<p>Take the limit as <span class="math-container">$\sigma \rightarrow 1$</span>,
<span class="math-container">\begin{align*}
\lim_{\sigma \rightarrow 1} \frac{C^{1-\sigma}-1}{1-\sigma}.
\end{align*}</span>
This will then results in log utility. Calculating the limit goes as follows.
Since plugging in <span class="math-container">$\sigma = 0$</span> would give <span class="math-container">$\frac 0 0$</span>, we use L'Hopital's rule.
This gives us
<span class="math-container">\begin{align*}
\lim_{\sigma \rightarrow 1} \frac{C^{1-\sigma}-1}{1-\sigma}
  &amp;= \lim_{\sigma \rightarrow 1} \frac{- C^{1-\sigma} \log C}{-1} \\
  &amp;= \log C.
\end{align*}</span></p>

CC BY-SA 4.0
2017-09-29T05:03:45.657Z
2024-12-04T17:47:01.567Z
2024-12-04T17:47:01.567Z
jmbejara
<p>How to compute the utility function when risk aversion is equal to 1?
If the utility function of consumption is set as,</p>

<p>$\frac{C^{1-\sigma} - 1}{1-\sigma}$.
Is it meaningful as $\sigma$ is equal to 1?
How to prove it if so? Thanks a lot!</p>

<utility>
How to compute the utility function when risk aversion is equal to 1?
Carl
# How to compute the utility function when risk aversion is equal to 1?

(Asked by: username_0 on 2017-09-29)

How to compute the utility function when risk aversion is equal to 1?
If the utility function of consumption is set as,

$\frac{C^{1-\sigma} - 1}{1-\sigma}$.
Is it meaningful as $\sigma$ is equal to 1?
How to prove it if so? Thanks a lot!

**Tags:** utility

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## 1 answers

### Answer by username_1 on 2017-09-29. Score: 6 | Accepted answer

Take the limit as $\sigma \rightarrow 1$,
\begin{align\*}
\lim\_{\sigma \rightarrow 1} \frac{C^{1-\sigma}-1}{1-\sigma}.
\end{align\*}
This will then results in log utility. Calculating the limit goes as follows.
Since plugging in $\sigma = 0$ would give $\frac 0 0$, we use L'Hopital's rule.
This gives us
\begin{align\*}
\lim\_{\sigma \rightarrow 1} \frac{C^{1-\sigma}-1}{1-\sigma}
&= \lim\_{\sigma \rightarrow 1} \frac{- C^{1-\sigma} \log C}{-1} \\
&= \log C.
\end{align\*}


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