Cournot Equilibrium with N firms

Transcription

1 Recap Last class (September 8, Thursday) Examples of games wth contnuous acton sets Tragedy of the commons Duopoly models: ournot o class on Sept. 13 due to areer Far Today (September 15, Thursday) Duopoly models ournot, Bertrand omparson: ournot, Bertrand, and Monopoly Stackelberg Multstage games wth observed actons ournot Eulbrum wth frms max (, Frst order condtons: ) [ a b b j ] j c a 2b b j c 0 1,..., j Substtute Q= =1,.., : a b bq c Sum over =1,..,: a bq bq 0 1,..., c 0 (**) Q s the only unknown n ths euaton! 1

2 2 ournot Eulbrum wth frms from Euatons (**) 1,..., 1 1 1) ( 1) ( n b c bq a c a P b c b a Q If each frm has the same cost c =c: 1 1) ( c a P b c a Q ournot Eulbrum wth frms What happens to the prce as ncreases? As, P c What happens to the total uantty as ncreases? Try =2 Q=?, =10 Q=?,, Q?? If each frm has the same cost c =c: 1 1) ( 1) ( ) ( c a P b c a Q b c a Q

3 ournot Eulbrum wth frms Q P a c ( 1) b ( 1) b a 1 c 1 c c where c s the average cost What happens to the total uantty sold as the number of frms ncreases? Goes up, approaches to a c as b ournot Eulbrum wth frms Q P a c ( 1) b ( 1) b a 1 c 1 c c where c s the average cost What happens to the prce as the number of frms ncreases? Decreases. Approaches average cost. 3

4 ournot Eulbrum wth frms Q a bq c b a c ( 1) b ( 1) b 1,..., n P a 1 c 1 from Euatons (**) What happens to the uantty of Frm, f ts cost goes up? Decreases. Lower cost frms grab a larger porton of the market share Bertrand Eulbrum Model Frms set prces rather than uanttes P=a-bQ, or euvalently, Q=(a-P)/b ustomers buy from the frm wth the cheapest prce The market s splt evenly f frms offer the same prce 4

5 Best response Frm 1 s proft functon: (P 1 )=(P 1 -c 1 ) 1 To ensure 1 >0 (Recall: P=a-bQ and Q=(a-P)/b) P 1 a To ensure nonnegatve profts P 1 c 1 Frm 1 should choose c 1 P 1 a Smlarly, frm 2 should choose c 2 P 2 a Best response (cont.) Frm s demand depends on the relatonshp between P 1 and P 2 0, a P, b a P, 2b f P P f P Pj f P P j j P Frm 1 should choose c 1 P 1 P 2 (f possble) Frm 2 should choose c 2 P 2 P 1 (f possble) 5

6 Bertrand eulbrum For both frms to sell postve uanttes proftably c 1 P 1 P 2 and c 2 P 2 P 1 Suppose c= c 1 = c 2 P=c 1 = 2 = (a-c)/2b Suppose c 1 < c 2 P 1 =c 2 - P 2 c 2 1 = (a- c 2 +)/b 2 = 0 Example P = 130-( ) (a=130, b=1) c 1 = c 2 = c = 10 P=10 1 = 2 = (a-p)/2b = 60 Q=120 Frms profts: 1 = 2 = 0 6

7 Monopoly vs. ournot vs. Bertrand Frms sell at cost ompettve Bertrand ournot Monopoly Prce Quantty Total Frm Profts oncdence for ths example? Frm profts and prces: ompettve Bertrand ournot Monopoly Quanttes: ompettve Bertrand ournot Monopoly Monopoly vs. ournot vs. Bertrand ompettve Bertrand ournot Monopoly Prce c c (a+2c)/3 (a+c)/2 7

8 Monopoly vs. ournot vs. Bertrand ompettve Bertrand ournot Monopoly Quantty (a-c)/b (a-c)/b 2(a-c)/3b (a-c)/2b Monopoly vs. ournot vs. Bertrand ompettve Bertrand ournot Monopoly Total Frm Profts 0 0 2(a-c) 2 /9b (a-c) 2 /4b 8

9 Monopoly vs. ournot vs. Bertrand ompettve Bertrand ournot Monopoly Prce c c (a+2c)/3 (a+c)/2 Quantty (a-c)/b (a-c)/b 2(a-c)/3b (a-c)/2b Total Frm Profts 0 0 2(a-c) 2 /9b (a-c) 2 /4b Frm profts and prces: ompettve Bertrand ournot Monopoly Quanttes: ompettve Bertrand ournot Monopoly ournot competton 130 P onsumer s surplus: Dfference between wllngness to pay and the prce onsumer surplus=3200 P=130-Q Frm profts=3200 Deadweght loss=800 M=

10 Bertrand competton 130 P onsumer surplus=7200 P=130-Q M=10 Monopoly 130 P onsumer surplus= P=130-Q Frm profts= Deadweght loss=1800 M=

11 Monopoly vs. ournot vs. Bertrand ompettve Bertrand ournot Monopoly onsumer surplus Deadweght loss Total Frm Profts Stackelberg Model Two competng frms, sellng a homogeneous good The margnal cost of producng each unt of the good: c 1 and c 2 The market prce, P s determned by (nverse) market demand: P=a-bQ f a>bq, P=0 otherwse. Q= total market demand Both frms seek to maxmze profts So far, same as ournot 11

12 Stackelberg Model Two competng frms, sellng a homogeneous good The margnal cost of producng each unt of the good: c 1 and c 2 The market prce, P s determned by (nverse) market demand: P=a-bQ f a>bq, P=0 otherwse. Q= total market demand Both frms seek to maxmze profts Frm 1: Leader Frm 2: Follower Frm 1 moves frst and decdes on the uantty to sell: 1 Frm 2 moves next and after seeng 1, decdes on the uantty to sell: 2 Stackelberg Model Q j : the space of feasble j s, j=1,2 Strateges of frm 1: 1 Q 1 Strateges of frm 2: s 2 : Q 1 Q 2 Possble strateges of Frm 2 are functons of 1 Outcomes and payoffs n pure strateges ( 1, 2 ) = ( 1, s 2 ( 1 )) j ( 1, 2 ) = [a-b( )- c j ] j We have dentfed all three elements of the game: strategy space, outcomes, and payoffs ash eulbrum: A strategy profle such that nether player can gan by swtchng to a dfferent strategy. 12

13 Stackelberg Model: Strategy of Frm 2 Suppose frm 1 produces 1 Frm 2 s profts, f t produces 2 are: 2 = (P-c) 2 = [a-b( )] 2 c 2 2 = (Resdual) revenue ost Frst order condtons: d π 2 /d 2 = a - 2b 2 b 1 c 2 = = RMR M = 0 2 =(a-c 2 )/2b 1 /2= R 2 ( 1 ) Famlar?? s 2 = R 2 ( 1 ) Strategy of frm 2 Stackelberg Model: Frm 1 s decson Frm 1 s profts, f t produces 1 are: 1 = (P-c) 1 = [a-b( )] 1 c 1 1 We know that from the best response of Frm 2: 2 =(a-c 2 )/2b 1 /2 Substtute 2 nto 1 : 1 = [a-b( 1 + (a-c 2 )/2b- 1 /2)] 1 c 1 1 = [(a+ c 2 )/2-(b/2) 1 -c 1 ] 1 From FO: dπ 1 /d 1 = (a+ c 2 )/2-b 1 -c 1 = 0 1 = (a-2c 1 +c 2 )/2b 13

14 Stackelberg Eulbrum We have Frm 1 s strategy: 1 = (a-2c 1 +c 2 )/2b And frm 2 s best response 2 =(a-c 2 )/2b 1 /2 Therefore: 2 =(a+2c 1-3c 2 )/4b If c 1 = c 2 = c 1 = (a-c)/2b 2 = (a-c)/4b Q = 3(a-c)/4b Stackelberg Eulbrum 1 = (a-2c 1 +c 2 )/2b 2 =(a+2c 1-3c 2 )/4b If c 1 = c 2 = c 1 = (a-c)/2b, 2 = (a-c)/4b, Q = 3(a-c)/4b When c 1 = c 2 = c, whch one s larger, 1 or 2? 14

15 Stackelberg Eulbrum 1 = (a-2c 1 +c 2 )/2b 2 =(a+2c 1-3c 2 )/4b If c 1 = c 2 = c 1 = (a-c)/2b, 2 = (a-c)/4b, Q = 3(a-c)/4b What happens to the uanttes as c 1 goes down? 1 goes up at the rate 1/b, 2 goes down at the rate 1/2b. So, the overall uantty goes up at the rate 1/2b. Recall: Under ournot, 1 ncreases at the rate 2/3b as c 1 decreases Stackelberg Eulbrum 1 = (a-2c 1 +c 2 )/2b 2 =(a+2c 1-3c 2 )/4b If c 1 = c 2 = c 1 = (a-c)/2b, 2 = (a-c)/4b, Q = 3(a-c)/4b What happens to the uanttes f c 2 goes down? 1 goes down at the rate 1/2b, 2 goes up at the rate 3/4b. So, the overall uantty goes up at the rate 1/4b. 15

16 Stackelberg Eulbrum 1 = (a-2c 1 +c 2 )/2b 2 =(a+2c 1-3c 2 )/4b If c 1 = c 2 = c 1 = (a-c)/2b, 2 = (a-c)/4b, Q = 3(a-c)/4b Whch one benefts the customers more? A decrease n c 1 or c 2? ournot vs. Stackelberg vs. Bertrand Bertrand Stackelberg ournot Monopoly Prce c (a+3c)/4 (a+2c)/3 (a+c)/2 Quantty 3(a-c)/4b (a-c)/b ((a-c)/2b+ 2(a-c)/3b (a-c)/2b (a-c)/4b) Total Frm Profts 0 3(a-c) 2 /16b 2(a-c) 2 /9b (a-c) 2 /4b Total output Stackelberg? ournot Stackelberg output s hgher than the output n ournot 16

17 ournot vs. Stackelberg vs. Bertrand Bertrand Stackelberg ournot Monopoly Prce c (a+3c)/4 (a+2c)/3 (a+c)/2 Quantty 3(a-c)/4b (a-c)/b ((a-c)/2b+ 2(a-c)/3b (a-c)/2b (a-c)/4b) Total Frm Profts 0 3(a-c) 2 /16b 2(a-c) 2 /9b (a-c) 2 /4b Prce Stackelberg? ournot Prce n Stackelberg s less than the Prce n ournot ournot vs. Stackelberg vs. Bertrand Bertrand Stackelberg ournot Monopoly Prce c (a+3c)/4 (a+2c)/3 (a+c)/2 Quantty 3(a-c)/4b (a-c)/b ((a-c)/2b+ 2(a-c)/3b (a-c)/2b (a-c)/4b) Total Frm Profts 0 3(a-c) 2 /16b 2(a-c) 2 /9b (a-c) 2 /4b Total proft Stackelberg? ournot Would the frst frm prefer ournot or Stackelberg? What about the second frm? Total proft s less n Stackelberg versus ournot 17

18 Example: Stackelberg ompetton P = 130-( ), so a=130, b=1 c 1 = c 2 = c = 10 Frm 2: 2 =(a-c 2 )/2b 1 /2 = 60-1 /2 Frm 1: Resdual demand: a-b( ) = 70-1 /2 RMR = (a+ c 2 )/2-b 1 = 70-1 Set RMR=M 70-1 = 10 1 = 60 Market prce and demand Q=90 P=40 Stackelberg ompetton: Frm 1 strategy 130 P P=130-Q 60 M=10 18

19 Stackelberg competton 130 P onsumer surplus=4050 P=130-Q Frm profts= Deadweght loss=450 M= Monopoly vs. ournot vs. Bertrand vs. Stackelberg Bertrand Stackelberg ournot Monopoly Prce Quantty (60+30) Total Frm Profts ( ) Frm profts and prces: Bertrand Stackelberg ournot Monopoly 19

20 Monopoly vs. ournot vs. Bertrand vs. Stackelberg Bertrand Stackelberg ournot Monopoly onsumer surplus Deadweght loss Total Frm Profts ( ) Mult-Stage Games wth Observed Actons These games have stages such that In each stage k, every player knows all the actons (ncludng those by ature) that were taken at any prevous stage Players move smultaneously n each stage k Some players may be lmted to acton set do nothng n some stages Each player moves at most once wthn a gven stage Informaton set Other player s choces up to stage k 20

21 Stackelberg game Stage 1 Frm 1 chooses ts uantty 1 ; Frm 2 does nothng Stage 2 Frm 2, knowng 1, chooses ts own uantty 2 ; Frm 1 does nothng Mult-Stage Games wth Observed Actons h k : Hstory at the start of stage k h k ( a 0 1, a,..., a k 1 ), k 1,..., K Set of actons taken by the players untl stage k Example: In the Stackelberg game, hstory at the begnnng of stage 1 s empty stage 2 s the uantty choce of Frm 1. 21

22 Mult-Stage Games wth Observed Actons h s k A ( h k : ) : : Hstory at the start of stage k h k ( a 0 1, a,..., a Set of actons avalable to player n stage k gven hstory h Pure strategy for player that specfes an acton each hstory h a A ( h k k 1 ), k 1,..., K k k ) for each k and A pure strategy s a functon that maps hstores to actons s: H A Mxed strateges specfy probablty dstrbutons over the actons n each stage. 22