Consider a market for a homogenous product with n identical price-setting stores, where n is determined by free entry. Each store has a cost function C (q) = √q, where q is the number of customers the store serves. There are M + 15 consumers in the market, each of whom wishes to buy up to one unit and is willing to pay for it up to r = 1. The number of 15 consumers know the prices charged by all the stores in the market (i.e., have zero search costs), while M consumers do not know the prices at all (i.e., have prohibitively high search costs). Of the latter M=n visit store i and none of the other stores, i = 1,…., n.

1. Prove that there cannot exist a symmetric pure-strategy equilibrium in this market.

2. Suppose that all the stores in the market use the same mixed strategy. What is the support of the mixed strategy as a function of n?

3. Write the profit of a store when it charges p = 1 Prove that this profit is zero. Use the zero profit condition to compute the equilibrium number of firms, n^*. Given n^*, write the support of the equilibrium mixed strategy of prices.

4. Compute the profits of a store when it happens to be charging the lowest price in the market and when it does not. Using these expressions, compute the equilibrium distribution of prices at each store, F^*(p).

5. What happens to the distribution of prices when the number of uninformed consumers, M, increases? What does this result mean for the uninformed consumers? Give an intuition for this.

6. What happens to the distribution of price paid by informed consumers when the number of uninformed consumers, M, increases? Provide an intuition for this result.