Question

1. Suppose a short-run production function is described as Q = 2L – (1/800)L^2 where L is the number of labors used each hour. The firm’s cost of hiring (additional) labor is $20 per hour, which includes all labor costs. The finished product is sold at a constant price of $40 per unit of Q.

a. How many labor units (L) should the firm employ per hour

b. Given your answer in a, what is the output (Q) per hour

c. Given your answer in b, what is the resulting profit per hour assuming only labor costs?

d. Suppose that labor costs remain unchanged but that the price received per unit of output increases to $50. How many labor units (L) will the firm now employ?

e. Suppose instead that the price of the product is unchanged at $40, but that the cost of hiring labor increases to $24 per hour. How many labor units (L) will the firm employ?

f. In terms of the demand (curve) for labor, how would we visually see (what is the difference between) the changes in parts d and e? 2 points

g. Using the production function Q = 2L – (1/800)L^2. How many units of L would be required to produce 800 units (Q=800). Use the quadratic formula to solve for L.

h. Suppose that management increases the size of its plant, providing each worker with additional capital. What is the most likely impact on the marginal products of its labor input?

Answer 1

Q=2L - (1/800)L²

TR = PQ = 80L - 40(1/800)L²

MR = Dq/Dl = 80 - (1/10)L = Mc = 20

60 = (1/10)L

L = 600

B). When L = 600

Q = 2(600) - (1/800)(600)² = 1200 - 450 = 750

C). Profit = Revenue - Cost = 750(40) - 600(20) = 30,000 - 12,000 = 18,000

D). P = 50

QP = 100L - (1/800)50L²

MR = 100 -( 1/8)L = MC = 20

80 = (1/8)L

L = 640

E). MR = 80 - (1/10)L = 24

56 = (1/10)L

560 = L

F) In part D, the Demand curve of labour will shift to the right while in part e, the curve will shift to the left.

G) 800 = 2L - (1/800)L²

  Multiplying the entire equation by 800

640000 = 1600 L - L²

L² -1600L + 640000= 0

L = 800