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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:


Proof:


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)
 

Thus


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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