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. So we can differentiate (iii) the firm’s value function in a capital adjustment problem with fixed costs, and (iv) the households’ value functions in insurance arrangements with indivisible goods.
Andrew Clausen and Carlo Strub (2017). A General and Intuitive Envelope Theorem. mimeo.
Previous versions of this paper circulated under the name Envelope Theorems for Non-Smooth and Non-Concave Optimization (2012, with A. Clausen)