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(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[/x1] + x2 [/ x2] + ...... + xn[ /dxn]
or simply:
(x) = E/font>i=1n [ (x)/xi]xi
There is a corollary to this:
.Corollary: if :Rn R is homogenous of degree 1, then:
E/font> ni=1[2(x)/ xixj]xi = 0 for any j.
For proofs, see our mathematical section.