BEE2002 Microeconomics Examination Paper May/June Duration 3 hours. Calculators permitted.

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1 BEE2002 Microeconomics Examination Paper May/June Duration 3 hours. Calculators permitted. Choose 6 questions from Section One and 3 questions from Section Two. Each section is worth 50% of the total marks. Questions within a section carry equal weight. Unjustified answers will receive very low marks. If you answer more than the minimum number, your grade will reflect the best 6 marks from Section One and yourbest 3 marks from Section Two. Please write in Ink/BallPoint. Section One 1.1 Budget Tom likes to eat apples and oranges and live in his rented flat. The mean landlord always makes rent half of his income (that is the landlord changes the rent when Tom s income changes). Apples are twice as expensive as oranges. When Tom has 1000 pounds in income, he pays the rent on his flat and eats 10 apples and 5orangesaday. Howmuchmusthisincomebetoeat5applesadayand10 oranges a day? 1.2 Adding Costs Exeter Motors (EM) produces cars with three plants: one in St. Thomas, one in Marsh Barton and the other in Exwick. The costs of each plant are c s (y s )=ys, 2 c m (y m )=2ym 2,andc e(y e )=4ye 2, respectively. If EM wishes to keep only two plants open, which will it choose and what will its costs be? What will its costs be if it keeps all three plants open? Which is cheaper? 1.3 Cost Functions and Returns to Scale A company has production technology y =min{x 2 1,x 2 2}. What is its cost function? What is the returns to scale? Show this is the case with both the cost function and production technology. 1.4 Demand Ann has utility function u(x 1,x 2 )=x 5 1x 5 2. What is Ann s demand for good x 1 as a function of prices and income? (Derive this from the consumer s problem rather than just use a formula.) If price of good 2 doubles, what should happen to the price of good 1 for this demand to remain unchanged? If the price of good 1 BEE2002

2 1 doubles, what should happen to income such that the total amount spent on good 1 remains unchanged? 1.5 Homotheticity Prove that the following utility u(x, y) =2x 2 +4y is not homothetic with the following example a =(1, 2), b =(2, 1) (choose your own k). Make sure you include a definition of homotheticity. 1.6 Compensation Arnold lives in Exeter and has utility function of 3x +4y. Prices are both 1 and his income is 24. (i) What is his utility? (ii) Suddenly the government decides to tax good y one hundred percent (net price doubles). What is his new utility? (iii) How much additional money must he earn after the tax in (ii) to restore his utility to that in (i)? (iv) If instead the government decides to tax his income, how much must it tax to cause his utility to be that in (ii)? 1.7 Price competition Flybe and Easyjet are competing for passengers on the Exeter-Paris route. Assume marginal cost is 4 and demand is Q = 18 P. If they choose prices simultaneously, what will be the Bertrand equilibrium? If they can collude together and fix prices, what would they charge. In practice with such competition under what conditions would you expect collusion to be strong and under what conditions would you expect it to be weak. 1.8 Quantity Competition Firm A has better quality and consumers are willing to pay 1 pound more for its product than Firm B s product. The inverse demand curves are given as follows: p A =5 (q A + q B )andp B =4 (q A + q B ). Assume both firms face constant marginal costs of one. What is the Cournot-Nash equilibrium? What outputs would maximize the firms joint profit? (Hint: Remember one firm s product is of better quality.) If one firm were not able to pay the other, what would make this difficult to collude upon. Instead, the firms decide to produce the same amount of output (q A = q B ). What common output would maximize their joint profits? Again, would this be difficult to agree upon? 2 BEE2002

3 1.9 ipod and Network Externalities The ipod and its website itunes have sent its manufacturer s (Apple) stock soaring. The ipod plays mp3 music filesanditunessellsthesefiles only for ipod mp3 players. Ipod users cannot buy music from other sites. Discuss what network externalities are and how they may be related to the ipod and itunes. What are some of the advantages for apple to keep this exclusivity? What would happen if Apple opens itunes to be compatible with other mp3 players and the ipod to be open to other sites? Are there any advantages to either of these possibilities? Section Two 2.1 Bundling in Auctions University Exeter Charity is seeking to raise funds for to rebuild the Roman baths near the Cathedral. They have decided to have a charity auction. Their two main items that will go under the hammer are a lunch with the VC and a special reserved parking spot. Three of the lecturers are interested. Their reservation values for either item are listed in the table below. Lunch Spot Anna $80 $40 Wei $40 $80 Todd $20 $20 They decide to run an English auction by auctioning each item separately. (i) How much revenue would they raise? Now they decide to auction both items in a bundle. (ii) What are their combined valuations? (iii) How much revenue would they raise? Now say that the lecturers values are different from the above. Lunch Spot Anna $80 $0 Wei $0 $80 Todd $50 $50 They decide to run an English auction by auctioning each item separately. (iv) How much revenue would they raise? Now they decide to auction both items in a bundle. (v)whataretheircombinedvaluations? (vi) How much revenue would they raise? (vii) From the outcomes of parts (i), (iii), (iv), and (vi) which are Pareto optimal? (viii) Do we see bundling in auctions in practice? Why do you think this is? 3 BEE2002

4 2.2 Battle of the bunnies It is Sunday morning and Mark wants to go drinking Cider with his buddies. Ruth wants to go to the country club with her friends. Unfortunately, someone has to look after the children. To decide this, they agreed to have a contest between their children s toys and pets. The top 3 are a pet rat, a pet dog, and a battery-operated bunny. (For simplicity, assume there are two of each.) They put them each in the same ring and let them battle it out. The last in the ring wins. There are 3 categories of skills that each possess: strength, stamina, and quickness. The contestant with the stronger two of the three categories wins the battle. For quickness, the order is (starting with the fastest) rat, dog, bunny. For stamina, the order is (starting with the longest) bunny, rat, dog. For strength, the order is (starting with the strongest) dog, bunny, rat. (i) There are 3 possible matches between different types of contestants. Who will win each? (ii) What property do these matches between contestants violate (this property is normally assumed about preferences)? Assume that Mark and Ruth choose simultaneously. Also assume that they value a win at 10 and a loss at 0. If both choose the same type of contestant, it is worth 5. (iii) Write the normal form game. (iv) Are we certain that there is an equilibrium? (v) Is Mark and Ruth choosing the same type of contestant an equilibrium of this game? Why or why not? Find all pure strategy Nash equilibria of this game. (vi) Is choosing each contestant with the same probability (1/3) a mixed-strategy Nash Equilibrium? Explain why or why not by looking at the possible payoffs of the other player. 2.3 Price Discrimination The university has decided that only special calculators can be used during the exam. The Student Guild has a monopoly on the calculators that are sold before and not during the exam. Ari and Jodi are both students taking the microeconomics exam. Ari feels that he can get by with only one calculator. Jodi feels that a second calculator would be helpful in case the first calculator breaks down. The following table shows their valuations (note the number under the heading 2 calculators means the valuation for two calculators rather than for the 2nd calculator). Assume it costs them 5 pounds per calculator. (Assume if indifferent in valuation terms to buying or not, Jodi and Ari buy.) 1calculator 2calculators Ari $20 $20 Jodi $30 $40 (i) If the Guild could only charge one price per calculator, independent of who 4 BEE2002

5 buys it or how many, what would they charge? (ii) If the Guild could fully price discriminate (tell who is who and charge based on quantity), what would they charge for the various combinations? (iii) If the Guild could not tell who is who, but can charge different prices for different quantities what would they charge? (iv) If the Guild could tell who is who, but must charge a constant price per calculator, what would they charge? (v) How do the profits compare in all the cases? 2.4 Pareto Optimality and Envy Free There are three young children: Dorelle, Anaelle and Joel. Assume each parent (mommy or daddy) can only hold one child at a time. (i) What are the feasible allocations for the children (the parents are merely the objects)? Assume each child wants to be held and has preferences over which parent holds them. (ii) Find an example for the utilities of the three children where there are six Pareto optimal allocations? (iii) Find an example for the utilities of the three children where there are only four Pareto optimal allocations? (iv) Is either of your examples envy free? If not can you come up with an envy free example? (v) Now say Joel would rather play with his cars than be held. Can you come up with an envy free example now? 5 BEE2002 end of paper