Popular Support for Rank-dependent Social Evaluation Functions 1

Transcription

1 Popular Support for Rank-dependent Social Evaluation Functions 1 (First Draft) Juan Gabriel Rodríguez Universidad Complutense de Madrid, Campus de Somosaguas, Madrid, Spain Tel: juangabriel.rodriguez@ccee.ucm.es and Rafael Salas Universidad Complutense de Madrid, Campus de Somosaguas, Madrid, Spain Tel: r.salas@ccee.ucm.es East Carolina University (Visiting Fellow), Greenville, NC USA ABSTRACT This paper finds conditions on income distributions under which the class of rankdependent social evaluation functions à la Kakwani Lambert (Kakwani, 1985 and Lambert, 1985) is consistent with majority voting. Under these conditions the equally distributed equivalent income is equal to the median of the distribution. In addition, we are able to deduce the parameter endorsed by that family of rank-dependent social evaluation functions which is consistent with the Gini coefficient. We try to confirm empirically that the required conditions are generally accepted with an illustration based on the Survey on Income and Living Conditions dataset for the European Union in the period JEL Classification: D31, D63, P16. Key Words: social evaluation function; the Gini coefficient; majority voting; symmetric distribution. 1 We are grateful to Juan Prieto Rodríguez for his assistance with the database. This research has benefited from Spanish Ministry of Science and Technology Project ECO The usual disclaimer applies.

2 1. Introduction In a recent paper Salas and Rodriguez (2011) have shown that if two distributions are symmetric under the same strictly increasing transformation such as the family of lognormal income distributions the equally distributed equivalent income (EDE) of a utilitarian social evaluation function (SEF) will be equal to median income. As a result, the utilitarian SEF will be ordinal equivalent to median income and both, the utilitarian SEF and median, will rank distributions in the same way. Furthermore, these authors have shown that the Condorcet winner among those distributions that are symmetric under the same strictly increasing transformation is the distribution with the highest median income. 2 By linking these two results, they have proved that an additive utilitarian social evaluation function is consistent with the outcome of majority voting if the income distributions are symmetric under the same strictly increasing transformation. As a practical corollary, Salas and Rodríguez (2011) by making use of the power transformation have been able to prove that a utilitarian social evaluation function à la Kolm Atkinson (Kolm, 1969 and Atkinson, 1970) consistent with the Kolm-Atkinson index of inequality is indeed an increasing transformation of median income. However, the extension of these results to a utilitarian SEF consistent with the Gini coefficient seems to be impossible given the result in Newbery (1970). According to Newbery (1970) there is no differentiable strictly concave utility function u( ) such that an utilitarian SEF W accords with the Gini coefficient. Worst still, Dasgupta et al. (1973) generalized Newbery s result from W to any strictly quasi-concave SEF and, 2 A Condorcet winner is an alternative which can beat every other alternative in a two-way vote. 2

3 later on, Lambert (1985) directly generalized Newbery s result from W to any differentiable SEF. Fortunately, some authors (see, Sheshinski, 1972, Sen, 1973, Kakwani, 1985 and Lambert, 1985) have defended that a convincing rationale for the use of a SEF consistent with the Gini coefficient could still be given if we abandon the class of individualistic social evaluation functions. 3 In particular, Kakwani (1985) and Lambert (1985) provided a positive result by widening the domain for personal preferences to incorporate envy or altruism. We extend in this paper the results in Salas and Rodríguez (2011) to the class of rankdependent SEFs à la Kakwani Lambert (Kakwani, 1985 and Lambert, 1985) that are consistent with the Gini coefficient. For this purpose, we look for the transformation that makes the EDE of the class of rank-dependent SEFs à la Kakwani Lambert to be equal to median income. We illustrate our main result with data drawn from the Survey on Income and Living Conditions (EU-SILC) dataset for the European Union in the period The organization of the paper is as follows. In Section 2, we provide the extension of Salas and Rodriguez s results for the class of rank-dependent SEFs à la Kakwani Lambert. Section 3 provides an empirical illustration, while Section 4 presents some concluding remarks. 3 In 1978, Blackorby and Donaldson gave a list of properties that characterized (though not completely) the social evaluation functions which accord with the Gini coefficient. They should be homothetic, quasiconcave and additive but not separable. 3

4 2. The extension to rank-dependent SEFs Assume a number of homogeneous income recipients!! where! is the set of natural numbers greater than one. An income distribution in this population is represented by a positive ascending-ranked vector! = (!!,!!,,!! ), so that 0 <!!!!!!. The set of all (positive ascending-ranked) income distributions is denoted by! =!!!!. For all!! we define a continuous Social Evaluation Function (SEF)!! from!! to ℝ: (!!,!!,,!! )!! (!!,!!,,!! ), for all!! that ranks the set of all income distributions according to welfare. The set of all continuous, increasing and S-concave SEF!! for all!! is denoted by Ω. We also define the equally distributed equivalent income (EDE)! corresponding to an income distribution (!!,!!,,!! )!! for all!! as the value of! that solves:!!!,!,,! =!!!1,!2,,!! (1) Continuity of!! guarantees unique solution for! = Ξ! (!1,!2,,!! ). The function Ξ! is a particular numerical representation of!!. More generally, any monotone transformation of Ξ! represents same ordinal preferences as in!!. Assuming the inequality index!! from!! to ℝ: (!!,!!,,!! )!! (!!,!!,,!! ) for all!! that ranks the set of all income distributions. The set of all continuous and Sconvex!! for all!! is denoted by Γ. We concentrate on a particular subset of the inequality indices! Γ, that was introduced by Donaldson and Weymark (1980) an!! =! d denoted single- 4

5 parameter Gini (S-Gini). They are defined as!! from!!!(1, ) to ℝ: (!!,!!,,!! ;!)!! (!!,!!,,!! ;!) by4!+1!!!!!!!!,!"!!,!=1!! 1!!!;! = 1!!!"#,! > 1, (2) where!(!) is the mean of the distribution!. This family of indices belongs to the linear rank-dependent inequality indices proposed by Mehran (1976) and coincides with the standard Gini coefficient when! = 2. Donaldson and Weymark (1980) proved that this family could be derived as an AKS inequality index (Atkinson 1970, Kolm, 1969, Sen, 1973) directly from a SEF as follows:!! (!) = 1 Ξ!!!! for all!"!! and!"#. (3) For the S-Gini indices we obtain this AKS-based particular representation of the SEF:!!!;! = Ξ!!;! =!(!)(1!!!;! ) =!+1!!!!!!!!!=1!! (4) It belongs to the Yaari family of rank-dependent SEFs, Yaari (1987, 1988). More generally, Kakwani (1985) and (1986) and Lambert (1985) proposed the following class of SEFs:!!!;! = Ξ!!;! =!! 1!!!!,!"!!,!"#, 0 <! 1. (5) Note that for this family of SEFs, we can derive the following family of inequality indices (we shall call it the Kakwani-Lambert family of inequality indices): 1!!! =! 1 Ξ!!;!!!,!"!!,!"#, 0 <! 1. (6) 4 Yitzhaki (1983) and Donaldson and Weymark (1983) provided a continuous version of the same family of indices. 5

6 By assuming that the S-Gini indices belong to the Kakwani-Lambert family of inequality indices, equation (5) becomes:!!!;!;! = Ξ!!;!;! =!! 1!!!!;!,!"!!,!"#,! > 1,0 <! 1 (7) From equations (2) and (7), we obtain this particular representation of the SEF:!!!;!;! = Ξ!!;!;! = 1!!! +!!+1!!!!!!!!,!=1!! (8) Note the right-hand side term of equation (8) becomes a weighted average of the (extreme) utilitarian SEF and the AKS-based SEF in equation (4). For this particular family, in order to get the EDE value as the median of the distribution, we propose the following monotone transformation of the initial distribution!:!! =!!!!! = 1! 1!!! +!!+1!!!!!!!,!!! = 1,,!;! > 1,0 <! 1. (9) As a result, we obtain:!!=1!! = Ξ!!;!;! =!. (10) If! = (!1,! 2,,!! ) is symmetrically distributed, the mean and median of! equals!/!. Moreover, since!!! is a monotone transformation, we can get the median of the original distribution!,!!, as the inverse of the median of the transformed distribution! :!!!!!!!!!! =!!! (11) where!!! is the value of!!! at the median value which, without lost of generality, is! = (! + 1)/2. Then, the value of!!! is equal to 1/! if the inequality aversion parameter! is 2. For this case, the EDE is equal to median income, that is,!! =!. Provided that the EDE of the class of rank-dependent SEFs à la Kakwani Lambert is 6

7 equal to median income both will rank distributions in the same way. Furthermore, following Salas and Rodríguez (2011), we can prove that the class of rank-dependent SEFs à la Kakwani Lambert is consistent with the outcome of majority voting if the income distributions under analysis are symmetric under the same transformation in (8). In this manner, we generalize the results in Salas and Rodríguez (2011) to a class of rank-dependent set of SEFs that are consistent with the Gini index. 3. An empirical illustration We illustrate our main result with data drawn from the Survey on Income and Living Conditions (EU-SILC) dataset for the European Union in the period This database is a panel data of 26 European countries for three consecutive years. In this manner, we can contrast the symmetry hypothesis with a set of heterogeneous countries not only in economic terms but also in political, cultural and institutional terms. Aftertax and transfer household income is the variable under analysis. Post-tax income is calculated as pre-tax income plus cash transfers from the government, minus income tax payments and social security contributions. Moreover, incomes are normalized by an adult-equivalence scale defined as e 0.5, where e is household size. Observations with negative incomes are removed and household observations are weighted by the sample weights. First of all, we apply to the data the transformation proposed in the previous section. For this transformation, we fix the parameter v to two (v = 2) and consider that k [0, 1] with two decimal points of accuracy. We then test formally each transformed distribution for symmetry using a consistent nonparametric kernel-based test developed 7

8 by Ahmad and Li (1997). The procedure we use tests the hypothesis that a distribution is symmetric about the median. In particular, the null hypothesis is expressed as H : f ( x) = f ( ) for all x R, whereas the alternative is that H 0 is false, i.e. 0 x H : f ( x) f ( ) for some x R. This is an intuitively appealing and very direct way 1 x of constructing a test statistic for the symmetry of a distribution. An advantage of the proposed method is that it is based on distributions and consequently, it takes into account all the information that may be forthcoming from many moments without assuming the existence of such moments, and without much more difficulty in estimation. 5 For the same reason, the method is also less likely to suffer from inadvertent introduction of special relations between moments that only hold for special distributions (i.e. the Gaussian distribution). Finally, large samples to estimate high order moments, assuming they exist, are not needed. Assume a random sample of n i.i.d. observations of income Y i, i = 1,, n, drawn from the distribution F and ordered such that Y Y Y 1 2 n. From Ahmad and Li (1997), we know that n h I 2n converges to a normal distribution with mean 0 and variance 2 4σ, where h is the smoothing parameter and I 2n is as follows: I 2n 1 = 2 n h n n i = 1 j i Yi Y K h j + Yi Y K h j, (12) 5 Many of the existing tests of symmetry examine high order moments (see for example Premaratne and Bera, 2005). Bai and Ng (2005) report the difficulties with estimating higher order moments, such as kurtosis, as well as the greater power of test which are based simultaneously on several odd order moments. 8

9 2 where K( ) is the Gaussian kernel function. We estimate the variance σ according to the following term: 2 n n 1 1 Yi Y j σ = K. (13) 2 2 π n h i = 1 j = 1 h Following Ahmad and Li (1997), we chose h = δ n 1 γ, where δ denotes the standard deviation of the sample data, and the value γ = 6. This test is one-sided as the alternative hypothesis states that the statistic I 2n is positive. Therefore, the critical value is 2.33 whether a 1% level of significance is adopted. Assuming that v = 2, we compute the critical values of the parameter k* for which symmetry is not rejected. In Table 1, we reject symmetry when the values k min and k max are unspecified. We can see that the symmetry hypothesis is not rejected in the 86% of cases. Consequently, we can state that the symmetry condition is not too restrictive in practice. 6 After testing for symmetry, we compute the optimal value k* as the most probable k for which symmetry is not rejected. We show in Figure 1 the distribution of estimated k*. We observe that the parameter k* ranges from 0.1 to 0.6 with a value around 0.4 as the most frequent. In Table 1 we also provide the level of welfare (W) for the corresponding optimal inequality aversion parameter k*, the median income (m), the mean income (µ) and the Gini coefficient (G). We observe in Figures 2 and 3 (for those cases whose symmetry condition has not been rejected) that both median and mean incomes track welfare very 6 Salas and Rodríguez (2011) found for a panel of 116 countries (the World Bank s POVCAL database) that the symmetry condition is not rejected for the 92.14% of the cases. 9

10 well, though mean income behaves a bit worse because the coefficient of determination R 2 is for median income, while it is for mean income. Moreover, the slope of the regression for the median income (1.032) is closer to one than the slope of the regression for the mean income (0.925). Therefore, we observe like Salas and Rodríguez (2011) that median income is a better proxy for social welfare than mean income. Finally, we observe in Figure 4 the positive relationship between the optimal parameter k* and the standard Gini coefficient (the coefficient of determination R 2 is 0.529). It seems that the larger the Gini coefficient, the higher the optimal parameter k*. This relationship is reinforced by the positive correlation between the Gini coefficient and the interval [k min, k max ] for which an income distribution is symmetrizable. 4. Concluding Remarks Traditionally, two different strategies have been adopted by scholars to aggregate individual preferences and, hence, to rank distributions: a political process like the majority voting mechanism and a set of desirable assumptions, as represented by a social evaluation function. Salas and Rodríguez (2011) by using the family of power transformations were able to link social evaluation functions à la Kolm Atkinson with the majority voting procedure. In this paper we extend that result to the class of rank-dependent social evaluation functions à la Kakwani Lambert. By doing this, we find the conditions under which the EDE of a SEF that is consistent with the Gini coefficient is equal to the median of the distribution. In addition, the proposal allows us to deduce the parameter endorsed by 10

11 that family of rank-dependent social evaluation functions. We confirm empirically that the required conditions are generally accepted with an illustration based on the EU- SILC dataset for the European Union in the period

12 Table 1. The optimal inequality aversion parameter k* in Europe ( ). Country Year k min k max k* W m µ G Austria Austria Austria Belgium Belgium Belgium Cyprus Cyprus Cyprus Czech Rep Czech Rep Czech Rep Denmark Denmark Denmark Estonia Estonia Estonia Finland Finland Finland France France France Germany Germany Germany Greece Greece Greece Hungary Hungary Hungary Iceland Iceland Iceland Ireland Ireland Ireland Italy Italy Italy Latvia Latvia Latvia Lithuania Lithuania Lithuania Luxembourg Luxembourg Luxembourg Norway Norway Norway Poland Poland Poland Portugal Portugal Portugal Slovakia Slovakia Slovakia Slovenia Slovenia Slovenia Spain Spain Spain Sweden Sweden Sweden The Netherlands The Netherlands The Netherlands United Kingdom United Kingdom United Kingdom

13 Figure 1. Relative frequencies of k* in Europe ( ) Relative frequency k* value Figure 2. Correlation between median income and welfare in Europe ( ) Welfare y = x R 2 = Median Figure 3. Correlation between mean income and welfare in Europe ( ). 13

14 Welfare y = x R 2 = Mean Figure 4. Correlation between the k*, k min, k max and the Gini coefficient (v = 2) k values k* kmin kmax Gini Coefficient 14

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