Firms and Markets Assignments and Problems Price Discrimination and Auctions Practice Problems 2 1. Many states sell vanity license plates (with personalized initials) to drivers, for extra fees. The cost of producing a vanity plate is the same as the cost of a normal license plate. Three states have (separately) hired you as a consultant to advise them on their pricing of these license plates. They have given you the following information on the current prices (i.e., the extra fees) that they are charging and the estimated price elasticity of demand in each state for vanity plates at those prices: State Price Elasticity Massachussets $50-1.00 Ohio $60-0.75 Utah $30-1.50 Each state wants to maximize its revenue from the sale of the vanity plates. What general pricing advice (higher? lower? no change?) would you give to each state? Explain. (If you need to make additional assumptions, explain what they are.) 2. Your software company has just completed the first version of SpokenWord, a voiceactivated word processor. As marketing manager, you have to decide on the pricing of the new software. You commissioned a study to determine the potential demand for SpokenWord. From this study, you know that there are essentially two market segments of equal size, professionals and students (one million each). Professionals would be willing to pay up to $400 and students up to $100 for the full version of the software. A substantially scaled-down version of the software would be worth $50 to consumers and worthless to professionals. It is equally costly to sell any version. In fact, other than the initial development costs, production costs are zero. (a) What are the optimal prices for each version of the software? Suppose that, instead of the scaled-down version, the firm sells an intermediate version that is valued at $200 by professionals and $75 by students. (b) What are the optimal prices for each version of the software? Is the firm better off by selling the intermediate version instead of the scaled-down version?
3. We had the baby Mac, then the imac; it s now time for the baby imac. As head of marketing of Apple Computer, you decided you could do better than the current situation. Last year, the company sold 1 million imacs for $1,500 each. This is the most you can get from the market segment that currently buys the imac. According to a marketing study, there is a second market segment of 2 million people willing to pay up to $500 for a stripped-down version of the imac. Your market researchers also tell you that (i) the first segment would be willing to pay up to $800 for the stripped down version, (ii) the second segment would not be willing to pay more than $1,000 even for the full-fledged version of the imac. Finally, your production people tell you that it costs $300 to produce an imac, be it the standard version or the stripped-down version. What is your optimal pricing policy? 4. Your company (a large diversified conglomerate) has decided to spin off a unit that produces TV content. You know there are several potential buyers and would like to receive as much as possible from this sale. Which sale mechanism would you chose, and why? 5. Compare the performance of auctions and posted prices as alternative selling mechanisms. When would you use one of the other? Games 6. Consider the following two-player game, where Player 1 chooses row (T or B) and Player 2 chooses column (L or R). Player 2 Player 1 L R 8 9 T 5 100 6 7 B 6 7 Regarding the players strategies, which of the following is true: (a) Player 1 has a dominant strategy, but Player 2 does not. (b) Player 2 has a dominant strategy, but Player 1 does not. (c) Both Player 1 and Player 2 have dominant strategies. (d) No player has dominant strategies. Regarding the Nash equilibrium(a) of the game, which of the following is true: (a) There exists no Nash equilibrium in this game. (b) There exists only one Nash equilibrium: (B,R). (c) There exists only one Nash equilibrium: (T,R). (d) There exist exactly two Nash equilibria. Practice Problems 2 Page 2
(e) None of the above. 7. Consider the following two-player, three-strategy game: Player 2 Player 1 L C R 2 0 1 T 1 1 1 1 2 1 M 2 2 2 2 4 3 B 10 0 3 Regarding the strategies of this game: (a) T is a dominant strategy for Player 1 (b) C is a dominant strategy for Player 2. (c) R is a dominated strategy for Player 2. (d) None of the above. The Nash equilibrium(a) of this game is(are) given by: (a) (B,L). (b) (M,C) and (B,R). (c) (B,R). (d) (M,C). (e) None of the above 8. The tables below represent a series of two-player games which illustrate the rivalry between Time magazine and Newsweek. Each magazine's strategy consists of choosing a cover story: Impeachment or Financial crisis are the two choices. (Time s strategy is in row, Newsweek s in column. In each cell, the first value is Time s payoff, the second Newsweek s payoff.) The first version of the game corresponds to the case when the game is symmetric (Time and Newsweek are equally well positioned). As the payoff matrix suggests, Impeachment' is a better story but payoffs are lower when both magazines choose the same story. The second version of the game corresponds to the assumption that Time is a more popular magazine (Time's payoff is greater then Newsweek's when both magazines cover the same story). Finally, the third version of the game illustrates the case when the magazines are sufficiently different that some readers will buy both magazines even if they cover the same story. For each of the three versions of the game, Practice Problems 2 Page 3
(a) Determine whether the game can be solved by dominant strategies; (b) Determine all Nash equilibria; (c) Indicate clearly which assumptions regarding rationality are required in order to reach the solutions in (a) and (b). Impeachment Financial crisis Impeachment 35,35 70,30 Financial crisis 30,70 15,15 (i) Time and Newsweek are evenly matched Impeachment Financial crisis Impeachment 42,28 70,30 Financial crisis 30,70 18,12 (ii) Time is more popular than Newsweek Impeachment Financial crisis Impeachment 42,28 70,50 Financial crisis 50,70 30,20 (iii) Some readers will buy both magazines 9. You are currently the sole seller of ByeByeCold, a revolutionary drug that almost instantly eliminates cold symptoms. Although the production cost is only $.10 per dose, you sell ByeByeCold for $1.39 per dose, for a total profit of $900m a year. You are currently considering licensing ByeByeCold to a second producer. One of your managers suggested, since the firm would be sharing the market with a competitor, it would be appropriate to charge a flat fee that covers half the current profits plus a generous margin; the value of $700m was suggested. An alternative proposal would be to set a royalty fee of $.50 per dose. What is your opinion? 10. Fifteen million ZZ894 microprocessors are produced annually. The unit production cost is $120. Aggressive competition between the three main producers implies an equilibrium price very close to marginal cost. An independent laboratory in Northern California has recently discovered and patented a novel production process that will slash production costs by 50%. How much would you be willing to pay for the patent if you were granted exclusivity? What if exclusivity were not guaranteed? Strategy Problems 11. Two paper pulp firms compete in the strategic variable production capacity. In each period, the market price is a function of demand and total industry capacity (that Practice Problems 2 Page 4
is, the equilibrium price is such that demand equals total capacity). (a) What shape would you expect the reaction curves to have? (b) Determine the impact of an increase in one of the firm s cost on market shares and industry output. 12. The following normal form game depicts the situation of two firms deciding where to position their products. There are five possible positions: Far Left (FL), Middle Left (ML), Center (C), Middle Right (MR), and Far Right (FR). (a) What do the numbers suggest about the nature of price competition in this industry? (b) Determine each firm s reaction function (best responses). (c) Determine the Nash equilibrium (equilibria) of the game. Firm 1 Firm 2 FL ML C MR FR 0 10 30 40 35 FL 0 5 15 20 35 5 0 20 32 10 ML 10 0 10 28 40 15 10 0 10 15 C 30 20 0 20 30 20 28 20 0 5 MR 40 32 10 0 10 35 40 30 10 0 FR 35 20 15 5 0 13. Consider a two-firm pricing game where firms sell differentiated products. (a) What shape would you expect reaction curves to have? (Hint: remember the Bertrand model is an extreme case when product differentiation vanishes.) (b) Determine the equilibrium of the model graphically. (c) Suppose one of the firm s cost goes down. What do you expect will happen to equilibrium prices? 14. The following normal form game depicts the pricing rivalry between Intel and AMD for their newest top-of-the-line chip. Each company considers five possible price Practice Problems 2 Page 5
levels: $449, $399, $349, $299 and $249. Payoffs are monthly profits in million dollars. (a) Determine each firm s reaction functions. (b) Determine the Nash equilibrium (equilibria) of the game. (c) What aspects do you think are left out in this game? What other strategies would you advise Intel or AMD to follow? (This is an open question; please limit your answer to maximum one page.) Intel AMD 449 399 349 299 249 50 70 55 45 30 449 200 180 150 120 95 35 45 50 40 25 399 230 210 160 130 110 15 25 30 35 20 349 210 190 180 140 120 5 15 20 25 15 299 180 170 150 120 100-5 10 15 20 10 249 120 110 105 95 90 15. Consider a market for used cars. One third of the cars are lemons. Lemons are worth zero to both buyers and sellers. Two thirds of the cars are in good conditions. Sellers value these cars at $2,000; buyers value them at $2,500. Although sellers know the quality of the cars, buyers cannot observe quality. Show that, in equilibrium, no sales take place in this market. 16. Unfriendly Airlines has a lot of cash. Friendly Airlines does not. Currently, both firms are setting high fares and making profits of $50m and $40m a month, respectively. Unfriendly Airlines could start a price war, in which case each airline would lose $10m a month. Unfriendly Airlines is uncertain about Friendly Airlines financial position. Their conjecture is that, with probability 40%, Friendly Airlines would not survive a price war, with probability 60% it would; it would find out within a month of price war. If Friendly Airlines does not survive a price war, Unfriendly Airlines could jack up the fares and make profits of $160m a month; or it could maintain the initial fares and make profits of $100m a month. The only problem is that Friendly Airlines might then take them to court and allege predatory pricing. Taking Unfriendly Airlines to Practice Problems 2 Page 6
court would cost Friendly Airlines $50m. If Unfriendly Airlines jacks up the fares, the probability that Friendly Airlines wins the case is 80%, whereas if they don t that probability is zero. By losing the case, Unfriendly Airlines would have to pay damages to the tune of $200m and be forced to return to their original fares; also, Friendly Airlines would then be back in business. Derive the tree corresponding to this game. For this purpose, assume that the future consists of six months: the next month, where a price war might or might not take place, and the remaining five months. Solve the game for its Nash equilibrium. 17. The following game describes a situation in which Firms 1 and 2 decide which standard to adopt, A or B. Firm 2 Firm 1 A B 80 70 120 85 60 140 55 100 (a) Determine the Nash equilibrium(a) of this game. (b) Consider an alternative game where Player 1 moves first. Represent this alternative game and determine the Nash equilibrium. (c) Based on your analysis of (b), what is Player 1 s value from being a first mover (i.e., what is the value of commitment)? 18. More concentrated industries are more prone to collusion. However, concentration is by no means the only factor on which market power depends. Discuss (maximum two pages). 19. Discuss the following quote (maximum two pages): When it comes to deciding which version of a new technology to adopt, the market tends not to work well; government intervention is therefore called for. 20. Explain why each of the following statements is true, false, or ambiguous. (a) Reaction curves always slope upwards: the more my rival chooses of a given variable, the more I want to choose of it too. (b) The better information there is, the more efficient markets are. (c) Network effects imply that demand is more inelastic. (d) If a strategy is not dominant, then it is dominated. Practice Problems 2 Page 7
(e) When choosing a new product s characteristics, the rule should be to find a product design that is as close as possible to where the demand is. (f) When network effects are important, firms should always aim at producing products that are compatible with their rivals. Written by Luís Cabral and David Backus. 2001 David Backus and Luís Cabral. Practice Problems 2 Page 8