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The '''envelope theorem''' is a theorem about optimization problems ([[utility maximization problem|max]] & min) in [[microeconomics]].  It may be used to prove [[Hotelling's lemma]], [[Shephard's lemma]], and [[Roy's identity]]. It also allows for easier computation of [[comparative statics]] in generalized economic models.
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The theorem exists in two versions, a regular version (unconstrained optimization) and a generalized version (constrained optimization). The regular version can be obtained from the general version because unconstrained optimization is just the special case of constrained optimization with no constraints (or constraints that are always satisfied, i.e. constraints that are identities such as <math>0 = 0</math> or <math>(x+1)^2=x^2+2x+1</math>.
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The theorem gets its name from the fact that it shows that a less constrained maximization (or minimization) problem (where some parameters are turned into variables) is the upper (or lower for min) envelope of the original problem. For example, see [[cost curve|cost minimization]], and compare the long-run (less constrained) and short-run (more constrained &ndash; some factors of production are fixed) minimization problems.
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For the theorem to hold, the functions being dealt with must have certain [[well-behaved]] properties. Specifically, the correspondence mapping parameter values to optimal choices must be differentiable, with it being single-valued (and hence a function) a necessary but not sufficient condition.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


The theorem is described below. Note that bold face represents a vector.
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


==Envelope theorem==
'''source'''
A curve in a two dimensional space is best represented by the parametric equations like x(t) and y(t).
:<math forcemathmode="source">E=mc^2</math> -->
The family of curves can be represented in the form <math>g(x,y,c) = 0 </math> where c is the parameter.
Generally, the envelope theorem involves one parameter but there can be more than one parameter involved as well.


The envelope of a family of curves g(x,y,c) = 0 is a curve such that at each point on the curve there is some member of the family that touches that particular point tangentially.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
This forms a curve or surface that is tangential to every curve in the family of curves forming an envelope.


Consider an arbitrary maximization (or minimization) problem where the objective function <math>f(\bold x,\bold r)</math> depends on some parameters <math>\bold r</math>:
==Demos==


:<math>f^*(\bold r) = \max_{\bold x} f(\bold x,\bold r)\,</math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


The function <math>f^*(\bold r)</math> is the problem's optimal-value function &mdash; it gives the maximized (or minimized) value of the objective function <math>f(\bold x,\bold r)</math> as a function of its parameters <math>\bold r</math>.


Let <math>\bold x^*(\bold r) </math> be the (arg max) value of <math>\bold x</math>, expressed in terms of the parameters, that solves the optimization problem, so that <math>f^*(\bold r) = f(\bold x^*(\bold r), \bold r)</math>. The envelope theorem tells us how <math>f^*(\bold r)</math> changes as a parameter changes, namely:
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** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


:<math>\frac{d\ f^*(\bold r)}{d\ r_i} = \frac{\partial f(\bold x,\bold r)}{ \partial r_i} \Bigg|_{\bold x = \bold x^*(\bold r)}</math>
==Test pages ==


That is, the derivative of <math>f^*(\bold r)</math> with respect to <math>r_i</math> is given by the partial
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
derivative of <math>f(\bold x,\bold r)</math> with respect to <math>r_i</math>, holding <math>\bold x</math> fixed, and then evaluating at the optimal choice <math>\bold x = \bold x^*(\bold r)</math>.
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==General envelope theorem==
*[[Inputtypes|Inputtypes (private Wikis only)]]
There also exists a version of the theorem, called the '''general envelope theorem''', used in constrained optimization problems which relates the partial derivatives of the optimal-value function to the partial derivatives of the [[Lagrangian]] function.
*[[Url2Image|Url2Image (private Wikis only)]]
 
==Bug reporting==
We are considering the following optimization problem in formulating the theorem (max may be replaced by min, and all results still hold):
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
 
:<math>\max_{\bold x} f(\bold x,\bold r) \;\; s.t. \;\; \bold g(\bold x,\bold r) = \bold 0</math>
 
Which gives the Lagrangian function:
 
:<math>\mathcal{L}(\bold x,\bold r) = f(\bold x,\bold r) - \boldsymbol{\lambda} \cdot \bold g(\bold x,\bold r)</math>
 
Where:
 
:<math>\boldsymbol{\lambda} = (\lambda_{1},\dots,\lambda_{n})</math>
 
:<math>\bold g(\bold x,\bold r) = (g_{1}(\bold x,\bold r),\dots,g_{n}(\bold x,\bold r))</math>
 
:<math>\bold 0 = (0,\dots,0) \in \mathbb{R}^n</math>
 
:<math>\cdot</math> is the [[dot product]]
 
Then the '''general envelope theorem''' is:
 
:<math>\frac{d f^*(\bold r)}{d r_i} = \frac{\partial \mathcal{L}(\bold x,\bold r)}{\partial r_i} \Bigg|_{ \bold x = \bold x^*(\bold r), \ \boldsymbol{\lambda} = \boldsymbol{\lambda}(\bold r) }</math>
 
Note that the Lagrange multipliers <math>\boldsymbol{\lambda}</math> are treated as constants during differentiation of the Lagrangian function, then their values as functions of the parameters are substituted in afterwards.
 
==Envelope theorem in generalized calculus==
In the [[calculus of variations]], the envelope theorem relates [[evolute]]s to single [[path (topology)|paths]].  This was first proved by [[Jean Gaston Darboux]] and [[Ernst Zermelo]] (1894) and [[Adolf Kneser]] (1898).  The theorem can be stated as follows:
 
"''When a single-parameter family of external paths from a fixed point ''O'' has an [[envelope (mathematics)|envelope]], the integral from the fixed point to any point ''A'' on the envelope  equals the integral from the fixed point to any second point ''B'' on the envelope plus the integral along the envelope to the first point on the envelope'', ''J''<sub>''OA''</sub> = ''J''<sub>''OB''</sub> + ''J''<sub>''BA''</sub>."<ref>{{cite book |last=Kimball |first=W. S. |title=Calculus of Variations by Parallel Displacement |location=London |publisher=Butterworth |page=292 |year=1952 }}</ref>
 
==See also==
*[[Arg max]]
*[[Optimization problem]]
*[[Random optimization]]
*[[Simplex algorithm]]
*[[Topkis's Theorem]]
*[[Variational calculus]]
 
==Notes==
{{Reflist}}
 
==References==
*{{cite journal |last=Sydsaeter |first=Knut |last2=Hammond |first2=Peter |title=Essential Mathematics for Economic Analysis |edition=3rd |location=Harlow |publisher=Prentice Hall |year=2008 |isbn=978-0-273-71324-1 }}
 
[[Category:Underlying principles of microeconomic behavior]]
[[Category:Calculus of variations]]
[[Category:Economics theorems]]
[[Category:Theorems in analysis]]
 
[[de:Umhüllungssatz]]
[[it:Teorema dell'inviluppo]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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