Consider the following variant of the Spatial Bertrand Model: There is a line of length 4 along which 800 consumers are uniformly distributed. There are two firms in this market located at the two endpoints of the line, at x = 0 and x = 4. Firm 1 is charging price p1 for good 1 and firm 2 is charging a price p2 for good 2. Both firms have a constant marginal cost c = 20. Define a consumer's location x as her most preferred product. Each consumer is willing to pay $100 for their most preferred product. However, there is a disutility associated with purchasing a good that is not your most preferred. Specifically, for a consumer located at x, the disutility of purchasing good 1 is 10x and the disutility of purchasing good 2 is 10(4 x)

a. Where is the marginal consumer located? Derive each firm's demand and profit as a function of prices p1 and p2 (5 pts)

b. What price will both firms set in equilibrium? How much profit is each firm making in equilibrium? (5 pts)

c. Suppose that instead of both firms choosing their prices simultaneously, firm 1 chooses its price p1 first and then firm 2, after observing firm 1's price, chooses its own price p2. What are the new Nash equilibrium prices of this game? How much profit is each firm making in equilibrium? (10 pts)