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.
翻译:我们根据最佳运输理论的多变重新排列,提出洛伦茨曲线的多变扩展。我们定义了矢量洛伦兹地图,作为与多变资源分配相关的矢量四分位分布图的组成部分。洛伦兹地图的每个组成部分都是每种资源的累积份额,如传统的单轨情况。这种洛伦斯地图的点排序定义了一个新的多变主要顺序。我们定义了多归性吉尼指数,并根据洛伦兹地图进行了完整排序。我们制定了收入平等主义,并表明平等分配等级对于我们为大类分配订购的不平等是最大的。我们提议了逆向洛伦兹函数的等级,作为在两个方面可视化和比较不平等的实用工具,并将其应用于1989至2019年美国的收入-财富不平等。