= Solution
Define the positive constant
$$
k=\rho+(R-1)\left(\alpha-\frac12R\sigma^2\right).
$$
Since $S=\delta/k$, the <stock> price has capital-appreciation dynamics $dS/S=\alpha dt+\sigma dW$. But each share also pays the flow $\delta=kS$. Its <dividend-inclusive portfolio return> therefore has <drift>
$$
\mu_{\mathrm{total}}=\alpha+k.
$$
This is the <drift> to use in the <Merton consumption-investment problem>; using only capital appreciation would omit an actual source of <portfolio wealth>.
At the proposed bank rate,
$$
\mu_{\mathrm{total}}-r^*
=\alpha+k-\left[\rho+R\alpha-\frac12(1+R)R\sigma^2\right]
=R\sigma^2.
$$
Thus the optimal <stock> fraction from part (i) is
$$
\boxed{\pi_M=1.}
$$
Moreover,
$$
r^*+\frac{(\mu_{\mathrm{total}}-r^*)^2}{2R\sigma^2}
=\rho+R\alpha-\frac12R^2\sigma^2.
$$
Substituting this into the optimal <consumption> coefficient yields
$$
\gamma_M=\frac{\rho+(R-1)\left(\rho+R\alpha-\frac12R^2\sigma^2\right)}{R}
=\rho+(R-1)\left(\alpha-\frac12R\sigma^2\right)=k>0.
$$
Hence the finite-value condition in part (i) is automatically satisfied.
The agent's endowment of the single productive share has <portfolio wealth> $w=S$. Investing fraction one in the <stock> means holding $w/S=1$ share, leaving $w-S=0$ in the <bank account>. The optimal <consumption> rate is
$$
\boxed{c_t=\gamma_Mw_t=kS_t=\delta_t.}
$$
This policy also satisfies the <self-financing portfolio> equation: with one share and no bank balance,
$$
dw_t=dS_t+\delta_tdt-c_tdt=dS_t.
$$
Thus $w_t=S_t$ remains valid at all times. \b[The agent holds the entire one-share supply, has zero net borrowing or lending, and consumes exactly the supplied goods.] The <stock>, bank-account and consumption-good markets all clear.
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