Economics 431 Fall nd midterm Answer Key


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1 Economics 431 Fall nd midterm Answer Key 1) (20 oints) Big C cable comany has a local monooly in cable TV (good 1) and fast Internet (good 2). Assume that the marginal cost of roducing either good is zero. There are three customers, A, B and C with different reservation rices for the two goods. Consumer utility from a roduct equals their reservation rice minus the rice of the roduct. Consumers do not buy unless they get ositive utility. The reservation rices are as follows Customer Reservation rice for good 1 Reservation rice for good 2 A 2 5 B 4 4 C 6 2 a) (7 oints) Suose the goods are offered searately. Comute the monooly rices for good 1 and good 2. Good 1: Price Quantity Profit , Good 2: Price Quantity Profit Monooly rices (4, 4). b) (5 oints) Now assume that both goods are offered only as a bundle, but not searately. Comute the rofitmaximizing bundle rice. Good 1: Price Quantity Profit The bundle is offered for rice 7 c) (8 oints) Let the two goods be offered searately at rices you have found in a), andalso as a bundle at rice you have found in b). Determine which customers buys only individual goods or both as a bundle. Determine the monoolist s rofit. What is the most rofitable way to bundle the goods? Customer Utility from bundle Utility from good 1 Utility from good 2 Decision A buy good 2 B buy bundle C buy good 1 1
2 Profit: 4+4+7=15 Pure bundling is the most rofitable. 2) (15 oints) A local market for milk is served by several farms. Assume that you can use Cournot model with linear demand and constant marginal costs that may differ across farms to describe how this market oerates. The rice of milk is $2.50 agallon. FarmA has marginal cost of $1.50 er gallon, has a 25% marketshareandmakesarofit of$1000 er day. a) (8 oints) Farm B has a 20% share of this market. Calculate its marginal cost c i = s i η c B = s B c A s A 2.50 c = c =0.8, c =1.70 b) (7 oints) Calculate the rofitoffarmb (Hint: can the ratio of rofits be exressed through market shares?) c i = Bq i π i =( c i ) q i = Bq 2 i π B π A = Bq2 B Bq 2 A = s2 B s 2 A π B =$640er day. 3) (15 oints) Assume that consumer tastes for soft drinks can be described by a location model of roduct differentiation. Let manufacturers have identical marginal costs and set drink rices strategically to maximize rofits. a) (10 oints) Consider three drinks: Coke, Srite (both made by Coca Cola Comany) and Uncle Al s Lemon Mixer. Suose that consumers erceive Coke and Srite as different roducts, but cannot tell the difference between Srite and Uncle Al s Lemon Mixer. Restaurant A serves only Coke and Srite. Restaurant B serves only Coke and Uncle Al s Lemon Mixer. Restaurant C serves only Srite and Uncle Al s Lemon Mixer. All three restaurants have consumers with identical distribution of tastes and willingness to ay. Which drinks at which restaurants will be riced above their marginal cost? Which restaurant will have a higher rice for Coke and why? Drinks at A and B will be riced above marginal cost, drinks at C will be riced at marginal cost. Satial Bertrand vs. Bertrand. Restaurant A will have a higher rice for Coke, because Coca Cola is a monoolist in restaurant A. 2
3 b) (5 oints) Using your results in art a), exlain why we observe that restaurants serving Coke do not serve Pesi, and vice versa. If consumers make little difference between Coke and Pesi, and they are both offered at the same restaurant, they will have to be riced close to marginal cost, and rofit islow. Beinga monoolist in some locations is more rofitable than being a Bertrand duoolist in all locations. 3
4 4) (35 oints) Consider a game between two airlines. Players simultaneously choose whether to set a high rice or a low rice for the tickets. The ayoffs are given by the following matrix L H L 1, 1 6, 0 H 0, 6 5, 5 a) (10 oints) Find all the Nash equilibria of this game. Are equilibrium ayoff allocations Pareto otimal? Suose the game above is reeated a finite number of times. Is there a subgameerfect equilibrium strategy that can lead to Pareto otimal ayoffs at least in some rounds of the game? Exlain. The unique Nash equilibrium is (L, L). It is not Pareto otimal: the allocation that Pareto dominates (1, 1) is (5, 5). By Selten s theorem, all subgame erfect equilibria of the reeated game have layers choose (L, L) action rofile in every round. Therefore, none of the Pareto otimal ayoff allocations (0, 6), (6, 0) or (5, 5) can be realized in any of the rounds. b) (10 oints) Suose instead that both airlines can choose among three rices {L, M, H}. The ayoff matrix for the new game is L M H L 1, 1 2, 0 6, 0 M 0, 2 3, 3 2, 0 H 0, 6 0, 2 5, 5 Find all the Nash equilibria of this stage game. Suose that the stage game is reeated twice, and the ayoff in the second eriod is discounted by δ<1. Find the range of δ for which the following strategy rofile is a subgame erfect equilibrium. Player 1 lays H in the first eriod. If the outcome of the first eriod is (H, H), thenhelaysm in the second eriod, otherwise he lays L in the second eriod. Player 2 has the same strategy. The stage game has two Nash equilibria (M,M) and (L, L). In the second eriod, the strategy tells the layers to choose a Nash equilibrium action after every history, therefore, both layers lay their best resonse in any subgame that leads to the second eriod. In the first eriod, a layer can have a otential deviation to L instead of H. For this deviation to be unrofitable, we must have 5+3 δ>6+1 δ δ> 1 2 c) (15 oints) Now suose that the game in art b) is reeated three times. Write down the comlete descrition of the strategies (that tell what action to take initially and after every ossible history) that can sustain the outcome (H, H) in the firsttwoeriodsasasubgameerfect equilibrium. Find the range of δ for which the strategy rofile you described is a subgame erfect equilibrium. 4
5 Player 1. Play H in the first eriod. In subsequent eriods, lay M if the history has only (H, H) in the ast. For all other histories, lay L in the current eriod. If the history in the first eriod is (H, H), fromthenonthegameisidenticaltothatofartb). The strategy rofile is an equilibrium in this subgame for δ> 1. For any other history after the 2 first eriod, the strategy rescribes to lay a Nash equilibrium of the stage game, therefore, both layers lay their best resonse in any subsequent subgame. In the first eriod, a layer can also have a otential deviation to L instead of H. However,ifit is unrofitable to deviate in the second eriod, it must also be unrofitable to deviate in the first eriod, because the unishment for the same deviation is then two eriods long. 5) (20 oints) Consider the following sequential game of entry deterrence on a market with linear demand given by =9 Q The incumbent (firm 1) first chooses whether or not to invest in new technology. The investment costs K. If the incumbent invests, his marginal cost is c L =1, and if he does not invest, it is c H =6. Then the incumbent and the entrant (firm 2), whose marginal cost is c H =6lay a Cournot quantity game. The rofit functions are ½ (A cl B (q π 1 = 1 + q 2 )) q 1 K if invests (A c H B (q 1 + q 2 )) q 1 if does not invest π 2 =(A c B (q 1 + q 2 )) q 2 a) (10 oints) Show that if the incumbent invests, the Nash equilibrium of the subsequent Cournot game has the entrant roduce zero. How much does the incumbent roduce if the entrant roduces zero? (Hint: will the entrant be better off roducing zero if making q 2 > 0 dros the rice below his marginal cost?) If the entrant roduces zero, the incumbent s best resonse is to roduce monooly outut that corresonds to his marginal cost c L q M = A c L 2B =4 If at this outut level the market rice is below c H, = A + c L =5< 6=c H, 2 then the entrant will have a negative rofit fromanyq 2 > 0. Therefore, entrant chooses to roduce zero as a best resonse. The strategies that are best resonses to each other constitute a Nash equilibrium. b) (10 oints) What are the equilibrium rofits of the two firms if the incumbent does not invest? Find all values of K for which the incumbent will choose to invest in new technology. If the incumbent does not invest, both firmshaveequalmarginalcostsc H,andtheyeachget π 1 = π 2 = 1 (A c H ) 2 =1 9 B 5
6 If the incumbent invests, he gets 1 (A c L ) 2 K =16 K 4 B The incumbent invests if 16 K>1, i.e. K<15. 6
7 Reference Guide Cournot Oligooly with linear demand Elasticity = A BQ η = dq d Q = 1 B Q. Firm i chooses q i to maximize its rofit given the oututs of all other firms: max (A c i BQ i Bq i ) q i q i In equilibrium, each firm s outut must satisfy the condition for rofit maximization: A c i BQ i 2Bq i =0 or or where s i is market share of firm i. c i = Bq i c i = s i η, 7
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