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[50 points] Sir Toppem Hat's Rail is the only train line operating between the coastal towns of St. Thomas and Percy. Forty (40) percent of Toppem Hat's potential customers are business passengers, who need certain train service features and are willing to pay a substantial premium for them. The other sixty (60) percent are casual passengers, who do not require these additional services. Toppem Hat's can offer seating in two types of train car to its passengers: rst-class or economy. The table below describes the willingness to pay for each category of seating by each type of passenger. The table also presents the marginal costs of each type of seat for Toppem Hat's. Throughout this question, assume that Toppem Hat's can offer as many seats of each type as it wants. All gures are in dollars. Willingness to pay Car Type Costs Casual Business First-class 50 60 100 Economy 20 40 50 Each type of passenger calculates the net payoff (benets minus price) that she would get from a seat in each type of train car and buys the one that would give the higher net payoff, provided that this payoff is non-negative. If both types of car give equal, nonnegative net payoffs for a passenger, she buys rst-class; if both types of car have strictly negative net payoff for a passenger, she does not buy either of them. Assume that Sir Toppem Hat wants to maximize his expected prots. (a) [7 points] Suppose that the rail line knows that business travelers always wear suits for train travel, while casual travelers never do the types of traveler are immediately identiable. Suppose also that travelers can only learn what types of seat are available to them by approaching the ticket counter, so that TOppem Hat's clerks can offer a certain type of seat and a unique price to each traveler. What type or types of seat would be made available to casual travelers, and at what price or prices? What about to business travelers? More realistically, Sir Toppem Hat's cannot tell one type of passenger apart from the other in advance. It just sets the prices of rst-class and economy seats, and lets each customer decide which category to buy. (b) [7 points] Suppose that Sir Toppem Hat's offers both categories of seat, charging pe for econ- omy and p f for rst-class. Write down the incentive-compatibility constraints and participation constraints on pa and p f that the rail line must satisfy if it wants the regular passengers to travel economy and the highend ones to buy rstclass tickets. (c) [7 points] What values of pe and p f will maximize expected prots when Sir Toppem Hat's offers both car types? What are the expected prots per 100 customers? ((1) [7 points] How much consumer surplus is generated by this pricing scheme, per 100 travelers
