If the variables and are multiplied by a positive number , the function value is multiplied by the factor .
Example 1: The function
is homogeneous of the degree 4:
For example, for
So, if and
are doubled, the
increases by a factor of 16.
Example 2: The function is not homogeneous:
So, here it is not possible to factor out
. Consequently, the
of a homogeneous function is not fulfilled.
In general, it can be said that a polynomial is homogeneous of the degree when the sum of the exponents in each summand is equal to .
Example 3: An important function in many economic models is the Cobb-Douglas function
This function is often used to describe production processes.
are called input
factors, is the number
of units produced, i.e.
is called a production function.
It is easy to show that the Cobb-Douglas function is homogeneous of the degree :