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Economics 51;Stanford University;Winter Quarter 2005-2006;Homework 3;Due at the beginning of your section on Friday, February 3, 2006;Work through some additional problems from Chapter 13 for extra practice!;Oh, and throughout this assignment, note that prices and quantities are not always integers.;Problem 1: Its a matter of quantity, not quality;Consider a market in which two firms behave as Cournot competitors and produce an identical;product. Firm #1 has total cost function TC1 (q 1) = q 1 2, and Firm #2 has total cost function;TC2 (q2) = 2q 2 2. The market demand curve is given by QD = 230 P.;a) Write down the equations for Firm #1 and Firm #2s Cournot reaction functions.;b) Solve for the Cournot equilibrium price, the quantity sold by each individual firm in the;Cournot equilibrium, the profits earned by each firm, and the total consumer surplus;earned by consumers when this equilibrium prevails. Show your work.;c) Now suppose Firm #1 is able to choose its quantity first [with full knowledge of Firm;#2s reaction function], and thereby can act as a Stackelberg leader. Solve for the;Stackelberg equilibrium price, the quantity sold by each firm in the Stackelberg;equilibrium, the profits earned by each firm, and the total consumer surplus earned by;consumers in this equilibrium. Show your work. Are consumers better off in the;Cournot or in the Stackelberg outcome? In one sentence, explain why this makes sense;intuitively.;d) Finally, compute the price that would prevail, the quantity produced by each firm, and the;total consumer surplus earned in the economically efficient outcome. Show your work.;e) In a single diagram, plot Firm #1 and Firm #2s reaction functions (carefully labeled so;that it is clear which is which!). On this diagram, indicate the Cournot, Stackelberg, and;efficient outcomes.;Problem 2: Always Low Prices;Now consider another market in which two firms produce an identical product. This time assume;that Firm #1 has total cost function TC1 (q 1) = 6400 + 0.5q1 2, and Firm #2 has total cost function;TC2 (q2) = 6400 + q 22. The market demand curve is given by QD = 420 P.;a) If the two firms are able to behave as a perfect cartel, what price will they choose? How;many units will each firm sell? What will be each firms profits? Show your work.;b) Now suppose that Firm #1 is able to act as a dominant firm (as defined by the price;leadership model), and that Firm #2 will accept whatever price the first firm establishes.;What price will Firm #1 choose? How many units will each firm sell? What will be each;firms profits? Show your work.;c) Finally, assume that Firm #2 decides to assert its own price-setting power as well.;Calculate the Bertrand equilibrium (i.e. the price that will prevail, the quantity that will;be produced by each firm, and the profits earned by each firm). Show your work.;Problem 3: Three companies, or threes company?;Now consider once again the market described in Problem 1 above, but suppose a third firm;enters, duplicating the technology of Firm #2: that is, TC3 (q 3) = 2q 32.;1;Economics 51;Stanford University;Winter Quarter 2005-2006;a) Assuming that these three firms now act as Cournot competitors, write down the;equations for their reaction functions. Compute the Cournot equilibrium price, the;quantity produced by each firm in this equilibrium, and the profits earned by each firm.;b) Now suppose that Firm #1 is able to act as a Stackelberg leader, and that it knows the;other two firms reaction functions. Solve for the Stackelberg equilibrium price, the;quantitities produced by each firm, and the profits earned by each firm. Show your work.;If you dont own a 2 nd edition of B&B, refer to the Appendix to this homework on;Coursework. For future reference: there are lots of good problems in the 2nd edition, so you;might want to take a trip to the bookstore or library, or make photocopies from a friend s book.;Problem 4: A differentiated kind of price competition;Besanko and Braeutigam, 2nd edition, problem 13.23 (page 518).;Problem 5: Are these skies more or less friendly?;Besanko and Braeutigam, 2nd edition, problem 13.25 (page 518), which is the same as B&B, 1st;edition, problem 13.11 (page 592).;Problem 6: How many will survive?;Besanko and Braeutigam, 2nd edition, problem 13.26 (pages 518-519).;2

 

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