Introduction

The question of the representation of a convex preference preorder by a concave utility function was first raised and answered by de Finetti (1949), and further studied by Fenchel (1953, 1956), Moulin (1974), and by Kannai in the forthcoming article “Concavifiability and constructions of concave utility functions” which also discusses the problem of least concave utility functions. To illustrate the value of such a concave representation by one example, we consider an exchange economy ℰ whose consumers have convex preferences, and, following Scarf (1967), we associate with the economy ℰ a game without side payments in coalition form. If the preferences of each consumer are represented by a concave utility function, then the characteristic set of utility vectors of each coalition is convex, as in the original definition of Aumann-Peleg (1960). The convexity of these characteristic sets permits, for instance, a simplification [Scarf (1965) and Ekeland (1974); see also the related article of Shapley (1969)] of the proof of the non-emptiness of the core of Scarf (1967).