Let us consider a three times differentiable utility function (u) : (mathbb{R} ightarrow mathbb{R}). (i) Prove that, if (u) is increasing and DARA, then

Let us consider a three times differentiable utility function \(u\) : \(\mathbb{R} \rightarrow \mathbb{R}\).

(i) Prove that, if \(u\) is increasing and DARA, then \(u^{\prime \prime \prime}(x)>0\) for every \(x \in \mathbb{R}\).

(ii) For an increasing and concave utility function \(u: \mathbb{R} \rightarrow \mathbb{R}\), define the degree of absolute prudence as \(p_{u}^{a}(x):=-u^{\prime \prime \prime}(x) / u^{\prime \prime}(x)\), for \(x \in \mathbb{R}_{+}\). Show that the utility function \(u\) is DARA if and only if \(p_{u}^{a}(x)>r_{u}^{a}(x)\), for every \(x \in \mathbb{R}\).

(iii) Show that \(p_{u}^{a}(x)>0\), for all \(x \in \mathbb{R}\), is a necessary condition to have a DARA utility function.