Finite-Horizon Portfolio Choice, Labor Supply, and Early Retirement under Borrowing Constraints

We study a finite-horizon optimal consumption and portfolio problem with labor supply flexibility and an irreversible early retirement option under a borrowing constraint. The agent chooses consumption, risky investment, and leisure before retirement, while after retirement labor income disappears and leisure is fixed at its maximal level. Preferences are described by a Cobb--Douglas utility, and wealth must remain nonnegative. {Using a dual martingale method, we transform the primal problem into a zero-sum stopper--singular-controller game. The associated dual value is characterized by a min--max parabolic variational inequality with obstacle and gradient constraints. We show that the maximal strong solution of the resulting variational inequality is the unique admissible strong solution whose gradient-constrained free boundary, namely the binding boundary, is monotone increasing in calendar time. A verification argument then identifies this strong solution with the value of the stopper--singular-controller game, and duality recovers the optimal retirement, consumption, leisure, and portfolio policies.} The numerical analysis recovers the value function and optimal policies, and illustrates how labor supply flexibility affects consumption, portfolio choice, and retirement timing under borrowing constraints.