= 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 $c(t)=Ce^{rt/(1-\alpha)}$. 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.
Back to article page
