Euler's Homogeneous Function Theorem — Definition, Formula & Examples

Euler's Homogeneous Function Theorem states that if a function is homogeneous of degree

n

, then the sum of each variable multiplied by its partial derivative equals

n

times the function itself.

Let $f(x_1, x_2, \ldots, x_k)$ be a continuously differentiable function that is homogeneous of degree $n$ , meaning $f(tx_1, tx_2, \ldots, tx_k) = t^n f(x_1, x_2, \ldots, x_k)$ for all $t > 0$ . Then $\sum_{i=1}^{k} x_i \frac{\partial f}{\partial x_i} = n \, f(x_1, x_2, \ldots, x_k)$ .

Key Formula

x_1 \frac{\partial f}{\partial x_1} + x_2 \frac{\partial f}{\partial x_2} + \cdots + x_k \frac{\partial f}{\partial x_k} = n \, f(x_1, x_2, \ldots, x_k)

Where:

$f$ = A differentiable function homogeneous of degree n
$x_i$ = The i-th independent variable
$\frac{\partial f}{\partial x_i}$ = Partial derivative of f with respect to x_i
$n$ = The degree of homogeneity

How It Works

First, confirm the function is homogeneous by checking whether

f(tx_1, \ldots, tx_k) = t^n f(x_1, \ldots, x_k)

for some integer or real number

n

. Then compute each partial derivative

\frac{\partial f}{\partial x_i}

. Multiply each partial derivative by its corresponding variable

x_i

and sum the results. The theorem guarantees this sum equals

n \cdot f

. The proof follows directly from differentiating the homogeneity condition

f(tx) = t^n f(x)

with respect to

t

and then setting

t = 1

Worked Example

Problem: Verify Euler's theorem for

f(x, y) = x^3 + 3x^2y

Step 1: Check homogeneity. Replace

x

with

tx

and

y

with

ty

f(tx, ty) = (tx)^3 + 3(tx)^2(ty) = t^3 x^3 + 3t^3 x^2 y = t^3 f(x,y)

Step 2: The function is homogeneous of degree

n = 3

. Now compute the partial derivatives:

\frac{\partial f}{\partial x} = 3x^2 + 6xy, \qquad \frac{\partial f}{\partial y} = 3x^2

Step 3: Form the Euler sum and simplify:

x(3x^2 + 6xy) + y(3x^2) = 3x^3 + 6x^2y + 3x^2y = 3x^3 + 9x^2y = 3(x^3 + 3x^2y) = 3f

Answer: The Euler sum equals

3f(x,y)

, confirming the theorem with

n = 3

Why It Matters

This theorem appears frequently in thermodynamics and economics, where extensive and intensive variables satisfy homogeneity conditions. In economics, it underlies the result that if a production function has constant returns to scale (degree 1), paying each input its marginal product exactly exhausts total output.

Common Mistakes

Mistake: Applying the theorem to functions that are not homogeneous, such as

f(x,y) = x^2 + y + 1

Correction: Always verify

f(tx, ty) = t^n f(x,y)

first. Additive constants or mixed-degree terms break homogeneity.