Question

Suppose there are five firms in an industry. Each must individually decide in which of two cities to locate its manufacturing plant. If firm i locates in city j, its profits are π × nj, where π > 0 and nj is the number of firms from this industry that locate in city j (including itself).

(a) Give a substantive explanation for why profits might look like this.

(b) Identify all of the Nash equilibria of this game. Now suppose that one of the cities (city 1) is better for this industry than the other (city 2). A firm that locates in city 1 makes profits π1 × n1 and a firm that locates in city 2 makes profits π2 × n2. Assume π2 < π1 < 5π2.

(c) Have the Nash equilibria of the game changed?

(d) Explain how your answer suggests that agglomeration economies can create coordination traps.

(e) Often local governments offer incentives to induce a few large firms in some industry to relocate to their city. Evaluate the likely efficacy of such a policy in light of this model.

Answer 1

(a) If there is more firm in a city then the cost of input might reduce which increases the profit of the firm.

(b) All firm located in either city is a Nash Equilibrium because given all other firm are located in city i, the payoff of a firm locating in city i is 5 while locating in other city is . Hence he will locate in city i.

There is no equilibrium in which firm locate in both city, because consider a city i has less firm located compare to othe city (j). Now, consider a firm in city i, if he chooses to locate in city j (locate in another city) his payoff will increase as there are more firm in city j. Hence this is not an equilibrium.

Hence, All firm located in either city is only Nash Equilibrium.

(c) By the similar logic, there is no equilibrium in which firm are located in both city as there will be city which is generating weakly less payoff than other city, and thus firm who is getting less will locate in other city.

All firm locate in city 1 is Nash Equilibrium, because if choose to locate in other city his payoff will be (current payoff).

All firm locate in city 2 is Nash Equilibrium, because if choose to locate in other city his payoff will be (current payoff).

Hence, equilibrium remains unchanged.

(d) Consider an equilibrium in which all firm are located in city 2.
Observe, if all firm locates in city 1 then everyone will be better off. But each firm doesn't has enough incentive to move in city 1, unless all other firm are moving to city 1.

Hence, there is a coordination trap, they stuck is low level of equilibrium, everybody want other of move in city 1 first. Although if everybody mover with coordinately there payoff will increase.

(e) If local government can create enough incentive for few firms to move in city 1, then they will move to city 1. Becuase some firm moved to city 1, which increase the profit of city 1 and decreases the profit of city 2. Which likely to give incentive to all other firms shifts to city 1. Hence, economy will reach better equilibrium.

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