Concavification of incentive utility ($\bar{F}=\text{least concave majorant of }F$)

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.

With zero interest and zero risk premium, a manager whose current wealth lies between common-tangent contacts $\ell ,h$ can improve expected utility maximization through a fair lottery paying those two values. The probability of $h$ is $(w_0-\ell )/(h-\ell )$. In a Brownian filtration, replicating a bounded terminal lottery gives a nonnegative wealth martingale throughout.

For $F(y)=-e^{-ay}$ below hurdle $w_0$ and $F(y)=-e^{(b-a)w_0-by}$ above it, with $b>a>0$, an interior common tangent touches at $\ell =w_0-[b/a-1-\log (b/a)]/(b-a)$ and $h=w_0+[\log (b/a)-1+a/b]/(b-a)$. The slope is $a{e}^{-a\ell }$ and $h-\ell =1/a-1/b$. The formula requires $\ell \ge 0$; a smaller hurdle gives a different endpoint-at-zero concavification.