$$f\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{ishomogeneousofthedegree}\phantom{\rule{1em}{0ex}}n\iff x{f}_{1}^{\prime}\left(x,y\right)+y{f}_{2}^{\prime}\left(x,y\right)=\mathit{nf}\left(x,y\right)\phantom{\rule{2em}{0ex}}\left(1\right).$$ |

$$f\left(\mathit{kx},\mathit{ky}\right)={k}^{n}f\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{forall}\phantom{\rule{1em}{0ex}}k\in {\mathbb{R}}_{0}^{+}$$ |

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

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

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)

$$\text{Income}=\text{unitprice*quantitysold}=\mathit{pq}=\mathit{pF}\left(K,L\right)$$ |

5)

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

(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