A firm produces output, y, by using capital, k, and labor, l, according to the production function.

y=k^at^b

The firm can purchase all the capital and labor it wants at prices r and w, respectively.

a) Use the method of Lagrange multipliers to find the cost function c(r,w,y). Find the average and marginal cost.

b) What is the interpretation of the Lagrange multiplier in part (a)?

c) What is the importance of the term (a+b) being less than, equal to, or greater than one?

Solution Preview

a) Use the method of Lagrange multipliers to find the cost function c(r,w,y). Find the average and marginal cost.

To minimize output given the budget is:
Min C = wL + rK
s.t. Y = K^a L^b
Lagrange function is
F = wL + rK - D( K^a L^b - Y)
Where D is the Lagrange multiplier.

First order condition:
dF/dK= r - D aK^(a-1) L^b = 0 (1)
dF/dL = w - D bK^a L^(b-1) = 0 (2)
dF/dD = - K^a L^b + Y = 0 (3)

from (1): r = D aK^(a-1) L^b (4)
from (2): w = D bK^a L^(b-1) (5)

(4)/(5):
r /w = a/b * L/K
K = a/b * w/r * L
substitute into (3):
-K^a L^b + Y = 0
K^a L^b = Y
(a/b * w/r * L)^a L^b = Y
(aw/br)^a L^(a+b) = Y ...

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