An application of Back's Theorem to an ordering of distributions of wellbeing.

Back's Theorem (K. Back, Concepts of similarity for utility functions. Journal of Mathematical Economics 15, 129-142 (1986)) proves the existence of jointly continuous utility functions for a family of closed partial preorder defined on closed subsets of a locally compact, second countable commodity set. In this paper we apply Back's Theorem to an ordering of distributions of wellbeing when the population varies. The preoders are assumed to be closed and not necessarily total and they are defined on closed subsets of distributions. In fact, pairs of distributions may not be comparable or their comparison could be not meaningful. In this setting, Back's Theorem allows us to establish the existence of jointly continuous utility functions, that is, utility functions that depend continuously on the population, the distribution set and the preference relation. Our theorem generalizes Broome's result (J.Broome, Representing an ordering when the population varies., Social Choice and Welfaire, 20, (2003) 243-246).