Alternate definitions of Gini, Hoover and Lorenz measures of inequalities and convergence with respect to the Wasserstein W₁ metric

Définitions alternatives des indices de Gini et de Hoover et de la courbe de Lorenz, et convergence par rapport à la distance de Wasserstein W₁

Valentin Melot
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Valentin Melot: IGF - Inspection générale des finances, ministère de l’économie et des finances

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Abstract: This article focuses on some properties of three tools used to measure economic inequalities with respect to a distribution of wealth µ: Gini coefficient G, Hoover coefficient or Robin Hood coefficient H, and the Lorenz concentration curve L. To express the distributions of resources, we use the framework of random variables and abstract Borel measures, rather than discrete samples or probability densities. This allows us to consider arbitrary distributions of wealth, e.g. mixtures between discrete and continuous distributions. In the first part (sections 1-4), we discuss alternate definitions of G, H and L that can be found in economics literature. The Lorenz curve is defined as the normalized integral of the quantile function [Gastwirth, 1971], which is not the same as saying "L(p) is the share of wealth owned by the 100p first centiles of the population" (proposition 1.4). The Gini and Hoover coefficients are introduced in terms of expectation of random variables. In section 3, we interpret Gini and Hoover as geometrical properties of the Lorenz curve (theorem 3.3 and corollary 3.10). In particular, we give a more general and straightforward proof of the main result of [Dorfman, 1979]. Section 4 gives two direct applications. We en route prove the (not trivial) fact that the Lorenz curve fully characterizes a distribution, up to a rescaling (proposition 2.4). The second part of the article (section 5-7) focuses on the consistency of G(µ), H(µ) and Lµ as µ is approximated or perturbated. The relevant tool to use is the Wasserstein metric W 1 , i.e. the L1 metric between quantile functions. W1 (µn , µ∞) → 0 if and only if underlying random variables converge in distribution and the total amount of wealth converges. In theorem 5.2 and proposition 5.5, we show that if and Lµn → Lµ∞ uniformly. Subsection 5.4 discusses topological implications of this fact. Thus, applications 6.1, 6.3, 6.7, 6.8 and 6.12 justify that the empirical Gini, Hoover indexes and Lorenz curves computed on a sample or rebuilt with partial information converge to the real Gini, Hoover indexes and Lorenz curve as information increases. Eventually, in section 7, we discuss the situations where the W1 convergence is not ensured, but weaker asumptions can be made (convergence in distribution in 7.1, convergence of means in 7.2.1 or uniform integrability in 7.2.2)

Keywords: Lorenz Curve; Gini index; Economic inequalities; Wasserstein convergence; abstract measure theory (search for similar items in EconPapers)
Date: 2024-09
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