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

TL;DR

The paper investigates interrelations among the Gini index $g$, Pietra index $p$, and Kolkata index $k$, all derived from the Lorenz function $L(x)$. It proves analytically that $p=2k-1$ and, under a minimal Lorenz expansion $L(x)=Ax+Bx^2$ with $A+B=1$, derives $g=B/3$, $p=B/4$ leading to $p=(3/4)g$, and in the small-$g$ limit $k=1/2+(3/8)g$. The authors test these relations against empirical data from US IRS incomes (1983–2022), Indian Bollywood box-office incomes (1999–2024), and Nobel laureate citation patterns (2020–2025), finding $p/(2k-1)$ to be slightly above unity (within ~5%) with notable deviations for larger $g$ or $k$. They also observe $k$ remaining near or above the Pareto value $0.80$ in several datasets, reflecting persistent inequality and potential self-organized criticality. Overall, the work offers a unified Lorenz-based framework linking $g$, $p$, and $k$ while highlighting the limitations of simple analytic forms in regimes of strong inequality.

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 the Pietra or Robin Hood index $p$ should equal to the excess wealth $2k-1$ possessed by the richest $1-k$ 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.

Figures (5)

• Figure 1: The Lorenz curve or function ($L(x)$, in red) shows the proportion of total wealth owned by a fraction ($x$) of people in ascending order of wealth. The black dotted-dashed line represents a scenario of perfect equality in which everyone possesses the same amount of wealth. The Gini index ($g$) is calculated from the area between the Lorenz curve and the equality line (shaded region), normalized by the total area (= 1/2) under the equality line. The Pietra index $p$ given by the length of the maximum vertical distance of the Lorenz curve from the equality line and indicated here by the solid vertical line in black. The complementary Lorenz function ($L_c(x) \equiv 1 - L (x)$ is) shown in green. The Kolkata index ($k$) is determined by the fixed point of the complementary Lorenz curve: $L_c (k) = k$ or $L(k) = 1 - k$. Geometrically it gives the point at which the Lorenz curve intersects the diagonal line perpendicular to the equality line and it gives the fraction $k$ of wealth that is possessed by the richest $1 - k$ fraction of people. As such $k = 0.8$ corresponds to Pareto's 80-20 law.
• Figure 2: Growths of Pietra ($p$) and Kolkata ($k$) indices against Gini ($g$) index values for US economy income data (IRS data irsYakovenko22 for the period 1982-2022). The values of $k$ and $g$ are growing with time, because of increasing rate of withdrawal of public welfare programs. The upper left inset, showing the values of $p/(2k - 1)$ against the years, indicate a value higher than unity as predicted by a rigid theoretical argument (relation (1)). The lower right inset shows the values of $p/g$ against years and show values considerably different from 3/4, as obtained from the theoretical relation (3) (obtained an additional assumption (2) of the minimal form of the Lorenz function). Details of the values of the inequality indices and and of their relations are given in Table I of the Appendix.
• Figure 3: Growths of Pietra ($p$) and Kolkata ($k$) indices against Gini ($g$) index values for US economy income tax return data (IRS data irsYakovenko22 for the period 1982-2022). The values of $k$ in the tax data (which can be argued to represent the prevailing inequality status better) seems to have grown a little beyond the Pareto value ($k = 0.80$) with increasing withdrawal of public welfare programs. The upper left and lower right insets, showing the values of $p/(2k-1)$ against $g$ and $k$ respectively, indicate values slightly higher than unity (as predicted by a rigid theoretical argument; see relation (1). The lower right inset shows the values of p/g against years and show values considerably different from 3/4, as obtained from the theoretical relation (3) (obtained an additional assumption (2) of the minimal form of the Lorenz function). Details of the values of the inequality indices and of their relations are given in Table II of the Appendix.
• Figure 4: Growths of Pietra ($p$) and Kolkata ($k$) indices against Gini ($g$) index values for movie income inequalities (Bolloywood India data 13bbollywood for the period 1999-2024). The values of $k$ in the tax data (which can be argued to represent the prevailing inequality status better) seems to have grown a little beyond the Pareto value ($k$ = 0.80) with increasing withdrawal of public welfare programs. The upper left inset, showing the values of $p/(2k - 1)$ against the years, indicate values higher than unity as predicted by a rigid theoretical argument (relation (1)). The lower right inset shows the values of $p/g$ against years and show values considerably different from 3/4, as obtained from theoretical relation (3) (obtained an additional assumption (2) of the minimal form of the Lorenz function). Details of the values of the inequality indices and and of their relations are given in Table III of the Appendix.
• Figure 5: Growths of Pietra ($p$) and Kolkata ($k$) indices against Gini ($g$) index values for the citation inequalities among the papers published by different Nobel Laureates in economics, physics, chemistry and medicine during the last six years (Google Scholar open-access data google-scholar for scientists, each having their own home page with verified e-mail address and more than 100 papers). The values of $k$ (and also of $g$ for the inequalities in citation distributions across the publications of individual Nobel laureates) have grown, because of extreme competition among the scientists, quite beyond the Pareto value ($k$ = 0.80, and closer to self-organized critical system value Manna22Banerjee23B) because of any public welfare kind of support system (for the producers) and the market being quite competitive. The upper left and lower right insets, showing the values of $p/(2k - 1)$ against $g$ and $k$ respectively, indicate values slightly higher than unity (as predicted by a rigid theoretical argument; see relation (1). The lower right inset shows the values of $p/g$ against years and show values considerably different from 3/4, as obtained from the theoretical relation (3) (obtained an additional assumption (2) of the minimal form of the Lorenz function). Details of the inequality indices and of their relations are given in Table IV of the Appendix.