Solutions to three functional equations arising from different ways of measuring utility

Utility of gains (losses) can be measured in four distinct ways: riskless vs risky choices and gains (losses) alone vs the gain-loss trade-off. Conditions forcing these measures all to be the same lead to functional equations, three of which are F-1[F(X) + F(-Y)]Z = F-1[F(XZ) + F(-YZ)] (i) (F:] -k, k'[-->] - K, K'[; k, k', K, K' > 0) F(X - R)[1 - F(Y)] + F(Y) = F[F-1(F(X)[1 - F(Y)] + F(Y)) - S] (ii) (F: [0,1[--> [0,1[) F-1[F(X) + F(Y) - F(X)F(Y)]Z = F-1[F(XZ) + F[YP(X, Z)] - F(XY)F[YP(X, Z)]] (iii) (F: [0, 1[--> [0, 1[, P: [0, 1[ x [0, 1] --> [0, 1]). We determine all strictly increasing, surjective (and thus continuous) solutions of (i) and (ii) and all strictly increasing, sujective solutions of (iii) that are differentiable on [0, 1[ as are their inverses (thus, F' not equal 0 on ]0, 1[). (C) 1996 Academic Press, Inc.