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

CC BY-SA 4.0
2022-04-25T10:22:31.843Z
2022-04-25T10:22:31.843Z
VARulle
<p>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:
<span class="math-container">$$U(W) = \frac{W^{1-r}}{1-r}$$</span>
But then (1) the utility function is not defined at <span class="math-container">$W=0$</span> whenever <span class="math-container">$r&gt;1$</span>, (2) utility flips from negative to positive when <span class="math-container">$W$</span> goes from positive to negative if <span class="math-container">$r$</span> is an even number.</p>
<p>I read how the definition of relative risk aversion was derived from <a href="https://economics.stackexchange.com/questions/5320/derivation-of-arrow-pratt-risk-aversion-measure">this post</a>, and I am not sure if its physical meaning would allow such a utility function.</p>
<p>So my question is (1) Does such utility function exist? (2) An example if it does.</p>

CC BY-SA 4.0
2022-04-21T12:06:18.083Z
You want to explicitly include negative wealth levels?
VARulle
CC BY-SA 4.0
2022-04-21T13:25:17.650Z
yes, from negative to positive.
Sara
CC BY-SA 4.0
2022-04-22T02:34:40.660Z
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
CC BY-SA 4.0
2022-04-26T11:54:09.967Z
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)
jpfeifer
CC BY-SA 4.0
2022-04-27T03:54:54.680Z
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
<utility><risk-aversion>
Constant relative risk aversion for wealth spanning from negative to positive
Sara
# Constant relative risk aversion for wealth spanning from negative to positive

(Asked by: username_0 on 2022-04-21)

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](https://economics.stackexchange.com/questions/5320/derivation-of-arrow-pratt-risk-aversion-measure), 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.

**Tags:** utility, risk-aversion

> **Comments:**
> **username_1**: You want to explicitly include negative wealth levels?  
> **username_0**: yes, from negative to positive.  
> **username_0**: 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.  
> **username_2**: 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)  
> **username_0**: 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?  

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## 1 answers

### Answer by username_1 on 2022-04-25. Score: 3 | Accepted answer

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 $W>0$.


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