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Extremal points of Lorenz curves and applications to inequality analysis

Author

Listed:
  • Amparo Ba'illo
  • Javier C'arcamo
  • Carlos Mora-Corral
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.

Suggested Citation

  • Amparo Ba'illo & Javier C'arcamo & Carlos Mora-Corral, 2021. "Extremal points of Lorenz curves and applications to inequality analysis," Papers 2103.03286, arXiv.org.
    • Handle: RePEc:arx:papers:2103.03286
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    References listed on IDEAS

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