A Theorem on the Additivity of the Quasi-concave Closure of an Additive Convex Function*

Most of utility theory depends on the assumption that the preference relation is convex, i.e., that for all x the set {ylykx} is a convex set. However, even if we do not assume that the preference relation is convex, some of the results of utility theory still hold. To obtain these results, it is convenient to define the ‘convexed’ relation 2‘ in the following way [see Hildenbrand (1974)]: Given a relation 2 define the convexed relation 2’ by x&‘y iff Vz [yEConv{wlw~z}] [ C => XE onv{wlwkz}], where ConvA means the convex hull of the set A. However, not all the properties of the unconvexed relation hold for the convexed one as well. In this paper it will be assumed that the relation 2 can be represented by an additive convex function of the form 1 ui(xi) (x,, . . . , x, E R), and it will be shown under what conditions the relation 2’ can be represented by an additive function as well. Note that since the utility function is not a quasi-concave one, we shall not be able to use the technique which was used by Houthakker (1960) to show the conditions under which both direct and indirect utility functions are additive.