#### 15.4.1 CES substitution elasticities

The CES- production function or CES- utility function is a production function for which the substitution elasticity always assumes the same value. Here, CES stands for c onstant e BeginExpansion $>$ EndExpansion XXX lasticity of substitution. This property is advantageous in many economic applications. The symmetrical XXX form is: ${ℝ}^{n}↦ℝ,f\left(v\right)={a}_{0}{\left({\sum }_{j=1}^{n}{c}_{j}{v}_{j}^{-\rho }\right)}^{-\frac{h}{\rho }},n\ge 1$, 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). 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 also doubled. For $h>1$ positive economies of scale apply, for $h>1$ XXX negative economies of scale apply.
u represents the production- or utility- level, a the technology factor, ${c}_{1}$ and ${c}_{2}$ the relative weights of the two input factors x and y.
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 ${c}_{1}$.
A selection of graphical illustrations can be found hier.

(c) by Christian Bauer
Prof. Dr. Christian Bauer
Chair of monetary economics
Trier University
D-54296 Trier
Tel.: +49 (0)651/201-2743
E-mail: Bauer@uni-trier.de
URL: https://www.cbauer.de