Envelope theorem
In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem.The envelope theorem is an important tool for comparative statics of optimization models.are given by: "Envelope theorems" describe sufficient conditions for the value functionTraditional envelope theorem derivations use the first-order condition for (1), which requires that the choice setHowever, in many applications such as the analysis of incentive constraints in contract theory and game theory, nonconvex production problems, and "monotone" or "robust" comparative statics, the choice sets and objective functions generally lack the topological and convexity properties required by the traditional envelope theorems.Paul Milgrom and Ilya Segal (2002) observe that the traditional envelope formula holds for optimization problems with arbitrary choice sets at any differentiability point of the value function,[5] provided that the objective function is differentiable in the parameter: Theorem 1: Let, Under the assumptions, the objective function of the displayed maximization problem is differentiable atWhile differentiability of the value function in general requires strong assumptions, in many applications weaker conditions such as absolute continuity, differentiability almost everywhere, or left- and right-differentiability, suffice.to be absolutely continuous,[5] which means that it is differentiable almost everywhere and can be represented as an integral of its derivative: Theorem 2: Suppose thatTheorem 2 ensures the absolute continuity of the value function even though the maximizer may be discontinuous.In a similar vein, Milgrom and Segal's (2002) Theorem 3 implies that the value function must be differentiable atdenote the indirect profit function of a price-taking firm with production setis restricted to a bounded interval) yields i.e. the producer surplusrepresent the "menu" of possible outcomes the agent could obtain in the mechanism by sending different messages.The integral condition (4) is a key step in the analysis of mechanism design problems with continuous type spaces.In particular, in Myerson's (1981) analysis of single-item auctions, the outcome from the viewpoint of one bidder can be described astakes the form (This equation can be interpreted as the producer surplus formula for the firm whose production technology for converting numeraireThis 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' probabilitiesFinally, this condition is a key step in Myerson's (1981) of optimal auctions.[6] For other applications of the envelope theorem to mechanism design see Mirrlees (1971),[7] Holmstrom (1979),[8] Laffont and Maskin (1980),[9] Riley and Samuelson (1981),[10] Fudenberg and Tirole (1991),[11] and Williams (1999).[12] 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).The details of these applications are provided in Chapter 3 of Milgrom (2004),[19] who offers an elegant and unifying framework in auction and mechanism design analysis mainly based on the envelope theorem and other familiar techniques and concepts in demand theory., Theorem 1 can be applied to partial and directional derivatives of the value function., Theorem 1 implies the envelope formula for their gradients:[citation needed]must be zero:[citation needed] This "integrability condition" plays an important role in mechanism design with multidimensional types, constraining what kind of choice rulesis continuously differentiable, this integrability condition is equivalent to the symmetry of the substitution matrix(In consumer theory, the same argument applied to the expenditure minimization problem yields symmetry of the Slutsky matrix.)Under these assumptions, it is well known that the above constrained optimization program can be represented as a saddle-point problem for the Lagrangianis the vector of Lagrange multipliers chosen by the adversary to minimize the Lagrangian.