Finite-horizon logarithmic investment and consumption ($c_t^{\ast }=\frac{b{V}_t}{a+b(T-t)},{\pi }_t^{\ast }=\frac{\mu -\rho }{{\sigma }^2}{V}_t$)

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.