A sufficient condition for additively separable functions

Additively separable functions on RN are of the form I;= 1 Ui(Xi). These functions have the obvious property that they are completely separable. That is, the induced orders on the (N 1)-dimensional sets {(x,, . . . , xN): xi0 = c} are independent of c. A natural question is whether the opposite is also true, that is, whether complete separability of a function V implies additive separability. Debreu (1954) proved this to be the case whenever the domain of I/ is a product of intervals rcl x ... x rcN c RN. Wakker (1989) extended this result to ordered cones (i.e., sets in RN where xi 2..* 2x, 20). He also pointed out the importance of the requirement that indifference surfaces are connected. In Segal (1991) I showed that one can relax the assumption that the domain S is an ordered cone and that it is sufficient to require that intersections of the domain of the function with parallel-to-the-axes hyperplanes are connected sets. Chateauneuf and Wakker (1993) extended these results to general product spaces. For further references, see this last paper. See also Blackorby et al. (1978) for a survey of applications of additively separable functions in economics. In this paper I offer a further relaxation of the above mentioned requirements. Intersections of parallel-to-the-axes hyperplanes with the