Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 39 / 3 / i

Choose the utility function primitive $U(c)=c^{1-R}/(1-R)$. Any other primitive with the same derivative differs by a constant $C$; this adds $C/\rho$ to the value function and does not alter the policy. Put $p=1-R$ and let ${\pi }_t$ be the stock fraction. The self-financing portfolio equation, including consumption withdrawals, isFor a time-independent infinite-horizon value function $V$, the Hamilton-Jacobi-Bellman equation isScaling initial portfolio wealth and all consumption rates by the same positive number preserves admissibility and scales utility function by its $p$th power. This homogeneity gives $V(w)=K{w}^p/p$. In particular,The consumption first-order condition $c^{-R}=V^{\prime }(w)$ and stock first-order condition $w(\mu -r)V^{\prime }(w)+{\sigma }^2\pi w^2{V}^{\prime \prime }(w)=0$ giveThe second derivatives of the optimized expressions are negative, so these are their global maximizers. Writing ${\gamma }_M=K^{-1/R}$ and substituting into the Hamilton-Jacobi-Bellman equation givesThereforeThe given inequality is precisely ${\gamma }_M>0$, so the Merton consumption-investment problem value is $V(w)={\gamma }_M^{-R}{w}^{1-R}/(1-R)$. This derives the requested policy; no additional verification argument is needed here.