Question

Consider the Hotelling model of the competition between two firms. They must choose to set up business on a line (=1). Select all that apply. (PLEASE EXPLAIN ANSWERS COMPLETELY)

	a.	If both firms are localized in position 1/2 (i.e., center of the line), neither firm has incentives to deviate and move to a different position.
	b.	If Firm 1 and Firm 2 localize at the same point along the line, they will each sell to 50% of the consumers.
	c.	In the Nash Equilibrium in pure strategies firms will localize together anywhere along the line.
	d.	If Firm 1 is located at position 1/2 (i.e., center of the line) and firm 2 is located somewhere else, then both firms have incentives to deviate and change their position along the line.

Answer 1

Assumption:

Customers are uniformly distributed along the line. The firms simultaneously select a position. Customers go to the closest vendor and split themselves evenly if the vendors choose an identical position. Each vendor wants to maximize its number of customers.

Explanation:

As outside observers, we can say that the best position for each seller would be a quarter of a mile from opposite ends of the beach, i.e, (1/4) & (3/4). That way, no one has to walk more than a quarter of a mile to get the product. People in the middle will be will be indifferent, since it’s a quarter of a mile to either shopkeeper.

But the sellers don’t think this way, they see that they are doing well, serving half of the clients. But what if were to place themselves a little toward the center of the line? All of the people on our end won’t change their behavior — we’re still the closest shop. But now we’re closer to the middle of the line than our competitor, so those people in the middle are no longer indifferent. They’ll come to us. In fact, the closer we move toward our competitor, the more customers we’ll get.

Of course, our competitor is thinking the same thing. So where do we eventually end up? It’s not hard to see that unless the two firms are situated side by side in the middle of the line, there’s economic incentive for one or the other to move. The middle is an equilibrium point — neither vendor can benefit by moving.

In the end, by moving to the middle the vendors haven’t gained anything — they’re still both serving just half of the customers. Hotelling’s Law states that there’s a natural tendency for competitors to be pulled toward a common middle ground.

Thus, the Nash equilibrium is for both vendors to select the median location (.5) or any other location at the same point along the line, doing this guarantees each firm serves exactly half the customers (because of the assumption that customers go to the closest vendor and split themselves evenly if the vendors choose an identical position). Any other position of any of the firms(for example one firm at position(.5) and the other at any other position) will lead to strictly less number of customers to at least one firm, which is not an acceptable situation. Repetitive games of this type will eventually lead the firms to close in and sell together.