Price-dispersed Preferences and C’ Mean Demand

The purpose of this article is to present a class of consumption sectors which exhibit a continuously differentiable mean demand although individual preferences are not assumed to be convex. As we have argued in an earlier paper [Dierker et al. (1980b, introduction)], several economic questions require mean demand to be a continuously differentiable function and not only to be a continuous one. The literature on smoothing demand by aggregation shows that Co mean demand functions can be obtained by methods which are not appropriate to obtain a C’ mean demand. Studying the differentiability of mean demand one encounters difficult conceptual problems which do not occur in the study of Co mean demand. In the present paper we present a class of consumption sectors, i.e., of distributions of consumers’ characteristics, with C’ mean demand. Presenting this class we use assumptions which are not generic. We think, though, that some of the methods used here are promising tools for the development of a more satisfactory theory of smoothing demand. It is essential for the purpose of smoothing demand by aggregation in the presence of preference non-convexities that preferences are sufficiently dispersed. Since in a Euclidean space Lebesgue measure is often used to formalize dispersion, one is tempted to use Lebesgue measure in order to formulate the notion of dispersion on the space of preferences. This idea has led to the socalled parametric approach in which consumers are supposed to have characteristics in a finite dimensional space [see, in particular, Araujo and Mas-Cole11 (1978), Hildenbrand (1980), Sondermann (1975, 1976, 1980) and Yamazaki (1980)].