Lorenz map, inequality ordering and curves based on multidimensional rearrangements

TL;DR

This paper introduces a multivariate Lorenz curve extension using optimal transport, establishing a new inequality ordering and tools for visualizing multidimensional inequality, with applications to income and wealth data.

Contribution

It develops a multivariate Lorenz map based on vector quantiles, creating a new majorization order and inequality measures for multidimensional resource allocation.

Findings

Proposes a multivariate Lorenz curve using optimal transport theory.

Defines a new multivariate majorization order based on Lorenz maps.

Introduces the inverse Lorenz function for visualizing multidimensional inequality.

Abstract

We propose a multivariate extension of the Lorenz curve based on multivariate rearrangements of optimal transport theory. We define a vector Lorenz map as the integral of the vector quantile map associated with a multivariate resource allocation. Each component of the Lorenz map is the cumulative share of each resource, as in the traditional univariate case. The pointwise ordering of such Lorenz maps defines a new multivariate majorization order, which is equivalent to preference by any social planner with inequality averse multivariate rank dependent social evaluation functional. We define a family of multi-attribute Gini index and complete ordering based on the Lorenz map. We propose the level sets of an Inverse Lorenz Function as a practical tool to visualize and compare inequality in two dimensions, and apply it to income-wealth inequality in the United States between 1989 and 2022.