Midterm Summer11 Solutions

Midterm Summer11 Solutions - Economics 109 Midterm Exam...

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Unformatted text preview: Economics 109 Midterm Exam Prof. Buzard, Summer 201 1 You have 70 minutes to complete this examination. You may not use notes, calculators, or any books during the examination. Write your answers, including all necessary derivations, and nothing more, in the spaces provided. You may use the scratch paper that has been distributed for all other work but will not turn it in. To receive full credit, you must fully justify your answers (except where noted). Proper justification may consist of showing your work in attaining the answer, if your work reveals the reasoning that supports your answer. Excess verbosity will not help your grade. Even if your answer is correct, citing incorrect or irrelevant justifications will reduce your grade. If a question has multiple correct answers, you must (at a minimum) identify all of them to receive full credit. Identifying incorrect answers will reduce your grade even if you also identify all the correct answers. Partial credit may be given for correct reasoning and explanation, even if your final answer is incorrect, and is awarded at the discretion of the grader. 1. Consider the following normal form game, \ a0 V CB a) Calculate BR2(91) Where 01= (1/3, 1/2. ,1/3). F ua(eu)b)- #515l+3-©7% (Adena) 2 4+2-s+-2-1=8/2 anaemia} Ma (9») P3 :é‘béswéri : t/g b) Name the Pareto efficient strategy profiles (answers only) QX\ b) (E E\ C C F5 c) Determine the rationalizable set. A 1/3 dwwrxmai Nag 58+:Q Once {\de bu) W E M 6M1 (Va moor/1m ijtimkep‘) (4‘07 1 370 H9, ”’1 a 373w utilgommmagwou M co R {BCBX XSEE} d) Show analysis of best response on the normal form and report all pure strategy Nash equilibria. (3 mm (c F) e) Compute any mixed- -strategy equilibria of the game. If none exist, explain why. We Can Auljxlxak outta (\wa to Wu family): M‘(B)%\ : ui(()%> “a (PJEB:£&9(P)F 395*“1, : ”$9933 3P*“\>= PM“? QCLJA _ Q, 3% thL 1 4-39 HSME : (<33 93B 5 z . ) ' ‘4ng P \mfib 12/ We 6.: "of/2%) other you choose but you must clearly state your choice). . . . i f: A 3, 0, 3 2. ConSIder the extensrve form game on the right. .<: a) How many strategy profiles are there in this game? S R 0‘ 2. 1 (answer only) ‘ G gngmg SpiMAQRGJZQi 36:33sz 9949M AEX B I o C‘ 1,—-1,2 b) Draw the normal form for this game (payoffs are in ,‘<: the order B,G,F; you may list them in this order or any N I,” Q 0, 23 1 F ’ C 1 SM = Monitor Kids "P a) Show an analysis of best response on the above normal form. b) Name all pure strategy Nash equilibria of this game. SM = Chores " P The/Fe am 013mg. 0) There is a mixed-strategy Nash equilibrium of the game in which B and S each play N with probability 1/2. Clearly state an indifference condition that is sufficient to determine the probability with which their mother must monitor them in this equilibrium and solve it. ugCN)‘/1)P3 ‘ ”‘8 (ENE (Cl “30””) 5"“3G'79193 ~- tsp wbvga-s help dank? + ngp) | .1 ”SI‘g‘P 339*23‘23? Co -2 3 l) ‘ a “WMWW_W-W~M Emu—2 mm PmbWH mam/3kg 4. Consider the following normal form game. 31 = [0.11.52 = [0,1], 111691.32) = 351 _ ZSISZ _ 28%, andu2(51, S2) = 52 + 28182 " 23g _ Find the following. a) The best response functions (the solution 1s interior so you can use calculus). 1* W 331 $599,933 my $9’W9S5- -3193 \ S s ‘ y- _ '1 \‘WDS 43*: 3&39' s —o - 1 ug:i§* $”** b) The Nash equilibri W 3 .2“; L 3.L_.‘. ~_,\. ._ 5.. § :_ S1 L1 531 1‘1 8 L1$[-8 *3 .7 gs. 35‘s: A 911“"; 9f»; ‘ifié‘filfifa ’1 Nb-C5,g) ts c) The set of strategies tha 1 :31 3 .~-\_ 1 - R‘ 3: $53. 3:1:{0‘1 \ g9. 4*33, S.eEO,1](subscripts forrounds, ~ - 3 3 S‘Vv’yo >L su erscrl tsfor la ers g .1. -1 9 ‘+ ‘1 P ‘P R13 3’ 1 ‘18-0 (f g‘Ll4JL-(13e R1: L :3. :Ce}? 3,} 1.19 ‘3 42— q 1 13.1 11,4 12 ,- 3. l- 5 L- 9 ‘L 5‘ ‘1 3 a 5 a 3'1-1'“? R1 m 1.1.3-5 R1: : 1 R2: - £3 3 $22,134.91 431112 2 ”J8 ‘518 1'4 5'71'8 d) From player 1’ 5 point of View, are S: ands, complements or substitutes? What about from player 2’s point of v1ew? :1: in mm C3 (1%“ £0“ $325 3 {NW (s9 Feed-11¢ (in °ES\ e) What IS the socially efficie t outcome? How does it differ from your answer to part (b) and how does this relate to your answer to part (d)? 1 D 1 mm 11‘ +uQ : 33‘~Qg\33/33?+33+ 33133'3-39 = 35"93. 15.34932 S\)Sg ’ 3U‘*M3 ‘ I5~S:O= :3 BMW - _ -1 33‘ LL‘ 731 11 x5; = \Hsawo A sg-q s wwm Maw 6‘4).§'WW+FD)_ 1/1 5. Consider a two—player contractual setting in whimhe players must decidew in a joint project. The underlying technology of the relationship 15 represented by the following normal form: a) Draw the normal form for the induced game with limited verifiability. b) If verifiability is limited, can a contract achieve the efficient outcome? Why or why not? Lo 7 8 ck) I W W (1’1)wa Wfimm5>b+o< Omol 61%‘201 (Wm dVngmmWedojt’fiuWhma flaw 0) Would the players jointly be willing to pay to transform the setting to one of full verifiability of“? m ’U 8 their investment decisions? Why? If so, what 5 the maximum they would pay? g0 W Mew w Wt£1ww 11101115131111 WEI/v3 14161 W (:17 N) M113 wt) (fluk‘fl admruvd. Mum mMwannwpa wmfliiwfiw Om fifiaj {NINE offlmijgww 130111212101 Mupfi ’L‘ 6. (Circle the correct answer) We can only perform the iterated dominance procedure if players have a) common knowledge of rationality. b) at least one dominated strategy. 0) common knowledge of the game. .aand c. 7. (Circle the correct answer) In a two-player game, if a strategy is undominated, it must be a best response to how many of the opponents strategies? a) all strategies. b) at least half the strategies. @ at least one strategy. d) no strategies. e) not enough information. 8. (True or false) A socially efficient strategy profile must also be Pareto efficient. If true, explain Why, based on the two definitions of efficiency. If false, give a counterexample; i.e., construct a normal form game in which the claim is false. “Ynez %OL ntmmryrdfin ta mt PM ngtw jaw WW dill/Mom” HMX’MW “Wax/[WWW Wm )LN GMCX‘S. {N ‘bm big SEW 1 (£1 eke/fl am 0:3 9. (Short answer) Why is it possible for the Nash equilibrium outcome to differ from the socially efficient outcome? /\)ru 31>me W Wt Wfim Pia/6%) . Lnobu/LOL tmfismflm Mm (30.31%; {NM (,0 mm fiww /W WW OWL “be hr WLmAPm‘ 10. (Short answer Describe one strategic tensioriS-Aat contgzgwn elp remedy. ...
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