located around the circle that is of unit circumference. Each consumer buys only one product, and pays the price for a product and transportation cost given by tx, where x is the distance between the consumer and the product, and t is the cost for each distance traveled. Assume that t = 4.Firms are symmetric, and so if there are n firms in the market, then the distance between two firms is 1 . The cost of each firm is given by cq + F, where c is a constant marginalncost, q is the quantity produced by each firm, and F is fixed cost that each firm needs to incur to enter the market. Firms enter the market as long as their profits are not negative. Assume that c = 1 and F = 1.Consider two-stage game where at the first stage, firms decide to enter, and the number of firms n is determined; at the second stage, they are engaged in price competition.7.1. Consider a symmetric equilibrium where each firm charges the same price p? in equilibrium, and determine the equilibrium number of firms n? in the market.(1) Start with the second stage for price competition given n. Suppose that firm i charges pi, while all other firms charge p?. Consider a critical consumer who is indifferent between firm i?s product and firm i + 1?s product. Let x denote the distance between firm i and the critical consumer. What is x in terms of n, t, p?, and pi?(2) What is firm i?s profit? (Note that firm i?s residual demand is 2x.)(3) Determine firm i?s best response for pi given that all other firms choose p?. (4) What is the equilibrium p?? What is the equilibrium profit ?? given n?(5) Now come back to the first stage. Recall that firms enter as long as their profitsare not negative. How many firms will enter the market in equilibrium? That is, what is n??
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