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

Each firm wants to set up a manufacturing plant in a particular city. From the profit function, we see that the more firms set up a plant in a city, the more fruitful it is for other firms. Why does the profit function look like this? To understand this, we need to know the concept of economies of agglomeration - cost savings arising from the assemblage of multiple firms/plants in one area. As more firms in related industries cluster together, their costs of production decline significantly (competing suppliers, division of labour, greater specialization etc.), thereby increasing profits. Thus, the profit of any firm is a function of the number of firms building plants in that city.

If we consider both cities to be equal in all manner (ceteris paribus), a firm will benefit from settling in a city where all other firms are setting up plants, thereby increasing every firm's profit.
Thus, if all firms build plants in one city, the profit for each firm will be .
Any other combination leads to lesser profits.

Now, one of the cities (City 1) is better for the industry, our condition for equality is gone. We can see that . Therefore, in the new Nash equilibrium, all firms will set up plants in City 1, and each firm will have a profit function of: .

In the above cases, we have observed the highest possible outcome, it offers only the possibility of higher productivity, that can only be realized through some kind of coordination. If one firm fails to coordinate, every firm loses out in profit while the firm that fails to coordinate gets only in profit. This is a coordination trap and any agglomeration economy is prone to fall to this trap since it's profits depend largely on all firms coordinating on their decisions.

Hope this helped!