Oligopoly Theory Made Simple

Transcription

1 Olgopoly Theory Made Smple Huw Dxon Chapter 6, Surfng Economcs, pp 5-60.

2 Olgopoly made smple Chapter 6. Olgopoly Theory Made Smple 6. Introducton. Olgopoly theory les at the heart of ndustral organsaton (IO) snce ts object of study s the nterdependence of frms. Much of tradtonal mcro-economcs presumes that frms act as passve prce-takers, and thus avods the complex ssues nvolved n understandng frms behavour n an nterdependent envronment. As such, recent developments n olgopoly theory cover most or all areas of theoretcal IO, and partcularly the new IO. Ths survey s therefore very selectve n the materal t surveys: the goal s to present some of the basc or core results of olgopoly theory that thave a general relevance to IO. The recent development of olgopoly theory s nextrcably bound up wth developments n abstract game theory. New results n game theory have often been appled frst n the area of olgopoly (for example, the applcaton of mxed strateges n the 950s see Shubk 959, and more recently the use of subgame perfecton to model credblty). The flow s often n the opposte drecton: most recently, the development of sequental equlbra by Kreps, Mlgrom, Roberts, and Wlson arose out of modellng reputatonal effects n olgopoly markets. Over recent years, wth the new IO, the relatonshp wth game theory has become closer. Ths chapter therefore opens wth a revew of the basc equlbrum concepts employed n the IO Nash equlbrum, perfect equlbrum, and sequental equlbrum. The basc methodology of the new IO s neo-classcal: olgopolstc rvalry s studed from an equlbrum perspectve, wth maxmsng frms, and uncertanty s dealt wth by expected proft of payoff maxmzaton. However, the subject matter of the new IO dffers sgnfcantly from the neo-classcal mcro-economcs of the standard textbook. Most mportantly, much of the new IO focuses on the process of competton over tme, and on the effects of mperfect nformaton and uncertanty. As such, t has expanded ts vson from statc models to consder aspects of phenomena whch Austran economsts have long been emphassng, albet wth a rather dfferent methodology. The outlne of the chapter s as follows. After descrbng the basc equlbrum concepts n an abstract manner n the frst secton, the subsequent two sectons explore and contrast the two basc statc equlbra employed by olgopoly theory to model product market competton Bertrand (prce) competton, and Cournot

3 Olgopoly made smple (quantty) competton. These two approaches yeld very dfferent results n terms of the degree of competton, the nature of the frst-mover advantage, and the relatonshp between market structure (concentraton) and the prce-cost margn. The fourth secton moves on to consder the ncentve of frms to precommt themselves n sequental models; how frms can use rreversble decsons such as nvestment or choce of managers to nfluence the market outcome n ther favour. Ths approach employs the noton of subgame perfect equlbra, and can shed lght on such ssues as whether or not olgopolsts wll overnvest, and why non-proft maxmzng managers mght be more proftable for ther frm than proft maxmzers. The ffth secton explores competton over tme, and focuses on the results that have been obtaned n game-theoretc lterature on repeated games wth perfect and mperfect nformaton. Ths analyss centres on the extent to whch collusve outcomes can be supported over tme by credble threats, and the nfluence of mperfect nformaton on a frm s behavour n such a stuaton. Alas, many areas of equal nterest have had to be omtted notably the lterature on product dfferentaton, advertsng, nformaton transmsson, and prce wars. References are gven for these n the fnal secton. Lastly, a word on style. I have made the exposton of ths chapter as smple as possble. Throughout the chapter I employ a smple lnearzed model as an example to llustrate the mechancs of the deas ntroduced. I hope that readers wll fnd ths useful, and I beleve that t s a vtal complement to general conceptual understandng. For those readers who apprecate a more rgorous and general mathematcal exposton, I apologse n advance for what may seem sloppy n places. I beleve, however, that many of the basc concepts of olgopoly theory are suffcently clear to be understood wthout a general analyss, and that they deserve a wder audence than a more formal exposton would receve. 6. Non-cooperatve equlbrum The basc equlbrum concept employed most commonly n olgopoly theory s that of the Nash equlbrum, whch orgnated n Cournot s analyss of duopoly (838). The Nash equlbrum apples best n stuatons of a one-off game wth perfect nformaton. However, f frms compete repeatedly over tme, or have mperfect nformaton, then the basc equlbrum concept needs to be refned. Two commonly

4 Olgopoly made smple used equlbrum concepts n repeated games are those of subgame perfecton (Selten 965), and f nformaton s mperfect, sequental equlbra (Kreps et al. 98). We shall frst ntroduce the dea of a Nash equlbrum formally, usng some of the termnology of game theory. There are n frms, =, who each choose some strategy a from a set of feasble actons A. The frm s strategy mght be one varable (prce/quantty/r&d) or a vector of varables. For smplcty, we wll take the case where each frm chooses one varable only. We can summarze what each and every frm does by the n-vector (a, a, a n ). The payoff functon shows the frm s profts π as a functon of the strateges of each frm: π = π (a, a,, a n ) () The payoff functon essentally descrbes the market envronment n whch the frms operate, and wll embody all the relevant nformaton about demand, costs and so on. What wll happen n ths market? A Nash equlbrum s one possblty, and s based on the dea that frms choose ther strateges non-cooperatvely. A Nash equlbrum occurs when each frm s choosng ts strategy optmally, gven the strateges of the other frms. Formally, the Nash equlbrum s an n-vector of strateges ( a, a..., a n ) such that for each frm, strateges of the n other frms a. That s, for each frm: a yelds maxmum profts gven the π ( a, a ) π ( a, a ) () for all feasble strateges a A. The Nash equlbrum s often defned usng the concept of a reacton functon. A reacton functon for frm gves ts best response gven what the other forms are dong. In a Nash equlbrum, each frm wll be on ts reacton functon. Why s the Nash equlbrum so commonly employed n olgopoly theory? Frstly, because no frm actng on ts own has any ncentve to devate from the equlbrum. Secondly, f all frms expect a Nash equlbrum to occur, they wll choose ther Nash equlbrum strategy, snce ths s ther best response to what they expect the other frms to do. Only a Nash equlbrum can be consstent wth ths ratonal antcpaton 3

5 Olgopoly made smple on the part of frms. Of course, a Nash equlbrum may not exst, and there may be multple equlbra. There are many results n game theory relatng to the exstence of Nash equlbrum. For the purpose of nductral economcs, however, perhaps the most relevant s that f the payoff functons are contnuous and strctly concave n each frm s own strategy then at least one equlbrum exsts. Unqueness s rather harder to ensure, although ndustral economsts usually make strong enough assumptons to ensure unqueness. 3 If market competton s seen as occurrng over tme, t may be napproprate to employ a one-shot model as above. In a repeated game the one-shot consttuent game s repeated over T perods (where T may be fnte or nfnte). Rather than smply choosng a one-off acton, frms wll choose an acton a t n each perod t =, T. For repeated games, the most commonly used equlbrum concept n recent olgopoly theory lterature s that of subgame perfecton whch was frst formalsed by Selten (965), although the dea had been used nformally (e.g. Cyert and De Groot 970). At each tme t, the frm wll decde on ts acton a t gven the past hstory of the market h t, whch wll nclude the prevous moves by all frms n the market. A frm s strategy n the repeated game 4 s smply a ruleσ whch the frm adopts to choose ts acton a t at each perod gven the hstory of the market up to then, h t : a t = σ (h t ) If we employ the standard Nash equlbrum approach, an equlbrum n the repeated game s smply n strateges( σ...,, σ σ n ) such that each frm s strategy σ s optmal gven the other frms strateges σ. Thus no frm can mprove ts payoff by choosng a dfferent strategy, gven the strateges of the other frms. However, a major crtcsm of usng the standard Nash equlbrum n repeated games s that t allows frms to make threats whch are not credble, n the sense that t would not be n ther nterest to carry out the threat. For example, consder the example of entry deterrence, wth two perods. In the frst perod, the entrant decdes whether or not to enter. In the second perod, the entrant and ncumbent choose outputs. The ncumbent could adopt the followng strategy: f entry does not occur, produce the monopoly output. If entry does occur, produce a very large output whch drves down the prce below costs at whatever output s chosen by the entrant. In effect, the entrant s posed wth a powerful threat by the ncumbent: f you enter, I ll 4

6 Olgopoly made smple flood the market and we ll both lose money. Clearly, wth ths powerful threat, the ncumbent wll be able to deter any entry. However, t s not a credble threat: f entry were to occur, then the ncumbent would not wsh to carry out ths potent threat. Thus the ncumbent s strategy s not credble, snce he would be makng unnecessary losses. Subgame perfecton was formulated to restrct frms to credble strateges. The basc dea of subgame perfecton s qute smple. In a Nash equlbrum the frm chooses ts strategy σ once and for all at the begnnng of the game, and s commtted to t throughout the play (as n the above example). To rule out noncredble threats, however, n a subgame-perfect equlbrum, at each pont n tme frms choose ther strategy for the rest of the game. The subgame at any tme t s smply the remander of the game from t through to the last perod T. Subgame perfecton requres that the strateges chosen are Nash equlbra n each subgame. Ths rules out non-credble threats, snce n effect t requres a frm to choose ts strategy optmally at each stage n the game. In our example, the ncumbent s threat to expand output s not credble : n the subgame consstng of the second perod, t s not a Nash equlbrum. Indeed, f the market s Cournot and there s a unque Cournot equlbrum, then the unque subgame-perfect strategy for the ncumbent nvolves producng the Cournot output f entry has occurred. One of the major attractons of subgame perfecton s that t narrows down the number of equlbra: there are often multple Nash equlbra n repeated games and mposng credblty on strateges can reduce the number consderably, at least n fntely repeated games. Wth mperfect nformaton, a commonly used equlbrum concept s that of a sequental equlbrum (Kreps et al. 98). Ths s formally a rather complex concept, but we shall provde a smple example n the secton competton over tme. The basc dea of subgame perfecton s employed, wth the added ngredent of Bayesan updatng of nformaton. 5 Frms may be uncertan about each other s payoff functons (e.g. they do not know each other s costs or each other s objectves). At the start of the game, frms have certan pror belefs, whch they then update through the game. Frms may be able to learn somethng about each other from each other s actons. In such a stuaton, frms of a certan type may be able to buld a reputaton by takng actons whch dstngush themselves from frms of another type. For example, n Mlgrom and Robets (98b) paper on entry deterrence, lowcost ncumbents are able to dstngush themselves from hgh-cost ncumbents by 5

7 Olgopoly made smple followng a lmt prcng strategy whch t s unproftable for hgh-cost frms to pursue. These reputatonal equlbra are very mportant snce they can explan how frms mght behave aganst ther short-run nterest n order to preserve ther reputaton ntact, for example as a low-cost frm or as an aggressve compettor. 6.3 Cournot and Bertrand equlbra wth homogeneous products The prevous secton consdered the concept of a Nash equlbrum n purely abstract terms. To make the concept concrete we need to specfy the exact nature of the strateges chosen and defne the payoff functon. Corporate strategy s, of course, very broad embracng all the actvtes of the frm prce, output, nvestment, advertsng, R & D and so on. In practce, olgopoly theory abstracts from the complexty of real-lfe corporate strategy and concentrates on just one or two strategc varables. There are two basc ways of modellng how frms compete n the market. The frst takes the vew that the frm s strategc varable s ts output and orgnates n Cournot (838). The second takes the vew that the frm s basc strategc varable s prce and, orgnates n the work of Bertrand (883), Edgeworth (95) and more recently n models of mperfect competton wth product dfferentaton (e.g. Chamberln 933; Dxt and Stgltz 977). As we shall see, whether prce or output s the strategy makes a dfference to the equlbrum outcome. For example, one of the basc ssues of nterest to ndustral economsts s the relatonshp between concentraton and the prce-cost margn. The standard noton that hgher concentraton leads to a hgher prce-cost margn s based on the Cournot vew, and does not hold n the Bertrand framework where there can be a perfectly compettve outcome wth two or more frms. The dstncton between prce and quantty settng n the context of olgopoly s not present n monopoly, where t makes no dfference whether the monopolst chooses a prce or quantty (the monopolst smply chooses a pont on ts demand curve). In order to capture the dstncton between the Cournot and Bertrand framework n ts starkest form, we wll frst consder the smplest case of homogeneous goods. We wll then dscuss what arguments there are for choosng between the two competng approaches to modellng product market competton. In the next secton we wll pursue ths fundamental dchotomy further n the context of the more realstc case of dfferentated commodtes. 6

8 Olgopoly made smple 6.3. Cournot-Nash equlbrum wth homogeneous goods The basc vew of the market taken by Cournot was that frms choose ther outputs and that the market then clears gven the total output of frms. There are n frms =, n, whch produce outputs x, ndustry output beng x = = x n. We wll make the smplest possble assumpton about demand and costs: n x = A Industry demand P = - (3) A Frm s costs c(x ) = cx Equaton (3) s called the nverse ndustry demand functon. Normally the ndustry demand curve s seen as arsng from the utlty-maxmzng behavour of consumers the market demand curve tells us how much households wsh to buy at a gven prce. The mathematcal operaton of takng the nverse, as n (3), has mportant economc mplcatons: t assumes that there can only be one market prce. Thus frms have no drect control over the prce of ther output, only an ndrect control va the effect that changes n ther own output have on the total ndustry output. Gven A A, we can defne the frm s payoff functon whch gves frm s profts as a functon of the outputs chosen: π ( x = x n, x,... xn) = x ( x j) j= x x j x j cx cx (4) Each frm has a reacton functon, whch gves ts proft-maxmzng output as a functon of the outputs chosen by the other frms. Snce frm treats the output of the other frm j? as fxed, the frst-order condton for maxmzng (4) wth respect to x s 6 : π x = j x j x c = 0 Solvng ths defnes the reacton functon for frm : 7

9 Olgopoly made smple n x j j c x = =,, n (5) Each frm has a smlar reacton functon, and the Nash equlbrum occurs when each frm s on ts reacton functon (.e. choosng ts optmal output gven the output of other frms). There wll be a symmetrc and unque Cournot-Nash equlbrum whch s obtaned by solvng the n equatons (5) for outputs (whch are all equal by symmetry), c c x = x = Cournot-Nash equlbrum output (6) n + whch results n equlbrum prce: n p c = + c (7) n + n + For example, f n = (monopoly) we get the standard monopoly soluton. For n =, x c = ( c) / 3, p c = /3 + (/3)c. The prce cost margn for each frm s: c c p c c µ = = (8) c p + nc There s a clear nverse relatonshp between the equlbrum prce-cost margn and the number of frms. As the number of frms become nfnte ( n ), the prce-cost margn tends to ts compettve level of 0: as the number falls to one, t tends to ts monopoly level ( c)/( +c), as predcted n Fgure. Fgure The prce-cost margn and the number of frms n the Cournot-Nash equlbrum What s the ntuton behnd ths relatonshp between number of frms and the prcecost margn? Very smply, wth more frms each frm s own demand becomes more elastc. Wth an nfnte number of frms the frm s elastcty becomes nfnte and hence the frms behave as compettve prce-takers. The representatve frm s elastcty? can be related to the ndustry elastcty? : p dx η = (9a) x dp 8

10 Olgopoly made smple p dx x p dx η = = (9b) x dp x x dp Under the Nash assumpton frms treat the other frms outputs as gven and the change n ndustry output x equals the change n frm s output. Hence dx /dp = dx/dp. Of course x /x s the th frm s market share whch, for our example, s n equlbrum /n. Hence (9b) becomes:? = n.? (0) In equlbrum each frm s elastcty s equal to n tmes the ndustry elastcty of demand. As n gets large, so does? leadng to approxmately prce-takng behavour Bertrand competton wth homogeneous products In hs famous revew, Bertrand crtcsed Cournot s model on several counts. One of these was the reasonable one that frms set prces not quanttes: the output sold by the frm s determned by the demand t faces at the prce t sets. What s the equlbrum n the market when frms set prces, the Bertrand-Nash equlbrum? If frms set prces the model s rather more complcated than n the Cournot framework snce there can be as many prces n the market as there are frms. In the Cournot framework the nverse ndustry demand curve mples a sngle market prce. In the Bertrand framework each frm drectly controls the prce at whch t sells ts output and, n general, the demand for ts output wll depend on the prce set by each frm and the amount that they wsh to sell at that prce (see Dxon 987b). However, n the case of a homogeneous product where frms have constant returns to scale, the demand facng each frm s very smple to calculate. Takng the case of duopoly, f both frms set dfferent prces then all households wll wsh to buy from the lower-prced frm whch wll want to meet all demand (so long as ts prce covers cost), and the hgher-prced frm wll sell nothng. If the two frms set the same prce, then the households are ndfferent between buyng from ether seller and demand wll be dvded between them (equally, for example). If frms have constant margnal cost there exsts a unque Bertrand-Nash equlbrum wth tow or more frms where each 9

11 Olgopoly made smple frm sets ts prce p equal to margnal cost the compettve equlbrum. Ths can be shown n three steps: Step. If both frms set dfferent prces then that cannot be an equlbrum. The hgher-prced frm wll face no demand and hence can ncrease profts by undercuttng the lower-prced frm so long as the lower-prced frm charges n excess of c. If the lower-prced frm charged c, t could ncrease profts from 0 by rasng ts prce slghtly whle undercuttng the hgher-prced frm. Hence any Bertrand-Nash equlbrum must be a sngle-prce equlbrum (SPE). Step. The only SPE s where all frms set the compettve prce. If both frms set a prce above c, then ether frm can gan by undercuttng the other by a small amount. By undercuttng, t can capture the whole market and hence by choosng a small enough prce reducton t can ncrease ts profts. Step 3. The compettve prce s a Nash equlbrum. If both frms set the compettve prce then nether can gan by rasng ts prce. If one frm rases ts prce whle the other contnues to set p = c, the lower-prced frm wll face the ndustry demand leavng the frm whch has rased ts prce wth no demand. The Bertrand-Nash framework yelds a very dfferent relatonshp between structure and conduct from the Cournot-Nash equlbrum: wth one frm the monopoly outcome occurs; wth two or more frms the compettve outcome occurs. Large numbers are not necessary to obtan the compettve outcome and, n general, prce-settng frms wll set the compettve prce-takng prce. Clearly t makes a dfference whether frms choose prces or quanttes. What grounds do we have for choosng between them? Frst, and perhaps most mportantly, there s the queston of the type of market. In some markets (for prmary products, stocks and shares) the people who set prces (brokers) are dfferent to the producers. There exsts what s essentally an aucton market: producers/supplers release a certan quantty nto the market and then brokers wll sell ths for the hghest prce possble (the market clearng prce). The Cournot framework would thus seem natural where there are aucton markets. Whle there are aucton markets, there are also many ndustral markets wthout brokers where the producers drectly set the 0

12 Olgopoly made smple prce at whch they sell ther produce. Clearly, the typcal sort of market whch concerns ndustral economsts s not an aucton market but a market wth prcesettng frms. How can the use of the Cournot framework be justfed n markets wth prce-settng frms? It s often argued that the choce of Bertrand or Cournot competton rests on the relatve flexblty of prces and output. In the Bertrand framework frms set prces and then produce to order. Thus, once set, prces are fxed whle output s perfectly flexble. In the Cournot framework, however, once chosen, outputs are fxed whle the prce s flexble n the sense that t clears the market. Thus the choce between the two frameworks rests on the relatve flexblty of prce and output. Ths s of course an emprcal queston but many would argue that prces are more flexble than quanttes (e.g. Hart 985) and hence the Cournot equlbrum s more approprate. A very nfluental paper whch explores ths vew s Kreps and Schenkman (983). They consder the subgame perfect equlbrum n a two-stage model. In the frst stage, frms choose capactes; n the second stage frms compete wth prce as n the Bertrand model and can produce up to the capacty nstalled. The resultant subgameperfect equlbrum of the two-stage model turns out to be equvalent to the standard Cournot outcome. Ths result, however, s not general and rests crucally on an assumpton about contngent demand (the demand for a hgher-prced frm gven that the lower-prced frm does not completely satsfy ts demand) see Dxon (987a). An alternatve approach s to allow for the flexblty of producton to be endogenous (Dxon 985; Vves 986). The Bertrand and Cournot equlbra then come out as lmtng cases correspondng to when producton s perfectly flexble ( a horzontal margnal cost curve yelds the Bertrand outcome) or totally nflexble (a vertcal margnal cost curve at capacty yelds Cournot). Another reason that the Cournot framework s preferred to the Bertrand s purely techncal: there s a fundamental problem of the non-exstence of equlbrum n the Bertrand model (see Edgeworth 95; Dxon 987a). In our smple example frms have constant average/margnal costs. If ths assumpton s generalzed for example to allow for rsng margnal cost non-exstence of equlbrum s a problem. 7 A common argument for the Cournot framework s ts plausblty relatve to the Bertrand framework. Many economsts beleve that numbers matter : t makes a dfference whether there are two frms or two thousand. Thus the predcton of the Bertrand model a zero prce-cost margn wth two or more frms s mplausble

13 Olgopoly made smple (see, for example, Hart 979; Allen and Hellwgg 986). The Cournot equlbrum captures the ntuton that competton decreases wth fewer frms. There are two ponts to be rased here: one emprcal, one theoretcal. Frstly, on the emprcal level there exsts lttle or no evdence that there s a smooth monotonc relatonshp between the level of concentraton and the prce-cost margn. Secondly, on the theoretcal level, the stark contrast n the Bertrand and Cournot formulatons has been exhbted here only n the case of a smple one-shot game. In a repeated game numbers may well matter. For example, Brock and Schenkman (985) consder a prce-settng super-game and show that there s a relatonshp between numbers and prces that can be sustaned n the ndustry (although the relatonshp s not a smple monotonc one). A related pont s that the Nash equlbrum s a non-cooperatve equlbrum. Numbers may well matter when t comes to mantanng and enforcng colluson and one of Bertrand s crtcsms of Cournot was that colluson was a lkely outcome wth only two frms. 6.4 Cournot and Bertrand equlbra wth dfferentated commodtes In ths secton we wll explore and contrast the Bertrand and Cournot approaches wthn a common framework of dfferentated products wth symmetrc lnear demands. As we shall see, there are agan sgnfcant contrasts between markets where frms compete wth prces and quanttes. Frstly, we wll compare the equlbrum prces and show that the Cournot equlbrum yelds a hgher prce than the Bertrand equlbrum. Thus, as n the case of homogeneous products, Cournot competton s less compettve than Bertrand competton although the contrast s less. Secondly, we contrast the Stackelberg equlbrum (where one frm moves before the other) and the correspondng frst-mover advantage. In the Cournot framework the leader ncreases hs own output and profts at the expense of the follower and total output ncreases, reflectng a more compettve outcome than the standard Nash equlbrum. In the Bertrand framework the Stackelberg leader wll rase hs prce and ncrease hs profts. The follower wll also rase hs prces and ndeed hs profts wll ncrease by more than the leaders. Unlke the Cournot case there s then a secondmover advantage n the Bertrand case. Overall, wth prce competton the Stackelberg equlbrum leads to hgher prces and profts and a contracton n total

14 Olgopoly made smple output. These dfferences between the behavour of markets wth prce and quantty competton have mportant polcy mplcatons whch wll be dscussed at the end of ths secton n the context of the recent lterature on strategc trade polcy. We contnue to assume that frms have constant average/margnal cost, A. However, we wll drop A and assume that there s a symmetrc lnear demand system; n the case of two frms wth dfferentated products we have: A3 For 0 < a < x x p + αp = (a) p + αp = (b) where a > 0 mples the two outputs are substtutes (e.g. margarne and butter): f a were negatve then they would be complements (e.g. personal computers and software). In the exposton we wll assume throughout that the frms produce substtutes and for techncal reasons that a < (.e. qute plausbly the frm s own prce has a greater absolute effect on ts demand than the other frm s prce). The above equatons express outputs (or more precsely, demands) as a functon of prces. 8 If we want to explore the Cournot framework wth dfferentated products we need to nvert () to gve the prces that wll clear the markets for chosen outputs. Invertng () we have: p p = a 0 = a 0 a x a x a a x x + α α where a 0 = ; a ; a α =. α α Snce a > 0 both prces are decreasng n both outputs. Thus an ncrease n x by one unt wll decrease p by a, and p by a (of course a > a for a < ). () 6.4. Cournot-Nash equlbrum There are two frms whch choose outputs, the resultant prces gven by the nverse demand system (). Frm s payoff functon s: π [( a a x a x ) c] = x 0 To obtan frm s reacton functon x s chosen optmally gven x : 3

15 Olgopoly made smple π x = a 0 a x ax c = 0 Solvng for x ths yelds: x ( x ) a c a x 0 = r = (3) a The slope of the reacton functon s gven by: dx dx a a α = r = < 0 Wth substtutes each frm s reacton functon s downward slopng n output space as n Fgure. The frms are dentcal and there s a unque symmetrc equlbrum at N wth x = x = x c : x c = a a c + a + α = + α 0 c ( α ) (4) wth resultant prce: p c = + c( α ) ( + α )( α ) Fgure Cournot reacton functons (5) 6.4. Bertrand-Nash equlbrum Turnng now to the Bertrand case frms choose prces so that we use the drect demand system (). Frm s profts are: ( p + αp ) c( p αp ) π = + (6) p π p = p + αp + c = 0 Hence frm s reacton functon n prce space s: + c +αp p = s ( p ) = (7) 4

16 Olgopoly made smple The slope s: dp dp s = α > 0 Thus the two frms reacton functons are upward slopng n prce space as depcted n Fgure 3. Fgure 3 Bertrand-Nash equlbrum There s a unque symmetrc equlbrum prce p = p = p b wth correspondng output, prce and prce-cost margns: p b x b + c = (8) α ( α ) c = α ( ) b c α µ = (0) + c How do the Cournot and Bertrand equlbra compare? Drect comparson of (8) wth (5) shows that p b < p c :.e. the Bertrand equlbrum prces are lower than Cournot prces. We formulate ths n the followng observaton: (9) Observaton. If frms demands are nterdependent a? 0 then: c b c b p > p ; x < x ; c µ > µ b If a = 0 then each frm s, n effect, a monopolst snce there are no cross-prce effects and the two outcomes are, of course, the same. It should be noted that the above observaton remans true when the goods are complements (- < a < 0). Wth dfferentated products, then, Bertrand competton wll be more compettve than Cournot competton although the dfference s less stark than n the case of homogeneous products. Wth product dfferentaton frms have some monopoly power even wth prce competton and do not have the same ncentves for undercuttng ther competton as n the homogeneous goods case. 5

17 Olgopoly made smple What s the ntuton behnd the above observaton that prce competton s more compettve than quantty competton? Clearly, for a monopoly, t makes no dfference whether prce or quantty s chosen; t smply chooses the proftmaxmzng prce-output pont on ts demand curve. There s a sense n whch ths s also true for the olgopolst: gven what the other frm s dong t faces a demand curve and chooses a pont on that demand curve. However, the demand curve facng frm wll be dfferent f frm keeps x constant (and hence allows p to vary) from when frm keeps p constant (and hence allows x to vary). From () f frm has prce as ts strategy and holds p constant, frm s demand s: wth slope ( + p ) x = α p () dx dp p = (a) and elastcty p η (b) p = x If, on the contrary, frm has output as ts strategy t allows ts prce p to vary as p vares (to keep x constant): p x + αp = (3) Substtutng (3) nto (a) we obtan frm s demand when x s held constant: x ( + α ) ( α ) p αx = (4) wth slope and elastcty: dx p x = ( α ); η ( ) x = α (5) dp x Clearly, comparng elastctes () and (5): η < p < η x 0 6

18 Olgopoly made smple Thus the demand facng frm s more elastc when frm holds p constant (and allows x to vary) than when x s held constant (and p allowed to vary). For example, suppose that frm consders movng up ts demand curve to sell one less unt of x wth substtutes (a > 0). If frm holds x constant then as frm reduces ts output and rases ts prce the prce for x wll rse (va (b)). Clearly the demand for frm wll be more elastc n the case where frm does not rase ts prce and expands output. We have derved the above observaton under very specal assumptons A, A3. How far can we generalse ths comparson of Cournot and Bertrand prces? Ths has been the subject of much recent research see for example Cheng (984), Hathaway and Rckard (979), Okuguch (987), Sngh and Vves (984), Vves (985a,b). Vves (985a) consders a more general dfferentated demand system whch need not be lnear or symmetrc (bd. 68) and derves farly general condtons for whch the Bertrand prce s less than the Cournot prce. Of course there need not be unque Cournot or Bertrand equlbra: wth multple equlbra the comparson becomes conceptually more complex. Vves (985b) has establshed a result that for very general condtons there exsts a Bertrand equlbrum whch nvolves a lower prce than any Cournot equlbrum. Of course there are other contrasts to be drawn between Cournot and Bertrand- Nash equlbra. For example, there s the queston of welfare analyss employng standard consumer surplus. A smple example employng the lnear demand system (), () s provded by Sngh and Vves (984: 76) whch shows that the sum of consumer and producer surplus s larger n Bertrand than n Cournot-Nash equlbrum both when goods are substtutes and complements Stackelberg leadershp and the advantages of movng frst The dfference between Cournot and Bertrand competton go deeper than the smple comparsons of the prevous secton. To llustrate ths we wll examne the advantages of movng frst n the two frameworks. The standard Nash equlbrum assumes that frms move smultaneously. However, Henrch von Stackelberg (934) suggested an alternatve n whch one frm (the leader) moves frst, the other (the follower) moves second. Thus when the follower chooses ts strategy t treats the leader s choce as gven. However, the leader wll be able to nfer the follower s choce and take ths nto account n ts decson. The explct algebrac analyss of the 7

19 Olgopoly made smple Stackelberg equlbrum s rather complcated and we wll rather employ the famlar so-proft loc. 9 In the followng analyss t s mportant to note that under A, A3 the model s perfectly symmetrc; thus whether n prce or quantty space the frms reactons functons are symmetrc n the sense that frm s reacton s a reflecton of frm s n the 45 o lne (see Fgures,3). Smlarly, frm s so-proft loc are smply reflectons of frm s n the 45 o lne and vce versa. Frstly we analyse the Stackelberg equlbrum n the Cournot case. The follower (frm ) wll smply choose ts output to maxmze ts profts gven x so that x =r (x ). The leader, however, wll choose x to maxmze ts profts gven that x depends on x va r. Thus by movng frst the leader can pck the pont on frm s reacton functon that yelds t the hghest profts: ths s represented n Fgure 4 by the tangency of so-proft loc p L to r at pont A. Fgure 4 Frst-mover advantage n Cournot model If frm were the leader and frm the follower then, by symmetry, frm s Stackelberg pont would be at pont B the reflecton of A n the 45 o lne (at ths pont frm s so-proft locus s tangental to frm s reacton functon). Comparng ponts A, B and the Nash equlbrum at pont N we can see that f frm s the leader t earns p L whch s greater than n the Nash equlbrum p N. If frm s a follower t wll end up at pont B and earn only p F whch s less than p N. Hence, n the Cournot framework we have: p L > p N > p F (Cournot) profts Nash profts of profts of leader follower There s thus a frst-move advantage n two senses: the leader earns more than n the smultaneous-move case (p L > p N ); the leader earns more than the follower (p L > p F ). The leader ncreases hs output and profts at the expense of the follower (n fact, the declne n the follower s profts from p N to p F leader s from p N to p L : ndustry profts fall). s larger than the ncrease n the In the Bertrand case the story s rather dfferent: there s a second-mover advantage. The reacton functons and so-proft loc of frm depcted n prce space are shown n Fgure 5 and agan are symmetrc. The so-proft loc for frm are 8

20 Olgopoly made smple hgher the further away they are postoned from the x-axs (frm wll earn hgher profts the hgher p s). N s the Nash equlbrum, A occurs f s the leader, B f s the leader. Fgure 5 Second-mover advantage n Bertrand model If frm s the leader t wll choose to rase ts prce movng along frm s reacton functon S from N to A and profts wll ncrease from p N to p L. However, f frm s the follower t wll end up at B wth profts p F. Note that the follower rases hs prce by less than the leader and that p F > p L. thus the leader wll set a hgher prce than the follower, produce a lower output and earn less profts: p F > p L > p N (Bertrand) There s an advantage to movng second. The second-mover advantage goes beyond Bertrand equlbra and extends to any game wth upward-slopng reacton functons (Gal-Or) 986). There s stll a frst-mover advantage n the sense that the leader earns more than n the smultaneous move case (p L > p N ). In contrast to the Cournot case, n the Bertrand case Stackelberg leadershp leads to hgher prces, profts and lower outputs Prces vs. quanttes Prce and quantty competton have very dfferent mplcatons for the nature of product market competton between frms. Most mportantly, from the frms pont of vew, prce competton leads to lower profts than does quantty competton n Nash equlbrum. As was dscussed earler, whether frms should be vewed as competng wth prce or quantty can be seen as dependng on structural or nsttutonal characterstcs of the market the flexblty of producton, whether the market s an aucton market etc. An alternatve approach s to treat the frm s decson to choose prce or quantty as tself a strategc decson (Klemperer and Meyer 986; Sngh and Vves 984). Whle t s perhaps not qute clear how frms mght acheve ths, t s at least a useful experment and wll reveal the ncentves whch frms have to acheve one or the other type of competton. 9

21 Olgopoly made smple Wthout uncertanty, ths experment s not frutful: frms are ndfferent between choosng prce or quantty. The reason s that from the ndvdual frm s perspectve t smply faces a demand curve and, lke a monopolst, chooses a pont on that demand curve. It can acheve any pont on the demand curve by choosng ether prce or quantty. The frm s own prce/quantty decson does not affect ths demand curve whch s rather determned by the other frm s choce. Frm s choce has a pure externalty effect on the demand faced by frm : f chooses prce, s demand s more elastc than f frm had chosen quantty. However, n the Nash framework, each frm wll gnore ths externalty: gven the other frm s choce, each frm wll face a partcular demand curve and wll be ndfferent between settng prce or quantty tself. In the case of duopoly there wll be four Nash equlbra n ths strategc game: one where both set quanttes (Cournot); one where both set prces (Bertrand); and two asymmetrc equlbra where one sets prce and the other quantty. Wth certanty, then, allowng frms to choose prce or quantty tells us nothng about whch may be more approprate. The presence of uncertanty (addng a stochastc term to A3, for example) can mean that frms have a strct preference between prce and quantty settng. The results depend very much on the exact assumptons made (s demand uncertanty addtve or multplcatve; s demand lnear n prces?). For the smple lnear demand system A3 wth an addtve stochastc term frms wll prefer quantty settng f margnal costs are ncreasng; they wll be ndfferent f margnal costs are constant (as n A); they prefer prce settng f margnal costs are decreasng (Klemperer and Meyer 986; proposton ). Whle ths and related results are at present rather specfc they do suggest that the presence and nature of uncertanty provde some nsghts nto how frms vew the alternatves of prce and quantty settng. 6.5 Precommtment: strategc nvestment and delegaton In the prevous secton we explored the nature of the frst-mover advantage n the Cournot and Bertrand framework. Clearly, f we start from the Nash equlbrum there s an ncentve for the frm to precommt ts output/prce to obtan ths frstmover advantage. By precommtment t s meant that the frm takes some acton pror to competng n the product market whch commts t to a certan course of acton. In the standard Cournot model t s not credble for one frm to produce the 0

22 Olgopoly made smple Stackelberg output n the smultaneous move game. For example, n terms of Fgure 6, frm s Stackelberg pont A s not on ts reacton functon so that gven frm s output x A frm would lke to produce x. The only credble equlbrum s the Nash equlbrum at N. In order to move towards ts Stackelberg pont the frm must be able to precommt ts output n some way. In the prevous secton we smply assumed that the leader was able to move frst. In some stuatons t s natural to assume a partcular sequence of moves (e.g. entrant/ncumbent, domnant frm). However, n the case of actve ncumbents whch are competng on even terms, smultaneous moves seem more natural. Fgure 6 Non-credblty of Stackelberg pont Gven that there s an ncentve for the frm to precommt how can ths be acheved? Ths secton wll look at two methods of precommtment whch have receved much recent attenton precommtment through nvestment and precommtment through delegaton. The basc dea s smple: the frm can take actons pror to competng n the market whch wll alter the Nash equlbrum n the market. Frms can take actons such as nvestment decsons 0 and choce of managers that are rreversble (n the sense of beng fxed over the market perod) and whch alter the frm s reacton functon thus shftng the Nash equlbrum n the market. We wll frst consder how nvestment by frms can be used strategcally to alter the market outcome. For a wde range of ndustral processes economsts snce Marshall have taken the vew that t s approprate to treat the captal stock decson as beng taken on a dfferent tme scale (the long run ) to prce/output decsons (the short-run ). When frms compete n the product market t follows that they treat ther captal stock as fxed. The captal stock chosen by the frm wll nfluence ts costs when t competes n the market. The fact that captal s commtted before output/prce decsons means that t can use nvestment strategcally to nfluence the market outcome. In essence, through ts choce of captal stock, the frm wll determne the short-run costs whch t wll have when t chooses output/prce; the frm s margnal costs wll determne ts reacton functon and hence the Nash equlbrum n the product market. Schematcally:

23 Olgopoly made smple Investment? short-run? reacton? market margnal cost functon equlbrum For example, n the Cournot case the frm can ncrease ts nvestment, reduce ts margnal cost and hence shft ts reacton functon out so that the product market equlbrum moves towards the Stackelberg pont. Of course, ths precommtment s not costless: captal costs money and, as we shall see, such use of captal leads to productve neffcency. Agan there s an mportant dchotomy between the Cournot and Bertrand approaches: f the product market s Cournot then the frm wll want to overnvest; f the product market s Bertrand then the frms wll want to undernvest. We wll brefly llustrate both stuatons. The structure of strategc nvestment models s very smple: there are two stages to capture the dstncton between the short and the long run. In the frst strategc stage, the frms choose ther captal stock; n the second market stage frms choose output/prce. The choce of captal stock n the frst stage wll determne the cost functon whch the frm has. In the prevous two sectons we assumed constant average/margnal cost at c (A). Ths can be conceved of as the long-run cost functon. In order to keep the exposton consstent we wll assume that frms have a producton functon of the form: A4 x = k L / / c ( x k ) x, = rk + (6) k where lnear ncreasng margnal cost: c = x k x (7) Thus an ncrease n nvestment lowers the margnal cost of producng output. The producton functon A4 dsplays constant returns to scale and hence the long-run cost functon has constant average/margnal cost n terms of A, mnmum average cost c = vr. We wll frst outlne the strategc nvestment model wth Cournot competton n the product market, a smple verson of Brander and Spencer s (983) artcle. If nvestment s used non-strategcally, then the frm smply operates on ts long run cost

24 Olgopoly made smple functon gven by A: captal and labour are chosen to mnmse producton costs. In the strategc nvestment framework, however, the frm s costs wll be gven by ts short-run cost functon (6). Turnng to the market stage, the frm s profts are: x π = x ( a0 ax ax j ) rk (8) k Fgure 7 Investment shfts from s reacton functon out The reacton functon whch the frm has n the market stage, condtonal on k, s derved by settng π / x = 0: x a0 ax j = r ( x j, k ) = (9) a + ( / k ) By ncreasng ts nvestment frm wll reduce ts margnal costs and from (9) t wll shft ts reacton functon out as n Fgure 7. Gven the level of nvestment by the two frms (k k ) are obtaned by solvng (9) for x and x (we leave ths as an exercse for the reader). In general form: x = x k, k j (30) + Ths notaton sgnfes that frm s equlbrum output n the market stage depends postvely on ts own nvestment and negatvely on nvestment by the other frm. Suppose we start off at pont A n Fgure 9: an ncrease n k to the market equlbrum goes from A to B, x rsng and x ' k shft out r so that fallng. Conversely, an ncrease n k shfts the equlbrum from A to C. Thus by alterng ther nvestment, the frms can alter ther reacton functons and hence the market stage equlbrum. Fgure 8 Market stage equlbrum gven nvestment k, k How s the optmal level of nvestment n the frst strategc stage determned? Frms choose nvestment levels k gven that the second-stage outputs wll be as n (30). We can see ther profts as a functon of captal stocks. For frm we have profts: 3

25 Olgopoly made smple π ( k, k ) = x ( k, k )[ a a x ( k, k ) a x ( k, k )] c( x ( k, k ) k ) 0, The RHS term n square brackets s the prce, whch s multpled by output to obtan revenue from whch are subtracted costs. The frm wll choose k to maxmze ts profts (3) hence: (3) π k x = k a 0 a x a x c x x c a = 0 k k - x (3) Snce frm chooses x to maxmze profts gven x and k n the second stage, the bracket on the rght-hand sde of (3) s zero (t s smply ts reacton functon (9)). Hence (3) becomes: c k = a x k x > 0 (33) What does (33) tell us? c / k gves the effect of nvestment on the total costs of producng x. If c / k = 0, as n the standard non-strategc case, then k mnmzes the cost of producng x. If c / k > 0, as n (33), then there s overcaptalzaton : more nvestment than would mnmze costs ( a reducton n k would reduce average costs). If c / k < 0, then there s undercaptalzaton : less nvestment than would mnmze costs. Fgure 9 Market stage equlbrum and changes n nvestment Wth a Cournot market stage, then, there s overcaptalzaton of producton n the market stage. The ntutve reason s qute smple. Gven the other frm s reacton functon, each frm can shft ts own reacton functon out towards ts Stackelberg pont. Of course there s a cost to: more nvestment leads to hgher captal costs and neffcent producton. The frms wll shft out ther reacton functons beyond ther nnocent level and the fnal product market equlbrum wll be at a pont such as S n Fgure 0. At the equlbrum level of nvestment the addtonal cost of nvestment equals the addtonal gans from movng out the reacton functon further. In the strategc nvestment equlbrum S then, both frms produce a larger output than n the non-strategc equlbrum N. 4

26 Olgopoly made smple In the Bertrand case an exactly analogous argument apples for strategc nvestment. However, there s the opposte result of undercaptalzaton. The Stackelberg equlbrum results n hgher prces and lower outputs than n the Bertrand case. Thus frms wll restrct nvestment relatve to the nnocent Bertrand equlbrum n order to shft ther reacton functon out n prce space, as n Fgure. Startng from the nnocent Bertrand equlbrum at N, f frm restrcts ts nvestment ts margnal costs rse and ts reacton functon shfts outwards to s ' (an outward shft because wth hgher margnal costs t wll wsh to set a hgher prce and produce a smaller quantty gven the prce chosen by the other frm). If both frms undernvest strategcally the resultant equlbrum wll be at S wth hgher prces and lower output. Fgure 0 Strategc nvestment equlbrum the Cournot case Fgure Undernvestment rases prces We wll brefly sketch the algebra underlyng ths result. Under A and A4 the frm s profts are: ( p αp p j ) rk π = p p αp p j (34) k Settng π / p = 0 frm s reacton functon p = s (p j ) s: P + ( / k ) = + + ( / k ) ( + ( / k )) Pj ( ) α + ( / k ) (35) Solvng for p to p j gven k and k j n general terms we have: P = p k, k j, j =,,? j (36) + The frms choose k to maxmze (34) gven that n the market subgame frms prces are gven by (36) (.e. a Bertrand-Nash equlbrum occurs). 5

27 Olgopoly made smple dπ dk π = p dp j π dp j π = 0 dk p dk k j (37) (0) (-) (+) (-) (+) Note that π / p = 0, snce frms are n ther market stage reacton functon (35) and, further, that π k = c / k. Hence (37) can be expressed as: / c k π dp = p j dk (+) (-) j < 0 (36) That s, undercaptalzaton of producton results from the strategc use of nvestment wth Bertrand product market competton. Clearly, the result of strategc nvestment models depends on the nature of product market competton. Other papers have made dfferent assumptons than the smple Cournot-Nash and Bertrand-Nash equlbra. Dxon (985) consders the case of a compettve product market; Eaton and Grossman (984) and Yarrow (985) a conjectural Cournot equlbrum; Dxon (986b) a consstent conjectural varaton equlbrum n the product market. Snce producton wll generally be neffcent n a strategc nvestment equlbrum, frms have an ncentve to try and precommt ther labour nput at the same tme as ther captal. By so dong, frms wll be able to produce any output effcently, whle beng free to precommt themselves to a wde range of outputs. In Dxon (986a), precommtment s treated as a strategc choce by the frm: the frm can precommt ether, nether, or both captal and labour n the strategc stage. Because of the strategc neffcency n producton that occurs when only captal s precommted, under almost any assumpton about the nature of product market competton, frms would prefer to precommt both factors of producton (Dxon 986a; Theorem and? p. 67). If frms precommt both factors of producton n the strategc stage then, n effect, they have chosen ther output for the market stage and the resultant equlbrum s equvalent to the standard Cournot equlbrum. How mght frms be able to precommt ther output n ths manner? One mportant method that may be avalable s choce of technology. More specfcally, the frm may have a choce between a 6

28 Olgopoly made smple putty-putty technology that allows for smooth substtuton of captal for labour n the market stage or an otherwse equvalent putty-clay technology that s Leontef n the market stage. If the frm chooses a putty-clay technology then ts choce of nvestment and technque n the strategc stage effectvely tes down ts output and employment n the market stage. If possble, then, frms would prefer to have totally nflexble producton n the market stage. Ths strong result gnores uncertanty of course. If demand or factor prces are uncertan there wll be a countervalng ncentve to retan flexblty. In strategc nvestment models t s frms themselves whch precommt. Governments, however, can undertake precommtments whch frms themselves cannot make. In the context of trade polcy there has been much recent research on how governments can mprove the poston of ther own frms competng n nternatonal markets (see Venables 985 for excellent surveys). If domestc frms are competng n foregn markets the net beneft to the home country n terms of consumer surplus s the repatrated profts total revenue less the producton costs (wth compettve factors markets producton costs represent a real socal cost to the exportng country). Government trade polcy may therefore be motvated by what s called rent extracton : that s, helpng ther own frms to make larger profts whch are then repatrated. Trade polcy, usually n the form of an export subsdy or tax, s a form of precommtment by the government whch enables domestc frms to mprove ther poston n foregn markets. Brander and Spencer (984) presented the frst model based on the rent-extracton prncple and argued for the use of export subsdes n the context of a Cournot-Nash product market. Subsdes have the effect of reducng the margnal costs faced by exporters and can thus be used to shft out ther reacton functons to the Stackelberg pont (the cost subsdes cost nothng from the pont of vew of the exportng country snce they merely redstrbute money from the taxpayers to shareholders). As Eaton and Grossman (983) argued, the exact form of the trade polcy wll be senstve to the nature of product market competton. Wth a Bertrand product market, of course, rent extracton arguments lead to the mposton of an export tax snce ths wll shft the Bertrand compettor s reacton functon outwards n prce space towards ts Stackelberg pont. The ncentve to precommt n olgopolstc markets also sheds lght on one of the perennal ssues of ndustral economcs what are the objectves of frms? The dvorce of ownershp from control can be vewed as an act of delegaton by 7