Question

2. Suppose a circuit manufacturer can use regular (human) workers or robots to produce its output. The quantity of workers is L and the quantity of robots is R. The production function is q = 50L - L2 + 100R - R2. The firm is a small firm in the input market for workers and robots, and a small firm in the market for circuits. The price per unit of labor is w. The price per robot is r. The firm can sell its output for price p= $10. Assume the firm is operating in the long-run. (a) Derive the firm's demand for workers as a function of w. (b) Derive the firm's demand for robots as a function of r. (c) If w = 100 and r = 100, how many workers and how many robots will the firm employ? Why are they different? Your answer should reference the marginal revenue products of labor and robots. (d) If w = 100, what value of r would induce the firm to employ the same number of workers and robots?

Answer 1

Given

q=50L-(1/2)L^2+100R-(1/2)R^2

a)

Marginal Product of labor=MPL=dq/dL=50-L

Marginal Revenue Product of labor=MRPL=MPL*Price=(50-L)*10=500-10L

Set MRPL=wage rate=w for profit maximization

500-10L=w

10L=500-w

L=50-0.1w

b)

Marginal Product of robot=MPK=dq/dK=100-K

Marginal Revenue Product of robot=MRPK=MPK*Price=(100-K)*10=1000-10K

Set MRPK=r for profit maximization

1000-10K=r

10K=1000-r

K=100-0.1r

c)

If w=100, L=50-0.1w=50-0.1*100=40

If r=100, L=100-0.1r=100-0.1*100=90

Firm will hire 40 units of labor and 90 units of robots to maximize profit

We observe that price of both inputs is same, but optimal employment is different. It is because of differnce in marginal product of labor (MPL) and marginal product of robots (MPK)

d)

If w=100, L=50-0.1w=50-0.1*100=40

Set K=40 i.e.

K=100-0.1r=40

0.1r=60

r=600

value of r should be 600