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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).
{{Infobox Polyhedron with net |
  Image_File=Bilunabirotunda.png|
  Polyhedron_Type=[[Johnson solid|Johnson]]<br>[[disphenocingulum|J<sub>90</sub>]] - '''J<sub>91</sub>''' - [[triangular hebesphenorotunda|J<sub>92</sub>]]|
  Face_List=2x4 [[triangle]]s<br>2 [[Square (geometry)|square]]s<br>4 [[pentagon]]s|
  Edge_Count=26|
  Vertex_Count=14|
  Symmetry_Group=''D''<sub>2h</sub>||
  Vertex_List=4(3.5<sup>2</sup>) <br> 8(3.4.3.5) <br> 2(3.5.3.5)|
  Dual=-|
  Property_List=[[Convex set|convex]]|
  Net_Image_File=Johnson solid 91 net.png
}}


The laws of returns to scale are a set of three interrelated and sequential laws:
In [[geometry]], the '''bilunabirotunda''' is one of the [[Johnson solid]]s (''J''<sub>91</sub>). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the [[Platonic solid|Platonic]] and [[Archimedean solid|Archimedean]] solids.
Law of Increasing Returns to Scale,
Law of Constant Returns to Scale,
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'').
The 92 Johnson solids were named and described by [[Norman Johnson (mathematician)|Norman Johnson]] in 1966.


==Example==
== Cartesian coordinates ==
<p>The following define the vertices of a bilunabirotunda centered at the origin with edge length 1: </p>
<dl>
<dd><math>\left(0, 0, \pm\frac{\varphi}{2}\right)</math></dd>
<dd><math>\left(\pm\frac{(\varphi+1)}{2}, \pm\frac{1}{2}, 0\right)</math></dd>
<dd><math>\left(\pm\frac{1}{2},\pm\frac{\varphi}{2},\pm\frac{1}{2}\right)</math></dd>
</dl>
<p>where <math>\varphi=\frac{1+\sqrt{5}}{2} </math> is the golden ratio.</p>


When all inputs increase by a factor of 2, new values for output will be:
==External links==
 
* {{MathWorld | urlname=JohnsonSolid | title=Johnson Solid}}
* Twice the previous output if there are constant returns to scale (CRS)
** {{MathWorld | urlname=Bilunabirotunda | title=Bilunabirotunda}}
 
* Less than twice the previous output if there are decreasing returns to scale (DRS)
 
* More than twice the previous output if there are increasing returns to scale (IRS)
 
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.
 
==Formal definitions==
 
Formally, a production function <math>\ F(K,L)</math> is defined to have:
*Constant returns to scale if (for any constant ''a''  greater than 0) <math>\ F(aK,aL)=aF(K,L) </math>
*Increasing returns to scale if (for any constant ''a'' greater than 1) <math>\ F(aK,aL)>aF(K,L), </math>
*Decreasing returns to scale if (for any constant ''a'' greater than 1) <math>\ F(aK,aL)<aF(K,L) </math>
where ''K'' and ''L'' are factors of production—capital and labor, respectively.
 
==Formal example==
 
The [[Cobb-Douglas]] functional form has constant returns to scale when the sum of the exponents adds up to one.
The function is:
:<math>\ F(K,L)=AK^{b}L^{1-b}</math>
where <math>A > 0</math> and <math>0 < b < 1</math>.  Thus
:<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>
 
But if the Cobb-Douglas production function has its general form


:<math>\ F(K,L)=AK^{b}L^{c}</math>
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}}
{{Polyhedron-stub}}
[[Category:Johnson solids]]


[[Category:Microeconomics]]
[[es:Bilunabirrotonda]]
[[Category:Production economics]]
[[eo:J91]]
[[fr:Birotonde bilunaire]]
[[it:Bilunabirotunda]]
[[nl:Bilunabirotonde]]
[[zh:雙新月雙罩帳]]

Revision as of 08:05, 13 August 2014

47 yr old Intelligence Officer Kepp from Alert Bay, really loves fencing, new commercial property launch in singapore developers in singapore and actions. Is enthused how vast the earth is after paying a visit to Blaenavon Industrial Landscape.

In geometry, the bilunabirotunda is one of the Johnson solids (J91). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.

The 92 Johnson solids were named and described by Norman Johnson in 1966.

Cartesian coordinates

The following define the vertices of a bilunabirotunda centered at the origin with edge length 1:

(0,0,±φ2)
(±(φ+1)2,±12,0)
(±12,±φ2,±12)

where φ=1+52 is the golden ratio.

External links



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Template:Polyhedron-stub

es:Bilunabirrotonda eo:J91 fr:Birotonde bilunaire it:Bilunabirotunda nl:Bilunabirotonde zh:雙新月雙罩帳

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