**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 :R^{n}ｮ R be a real-valued function. Then we say ｦ (x_{1}, x_{2}...., x_{n}) ishomogeneous of degree oneorlinearly homogeneousif lｦ(x) = ｦ (lx) where l ｳ 0 (x is the vector [x_{1}...x_{n}]).

Theorem: (Euler's Theorem) If the function ｦ :R^{n}ｮ R is linearly homogeneous of degree 1 then:

ｦ(x

_{1}, x_{2}, ...., x_{n}) = x_{1}ｷ[ｶｦ/ｶx_{1}] + x_{2}ｷ [ｶｦ/ｶ x_{2}] + ...... + x_{n}ｷ[ｶｦ /dｶx_{n}]

or simply:

ｦ(x) = ・/font>

_{i=1}^{n}_{ }[ｶｦ (x)/ｶx_{i}]ｷx_{i}

There is a corollary to this:

.

Corollary: if ｦ :R^{n}ｮ R is homogenous of degree 1, then:・/font>

^{n}_{i=1}[ｶ^{2}ｦ(x)/ｶ x_{i}ｶx_{j}]x_{i}= 0 for any j.

For proofs, see our mathematical section.