How is the utility function with constant relative risk-aversion obtained?

Question

In this slide, it says that constant relative risk-Aversion utility function have this form.

$u(x) = \frac{1}{1-b} x^{1-b}$ for $b≠1$

$u(x) = In(x)$ for $b=1$

When I tried to derive the utility function from $b = -x \frac{u''(x)}{u'(x)}$ (the representation of CRRA), I got $u(x) = x^{1-b}$.

While I understand that preference will not be changed by linear transformation of the utility function, I wonder why is it necessary to make the utility function be $u(x) = \frac{1}{1-b} x^{1-b}$ for $b≠1$.

Also how is the "$u(x) = In(x)$ for $b=1$" obtained?

Answer 1

In the slide, we're given the marginal utility (or the derivative of the utility function) as $m(x) = x^{-b}$. The utility function whose derivative is $m(x)$ is

\begin{eqnarray*} u(x) = \int m(x) dx = \int x^{-b} dx = \begin{cases} \frac{1}{1- b} x^{1-b} & \text{when } b \neq 1 \\ \ln x & \text{when } b = 1 \end{cases} \end{eqnarray*}

Verify that $u'(x) = m(x)$.