A homogeneous function has variables that increase by the same proportion. In other words, if you multiple all the variables by a factor λ (greater than zero), then the function’s value is multiplied by some power λn of that factor. The power is called the degree.
A couple of quick examples:
- First degree homogeneous function: you multiply all variables by 2 and the function output is multiplied by 2.
- Second degree homogeneous function: you multiply all variables by 2 and the function output is multiplied by 22.
Formally, a function f is homogeneous of degree r if (Pemberton & Rau, 2001):
f (λx1, …, λxn) = λr f (x1, …, xn)
In other words, a function f (x, y) is homogeneous if you multiply each variable by a constant (λ) → f (λx, λy)), which rearranges to λn f (x, y). The exponent n is called the degree of the homogeneous function.
How Do I Know if I Have a Homogeneous Function?
While it isn’t technically difficult to show that a function is homogeneous, it does require some algebra.
All linear functions are homogeneous of degree 1. For example, take the function f(x, y) = x + 2y.
Step 1: Multiply each variable by λ: f( λx, λy) = λx + 2 λy.
Step 2: Simplify using algebra.
- Factor out λ: f( λx, λy) = λ (x + 2y).
- Substitute function notation back in (because f(x, y) = x + 2y): λ f(x, y)
The idea is, if you multiply each variable by λ, and you can arrange the function so that it has the basic form λ f(x, y), then you have a homogeneous function.
The algebra is also relatively simple for a quadratic function. For example, let’s say your function takes the form
f (x, y) = ax2 + bxy + cy2
Where a, b, and c are constants.
Step 1: Multiply each variable by λ:
f (λx, λy) = a(λx)2 + b(λx)(λy) + c(λy)2
Step 2: Simplify, using algebra:
- Pull the constant λ in front:
a(λx)2 + b(λx)(λy) + c(λy)2 = ax2 + bxy + cy2
- Substitute the function in (because f (x, y) = ax2 + bxy + cy2):
λ2 f (x, y)
Other examples of homogeneous functions include the Weierstrass elliptic function and triangle center functions.
Watch this short video for more examples.
Pemberton, M. & Rau, N. (2001). Mathematics for Economists. An Introductory Textbook. Manchester University Press.