   Chapter 14.3, Problem 84E

Chapter
Section
Textbook Problem

Show that the Cobb-Douglas production function P = bLαKβ satisfies the equation L ∂ P ∂ L + K ∂ P ∂ K = ( α + β ) P

To determine

To show: The Cobb-Douglas production function, P=bLαKβ satisfies the given equation LPL+KPK=(α+β)P .

Explanation

Proof:

The Cobb-Douglas production function is, P=bLαKβ .

The equation LPL+KPK=(α+β)P is to be proved.

Obtain the value of LPL+KPK .

LPL+KPK=LL(bLαKβ)+KK(bLαKβ)=LbKβL(Lα)+KbLαK(Kβ)=LbKβ(αLα1)+KbLα(βKβ1)=αbK(L1+α1)+βbLα(K1+β1)

On further simplification, the required value is obtained as follows

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