# Constant relative risk aversion for wealth spanning from negative to positive

I am modeling scenarios that could involve wealth for all real numbers and I am assuming constant relative risk aversion. I need to model the scenarios for different risk aversion levels, but I can't find a utility function that is suitable. I read some sources and it seems with constant relative risk aversion, we usually use the utility function: $$U(W) = \frac{W^{1-r}}{1-r}$$ But then (1) the utility function is not defined at $$W=0$$ whenever $$r>1$$, (2) utility flips from negative to positive when $$W$$ goes from positive to negative if $$r$$ is an even number.

I read how the definition of relative risk aversion was derived from this post, and I am not sure if its physical meaning would allow such a utility function.

So my question is (1) Does such utility function exist? (2) An example if it does.

• You want to explicitly include negative wealth levels? Apr 21, 2022 at 12:06
• yes, from negative to positive.
– Sara
Apr 21, 2022 at 13:25
• The commonly seen utility function $U(W) = \frac{W^{1-r}}{1-r}$ is indeed undefined at $0$ if $1-r<0$. That's one of the reasons why I am asking for a utility function.
– Sara
Apr 22, 2022 at 2:34
• I don't understand why utility would depend on wealth. Usually, we derive utility from consumption that wealth enables. While wealth can be negative, consumption cannot be negative. It's the period income that allows having strictly positive consumption (and paying down negative wealth) Apr 26, 2022 at 11:54
• The paper cited in the post I referenced (and also the example in wikipedia) use bet/gample scenarios. I don't think in these scenarios consumption is required. It looks like we can derive a utility function if we can specify the amount of risk we are willing to bear for an increment in wealth?
– Sara
Apr 27, 2022 at 3:54

I'm afraid the answer is no. Let's take any continuous and strictly increasing utility function $$U(W)$$ which is twice differentiable almost everywhere on the reals. Constant relative risk aversion (CRRA) is defined as $$-W\frac{U''}{U'}\equiv r$$. Restricting to $$r>1$$ and $$W>0$$ for the moment implies $$\frac{U''}{U'}=-\frac{r}{W}$$ and thus $$\left(\log U'\right)'=-r(\log W)'=(\log W^{-r})'$$, implying $$\log U'=c+\log W^{-r}$$ for some constant $$c$$. This gives $$U'=CW^{-r}$$ for some constant $$C>0$$ and finally $$U=C\frac{W^{1-r}}{1-r}+K$$ for some constants $$C>0$$ and $$K$$.
Thus, up to a positive affine transformation, your "usual" utility function $$U(W)=\frac{W^{1-r}}{1-r}$$ is indeed the unique one exhibiting CRRA. But this function goes to $$-\infty$$ for $$W\searrow 0$$, and hence it cannot be continued to negative values of $$W$$. So you'll have to restrict your utility functions to either $$r\le 1$$ or to $$W>0$$.