Previous envelope theorems establish differentiability of value functions. Our techniques apply to all functions whose derivatives appear in first-order conditions. We derive first-order conditions involving the derivatives of (i) the Stackelberg follower’s policy in a Stackelberg leader’s problem, and (ii) a borrower’s value function and default cut-off policy function in an unsecured credit economy. Our techniques also accommodate optimization problems involving discrete choices, infinite horizon stochastic dynamic programming, and Inada conditions.
We study general dynamic programming problems with continuous and discrete choices and general constraints. The value functions may have kinks arising (1) at indifference points between discrete choices and (2) at constraint boundaries. Nevertheless, we establish a general envelope theorem: first-order conditions are necessary at interior optimal choices. We only assume differentiability of the utility function with respect to the continuous choices. The continuous choice may be from any Banach space and the discrete choice from any non-empty set.