 Common tangent for exponential incentive utility

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.