Aggregate units, within-group inequality and the decomposition of inequality measures

Abstract

The decomposition of inequality measures according to within-group and between-group contributions has been addressed up to now (a) to identifying measures which satisfy in a natural way such decomposition, being characterized by special regularity assumptions, and (b) to the preliminary determination of the between-group inequality as the most primitive one, letting the within-group term as residual with respect to total inequality. Both approaches reveal some drawbacks in the use and interpretation of inequality measures. Therefore an alternative approach is proposed, which is applicable—as well as (b)—to every inequality measure we start from a distribution reflecting only the within-group inequality, so that the corresponding measure Iw can be computed, leaving the between group term IB as a residual with respect to the total inequality I. This procedure assumes that the groups are composed by the same number m of aggregate units, or by some replicas of the same distribution of aggregate units; on this basis an "average percentile distribution" is defined as a distribution averaged over all groups, clearly responsible for the within-group inequality; actually the Lorenz curve of this distribution appears as a weighted mean of the group Lorenz curves. As a rule the application of this procedure to particular inequality measures requires the computation of the average percentile distribution; however, for the Gini ratio and the relative mean deviation with respect to the median Iw turns out to be the weighted mean of the measures computed on the groups. In the last paragraph the above procedure is applied to several Pareto distributions, composed of two or four groups.