Journal of Econorm Behavior and Brganizati~m 4 (1983) 249-264.North-Holland ENDQGEN6IuS CONJIWI’URA,L VARIATIQN!i IN OLIGOPOLY* Joel M. GUTTMAN Bar-lion Univmit y, Hatnat-C;avi , Israel Michael MILLER Unioersity of California, Los AngeLs. ( ‘A 90024, USA Conventional otigopoi!* theories ..yrovide no me& ?ism br ohgopolisks to learn from their experience of other fir& behavior in formiq; cctnj, *:tural variations. This pqer explores two approaches to the solution of this problem. In the mst (tame‘consistznnt conjectur& approach), oligopolists passively react to changes in other firms’ outputs, so as to keep their own profits at a maximum . Other firms correctly anticipate these reactions. In the second approach {the katching behavior’ approach), firms actively choose reactions whici, wi?l il-tduee other firms to r~zh~. their outputs. Again, these reactions are correctly anticipated by the other tirrr~.. The implications of t&se alternative models are derived and contrasted with the implications of the Cournot and join&profit mGmization models. 1. InmdIIctiQu A major difficulty - perhaps the major diffkulty - of I-;odeliq oligopoly is the problem of specifag how each oligopolist anticipates his rivals wdl react if he varies his price or output. Depending on what anticipation:s each oligopolist is assumed to have, one obtains quite diBerent soMaIns to ;l;e oligopoly probltm. It is widely recognized that the claskal oligopoly th~ori.es have a seno::s common flaw: they arbitrarily and nzechanically msme some yarticuler anticipation on the part of the oligopolists which may not be sonifirr;,ed by the oligopolists’ experience. The Coumo: solution, for exampie, simply assumes that each oligopakt anticipates no reactdorr at all by his Y,VBIS as he varies his own output. This zero konjectwal variation’ wc_mIdtx belietf b! the oligopolist’s experience, except in a p:.!rely ~y~~t~~~t~~a~ world irr which the determinittion of the ali~opoly’s output was a or,e=shot aSair. Ol67-2~58H/83,‘$3.QO 0 1983, Else~er Science Publishers B.V. (North-HollandE
Transcript
Journal of Econorm Behavior and Brganizati~m 4 (1983) 249-264. North-Holland
ENDQGEN6IuS CONJIWI’URA,L VARIATIQN!i IN OLIGOPOLY*
Joel M. GUTTMAN
Bar-lion Univmit y, Hatnat-C;avi , Israel
Michael MILLER
Unioersity of California, Los AngeLs. ( ‘A 90024, USA
Conventional otigopoi!* theories ..yrovide no me& ?ism br ohgopolisks to learn from their experience of other fir& behavior in formiq; cctnj, *:tural variations. This pqer explores two approaches to the solution of this problem. In the mst (tame ‘consistznnt conjectur& approach), oligopolists passively react to changes in other firms’ outputs, so as to keep their own profits at a maximum . Other firms correctly anticipate these reactions. In the second approach {the katching behavior’ approach), firms actively choose reactions whici, wi?l il-tduee other firms to r~zh~. their outputs. Again, these reactions are correctly anticipated by the other tirrr~.. The implications of t&se alternative models are derived and contrasted with the implications of the Cournot and join&profit mGmization models.
1. InmdIIctiQu
A major difficulty - perhaps the major diffkulty - of I-;odeliq oligopoly is the problem of specifag how each oligopolist anticipates his rivals wdl react if he varies his price or output. Depending on what anticipation:s each oligopolist is assumed to have, one obtains quite diBerent soMaIns to ;l;e oligopoly probltm.
It is widely recognized that the claskal oligopoly th~ori.es have a seno::s common flaw: they arbitrarily and nzechanically msme some yarticuler anticipation on the part of the oligopolists which may not be sonifirr;,ed by the oligopolists’ experience. The Coumo: solution, for exampie, simply assumes that each oligopakt anticipates no reactdorr at all by his Y,VBIS as he varies his own output. This zero konjectwal variation’ wc_mId tx belietf b! the oligopolist’s experience, except in a p:.!rely ~y~~t~~~t~~a~ world irr which the determinittion of the ali~opoly’s output was a or,e=shot aSair.
ifre igj~;~m&. We 41 these a~ti~pa~~~s enbog~o~ conjectural variatiolns t,CV& be+,2ause one can view the ~~ig~~~~jslt~ as hxrning from their eqxrience of gibDl[ r~&om by their tit&s. We djs~il~g~~h two types Of ohgopolistic iatefaeaions: (a) ~GB,T&, in wh&h all ~C~QF!i adjmt their owu ~utgut ievcls SO as to k~p their profits at a maximum, disregarding the efEet of the im@icd ~~~~ on their riva$$ uutputs (but eoraeetl~ a~~~~~ating the X&M% aefibrs’ resetions), and (b) QLT& in which a~tczrs actively choose reactions that have ar&cipated eff’eets cm other stws” levels of output, and abandon the sbjective uf ma:&&ing their own pro&s when ather actors’ outputs are nQt Pi1 their eC@hbrium levels.
We shcw that the passive ~~ter~~~~~s model (which has recently been rrxka.mmd and ehbomkd ply a g~owinp, ‘consistent ccmjectures hterature)’ t.1) canner e@ain vtshr~tary cx&rsioa, pi:edMing a Im::er output than that I,, .i ahe G~rnot model, at .‘east for identical ae&s, aad (b) predicts iradustry !srauilib+a virtualfy idemical to simple Cour sot equilibria as the number of SJms iIKreases -- ;xt least under the simglifyiag assumptions made here.
Tn order t42 expiain voluntary C&Z.&XI, ii model nf active iuteracticms is then devehqxd. 7 ‘he model ol active mteractions IF based on work by Guttman (197Q2 Active reactions abe simply tlafr:ats that if another ohgopvxist &an&% his output, on& own rutput wiii respond in a certain fashmn. L&Eke the czuopemtive game of Nash (l953), however, we explore the case - w&h if considerabgv mom relevant to ohggopoly as well as to p&tiu;p! behavior - that these threabs are not enfixp4, but rather are self- enforcing By ‘~#-enforcing’ we mew that, wBmt rhe other firms are ~r~u~g their q8tima: outputs given firm i’s matching commitment, firm i
is ah producing its spt%maI f it-mxCimiting~ cutput (again, ta’king account cbf its -rmdtiti remif the qther firm’s I,utputs). If, however, the &her Grms deviate from the&t _&it-maximizing outputs in order to ‘test’ firm r”s rea&x2, then firm i aIs0 will not be producing ;~t its optimal output, because crf its committed reaetiox In su& a situation, firm i’s behavior is, seem&&, irrz~ti~md. We pusMate, howeper, that firn,l I nevertheless will %&iw thruugb” and produce ai its ccxnmit&d output level, because the long- Pxr e&s t R ~~~~~~~g it s r,3mmitment outweigh the short-run gains of keepi2g its 02Fp? . 3: al: its opt&al Ievef. Thi: long-run benefits of ~u~~~~g it: ~~rn~~t are that the comzzitment will then be believalrle a& &IS other‘. firm tviff be indud to prorf~tce at lower output levels, raising the price of
the prOduCt. THUS, @VUI a long enough time horizon (more precisely, to follow the results of the supergame literature9 an infinite horizon .3r one with a sufficierstly unc&tin endpoint), firm i’s behavior is truly ration& It should be borne in rmiild, moreover, that firm. i know-s that the other firms’ tests cause them to produce suboptimal outputs, and thus the tests c:an only be temporary.
We have not, however, dtlreloped a general theory of non.caoperative threat-making. In order to obtain determinate results, we ;hxve 5~::~t forced to
make strong simglifying assumpt.ir,ns. It i, assumed thai t’tie reaction coefficients are chosen before the actual levels of output arc ,-ho:;en. [Levine (lQ81, ch. 3), derive> this two-step structure in a study which further develop:, and formalizes the idea of matching behavior as applied to o1ii;opoly.j The oligopolists’ ‘reacticn coeficients’, moreover, are assumed to 1c constants, i.e., each oligopolist’s rlutput is line.arly related to other firms’ ou~puts.~
Although- we have borrow:d the distir;ction between short-n:!n and iong-
run rationality and the associated iimitatiorr ‘to !infinite- a: uncertain-horizon games from the supergame literature, 4 the iIut:_Gti!on involved in matching behavior is quite dNerent from the Nash equilibrium assumed br/ that literature. In the supergame literature on :he Prisoner’s Dilemma, mtihiple equitibria typic&f result, &cause the only requirement of the Nash equilibrium without precommitments .s that a given strategy ‘hold its own’ against the opposing strategies of uno)nditional non-cooperation (defe.:t;on) and cooperation. Thiis cat be seen by referring to table 1, which depicts t1.2 payoff matrix of the two-persoal Prisoyler’s D:;emma extended t? include the tit-for-tat strategy. In this matrix, C is the unconditional cooperative strategy, N is the unconditional non-cooperative ?trattgj, and TFT is rhe tit- for-tat strategy which plays C in the first round of the game and the other player’s strategy nf the previous round in ali subsequent rouslds of the game. As can be easily verified, TFT is n:, worse 1han any c ’ er strategy when playing against TFT Thus (TFTlf TFTJ is a Nzsh equilibrium. When playing against N, however, N is no worse tharir TF7’ (and better than C;, SO
‘One basis for &rch a linearity assumptiGm would be an argument from the existence IA ,imperfe:t information and, perhaps, bounced rationality. See, e.g.. Simon ( 14591. (We are indebted to J. Hirshleifer for this point.) It nosy tw argued that in taks tme for each duopolist to learn his rival’s reactiou strategy, and that -.o impute hi$er-order terms in the nval’s reacti~ strategy would be ~~I~I~~o~~~~, given the Irecessity OF c,ntiru I :s*isiy rcvisina one’s CT in an> case. Set: l..evine: (1981) for a further justifica~ron of thir. lineatiky assumptirw See atso Tidemnn
(n.43.). for a ~a~c~iat~~~ that i%ei=e are 10’? strategies for rbr Pr:soner’s Dilemma repeated ~only P tlarneg and the ‘a csrmpukr the size of the i mt , GXIlQO &Xi Of ~~riC~OQKXX%S0~~ ON .!+d@SstWtIl
!lO-io meter, on a side, ea& capable of evaiuadng a ~rai-_gy in a nanoieco& I LO
wxm~). . _ wodd take about 2M million year F> to evaluate roll the strategies This ckarly su_~ests the ~‘~~sih~~~t~ of a l~rn~t~t~~~ on the strateg! space.
4Tr16.Z fX3piiiCisl :ckvZIXZ Of I’ f Cd-ga.iX &TCCk which lc& tC 121ir im~lalliw mLt\ 13;
que4oned, however, Even in gat les v;ith .f IC~~WR .mdp,jin,. redi wvor!,J rit~>rh ~ii;ib ~:i3i!icii
Leoutief (1936) showed IWW duop4ists can have mutt& ly consiste:at., ‘rational’ expectations. ’ Each duot>olist knows the reactim of his riva: if he: were to chaage his own output, and bases his. &oice of output on that expected reaction. Neither duupolcst actively ‘chooses* a. reaction function in the sense of an active matching 01 threat policy; rather, path passively rea:ts to the other firm’s choice of output.
A formal rnodei of such interactions was provided, in a more general context, by Marschak and Selten (1978). In I heir game, each firm chooses a level of output and a reaction strategy tha: shows how its output would change if the ot!rer firm were to depart from its l:urrent cutput. The equilibrium conjectural variations in Leontiers moclel are ‘stable’ and ‘restabilizing’ in the Marschak-Selten sense: i.e., no firm would c.hange its output, knowing the reaction of its rival, and no firm would have an incentive not to f?low through with its announced reaction if it is called upon to do so.
Leontief beiieved that *his generaiization of the Stackelberg model would only rarely yield an equilibrium. 6 This belief, however, appears to be incorrect. It is not difficult to show that under Leon tief’s assumptions of linear deman.d curves and upward-sloping, linear marg,.nal cost curves, t.wo identical duopohsts will always find an e.@ibrium pair of CVs and outputs. There is cnly one economicafly interesting equilibrium. As in the model of active interactions, we assume that each duopolist treats its rival’s output as responding linearly to its ovm output. [Bresnahan (1981) has shown that the only consistent conjecture2 in a model such as our passive interactions model, among the class of polynctiais, are linear conjectures.] Each firm’s ‘reaction coefficient’ (which is the CV anticipated by its rival) is its besi reaction soefficient given the reaction coefficient of its rival.
Let the demand curve facing each of two ;dentioal duopolists be
p = K - a(~~ + q2), where
P =the price of their output, 419% ==their outputs, and K,a = positive constants.
Moreover, let each duopolist’s mar-ginal cost curve be
Wegishi ad Okuguchi (19%) c&doped a similar model. Thcii ma&d, howevei, postula es It-r, each cfuopo!ist assumes t.hat its rival is a ~oilower’ in Stackelberg’s seme. The Wegis‘ Ii- Oktliguchi equilibrium simply &es the resultisg Cv”s consistent with each other
%emtief (1936, p. 556) QesmiM his equilibrium LU ‘not very probable, but neverthdzss t~~eoret~~~~y s~~~~~t~.
opportunity hxks kr Iim 1. Firm 1 chooses the matching rate which r.mx&&m its pro&$ as Miited hy 8 tmgency hfAween F and firm l’s higlmt attainable improB curve, here deuoted as 17;. Tihils the M shown in the fisre is firm I’s opt.&4 uxzt&ng liue$ and El the resulting equililxiusn. If f&m 2 had been the lirst to commit its& to a matchilitg po3icy, it Flvould have beea alble to ‘choose’ .a equililxiuxu more favorable to itseli& such as E,.
Notq however, that J& is not if joint 0ISimum for the two fhms, simx the two fir&s imp&t curves are not tangat at that poiut.8 But the pmcess me4 not Y&P ret El. From E,, a further matching cmmuitment wo?Gd be forthcomi~~g EroGz one of &e two &rm. and press would clearly p~‘omed until the I%.ret~~ *@oatfact curve’ was reached, if only asymrptol&ally. It is not para&xical that hrm P itself may maka the new ma:chiag cormiiiut after & is reached, because El is opt ml for firm f only
r-f&&w to the &art&g SKI& i?&. &tee El is reached, a new M, with a differewrt 5loI3e, !xm3mes ojptimd f& firm 1.
The m%%iztg behavior qu%briurn, given this asymmetric Sehavior of the two firms, is &us a Pareto-op&ml equiiihrium,“) The assumed asymmetry,
J.&f. Guttvmn nn;l M. Mih, Bndogmow conjecm,al vwiatians in oligopoly 259
however, may be theoretically unsatisfying, the final equilibrium depending on which firm first commits itself to a matching policy. What kind of cquihbriuw w~ub result if the two firms simultaneously announced matching strategies? We then would require b6th firms to have both autonomous (‘flat’) and matching components in their outputs. To describe the model formal@, let a1 brs the (flat’ (autonomous) output of firm i, and b, be its reaction coefficient. Then
4s = ai + bia,
is the output of firn. i, where aj is the flat output of firm j, its counterpart. To visualize hov such a process could occur wrthout sny coordination
between the duopoiists, consider a sequence of outputs by the duopolists continuing over time. Occasionally, one firm ‘tests’ the other by varying its Ylat’ output from its profit-maximizing level and watching the response of the other firm. With t&s information on the response of its rival, the fum adjusts its flat output 50 that, taking its rival’s reaction into account, it is maximizing its profits. Each firm determines its reaction so that its profits ;are ma ximized, *aking into account how its choice of reaction coefficient .affects its rival’s ‘Bat output and thus affects its own profits.
This maximization of profits by varying one’s reaction coefficient is .accomplished b:l predicting the equilibrium levels of !Jiat outputs of both iirms given any pair of reaction coefficients, b, and b2. The firm’s profits are :not maximized zrt any other pairs of outputs. When either firm ,s ‘tested, its rival has moved away from this equitibrium, and tht*.s the firm being tested &IS an incenti\x to fail to react in its earlier-revealed .manner. We assume.. however, that c: l.ch firm views such costs as being transitory and therefore: aegligible in arty long-run cakx&&on s,uch as the one #considered here. The knefits of reacting as previously revealed or ‘annoumxd’, in contrast, are permanent - they are maintaining one’s credibility and thus maintaining an equilibrium pair of outruts that maximizes the km’s long-run profits.
Although the process through which we envision reaction rates being revealed is tllu:; a multistage or repeated game, we model it here as a simple two-stage ga,ne, in which the reaction rates are first determined and t?en the ‘flat’ outputs are determined. We lose nothing by such a simplification, sirlse we are interested only in characterizing an equilibrium in which the react on rates and flat outputs repeat themselves mdefmitely over time. Gver! :he ‘automaticit y’ off the outr me or outcomes of the game dctermir’ng the lat xktputs, we redly are 6,. ordering ouly a one-stage game in which the firms choose a matching strategy, which they continue to use indefinitely. Ti a:107”f: det~~rn:~~~g the flat outputs serves simply to define the payoff funst~~n TCI!_ the parze ~ete~~~~~~~r~~ the reactio or ~a~~h~~g rates (the hi)* since tk! b, are &c>sen so 8s to ma#&ize ‘., e kr S gsrxK-.ts FiVen tk resxitin$
firm I’s r.!.
First, ?h%3 ldtfr $ma!leF marginal CoStS choose ]laFgeF reaction coefficients. This is an impli:a’.ion which is testable, in principle, e.g., with experiment,31 dile2. Second an interesting relationship between Q*, Q,, and Q, emerges. The, predicte~l -_quihbrium output (which, with non &xha~ ii~mr; invohes :,
Slight indcte!*rainacy12) iS We-t%Fd @f the way ffom the ihmclt t& to the
joint-profit maximizing equilibrium Q,. The questions of whethtv thls holds in all cases or whether sinnilar regulaktiea appear with more than two actors remain Co be investigated. Nevertheless, it is significant thsir the present model goes part of the wa;,t to explaining cooperation by cluc~polists without invoking the usual <>quirement of cooperation - enforceable agreements.
We have desctrbed UWQ alternative n&odeis of endogenous ronjeciural variations: a model of ‘passive interactions’ originally suggested by heontief, and a model of ‘a&z interactions’ based on the notion of matching behavior. 4)n the theor,etica; liont, a nuz~ber of improvements and generalizations could be ma&z. The ana!ysis of active interactior.s could be generalized to thr: case of more than 2 firms; non-linear demand curves and variable reaction coefficierrtu cor;ld be investigated. But perhaps a more promising line of futwe ixquiry would be empirical. The resrlks to date on this issue nave been mixed, On the one hand, she experinr.ental WCK’C of Hoggat: (i967, 196) indicates the existence of maWE :no, beh,d dior, supporting the mode1 of active in;teractioas. IOn the other i~a1~3, emrirkal work by Iwata (1974) on th.e Japanese flat glass industry founci bo,ih pziitive and negative reaction coeffscients. l3
