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 ...

Output, Profit, Fixed Costs and Perfect Competition. ... c. What is the value of average fixed cost at this ... d. At what output will average variable cost be minimized ...

... d. Determine the value of Q at which point the marginal cost function takes ... Books and computer purchases needed for his study will cost an average of $2,000 ...

... Thanks,. The expert solves the marginal cost, marginal revenues and total revenues. Average variable costs with respective values are provided. ...

... its Average revenue is greater than the average variable cost which is at 100 units. In short term it is able to meet the variable and some of the fixed costs. ...