# CES production function (1961)

Proposed by the American economist Kenneth Arrow (1921- ), Hollis Chenery (1918- ), BAGICHA S. MINHAS, and Robert Solow (1924- ), CES production function is also known as the constant elasticity of substitution function.

Constant elasticity of substitution (CES), in economics, is a property of some production functions and utility functions.

Specifically, it arises in a particular type of aggregator function which combines two or more types of consumption goods, or two or more types of production inputs into an aggregate quantity. This aggregator function exhibits constant elasticity of substitution

### CES production function

The CES production function is a neoclassical production function that displays constant elasticity of substitution. In other words, the production technology has a constant percentage change in factor (e.g. labour and capital) proportions due to a percentage change in marginal rate of technical substitution. The two factor (capital, labor) CES production function introduced by Solow, and later made popular by Arrow, Chenery, Minhas, and Solow is:

{\displaystyle Q=F\cdot \left(a\cdot K^{\rho }+(1-a)\cdot L^{\rho }\right)^{\frac {\upsilon }{\rho }}}

where

• {\displaystyle Q} = Quantity of output
• {\displaystyle F} = Factor productivity
• {\displaystyle a} = Share parameter
• {\displaystyle K}{\displaystyle L} = Quantities of primary production factors (Capital and Labor)
• {\displaystyle \rho } = {\displaystyle {\frac {\sigma -1}{\sigma }}} = Substitution parameter
• {\displaystyle \sigma } = {\displaystyle {\frac {1}{1-\rho }}} = Elasticity of substitution
• {\displaystyle \upsilon } = degree of homogeneity of the production function. Where {\displaystyle \upsilon } = 1 (Constant return to scale){\displaystyle \upsilon } < 1 (Decreasing return to scale){\displaystyle \upsilon } > 1 (Increasing return to scale).

As its name suggests, the CES production function exhibits constant elasticity of substitution between capital and labor. Leontief, linear and Cobb–Douglas functions are special cases of the CES production function. That is,

• If {\displaystyle \rho } approaches 1, we have a linear or perfect substitutes function;
• If {\displaystyle \rho } approaches zero in the limit, we get the Cobb–Douglas production function;
• If {\displaystyle \rho } approaches negative infinity we get the Leontief or perfect complements production function.

The general form of the CES production function, with n inputs, is:

{\displaystyle Q=F\cdot \left[\sum _{i=1}^{n}a_{i}X_{i}^{r}\ \right]^{\frac {1}{r}}}

where

• {\displaystyle Q} = Quantity of output
• {\displaystyle F} = Factor productivity
• {\displaystyle a_{i}} = Share parameter of input i, {\displaystyle \sum _{i=1}^{n}a_{i}=1}
• {\displaystyle X_{i}} = Quantities of factors of production (i = 1,2…n)
• {\displaystyle s={\frac {1}{1-r}}} = Elasticity of substitution.

Extending the CES (Solow) functional form to accommodate multiple factors of production creates some problems. However, there is no completely general way to do this. 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. This is true for any production function. This means the use of the CES functional 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 equilibrium and general equilibrium models. Different nests (levels) allow for the introduction of the appropriate elasticity of substitution.

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This is a linearly homogenous production function with a constant elasticity of input substitution, which takes on forms other than unity.

It replaced the Cobb-Douglas Production Function model which looked at physical output as a product of labor and capital inputs.

The equation for the CES production function model is:

Q = A(ak – b-b + (1 – c)L – b-b) – 1/b

where Q is output, K capital and L labor and a, b, c are constants.

### One thought on “CES production function (1961)”

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