### 10.3 Homogeneous functions of two variables

Definition:
A function $f:{ℝ}^{2}↦ℝ,\left(x,y\right)↦f\left(x,y\right)$ is called homogeneous of the degree $n\in ℝ$, if for all $\left(x,y\right)\in {ℝ}^{2}$ the following is valid:

If the variables $x$ and $y$ are multiplied by a positive number $k>0$, the function value is multiplied by the factor ${k}^{n}$.

Example 1: The function $f\left(x,y\right)=5{x}^{2}{y}^{2}+x{y}^{3}$ is homogeneous of the degree 4:

 $f\left(\mathit{kx},\mathit{ky}\right)=5{\left(\mathit{kx}\right)}^{2}{\left(\mathit{ky}\right)}^{2}+\left(\mathit{kx}\right){\left(\mathit{ky}\right)}^{3}=5{k}^{2}{x}^{2}{k}^{2}{y}^{2}+\mathit{kx}{k}^{3}{y}^{3}={k}^{4}\left(5{x}^{2}{y}^{2}+x{y}^{3}\right)={k}^{4}f\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{d.h.}\phantom{\rule{1em}{0ex}}n=4$

For example, for $k=2$ we get

 $f\left(2x,2y\right)={2}^{4}f\left(x,y\right)=16f\left(x,y\right)$

So, if $x$ and $y$ are doubled, the function value $f\left(x,y\right)$ increases by a factor of 16.

Example 2: The function $f\left(x,y\right)={x}^{2}y+\mathit{xy}$ is not homogeneous:

 $f\left(\mathit{kx},\mathit{ky}\right)={\left(\mathit{kx}\right)}^{2}\left(\mathit{ky}\right)+\left(\mathit{kx}\right)\left(\mathit{ky}\right)={k}^{3}{x}^{2}y+{k}^{2}\mathit{xy}={k}^{2}\left(k{x}^{2}y+\mathit{xy}\right)={k}^{3}\left({x}^{2}y+\frac{1}{k}\mathit{xy}\right)$

So, here it is not possible to factor out ${k}^{3}$ or ${k}^{a}$ for any $a$. Consequently, the definition equation $f\left(\mathit{kx},\mathit{ky}\right)={k}^{n}f\left(x,y\right)$ of a homogeneous function is not fulfilled.

In general, it can be said that a polynomial is homogeneous of the degree $n$ when the sum of the exponents in each summand is equal to $n$.

Example 3: An important function in many economic models is the Cobb-Douglas function

 $f\left(x,y\right)=C{x}^{a}{y}^{b}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}\left(x,y\right)\in {ℝ}^{+}×{ℝ}^{+},C>0,a>0,b>0$

This function is often used to describe production processes. $x$ and $y$ are called input factors, $F\left(x,y\right)$ is the number of units produced, i.e. $F$ is called a production function.
It is easy to show that the Cobb-Douglas function is homogeneous of the degree $a+b$:

 $f\left(\mathit{kx},\mathit{ky}\right)=C{\left(\mathit{kx}\right)}^{a}{\left(\mathit{ky}\right)}^{b}=C{k}^{a}{x}^{a}{k}^{b}{y}^{b}={k}^{a+b}C{x}^{a}{y}^{b}={k}^{a+b}f\left(x,y\right)\phantom{\rule{1em}{0ex}}\text{d.h.}\phantom{\rule{1em}{0ex}}n=a+b$

(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