Envelope theorem

Let $f(x,\backslash alpha)$ and $g\_\{j\}(x,\backslash alpha),\; j\; =\; 1,2,\; \backslash ldots,\; m$ be real-valued continuously differentiable functions on $\backslash mathbb\{R\}^\{n+l\}$, where $x\; \backslash in\; \backslash mathbb\{R\}^\{n\}$ are choice variables and $\backslash alpha\; \backslash in\; \backslash mathbb\{R\}^\{l\}$ are parameters, and consider the problem of choosing $x$, for a given $\backslash alpha$, so as to:

:$\backslash max\_\{x\}\; f(x,\; \backslash alpha)$ subject to $g\_\{j\}(x,\backslash alpha)\; \backslash geq\; 0,\; j\; =\; 1,2,\; \backslash ldots,\; m$ and $x\; \backslash geq\; 0$.

The Lagrangian expression of this problem is given by

:$\backslash mathcal\{L\}\; (x,\; \backslash lambda,\; \backslash alpha)\; =\; f(x,\; \backslash alpha)\; +\; \backslash lambda\; \backslash cdot\; g(x,\; \backslash alpha)$

where $\backslash lambda\; \backslash in\; \backslash mathbb\{R\}^\{m\}$ are the Lagrange multipliers. Now let $x^\{\backslash ast\}(\backslash alpha)$ and $\backslash lambda^\{\backslash ast\}(\backslash alpha)$ together be the solution that maximizes the objective function f subject to the constraints (and hence are saddle points of the Lagrangian),

:$\backslash mathcal\{L\}^\{\backslash ast\}\; (\backslash alpha)\; \backslash equiv\; f(x^\{\backslash ast\}(\backslash alpha),\; \backslash alpha)\; +\; \backslash lambda^\{\backslash ast\}(\backslash alpha)\; \backslash cdot\; g(x^\{\backslash ast\}(\backslash alpha),\; \backslash alpha),$

and define the value function

:$V(\backslash alpha)\; \backslash equiv\; f(x^\{\backslash ast\}(\backslash alpha),\; \backslash alpha).$

Then we have the following theorem.{{cite journal |first=S. N. |last=Afriat |year=1971 |title=Theory of Maxima and the Method of Lagrange |journal=SIAM Journal on Applied Mathematics |volume=20 |issue=3 |pages=343–357 |doi=10.1137/0120037 }}{{cite book |first=Akira |last=Takayama |title=Mathematical Economics |location=New York |publisher=Cambridge University Press |edition=Second |year=1985 |isbn=978-0-521-31498-5 |pages=[https://archive.org/details/mathematicalecon00taka/page/137 137]–138 |url=https://archive.org/details/mathematicalecon00taka |url-access=registration }}

Theorem: Assume that $V$ and $\backslash mathcal\{L\}$ are continuously differentiable. Then

:$\backslash frac\{\backslash partial\; V(\backslash alpha)\}\{\backslash partial\; \backslash alpha\_\{k\}\}\; =\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}^\{\backslash ast\}\; (\backslash alpha)\}\{\backslash partial\; \backslash alpha\_\{k\}\}\; =\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\; (x^\{\backslash ast\}\; (\backslash alpha),\; \backslash lambda^\{\backslash ast\}\; (\backslash alpha),\; \backslash alpha)\}\{\backslash partial\; \backslash alpha\_\{k\}\},\; k\; =\; 1,\; 2,\; \backslash ldots,\; l$

where $\backslash partial\; \backslash mathcal\{L\}\; /\; \backslash partial\; \backslash alpha\_\{k\}\; =\; \backslash partial\; f\; /\; \backslash partial\; \backslash alpha\_\{k\}\; +\; \backslash lambda\; \backslash cdot\; \backslash partial\; g\; /\; \backslash partial\; \backslash alpha\_\{k\}$.

Let $X$ denote the choice set and let the relevant parameter be $t\backslash in\; \backslash lbrack\; 0,1]$. Letting $f:X\backslash times\; \backslash lbrack\; 0,1]\backslash rightarrow\; R$ denote the parameterized objective function, the value function $V$ and the optimal choice correspondence (set-valued function) $X^\{\backslash ast\; \}$ are given by:

{{NumBlk|:|$V(t)\; =\backslash sup\_\{x\backslash in\; X\}f(x,t)$|{{EquationRef|1}}}}

{{NumBlk|:|$X^\{\backslash ast\; \}(t)\; =\backslash \{x\backslash in\; X:f(x,t)=V(t)\backslash \}$|{{EquationRef|2}}}}

"Envelope theorems" describe sufficient conditions for the value function $V$ to be differentiable in the parameter $t$ and describe its derivative as

{{NumBlk|:|$V^\{\backslash prime\; \}\backslash left(\; t\backslash right)\; =f\_\{t\}\backslash left(\; x,t\backslash right)\; \backslash text\{\; for\; each\; \}x\backslash in\; X^\{\backslash ast\; \}\backslash left(\; t\backslash right),$|{{EquationRef|3}}}}

where $f\_\{t\}$ denotes the partial derivative of $f$ with respect to $t$. Namely, the derivative of the value function with respect to the parameter equals the partial derivative of the objective function with respect to $t$ holding the maximizer fixed at its optimal level.

Traditional envelope theorem derivations use the first-order condition for ({{EquationNote|1}}), which requires that the choice set $X$ have the convex and topological structure, and the objective function $f$ be differentiable in the variable $x$. (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: Let $t\backslash in\; \backslash left(\; 0,1\backslash right)$ and $x\backslash in\; X^\{\backslash ast\; \}\backslash left(t\backslash right)$. If both $V^\{\backslash prime\; \}\backslash left(\; t\backslash right)$ and $f\_\{t\}\backslash left(x,t\backslash right)$ exist, the envelope formula ({{EquationNote|3}}) holds.

Proof: Equation ({{EquationNote|1}}) implies that for $x\backslash in\; X^\{\backslash ast\; \}\backslash left(\; t\backslash right)$,

:$\backslash max\_\{s\backslash in\; \backslash left[\; 0,1\backslash right]\; \}\backslash left[\; f\backslash left(\; x,s\backslash right)\; -V\backslash left(\; s\backslash right)\backslash right]\; =f\backslash left(\; x,t\backslash right)\; -V\backslash left(\; t\backslash right)\; =0.$

Under the assumptions, the objective function of the displayed maximization problem is differentiable at $s=t$, and the first-order condition for this maximization is exactly equation ({{EquationNote|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. In particular, Milgrom and Segal's (2002) Theorem 2 offers a sufficient condition for $V$ 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 $f(x,\backslash cdot\; )$ is absolutely continuous for all $x\backslash in\; X$. Suppose also that there exists an integrable function $b:[0,1]$ $\backslash rightarrow$ $\backslash mathbb\{R\}\_\{+\}$ such that $|f\_\{t\}(x,t)|\backslash leq\; b(t)$ for all $x\backslash in\; X$ and almost all $t\backslash in\; \backslash lbrack\; 0,1]$. Then $V$ is absolutely continuous. Suppose, in addition, that $f(x,\backslash cdot\; )$ is differentiable for all $x\backslash in\; X$, and that $X^\{\backslash ast\; \}(t)\backslash neq\; \backslash varnothing$ almost everywhere on $[0,1]$. Then for any selection $x^\{\backslash ast\; \}(t)\backslash in\; X^\{\backslash ast\; \}(t)$,

{{NumBlk|:|$V(t)=V(0)+\backslash int\_\{0\}^\{t\}f\_\{t\}(x^\{\backslash ast\; \}(s),s)ds.$|{{EquationRef|4}}}}

Proof: Using ({{EquationNote|1}})(1), observe that for any $t^\{\backslash prime\},t^\{\backslash prime\; \backslash prime\; \}\backslash in\; \backslash lbrack\; 0,1]$ with

:$|V(t^\{\backslash prime\; \backslash prime\; \})-V(t^\{\backslash prime\; \})|\; \backslash leq\; \backslash sup\_\{x\backslash in\; X\}|f(x,t^\{\backslash prime\backslash prime\; \})-f(x,t^\{\backslash prime\; \})|\; =\backslash sup\_\{x\backslash in\; X\}\backslash left\backslash vert\; \backslash int\_\{t^\{\backslash prime\; \}\}^\{t^\{\backslash prime\; \backslash prime\; \}\}f\_\{t\}(x,t)dt\backslash right\backslash vert\; \backslash leq\; \backslash int\_\{t^\{\backslash prime\; \}\}^\{t^\{\backslash prime\; \backslash prime$

}}\sup_{x\in X}|f_{t}(x,t)|dt\leq \int_{t^{\prime }}^{t^{\prime \prime }}b(t)dt.

This implies that $V$ is absolutely continuous. Therefore, $V$ is differentiable almost everywhere, and using ({{EquationNote|3}}) yields ({{EquationNote|4}}). 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 $t=t\_\{0\}$ and hence satisfy the envelope formula ({{EquationNote|3}}) when the family $\backslash left\backslash \{\; f\backslash left(\; x,\backslash cdot\; \backslash right)\; \backslash right\backslash \}\; \_\{x\backslash in\; X\}$ is equi-differentiable at $t\_\{0\}\backslash in\; \backslash left(\; 0,1\backslash right)$ and $f\_\{t\}\backslash left(X^\{\backslash ast\; \}\backslash left(\; t\backslash right)\; ,t\_\{0\}\backslash right)$ is single-valued and continuous at $t=t\_\{0\}$, even if the maximizer is not differentiable at $t\_\{0\}$ (e.g., if $X$ is described by a set of inequality constraints and the set of binding constraints changes at $t\_\{0\}$).

Theorem 1 implies Hotelling's lemma at any differentiability point of the profit function, and Theorem 2 implies the producer surplus formula. Formally, let $\backslash pi\; \backslash left(\; p\backslash right)$ denote the indirect profit function of a price-taking firm with production set $X\backslash subseteq\; \backslash mathbb\{R\}^\{L\}$ facing prices $p\backslash in\; \backslash mathbb\{R\}^\{L\}$, and let $x^\{\backslash ast\; \}\backslash left(\; p\backslash right)$ denote the firm's supply function, i.e.,

:$\backslash pi\; (p)=\backslash max\_\{x\backslash in\; X\}p\backslash cdot\; x=p\backslash cdot\; x^\{\backslash ast\; \}\backslash left(\; p\backslash right)\; \backslash text\{.\}$

Let $t=p\_\{i\}$ (the price of good $i$) and fix the other goods' prices at $p\_\{-i\}\backslash in\; \backslash mathbb\{R\}^\{L-1\}$. Applying Theorem 1 to $f(x,t)=tx\_\{i\}+p\_\{-i\}\backslash cdot\; x\_\{-i\}$ yields $\backslash frac\{\backslash partial\; \backslash pi\; (p)\}\{\backslash partial\; p\_\{i\}\}=x\_\{i\}^\{\backslash ast\; \}(p)$ (the firm's optimal supply of good $i$). Applying Theorem 2 (whose assumptions are verified when $p\_\{i\}$ is restricted to a bounded interval) yields

:$\backslash pi\; (t,p\_\{-i\})-\backslash pi\; (0,p\_\{-i\})=\backslash int\_\{0\}^\{p\_\{i\}\}x\_\{i\}^\{\backslash ast\; \}(s,p\_\{-i\})ds,$

i.e. the producer surplus $\backslash pi\; (t,p\_\{-i\})-\backslash pi\; (0,p\_\{-i\})$ can be obtained by integrating under the firm's supply curve for good $i$.

Consider an agent whose utility function $f(x,t)$ over outcomes $x\backslash in\; \backslash bar\{X\}$ depends on his type $t\backslash in\; \backslash lbrack\; 0,1]$. Let $X\backslash subseteq\; \backslash bar\{X\}$ represent the "menu" of possible outcomes the agent could obtain in the mechanism by sending different messages. The agent's equilibrium utility $V(t)$ in the mechanism is then given by (1), and the set $X^\{\backslash ast\; \}(t)$ of the mechanism's equilibrium outcomes is given by (2). Any selection $x^\{\backslash ast\; \}(t)\backslash in\; X^\{\backslash ast\; \}(t)$ is a choice rule implemented by the mechanism. Suppose that the agent's utility function $f(x,t)$ is differentiable and absolutely continuous in $t$ for all $x\backslash in\; Y$, and that $\backslash sup\_\{x\backslash in\; \backslash bar\{X\}\}|f\_\{t\}(x,t)|$ is integrable on $[0,1]$. Then Theorem 2 implies that the agent's equilibrium utility $V$ in any mechanism implementing a given choice rule $x^\{\backslash ast\; \}$ 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 $x=\backslash left(\; y,z\backslash right)$, where $y$ is the bidder's probability of receiving the object and $z$ is his expected payment, and the bidder's expected utility takes the form $f\backslash left(\; \backslash left(\; y,z\backslash right)\; ,t\backslash right)\; =ty-z$. In this case, letting $\backslash underline\{t\}$ denote the bidder's lowest possible type, the integral condition (4) for the bidder's equilibrium expected utility $V$ takes the form

:$V(t)-V(\backslash underline\{t\})=\backslash int\_\{0\}^\{t\}y^\{\backslash ast\; \}(s)ds.$

(This equation can be interpreted as the producer surplus formula for the firm whose production technology for converting numeraire $z$ into probability $y$ of winning the object is defined by the auction and which resells the object at a fixed price $t$). 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 $y^\{\backslash ast\; \}\backslash left(\; t\backslash right)$ of getting the object for all types $t$ as well as by the expected payoffs $V(\backslash underline\{t\})$ 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),{{cite journal | author=Mirrlees, James | title= An Exploration in the Theory of Optimal Taxation| journal=Review of Economic Studies | year=2002| volume=38 | issue= 2| pages=175–208 | doi=10.2307/2296779| jstor= 2296779}} Holmstrom (1979),{{cite journal | author=Holmstrom, Bengt | s2cid=55414969| title=Groves Schemes on Restricted Domains| journal=Econometrica| year=1979| volume=47 | issue=5| pages=1137–1144 | doi=10.2307/1911954| jstor=1911954}} Laffont and Maskin (1980),{{cite journal |author1=Laffont, Jean-Jacques |author2=Eric Maskin | title=A Differentiable Approach to Dominant Strategy Mechanisms| journal=Econometrica| year=1980| volume=48 |issue=6 | pages=1507–1520 | doi=10.2307/1912821|jstor=1912821 }} Riley and Samuelson (1981),{{cite journal |last1=Riley |first1=John G. |first2=William S. |last2=Samuelson | title=Optimal Auctions | journal=American Economic Review | year=1981| volume=71 | pages=381–392 | issue=3 |jstor=1802786 }} Fudenberg and Tirole (1991),{{cite book |last1=Fudenberg |first1=Drew |first2=Jean |last2=Tirole| title= Game Theory| year=1991 |location=Cambridge | publisher = MIT Press |isbn=0-262-06141-4 }} and Williams (1999).{{cite journal | author=Williams, Steven | title=A Characterization of Efficient, Bayesian Incentive Compatible Mechanism | journal=Economic Theory | year=1999| volume= 14| pages= 155–180 | doi=10.1007/s001990050286| s2cid=154378924 }} 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).{{cite book |last= Myerson |first=Roger |title=Game Theory| year=1991 |location=Cambridge | publisher =Harvard University Press |isbn=0-674-34115-5 }}) 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),{{cite journal

| last1=Myerson | first1=Roger B.

| s2cid=12282691

| title=Optimal Auction Design

| journal=Mathematics of Operations Research

| year=1981

| volume=6

| issue=1

| pages=58–73

| doi=10.1287/moor.6.1.58}} the revenue equivalence theorem (for auctions), the Green–Laffont–Holmstrom theorem (Green and Laffont, 1979; Holmstrom, 1979),{{cite book |last1=Green |first1=J. |last2=Laffont |first2=J. J. |title= Incentives in Public Decision Making| year=1979 | location = Amsterdam |publisher=North-Holland |isbn=0-444-85144-5 }} the Myerson–Satterthwaite inefficiency theorem (Myerson and Satterthwaite, 1983),{{cite journal |author1=Myerson, R. |author2=M. Satterthwaite| title=Efficient Mechanisms for Bilateral Trading | journal=Journal of Economic Theory| year=1983| volume=29 |issue=2| pages=265–281 | doi=10.1016/0022-0531(83)90048-0| url=http://www.kellogg.northwestern.edu/research/math/papers/469.pdf| hdl=10419/220829| hdl-access=free}} the Jehiel–Moldovanu impossibility theorems (Jehiel and Moldovanu, 2001),{{cite journal |last1=Jehiel |first1=Philippe |first2=Benny |last2=Moldovanu | title=Efficient Design with Interdependent Valuations| journal=Econometrica| year=2001| volume=69 | pages=1237–1259| issue=5 | doi=10.1111/1468-0262.00240|citeseerx=10.1.1.23.7639}} the McAfee–McMillan weak-cartels theorem (McAfee and McMillan, 1992),{{cite journal |author1=McAfee, R. Preston |author2=John McMillan | title=Bidding Rings| journal=American Economic Review | year=1992| volume=82 | pages=579–599| issue=3 |jstor=2117323 }} and Weber's martingale theorem (Weber, 1983),{{cite book | last= Weber |first=Robert |chapter=Multiple-Object Auctions |title=Auctions, Bidding, and Contracting: Uses and Theory | year=1983 |editor-first=R. |editor-last=Engelbrecht-Wiggans |editor2-first=M. |editor2-last=Shubik |editor3-first=R. M. |editor3-last=Stark |location=New York | publisher =New York University Press|pages=165–191 |isbn=0-8147-7827-5 |chapter-url=https://www.kellogg.northwestern.edu/research/math/papers/496.pdf }} etc. The details of these applications are provided in Chapter 3 of Milgrom (2004),{{cite book | author= Milgrom, Paul |title= Putting Auction Theory to Work| year=2004 | publisher = Cambridge University Press|url=https://books.google.com/books?id=AkeHTU7XW4kC|isbn= 9780521536721}} 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.

For a multidimensional parameter space $T\backslash subseteq\; \backslash mathbb\{R\}^\{K\}$, Theorem

1 can be applied to partial and directional derivatives of the value

function.{{Citation needed|date=January 2023}} If both the objective function $f$ and the value function $V$ are (totally) differentiable in $t$, Theorem 1 implies the envelope formula for their gradients:{{Citation needed|date=January 2023}} $\backslash nabla\; V\backslash left(\; t\backslash right)\; =\backslash nabla\; \_\{t\}f\backslash left(\; x,t\backslash right)$ for each $x\backslash in\; X^\{\backslash ast\; \}\backslash left(\; t\backslash right)$. 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 $t\_\{0\}$ and $t$.{{Citation needed|date=January 2023}} Namely, suppose that functions $f(x,\backslash cdot\; )$ are differentiable for all $x\backslash in\; X$ with $|\backslash nabla\; \_\{t\}f(x,t)|\backslash leq\; B$ for all $x\backslash in\; X,$ $t\backslash in\; T$. A smooth path from $t\_\{0\}$ to $t$ is described by a differentiable mapping $\backslash gamma\; :\backslash left[\; 0,1\backslash right]\; \backslash rightarrow\; T$ with a bounded derivative, such that $\backslash gamma\; \backslash left(\; 0\backslash right)\; =t\_\{0\}$ and $\backslash gamma\; \backslash left(\; 1\backslash right)\; =t$.{{Citation needed|date=January 2023}} 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 $\backslash nabla\; \_\{t\}f(x^\{\backslash ast\; \}(t),t)$ of the objective function along the path:{{Citation needed|date=January 2023}}

:$V(t)-V(t\_\{0\})=\backslash int\_\{\backslash gamma\; \}\backslash nabla\; \_\{t\}f(x^\{\backslash ast\; \}(s),s)\backslash cdot\; ds.$

In particular, for $t=t\_\{0\}$, this establishes that cyclic path integrals along any smooth path $\backslash gamma$ must be zero:{{Citation needed|date=January 2023}}

:$\backslash int\; \backslash nabla\; \_\{t\}f(x^\{\backslash ast\; \}(s),s)\backslash cdot\; ds=0.$

This "integrability condition" plays an important role in mechanism design with multidimensional types, constraining what kind of choice rules $x^\{\backslash ast\; \}$ can be sustained by mechanism-induced menus $X\backslash subseteq\; \backslash bar\{X\}$.{{Citation needed|date=January 2023}} In application to producer theory, with $x\backslash in\; X\backslash subseteq\; \backslash mathbb\{R\}^\{L\}$ being the firm's production vector and $t\backslash in\; \backslash mathbb\{R\}^\{L\}$ being the price vector, $f\backslash left(\; x,t\backslash right)\; =t\backslash cdot\; x$, and the integrability condition says that any rationalizable supply function $x^\{\backslash ast\; \}$ must satisfy

:$\backslash int\; x^\{\backslash ast\; \}(s)\backslash cdot\; ds=0.$

When $x^\{\backslash ast\; \}$ is continuously differentiable, this integrability condition is equivalent to the symmetry of the substitution matrix $\backslash left(\backslash partial\; x\_\{i\}^\{\backslash ast\; \}\backslash left(\; t\backslash right)\; /\backslash partial\; t\_\{j\}\backslash right)\; \_\{i,j=1\}^\{L\}$. (In consumer theory, the same argument applied to the expenditure minimization problem yields symmetry of the Slutsky matrix.)

Suppose now that the feasible set $X\backslash left(\; t\backslash right)$ depends on the parameter, i.e.,

:$V(t)\; =\backslash sup\_\{x\backslash in\; X\backslash left(\; t\backslash right)\; \}f(x,t)$

:$X^\{\backslash ast\; \}(t)\; =\backslash \{x\backslash in\; X\backslash left(\; t\backslash right)\; :f(x,t)=V(t)\backslash \}\backslash text\{,\; \}$

where $X\backslash left(\; t\backslash right)\; =\backslash left\backslash \{\; x\backslash in\; X:g\backslash left(\; x,t\backslash right)\; \backslash geq\; 0\backslash right\backslash \}$ for some $g:X\backslash times\; \backslash left[\; 0,1\backslash right]\; \backslash rightarrow\; \backslash mathbb\{R\}^\{K\}.$

Suppose that $X$ is a convex set, $f$ and $g$ are concave in $x$, and there exists $\backslash hat\{x\}\backslash in\; X$ such that $g\backslash left(\; \backslash hat\{x\},t\backslash right)\; >0$ for all $t\backslash in\; \backslash left[\; 0,1\backslash right]$. Under these assumptions, it is well known that the above constrained optimization program can be represented as a saddle-point problem for the Lagrangian $L\backslash left(\; x,\backslash lambda,t\backslash right)\; =f(x,t)+\backslash lambda\backslash cdot\; g\backslash left(\; x,t\backslash right)$, where $\backslash lambda\; \backslash in\; \backslash mathbb\{R\}\_\{+\}^\{K\}$ is the vector of Lagrange multipliers chosen by the adversary to minimize the Lagrangian.{{cite book | author= Luenberger, D. G. |title= Optimization by Vector Space Methods| year=1969 | publisher = New York: John Wiley & Sons|url=https://books.google.com/books?id=lZU0CAH4RccC|isbn= 9780471181170}}{{page needed|date=February 2017}} This allows the application of Milgrom and Segal's (2002, Theorem 4) envelope theorem for saddle-point problems,{{cite journal |author1=Milgrom, Paul |author2=Ilya Segal | title= Envelope Theorems for Arbitrary Choice Sets| journal=Econometrica| year=2002| volume=70 | pages=583–601 | issue=2 | doi=10.1111/1468-0262.00296|citeseerx=10.1.1.217.4736 }} under the additional assumptions that $X$ is a compact set in a normed linear space, $f$ and $g$ are continuous in $x$, and $f\_\{t\}$ and $g\_\{t\}$ are continuous in $\backslash left(\; x,t\backslash right)$. In particular, letting $\backslash left(\; x^\{\backslash ast\}(t),\backslash lambda^\{\backslash ast\; \}\backslash left(\; t\backslash right)\; \backslash right)$ denote the Lagrangian's saddle point for parameter value $t$, the theorem implies that $V$ is absolutely continuous and satisfies

:$V(t)=V(0)+\backslash int\_\{0\}^\{t\}L\_\{t\}(x^\{\backslash ast\; \}(s),\backslash lambda^\{\backslash ast\; \}\backslash left(\; s\backslash right)\; ,s)ds.$

For the special case in which $f\backslash left(\; x,t\backslash right)$ is independent of $t$, $K=1$, and $g\backslash left(\; x,t\backslash right)\; =h\backslash left(\; x\backslash right)\; +t$, the formula implies that $V^\{\backslash prime\; \}(t)=L\_\{t\}(x^\{\backslash ast\; \}(t),\backslash lambda^\{\backslash ast\; \}\backslash left(\; t\backslash right)\; ,t)=\backslash lambda^\{\backslash ast\}\backslash left(\; t\backslash right)$ for a.e. $t$. That is, the Lagrange multiplier $\backslash lambda^\{\backslash ast\}\backslash left(\; t\backslash right)$ on the constraint is its "shadow price" in the optimization program.{{cite book | author= Rockafellar, R. T. |title= Convex Analysis| year=1970 | publisher = Princeton: Princeton University Press|url=https://books.google.com/books?id=1TiOka9bx3sC|isbn= 0691015864 | page=280}}

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.

{{Div col|colwidth=20em}}

• Maximum theorem
• Danskin's theorem
• Hotelling's lemma
• Le Chatelier's principle
• Roy's identity
• Value function

{{Div col end}}

{{Reflist|30em}}

Category:Production economics

Category:Calculus of variations

Category:Economics theorems

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