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This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
'''Gossen's Second “Law”''', named for [[Hermann Heinrich Gossen]] (1810–1858), is the assertion that an [[Economics|economic]] agent will allocate his or her expenditures such that the ratio of the [[marginal utility]] of each [[Good (economics)|good]] or [[Service (economics)|service]] to its price (the marginal expenditure necessary for its acquisition) is equal to that for every other good or service. Formally,
:<math>\frac{\partial U/\partial x_i}{p_i}=\frac{\partial U/\partial x_j}{p_j}~\forall\left(i,j\right)</math>
where
* <math>U</math> is [[utility]]
* <math>x_i</math> is quantity of the <math>i</math>-th good or service
* <math>p_i</math> is the price of the <math>i</math>-th good or service


== Informal derivation ==
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* 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.


Imagine that an agent has spent money on various sorts of goods or services.  If the last unit of currency spent on goods or services of one sort bought a quantity with <em>less</em> marginal utility than that which would have been associated with the quantity of another sort that could have been bought with the money, then the agent would have been ''better off'' instead buying more of that other good or service.  Assuming that goods and services are continuously divisible, the only way that it is possible that the marginal expenditure on one good or service should not yield more utility than the marginal expenditure on the other (or ''vice versa'') is if the marginal expenditures yield ''equal'' utility.
Registered users will be able to choose between the following three rendering modes:


== Formal derivation ==
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


Assume that utility, goods, and services have the requisite properties so that <math>\partial U/\partial x_i</math> is [[Well-defined|well defined]] for each good or service. An agent then optimizes
<!--'''PNG''' (currently default in production)
:<math>U\left(x_1 ,x_2 ,\dots,x_n\right)</math>
:<math forcemathmode="png">E=mc^2</math>
subject to a [[budget constraint]]
:<math>W\geq\sum_{i=1}^n \left(p_i\cdot x_i \right)</math>
where
* <math>W</math> is the total available sum of money
Using the method of [[Lagrange multipliers]], one constructs the function
:<math>\mathcal{L}\left(x_1 ,x_2 ,\dots,x_n ,\lambda\right)=U\left(x_1 ,x_2 ,\dots,x_n\right)+\lambda\cdot\left[W-\sum_{i=1}^n \left(p_i\cdot x_i \right)\right]</math>
and finds the first-order conditions for optimization as
:<math>\frac{\partial\mathcal{L}}{\partial\lambda}=0</math>
(which simply implies that all of <math>W</math> will be spent) and
:<math>\frac{\partial\mathcal{L}}{\partial x_i}=0~~\forall i</math>
so that
:<math>\frac{\partial U}{\partial x_i}-\lambda\cdot p_i =0~~\forall i</math>
which is algebraically equivalent to
:<math>\frac{\partial U/\partial x_i}{p_i}=\lambda~~\forall i</math>
Since every such ratio is equal to <math>\lambda</math>, the ratios are all equal one to another:
:<math>\frac{\partial U/\partial x_i}{p_i}=\frac{\partial U/\partial x_j}{p_j}~\forall\left(i,j\right)</math>


(Note that, as with any maximization using first-order conditions, the equations will hold only if the utility function satisfies specific concavity requirements and does not have maxima on the edges of the set over which one is maximizing.)
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


== References ==
<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].


* Gossen, Hermann Heinrich; ''Die Entwicklung der Gesetze des menschlichen Verkehrs und der daraus fließenden Regeln für menschliches Handeln'' (1854).  Translated into English as ''The Laws of Human Relations and the Rules of Human Action Derived Therefrom'' (1983) MIT Press, ISBN 0-262-07090-1.
==Demos==


== See also ==
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


* [[Gossen's laws]]
* [[Hermann Heinrich Gossen]]
* [[Marginal utility]]
* [[Marginalism]]


{{DEFAULTSORT:Gossen's Second Law}}
* accessibility:
[[Category:Marginal concepts]]
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
[[Category:Consumer theory]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
[[Category:Utility]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** 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.


[[nl:Tweede wet van Gossen]]
==Test pages ==
 
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]
 
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==Bug reporting==
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 .

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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