(i) Show that a utility function (u: mathbb{R} ightarrow mathbb{R}) is characterized by a constant relative risk aversion coefficient different from 1 if and

(i) Show that a utility function \(u: \mathbb{R} \rightarrow \mathbb{R}\) is characterized by a constant relative risk aversion coefficient different from 1 if and only if \(u(x)=\alpha x^{1-b}+\beta\) for \(\alpha>0\) and \(\beta \in \mathbb{R}\), with \(b eq 1\).

(ii) Show that a utility function \(u: \mathbb{R} \rightarrow \mathbb{R}\) is characterized by a constant relative risk aversion coefficient equal to one if and only if \(u(x)=\alpha \log (x)+\beta\) for \(\alpha>0\) and \(\beta \in \mathbb{R}\).