2 Consider a firm that sells output and buys labor in a comp

2. Consider a firm that sells output and buys labor in a competitive market. The firm’s production function is given by q=K^1/2L^1/2 where q is the quantity of output, K is the amount of capital, and L is the number of labor hours.  The marginal product of labor is MPL=1/2K^1/2L^-1/2 .Suppose the amount of capital is fixed at 100 units (i.e., short-run).

a.) If the output can be sold for $10 per unit and labor can be purchased for $5 per unit, how many units of labor should be hired?

b.)  If the price of labor increases to $10 per unit, how many units of labor should be hired?

c.)  If labor can be purchased for $5 per unit and the output can be sold for $20 per unit, how many units of labor should be hired?

d.) How will the number of labor units hired change if the fixed amount of capital is increased from 100 units to 144 units (using the output and labor prices from part (a))?

Solution

(28)

For a firm in competitive market, its profit is maximized when the firm hires labor at the level for which following condition is satisfied:

Output price (P) x Marginal product of labor (MPL) = Marginal Cost of labor (Wage rate)

(a) When K = 100, MPL = (1/2) x (100)^1/2 / L^1/2 = (1/2) x 10 / L^1/2 = 5 / L^1/2

$10 x 5 / L^1/2 = $5

5 / L^1/2 = 1/2

L^1/2 = 10

L = 100

(b) When w = $10,

$10 x 5 / L^1/2 = $10

5 / L^1/2 = 1

L^1/2 = 5

L = 25

(c) When w = $5 and P = $20,

$20 x 5 / L^1/2 = $5

5 / L^1/2 = 1/4

L^1/2 = 20

L = 400

(d) When K = 144, MPL = (1/2) x (144)^1/2 / L^1/2 = (1/2) x 12 / L^1/2 = 6 / L^1/2

$10 x 6 / L^1/2 = $5

6 / L^1/2 = 1/2

L^1/2 = 12

L = 144