10.4 Important properties of homogeneous functions: The Euler theorem and the absence of profit for companies with linear economies of scale

$$f\left(x,y\right)\text{is homogeneous of the degree}n\iff x{f}_1^{\prime }\left(x,y\right)+y{f}_2^{\prime }\left(x,y\right)=\mathit{nf}\left(x,y\right)\left(1\right).$$

We first show that the right side of (2) is valid, if $f\left(x,y\right)$ is homogeneous of the degree $n$ (i.e. "$\Rightarrow$"). The definition of a homogeneous function says:

$$f\left(\mathit{kx},\mathit{ky}\right)=k^{n}f\left(x,y\right)\text{for all}k\in {ℝ}_0^{+}$$

We differentiate both sides with respect to $k$:

$$x{f}_x\left(\mathit{kx},\mathit{ky}\right)+y{f}_y\left(\mathit{kx},\mathit{ky}\right)=n{k}^{n-1}f\left(x,y\right)$$

If you set $k=1$, you get the right side of (1):

$${\mathit{xf}}_x\left(x,y\right)+{\mathit{yf}}_y\left(x,y\right)=\mathit{nf}\left(x,y\right)$$

For the proof of the opposite direction (i.e. "$\Leftarrow =$") we present $g\left(\mathit{tk}\right)=f\left(\mathit{kx},\mathit{ky}\right)$ as a differential equation and show that there is only one solution:

$$g^{\prime }\left(k\right)=\frac{d}{\mathit{dk}}f\left(\mathit{kx},\mathit{ky}\right)=x{f}_x\left(\mathit{kx},\mathit{ky}\right)+y{f}_y\left(\mathit{kx},\mathit{ky}\right)=k^{-1}\left(\mathit{kx}{f}_x\left(\mathit{kx},\mathit{ky}\right)+\mathit{ky}{f}_y\left(\mathit{kx},\mathit{ky}\right)\right)=\frac{1}{k}\mathit{nf}\left(\mathit{kx},\mathit{ky}\right)=\frac{n}{k}g\left(t\right)$$

This differential equation has exactly one solution $g\left(k\right)=k^{n}g\left(1\right)$, therefore

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

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 itself.
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 $\alpha +\beta =1$, i.e. $F\left(K,L\right)=K^{\alpha }{L}^{\beta }=K^{\alpha }{L}^{1-\alpha }$ or linear functions like $F\left(K,L\right)=\mathit{aK}+\mathit{bL}$. 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 $p$ does 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 quantity optimally.

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 $w=\mathit{pL}{F}_L\left(K,L\right)$ and the interest rate $r=\mathit{pK}{F}_K\left(K,L\right)$. 3) The production function is linear homogeneous, i.e.

$$L{F}_L\left(K,L\right)+K{F}_K\left(K,L\right)=F\left(K,L\right)$$

4)

$$\text{Income}=\text{unit price * quantity sold}=\mathit{pq}=\mathit{pF}\left(K,L\right)$$

5)

$$\text{Costs}=\mathit{wL}+\mathit{rK}=p{F}_L\left(K,L\right)L+p{F}_K\left(K,L\right)K=p\left(L{F}_L\left(K,L\right)+K{F}_K\left(K,L\right)\right)=\mathit{pF}\left(K,L\right)$$