Weakly Monotonic Preference Relations Representable by Concave Utility Functions - It's Easy

In this paper we provide conditions under which a preference relation can be represented by concave utility functions. The condition follows naturally from a proof of a theorem about representability of continuous and weakly monotonic preference relations by continuous and weakly increasing utility functions due to Wold (1943). For continuous and homothetic (hence weakly monotonic) preference relations our sufficient condition for the existence of a numerical representation by a concave utility function is also a necessary condition for such a numerical representation. In fact we are able to obtain a necessary condition for numerical representation of a homothetic preference relation by a concave utility function which we call global concavifiability and which is much stronger than concavifiability. On the way, we pick up the result, that all homothetic and continuous preference relations which are convex are numerically representable by concave utility functions.