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

Asim Ghosh; Bikas K. Chakrabarti

doi:10.1103/wb7q-78gb

paper

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

arXiv:2512.00754 · doi:10.1103/wb7q-78gb

Abstract

We study 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 $k$ of wealth 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 $2 k - 1$ , the Robin Hood index should strictly be equal to the excess wealth fraction $2 k - 1$ 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 show that $p / (2 k - 1)$ 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, by considering the Lorenz function appropriate for the generalized Pareto power law distribution, we obtain an extended range ( $1_{+} < p / g \leq e /2$ ) that captures most of the empirical data.

Accepted for publication in Physical Review E

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

Abstract

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