Representative endowments and uniform Gini orderings of multi-attribute inequality

For the comparison of inequality in multiple attributes the use of generalized Gini indices is proposed. Spectral social evaluation functions are used in the multivariate setting, and Gini dominance orderings are introduced that are uniform in attribute weights. Classes of spectral evaluators are considered that are parameterized by their aversion to inequality. Then a set-valued representative endowment is defined that characterizes $d$-dimensioned inequality. It consists of all points above the lower border of a convex compact in $R^d$, while the pointwise ordering of such endowments corresponds to uniform Gini dominance. Properties of uniform Gini dominance are derived, including relations to other orderings of $d$-variate distributions such as usual multivariate stochastic order and convex order. The multi-dimensioned representative endowment can be efficiently calculated from data; in a sampling context, it consistently estimates its population version.