Consider a linear city that stretches along a 1 km distance. Suppose buyers are distributed uniformly along the city which can be best represented by an interval [0,1]. There are two firms, A and B, producing the same good. Firm A is located at the end point 0. Firm B is located at the other end point 1. Each firm has zero marginal cost and zero fixed cost. Each buyer decides which firm to buy from according to the smallest overall cost: sum of the firm's price and buyer's transportation cost of going to and coming back from the firm.

A buyer, who is at a distance d from a firm, has a transportation cost of going to and coming back from this firm is given by 2td. Firms announce their prices simultaneously, PA and PB.

(a) Suppose that, after firms announce their prices simultaneously, each buyer decides which firm to buy from. But, each buyer who already decided to buy from firm B, when arrived at the firm, buys exactly 2 units with probability 0 < q < 1, and does not buy at all with probability 1 - q. Those who decide to buy from firm A buy exactly one unit from A with probability 1. Firms know q, but buyers are not aware of this uncertainty until they arrive at firm B. Find the equilibrium prices, PA and PB. How does firm A's equilibrium price depend on q? Provide intuition.

(b) Now, suppose that each buyer who decides to buy from firm B, buys exactly 1 unit with probability 1, so there is no such demand shock as in (a) above. However, the transportation costs to the firm A are now different for different buyers: For all the buyers with a location x < 1/2 the cost of round trip to go to and come back from firm A is still 2tx, but for those buyers with a location x 1/2, the round trip to and from firm A costs only tx (there is an express free bus from firm A on the way back only for those with x 1/2). The cost of round trip to and from firm B is still 2td for any buyer with a distance d from firm B. Find the equilibrium prices, PA and PB.