128.250.24.88: clarified first sentence

2014-03-31T22:34:47Z

clarified first sentence

183.157.160.21 at 14:36, 13 December 2013

2013-12-13T14:36:05Z

← Older revision		Revision as of 16:36, 13 December 2013
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	~~Hi there~~. ~~Let me begin~~ by ~~introducing~~ the ~~author~~, ~~her~~ name is ~~Sophia~~. ~~I am an invoicing officer~~ and ~~I'll~~ be ~~promoted soon~~. ~~Some time~~ in the ~~past she chose to reside~~ in ~~Alaska~~ and ~~her mothers~~ and ~~fathers live nearby~~. ~~To climb~~ is ~~something she would never give up~~.<br><br>~~Here~~ is ~~my blog -~~ [http://~~cartoonkorea~~.~~com~~/~~ce002~~/~~1093612 online psychic reading~~]		'''Constant elasticity of substitution''' ('''CES'''), in [[economics]], is a property of some [[production function]]s and [[utility function]]s.

			More precisely, it refers to a particular type of aggregator function which combines two or more types of consumption, or two or more types of productive inputs into an aggregate quantity. This aggregator function exhibits constant [[elasticity of substitution]].

			==CES production function==
			The CES [[production function]] is a type of production function that displays constant [[elasticity of substitution]]. In other words, the production technology has a constant percentage change in factor (e.g. [[Manual labour\|labour]] and [[Capital (economics)\|capital]]) proportions due to a percentage change in [[marginal rate of technical substitution]]. The two factor (Capital, Labor) CES production function introduced by [[Robert Solow\|Solow]],<ref name="Solow1956">{{cite journal \|last=Solow\|first=R.M \|year=1956 \|month= \|title=A contribution to the theory of economic growth \|journal=The Quarterly Journal of Economics \|volume=70 \|pages=65–94 \|doi=10.2307/1884513}}</ref> and later made popular by [[Kenneth Arrow\|Arrow]], [[Hollis B. Chenery\|Chenery]], [[Bagicha Singh Minhas\|Minhas]], and [[Robert Solow\|Solow]] is:<ref name="Arrow1961">{{cite journal \|last=Arrow \|first=K. J. \|authorlink= \|coauthors=Chenery, H. B.; Minhas, B. S.; Solow, R. M. \|year=1961 \|month= \|title=Capital-labor substitution and economic efficiency \|journal=Review of Economics and Statistics \|volume=43 \|issue=3 \|pages=225–250 \|jstor=1927286\|doi=10.2307/1927286 \|publisher=The MIT Press }}</ref><ref name="Jorgensen2000">{{cite book \|title=Econometrics, vol. 1: Econometric Modelling of Producer Behavior \|last=Jorgensen \|first=Dale W. \|authorlink= \|coauthors= \|year=2000 \|publisher=MIT Press \|location=Cambridge, MA \|isbn=0-262-10082-7 \|page=2 }}</ref><ref name="Klump">{{cite journal \|last=Klump \|first=R \|authorlink= \|coauthors=McAdam, P; Willman, A. \|year=2007 \|month= \|title=Factor Substitution and Factor Augmenting Technical Progress in the US: A Normalized Supply-Side System Approach \|journal=Review of Economics and Statistics \|volume=89 \|issue=1 \|pages=183–192\|publisher=The MIT Press }}</ref>

			:<math> Q = F \cdot \left(a \cdot K^r+(1-a) \cdot L^r\right)^{\frac{1}{r}} </math>

			where

			* <math>Q</math> = Output
			* <math>F</math> = Factor productivity
			* <math>a</math> = Share parameter
			* <math>K</math>, <math>L</math> = Primary production factors (Capital and Labor)
			* <math>r</math> = <math>{\frac{(s-1)}{s}}</math>

			* <math>s</math> = <math>{\frac{1}{(1-r)}}</math> = Elasticity of substitution.

			As its name suggests, the CES production function exhibits constant elasticity of substitution between capital and labor.
			[[Leontief production function\|Leontief]], linear and [[Cobb–Douglas production function\|Cobb–Douglas]] production functions are special cases of the CES production function. That is, if <math>r=1</math> we have a linear function 1, if <math>r</math> approaches zero, in the limit we get the Cobb–Douglas function; and as <math>r</math> approaches negative infinity we get the Leontief function. The general form of the CES production function is:<ref>http://www.econ.ucsb.edu/~tedb/Courses/GraduateTheoryUCSB/elasticity%20of%20substitutionrevised.tex.pdf</ref>

			:<math> Q = F \cdot \left[\sum_{i=1}^n a_{i}X_{i}^{\frac{(s-1)}{s}}\ \right]^{\frac{s}{(s-1)}}</math>

			where

			* <math>Q</math> = Output
			* <math>F</math> = Factor productivity
			* <math>a_{i}</math> = Share parameter of input i, <math> \sum_{i=1}^n a_{i} = 1 </math>
			* <math>X</math> = Production factors (i = 1,2...n)
			* <math>s</math> = Elasticity of substitution.

			Extending the CES (Solow) form to accommodate multiple factors of production creates some problems, however. There is no completely general way to do this. [[Hirofumi Uzawa\|Uzawa]] showed the only possible n-factor production functions (n>2) with constant partial elasticities of substitution require either that all elasticities between pairs of factors be identical, or if any differ, these all must equal each other and all remaining elasticities must be unity.<ref name="Uzawa1962">{{cite journal \|last=Uzawa\|first=H\|year=1962 \|month= \|title=Production functions with constant elasticities of substitution \|journal=Review of Economic Studies\|volume=9 \|pages=291–299 }}</ref> This is true for any production function. This means the use of the CES form for more than 2 factors will generally mean that there is not constant elasticity of substitution among all factors.

			Nested CES functions are commonly found in partial/general equilibrium models. Different nests (levels) allow for the introduction of the appropriate elasticity of substitution.

			The CES is a ''neoclassical production function''.

			==CES utility function==
			{{see also\|Isoelastic utility}}
			The same functional form arises as a utility function in [[consumer theory]]. For example, if there exist <math>n</math> types of consumption goods <math>c_i</math>, then aggregate consumption <math>C</math> could be defined using the CES aggregator:

			:<math> C = \left[\sum_{i=1}^n a_{i}^{\frac{1}{s}}c_{i}^{\frac{(s-1)}{s}}\ \right]^{\frac{s}{(s-1)}}</math>

			Here again, the coefficients <math>a_i</math> are share parameters, and <math>s</math> is the elasticity of substitution. Therefore the consumption goods <math>c_i</math> are perfect substitutes when <math>s </math> approaches infinity and perfect complements when <math>s</math> approaches zero. The CES aggregator is also sometimes called the ''Armington aggregator'', which was discussed by [[Paul Armington\|Armington]] (1969).<ref>{{cite journal \|last=Armington \|first=P. S. \|authorlink= \|coauthors= \|year=1969 \|month= \|title=A theory of demand for products distinguished by place of production \|journal=IMF Staff Papers \|volume=16 \|issue= \|pages=159–178 \|url= \|accessdate= }}</ref>

			A CES utility function is one of the cases considered by [[Avinash Dixit\|Dixit]] and [[Joseph Stiglitz\|Stiglitz]] in their study of optimal product diversity in a context of [[monopolistic competition]].<ref>{{cite journal \|last=Dixit \|first=Avinash \|last2=Stiglitz \|first2=Joseph \|year=1977 \|title=Monopolistic Competition and Optimum Product Diversity \|journal=[[American Economic Review]] \|volume=67 \|issue=3 \|pages=297–308 \|jstor=1831401 }}</ref>

			==References==
			{{reflist}}

			==External links==
			*[http://www2.hawaii.edu/~fuleky/anatomy/anatomy2.html Anatomy of CES Type Production Functions in 3D]

			[[Category:Production economics]]
			[[Category:Consumer theory]]
			[[Category:Econometrics]]
			[[Category:Elasticity (economics)]]
			[[Category:Utility]]

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2012-05-09T09:46:09Z

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Hi there. Let me begin by introducing the author, her name is Sophia. I am an invoicing officer and I'll be promoted soon. Some time in the past she chose to reside in Alaska and her mothers and fathers live nearby. To climb is something she would never give up.<br><br>Here is my blog - [http://cartoonkorea.com/ce002/1093612 online psychic reading]

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128.250.24.88: clarified first sentence

183.157.160.21 at 14:36, 13 December 2013

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