Consider a market for a homogenous product with n identical stores, where n is determined by free entry. Each store has a cost function C (q) = 4 + q, for q ≤ 4 and c (q) = ∞ for q > 4 (in other words, each store can sell up to 4 units and its cost of selling the first q units is 4 + q). There are L consumers in the market, each of whom wishes to buy up to 1 unit and is willing to pay for it up to r = 5. Suppose that a fraction of all the consumers is fully informed about the prices that the different stores charge. The remaining (1 – α)L consumers are uninformed and have to pay a cost z in order to learn the prices that different stores charge. If an uninformed consumer does not pay z, she knows only the distribution of prices but not the actual prices charged by each store. Such a consumer then picks a store at random. However, once an uninformed consumer pays z, she becomes completely informed and knows all prices charged by all stores.

1. Compute the marginal and average costs of stores and illustrate it in a figure.

2. Suppose that z = 0. Solve for the long-run competitive equilibrium in the market.

3. Now suppose that z > 0. Prove that there can be at most 2 prices in a Nash equilibrium.

4. Assume that there are two prices being charged in equilibrium. What is the low price, p_l? Given your answer, compute the high price, p_h. 

5. Compute the demand faced by low and high price stores (note that uninformed consumers pick stores at random so each store gets an equal share of the (1- α)L uninformed consumers; informed customers are indifferent among all stores that charge low prices, so each one of these stores gets an equal share of the αL informed consumers).

6. Use your answers in (4) and (5) to express the zero profit conditions for high and low price stores (recall that there is a free entry so in equilibrium, each store must earn a zero profit).

7. Solve the conditions you wrote in (6) for λ and n.

8. How do the equilibrium values of λ and n vary with z? Explain the intuition for your result.

9. Compute the average price on the market and the standard deviation of prices. Using these calculations, let PD = SD / AP be a measure of price dispersion, where SD is the standard deviation of prices, and AP is the average price. How is PD affected by s? How is PD affected by α? Are these results intuitive? Explain.