Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 35 / 2 / b / Solution

Constant relative risk aversion coefficient means that $\alpha =-W{U}^{\prime \prime }(W)/U^{\prime }(W)$ does not depend on wealth, for $W>0$ and $U^{\prime }(W)>0$. The defining differential equation integrates asA second integration therefore givesThe additive constant and positive scale do not affect expected utility preferences. Since $U^{\prime \prime }(W)=-\alpha c{W}^{-\alpha -1}$, positive $\alpha$ gives risk aversion, zero gives risk neutrality, and negative $\alpha$ gives risk seeking. These are respectively concave, affine and convex utility functions. The usual CRRA utility class is the positive-$\alpha$ case.