Abstract 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.

This paper derives conditions under which preferences and technology are nonparametrically identified in hedonic equilibrium models, where products are differentiated along more than one dimension and agents are characterized by several dimensions of unobserved heterogeneity. With products differentiated along a quality index and agents characterized by scalar unobserved heterogeneity, single crossing conditions on preferences and technology provide identifying restrictions in Ekeland, Heckman and Nesheim (2004) and Heckman, Matzkin and Nesheim (2010). We develop similar shape restrictions in the multi-attribute case. These shape restrictions, which are based on optimal transport theory and generalized convexity, allow us to identify preferences for goods differentiated along multiple dimensions, from the observation of a single market. We thereby derive nonparametric identification results for nonseparable simultaneous equations and multi-attribute hedonic equilibrium models with (possibly) multiple dimensions of unobserved heterogeneity. One of our results is a proof of absolute continuity of the distribution of endogenously traded qualities, which is of independent interest.