Envelope Theorem, Euler, and Bellman Equations without Differentiability

We extend the envelope theorem, the Euler equation, and the Bellman equation to dynamic constrained optimization problems where binding constraints can give rise to non-differentiable value functions. The envelope theorem -- an extension of Milgrom and Segal (2002) theorem for concave functions -- provides a generalization of the Euler equation and establishes a relation between the Euler and the Bellman equation. For example, we show how solutions to the standard Belllman equation may fail to satisfy the respective Euler equations, in contrast with solutions to the infinite-horizon problem. In standard maximisation problems the failure of Euler equations may result in inconsistent multipliers, but not in non-optimal outcomes. However, in problems with forward looking constraints this failure can result in inconsistent promises and non-optimal outcomes. We also show how the inconsistency problem can be resolved by a minimal extension of the co-state. As an application we extend the theory of recursive contracts of Marcet and Marimon (1998, 2015) to the case where solutions are not unique, resolving a problem pointed out by Messner and Pavoni (2004).