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!

  • $\begingroup$ absolute or relative risk aversion? $\endgroup$ – 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*}

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