Dominick Salvatore Schaums Outline of Microeconomics, 4th edition Schaums Outline Series 2006

CHAP. 6] THEORY OF PRODUCTION 145
Forming function Z, which incorporates the production function to be maximized subject to the given cost
outlay set equal to zero, and get
Z ¼ f (L, K) þ l (C wL rK)
where l is the Lagrangian multiplier. Taking the first partial derivative of Z with respect to L and K, we get
@Z
@L ¼ @f
@L
l w ¼ 0
Dividing the first equation by the second, we get
and
@Z
@K ¼ @f
@K l r ¼ 0
MP L
MP K
¼ w r
or
MP L
w
¼ MP K
r