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In [[abstract algebra]], an '''adelic algebraic group''' is a [[semitopological group]] defined by an [[algebraic group]] ''G'' over a [[number field]] ''K'', and the [[adele ring]] ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; the definition of the appropriate [[topological space|topology]] is straightforward only in case ''G'' is a [[linear algebraic group]]. In the case of ''G'' an [[abelian variety]] it presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawa numbers. Adelic algebraic groups are widely used in [[number theory]], particularly for the theory of [[automorphic representation]]s, and the [[arithmetic of quadratic form]]s.
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).


In case ''G'' is a linear algebraic group, it is an [[affine algebraic variety]] in affine ''N''-space. The topology on the adelic algebraic group <math>G(A)</math> is taken to be the [[subspace topology]] in ''A''<sup>''N''</sup>, the [[Cartesian product]] of ''N'' copies of the adele ring.
The laws of returns to scale are a set of three interrelated and sequential laws:
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}}


==Ideles==
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'').
An important example, the '''idele group''' ''I''(''K''), is the case of <math>G = GL_1</math>. Here the set of '''ideles''' (also ''idèles'' {{IPAc-en|ɪ|ˈ|d|ɛ|l|z}}) consists of the invertible adeles; but the topology on the idele group is ''not'' their topology as a subset of the adeles. Instead, considering that <math>GL_1</math> lies in two-dimensional [[affine space]] as the '[[hyperbola]]' defined parametrically by


:{(''t'', ''t''<sup>−1</sup>)},
==Example==


the topology correctly assigned to the idele group is that induced by inclusion in ''A''<sup>2</sup>; composing with a projection, it follows that the ideles carry a [[finer topology]] than the subspace topology from ''A''.
When all inputs increase by a factor of 2, new values for output will be:


Inside ''A''<sup>''N''</sup>, the product ''K''<sup>''N''</sup> lies as a [[discrete subgroup]]. This means that ''G''(''K'') is a discrete subgroup of ''G''(''A''), also. In the case of the idele group, the [[quotient group]]
* Twice the previous output if there are constant returns to scale (CRS)


:''I''(''K'')/''K''<sup>×</sup>
* Less than twice the previous output if there are decreasing returns to scale (DRS)


is the '''idele class group'''. It is closely related to (though larger than) the [[ideal class group]]. The idele class group is not itself compact; the ideles must first be replaced by the ideles of norm 1, and then the image of those in the idele class group is a [[compact group]]; the proof of this is essentially equivalent to the finiteness of the class number.
* More than twice the previous output if there are increasing returns to scale (IRS)


The study of the [[Galois cohomology]] of idele class groups is a central matter in [[class field theory]]. [[Character (group theory)|Characters]] of the idele class group, now usually called [[Hecke character]]s, give rise to the most basic class of [[L-function]]s.
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.


==Tamagawa numbers==
==Formal definitions==
{{see also|Weil conjecture on Tamagawa numbers}}


For more general ''G'', the '''Tamagawa number''' is defined (or indirectly computed) as the measure of
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.


:''G''(''A'')/''G''(''K'').
==Formal example==


[[Tsuneo Tamagawa]]'s observation was that, starting from an invariant [[differential form]] ω on ''G'', defined ''over K'', the measure involved was [[well-defined]]: while ω could be replaced by ''c''ω with ''c'' a non-zero element of ''K'', the [[product formula]] for [[valuation (algebra)|valuation]]s in ''K'' is reflected by the independence from ''c'' of the measure of the quotient, for the product measure constructed from ω on each effective factor. The computation of Tamagawa numbers for [[semisimple group]]s contains important parts of classical [[quadratic form]] theory.
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>


==History of the terminology==
But if the Cobb-Douglas production function has its general form
Historically the ''idèles'' were introduced  by {{harvs|txt|last=Chevalley|authorlink=Claude Chevalley|year=1936}} under the name "élément idéal", which is "ideal element" in French, which {{harvtxt|Chevalley|1940}} then abbreviated to "idèle". (In these papers he also gave the ideles a rather non-[[Hausdorff topology]].) This was to formulate [[class field theory]] for infinite extensions in terms of topological groups. {{harvtxt|Weil|1938}} defined (but did not name) the ring of adeles in the function field case and pointed out that Chevalley's group of ''Idealelemente''  was the group of invertible elements of this ring. {{harvtxt|Tate|1950}} defined the ring of adeles as a restricted direct product, though he called its elements "valuation vectors" rather than adeles.


{{harvtxt|Chevalley|1951}} defined the ring of adeles in the function field case, under the name "repartitions". The term ''adèle'' (short for additive idèles, and also a French girls' name) was in use shortly afterwards {{harv|Jaffard|1953}} and may have been introduced by [[André Weil]].  The general construction of adelic algebraic groups by {{harvtxt|Ono|1957}} followed the algebraic group theory founded by [[Armand Borel]] and [[Harish-Chandra]].
:<math>\ F(K,L)=AK^{b}L^{c}</math>
<!--
 
==Notes==
with <math>0<c<1,</math> then there are increasing returns if ''b'' + ''c'' > 1 but decreasing returns if ''b'' + ''c'' < 1, since
{{reflist}}-->
 
:<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==
==References==
*{{Citation | last1=Chevalley | first1=Claude | title=Généralisation de la théorie du corps de classes pour les extensions infinies. | language=French | jfm=62.1153.02  | year=1936 | journal=Journal de Mathématiques Pures et Appliquées  | volume=15 | pages=359–371}}
{{reflist}}
*{{Citation | last1=Chevalley | first1=Claude | title=La théorie du corps de classes | jstor=1969013 | mr=0002357  | year=1940 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=41 | pages=394–418}}
 
*{{Citation | last1=Chevalley | first1=Claude | title=Introduction to the Theory of Algebraic Functions of One Variable | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys, No. VI | mr=0042164  | year=1951}}
==Further reading==
*{{Citation | last1=Jaffard | first1=Paul | title=Anneaux d'adèles (d'après Iwasawa) | url=http://www.numdam.org/item?id=SB_1954-1956__3__23_0 | publisher=Secrétariat mathématique, Paris | series=Séminaire Bourbaki, | mr=0157859  | year=1953}}
* 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.]
*{{Citation | last1=Ono | first1=Takashi | title=Sur une propriété arithmétique des groupes algébriques commutatifs | url=http://www.numdam.org/item?id=BSMF_1957__85__307_0 | mr=0094362  | year=1957 | journal=Bulletin de la Société Mathématique de France | issn=0037-9484 | volume=85 | pages=307–323}}
* [[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.]
*{{Citation | last1=Tate | first1=John T. | title=Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) | publisher=Thompson, Washington, D.C. | isbn=978-0-9502734-2-6 | mr=0217026 | year=1950 | chapter=Fourier analysis in number fields, and Hecke's zeta-functions | pages=305–347}}
* John Eatwell (1987). "Returns to scale," ''[[The New Palgrave: A Dictionary of Economics]]'', v. 4, pp.&nbsp;165–66.
*{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=Zur algebraischen Theorie der algebraischen Funktionen. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002174502 | language=German | doi=10.1515/crll.1938.179.129 | year=1938 | journal=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | volume=179 | pages=129–133}}
* [[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==
==External links==
*{{springer|first=A.S. |last=Rapinchuk|id=T/t092060|title=Tamagawa number}}
* [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:Topological groups]]
[[Category:Microeconomics]]
[[Category:Algebraic number theory]]
[[Category:Production economics]]
[[Category:Algebraic groups]]

Revision as of 08:03, 13 August 2014

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 input levels including physical 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).

The laws of returns to scale are a set of three interrelated and sequential laws: 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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

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).

Example

When all inputs increase by a factor of 2, new values for output will be:

  • Twice the previous output if there are constant returns to scale (CRS)
  • 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 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.[1][2][3] 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 F(K,L) is defined to have:

  • Constant returns to scale if (for any constant a greater than 0) F(aK,aL)=aF(K,L)
  • Increasing returns to scale if (for any constant a greater than 1) F(aK,aL)>aF(K,L),
  • Decreasing returns to scale if (for any constant a greater than 1) F(aK,aL)<aF(K,L)

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:

F(K,L)=AKbL1b

where A>0 and 0<b<1. Thus

F(aK,aL)=A(aK)b(aL)1b=Aaba1bKbL1b=aAKbL1b=aF(K,L).

But if the Cobb-Douglas production function has its general form

F(K,L)=AKbLc

with 0<c<1, then there are increasing returns if b + c > 1 but decreasing returns if b + c < 1, since

F(aK,aL)=A(aK)b(aL)c=AabacKbLc=ab+cAKbLc=ab+cF(K,L),

which is greater than or less than aF(K,L) as b+c is greater or less than one.

See also

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Further reading

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