= Finite-horizon power-utility investment and consumption
{title2=$c_t^*=W_t/b(T-t),\quad\pi_t^*=(\mu-r)W_t/(\gamma\sigma^2)$}
For <constant relative risk aversion utility> with parameter $\gamma>0$, terminal weight $\eta>0$ and utility discount rate $\delta$, in a <Black-Scholes model> put $\lambda=(\mu-r)/\sigma$, $k=[\delta+(\gamma-1)(r+\lambda^2/(2\gamma))]/\gamma$ and $b(\tau)=\int_0^\tau e^{-ks}\,ds+\eta^{1/\gamma}e^{-k\tau}$. The displayed optimal <consumption> and dollar stock position follow by <inverse marginal utility> and the <state-price budget constraint>. For $\gamma\ne1$ the value is $b(T-t)^\gamma W_t^{1-\gamma}/(1-\gamma)$ with discount measured from $t$. Finite horizons do not require $k>0$. The <concave supporting-tangent inequality> and replication of the optimal budget prove global optimality, rather than only a formal differential-equation solution.
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