A Unified Approach to Measuring Poverty and Inequality
defined with reference to an underlying variable such as schooling, the analysis can help identify the extent to which the variable explains inequality. To analyze changes in inequality over time, one can separate the effect of changes in population sizes across subgroups (for example, arising from demographic factors) from the fundamental shifts in subgroup income distributions. This can be combined with regression analysis to model income changes and to pinpoint the variables that appear to be driving inequality. The generalized entropy measures are the only inequality measures satisfying the usual form of additive decomposability, with the Theil measures (a = 0 and a = 1) and half the squared coefficient of variation (a = 2) being most commonly used in empirical evaluations. The second Theil measure, also called the mean log deviation, has a particularly simple decomposition in which the within-group term is a population-share weighted average of subgroup inequality levels. This streamlined weighting structure can greatly simplify interpretation and application of decomposition analyses. The allied property of subgroup consistency is helpful in ensuring that regional changes in inequality are appropriately reflected in overall inequality. Suppose there is no change in the population sizes and mean income levels of the subgroups. If inequality rose in one subgroup and was unchanged or rose in each of the other subgroups, it would be natural to expect that inequality overall would rise. For additively decomposable measures, this rise in inequality is assured: because the smoothed distribution is unchanged, the between-group term is unaffected. Because the weights on subgroup inequality levels are fixed (when subgroup means and population sizes do not change), the within-group term must rise. Subgroup consistency is a more lenient requirement, because it does not specify the functional form that links subgroup inequality levels and overall inequality. Consequently, on the one hand we find that the Atkinson measures (which are transformations of the generalized entropy measures) are all subgroup consistent without being additively decomposable. On the other hand, the Gini coefficient is not subgroup consistent. The problem with the Gini coefficient arises when the income ranges of the subgroup distributions overlap. In that case, the effect of a given distributional change on subgroup inequality can be opposite to its effect on overall inequality. The Gini coefficient can be broken into a within-group term, a between-group term, and an overlap term—and it is the overlap term that can override the within-group effect to generate subgroup inconsistencies.
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