Common tangent for exponential incentive utility
= Common tangent for exponential incentive utility
{title2=$h-\ell=1/a-1/b$}
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 $ae^{-a\ell}$ and $h-\ell=1/a-1/b$. The formula requires $\ell\geq0$; a smaller hurdle gives a different endpoint-at-zero concavification.