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 :Rn R be a real-valued function. Then we say (x1, x2 ...., xn) is homogeneous of degree one or linearly homogeneous if lヲ(x) = (lx) where l 0 (x is the vector [x1...xn]).

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

(x1, x2, ...., xn) = x1キ[カヲ/x1] + x2キ [カヲ/ x2] + ...... + xnキ[カヲ /dxn]

or simply:

(x) = ・/font>i=1n [カヲ (x)/xi]キxi

There is a corollary to this:

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

・/font> ni=1[2(x)/ xixj]xi = 0 for any j.

For proofs, see our mathematical section.