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

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!

• absolute or relative risk aversion? – user14471 Sep 29 '17 at 19:52

Take the limit as $\sigma \rightarrow 1$, \begin{align*} \lim_{\sigma \rightarrow 1} \frac{C^{1-\sigma}}{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-\sigma} &= \lim_{\sigma \rightarrow 1} \frac{- C^{1-\sigma} \log C}{-1} \\ &= \log C. \end{align*}