On the Kolkata index as a measure of income inequality

Abstract

We study the mathematical and economic structure of the Kolkata (k) index of income inequality. We show that the k-index always exists and is a unique fixed point of the complementary Lorenz function, where the Lorenz function itself gives the fraction of cumulative income possessed by the cumulative fraction of population (when arranged from poorer to richer). We argue in what sense the k-index generalizes Pareto’s 80/20 rule. Although the k and Pietra indices both split the society into two groups, we show that k-index is a more intensive measure for the poor–rich split. We compare the normalized k-index with the Gini coefficient and the Pietra index and discuss when they coincide. Specifically, we identify the complete family of Lorenz functions for which the three indices coincide. While the Gini coefficient and the Pietra index are affected by transfers exclusively among the rich or among the poor, the k-index is only affected by transfers across the two groups.