Calculus How To

Homogeneous Function

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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):

fx1, …, λxn) = λr f (x1, …, xn)

In other words, a function f (x, y) is homogeneous if you multiply each variable by a constant (λ) → fx, 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 x, 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.

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

Stephanie Glen. "Homogeneous Function" From Calculus for the rest of us!

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