained popularity because of their decomposability property. Shorrocks (1980) has derived the entire class of measures which are decomposable. These measures are given by ./ For welfare implications of the Cini index see Sen (1973) and Kakwani (1980). -4- I C TiT3 [ MC - 1, c 0,1 (2.2) Io = logp - log(g) I = v logp where log(g) = E(logx) = fJ logx f(x)dx m = E(xc) = fr xCf(x)dx v = E(xlogx) = x logx f(x)dx E stands for the expected value and g for the geometric mean of the distribution; Io and I, are the two inequality measures proposed by Theil. Atkinson (1970) proposed a class of inequality measures which are derived from the notion of utility function. He assumed that the social welfare function is utilitarian and every individual has exactly the same utility function. His class of measures are under the assumption that the individual function is homothetic given by 1 A(e) = 1 - [mine] 1-, E 1 where c is a measure of the degree of inequality aversion or the relative - 5 - sensitivity to income transfers at different income levels. The larger the value of c, the greater the weight which is attached to the individuals at the lower end of the distribution. If c = 0, it reflects an inequality neutral attitude, in which case the society does not care about the inequality at all. When E = 1.0, Atkinson's measure is given by A(l.0) = 1 = where g given by log(g) = .t logxf(x)dx is the geometric mean of the distribution. -6- 3. CONFIDENCE IWNERVAL AND HYPOTHESIS TESTING Let xl, x2, ..., xn be a random sample of n observations drawn from a population with mean income p and variance a . Suppose e is an inequality measure defined in terms of the population distribution and 8 is its sample estimate based on n observations, then it will be demonstrated below that ,n. (e - e) is asymptotically normally distributed with zero mean and variance a2 A A 2 A 2 (e). If a 2() is a consistent sample estimator of a (e), then ni = {n- ( 3 9 )(3.1) a(e~) is an asymptotic normal variate with zero mean and unit variance.2/ Thus, nI can be used to form distribution free confidence interval for inequality measures. Further suppose 91and Q are estimates of an inequality measure 6 computed on the basis of two independently drawn random samples of sizes n - 2 A 2 and n2, respectively. Let al and a2 be the sample estimators of the variances of /n 18 and /n282, respectively, then statistics * 01 02 -^ 2 ^ 2 (3.2) a1 a2 / n1 n2 2/ It is customary to refer a(a) as the standard error of A. /1 - 7 - have asymptotic normal distribution with zero mean and unit variance. Thus, n can be used to test the null hypothesis that the observed inequality differences in any two samples are statistically significant.3/ 3/ Note that the expression in the denomination of (3.2) is the standard error of (01 a2). 4. HYPOTHESIS TESTING BASED ON THE LORENZ CURVE, GINI INDEX AND RELATIVE MEAN DEVIATION Lorenz curve Let xp be the income level corresponding to the pth percentile of the distribution and i be the corresponding sample estimate, then for a sample p xl, 2 .**.., xn, defining a. = 1, if x.< x 1 1 p = 0, otherwise provides a sample estimate of the Lorenz curve ordinate L(p) as px L(p) =_ x where n Z a.x. - i=1 x p n E a. i=l and s being the sample mean (noting that x = x when p = 1.0). Thus, x is a sample estimate of uL(p)/p. Beach and Davidson (1983) have proved that /ni (L(p) - L(p)) follows an asymptotic normal distribution with zero mean and variance a2= S [X2+(1_p)(x _ pL(p)) 2+(IP2) 2_2 (pL(P))[X2+ -(P)( __Lp))]_ L 12 p p p 11 u p p p p where a2 is the variance of the entire x distribution and Ap is the variance p 2 2 of x conditional on x < x* Note that =a when p = 1.0. A sample p ~~~~p -9- estimate of a2L is given by 2.2 2- A2=.. p ~A2 2,+(xp 2 2(P xp)rA2 +( aL = 2 1 +(l-p)(x - x) -+ ) a - 3 1 +(x-x )(x -x) x where 2 1 n 2 -2 a Z X. -X n i=l n £ 2 2 = i=l 'i'i -2 X = - x p n p £ a. i=l 2 It can be seen that aL =0 when p = 1.0 which happens because L(p) 1.0 for p = 1.0. Because p is fixed, the sample estimate of the inequality measure *(p) in (2.1) will have the same variance as L(p) and, therefore, Ti = i(P) - ¢(p) (aL//n ) will be an asymptotic normal variate with zero mean and unit variance. Thus, for instance p = 0.40, which provides distribution free statistical inference for an inequality measure based on an income share of the bottom 40 percent population. Cini index Let us define - 10 - Ax = fo Ix-ylf(y)d(y) which can also be written as A = 2xF(x) - 2PF (x) - (x-P) K where F(s) is the distribution function and Fj(x) is given by F (x) = Xf(X)dX 1 0 ~fxd An unbiased estimator of Ax is given by di - (nl [2(xi-ii)pi-(.i-x) i - where pi = a nd xi is the mean of the first i individuals in the sample (when individuals are ranked in ascending order). Because E(VA) = A, 1n d = E d. n provides an unbiased estimator of A, Hoeffding's theorem on order statistics, /n(d-A), has a limiting normal distribution with zero mrea-n and variance (Fraser 1957): - 11 - 2 ~~2 d~2 °t = 4 [fW A f(x)dx-A d Ox X A sample estimator of the Gini index is G - d 2x Applying then 6 method given in Rao (1965), it follows that AE(G-G) has a limiting normal distribution with zero mean and variance a2 = 2.a -4Gncov(d,x) + 4G 2a2 C 2 'd where = var(x), cov(d,s) is given by (Fraser 1957). cov(,d) = 2 [.f xA f(x)dx - vA]. Because a is in terms of population parameters, it has to be G estimated from sample observations, a consistent estimator of which may be obtained as ~2. 1 ['2 - 2~21 a =x 2a 4Gncov(d,x) + 4G a G 4-2 'd where o2 1 Z X2 X2 n 3. - 12 - cov(x,d) = n [n x.d.- xd] nni= I1 °d2= 4 [n £ di _ d Relative mean deviation The relative mean deviation may be estimated from a sample as 2x n i-l1 which as demonstrated by Gastwirth (1974), is asymptotically normally distributed with mean R and variance n Ra where a2 = 12 [o2(F(p)-R)2+ 2h2(F(u)-R) + h2- P2R21 R 2 where h= (x_-)2f(x)dx 0 and F(p) being the proportion of individuals having income less than or equal to P. If in a sample of n observations, n, individuals have income less than or equal to the sample mean income x, then the sample estimates of F(p) and h2 may be obtained as n 1 - x)2 F(i.d-and h (xE n n ~~i=1 - 13 - 5. ASYMPTOTIC DISTRIBUTION OF A GENERAL CLASS OF INEQUALITY MEASURES Except for inequality measures related to income shares, the Gini index and the relative mean deviation, all the remaining inequality measures considered in Section 3 can be written in a general form: e = H[E(h(x)), 1] (5.1) where E(h(x)) is the expected value of a function h(x): E[h(x)] = f h(x)f(x)dx 0 A sample estimate of e is given by 8 = H[E(h(x)), x (5.2) where 1n E(h(x)) - l h(x and s being the sample mean. Following Cramer (1946), it can be proved under fairly general conditions that /n(6-0) is asymptotically normally distributed with zero mean and variance var(O) = var[E(h(x))]H 1 + 2cov[E(h(x)),J]H1H2+ var(x)H2 (5.3) - 14 - where var stands for variance and cov for covariance, and H1 and H2 are the values of the first order partial derivatives of H at the points E(h(x)) and i, respectively. For this result to hold, the function H must be continuous and must have continuous derivatives of the first and second order with respect to its arguments. These conditions will always hold for all the inequality measures considered in this paper.4/ Further, it can be verified that (Cramer, 1946): varE(h(x)) = - [E(h(x))2- [E(h(x))2] 2 var )= n then substituting these results in (5.3) yields the variance of the asymkptotic distribution of 8. 41 The result in (5.3) is also known as 6 method (see Rao, 1965). - 15 - 6. SPECIFIC INEQUALITY MEASURES In this section we derive the asymptotic distributions of specific inequality measures using the general results presented in Section 5. Coefficient of variation A sample estimate of the coefficient of variation is - 2 I/ K Cn 2 where m2 = x. z., which is obtained from (5.2) when h(x) = x . It follows 2 ni=l1 immediately from (5.3) and (5.4) that /n(C-C) is asymptotically normally distributed with zero mean and variance 2 2 2_ - (m4-m2) + 4m2(m2 -Pm3) var(C) = 4 2 2 (6.1) 161i (m_- ) where m_ = E(xc), c = 2, 3 and 4. A sample estimate of var(C) can be obtained - 1 nc by substituting, x for p and m = n I xi for mc in (6.1), which immediately provides the standard error of C. The class of decomposable measures A sample estimate of Ic in (2.2) is given by Ac = 1 m - 1], c s O, 1 C (C-l) [-cC x which is obtained from (5.2) when h(x) = xc. Again, it follows from (5.3) and (5.4) that in(Ic- IC) is asymptotically normally distributed with zero mean and variance - 16 - 1 1 2 m a2 a2m(m+- pm) 22 -2-c2c c 2 2c+2 2 2c+l c (c-i) .i(C-i) u± c(c-1) Next, we consider Theil's two inequality measures Io and Il, the sample estimates of which are I= logx - log(g) and I = - logs x 1 in where log(g) = - E logx. (6.2) n~~~ i=l and A I n v = nz Xlogx. (6.3) n=l Because I0 and I1 can be obtained from (5.2) by substituting h(x) = logx and h(x) = xlogx, respectively. It immediately follows from (5.3) and (5.4) that Vn(10 - Io) and /n(d1- I1) are asymptotically distributed with zero mean and variances5/ 5/ The asymptotic distributions of Theil's measures have been derived earlier (Nygird and Sandstrom, 1981) by a rather complicated method. Moreover, their results are not presented in a fashion that makes them easily applicable to practical problems. - 17 - var(I ) = a + E(logx)2 - (log(g))' - . (v-4og(g)) 0 2 i and var(I ) = 2 [E(xlogx)_- v2] + (jv) a _ 2(.iv) rE(21) - vp 1 2 ~~~~4 3 xg vi, respectively. The sample estimates of g and v are given in (6.2) and (6.3), respectively and those of E(xlogx)2 and E(x2logx) are given by E(xlogx)2= nz (x.logx.) and 2 1in 2 E(x logx) = n E xilogri, respectively. Atkinson's inequality measures A sample estimator of Atkinson's class of estimators is 1 A(£) = 1 - - (min) , c * 1 where ^ 1 ~n 1-6 1 e n i-l 1 (6.4) 1 ni=1 I - 18 - Because this class of inequality measures belongs to the more general class (Section 5) and can be obtained from (5.2) when h(x) = x 1, it follows that /n(A(e)-A(c)) is asymptotically normally distributed with zero mean and variance 2c 2 21-C 1-c 2-2 1 1-c 2 ( 2 (in1 ) £(m22-m1 m_) 4 (ml-£) ° 1+C 2(m 1-e 13 2-cJi 3 (1-C) 2- 1-) the sample estimate of which is obtained using (6.4). For c = 1.0, a sample estimator of Atkinson's measure is given by A(l.0) = 1 - x in where log(g) = n E logxi. i=1 It can be easily shown that A0.0) = 1-e 0 where Io is an estimator of Theil's first inequality measure. This equation gives var(A(l.0)) = e210 var(IO) which immediately provides standard error of Atkinson's measure for e = 1.0. - 19 - 7. METHODOLOGY: APPLICATION TO COTE D'IVOIRE The methodology discussed here is applied to the data obtained from the C8te d'Ivoire Living Standards Survey (CILSS), conducted in 1985 by the World Bank's Living Standards Unit and the Direction de la Statistique, Ministere de l'Economie et des Finances of the Republic of C6te d'Ivoire. To analyze inequality, we need to measure the economic welfare of each individual in the society. Although income is widely used to measure economic welfare, it has many serious drawbacks.6/ One major drawback of using income as a measure of economic welfare is that it may have substantial fluctuations which are averaged out in the long run. Therefore, it has been suggested that consumption is a better indicator of the actual economic position of a household than its current income. In this paper we have used per capita adjusted consumption as a measure of economic welfare. This measure, constructed by Clewwe (1987), takes into account the imputed value of owner-occupied dwelling, the depreciated value of consumer durables and the regional price variations. To take account of the differing needs of various household members, Glewwe divided the total household consumption by the number of equivalent adults. In his formulation of equivalent adults, children were given a smaller weight than adults: children less than 7 years old were given a weight of 0.2; between the ages of 7 and 13, 0.3; and between ages the 13 and 17, 0.5. The estimated values of various inequality measures together with their standard errors (derived in Section 6) are presented in Table 1, columns 1 and 2. The last column in the table gives the numerical value of the 61 For a detailed discussion of this issue, see Kakwani (1986). - 20 - statistic n as defined in (3.1). This statistic follows an asymptotic normal distribution with zero mean and unit standard deviation. If n exceeds 1.96, it means that the hypothesis of zero inequality is rejected at 5 percent level of significance. This method is valid subject to the condition that-,gur samples are large. In practice, it is often difficult to know whether our samples are large enough for these approximations to be valid. However, the approximation is usually good for samples larger than 30 (Cramer 1946). Because our analysis of inequality is based on sample sizes larger than 250, the statistical inference based on asymptotic distributions is appropriate. Table 1 shows that the values of n are considerably larger than 1.96 which lead to the obvious conclusion that a large inequality in the distribution of economic welfare exists in Cote d'Ivoire. However, an important observation is that the numerical values of n differ considerably for different inequality measures. The magnitude of 1T indicates how large the standard error of an inequality measure is relative to its value. Thus, the larger the value of n the greater is the precision with which the inequality measure can be estimated from a given sample. It is observed that among all the single inequality measures presented in the table, the Gini index gives the smallest confidence interval relative to its value. Theil's second inequality measure performs the worst on the basis of this criterion. thus, these observations tend to suggest that the n statistic provides an acuitional criterion for selecting an inequality measure.s/ 71 The problem of choice of an inequality measure among several alternatives has been extremely discussed in the literature; see for instance Atkinson (1970), Newbery (1970), Sheshinski (1972), Dasgupta, Sen and Starett (1973), Rothschild and Stiglitz (1973), Sen (1973) and Kakwani (1980). - 21 - TABLE 1: Inequality Measures and Standard Errors: C6te d'Ivoire, 1985* Value of Inequality inequality Standard Measures measure error n Income share of bottom 10 percent 2.02 0.08 25.25 20 percent 5.36 0.18 29.78 30 percent 9.84 0.29 33.93 40 percent 15.34 0.41 37.41 Gini index 42.87 1.06 40.44 Relative mean deviation 30.77 1.19 25.86 Coefficient of variation 52.93 3.08 17.19 Theil's inequality measures Io 31.61 1.57 20.13 Il 34.45 2.29 15.04 Atkinson's inequality measures A (1.0) 27.10 1.14 23.77 A (1.5) 36.92 1.29 28.62 A (2.0) 45.27 1.37 33.04 * The inequality measures and their corresponding standard errors have been multiplied by 100 in order to express them in percentages. Table 2 presents inequality comparisons by sex of household head. In the sample, 7.9 percent of households were headed by women. It is interesting to note that the mean consumption (adjusted) of female-headed households was found to be about 20 percent larger than those of male-headed households. The difference between the mean consumption of the two groups of households is statistically significant at the 5 percent level (see the last row in the table). This is a surprising result because in many developing countries female-headed households are often poorer than those headed by males. Part of - 22 - the explanation of the situation in Cote d'Ivoire has been provided by Glewwe (1987). He observed that female-headed households are disproportionately located in Abidjan and other urban areas which are considerably richer than the rural areas. The numerical results in Table 2 tend to suggest that inequality of consumption among female-headed households is lower than that among male- headed households. For instance, the Gini index among female-headed households has a value of 42.16 whereas among male-headed households the value is 42.84. The difference in inequality is even larger when we compare the values of coefficient of variation. To test whether the observed inequality differences are statistically significant, we need to compute the values of n* given in (3.2). n* follows asymptotic normal distribution with zero mean and unit variance. If the absolute value of n* exceeds 1.96, we reject the null hypothesis of equal inequality at the 5 percent significance level. The numerical values of n* presented in the last column of Table 2 lead tus to conclude that inequality differences between the female and male-headed households are not statistically significant. This conclusion is valid for all the inequality measures presented in the table. Thus, the observed differences in the values of inequality measures may lead to misleading conclusions without the statistical tests. Several nationalities live in CMte d'Ivoire, but Ivorians are the most dominant comprising 85.7 percent of the surveyed population. For our inequality comparisons we have put together all other nationalities in one group. These comparisons are provided in Table 3. It is interesting to note that the adjusted per capita consumption is almost identical in the two - 23 - TABLE 2: Inequality Comparisons by Sex of Household Head: C8te dlvoire, 1985 Female-Headed Households Male-Headed Households Inequality Value of Value of Measures inequality Standard n1 inequality Standard n n measure error _ measure error l Income share of bottom 40% 15.67 1,29 12.15 15.32 0.43 35.63 0.26 Gini Index 42.16 2.87 14.69 42.84 1.10 38.95 -0.22 Relative mean deviation 30.85 4.02 7.67 30.71 1.25 24.57 0.03 Coefficient of variation 46.80 6.04 7.75 53.34 3.32 16.07 -0.95 Thell's inequality measures 10 30.26 4.25 7.12 31.58 1.67 18.91 -0.29 11 31.39 5.13 6.12 34.57 2.46 14.05 -0.56 Atkinson's measures A (1.0) 26.11 3.14 8.32 27.08 1.22 22.20 -0.29 A (1.5) 35.97 3.85 9.34 36.87 1.36 27.11 -0.22 A (2.0) 44.44 4.41 10.08 45.20 1.44 31.39 -0.16 Adjusted per capita mean consumption 406.96 34.21 11.90 337.64 9.47 35.65 1.95* * Significant at 5% level. groups. The difference is insignificant at the 5 percent level. Can we arrive at a similar conclusion about the income inequality between the two groups? The answer is yes as well as no. The values of n* in the last column of Table 3 show that inequality difference is significant at the 5 percent level on the basis of Theil's first inequality measure Io and Atkinson's family (when £ = 1.5 and 2.0) whereas the - 24 - remaining measures show insignificant differences. The inequality measures which show significant differences are those which have a high value of inequality aversion parameters. These measures tend to give greater weight to income transfers at the lower end of the income distribution. It can, therefore, be concluded that inequality among the poor of Ivorian nationality is significantly higher than that among the poor of the remaining nationalities. The result clearly has an important implication for the poverty in the two groups. C6te d'Ivoire was divided into five survey regions: Abidjan, Other Urban, West Forest, East Forest and Savannah. Table 4 presents the mean adjusted consumption for each region with its standard error. It can be seen that Abidjan (the colonial capital and largest city) has the highest adjusted per capita mean consumption - almost twice that of the whole population. The Other Urban areas are also fairly wealthy. The remaining three rural areas are poor. The lowest average consumption is in Savannah, almost half that of the whole population, followed by the densely populated East Forest region. To see if the mean consumption differs significantly between the regions, the statistics given by _ _ iJ ^ 2 ^ 2 o. + a. = 1 1 ni nj were computed, where x. is the per capita adjusted mean consumption of the ith 1 ^ ~~~2 region with its estimated variance -. The numerical values are presented n. - 25 - TABLE 3: Inequality Comparisons by Nationality of Household Head: C8te d4lvoire, 1985 Ivorians Others Inequality Value of Value of Measures inequality Standard n1 inequality Standard n n measure error measure error Income share of bottom 40 % 15.07 0.45 33.49 17.01 0.94 18.10 -1.86 Gini index 43.47 1.18 36.84 39.05 2.35 16.62 1.68 Relative mean deviation 31.28 1.30 24.06 27.71 2.89 9.59 1.13 Coefficient of variation 53.93 3.48 15.50 46.50 5.00 9.30 1.22 Theil's measures I0 32.59 1.78 18.31 25.70 3.04 8.45 1.96* 11 35.46 2.60 13.64 28.42 4.16 6.83 1.44 Atkinson's measures A (1.0) 27.81 1.22 22.20 26.11 3.14 8.32 0.29 A (1.5) 37.83 1.44 26.27 30.93 2.72 11.37 2.24* A (2.0) 46.32 1.51 30.68 37.90 2.87 13.21 2.60* Adjusted consumption per capita 341.52 10.22 33.42 343.87 19.43 17.70 -0.11 * Inequality difference is significant at 5% level. in Table 5. Since Fi. follows an asymptotic normal distribution with zero mean and unit variance, its absolute value greater than 1.96 will indicate significant average consumption differences among the regions. It can be seen from Table 5 that all the mean consumption differences are significant at the 5 percent level. Thus, the observed differences in mean consumption among the the regions are not due to sampling errors. There are some other elements which must be analyzed to explain such wide differences. - 26 - TABLE 4: Mean Consumption and its Standard Error by Region: C6te d'Ivoire, 1985 mean income Mean Standard n= m i Region Consumption Error standard error Abidjan 614.39 32.11 19.13 Other urban 392.23 17.25 22.74 West Forest 295.96 13.00 22.77 East Forest 244.63 11.49 21.29 Savannah 175.40 8.19 21.42 All regions 341.85 9.14 37.40 TABLE 5: Statistics for Testing Significance of Differences in Mean Income Between Regions: C&te d'Ivoire, 1985 Region Abidjan Other Urban West East Savannah Abidjan 0.00 6.09 9.19 10.84 13.25 Other urban -6.09 0.00 4.46 7.12 11.36 West Forest -9.19 -4.46 0.00 2.96 7.85 East Forest -10.84 -7.12 -2.96 0.00 4.91 Savannah -13.25 -11.36 -7.85 -4.91 0.00 - 27 - Because welfare depends on both the size and the distribution of consumption, it will be interesting to see how inequality varies among the regions. The numerical values of various inequality measures with their standard errors are presented for each region in Table 6. The results suggest that inequality varies widely among the various regions. The highest inequality is observed in Abidjan, the wealthiest region in terms of per capita average consumption. The lowest inequality is observed in the West Forest region, which has more recent settlements and is less heavily populated than the East Forest region (Glewwe 1987). To test whether observed inequality differences between the regions are statistically significant, we computed the values of n* given in (3.2) for all possible pairs of regions. The numerical values are presented in Table 7. The conclusions emerging from the table are summarized below. First, the significance of inequality differences among the regions depends on which inequality measure is selected. For instance, comparing Abidjan and Other Urban cities (see column 1, Table 7) we find that inequality differences are significant at the 5 percent level on the basis of the Gini index, Theil's first inequality measure Io and all measures of the Atkinson family, though the remaining measures show insignificant differences. Thus, we cannot infer that the inequality differences between these two regions are statistically significant or insignificant. The fact that some measures show significant differences in inequality between two groups while others do not, may depend on what the different measures judge inequality to be. For example, one that is very sensitive at the positive tail of the distribution may find significant - 28 - differences because one of the two groups contains some rich people, although a measure that is insensitive to positive outliers may judge that the differences in inequality are insignificant. Thus, if we think a difference in inequality exists it depends on what we judge inequality to be, as reflected in the inequality measure that most matches our value judgements. Second, on the basis of the Gini index, the inequality differences between Abidjan, Savannah and East Forest regions are statistically insignificant. Similarly, the inequality differences between Other Urban areas and West Forest are statistically not significant. The statisticaLly significant inequality differences are observed between Abidjan and Other Urban areas; Abidjan and West Forest; and West Forest and East Forest. From these observations we cannot make a general inference that the inequality in urban areas is higher than that in rural areas. Also, there is no significant relationship between inequality and mean consumption levels of the regions. The inequality difference between the richest region of Abidjan and the poorest region of Savannah is statistically not significant. TABLE 6: Inequality Within Phgions: Weto dlvoire, 19B5 Abidjan Other Ur in West Forest East Forest Savannah Inequality Measures Value Standard ni Value S ai,dard ri Value Standard Ti Value Standard ni Value Standard Ti error error error error error Income share of bottom 40S 16.94 0.86 19.70 19.06 0.79 24.13 20.92 0.90 23.24 18.12 0.83 21.83 17.33 0.78 22.22 Gini Index 40.93 1.92 21.32 35.15 1.88 18.70 31.71 1.88 16.87 37.32 2.03 18.38 37.46 1.80 20.81 Relative mean deviation 30.06 2.67 11.26 25.52 2.55 10.01 22.60 3.26 6.93 26.74 2.63 10.17 27.21 2.60 10.47 Coefficient of variation 47.76 3.28 14.56 40.12 6.57 6.11 33.88 3.45 9.82 44.63 4.56 9.79 40.63 4.52 8.99 Thell's measures 10 27.79 2.64 10.53 20.40 2.28 8.95 17.03 2.02 8.43 23.44 2.55 9.19 23.60 2.30 10.26 I1 30.90 3.22 9.60 22.15 3.60 6.15 17.73 2.54 6.98 26.08 3.60 7.24 24.36 3.14 7.76 Atkinson's measures A (1.0) 24.26 2.00 12.13 18.45 1.86 9.92 15.66 1.70 9.21 20.89 2.02 10.34 21.02 1.81 11.61 A (1.5) 32.66 2.39 13.67 25.67 2.18 11.78 22.40 2.18 10.28 28.78 2.39 12.04 29.48 2.18 13.52 A (2.0) 39.60 2.62 15.11 32.01 2.43 13.17 28.71 2.57 11.17 35.79 2.68 13.35 36.84 2.43 15.16 TABLE 7: Testing for Significance of Inequality Differences Between Regions: Cate d'lvoire, 1985 Other Other West Inequality Abidjan Abidjan AbidJan Urban Urban Other Forest West East Measures Other West East Abidjan West East Urban East Forest Forest Urban Forest Forest Savannah Forest Forest Savannah Forest Savannah Savannah Income share of bottom 40% population -1.81 -3.20 -0.99 -0.34 -1.55 0.82 1.56 2.29 3.01 0.69 Gini index 2.15* 3.43* 1.29 1.32 1.29 -0.78 -0.89 -2.03* -2.21* -0.05 Relative mean deviation 1.23 1.77 0.89 0.76 0.71 -0.33 -0.46 -0.99 -1.11 -0.13 Coefficient of 0 variation 1.04 2.92* 0.56 1.28 0.84 -0.56 -0.06 -1.88 -1.19 0.62 Thei ls inequality measures 10 2.12* 3.24* 1.19 1.20 1.11 -0.89 -0.99 -1.97* -2.15* -0.05 12 1.81 3.21* 1.00 1.45 1.00 -0.77 -0.46 -1.90 -1.64 0.36 Atkinson's measures A (1.0) 2.13* 3.28* 1.19 1.20 1.11 -0.89 -0.99 -1.98 -2.16* -0.05 A (1.5) 2.16* 3.17* 1.15 0.98 1.06 -0.96 -1.24 -1.97 -2.30* -0.22 A (2.0) 2.12* 2.97 1.02 0.77 0.93 -1.04 -1.41 -1.91 -2.30* -0.29 * Inequality difference is significant at 5% level. - 31 - 8. CONCLUSIONS The main purpose of the paper has been to provide distribution free statistical inference for income or consumption inequality. The results are derived in a simple manner and presented so that they are easily applicable to empirical work. The empirical application to C6te d'Ivoire suggests that observed differences in the values of inequality measures may lead to misleading conclusions without the statistical tests. The standard errors of the estimated inequality measures can be so large that the observed differences in the values of inequality measures can be statistically insignificant. Further, some measures may show significant differences in inequality while others may show insignificant differences. Thus, alternative inequality measures may lead to conflicting conclusions. This is an important finding which suggests that we should select an appropriate measure before embarking on the analysis of inequality differences between populations. Because inequality measures are estimated on the basis of sample observations, the importance of using statistical tests cannot be over- emphasized. However, these tests are based on the assumption that the samples used are representative of the population they are drawn from. In practice this assumption may be violated due to non-response errors.8- Moreover, non- sampling errors may be so large that it may make little sense to worry about sampling errors. In future work greater attention should be paid to the non- sampling errors. 8/ The Living Standards Survey data for Cote d'Ivoire, 1985 used in the present paper has a 92 percent response rate, therefore, the possibility of large non-sampling errors is very small. - 33 - REFERENCES Ahlutwalia, M.S. (1974), "Income Inequality: Some Dimensions of the Problem", in Redistribution with Growth, New York: Oxford University Press. Atkinson, A.B. (1970), "On the Measurement of Inequality", Journal of Economic Theory, Vol. 2. Beach, C.M. and R. Davidson (1983), "Distribution-Free Statistical Inference with Lorenz Curves and Income Shares", Review of Economic Studies, L, pp. 723-735. Cramer, H. (1946), Mathematical Methods of Statistics, Princeton: Princeton University Press. Dasgupta, P., A.K. Sen and D. Starrett (1973), "Notes on the Measurement of Inequality", Journal of Economic Theory, Vol. 6, pp. 180-187. Fraser, D.A. (1957), Non-parametric methods in statistics, New York: Wiley. Gastwirth, J.L. (1974), "Large Sample Theory of Some Measures of Inequality", Econometrica, Vol. 42, pp. 191-196. Glewwe, Paul (1987), "The Distribution of Welfare in the Republic of Cate d'Ivoire in 1985", Living Standards Measurement Study Working Paper No. 29, The World Bank, Washington, D.C. Kakwani, N. (1980), Inequality and Poverty: Methods of Estimation and Policy Applications, New York: Oxford University Press. Kakwani, N. (1986), Analyzing Redistribution Policies: A Study Using Australian Data, New York: Cambridge University Press. Newbery, D. (1970), "A Theorem on the Measurement of Inequality", Journal of Economic Theory, Vol. 2, pp. 264-266. Nygard, F. and Sandstrom (1981), Measuring Income Inequality, Stockholm, Sweden: Almqvist & Wiksell International. Rao, C.R. (1965), Linear Statistical Inference and Its Applications, New York: John Wiley & Sons. Rothschild, M. and J.E. Stiglitz (1973), "Some Further Results on the Measurement of Inequality", Journal of Economic Theory, Vol. 6, pp. 188-204. Sen, A.K. (1973), On Economic Inequality, Oxford: Clarendon Press. - 34 - Sheshinski, E. (1972), "Relation Between a Social Welfare Function and the Gini Index of Inequality", Journal of Economic Theory, Vol. 4, pp. 98-100. Sendler, W. (1979), "On Statistical Inference in Concentration Measuremient", Metrika, Vol. 26, pp. 109-122. Shorrocks, A.F. (1980), "The Class of Additively Decomposable Inequality Measures", Econometrica, Vol. 48. Theil, H. (1967), Economics and Information Theory, Studies in Mathematical and Managerial Economics, Vol. 7, Amsterdam: North-Holland. Distributors of World Bank Publications ARGMNTINA FRANCE MICO WAIN Cute. Hinh, EEL WouIdSNk Publlcttu INPOME M,e,d.Ftee Libn% SA GCdiu G~ 66, eruu d'iba Apytade PC"Il.6 Ct2lk,S7 Florida 165,4th Ro,rOfe-043/465 75116 Puts 14AMlna4auz,LM-dDF. 20 Madrid 1333 _ emAhe GERMANY FEDERALR11PUJELICOI' MOROCCO tfru4a lternadenat ASOCE AUSTRALIA PAPUA NEW GUINEA, UNO-Vartag Sod Eltrdet MumeAe rc M e CeeaeldeCeel391 MA SOLNMON ISLANDS, Pepiedmiw AUleS l2meMblesiMtd.d'Ants OM= 9 Uuu VANUATU, AND WESTERN SAMOA D0.30 B_n I Cuiklce DA.Boks&jownurMals SR LANKA AND THE MALDIVES 11413 Stai Se GYEECE NETHERLANDS LikeHetuwelenehp Mt:hamn 3132 KERE htOr-Publlkatteeb.c. P.O.E DM144 Vttaede 24 Ippedereut PASh a PladYm P.O. Y 14 1C0, SeGiEmpim A. 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Em 41096 smasl Crig Hddold 10 jlhboube 3824 LSMS Working Papers (continued) No. 36 Labor Market Activity in C6te d'Ivoire and Peru No. 37 Health Care Financing and the Demand for Medical Care No. 38 Wage Determinants and School Attainment among Men in Peru No. 39 The Allocation of Goods within the Household: Adults, Children, and Gender No. 40 The Effects of Household and Community Characteristics on the Nutrition of Preschool Children: Evidence from Rural Cote d'Ivoire No.41 Public-Private Sector Wage Differentials in Peru, 1985-86 No. 42 The Distribution of Welfare in Peru in 1985-86 No.43 Profits from Self-Employment: A Case Study of C6te d'Ivoire No. 44 The Living Standards Survey and Price Policy Reform: A Study of Cocoa and Coffee Production in C6te d'Ivoire No.45 Measuring the Willingness to Payfor Social Services in Developing Countries No. 46 Nonagricultural Family Enterprises in C6te d'Ivoire: A Descriptive Analysis No. 47 The Poor during Adjustment: A Case Study of C6te d'Ivoire No. 48 Confronting Poverty in Developing Countries: Definitions, Information, and Policies No.49 Sample Designs for the Living Standards Surveys in Ghana and Mauritania/Plans de sondage pour les enquetes sur le niveau de vie au Ghana et en Mauritanie No. 50 Food Subsidies: A Case Study of Price Reform in Morocco (also in French, 50F) No. 51 Child Anthropometry in C6te d'Ivoire: Estimates from Two Surveys, 1985 and 1986 No. 52 Public-Private Sector Wage Comparisons and Moonlighting in Developing Countries: Evidence from C6te d'Ivoire and Peru No. 53 Socioeconomic Determinants of Fertility in C6te d'Ivoire No. 54 The Willingness to Pay for Education in Developing Countries: Evidence from Rural Peru No. 55 Rigiditedes salaires: Donnees microeconomiques et macroeconomiques sur l'ajustement du marche'du travail dans le secteur moderne (in French only) No. 56 The Poor in Latin America during Adjustment: A Case Study of Peru No. 57 The Substitutability of Public and Private Health Care for the Treatment of Children in Pakistan No. 58 Identifying the Poor: Is "Headship" a Useful Concept? No. 59 Labor Market Performance as a Determinant of Migration No. 60 The Relative Effectiveness of Private and Public Schools: Evidence from Two Developing Countries No. 61 Large Sample Distribution of Several Inequality Measures: With Application to C6te d'Ivoire No. 62 Testing for Significance of Poverty Differences: With Application to C6te d'Ivoire No. 63 Poverty and Economic Growth: With Application to C6te d'Ivoire No. 64 Education and Earnings in Peru's Informal Nonfarm Family Enterprises No. 65 Formal and Informal Sector Wage Determination in Urban Low-Income Neighborhoods in Pakistan No. 66 Testing for Labor Market Duality: The Private Wage Sector in Cote d'Ivoire No. 67 Does Education Pay in the Labor Market? The Labor Force Participation, Occupation, and Earnings of Peruvian Women No. 68 The Composition and Distribution of Income in C6te d'Ivoire No. 69 Price Elasticities from Survey Data: Extensions and Indonesian Results The World Bank s Headquarters European Office Tokyo Office E 1 1818 H Street, N.W. 66, avenue d'Iena Kokusai Building Washington, D.C. 20433, U.S.A. 75116 Paris, France 1-1, Marunouchi 3-chome Chiyoda-ku, Tokyo 100, Japan Telephone: (202) 477-1234 Telephone: (1) 40.69.30.00 Facsimile: (202) 477-6391 Facsimile: (1) 40.69.30.66 Telephone: (3) 3214-5001 Telex: WUI 64145 WORLDBANK Telex: 640651 Facsimile: (3) 3214-3657 RCA 248423 WORLDBK Telex: 26838 Cable Address: INTBAFRAD WASHINGTONDC