Relations Between the Inequality Indices Gini, Pietra and Kolkata: Theory and Data Analysis

TL;DR

This paper explores the mathematical relationships between three inequality indices—Gini, Pietra, and Kolkata—using theory and data analysis across income, entertainment, and citation datasets, revealing consistent proportional relationships.

Contribution

It provides rigorous theoretical bounds and empirical validation of relationships among the Gini, Pietra, and Kolkata inequality indices, including new derived formulas.

Findings

The Pietra index p is always greater than or equal to 2k-1.

The Robin Hood index equals exactly 2k-1.

Empirical data shows p/(2k -1) exceeds 1 by less than 5%."

Abstract

We study here relations between three inequality indices, namely the Gini ($g$), Pietra ($p$) and Kolkata ($k$) introduced in 1912, 1915 and 2014 respectively and all are derived from the Lorenz function $L(x)$ introduced in 1905. The Kolkata index (which corresponds to a fixed point of the complementary Lorenz function $L_c(x) \equiv 1-L(x)$) gives the fraction of wealth $k$ possessed by the richest $1-k$ fraction of people ($k$ = 0.8 corresponds to Pareto's 80-20 law from 1896). We show rigorously that while the Pietra index value $p$ should be greater than or equal to $2k-1$, the Robin Hood index should strictly be equal to the excess wealth fraction $2k-1$ possessed by the richest $1-k$ fraction of people. Our numerical data analysis for US IRS Income data (1983-2022), Bollywood (India) movie income data (1999-2024) and the citation inequalities across the publications by forty Nobel Laureates (2020-2025) in Economics, Physics, Chemistry and Medicine clearly shows that $p/(2k -1)$ is always greater than unity but the deviation is never more than five percent. Assuming some simple analytic form for the Lorenz function, we also derived the relations $k = (1/2) + (3/8)g$ for small $g$ values and $p/g = 3/4$. However, these relations generally deviate significantly for larger $g$ or $k$ values when compared with observations.