WIPS 249l POLICY RESEARCH WORKING PAPER 2499 The Treatment of One problem when estimating a Cobb-Douglas Non-Essential Inputs in a production function with Cobb-Douglas Technoloy 1 micro data is how to deal Cobb - Douglas Technology with the observations that show positive output but do An Application to Mexican Rural not use some of the inputs. As Household-Level Data the log of zero is not defined, one standard procedure is to arbitrarily replace those zero Isidro Soloaga values with 'sufficiently small numbers. But can we do better than that? An alternative approach is presented and applied to Mexican farm-level data. The World Bank Development Research Group Trade December 2000 I POLiCY RESEARCH WORKING PAPER 2499 Summary findings The standard approach for fitting a Cobb-Douglas functional form to capture the fact that the data have a production function to micro data with zero values is to positive output even when some of the inputs are not replace those values with "sufficiently small" numbers to used. facilitate the logarithmic transformation. In general, the To highlight the empirical importance of the approach, estimates obtained are extremely sensitive to the he applies it to Mexican farm-level production data that transformation chosen, generating doubts about the use he gathered. of a specification that assumes that all inputs are essential Many households did not use family or hired labor in (as the Cobb-Douglas does) when that is not the case. farm production, or had different capital composition Soloaga presents an alternative method that allows one (that is, zero value for non-land farm assets). to estimate the degree of essentiality of the production The estimations provide a clear measurement of the inputs while retaining the Cobb-Douglas specification. degree of essentiality of potentially non-essential inputs. By using the properties of translatable homothetic They also indicate the size of the error introduced by the functions, he estimates by how much the origin of the common "trick" of adding a "small" value to zero input input set should be translated to allow the Cobb-Douglas values. This paper-a product of Trade, Development Research Group-is part of a larger effort in the group to explore conceptual and practical issues relative to the effects of international trade policy on individual producers. Copies of the paper are available free from the World Bank, 1818 H Street NW, Washington, DC 20433. Please contact Lili Tabada, room MC3- 333, telephone 202-473-6896, fax 202-522-1159, email address Itabada@worldbank.org. Policy Research Working Papers are also posted on the Web at www.worldbank.org/research/workingpapers. The author may be contacted at isoloaga@worldbank.org. December 2000. (18 pages) The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findipsgs out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors, or the countries they represent. Produced by the Policy Research Dissemination Center The treatment of non-essential inputs in a Cobb-Douglas technology. An application to Mexican rural household level data Isidro Soloaga* *World Bank, Development Research Group, Trade Research Team. 1818 H St. NW, Washington DC, 20433, Room MC3-307. Tel.: (202) 473-8085; Fax: (202) 522-1159; e- mail: isoloaga~worldbank.org The author is grateful to Marc Nerlove, R. Aggarwal, R. Betancourt, B. Gardner and R. L6pez, all of them from the University of Maryland, and to an anonymous referee for comments and suggestions. Key Words: Translation homotheticity, production function, non-essential inputs. 1. Introduction Cobb-Douglas functions are among the best known production functions utilized in applied production analysis'. The most general form for a Cobb-Douglas is: f(x) = AH xf6', ,B > O, i =1,2,...,n. r=1 This functional form has the properties of: i) strict monotonicity: if x'> x then f (x') > f (x); III) quasi-concavity: V(y) = {x: f (x) 2 y} is a convex set; IV) strict essentiality: of f(x,...x; A Xi+a x,,) = 0 for all x, > 0; iv) the set V(y) is closed and nonempty for all y>O; v) f (x) is finite, nonnegative, real valued, and single valued for all nonnegative and finite x; it is also continuos and everywhere twice-continuously differentiable. n vi) f (x) is homogenous of degree k = Ei,, i=1 Property IV) indicates the Cobb-Douglas technology requires all inputs to be essential in production: all must be used in strictly positive amounts to obtain a positive output (i.e., the input requirement sets do not intersect the axis). This requirement of the production function is easily fulfilled when aggregated data-say country or industry level-are used. But, when a more micro level analysis is required, the researcher may well end up having some observations with positive levels of output, even when some of the inputs have zero values. This situation is typically found in analysis of labor supply in rural settings where, for instance, researchers need to differentiate household labor 2 supply for farming by type of household member (e.g., male/female). As not all households use both types of labor for farming activities, some observations have positive level of output but zero use of one (or both) of these inputs. That is to say, one (or both) of these particular inputs is non-essential for production. The same situation may show up when the researcher wants to concentrate his/her analysis in other inputs, as some farmers may not use them in production (e.g., hired labor, children labor, fertilizers, machinery). For these cases, a Cobb-Douglas (or the more general translog) can be used only if we make some transformation to the zero-value arguments2. Researchers in general estimate a logarithmic transformation of (1) in the form: (2) ln(y(x)) = In 4 + EA Iln(x, ), 6, > 0, i = 1,2,..., n. i=1 and modified zero-value arguments by either replacing them by 1--that is ln(x,) = 0 when xi = 0--or with "small" values (see, for instance, MaCurdy and Pencavel, 1986, and Jacoby, 1992). In other words, whenever they find inputs that are non-essential (i.e., for some observation i y, >0 but xki = 0) they replace Xk, by xki = Xki + ai, with a, equal to 1-or to a "sufficiently small" value-using the same value for all i (i.e., a, = c). Obviously, these procedures are arbitrary and are forcing the production function to include input quantities that are not actually observed. I show in the following empirical section of the paper that changes in the cc values adopted may cause the estimated regression coefficients and their standard errors to vary significantly, ' The present analysis is centered in the Cobb-Douglas functional form only for exposition purposes. The same analysis carries over other, more general, functional forms (e.g., the translog, that can be restricted to obtain the Cobb-Douglas) (see Chambers, 1988). 2 The estimation of production functions in general, and Cobb-Douglas production functions in particular, presents many additional problems. See Varian, 1984, Chapter 4, Econometrics and Economic Theory, for a discussion. 3 generating doubts about the "tricks" used to retain a specification that implies that all inputs are essential (as the Cobb-Douglas does) when that is not the case. This paper proposes an alternative method which uses the properties of translation homotheticity, and translates the origin of coordinates of the production space in the direction of the non-essential inputs. The translation coefficients are estimated by maximum likelihood. I highlight the empirical importance of the approach by applying it to farm level production data coming from a World Bank 1995 survey I conducted in rural Mexico. Table 1 presents the mean value of key variables of the data. As many households in the data did not use family labor on farm production, or did not use hired labor, and had different capital composition (some zero non-land farm assets), the sample provides good testing ground to see the effect of the alternatives ways of "solving" the problem posit by inputs with zero values. An important feature of the estimations is that they provide a clear measurement of the degree of non-essentiality of all non-land inputs. In what follows, I assess the impact on the estimates of different assumptions about a, when a Cobb-Douglas production function is estimated with farm level data. I then apply the new procedure developed in this paper to the same data set and compare them with those of the previous sections. The last part of the paper summarizes the findings. 4 Table 1. Descriptive statistics STATE Variable Units Guanaj Sonora Puebla Tlaxcala uato Value of output per hectare US$ of 1994 691 974 482 363 Production factors Land planted Hectares 15.1 35.7 3.6 4.6 Value of non-land assets US$ of 1994 51911 15124 13729 9010 6 Value of animal assets US$ of 1994 8378 89466 5013 4589 Expenditures on hired labor US$ of 1994 16993 29012 1493 1796 Expenditures on other inputs US$ of 1994 18680 33508 801 1447 Family labor, adult male>12 % 1.6 1.0 1.5 1.4 Family labor, adult female % 0.9 0.1 0.9 0.7 Household's demographics # Children<13/# adults % 0.47 0.4 0.45 0.46 Education male family labor Years 4.07 6.13 5.87 5.18 Education female family labor Years 3.32 3.88 4.87 4.22 Age HH head Years 59.54 58.06 58.1 57.04 Formal education HH head Years 1.88 4.38 3.6 2.97 Proportion of male hh head ___ ° 0.90 0.87 0.86 0.86 Proportion of HH head wl off-farm % 0.12 0.25 0.37 0.38 jobs _ Proportion of HH w/secure title on % 0.89 0.9 0.83 0.94 land Location factors Distance to market Km 8.89 23.16 14.2 8.71 Celaya I % hh in State 0.34 0 0 0 Celaya 2 % hh in State 0.34 0 0 0 Irapuato 1 % hh in State 0.17 0 0 0 Irapuato 2 % hh in State 0.16 0 0 0 Puebla 1 % hh in State 0 0 0.15 0 Puebla 2 % hh in State 0 0 0.27 0 Puebla 3 % hh in State 0 0 0.35 0 Puebla 4 % hh in State 0 0 0.23 0 Tlaxcala 1 % hh in State 0 0 0 0.48 Tlaxcala 2 % hh in State 0 0 0 0.52 Navojoa % hh in State 0 0.48 0 0 Obregon % hh in State 0 0.52 0 0 5 2. An example for farm level production data. A more general form of expressing equation (2) would be k n (3) ln(y) = 60 + EAj ln(xi + a, ) + E ,Bj ln(xj + aj) + ,u j-1 where, for a given sample of data, inputs of type xi are assumed to be positive for all observations and inputs of type xi are assumed to take the value zero for some observations. That is to say, for a particular sample of data, " x, type" of inputs are essential for production whereas " xj type" are not. The econometric issue is to estimate the parameters o,/, 8's,,6j's and 2'. In order to do that, the common procedure is to assume values for all the a.'s since otherwise the logarithm will not be defined for those observations with zero value for xi . The key contribution of this paper is, instead of choosing beforehand the value for the aj's, to "let the data tell us" what those values are by estimating them with a maximum likelihood technique. Although all x, are assumed to be positive (which implies that ln(x5) is always defined, even if it is assumed that all a, s are zeroes ), all the inputs, not just the xi 's, have the possibility of being non- essential. To incorporate this, the methodology developed here allows also to estimate the values for the a, 's. The following estimation uses 399 observations of the 1995 survey for which all the information required for estimating an agricultural Cobb-Douglas production function were available. 6 Descriptive statistics of this sub-sample of the data are presented in Table 1, and the proportion of the observations with some factors having zero values is detailed in Table2. Table 2. Importance of inputs with zero values Variable # of times have % of total samnple value=O Expenditures in hired labor 37 9 Family labor used in farming activities, males 45 11 Family labor used in farming activities, female 238 60 Non-Land Productive Assets (Machinery, etc.) 53 13 Assets in Animals 0 0 Expenditures in other inputs 0 0 Table 3 presents the results coming from estimating equation (3) under different assumptions about the translation parameters a, and a,, and highlights the problems of using the ad-hoc solutions indicated in the introduction to this paper. The table has four sets of estimates: the first after adding "I" to the variables with some zero values, the second after adding "0.1 ", the third after adding "0.01" and the fourth after adding "0.00 1"3. The R's-squared indicate a good fit of the model (around 83%), and the sign of the "production factors" variables are positive as expected. The quantity of hectares planted, the non-land assets, the expenditures on hired labor and expenditures on other inputs were statistically significant in the four estimations as well as male family labor applied to agriculture (except in the first regression). Returns to scale are about constant in the four regressions, which is in line with other studies done on agricultural production (Lopez and Valdes, 1998). 7 The last three columns of the table summarize the results that are of importance for the purpose of this paper. Under the heading of "Range" I calculated the difference between the highest and the lowest of the parameters estimated in the four regressions. In the penultimate and the last column of the table I calculated the ratio of the "Range" to the Max and the min values of each regressor respectively. It is clear from these columns that "important" coefficients of the regression vary significantly according to the value of the o chosen: the marginal productivity of non-land assets ranges from a minimum of 0.024 (when a =0.001) to a maximum of 0.045 (when a =1), which implies 47% of the maximum value of the parameter (or 88% of the minimum). Similar percentages can be found in the case of the estimates for expenditures on hired labor. For the case of male family labor applied to agriculture the coefficient was not significant in the first regression and turned out statistically significant in the other three with a wide range of variation in the estimated value of the parameter (from a Max of 0.113 to a min of 0.049. 3The aj 's are equal to a, that is, the same for all fs, whereas the a, s are implicitly assumed to be zeroes. Researchers' choice of a is acknowledged to be arbitrary. For the purpose of this paper, I present here a set of four "small" values, which are those usually found in empirical papers. 8 Table 3. Alternative procedures generally used for non-essential inputs If some xko., if some Xk,-n If some Xk, If some x,,=1 Max Min Range Range/ Range/ log(xk)=log(xk+l) og(xk)=log(xk+0. 1) log(xk)=log(xk+0l01) log(xk)=Iog(xk+0O00 ) Max Min Variable Estim. |Std.err S Std.err Sig.Std.err Sig. Estim. |Std.er Sig. (a) (b) (c)= (c)/(a) (c)/(b) Intercept 3.218 0.905 *** 3.085 0.904 ** 3.150 0.907 ** 3.207 0.910 * .218 3.085 0.133 4% 4% Production factors Fain-lab-adult male 0.066 0.092 0.113 0.052 ** 0.070 0.031 ** 0.049 0.021 ** 0.113 0.049 0.065 57% 133% Fam-lab-adult-female 0.001 0.126 0.007 0.062 0.007 0.035 0.006 0.024 0.007 0.001 0.006 83% 479% Ha planted 0.386 0.060 *** 0.390 0.060 OA 0400 0.060 *** 0.407 0.060 *t 0.407 0.386 0.021 5% 5% Assets 0.045 0.013 ** 0.036 0.011 *** 0.029 0.009 ** 0.024 0.008 *5* 0.045 0.024 0.021 47% 88% Animal stock 0.007 0.011 0.006 0.011 0.006 0.011 0.006 0.011 0.007 0.006 0.002 25% 33% Expend-labor 0.093 0.019 ** 0.073 0.015 *+ 0.057 0.012 t 0.047 0.010 ** 0.093 0.047 0.046 490/o 97% Expend-inputs 0.411 0.042*+ 0.429 0.042 * 0.437 0.042*5* 0.443 0.042 + 0.443 0.411 0.031 7% 8% Household's demographics #Children/# adults 0.212 0.084 + 0.197 0.084 * 0.189 0.085 ** 0.186 0.085 ** 0.212 0.186 0.027 13% 14% Education wkland male 0.029 0.057 -0.095 0.081 -0.113 0.083 -0.116 0.083 0.029 -0.116 0.145 497% -125% Education wkland fem- -0.151 0.064 ** -0.160 0.101 -0.165 0.104 -0.167 0.104 -0.151 -0.167 0.016 -10% -9YO Hh head has otherjob -0.022 0.093 -0.020 0.092 -0.019 0.093 -0.017 0.093 -0.017 -0.022 0.005 -29% -22% Education hh head 0.127 0.067 * 0.175 0.069 + 0.186 0.069 t** 0.190 0.069 ' 0.190 0.127 0.063 33% 49% Age hh head -0.064 0.199 0.053 0.205 0.075 0.207 0.080 0.207 0.080 -0.064 0.144 181% -224% HhJismaleheaded(dumrny) 0.197 0.130 0.169 0.130 0.173 0.130 0.176 0.130 0.197 0.169 0.029 15% 17% Land istitled (dummy) 0.212 0.123 * 0.210 0.122 t 0.206 0.122 * 0.204 0.123 * 0.212 0.204 0.008 4% 4% Location factors Distance to market -0.001 0.028 -0.006 0.028 -0.00 0.028 -0.008 0.028 -0.001 -0.008 0.006 484% -83% Puel (dummy) -0.05 0.255 -0.092 0.253 -0.076 0.254 -0.064 0.254 -0.064 -0.092 0.028 -43% -300% Pue2 (dummy) 0.722 0.182 * 0.755 0.182 *q 0.773 0.182 *5* 0.783 0.182 * 0.783 0.722 0.061 8% S% PuN3(dummy) 0.222 0.170 0.213 0.170 0.221 0.170 0.227 0.170 0.227 0.213 0.014 6% 7% Pue4 (dummy) 0.659 0.203 + 0.692 0.202 4 0.695 0.203 ** 0.698 0.203 * 0.698 0.659 0.039 6% 6% Irapual (dummy) 0.063 0.180 0.047 0.180 0.045 0.180 0.044 0.181 0.063 0.044 0.019 30% 43% Irapua2 (dummy) 0.681 0.199 $ 0.674 0.200 * 0.683 0.200 * 0.690 0.200 * 0.690 0.674 0.016 2% 2% Celayal (dummy) 0.728 0.152 I 0.710 0.152 * 0.709 0.153 + 0.710 0.153 t 0.728 0.709 0.019 3% 3% Celaya2 (dummy) 0.497 0.149 0.486 0.148 * 0.494 0.149*** 0.500 0.149 + 0.500 0.486 0.014 3% 3% Obregon (dummy) 0.662 0.186 0.667 0.186 * 0 0.670 O.I0.16 0.673 0.187 ** 0.673 0.662 0.011 2% 2% Navojoa (dummy) 0.903 0.173 ++ 0.915 0.173 * 0.919 0.173 * 0.921 0.174 ** 0.921 0.903 0.018 2% 2% Returns to scale 1.010 1.054 1.006 0.981 1.054 0.981 0.073 7% 7% Adj-r-square 0.830 0.831 0.830 0.829 All variables are in logs, except dummies. (I) Significance levels: +** at 99%, +* at 95%, * at 90% 9 The methodology used in this paper overcomes this problem by "letting the data tell us" what the values of the ei' s are. The basic principle behind the approach is Marc Nerlove's dictum: "If it matters, it can be estimated". This was done in the empirical part of the model by obtaining maximum likelihood estimates of the a, 's and a,'s, in addition to ,8l , ,B' s,flj's and a2. That is, I estimated those values of the unknown parameters that would, under a multivariate normal specification, maximize the probability of obtaining the sample actually observed (Judge et al., 1988, p. 222).The estimated parameters Bo, i, 's, and ,lj's are the usual ones for a Cobb-Douglas technology, and a, 's and a} 's are the translation parameters for this particular case. Results of maximum likelihood joint estimation, are presented in Table 4. The last two columns of the table highlights the differences with the estimates presented in Table 4. With arbitrary cr 's, some "production factors" estimates are always above what they should be: the coefficient for male family labor in agriculture is between 1.59 and 3.67 times bigger, and the one for hectares planted is more than 1.87 times bigger. As the returns to scale are about constant also for this specification, the coefficients for the other "production factors", female family labor, non-land assets, expenditures on hired labor, and expenditures on other inputs, are smaller- between 12% and 95% of the value of the estimates coming from our maximum likelihood method. 10 Table 4. Maximum likelihood estimation of all alphas. Variables Estimates Std.error Signific (a) from table (b) from (a) lance (1) 3)/(a), in% table 3)1(a), L~~~~~~~~~~~~ I I in % Intercept 1.754 0.909 184 176 Production Factors Fam-Lab-Adult Male 0.031 0.009 367 159 Fain-Lab-Adult-Female 0.009 0.015 82 12 Ha Planted 0.207 0.051 197 187 Assets 0.073 0.019 62 33 Animal Stock 0.009 0.012 78 67 Expend-Labor 0.188 0.035 49 25 Expend-inputs 0.466 0.043 95 88 Household's Demographics #Children/# Adults 0.262 0.083 81 71 Education Wkiand Male -0.096 0.080 -30 121 Education Wkland Female -0.164 0.099 92 102 Hh Head Has Other Job -0.057 0.089 30 39 Education Hh Head 0.174 0.068 t 109 73 Age Hh Head 0.180 0.203 44 -36 Hh Is Male Headed (Dummy) 0.244 0.124 81 69 Land Is Titled (Dummy) 0.187 0.118 113 109 Location Factors Distance To Market -0.002 0.032 44 351 Puel (Dummy) -0.155 0.241 41 59 Pue2 (Dummy) 0.571 0.179 137 126 Pue3 (Dummy) 0.176 0.164 129 121 Pue4 (Dummy) 0.465 0.193 150 142 Irapual (Dummy) 0.014 0.172 - 440 307 Irapua2 (Dummy) 0.615 0.194 112 110 Celayal (Dummy) 0.666 0.145 105 106 Celaya2 (Dummy) 0.458 0.144 109 106 Obregon (Dummy) 0.689 0.178 98 96 Navojoa (Dummy) 0.902 0.166 t** 102 100 Alphas Alpha-Family Male Labor 0.000000107 0.000000039 Alpha-Family Female Labor 0.00001317 0.0000519 Alpha-Non-Land Assets 72.220 97.067 Alpha-Expenditures hired labor 102.560 56.348 Alpha-Animal assets 18.329 19.468 Alpha-Other expenditures 4.990 36.863 0.983 Returns To Scale 107 100 Adj-R-Square (From OLS) 82.98 (I) Significance levels: *** at 99%, * at 950, * at 90% 11 Thus, if, for example, we use the marginal productivity of family labor force in agriculture activities to assess family labor allocation to off-farm activities (as in Jacoby- 1992- for instance), we are going to overestimate its true on-farm productivity (by 59% or 267%, depending on the a chosen). Likelihood ratio tests rejected at the 99%,99%,95% and 90% significance level the null hyphoteses that the estimated alphas of table 5 are statistically the same to those used in any of the four exercises of table 3, respectively4. The alphas estimated are an indication of the degree of essentiality of the production inputs. This can be seen more clearly in Table 5, which contains the estimated value of the alphas, the sample mean of the variable they are attached to, and the ratio of these two values. Results shows that the ordering of the inputs taking into account their degree of essentiality is: Male family labor, female family labor, other expenditures, farm assets, expenditures on hired labor, and, animal assets5. That means that, for instance, it is "more difficult" to have some positive level of production without male family labor in agriculture than without female family labor in agriculture. In turn, it is relatively easier to get some production when animal assets are zero than when the other forms of non- land farm assets are zero, since the origin of the input set was translated in the direction of the latter inputs by 0.138% whereas for animal assets the translation was 0.491%. It is important to notice that since only the alpha for hired labor is significantly different from 4 When the alphas are estimated, the maximized value of the log likelihood function is 435.693. When the alphas are assumed to be fLxed, the values are -445.982,-443.771, -442.386 and -441.093 for values of alpha 1, 0.1, 0.01 and 0.001 respectively. 5 The ranking starts with the smaller translation of the input set in the direction of the input. For instance, for male family labor the translation is only 0.00001% of the sample mean of the variable, closer to the origin than, for example, female family labor (translated by 0.002% of its sample mean). 12 zero, in an statistical sense only this input is truly "non-essential". As they are data- specific, alpha values are most likely going to vary when this procedure is applied to different a different data set. Table 5. Relative importance of the alphas Alpha: Estimated Sample mean (c)=(a)/(b), in Ranking coeff. of the variable % (a) alpha is attached to. (b) Malefamilylaborin 0.00000011 1.4 0.00001 1 agriculture Female family labor 0.000013 0.7 0.002 2 in agriculture Non-land, non- 72.2 52364 0.138 4 animal farm assets Animal assets 102.6 20893 0.491 6 Expenditures on 18.3 12732 0.144 5 hired labor __ Other expenditure . 4.99 13990 0.036 3 3. Conclusions The standard approach for fitting a Cobb-Douglas production function to micro-data with zero values is to transform zero-values to facilitate the logarithmic transformation. In general the estimates obtained are extremely sensitive to the transformation chosen, generating doubts about the use of a specification that assumes all inputs are essential (as the Cobb-Douglas does) when that is not the case. I propose here an alternative method which allows to actually estimate the degree of essentiality of the various production inputs, retaining at the same time the Cobb-Douglas specification. By utilizing the properties of translatable homothetic functions, I estimate by how much the origin of the input set should be translated to allow for the Cobb-Douglas functional form to capture 13 the fact that the data have positive amount of output even when some of the inputs are not being used. To highlight the empirical importance of the approach, it is applied to farm level production data collected in rural Mexico. Many households did not use family labor on farm production, did not use hired labor, or had different capital composition (i.e., zero value for non-land farm assets). An important feature of the estimations is that they provide a clear measurement of the degree of essentiality of potentially non-essential inputs and also an indication of the size of the error introduced by the common "trick" of adding a "small" value to zero input values. 14 Appendix: Translation homotheticity Chambers and Fare (p. 632) introduced the concept of translation homotheticity. The technology structure is translation homothetic if L(y) can be written as (3) L(Y) = H(y; gj )g. + L(l), y E 94, where L(y)={x:xcanproducey}, yE9I'; L(l) = {x: x can produce 1, which is the reference output vector}, and H(y;.) is a nondecreasing fimction consistent with the following properties: a.-Di(y,x-ag,.;g,)=Di(y,x;gx) - a, aeXl; b.- bi(Y,X; Ag.) = A' D((Y, X; gx), A > °; -(x' Y,-') >: (x,-y) AD (y', x'; g. ) 2 Aj (y, x; gx ); i.e., non decreasing in inputs and nonincreasing in output; d.- Di(y,x;gx) is concave in x; e.- x E L(y) <* D(y, x; g.) > O. Where D, is the directional input distance function developed by Chambers, Chung and Fare (1996), as is defined Dj: 91" x 9INX 9N 1 9 by Di(y,x;g) =sup{aE91:x-ageL(y)}; a =sup{a 91:x e ag + L(y)} a Translation homotheticity can be visualized as having inputs sets for different output levels that are generated by taking a common reference set L(l), and then translating that reference set in the direction of the vector g,. A movement out from any 15 point on L(l) in the direction of g. will cut isoquants at points having the same marginal rate of substitution as at the point on L(1). Figures la and lb of Chambers, et al. (1996 p.410) are reproduced here to illustrates the concept. In Figure Ia x e L(y)andD,(y,x;g) is given by the ratio Figure 1a. Figure lb. L(y) x2 x2 L(y) xi ~~~~~~~~~~~~~xi -gx -g ||g'i|l||g|| > 0, where Ilkil denotes the norrn of vector k In Figure lb x z L(y) but moving x in the direction of g eventually encounters L(y) . Here D (y, x;g) is given by - llg 01 0 the inputs x, and Xk are non-essential. I can choose the reference vector gx in the direction of these non-essential inputs, and translate the origin of coordinates x = x(0,..., 0,0,,... 0) to x =x(O,...,aj, akI...O) in such awaythat we can obtain positive amount of output with zero quantities of the inputs xi and xk . and while doing this, still conserve the Cobb-Douglas functional form. The new input set for the tth observation will be defined by x, = x(x1, ,..., xi, + a, Xkt + ak, ..., x, ) . The a 's and ak's would provide a measurement of how non-essential are these non- essential inputs. 17 References Chambers, R.G. (1988) Applied production analysis. A dual approach. Cambridge University Press. N.Y. Chambers, R.G., Y.Cheung, and R. Fare (1996) "Benefit and Distance Functions". Journal of Economic Theory 70, pp. 407-419. Chambers, R.G., and R. Fare (1998) "Translation homotheticity". Economic Theory 11, pp. 629-641. Jacoby, H. G. (1992) "Productivity of Men and Women and the Sexual Division of Labor in Peasant Agriculture of the Peruvian Sierra" Journal of Development Economics 37, pp. 265-287. Judge, G.G. et al. (1988) Introduction to the theory and practice of econometrics. Second Edition. J. Wiley & Sons. N.Y. L6pez, R. and A. Valdes (1997) Rural Poverty in Latin America: Analytics, new empirical evidence, and Policy. The World Bank. Washington, DC. MaCurdy, T. E. and J. H. Pencavel (1986). "Testing Between Competing Models of Wage and Employment Determination in Unionized Markets". Journal of Political Economy, 94(3): S3-39 Varian, A. 1984. Microeconomic Analysis. Norton, New York. 18 Policy Research Working Paper Series Contact Title Author Date for paper WPS2480 Productivity Growth and Resource Mubarik Ali November 2000 D. Byerlee Degradation in Pakistan's Punjab: Derek Byerlee 87287 A Decomposition Analysis WPS2481 Foreign Direct Investment in Africa: Jacques Morisset November 2000 N. Busjeet Policies Also Malter 33997 WPS2482 Can Institutions Resolve Ethnic William Easterly November 2000 K. Labrie Conflict? 31001 WPS2483 The Credit Crunch in East Asia: Pierre-Richard Agenor November 2000 M. Gosiengfiao What Can Bank Excess Liquid Joshua Aizenman 33363 Assets Tell Us? Alexander Hoffmaister WPS2484 Banking Crises in Transition Helena Tang November 2000 A. 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