Discrete-time dynamic principal-agent models: Contraction mapping theorem and computational treatment

We consider discrete-time dynamic principal-agent problems with continuous choice sets and potentially multiple agents. We prove the existence of a unique solution for the principal's value function only assuming continuity of the functions and compactness of the choice sets. We do this by a contraction mapping theorem and so also obtain a convergence result for the value function iteration. To numerically compute a solution for the problem, we have to solve a collection of static principal-agent problems at each iteration. As a result, in the discrete-time setting solving the static problem is the difficult step. If the agent's expected utility is a rational function of his action, then we can transform the bi-level optimization problem into a standard nonlinear program. The final results of our solution method are numerical approximations of the policy and value functions for the dynamic principal-agent model. We illustrate our solution method by solving variations of two prominent social planning models from the economics literature.