By Lars Tyge Nielsen

*Economic Theory* 25 (2005), 203-215

### Abstract

This paper defines decreasing absolute risk aversion in purely behavioral terms without any assumption of differentiability and shows that a strictly increasing and risk averse utility function with decreasing absolute risk aversion is necessarily differentiable with an absolutely continuous derivative. A risk averse utility function has decreasing absolute risk aversion if and only if it has a decreasing absolute risk aversion density, and if and only if the cumulative absolute risk aversion function is increasing and concave. This leads to a characterization of all such utility functions. Analogues of these results also hold for increasing absolute and for increasing and decreasing relative risk aversion.

Keywords and Phrases: Absolute risk aversion, relative risk aversion, decreasing risk aversion, increasing risk aversion, cumulative absolute risk aversion, cumulative relative risk aversion.

JEL Classifiction Numbers: D81