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

Philipp Renner, Karl Schmedders*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1215-1251
Number of pages37
JournalQuantitative Economics
Issue number4
StatePublished - Nov 2020


  • C63
  • D80
  • D82
  • Optimal unemployment tax
  • principal–agent model
  • repeated moral hazard

ASJC Scopus subject areas

  • Economics and Econometrics


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