High School

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Suppose that a firm uses both labor (L) and capital (K) as inputs and has the long-run production function [tex]Q = F(L, K) = 10\sqrt{L}\sqrt{K}[/tex].

(a) If its capital is fixed at [tex]K = 9[/tex] in the short run (assuming efficient production methods), what is its short-run production function?

(b) How much does it produce in the short run if it hires 1 worker? 4 workers? 9 workers?

(c) If its capital is fixed at [tex]K = 9[/tex] in the short run, what is the average product of labor when there is 1 worker? When there are 4 workers? When there are 9 workers?

(d) If its capital is fixed at [tex]K = 9[/tex] in the short run, what is the marginal product of labor when there is 1 worker? When there are 4 workers? When there are 9 workers?

Please explain the methods used to complete the question.

Answer :

Final answer:

The short-run production function when the capital is fixed at K=9 is Q = 30√L. The firm produces 30 units with 1 worker, 42.4 units with 4 workers, and 52 units with 9 workers. The average product of labor is 30 when there is 1 worker, 10.6 when there are 4 workers, and 5.8 when there are 9 workers. The marginal product of labor is 30 when there is 1 worker, 12.4 when there are 4 workers, and 9.6 when there are 9 workers.

Explanation:

(a) In the short run, when the capital is fixed at K = 9, the short-run production function is Q = F(L,9) = 10√L√9 = 10√9L = 30√L. So, the short-run production function is Q = 30√L.

(b) To find how much the firm produces in the short run, we substitute the values of L into the short-run production function. If it hires 1 worker, Q = 30√1 = 30. If it hires 4 workers, Q = 30√4 = 30√2 ≈ 42.4. If it hires 9 workers, Q = 30√9 = 30√3 ≈ 52.0. So, the firm produces 30 units with 1 worker, 42.4 units with 4 workers, and 52 units with 9 workers.

(c) The average product of labor is the total output divided by the number of workers. When there is 1 worker, the average product of labor is Q/L = 30/1 = 30. For 4 workers, the average product of labor is Q/L = 42.4/4 = 10.6. And for 9 workers, the average product of labor is Q/L = 52/9 ≈ 5.8.

(d) The marginal product of labor is the additional output produced when one more worker is hired. When there is 1 worker, the marginal product of labor is the same as the average product of labor, which is 30. When there are 4 workers, the marginal product of labor is the difference in output between 4 workers and 1 worker, which is Q(4) - Q(1) = 42.4 - 30 = 12.4. When there are 9 workers, the marginal product of labor is the difference in output between 9 workers and 4 workers, which is Q(9) - Q(4) = 52 - 42.4 ≈ 9.6.

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