bg 0. 2 Cournot Oligopoly The Cournot Model q i

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1 7 Courot Olgopoly The Courot Model As metoed Chap., Courot s olgopoly model was oe of the frst mathematal models proposed the feld of eooms See Courot (838). It addresses the futog of a market wth umerous atomst demaders versus few relatvely large supplers. Ths mples that all the supplers fluee market pre appreably, ad hee, lke moopolsts, take aout of the demad futo of the osumers o the market order to alulate ther best moves. As a rule, demad s a dereasg futo of pre. I equlbrum demad equals supply, ad oe a also speak of the verse demad futo whh states how market pre depeds o supply. I the ase of Courot, t s most oveet to speak of ths verse demad futo, p = f ( Q), (.) bg 0. where p deotes pre, Q deotes market supply, ad f Q < Courot takes the quatty of supply for eah olgopolst as the proper deso varable, so the pre somehow results through a ot spefed market learg mehasm. Ths was later the goal for rtsm, as a few bg frms would rather set supply pres tha quattes suppled; ad ths would ope up for the possblty of uttg out ompettors ase the ommodty marketed s oeved as homogeous by the osumers. Supposg there are supplers whose dvdual supples are deoted q, market supply beomes, Q= = q =. (.) T. Puu, Olgopoly: Old Eds New Meas, DOI 0.007/ _, Sprger-Verlag Berl Hedelberg 0 7

2 8 Courot Olgopoly There s a pot defg resdual supply, Q = Q q, (.3) the supply of all the other frms, whh s ot uder the otrol of the th frm. The profts, reveue mus ost for the th frm ow beome b g b g, Π = f q + Q q C q (.4) where Cbg q s the ost futo. Dfferetatg partally wth respet to q, results the frst order odto, b g b g b g. f q + Q + f q + Q q = C q (.5) Ulke the ase of moopoly, the optmum odto does ot determe the value of q ; the soluto depeds o Q, the resdual supply by the ompettors about whh the th frm has o erta kowledge. It a oly have more or less sophstated expetatos, based o past experee, ad alulate the best desos uder eah suh expeted Q. The outome s the reato futo, q = φ bg, Q whh redues the optmalty odto to a detty,.e., bg h bg h bg bg h f φ Q + Q + f φ Q + Q φ Q C φ Q. (.6) (.7) To fd suh reato futos φ bg Q e losed form would be desrable for ay modeller. Ufortuately, there are very few demad ad ost futos that allow oe to do so. Next follow a few examples where ths programme a be aomplshed,.e., the tradtoal ase of a lear demad futo, ad the ase of a so-elast, or hyperbola shaped demad futo.

3 The Courot Model 9 Example : Lear Demad Suppose we have the verse demad futo p= a bq, (.8) where a, b are two postve ostats. Further, assume the ost futos are C = q, where are ostat margal, equal to average varable osts. The the proft of the th frm beomes, b gh b g, Π = a b q + Q q q = a bq q bq (.9) ad the maxmum odto, a bq = bq. (.0) Solvg, the reato futo s readly obtaed, q a = b Q. (.) These reato futos are straght les wth the ostat slope. Obvously we must have a >,.e., the maxmum pre obtaable must exeed the ut ost, otherwse the frm ould ot obta ay proft. Further, the reato futo would retur a egatve output q, uless Q a < b (.) holds. As ths s meagless, ay egatve q, would be replaed by zero, whh meas that f the ompettors supply too muh, the the frm drops out.

4 0 Courot Olgopoly To get a omplete pture t s better to expltly hek for oegatvty of profts. Substtutg the reato futo the proft expresso oe obtas b gf HG a bq a b Π = I K J F HG Q b a b Q I K J, (.3) whh equals Π = F a Q 4H G b I b K J. (.4) It may seem that ths expresso s always oegatve, but whe the reato q s egatve, ths s due to the fat that egatve osts outwegh egatve reveues, whh has o fatual meag. Aordgly, osderatos of profts add othg ew to the ostrat for a postve reato already stated, so the omplete reato futo reads R T a Q Q a < q b b = S Q a, 0., b (.5) Note that the resultg reato futo s peewse lear,.e., olear, eve ths smplest ase wth a lear demad futo ad ostat ut osts. I settg up these reato futos, all the steps eeded to be osdered at settg up reato futos further examples were already eoutered. Observe that the o-egatvty ostrat refers to alulated profts based o the ompettors expeted moves. If they tur out wrog, the a dyam proess the frms may stll experee atual egatve profts.

5 The Courot Model Example : Isoelast Demad Lear demad futos are just smple frst approxmatos. If oe assumes dvdual osumers or groups of osumers to have dfferet lear demad futos, the market demad evtably results as a broke tra of le segmets, as proposed by Robso (933). Used olgopoly models t results the terestg pheomea desrbed by Palader (936, 939), but the aalyss beomes more messy tha s sutable for ths smple exemplfato, so we defer the aalyss of ths ase to a later hapter. I stead we ow assume that the osumers maxmze utlty futos of a Cobb Douglas type, a produt of fratoal powers of the quattes of dfferet ommodtes osumed. Wthout loss of geeralty suh utlty futos a be resaled so that the sum of these power fratos equals uty. Maxmzg suh utlty futos uder gve budget ostrats, the result ome beg splt budget shares where the power fratos from the utlty futo provde the weghts. Fxg budget shares of a gve ome meas that quatty demaded ad pre are reproal. See Puu (99, 004). Passg from the dvdual demad futos to the market, as all dvdual futos are of the same form results market demad as well beg reproal to pre, the umerator just beg the sum of all the dvdual budget shares. As the measuremet ut for the ommodty s optoal, oe a hoose t suh a way as to ormalze the umerator to uty, ad thus obta the verse demad futo α α α Assume a utlty futo U = q q... q for a represetatve osumer, where the q for the momet represet quattes of dfferet ommodtes osumed, s the umber of ommodtes, ad α are some ostats, suh that α + α +... α = holds. The osumer, p = (.6) Q maxmzg utlty uder the budget ostrat y = pq + pq +... p q, the hooses pq = α y, whh s the same as q = α y/ p, where we drop the des as oly oe ommodty s of terest. Summg over all osumers Q= α y/ p. The osumer detfato des for the α :s ad y:s have bee suppressed order ot to overload otato, but they are mplt, ad a be dfferet for all the osumers. Market pre p, o the otrary, s the same for all osumers.

6 Courot Olgopoly The profts of a represetatve frm ow beome Π = q +, q Q q (.7) aga assumg ostat ut osts, ad the proft maxmum odto reads, Q = bq + Qg. (.8) As the umerator, resdual demad, as well the rght had sde, ut ost, are postve, oe a take roots, ad solve for the smple reato futo, q Q = Q (.9) Ths futo has a leadg square root term of Q, ad a evetually domat egatve lear term, so ts shape s umodal, startg the org, reasg to a maxmum, ad the dereasg to zero. The reato q remas postve as log as Q < (.0) holds. If ot, the aga the egatve outome wll have to be replaed by zero. Aordgly, the omplete reato futo reads q = R S T Q Q, Q <. 0, Q (.)

7 Courot Equlbra 3 Oe should hek profts for oegatvty. Substtutg the ozero brah of the reato futo the proft expresso yelds Π = Q Q Q F HG Q Q I KJ = d. Q (.) Aga proft s oegatve, but aga, whe the reato s egatve, t s due to egatve osts outweghg egatve reveues, whh s osese. Eve ths tme we, however, do ot eed to trodue ay more ostrats tha already represeted by the zero brah odto. Ths ase wth soelast demad s qute useful for dsplayg some of the terestg dyams possbltes of olgopoly theory, but t has ts lmtatos: I partular, t s ot sutable for the dsusso of moopoly or olluso. Ths s beause f there s just oe frm, the moopolst, the t reeves a ostat (ut) reveue, pq =. O the other had, produto osts rease wth output, ad all varable osts may be avoded f the moopolst produes othg. It the sells ths othg at a fte pre, as may be see from the demad futo. I terms of substae ths s osese, so oe just aot use the model for the moopoly ase. I dyam olgopoly, the presee of other ompettors as a rule keeps the system from ladg zero output for all frms, a ase of mplt olluso, whh has the same haratersts as moopoly. Further the derved reato futos all have fte slope at the org, so uder ay perturbato, the system wll ever stk there. But, a omplete haraterzato of possble orbts, there rema degeerate orbts whh eve have ther bass of attrato ad are stable a weak Mlor sese. See Tramotaa et al. (00). Courot Equlbra Next osder the equlbra of the geeral Courot model. The optmum odtos bg bg bg f Q f Q q = C q (.3)

8 4 Courot Olgopoly wth the defto Q= = q = added provde a set of ( + ) equatos the same umber of ukows, q ad Q. The soluto(s) provde the oordates of the Courot equlbra. Ths also meas that all the reato futos q = φ bg Q wth defto Q = qj are satsfed. Whe the ompettors are so few that oe a dsplay the phase spae graphally, for stae two, the duopoly, the the Courot equlbra are tersetos of the reato urves. Wth three ompettors, already dffult to vsualze, they are tersetos three-spae of three surfaes. j Example : Lear Demad I the two exemplfyg ases t s possble to alulate the oordates for the Courot pot. Startg wth the lear, the optmalty odtos read q a = b Q. (.4) Subtratg q from both sdes, ad multplyg wth, q a = Q, b (.5) The, takg the sum over dex, s obtaed, whh solves for a Q = b = = Q (.6) a Q =, + b (.7)

9 Courot Equlbra 5 where = = = (.8) deotes average ut ost for the ompettors. Substtutg for total supply the Courot equlbrum pot, oe gets q a a = b + b (.9) for the supples of the dvdual frms. It s also easy to alulate the equlbrum market pre, through substtutg for market supply Q the verse demad futo. Hee p= a bq= a (.30) Ths s a well kow result from work wth lear demad futos: Equlbrum pre equals a weghted average of maxmum pre a, ad ut produto ost. The more frms that stay atve the olgopoly, the more weght has the seod term ad the less has the frst. I moopoly, pre lads halfway betwee maxmum pre ad ut ost, duopoly the weght for the ost term s two thrds ad oe thrd for maxmum pre. Wth a reasg umber of ompettors the maxmum pre term teds to lose all mportae, ad oe approahes margal ost prg, as presrbed perfet ompetto. To osder whh frms wll stay the olgopoly market, oe ould ether hek the Courot equlbrum supply oordate for postvty, or hek that ut ost be lower tha market pre,.e., < a (.3) It must be kept md that the defto of otas also the value of. See Caovas et al. (009). To squeeze out some more formato, suppose the frms are umbered after reasg ut osts,.e.,

10 6 Courot Olgopoly < <..., (.3) whh a be doe wthout loss of geeralty. The the odtos a a < a, < +, 3 < + +, & 3 (.33) or, geeral, F HG < a+ j= j= j I KJ, (.34) are obtaed from (.3). Hee, how may ompettors that may be aommodated a Courot equlbrum depeds o the prese ost struture that olgopoly brah. I the speal ase where all the frms are detal,.e., =, t s suffet that = < a. If, o the other had, the ost stuato s very uequal, for stae, > ba+ g/, the oly a moopoly of the frst frm a make postve profts, f 3 > ba+ + g/ 3, the a duopoly a persst, et. Example : Isoelast Demad Also for the seod example t s possble to obta the losed form expressos for the Courot pot oordates ad the rest of the results just dsussed. Reall the ma brahes (.9) of the reato futos, q Q = Q. (.35) Addg Q to both sdes,

11 Courot Equlbra 7 Q Q =, (.36) or, takg squares, Q Q =, (.37) s obtaed. Multplyg by, reallg that Q = Q q, ad reorgazg slghtly, q = Q Q, (.38) Takg the sum over, oe gets Q= Q Q, (.39) where, lke the prevous ase, = = = (.40) deotes average ut ost. As Q > 0, t s permtted to ael oe power equato (.39), so Q =, (.4) ad, osequetly, q F b g b g. b g = I HG K J = (.4)

12 8 Courot Olgopoly Several fats are worth beg oted. As dated the troduto, the model s ot sutable for dsussg moopoly. The odtos (.4) ad (.4) result zero output whe =. Oe a also easly alulate olgopoly pre the Courot equlbrum: p =. (.43) Dsregardg aga the degeerate moopoly ase where pre s fte, equlbrum pre s twe the ut ost for a duopoly, oe ad a half tmes ost a tropoly, ad so forth. Aga, as the umber of ompettors reases, oe approahes margal ost prg ad the ase of perfet ompetto. As for the questo whh frms may obta postve profts the Courot equlbrum, oe a aga ether osder postvty of the reato, or that ut ost be lower tha market pre,.e., <. (.44) Aga, t should be oted that s the average of all the ut osts, ludg that o the left, ad aga, oe a assume the frms to be umbered order of reasg ut osts. I ths way, gorg the degeerate ase of moopoly, oe has < + for duopoly, whh s always fulflled wth postve ut osts, 3 < for tropoly, whh s the same as 3 < +, or, geeral. j= < j= j (.45) Hee, for two ompettors a Courot olgopoly, there are o ostrats for the osts; for three, the ut ost of the thrd must, however, ot exeed the sum of the osts of the two frms wth the lowest osts. A fourth frm may be added provded ts ut ost does ot exeed oe half of the sum of the osts for the three wth the lowest, ad so forth. Aga, the speal ase =, othg s ostraed.

13 The Courot Iteratve Map 9 The Courot Iteratve Map It s ow tme to retur to the geeral ase, ad osder the dyams of Courot ato. To ths ed t s eessary to state how expetatos for the resdual supples Q are formed a evolvg system order that eah ompettor be able to alulate the proper respose aordg to q = φ bg Q as stated (.6) ed exemplfed (.) ad (.9). Somehow real ompettors ad modellers alke have to form suh expetatos, ad, as already stated, the assumpto losest at had s that eah ompettor assumes the others to mata ther prevous moves, eve f the assumpto ostatly shows up as beg wrog. Ths lks future to past, so oe a wrte b g b gh q t+ = φ Q t wth tme perod detfatos for the varables. Of ourse (.46) = bg bg bg bg bg = Qt = q t, Q t = Qt q t, (.47) ad lkewse for tme perod t +. These rules date how all the quatty varables, dvdual supply, total supply, ad resdual supply for eah followg perod are obtaed from those of the urret. Hee the whole followg orbt of the system ould prple be alulated. For ay suh system oly four possble orbt types whh are approahed asymptotally after a traset exst. Fxed pots (or equlbra), perod orbts (whh always retur to the same state after a fte umber of steps), quasperod orbts (those that have a fte perod, omg lose to prevous states perodally but ever qute so fte tme), ad haot orbts (suh that are upredtable, as earby orbts are separated at a expoetal rate, though bet bak ad kept a fte area of the phase spae). All these are possble attrators. They a be loely, or they a oexst. A fxed pot a oexst wth aother fxed pot, as demostrated by Palader (936, 939) ad Wald (936), wth a perod orbt (as also show by Palader), or eve wth a haot attrator. I the ase of oexstee, eah attrator has ts proper bas of attrato, ad the separatg bas boudares dvde the total phase spae suh attrato bass. Bass ad bas boudares

14 30 Courot Olgopoly a be smple, smply oeted areas ad urves of fte legth, but they a also, eve wth very smple systems, be very omplated, so alled fratals. These potetal attrators, fxed pots, perod, quasperod, ad haot, a also be ustable,.e., repellors. They yet rema terestg for desrbg the dyams, as they exst ad repel all trajetores that ome ear. Further, attrators a tur to repellors due to some slght hage of a parameter of the model, whh ase oe speaks of a bfurato. Moder theory for dyamal systems has got very far studyg ad lassfyg types of bfuratos, as well as of attrators ad bass. Formato of Expetatos I formulatg the teratve dyam system, or map as s the mathematal term, t was assumed that the ompettors just assumed all others to reta ther prevous moves. Ths s a qute problemat ase a evolvg system, exept whe t s a stable fxed pot. Observg Chaot Orbts So, what a the ompettors lear from the atually observed orbt? A haot orbt s by defto upredtable, though determst. Ths s due to the magfato of omputatoal roudg off errors. Determsm just meas that the future of the system s totally determed by the tal odtos; but these tal odtos must be kow exatly, umeral values wth a fte umber of demals. Suh preso s ever possble realty. Now, a slght measuremet error a have dfferet osequees. I a system suh as foused tradtoal dyams, t just leads to a slght dsplaemet of the orbt. I a haot system, orbts startg from dfferet tal odtos, o matter how slght the dfferee s, eed o more tha a few moves to ed up ompletely dfferet parts of the phase spae. Ths s what upredtablty meas. For the preset otext t s obvous that orbts that are ever predtable are useless for observato ad use formg expetatos. Ufortuately, moder dyams has show that ths pheomeo teds to be preset eve the smplest olear dyam systems.

15 Formato of Expetatos 3 Observg Perod Orbts Startg at the smplest ed, dsregardg the obvous ase of a stable fxed pot, suppose the system atually produes a perod orbt, suh as may be observed ad leared by the ompettors. Would t the be possble to adapt to ths, ad produe a orbt of exatly that perodty? Suh a dea s lose to the favourte eooms dea of ratoal expetatos. As wll ow be show ths s always mpossble, beause suh learg ad adaptg to ay perodty s boud to produe a perodty dfferet from the oe leared, uless the orbt s a fxed pot. See Puu (008). So, suppose there exsts a perod orbt of perod T as soluto to the dyam Courot system. The, for the fatual orbt, b g bg, q t + T = q t (.48) holds for all tme perods t. As the perodty s assumed to be leared ad adjusted to by the ompettors, they wll form the expetatos orretly ad respod to observed fats ot oe perod bak, but T perods bak. Hee, b g b gh q t+ T = φ Q t. (.49) But, from equatos (.48) ad (.49), the b g q t = φ Q t, b gh (.50) whh relates the varables the same tme perod, ad s hee a defto for the Courot equlbrum fxed pot. Ths relato holds, as assumed, for all tme perods t, whh shows that the assumed perod orbt s a fxed pot. A fxed pot, of ourse, s a perod pot of all perodtes. But ths smple proof shows that a perod pot whh s ot a fxed pot aot be leared, adapted to, ad yet result a perod orbt of the assumed perod, uless t s a fxed pot. For the reader who s very fod of the dea of ratoal expetatos, we a add a dfferet argumet. Suppose the dyam system produes a orbt of, say, perod 3. Ths meas that the system evetually vsts three pots here alled A, B, C over ad over. Now, suppose the ompettors lear ths, ad adapt. As the system s determst, ad the same as before, the outome

16 3 Courot Olgopoly would be the same sequee, though wth two udetermed pots terveg perods. Assumg oe aga starts A, the system produes A B C. Now, try to make ths the orgal 3-perod orbt. I that ase B should follow the frst etry A, ad fll out the frst blak, but the the thrd ad ffth blaks wll be flled out by C ad A that order, produg A B_ BC_ C A_. Fally, the remag blaks eed to be flled out, by C, A, B, produg ABC BCACAB. Ths starts qute orretly wth A, B, C, but the omes B, C, A, ad fally, C, A, B. It s thus mpossble to reogze the sequee as aythg but a 9-perod orbt. After the last etry the sequee starts all over aga. Ths shows that f the ompettors lear the 3-perodty they produed before, the they, fat, produe a 9-perod orbt, whh, f leared ad adapted to, ts tur produes aother perodty. If all the ompettors do ot lear the perodty smultaeously, but some are slower learg ad adaptg tha others, the the result beomes eve muh more omplated. The oluso s that learg ad adaptg to a observed perodty by the agets s always boud to hage ths very perodty. I a loose way t s smlar to the mpossblty prple physs, where the very at of observg ad measurg postos ad mometa of partles hages the fats to be observed. It s self-evdet, that the hage of perodty does ot our whe the perod orbt s a fxed pot wth just oe pot of phase spae A vsted over ad over. As a geeral oluso; some orbts, suh as the haot, are too omplated to lear ad adapt to; some, suh as the perod, are smple to observe, lear, ad adapt to, but whe the agets do ths, they evtably hage the outome. To adapt to a observed orbt ad make the system produe that very orbt seems to be mpossble, uless the orbt s a fxed pot. What the a the ompettors do to esape from expetg the others to reta ther prevous moves, sometmes alled ave expetatos, whh

17 Stablty 33 always shows up to be wrog? They a observe treds, reases or dereases from two subsequet perod observatos, or eve urvatures a attempt to detfy approahg turg pots. However, aalysg suh systems s boud to beome extremely omplex ad messy as t multples up the order of the system. I the sequel ths smple assumpto oerg the formato of expetatos wll be kept, exept whe dsussg oservatve moves terms of adaptve systems, or Stakelberg leadershp, whh represets aother type of learg, ot of the atual orbt, but of the reato futos of the ompettors. Stablty Courot equlbra, fxed pots of the Courot teratve map, a, as stated above, be stable or ustable. If they are ustable, they a yet be of terest haraterzg the dyam system, but they are o attrators. For establshg the stablty of a Courot equlbrum pot of the geeral teratve system, b g b gh q t+ = φ Q t, oe frst eeds to alulate the dervatves of the reato futos, The the Jaoba matrx reads, dq t + b g bg dq t = φ. (.5) (.5) J = L N M 0 φ L φ φ 0 L φ φ φ M M O M L 0 O Q P. (.53)

18 34 Courot Olgopoly Obvously, as the supples of all the ompetg frms eter the sum defed as resdual supply, the same etres appear as off-dagoal elemets eah row. As the supply of eah frm tself s abset ths defto of resdual supply, the dagoal elemets are zero. The ext move would be to form the matrx J λi = L N M λ φ L φ φ λ L φ φ φ L λ M M O M O Q P, (.54) where I deotes the by detty matrx, ad the haraterst equato, J λi = 0. (.55) Ths th degree polyomal equato λ determes the egevalues, real or omplex, ad for stablty oe must esure that all these λ, λ,... λ are the ut rle the omplex plae. Ths s a formdable programme, ad gve the geeral equatos for the Courot system, b g bgh b g bgh b g b g f q t+ + Q t + f q t+ + Q t q t+ = C q t+ h, mplt dfferetato results (.56) f + φ f φ = f + φ f C, (.57) whh seems to be exeedgly messy. Oe would have to alulate the Courot pot oordates from (.3), substtute those the dervatves of the reato futos (.57), the those the Jaoba (.53), ad fally solve the polyomal haraterst equato (.55). Several of these steps are just mpossble vew of the geeral proedures of mathematal eooms.

19 Stablty 35 That muh just to determe the stablty of a sgle fxed pot! How ould oe at all deal wth the orbts resultg from a uspefed teratve map wth just a few qualtatve propertes? These fats emphasze the absolute eed to work wth spefed global models resultg redble losed form reato futos. Example : Lear Demad It s therefore good to retur to the two examples. I the lterature Theohars (959) s geerally stll redted for havg bee the frst to show that a Courot equlbrum a olgopoly wth a lear demad futo ad ostat ut osts for the ompettors s destablsed whe ther umber exeeds three, whereas stablty beomes eutral wth three ompettors. The the (uque) fxed pt s ot yet ustable, but ay slght devato from t puts up a edless osllatory moto; eah suh moto beg eutrally stable as well. Palader (939), whose more mportat otrbutos wll be dealt wth a later hapter, stated exatly the same 0 years earler. Ufortuately, the artle was wrtte Swedsh, a laguage eve preset day Aglo-Saxo oreted Swedsh eoomsts ether read or te. The stablty ssue, whh was extremely omplated the geeral ase, s ow equally smple. From the reato futos, The dervatves are always b g b gh b g a q t+ = Q t = φ Q t. b b g bg dq t + = φ =, dq t (.58) (.59).e., ostat, so that oe does ot eve eed the partular oordates of the Courot equlbrum pot to put up the by Jaoba matrx

20 36 Courot Olgopoly J = L N M 0 0 L L M M O M L 0 O Q P. (.60) The haraterst equato s easly obtaed as a th degree polyomal that fatorzes to The egevalues are bg b g b g J λi = P λ = λ λ+. h (.6) λ,... λ =, λ = b g. (.6) The frst are equal to ad are well wth the ut rle of the omplex plae, but the last oe b g falls outsde t whe > 3. For = 3, λ =, so ths s a boudary ase. The multple egevalues are assoated wth dfferees of the varables. They always date stablty, eve whe the system dverges. Ths meas that, ay moto, dfferees betwee the behavour of the frms are eveed out, so that the atos are asymptotally oordated. The last egevalue s assoated wth total market supply, ad t s t that brgs stablty to the system whe the umber of ompettors grows. Theohars also gave the omplete global soluto to the system, but there s o eed to preset t here, beause a ustable lear system always just explodes, resultg fte devatos, egatve supples, ad egatve pres. Hs soluto s therefore ot fatually relevat. The global dyams of the peewse lear ase, where aout s take of oegatvty of pre, supples, ad profts, was oly aalysed a few years ago. See Caovas et al. (008). Palader too, was oly oered about the stablty, but he stated the results ompletely, as metoed, 0 years before Theohars. I 939 he wrote as a odto for equlbrum wth a erta umber of ompettors to be stable to exogeous dsturbaes, oe a stpulate that the

21 Stablty 37 dervatve of the reato futo φ must be suh that the odto b gφ < holds. If ths rtero s appled to, for stae, the ase wth a lear demad futo ad ostat margal osts, the equlbra beome ustable as soo as the umber of ompettors exeeds three. Not eve the ase of three ompettors wll equlbrum be restored, rather there remas a edless osllato. As for the global dyams of the peewse lear ase, ot muh extg ours. There are just the possblty of a stable moopoly, or a duopoly; otherwse two-perod osllatos where all the frms move phase are the attrators. If the umber of ompettors exeeds three, all frms drop out every seod perod. Ths mght seem to preset a exellet ase for learg the orbt by some ompettors who mght try to move out of phase, eve attemptg to beome moopolsts every seod perod. But the oe eouters the problem stated above that ths learg ad adaptato alters the perodty tself. Whh frms wll stay o the market depeds, as was show the seto o Courot equlbra, o the ost struture. It may our that, eve wth very umerous frms, the ost struture amog them mght be suh that oly oe or two frms wth ost advatages stay the Courot equlbrum; all the other beg boud to drop out the dyam proess. Example : Isoelast Demad From (.35) the ma brahes of the reato futos the ase of soelast demad were b g bgh b g bg Q t q t+ = φ Q t = Q t, (.63) so the dervatves, b g bg dq t + = φ = dq t Q t b g (.64) are easly obtaed. Further, from (.37), Courot equlbrum,

22 38 Courot Olgopoly Q = Q. (.65) Hee, φ = Q (.66) the Courot fxed pot. Further, t was true that Q =, (.67) where = = = (.68) was average margal ost. Usg (.67) for Q, φ = (.69) Courot equlbrum. Agza (998) ad Ahmed et al. (998) wated to demostrate that the Courot equlbrum s destablsed ths ase whe the umber of ompettors exeeds four, ad s eutrally stable whe the umber s exatly four. The froter of destablsato s hee just pushed oe ompettor further as ompared to the ase of a lear demad futo. Ther proof ould ot be ompleted wthout hagg the demad futo to p = Q + p 0, (.70) through addg a ostat. Ths, however, ulke the orgal ase of the soelast futo, s ot derved from ay bas mroeooms of osumers

23 Stablty 39 maxmzg ther utlty futos. Further, ote that ths hage has o oeto wth the problem dsussed that the soelast demad futo s usutable for dealg wth the ase of moopoly; to avod problems that oeto, oe would have to add a ostat to supply, rather tha to pre. But, ufortuately, ths s ot based o the theory of the osumer ether. However, there s a smple way to arrve at the olusos of Agza (998) ad Ahmed et al. (998) wth the orgal demad futo. That s to assume that the ompettors are detal, so that all =. The φ =, (.7) whh faltates thgs a lot. The by Jaoba matrx for the Courot equlbrum pot ow beomes J = L NM ad the haraterst equato O QP 0 L 0 L M M O M, L 0 F b g b g I J J = P = HG K J I λ λ λ λ+ HG K J = 0. F (.7) (.73) Aga, the polyomal fatorzes, ad amog the egevalues, λ,... λ =, λ b g, = (.74) the frst, of multplty, are sde the ut rle the omplex plae, whereas the last oe omes outsde for > 4, ad s hee assoated wth stablty.

24 40 Courot Olgopoly The preset model wth soelast demad offers better perspetves for terestg dyams global aalyss tha the ase wth lear demad, but, qute as the lear ase, the Courot equlbrum s destablsed whe the umber of ompettors exeeds a very small umber, though ow four stead of three. The sgfae of the result, attrbuted to Theohars (959), has relevae for the ssue of reasg ompetto trasformg the market from moopoly over olgopoly to perfet ompetto. Ths meas that wth a reasg umber of ompettors, the Courot equlbrum approahes margal ost prg ad elmato of profts for the margal frm (the oe wth hghest ut ost). But, ths, whh, as was show above, happes both models, s of lttle terest f ths same proess equlbrum s destablsed. If the system s o loger attrated to the Courot equlbrum t does ot matter that t trasforms to a perfet ompetto equlbrum. Yet, tutvely, oe would lke to keep the possblty of olgopoly seamlessly trasformg to perfet ompetto. The questo, whh seems, ot to have bee addressed s; why does ths happe wth dfferet reasoable demad futos? The lue les the assumptos o the ost sde, ostat margal ad average varable osts. These emerge from produto uder ostat returs. But a frm produg uder ostat returs has prple fte apaty. I perfet ompetto, a market pre, take as ostat by the dvdual frms, whh exeeds a ostat ut ost by the test frato, makes t possble to blow up the proft to ay value just through multplyg up the sale of operato. Admttg ths, oe may say that destablsato due to addg more ad more fte szed frms s ot very surprsg; or s t relevat to the ssue of trasformg olgopoly to perfet ompetto. It has bee mplt that the omparso should be betwee ases of few large frms versus may small frms. But large ad small aot be modelled wth ostat ut osts. Oe would eed varable, ot ostat, returs, preferably ludg apaty lmts for suh modellg. Ths a, as show a followg hapter, be modelled dfferet ways. I the peewse lear model, oe ould keep ostat ut osts, but at a blut apaty lmt let the ost jump up to fty. It s also possble to use the CES produto futo, wth aptal fxed through a at of vestmet, to obta a ost futo that asymptotally goes to fty at a apaty lmt due to the fxed aptal. Ths makes t eessary to lude assumptos of aptal formato, ad paves the groud for edogeous modellg of how aptal formato ad ompetto may evolve a brah. See Puu ad Pahuk (009).

25 Referees 4 Referees Agza, H. N. (998). Explt stablty zoes for Courot games wth 3 ad 4 ompettors. Chaos, Soltos & Fratals, 9, Ahmed, E., & Agza H. N. (998). Dyams of a Courot game wth ompettors. Chaos, Soltos & Fratals, 9, Caovas, J. S., Puu, T., & Ruz, M. (009). The Courot Theohars problem reosdered. Chaos, Soltos & Fratals, 37, Courot, A. (838). Réheres sur les prpes mathématques de la théore des rhesses. Pars: Duot. Palader, T. F. (936). Istablty ompetto betwee two sellers, Abstrats of papers preseted at the researh oferee o eooms ad statsts held by the Cowles Commsso at Colorado College, Colorado College Publatos, Geeral Seres No. 08, Studes Seres No.. Palader, T. F. (939). Kokurres oh markadsjämvkt vd duopol oh olgopol. Ekoomsk Tdskrft, 4, 4 45, 50. Puu, T. (99, 004). Chaos duopoly prg. Chaos, Soltos & Fratals,, , republshed : J. B. Rosser (Ed.), Complexty Eooms: The Iteratoal Lbrary of Crtal Wrtgs Eooms, Vol. 74. Chelteha Edward Elgar. Puu, T. (008). Ratoal expetatos ad the Courot-Theohars problem, Dsrete Dyams Nature ad Soety ID, 303,. Puu, T., & Pahuk, A. (009). Olgopoly ad stablty. Chaos, Soltos, & Fratals, 4, Robso, J. (933). The Eooms of Imperfet Competto. Lodo: Mamlla. Theohars, R. D. (959). O the stablty of the Courot soluto o the olgopoly problem. Revew of Eoom Studes, 7, Tramotaa, F., Gard, L., & Puu, T. (00). New propertes of the Courot duopoly wth soelast demad ad ostat ut osts. Chaos, Soltos, & Fratals ( press). (submtted). Wald, A. (936). Über ege Glehugssysteme der mathematshe Ökoome. Zetshrft für Natoalökoome, 7,

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