Inequality decomposition analysis, the Lorenz curve and the Gini coefficient

We compare two alternative procedures for decomposing the Lorenz curve and the Gini coefficient into within-groups and between-groups contributions: the standard additive decomposition (Bhattacharya and Mahalanobis in J Am Stat Assoc 62(317):143–161, 1967) and another inspired by the path-independent decomposition (Foster and Shneyerov in J Econ Theory 91(2):199–222, 2000). We show that the former approach offers a clean measure of the between-group inequality, which is insensitive to changes in the distribution that do not alter the relative difference between the groups' averages. On the other hand, the latter approach offers an unbiased measure of the within-group inequality, which is independent of the between-group component. Hence, we propose and interpret a new decomposition of the Gini index that is based on the path-independent approach. Finally, we explore the implications of Lorenz between-group dominance combined with Lorenz within-group dominance. We show the difficulty of defining sufficient conditions for two Lorenz curves not to intersect and suggest an alternative partial order based on concentration curves which does not account for the overlap between groups.