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### S2010_final

Course: ECON 171, Fall 2009
School: UCSB
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Word Count: 740

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171 Econ Spring 2010 Final Exam June 8 You have three hours to take this exam. Please answer all 5 questions, for a maximum of 100 points. Point totals and subtotals are indicated in brackets. To obtain credit, you must provide arguments or work to support your answer. 1. [10] Find all Nash equilibria of the following game. A B C X 2, 1 3, 3 1, 2 Y 4, 2 0, 0 2, 8 1 Z 2, 0 1, 1 5, 1 2. [20] Consider the...

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171 Econ Spring 2010 Final Exam June 8 You have three hours to take this exam. Please answer all 5 questions, for a maximum of 100 points. Point totals and subtotals are indicated in brackets. To obtain credit, you must provide arguments or work to support your answer. 1. [10] Find all Nash equilibria of the following game. A B C X 2, 1 3, 3 1, 2 Y 4, 2 0, 0 2, 8 1 Z 2, 0 1, 1 5, 1 2. [20] Consider the extensive-form game represented below. 1 A C B 2 X 5, 2 Y 2, 6 X 6, 2 Y 6, 2 2, 6 (a) [5] Which solution concept is the appropriate one to apply to this game? Find the set of equilibria using that concept. (b) [5] Suppose that we delete the information set between player 2s two decision nodes. (Plugging the values (a, b, c) = (5, 6, 2) into the gure below shows the extensive form of this game.) In other words, suppose that 2 can actually observe whether 1 chose B or C . What solution concept should we apply now? Find the unique equilibrium under that concept. (c) [5] Now let (a, b, c) = (3, 1, 3). Find all PSNE of this game. (d) [5] Which of these are subgame-perfect? 1 A a, 2 C B 2 X 2 Y 2, 6 6, 2 2 X b, 2 Y c, 6 3. [10] Suppose two players play one of the two normal-form games shown below. Player 1 knows which game is being played, but player 2 thinks that it is Game (a) with probability 2/3 and Game (b) with probability 1/3. Find a pure-strategy Bayesian Nash equilibrium of this Bayesian game. U D L 0, 1 1, 0 R 1, 0 2, 1 U D Game (a) L 0, 1 1, 0 R 1, 0 0, 1 Game (b) Figure 1: Player 1 knows which game is being played, but Player 2 does not. 3 4. [30] In the signaling game represented below, there are two types of Player 1, smart and dumb, the probabilities of which are 0.4 and 0.6, respectively. Player 1 is in college and can either ((Q)uit or (G)raduate. Player 2 is a prospective employer and can either (N)ot hire or (H)ire Player 1. Player 2s payo does not depend upon 1s education, only her intelligence. Player 1s payo depends partly on her education: both types benet from completing their education, but the smart type gets more out of it. Player 1s payo also depends on 2s hiring decision: the smart type wants a job but the weak type does not. 0, 0 1, 1 N H .4 Q 1s G 2, 0 2 N H c .6 2 Q 1d G N H N H 2, 1 0, 0 3, 1 3, 1 1, 0 (a) [10] Find a separating PBE. (b) [10] Find a pooling PBE. (c) [10 ] Find an equilibrium in which one type of player 1 mixes, playing both Q and G with positive probability. 4 5. [30] The government decides to auction o the rights to drill all of the oil under Sulphur Mountain. Ilse and Junjun decide to participate in the auction and, not knowing exactly how much oil is underground, each hires a consultant to estimate the size of the oil reserves. Consultants are expensive, though, and Ilse and Junjun can each only aord to pay the consultant to estimate the oil under one side of the mountain. Ilses consultant estimates that there are ei dollars worth of oil under the north side of the mountain and Junjuns consultant estimates that there are ej dollars worth of oil under the south side of the mountain, where ei and ej are both drawn uniformly from the interval [0, 1]. Thus, the total value of the oil is v = ei + ej , but because each person keeps her own estimate secret from the other, they each only know their own estimate. The only thing that each person knows about the other persons estimate is that it is drawn uniformly from the unit interval. (a) [20] Suppose that the government holds a second-price auction for the oil rights. I claim that there is a (Bayesian) Nash Equilibrium in which each player bids twice her estimate. Verify this claim, by showing that if Junjuns strategy is bj = 2ej , then Ilses best response is bi = 2ei . To help you, Ive broken down the process into the following steps: 1. [4] Write down the probability that Ilse wins as a function of her bid, bi (given Junjuns strategy). 2. [4] Write down the expected price that Ilse pays if she wins. 3. [4] Write down the expected value of Junjuns estimate, ej if Ilse wins. 4. [4] Using these three calculations, write down Ilses expected payo for bidding bi , given ei . 5. [4] Show that the bi that maximizes this expected payo is bi = 2ei . (b) [10 ] Now suppose that the auction is a rst-price auction. Is the Nash equilibrium bidding strategy going to result in higher or lower bids? Find the NE bidding strategy and show why it works. 5
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University of Florida - MAC - 2233
Project 22/23MAC 22331. Examine the function h(x) =27x2. What is its domain ?x (x + 1)3Determine the horizontal asymptotes and vertical asymptotes, if any, for the graph of h(x) .Does the function have any removable discontinuities?What are the in
UCSB - ECON - 171
Class InfoGame TheoryNormal Form GamesEcon 171 Spring 2010Class 1 - March 30, 2010Index cards: Name, pronunciation, perm no.,major, nativelanguage, Math background, Reason for interest in class,Economics is. . .Web: http:/econ.ucsb.edu/~grossman/E
University of Florida - MAC - 2233
Project 22/23MAC 22331. Examine the function h(x) =27x2. What is its domain ? x = 0, 1x (x + 1)3Determine the horizontal asymptotes and vertical asymptotes, if any, for the graph of h(x) .Does the function have any removable discontinuities?27xan
UCSB - ECON - 171
Introduction to Game TheoryEconomics 171 - Spring 2010SyllabusInstructor:Zack Grossmangrossman[at]econ.ucsb.eduOffice: NH 3049Office Hours: R 12:30-1:30Course Homepage: http:/econ.ucsb.edu/~grossman/Econ171S10Lectures: TR 2 3:15, PHELPS 1260Cour
University of Florida - MAC - 2233
Project 24MAC 22331. (From lecture) You are required to construct a closed rectangular box with a surface area of 48square feet so that the length is twice the width. What dimensions will maximize the volumeof the box?2. Sketch the graph of the funct
University of Florida - MAC - 2233
Project 24MAC 22331. (From lecture) You are required to construct a closed rectangular box with a surface area of 48square feet so that the length is twice the width. What dimensions will maximize the volumeof the box?Max volume when width is 2, leng
UCSB - ECON - 177
Professor GarrattEconomics 177AuctionsMW 11:00-12:15, 387 103www.econ.ucsb.edu/~garratt/Econ177Auctions have been used to allocate goods for thousands of years, but in the past 25 years there hasbeen a surge in popularity. Auctions are now routinely
University of Florida - MAC - 2233
Project 25/26MAC 22331. Carefully sketch the graph of f (x) = 4 + e(x2) below , including any intercepts and asymptotal lines, AND state its domain and range in interval notation.Domain:Range:Formula: g (x)=Now suppose g (x) is the function obtained
UCSB - ECON - 177
Cummulative Distribution Function of Bids for All Pay AuctionSpring 2011100.00%90.00%80.00%70.00%60.00%2 bidders5 bidders10 bidders50.00%40.00%30.00%20.00%10.00%0.00%05101520253035404550Bid556065707580859095100
University of Florida - MAC - 2233
Project 28MAC 22331. A population of wolves increases so that the number of wolves x years from now is given bythe function W (x) = 200(1 ex ). Calculate W (x).At what rate does the population change after 1 year? 3 years? (Include units and round to
UCSB - ECON - 177
Assignment 1: Econ 177Professor GarrattYour assignment is to estimate your own private bid functions that characterize your biddingbehavior in the 2-bidder and 5-bidder first- and second price auctions. We will only consider thecase of no entry fee an
University of Florida - MAC - 2233
Project 29MAC 22331. Examine the function h(x) =3 ln(x). What are its domain and intercepts ?xCalculate h (x), and h (x) . Use a number line to determine the interval(s) on which thefunction is increasing/decreasing, concave up/down. Are there any
UCSB - ECON - 177
Assignment 2: Econ 177Professor GarrattConsider the following two payment options:Scenario A: A gamble that pays \$100 with probability p and \$0 with probability (1-p).Scenario B: A payment of \$x with probability 1.In the table below, for each cash am
University of Florida - MAC - 2233
Project 29MAC 22331. Examine the function h(x) =3 ln(x). What are its domain and intercepts ?xDomain x &gt; 0;intercept ( 1, 0 )Calculate h (x), and h (x) . Use a number line to determine the interval(s) on which thefunction is increasing/decreasing
UCSB - ECON - 177
Suboptimal Bidding and Profits in 2-Bidder First-Price Auction with \$25 Entry Fee4030Dollars2010012345678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40-10-20RoundAverage deviation from equ
University of Florida - MAC - 2233
Project 30MAC 22331. Suppose that the number of worldwide users of a certain internet service is growing exponentially. Suppose that, after assessing the data, there were 100 million users originally, and140 million users at the end of the second year.
UCSB - ECON - 177
Chapter 12AppendixBelow are brief descriptions and instructions for the experiments used in thiscourse. In each instance I use the following notation:v = your valueb = your bidB = the highest bid among everyone elser = reserve pricec = entry feeE
University of Florida - MAC - 2233
Project 31MAC 22331. Recall the formula and graph for the logistic function; note that one example is the function100P (t) =. Where does the inection point for the graph occur?1 + 2e0.2tNow sketch the function P (t) on the axes above. Include the i
UCSB - ECON - 177
Auction Theory With ExperimentsbyRodney J. Garratt1April 7, 20111 Departmentof Economics, University of California at Santa Barbara, garratt@econ.ucsb.edu.2OverviewThe goal of this text, its supplement, and software is to provide a meansof teachi
University of Florida - MAC - 2233
Project 32/33MAC 22331. Evaluate each of the following integrals using a basic substitution.8x3dx4 3x43x2 e2x dx1exdx4x21 + 2 ln(x)dx4x2. Demonstrate that F (u) = (u 1)eu is an antiderivative for the function f (u) = ueu .Use the substitut
UCSB - ECON - 177
Entry Decisions in 2-Bidder First-Price Auction with \$25 Entry Fee10.90.80.7Percent0.60.50.40.30.20.1012345678910 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40RoundPercent correct entryP
University of Florida - MAC - 2233
Project 32/33MAC 22331. Evaluate each of the following integrals using a basic substitution.8x3dx4 3x43x2 e2x dxeach of these can be fairly easilyby taking the derivative of your answer1exdx4x21 + 2 ln(x)dx4x2. Demonstrate that F (u) = (u
UCSB - ECON - 177
FPA 2-bidder Spring 2011100Bids outside target range: 13Percent outside target: 0.056522Intercept 0.2717 s.e.: 1.3539Slope: 0.7812 s.e.: 0.012349080Average revenue:Average earnings:7062.92565Highest earnings: 124.84 Subject: 7560Bids52.4174
University of Florida - MAC - 2233
Project 34/35MAC 22331. Examine the parabolic arch formed by the function f (x) = 3x 1 x2 on [ 0, 6 ] . We would2like to approximate its area using rectangles.Sketch the parabola below on [ 0, 6 ] and shade the area beneath it.Now for some approxima
UCSB - ECON - 177
FPA 5-bidder Spring 2011100Bids outside target range: 14Percent outside target: 0.06087Intercept -0.4221 s.e.: 0.7361Slope: 0.8983 s.e.: 0.0139080Average revenue:74.7235170Bids605040302010001020304050Values60708090100
University of Florida - MAC - 2233
MAC 2233 Quiz 1 Solutions1. Find the equation of the line that passes through the point (2, 4) and is perpendicularto the line 3x + 4y 22 = 0.SOLUTION Solve the given equation for y to identify the slope of the given line.The slope of the given line i
UCSB - ECON - 177
Introduction to Auction TheoryEcon 177Spring 2011Introduction: A Historical PerspectiveHerodotus reports that auction were used in Babylon as early as 500B.C.193 A.D. the Pretorian Guard sold the Roman Empire by means ofan auctionWide array of com
University of Florida - MAC - 2233
MAC 2233 Quiz 2 SolutionsJune 5, 20091. Let f be a function which is dierentiable at a point x. Write the limit denition of f (x),the derivative of f at x.Solution: limx0f (x+h)f (x)h2. (a) Write the power rule for the dierentiating the function f
University of Florida - MAC - 2233
MAC 2233 Quiz 3 SolutionsJune 12, 20091. Let f be a function which is dierentiable at a point x. Write the limit denition off (x), the derivative of f at x.Solution: limx0f (x+h)f (x)h2. Find and simplify f (x) if f (x) =Solution: f (x) = 3x+3 3
UCSB - ECON - 177
Getting StartedWe seek a theory of bidding behavior in auctions.Our theory will attempt to explain how peoples bids are related to theirindividual valuations, or simply values, for the item being auctioned.In mathematical terminology, we want a mappin
University of Florida - MAC - 2233
MAC 2233 Quiz 4 SolutionsJuly 1, 2009You must show all work to receive credit1. Let f (x) = 3x4 6x3 + x 8. Find the open intervals where f is concave down.Solution It suces to nd all open intervals where f (x) &lt; 0.Begin by computing f and f . We have
UCSB - ECON - 177
Bidding BehaviorWe assume people attempt to maximize their payo from participating inan auction.Hence, we are in a sense trying to determine their optimal bids.However, an auction is a game in which the payo an individual earnsfrom any given bid depe
University of Florida - MAC - 2233
MAC 2233 Quiz 5May 10, 20091. Let f (x) = 3x4 + 8x3 . Find all absolute extrema of f on the interval [1, 1].Solution: First, nd the critical points of f that are in the given interval:f (x) = 12x3 + 24x2= 12x2 (x + 2)so the critical points are x = 0
UCSB - ECON - 177
Expected RevenueHere we calculate the expected revenue under the ecient equilibriumbidding strategies for the rst- and second-price auction formats.In a rst-price auction with F () uniform on [0,100], the symmetricequilibrium bidding strategy has each
University of Florida - MAC - 2233
MAC 2233 Quiz 6 SolutionsJuly 24, 20091. Let f (x) = (x 1)e3x+2 . Determine the intervals where f is increasing.Solution: We want to determine where f &gt; 0. First, nd f :f (x) = 1 e3x+2 + (x 1) 3 e3x+2= e3x+2 (1 + 3(x 1)= e3x+2 (3x 2).22Since e3x+
University of Florida - MAC - 2233
UCSB - ECON - 177
Professor Rod GarrattECON 177MidtermApril 27, 2011The exam is worth a total of 30 points. You have 1 hour and 15 minutes to complete thisexam. Good Luck!In all the questions that follow you may assume each of the i=1,.,n bidders' values aredrawn in
University of Florida - MAC - 2233
MAC 2233 Review Worksheet Answers1. a. 5/8 grams1. b. 20 years2. m = 7/23.x 8 x 4 ln xC4.23/ 2 x 1 2 x 1 C35.1C3x3 e 3x 6.13/ 2ln4x 2 x 2C217. e x 2C8.ln 229. 838 fish10. 56/9
UCSB - ECON - 177
Discrete BidsAverage Bid if saw 0: 1.364Average Bid if saw 3: 2.73Average Profit = .462 millionNumber positive = 6, Number negative = 4, Number zero = 3Continuous BidsAverage Bid if saw 0: 1.316Average Bid if saw 3: 2.605Average Profit = .649 mill
University of Florida - MAC - 2233
MAC2233 Chapter 2 Review1. For the function f (x) = x2 3x 4 state the domain and evaluate thefunction at x = 1, 0, a + h.2. For the function y = t2 +1 2 state its domain as well as the dependenttand independent variables and evaluate the function at
UCSB - ECON - 177
PDFs for Revenue of First- and Second-Price Auctions (2-bidders)10.90.8Probability0.70.60.50.40.30.20.10051015202530354045505560Revenue1st price2nd price65707580859095100
University of Florida - MAC - 2233
MAC2233 Chapter 3 Review1. Find the derivative of h(x) = (x 2)(2x + 3).2. Find the derivative of f (t) = 3t4 2 + t5 .2t24x3. Find the derivative of f (x) = 3x x+2+2 .22t4. Find the equation of the tangent line to y = t4 3+2+1 at t = 0.t25. Th
University of Florida - MAC - 2233
MAC2233 Chapter 4 Review1. Find the intervals where the function g (t) = t22t is decreasing and+1increasing.2. Use the First Derivative Test to nd the relative extrema of the function f (x) = x3 3x2 .3. Sales in the Web-hosting industry are projected
UCSB - ECON - 177
Popcorn Experimentslope 2 bidder = .802 s.e. = 0.061slope 27 bidder = .834 s.e. = .08400030002000100000100020003000guess27bidderPredicted 27bidderPredicted 2bidder2bidder4000
University of Florida - MAC - 2233
MAC2233 Chapter 5 Review1. Evaluate 23/4 43/2 .2. Simplify the expression x3/5 x8/3 .3. Solve the following equation for x: 5x2 3= 51x .24. Jada deposited an amount of money in a bank 5 years ago. If the bankhad been paying interest at the rate of
UCSB - ECON - 177
2 Bidder, V=0 and V=3060Average Reserve5040V=0V=303020100123456789 10 11 12 13 14 15 16 17 18 19 20Round
University of Florida - MAC - 2233
MAC2233 Chapter 6 Review1. Find the indenite integral7x3 dx.2. Find the indenite integral6 x 9x2 + 5ex dx.3. Find the indenite integral2x2 7dx.x34. Find the indenite integral(x + 2)3 dx.5. Given f (x) = ex 2x and f (0) = 2, nd f (x).6. Find th
UCSB - ECON - 177
5 bidder, V=0 and V=308070Average Reserve6050V=0V=30403020100123456789 10 11 12 13 14 15 16 17 18 19 20Round
University of Florida - MAC - 2233
MAC2233 / Section 1834/ MWF4 TUR L007Instructor: Scott KeeranOce: 467 Little HallE-mail: keeran@u.eduWebsite: www.math.u.edu\ keeranOce Hours: M 3rd, W,F 5th, Th 6thText: Applied Calculus for the Managerial, Life, and Social Science,Volume 1, Unive
University of Florida - MAC - 2233
NAME:MAC 2233 EXAM 1 MAY 27, 2010PART 1: SHORT ANSWER1. (3 points) Give the denition of a function.2. (3 points) Let f be a function. Give the three conditions f must satisfy to be continuousat x = a.3. (2 points) Let f be a function. Write the limi
UCSB - ECON - 177
2 and 5 bidder, V=308070Average Reserve60505 bidderoptimum2 bidder403020100123456789 10 11 12 13 14 15 16 17 18 19 20Round
University of Florida - MAC - 2233
Name:MAC 2233 EXAM 3July 15, 2010PART 1: Denitions1. (3 points) Complete the denition:A function f is concave upward on the interval (a, b) if2. (3 points) Dene the exponential function f with base b. Include all restrictions on b.3. (3 points) Com
UCSB - ECON - 177
2 and 5 bidder, V=070Average Reserve6050405 bidderoptimum2 bidder3020100123456789 10 11 12 13 14 15 16 17 18 19 20Round
University of Florida - MAC - 2233
MAC2233 Test 1(7 pts) 1. A function f has domain (, ) and a function g has domain [2, ); the domainof (f + g )(x) is given by:A. (, )B. (2, )C. (, ) [2, )D. [2, )E. (, 2)x(7 pts) 2. The domain of the function h(x) = x2+2 is given by:9A. (, )B.
University of Florida - MAC - 2233
MAC2233 Test 2 A(7 pts) 1. The equation of the tangent line to the function y =A. y = 3 x + 5441B. y = 7 x + 443C. y = 5 x + 44x2 + 3x at x = 1 is given by:D. y = 1 x + 744E. y = 9 x 144x(7 pts) 2. If f (x) = x2+1 , then f (2) is equal
UCSB - ECON - 177
Econ 177Sample Questions Part IIn all the questions that follow you may assume each of the i = 1, ., nbidders values are drawn independently from the uniform distribution on[0,100], which is dened as followsF (v ) = Pr[i v ] =vv.1001. What is th
University of Florida - MAC - 2233
MAC2233 Test 3 A(7 pts) 1. The interval(s) over which g (t) = t22t is increasing is (are) given by:+1A. (, )B. (, 1)C. (1, )D. (, 1) (1, )E. (1, 1)(7 pts) 2. The function f (x) = x + 9/x + 2 has a local max at:A. x = 3B. x = 2C. x = 0D. x = 3
UCSB - ECON - 177
Econ 177Sample Questions Part II1. Consider a second-price auction with a single private value bidder whovalue is drawn from the uniform distribution on [0,100]. You are theseller.(a) Compute the optimal reserve price assuming your value for theitem
University of Florida - MAC - 2233
MAC2233 Test 4 A(7 pts) 1. Given f (x) = ln x2 5, f (3) is equal to:3B. 83A. 4C. 32D. 233E. 16(7 pts) 2. The equation of the tangent line to g (x) = 2xe3x at x = 1 is given by:A. y = 3e3 x + 5e3B. y = 4e3 x + 6e3D. y = 5e3 x 3e3C. y = 2e3
UCSB - ECON - 177
A signalCaseCaseCaseCase1234Charlie RiccoB signal0303A's bid0033B's bid1.54.51.54.51.51.54.54.5A's profitResults from-0.75 1000-1.5 Trials00.75-0.375-0.3795Willliam RussellWillliam Russell
University of Florida - MAC - 2233
MAC2233 Final A(7 pts) 1. Given y 2 x2 + 3x = 3y 3 , y is given by:2xy 2 +3A. 9y 2 2yx22xy 2 +32xy 2 3B. 9y 2 +2yx22xy 2 +3C. 9y 2 2yx2D. 9y 3 +2yx22xy 2 +3E. 9y 2 2y 2 x2(7 pts) 2. Given h(x) = (4x2 3x ln x)3 , h (1) is equal to:A. 120B. 80