Multivariate Constant Risk Posture

A characterization is given of multivariate utility functions having constant additive risk posture, i.e., those utility functions for which it is attractive to add a random vector at all wealth levels once this is true at some wealth level. It turns out this is so if and only if there exist prices converting all commodities to money and the decision maker has constant absolute risk posture toward money (or equivalently his utility toward money is exponential or linear). The approach is axiomatic and considers substantially weaker regularity conditions of the utility functions than have been considered heretofore, namely measurability or boundedness on some open set instead of differentiability or continuity and strict monotonicity. Similar results are achieved for constant proportional risk posture. (Author).