Chapter 12
CES production function

The CES- production function is a production function whose substitution elasticity always assumes the same value. Here, CES stands for c onstant e lasticity of s ubstitution. This property is advantageous in many economic applications. The symmetrical form is: ${ℝ}^n\mapsto ℝ,f\left(v\right)=a_0{\left({\sum <!--\; FUNCTION\; APPLICATION\; -->}_{j=1}^n{c}_j{v}_j^{-\rho }\right)}^{-\frac{h}{\rho }},n\ge 1$, where the elasticity of substitution is $\sigma =\frac{1}{1+\rho }$.
By variation of $\rho$ the type of utility function can be changed from Leontief to Cobb-Douglas to perfect substitution (linear utility) function. This is illustrated in the following graph "Substitutionality of production factors". In the present diagram, further parameters can be changed.
represents the production level. The higher u is chosen, the more is produced and the more input is needed. With increasing u the isoquant shifts outwards.
a represents the technology factor. The higher a is chosen, the more can be produced with the same amount of production factors. Thus, for a constant output quantity, with better technology less input is required. With increasing a the isoquant shifts inwards.
Instead of production level u and technology factor a, the effective output $\frac{u}{a}$ can also be considered.
c${}_1$ and c${}_2$ represent the relative weights of the two production factors (inputs) $\mathit{Factor}1$ and $\mathit{Factor}2$. The higher the weight c${}_i$ of the production factor $\mathit{Factor}1$, the more output can be produced per unit of Factor i.
h indicates the degree of homogeneity. If h = 1, the function is linearly homogeneous, i.e. if all input factors are doubled, the output is doubled as well. For $h>1$ positive economies of scale apply, for $h<1$ negative economies of scale apply.
For an effective total output $\frac{u}{a}$ greater than 1 the following applies: The greater h is, the higher c.p. the total output, i.e. less input is needed to achieve a certain output u. With increasing h, the isoquant slips inwards. At $\frac{u}{a}=1$ h has no effect on the isoquant and at $\frac{u}{a}<1$ the effect of h is inverse, because the power function is decreasing to a base smaller than 1. This case is usually not considered.
In the above graph, for n=2 a graph of the CES function is

$$u=a{\left(c_1{x}^{-\rho }+c_2{y}^{-\rho }\right)}^{-\frac{h}{\rho }}.$$

Since this representation is overparameterized, the parameter was set $c_2=1$, so that the relative weight of the two goods is represented only by $c_1$.