Envelope theorem on arvau

Wikipedia / 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.

that are optimized.

Statement

be real-valued continuously differentiable functions, so as to:

The Lagrangian expression of this problem is given by

are the Lagrange multipliers 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: are continuously differentiable. Then

For arbitrary choice sets

are given by:

and describe its derivative as

denotes the partial derivative holding the maximizer fixed at its optimal level.

Traditional envelope theorem derivations use the first-order condition for (. (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 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, provided that the objective function is differentiable in the parameter:

Theorem 1: exist, the envelope formula () holds.

Proof: Equation (,

, 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 to be absolutely continuous, which means that it is differentiable almost everywhere and can be represented as an integral of its derivative:

Theorem 2:,

Proof: Using (,

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 and hence satisfy the envelope formula ().

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 denote the firm's supply function, i.e.,

is restricted to a bounded interval) yields

Applications to mechanism design and auction theory

must satisfy the integral condition (4).

The integral condition (4) is a key step in the analysis of mechanism design takes the form

). This condition in turn yields Myerson's (1981) celebrated revenue equivalence theorem 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

, Theorem 1 can be applied to partial and directional derivatives of the value function., Theorem 1 implies the envelope formula for their gradients:.. Theorem 2 implies that for any such smooth path, the change of the value function can be expressed as the path integral of the objective function along the path:

must be zero:

. must satisfy

is 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.)

Applications to parameterized constraints

depends on the parameter, i.e.,

is a convex set. Under these assumptions, it is well known that the above constrained optimization program can be represented as a saddle-point problem 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, is absolutely continuous and satisfies

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.