A firm’s production technology is given by the production function where q = L 2
ID: 1216213 • Letter: A
Question
A firm’s production technology is given by the production function where
q = L2K
L represents labor hours, K machine hours and q the amount of output. The market wage and rental rates are, w= $64 and r = $128. The firm is operating in the long run where it can adjust both inputs.
(a) Suppose that the firm currently is using twice as many units of labor as capital. Is it minimizing its long run total cost? If so why so and if not why not? Explain. If it is not minimizing its long run cost, how should it adjust its input usage? Explain. Provide appropriate calculations.
(b) Suppose that the firm wants to produce 128 units of output. Determine the cost minimizing combination of L and K. Calculate the resulting long run total cost. Show and explain all calculations.
(c) Calculate the short run total cost if q =128 and w= $64 and r = $128, but capital, K is fixed at 1.
(d) Without assuming a specific numerical production target, but using w= $64 and r = $128 calculate the equation for the long run total cost function (in terms of q). (Hint: Assume that the level of output is q. Using the above w, r values first determine the least cost combinations of L and K)
(e) Using Excel- Solver verify your answers to (b) above. (Show your work. Show the spreadsheets in detail. Show the Solver window embedded on the relevant worksheet so that the commands in the Solver window become directly visible and are linked to the cells of the worksheet. To show the solver window, use print screen command on your key board and then create a MS Word document using paste.
Explanation / Answer
q = L2K
Long run total cost, LTC = wL + rK = 64L + 128K
(a) L = 2K
LTC is minimum when
MPL / MPK = w / r = 64 / 128 = 1/2
MPL = dq / dL = 2LK
MPK = dq / dK = L2
MPL / MPK = 2LK / L2 = 2K / L = 1 / 2
L = 4K
This is the required condition for LTC to be minimized. But given L = 2K, long run TC is not minimized.
To ensure LTC is minimized, labor has to be reduced and capital has to be increased until MPL / MPK = 1/2.
(b) q = 128
From part (a), L = 4K. Substituting in production function,
128 = 4K x 4K x K = 16 x K3
K3 = 8
K = 2
L = 4K = 4 x 2 = 8
So, LTC = (64 x 8) + (2 x 128) = 512 + 256 = 768
(c) q = 128, K = 1, w = 64, r = 128
128 = L2 x 1 = L2
L = 11.31
Short run TC = 64 x 11.31 + 128 x 1 = 724 + 128 = 852
(d)
From part (a), L = 4K. Substituting in production function:
q = 4K x 4K x K = 16 x K3
K3 = q / 16
K = (q / 16)1/3
L = 4K = 4 x (q / 16)1/3
LTC = 64L + 128K = 64 x 4 x (q / 16)1/3 + 128 x (q / 16)1/3 = (q / 16)1/3 x (256 + 128) = 384 x (q / 16)1/3
Note: First 4 sub-parts are answered.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.