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The Converse of Euler's Theorem

A function F(L,K) is homogeneous of degree n if

F(λL,K)λ) = λnF(L,K)
for all λ.

Euler's Theorem is that if F(L,K) is homogeneous of degree n then

(∂F/∂L)L + (∂F/∂K)K = nF(l,K).

The converse of Euler's Theorem is:


Since G(L,K) = (1/n)[(∂G/∂L)L + (∂G/∂K)K]
G(λL,λK) = (∂G/∂(λL))λL + (∂G/∂(λK)λK
= (λ/n)[(∂G/∂(λL))L + (∂G/∂(λK)K]

Note that

dG(λL,λK)/dλ = (∂G/∂(λL))L + (∂G/∂(λK))K .

An inspection of the above equations reveals that

dG(λL,λK)/dλ = (n/λ) G(λL,λK)

This is a differential equation for G in terms of λ, which when expressed in differential form is:

dG/G = n(dλ/λ)

The solution to the above equation is

ln(G(λL,λK)) = c' + nln(λ) = c' + ln(λn)


G(λL,λK) = cλn

The integration constant c is evaluated by setting λ equal to unity; i.e.,

G(L,K) = c
and hence
G(λL,λK) = G(L,K)λn

Thus G(L,K) is homogeneous of degree n.

This result of course generalizes to a function of an arbitrary number of arguments.

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