where n is the total population, yi is the outcome (in our case income) of individual i, µ is the mean income, and c is a parameter chosen by the researcher.
As the value of c increases, the sensitivity to inequality among those in the upper end of the distribution increases. While Theil entropy measure is obtained from a c-value of 1, a c-value of 0 gives Theil L or mean log deviation. GE (2) is ordinally equivalent to the squared coefficient of variation (Elbers et al., 2008).
The general entropy class of measures can be conveniently decomposed into between-group and within-group components (Shorrocks, 1984) as follows:
where j is the population sub-group, gj is the population share of the jth subgroup, and GEj is the inequality within the jth subgroup.
While the first term measures the between-group component of total inequality, the second term denotes inequality within the subgroups. The between-group component gives the level of inequality pertaining to a distribution where everyone within each sub-group has the same outcome µj. The between-group component can be summarized as
for any population partition ∏, where IB(∏) is the between-group component and I is total inequality.
Empirical applications of decomposable measures to the data on income distribution typically do not find significant inter-group inequality. For example, one of the earliest studies in this direction which made significant impact on the subsequent empirical literature was done by Anand (1983) on Malaysia, who found that inequality between the native Malays and Chinese Malaysians accounted for only 15% of the overall inequality in the country in the early 1970s. Based on this finding, he recommended that the strategy of the government should be directed to the sources of inequality among the people within the same ethnic group rather than focus on between-group inequality. Studies in India have also found a small contribution of between-group inequality to total inequality, where the groups are SCs, STs, and others. Mutatkar (2005) found less than 5% contribution of inequality between groups in three rounds of National Sample Survey (NSS) data in the 1980s and 1990s. Using data from the 1993–1994 round of NSS, Deshpande (2000) finds an even lower contribution between these three social groups in Kerala. Part of the reason for this observed low between-group inequality is inherent in the nature of the standard inequality decomposition method itself. The paper by Elbers et al. (2008) points out that the standard procedure for decomposing inequality into a between-group and a within-group component fails to capture the true extent of between-group inequality as it is compared with the overall inequality which can be seen as inter-group inequality when each individual constitutes a group. Therefore, overall inequality tends to be way above between-group inequality as in the former the number of groups is exactly equal to the number of individuals, which is large.
In the standard procedure, between-group inequality tends to increase as the number of groups increases, and such inequality is also sensitive to the relative population composition of the groups. Ray Chaudhury (2014) finds that in 2009–2010 the shares of inequality in the distribution of consumer expenditure attributable to the differences between two broadly defined social groups — SC/ST (comprising Scheduled Castes and Scheduled Tribes) and non-SC/ST (comprising ‘other backward castes’ and ‘others’) — in two Indian states of Kerala and Punjab were 1% and 3.1%, respectively, even though there was no significant difference in the relative disparity in mean expenditures between non-SC/ST and SC/ST in these two states. Therefore, the difference in the degree of between-group inequality in these two states, as measured by the between-group component of the overall inequality, might largely be due to the difference in the population shares of the social groups (non-SC/ST and SC/ST), rather than the difference in relative mean expenditures of these social groups.
We first use the traditional method of inequality decomposition and find out how the between-group component differs when we consider different groupings, namely caste, class, and religion. However, since the traditional method of inequality decomposition is sensitive to the relative sizes and the number of groups under question, the decompositions are not comparable across alternative groupings. For instance, by the conventional method, the shares of between-group inequality in income (groups defined in terms of racial identities) in three countries infamous for racial inequality (namely United States, Brazil, and South Africa) have been shown by Elbers et al. (2008) to be 8%, 16%, and 33%, respectively. They question if ‘these numbers provide a good yardstick with which to judge the relevance of race to an understanding of inequality in these countries’. They further point out that while the mean difference in income between the white and non-white groups is stark in all three countries, the population shares of the white versus non-white groups vary widely (with non-whites comprising 80%, 50%, and 28% of the population in South Africa, Brazil, and the United States, respectively). Furthermore, the number of racial groups is also not invariant across the countries (four for Brazil and South Africa and five for the United States). Elbers et al. (2008) illustrate that the difference in the share of the between-group component may not be reflective of the differences in relative mean incomes alone, since it is not normalized for differences in the number and the relative size of groups. As we indicated above, the share of the between-group component in total inequality, as decomposed by the traditional method, has been typically low since it is taken to be the ratio between observed group inequality and total (or interpersonal) inequality.
The latter may be looked upon as a particular type of between-group inequality, where every household (or individual, depending upon the unit of analysis) constitutes a separate group. Elbers et al. (2008) argue that it is perhaps unrealistic to compute the share of observed between-group inequality against the benchmark of total interpersonal inequality, since the actual number of social groups considered in a decomposition exercise is too small compared to the total population. They suggest an alternative measure of the share of between-group inequality that is normalized with respect to the number and relative size of groups. They replace total inequality in the denominator of the conventional ratio with the ‘maximum between-group inequality that could be obtained if the number of groups and their sizes were restricted to be the same as for the numerator’.
Elbers et al. (2008) compare the extent of between-group inequality with a differently constructed benchmark, which is obtained by partitioning the individual incomes into two non-overlapping groups. If there are two groups are of sizes n1 and n2, the incomes are rearranged in such a manner that the richest person of the poorer group is poorer than the poorest person in the richer group. The between-group inequality between these reconstructed groups can now be seen as the maximum possible between-group inequality given the relative sizes of the groups. The modified measure allows meaningful comparison of between-group inequality across different social settings, where the number and relative size of groups are different. Thus they propose a seemingly small adaptation of the conventional procedure to produce an alternative statistic that overcomes some of these limitations of the conventional decomposition procedure. Elbers et al. (2008) illustrate this point with reference to South Africa. They show that when inequality is decomposed by racial group defined in terms of a ‘white/non-white’ classification, the conventional decomposition suggests that only about 27% of inequality is attributable to between-group differences. Their alternative statistic, on the other hand, shows that two groups are 80% of the way towards a completely partitioned South African income distribution.
The alternative index proposed by Elbers et al. (2008) is given by
where the denominator gives ‘the maximum between-group inequality that could be obtained by reassigning individuals