= Binomial power-utility terminal wealth
{title2=$X^*=R^nw_0Z^{-\eta}/\mathbb E Z^{1-\eta}$}
In a complete $n$-period <binomial market> with riskless gross return $R$, pricing density $Z$ and <utility function> $U(x)=\gamma x^{1/\gamma}$, assume $\gamma>1$ or $\gamma<0$ and put $\eta=\gamma/(\gamma-1)$. The optimal terminal payoff is $X^*=R^nw_0Z^{-\eta}/\mathbb E[Z^{1-\eta}]$. Its marginal utility is proportional to $Z$; the supporting-line inequality for the strictly concave <utility function> proves optimality using the pricing <budget constraint>.
This parameter restriction makes the utility increasing and strictly concave on positive wealth: $U\prime(x)=x^{1/\gamma-1}>0$ and $U\prime\prime(x)=(1/\gamma-1)x^{1/\gamma-2}<0$. For $0<\gamma<1$ the curvature reverses, and the supporting-tangent maximum argument no longer applies.
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