4) HOTELLING PRICES: MOVIE TICKETS Movie theatre firms F1 and F2 are showing the same movie and have located at L1=0 and L2=1, the two ends of a one-mile-long street. On this street there are signposts every 1/5th mile (at 0, 0.2, 0.4, 0.6, 0.8, and 1), and there are 100 movie fans unevenly distributed as shown in the diagram below. Each group of fans sits halfway between two adjacent signposts, so there are 10 fans at 0.1, 30 at 0.3, and so on. Each movie fan will buy a ticket from the nearest theatre, but only if the following buying condition holds: v≥p+ tlx – yl, where v is the customer's valuation (or willingness to pay), x is the customer's location, y is a theatre's location, and t/x – y/ is a measure of the physical travel cost from x to y. If both theatres are within a fan group's acceptable travel distance, the fans choose the theatre that maximizes consumer surplus: CS = v − p − t|x − y| = valuation - total cost. Assume that v = 10, t = 10, and F2's price is P2 = 5 in this Hotelling pricing model (that has fixed locations). Assume prices are integers between 3 and 7: Pi = {3,4,5,6,7}, for i = 1,2. a) Find F1's profit (revenue) if it undercuts F2's price by setting price P1 = 3. b) Find F1's profit if it matches F2's price by setting P1 = 5. Find F1's profit if it sets a relatively high price of P1 = 7. - Of the three prices we tested in parts (a)-(c), what is P1*, F1's best response to its rival's price of P2 = 5? We have only analyzed F1's choices of this market. Does F2 charge P2 = 5 in a Nash Equilibrium? HINT: For (P1=7, P2=5) to be a Nash Equilibrium, each price must be a best response to the other. Clearly show that P2=5 generates higher revenue for F2 than the other prices (or that it does not!). 10 fans at 0.1 30 fans at 0.3 20 fans at 0.5 30 fans 10 fans at 0.7 at 0.9 0 0.2 0.4 0.6 0.8 8:0 L1=0 1 L2=1