Euler's Theorem

Euler's Theorem

Euler’s Theorem states that if we have a function which is homogeneous of degree 1 (e.g. constant returns to scale, if a production function), then we can express it as the sum of its arguments weighted by their first partial derivatives.

Definition: (Linear Homogeneity) Let і :Rⁿ ў R be a real-valued function. Then we say і (x₁, x₂ ...., x_n) is homogeneous of degree one or linearly homogeneous if lі(x) = і (lx) where l Ё 0 (x is the vector [x₁...x_n]).

Theorem: (Euler's Theorem) If the function і :Rⁿ ў R is linearly homogeneous of degree 1 then:

і(x₁, x₂, ...., x_n) = x₁Ї[Іі/Іx₁] + x₂Ї [Іі/І x₂] + ...... + x_nЇ[Іі /dІx_n]

or simply:

і(x) = Е_i=1ⁿ[Іі (x)/Іx_i]Їx_i

There is a corollary to this:

.Corollary: if і :Rⁿ ў R is homogenous of degree 1, then:

Е ⁿ_i=1[І²і(x)/І x_iІx_j]x_i = 0 for any j.

For proofs, see our mathematical section.