Economies of scale

When analyzing production functions, four aspects are of particular importance:
(1) the effect of individual production factors, (2) the interchangeability of
production factors, (3) costs, and (4) production volume. Here, we will focus in
particular on the last aspect.

A naive assumption is that with optimal factor input, doubling the input produces
twice as much output. A production function with this property is called a
production function with constant economies of scale. However, this is
not always the case. On the one hand, it is possible that the optimal
factor input ratio changes with different input quantities. For example, for
reasons of space or legal regulations, a multiplication of the machine-
or personnel- input may be connected with certain obstacles and costs,
resulting in a shift towards relatively more personnel- or capital- input.
Production functions in which the optimal factor input ratio always remains
constant are called homothetic. Homogeneous production functions, like the
Cobb-Douglas production function used here, are a special case of this.
With homogeneous production functions, also the factor that indicates
by how much the output quantity increases when all input factors are
doubled remains constant, independent of the current quantity of input. This
factor is called degree of homogeneity. In the above diagram it is called
$\lambda $.
The input times x results in an increase (or decrease for
$x<1$) of the output
to ${x}^{\lambda}$-times
as much.

If $\lambda <1$,
then with an even increase of all inputs by x%, the output increases by less than
x%. This is called decreasing economies of scale.

If $\lambda =1$,
then with an even increase of all inputs by x% the output increases by exactly x%.
This is called constant economies of scale.

If $\lambda >1$,
then with an even increase of all inputs by x% the output increases by more than
x%. This is called increasing economies of scale.

The degree of homogeneity can be adjusted with the slider. Since in the graph
the isoquants for an output of 5, 10 and 15 are shown, the following results are
obtained: With decreasing economies of scale, more than twice of all input factors
are needed to double the output from 5 to 10. The isoquants move away from each
other. With increasing economies of scale less than twice the input factors are
needed to double the output from 5 to 10. The isoquants move towards each
other.

The production function used in the graph is

$$\mathit{Output}={x}^{\frac{1}{2}\lambda}{y}^{\frac{1}{2}\lambda},$$ |

which has a degree of homogeneity of
$\lambda $. For
a better visualization of the effect we have dynamized the function, so that the
output is always normalized to 5 when x=5 and y=5.

Definition: Homogeneous function:

A function $f\left(x,y\right)$ is called
homogeneous of degree $\lambda $,
if

$$f\left(\mathit{kx},\mathit{ky}\right)={k}^{\lambda}f\left(x,y\right).$$ |

For functions with more than two input variables, the definition is
accordingly.

Definition: Homothetic function:

A function $h\left(x,y\right)$
is called homothetic, if there is a homogeneous function
$f\left(x,y\right)$ and a monotone
transformation $g$
(e.g.$g\left(x\right)=\mathit{ln}\left(x\right)$
or $g\left(x\right)={x}^{2}$ for
$x>0$) so
that

$$h\left(x,y\right)=g\left(f\left(x,y\right)\right)$$ |

for all x,y.

$$f\left(\mathit{kx},\mathit{ky}\right)={k}^{\lambda}f\left(x,y\right).$$ |

For functions with more than two input variables, the definition is
accordingly.

(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