Multiple-Attribute Lorenz Functions and Gini Indices: A Measure Transportation Approach

Based on measure transportation ideas and the related concepts of quantile functions and regions, we propose multiple-output generalizations of the traditional univariate concepts of Lorenz and concentration functions, and the related Gini and Kakwani coefficients. These new concepts have a natural interpretation, either in terms of contributions of quantile regions to the expectation of some variable of interest, or in terms of the physical notions of work and energy, which sheds new light on the nature of economic and social inequalities. When based on center-outward quantile regions, the proposed concepts pave the way to a statistically sound definition, based on multiple attributes, of the notion, so far limited to bivariate characterizations, of <i>middle-class</i>—a notion of practical importance in various socio-economic and political contexts.