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. 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 important tool for comparative statics of optimization models.
The term envelope derives from describing the graph of the value function as the "upper envelope" of the graphs of the parameterized family of functions formula_1 that are optimized.
Statement.
Let formula_2 and formula_3 be real-valued continuously differentiable functions on formula_4, where formula_5 are choice variables and formula_6 are parameters, and consider the problem of choosing formula_7, for a given formula_8, so as to:
The Lagrangian expression of this problem is given by
where formula_13 are the Lagrange multipliers. Now let formula_14 and formula_15 together be the solution that maximizes the objective function "f" subject to the constraints (and hence are saddle points of the Lagrangian),
and define the value function
Then we have the following theorem.
Theorem: "Assume that formula_18 and formula_19 are continuously differentiable. Then"
"where formula_21".
For arbitrary choice sets.
Let formula_22 denote the choice set and let the relevant parameter be formula_23. Letting formula_24 denote the parameterized objective function, the value function formula_18 and the optimal choice correspondence (set-valued function) formula_26 are given by:
"Envelope theorems" describe sufficient conditions for the value function formula_18 to be differentiable in the parameter formula_28 and describe its derivative as
where formula_29 denotes the partial derivative of formula_30 with respect to formula_28. Namely, the derivative of the value function with respect to the parameter equals the partial derivative of the objective function with respect to formula_28 holding the maximizer fixed at its optimal level. 
Traditional envelope theorem derivations use the first-order condition for (), which requires that the choice set formula_22 have the convex and topological structure, and the objective function formula_30 be differentiable in the variable formula_7. (The argument is that changes in the maximizer have only a "second-order effect" at the optimum and so can be ignored.) However, 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 Segal (2002) observe that the traditional envelope formula holds for optimization problems with arbitrary choice sets at any differentiability point of the value function, provided that the objective function is differentiable in the parameter:
Theorem 1: Let formula_36 and formula_37. If both formula_38 and formula_39 exist, the envelope formula () holds.
Proof: Equation () implies that for formula_40,
Under the assumptions, the objective function of the displayed maximization problem is differentiable at formula_42, and the first-order condition for this maximization is exactly equation (). Q.E.D.
While 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. In particular, Milgrom and Segal's (2002) Theorem 2 offers a sufficient condition for formula_18 to be absolutely continuous, which means that it is differentiable almost everywhere and can be represented as an integral of its derivative:
Theorem 2: Suppose that formula_44 is absolutely continuous for all formula_45. Suppose also that there exists an integrable function formula_46 formula_47 formula_48 such that formula_49 for all formula_45 and almost all formula_23. Then formula_18 is absolutely continuous. Suppose, in addition, that formula_44 is differentiable for all formula_45, and that formula_55 almost everywhere on formula_56. Then for any selection formula_57,
Proof: Using ()(1), observe that for any formula_58 with formula_59,
This implies that formula_18 is absolutely continuous. Therefore, formula_18 is differentiable almost everywhere, and using () yields (). Q.E.D.
This result dispels the common misconception that nice behavior of the value function requires correspondingly nice behavior of the maximizer. Theorem 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 at formula_63 and hence satisfy the envelope formula () when the family formula_1 is equi-differentiable at formula_65 and formula_66 is single-valued and continuous at formula_63, even if the maximizer is not differentiable at formula_68 (e.g., if formula_69 is described by a set of inequality constraints and the set of binding constraints changes at formula_68).
Applications.
Applications to producer theory.
Theorem 1 implies Hotelling's lemma at any differentiability point of the profit function, and Theorem 2 implies the producer surplus formula. Formally, let formula_71 denote the profit function of a price-taking firm with production set formula_72 facing prices formula_73, and let formula_74 denote the firm's supply function, i.e.,
Let formula_76 (the price of good formula_77) and fix the other goods' prices at formula_78. Applying Theorem 1 to formula_79 yields formula_80 (the firm's optimal supply of good formula_77). Applying Theorem 2 (whose assumptions are verified when formula_82 is restricted to a bounded interval) yields
i.e. the producer surplus formula_84 can be obtained by integrating under the firm's supply curve for good formula_77.
Applications to mechanism design and auction theory.
Consider an agent whose utility function formula_86 over outcomes formula_87 depends on his type formula_23. Let formula_89 represent the "menu" of possible outcomes the agent could obtain in the mechanism by sending different messages. The agent's equilibrium utility formula_90 in the mechanism is then given by (1), and the set formula_91 of the mechanism's equilibrium outcomes is given by (2). Any selection formula_57 is a choice rule implemented by the mechanism. Suppose that the agent's utility function formula_86 is differentiable and absolutely continuous in formula_28 for all formula_95, and that formula_96 is integrable on formula_56. Then Theorem 2 implies that the agent's equilibrium utility formula_18 in any mechanism implementing a given choice rule formula_99 must satisfy the integral condition (4).
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 as formula_100, where formula_101 is the bidder's probability of receiving the object and formula_102 is his expected payment, and the bidder's expected utility takes the form formula_103. In this case, letting formula_104 denote the bidder's lowest possible type, the integral condition (4) for the bidder's equilibrium expected utility formula_18 takes the form
(This equation can be interpreted as the producer surplus formula for the firm whose production technology for converting numeraire formula_102 into probability formula_101 of winning the object is defined by the auction and which resells the object at a fixed price formula_28). 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 formula_110 of getting the object for all types formula_28 as well as by the expected payoffs formula_112 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 Myerson–Satterthwaite inefficiency theorem (Myerson and Satterthwaite, 1983), the Jehiel–Moldovanu impossibility theorems (Jehiel and Moldovanu, 2001), the McAfee–McMillan weak-cartels theorem (McAfee and McMillan, 1992), and Weber's martingale theorem (Weber, 1983), etc. The details of these applications are provided in Chapter 3 of Milgrom (2004), 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.
Applications to multidimensional parameter spaces.
For a multidimensional parameter space formula_113, Theorem
1 can be applied to partial and directional derivatives of the value
function. If both the objective function formula_30 and the value function formula_18 are (totally) differentiable in formula_28, Theorem 1 implies the envelope formula for their gradients: formula_117 for each formula_40. While total differentiability of the value function may not be easy to ensure, Theorem 2 can be still applied along any smooth path connecting two parameter values formula_68 and formula_28. Namely, suppose that functions formula_44 are differentiable for all formula_45 with formula_123 for all formula_124 formula_125. A smooth path from formula_68 to formula_28 is described by a differentiable mapping formula_128 with a bounded derivative, such that formula_129 and formula_130. Theorem 2 implies that for any such smooth path, the change of the value function can be expressed as the path integral of the partial gradient formula_131 of the objective function along the path:
In particular, for formula_63, this establishes that cyclic path integrals along any smooth path formula_134 must be zero:
This "integrability condition" plays an important role in mechanism design with multidimensional types, constraining what kind of choice rules formula_99 can be sustained by mechanism-induced menus formula_89. In application to producer theory, with formula_138 being the firm's production vector and formula_139 being the price vector, formula_140, and the integrability condition says that any rationalizable supply function formula_99 must satisfy
When formula_99 is continuously differentiable, this integrability condition is equivalent to the symmetry of the substitution matrix formula_144. (In consumer theory, the same argument applied to the expenditure minimization problem yields symmetry of the Slutsky matrix.)
Applications to parameterized constraints.
Suppose now that the feasible set formula_145 depends on the parameter, i.e.,
where formula_148 for some formula_149
Suppose that formula_22 is a convex set, formula_30 and formula_152 are concave in formula_7, and there exists formula_154 such that formula_155 for all formula_156. Under these assumptions, it is well known that the above constrained optimization program can be represented as a saddle-point problem for the Lagrangian formula_157, where formula_158 is the vector of Lagrange multipliers chosen by the adversary to minimize the Lagrangian. This allows the application of Milgrom and Segal's (2002, Theorem 4) envelope theorem for saddle-point problems, under the additional assumptions that formula_22 is a compact set in a normed linear space, formula_30 and formula_152 are continuous in formula_7, and formula_29 and formula_164 are continuous in formula_165. In particular, letting formula_166 denote the Lagrangian's saddle point for parameter value formula_28, the theorem implies that formula_18 is absolutely continuous and satisfies
For the special case in which formula_170 is independent of formula_28, formula_172, and formula_173, the formula implies that formula_174 for a.e. formula_28. That is, the Lagrange multiplier formula_176 on the constraint is its "shadow price" in the optimization program.
Other applications.
Milgrom and Segal (2002) demonstrate that the generalized version of the envelope theorems can also be applied to convex programming, continuous optimization problems, saddle-point problems, and optimal stopping problems.