A duality approach to continuous-time contracting problems with limited commitment

We propose a duality approach to solving contracting models with either one-sided or two-sided limited commitment in continuous time. We establish weak and strong duality theorems and provide a dynamic programming characterization of the dual problem. The dual problem gives a linear Hamilton–Jacobi–Bellman equation with a known state space subject to free-boundary conditions, making analysis much more tractable than the primal problem. We provide two explicitly solved examples of a consumption insurance problem. We characterize the optimal consumption allocation in terms of the marginal utility ratio. We find that neither autarky nor full risk sharing can be an optimal contract with two-sided limited commitment, unlike in discrete-time models. We also derive an explicit solution for the unique long-run stationary distribution of consumption relative to income.