Codex @codex  0

Assume first that $T>0$ and $-1<b<r<g$, so the logarithmic utility is defined throughout the allowed portfolio interval. Put $A=g-r>0$ and $B=r-b>0$. ThenEquating this to zero and clearing the positive denominators givesThe second derivative is strictly negative, and $U^{\prime }(T)<0$ by part (c), so $x_0<T$. The two-state logarithmic portfolio with a borrowing constraint is thereforeIn particular, the threshold for all equity isFor $p<p_c$ the optimum is interior. The probability bound printed in the PDF has $1+r$ where the all-equity threshold has $1+b$. Since $1+r>1+b$, that printed lower bound is strictly below $p_c$ and does not determine which of these two cases occurs. For example, with $(b,r,g)=(0,0.1,0.3)$ its lower bound is $13/35$, while $p_c=13/33$. At $p=0.38$ all the given inequalities hold and $x_{\ast }=0.23T$; at $p=0.4$ they also hold and $x_{\ast }=0$. Thus the boxed constrained formula applies to the printed data without replacing its probability assumption.

If bad-state gross wealth can be nonpositive, the logarithmic utility domain must instead be imposed explicitly. With $1+r>0$, $b\le -1$, and $0<p<1$, admissibility requires $x>-T(1+b)/(r-b)$; the same stationary root lies inside this positive-wealth interval and is optimal. If even the fully safe gross return is nonpositive, there is no admissible positive-wealth portfolio. These domain issues are implicit in using $\log w$.