Extremal points of Lorenz curves and applications to inequality analysis

TL;DR

This paper characterizes extremal Lorenz curves with fixed Gini indices, introduces a new two-dimensional inequality and dissimilarity index, and demonstrates its practical application to income distribution analysis.

Contribution

It provides a mathematical characterization of extremal Lorenz curves, develops a novel bidimensional inequality index, and applies it to real income data for inequality analysis.

Findings

Maximal L1-distance between Lorenz curves with given Gini coefficients computed.

A new index measuring relative inequality and dissimilarity introduced.

Application to EU-SILC income data demonstrates practical utility.

Abstract

We find the set of extremal points of Lorenz curves with fixed Gini index and compute the maximal $L^1$-distance between Lorenz curves with given values of their Gini coefficients. As an application we introduce a bidimensional index that simultaneously measures relative inequality and dissimilarity between two populations. This proposal employs the Gini indices of the variables and an $L^1$-distance between their Lorenz curves. The index takes values in a right-angled triangle, two of whose sides characterize perfect relative inequality-expressed by the Lorenz ordering between the underlying distributions. Further, the hypotenuse represents maximal distance between the two distributions. As a consequence, we construct a chart to, graphically, either see the evolution of (relative) inequality and distance between two income distributions over time or to compare the distribution of income of a specific population between a fixed time point and a range of years. We prove the mathematical results behind the above claims and provide a full description of the asymptotic properties of the plug-in estimator of this index. Finally, we apply the proposed bidimensional index to several real EU-SILC income datasets to illustrate its performance in practice.