# Ucla Econ 101 Final Spring 2011

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• Topic: Nash equilibrium, Game theory, Trembling hand perfect equilibrium
• Pages : 5 (1009 words )
• Published : November 19, 2012

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Final Exam (VERSION 1): Econ 101
• Please write your name at the top of every page of this mideterm • Please write your name, TA’s name, and the time of your discussion section here

TA’s Name:

Discussion Time:

• The exam has one parts: Written Questions. • There should be 16 total pages (front and back). Quickly read through the exam before beginning. • There are 100 total points available. Point values are listed next to each problem part. Please allocate your time accordingly

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Written Questions
1. Consider the following payoﬀ matrix Player L M T 2, 0 3, 1 Player 1 C 3, 4 1, 2 B 1, 3 0, 2 2 R 4,2 2,3 3,0

a. (5pnts) Find the pure strategy Nash equilibria of the simultaneous game b. (5pnts) Now suppose the game is played sequentially. Find the subgame perfect equilibrium if player 1 goes ﬁrst and if player 2 goes ﬁrst. c. (5pnts) Discuss whether each of the players would want to go ﬁrst or second. d. (5pnts) Write down a system of equations such that the solution to the system would give a completely mixed strategy equilibrium of this game (please clearly deﬁne all of your notation). Can this system of equations be solved? (Hint: think about the condition requiring player 1 to play B with positive probability). Explain what the answer means.

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2. Suppose Player 1 and Player 2 are playing a simultaneous move game with the following payoﬀ matrix: Player 2 L R T 0, 4 α, 3 Player 1 B 3, 3 4, 6 where α ≥ 0 a. (5pnts) Deﬁne a dominant strategy equilibrium. Is there any value of α for which there is a dominant strategy equilibrium. If so, ﬁnd the values of α. If not, show why. b. (5pnts) Describe all the pure and mixed strategy equilibria of the game as a function of α c. (5pnts) Suppose α = 5. What would the outcome be if the players could cooperate?

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3. Billy has just inherited a horse ranch from his uncle. The ranch is located in Oshkosh, WI and rents horses. A unique feature of the stable is the nearby riding trails that overlook Lake Winnebago. Billy has two types of potential customers: novice riders (N) and serious riders (S). The (per customer) demand for horse rides on the ranch is qS = 75 − 1.25PS , where qS is the number of hourlong rides a serious rider makes per year. The demand for novice riders is qN = 57 − 1.25PN . Assume there are 75 riders of each type in the town. Billy’s cost function is T C = 12q, where q is the total number of hours the horses are ridden per year. a. (5pnts) Suppose Billy does not price discriminate. Find prices, quantities, and Billy’s proﬁt. b. (5pnts) Suppose Billy can tell who’s a serious rider because of the types of hat they ware. Find the 3rd degree price discriminating prices, quantities and proﬁts. c. (5pnts) Suppose Billy is not able to tell the diﬀerence between the two types of rider. He decides to start charging a yearly membership fee, T , as well as an hourly price, p. Find the optimal choices of T and p d. (5pnts) Suppose Billy IS able to tell the diﬀerence between the two types of but still thinks the 2-part tariﬀ is a good idea. Find the annual fee and per hour price that Billy would charge to each group

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4. (16pnts) Boeing and Airbus are the 2 ﬁrms that produce commercial aircraft. The demand for airplanes is given by: Q = 10 − P . Boeing’s costs are given by T CB = cB qB and Airbus’ costs are given by: T CA = cA qA where cA , cB are constants. a. (5pnts) Find the Cournot quantities, prices and proﬁts. Find Stackelberg quantities, prices, and proﬁts assuming Boeing chooses output ﬁrst b. (5pnts) Suppose that right now cB = cA = 5. Boeing has access to a process innovation that will lower marginal costs from 5 to 0. How much would Boeing be willing to invest to implement the innovation. (Assume Cournot Competition from here on) c. (5pnts) Suppose that the innovation is such that Airbus can...