Each consumer wants to buy at most one unit of the good, and receives net utility 1-p when he buys one unit at price p. • Firms can produce the good at a marginal cost of zero, and hence make a profit of p. • A fraction of consumers have a discount factor &1 (type-l consumers), and a fraction 1- of the consumers have a discount factor 82 (type-2). • In each period, each consumer visits one firm and finds the price it charges. Then the consumer decides whether to buy from that firm or to visit another firm the following period. A consumer who make a purchase leaves the market and is replaced by an identical new consumer, so the proportions of the two types of consumers remain constant across periods. • You are told that in equilibrium, a fraction 7 of the firms charge a price of pı and the remaining 1 – a charge a higher price p2 > p1. a. Suppose type-2 consumers are indifferent between stopping rules that correspond to the two prices. Use this information to find a relationship that must hold between pı and p2. (This will involve 8 and 7 as well.) b. Suppose 81 > 82. What must the optimal stopping rule for type 1 consumers be? Now assume that type-2 consumers buy from the first firm that they encounter (note that this is an optimal rule given they are indifferent). Type-l consumers follow their optimal stopping rule. c. Consider a firm that sells at p2. What is the probability that a consumer that arrives at this firm will buy from it? d. In this market, (< 1 <1, i.e., there are firms that sell at both prices p1 and p2. What is the relation between the two prices that must hold for this to be true? e. Using your answers above, find the equilibrium value of 7 in terms of the 8s, 0 and the prices.

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