CES function

In economics , the CES function (short for constant elasticity of substitution - “ constant elasticity of substitution ”) is a class of functions that are characterized by the fact that they have the same elasticity of substitution at every point in their domain . This property is advantageous in a large number of economic applications - be it in the micro or in the macroeconomic area. For certain parameter constellations, the general CES function also gives rise to special function classes that are also widely used.

In scientific practice, see CES functions including as demand functions (CES demand function), utility functions (CES utility function) and production function (CES production function) use.

definition

A function is generally referred to as a CES function

${\displaystyle z=\beta \cdot \left[\alpha _{1}x_{1}^{-\rho }+\alpha _{2}x_{2}^{-\rho }+\dotsb +\alpha _{n}x_{n}^{-\rho }\right]^{-\gamma /\rho }}$

with ; and for everyone as well . ${\displaystyle \beta ,\gamma >0}$ ${\displaystyle x_{i}>0}$ ${\displaystyle \alpha _{i}>0}$ ${\displaystyle i=1,\dotsc ,n}$ ${\displaystyle \rho \neq 0}$

This is (for reasons to be explained) the degree of homogeneity. You almost always bet, and usually also . ${\displaystyle \gamma }$ ${\displaystyle \beta =1}$ ${\displaystyle \gamma =1}$

Different uses

If the function is used as a production function, it is regularly referred to as (instead of ) to express that it shows the quantity of a good produced. They then stand for the amount of the input factor used , whereby there are input factors. For example, the two-factor CES production function is often used (sometimes with the default ), where the capital and labor input are used; in a version introduced by Robert Solow in the field of growth theory . ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle x_{i}}$ ${\displaystyle i}$ ${\displaystyle n}$ ${\displaystyle y=\left[\alpha _{1}K^{-\rho }+\alpha _{2}L^{-\rho }\right]^{-1/\rho }}$ ${\displaystyle \alpha _{1}+\alpha _{2}=1}$ ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle y=(\alpha K^{\rho }+L^{\rho })^{1/\rho }}$

When used as a utility function (usually ) denotes the amount of the good consumed . ${\displaystyle u}$ ${\displaystyle x_{i}}$ ${\displaystyle i}$

properties

It can be shown that the CES function in the defined sense is homogeneous in degree . Furthermore it is for quasi-convex , for quasi-concave . For and at the same time it is concave and for , even strictly concave. ${\displaystyle \gamma }$ ${\displaystyle \rho \leq -1}$ ${\displaystyle \rho \geq -1}$ ${\displaystyle 0<\gamma \leq 1}$ ${\displaystyle \rho \geq -1}$ ${\displaystyle 0<\gamma <1}$ ${\displaystyle \rho >-1}$

Special cases

It can be shown that the CES function changes into a function of the Cobb-Douglas type (with substitution elasticity ) and into a Leontief function ( ) for. ${\displaystyle \rho \rightarrow 0}$ ${\displaystyle \sigma =1}$ ${\displaystyle \rho \rightarrow \infty }$ ${\displaystyle \sigma =0}$

Specific parameter constellations allow further details. For example, with the CES type with elasticity of substitution . For converges and reduces to the linear-homogeneous Cobb-Douglas function . For it follows again and the Leontief function results in the limit value . ${\displaystyle z=\left[\alpha _{1}x_{1}^{-\rho }+\dotsb +\alpha _{n}x_{n}^{-\rho }\right]^{-1/\rho }}$ ${\displaystyle \alpha _{1}+\dotsb +\alpha _{n}=1}$ ${\displaystyle \sigma =1/(1+\rho )}$ ${\displaystyle \rho \rightarrow 0}$ ${\displaystyle \sigma \rightarrow 1}$ ${\displaystyle z}$ ${\displaystyle z=x_{1}^{\alpha _{1}}\dotsm x_{n}^{\alpha _{n}}}$ ${\displaystyle \rho \rightarrow \infty }$ ${\displaystyle \sigma \rightarrow 0}$ ${\displaystyle z=\min\{x_{1},\dotsc ,x_{n}\}}$

literature

• Kenneth Arrow , HB Chenery, BS Minhas and Robert Solow : Capital-Labor Substitution and Economic Efficiency. In: Review of Economics and Statistics. 43, No. 3, 1961, pp. 225-250.
• Geoffrey A. Jehle and Philip J. Reny: Advanced Microeconomic Theory. 3rd ed. Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 .
• Andreu Mas-Colell, Michael Whinston, and Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-195-07340-1 .
• Carl P. Simon and Lawrence Blume: Mathematics for Economists. WW Norton, New York and London 1994, ISBN 0-393-95733-0 .
• Knut Sydsæter, Arne Strøm and Peter Berck: Economists' mathematical manual. 4th edition. Springer, Berlin a. a. 2005, ISBN 978-3-540-26088-2 (also as an e-book: doi : 10.1007 / 3-540-28518-0 ).
• Knut Sydsæter u. a .: Further mathematics for economic analysis. 2nd ed. Financial Times / Prentice Hall, Harlow 2008, ISBN 978-0-273-71328-9 .
• Hal Varian : Microeconomic Analysis. WW Norton, New York and London 1992, ISBN 0-393-95735-7 .
• Susanne Wied-Nebbeling and Helmut Schott: Fundamentals of microeconomics. Springer, Heidelberg a. a. 2007, ISBN 978-3-540-73868-8 . [S. 127-131]

Individual evidence

1. ↑ The definition here follows Sydsæter u. a. 2008, p. 72 and Simon / Blume 1994, p. 275. It is a generalized form; in many cases, certain properties are already required. The vast majority of the literature is limited to the case with (Sydsaeter / Strøm / Berck 2005, p. 166; Varian 1992, p. 19; Jehle / Reny 2011, p. 130; Mas-Colell / Whinston / Green 1995, P. 97) and usually is also (Varian 1992, p. 19; Jehle / Reny 2011, p. 130; Mas-Colell / Whinston / Green 1995, p. 97); sometimes only the case is considered (Jehle / Reny 2011, p. 130). The function is also regularly used instead of (Varian 1992, p. 19; Jehle / Reny 2011, p. 130; Mas-Colell / Whinston / Green 1995, p. 97; as here Sydsaeter / Strøm / Berck 2005, p. 166 and Wied-Nebbeling / Schott 2007, p. 128); However, this only results in an interpretative difference as a result of differing definitions of elasticity. ${\displaystyle \beta =1}$ ${\displaystyle \gamma =1}$ ${\displaystyle \alpha _{i}=1}$ ${\displaystyle \rho }$ ${\displaystyle -\rho }$
2. ↑ See, for example, Wied-Nebbeling / Schott 2007, p. 128.
3. ^ Robert M. Solow: A Contribution to the Theory of Economic Growth. In: The Quarterly Journal of Economics. 70, No. 1, 1956, pp. 65-94 ( [1] (PDF; 2.2 MB); JSTOR 1884513 ).
4. ↑ For this and the following properties cf. Sydsæter u. a. 2008, p. 72 (there also with evidence) and Sydsaeter / Strøm / Berck 2005, p. 166.
5. ↑ This property is of course not of any practical relevance to speak of; in this case , the CES technology would then imply concave isoquants , which does not seem plausible. ${\displaystyle \rho <-1}$
6. ↑ See, for example, Wied-Nebbeling / Schott 2007, pp. 128 ff. For the case and . ${\displaystyle n=2}$ ${\displaystyle \alpha _{1}+\alpha _{2}=1}$
7. ↑ See Jehle / Reny 2011, p. 131.