Dimex Fabrication Co., a small manufacturer of sheet-metal body parts for a majo

Question

Dimex Fabrication Co., a small manufacturer of sheet-metal body parts for a major U.S. automaker, estimates its long-run production function to be:

where Q is the number of body parts produced daily, K is the number of sheet-metal presses in its manufacturing plant, and L is the number of labor-hours per day of sheet-metal workers employed by Dimex. Dimex is currently operating with 8 sheet-metal presses.

Please show your analytical work when applicable.

d. If you were the production manager at Dimex, to maximize your manufacturing production (i.e. max TP), you can expect to use how many labor-hour(s)? Explain. (4 pts)

e. Dimex plans to employ 10 labor-hours for its current production. Calculate its total product (TP). (4 pts)

Explanation / Answer

When K = 8,

Q = - 0.015625 x (8)3 x L3 + 10 x (8)2 x L2 = - 0.015625 x 512 x L3 + 10 x 64 x L2 = - 8L3 + 640L2

(a) Production is maximized when dQ/ dL = 0

- 24L2 + 1,280L = 0

24L2 = 1,280L

L = 53.33 units (Assuming L is non-zero, dividing both sides by 24L)

(e) When L = 10,

TP = Q = - 0.015625 x (8)3 x (10)3 + 10 x (8)2 x (10)2 = - 0.015625 x 512 x 1,000 + 10 x 64 x 10 = 8,000 + 6,400

= 14,400

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