Constant elasticity of substitution

In economics, more specifically econometrics or mathematical economics, there are production functions that describe the output given a certain combination of inputs (e.g. labour and capital). In describing these functions, the elasticity of substitution describes the size of the change in input substitution given a change in the price of the inputs.

The two factor (Capital, Labor) Constant Elasticity of Substitution (CES) production function introduced by Arrow, Chenery, Minhas, and Solow, (1961), is:


 * $$ Q = F \cdot \left(a \cdot K^r+(1-a) \cdot L^r\right)^{\frac{1}{r}} $$

where


 * $$Q$$ = Output
 * $$F$$ = Factor productivity
 * $$a$$ = Share parameter
 * $$K$$, $$L$$ = Primary production factors (Capital and Labor)
 * $$r$$ = $${\frac{(s-1)}{s}}$$
 * $$s$$ = Elasticity of substitution. $$s$$ = $${\frac{1}{(1-r)}}$$

As its name suggests, the CES production function exhibits constant elasticity of substitution between capital and labor. Leontief and Cobb-Douglas production functions are special cases of CES production function. That is, in the limit as $$s$$ approaches 1, we get the Cobb-Douglas function; as $$s$$ approaches infinity we get the linear (perfect substitutes) function; and for $$s$$ approaching 0, we get the Leontief (perfect complements) function.

The general form of the CES production function is:


 * $$ Q = \left[\sum_{i=1}^n a_{i}^{\frac{1}{s}}X_{i}^{\frac{(s-1)}{s}}\ \right]^{\frac{s}{(s-1)}}$$

where


 * $$Q$$ = Output
 * $$F$$ = Factor productivity
 * $$a$$ = Share parameter
 * $$X$$ = Production factors (i = 1,2...n)
 * $$s$$ = Elasticity of substitution.

Nested CES functions are commonly found in partial/general equilibrium models. Different nests (levels) allow for the introduction of the appropriate elasticity of substitution.

The CES is a neoclassical production function.

The same functional form arises as a utility function in consumer theory. More generally, this functional form acts as an aggregator function with constant elasticity of substitution. That is, it combines two or more types of consumption, or two or more types of productive inputs into an aggregate quantity. For example, if there exist $$n$$ types of consumption goods $$c_i$$, then aggregate consumption $$C$$ could be defined using the CES aggregator:


 * $$ C = \left[\sum_{i=1}^n a_{i}^{\frac{1}{s}}c_{i}^{\frac{(s-1)}{s}}\ \right]^{\frac{s}{(s-1)}}$$

Here again, the coefficients $$a_i$$ are share parameters, and $$s$$ is the elasticity of substitution. Therefore the consumption goods $$c_i$$ are perfect substitutes when $$s $$ approaches infinity and perfect complements when $$s=0$$. The CES aggregator is also sometimes called the Armington aggregator, which was discussed by Armington (1969).