Lorenz Surfaces Based on the Sarmanov–Lee Distribution with Applications to Multidimensional Inequality in Well-Being

José María Sarabia,Vanesa Jorda

doi:10.3390/math8112095

Abstract

The purpose of this paper is to derive analytic expressions for the multivariate Lorenz surface for a relevant type of models based on the class of distributions with given marginals described by Sarmanov and Lee. The expression of the bivariate Lorenz surface can be conveniently interpreted as the convex linear combination of products of classical and concentrated univariate Lorenz curves. Thus, the generalized Gini index associated with this surface is expressed as a function of marginal Gini indices and concentration indices. This measure is additively decomposable in two factors, corresponding to inequality within and between variables. We present different parametric models using several marginal distributions including the classical Beta, the GB1, the Gamma, the lognormal distributions and others. We illustrate the use of these models to measure multidimensional inequality using data on two dimensions of well-being, wealth and health, in five developing countries.

Highlights

Even though a consensus has slowly emerged among scholars that economic inequality is a multidimensional construct, most of the early work in this area relies almost exclusively on income variables
In this paper, using the definition proposed by Arnold [16], we derive analytic expressions for the bivariate Lorenz surface for different formulations of the underlying bivariate distribution
Using the methodology described in the previous section we have estimated the parameters of the Sarmanov–Lee distribution considering the beta distribution for modelling marginal distributions (Table 1)

Summary

Introduction

Even though a consensus has slowly emerged among scholars that economic inequality is a multidimensional construct, most of the early work in this area relies almost exclusively on income variables. Multidimensional measures that consider both types of inequalities have been proposed by [6,7,8,9,10,11,12,13]. Depending on their properties, different measures might reveal completely different results. When this happens, graphical methods can be a powerful tool to explore where the redistributive movements take place at different parts of the distribution. The three existing definitions of the multivatiate LC were proposed by Taguchi [14,15], Arnold [16] and Koshevoy and Mosler [17], who introduced the concepts of Lorenz zonoid and Gini zonoid index

Objectives

Results

Conclusion