A couple of functional equations applied to utility theory

Abstract

We show how characterizations of separable and additive representations of rank-dependent expected utility and of rank- independent expected utility lead to the functional equation f(v) = f(vw) + f(g-1[g(v)q(w)]) (v ϵ [0, k[, w ϵ [0, 1]) or to its particular case g = identity, respectively. These equations are solved here without any regularity condition other than the nonnegativity of f. Nothing needs to be assumed about q. In order tahat g[-1] be defined, one postulates in the general case that g is strictly monotonic and continuous.