The production function of a typical food manufacturer has been estimated to be
ID: 1216674 • Letter: T
Question
The production function of a typical food manufacturer has been estimated to be q = K^1/3 L^1/2 where K denotes capital and L denotes labor. In the short run, the firm’s capital is fixed at K =1000. The prices of capital and labor are fixed at r=1 and w=2 in the short run and the long run. 1. Derive the short run cost function of the firm. 2. Derive the marginal cost and average variable cost functions of the firm in the short run and represent them on a graph with price on the vertical axis. 3. Does the firm shut down at any positive output price in the short run? Explain your reasoning. 4. Derive the long run cost function of the firm using the method of your choice. 5. Derive the marginal cost and average cost functions of the firm in the long run. 6. Does the firm decide to shut down at any positive price in the long run? Explain your reasoning.Explanation / Answer
q = K1/3L1/2
Total cost, TC = wL + rK = 2L + K
(1) K = 1000
q = (1000)1/3L1/2 = 10 x L1/2
So, L1/2 = q / 10
Squaring both sides,
L = q2 / 100
Substituting in TC function,
TC = 2 x (q2 / 100) + K = (q2 / 50) + 1000 [Short run cost function]
(2)
MC = dTC / dq = (2q / 50) = q / 25
AVC = TVC / q = [q2 / 50] / q = q / 50
(3)
The firm will shut down in short run if its price is lower than AVC. Since price information is not provided, we cannot conclude whether output price is higher or lwoer than AVC, so cannot conclude if firm shuts down.
(4)
Input combination is optimal if (MPL / MPK) = w / r
MPL = dq / dL = (1/2) x K1/3 / L1/2
MPK = dq / dK = (1/3) x L1/2 / K2/3
So, MPL / MPK = (3/2) x (K / L) = 2 / 1 = 2
3K / 2L = 2
3K = 4L
K = 4L / 3
Substituting in production function:
q = (4L / 3)1/3L1/2 = (4/3)1/3 x L5/6
L5/6 = q x (3/4)1/3
Raising each side to the power of (6/5), we get
L = q6/5 x (3/4)2/5
Substituting in TC,
TC = 2L + K = 2L + (4L / 3) = 10L / 3 = (10 / 3) x q6/5 x (3/4)2/5 [Long run cost function]
Note: First 4 questions are answered.
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