A General and Intuitive Envelope Theorem

Previous envelope theorems establish differentiability of value functions in convex settings. Our envelope theorem applies to all functions whose derivatives appear in first-order conditions, and in non-convex settings. For example, in Stackelberg games, the leader’s first-order condition involves the derivative of the follower’s policy. Similarly, we differentiate (i) the borrower’s value function and default cut-off policy function in an unsecured credit economy, (ii) the firm’s value function in a capital adjustment problem with fixed costs, and (iii) the households’ value functions in insurance arrangements with indivisible goods. Our theorem accommodates optimization problems involving discrete choices, infinite horizon stochastic dynamic programming, and Inada conditions.