10.3 Homogeneous functions of two variables

A function f : 2, (x,y)f(x,y) is called homogeneous of the degree n , if for all (x,y) 2 the following is valid:
f(kx,ky) = knf(x,y)for allk 0+

If the variables x and y are multiplied by a positive number k > 0, the function value is multiplied by the factor kn.

Example 1: The function f(x,y) = 5x2y2 + xy3 is homogeneous of the degree 4:

f(kx,ky) = 5(kx)2(ky)2+(kx)(ky)3 = 5k2x2k2y2+kxk3y3 = k4(5x2y2+xy3) = k4f(x,y)d.h.n = 4

For example, for k = 2 we get

f(2x, 2y) = 24f(x,y) = 16f(x,y)

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

Example 2: The function f(x,y) = x2y + xy is not homogeneous:

f(kx,ky) = (kx)2(ky)+(kx)(ky) = k3x2y+k2xy = k2(kx2y+xy) = k3(x2y+1 kxy)

So, here it is not possible to factor out k3 or ka for any a. Consequently, the definition equation f(kx,ky) = knf(x,y) 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(x,y) = Cxaybfor(x,y) + × +,C > 0,a > 0,b > 0

This function is often used to describe production processes. x and y are called input factors, F(x,y) 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(kx,ky) = C(kx)a(ky)b = Ckaxakbyb = ka+bCxayb = ka+bf(x,y)d.h.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