Homogenous Functions

A homogenous function of order $n$ satisfies:
$$
f(\lambda x_1, \lambda x_2, \ldots, \lambda x_m) =\lambda^n f(x_1, x_2, \ldots, x_m)
$$
For e.g.
$$
f(x,y) = \sqrt{x} y^2
$$
is homogenous of order $n = 3/2$ . However:
$$
f(x,y) = \sqrt x y^2 + x^2 y^2
$$
is not a homogenous function because it is the sum of two terms which transform differently under scaling.

Euler’s Theorem

Let $f(x_1, x_2, \ldots, x_m)$ be a homogenous function of order $n$ in $m$ variables $x_1, x_2, \ldots, x_m$. Then the following is true ¹ :
$$
n f(x_1, x_2, \ldots, x_m) = x_1 \frac{\partial f}{\partial x_1} + x_2 \frac{\partial f}{\partial x_2} + \ldots + x_m \frac{\partial f}{\partial x_m}
$$

Proof

It is sufficient to use a function of two variables to prove the theorem as the proof extends trivially to greater numbers of variables.
Let:
$$
f(\lambda x, \lambda y) = \lambda^n f(x,y)
$$
Taking the derivative of both sides w.r.t $\lambda$:
\begin{eqnarray}
n \lambda^{n-1} f(x,y) & = \frac{\partial f}{\partial x’} \frac{\partial x’}{\partial \lambda} + \frac{\partial f}{\partial y’} \frac{\partial y’}{\partial \lambda} \\
& = x \frac{\partial f}{\partial x’} + y \frac{\partial f}{\partial y’}
\end{eqnarray}
where $x’ = \lambda x, y’ = \lambda y$. Setting $\lambda = 1$ in the above we obtain:
$$
n f(x,y) = x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y}
$$
q.e.d.

Extension to arbitrary scaling parameters

If often happens that all dependent variables scale do not scale in the same way. In such cases, the general scaling behavior of a function is given as ² :
$$ \lambda^n f(x_1, \ldots, x_m) = f(\lambda^{\alpha_1} x_1, \ldots, \lambda^{\alpha_m} x_m) $$
In such cases, one can repeat the procedure in the previous section (differentiate both sides w.r.t, $\lambda$ and finally set $\lambda = 1$) to obtain:
\begin{equation}\label{eqn:euler-extended}
n f(x_1, \ldots, x_m) = \alpha_1 x_1 \frac{\partial f}{\partial x_1} + \ldots + \alpha_m x_m \frac{\partial f}{\partial x_m}
\end{equation}

Existence of Converse

The converse is also holds true, i.e. if \ref{eqn:euler-extended} is true then $f(x_1,\ldots, x_m)$ is a homogenous function of degree $n$.