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Consumers are distributed uniformly over the interval [0,1]. There are N firms. The firms simultaneously each choose one location along the interval. Once a location is chosen, each firm faces a marginal cost of c for each unit it sells, and earns a price of p>c for each unit. The price p is fixed through regulation, and does not depend upon the locations of the firms. Because price is exogeneous and identical across firms, each consumer simply purchases from the closest firm. (a) Suppose N = 2. Find all pure strategy Nash equilibrium locations, and explain your answer. (b) Suppose N = 3. Are there any Nash equilibria in pure strategies (locations)? If so, find one and demonstrate that it is a Nash equilibrium. If not, explain why not. (c) Finally, suppose N=4. Show that there is a Nash equilibrium in which two firms locate together at one point, and two firms locate together at another point.
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