Solving for the Retirement Age in a Continuous-time Model with Endogenous Labor Supply

This paper studies a continuous-time life-cycle model with a consumption-leisure choice made by a finitely-lived agent with a random lifetime. We explicitly account for the leisure constraint in the corresponding constrained optimal control problem with a commonly postulated solution structure, and provide a complete analytic solution of the problem. By solving both the constrained and unconstrained control formulations, we demonstrate the inaccuracy of the latter formulation (where an unconstrained leisure path gets truncated once it exceeds the time endowment limit) can indeed be significant. For cases when the subjective discount rate is quite close to or exceeds the interest rate, the optimal control path would not be consistent with the structure of the optimal solution so commonly postulated in applied studies.