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 λ^{n} 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 2^{2}.

## Formal Definition

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

*f* (λ*x*_{1}, …, λ*x*_{n}) = λ^{r} *f* (*x*_{1}, …, *x*_{n})

In other words, a function *f* (*x*, *y*) is homogeneous if you multiply each variable by a constant (λ) → *f* (λ*x*, λ*y*)), which rearranges to λ^{n} 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*) = *ax*^{2} + *bxy* + *cy*^{2}

Where a, b, and c are constants.

Step 1: Multiply each variable by λ:

*f *(λ*x*, λ*y*) = *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}=*ax*^{2}+*bxy*+*cy*^{2} - Substitute the function in (because
*f*(*x*,*y*) =*ax*^{2}+*bxy*+*cy*^{2}):

λ^{2}*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.

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