The envelope theorem, Euler and Bellman equations, without differentiability

Ramon Marimon, Jan Werner

Research output: Contribution to journalArticlepeer-review


We extend the standard Bellman's theory of dynamic programming and the theory of recursive contracts with forward-looking constraints of Marcet and Marimon (2019) to encompass non-differentiability of the value function associated with non-unique solutions or multipliers. The envelope theorem provides the link between the Bellman equation and the Euler equations, but it may fail to do so if the value function is non-differentiable. We introduce an envelope selection condition which restores this link. In standard single-agent dynamic programming, ignoring the envelope selection condition may result in inconsistent multipliers, but not in non-optimal outcomes. In recursive contracts it can result in inconsistent promises and non-optimal outcomes. Planner problems with recursive preferences are a special case of recursive contracts and, therefore, solutions can be dynamically inconsistent if they are not unique. A recursive method of solving dynamic optimization problems with non-differentiable value function involves expanding the co-state and imposing the envelope selection condition.

Original languageEnglish (US)
Article number105309
JournalJournal of Economic Theory
StatePublished - Sep 2021

Bibliographical note

Publisher Copyright:
© 2021 Elsevier Inc.


  • Bellman equation
  • Envelope theorem
  • Euler equation
  • Recursive contracts
  • Recursive preferences
  • Value function


Dive into the research topics of 'The envelope theorem, Euler and Bellman equations, without differentiability'. Together they form a unique fingerprint.

Cite this