Recap. Duopoly models. ! Last class (January 13, 2004) ! Today (January 15, 2004) ! Two competing firms, selling a homogeneous good

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1 Reap! Last lass (January 13, 24)! Domnant and domnated atons! Best response! ash eulbrum! Mxed strateges! Pareto domnane! Today (January 15, 24)! Examples of games wth ontnuous aton sets! Duopoly models: Cournot and Bertrand Duopoly models! Two ompetng frms, sellng a homogeneous good! The margnal ost of produng eah unt of the good: 1 and 2! The market pre, P s determned by (nverse) market demand:! Pa-bQ f a>bq, P otherwse.! Both frms seek to maxmze profts! Cournot: Frms set uanttes smultaneously! Bertrand: Frm set pres smultaneously! Stakelberg: Frms set uanttes, frm 1 followed by frm 2 1

2 Cournot Competton! The market pre, P s determned by (nverse) market demand:! Pa-bQ f a>bq, P otherwse.! Eah frm dedes on the uantty to sell (market share): 1 and 2! Q 1 2 total market demand! Both frms seek to maxmze profts Cournot Competton: Best response of Frm 1! Suppose frm 2 produes 2! Frm 1 s profts, f t produes 1 are: π 1 (P- 1 ) 1 [a-b( 1 2 )] (Resdual) revenue Cost! How to hoose 1 to maxmze π 1?! Frst note that π 1 s onave: d 2 π 1 /d 1 2-2b <! Frst order ondtons (FOC): d π 1 /d 1 a - 2b 1 -b 2 1 Resdual margnal revenue Margnal ost 1 (a- 1 )b 2 R 1 ( 2 ) 2

3 Cournot Competton: Best response of Frm 2! Suppose frm 1 produes 1! Frm 2 s profts, f t produes 2 are: π 2 (P- 2 ) 2 [a-b( 1 2 )] (Resdual) revenue Cost! Frst order ondtons: d π 2 /d 2 a - 2b 2 b 1 2 RMR MC 2 (a- 2 )b 1 R 2 ( 1 ) Example: Cournot Competton! P 13-( 1 2 ), so a13, b1! 1 2 1! Suppose Frm 2 thnks that Frm 1 wll set 1 4! Resdual demand of Frm 2: P 9-2! Resdual revenue of Frm 2: RR[9-2 ] 2! Resdual margnal revenue: RMR9-2 2! Settng RMRMC

4 Cournot Competton: Graphal soluton 13 P 9 Resdual demand: P9-5 RMR 9-2 P13-Q MC Cournot Eulbrum! 1 (a- 1 )b 2! 2 (a- 2 )b 1! Solvng together for 1 and 2 : C 1 (a )/3b C 2 (a )/3b! Market demand and pre: Q C C 1 C 2 (2a- 1-2 )/3b P a - bq C (a 1 2 )/3 4

5 Example: Cournot Competton! P 13-( 1 2 ), so a13, b1! 1 2 1! The frms best response funtons: 1 (a-b 2 -)b ( )6-2 2 (a-b 1 -)b ( )6-1! Solvng for 1 and 2 : Q8 P 5! Frms profts: π 1 π 2 (5-1)4 16 Cournot Competton: Graphal soluton! 1 R 1 ( 2 ) R 2 ( 1 ) R 2 ( 1 ) 6-1 Cournot eulbrum R 1 ( 2 )

6 6 Cournot Eulbrum wth frms j j b b a ] [ ), ( max π Frst order ondtons: b b a j j,..., 1 2 Substtute Q : bq b a,..., 1 Sum over : bq bq a Cournot Eulbrum wth frms 1 1 1) ( 1) ( a P b b a Q C C If eah frm has the same ost : 1 1) ( a P b a Q C C C

7 Bertrand Eulbrum Model! Frms set pres rather than uanttes! Pa-bQ! Customers buy from the frm wth the heapest pre! The market s splt evenly f frms offer the same pre Best response! Frm 1 s proft funton: π(p 1 )(P 1-1 ) 1! To ensure 1 > (reall: Pa-bQ and Q(a-P)/b) P 1 a! To ensure nonnegatve profts P 1 1! Frm 1 should hoose 1 P 1 a! Smlarly, frm 2 should hoose 2 P 2 a 7

8 Best response (ont.)! Frm s demand depends on the relatonshp between P 1 and P 2, f P > Pj a P, f P < Pj 1,2 b a P, f P Pj P 2b! Frm 1 should hoose 1 P 1 P 2 (f possble)! Frm 2 should hoose 2 P 2 P 1 (f possble) Bertrand eulbrum! For both frms to sell postve uanttes proftably 1 P 1 P 2 and 2 P 2 P 1! Suppose 1 2 P 1 2 (a-)b! Suppose 1 < 2 P 1 2 -ε P (a- 2 ε)/b 2 8

9 Example! P 13-( 1 2 ) (a13, b1)! 1 2 1! P1! 1 2 (a-p)b 6 Q12! Frms profts: π 1 π 2 Quantty-settng monopolst! Sngle frm (monopolst), sellng a sngle good! The margnal ost of produng eah unt of the good:! The frm dedes on the uantty to sell: Q (market demand! The market pre, P s determned by (nverse) market demand:! Pa-bQ f a>bq, P otherwse.! The frm seeks to maxmze profts 9

10 Quantty-settng monopolst! The frm s profts, f t produes Q are: π (P-)Q [a-bq]q Q Revenue Cost! How to hoose Q to maxmze π?! Frst note that π s onave: d 2 π/dq 2-2b <! Frst order ondtons (FOC): d π/dq a - 2bQ Margnal revenue Margnal ost Q (a-)b P (a) Example! P 13-Q (a13, b1)! 1! Q (a-)b 6 P (a) 7! Monopolst s profts: π (7-1)6 36 1

11 Monopoly vs. Cournot vs. Bertrand Compettve Bertrand Cournot Monopoly Pre Quantty Total Frm Profts 32 36! Frm profts and pres: Compettve Bertrand Cournot Monopoly Monopoly vs. Cournot vs. Bertrand Compettve Bertrand Cournot Monopoly Pre (a2)/3 (a) Quantty (a-)/b (a-)/b 2(a-)/3b (a-)b Total Frm Profts 2(a-) 2 /9b (a-) 2 /4b! Frm profts and pres: Compettve Bertrand Cournot Monopoly 11

12 Cournot ompetton 13 P 9 5 Consumer surplus32 P13-Q Frm profts Deadweght loss8 MC1 13 Bertrand ompetton 13 P Consumer surplus72 P13-Q MC1 12

13 Monopoly 13 P Consumer surplus18 7 P13-Q Frm profts36 6 Deadweght loss18 MC1 13 Monopoly vs. Cournot vs. Bertrand Compettve Bertrand Cournot Monopoly Consumer surplus Deadweght loss 8 18 Total Frm Profts

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