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Define the positive constantSince $S=\delta /k$, 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 driftThis 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,Thus the optimal stock fraction from part (i) isMoreover,Substituting this into the optimal consumption coefficient yieldsHence 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 isThis policy also satisfies the self-financing portfolio equation: with one share and no bank balance,Thus $w_t=S_t$ remains valid at all times. 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.