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The Envelope Theorem is a a result about the differentiability properties of the objective function of an 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.
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==Envelope theorems for arbitrary choice sets==
 
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Let <math>X</math> denote the choice set and let the relevant parameter be <math>t\in
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\lbrack 0,1]</math>. Letting <math>f:X\times \lbrack 0,1]\rightarrow R</math> denote the
parameterized objective function, the value function <math>V</math> and the optimal
choice correspondence (set-valued function) <math>X^{\ast }</math> are given by:


:<math> V(t) =\sup_{x\in X}f(x,t)</math(1)
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:<math> X^{\ast }(t) =\{x\in X:f(x,t)=V(t)\}</math(2)
 
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"Envelope theorems" describe sufficient
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conditions for the value function <math>V</math> to be differentiable in the parameter <math>
t</math> and describe its derivative as


:<math> V^{\prime }\left( t\right) =f_{t}\left( x,t\right) \text{ for each }x\in
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X^{\ast }\left( t\right) .</math(3)  
 
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Namely, the derivative of the value function with respect to the parameter
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equals the partial derivative of the objective function with respect to <math>t</math>
holding the maximizer fixed at its optimal level. (The term derives from
describing the graph of <math>V</math> as the "upper
envelope" of the graphs of the parameterized family of
functions <math>\left\{ f\left( x,\cdot \right) \right\} _{x\in X}</math>.)


Traditional envelope theorem derivations use the first-order condition for (1), which requires that the choice set <math>X</math> have the convex and
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topological structure, and the objective function <math>f</math> be differentiable in
 
the variable <math>x</math>. (The argument is that changes in the maximizer have only a
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"second-order effect" at the optimum and
 
so can be ignored.) However, in many applications such as the analysis of
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incentive constraints in contract theory and game theory, nonconvex
 
production problems, and "monotone" or
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"robust" comparative statics, the choice
 
sets and objective functions generally lack the topological and convexity
  <li>x</li>
properties required by the traditional envelope theorems.
 
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</ul>


[[Paul Milgrom]] and Segal (2002) observe that the traditional envelope formula holds
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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 <math>t\in \left( 0,1\right) </math> and <math>x\in X^{\ast }\left(
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t\right) </math>. If both <math>V^{\prime }\left( t\right) </math> and <math>f_{t}\left(
 
x,t\right) </math> exist, the envelope formula (3) holds.
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'''Proof:''' (1) implies that for <math>x\in X^{\ast }\left( t\right) </math>,
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:<math> \max_{s\in \left[ 0,1\right] }\left[ f\left( x,s\right) -V\left( s\right)\right] =f\left( x,t\right) -V\left( t\right) =0. </math>
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Under the assumptions, the objective function of the displayed maximization
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problem is differentiable at <math>s=t</math>, and the first-order condition for this
maximization is exactly (3). Q.E.D.


While differentiability of the value function in general requires strong
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assumptions, in many applications weaker conditions such as absolute
 
continuity, differentiability almost everywhere, or left- and right-
  <li>http://forum.dvud.de/activity/p/8241/</li>
differentiability, suffice. In particular, Milgrom and Segal 's (2002)
 
Theorem 2 offers a sufficient condition for <math>V</math> to be absolutely continuous,
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which means that it is differentiable almost everywhere and can be
 
represented as an integral of its derivative:
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'''Theorem 2:''' Suppose that <math>f(x,\cdot )</math> is absolutely continuous for
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all <math>x\in X</math>. Suppose also that there exists an integrable function <math>b:[0,1]</math>
<math>\rightarrow </math> <math>\mathbb{R}_{+}</math> such that <math>|f_{t}(x,t)|\leq b(t)</math> for all <math>
x\in X</math> and almost all <math>t\in \lbrack 0,1]</math>. Then <math>V</math> is absolutely
continuous. Suppose, in addition, that <math>f(x,\cdot )</math> is differentiable for
all <math>x\in X</math>, and that <math>X^{\ast }(t)\neq \varnothing </math> almost everywhere on <math>
[0,1]</math>. Then for any selection <math>x^{\ast }(t)\in X^{\ast }(t)</math>,


:<math> V(t)=V(0)+\int_{0}^{t}f_{t}(x^{\ast }(s),s)ds. </math(4)
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'''Proof:''' Using (1), observe that for any <math>t^{\prime
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},t^{\prime \prime }\in \lbrack 0,1]</math> with <math>t^{\prime }<t^{\prime \prime }</math>,


:<math> |V(t^{\prime \prime })-V(t^{\prime })| \leq \sup_{x\in X}|f(x,t^{\prime\prime })-f(x,t^{\prime })|
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=\sup_{x\in X}\left\vert \int_{t^{\prime }}^{t^{\prime \prime
 
}}f_{t}(x,t)dt\right\vert \leq \int_{t^{\prime }}^{t^{\prime \prime
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}}\sup_{x\in X}|f_{t}(x,t)|dt\leq \int_{t^{\prime }}^{t^{\prime \prime
 
}}b(t)dt.</math>
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This implies that <math>V</math> is absolutely continuous. Therefore, <math>V</math> is
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differentiable almost everywhere, and using (3) yields (4). Q.E.D.


This result dispels the common misconception that nice behavior of the value
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function requires correspondingly nice behavior of the maximizer. Theorem 2
 
ensures the [[absolute continuity]] of the value function even though the
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maximizer may be discontinuous. In a similar vein, Milgrom and Segal's
 
(2002) Theorem 3 implies that the value function must be differentiable at <math>
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t=t_{0}</math> (and hence satisfy the envelope formula (3) when the
 
family <math>\left\{ f\left( x,\cdot \right) \right\} _{x\in X}</math> is
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equi-differentiable at <math>t_{0}\in \left( 0,1\right) </math> and <math>f_{t}\left(
 
X^{\ast }\left( t\right) ,t_{0}\right) </math> is single-valued and continuous at <math>
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t=t_{0}</math>, even if the maximizer is not differentiable at <math>t_{0}</math> (e.g., if <math>
 
X </math> is described by a set of inequality constraints and the set of binding
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constraints changes at <math>t_{0}</math>).
 
</ul>


==Applications of Envelope Theorems==
== hair comb Louis Vuitton Factory Outlet Usa ==


===Applications to Producer Theory===
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Theorem 1 implies [[Hotelling's lemma]] at any differentiability point of the
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profit function, and Theorem 2 implies the [[producer surplus]] formula.
Formally, let <math>\pi \left( p\right) </math> denote the profit function of a
price-taking firm with production set <math>X\subseteq \mathbb{R}^{L}</math> facing
prices <math>p\in \mathbb{R}^{L}</math> , and let <math>x^{\ast }\left( p\right) </math> denote
the firm's supply function, i.e.,


:<math> \pi (p)=\max_{x\in X}p\cdot x=p\cdot x^{\ast }\left( p\right) \text{.} </math>
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Let <math>t=p_{i}</math> (the price of good <math>i</math>) and fix the other goods' prices at <math>
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p_{-i}\in \mathbb{R}^{L-1}</math>. Applying Theorem 1 to <math>f(x,t)=tx_{i}+p_{-i}\cdot x_{-i}</math> yields <math>\partial \pi (p)\partial p_{i}=x_{i}^{\ast }(p)</math> (the
firm's optimal supply of good <math>i</math>). Applying Theorem 2 (whose assumptions
are verified when <math>p_{i}</math> is restricted to a bounded interval) yields


:<math>  
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\pi (t,p_{-i})-\pi (0,p_{-i})=\int_{0}^{p_{i}}x_{i}^{\ast }(s,p_{-i})ds, </math>
 
i.e. the producer surplus <math>\pi
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(t,p_{-i})-\pi (0,p_{-i})</math> can be obtained by integrating under the firm's
 
supply curve for good <math>i</math>.
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===Applications to Mechanism design and Auction Theory===
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Consider an agent whose utility function <math>f(x,t)</math> over outcomes <math>x\in \bar{X}
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</math> depends on his type <math>t\in \lbrack 0,1]</math>. Let <math>X\subseteq \bar{X}</math>
 
represent the "menu" of possible outcomes
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the agent could obtain in the mechanism by sending different messages. The
 
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and the set <math>X^{\ast }(t)</math> of the mechanism's equilibrium outcomes is given
 
by (2). Any selection <math>x^{\ast }(t)\in X^{\ast }(t)</math> is a choice rule
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implemented by the mechanism. Suppose that the agent's utility function <math>  
 
f(x,t)</math> is differentiable and absolutely continuous in <math>t</math> for all <math>x\in Y</math>,
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and that <math>\sup_{x\in \bar{X}}|f_{t}(x,t)|</math> is integrable on <math>[0,1]</math>. Then
 
Theorem 2 implies that the agent's equilibrium utility <math>V</math> in any mechanism
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implementing a given choice rule <math>x^{\ast }</math> must satisfy the integral
 
condition (4).
</ul>


The integral condition (4) is a key step in the analysis of mechanism design
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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 <math>x=\left( y,z\right) </math>, where <math>y</math> is the bidder's
probability of receiving the object and <math>z</math> is his expected payment, and the
bidder's expected utility takes the form <math>f\left( \left( y,z\right)
,t\right) =ty-z</math>. In this case, letting <math>\underline{t}</math> denote the bidder's
lowest possible type, the integral condition (4) for the bidder's
equilibrium expected utility <math>V</math> takes the form


:<math> V(t)-V(\underline{t})=\int_{0}^{t}y^{\ast }(s)ds. </math>
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(This equation can be interpreted as the producer surplus
== etc. Mont Blanc Australia Pens ==
formula for the firm whose production technology for
converting numeraire <math>z</math> into probability <math>y</math> of winning the object is
defined by the auction and which resells the object at a fixed price <math>t</math>).
This condition in turn yields Myerson's (1981) celebrated [[revenue equivalence|Revenue
Equivalence Theorem]]: the expected revenue generated in an auction in which
bidders have independent private values is fully determined by the bidders'
probabilities <math>y^{\ast }\left( t\right) </math> of getting the object for all
types <math>t</math> as well as by the expected payoffs <math>V(\underline{t})</math> 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
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Mirrlees (1971), Holmstrom (1979), Laffont and Maskin (1980), Riley and
 
Samuelson (1981), Fudenberg and Tirole (1991), and Williams (1999). While
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these authors derived and exploited the envelope theorem by restricting
 
attention to (piecewise) continuously differentiable choice rules or even
  <li>[http://www.xiaojinmixian.com/news/html/?91836.html http://www.xiaojinmixian.com/news/html/?91836.html]</li>
narrower classes, it may sometimes be optimal to implement a choice rule
 
that is not piecewise continuously differentiable. (One example is the class
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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
  <li>[http://www.nayears.com/home.php?mod=space&uid=20525 http://www.nayears.com/home.php?mod=space&uid=20525]</li>
and implies such important results as Holmstrom's lemma (Holmstrom, 1979),
 
Myerson's lemma (Myerson, 1981), the revenue equivalence theorem (for
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auctions), the Green-Laffont-Holmstrom theorem (Green and Laffont, 1979;
 
Holmstrom, 1979), the Myerson-Satterthwaite inefficiency theorem (Myerson
</ul>
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), etcThe
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===
== Or the baby. I am a creationist Buy Longchamp ==


For a multidimensional parameter space <math>T\subseteq \mathbb{R}^{K}</math>, Theorem
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1 can be applied to partial and directional derivatives of the value
 
function. If both the objective function <math>f</math> and the value function <math>V</math> are
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(totally) differentiable in <math>t</math>, Theorem 1 implies the envelope formula for
 
their gradients: <math>\nabla V\left( t\right) =\nabla _{t}f\left( x,t\right) </math>
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for each <math>x\in X^{\ast }\left( t\right) </math>. While total differentiability of
 
the value function may not be easy to ensure, Theorem 2 can be still applied
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along any smooth path connecting two parameter values <math>t_{0}</math> and <math>t</math>.
 
Namely, suppose that functions <math>f(x,\cdot )</math> are differentiable for all <math>  
  <li>[http://ukvotism.com/activity/p/422817/ http://ukvotism.com/activity/p/422817/]</li>
x\in X</math> with <math>|\nabla _{t}f(x,t)|\leq B</math> for all <math>x\in X,</math> <math>t\in T</math>. A
 
smooth path from <math>t_{0}</math> to <math>t</math> is described by a differentiable mapping <math>
  <li>[http://www.ewmsj.com/news/html/?230589.html http://www.ewmsj.com/news/html/?230589.html]</li>
\gamma :\left[ 0,1\right] \rightarrow T</math> with a bounded derivative, such
 
that <math>\gamma \left( 0\right) =t_{0}</math> and <math>\gamma \left( 1\right) =t</math>.
</ul>
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 <math>  
\nabla _{t}f(x^{\ast }(t),t)</math> of the objective function along the path:


:<math> V(t)-V(t_{0})=\int_{\gamma }\nabla _{t}f(x^{\ast }(s),s)\cdot ds. </math>
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In particular, for <math>t=t_{0}</math>, this establishes that cyclic path integrals
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along any smooth path <math>\gamma </math> must be zero:
 
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</ul>


:<math>\int \nabla _{t}f(x^{\ast }(s),s)\cdot ds=0. </math>
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This "integrability condition" plays an
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important role in mechanism design with multidimensional types, constraining
 
what kind of choice rules <math>x^{\ast }</math> can be sustained by mechanism-induced
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menus <math>X\subseteq \bar{X}</math>. In application to producer theory, with <math>x\in
 
X\subseteq \mathbb{R}^{L}</math> being the firm's production vector and <math>t\in
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\mathbb{R}^{L}</math> being the price vector, <math>f\left( x,t\right) =t\cdot x</math>, and
 
the integrability condition says that any rationalizable supply function <math>  
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x^{\ast }</math> must satisfy
 
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</ul>


:<math> \int x^{\ast }(s)\cdot ds=0. </math>
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When <math>x^{\ast }</math> is continuously differentiable, this integrability
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condition is equivalent to the symmetry of the [[substitution matrix]] <math>\left(
 
\partial x_{i}^{\ast }\left( t\right) /\partial t_{j}\right) _{i,j=1}^{L}</math>.
  <li>[http://enseignement-lsf.com/spip.php?article66#forum18522555 http://enseignement-lsf.com/spip.php?article66#forum18522555]</li>
(In [[consumer theory]], the same argument applied to the expenditure
 
minimization problem yields symmetry of the [[Slutsky matrix]].)
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===Applications to Parameterized Constraints===
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:<math> V(t) =\sup_{x\in X\left( t\right) }f(x,t) </math>
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Suppose that <math>X</math> is a convex set, <math>f</math> and <math>g</math> are concave in <math>x</math>, and there
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For the special case in which <math>f\left( x,t\right) </math> is independent of <math>t</math>, <math>
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Milgrom and Segal (2002) demonstrate that the generalized version of the
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*{{cite journal | author=Laffont, Jean-Jacques and Eric Maskin | title=A Differentiable Approach to Dominant Strategy Mechanisms| journal=Econometrica| year=1980| volume=48 | pages=1507–1520}}
 
 
</ul>
*{{cite book | author= Luenberger, D. G. |title= Optimization by Vector Space Methods| year=1969 | publisher = New York: John Wiley & Sons}}
 
*{{cite journal | author=McAfee, R. Preston and John McMillan | title=Bidding Rings| journal=American Economic Review | year=1992| volume=82 | pages=579–599| issue=3}}
 
*{{cite book | author= Milgrom, Paul |title= Putting Auction Theory to Work| year=2004 | publisher = Cambridge University Press}}
 
*{{cite journal | author=Milgrom, Paul and Ilya Segal | title= Envelope Theorems for Arbitrary Choice Sets| journal=Econometrica| year=2002| volume=70 | pages=583–601 | issue=2}}
 
*{{cite journal | author=Mirrlees, James | title= An Exploration in the Theory of Optimal Taxation| journal=Review of Economic Studies | year=2002| volume=38 | pages=175–208}}
 
*{{cite journal | author=Myerson, Roger | title= Optimal Auction Design| journal=Mathematics of Operations Research | year=1981| volume=6 | pages=58–73}}
 
*{{cite book | author= Myerson, Roger |title=Game Theory| year=1991 | publisher = Cambridge, Harvard University Press}}
 
*{{cite journal | author=Myerson, R. and M. Satterthwaite| title=Efficient Mechanisms for Bilateral Trading | journal=Journal of Economic Theory| year=1983| volume=29 | pages=265–281}}
 
*{{cite journal | author=Riley, John G. and William S. Samuelson | title=Optimal Auctions | journal=American Economic Review | year=1981| volume=71 | pages=381–392 | issue=3}}
 
*{{cite book | author= Rockafellar, R. T. |title= Convex Analysis| year=1970 | publisher = Princeton: Princeton University Press}}
 
*{{cite book | author= Weber, Robert |title="Multiple-Object Auctions" in Auctions, Bidding, and Contracting: Uses and Theory| year=1983 | publisher = R. Engelbrecht-Wiggans, M. Shubik and R. M. Stark. New York, New York University Press|pages=165–191}}
 
*{{cite journal | author=Williams, Steven | title=A Characterization of Efficient, Bayesian Incentive Compatible Mechanism | journal=Economic Theory | year=1999| volume= 14| pages= 155–180}}
 
 
 
[[Category:Underlying principles of microeconomic behavior]]
[[Category:Calculus of variations]]
[[Category:Economics theorems]]
[[Category:Theorems in analysis]]

Latest revision as of 10:00, 11 November 2014

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The main entry way to the Old Town begins from the Lokietek Square (Plac okietka), named after the medieval Polish King who granted city privilages to Lublin in 1317. The Old Town is guarded through the Cracow Gate (Brama Krakowska), which has been erected in 14th century included in the walls surrounding the whole city, after which renovated in18th centure, when it gained its current baroque appearance. After just a few steps you can get to the Market Square (Rynek) with an impressive building of Royal Tribunal (Trybuna Koronny) in the centre, built in 14th century as a City Hall and renovated in 18th century by a Nike Shox Nz Size 13 famous architect Merlini. It was functioning as a Supreme Court for the entire Eastern half Poland between 16th and 18th century.

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