Problem Set 5

Oligopoly in Differentiated Good Markets

1. In Tuftsville, everyone lives along Main Street which is 10 miles long. There are 1000 people uniformly spread up and down Main Street, and each day they each buy one fruit smoothie from one of the two stores located at either end of Main Street. Customers ride their motor scooters to and from the store and the motor scooters use $0.50 worth of gas per mile. Customers buy their smoothies from the store offering the lowest price, which is the store's price plus the customer's travel costs getting to and from the store. Ben owns the store at the west end of Main Street and sells smoothies at a price of p1 per smoothie. Will owns the store at the east end of Main Street and sells smoothies at a price of p2 per smoothie. The marginal cost of a smoothie is $1. In addition, each owner pays the city $250 per day for the right to sell smoothies. Assume that prices are chosen simultaneously.

(a) Write down the equation that determines the location of the consumer who is indifferent between buying from Ben or Will.

(b) Formulate Ben's optimization problem and derive his best reply to p2.

(c) Formulate Will's optimization problem and derive his best reply to p1.

(d) Find the equilibrium prices, quantities and profits.

2. George is attracted by the profits that Ben and Will are earning and decides to open a store at the midpoint of Main Street. His costs are the same as Ben and Will.

(a) Write down equations that determine the location of the consumer who is indifferent between buying from Ben or George and the location of the consumer who is indifferent between buying from George and Will.

(b) If Will and Ben do not change their prices (your answer to (d) in problem 2), what is George's best reply? How much profit would he earn?

(c) Derive the best replies for Will and Ben and find the equilibrium prices and quantities.

3. Suppose George is actually Ben's partner and they set prices to maximize profits at both locations.

(a) How do you think prices will change relative to your answer to (c) of question 3? Provide intuition.

(b) Confirm your intuition by deriving best replies and computing equilibrium prices.