Homework 3, Solutions Managerial Economics: Eco 685


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1 Homework 3, Solutions Managerial Economics: Eco 685 Question 1 a. Second degree since we have a quantity discount. For 2 GB the cost is $15 per GB, for 5 GB the cost is $10 per GB, and for 10 GB the cost is $8 per GB. b. We have 3rd degree price discrimination here, the drink is the same good, but the price is different for happy hour customers and night customers. Apparently, the after work group is more price sensitive than the group going out later. Note that one would think that the after work group is wealthier and therefore more price sensitive, but there are many reasons other than income why a group might be more price sensitive. For example, the after work group may have less free time. c. Third degree price discrimination, since different income levels have different prices. d. First degree price discrimination, since each customer is charged a potentially different price. e. One could argue either third degree price discrimination, or argue that women s hair cuts are a different good, for example by taking longer to cut. Question 2 a. This is the popcorn problem : upcharging. Ritz would like to charge it s less price sensitive (probably business) customers more. It therefore makes WIFI cost extra, with a large markup. More price sensitive customers pay less by skipping the WIFI. Of course, this is not truly price discrimination as the two customers are buying slightly different products. b. To solve this problem, we first pretend we can tell the customers apart, and figure out the ideal prices. For the consumers we would like to charge: P c = e p MC = , (1) P c = $135. (2) For business travelers, we have: P b = e p MC = (90+1), (3) 1
2 P b = $273. (4) Note that the marginal cost of the WIFI is included, since business travelers use WIFI. Now implement these prices. (a) Charge $135 for the room, which ensures we capture all of the consumer travelers. (b) The WIFI charge is then the difference between the two ideal prices: $273 $135 = $138(this number is a little unrealistic, but the principle holds). Now the business customers will pay $135 for the room plus $138 for the WIFI for a total of $273, which is the ideal price. (c) The markup on the WIFI is then: Markup = P cost cost = $ = 13,700%. (5) c. Motel 6 may offer free WIFI for several reasons. First, the market may be more competitive for low price hotel rooms. Price Discrimination and similar strategies require a lack of competition to work. Second, business and consumer travelers may have more similar price elasticities in the low end market. If business travelers are using the Motel 6, probably they are trying to save money too. Or Motel 6 may just not have any business travelers. Question 3 a. First we need the prices. Using the demand curves: Q F = 100 2P F, (6) P F = Q F. (7) Q N = 100 P N, (8) P N = 100 Q N. (9) Profits are then: π = P F Q F +P N Q N TC = ( 50 1 ) 2 Q F Q F +(100 Q N )Q N 50 10(Q F +Q N ) = 40Q F 1 2 Q2 F +90Q N Q 2 N 50 (10) 2
3 To find the optimal quantity (which maximizes profits), we set the derivative or slope equal to zero: dπ dq F = 40 Q F = 0, Q F = 40 (11) dπ dq N = 90 2Q N = 0, Q N = 45 (12) Using the demand curves (7) and (9), the optimal prices are: P F = = $30 (13) 2 P N = = $55. (14) b. The price elasticity for the Florida residents is: e p = e p = ( )( ) P dq = 30 Q dp 40 ( 2) = 3 2 ( )( ) P dq = 55 Q dp 45 ( 1) = 11 9 (15) (16) Here the derivatives are computed using the demand curves (6) and (8). The Florida residents are more price sensitive. The price is therefore lower for Florida residents. c. One would charge $55 for the cruise, and then offer a $10 Florida resident discount (probably one would require a Florida id or address). d. Several answers are possible. One might worry about cheating, for example using a friend or relative s Florida address (or having them buy the tickets for a nonresident). Arbitrage is also possible, Florida residents could buy tickets at a low price and then resell them. To counter the arbitrage, Carnival would take steps to make sure the name on the ticket matches the id of the traveler. e. Now we must charge the same price to both consumers: Q F = 100 2P, (17) 3
4 Q N = 100 P, (18) So the total demand is: Q = Q F +Q N = 100 2P +100 P = 200 3P. (19) Q = 200 3P. (20) Solve for the price: P = Q. (21) Here we can do the usual MR = MC: ( 200 TR = P Q = 3 1 ) 3 Q Q = Q 1 3 Q2. (22) MR = dtr dq = Q. (23) 3 Marginal cost is found via: TC = 50+10Q. (24) MC = dtc dq = 10. (25) Therefore, MR = MC, (26) Q = 10, (27) Q = 30, Q = 85. (28) 4
5 Therefore, P = = (29) 3 The price is between the two discrimination prices as expected. Question 4 a. If B holds, C should cut (10 > 8) and if B cuts, C should cut (6 > 4). Thus the dominant strategy is for C to cut. By symmetry, the dominant strategy is for B to cut. b. The Nash Equilibria is the dominant strategy. Both players cut. c. If both players hold the social benefit is SB = 8+8 = 16. If one player cuts and the other holds we have SB = 10+4 = 14. If both cut then SB = 6+6 = 12. Thus both players holding is the best outcome for the two firms. The Nash equilibrium is the worst possible outcome. We have a prisoner s dilemma. If the two firms could sign a contract, they would sign a contract to hold prices and split the $16 million. However, since such contracts are illegal and in any event, each has an incentive to cheat on any agreement. Cheating results in a price war. d. No. Even though (B) has promised to keep prices high, (B) has a strong incentive to renege on the promise and cut prices. Firm (C) should respond by ignoring firm (B) and cutting prices. Question 5 a. We have: Agency 3 Rates Good Agency 2 Good Bad Agency 1 Good 8, 8, 8 8,6, 8 Bad 6, 8, 8 4,4,3 Agency 3 Rates Bad Agency 2 Good Bad Agency 1 Good 8, 8,6 3,4,4 Bad 4,3,4 4, 4, 4 Table 1: Herding game solution. 5
6 b. No player has a dominant strategy: all players rate good, except when the other two players rate bad. So the best response depends on the other players actions. c. From the circles in table 1, there are two Nash equilibria, one where all rate good and one where all rate bad. d. Yes, both Nash equilibria have all players agreeing. Therefore, in the world one will observe either all players rating good or all players rating bad. e. We have: Agency 3 Rates Good Agency 2 Good Bad Agency 1 Good 8, 8, 4 8,6,4 Bad 6, 8,4 4,4,6 Agency 3 Rates Bad Agency 2 Good Bad Agency 1 Good 8, 8,3 3,4, 8 Bad 4,3, 8 4, 4, 8 Table 2: Herding game solution. f. Yes, both Nash equilibria have all players agreeing. Player 3 believes the correct rating is bad. But player 3 is not completely sure. Therefore, if both other players rate good, and player 3 happens to be wrong, the penalty is very high. So player 3 rates good in this case even though player 3 suspects the correct rating is bad. In the book, very few investors were able to maintain a short position long enough to cash in on the mortgage crises. Clients would leave because, until the crises occurred, everything looked fine and the short positions were losing money. Question 6 a. The best responses are: 6
7 Pepsi Spend 0 Spend 2 Spend 4 Spend 6 Spend 10 Coke Spend 0 60,49 58,50 45,35 20,40 18,45 Spend 2 50,35 65,30 30,25 22,32 18, 36 Spend 4 45,10 60,20 50,40 24, 44 20,40 Spend 6 50,5 60,10 52,16 30,20 22,18 Spend 10 60,0 62,12 52,18 28,20 24,21 Table 3: Advertising Game. All payoffs and strategies are in millions of dollars. Therefore, two Nash equilibria exist, one in which Pepsi and Coke each spend 6 (30,20) and one in which both companies spend 10 (24,21). Suppose Coke forecasts Pepsi will spend six. Then Coke s best response is to spend 6 (30 is better than any other outcome for Coke in the column spend 6 ). Coke predicts Pepsi will advertise, so Coke must respond by advertising or lose customers. If Coke spends 6, then Pepsi s best response is to spend 6 (Pepsi must match Coke s advertising or lose customers), which equals the prediction of Coke. Therefore we have a Nash equilibrium. Suppose Coke forecasts Pepsi will spend ten. Then Coke s best response is to spend 10 (24 is better than any other outcome for Coke in the column spend 10 ). If Coke spends 10, then Pepsi s best response is to spend 10, which equals the prediction of Coke. Again, each player must match the advertising of the other or lose customers and each believes the other will advertise. Therefore we have a Nash equilibrium. These are bad outcomes for both firms. Much of advertising tends to simply cancel out competitors advertising. But then both firms have spent much advertising dollars without any extra sales. b. With collusion, firms choose the cell with the highest social benefit. Here both firms agree to spend 0 has the highest social benefit (60+49 = 109). c. No. Under a collusive agreement to spend nothing on advertising, Pepsi has an incentive to increase advertising to 2 in order to steal business from Coke (50 > 49). Once Pepsi does this, Coke will have an incentive to increase spending to 2, and so on until the Nash equilibrium is reached. d. Pepsi prefers both firms spend 10 (21 > 20), whereas Coke preferes both firms spend 8 (30 > 24). Question 7 No Firms use pricematching to deter other firms from cutting prices. Both firms will offer a high price, and offer to match competitors prices. However, since the competitors both are 7
8 offering a high price, the consumer is unable to take advantage of the match and has only high prices to choose from. 8
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