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Lorenz Map, Inequality Ordering and Curves Based on Multidimensional Rearrangements

Y. FAN, M. Henry, B. Pass, J.A. Rivero


Montreal Econometrics Seminar 2022-2023
joint with the departments of economics of the universities of Montréal, Québec à Montréal, Concordia and McGill and with CIRANO

Organizers : Marine Carrasco (U. of Montreal) and Saraswata Chaudhuri (McGill U.)

* Virtual Seminar. Please contact an organizer for Zoom login information.

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 to 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. We define a multi-attribute Gini index and complete ordering based on the Lorenz map. We formulate income egalitarianism and show that the class of egalitarian allocations is maximal with respect to our inequality ordering over a large class of allocations. 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 2019.


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