by the mechanism. Suppose that the agent's utility function f ( x , t ) {\displaystyle f(x,t)} is differentiable and absolutely continuous in t {\displaystyle In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of the objective, the envelope theorem shows that, in a certain sense, changes in the optimizer of the objective do not contribute to the change in the objective function. The envelope theorem is an import (This equation can be interpreted as the producer surplus formula for the firm whose production technology for converting numeraire z {\displaystyle z} into probability y {\displaystyle y} of winning the object is defined by the auction and which resells the object at a fixed price t {\displaystyle t} ). This condition in turn yields Myerson's (1981) celebrated revenue equivalence theorem: the expected revenue generated in an auction in which bidders have independent private values is fully determined by the bidders' probabilities y ∗ ( t ) {\displaystyle y^{\ast }\left(t\right)} of getting the object for all types t {\displaystyle t} as well as by the expected payoffs V ( t _ ) {\displaystyle V({\underline {t}})} of the bidders' lowest types. Finally, this condition is a key step in Myerson's (1981) of optimal auctions. For other applications of the envelope theorem to mechanism design see Mirrlees (1971), Holmstrom (1979), Laffont and Maskin (1980), Riley and Samuelson (1981), Fudenberg and Tirole (1991), and Williams (1999). While these authors derived and exploited the envelope theorem by restricting attention to (piecewise) continuously differentiable choice rules or even narrower classes, it may sometimes be optimal to implement a choice rule that is not piecewise continuously differentiable. (One example is the class of trading problems with linear utility described in chapter 6.5 of Myerson (1991).) Note that the integral condition (3) still holds in this setting and implies such important results as Holmstrom's lemma (Holmstrom, 1979), Myerson's lemma (Myerson, 1981), the revenue equivalence theorem (for auctions), the Green–Laffont–Holmstrom theorem (Green and Laffont, 1979; Holmstrom, 1979), the…