Multidimensional Inequality Measurement via Optimal Transport

The Lorenz curve and Gini index are standard tools for the evaluation of inequality in one dimension. However, inequality is inherently multi-dimensional. Extending the Lorenz curve and Gini index to a multidimensional context has proved controversial. This paper proposes a new multivariate extension based on multivariate rearrangements of optimal transport theory, which shares many of the desirable properties of their univariate counterparts. In particular, the corresponding multivariate inequality ordering is equivalent to preference by any social planner with inequality averse multivariate rank dependent social evaluation functional.