 Work-then-consume optimum for a nonconcave consumption-leisure utility

For wealth dynamics $\dot{x}=rx+\ell -c$ and utility $c^{\alpha }(1-\ell {)}^{\beta }$, with $0<\alpha ,\beta <1$ and $\alpha +\beta >1$, the unconstrained interior control equations do not give the maximum. Put $s=1-\ell$. At positive shadow price $p$, maximization over $c$ leaves a function of $s$ proportional to $s^{\beta /(1-\alpha )}-ps$, up to constants. It is convex and hence maximal at $s=0$ or $s=1$. The optimal controls work with zero consumption first, then cease work and consume at $c(t)=C{e}^{rt/(1-\alpha )}$. The switch occurs where this consumption reaches $\alpha /(1-\alpha )$; the full discounted budget fixes $C$ and the switch time. The decreasing-consumption interior stationary candidate violates the Hamiltonian maximization condition.