Using the formulas for ARA and RRA

(a) Show that exponential utility: \(U(x)=1-e^{-c x}\) for some \(c>0\), has constant ARA (CARA).

(b) For power utility: \(U(x)=\left(x^{1-\alpha}-1ight) /(1-\alpha)\) for \(\alpha eq 1\)

i. Show that power utility is concave (risk averse) when \(\alpha>0\) and convex (risk seeking) when \(\alpha<0\).

ii. What type of behavior is modeled when \(\alpha=0\) ?

iii. Show that power utility has constant RRA (CRRA).

iv. Using L'Hôpital's rule, show that power utility reduces to \(\log\) utility, \(U(x)=\ln (x)\), when \(\alpha=1\).