# On the Kolkata index as a measure of income inequality

## TL;DR

This paper explores the mathematical properties of the Kolkata (k) index as a measure of income inequality, showing its relation to Lorenz functions, its uniqueness, and how it compares to Gini and Pietra indices.

## Contribution

It introduces the k-index as a fixed point of the Lorenz function, compares it with existing inequality measures, and analyzes its sensitivity to income transfers across groups.

## Key findings

- The k-index always exists and is a unique fixed point of the Lorenz function.
- The k-index generalizes Pareto's 80/20 rule.
- The k-index is only affected by income transfers across poor and rich groups.

## 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 show that 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. We establish that for any income distribution the value of Gini coefficient is no less than that of the Pietra index and the value of the Pietra index is no less than that of the normalized k-index. 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.