Nash social welfare orderings

Abstract

The paper considers the problem of description of social welfare orderings (SWOs) on the entire utility (Euclidean) space R n satisfying Scale Independence. These orderings and the functions representing them are called the Nash social welfare orderings and the Nash social welfare functions (SWFs), respectively. The more properties of the SWOs into consideration are Strong Pareto and two variants of weakening for continuity: the ‘orthant’ continuity, meaning continuity of a SWO on each separate orthant of the utility space, and ‘upper preserving in the limit’ (UPL) that is equivalent to existence of maximal elements for a SWO on each compact set. The complete characterization of the Nash SWOs satisfying these continuity conditions is given. For a fixed arbitrary orthant of the utility space each Strong Pareto Nash SWO is representable by a Nash (Cobb–Douglas) type SWFs (if the SWO is orthantly continuous) or by a lexicographical ordering defined by a collection of such functions (if the SWO satisfies only UPL). Vectors from the different orthants are compared either by the orthant rules (if they coincide for the orthants), or by a linear ordering on the set of orthants.