A univariate utility function u(z) is usually required to satisfy the conditions u′(z)≥0, u′′(z)≤0, i.e. the function should be nondecreasing and concave. Some authors, however, require that the more restrictive conditions: u′(z)≥0, u″(z)≥0, u″′(z)≥0, u4≤0, … should be satisfied. Given a multivariate utility function u(z1,…zs), we may require, in agreement with the above weaker conditions, that it be nondecreasing in each variable and concave in all variables. On the other hand, maintaining these conditions, we may impose on it the additional requirement that all of its partial derivatives of odd (even) order should be nonnegative (nonpositive). In this paper our objective is twofold. First, we construct a multivariate utility function that satisfies the above-mentioned stronger conditions. Secondly, given the random wealths X1, …,Xs, we give lower and upper bounds for the expectation of u(X1, …, Xs) when the random variables are discrete with finite supports, their joint distribution is unknown but known are some of their multivariate moments.
