Simon Bowmaker

Problem Set 6

Submit at lecture (Monday, November 10)

Write your answers on separate sheets of paper. Please include: your name your recitation teacher’s name day and time of the recitation

NB: if your recitation takes place on Monday morning, you must submit your assignment to your teacher at the beginning of the recitation. 1. Assume a monopolist faces the following market demand: Q = 100 - 2P. The monopolist’s total cost function is TC = 5+8Q2. What is the monopolist’s profit-maximizing level of output and price?

Answer

Step 1. Derive the MR and MC functions.

Since Q = 100 - 2P, P = 50 – (Q/2)

TR=P*Q=(50-1/2Q)Q=50Q-1/2Q2

MR=dTR/dQ=50-Q

MC=dTC/dQ=16Q

Step 2. Set MR=MC and solve for P*, Q*

MR=MC

50-Q=16Q

Q*=2.94 or 3 (rounded)

P*=50-1/2Q*=50-1/2*3=48.5

2. Assume a monopolist faces a market demand curve P = 130 – 2Q, and has the short-run total cost function TC = 350 + 10Q.

(a) What is the profit-maximizing level of output and price? What are profits?

(b) Graph the marginal revenue, marginal cost, and demand curves.

(a) Step 1. Derive the MR and MC functions.

P 130 – 2Q

TR=P*Q=(130 – 2Q)Q=130Q – 2Q2

MR=dTR/dQ=130 – 4Q

MC= dTC/dQ=10

Step 2. Set MR=MC and solve for P*, Q*, and π.

MR=MC

130 – 4Q=10

4Q=120

Q*=30

P*=130 – 2Q=130 – 2*30=130-60=70

Π=TR-TC=P*Q-TC=70*30-(350 10*30)=2100-650=1450

(b) Deadweight loss is equal to area abc.

3. If a monopoly firm sells a product with price $100, and marginal cost of $20, what is the Lerner Index? Interpret the Lerner Index here. What is the demand elasticity?

The price/marginal cost ratio will be 100/20 5. Its Lerner Index is (100-20)/100 80/100=0.8. As the number is close to 1, this indicates significant market power. For every dollar given to the firm by the consumer, 80 cents is mark-up. The firm believes it faces a demand elasticity of –1.25 (the Lerner Index is the inverse of the price elasticity of demand).

4. When the iPod was