Assignment: Oligopoly in Differentiated Good Markets
1. Consider the following static game, originally proposed by Cournot. There are two monopolists, one controlling the supply of zinc and the other the supply of copper, who sell to the brass industry, which is perfectly competitive. Suppose one unit of copper and one unit of zinc are needed to produce one unit of brass. Thus, PB = PC + PZ, where PB, PC, and PZ denote the prices of brass, copper, and zinc respectively. Assume that the demand for brass is given by D(PB) = 1 - PB. Finally, assume that the unit costs of producing copper and zinc are zero.
(a) Suppose that the copper and zinc monopolies choose their prices simultaneously and non-cooperatively. Derive the best replies of the two monopolists. Are the products strategic complements or substitutes?
(b) Compute the Nash equilibrium prices.
(c) Suppose the two monopolists merged. What are the joint profit maximizing prices of copper and zinc?
(d) How does your answer to (c) compare to that in (b)? Explain the difference.
2. Suppose there are two coffee houses along Main Street. The street is one mile. One hundred residents are uniformly distributed along this stretch. Each resident buys one cup of coffee per day. A cup of coffee differs only in its location and price and not in any other way. Each customer derives a utility of $3.00 from a cup of coffee.
Starbucks Coffee House is located at the left end of the street (i.e., at 0) and Esquire Coffee House is located at the midpoint of one mile stretch (i.e., at ½). Each consumer incurs a round trip cost of $1.00 per mile from traveling from their most preferred location to the location of either coffee house. The price of coffee at Starbucks is p and at Esquires, it is q. The marginal cost of a cup of coffee is zero.
(a) Given any pair of prices (p, q), write down an equation which determines the location of the marginal consumer who is indifferent between purchasing from Starbucks or from Esquire.
(b) Derive the best reply functions to the pricing game in which the coffee houses choose prices simultaneously and determine the equilibrium prices and market shares.
(c) Is the market covered?
3. Esquire decides to open another coffee house at the right end of Main Street (i.e., location 1). Assume that Esquire can set different prices, q1 and q2, at its locations at ½ and 1 respectively.
(a) Given prices (p, q1, q2), determine the locations of the two marginal consumers - the one who is indifferent between purchasing from Esquires at location ½ and the Starbucks and the one who is indifferent between purchasing from Esquires at location ½ and from Esquires at location 1.
(b) Derive the best reply functions to the pricing game in which the coffee houses choose prices simultaneously.
(c) Determine the equilibrium prices and market shares. What is the impact of Esquires expansion on Starbucks profits? Explain your answer.