NOTE: this problem follows the development from class, but here there are 3 potential buyers. A monopolist—with zero costs of production—faces 3 potential buyers. Customers 1, 2, and 3 have gross values for the product of v1 = 12, v2 = 10, and v3 = 7, respectively. Consumers buy at most one unit of the good over the 2 periods. There are only 2 selling periods. The seller cannot commit to a price path; rather, given what has occurred in any earlier period, it posts a price at the beginning of each period and consumers decide whether to buy. A player’s net payoff in the period she consumes is v − p, where v is here value and p is the price she paid for the good. Buyers and the monopolist discount future payoffs by the discount factor δ = 0.80. Assume consumers are sophisticated, i.e., forward-looking.
(1) If the monopolist makes no sales in the first period, what price does it set in period 2? Calculate the present value (from the first period) of the associated profit.
(2) Suppose the monopolist wants to sell exactly 1 unit in period 1. What price will it set in period 1, and what will be the subsequent price in period 2? Calculate the present value (from the first period) of the associated profit.
(3) Suppose the monopolist wants to sell exactly 2 units in period 1. What price will it set in period 1, and what will be the subsequent price in period 2? Calculate the present value (from the first period) of the associated profit.
(4) Suppose the monopolist wants to sell exactly 3 units in period 1. What price will it set in period 1? (There’s no one left to buy the good in period 2.) Calculate the present value (from the first period) of the associated profit.
(5) What is the most profitable price for the monopolist to set in period 1?