On the Separability of the Quasi Concave Closure of an Additively Separable Function

Let 2 be a binary relation on R’!+. The convexed relation 2 is defined by xky iff Vz[y~conv{w:w~z}]*[x~conv{w:w~z)] [see Hildenbrand (1974)]. Starr (1969), first formally presented this relation, which was used by him and by others in equilibrium analysis of markets with non-convex preference relations [see for example Shaked (1976)]. The question arises which properties of the unconvexed relation hold also for the convexed one. Segal (1983) proved that if the relation 2 on R’!+ can be represented by an additively separable convex function of the form U(x,, . . . , x,) =I ui(xi) where Ul,. . . > u, are all convex functions, then the convexed relation 2 can be represented by an additively separable function if and only if there are U: R+R, d, ,..., d,>Q, and b, ,..., b, E R such that for every i, Ui(xi) = u(d,x,) + bi. Let U be a representation of 2. Denote by U” the quasi concave closure of U, the function which represents 2 and which is equal to U where possible (for an exact definition see section 2 below). This paper gives a full characterization of all the additively separable functions whose quasi concave closure is a transformation of an additively separable function. The above conditions are necessary and sufficient ones if U has first order derivatives and if it is not quasi concave, even if it is not a convex function. In the general case these conditions are not necessary. Necessary and sufficient conditions for this case are presented in section 5.