Representative endowments and uniform Gini orderings of multi-attribute welfare

For the comparison of inequality and welfare in multiple attributes the use of generalized Gini indices is proposed. Individual endowment vectors are summarized by using attribute weights and aggregated in a spectral social evaluation function. Such functions are based on classes of spectral functions, ordered by their aversion to inequality. Given a spectrum and a set $P$ of attribute weights, a multivariate Gini dominance ordering, being uniform in weights, is defined. If the endowment vectors are comonotonic, the dominance is determined by their marginal distributions; if not, the dependence structure of the endowment distribution has to be taken into account. For this, a set-valued representative endowment is introduced that characterizes the welfare of a $d$-dimensioned distribution. It consists of all points above the lower border of a convex compact in $\R^d$, while the set ordering of representative endowments corresponds to uniform Gini dominance. An application is given to the welfare of 28 European countries. Properties of $P$-uniform Gini dominance are derived, including relations to other orderings of $d$-variate distributions such as convex and dependence orderings. The multi-dimensioned representative endowment can be efficiently calculated from data. In a sampling context, it consistently estimates its population version.