Discrete-Time Dynamic Principal-Agent Models : Contraction Mapping Theorem and Computational Treatment

Renner, Philipp and Schmedders, Karl (2020) Discrete-Time Dynamic Principal-Agent Models : Contraction Mapping Theorem and Computational Treatment. Quantitative Economics, 11 (4). pp. 1215-1251. ISSN 1759-7323

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Abstract

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 we 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.