Working Paper: CEPR ID: DP1651
Authors: Lars Tyge Nielsen
Abstract: This paper shows that a strictly increasing and risk averse utility function with decreasing absolute risk aversion is necessarily differentiable with a positive and absolutely continuous derivative. The cumulative absolute risk aversion function, which is defined as the negative of the logarithm of the marginal utility, will also be absolutely continuous. Its density, called the absolute risk aversion density, is a generalization of the coefficient of absolute risk aversion, and it is well defined almost everywhere. A strictly increasing and 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 convenient characterization of all such utility functions. Analogues of all the results also hold for increasing absolute risk aversion, as well as for increasing and decreasing relative risk aversion.
Keywords: risk aversion; decreasing absolute risk aversion
JEL Codes: D81
Edges that are evidenced by causal inference methods are in orange, and the rest are in light blue.
| Cause | Effect |
|---|---|
| strictly increasing and risk-averse utility function with decreasing absolute risk aversion (D11) | differentiability of the utility function (D11) |
| differentiability of the utility function (D11) | characteristics of the utility function (D11) |
| cumulative absolute risk aversion function is increasing and concave (D11) | characterization of utility functions (D11) |
| decreasing absolute risk aversion (D11) | decreasing absolute risk aversion density (D11) |
| wealth increases (D31) | decreasing risk aversion behavior of investors (G40) |