Income stratification and between-group inequality

• I generalize a result on the decomposition of the index to more than two groups. • It is shown how overlapping of groups impacts between-group inequality. • stratification for the • the pairwise contributions of to stratification. The paper shows explicitly how the overlapping of groups impacts between-group inequality by generalizing a result on the group-wise decomposition of the Gini index to more than two groups. It is demonstrated that the ratio of Yitzhaki’s measure of between-group inequality to the conventional measure is in general equal to one minus twice the weighted average probability that a random member of a richer (on average) group is poorer than a random member of a poorer (on average) group, and may therefore be interpreted as an overall index of income stratification in the population. The results are used to tabulate the contribution of each pair of regions in the world to the overall level of global income stratification.


Introduction
It is well known that the standard decomposition of the Gini index G by population groups does not yield an exact partition into between-group and within-group components, G B and G W respectively, unless the income ranges of the groups are non-overlapping (e.g., Mookherjee and Shorrocks, 1982). This has led both to an extensive literature exploring the nature of the ''residual'' from the standard decomposition (e.g. Lambert and Aronson, 1993;Lambert and Decoster, 2005) and to a parallel search for alternative decompositions that might prove more amenable to analysis and interpretation. In the latter vein, Yitzhaki and Lerman (1991) provides a partition of the Gini into between-group, within-group and overlapping components -G b , G w and G o respectively -where overlapping is considered as the inverse of the sociological concept of 'stratification'. Yitzhaki (1994) subsequently combines the latter two elements into a single within-group measure G wo that is * Tel.: +44 01382 384377.
E-mail address: p.f.allanson@dundee.ac.uk. explicitly written as a function of the degree of inequality within groups and the degree of overlapping between each pair of groups, but G b is also affected by overlapping and it remains to be shown how this measure relates to the conventional between-group index G B if there are more than two groups. 1

Group-wise decomposition of the Gini index
Consider a population divided into K ≥ 2 mutually exclusive and exhaustive groups that are ordered by mean income from the poorest to the richest group. Let Y k , F k (Y k ), µ k , p k and q k represent respectively the income (or some other relevant aspect of wellbeing) variable, cumulative distribution function, expected value, population share and income share of group k. The overall the group l and overall income distributions are given as F l (Y k ) and F u (Y k ) respectively, with corresponding mean ranksF kl andF ku .
Following Mookherjee and Shorrocks (1982), the conventional group-wise decomposition of the Gini index may be writ- ) /µ k denoting the Gini index of group k; and the residual R is interpreted as an 'interaction effect'. The alternative approach of Yitzhaki (1994) yields the exact decomposition in the open interval [0, 2] and is an increasing function of the fraction of group l that is located in the income range of group k, taking a value of zero when there is no overlap between the two groups and of one if the income distributions of the two groups are iden- the income ranges of the various groups overlap to any extent with the difference R W = G wo − G W given as: (1) Yitzhaki and Lerman (1991, p. 323) conclude that ''inequality and stratification are inversely related'', arguing that this relationship is consistent with relative deprivation theory in that ''stratified societies can tolerate higher inequality than unstratified societies'' since ''As people become more (less) engaged with each other, they have less (more) tolerance for a given level of inequality''. However, as Monti and Santori (2011) observe, this conclusion ignores the effect of overlapping on the between-group component G b , which will also affect the overall level of inequality perceived by the society. Yitzhaki and Lerman (1991, p. 322 Monti and Santori (2011) further demonstrate in the two group case that the ratio of G b to G B is equal to: where Prob(Y 1 > Y 2 ) is the probability of transvariation, i.e. the probability that the income of a random member of the poorer (on average) group is more than that of a random member of the richer (on average) group. To extend this result to the general case of K ≥ 2 groups, note that G b may also be expressed as: where the first line follows sinceF ku =  l p lFkl andF kk = 0.5, while G kl b denotes the Yitzhaki (1994) between-group index in the sub-population consisting only of groups k and l. Similarly, G B can be written as: where G kl B denotes the between-group Gini in the sub-population consisting of groups k and l only. Using (2) and (4), (3) may be rewritten as: from which it follows immediately that I will in general be equal to: Hence I is in general equal to one less twice the weighted average probability of transvariation between the various pairs of groups in the population. In his study of earnings differentials Gastwirth (1975) proposes TPROB = 2Prob(Y 1 > Y 2 ) as an index of overlapping between two groups, taking an ''ideal'' value of one when the two distributions are identical since Prob(Y 1 > Y 2 ) = 0.5 in this case. Thus I in (2) may be interpreted as the complementary index of non-overlapping or stratification, with (6) providing a generalization to two or more groups. I is a unit-free index that will take a maximum value of one when there is no overlap between any of the groups such that Prob(Y k > Y l ) = 0 ∀k, l > k; and will equal zero when the income distributions of all the groups are identical. 2 For K > 2, the extent to which nonoverlapping between any pair of groups contributes to the overall level of stratification is an increasing function of their population shares and the difference in mean incomes between them. I is invariant to both the scaling and translation of incomes. It is also invariant to replication both of the population within existing groups and of groups.
I has previously been identified by Milanovic and Yitzhaki (2002, p. 161) ''as an index indicating the loss of between group inequality due to overlapping''. The difference R B = G b − G B can be written from (6) as: 2 Negative values of I are also possible when mean incomes by group are negatively correlated with mean ranks. on which basis it may be argued, in contrast to Yitzhaki and Lerman (1991), that unstratified societies can tolerate more between-group inequality than stratified societies because individuals' positions within society are less narrowly determined by group membership. Nevertheless, holding G W and G B constant, overlapping per se must increase overall inequality since R ≥ 0 by definition (Pyatt, 1976), with (1) and (7) yielding a novel expression for R = R W + R B as: where the final line makes use of the expression for R presented in Lambert and Decoster (2005) for the two group case. 3

Empirical illustration
By way of illustration, this section elaborates the empirical analysis presented in Milanovic and Yitzhaki (2002) of world inequality in 1993 by regions. 4 The top panel in Table 1 presents estimates from their Tables 4 and 7 of population shares, p k ; mean incomes, µ k ; and mean rankings in the income distributions of each region,F kl , and the worldF ku . The lower panel reports the pairwise components of I identified in the final line of (6), where the calculation of these estimates makes use of the identityF kl = Prob(Y k > Y l ). The components sum to give the value of the stratification index I = 0.776, which is equal to the ratio of their reported estimates of G b = 0.309 and G B = 0.398. 5 Examination of the individual entries shows that the main contribution to stratification, accounting for as much as two thirds of the total, is due to the Asia and Western Europe/North America/Oceania (WENAO) pair as a result of a combination of the low degree of income overlap, the populousness of the two regions and the large 3 Lambert and Decoster (2005) state that their attention is confined to the case of two population subgroups ''for ease of presentation, but the results can clearly be extended''. 4 These regions are described as 'continents' in Milanovic and Yitzhaki (2002) though the correspondence is not exact. 5 Note that this is not the case with the results presented in Monti and Santori (2011) who base their analysis on country-level mean income data. difference in mean incomes between them. In contrast, the Africa and Asia pair contributes negatively to stratification, although the magnitude of this effect is negligible, because an African chosen at random is likely to be better off than a randomly chosen Asian despite the fact that average incomes are lower in Africa. Given that the value of I implies a weighted average probability of transvariation of 11.2%, only the Africa and WENAO pair and Asia and WENAO pair contribute more to R B than to G B .

Conclusion
The paper demonstrates how the residual from the conventional decomposition of the Gini index is fully absorbed into the between-group and within-group components proposed by Yitzhaki (1994). In particular, it is established that I = G b /G B is in general equal to one minus twice the weighted average probability of transvariation and may therefore be interpreted as an overall index of income stratification in the population. Using this result it is shown that the main source of stratification between regions of the world in 1993 was the limited overlap between the income distributions of Asia and WENAO given the relative populousness of the two regions and the difference in mean incomes between them. High per capita growth rates in some poorer Asian countries, most notably China and India, may be expected to have reduced levels of both stratification and inequality between regions in more recent years. 6