Problem Set 6 Answers 
Microeconomics (Fall 2014)
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
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