Dynamic Principal Agent Models∗

This paper contributes to the theoretical and numerical analysis of discrete time dynamic principal agent problems with continuous choice sets. We first provide a new and simplified proof for the recursive reformulation of the sequential dynamic principal agent relationship. Next we prove the existence of a unique solution for the principal’s value function, which solves the dynamic programming problem in the recursive formulation, by showing that the Bellman operator is a contraction mapping. Therefore, the theorem also provides a convergence result for the value function iteration. To compute a solution for the problem we have to solve a collection of static principal agent problems at each iteration. Under the assumption that the agent’s expected utility is a rational function of his action, we can transform the bilevel optimization problem into a standard nonlinear program (NLP). We can then solve these nonlinear problems with a standard NLP solver. 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.