Question

Suppose a small competitive firm operates a technology that the firm’s owner knows from experience to work as follows: “Weekly output is the square root of the minimum of the number of units of capital and the number of units of labor employed that week.” Suppose that in the short run this firm must use 16 units of capital but can vary its amount of labor freely. (a) Write down a formula that describes the marginal product of labor in the short run as a function of the amount of labor used. (Be careful at the boundaries, i.e. think about the ‘corner’ cases) (b) If the wage is w = $1 and the price of output is p = $4, how much labor will the firm hire in the short run? (c) What if w = $1 and p = $10? (d) Derive an equation for the firm’s short-run demand for labor as a function of w and p.

(the other answers posted on Chegg for this question were incorrect)

Answer 1

We are given that

Weekly output Q is the square root of the minimum of the number of units of capital and the number of units of labor employed that week.This implies that production function is

Q = (minimum (K, L))^0.5

or

Q^2 = minimum (K, L)

Suppose that in the short run this firm must use 16 units of capital but can vary its amount of labor freely This makes the short run function

Q^2 = minimum (16, L)

a) MPL = 0. This is because addition of one unit of labor does not increases output unless capital is also increased by 1 unit. This makes Q^2 = 16 or Q^2 = L. Hence we have 16 = L This again implies that labor is fixed at 16 and does not depend on wage rate and price level

b) L is used in the proportion of capital so L is 16. This makes output = 4 units.

c) Labor demand does not depend upon the price or wage rate as the firm makes sure that K = L and when K is 16, L is fixed at 16

d) Short run demand for labor is L = K.