In a market there are many buyers (consumers) and many sellers (firms), and one homogeneous good.

a Each consumer wants to buy at meet one unit of the good, and receives net utility 1 p when he buys one unit at price p. Ir Firms can produce the good at a marginal cost of zero, and hence make a prot of p. I A fraction 5' of consumers have a discount factor 51 (type-1 consumers}, and a fraction 1 3 of the consumers have a discount factor 52 [type-2]. I In each period, each consumer visits one rm and nds the price it charges. Then the consumer decides whether to buy from that rm or to visit another rm 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. e You are told that in equilibrium, a fracticn 11' of the rms charge a price of 3:1 and the remaining 1 - '' charge a higher price p; 2: P1-1 a. Suppose type-2 consumers are indifferent between stopping rules that correspond to the two prices. Use this information to nd a relationship that must hold between 1:11 and 112. (This will involve 5 and 11' as well.) b. Suppose 61 2::- 62. 1What must the optimal stopping rule for type 1 consumers be? Now assume that type-2 consumers buy from the rst rm that they encounter (note that this is an optimal ruie given they are indifferent}. Type-l consumers follow their optimal stopping rule. c. Consider a rm that sells at p2. What is the probability that a consumer that arrives at this rm will buy from it? :1. In this market, 0 {I s a: 1, La, there are rms that sell at both prices p1 and pg. What is the relation between the two prices that must hold for this to be true? e. Using your answers above, nd the equilibrium value of 11' in terms of the 6s, I9 and the prices