Euler's homogeneous function theorem

Formal statement

Let f: U \subset R^n \to R be continuously differentiable on an open set U that is invariant under positive scalar multiplication and suppose f is homogeneous of degree k. Then for every x in U,

x1 (∂f/∂x1) + x2 (∂f/∂x2) + ... + xn (∂f/∂xn) = k f(x). This identity is often written compactly as sum_i xi ∂_i f(x) = k f(x).

Proof idea

A short derivation uses a one‑variable auxiliary function g(t) = f(tx). Differentiating g(t) = t^k f(x) with respect to t and then evaluating at t = 1 gives g'(1) = k f(x). By the chain rule g'(1) = sum_i xi ∂_i f(x), which yields the stated equality. The argument requires differentiability of f and the homogeneity assumption but not convexity or other stronger properties.

Examples and applications

• Monomials and polynomials: f(x,y)=x^a y^b is homogeneous of degree a+b. Euler's identity recovers the expected weighted sum of partials.
• Production functions in economics: a Cobb–Douglas function f(K,L)=A K^α L^β is homogeneous of degree α+β. When α+β=1 (constant returns to scale), Euler's theorem implies output can be decomposed as K·MP_K + L·MP_L, a key step in showing that competitive payments to factors can exhaust product.
• Geometry and physics: homogeneity and Euler's relation appear when studying homogeneous potentials, scaling laws, and dimensional analysis.

Conditions, variants and history

The usual hypotheses are continuous differentiability on a ray‑invariant domain and exact homogeneity for positive scalars. There are extensions to functions homogeneous for all real t, to distributions, and to complex variables. The result is named after Leonhard Euler, who studied homogeneous forms; it is a standard tool across analysis and applied fields. For a succinct reference page about the theorem see related material, and for its role in economics consult economic applications.

Notable remarks

Euler's theorem is sometimes called Euler's identity for homogeneous functions; it should not be confused with other famous formulas that also bear Euler's name. In practice the identity gives a convenient check on homogeneity and supplies conserved or balanced decompositions in models where factors are paid marginal products.