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

41 Pages Posted: 14 Apr 2016 Last revised: 8 Oct 2019

See all articles by Philipp Renner

Philipp Renner

Lancaster University

Karl Schmedders

University of Zurich

Date Written: November 14, 2016


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. Thus 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. This means that in the discrete time setting solving the static problem is the difficult part. 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.

Keywords: Optimal unemployment tax, principal-agent model, repeated moral hazard

JEL Classification: C63, D80, D82

Suggested Citation

Renner, Philipp Johannes and Schmedders, Karl, Discrete Time Dynamic Principal--Agent Models: Contratction Mapping Theorem and Computational Treatment (November 14, 2016). Swiss Finance Institute Research Paper No. 16-26. Available at SSRN: https://ssrn.com/abstract=2764140 or http://dx.doi.org/10.2139/ssrn.2764140

Philipp Johannes Renner

Lancaster University ( email )

Managment School
Department of Economics
Lancaster LA1 4YX, Lancashire LA1 4YX
United Kingdom

Karl Schmedders (Contact Author)

University of Zurich ( email )

Moussonstrasse 15
Zürich, CH-8044
+41 (0)44 634 3770 (Phone)

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