Despite having several factors of production in substitutability, the most common are the forms of elasticity of substitution. On the contrary of restricting direct empirical evaluation, the constant Elasticity of Substitution are simple to use and hence are widely used. McFadden states that; The constant E.S assumption is a restriction on the form of production possibilities, and one can characterize the class of production functions which have this property. This has been done by Arrow-Chenery-Minhas-Solow for the two-factor production case. and later made popular by
Arrow,
Chenery,
Minhas, and
Solow is: :Q = F\cdot \left(a \cdot K^\rho+(1-a) \cdot L^\rho\right)^{\frac{\upsilon}{\rho}} where • Q = Quantity of output • F =
Total Factor Productivity • a = Share parameter • K, L = Quantities of primary production factors (Capital and Labor) Calculating the Rate of Technological Substitution between K and L: RTS_{L,K}=\frac{F ({\frac{\upsilon}{\rho}-1})\cdot \left(a \cdot K^\rho+(1-a) \cdot L^\rho\right)^{\frac{\upsilon}{\rho}-1}\cdot a \cdot\rho\cdot L^{\rho-1}}{{F ({\frac{\upsilon}{\rho}-1})\cdot \left(a \cdot K^\rho+(1-a) \cdot L^\rho\right)^{\frac{\upsilon}{\rho}-1}\cdot a \cdot\rho \cdot K^{\rho-1}}}= (\frac{L}{K})^{\rho-1} Now we will derive the
elasticity of substitution: \frac{K}{L} = RTS_{L,K}^\tfrac{1}{1-\rho}\Rightarrow ln(\frac{K}{L})=\tfrac{1}{1-\rho} ln(RTS_{L,K}) \frac{dln(\frac{K}{L})}{dln(RTS_{L,K})}=\frac{1}{1-\rho} • \rho = {\frac{\sigma-1}{\sigma}} = Substitution parameter • \sigma = {\frac{1}{1-\rho}} =
Elasticity of substitution • \upsilon = degree of homogeneity of the production function. Where \upsilon = 1
(Constant return to scale), \upsilon \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 \rho approaches 1, we have a
linear or perfect substitutes function; • If \rho approaches zero in the limit, we get the
Cobb–Douglas production function; • If \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: : Q = F \cdot \left[\sum_{i=1}^n a_{i}X_{i}^{r}\ \right]^{\frac{1}{r}} where • Q = Quantity of output • F = Total Factor Productivity • a_{i} = Share parameter of input i, \sum_{i=1}^n a_{i} = 1 • X_i = Quantities of factors of production (i = 1,2...n) • 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. ==CES utility function==