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In [[economics]], '''returns to scale''' and '''[[economies of scale]]''' are related but different terms that describe what happens as the scale of production increases in the long run, when all [[factor of production|input]] levels including physical [[capital (economics)|capital]] usage are variable (chosen by the firm). The term '''returns to scale''' arises in the context of a firm's [[production function]].  It explains the behaviour of the rate of increase in output (production) relative to the associated increase in the inputs (the factors of production) in the long run. In the long run all factors of production are variable and subject to change due to a given increase in size (scale).
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


The laws of returns to scale are a set of three interrelated and sequential laws:
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Law of Increasing Returns to Scale,
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Law of Constant Returns to Scale,
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and Law of Diminishing returns to Scale.
If output increases by that same proportional change as all inputs change then there are '''constant returns to scale''' (CRS). If output increases by less than that proportional change in inputs, there are '''decreasing returns to scale''' (DRS). If output increases by more than that proportional change in inputs, there are '''increasing returns to scale''' (IRS). A firm's production function could exhibit different types of returns to scale in different ranges of output.  Typically, there could be increasing returns at relatively low output levels, decreasing returns at relatively high output levels, and constant returns at one output level between those ranges.{{Citation needed|date=January 2012}}


In mainstream microeconomics, the returns to scale faced by a firm are purely technologically imposed and are not influenced by economic decisions or by market conditions (i.e., conclusions about returns to scale are derived from the specific mathematical structure of the production function ''in isolation'').
Registered users will be able to choose between the following three rendering modes:


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


When all inputs increase by a factor of 2, new values for output will be:
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


* Twice the previous output if there are constant returns to scale (CRS)
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


* Less than twice the previous output if there are decreasing returns to scale (DRS)
<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].


* More than twice the previous output if there are increasing returns to scale (IRS)
==Demos==


Assuming that the factor costs are constant (that is, that the firm is a perfect competitor in all input markets), a firm experiencing constant returns will have constant [[cost curve|long-run average costs]], a firm experiencing decreasing returns will have increasing long-run average costs, and a firm experiencing increasing returns will have decreasing long-run average costs.<ref>{{cite journal |last=Gelles |first=Gregory M. |last2=Mitchell |first2=Douglas W. |title=Returns to scale and economies of scale: Further observations |journal=[[Journal of Economic Education]] |volume=27 |year=1996 |issue=3 |pages=259–261 |jstor=1183297 }}</ref><ref>{{cite book |last=Frisch |first=R. |title=Theory of Production |location=Dordrecht |publisher=D. Reidel |year=1965 }}</ref><ref>{{cite book |last=Ferguson |first=C. E. |title=The Neoclassical Theory of Production and Distribution |location=London |publisher=Cambridge University Press |year=1969 |isbn=0-521-07453-3 }}</ref>  However, this relationship breaks down if the firm does not face perfectly competitive factor markets (i.e., in this context, the price one pays for a good does depend on the amount purchased). For example, if there are increasing returns to scale in some range of output levels, but the firm is so big in one or more input markets that increasing its purchases of an input drives up the input's per-unit cost, then the firm could have diseconomies of scale in that range of output levels. Conversely, if the firm is able to get bulk discounts of an input, then it could have economies of scale in some range of output levels even if it has decreasing returns in production in that output range.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


==Formal definitions==


Formally, a production function <math>\ F(K,L)</math> is defined to have:
* accessibility:
*Constant returns to scale if (for any constant ''a''  greater than 0) <math>\ F(aK,aL)=aF(K,L) </math>
** 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]]
*Increasing returns to scale if (for any constant ''a'' greater than 1) <math>\ F(aK,aL)>aF(K,L), </math>
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
*Decreasing returns to scale if (for any constant ''a'' greater than 1) <math>\ F(aK,aL)<aF(K,L) </math>
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
where ''K'' and ''L'' are factors of production—capital and labor, respectively.
** [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.


==Formal example==
==Test pages ==


The [[Cobb-Douglas]] functional form has constant returns to scale when the sum of the exponents adds up to one.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
The function is:
*[[Displaystyle]]
:<math>\ F(K,L)=AK^{b}L^{1-b}</math>
*[[MathAxisAlignment]]
where <math>A > 0</math> and <math>0 < b < 1</math>.  Thus
*[[Styling]]
:<math>\ F(aK,aL)=A(aK)^{b}(aL)^{1-b}=Aa^{b}a^{1-b}K^{b}L^{1-b}=aAK^{b}L^{1-b}=aF(K,L).</math>
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


But if the Cobb-Douglas production function has its general form
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
:<math>\ F(K,L)=AK^{b}L^{c}</math>
==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 .
with <math>0<c<1,</math> then there are increasing returns if ''b'' + ''c'' > 1 but decreasing returns if ''b'' + ''c'' < 1, since
 
:<math>\ F(aK,aL)=A(aK)^{b}(aL)^{c}=Aa^{b}a^{c}K^{b}L^{c}=a^{b+c}AK^{b}L^{c}=a^{b+c}F(K,L),</math>
 
which is greater than or less than <math>aF(K,L)</math> as ''b''+''c'' is greater or less than one.
 
==See also==
{{Portal|Economics}}
*[[Diseconomies of scale]]
*[[Economies of agglomeration]]
*[[Economies of scope]]
*[[Experience curve effects]]
*[[Ideal firm size]]
*[[Homogeneous function]]
*[[Mohring effect]]
*[[Moore's law]]
 
==References==
{{reflist}}
 
==Further reading==
* Susanto Basu (2008). "Returns to scale measurement," ''[[The New Palgrave Dictionary of Economics]]'', 2nd Edition. [http://www.dictionaryofeconomics.com/article?id=pde2008_I000297&edition=current&q=Increasing%20Returns&topicid=&result_number=5 Abstract.]
* [[James M. Buchanan]] and Yong J. Yoon, ed. (1994) ''The Return to Increasing Returns''. U.Mich. Press. Chapter-preview [http://books.google.com/books?id=d4yFu-yVn1AC&printsec=find&pg=PR5=false#v=onepage&q&f=false links.]
* John Eatwell (1987). "Returns to scale," ''[[The New Palgrave: A Dictionary of Economics]]'', v. 4, pp.&nbsp;165–66.
* [[Joaquim Silvestre]] (1987). "Economies and diseconomies of scale," ''The New Palgrave: A Dictionary of Economics'', v. 2, pp.&nbsp;80–84.
* Spirros Vassilakis (1987). "Increasing returns to scale,"  ''The New Palgrave: A Dictionary of Economics'', v. 2, pp.&nbsp;761–64.
 
==External links==
* [http://internationalecon.com/v1.0/ch80/80c020.html Economies of Scale and Returns to Scale]
* [https://www.youtube.com/watch?v=AttvGU47Eg8 Video Lecture on Returns to Scale in Macroeconomics]
 
{{microeconomics}}
 
[[Category:Microeconomics]]
[[Category:Production economics]]

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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