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A state-price density is a strictly positive random variable $Z$ withOn the stock-state space this means a positive integrable function $z(s)$ satisfying ${\int }_0^{C}z(s)ds=C/R$ and ${\int }_0^{C}sz(s)ds=C$. Equivalently $dQ/dP=RZ$ is an equivalent probability measure with ${\mathbb{E}}_Q{S}_1=R$. Such measures exist because $0<R<C$, for example exponential tilts of the uniform law with mean tuned to $R$.

By the fundamental theorem of asset pricing, any arbitrage-free claim price $p$ is represented by such a density in the enlarged market. For payout $S_1^2$,with strictness because equivalence preserves the stock's nondegenerate distribution. Also $S_1^2<C{S}_1$ almost surely, so $p<C$. Hence $R<p<C$.

When $C=3R$, nonnegative wealth for all stock states requires $-1/2\le \pi \le 1$. At $\pi =1$ wealth is $S_1$. The tangent inequality for the strictly concave square root gives, for $s>0$,Uniform integration gives $\mathbb{E}\sqrt{S_1}=2\sqrt{C}/3$ and $\mathbb{E}{S}_1^{-1/2}=2/\sqrt{C}$, so the expected linear term vanishes exactly when $C=3R$. Strict concavity makes equality possible only for $\pi =1$. Thus the unique optimal holding is ${\pi }^{\ast }=1$, including this boundary optimum without assuming an interior maximizer.

The marginal utility price density isIt is integrable and satisfies both pricing constraints: $\mathbb{E}{Z}_{\ast }=1/R$ and $\mathbb{E}(Z_{\ast }{S}_1)=C/(3R)=1$. Therefore