Optimal contract design with labor–leisure choice under limited commitment: A free boundary approach

We study a continuous-time optimal contracting problem involving labor–leisure choices under limited commitment. A principal offers a contract to a risk-averse agent whose wage follows a geometric Brownian motion. The agent derives utility from both consumption and leisure, modeled through a Cobb–Douglas utility function. Due to limited commitment, the agent’s participation and promised utility constraints must be satisfied throughout the contracting horizon. By employing a dual approach and dynamic programming principles, we transform the problem into a singular stochastic control problem associated with a variational inequality and a free boundary. We provide an explicit closed-form solution to the variational inequality and characterize the optimal contract in terms of consumption and leisure processes. Numerical simulations illustrate the dynamic behavior of the optimal consumption, leisure, and continuation utility processes. Our approach demonstrates the effectiveness of duality methods and singular control techniques in solving nonlinear stochastic optimization problems with state-dependent constraints, contributing to the computational aspects of optimal control and contract theory.