= Solution
The <power-wealth investment with running utility> objective is homogeneous of degree $\gamma$ in wealth: scaling both wealth and positions by $a>0$ scales the reward by $a^\gamma$. Accordingly set $J(w,t)=A(t)w^\gamma$. Then
$$
\frac{J_w}{J_{ww}}=\frac{w}{\gamma-1},\qquad
\boxed{\frac{B^*}{w}=1-\frac{\mu-r}{(1-\gamma)\sigma^2},\qquad\frac{\pi^*}{w}=\frac{\mu-r}{(1-\gamma)\sigma^2}.}
$$
For completeness the remaining equation is $A'+\kappa A+e^{-\rho t}=0$, $A(T)=1$, where
$$
\kappa=\gamma r+\frac{\gamma(\mu-r)^2}{2(1-\gamma)\sigma^2},\qquad
A(t)=e^{\kappa(T-t)}+\int_t^T e^{\kappa(s-t)-\rho s}ds>0.
$$
This confirms $J_{ww}<0$. The constant optimal risky fraction also makes wealth a positive <geometric Brownian motion> when started positive.
Financially the <relative risk aversion coefficient> is $1-\gamma$. Increasing $\gamma$ reduces <risk aversion>, raises the risky fraction, and decreases the bond fraction: its derivative is $-(\mu-r)/[\sigma^2(1-\gamma)^2]<0$. If the risky fraction is below one, some wealth is lent; at one, none is invested in bonds; above one, $B^*<0$ represents borrowing to leverage the risky position. As $\gamma\uparrow1$ the risky fraction becomes unbounded, consistent with the absence of a finite unconstrained optimum for linear utility and positive excess drift. As $\gamma\downarrow0$ the limiting fraction is $(\mu-r)/\sigma^2$, the logarithmic-utility allocation obtained by the appropriately rescaled power-utility limit; the literal objective at $\gamma=0$ would be constant and would not select a unique strategy.
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