The converse envelope theorem

TL;DR

This paper establishes a converse envelope theorem linking the envelope formula to a first-order condition, extending its application to general outcomes and preferences in mechanism design and information selling.

Contribution

It introduces a converse envelope theorem that requires no structure on the choice set, broadening the theorem's applicability in economic theory.

Findings

The envelope formula is equivalent to a first-order condition.

Any increasing allocation is implementable under general outcomes.

Application to selling information demonstrates practical relevance.

Abstract

I prove an envelope theorem with a converse: the envelope formula is equivalent to a first-order condition. Like Milgrom and Segal's (2002) envelope theorem, my result requires no structure on the choice set. I use the converse envelope theorem to extend to general outcomes and preferences the canonical result in mechanism design that any increasing allocation is implementable, and apply this to selling information.