Merton consumption constant (${\gamma }_M=[\rho -(1-R)(r+{\kappa }^2/(2R))]/R$)

In the infinite-horizon Merton consumption-investment problem with constant relative risk aversion utility, this is the optimal ratio of consumption to wealth. A finite value requires ${\gamma }_M>0$. The state-price budget constraint then fixes consumption's multiplier because $\mathbb{E}[{\xi }_t(y{e}^{\rho t}{\xi }_t{)}^{-1/R}]=y^{-1/R}{e}^{-{\gamma }_{M}t}$, whose time integral is $y^{-1/R}/{\gamma }_M$. When this integral diverges, the value is $+\infty$ for $0<R<1$ and $-\infty$ for $R>1$.