 Complete-market terminal utility optimizer

For an increasing strictly concave utility function satisfying the Inada conditions, put $I=(U^{\prime }{)}^{-1}$ and let $\zeta$ be the positive terminal state-price density. In a finite complete market, select $y>0$ so that $\mathbb{E}[\zeta I(y\zeta )]$ equals initial wealth. The displayed payoff is optimal under that budget: the supporting-line inequality for utility has marginal slope $y\zeta$, and its expected linear error vanishes on the budget constraint. Market completeness supplies its replicating strategy.