Representative endowments and uniform Gini orderings of multi-attribute welfare (2103.17030v4)

Abstract: 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.