Consider the HARA class of utility functions introduced in Sect. 2.2 (left(r_{u}^{a}(x)=1 /(a+b x) ight.), for some

Consider the HARA class of utility functions introduced in Sect. 2.2 \(\left(r_{u}^{a}(x)=1 /(a+b x)\right.\), for some \(\left.a, b \in \mathbb{R}\right)\). Verify that:

(i) a HARA utility function shows a decreasing coefficient of absolute risk aversion if and only if \(b>0\) and a decreasing coefficient of relative risk aversion if and only if \(a<0\);

(ii) the three specifications of utility function \(\log (x+a)\) for \(b=1,-a \exp (-x / a)\) for \(b=0\) and \(\frac{1}{b-1}(a+b x)^{\frac{b-1}{b}}\) in the remaining cases provide an exhaustive representation of the HARA class of utility functions. Determine the domain of these functions.