Question

A firm has production function q = 100 L + KL− L^2 − K^2 The price of the good is $1. The wage is $10, and the price of capital is $30. Assume that the firm is a price - taker in a perfectly competitive market.

a. What will the firm’s profit maximizing choice of capital and labor be?

b. Suppose that the firm’s capital is fixed in the short-run and wage rises to $20. What is the firm’s new profit maximizing choice of labor?

c. In the long-run wages are still $20. What is the firm’s new choice of labor and capital?

d. Did labor fall more in the short-run or long-run? Explain why.

Answer 1

a)

Given, price of good=P=$1

Wage rate=w=$10

Price of capital=r=$30

q=100L+KL-L^2-K^2

Marginal Product of labor=MPL=dq/dL=100+K-2L

Marginal Product of capital=MPK=dq/dK=L-2K

For profit maximization,

MPL/MPK=w/r

(100+K-2L)/(L-2K)=10/30

300+3K-6L=L-2K

5K+300=7L

K=(7L-300)/5

We know that

Marginal revenue product of capital=price of capital

P*MPK=r

1*(L-2K)=30

Put K=(7L-300)/5

L-(2/5)*(7L-300)=30

L-2.8L+300=30

-1.8L=-270

L=150 (optimal amount of labor)

K=(7L-300)/5=(7*150-300)/5=150 (optimal amount of capital)

b)

Now K=150 and w=$20

We know that

Marginal revenue product of labor=price of labor

P*MPL=w

1*(100+K-2L)=20

100+150-2L=20

250-2L=20

L=115 (Short run choice of labor)

c)

For long run, set

MPL/MPK=w/r

(100+K-2L)/(L-2K)=20/30

300+3K-6L=2L-4K

7K+300=8L

L=(7K+300)/8

We know that

Marginal revenue product of capital=price of capital

P*MPK=r

1*(L-2K)=30

Put L=(7K+300)/8

(7K+300)/8-2K=30

7K+300-16K=240

-9K=-60

K=20/3=6.67

L=(7K+300)/8=(7*20/3+300)/8=43.33

d)

We can observe labor fall more in long run as capital has also decreased to maintain optimal long run combination of inputs.