Consider the following version of the Hotelling model. There is a city on the interval [0,1], and consumers are uniformly distributed along the interval. There are two firms i = A, B selling identical products with identical cost functions ci (qi ) = 1 qi . Firm A is located at point a [0, 1], and firm B is located at (1b) [0,1].

Consumers purchase only one unit of the good from one firm. Consumers pay a price pi and also have a quadratic travel cost. Consumer surplus is as follows:

v pA t(a x)2 if purchasing from ACS= vpBt((1b)x)2 ifpurchasingfromB

We derived the following demand functions implied by these preferences in class:DA(pA, pB) = pB pA + 1 (1 b a) + a

2t(1ba) 2 DB(pB, pA) = pA pB + 1 (1 b a) + b.

2t(1ba) 2

1. Assume that the government imposes a regulated price pA = pB = p > 1. Show that the consumer at x will purchase from firm A if the cost of traveling to firm A is less than the cost of traveling to firm B. (Do not assume that a = b for this or any of the following questions.)

2. Continue assuming that the government imposes a regulated price pA = pB = p > 1. What is the quantity sold by firm i, qi? (Hint: Use the demand functions.)

3. What is firm is profits if the regulated price is p = 2?

4. Continue assuming that p = 2. In part c, you found the profits of the firms in terms of product characteristics. Notice that because the government sets the price, the firms do not engage in price competition. Now assume that b = 0, meaning that firm B is located at 1. Show that profits for firm A increase as A moves closer to B. (Hint: You can use a graph ofAs profits with respect to a, or you can show this with a short proof using derivatives.)