 Solution

Substitution of $F(n+1,x)=B_{n+1}{e}^{-A_{n+1}x}$ separates the infima over investment and consumption. The investment contributes $\gamma ={\inf }_{h}E(e^{-hR})$ after rescaling, and the first-order condition preserves exponential form, with $A_n=A_{n+1}/(1+A_{n+1})$.