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 ｦ :Rn ｮ R be a real-valued function. Then we say ｦ (x1, x2 ...., xn) is homogeneous of degree one or linearly homogeneous if lｦ(x) = ｦ (lx) where l ｳ 0 (x is the vector [x1...xn]).
Theorem: (Euler's Theorem) If the function ｦ :Rn ｮ R is linearly homogeneous of degree 1 then:
ｦ(x1, x2, ...., xn) = x1ｷ[ｶｦ/ｶx1] + x2ｷ [ｶｦ/ｶ x2] + ...... + xnｷ[ｶｦ /dｶxn]
or simply:
ｦ(x) = ・/font>i=1n [ｶｦ (x)/ｶxi]ｷxi
There is a corollary to this:
.Corollary: if ｦ :Rn ｮ R is homogenous of degree 1, then:
・/font> ni=1[ｶ2ｦ(x)/ｶ xiｶxj]xi = 0 for any j.
For proofs, see our mathematical section.