Characterizing multidimensional inequality measures which fulfil the Pigou-Dalton bundle principle

In the unidimensional setting, the well known Pigou-Dalton transfer principle is the basic axiom to order distribution in terms of inequality. This axiom has a number of generalizations to the multidimensional approach which have been used to derive multidimensional inequality measures. However, up to now, none of them has assumed the Pigou-Dalton bundle dominance criterion, introduced by Fleurbaey and Trannoy (2003) although this principle captures the basic idea of the original Pigou-Dalton transfer principle, demanding that also in the multidimensional context “a transfer from a richer person to a poorer one decreases inequality”. Assuming this criterion the aim of this paper is to characterize multidimensional inequality measures. For doing so, firstly we derive the canonical forms of multidimensional aggregative inequality measures, both relative and absolute, which fulfil this property. Then following the Atkinson and Kolm-Pollak approaches we identify sub-families whose underlying social evaluation functions are separable. The inequality measures we derive share their functional forms with other parameter families already characterized in the literature, the major difference being the restrictions upon the parameters. Nevertheless, we show that it is not necessary to give up any of the usual requirements to assume the Pigou-Dalton bundle criterion. Thus, in empirical applications it makes sense to choose measures that also fulfil this principle.