Chapter 14
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

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

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

for all x,y.

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