Constant elasticity of substitution

The vital economic element of the measure is that it provided the producer a clear picture of how to move between different modes or types of production, for example between modes of production relying on more labour. Several economists have featured in the topic and have contributed in the final finding of the constant. They include Tom McKenzie, John Hicks and Joan Robinson.

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

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.

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
• ${\displaystyle F}$ = Total factor productivity
• ${\displaystyle a}$ = Share parameter
• ${\displaystyle K}$, ${\displaystyle L}$ = Quantities of primary production factors (capital and labor)

Calculating the rate of technological substitution between K and L:

${\displaystyle RT{S}_{L,K}=\frac{F(\frac{\upsilon }{\rho }-1)\cdot {\left(a\cdot K^{\rho }+(1-a)\cdot L^{\rho }\right)}^{\frac{\upsilon }{\rho }-1}\cdot a\cdot \rho \cdot L^{\rho -1}}{F(\frac{\upsilon }{\rho }-1)\cdot {\left(a\cdot K^{\rho }+(1-a)\cdot L^{\rho }\right)}^{\frac{\upsilon }{\rho }-1}\cdot a\cdot \rho \cdot K^{\rho -1}}=(\frac{L}{K}{)}^{\rho -1}}$

Now we will derive the elasticity of substitution:

${\displaystyle \frac{K}{L}=RT{S}_{L,K}^{\frac{1}{1-\rho }}\Rightarrow ln(\frac{K}{L})=\frac{1}{1-\rho }ln(RT{S}_{L,K})}$

${\displaystyle \frac{dln(\frac{K}{L})}{dln(RT{S}_{L,K})}=\frac{1}{1-\rho }}$

• ${\displaystyle \rho }$ = ${\displaystyle \frac{\sigma -1}{\sigma }}$ = 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
• ${\displaystyle F}$ = Total 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.