Advertising in a Differential Oligopoly Game *

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1 dversg a Dffereal Olgopoly Game ROBERTO CELLINI Uversà d Caaa, Facolà d Ecooma, Dparmeo d Ecooma e Meod Quaav ad LUC LMBERTINI Uversà d Bologa, Facolà d Sceze Polche, Dparmeo d Sceze Ecoomche Ocober 00 bsrac - We llusrae a dffereal olgopoly game where frms compee à la Couro homogeeous goods he mare phase, ad ves adversg acves amed a creasg cosumers reservao prce. Such vesmes produce exeral effecs, characerzg he adversg acvy as a publc good. We derve he ope-loop ad he closed-loop Nash eulbra, ad show ha he properes of he eulbra deped o he curvaure of he mare demad fuco. The comparave assessme of hese eulbra shows ha frms adversg effors are larger he ope-loop ha he closed-loop eulbrum. We also show ha a carel volvg all frms, seg boh uaes ad adversg effors so as o maxmze jo profs, may produce a seady sae where socal welfare s hgher ha he socal welfare levels assocaed wh boh he o-cooperave segs. Keywords: adversg, dffereal games, capal accumulao, ope-loop eulbra, closed-loop eulbra. JEL Classfcao: D43, D9, L3, M37. We ha Vcezo Decolò, Emlo Garda ad Gapaolo Ross for helpful dscusso ad suggesos. The usual dsclamer apples. ddress: Corso Iala 55, 959 Caaa, Ialy. e-mal: cell@uc. ; phoe: ex. 37; fax: ddress: Srada Maggore 45, 405 Bologa, Ialy. e-mal: lamber@spbo.ubo. ; phoe: ; fax

2 dversg a Dffereal Olgopoly Game. Iroduco The exsg leraure o dyamc models of adversg ca be broadly paroed o wo ma caegores. The frs orgaes from Vdale ad Wolfe 957, ad s characerzed by a drec relao bewee he rae of chage sales ad he adversg effors of frms. The secod daes bac o Nerlove ad rrow 96, ad cosders adversg as a srume o crease he soc of goodwll or repuao, summarsg he effecs of pas ad curre adversg expedures carred ou by a frm, o he curre demad for her goods. releva resul emergg from he Nerlove-rrow model s he dyamc verso of he well ow Dorfma-Seer 954 codo, esablshg ha he adversg vesme s proporoal o sales. We prese a dyamc model of olgopoly wh homogeeous producs, where frms compee à la Couro he mare phase, ad ves adversg acves amed a creasg cosumers' reservao prce over me. The adversg vesme carred ou by ay frm splls over o he rvals as log as he mare demad fuco he same for all frms shfs ouwards. ccordgly, hs d of adversg s a publc good. We cosder a very geeral formulao of he demad fuco, accoug for cocavy, leeary ad covexy. Ideed, here emerges ha he dyamc properes of he eulbra drascally deped upo he degree of he mare demad curvaure. I he sadard case of lear demad, o sesble eulbra exs. If demad s cocave, oe saddle-pah eulbrum emerges. If demad s covex, he eulbrum s usable. The deermas of he seady sae vesme ad reservao prce are frs aalyzed he o-cooperave game, usg aleravely he ope-loop ad he closedloop eulbra as soluo coceps. The, we aalyze he opmal decsos by a carel made up by all frms he mare, amg a he maxmum jo profs wh respec o For he frs class of games, see Lema ad Schmedorf 978, Fechger 983 ad Ercso 985; for he secod, see Seh 977, Fershma 984 ad Jørgese ad Zaccour 999, er ala. For exhausve surveys, see Jørgese 98, Fechger, Harl ad Seh 994 ad Docer, Jørgese, Va Log ad Sorger 000, ch..

3 boh oupu ad adversg effor, ad he opmal behavor for a socal plaer choosg oupu levels ad adversg vesmes of all frms so as o maxmze socal welfare. lso hese cases, he dyamc properes of he seady sae deped o he curvaure degree of he mare demad fuco. Reservao prces, vesmes ad produco decsos are compared across he seady saes of all he aforemeoed regmes. releva resul s ha socal welfare seady sae may well be larger uder he carel eulbrum, ha he o-cooperave Couro eulbrum rrespecvely of wheher he laer s derved uder he ope-loop or he closed-loop formao srucure. Ths s due o he exeraly effec affecg frms adversg acves, ad he assocaed effcecy affecg he Couro seg. The srucure of he paper s as follows. Seco llusraes he basc seup. Secos 3 ad 4 solve he model uder he ope-loop ad he closed-loop formao srucure, respecvely. Seco 5 compares he seady saes derved seco 4 ad evaluaes hem agas he exsg leraure. Seco 6 deals wh of he opmal carel behavour ad he frs bes soluo ha a socal plaer would mpleme. The comparave assessme of all eulbra s Seco 7. Seco 8 brefly cocludes.. The basc model We cosder a olgopoly uder full formao. ay me 0,, frms sell a homogeous good. Margal produco coss are assumed o be decal across frms ad ormalzed o zero for he sae of smplcy. The mare demad curve s borrowed from derso ad Egers 99, 994 : P [ Q ], The curre leraure o olgopoly heory usually adops a lear mare demad, wh eher homogeeous or dffereaed goods; for exhausve surveys, see Trole 988, Mar 993, ad Vves 000 er ala. relavely scay aeo has bee devoed o he aalyss of he effecs of mare demad curvaure o frms' sraegc behavor; such a problem s suded by derso ad Egers 99, 994 ad Lamber 996.

4 where P deoes he prce, reservao prce, ad 0, Q he aggregae uay of he good, s he s a posve parameer deermg he curvaure of demad. The verse demad fuco s covex f 0,, s cocave f, ad s lear f., I s easy o verfy ha he prce elascy of demad s, absolue value, ε Q, P Q / Q, whle he cosumers' surplus evaluaed a Q,.e., he oal area / below he verse demad curve, s CS /, whch also measures socal Q welfare a he perfecly compeve eulbrum. Ths meas ha he curvaure parameer affecs he prce elascy; mare sze; ad oal surplus. Moreover, for a gve, he hgher s, he larger s mare sze; lewse, gve, he hgher s, he larger s mare sze. Frms are able o crease over me hrough vesme adversg campags. Formally, he followg dffereal euao descrbg he dyamcs of varable amous o sayg ha he mare sze creases as resul of he sum of adverseme acves of all frms, ad, furhermore, s subjec o a cosa deprecao rae : d. d Ths ype of adversg s deed a pure publc good, ha he effor carred ou by ay frm beefs all frms ale see Fershma, 984; Fershma ad Nza, 99; accordgly, s somemes referred o as cooperave, wh he mplc cavea ha frms do o ecessarly cooperae he sese of jo prof maxmsao. 3 The adverseme acvy of each frm eals a uadrac cos c b, wh b>0. Of course, he spllover effec characersg he adversg acvy rvally eals ha he dvdually opmal adversg effor s uderszed from he sadpo of he whole populao of frms. 3 Ths labellg daes bac o Fredma 983; see also Mar 993, ch. 6. For a model where adversg s boh cooperave ad predaory, see Pga 000, pp

5 The dyamc problem ca be summarzed as follows: he objecve of each frm s o acheve he maxmum prese value of he flows of fuure profs, ag o cosderao ha he prce depeds o he produco decsos of all frms, he coss are represeed by he dvdual vesme coss adverseme whch splls over o all rvals, ad ha a posve dscoug rae ρ, commo o all frms, apples o fuure coss ad beefs. Formally, each dvdual frm faces he problem: 3 Max J s.. 0 e ρ d K d [ Q ] b d where Q j j ad j K. j The choce varables of frm are ad. s he sae varable, commo o all ages, subjec o he al codo 00>0. The soluo of hs dyamc opmzao problem by player s based o he Hamloa fuco H. Le µ be he co-sae varable assocaed o he sae varable by player, ad le λ ρ µ e be he curre-value co-sae varable. The Hamloa fuco s herefore: ρ 4 H [ Q ] b λ [ K ] e. The ex secos solve he fully o-cooperave problem, accordg o he opeloop Nash eulbrum cocep, ad he closed-loop memoryless Nash eulbrum cocep, respecvely. The dfferece bewee hese wo coceps ress o he possbly of modfyg he plas, oce he game has sared. I parcular, uder he former soluo cocep, frms precomm her decsos o he corol varables o a gve me pah: hey desg he opmal pla a he al me ad he sc o forever. Uder he laer soluo cocep, frms do o precomm o ay pah ad her decsos a ay sa deped o all he precedg hsory, ad specfcally o he value of he sae varable a ha sa. 4

6 5 The closed-loop soluo s srogly me cosse ad herefore subgame perfec, whle he ope-loop soluo s oly wealy me-cosse,.e., s o sub-game perfec The ope-loop Nash eulbrum The frs order ecessary codos o characerze he ope-loop Nash eulbrum are: d d H H H ρλ λ alog wh he al codo ad he rasversaly codo 0 lm e ρ λ. The frs codo of sysem 5 urs ou o be: 6 [ ] [ ] 0 Q Q H, from whch he Couro reaco curve ca be obaed. Uder he symmery codo, whch mples Q--, he opmal uay o be produced by each frm s: 4 oher srogly me-cosse ad herefore subgame perfec soluo cocep s he feedbac eulbrum usg Bellma s euao. The feedbac eulbrum s a closed-loop eulbrum, whle he oppose s o rue geeral. For a clear exposo of he dfferece amog hese eulbrum soluos see Basar ad Olsder 98, pp , ad chaper 6, parcular Proposo 6.. There exs classes of games where he closed-loop ad he ope-loop soluos cocde see Mehlma, 988, ch. 4; Regaum, 98; Fershma, 987; Fershma, Kame ad Muller, 99; Docer, Jørgese, Va Log ad Sorger, 000, ch. 7, Cell ad Lamber, Secod order codos are omed hroughou he paper, for he followg reaso. Whe demad s cocave, hey are always me ha he problem s cocave ad sgle-peaed; whe demad s eher lear or covex, secod order codos are rreleva ha here exss o sable eulbrum, as wll become clear he remader of he aalyss.

7 6 6'. The secod codo of sysem 5 s 7 0 b H λ, from whch we ca easly derve b λ ad, by dffereag w.r.. me: 7' d d b d d λ. Fally, from he hrd codo of sysem 5 we derve: 8 [ ] λ ρ λ Q d d. Cosderg smulaeously codos 7' ad 8 we oba: 9 [ ] ρ Q b d d. Now we are a poso able o compleely characerze he dyamc sysem uder opmal decso rules by par of Couro frms, ad uder symmery codos, : 0 ρ b d d d d

8 phase dagram ca be easly draw o represe hs dyamc sysem. I seady / / sae, d/d0 mples / whle d/d0 mples /[b ρ ]. I he space,, he former relao correspods o a sragh le wh posve slope eual o /, whle he laer correspods o a curve whch may be covex f 0,, cocave f,, or lear f. The hree paels of Fgure show he alerave cases. The lear case s show uder he hypohess ha he locus d/d0 les above he locus d/d0, bu he oppose case could be eually possble, as well as he case ha he wo loc cocde. The arrows are draw ag o accou ha above below he le d/d0, he sae varable creases decreases, ad above below he curve d/d0, varable creases decreases. The doed le defes he saddle pah. The dyamc sysem adms oe meagful seady sae soluo, besdes he rval oe 0, 0 : 0 ; /. b ρ Ths soluo does o exs he parcular case where, as clear he graphcal represeao. Correspodgly, s mmedae o fd he amou of he dvdual produco level seady sae: /. The seady sae values of he releva varables are also repored Table Seco 7, order o ease comparsos across dffere regmes. INSERT FIGURE HERE 7

9 8 The learzao of he dyamc sysem s as follows: 0' Ω Ω, / / ρ b d d d d. The Jacoba marxω, evaluaed a he seady sae po, has he followg race ad deerma: ρ ρ de Ω Ω r I s sraghforward o verfy ha he race s always posve, whle he deerma s posve, egave or l, f 0< <, > or, respecvely. Thus, he followg holds: Proposo. The seady sae of he o-cooperave game, where b / ; ρ ; s usable whe he mare demad curve s covex, does o exs whe he mare demad s lear, ad s a saddle whe he mare demad s cocave. Proposo saes ha here exss o ecoomcally meagful saddle po whe. The uo behd hs resul s as follows. Whe mare demad s eher lear or covex, he mare sze.e., he area below he demad curve s oo small o jusfy frms vesme a publc good. The saddle po where 0 ca obvously be dsregarded. Focusg our aeo o he case of cocave mare demad.e., o he saddle pah eulbrum of he dyamc game, he deermas of he seady sae varables, as show by e. 0, are as follows:

10 < 0, b < 0, < 0, ρ < 0, < 0, b < 0, < 0, ρ < 0. The ecoomc meag s easy o erpre. Frs of all, here exss a drec relaoshp bewee he seady sae levels of ad. The larger s he seady sae value of he mare sze, he larger s he vesme effor reured o euralze he deprecao of ad o rema seady sae. Secod, he larger s he cos of vesme effor capured by parameer b, he smaller he seady sae opmal level of vesme effor. Smlarly, he hgher he deprecao rae, he lower he opmal seady sae level of mare sze ad vesme effors. Furhermore, he hgher he dscoug facor, he less mpora are fuure profs, ad he he lower he opmal mare sze he seady sae. Fally, he larger s,.e., he wder s he mare ceers parbus, he less covee s o ves demad-creasg adversg. The effec of he umber of frms,, upo he opmal seady sae values s o clearcu, because wo oppose effecs are operao. O he oe had, he larger s, he harsher s mare compeo, whch maes more covee o have a larger mare sze. O he oher had, he larger s, he larger s he spllover effec comg from he vesmes operaed by oher frms. 4. The closed-loop Nash eulbrum I hs seco, we vesgae he closed-loop soluo of he dffereal game, ha s, he opmal soluo whe each player aes o accou ha a ay po me he corol varables of oher players affec he sae varables, reurg a revso of opmal produco ad vesme plas. I parcular, here we adop he so-called "memoryless" closed-loop soluo cocep, accordg o whch every player s reured o ow oly he curre value of he sae varable, ad he owledge of he complee pas hsory s o ecessary o compue he opmum. Ths d of closed-loop soluo s 9

11 0 srogly me cosse. alycally, he frs order codos o characerze he closedloop memoryless Nash eulbrum are: 0 0 d d H H H H H j j j j j j ρλ λ where H deoes he Hamloa fuco of player, as defed by euao 4. The frs ad secod codos of sysem are he same as he ope-loop soluo represeed sysem 5, whle he hrd codo aes o accou he eraco bewee he oher players' corols ad he curre level of he sae varable. s a coseuece, euaos 6 ad 7, yeldg he reaco curves, hold also hs problem, whle he co-sae euao s ow: 3 [ ] λ ρ λ Q d d. Of course, he codos derved from sysem have o be cosdered alog wh he usual al codos ad he rasversaly codos. Uder he symmery assumpos ad, ad followg he same procedure as he ope-loop problem, euaos 6, 7 ad 3 lead o he followg dyamc sysem ad : 4 ρ b d d d d or, marx form:

12 4' d d Ψ d d, Ψ b / / ρ lso hs case, he race of he Jacoba marx s always posve, whle he deerma evaluaed a he seady sae value of ad s posve, l or egave, accordg o he value of parameer. Coseuely, he properes of he seady sae aga deped o he curvaure of he mare demad fuco. More specfcally, he locus d/d0 cocdes wh s couerpar he opeloop soluo, whle he locus d/d0 uder he closed-loop decso rule s a curve / / wh euao [ ]/[b ρ ]. Ths ca be sragh, covex or cocave, accordg o he value of, as he ope-loop soluo, excep ha always les below s couerpar he ope-loop soluo. s a coseuece, uder he closed-loop soluo, he seady sae soluo of he dyamc sysem,.e., he erseco bewee he wo loc obvously dsregardg he org, occurs he correspodece of hgher respecvely, lower levels of boh ad ha he ope-loop eulbrum, whe < >. Ths s evde from he aalycal soluo of he seady sae eulbrum uder he closed-loop decso rule, whch gves: [ ] 5 ; / b ρ Qualavely speag, he dyamc properes of he closed-loop seady sae are he same as saed Proposo for he ope-loop eulbrum. Ths promps for a comparave evaluao of he wo eulbra, whch we carry ou he ex seco.

13 5. Seady saes uder ope-loop Nash eulbrum ad closed-loop Nash eulbrum uc comparso bewee he seady sae levels of he hghes reservao prce uder he Nash ope-loop eulbrum OL ad he Nash closed-loop eulbrum CL, as gve by euaos 0 ad 5 respecvely, perms us o wre: 5' CL OL where s evde ha he closed-loop seady sae eulbrum value of he hghes reservao prce s larger uder he closed-loop soluo cocep ha uder he opeloop oe f ad oly f <. I he more eresg case where >, he saddle eulbra are such ha he seady sae eulbrum value of he reservao prce s smaller uder he close-loop soluo ha uder ope-loop oe. graphcal represeao of he case wh, s gve fgure, where he locus perag o he closed-loop memoryless formao srucure s deoed by CL. Our ma resul may be summarzed as follows: Proposo. Whe he mare demad fuco s cocave, ad he eulbrum s a saddle, he seady sae levels of ad uder he closed-loop decso rule are smaller ha he seady sae levels of ad uder he ope-loop decso rule. Oherwse, whe he mare demad fuco s covex, ad he eulbrum s usable, he seady sae levels of ad uder he closed loop rule are larger ha he seady sae levels of ad uder he ope-loop rule. Fally, whe demad s lear, oly he degeerae seady sae wh zero vesme ad l mare sze does exs. Ths resul s seemgly coras wh he acured wsdom. I fac, he avalable models of dyamc olgopoly sugges ha he closed-loop soluo eals larger seady sae levels of vesme e.g., vesme produco capacy or R&D ha he capacy assocaed wh he seady sae uder he ope-loop Nash soluo cocep. The reaso s ha, uder he closed-loop soluo cocep, all players ae o accou he smulaeous aco of rvals, ad reac by vesg more ha hey would do o he bass of he plas desged a he begg of he game accordg o he ope-loop Nash soluo see, e.g., Fudeberg ad Trole, 983, 99 ch. 3; Reyolds, 987, 99. The

14 erpreao of hese resuls reles upo he aemp a pre-empmg rvals by acurg a large capacy or by vesg a large amou of resources R&D acves. By coras, he prese model, adversg vesmes produce a recprocal spllover, whch frms ae o accou whe playg he closed-loop eulbrum. Therefore, subgame perfeco eals lower adversg effors ad herefore lower spllover effecs o he beef of rvals as compared o wha emerges a he wealy me cosse soluo Full carelsao ad socal plag I hs seco, we sudy he model uder wo alerave hypoheses: all he frms he mare buld a carel where all decso varables are se cooperavely, order o maxmze he prese value of he jo profs. We label hs case as full carelsao; he decsos cocerg vesme ad oupu levels are ae order o maxmze he prese value of he socal welfare,.e., as f he mare were ruled by a socal plaer amg a he maxmzg oal surplus. We wll refer o hs case as socal plag. I he case of a carel, he dyamc problem ca be summarzed as follows. We have o fd he opmal plas of each of he symmerc frms cocerg dvdual produco ad vesme, order o acheve he maxmum prese value of he flows of fuure jo profs. Formally, he problem s 6 Max J s.. e 0 ρ [ ] d d b d The correspodg Hamloa fuco H FC s: FC ρ 7 H [ ] b λ[ ] e. 6 aalogous resul s obaed by Beruzz ad Lamber 00 a model of horzoal produc dffereao where frms adversg campags are amed a creasg he desy of cosumers alog Hoellg s lear cy see also Pga,

15 The frs order codos are: 8 FC H 0 FC H 0 λ b FC H dλ ρλ d [ ] [ ] 0 alog wh he al codo 00 ad he rasversaly codo ρ lm λ e 0. The frs codo of sysem 8 yelds: 9, whch s clearly smaller as compared o s couerpar he case of Couro olgopoly. By he usual mapulao of he remag codos of he sysem we oba: 0 d d / / ρ [ ], b ad we ca characerze he dyamc sysem uder he carel rules. I parcular, seady sae, d/d0 mples ha / ad d/d0 mples / /[b ρ / ]. I he space,, he opmal vesme fuco correspods o a curve whch may be covex f 0,, cocave f,, or lear f. I ay case, hs curve les above s couerpar uder he ope-loop decso rule, ad a foror above he releva curve obaed uder he closed-loop decso rule he o-cooperave game. s a coseuece, a seady sae po exss also he carel case ad s dyamc properes are he same as he Couro cases. I parcular, hs seady sae s a saddle uder he case, wha follows:. Moreover, s also easy o verfy 4

16 Proposo 3. For all,, he carel eulbrum s a saddle po, ad he seady sae levels of ad uder full carelsao are boh larger ha he seady sae levels of ad he ope-loop ad closed-loop Couro-Nash eulbra. Needless o say, f 0, seady sae values of ad uder he charel case are boh smaller ha he respecve varables uder Couro behavour, whle he seady sae does o exs f. very smlar procedure ca be adoped o fd he decso of a socal plaer amg a he maxmum prese value of he socal welfare flows. Socal welfare a me s defed as he sum of he cosumer surplus ad e profs of frms a ha me. Uder he symmery hypohess, cosumers' surplus a me ad aggregae profs are respecvely: CS [ s ] ds Q 0 / [ Q ] b π. We assume ha he plaer dscous fuure welfare flows a he same cosa rae ρ > 0 as frms, so ha hs dyamc problem s: 3 ρ Max SW e 0 d s.. d Q [ Q ] / b d / Solvg he plaer s problem, we fd ha d/d0 mples /[b ρ ], whch s a locus ha les always above all s couerpars he prevous games, he space,, boh whe s cocave, ad whe s covex. The erseco wh he 5

17 sragh le d/d0 gves he seady sae see, oce aga, fgure, where he le he case of socal plag s labeled as SP. Therefore, we have: Proposo 4. For all, ha ay oher regme., he socally opmal adversg effor seady sae s hgher Oce aga, f 0, seady sae values of ad uder he socal opmum case are boh smaller ha he respecve varables uder Couro behavour, whle he seady sae does o exs f. 7. Comparsos across dffere regmes Table gahers he seady sae values of he releva varables uder he dffere cosdered regmes. To ease he exposo, we defe: β b ρ. Table. - Releva seady sae varable values across regmes. Reservao prce Ivesme Produco Ope-loop olgopoly OL β β OL β OL Closedloop olgopoly CL β CL β CL β Full Carelsao FC β β FC β FC Socal plag SP β SP β SP β 6

18 I he remader of hs seco, we cofe our aeo o he eresg case where saddle pahs lead o he seady sae,.e., we cofe o he case,, ad we proceed o compare he dffere seady saes. The aalogous exercse uder he case 0, s rval ad s omed for he sae of brevy. We ca easly draw he graphcal represeao of he dyamc sysem uder he four dffere regmes uder cosderao: he ope-loop Couro-Nash eulbrum labeled by OL, he closed-loop Couro-Nash eulbrum CL, full carelsao FC ad socal plag SP. I all cases, he locus d/d0 s represeed by he sragh le /, whle he locus d/d s a cocave curve; we have show ha he poso of such curves he space, s clear-cu. I parcular, he releva curve uder he closed-loop olgopoly les below he curve correspodg o he ope-loop olgopoly, ha ur les below he curve assocaed o he carel case, ha fally les below he curve of he socal plag problem. INSERT FIGURE HERE The graphcal represeao, as well as he aalycal sudy based o Table, show ha reservao prces ad vesme seady sae are such ha CL<OL<FC<SP ad CL<OL<FC<SP. s o he amou of he dvdual frm's produco seady sae, he order s o uvocal. Ideed, he followg cha of euales urs ou o hold: CL<OL<SP, whle he produco uder he carel s smaller ha he produco of he plaer, bu may fall dffere posos wh respec o he levels of produco assocaed wh he Couro-Nash eulbra. I parcular, hree cases are possble: CL CL < < FC FC OL < < < OL CL FC ff < OL ff ff 0 < < < < < 7

19 ccordgly, he rag of he socal welfare levels across regmes s o uue. Pu dfferely, he socal welfare level s o a moooc fuco of. Ideed, a larger value of mples larger cosumer surplus ad frms operave profs, bu also reures a larger vesme. Neverheless, s possble ha he rag of socal welfare levels seady sae replcaes he rag of he eulbrum values of. I such a case, he maxmum socal level s obvously aaed uder socal plag, followed by he allocao chose by he carel, ad he by he allocaos assocaed wh he ope-loop Couro-Nash eulbrum ad he closed-loop Couro-Nash eulbrum. 7 I s also worh sressg ha, hs dyamc game, he full carelsao amog frms s o ecessarly dermeal o socal welfare, as compared o he eher of he o-cooperave games played à la Couro. The po ha colluso may have posve effecs, able o more ha compesae cosumers for he egave effecs of collusve prces has bee already made, dffere models. Recely, Fershma ad Paes 000 ad Paes 00 show ha colluso may affec he varey, cos, ad ualy of he producs mareed by a dusry, ad hs may affec he welfare as do he prce effecs of colluso. Our prese model adds he effec of adverg o he ls of facors leadg o he cocluso ha - a dyamc framewor - socey s o ecessarly beer off whe carels are forbdde. Ths amous o sayg ha here exs releva crcumsaces where he per se rule mgh profably be replaced by a more elasc rule of reaso. 8. Coclusos We have aalyzed a dffereal olgopoly game, where frms ves order o crease he cosumers' reservao prces a mare where he demad fuco s olear. The cosderao of o-lear mare demad has allowed us o hghlgh ha he curvaure of he mare demad affecs he sably properes of he seady sae eulbra. I parcular, he prese model, whe he mare demad s lear o meagful seady sae exss, besdes he rval suao where reservao prce ad 7 Ths ca be easly verfed by umercal calculaos, whch are avalable from he auhors upo reues. 8

20 vesme are zero. Whe he mare demad s cocave or covex, a meagful seady sae does exs. I s a saddle he former case, whle s usable he laer. We have also show ha seady sae vesmes ad oupu levels sae are sesve o he soluo cocep. I parcular, whe he mare demad s covex, he eulbrum vesmes by frms ad he assocaed cosumer reservao prce seady sae are larger uder he closed-loop decso rule ha uder he ope-loop oe. I hs case, however, he seady sae s usable. I he case where he mare demad fuco s cocave, ad he seady sae s a saddle, he closed-loop eulbrum vesmes ad he reservao prce are smaller ha he opmal vesmes ad reservao prce emergg a he ope-loop eulbrum. Ths case s he mos eresg oe, o oly because a saddle pah leads he sysem o he seady sae eulbrum, bu especally because he cocluso coradcs he wdespread dea ha closed-loop eulbrum eals larger vesmes o he par of frms, as compared o he ope-loop eulbrum. Ths s o rue he prese model, because adversg vesmes produce a recprocal spllover whch frms ae o accou whe payg accordg o he closed-loop Nash eulbrum. Eulbrum allocaos uder Couro olgopoly have bee compared o he opmal allocao uder full carelsao ad o he opmal behavour of a socal plaer amg a he maxmzao of dscoued socal welfare. I hs respec, he ma resul s ha he socal welfare level assocaed o he seady sae allocao of he carel may well be larger ha he socal welfare level geeraed by he fully o-cooperave Couro olgopoly, boh he ope-loop ad he closed-loop case. Ths s due o he fac ha a full exeraly characerzes he adversg campag, ad he carel vess cosderably more ha he populao of o-cooperave Couro frms. The beefs from hs fac s larger ha he damage dervg from he resrco of uay uder he carel agreeme. 9

21 REFERENCES derso, S.P. ad M. Egers 99, Sacelberg vs Couro Olgopoly Eulbrum, Ieraoal Joural of Idusral Orgazao, 0, derso, S.P. ad M. Egers 994, Sraegc Ivesme ad Tmg of Ery, Ieraoal Ecoomc Revew, 35, Basar, T. ad G. J. Olsder 98, 995 d, Dyamc Nocooperave Game Theory, Sa Dego, cademc Press. Beruzz, G. ad L. Lamber 00, dversg a Dffereal Game of Spaal Compeo, worg paper o. 400, Dparmeo d Sceze Ecoomche, Uversà d Bologa. Cell, R. ad L. Lamber 00, Dffereal Olgopoly Games where he Ope-Loop Memoryless ad Ope-Loop Eulbra Cocde, worg paper o. 40, Dparmeo d Sceze Ecoomche, Uversà d Bologa. Dorfma, R. ad P.O. Seer 954, Opmal dversg ad Opmal Qualy, merca Ecoomc Revew, 44, Docer, E.J, S. Jørgese, N. Va Log ad G. Sorger 000, Dffereal Games Ecoomcs ad Maageme Scece, Cambrdge, Cambrdge Uversy Press. Ercso, G.M. 985, Model of dversg Compeo, Joural of Mareg Research,, Fechger, G. 983, The Nash Soluo of a dversg Dffereal Game: Geeralzao of a Model by Lema ad Schmedorf, IEEE Trasacos o uomac Corol, 8, Fechger, G., R.F. Harl ad S.P. Seh 994, Dyamc Opmal Corol Models dversg: Rece Developmes, Maageme Scece, 40, Fershma, C. 984, Goodwll ad Mare Shares Olgopoly, Ecoomca, 5, 7-8. Fershma, C. ad S. Nza 99, Dyamc Voluary Provso of Publc Goods, Europea Ecoomc Revew, 35, Fershma, C. ad. Paes 000, Dyamc Olgopoly wh Colluso ad Prce Wars, Rad Joural of Ecoomcs, 3, Fredma, J.W. 983, dversg ad Olgopolsc Eulbrum, Bell Joural of Ecoomcs, 4,

22 Fudeberg, D. ad J. Trole 983, Capal as a Commme: Sraegc Ivesme o Deer Mobly, Joural of Ecoomc Theory, 3, Fudeberg, D. ad J. Trole 99, Game Theory, Cambrdge, M, MIT Press. Jørgese, S. 98, Survey of Some Dffereal Games dversg, Joural of Ecoomc Dyamcs ad Corol, 4, Jørgese, S. ad G. Zaccour 999, Eulbrum Prcg ad dversg Sraeges a Mareg Chael, Joural of Opmzao Theory ad pplcaos, 0, -5. Lamber, L. 996, Carel Sably ad he Curvaure of Mare Demad, Bulle of Ecoomc Research, 48, Lema, G. ad W.E. Schmedorf 978, Prof Maxmzao hrough dversg: Nozero Sum Dffereal Game pproach, IEEE Trasacos o uomac Corol, 3, Mar, S. 993, dvaced Idusral Ecoomcs, Oxford, Blacwell. Mehlma,. 988, ppled Dffereal Games, New Yor, Pleum Press. Nerlove, M. ad K.J. rrow 96, Opmal dversg Polcy uder Dyamc Codos, Ecoomca, 9, 9-4. Paes. 00, Framewor for ppled alyss Dyamc I.O., Pleary Lecure a he 8 h ual Coferece of he Europea ssocao for Research Idusral Ecoomcs, Dubl, ugus 30 Sepember, 00. Pga, C. 998, Dyamc Model of dversg ad Produc Dffereao, Revew of Idusral Orgazao, 3, Pga, C. 000, Compeo a Duopoly wh Scy Prce ad dversg, Ieraoal Joural of Idusral Orgazao, 8, Regaum, J. 98, Class of Dffereal Games for Whch he Closed Loop ad Ope Loop Nash Eulbra Cocde, Joural of Opmzao Theory ad pplcaos, 36, Reyolds, S.S. 987, Capacy Ivesme, Preempo ad Commme a Ife Horzo Model, Ieraoal Ecoomc Revew, 8, Reyolds, S.S. 99, Dyamc Olgopoly wh Capacy djusme Coss, Joural of Ecoomc Dyamcs ad Corol, 5, Seh, S.P. 977, Opmal dversg for he Nerlove-rrow Model Uder a Budge Cosra, Operaos Research Quarerly, 8,

23 Trole, J. 988, The Theory of Idusral Orgazao, Cambrdge, M, MIT Press. Vdale, M.L. ad H.B. Wolfe 957, Operaos Research Sudy of Sales Respose o dversg, Operaos Research, 5, Vves, X. 000, Olgopoly Prcg: Old Ideas ad New Tools, Cambrdge, M, MIT Press.

24 Fgure : Ope-loop dyamcs " ¾6 """ " """ 6 - " """ " """ " """ -? " """ d d 0 " """ " """»»»»: " """ - " "" "? " """ d d 0 " """ ¾ - "? "" " ¾ " "" " ¾6»»9»» " """ ¾??? d d 0 d d 0 d d 0 d d 0

25 Fgure : Seady saes uder d ere soluo coceps 6 dd0 dd0sp dd0fc dd0ol dd0cl -