Marginal utility price ($p_H=\mathbb{E}[U^{\prime }(W^{\ast })H]/(R\mathbb{E}{U}^{\prime }(W^{\ast }))$)

In a one-period utility problem with gross riskless return $R$, the marginal utility price is the infinitesimal claim price leaving optimized expected utility unchanged. If optimal terminal wealth is $W^{\ast }$, the pricing density is $Z=U^{\prime }(W^{\ast })/(R\mathbb{E}{U}^{\prime }(W^{\ast }))$, and $p_H=\mathbb{E}[ZH]$. The denominator is the marginal utility of initial wealth. Under the standard first-order optimality conditions this density prices the traded assets and gives the state-price density associated with the investor.