Price majorization and the inverse Lorenz function

Abstract

The paper presents an approach to the measurement of economic disparity in several commodities. We introduce a special view on the usual Lorenz curve and extend this view to the multivariate situation: Given a vector of shares of the total endowments in each commodity, the multivariate inverse Lorenz function (ILF) indicates the maximum percentage of the population by which these shares or less are held. Its graph is the Lorenz hypersurface. Many properties of the ILF are studied and the equivalence of the pointwise ordering of ILFs and the price Lorenz order is established. We also study similar notions for distributions of absolute endowments. Finally, several disparity indices are suggested that are consistent with these orderings. --Multivariate Lorenz order,directional majorization,price Lorenz order,generalized Lorenz function,multivariate disparity indices