Envelope theorem

Envelope theorem This article may be too technical for most readers to understand. Por favor, ajude a melhorá-lo para torná-lo compreensível para não especialistas, sem remover os detalhes técnicos. (novembro 2021) (Saiba como e quando remover esta mensagem de modelo) In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem.[1] 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.[2] 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 {estilo de exibição à esquerda{abandonou(x,cdot right)certo}_{xin X}} that are optimized.

Conteúdo 1 Declaração 2 For arbitrary choice sets 3 Formulários 3.1 Applications to producer theory 3.2 Applications to mechanism design and auction theory 3.3 Applications to multidimensional parameter spaces 3.4 Applications to parameterized constraints 3.5 Other applications 4 Veja também 5 References Statement Let {estilo de exibição f(x,alfa )} e {estilo de exibição g_{j}(x,alfa ),j=1,2,ldots ,m} be real-valued continuously differentiable functions on {estilo de exibição mathbb {R} ^{n+l}} , Onde {displaystyle xin mathbb {R} ^{n}} are choice variables and {displaystyle alpha in mathbb {R} ^{eu}} are parameters, and consider the problem of choosing {estilo de exibição x} , for a given {alfa de estilo de exibição } , so as to: {estilo de exibição max _{x}f(x,alfa )} subject to {estilo de exibição g_{j}(x,alfa )geq 0,j=1,2,ldots ,m} e {estilo de exibição xgeq 0} .

The Lagrangian expression of this problem is given by {estilo de exibição {matemática {eu}}(x,lambda ,alfa )=f(x,alfa )+lambda cdot g(x,alfa )} Onde {lambda de estilo de exibição em mathbb {R} ^{m}} are the Lagrange multipliers. Now let {estilo de exibição x^{ast }(alfa )} e {displaystyle lambda ^{ast }(alfa )} together be the solution that maximizes the objective function f subject to the constraints (and hence are saddle points of the Lagrangian), {estilo de exibição {matemática {eu}}^{ast }(alfa )equiv f(x^{ast }(alfa ),alfa )+lambda ^{ast }(alfa )cdot g(x^{ast }(alfa ),alfa ),} and define the value function {estilo de exibição V(alfa )equiv f(x^{ast }(alfa ),alfa ).} Then we have the following theorem.[3][4] Teorema: Assuma isso {estilo de exibição V} e {estilo de exibição {matemática {eu}}} are continuously differentiable. Então {estilo de exibição {fratura {partial V(alfa )}{partial alpha _{k}}}={fratura {parcial {matemática {eu}}^{ast }(alfa )}{partial alpha _{k}}}={fratura {parcial {matemática {eu}}(x^{ast }(alfa ),lambda ^{ast }(alfa ),alfa )}{partial alpha _{k}}},k=1,2,ldots ,eu} Onde {estilo de exibição parcial {matemática {eu}}/partial alpha _{k}=partial f/partial alpha _{k}+lambda cdot partial g/partial alpha _{k}} .

For arbitrary choice sets Let {estilo de exibição X} denote the choice set and let the relevant parameter be {displaystyle tin lbrack 0,1]} . Letting {estilo de exibição f:Xtimes lbrack 0,1]rightarrow R} denote the parameterized objective function, the value function {estilo de exibição V} and the optimal choice correspondence (set-valued function) {estilo de exibição X^{ast }} are given by: {estilo de exibição V(t)=sup_{xin X}f(x,t)} (1) {estilo de exibição X^{ast }(t)={xin X:f(x,t)=V(t)}} (2) "Envelope theorems" describe sufficient conditions for the value function {estilo de exibição V} to be differentiable in the parameter {estilo de exibição t} and describe its derivative as {estilo de exibição V^{melhor }deixei(tright)=f_{t}deixei(x,tright){texto{ para cada }}xin X^{ast }deixei(tright),} (3) Onde {estilo de exibição f_{t}} denotes the partial derivative of {estilo de exibição f} em relação a {estilo de exibição t} . Nomeadamente, the derivative of the value function with respect to the parameter equals the partial derivative of the objective function with respect to {estilo de exibição t} holding the maximizer fixed at its optimal level.

Traditional envelope theorem derivations use the first-order condition for (1), which requires that the choice set {estilo de exibição X} have the convex and topological structure, and the objective function {estilo de exibição f} be differentiable in the variable {estilo de exibição x} . (The argument is that changes in the maximizer have only a "second-order effect" at the optimum and so can be ignored.) No entanto, in many applications such as the analysis of incentive constraints in contract theory and game theory, nonconvex production problems, e "monotone" ou "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,[5] provided that the objective function is differentiable in the parameter: Teorema 1: Deixar {displaystyle tin left(0,1certo)} e {displaystyle xin X^{ast }deixei(tright)} . If both {estilo de exibição V^{melhor }deixei(tright)} e {estilo de exibição f_{t}deixei(x,tright)} exist, the envelope formula (3) detém.

Prova: Equation (1) implies that for {displaystyle xin X^{ast }deixei(tright)} , {estilo de exibição max _{sin left[0,1certo]}deixei[abandonou(x,sright)-Vleft(sright)certo]=fleft(x,tright)-Vleft(tright)=0.} Under the assumptions, the objective function of the displayed maximization problem is differentiable at {displaystyle s=t} , and the first-order condition for this maximization is exactly equation (3). 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. Em particular, Milgrom and Segal's (2002) Teorema 2 offers a sufficient condition for {estilo de exibição V} to be absolutely continuous,[5] which means that it is differentiable almost everywhere and can be represented as an integral of its derivative: Teorema 2: Suponha que {estilo de exibição f(x,cdot )} is absolutely continuous for all {estilo de exibição xin X} . Suppose also that there exists an integrable function {estilo de exibição b:[0,1]} {displaystyle rightarrow } {estilo de exibição mathbb {R} _{+}} de tal modo que {estilo de exibição |f_{t}(x,t)|leq b(t)} para todos {estilo de exibição xin X} and almost all {displaystyle tin lbrack 0,1]} . Então {estilo de exibição V} is absolutely continuous. Suponha, além do que, além do mais, este {estilo de exibição f(x,cdot )} is differentiable for all {estilo de exibição xin X} , and that {estilo de exibição X^{ast }(t)neq varnothing } almost everywhere on {estilo de exibição [0,1]} . Then for any selection {estilo de exibição x^{ast }(t)em X^{ast }(t)} , {estilo de exibição V(t)=V(0)+int_{0}^{t}f_{t}(x^{ast }(s),s)ds.} (4) Prova: Using (1)(1), observe that for any {displaystyle t^{melhor },t^{prime prime }in lbrack 0,1]} com {displaystyle t^{melhor }0} para todos {displaystyle tin left[0,1certo]} . Under these assumptions, it is well known that the above constrained optimization program can be represented as a saddle-point problem for the Lagrangian {displaystyle Lleft(x,lambda ,tright)=f(x,t)+lambda cdot gleft(x,tright)} , Onde {lambda de estilo de exibição em mathbb {R} _{+}^{K}} is the vector of Lagrange multipliers chosen by the adversary to minimize the Lagrangian.[20][page needed][21][page needed] This allows the application of Milgrom and Segal's (2002, Teorema 4) envelope theorem for saddle-point problems,[5] under the additional assumptions that {estilo de exibição X} is a compact set in a normed linear space, {estilo de exibição f} e {estilo de exibição g} are continuous in {estilo de exibição x} , e {estilo de exibição f_{t}} e {estilo de exibição g_{t}} are continuous in {estilo de exibição à esquerda(x,tright)} . Em particular, de locação {estilo de exibição à esquerda(x^{ast }(t),lambda ^{ast }deixei(tright)certo)} denote the Lagrangian's saddle point for parameter value {estilo de exibição t} , the theorem implies that {estilo de exibição V} is absolutely continuous and satisfies {estilo de exibição V(t)=V(0)+int_{0}^{t}EU_{t}(x^{ast }(s),lambda ^{ast }deixei(sright),s)ds.} For the special case in which {estilo de exibição esquerdo(x,tright)} is independent of {estilo de exibição t} , {displaystyle K=1} , e {displaystyle gleft(x,tright)=hleft(xdireita)+t} , the formula implies that {estilo de exibição V^{melhor }(t)=L_{t}(x^{ast }(t),lambda ^{ast }deixei(tright),t)=lambda ^{ast }deixei(tright)} for a.e. {estilo de exibição t} . Aquilo é, the Lagrange multiplier {displaystyle lambda ^{ast }deixei(tright)} on the constraint is its "shadow price" in the optimization program.[21][page needed] 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.[5] See also Maximum theorem Danskin's theorem Hotelling's lemma Le Chatelier's principle Roy's identity Value function References ^ Border, Kim C. (2019). "Miscellaneous Notes on Optimization Theory and Related Topics" (PDF). Lecture Notes. California Institute of Technology: 154. ^ Carter, Michael (2001). Foundations of Mathematical Economics. Cambridge: Imprensa do MIT. pp. 603–609. ISBN 978-0-262-53192-4. ^ Afriat, S. N. (1971). "Theory of Maxima and the Method of Lagrange". SIAM Journal on Applied Mathematics. 20 (3): 343–357. doi:10.1137/0120037. ^ Takayama, Akira (1985). 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