Extremal points of Lorenz curves and applications to inequality analysis
Abstract
We find the set of extremal points of Lorenz curves with fixed Gini index and compute the maximal -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 -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.
Keywords
Cite
@article{arxiv.2103.03286,
title = {Extremal points of Lorenz curves and applications to inequality analysis},
author = {Amparo Baíllo and Javier Cárcamo and Carlos Mora-Corral},
journal= {arXiv preprint arXiv:2103.03286},
year = {2021}
}