Advanced Microeconomics (ES30025)

Transcription

1 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () - Colluson Outlne:. Introducton. Isoproft Contours 3. Colluson 4. Olgopoly as a Repeated Gae 5. Trgger Strateges Appendx A: Appendx B: Cournot / Bertrand Trgger Strateges wth Many frs Present Value of an Infnte Strea. Introducton Recall that (total) output s lower under onopoly than under Cournot-Nash copetton, whch s tself lower than under perfect copetton: q θ < qn 3 θ < qc θ () q q q q q q n 0 q q R R q Fgure Gven our deand functon, p α βq, t ust be the case then that:

2 Advanced Mcroeconocs (ES3005) p α β θ α + c p n α β 3 θ α + c 3 p c α β ( θ) c 6c 6 3α + 3c 6 α + 4c 6 Topc Two: Olgopoly () Colluson () Thus p > p n > p c snce α > c. Output s therefore lower (and so prces hgher) n the Cournot soluton relatve to the copettve stuaton. Thus frs earn postve profts. But output s stll hgher than n the onopoly stuaton, whch ples that both prces and profts are lower than n the onopoly case, when frs act so as to axse jont profts. Indeed, t can be shown that Nash-Cournot proft les n between onopoly and copettve proft: π ( p c)q ( ) π α c α c β 3α + 3c 6 c α c β 4 β α c β 4 βθ (3) And: π n ( p n c)q n ( ) 3 π n 3 α c α + 4c 6 α c β c 3 α c β 9 β α c β 9 βθ (4) And: π c ( p c c)q c 0 (5) Such that: π 4 βθ > π n 9 βθ > π c 0 (6)

3 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson. Isoproft Contours We can see the Pareto neffcency of the Cournot-Nash Soluton ore clearly by dervng the soproft contours of each fr. Consder fr. It s apparent that fr s proft ncreases (decreases) as we ove down (up) fr s reacton functon θ q q * R α c q β π Decreasng π Increasng R 0 θ q Consder fr s proft functon: ( ) α β ( q + q ) π q,q Fgure q cq (7) Totally dfferentatng wth respect to q and q and settng to zero ples: ( ) ( α c) βq βq dπ q,q Whch ples: dq βq dq 0 (8 dq dq dπ 0 ( α c ) βq βq βq (9 Ths s the equaton of fr s soproft contour. Provdng ths equaton s satsfed then the level of fr s proft reans constant. Now, recall fr s reacton functon: 3

4 Advanced Mcroeconocs (ES3005) ( ) q * α c β q Topc Two: Olgopoly () Colluson (0) Thus along fr s reacton functon t ust be the case that: βq * ( α c) βq () Thus dq dq dπ 0 0 on fr s reacton functon. Thus, t ust be the case that dq dq > 0 and dq dπ dq 0 dπ < 0 to the left and rght of the reacton functon 0 respectvely (.e. as q s decreased (ncreased) below (above) q * respectvely). θ q q * R α c q β ˆq π < π 0 π 0 * q θ q Fgure 3 Intutvely, q * s fr s best response to ˆq. Thus pont E B ust denote the axu proft avalable to fr gven ˆq, and ust thereby denote a pont on fr s reacton functon. Snce any oveent away fro pont E B ust represent a lower level of proft for fr, there ust be two ponts ether sde of E B, say E A and E C, at whch fr s proft s dentcal such that the two ponts le on the sae soproft contour. Gven that fr s proft falls as we ove up fr s reacton functon, the soproft contour connectng E A and E C cannot be horzontal or u-shaped - by default they ust be n-shaped. A slar arguent can be ade n ters of fr - see Fgures

5 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson θ q q * R α c q β π Increasng π Decreasng R 0 θ q Fgure 4 q * θ q R α c q β * q π 0 π < π 0 ˆq θ q Fgure 5 5

6 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson Gven the nature of the two frs soproft contours t apparent that the Nash equlbru pont s not Pareto effcent - the contours ntersect at the Nash pont plyng that at least one of the two frs could be ade better off wthout the other fr beng ade worse off. q θ R θ π θ 3 q n π R 0 θ 3 θ θ q Fgure 6 Indeed, any pont along CC n Fgure 7 followng would render at least one of the fr s better off. CC s denoted the contact curve and defnes the ponts along whch the two frs soproft contours are tangent. Ponts along the contract curve are Pareto effcent - once on the curve one fr can only be ade better off at the expense of the other. It can be shown that the contract curve corresponds to the levels of onopoly outputθ. The queston s, then, why do frs not collude to restrct output to ths level and thereby axse jont profts. Assung, for exaple, each frs akes half the total level of onopoly output, they would be able to each enjoy: π q (,q ) 8 βθ > π n q n n (,q ) 9 βθ c > π q c c (,q ) 0 () The queston s, why are frs so unwllng, or unable, to collude and thereby axse jont profts? 6

7 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson q θ θ R π π θ 3 q n c π c π 0 θ 3 θ R θ q Fgure 7 3. Colluson The answer les n the fact that colluson s not a Nash equlbru wthn a one shot gae. The two frs wll only ever agree to the onopoly equlbru wthn such a gae f they can ake a bndng agreeent to keep to the agreed outputs. Otherwse the attept wll fal because q q,q ( ) s not a Nash Equlbru - gven that the other fr adheres to the agreeent, t s always n the other fr s to renege n a one-shot gae. Assue, for exaple, each fr has agreed to produce ffty percent of the total onopoly output. Assung fr beleved that fr was gong to stck to ths agreeent, fr s optal level of output s derved fro ts reacton functon. ( ) θ θ 4 q * θ q 3 8 θ q r > q 8 θ (3) r where q denotes fr s optal renege level of output. Essentally, fr ncreases ts output because p > MC at the onopoly-quota level of output. The prce level under renegton s: p r α β q r + q ( ) α β θ α 5 8 β α c β 3α + 5c 8 (4) And proft under renegton s equal to: 7

8 Advanced Mcroeconocs (ES3005) π r ( p r c)q r ( ) 3 8 π r 3 8 α c 3α + 5c 8 α c β 3 c 8 α c β 9 64 β α c β 9 64 βθ Topc Two: Olgopoly () Colluson (5) Such that: π r q r (,q ) 9 64 βθ > π q,q ( ) 8 64 βθ (6) The renege level of output s llustrated n Fgure 8 followng. But fr s no dfferent to fr. Fr wll face the sae teptatons as fr and would be reluctant to supply the onopoly level of output snce hs profts n ths case would be gven by: π q r (,q ) 6 64 βθ < π n q n n,q ( ) 9 βθ < π q (,q ) 8 βθ (7) Proof: π q r (,q ) α β θ 4 θ c α c ( ) 4 θ 5 8 βθ 8 βθ 0 64 βθ 6 64 βθ (8) q R π π R q n q C q C q r π π r π π 0 q q r q Fgure 8 8

9 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson 4. Olgopoly as a Repeated Gae We retan the basc odel set out n Secton II but now allow frs to choose prces or outputs n a sequence of te perods. The gae played n each te perod n defned as the consttuent gae and t s coon knowledge to the frs that they are engaged n a sequence of repettons of ths gae. They therefore forulate strateges for the repeated gae, not just for the one-perod consttuent gae n solaton fro any other perod. In such a context t s possble to ratonalse collusve behavour n the absence of bndng (.e. legally enforceable) agreeents. If a frs reneges on soe prevously agreed (explct or plct) colluson, then the other fr can punsh t by nstgatng, for exaple, a prce war (.e. output expanson) or carryng out soe other retalatory acton n the next perod. The threat of an antcpated future punshent can ake t ratonal for each fr to adhere to the agreeent such that agreeents between the can be sustaned by self-nterest. Such agreeents are defned as self-enforcng. Despte the apparent appeal of such an dea soe cauton s approprate. Frst, any gans fro devaton wll be realsed now whlst any losses fro punshent wll occur n the future. Thus, wll t always be the case that suffcently large future losses can be threatened to offset the gans fro edate devaton? Ths wll depend upon such factors as the echanss by whch punshent s nflcted, the rate at whch frs dscount future profts and the length of te for whch the devant can gan fro breakng the agreeent before punshent begns. Secondly, any punshent has to be credble, and gven that nflctng punshent s often harful to the nflctor (e.g. prce war), ths s not trval. The threat of punshent wll only be an effectve deterrent f potental devants beleve that t wll be carred out. Before we analyse these ssues n detal we ust frst address the queston as to whether the consttuent gae s repeated a fnte or nfnte nuber of te. If there s a known last perod of the gae, then the process of backward nducton shows that the above ntutve arguent for colluson ay break down. The equlbru repeated gae strategy ay then sply consst of repeated plays of the one-shot Nash-equlbru. Consder the Cournot odel. Colluson n the last perod s not a Nash-equlbru gven the gans fro devaton and the possblty of future punshent. The only Nash-equlbru n the last perod s thus the (unque) Cournot equlbru outputs. There s no next perod n whch to punsh devaton and colluson s not a Nash-equlbru of ths one-shot gae. It s therefore possble to support any colluson n the penultate perod wth credble threats of punshent n the last perod. Thus, for the sub-gae of the repeated gae consstng of the last-perod one-shot gae, the only Nash-equlbru s the one-shot Cournot equlbru. In the penultate perod both frs realse that ths s the case and so the threat of punshent by settng non Nash-equlbru Cournot outputs s not credble and cannot sustan colluson n the penultate perod. Thus Cournot outputs are the only Nash-equlbru n the penultate perod. The arguent extends by backward nducton to the frst perod such that the only credble Nash-equlbru of the fntely repeated gae has the frs choosng the Nash equlbru of the one-shot gae n every perod. If the gae s repeated forever, however, there s no last perod n whch to start the backward nducton process and so the repeated gae wll look exactly the sae fro whatever pont n te t s consdered. In such a stuaton colluson can be ratonalsed as a Nash equlbru of the repeated gae. Note that even f the gae s fnte but that there s soe probablty of there beng a next perod then colluson s possble. 9

10 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson In what follows we wll assue an nfnte gae such that t 0,,..., wth all frs facng the sae nterest rate r > 0 and all wshng to axse the present value of proft: V δ t π t (9) t0 where δ ( + r) s the dscount factor of fr and π t s the proft of fr n perod t. In what follows we wll focus on colluson over output and wll assue that frs agree to a jont proft axsng allocaton such that there objectve s to: axπ π q,q,π ( ) (0) The jont proft axsng allocaton s an appealng assupton snce t yelds the axu gan fro colluson. It s not, however, the only possble collusve outcoe. Indeed, t s not dffcult to construct odels n whch the Cournot-Nash equlbru yelds a hgher proft for one of the frs that t would obtan fro the output t would produce at the jont proft axsng equlbru. Such a fr would not then agree to the forer unless sde-payents (.e. lup-su redstrbutons of proft) between the two frs are feasble. If they are then frs axse ther gans fro colluson by producng q ( q ) n the dfferentated (hoogenous) product case to generate a total proft of π and then akng whatever sde-payent fro one to the other s necessary to acheve agreeent. The relatonshp between ther actual profts π L after sde-payents s: π L π π L () In any countres, however, sde-payents are seen as ndcatve of colluson and are therefore prohbted. Is colluson stll possble n such stuatons? 5. Trgger Strateges Nash-Cournot Reverson One ethod of sustanng any collusve agreeent s through the use of a trgger strategy. Suppose that the two frs agree to produce the onopoly output par q q (,q ), such that π ( q ) > π ( q n ),,. Assue further that f nstead they behaved non-collusvely, then n each perod of the repeated gae they would be at the Cournot-Nash equlbru (.e. they are quantty setters). To sustan ther collusve agreeent they agree on the followng trgger strateges: For any t 0,,,...,, f fr produces q n perod t, then fr j wll produce q j n perod t+. However, f fr reneged by producng q r q n perod t, then j wll produce ts Cournot-Nash equlbru output, q j n, n perod t+ and n every succeedng perod. Ths dscountng forulae assues that profts accrue at the begnnng of each perod. 0

11 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson Thus, a devaton by one fr trggers a peranent swtch by the other to ts Cournot-Nash equlbru output. Suppose then that at t 0, fr s consderng renegng on the agreeent. Snce t expects fr to produce q, ts best renegng output s: q r argaxπ ( q ;q ) θ q ( ) () That s, ts best response to q. Defne π r π q r (,q ). The edate gan to fr s thus π r π > 0, whch s postve snce q s not the best response to q. However, under the ters of the trgger strategy, fr wll then be faced wth q n n every future perod. Its best response to ths s q n, yeldng proft π n π q n n (,q ). Thus relatve to the case n whch t does not renege at t 0, fr wll loose an nfnte proft strea of π π n present value of π π n π r π π n ( π ) r ( ) r, and so t wll not pay fr to renege at t 0 f: ( ) wth a (3) or: ( ) ( ) r n crt r π n π π r π (4) Snce the repeated gae s the sae regardless of the te perod t at whch t begns, f (4) s satsfed at one t t s satsfed at all t, and so the trgger strateges wll support the collusve ( ) forever. Equaton (3) says that fr wll not renege f the edate proft gan fro so outputs q q,q dong s outweghed by the present value of the future losses of proft. Equaton (4) expresses ths condton n ters of an upper bound on the nterest rate. Gven the deand and cost paraeters that deterne the relatonshp between π r,π,π n ( ), colluson wll be sustanable provded the frs do not dscount the future too heavly - thereby weakenng the forces of future punshent. In general, f (4) s satsfed then the trgger strateges sustanng colluson represent a Nash equlbru of the repeated gae. If beleves that j wll play ts trgger strategy, then s best response s to play ts trgger strategy. Thus the equlbru output path wll be ( ) n every perod. The queston then arses - are the threats underlyng these q q,q trgger strateges (.e. of playng the one-shot Nash equlbru forever followng a devaton) credble? Whether we consder Cournot or Bertrand reverson, the trgger strateges sustanng colluson represent a Nash equlbru of the repeated gae. If fr beleves that fr j wll play hs trgger strategy, then fr s best response s to play hs trgger strategy as well. In ters of Cournot reverson, the equlbru output path wll be q,q ( ) n every perod. The

12 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson queston then arses, are the threats underlyng these trgger strateges -.e. of playng the one-shot Nash equlbru forever followng a devaton credble? Colluson supported by trgger strateges ebodyng punshent by Cournot-Nash reverson s a sub-gae perfect equlbru and n that sense the threat of punshent s credble. To llustrate, consder Cournot reverson and assue that fr has observed that fr j has reneged at perod t. fr s trgger strategy prescrbes that t should produce q n n every perod fro t + onwards. Fr j s best response to ths s to produce ts Cournot-Nash output, q j n. But the choce of the output pars q n,q n ( ) n every perod s a Nash equlbru of ths sub-gae (.e. the outputs are utually best responses) and so the punshent strateges satsfy the requreent of sub-gae perfecton. We ay, however, stll doubt the reasonableness of these trgger strateges. Eternal punshent would appear to be excessvely gr and we ght feel that a punshent that better fts the cre s ore approprate. There s also the ssue of the punshent hurtng the punsher snce π n < π. Perhaps ths would encourage fr j to propose that fr forgves and forgets and reverts to colluson? The proble here s that f ex ante such renegotaton of the trgger strateges were antcpated to be successful, then the credblty of the threat of punshent would be underned. 3 Another dffculty s that the punshent by Cournot reverson ay not be partcularly severe f π π n (.e. collusve profts are close to Cournot profts). 4 If ths s the case, then the nuerator of equaton (4) s such that crtcal nterest rate s soewhat n lted. Colluson could be ade ore sustanable f a ore sever punshent than π could be nflcted. We therefore turn to punshent strateges that expand draatcally the possbltes of colluson. The (Abreu) Carrot-and-Stck Approach A sple, but ngenous, dea developed by Abreu (986, 988) perts ore severe punshent than Cournot-Nash reverson but also yelds sub-gae perfect strateges. 5 It also dspenses wth the dea of eternal punshent, replacng t wth the ore appealng dea that colluson would be resued once a short- sharp shock for devaton has been nflcted. Colluson s sustaned by the stck of a proft reducng output expanson to punsh devaton, and by the carrot of subsequent reverson to the collusve outputs, whch lays an portant role of nducng frs to accept the loss of proft requred by the punshent phase. We agan denote the output and proft pars that the two frs chose as ther collusve allocaton as q,q are: ( ) and π (,π ) respectvely. The trgger-strateges defned by Abreu For any t 0,,,...,, f fr produce q n perod t, then fr j wll produce q j n perod t+. 3 Ths possblty has led to the proposal of an alternatve crteron of credblty of punshent strateges vs. renegotaton-proofness. See Farell, J. and E. Maskn. (989). Renegotaton n Repeated Gaes. Gaes and Econoc Behavour,, Recall that n our lnear deand exaple we have π βθ 8 and π n βθ 9 such that Δπ π π n βθ 7. 5 See: Abreu, D. J. (986). Extreal Equlbra of Olgopolstc Supergaes. Journal of Econoc Theory, 39 pp. 9 5; and Abreu, D. J. (988). On the Theory of Infntely Repeated Gaes wth Dscountng, Econoetrca, 56, pp

13 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson However, f fr reneges by producng q r q n perod t, then frs and j produce ther punshent outputs q p p (,q j ) n perod t+. These punshents outputs n general depend upon q (,q j ) and on whch fr has devated at t. Gven, the syetry of our odel, we can however restrct our attenton to punshent outputs that do not depend upon whch fr has devated at t. If the frs produce the punshent outputs at t + then they revert back to q (,q ) at t + and contnue wth these collusve outputs unless one of the frs devates agan. If fr devates fro ts punshent output at t + then the punshent outputs q p p (,q j ) are agan to be produced at t +, and so on. Devaton n the punshent phase (by ether fr) results n the re-poston of punshent, whlst acceptance of punshent (whch s costly to both frs) results n reverson to colluson. ( ) and, as before, consder the Denote punshent proft for fr as π p π q p p,q gans and losses to fr fro renegng on the agreed output allocaton q at t. Its edate proft gan s π r π where agan π r π q r,q j ( ) denotes the renege proft to fr. ( ) Assue that n the followng perod both frs do produce the punshent outputs q p,q j p so that fr earns π p (we justfy ths assupton of non-devaton n the punshent phase below). If renegng at te t s proftable, so wll be renegng at te t +, because the gae s dentcal at very possble startng pont. Thus, f fr reneges at t, t be punshed at t +, wll renege at t +, be punshed at t + 3, and so on. The proft strea fro renegng s thus ( ) whlst that fro not renegng s the constant ( ). Thus, fr wll not renege at te t f: the alternatng nfnte strea π r,π p,π r,π p,... nfnte proft strea π,π,π,π,... π r + δ π p + δ π r + δ 3 π p + δ 4 π r + δ 5 π p +... π + δ π + δ π + δ 3 π + δ 4 π + δ 5 π +... π r π δ π p ( π ) + δ π r π ( ) + δ 3 π p ( π ) + δ 4 π r ( π ) + δ 5 π p π ( )... (38) The left-hand sde of (38) s the edate gan fro renegng and the rght-hand sde s the dscounted value at te t of the dfference n proft streas fro not renegng and renegng. Notce that there s no guarantee that the rght-hand sde s even postve, let alone that t exceeds the left-hand sde, snce the π r π ters are all negatve. Now, fro Appendx C we have: δ t t E δ t δ t D E { 0,,4,6,... } (39) δ δ D,3,5,7,... Thus we can rewrte (38) as: { } (40) 3

14 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson δ 0 t E π r ( π ) + δ π r π δ t δ ( ) +... δ π p ( π ) + δ 3 π p ( π ) +... π r ( π ) δ t π p π t D ( ) π r ( π ) δ π p ( δ π ) (4) whch ples: π r ( π ) δ π p ( π ) (4) such that: π r ( π ) ( + r) π r π ( ) + r π p π ( ) π p ( π ) ( ) ( ) r π p π π r π Snce r > 0, a necessary (but not suffcent) condton for a collusve proft π sustanable (gven that there s no devaton n the punshent phase) s that: π r π π π p (43) to be (44) That s, that there exsts suffcently large outputs q p p to generate suffcently sall proft π that the one-perod gan fro renegng can be offset by a one-perod punshent. Whether ths wll hold depends upon the arket structure the deand and cost functons whch deterne the relatonshps aong these proft values. If condton (43) s satsfed, then t s clearly n the frs nterest to chose π p to so as to equate the rght-hand sde of (43) wth r, snce the larger s π p, the saller s the loss of proft n the punshent phase. On the other hand, for the sallest possble π p feasble, n the arket, condton (43) defnes the hghest nterest rate for whch colluson s sustanable. All ths assues that there s no defecton n the punshent phase. To justfy ths assupton we consder the sub-gae begnnng at t,, 3, when a fr has reneged at p t. Fr,, s to produce q and earn π p. If t does ths and condton (43) s satsfed, so that colluson fro t + onwards s antaned, then t earns the proft strea p consstng of π at t and π fro t + onwards. Ths has a dscounted value at t of 4

15 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson ( ) the proft π p + ( π r). Let q pp denote s best response output to q p j and π pp π q pp p,q j t wll ake f t reneges n the punshent phase n perod t. If t reneges at t, under the strategy descrbed above punshent s re-posed at t +. But f t pays to renege at t, then t wll pay to renege at t + and n every future perod when punshent s posed. Thus, assocated wt the decson to renege n the punshent phase s the nfnte strea of proft consstng of π pp forever. Ths has a dscounted value at t of π pp pp + ( π r). Thus, for fr to keep to the agreeent n the punshent phase and not to renege we requre: π p + π r π pp + π pp r π pp π p π pp π r (45) Thus the one perod gan fro renegng n the punshent phase ust be ore than offset by the present value of loss of proft resultng fro havng the punshent phase contnually reposed rather than restorng colluson. We can express ths n ters of the nterest rate: r π pp π π pp π (46) p If condtons (43) and (46) are sultaneously satsfed, then the collusve proft par q (,q ) s sustanable by the carrot-and-stck trgger strategy so descrbed. Note that the two condtons are utually renforcng and ust hold sultaneously. Condton (43) ensures that t never pays to devate gven that punshent wll be nflcted whlst condton (46) ensures that punshent wll be nflcted (even though t hurts the punsher) gven that n the perod followng punshent colluson wll be renstated and antaned. We can show that the strateges are sub-gae perfect equlbru strateges, so that the threats nherent n the are credble on ths crteron. There are four knds of sub-gae:. The gae tself, begnnng at t 0. In ths, f fr expects fr j to adhere to the specfes strategy, then gven that condtons (43) and (46) are satsfed, fr s best response s also to adhere to the collusve agreeent such that the strateges are a Nash equlbru for the entre gae;. A (proper) sub-gae begnnng at t,,, n whch nobody has reneged at t. Snce ths gae s dentcal to the gae at t 0, the strateges are a Nash equlbru for these sub-gaes; 3. A (proper) sub-gae begnnng at t,,, n whch one fr reneged at t. Gven condtons (45) and (46), fr s best response to q j p at t +,, s tself to produce q p at t and q at t +,. Thus the strateges nduce a Nash equlbru n these sub-gaes; 5

16 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson 4. A (proper) sub-gae begnnng at t,, n whch one fr reneged at t and q p p (,q ) was produced at t. Snce ths gae s dentcal to the gae begnnng at t 0, we agan have a Nash equlbru. Thus, snce the strateges nduce Nash equlbru n all possble sub-gaes, they are subgae perfect equlbra. 6

17 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson Appendx A: Trgger Strateges wth Bertrand Reverson Consder the followng trgger strategy: For any t 0,,,...,, f fr sets π r π p p n perod t, then fr j wll set p j p n perod t+. However, f fr reneges by settng p r p ε n perod t, then j wll set ts Bertrand equlbru prce, p j b c, n perod t+, and n every succeedng perod. Under Bertrand frs are choosng prces, thus we can envsage the strategy as follows: The frs collusve onopoly level of output occurs when both frs set p p such that q q q and π π π. If a fr s ntendng to reengage on ths agreeent, t wll set p p j ε, such that q q and q j 0 such that π π and π j 0. Such behavour wll trgger a peranent reverson to Bertrand prcng by fr j vs p j p j b c, to whch fr s s best response s to adopt Bertrand prcng also vs. p p b c Suppose then that at t 0 fr s consderng renegng on the agreeent. Snce t expects fr to set p p, t can take (alost) the entre arket by settng p p ε, and thus earn (alost) the entre onopoly proft vs. However, under the ters of the trgger strategy, fr wll then be faced wth p b c n every future perod. Its best response to ths s p b c, yeldng proft π b 0. Thus relatve to the case n whch t does not renege at t 0, fr wll loose an nfnte proft strea of π b ( π ) ( π 0) wth a present value ( ) r, and so t wll not pay fr to renege at t 0 f: of π 0 π r π π b π r r π b π π r π ( ) 0 ( ) r crt r π π π b (A) Recall that the crtcal rate of nterest under Nash-Cournot Reverson was 8 9 such b that r crt > r n crt, plyng that colluson s ore easly sustaned under Bertrand reverson that under Nash-Cournot reverson. Intutvely, the punshent under Bertrand duopoly s so uch greater (.e. π b 0 ) and so frs wll be less nclned to renege ceters parbus. 7

18 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson Appendx B: Cournot / Bertrand Trgger Strateges wth Many frs Cournot-Nash N ( ) α β q ax q π q ;q,q 3...q n dπ ( q ;q,q 3...q n ) α c dq q cq N ( ) βq β q 0 (B) (B) q * α c R β n q (B3) Snce the odel s syetrc, each fr wll have an analogous reacton functon such that q q n equlbru. Thus: * q α c R β ( N * )q α c q n + N β (B4) Wth: N α c q n Nq n + N β (B8) N α c p n α β + N β + N α + Nc Such that: ( ) (B9) π n ( p n c)q n π n ( p n c)q n + N α + Nc ( ) c ( ) + N α + Nc c + N + N α c β N + N α c β + N ( ) βθ (B0) 8

19 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson Monopoly q θ q θ N p α β θ α + c π ( p c)q α + c c θ N 4N βθ (B) (B) (B3) Renege q R θ ( N )q θ ( N ) θ N q r θ + N 4 N θ p r α β ( N ) N + θ + N 4 N ( ) + c( 3N ) π r ( p r c)q r α + N 4N ( ) + c( 3N ) α + N 4N c θ 4 + N N ( ) + N 6N βθ (B4) (B5) (B6) n r crt n r crt π n π π r π 4N + N 4N ( ) ( + N ) 4N 6N βθ N + N ( + N ) 6N βθ + N 4N + N ( ) 4N βθ βθ ( ) 4N 6N ( ) ( + N ) 4N 4N + N ( ) ( + N ) 6N 4N 4N ( + N ) (B7) 9

20 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson Bertrand b r crt π b π π r π π N 0 π π N π π N N ( ) N N (B8) Coparson Δr crt Δr crt n ( N ) r crt ( N ) b ( N ) r crt ( N ) 4N ( + N ) N (B9) Note: l N Δr crt l Δr N crt ( N ) < 0 ( N ) 0 (B0) Moreover: n r crt ( N ) b r crt ( N ) 4N ( + N ) N (B) ( ).547 N Thus t s only for the two-fr case that the ntal rate of nterest s hgher under Bertrand crt crt than under Cournot. Once we ove beyond two frs r cournot > r bertrand such that n general, collusve agreeents are relatvely ore stable under quantty than prce copetton - see Fgure B: 0

21 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson Fgure B: Crtcal Rates of Interest

22 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson Appendx C: Present Value of an Infnte Strea All Perods The future value (.e. next year) of $x today s $z $ ( + r)x such that the present value of $z $ ( + r)x next year s $x $ ( + r) z. Thus, agne recevng $x annually fro next year n perpetuty. The present value of ths su s: Z x( δ + δ δ ) n (C) where δ ( + r) and n. Now, δz x( δ + δ δ ) n+ such that: Z δz Z ( δ ) x( δ δ ) n+ Z x δ δ n+ δ (C) Thus: l n δ Z x δ x + r + r x + r + r + r x r x r (C3) Even Perods Now let: Z E δ 0 + δ + δ δ N (C4) Thus: δ Z E δ + δ 4 + δ δ N + (C5) Such that: Z E ( δ ) δ 0 δ N + Z E δ 0 δ N + δ (C6) And: l Z N E δ (C7) Such that:

23 Advanced Mcroeconocs (ES3005) Topc Two: Olgopoly () Colluson δ t t E Odd Perods Now let: E { 0,,4,6,... } (C8) δ Z D δ + δ 3 + δ δ N (C9) Thus: δ Z D δ 3 + δ 5 + δ δ N + (C0) Such that: Z D ( δ ) δ δ N + Z D δ δ N + δ (C) And: l Z N D δ δ (C) Such that: δ t δ t D δ D,3,5,7,... { } (C3) 3