= Solution
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, is
$$
\boxed{dw_t=\bigl[rw_t+\pi_tw_t(\mu-r)-c_t\bigr]dt+\sigma\pi_tw_t\,dW_t.}
$$
For a time-independent infinite-horizon <value function> $V$, the <Hamilton-Jacobi-Bellman equation> is
$$
\rho V(w)=\sup_{c>0,\,\pi}\left\{\frac{c^p}{p}+\bigl[rw+\pi w(\mu-r)-c\bigr]V'(w)+\frac12\sigma^2\pi^2w^2V''(w)\right\}.
$$
Scaling 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 $V(w)=Kw^p/p$. In particular,
$$
V'(w)=Kw^{-R},\qquad V''(w)=-RKw^{-R-1}.
$$
The <consumption> <first-order condition> $c^{-R}=V'(w)$ and <stock> <first-order condition> $w(\mu-r)V'(w)+\sigma^2\pi w^2V''(w)=0$ give
$$
\pi_M=\frac{\mu-r}{R\sigma^2},\qquad c=K^{-1/R}w.
$$
The <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> gives
$$
\rho=R\gamma_M+(1-R)\left(r+\frac{(\mu-r)^2}{2R\sigma^2}\right).
$$
Therefore
$$
\boxed{\pi_M=\frac{\mu-r}{R\sigma^2},\qquad
\gamma_M=\frac{\rho+(R-1)\left(r+\frac{(\mu-r)^2}{2R\sigma^2}\right)}{R},\qquad
c_t^*=\gamma_Mw_t.}
$$
The 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.
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