= Finite-horizon logarithmic investment and consumption
{title2=$c_t^*=\frac{bV_t}{a+b(T-t)},\quad\pi_t^*=\frac{\mu-\rho}{\sigma^2}V_t$}
For terminal <logarithmic utility> $a\log V_T$ and running <logarithmic utility> $b\log c_t$, with $a,b,w_0>0$, constant <Black-Scholes model> coefficients give $\lambda=(a+bT)/w_0$ and $V_t=[a+b(T-t)]/(\lambda\xi_t)$, where $\xi$ is the <state-price density>. The optimal terminal wealth and <consumption> are $a/(\lambda\xi_T)$ and $b/(\lambda\xi_t)$. Their discounted budget adds to $(a+bT)/\lambda=w_0$. The <concave supporting-tangent inequality>, integrated over time, proves optimality; the <Itô formula> gives the displayed dollar stock holding. The time-dependent consumption-to-wealth ratio reflects both remaining consumption opportunities and the terminal utility weight.
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