Teorema dell'inviluppo

Envelope theorem This article may be too technical for most readers to understand. Aiutaci a migliorarlo per renderlo comprensibile ai non esperti, senza rimuovere i dettagli tecnici. (novembre 2021) (Scopri come e quando rimuovere questo messaggio modello) 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 {stile di visualizzazione a sinistra{deviato(X,cdot right)Giusto}_{xin X}} that are optimized.

Contenuti 1 Dichiarazione 2 For arbitrary choice sets 3 Applicazioni 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 Guarda anche 5 Riferimenti Dichiarazione Let {stile di visualizzazione f(X,alfa )} e {stile di visualizzazione g_{j}(X,alfa ),j=1,2,ldots ,m} be real-valued continuously differentiable functions on {displaystyle mathbb {R} ^{n+l}} , dove {displaystyle xin mathbb {R} ^{n}} are choice variables and {displaystyle alpha in mathbb {R} ^{l}} are parameters, and consider the problem of choosing {stile di visualizzazione x} , for a given {displaystyle alfa } , so as to: {stile di visualizzazione massimo _{X}f(X,alfa )} subject to {stile di visualizzazione g_{j}(X,alfa )geq 0,j=1,2,ldots ,m} e {displaystyle xgeq 0} .

The Lagrangian expression of this problem is given by {stile di visualizzazione {matematico {l}}(X,lambda ,alfa )=f(X,alfa )+lambda cdot g(X,alfa )} dove {displaystyle lambda in mathbb {R} ^{m}} are the Lagrange multipliers. Now let {stile di visualizzazione 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), {stile di visualizzazione {matematico {l}}^{ast }(alfa )equiv f(x^{ast }(alfa ),alfa )+lambda ^{ast }(alfa )cdot g(x^{ast }(alfa ),alfa ),} and define the value function {stile di visualizzazione V(alfa )equiv f(x^{ast }(alfa ),alfa ).} Then we have the following theorem.[3][4] Teorema: Supponi che {stile di visualizzazione V} e {stile di visualizzazione {matematico {l}}} are continuously differentiable. Quindi {stile di visualizzazione {frac {partial V(alfa )}{partial alpha _{K}}}={frac {parziale {matematico {l}}^{ast }(alfa )}{partial alpha _{K}}}={frac {parziale {matematico {l}}(x^{ast }(alfa ),lambda ^{ast }(alfa ),alfa )}{partial alpha _{K}}},k=1,2,ldots ,l} dove {stile di visualizzazione parziale {matematico {l}}/partial alpha _{K}=partial f/partial alpha _{K}+lambda cdot partial g/partial alpha _{K}} .

For arbitrary choice sets Let {stile di visualizzazione X} denote the choice set and let the relevant parameter be {displaystyle tin lbrack 0,1]} . Letting {stile di visualizzazione f:Xtimes lbrack 0,1]rightarrow R} denote the parameterized objective function, the value function {stile di visualizzazione V} and the optimal choice correspondence (set-valued function) {stile di visualizzazione X^{ast }} are given by: {stile di visualizzazione V(t)= sup _{xin X}f(X,t)} (1) {stile di visualizzazione X^{ast }(t)={xin X:f(X,t)=V(t)}} (2) "Envelope theorems" describe sufficient conditions for the value function {stile di visualizzazione V} to be differentiable in the parameter {stile di visualizzazione t} and describe its derivative as {stile di visualizzazione V^{primo }sinistra(tright)=f_{t}sinistra(X,tright){testo{ per ciascuno }}xin X^{ast }sinistra(tright),} (3) dove {stile di visualizzazione f_{t}} denotes the partial derivative of {stile di visualizzazione f} riguardo a {stile di visualizzazione t} . Vale a dire, the derivative of the value function with respect to the parameter equals the partial derivative of the objective function with respect to {stile di visualizzazione 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 {stile di visualizzazione X} have the convex and topological structure, and the objective function {stile di visualizzazione f} be differentiable in the variable {stile di visualizzazione x} . (The argument is that changes in the maximizer have only a "second-order effect" at the optimum and so can be ignored.) Tuttavia, in many applications such as the analysis of incentive constraints in contract theory and game theory, nonconvex production problems, e "monotone" o "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: Permettere {displaystyle tin left(0,1Giusto)} e {displaystyle xin X^{ast }sinistra(tright)} . If both {stile di visualizzazione V^{primo }sinistra(tright)} e {stile di visualizzazione f_{t}sinistra(X,tright)} exist, the envelope formula (3) tiene.

Prova: Equation (1) implies that for {displaystyle xin X^{ast }sinistra(tright)} , {stile di visualizzazione massimo _{sin left[0,1Giusto]}sinistra[deviato(X,sright)-Vleft(sright)Giusto]=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). QED.

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 particolare, Milgrom and Segal's (2002) Teorema 2 offers a sufficient condition for {stile di visualizzazione 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: Supporre che {stile di visualizzazione f(X,cdot )} is absolutely continuous for all {stile di visualizzazione xin X} . Suppose also that there exists an integrable function {stile di visualizzazione b:[0,1]} {displaystyle rightarrow } {displaystyle mathbb {R} _{+}} tale che {stile di visualizzazione |f_{t}(X,t)|leq b(t)} per tutti {stile di visualizzazione xin X} and almost all {displaystyle tin lbrack 0,1]} . Quindi {stile di visualizzazione V} is absolutely continuous. Supponiamo, Inoltre, Quello {stile di visualizzazione f(X,cdot )} is differentiable for all {stile di visualizzazione xin X} , e quello {stile di visualizzazione X^{ast }(t)neq varnothing } almost everywhere on {stile di visualizzazione [0,1]} . Then for any selection {stile di visualizzazione x^{ast }(t)in X^{ast }(t)} , {stile di visualizzazione V(t)=V(0)+int _{0}^{t}f_{t}(x^{ast }(S),S)ds.} (4) Prova: Usando (1)(1), observe that for any {stile di visualizzazione t^{primo },t^{prime prime }in lbrack 0,1]} insieme a {stile di visualizzazione t^{primo }0} per tutti {displaystyle tin left[0,1Giusto]} . 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)} , dove {displaystyle lambda in mathbb {R} _{+}^{K}} is the vector of Lagrange multipliers chosen by the adversary to minimize the Lagrangian.[20][pagina necessaria][21][pagina necessaria] This allows the application of Milgrom and Segal's (2002, Teorema 4) envelope theorem for saddle-point problems,[5] under the additional assumptions that {stile di visualizzazione X} is a compact set in a normed linear space, {stile di visualizzazione f} e {stile di visualizzazione g} are continuous in {stile di visualizzazione x} , e {stile di visualizzazione f_{t}} e {stile di visualizzazione g_{t}} are continuous in {stile di visualizzazione a sinistra(X,tright)} . In particolare, letting {stile di visualizzazione a sinistra(x^{ast }(t),lambda ^{ast }sinistra(tright)Giusto)} denote the Lagrangian's saddle point for parameter value {stile di visualizzazione t} , the theorem implies that {stile di visualizzazione V} is absolutely continuous and satisfies {stile di visualizzazione V(t)=V(0)+int _{0}^{t}L_{t}(x^{ast }(S),lambda ^{ast }sinistra(sright),S)ds.} For the special case in which {stile di visualizzazione deviato(X,tright)} is independent of {stile di visualizzazione t} , {displaystyle K=1} , e {displaystyle gleft(X,tright)=hleft(xdestra)+t} , the formula implies that {stile di visualizzazione V^{primo }(t)=L_{t}(x^{ast }(t),lambda ^{ast }sinistra(tright),t)=lambda ^{ast }sinistra(tright)} for a.e. {stile di visualizzazione t} . 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