CES production function

12.1 Costs

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

(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