10.3 Homogeneous functions of two variables

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

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

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

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

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

$$f\left(x,y\right)=C{x}^a{y}^b\text{for}\left(x,y\right)\in {ℝ}^{+}\times {ℝ}^{+},C>0,a>0,b>0$$

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 $a+b$:

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