The envelope theorem, Euler and Bellman equations, without differentiability

• 成果类型: Article
• 署名作者: Marimon, Ramon; Werner, Jan
• 署名单位: European University Institute; Pompeu Fabra University; Centre for Economic Policy Research - UK; National Bureau of Economic Research; University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: JOURNAL OF ECONOMIC THEORY
• ISSN/ISSBN: 0022-0531
• DOI: 10.1016/j.jet.2021.105309
• 发表日期: 2021
• 关键词: Bellman equation; Euler equation; Envelope theorem; value function; Recursive contracts; recursive preferences

摘要

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. (c) 2021 Elsevier Inc. All rights reserved.