The profit-maximizing levels of labor and output are $250.
Q = K1/2L1/2 = L1/2 [Since K = 1]
(a) Average product of labor = Q / L = 1 / L1/2
= 1 / (9)1/2 = 1 / 3 = 0.33
(b) Marginal product of labor (MPL) = dQ / dL = 1 / 2L1/2
= 1 / [2 x (9)1/2] = 1 / (2 x3) = 0.17
(c) Profit is maximized when (Output price x MPL = Wage rate)
$100 x [1 / 2L1/2] = $10
1 / 2L1/2 = 1 / 10
2L1/2 = 1
L1/2 = 5
L = 25
Q = L1/2 = 5
(d) Let the maximum allowable price of capital be r.
Total cost (TC) = wL + rK = 10 x 25 + r = 250 + r [Since L = 25 (computed above) and K = 1]
When L = 25, Q = 5
Total revenue (TR) = $100 x 5 = $500
Profit ($) = TR – TC = 500 – (250 + r)
Profit = 500 – 250 – r
Profit = 250 – r
Therefore, profit = 0 if r = $250 (Maximum permissible price of capital)
The profit maximization rule states that if a firm wants to maximize profit, it should choose a level of output where marginal cost (MC) equals marginal revenue (MR) and the marginal cost curve is upward. .
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