Thus, if the two coffee shops collude, then they will produce a level of output q m

Transcription

1 Introductory Mcroeconomcs (ES10001) Exercse 9: Suggested Solutons 1. There are two coffee shops on campus facng an aggregate demand functon for cups of coffee of q d = 00 0 p. The coffee s dentcal and consumers wll always buy from the lowest prce shop. And f the two shops set the same prce, then each wll supply one half of the total demand. The two shops can purchase coffee at c = per cup wthout wastage.e. any unsold cups can be returned to the suppler at. It s the case that the equlbrum prce f the two shops collude over prce then prce s hgher than the equlbrum prce f they do not collude: TRUE / ALSE. Soluton: rst note that: q d ( p) = 00 0 p p d ( q) AR q MR q = 10 0q = q Thus, f the two coffee shops collude, then they wll produce a level of output q m such that: MR( q ) m = 10 0q m = = Mc( q ) m q m = 80 whch mples: p m = p d ( 80) = = 6 If they compete over prces then the Bertrand equlbrum mples: p 1 b = p 1 b = p b = = MC = AC Thus: p m = p b = 6 = 4 1

2 . What dd Chamberlan mean by the statements: Too many frms each producng too lttle output. And Excess capacty s the cost of dfferentness.? Soluton: Too many frms such that each makes normal proft n long run equlbrum and as such, unlke most monopolsts, has no resources to nvest n R&D. And f we have ushaped long-run average cost curves, then each frm wll be producng at below mnmum effcent scale. 3. Two frms decde to form a cartel and collude n a way that maxmzes ndustry profts. Each frm has zero producton costs and each frm s gven a postve output quota by the cartel. Whch of the followng statements s not true? A. Each frm would want to produce more than ts quota f t knew that the other would contnue to produce at ts quota B. The prce elastcty of demand wll be -1 at the output level chosen C. Output would be lower than f the frms behaved as Cournot compettors D. Output would be lower than f the frms behaved as Stackelberg compettors E. All of the other statements are false Soluton: Maxmum profts n an ndustry arse when output s at monopoly level whch requres p > MR = MC. Thus, all of the above statements apart from E are not true. 4. The nverse demand functon for mulled wne s p d ( q) = 0 q. There are constant returns to scale n ths ndustry wth unt costs of 8. Whch of the followng sets of statements s completely true? A. Monopoly output s 6, Cournot duopoly total output s 8, Bertrand duopoly total output s 4 B. Monopoly output s 8, Cournot duopoly total output s 8, Bertrand duopoly total output s 3 C. Monopoly output s 6, Cournot duopoly total output s 6, Bertrand duopoly total output s D. Monopoly output s 6, Cournot duopoly total output s 8, Bertrand duopoly total output s 1 E. one of the above Soluton: rst, note that: p d ( q) = AR q MR q = 0 q = 0 q Thus, monopoly output, q m, s derved from: MR( q ) m = 0 q m = 8 = MC ( q ) m q m = 6

3 or Cournot output, we need to derve the reacton functons for the two frms. rst, derve the margnal revenue functon for rm 1: AR 1 ( q 1 ) = 0 ( q 1 + q ) MR 1 q 1 = 0 q q 1 Thus, rm 1 wll maxmse proft when: MR 1 ( q 1 ) = 0 q q 1 = 8 = MC q 1 q 1 = 1 1 q Ths s rm 1 s reacton functon.e. rm 1 s optmal level of output gven the output of rm. ote that f rm dd not exst (.e. q = 0 ) then rm 1 would maxmse proft by settng q 1 * = 6, whch s the monopoly level of output. By symmetry, rm s reacton functon s: q = 1 ( 1 q 1 ) The two reacton functons defne each rm s optmal level of output gven the output of the other frm. In the ash-cournot equlbrum, both frms must be dong the best they can gven what the other frm s dong. Defnng these mutually optmal (.e. proft maxmsng) levels of output as q 1 n,q n = 1 n ( 1 q ) And: q n = 1 n ( 1 q 1 ) Thus:, then t must be the case that: 3

4 = n 1 q 1 = n q 1 3 = 6 q 1 n = 4 = q n q n = 8 nally, under Bertand prce wll go to MC = AC such that: p ( b q ) b = 0 q b = 8 = MC q b q b = 1 = AC( q ) b 5. A game has two players and each has two strateges. The strateges are Be ce and Be Mean. If both players play Be ce, both get a payoff of 5. If both players play Be Mean, both get a payoff of -3. If one player plays Be ce and the other plays Be Mean, the player who played Be ce gets 0 and the player who played Be Mean gets 10. Playng Be Mean s a domnant strategy for both players: TRUE / ALSE. Soluton: See Table 1: Player B Be ce Be Mean Player A Be ce 5, 5 0, 10 Be Mean 10, 0-3, -3 Table 1 6. A game has two players. Each player has two possble strateges. One strategy s Cooperate, the other s Defect. Each player wrtes on a pece of paper ether a C for cooperate or a D for defect. If both players wrte C, they each get a payoff of 100. If both players wrte D, they each get a payoff of 0. If one player wrtes C and the other player wrtes D, the cooperatng player gets a payoff of S and the defectng player gets a payoff of T. To defect wll be a domnant strategy for both players f: A. S + T > 100 B. T > S C S < 0 and T > 100 D. S < T and T > 100 E. S and T are any postve numbers Soluton: See Table : 4

5 Player B Cooperate Defect Player A Cooperate 100, 100 S, T Defect T, S 0, 0 Table 1 or Defect to be a domnant strategy for both players we requre T >100 and 0 > S. 7. In the town of Bathos, each of the > nhabtants has 100. They are told that they can all voluntarly contrbute to a fund that wll be evenly dvded among all resdents. If are contrbuted to the fund, the local Tesco wll match the prvate contrbutons so that the total amount to be dvded s. That s, each resdent wll get back a payment of / when the fund s dvded. If the people n town care only about ther own net ncomes, n ash equlbrum, how much wll each person contrbute to the fund? A. 0 B. 10 C. 0 D. 50 E. 100 Soluton: rst, denote the contrbuton of ndvdual as and the contrbuton of all the other (dentcal) ndvduals apart from ndvdual as such that total donatons are gven by = +. It s apparent that none of the (dentcal ndvduals has an ncentve to contrbute to the fund such that non-contrbutng denotes a ash equlbrum. To see ths, frst assume that < such that the total fund to be dvded amongst the resdents s. If everyone contrbutes to the fund, then ndvdual s return net of hs contrbuton s: ( ) = + If, however, everyone but ndvdual contrbutes then ndvdual s net return s: π = The beneft to ndvdual from not contrbutng when everyone else does s: π π = + + π π π π = + ( = 1 ) > 0 If no one else contrbutes to the fund, then ndvdual s net return f he contrbutes s: 5

6 π = If, however, nether ndvdual nor anyone else contrbutes, then ndvdual s return s: ( 0) = 0 = 0 The beneft to ndvdual from not contrbutng when no one else does s thus: ( 0) ( ) = 0 ( 0) π ( = 1 ) > 0 Thus, rrespectve of what anyone else does, each (dentcal) ndvdual has a unque postve beneft from not contrbutng. ow assume and that Tesco matches the fund such that the total fund to be dvded amongst the resdents s. If everyone contrbutes to the fund, then ndvdual s return net of hs contrbuton s: ( ) = + If, however, everyone but ndvdual contrbutes then ndvdual s net return s: ( 1 ) = The beneft to ndvdual from not contrbutng when everyone else does s now: ( ) ( ) ( ) = ( + ) + = + ( = ) > 0 If, however, no one else contrbutes to the fund, then ndvdual s net return net f he contrbutes s: 6

7 ( ) = If, however, nether ndvdual nor anyone else contrbutes, then ndvdual s return s: ( 0) = 0 = 0 The beneft to ndvdual from not contrbutng when no one else does s thus: ( 0) ( ) = 0 ( 0) π ( = ) > 0 Thus agan, rrespectve of what anyone else does, each (dentcal) ndvdual has a unque postve beneft from not contrbutng. A smpler way of seeng the result s to consder ndvdual s choce as to what 0 to contrbute. Let denote ndvdual s net proft from contrbutng. Then: π = + = ote that > 0 and s an (unknown) fxed amount. So (for ndvdual ) when π contrbutng, rrespectve of what anyone else does. ( ) s maxmsed = 0. So each (dentcal) ndvdual benefts from not 8. Arthur and Beatrce met at the resher s Ball. They agreed to meet for a date at a local pub the next week. Regrettably, they were so fraught wth passon that they forgot to agree on whch pub would be the ste of ther rendezvous. Luckly, the town of Bathos has only two pubs, The Red Lon and the The Old Red Lon. Havng dscussed ther tastes n pubs at the reshers Ball, both are aware that Arthur prefers The Red Lon to The Old Red Lon and Beatrce prefers The Old Red Lon to The Red Lon. In fact, the payoffs are as follows. If both go to The Old Red Lon, then Beatrce s utlty s 3 and Arthur s utlty s. If both go to The Red Lon, then Arthur s utlty s 3 and Beatrce s utlty s. If they go to dfferent pubs, then both have a utlty of 0. A. Ths game has no ash equlbra B. Ths game has a domnant strategy equlbrum C. There are two ash equlbra D. Ths game has one ash equlbrum E. Ths game two domnant strategy equlbra Soluton: See Table 3: 7

8 Beatrce The Red Lon The Old Red Lon Arthur The Red Lon 3, 0, 0 The Old Red Lon 0, 0, 3 Table 1 9. Ben and Joe have taken ther cars out on a lonely road and are engaged n a game of Chcken. Ben has a 4X4 Turbo utter B and Joe has a dodgy lttle hatch-back. Each of the players can choose ether to Swerve or to ot Swerve. If both choose Swerve, both get a payoff of zero. If one chooses Swerve and the other chooses ot Swerve, the one who chooses ot Swerve gets a payoff of 10 and the one who chooses Swerve gets zero. If both choose ot Swerve, the damage to Ben s car s farly mnor and he gets a payoff of -5, whle for Joe the results are dsastrous and he gets a payoff of A. Ths game has a domnant strategy equlbrum n whch Ben does not swerve and Joe swerves. B. Ths game has a domnant strategy equlbrum n whch Joe does not swerve and Ben swerves.. C. Ths game has two ash equlbra. D. The one and only ash equlbrum n ths game s where Ben does not swerve and Joe swerves. E. Ths game has three ash equlbra. Soluton: See Table 3: Joe Swerve ot Swerve Ben Swerve 0, 0 0, 10 ot Swerve 10, 0-5, -100 Table A duopoly faces the nverse demand curve p d = 160 q, mplyng margnal revenue functons of MR 1 = 160 q 4q 1 and MR = 160 q 1 4q. Both frms n the ndustry have constant costs of 10 per unt of output. In a Cournot equlbrum how much output wll each duopolst sell? A. 75 B. 54 C. 5 D. 35 E. 48 Soluton: rm 1 wll maxmse proft when: MR 1 ( q 1 ) = 160 q 4q 1 = 10 = MC 1 q 1 q 1 = q 8

9 And by symmetry: q = 1 4 ( 150 q 1 ) In Cournot equlbrum we have: = 1 n 4 ( 150 q ) And: q n = 1 n 4 ( 150 q 1 ) Thus: = n q 1 4q 1 n = q 1 n 3q 1 n = 75 q 1 n = 5 = q n 9