The envelope theorem, Euler and Bellman equations, without differentiability

Ramon Marimon and Jan Werner

Journal of Economic Theory, 2021, vol. 196, issue C

Abstract: 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.

Keywords: Bellman equation; Euler equation; Envelope theorem; Value function; Recursive contracts; Recursive preferences (search for similar items in EconPapers)
JEL-codes: C60 C61 D86 E00 (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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Related works:
Working Paper: Envelope Theorem, Euler, and Bellman Equations without Differentiability (2015)
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Persistent link: https://EconPapers.repec.org/RePEc:eee:jetheo:v:196:y:2021:i:c:s0022053121001265

DOI: 10.1016/j.jet.2021.105309

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