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Concavification replaces a nonconcave incentive utility function by its least concave majorant. In a complete market with an atomless state-price density, the optimizing payoff may avoid all intervals where the majorant strictly exceeds the original utility, making the relaxed optimum exactly attainable. At a pricing atom corresponding to a linear segment, a lottery between contact points can attain the same value at unchanged cost.

A consumption satisfaction stock smooths past consumption by ${\xi }_t=e^{-\lambda t}{\xi }_0+{\int }_0^t{e}^{-\lambda (t-s)}{c}_{s}ds$. Unlike multiplicative habit utility, the running utility function may depend on this stock alone. A unit of present consumption raises future consumption satisfaction while costing one unit of financial portfolio wealth.

For $X=\mu +\sigma Z$ with $Z$ having the standard normal distribution and an increasing concave utility function $U$, define $f(\mu ,\sigma )=\mathbb{E}[U(X)]$. Then $f_{\mu }=\mathbb{E}[U^{\prime }(X)]>0$ and Gaussian integration by parts gives $f_{\sigma }=\sigma \mathbb{E}[U^{\prime \prime }(X)]\le 0$. Moreoveris negative semidefinite, so $f$ is jointly concave in $\mu$ and $\sigma$.

An incentive fee above a hurdle changes a manager’s utility function into $F(y)=U(\epsilon y+\alpha (y-w_0{)}_{+})$. For constant absolute risk aversion utility, each branch is strictly concave, but the marginal reward jumps upward at the hurdle. Thus the effective terminal-wealth utility is not concave, and concavification of incentive utility can reveal optimal risk-taking lotteries.

The Inada conditions specify infinite marginal utility at zero consumption or wealth and zero marginal utility at infinity. For an increasing strictly concave utility function, they make the inverse marginal utility map cover all positive shadow prices, facilitating the state-price budget constraint method.

For an increasing differentiable strictly concave utility function satisfying the Inada conditions, the inverse marginal utility maps each positive shadow price to the unique positive payoff with that marginal utility. For normalized CRRA utility, $I(y)=y^{-1/R}$.