UnsolvedExercises_Chapter14

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Apr 3, 2024

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UNSOLVED EXERCISES U2. MicroStuff is a software company that sells two popular applications, WordStuff and ExcelStuff. It doesn’t cost anything for MicroStuff to make each additional copy of its applications. MicroStuff has three types of potential customers, represented by Ingrid, Javiera, and Kathy. There are 100 million potential customers of each type, whose valuations for each application are as follows: (a) If MicroStuff sets separate prices for WordStuff and ExcelStuff, what price should it set for each application to maximize its profit? How much profit does MicroStuff earn with these prices? (b) What does each type of customer (Ingrid, Javiera, Kathy) buy when MicroStuff sets separate profit-maximizing prices for WordStuff and ExcelStuff? (c) Instead of selling the applications separately, MicroStuff decides always to sell WordStuff and ExcelStuff together in a bundle, charging a single price for both. What single price for the bundle would maximize its profit? How much profit does MicroStuff make selling its software only in bundles? (If multiple prices maximize its profit, consider the highest price.) (d) What does each type of customer buy when MicroStuff sets a single profit-maximizing price for a bundle containing WordStuff and ExcelStuff? How does this answer compare with your answer in part (b)? (e) Which pricing scheme does each customer type prefer? Why? (f) If MicroStuff sold the applications both as a bundle and separately, which products (WordStuff, ExcelStuff, or the bundle) would it want to sell to each customer type? How can MicroStuff make sure that each customer type purchases exactly the product that it intends for them to purchase? (g) What prices—for WordStuff, ExcelStuff, and the bundle—would MicroStuff set to maximize its profit? How much profit does MicroStuff make selling the products at these three prices? (h) How do your answers to parts (a), (c), and (g) differ? Explain why. U3. Consider a managerial effort example similar to the one in Section 5. The value of a successful project is $420,000 ; the probabilities of success are 0.5 with good supervision and 0.25 without. The manager’s expected payoff equals his expected income minus the cost of his effort. His current job pays $90,000 , and his cost for exerting the extra effort for good supervision on your project is $100,000 . (a) Show that inducing extra effort would require the firm to offer a compensation scheme with a negative base salary; that is, if the project fails, the manager pays the firm an amount stipulated in the scheme. (b) How might a negative base salary be implemented in reality?
(c) Show that if a negative base salary is not feasible, then the firm does better to settle for low pay and no extra effort. U4. Cheapskates is a very minor-league professional hockey team. Its facilities are large enough to accommodate all of the 1,000 fans who might want to watch its home games. It can provide two kinds of seats—ordinary and luxury. There are also two types of fans: 60% of the fans are blue-collar fans, and the rest are white-collar fans. The costs of providing each kind of seat and the fans’ willingness to pay for each kind of seat are given in the following table (measured in dollars): Each fan will buy at most one seat, depending on the consumer surplus he would get (maximum willingness to pay minus the actual price paid) from each kind. If the surplus for both kinds is negative, then he won’t buy any. If at least one kind gives him a nonnegative surplus, then he will buy the kind that gives him the larger surplus. If the two kinds give him an equal nonnegative surplus, then the blue-collar fan will buy an ordinary seat, and the white-collar fan will buy a luxury seat. The team owners provide and price their seating to maximize profit, measured in thousands of dollars per game. They set a price for each kind of seat, sell as many tickets as are demanded at these prices, and then provide the numbers of seats of each kind for which the tickets have sold. (a) First, suppose the team owners can identify the type of each individual fan who arrives at the ticket window (presumably by the color of his collar) and can offer him just one kind of seat at a stated price, on a take-it-or-leave-it basis. What is the owners’ maximum profit, 𝜋 , under this system? (b) Now, suppose that the owners cannot identify the type of any individual fan, but they still know the proportion of blue-collar fans. Let the price of an ordinary seat be ? and the price of a luxury seat be ? . What are the incentive-compatibility constraints that will ensure that the blue- collar fans buy the ordinary seats and the white-collar fans buy the luxury seats? Graph these constraints on an ?– ? coordinate plane. (c) What are the participation constraints for the fans’ decisions on whether to buy tickets at all? Add these constraints to the graph in part (b). (d) Given the constraints you found in parts (b) and (c), what prices ? and ? maximize the owners’ profit, 𝜋 2 , under this price system? What is 𝜋 2 ? (e) The owners are considering whether to set prices so that only the white-collar fans will buy tickets. What is their profit, 𝜋 𝑤 , if they decide to cater to only the white-collar fans?
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