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We propose a multivariate extension of the Lorenz curve based on multivariaterearrangements of optimal transport theory. We define a vector Lorenz map asthe integral of the vector quantile map associated to a multivariate resourceallocation. Each component of the Lorenz map is the cumulative share of eachresource, as in the traditional univariate case. The pointwise ordering of suchLorenz maps defines a new multivariate majorization order. We define amulti-attribute Gini index and complete ordering based on the Lorenz map. Weformulate income egalitarianism and show that the class of egalitarianallocations is maximal with respect to our inequality ordering over a largeclass of allocations. We propose the level sets of an Inverse Lorenz Functionas a practical tool to visualize and compare inequality in two dimensions, andapply it to income-wealth inequality in the United States between 1989 and2019.

Lorenz map, inequality ordering and curves based on multidimensional rearrangements