Constant relative risk aversion for wealth spanning from negative to positive

Question

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.

Answer 1

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 $W>0$.