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:
${\mathbb{R}}^{n}\mapsto \mathbb{R},f\left(v\right)={a}_{0}{\left({\sum}_{j=1}^{n}{c}_{j}{v}_{j}^{-\rho}\right)}^{-\frac{h}{\rho}},n\ge 1$, 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