Homogeneous Function

Types of Functions >

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:

1. First degree homogeneous function: you multiply all variables by 2 and the function output is multiplied by 2.
2. Second degree homogeneous function: you multiply all variables by 2 and the function output is multiplied by 2².

Formal Definition

Formally, a function f is homogeneous of degree r if (Pemberton & Rau, 2001):

f (λx₁, …, λx_n) = λ^r f (x₁, …, x_n)

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.

1. Factor out λ: f( λx, λy) = λ (x + 2y).
2. 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:

1. Pull the constant λ in front:
   a(λx)² + b(λx)(λy) + c(λy)² = ax² + bxy + cy²
2. 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.

References

Pemberton, M. & Rau, N. (2001). Mathematics for Economists. An Introductory Textbook. Manchester University Press.