Chapter 3: Oligopoly

Transcription

1 Notes on Chapter 3: Olgopoly Mcroeconoc Theory IV 3º - LE-: Iñak Agurre Departaento de Fundaentos del Análss Econóco I Unversdad del País Vasco

2 Mcroeconoc Theory IV Olgopoly Introducton.. The Cournot odel.... Duopoly.... Olgopoly (n frs)...3. Welfare analyss... The Bertrand odel.... Hoogeneous product.... Heterogeneous product..3. Leadershp n the choce of output. The Stackelberg odel..4. Colluson and the stablty of agreeents..4.. Short-ter colluson..4.. The stablty of agreeents under a fnte teporal horzon and under an nfnte teporal horzon.

3 Mcroeconoc Theory IV Olgopoly Introducton Non-Cooperatve Gae Theory s very useful for odellng and understandng the ultagent econoc probles characterzed by strategc nterdependency, n partcular for analyzng copetton between frs n a arket. Perfect copetton and pure onopoly (not threatened by entry) are specal non-realstc cases. It s ore frequent n the real lfe to fnd ndustres wth not any frs or wth a lot of frs but wth a saller nuber of the producng a large proporton of the total producton. Wth few frs, copetton s characterzed by strategc consderatons: each fr takes ts decsons (prce, output, advertsng, etc.) takng nto account or conecturng the behavour of the others. Copetton n an olgopoly can be seen as a gae where frs are the players. So we shall adopt a Gae Theory perspectve to analyze the dfferent odels of olgopoly. For each case, we shall wonder what gae frs are playng (nforaton, order of playng, strateges, etc.), and what the equlbru noton s. An portant dfference between the gaes of the prevous chapter and the gaes we shall solve n ths chapter s that the forer were fnte gaes whle the latter are nfnte gaes. 3

4 Mcroeconoc Theory IV Olgopoly.. The Cournot odel... Duopoly () Contet. () Representaton of the gae n noral for. () Noton of equlbru. (v) Best response functon. Characterzaton of equlbru. (v) Eaple. Graphc representaton. () Contet The Cournot odel has four basc characterstcs: a) We consder a arket served by frs. b) Hoogeneous product. That s, fro the consuers pont of vew, the goods produced by the two frs are perfect substtutes. c) Quantty copetton. The varable of choce of each fr s output level. Denote and the producton levels of fr and fr, respectvely. by d) Sultaneous choce. The two frs have to choose ther outputs sultaneously. That s, each fr has to choose ts output wthout knowng the rval s choce. Sultaneous Choce does not ean that choces are ade at the sae nstant n te. An equvalent contet would be one where one fr chooses ts output frst and tthe other fr chooses ts output afterwards but wthout observng the decson adopted by the frst fr. In other words, sequental choce together wth perfect nforaton (the player ovng second does not observe what the frst over does) s equvalent to sultaneous choce. The nverse deand functon s p( ), where p ( ) < 0. As the product s hoogeneous, the prce at whch a fr can sell wll depend on the total output: p( ) = p( + ). 4

5 Mcroeconoc Theory IV Olgopoly The producton cost of fr s C ( ), =,. () Representaton of the gae n noral for ) =,. (Players) ) 0. Any non-negatve aount would serve as a strategy for player. Equvalently, we can represent the set of strateges of player I as [0, ), =,. 3) The payoff obtaned by each fr at the cobnaton of strateges (, ) s: (, ) p( ) C ( ) Π = + Π = + = Π (, ) = p( + ) C( ) (, ) p( ) C ( ),,,,. () Noton of equlbru. Cournot-Nash equlbru It s very easy to adapt the defnton of the Nash equlbru n the prevous chapter to ths new contet. (,.., ) s a Nash equlbru f : Π (s, s ) Π (s, s ) s S,, =,...,n. s s s n For the Cournot duopoly gae we say: (, ) s a Cournot-Nash equlbru f Π Π =. (, ) (, ) 0,,,, The second defnton based on best responses proves ore useful. s ( s,.., s n ) s a Nash equlbru f: s MR( s ), =,.., n where MR (s ) = s S : Π (s,s ) Π (s,s { ), s S,s s }.. For the Cournot duopoly gae: 5

6 Mcroeconoc Theory IV Olgopoly (, ) s a Cournot-Nash equlbru f = f ( ),, =,,, where f ( ) s the best-response functon of fr to fr s output. (v) Best response functon. Characterzaton of equlbru The proceedure that we follow to obtan the Nash equlbru s slar to that used n the prevous chapter. Frst we calculate the best response of each player to the possble strateges of ts rval and then we look for cobnatons of strateges whch are utually best responses to each other. Gven a strategy of fr we look for the strategy that provdes ost proft for fr. That s,, gven the strategy 0 fr s best response s to choose a strategy such that: a Π (, ) p( + ) C ( ) 0 Π = p( + ) + p( + ) C( ) = 0 () f( ) Π = p + + p + C < ( ) ( ) ( ) 0 Takng nto account the non-negatvty constrant, 0, or n ters of gae theory takng nto account that the best response ust belong to the strategy space of the player, the best response functon s: f = { f } ( ) a ( ),0. The Cournot-Nash equlbru s a strategy profle (, ) such that the strategy of each player s ts best response to the rval s strategy. That s, { } = f( ) = a f( ),0 = f( ) = a { f( ),0 },, =,,. = f( ) = a { f( ),0} 6

7 Mcroeconoc Theory IV Olgopoly Let us now forget the non-negatvty constrant and we are gong to assue that the best response functon s broadly characterzed by condton () (nteror soluton). By defnton, the best response ust satsfy the frst order condton: Π( f( ), ) = 0 fr s best response to 0 s f( ). The Cournot-Nash equlbru satsfes Π (, ) = 0 gven that = f ( ), =,. We have a sple way of checkng whether a cobnaton of strateges s a Nash equlbru: by calculatng each fr s argnal proft correspondng to that strategy profle. If any one of the s other than zero the equlbru condton s not et. Π ( ˆ, ˆ ) > 0 f ( ˆ ) ˆ ( ˆ, ˆ > ) s not a Cournot-Nash equlbru. Π ( ˆ, ˆ ) < 0 f ( ˆ ) ˆ ( ˆ, ˆ < ) s not a Cournot-Nash equlbru. (v) Eaple. Graphc representaton Consder the case of lnear deand and constant argnal cost: p( ) = a b and C ( ) = c, =,. Assue for the sake of splcty that the argnal cost s the sae for the two frs: c = c> 0, =,. ( a > cfor the eaple to ake sense). We frst obtan the best response functon for fr, =,. a Π (, ) p( + ) C ( ) [ a b( + )] c [ a c b( + )] 0 Π = p( + ) + p ( + ) C( ) = a b b c = 0 f( ) Π = < b 0 So the best response functon s: a c b f( ) = a { f( ),0} = a,0. b 7 a c b b

8 Mcroeconoc Theory IV Olgopoly The Cournot-Nash equlbru satsfes: a c b = f ( ) = a,0 > 0 b gven that a> c By solvng the syste: a c b = f( ) = a,0 > 0 b gven that a> c = f( ) = f( f( )) a c b a c+ b a c b a c+ b a c = = = = =. b b b 4b 3b a c b b a c a c b a c b 3b ( a c) a c = = = =. b b 6b 3b The Cournot-Nash total output s: ( a c) = + = and the equlbru prce 3b ( a c) a+ c p = p( + ) = a b =. Profts are: 3b 3 a c a c ( a c) Π =Π (, ) = [ p( + ) c] = = 3 3b 9b a c a c ( a c) Π =Π (, ) = [ p( + ) c] = =. 3 3b 9b 8

9 Mcroeconoc Theory IV Olgopoly Graphc representaton e f( ) 45º Cournot-Nash equlbru f( ) e 9

10 Mcroeconoc Theory IV Olgopoly... Olgopoly () Representaton of the gae n noral for. () Noton of equlbru. Best response functon. Cournot-Nash equlbru. () Lerner nde. (v) Specal cases. Constant argnal cost. () Representaton of the gae n noral for ) =,,..., n. (Players) ) 0. Slarly, [0, ), =,,.., n. 3) The proft of each fr correspondng to strategy profle (, ) s: Π (, ) = p( + ) C( ), =,,..., n. The way of representng the gae n noral for has changed slghtly. Gven the strategy profle (,,..., ) the relevant pont for fr, =,,..., n, s the total output produced n by the other frs, =. Therefore, (, ) s not really a strategy profle and Π (, ) s fr s proft assocated wth every cobnaton of strateges where fr produces and the other frs produce n aggregate that producton s dstrbuted aong the n - frs). (t beng rrelevant to fr how () Noton of equlbru. Best response functons. Cournot-Nash equlbru In the Cournot olgopoly gae we say that (,,.., n) (, ) s a Cournot-Nash equlbru f: Π Π =. (, ) (, ) 0,,,,..., n. 0

11 Mcroeconoc Theory IV Olgopoly In ters of best response functons the defnton s: (,,.., n) (, ) s a Cournot-Nash equlbru f = f ( ),, =,,.., n., where f( ) s fr s best response functon to all those cobnatons of strateges whose total output s. We net obtan the best response of fr to all those cobnatons of strateges (of the other frs) whose total output s. Fr I s best response s to choose a strategy such that: a Π (, ) p( + ) C ( ) 0 Π = p( + ) + p( + ) C( ) = 0 () f( ) Π = p + + p + C < ( ) ( ) ( ) 0 Takng nto account the non-negatvty constrant, 0, or n ters of gae theory that the best response ust belong to the player s strategy space, the best response functon s: { f } f ( ) = a ( ),0. The Cournot-Nash equlbru s a strategy profle (,,.., n) (, ) such that = f ( ),, =,,.., n.. Let us now forget the non-negatvty constrant and assue that the best response functon s broadly characterzed by condton () (nteror soluton). By defnton, the best response ust satsfy the frst order condton: Π( f( ), ) = 0 fr s best response to

12 Mcroeconoc Theory IV Olgopoly 0 s f ( ). The Cournot-Nash equlbru satsfes Π (, ) = 0 gven that = f ( ), =,,..., n. Here agan there s a sple way of checkng whether a cobnaton of strateges s a Nash equlbru: calculatng each fr s argnal proft correspondng to that strategy profle. If any one s other than zero the equlbru condton s not et. Π ( ˆ, ˆ ) > 0 f ( ˆ ) ˆ ( ˆ, ˆ > ) s not a Cournot-Nash equlbru. Π ( ˆ, ˆ ) < 0 f ( ˆ ) ˆ ( ˆ, ˆ < ) s not a Cournot-Nash equlbru. () Lerner nde By assung an nteror soluton we are gong to transfor condton () to obtan the Lerner nde of arket power. p p C ( + ) + ( + ) ( ) = 0 p ( ) p ( )[ + ] C( ) = 0 p ( ) p ( ) ( )[ + ] C( ) = 0 p p( ) ε ( ) Defnng fr s share as s = we get: s ε ( ) ( )[ ] C( ) = 0 p Then the Lerner nde of arket power for fr s p( ) C( ) s = p( ) ε ( )

13 Mcroeconoc Theory IV Olgopoly Then the Cournot odel s located between the case of pure onopoly ( s = ) and perfect p C copetton ( l = 0 ). s 0 p (v) Specal cases. Constant argnal costs a) Constant argnal cost: c > 0, =,.., n. At equlbru the frst order condton for each fr (nteror soluton) ust be satsfed: p p c n ( + ) + ( + ) = 0 =,..,. By addng up the n frst order condtons: That s n np( ) + p ( ) c = 0 = = n n np( ) + p ( ) = c = Then the total output n Cournot-Nash equlbru depends eclusvely on the su of the argnal costs, not on ther dstrbuton across frs (n an nteror soluton wth all n frs producng postve quanttes). b) Coon constant argnal cost: c = c> 0, =,.., n. The Lerner nde s: s p ( ) c = p( ) ε ( ) 3

14 Mcroeconoc Theory IV Olgopoly Takng nto account that f the product s hoogeneous and the argnal cost s the sae for all frs then the Cournot-Nash equlbru should be syetrc: s = = =, =,.., n. n n If the prce-elastcty s constant then: p ( ) c = p( ) n ε Therefore, when the nuber of frs ncreases the relatve prce-argnal cost argn (the Lerner nde) decreases and at the lt when n, p c. 4

15 Mcroeconoc Theory IV Olgopoly..3. Welfare analyss We consder the splest case where argnal cost s constant and coon to all frs. ( + ) + ( + ) = 0 =,..,. p p c n By addng the n frst order condtons: ( ) + ( ) = 0 np p nc We follow a slar approach to that n the chapter on onopoles to copare the Cournot output wth the effcent output. (Revew the obtanng of the welfare functon and the proble of azng socal welfare) a W( ) a u( ) C( ) 0 0 W (0) = u (0) C (0) > 0 p(0) > C (0) ( ) = ( ) ( ) = 0 ( ) = 0 Frst order condton. e W u C W ( ) = ( ) ( ) < 0 Strctly concave welfare functon. W u C e W ( ) = 0 W ( )? W ( ) < 0 u ( ) W ( ) = u ( ) C ( ) = p( ) > 0 p( ) n < 0 By defnton of Cournot output. e W ( ) = 0 W ( ) > 0 W ( ) < W ( ) > W ( ) < 0 e e dw ( ) ( ) 0 0 ( ) d W < < W 5

16 Mcroeconoc Theory IV Olgopoly W e W ( ) = 0 W ( ) > 0 e 6

17 Mcroeconoc Theory IV Olgopoly.. The Bertrand odel... Hoogeneous product () Contet. () Resdual deand. () Representaton of the gae n noral for. Noton of equlbru. (v) The Bertrand Parado. Characterzaton of equlbru and unqueness. () Contet The Bertrand odel s characterzed by the followng eleents ) We consder a arket wth frs. ) Frs sell a hoogeneous product. 3) Prce copetton. 4) Sultaneous choce. Each fr has to choose ts prce wth no knowledge of the rval s decson. Agan, sultaneous choce does not ean that choces are ade at the sae nstant n te; the relevant pont s that although one fr ay play frst the other does not observe ts decson. 5) Constant argnal cost and coon: c = c = c> 0. () Resdual deand Frs produce a hoogeneous product and copete on prce. Then fro the consuers pont of vew the only relevant pont s the relatonshp between the prces of the two frs; consuers buy the product fro the fr that sets the lower prce. That s, f one fr charges a lower prce than the other, t captures the entre arket and the second fr sells nothng. If both frs gve the sae prce then consuers are ndfferent between buyng 7

18 Mcroeconoc Theory IV Olgopoly fro one or the other. For the sake of splcty we assue that n the case of equal prces each fr sells to the half of the arket. Fr s resdual deand,, =,,, s: D( p ) p < p D( p, p ) = D( p ) p = p 0 p > p p D ( p, p ) p D( p) D ( p ) () Representaton of the gae n noral for. Noton of equlbru The gae n noral for s: ) =,. (Players) ) p 0. Any non negatve prce serves as a strategy for player. Equvalently, we can represent the player s strategy space as p [0, ), =,. 3) The frs profts correspondng to the strategy profle ( p, p ) are: ( p, p ) ( p c) D ( p, p ) Π = Π = = Π ( p, p) = ( p c) D( p, p) ( p, p ) ( p c) D( p, p ),,,,, where the resdual deand for fr,, =,,, s: D( p) p < p D( p, p) = D( p) p = p. 0 p > p 8

19 Mcroeconoc Theory IV Olgopoly In the Bertrand gae we say that ( p, p ) s a Bertrand-Nash equlbru f: Π p p Π p p p =. (, ) (, ) 0,,,, To splfy the analyss we use ths defnton eclusvely because the fact that the resdual deand of each fr s a dscontnuos functon of ts own prce eans that we cannot use standard optzaton technques (n fact nstead of havng best response functons we would have best response correspondences and the analyss would be ore cople). (v) The Bertrand Parado. Characterzaton of the equlbru and unqueness We deonstrate here that the unque Bertrand-Nash equlbru s: p = p = c Ths result s known as the Bertrand Parado: Two frs copetng on prces suffce to obtan a copettve outcoe Deonstraton We deonstrate that the strategy profle p = p = c: a) s a Nash equlbru. b) s the unque Nash equlbru. a) The proft of each fr under strategy profle (,) cc s: If fr unlaterally devates by chargng a prce p t wll ake no sales. By reducng ts prce below argnal cost p entre arket but ncur losses. Therefore, Π (,) cc Π ( p,) c p 0,, =,, Π (,) cc = ( c c) Dc (), =,. > c then ts proft wll be zero because < c t would capture the 9

20 Mcroeconoc Theory IV Olgopoly b) We deonstrate that no other cobnaton of strateges can be a Nash equlbru.the graph below shows the dfferent types of strategy profle. We check whether a strategy profle s an equlbru or not by calculatng the proft of each player correspondng to ths cobnaton of strateges and we wonder f any of the players has an ncentve to unlaterally devate. To elnate a strategy profle as an equlbru t suffces to show that at least one player can prove by devatng unlaterally. p p p > p p p < p = p c c p p ) Equal prces: p = p a) p = p > c NE? NO. In a strategy profle lke ths the proft of each fr would be: Π ( p, p) = ( p c) D( p, p) = ( p c) D( p). Fr =, would have an ncentve to devate unlaterally. For eaple, we can choose a prce p = p ε (where ε s an arbtrary postve aount as sall as requred): ( p c) D( p ) ( p c) D( p, p ) ( p, p ) ( p, p ) ( p c) D( p, p ) ( p c) D( p ). In fact there would be nfnte devatons such that fr proves wth a unlateral = =Π >Π = = devaton. 0

21 Mcroeconoc Theory IV Olgopoly b) p = p < c NE? NO. Fr s proft n such a strategy profle would be: Π ( p, p) = ( p c) D( p, p) = ( p c) D( p) < 0. < 0 Fr would have an ncentve to devate unlaterally. For eaple, any prce p > p : = p c D p p =Π p p >Π p p = p c D p p = p c D p 0 ( ) (, ) (, ) (, ) ( ) (, ) ( ) ( ). = 0 ) Dfferent prces: p p c) p > p > c NE? NO. Fr s proft n such strategy profle would be zero: Π ( p, p ) = ( p c) D( p, p ) = 0 and the proft of the other fr, fr, would be Π ( p, p ) = ( p c) D ( p, p ) = ( p c) D( p ) > 0. For fr any unlateral devaton p such that c< p p ncreases profts: ( p c) D( p ) ( ) (, ) (, ) (, ) ( ) (, ) ( )0 0. = p c D p p =Π p p >Π p p = p c D p p = p c = s p< p Although we have already proved that the strategy profle ( p, p ), wth p > p > c cannot be an equlbru we can also show that n any cases fr would also have an ncentve to devate unlaterally. (For eaple, f p p p c > > any unlateral devaton p > p > p ncreases the profts of fr. For the cases p p p c > > > and p p p c > > > t s also straghtforward to fnd devatons such the profts of fr ncrease. The only stuaton where fr would have no ncentve to devate would be one such that p p p c > = > ). d) Other cases: - p > c p NE? NO. Fr would have no ncentve to devate unlaterally whle for fr when p > c> p any p > p ncreases profts and when p > c= p fr

22 Mcroeconoc Theory IV Olgopoly ncreases profts by convenently ncreasng the prce. For eaple, f p p c p > = any prce p > p > c ncreases the profts of fr. When p p c p > > = any prce p p c > > (and others) ncreases the profts of fr. - c p > p NE? NO. Fr would have no ncentve to devate whle for fr any prce p > p ncreases profts.... Heterogeneous products (dfferentated products) () Heterogeneous product. Resdual deand. () Representaton of the gae n noral for. () Noton of equlbru. Best response functon. Bertrand-Nash equlbru. () Heterogeneous product.resdual deand We antan the rest of the assuptons of the Bertrand odel (two frs, sultaneous choce, constant and coon argnal cost, prce copetton) but now we consder that the two frs sell heterogeneous products. That s, frs sell products that are close but perfect substtutes. The deand for the product of fr, the resdual deand, s gven by D( p, p ). Assue D D that < 0, > 0 p p D D and > p p ; that s, the deand for product s a decreasng functon of ts own prce, products are substtutes and the own effect s larger than the cross effect.

23 Mcroeconoc Theory IV Olgopoly ()Representaton of the gae n noral for. Noton of equlbru The gae n noral for s: ) =,. (Players) ) p 0 or equvalently p [0, ), =,. 3) The proft of each fr correspondng to ( p, p) s: p p p c D p p Π (, ) = ( ) (, ) Π (, ) = ( ) (, ),, =,, Π ( p, p) = ( p c) D( p, p) p p p c D p p Now the resdual deand of each fr s a contnous functon of ts prce. (v) Noton of equlbru. Best response functon. Bertrand-Nash equlbru In ters of best responses the defnton of the Bertrand-Nash equlbru s: ( p, p ) s a Bertrand-Nash equlbru f p = g ( p ),, =,,., where g( p) s fr s best response to the prce p of ts rval. The best response of fr conssts of choosng p such that: a Π ( p, p ) ( p c) D( p, p ) p 0 Π p D = D( p, p) + ( p c) = 0 () g( p) p Π D D = + ( ) < 0. p c p p p 3

24 Mcroeconoc Theory IV Olgopoly.3. Leadershp n the choce of output. The Stackelberg odel () Contet. () Two-stage gae. Perfect nforaton. Noton of strategy. () Backward nducton. Subgae perfect equlbru. (v) Eaple: lnear deand and constant argnal cost. (v) Other Nash equlbra whch are not subgae perfect. () Contet The Stackelberg duopoly odel has four basc characterstcs: a) We consder a arket wth frs. b) Hoogeneous product. That s, fro the consuers pont of vew the two frs produce products whch are perfect substtutes. c) Quantty copetton. Let and be the outputs of fr and, respectvely. d) Sequental choce. One of the frs (the leader), fr, chooses ts output level frst. Net the other fr (the follower), fr, chooses ts output after observng the output chosen by fr. Fro a gae theory perspectve ths s a perfect nforaton gae. () A two-stage gae. Perfect nforaton. Noton of strategy Frs play a two-stage gae: Stage : fr chooses ts output 0. Stage : fr chooses ts output 0 after observng the output chosen by fr. Gven that both players ust have the sae percepton of the gae not only does player observes the choce ade by player but also player knows that player observes ts choce. That s, there s perfect nforaton and both players have the sae percepton of the gae. 4

25 Mcroeconoc Theory IV Olgopoly (Note: a gae wth two stages,.e. a sequental gae, n whch the second over does not observe the output chosen by the frst over.e. an perfect nforaton gae would be equvalent to a sultaneous gae, such as the Cournot gae). The strategy spaces of the players are as follows: - 0: any non-negatve quantty serves as a strategy for player ; equvalently [0, ). - The descrpton of the strateges for player s ore cople. Recall that a strategy s a coplete descrpton of what a player would do f he/she were called on to play at each one of hs/her decson nodes, ndependently of whether they are attanable or not gven the current behavor of the other(s) player(s). In the Stackelberg gae, each possble output of fr generates a dfferent decson node for fr. Therefore, fr s strategy s a functon ( ) whch tells us how uch fr s gong to produce for each possble producton of fr. 5

26 Mcroeconoc Theory IV Olgopoly () Backward nducton. Subgae perfect equlbru Although the gae sees too cople to be solved, we know that n perfect nforaton gaes wthout tes the backward nducton crteron proposes a unque soluton whch concdes wth the unque subgae perfect equlbru. The proceedure s slar to that used wth fnte gaes n the prevous chapter. We start fro the last subgaes, that s, at stage. Stage We elnate the non credble threats or donated actons n each subgae. Gven an output of fr (a subgae) the only credble threat s for fr to choose a proft azng output level: a Π (, ) p( + ) C ( ) 0 Π = p + + p + C = f ( ) ( ) ( ) 0 () ( ) Π = p + + p + C < ( ) ( ) ( ) 0 Takng nto account the non-negatvty constrant, 0, we have: { } f ( ) = a f ( ),0 Fr s strategy n the subgae perfect equlbru. In fnte gaes, the proceedure contnues by elnatng all the non credble threats and coputng the reduced gae. In the Stackelberg gae elnatng all the ncredble threats s equvalent to elnatng player s strateges other than f = { f } ( ) a ( ),0. Stage Player antcpates that fr wll behave at each subgae accordng to the strategy { } f ( ) = a f ( ),0. Fr s proft functon n reduced for s: 6

27 Mcroeconoc Theory IV Olgopoly Π (, f ( )) p( + f ( )) C ( ). Therefore, the proble for fr becoes: a Π(, f( )) p( + f( )) C( ). 0 d d Π = L p( + ) + [ + f( )] p( + ) C( ) = 0 () d Π < 0 d Therefore, the subgae perfect equlbru s the strategy profle (, f ( )). L (v) Eaple: lnear deand and constant argnal cost. Stage We elnate the non credble threats or donated actons n each subgae. Gven an output of fr (a subgae) the only credble threat s for fr to choose a proft azng output level: a Π (, ) p( + ) C ( ) [ a b( + )] c 0 Π a c b = p ( + ) + p( + ) C( ) = 0 () f( ) = b Π = p + + p + C = b < ( ) ( ) ( ) 0 Takng nto account the non-negatvty constrant, 0, we get: a c b f( ) = a { f( ),0} = a,0 Fr s strategy n the SPE. b 7

28 Mcroeconoc Theory IV Olgopoly Stage Player antcpates that fr wll behave at each subgae accordng to the strategy a c b f( ) = a { f( ),0} = a,0. b Fr s proft functon n reduced for s: Π (, f ( )) p( + f ( )) C ( ). Then fr s proble s: a c b a c b a Π (, f ( )) [ a c b( + f ( ))] [ a c b( + )] [ ] 0 b dπ L a c = p ( + ) + [ + f( )] p( + ) C( ) = a c b = 0 () = d b d Π < 0 d L Therefore, the subgae perfect equlbru s (, f( )). e 45º s L SPE: (, f( )) f( ) L e In order to obtan the profts of the frs we have to play the gae. 8

29 Mcroeconoc Theory IV Olgopoly a c L a c b s L a c b b a c = f( ) = = = b b 4b s L s a c a c 3( a c) = + = + = b 4b 4b 3( ) 3 ( ) a c a + p s = p c s = a b s = a b = ; 4b 4 a p c s c= 4 L s L ( a c) ( a c) ( a c) s s s ( a c) ( a c) ( a c) Π = ( p c) = = = ; Π = ( p c) = =. 4 b 8b 4 4b 6b (v) Other Nash equlbra whch are not subgae perfect e 45º { } ( ) = EN: (, ( )) NO SPE s f( ) L SPE: (, f( )) L e 9

30 Mcroeconoc Theory IV Olgopoly.4. Colluson and stablty of agreeents.4.. Short-ter colluson () Cournot odel. The colluson agreeent s not a short-ter equlbru. () Bertrand odel. The colluson agreeent s not a short-ter equlbru. () Cournot odel. The colluson agreeent s not a short-ter equlbru If frs colluded they would be nterested n azng aggregate profts. a Π (, ) +Π (, ) p ( + ) C( ) + p ( + ) C( ), Π = p ( + ) + ( + ) p( + ) C( ) = 0 () MRI = C = C Π = p ( + ) + ( + ) p( + ) C( ) = 0 () When argnal costs are constant and equal across frs condtons () and () are dentcal. The two-equaton syste would have nfnte solutons: any par of outputs such that + = would aze the ndustry proft. For these cases we also refer to the syetrc colluson agreeent where each fr produces a half of the onopoly output: =, =,. We now show that the colluson agreeent (plct of course) cannot be supported as an equlbru when the gae s played once. In other words, we deonstrate that the strategy profle (, ) s not a Nash equlbru n the Cournot gae. Gven the strategy 0 the best response of fr s to choose a strategy such that: a Π (, ) p( + ) C ( ) 0 Π = p( + ) + p( + ) C( ) = 0 () f( ) Π = p + + p + C < ( ) ( ) ( ) 0 30

31 Mcroeconoc Theory IV Olgopoly So the best response functon s: f = { f } ( ) a ( ),0. To check that the cobnaton of strateges (, ) s not a Nash equlbru we calculate the argnal proft for each fr: Π (, ) = p ( + ) + p( + ) C( ) = p( + )>0 < 0 By defnton of colluson agreeent. Then startng fro the colluson agreeent an ncrease n output leads to an ncrease n the fr s proft and, therefore, fr would have an ncentve to break the colluson agreeent. Put dfferently, gven the defnton of best response functon Π( f( ), ) = 0 and as Π(, ) >0 then f ( ) >. As we know what the optal devaton for fr s f t decdes to break the colluson agreeent, f( ), we denote by Π the proft that would obtan f t devates optally and the other frs do not devate. That s, Π =Π ( f ( ), ). 3

32 Mcroeconoc Theory IV Olgopoly Graphc analyss: lnear deand and constant argnal cost e f( ) 45º C-N Colluson lne: + = Syetrc colluson agreeent: =, =, f( ) e Olgopoly It s easy to generalze the above result to the case of n frs. The condton whch defnes the colluson agreeent (the strategy profle azng the aggregate proft) s: p p C n ( + ) + ( + ) ( + ) ( ) = 0 =,..,. To show that the strategy profle (,.., ) s not a Nash equlbru we obtan the argnal proft of each fr: n Π(, ) = p + + p + C = p + ( ) ( ) ( ) ( )>0 < 0 By defnton of the colluson agreeent. 3

33 Mcroeconoc Theory IV Olgopoly Then startng fro the colluson agreeent an ncrease n the output of fr also ncreases ts proft and, therefore, fr I would have an ncentve to break the colluson agreeent. In other words, gven the defnton of best response functon Π( f( ), ) = 0 and as Π(, ) >0 then f ( ). > As we know what the optal devaton for fr s f t decdes to break the colluson agreeent, f ( ), we denote by Π the proft that would obtan f t devates optally and the other frs do not devate. That s, Π =Π ( f ( ), ). 33

34 Mcroeconoc Theory IV Olgopoly () Bertrand odel. The colluson agreeent s not a short-ter equlbru Consder the Bertrand odel wth hoogeneous product and constant (and coon) argnal cost. The strategy profle whch represents the syetrc colluson agreeent s ( p, p ). The proft of each fr s: Π =Π ( p, p ) = ( p c) D( p ) = Π We know (t has been deonstrated) that a cobnaton of strateges of the type p = p > c s not a Nash equlbru. Any fr would have an ncentve to devate unlaterally. For eaple, we can choose p = p ε (where ε s an arbtrary postve aount as sall as necessary). Of course, there are nfnte devatons such that fr s better off. Fndng the optal devaton for fr s ore probleatcal. The best that t can do s to undercut ts rval s prce by the lowest aount possble, ε > 0, ε 0. Although we do not have that optal devaton well-defned we can be as near as we wsh to onopoly prce. Let Π be fr s proft when t optally breaks the colluson agreeent and the rval keeps t. That s, Π =Π( p ε, p ) = ( p ε c) D( p ε) ( p c) D( p ) =Π ε 0 34

35 Mcroeconoc Theory IV Olgopoly.4.. Stablty of agreeents. Fnte teporal horzon and nfnte teporal horzon We have shown that n the short ter the colluson (cooperaton) agreeent cannot hold as an equlbru n ether the Cournot gae or the Bertrand gae. In ths secton, we study the possbltes of colluson or cooperaton when the gae s repeated. () Fnte teporal horzon Backward nducton arguent colluson (cooperaton) cannot be supported as an equlbru (at each stage frs behave as n the one-shot gae). The reasonng s slar to that n the Prsoner s Dlea. () Infnte teporal horzon There are two ways of nterpretng an nfnte teporal horzon: () Lteral nterpretaton: the gae s repeated an nfnte nuber of tes. In ths contet, to copare two alternatve strateges a player ust copare the dscounted present value of the respectve gans. Let δ be the dscount factor, 0 < δ <, and let r be the dscount rate (0 < r < ) where δ = + r. () Inforatonal nterpretaton: the gae s repeated a fnte but unkonwn nuber of tes. At each stage, there s a probablty 0 < δ < of the gae contnung. In ths settng, each player ust copare the epected value (whch ght be also dscounted) of the dfferent strateges. We shall see that the estence of plct punshent threats ay serve to antan colluson as an equlbru of the repeated gae. 35

36 Mcroeconoc Theory IV Olgopoly Note frst that there s a subgae perfect equlbru of the nfntely repeated gae where each player plays hs/her short-ter Nash equlbru strategy n each perod. In the Cournot odel such a strategy would consst of producng n each perod the Cournot quantty ndependently of past hstory. In the Bertrand odel that strategy would consst of chargng a prce equals to argnal cost ndependently of past hstory. We net study the possblty that there ay be another subgae perfect equlbru where players cooperate wth each other. Consder the followng cobnaton of long ter strateges: s c {s t (H t )} t=, =,. where, to collude c s ( "cooperate" f all eleents of ) are equal to ("cooperate","cooperate") or t H H t t t = = "not cooperate"(the short-ter NE strategy) otherwse (n Cournot: f all eleents of H are equal to (, ) or t = = ) otherwse c t t ( Ht ) s (n Bertrand: c p f all eleents of Ht are equal to ( p, p ) or t = st ( Ht ) = ) c otherwse Note that these long ter strateges ncorporate plct punshent threats n case of breach of the (plct) cooperaton agreeent. The threat s credble because confess n each perod (ndependently of the past hstory) s a Nash equlbru of the repeated gae. 36

37 Mcroeconoc Theory IV Olgopoly To check whether t s possble n ths contet to antan cooperaton as an equlbru, we ust check that players have no ncentve to devate; that s, we ust check that the cobnaton of strateges (s c,s c ) consttutes a Nash equlbru of the repeated gae. Notaton Π Fr s proft under colluson at each stage of the gae. Π Fr s proft n the short-ter Nash equlbru at each stage of the gae. Π Fr s proft f the other frs cooperate and t optally devates. Π >Π >Π The dscounted present value for fr n the strategy profle (s c,s c ) s gven by: c c Π π( s, s) =Π + δπ + δ Π +... =Π ( + δ + δ +...) = δ If fr devates n the frst perod ts gans are: c Π π( s, s) =Π + δπ + δ Π +... =Π + δ( + δ + δ +...) Π =Π + δ δ Cooperaton s supported as a Nash equlbru f no player has any ncentve to devate; that s, when π (s c,s c ) π (s, s c ). It s straghtforward to check that f δ δ no fr has an ncentve to break the colluson agreeent, where δ Π Π =. Π Π 37

38 Mcroeconoc Theory IV Olgopoly Basc Bblography Varan, H. R., 99, Mcroeconoc Analyss, 3th edton, Norton. Chapter 4, sectons: 6., 6.3, 6.4, 6.5, 6.6, 6.0 and 6.. Copleentary Bblography Kreps, D. M., 994, A course n croeconoc theory, Harvester Wheatsheaf. Trole, J., 990, The Theory of Industral Organzaton, MIT Press. Varan, H. R., 998, Interedate Mcroeconocs: A Modern Approach, Norton. 38