What do normative indices of multidimensional inequality really measure?

• Normative multidimensional inequality indices measure both inequity and inefficiency.
• We provide a decomposition of inequality into inequity and inefficiency.
• Strikingly, the inequity component turns out to be a so-called two-stage measure.
• The decomposition clarifies controversies surrounding two prominent transfer axioms.
• An application to inequality in human development illustrates the analysis.

We argue that normative indices of multidimensional inequality do not only measure a distribution's extent of inequity (i.e., the gaps between the better-off and the worse-off), but also its extent of inefficiency (i.e., the non-realized mutually beneficial exchanges of goods). We provide a decomposition that allows us to quantify these two parts of inequality. Strikingly, the inequity component turns out to be a two-stage measure, that is, a measure that applies a unidimensional inequality measure to the vector of individual well-being levels. The decomposition also clarifies existing controversies surrounding two prominent transfer axioms, viz., uniform majorization and correlation increasing majorization. An application to inequality in human development illustrates the analysis.