10.4 Important properties of homogeneous functions: The Euler
theorem and the absence of profit for companies with linear economies of
We first show that the right side of (2) is valid, if
of the degree
The definition of a homogeneous function says:
We differentiate both sides with respect to
If you set ,
you get the right side of (1):
For the proof of the opposite direction (i.e.
as a differential equation and show that there is only one solution:
This differential equation has exactly one solution
Profits of companies with constant economies of scale
For companies with constant economies of scale, which compete both
on the factor market and on the goods market, the total revenue is divided
between the production factors, e.g. in the form of wages for work performed and
interest on equity or debt capital. No profit remains with the company
Instead of the expression "constant economies of scale", one could
also use equivalent terms such as "linear homogeneous production
function" or "a production function of the homogeneity degree 1".
Examples for such production functions are Cobb-Douglas functions with
We perform the proof for illustration purposes with only two production factors K
and L. The generalization to any number of production factors is trivial. Of
course, other factors of production, such as high and low skilled labor, can be used
without changing the result of the theorem.
The proof of the statement happens in five simple steps, which also explain, why
the conditions are necessary to reach the conclusion.
1) Competition on the goods market: The enterprise is a price taker, i.e.
(1) the quantity produced by the enterprise can be sold and (2) the market price
not change in response to a change in the production quantity of the enterprise.
Thus, the enterprise takes the market price as given and adjusts its production
2) Competition on the factor market: The company remunerates the factors of
production according to the marginal revenue product, i.e. the wage rate is
interest rate .
3) The production function is linear homogeneous, i.e.
(c) by Christian Bauer
Prof. Dr. Christian Bauer
Chair of monetary economics
Tel.: +49 (0)651/201-2743