Consider the production function Q = 4 K0.5L0.5 where K represents capital equip
ID: 1116410 • Letter: C
Question
Consider the production function Q = 4 K0.5L0.5 where K represents capital equipment and L is labor. The firm has already spent a total of $12,000 on the 4 units of capital equipment it owns. Currently the company does not have the flexibility needed to acquire additional equipment. If workers at the firm are paid a competitive wage of $100 per day and the firm’s product can be sold for $200 each. a) What is the firm labor usage? b) What is the firm profit maximizing level of output? c) What is the company maximum profit? Please show all steps
Explanation / Answer
Q = 4K0.5L0.5
When K = 4,
Q = 4 x (4)0.5L0.5 = 4 x 2 x L0.5 = 8L0.5
(a) Profit is maximized when MPL x Output price = Wage rate
MPL = dQ / dL = 8 x 0.5 / L0.5 = 4 / L0.5
(4 / L0.5) x $200 = $100
4 / L0.5 = 0.5
L0.5 = 4 / 0.5 = 8
Squaring both sides,
L = 64
(b) When K = 4 and L = 64,
Q = 8L0.5 = 8 x 8 = 64
(c)
Total cost ($) = Cost for capital + Total labor cost = 12,000 + (100 x 64) = 12,000 + 6,400 = 18,400
Total revenue ($) = Output price x Q = 200 x 64 = 12,800
Profit ($) = TR - TC = 18,400 - 12,800 = 5,600
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