Economics at Zero Marginal Cost: Concept and Properties of CES Production Function

Constant Elasticity of Substitution (CES) Production Function (1961)

A very interesting special class of production functions is those for which the elasticity of substitution (σ) is a constant. These have come to be known as CES production functions. This class of functions was first explored in a famous paper published in 1961 by Arrow, Chenery, Minhas, and Solow. These authors prove that a production function with ‘n’ inputs has constant elasticity of substitution σ between every pair of inputs if and only if the production function is either of the functional form:

$Q=A\left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }}$

Where,

Q = the output quantity

L and K = the inputs quantities

$A\ \epsilon\ (0,\infty )$ = determines the productivity

$\alpha \ \epsilon\ (0,1 )$ = determines the optimal distribution of the inputs, K and L

$\rho \ \epsilon\ \left [ (-1,0 )\cup (0,\infty )\right ]$ = determines the constant elasticity of substitution, which is $\sigma \ =\frac{1}{\left ( 1+\rho \right )}$ .

A production function that belongs to the CES class has two major characteristics:

a. It is homogeneous of degree one, and

b. It has constant elasticity of substitution.

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

$\\a)\ for\ \rho \rightarrow 0,\ \sigma \ approaches\ 1\ and\ the\ CES\ turns\ to\ the\ Cobb-Douglas\ form;\\ b)\ for\ \rho \rightarrow \infty ,\ \sigma \ approaches\ 0\ and\ the\ CES\ turns\ to\ the\ Leontief\ production\ function ;\ and \\ c)\ for\ \rho \rightarrow -1,\ \sigma \ approaches\ \infty \ and\ the\ CES\ turns\ to\ a\ linear\ function\ \left ( Perfect\ Substitutes \right )\\$

Properties of CES Production Function:

1. The CES production function is a homogeneous production function of degree one (constant returns to scale). If we increase the inputs L and K in the CES function by n-fold, output Q will also increase by n-fold.

2. In the CES production function, the marginal product can be expressed as a function of the average product.

3. In the CES production function, the elasticity of substitution is equal to $\sigma \ =\frac{1}{\left ( 1+\rho \right )}$ . The parameter in the CES production function determines the elasticity of substitution. This shows that the elasticity of substitution is a constant whose magnitude depends on the value of the parameter.

4. In the CES production function, the elasticity of substitution is constant but not necessarily equal to unity. It ranges from 0 to $\infty$ .

5. In the CES production function, the average and marginal products in the variables L and K are homogeneous of degree zero like all linearly homogeneous production functions.

6. The CES production function exhibits positive and diminishing returns to inputs.

Mathematical Proof of the Properties:

Proof:

The CES production function is given by:

$Q=f(L,K)=A\left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }}$

Multiplying both inputs L and K by $\lambda$ , we have

$\\Q=f(\lambda L,\lambda K)=A\left [ (\alpha \lambda L)^{-\rho } +(1-\alpha)(\lambda K)^{-\rho }\right ]^{-\frac{1}{\rho }}\\ Q=f(\lambda L,\lambda K)=A\left [ \alpha \lambda^{-\rho } L^{-\rho } +(1-\alpha) \lambda^{-\rho } K^{-\rho }\right ]^{-\frac{1}{\rho }}\\ Q=f(\lambda L,\lambda K)=A\left [\lambda^{-\rho } (\alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right )]^{-\frac{1}{\rho }}\\ Q=f(\lambda L,\lambda K)=A\ (\lambda^{-\rho })^{-\frac{1}{\rho }} \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }}\\ Q=f(\lambda L,\lambda K)= \lambda^{-\rho .{-\frac{1}{\rho }}}.\ A \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }}\\ Q=f(\lambda L,\lambda K)= \lambda^{1}.Q\\$

$\therefore$ The CES production function is homogeneous of degree 1 exhibiting constant returns to scale.

2. In the CES production function, the marginal product can be expressed as a function of the average product.

Proof:

The CES production function is given by:

$Q=f(L,K)=A\left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }}$

By definition,

$MP_{L}=\frac{\partial Q}{\partial L}\ and\ MP_{K}=\frac{\partial Q}{\partial K}\\$

Marginal Product of Labour:

$\\MP_{L}=\frac{\partial Q}{\partial L}\\ MP_{L}=\frac{\partial }{\partial L}\left ( A\left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }} \right )\\ MP_{L}=A \frac{\partial }{\partial L}\left ( \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }} \right )\\ MP_{L}=A.-\frac{1}{\rho }.\ \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}.\frac{\partial }{\partial L}(\alpha L^{-\rho } +(1-\alpha)K^{-\rho })\\ MP_{L}=A.-\frac{1}{\rho }.\ \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}.\left ( \frac{\partial }{\partial L} \alpha L^{-\rho } +\frac{\partial }{\partial L} (1-\alpha)K^{-\rho } \right )\\ MP_{L}=A.-\frac{1}{\rho }.\ \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}.\left (\alpha \frac{\partial }{\partial L} L^{-\rho } +0 \right )\\$

$\\MP_{L}=A.\ A^{\rho }\ A^{-\rho }.-\frac{1}{\rho }.\ \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}.\ \alpha .-\rho .\ \ L^{-\rho-1 }\\ \\MP_{L}= A^{1+\rho }\ \frac{1}{A^{\rho }} .\ \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}.\ \alpha .\ \ L^{-(\rho+1) }\\ \\MP_{L}= A^{1+\rho }\ \frac{1}{A^{\rho }} .\ \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}\ .\frac{\alpha }{L^{\rho+1}}\\ \\MP_{L}= A^{1+\rho }\ \frac{1}{A^{\rho }} .\ \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}\ .\frac{\alpha }{L^{1+\rho}}\\ \\MP_{L}=\frac{A^{1+\rho } \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}\ }{L^{1+\rho}} \frac{\alpha }{A^{\rho }}\\$

$\\MP_{L}=\frac{\left ( A \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ])^{-\frac{1}{\rho }} \right )^{1+\rho }\ }{L^{1+\rho}}. \frac{\alpha }{A^{\rho }}\\ \\MP_{L}=\frac{ Q^{1+\rho }\ }{L^{1+\rho}}. \frac{\alpha }{A^{\rho }}\\ \\MP_{L}=\left (\frac{ Q }{L} \right )^{1+\rho }. \frac{\alpha }{A^{\rho }}\\ \\MP_{L}=f\left (\frac{ Q }{L} \right )\\ \\MP_{L}=f (AP_{L})$

Marginal Product of Capital:

$\\MP_{K}=\frac{\partial Q}{\partial K}\\ MP_{K}=\frac{\partial }{\partial K}\left ( A\left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }} \right )\\ MP_{K}=A \frac{\partial }{\partial K}\left ( \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }} \right )\\ MP_{K}=A.-\frac{1}{\rho }.\ \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}.\frac{\partial }{\partial K}(\alpha L^{-\rho } +(1-\alpha)K^{-\rho })\\ MP_{K}=A.-\frac{1}{\rho }.\ \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}.\left ( \frac{\partial }{\partial K} \alpha L^{-\rho } +\frac{\partial }{\partial K} (1-\alpha)K^{-\rho } \right )\\ MP_{K}=A.-\frac{1}{\rho }.\ \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}.\left (0 +(1-\alpha) \frac{\partial }{\partial L} K^{-\rho } \right )\\$

$\\MP_{K}=A.\ A^{\rho }\ A^{-\rho }.-\frac{1}{\rho }.\ \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}.\ (1-\alpha ).-\rho .\ \ K^{-\rho-1 }\\ \\MP_{K}= A^{1+\rho }\ \frac{1}{A^{\rho }} .\ \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}.\ (1-\alpha )\ \ K^{-(\rho+1) }\\ \\MP_{K}= A^{1+\rho }\ \frac{1}{A^{\rho }} .\ \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}\ .\frac{1-\alpha }{K^{\rho+1}}\\ \\MP_{K}= A^{1+\rho }\ \frac{1}{A^{\rho }} .\ \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}\ .\frac{1-\alpha}{K^{1+\rho}}\\ \\MP_{K}=\frac{A^{1+\rho } \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }(1+\rho )}\ }{K^{1+\rho}} \frac{1-\alpha }{A^{\rho }}\\$

$\\MP_{K}=\frac{\left ( A \left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ])^{-\frac{1}{\rho }} \right )^{1+\rho }\ }{K^{1+\rho}}. \frac{1-\alpha }{A^{\rho }}\\ \\MP_{K}=\frac{ Q^{1+\rho }\ }{K^{1+\rho}}. \frac{1-\alpha }{A^{\rho }}\\ \\MP_{K}=\left (\frac{ Q }{K} \right )^{1+\rho }. \frac{1-\alpha }{A^{\rho }}\\ \\MP_{K}=f\left (\frac{ Q }{K} \right )\\ \\MP_{K}=f (AP_{K})$

3. In the CES production function, the elasticity of substitution is equal to $\sigma \ =\frac{1}{\left ( 1+\rho \right )}$ .

The CES production function is given by:

$Q=f(L,K)=A\left [ \alpha L^{-\rho } +(1-\alpha)K^{-\rho }\right ]^{-\frac{1}{\rho }}$

By definition,

Elasticity of substitution ( $\sigma$ )

$\\\sigma =\frac{\frac{\mathrm{d} \frac{K}{L}}{ \frac{K}{L}}}{\frac{\mathrm{d}MRTS_{LK}}{ MRTS_{LK}}}\\\\ \\\sigma =\frac{\frac{\mathrm{d} \frac{K}{L}}{ \frac{K}{L}}}{\frac{\mathrm{d} \frac{MP_{L}}{MP_{K}}}{ \frac{MP_{L}}{MP_{K}}}}\\\\ \sigma ={\frac{\mathrm{d} \frac{K}{L}}{ \frac{K}{L}}}.{\frac{\frac{MP_{L}}{MP_{K}}}{ \mathrm{d}\frac{MP_{L}}{MP_{K}}}}\\\\$

$\sigma ={\frac{\mathrm{d} \frac{K}{L}}{ \mathrm{d}\frac{MP_{L}}{MP_{K}}}}.{\frac{\frac{MP_{L}}{MP_{K}}}{ \frac{K}{L}}}-------(1)\\$

Marginal Product of Labour:

Marginal Product of Capital:

Along an expansion path,

$\frac{MP_{L}}{MP_{K}}=\frac{P_{L}}{{P_{K}}}-------(2)$

So, equation (1) becomes

$\sigma ={\frac{\mathrm{d} \frac{K}{L}}{ \mathrm{d}\frac{P_{L}}{P_{K}}}}.{\frac{\frac{P_{L}}{P_{K}}}{ \frac{K}{L}}}-------(3)\\$

$Substituting\ MP_{L}\ and\ MP_{K}\ in\ equation\ (2)$

$\\\frac{\frac{ Q^{1+\rho }\ }{L^{1+\rho}}. \frac{1-\alpha }{A^{\rho }}}{\frac{ Q^{1+\rho }\ }{K^{1+\rho}}. \frac{\alpha }{A^{\rho }}}=\frac{P_{L}}{{P_{K}}}\\\\ \\\frac{\frac{1-\alpha }{L^{1+\rho}}}{\frac{\alpha }{K^{1+\rho}}}=\frac{P_{L}}{{P_{K}}}\\\\ \\\frac{1-\alpha }{L^{1+\rho}}.\frac{K^{1+\rho}}{\alpha }=\frac{P_{L}}{{P_{K}}}\\ \\\frac{P_{L}}{{P_{K}}}=\left (\frac{1-\alpha }{\alpha } \right ).\frac{K^{1+\rho}}{L^{1+\rho} }-------(4)\\ \\\frac{P_{L}}{{P_{K}}}=\frac{1-\alpha }{\alpha }.\left (\frac{K}{L}\right )^{1+\rho}-------(5)\\$

Differentiating (5) both sides with to respect to K/L

$\\\frac{\mathrm{d} }{\mathrm{d} \frac{K}{L}}\left (\frac{P_{L}}{{P_{K}}} \right )=\frac{\mathrm{d} }{\mathrm{d} \frac{K}{L}} \left (\frac{1-\alpha }{\alpha }.\left (\frac{K}{L}\right )^{1+\rho} \right )\\ \\\frac{\mathrm{d} \frac{P_{L}}{{P_{K}}}}{\mathrm{d} \frac{K}{L}}=\left (\frac{1-\alpha }{\alpha } \right ) . \frac{\mathrm{d} }{\mathrm{d} \frac{K}{L}} \left (\frac{K}{L}\right )^{1+\rho}\\ \\\frac{\mathrm{d} \frac{P_{L}}{{P_{K}}}}{\mathrm{d} \frac{K}{L}}=\left (\frac{1-\alpha }{\alpha } \right ).\left (1+\rho \right ) \left (\frac{K}{L}\right )^{1+\rho-1}\\ \\\frac{\mathrm{d} \frac{P_{L}}{{P_{K}}}}{\mathrm{d} \frac{K}{L}}=\left (\frac{1-\alpha }{\alpha } \right ).\left (1+\rho \right ) \left (\frac{K}{L}\right )^{\rho}\\$

Taking reciprocal on both sides, we have

$\\\frac{\mathrm{d} \frac{K}{L}}{\mathrm{d} \frac{P_{L}}{P_{K}}}=\left (\frac{\alpha }{1-\alpha } \right ).\left (\frac{1}{1+\rho} \right ) \left (\frac{L}{K}\right )^{\rho}\\$

Substituting equation (4) and (5), we have

$\\\frac{\mathrm{d} \frac{K}{L}}{\mathrm{d} \frac{P_{L}}{P_{K}}}=\left (\frac{\alpha }{1-\alpha } \right ).\left (\frac{1}{1+\rho} \right ) \left (\frac{L}{K}\right )^{\rho}-------(6)\\$

Substituting equations (4) and (6) in (3), we have

$\\\sigma ={\frac{\mathrm{d} \frac{K}{L}}{ \mathrm{d}\frac{P_{L}}{P_{K}}}}.{\frac{\frac{P_{L}}{P_{K}}}{ \frac{K}{L}}}\\\\ \\\sigma =\left (\frac{\alpha }{1-\alpha } \right ). \left (\frac{1}{1+\rho} \right ).\left (\frac{L}{K}\right )^{\rho}.\ \left (\frac{1-\alpha }{\alpha } \right ).{\frac{\left (\frac{1-\alpha }{\alpha } \right ).\frac{K^{1+\rho}}{L^{1+\rho} }}{ \frac{K}{L}}}\\ \\\sigma =\left (\frac{\alpha }{1-\alpha } \right ).\left (\frac{1}{1+\rho} \right ) \left (\frac{L}{K}\right )^{\rho}.\ \left (\frac{1-\alpha }{\alpha } \right ).\frac{K^{1+\rho}}{L^{1+\rho} }.\ \frac{L}{K}\\ \\\sigma =\left (\frac{1}{1+\rho} \right ).\ \frac{L^{\rho}}{K^{\rho}}.\ \frac{K^{1+\rho}}{L^{1+\rho} }.\ \frac{K^{-1}}{L^{-1}}\\ \\\sigma =\left (\frac{1}{1+\rho} \right ).\ \frac{L^{\rho}}{K^{\rho}}.\ \frac{K^{1+\rho-1}}{L^{1+\rho-1} }\\ \\\sigma =\left (\frac{1}{1+\rho} \right ).\ \frac{L^{\rho}}{K^{\rho}}.\ \frac{K^{\rho}}{L^{\rho} }\\ \\\sigma =\frac{1}{1+\rho}$

$\therefore$ In the CES production function, the elasticity of substitution is equal to $\sigma \ =\frac{1}{\left ( 1+\rho \right )}$ .

Additional Notes:

A function $f\left ( x_{1},\ x_{2},\ x_{3},...,\ x_{n} \right )$ is said to be homogenous of degree ‘k’, if multiplication of each of its independent variables by a constant $\lambda$ will alter the value of the function by the proportion, $\lambda^{k}$ .i.e., $f\left ( \lambda x_{1},\ \lambda x_{2},\ \lambda x_{3},...,\ \lambda x_{n} \right )=\lambda^{k}f\left ( x_{1},\ x_{2},\ x_{3},...,\ x_{n} \right )$ or, i.e., $f\left ( \lambda x \right )=\lambda^{k}f\left ( x \right )$ . Note that,, may be negative.
The CES function includes three special cases:

Reading Lists:

1) Samuelson, P. A., & Nordhuas, W.D (1992), Economics (14th edition). McGraw Hill International edition, U.S.

2) Samuelson, P. A., & Nordhuas, W.D (2013), Microeconomics (19th edition). McGraw Hill Education (India) Pvt. Ltd.

3) Koutsoyiannis, A. (1990), Modern Microeconomics (2nd edition). Macmillan, London.

4) Henderson, J. M. and R. E. Quandt (1980), Microeconomic Theory: A Mathematical Approach (3rd edition). McGraw Hill, New Delhi.

5) Dwivedi, D. N. (2016), Microeconomics: Theory and Applications (3rd edition). Vikas Publication House Pvt. Ltd. Noida (UP), India.

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Economics at Zero Marginal Cost

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Tuesday, July 20, 2021

Concept and Properties of CES Production Function

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