Relation Between Two Income Inequality Measures: The Gini coefficient and the Robin Hood Index

Abstract

The objective of this investigation is to study the relation between two common measures of income inequality, the Gini coefficient and the Robin Hood index. An approximate formula for the Robin Hood index in terms of the Gini coefficient is developed from 100,000 Lorenz curves that are randomly generated based on 100 twenty-parameter families of income distributions. The approximate formula is tested against Robin Hood indexes of commonly-used one-parameter Lorenz curves, income data of several countries, and reported results of Robin Hood indexes. The approximate formula is also tested against results of a stochastic income-wealth model that is introduced in the present investigation. The formula is useful conceptually in understanding why Gini coefficients and Robin Hood indexes are correlated in distribution data and is useful practically in providing accurate estimates of Robin Hood indexes when Gini coefficients are known. The continuous piecewise-linear approximation is generally within 5% of standard one-parameter Lorenz curves and income distribution data and has the form: R ≈ 0.74G for 0 ⩽ G ⩽ 0.5, R ≈ 0.37+0.90(G-0.5) for $$0.5 ⩽ G ⩽ 0.8 and $$ R ≈ 0.64 + 1.26 (G- 0.8) for 0.8 ⩽ G ⩽ 0.95$$ where R is the Robin Hood index and G is the Gini coefficient.