ANSWER KEY 7 GAME THEORY, ECON 395

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1 ANSWER KEY 7 GAME THEORY, ECON 95 PROFESSOR A. JOSEPH GUSE 1 Gbbos.1 Recall the statc Bertrad duopoly wth homogeeous products: the frms ame prces smultaeously; demad for frm s product s a p f p < p j, s 0 f p > p j ad s a p / f p = p j. Margal costs are c < a. Show that the frms ca use trgger strateges that swtch forever to the stage-game Nash equlbrum after ay devato to susta the moopoly prce level a SPNE f ad oly f δ < 1. ANSWER. Let VNE represet the value of playg the stage game NE every perod for player. Sce the proft of each frm s zero every stage where they play the stage game NE, we have VNE 1 = V NE. Suppose that a gve stage, t, the two frms collude to maxmze total proft whch they splt. The maxmzato problem s max p a pp c where a p s the total demad for both frms output at prce p ad p c s the per ut proft. fd the frst order codto o the proft-maxmzg level of prce, p M, by takg the dervatve of proft ad settg t equal to zero: p M = [ a+c 5 a+c p ] Now use ths to calculate the total proft the frms ear whe they each set prce at p M. Call ths level of proft π M : π M = a p Mp M c = a a+c a+c = a c 4 c If both frms were to cooperatvely set ther prce to, p M, they would splt the demad half by assumpto the set-up ad therefore splt ths proft gag a c 8 each. Now cosder the most a frm could ga a sgle perod by devatg from the cooperatve prce level. If the other frm set ther prce equal to p M = a+c, the other frm could 1

2 PROFESSOR A. JOSEPH GUSE just udercut t, chargg p M ǫ ad ga the etre proft. Makg ǫ as small as possble, words, a devatg frm could ear as much as a c 4. Suppose that both frms adopt the followg grm trgger strategy: p t = { p M f p j t 1 = p M AND p t 1 = p M p B otherwse I words, play the cooperatve total proft maxmzg prce each perod as log as both players set that same prce the prevous perod. Otherwse set prce equal to the oe-shot Bertrad prce, p B, whch we kow from earler to be equal to c, the margal cost. To whether both frm adoptg ths strategy s a NE we eed to ask whether ether frm should devate. By cooperatg each frm wll ear π m / = a c 8 every perod. By devatg a frm wll ear a c 4 the perod they devate followed by 0 ever after. Hece devato from the grm trgger stragegy profle s worthwhle f PV dev PV coop a c 4 + δ0 1 δ a c 81 δ a c a c 1 δ 1 δ 1 1 δ Gbbos.15 Suppose there are frms a Courot olgopoly. Iverse demad s gve by PQ = a Q, where Q = =1 = q. Cosder the ftely repeated game based o ths stage game. a Whatsthelowestvalueofδ suchthatthefrmscausetrggerstrategestosustathe moopoly output level a subgame-perfect Nash equlbrum? How does the aswer vary wth, ad why? ANSWER We wll be terested the followg three strateges quattes ad assocated proft levels.

3 ANSWER KEY 7 GAME THEORY, ECON 95 Strategy Profle Descrpto Quatty Idvdual Proft All Frms Cooperate to Maxmze Proft q M = a c All Frms Play Courot Eqm stage game q C = a c Oe frm devates π M = a c 4 +1 π C = a c +1 q D = +1a c 4 π D = +1 a c 4 1 We eed to ask whether for a gve value of δ whether all frm playg grm trgger s a Nash Equlbrum the repeated game. I ths case grm trgger would mea each frm producg q M every perod as log as all frms dd ths the prevous perod ad producg q C ay perod where t was observed the prevous perod that someoe dd somethg other tha q M. Ths wll be a NE f PV colluso > PV devato π M 1 δ π D + δπ C 1 δ where colluso here meas playg the strategy ecouraged by the grm trgger strategy ad devatato meas takg advatage of the other players by maxmzg proft some perod. The preset value of colluto s the preset value of the fte stream of dvdual frm profts from playg the collusve strategy whe all other frms play t π M. The preset value of devato s the oe-tme proft of devatg π D the stage game plus the fte stream of subsequet profts from playg the ocooperatve courot equlbrum, π C. Pluggg the values we have for π M, π C ad π D we get 1 Dervg qm: Maxmze total proft, a Q cq w.r.t to Q. F.O.C. s Q = a c. So dustry total quatty to maxmze dustry total proft s Q M = a c. qm s just oe frm s share of QM. Dervg q C: See AK 1 Gbbos 1.4 Dervg q D: Suppose 1 frms all but frm cooperate by each producg q M; q D s the output level that wll maxmze s proft a sgle stage game. Hece t solves max q a c a c q q. F.O.C s a c a c q D.

4 4 PROFESSOR A. JOSEPH GUSE PV colluso > PV devato a c 41 δ δ +1 + δ δ a c + δ a c δ δ δ+1 4 +δ16 1 δ δ δ δ δ Note that s always egatve for all 1 sce the deomator s always egatve ad the umerator s always postve. Note also that the largest power the umeratve s O whle the the deomator we have O 4. Hece s a egatve umber approachg 0 as grows large. Whch meas that the mmum value of δ ecessary order to support colluso equlbrum s less tha 1 ad approaches 1 as grows large. b If δ s too small for the frms to use trgger strateges to susta the moopoly output, what s the most-proftable symmetrc subgame-perfect Nash equlbrum that ca be sustaed usg trgger strateges? OPTIONAL. Gbbos. Cosder a Courot duopoly operatg a market wth verse demad PQ = a Q, where Q = q 1 +q s the aggregate quatty o the market. Both frms have total costs c q = cq, but demad s ucerta: t s hgh a = a H wth probablty θ ad low a = a L wth probablty 1 θ. Furthermore, formato s asymetrc: frm 1 kows whether demad s hgh or lwo, but frm does ot. All of ths s commo kowledge. The two frms smulataeously chooose quattes. a What are the strategy spaces for the two frms. ANSWER. Frm 1: q 1 : {a L,a H } R Frm : q R + Sce Frm 1 observes the demad parameter Frms 1 s strategy s a fucto, mappg from the set of possble demad parameters, {a L,a H } to the postve real umbers feasble output levels. Sce there are oly two possble demad types, ths bols

5 ANSWER KEY 7 GAME THEORY, ECON 95 5 dow to Frm 1 s strategy beg a par or postve umbers - oe to be played f a = a L ad the other to be played f a = a H. Sce Frm observes ether the demad parameter or Frm 1 s acto, there s othg to make ts choce of cotget o; ts strategy s smply a choce of output quatty, q. b Make assumptos o a H, a L, θ, ad c such that all equlbrum quattes are postve. What s the Bayesa Nash equlbrum of ths game? ANSWER. If frm 1 s to playg a best respose agat a strategy of q by frm, the frm 1 should solve max q 1 a c q q 1 q 1 Frm 1 s best respose fucto ca be derved by wrtg dow the F.O.C. ad solvg for q 1 the usual way q 1q,a = a c q Frm s best respose fucto solves max q a H c q q 1 a H q θ H +a L c q 1 a L q q 1 θ H Note that the frst group of terms ths objectve fucto s the cotrbuto to F s expected proft whe demad s hgh ad F1 plays some strategy q 1 a H. Frm wo t kow whether demad s H or L, but wll kow that whe t s H that F1 wll plat q 1 a H. Hece ths frst group of terms s multpled by the probablty that demad s H, θ H. Dfferetatg w.r.t q gves us the followg F.O.C. a H q 1 a H θ H +a L q 1 a L 1 θ H c q To solve for the equlbrum we ca substtute the epresso we derved for F1 s BR futo to ths, whch yelds a H a H c q ah al θ H + θ H + a L a L c q 1 θ H c q 1 θ H c q a H θ H +a L 1 θ H c q q = a Hθ H +a L 1 θ H c q = Ea c

6 6 PROFESSOR A. JOSEPH GUSE Surprsgly?, ths meas that Frm s equlbrum strategy s to produce a output level as though t were a Courot Equlbrum ad demad s the expected demad. ote that Ea = a H θ H + a L 1 θ H. Sce Frm wll always play the average Courot quatty, Frm 1 wll wat to produce above ths quatty whe a = a H ad below ths quatty whe a = a L. Specfcally, q1a L = a L c a Hθ H +a L 1 θ H c q1a L = a L a H θ H a L 1 θ H c 6 q 1a L = a L Ea c 6 = a L Ea c Note that ths s less the F wll produce whe a+a L sce a L Ea c < Ea c. Smlarly, F1 s output whe t observes a = a H ca be calculated: q1a H = a H c a Hθ H +a L 1 θ H c q1a H = a H a H θ H a L 1 θ H c 6 q 1a H = a H Ea c 6 = a H Ea c Note that ths s less the F wll produce whe a+a L sce a H Ea c > Ea c. The aswer to the questo about what codtos the parameters have to meet order to guaratee that all quattes are postve s that we must have a L Ea c > 0 AND Ea c > 0 Notce that f ths codto hold the we see drectly that q 1 a L > 0. Also sce a H > a L, we kow that a L Ea c > 0 a H Ea c > 0 so q 1 a H > 0 as well. The secod codto guaratees that q > 0 Irocally, perhaps, Frm 1, who kows more tha Frm, wll ear less proft whe demad s low. Sce they wll face the same prce ad margal cost ad F wll produce more. 4 Gbbos.6 Cosder a frst-prce sealed-bd aucto whch the bdders valuatos are depedetly ad uformly dstrbuted o [0, 1]. Show that f there are bdders, the the strategy of bddg tmes oe s valuato s a symmetrc Bayesa ash equlbrum of ths aucto.

7 ANSWER KEY 7 GAME THEORY, ECON 95 7 ANSWER. Let v be the valuato of bder. To show that v s a symmetrc BNE, we eed to show that f all players except oe employ ths strategy the t s a best respose for the last oe. Suppose that all bdders j bd v j, bdder s problem s to maxmze expected utlty. maxeu b = Prw bv b+prlose b0 b Let s look more closely at Prw b Prw b = Prb > max {b j} j { } vj = Pr b > max j = Pr b > v 1 = Pr v 1 < b = j Pr &...&b > v 1 &b > v +1 &...&b > v &...&v 1 < b &b > v +1 < b &...&b > v < b v j < b Note the last step ca be take because each v j s a depedetly draw value whch meas the probablty of the jot statemet coceg each v j s equal to the product of each statemet. Usg the fact that each j s draw from the uform dstrbuto, we kow that, geeral Prv j < x = x, so we ca be eve more precse about Prw b: Prw b = j = b b Pluggg ths back to s objecve fucto ad dfferetatg w.r.t b, we get the followg frst order codto.

8 8 PROFESSOR A. JOSEPH GUSE b b v b b v b b b v b b v b b v b v b b b = v To be thorough, we should also check the secod-order codto SOC: d Eu b db d Eu b db / b b v b b v b [ [ b ] b ] b ] b b v b [ b v b [ v b b v b 1 ] Evaluatg ths last epressto at b = v we get

9 ANSWER KEY 7 GAME THEORY, ECON 95 9 d Eu b db v v v v v whch s true by our assumpto that v s draw o U0,1.