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Strategic firm behavior and entry deterrence: three essays Yong, Jong-Say 1993

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STRATEGIC FIRM BEHAVIOR AND ENTRY DETERRENCE:THREE ESSAYSbyJong-Say YONGB.A., National University of Singapore, 1985M.A., University of British Columbia, 1989A Thesis Submitted in Partial Fulfillment of The Requirements forthe Degree of Doctor of PhilosophyinThe Faculty of Graduate Studies(Department of Economics)We accept this thesis as conformingt^- required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust 1993© Jong-Say Yong, 1993.In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of 6-Cr 11, 12/17 / The University of British ColumbiaVancouver, CanadaDate^.DE-6 (2/88)ABSTRACTThis thesis consists of three independent chapters on entry deterrence. The first two chaptersconsider the use of contracts as a barrier to entry, while the final chapter examines thepossibility of firms expanding their product lines to deter entry in a vertical differentiationmodel. In Chapter 1, the role of exclusive dealing contracts in the liner shipping industryis investigated. It is shown that if the entrant is capacity-constrained, exclusive dealingcontracts can be an effective entry barrier, even if the entrant has a lower cost. Chapter 2considers an industry with two stages of production. It is shown that an upstream incumbentis able to deter the entry of a more efficient producer by establishing long-term contractualrelations with downstream firms, provided the downstream firms are in direct competitionagainst each other. Chapter 3 considers the question of entry deterrence in a one-dimensionalmarket where goods are differentiated by quality. It is shown that an incumbent firm maydecide to produce several products solely for the purpose of deterring entry. Again, it ispossible that a lower-cost entrant is deterred. In all three chapters, the welfare consequenceis clear: social welfare is lower, since more efficient entrants are excluded from the market.ContentsAbstractTable of ContentsList of FiguresAcknowledgement^ viINTRODUCTION^ 1Chapter 1 Exclusive Dealing Contracts As a Barrier to Entry in Liner Ship-ping^ 71.1 Introduction  ^71.2 The Shipping Industry  ^91.3 Related Literature  ^101.4 The Model ^  121.5 Discussion  ^221.6 Conclusion  ^26Chapter 2 Vertical Contracts and Entry Deterrence^ 282.1 Introduction  ^282.2 Related Literature  ^292.3 Basic Setting  ^312.4 Cournot Competition in the Downstream Market ^  382.5 Discussion  ^492.6 Conclusion  ^52Chapter 3 Quality Differentiation and Strategic Product Line Expansion^533.1 Introduction  ^533.2 Related Literature  ^553.3 Basic Model ^  573.4 Entry Deterrence  ^673.5 Technological Advances ^  723.6 Conclusion  ^77Bibliography^ 79Appendix 1^ 83lA Proof of Proposition 1.2 ^  831B Cournot Competition  871C Derivation of the Entry Deterrence Condition ^  91Appendix 2^ 922A Proofs of Results in Chapter 2 ^  922B Bertrand Competition in a Differentiated Product Market: An Example .^96Appendix 3^ 1003A Proofs of Results in Chapter 3 ^  100ivList of Figures1.1^Timing of game ^ 131.2^Graphical representation of entry deterrence condition, with a = 2, b 1^.^. 252.1^Timing of game ^ 352.2^Firm U's objective function under cases (i) and (ii) ^ 432.3^Normal form of the 'Choose a Supplier' game 442.4^Condition for (E, E) as a Nash equilibrium outcome ^ 473.1^Firm A introduces a second product ^ 643.2^Firm B introduces a second product 653.3^Incentive of incumbent to deter entry ^ 703.4^Equilibrium market structures 72A2.1 Reaction functions of firms U and E under case (ii) ^ 94ACKNOWLEDGEMENTIn the course of completing this dissertation, I have incurred an enormous debt to the manypeople who have helped in various ways. In particular, I thank my research supervisor HughNeary for his comments on earlier versions, and his support and encouragement. Withouthis help, this dissertation would not have been completed for many more months to come.I am also grateful to my dissertation committee members, Mukesh Eswaran, Trevor Heaver,and Margaret Slade. Their helpful comments and suggestions have greatly improved thesubstance as well as the presentation of the material. I also thank Ken Hendricks, Tae Oum,Scott Taylor, and Bill Waters for their comments on an earlier version, and John Cragg andBill Schworm for their advice and pointers on completing a dissertation. I have also benefitedfrom discussions and comments from ECON 640 workshop participants. Finally, I thank mywife, Lian 0i, for her forbearance and support.viINTRODUCTIONAn important question in industrial organization is whether incumbent firms are able to prof-itably exclude the entry of other firms, since entry barriers are the main source of monopolyand oligopoly power. Incumbent firms have little or no power over prices when entry barriersare nonexistent. By erecting entry barriers, incumbent firms often earn supranormal profits.Social welfare is usually lower in comparison to the free-entry case. The question also carriesimportant legal implications, since the courts in many instances have to decide whether cer-tain practices by incumbent firms restrict entry and hence limit competition in a particularindustry. Among well-known cases in the United States are Standard Oil of 1911, Alcoa of1941, and United Shoe Machinery of 1953.1-Following Bain (1956), we define an entry barrier as anything that allows incumbent firms toearn supranormal profits without the threat of entry. Broadly speaking, there are two forcesinteracting in creating entry barriers: the structural features of markets and the behaviorof incumbent firms. Bain identified four features of market structure that enable incumbentfirms to erect entry barriers. They are: economies of scale, absolute cost advantages, product-differentiation advantages, and capital requirements. In addition, Bain also suggested threekinds of behavioral responses by incumbent firms in the face of an entry threat:(i) Blockaded entry The incumbent firms behave as if there is no threat of entry. Noentry occurs, because the market is not attractive enough for potential entrants.(ii) Deterred entry Entry cannot be blockaded. The incumbent firms alter their behaviorto successfully thwart entry.(iii) Accommodated entry Entry occurs, because it is more costly for the incumbent firms'For a more detailed discussion of these and other cases, see Scherer (1980), Chapters 20 and 21.1to erect entry barriers than simply to allow entry.Bain's ideas were formalized in a game theoretic context by Spence (1977), Dixit (1979,1980),and Milgrom and Roberts (1982). The Spence-Dixit model says that firms compete throughthe accumulation of production capacity in the long run. An incumbency advantage leadsthe incumbent firm to accumulate a large capacity (and hence to charge a low price) in orderto deter or limit entry. The Milgrom-Roberts model, on the other hand, is based on theasymmetry of information between the incumbent firms and entrants. The incumbent firmscharge a low price to convey the information that either demand or their own marginal costis low, thus signaling low profitability for potential entrants.More recently, it has also been suggested that contractual arrangements (e.g., exclusive deal-ing contracts) between incumbent firms and customers in oligopolistic markets can serve asa barrier to entry. (See, for examples, Aghion and Bolton 1987, and Rasmusen et al. 1991.)Unlike the cases of capacity and prices, however, the idea of contracts as a barrier to entryappears to be considerably more controversial. In the antitrust literature, a school of thoughtthat is often referred to as the Chicago School holds that firms establish contractual relationspurely for efficiency reasons. (See, for examples, Bork 1978, Posner 1976, Marvel 1982 andOrnstein 1989.) These contractual arrangements, it is argued, simply cannot deter the entryof more efficient (i.e., lower-cost) entrants, the reason being that it is not in the best interestof the customers to limit competition for the incumbent firms. It is therefore not likely thatthe customers will agree to these arrangements, unless they are sufficiently compensated. Tosuccessfully deter lower-cost entrants, the incumbent firms need to compensate the customersfor their loss of alternative and less costly sources of supply. However, this cannot be prof-itable for the incumbent firms given that they are less efficient. Thus, in the case of linearpricing, the incumbent firms must price at or lower than potential entrants' marginal costs,but this cannot be profitable for the incumbent firms given that their costs are higher (Bork,21978).However, in industrial organization, the anticompetitive effects of contractual arrangements inoligopolistic markets remain a concern among many economists. Comanor and Frech (1985),Krattenmaker and Salop (1986), and Mathewson and Winter (1987) show that exclusivedealing arrangements between a manufacturer and dealers may reduce the competitiveness ofrival manufacturers or even eliminate the rivals altogether from the markets. Mathewson andWinter further show that such arrangements can have a welfare-enhancing property, sincewholesale prices may be lower under exclusive dealing. However, as pointed out by Bernheimand Whinston (1992), the result is restricted to the case of linear pricing. The possibilityof firms using two-part franchise contracts to avoid the problem of double marginalization isnot considered in these studies.Aghion and Bolton (1987) and Rasmusen et al. (1991) show that, under certain institutionalsettings, a monopolist seller is able to erect entry barriers through signing exclusive dealingagreements with customers. Rasmusen et al. assume the existence of a minimum efficientscale of production, such that an entrant will need a minimum number of customers, sayx, in order for entry to be worthwhile. Thus, the incumbent monopolist needs only to signup enough customers so that there are less than x customers for the entrant. In particular,Rasmusen et al. show that all customers entering the agreement is a Nash equilibrium.Aghion and Bolton, on the other hand, consider a type of contract that consists of twoelements: a contract price and liquidated damages, the latter are to be levied on the buyerif a breach of contract occurs. Entry is uncertain in the model, with the probability of entrydepending on the liquidated damages agreed upon between the seller and buyer. A marginallymore efficient entrant will find it unprofitable to induce the buyer to breach the contract. Incontrast, a very efficient entrant may find it profitable to offer a sufficiently low price so thatthe buyer is fully compensated for breaching the contract. The liquidated damages thus act3as an entry fee, which is divided between the seller and buyer. The authors show that, bysetting the liquidated damages optimally, the seller can exclude the entry of some, thoughnot all, lower-cost entrants.This thesis consists of three independent chapters, namely(1) "Exclusive Dealing Contracts as a Barrier to Entry in Liner Shipping,"(2) "Long-term Vertical Contracts and Entry Deterrence," and(3) "Quality Differentiation and Strategic Product Line Expansion."The first two chapters consider the question of entry deterrence through contracts along thesame line as Aghion and Bolton, and Rasmusen et al. The third chapter considers productline expansion as an entry barrier in a vertical differentiation model. These chapters aresummarized as follows.Chapter 1 considers the role of exclusive dealing contracts, known as loyalty contracts, inthe liner shipping industry. A salient feature of the industry is the existence of cartel-like associations known as liner conferences. In many trade routes, conferences face severecompetition from independent lines as well as tramps. The latter are in the ship charteringbusiness, but often enter the liner market when there is empty space left after securing amajor cargo. To defend their market share, conferences often offer shippers (i.e., customers)loyalty contracts, which essentially guarantee a lower price in return for shippers' 'loyalty' innot using nonconference lines. Since a conference usually consists of several members who arethe well-established lines in the business, it enjoys a distinct advantage in size as comparedto independent lines and tramps. In the model, we assume that the conferences do not facecapacity constraints as independent lines and tramps do.To analyse the competitive effects of loyalty contracts, we consider a two-stage game. Thereare three players: a conference, an entrant who is capacity constrained, and a shipper. Theentrant is assumed to be more efficient, i.e., to have a lower cost, than the conference. In the4first stage, the conference offers the shipper a loyalty contract, which specifies a price andcontains a clause that states that the shipper will not ship through nonconference lines. Ifthe shipper accepts the contract, production takes place in the second stage and no entryoccurs. On the other hand, entry occurs in the second stage if the contract is rejected. Theconference and entrant then play a price game in the second stage, and production takes placeafter prices are announced. It is shown that, under these circumstances, loyalty contractsmay represent an effective entry barrier—the entrant is excluded from the industry despiteits lower cost. Clearly, the outcome is not socially efficient. This result is in sharp contrastto the Chicago School argument that such contracts cannot be effective in deterring moreefficient entrants.Critical to the entry deterrence result in Chapter 1 is the assumption that the entrant iscapacity-constrained. In Chapter 2, we relax this assumption and examine the case whereno such constraint exists. The central question in Chapter 2 is whether firms in a verticalrelationship are able to deter entry by entering into long-term contracts. We consider a two-stage game. In stage 1, an upstream monopolist offers a long-term contract to each of twodownstream firms. If the contract is accepted by both downstream firms, it is implementedin stage 2, and no entry occurs. On the other hand, if the contract is rejected, the upstreammonopolist then faces a potential entrant, who must decide whether to invest in some researchand development activities. If the entrant invests in the R & D and succeeds, it will be ableto displace the monopolist as the sole upstream supplier. The probability that this occurs isexogenously given as 0 < < 1, which is known to all parties. Throughout, we allow firmsto use two-part franchise contracts in order to avoid the problem of double marginalization.The main conclusion of this chapter is that entry deterrence is not possible when the twodownstream firms are not in competition against each other (e.g., they operate in unrelateddownstream product markets). However, entry deterrence is a likely outcome if the twodownstream firms are in direct competition against each other. The downstream firms in this5case are willing to sign the long-term contract to avoid playing a Prisoner's Dilemma gamein stage 2. Social welfare is lower in this case, since the less efficient upstream monopolistis allowed to continue to operate instead of being displaced, with probability a, by a moreefficient entrant.Chapter 3 considers the question of entry deterrence in a one-dimensional market wheregoods are differentiated in quality. Specifically, we examine whether incumbent firms in thismarket are able to expand their product lines strategically for the purpose of entry deterrence.In the model, firms must decide how many products to introduce, as well as the price foreach product. A sunk cost is incurred for each quality a firm chooses to produce. Firms thencompete in prices, given their quality choices. We assume that firms are not able to alter theircapacity choices once the sunk cost is incurred. Thus the sunk costs enable firms to crediblycommit to a particular product quality. We show that a protected monopolist producesonly a single good, even if there are consumers who are not served by the monopolist. Ina duopoly where each duopolist produces a single good, firms choose the maximum degreeof differentiation to minimize price competition. Further, neither firm has any incentive toexpand its product line in the single-good duopoly equilibrium. However, if one firm entersthe market first, then it may wish strategically to expand its product line to deter potentialentrants who have a lower sunk cost. We also examine the case of technological advance whichenables firms to introduce higher-quality products than previously possible. We show that,when compared to a monopolist who faces the threat of entry, a protected monopolist is lesswilling to introduce a new product. Also, if given the exclusive right to the new technology,the high-quality producer in a duopoly has less incentive to make use of the new technologyas compared to the low-quality producer.6Chapter 1Exclusive Dealing Contracts As a Barrierto Entry in Liner Shipping1.1 IntroductionThis paper considers whether loyalty contracts, a form of exclusive dealing contracts widelyused in liner shipping, can deter lower cost entrants who are capacity-constrained. Thequestion of whether the use of these contracts is socially efficient is also considered.These issues have concerned governments for many decades. As early as 1906, the Britishgovernment appointed a royal commission (the Royal Commission on Shipping Rings) toinvestigate, among other matters, whether the practice of deferred rebates (a popular typeof loyalty contracts at the time) was detrimental to colonial trade. The Commission, how-ever, was unable to reach a consensus and two reports were issued. The majority reports,signed by eleven commissioners, concluded that the practice of deferred rebates did not createexcessive market power for the shipping conferences.' The minority report, signed by fivecommissioners, stated that the practice created an artificial barrier to entry and thus gaveconferences too much market power.'Some fifty years later, another landmark case occurred in the United States. The Japan-'A shipping conference is a cartel-like association of shipping lines. See Section 1.2 below.2Royal Commission on Shipping Rings (1909), von p.37 and pp.95-114.7Atlantic and Gulf Freight Conference filed a proposed dual rate system, another form ofexclusive dealing contract widely used in the United States, with the Federal Maritime Boardin 1952. The Board was in favor of the proposal, but an independent line, Isbrandtsen,joined by the United States Department of Justice and Department of Agriculture, protestedformally. A hearing was held, but the parties involved were unable to reach an agreement.The case then went before the United States Supreme Court in 1958 and the Court ruled infavor of Isbrandtsen on the ground that the dual rate contracts employed by the conferencestifled outside competition.3This and other similar rulings have been criticized by many economists associated with theChicago School. See, for examples, Bork (1978), Posner (1981), and Ornstein (1989). Theseauthors argue that contractual arrangements between firms cannot have any anticompet-itive effects, rather they are adopted for efficiency reasons. Many industrial organizationeconomists, however, do not share this view. The purpose of this paper is to show that, un-der plausible conditions pertaining to the shipping industry, liner conferences can effectivelyand profitably use loyalty contracts to limit entry of smaller lines and tramps.The plan of this paper is as follows. Section 1.2 gives a brief account of the shipping industry.Section 1.3 briefly reviews the literature on exclusive dealing contracts in the areas of shipping,antitrust and industrial organization. A formal model and the main results are presentedin Section 1.4. Some remarks and discussion of the results are contained in Section 1.5.Section 1.6 concludes the paper.3Federal Maritime Board v. Isbrandtsen Co., 356 U.S. 481 (1958). See McGee (1960), pp.252-60 for asummary.81.2 The Shipping IndustryBroadly speaking, the shipping industry consists of two distinct markets: liner and trampshipping. Liner companies provide regular shipping services between designated ports ac-cording to fixed schedules. The commodities liners transport are usually manufactured andsemimanufactured goods. These cargoes are typically originated by many shippers at severalports and destined for many consignees at several ports. Tramp shipping, in contrast, pro-vides vessel services on a time or trip chartered basis. There is no regular schedule nor fixedroute. The commodities transported by tramp ships are usually grains and other low-valuedgoods in shipload quantity, originated by one or a few shippers.A salient feature of liner shipping is the existence of cartel-like associations known as linerconferences.4 The primary objective of conferences is to limit competition between memberlines. This is usually achieved through setting common freight rates and other terms ofcarriage for all member lines. Sometimes conferences also allocate output among membersand divide revenues from joint operations. There are usually two conferences on a given traderoute, one for each direction of trade. Most conferences have less than ten members, althoughthere are some conferences with as many as fifty members. However, not all liner companiesare members of conferences. In some routes, independent and conference lines coexist.Conferences face competition from independent lines and tramps on many trade routes. Thelatter often enter the liner market if there is empty space left after securing a major cargo.The independents and tramps usually have limited capacities, smaller fleet sizes and provideless frequent and lower quality of services. Many liner conferences defend their market shareby signing a form of exclusive dealing contract known as loyalty contract with shippers (i.e.,customers). Broadly, loyalty contracts are of two types: deferred rebate and dual rate. Underthe deferred rebate system, shippers who use conference services exclusively for two successive4See Heaver (1991) for an account of some of the issues relating to liner conferences.9periods receive a rebate for the first period at the end of the second period. This practice isprohibited by the United States Shipping Act of 1916 for all international trade involving theUnited States. The dual rate contract, on the other hand, give shippers a discount of 10%to 20% off "noncontract" rates. In return, shippers agree not to ship through nonconferencelines. Note that these are contracts between a conference and shippers, so that shippers whosign the contract may use the services of any member lines without violating the terms ofcontract. In the absence of enforcement costs, the two forms of loyalty contracts are identical(McGee, 1960, pp.233-5).1.3 Related LiteratureNumerous economists, legal professionals and shipping researchers have investigated whetherloyalty contracts represent an artificial barrier to entry into liner shipping. However, opinionsremain at least as divided as in the days of the Royal Commission at the turn of the century.The following is a brief survey of some of their studies.McGee (1960, pp.249-50) believes that loyalty contracts are the most important device whichliner conferences use to deter entry. He observes that in numerous occasions the discountrates of dual rate contracts rose as competition from independent lines and tramps becamemore intense. Furthermore, no loyalty arrangements exist for certain commodity groups forwhich the tramps pose no challenge whatsoever. Bennathan and Walters (1969, pp.39-40)argue that loyalty contracts raise the "strategic scale of entry" in that an entrant has to offerat least some of the shippers a full substitute for the services of the conference. More recently,Sjostrom (1988) argues that a conference can profitably exclude a lower cost entrant who isconstrained in the frequency of services it can offer. His argument, however, is flawed.55With reference to Figure 1 of Sjostrom (1988, P. 343), it is not true that the conference can alwaysprofitably exclude a lower cost entrant. When the conference offers a lower price in the contract, it applies toall units supplied, not just to those at the margin, as claimed by Sjostrom (1988). Further, the equilibriumon which the comparative static exercises are performed is neither unique nor subgame perfect.10In contrast, Sletmo and Williams (1981, pp.209-10) and Davies (1986) do not think thatloyalty contracts are effective in deterring entry of independent lines or tramps. They pointto the fact that there have been frequent entry and exit of lines in the industry. In particular,Davies (1986) believes that the liner market is close to being perfectly contestable.In the antitrust literature, exclusive dealing contracts are often regarded as ineffective indeterring lower-cost entrants (see, for examples, Bork, 1978, Chapter 15; Posner, 1981; Orn-stein, 1989). The reason being that in order to induce customers to sign a loyalty contract,the incumbent firm must compensate the customers for what they would otherwise have got-ten from the entrant. To do this when the entrant has a lower cost must necessarily resultin losses for the incumbent, and is thus inconsistent with profit maximizing behavior. Infact, many authors believe that exclusive dealing contracts are socially efficient in that theyhelp to define property rights (Marvel, 1982), or reduce transaction costs and avoid free-riderproblems (Ornstein, 1989). Consequently, they advocate that the legal status of exclusivedealing contracts be changed from rule of reason to per se legal.In industrial organization, however, many economists have challenged this view. Comanorand Frech (1985), Kranttenmaker and Salop (1986), and Mathewson and Winter (1987)show that exclusive dealing arrangements between a manufacturer and its dealers may injurethe competitiveness of rival manufacturers.6 Mathewson and Winter further show that, insome cases, such arrangements may enhance social welfare. However, Bernheim and Whin-ston (1992) show that this result depends critically on the assumption that firms do not usetwo-part franchise contracts to eliminate double marginalization.On the question of entry deterrence, Aghion and Bolton (1987) show that exclusive dealingcontracts with liquidated damages can reduce the probability of entry of some lower-costentrants, although it does not preclude entry completely. Central to this result is their6However, Schwartz (1987) points out that the equilibrium outcomes obtained by Comanor and Frech arenot subgame perfect.11assumption that the incumbent firm and the buyer do not observe the potential entrant's cost,they know only its distribution function. Furthermore, there is a strictly positive probabilitythat the entrant's cost is higher than that of the incumbent, in which case no entry will occur.The buyer is willing to accept a contract that charges a price lower than the monopoly priceprecisely because there is a probability that no entry occurs. The result does not hold for thecase where all potential entrants have lower cost, nor for the case where a lower-cost entrantwill appear for certain.Rasmusen et al. (1991) show that if there exists a minimum efficient scale of production,exclusive dealing contracts can be profitable and effective in deterring entry. The intuitionis as follows. Suppose there are 10 customers, and the entrant needs to serve at least 3 inorder to reach the minimum efficient scale. The incumbent firm can deter entry by lockingup at least eight customers. If customers behave noncooperatively, then the outcome "allcustomers signing the contract" is clearly a Nash equilibrium. Implicit in their formulationis the assumption that the entrant is unable to offer similar contracts to the customers.1.4 The ModelConsider a two-stage model. There are three players: a conference, a potential entrant whohas a lower cost but is capacity constrained, and a shipper. The conference is assumed tobehave like a dominant firm. 7 The issues of how members of a cartel set prices and divideprofits are avoided. In the first stage, the conference offers the shipper a loyalty contract,which specifies a price and contains a clause which states that the shipper will not shipthrough non-conference lines. If the shipper accepts the contract, production takes placein the second stage and no entry occurs. On the other hand, entry occurs if the contract is7This assumption effectively requires the conference to behave in such a way as to maximize profits for allits member lines. However, not all writers agree that this is an appropriate assumption for liner conferences.See, for example, Sletmo and Williams (1981).12Conferenceoffers a loyalty contractShipperaccepts^Y,vZ rejectsProduction takes place^Entry occurs, conference andno entry occurs entrant play a price gameProduction takes place afterprices are announcedrejected, and the conference and entrant play a price subgame in the second stage. Productiontakes place after prices are announced. Information is perfect and there is no uncertainty.8The timing of the game is depicted in Figure 1.1.Figure 1.1: Timing of gameThe shipper produces a single output x, which is produced at zero cost but has to be shippedto the market at a per unit shipping cost of p. The shipper is assumed to be a price taker inthe input market. The shipper's demand for shipping services is9q(p) = a — bp.^ (1.1)The inverse demand function is p(q) = (a — q)/b.The capacities of the conference and the entrant are denoted as K and Ke, respectively.811 the conference is prohibited from offering the loyalty contract, then the game consists only of the secondstage; since, as shown later, both the conference and the entrant supply positive levels of output in equilibrium,entry deterrence is not possible.9This demand function is appropriate, for example, if the demand for the final product x is linear and themarket is characterized by oligopolistic competition between sellers who compete in quantities. Specifically,suppose the demand for the final product is p s = ho — h1 E, x„ where x, is the quantity supplied by firm i.Then, if firms compete in quantities, the input demand by the shipper is of the form in (1.1).13Capacity of each firm is exogenously given and cannot be changed. Each firm i is assumedto produce at a constant marginal cost, denoted ai, i = c,e, as long as output is less thancapacity. Formally, the cost function for firm i isfaiqi if qi <Ci oo^otherwise.For simplicity, let a, = 0. This assumption does not affect the results that follow since itmerely shifts the origin of a,.The conference is assumed to face no capacity constraint. The entrant, on the other hand,has a small capacity such that it would not be able to supply the monopoly output had allincumbents exited the market.1° Note that the monopoly output for the entrant is qrenarg maxq qp(q) = a/2. The capacity assumption is stated below.Assumption 1.1: K, > a, If, < a/2.It follows then K, > Ke, and that only the conference is in a position to offer a loyaltycontract. It is further assumed that the conference has a higher cost, although not so high asto render it uncompetitive against the entrant. Specifically, the conference's unit cost is nomore than p(K) = (a— 1(e)/b, the highest price at which the entrant can sell all its capacity.This is stated in assumption 1.2.Assumption 1.2: 0 < a, < (a — 1(e)/b.This assumption ensures that if entry is deterred, it is not because the conference is moreefficient. As in Deneckere and Kovenock (1990), strategies which involve a firm pricing belowits marginal cost will not be considered since these are weakly dominated strategies.Because the entrant is capacity constrained, it is necessary to derive the residual demand the10This is a realistic assumption for the liner shipping industry if capacity is interpreted as frequency ofservices over, say, a year. On most trade routes, independent lines and tramps are capable of providingfar less frequent services than conferences because of their limited fleet sizes. Further remarks about thisassumption can be found in Section 1.5.14conference faces when it is undercut. For this purpose, the efficient rationing rule is used.That is, the shipper is assumed to buy from the low-price firm first. If the low-price firmcannot supply all her demand, she then buys from the high-price firm. If the conferenceand the entrant charge the same price p and the entrant is not capacity constrained, theentrant supplies A(a — bp) and the conference supplies (1 — A)(a — bp), where A E [0, 1] isfixed exogenously. The payoff function for firm i, i= c,e is given below.Li(m) E (pi — a,)min(Ki, a — bpi)^if p < P3ll =^Ti(P2) E (pi — ai)Si^ if 132 = P3Hi(pi) _=-(p — ct,) max(0, a— IT 7 — bpi) if pi > p3.whereS, = min(K,, A(a — bp)),^ (1.2)S, = max(a — Ke — bp, (1— A)(a — bp)).^(1.3)Definearg max Hi(pi)PiII ^H2 (p)^pf = min{p :^= Li(p)}.In words, pfl. is the optimal price for firm i if it is to be the high price firm; the profit it getsis II:. Next, pf is such that firm i is indifferent between being the low price firm charging pfand the high price firm earning H. Further, let 141 = 0 if Hi = 0. It can be easily verifiedthat^Pc = —2b (a —^ba,),H^1(1.4)which, after substitutions, givesHc* = —41b(a — Ke — ba,)2,^ (1.5)and1Pc = y[(a-F be:0— \12(a— bac)Ke — Kg].^(1.6)15Note that the expression in the square root in (1.6) is positive since by Assumption 1.2,< a — bac. Furthermore, peli > a, due to Assumption 1.2.Notice that pH; = H =^= 0. That is, if the entrant is the high-price firm, it earns nothingsince the conference will supply the whole market.A subgame perfect equilibrium is sought, in which strategies constitute a Nash equilibriumfor each subgame. As usual, the logic of backward induction is employed to solve the game.Under a similar framework, Sjostrom (1988) argues that, in equilibrium, the entrant suppliesall its capacity by charging its marginal cost, while the conference acts as a dominant firmby supplying the residual demand. This is, however, not an equilibrium for the price sub-game. Consequently, the results he obtains from comparative static exercises are not valid.Proposition 1.1 states thatProposition 1.1 Given Assumptions 1.1 and 1.2, there does not exist a pure-strategy equi-librium for the price subgame.Proof Suppose not, i.e., suppose there exists a pure-strategy equilibrium (pc*,p). First, Ishow that p > a, for i = c, e. Since pricing below marginal costs is ruled out by assumption,p*, > a,. Thus it remains to show that p*, > etc. Suppose the contrary, p: < a,. Then,by Assumption 1.2, qe = K e, and this implies g, = a — Ke — bp,. Profit maximization bythe conference requires that p,* = pH, > cec. But if p,* > etc, then p,* < eke is not optimal, acontradiction.Next, consider three cases: p: > p , p: <p , and p: = p.Case 1: p: >In this case the conference supplies the whole market since it is the low-price firm. That is,g, = a — bp,* and ge = 0, which means that the entrant makes zero profit. But if the entrant16undercuts the conference slightly by charging (p*, — E), e arbitrarily small, it gets a profit of(p,* — E)min(If e, a — b(p,* — E)) > 0, a contradiction.Case 2: p: < p.In this case the entrant charges the lower price, thus it either sells up to capacity or suppliesthe whole market, i.e., qe = min(Ke, a — bp:). Consider two cases. First, suppose the entrantis capacity constrained, i.e., qe = K. Then p: < (a — /GO. Further, it must be truethat p: < p. Suppose not, then p: > p: > pH, . But in equilibrium, the highest pricethe conference charges is p, a contradiction. Thus p: < p,11 , but then there exists a pricep E (p: , ) such that He(p) pi( > p:  ll, which contradicts the hypothesis that p:is optimal. Next, suppose the entrant is not capacity constrained, i.e., qe = a — bp:. Thisimplies pc* > (a — > ac, where the latter inequality follows from Assumption 1.2. Sincethe entrant is not capacity constrained, the conference makes no sales and earns zero profit.However, by undercutting the entrant slightly, the conference can make a strictly positiveprofit. This again contradicts equilibrium.Case 3: p: p*In this case firm i gets to supply S„ i = c, e, as given in (1.2) and (1.3). If p* = ac, theconference makes zero profit, but by raising the price to pH, , it gets (pf,1- — cx,)(a — If e—bp1-) > 0,a contradiction. If p* > ac, then at least one firm has an incentive to undercut its rival.Suppose A -= 0, then the entrant is making zero profit, but by undercutting slightly it canmakes a strictly positive profit. If A > 0, then the conference has an incentive to undercut,since by charging p* it supplies S, < a — bp* while by undercutting slightly it gets to supplythe whole market. This contradicts the hypothesis that p* is optimal. irThis non-existence result is driven by the assumption that the entrant has a limited capacity.Without this capacity constraint, the standard Bertrand result obtains; that is, a pure-strategy equilibrium exists in the form of pc* = p: = ac, with the entrant displacing the17conference as the sole supplier. However, this equilibrium breaks down when the entrantis capacity-constrained. To see this, suppose the entrant charges the price pc = ac. Theconference then faces a residual demand and the optimal response is to charge piel > a,. Butif this is the case, the entrant, who is charging Pe = ac, will want to charge a price that isjust below plic . Thus, any pair of prices with Pe = a, is not an equilibrium. Further, theconference has an incentive to undercut the entrant if the entrant charges any price that isabove a,. Note that the entrant makes no sales if it is undercut. Thus, any pair of prices thatare both above a, cannot be an equilibrium. There is, therefore, no pure-strategy equilibriumfor the price subgame.Since there does not exist a pure-strategy equilibrium, it is necessary to look for mixed-strategy equilibria. A mixed strategy for firm i is a distribution function Gi with a support[pi,pi], where pi > pi. A mixed-strategy equilibrium is defined as a pair of distributionfunctions (G7, G;) such thatHi(G7, > Hi(Gi, G) V Gi, i = c, e,where 1-1,(G1, G3) is the expected profit of firm i when a pair of mixed strategies (G G) isplayed. A mixed-strategy equilibrium in this context may be interpreted as a situation inwhich firms randomly hold sales, as in Varian (1980). Casual empirical observations indicatethat freight rates in the liner shipping market are indeed quite volatile, particularly whenthere is new entry. (See, for example, Stopford, 1988.) Thus a mixed-strategy equilibriumappears to be an appropriate description.Proposition 1.2 gives the supports of the two firms' strategies and their corresponding equi-librium profits for the price subgame.Proposition 1.2 In a mixed strategy equilibrium, the conference and the entrant share acommon support [pc' ,p,I1], with their respective equilibrium profits as II,* and Le(pc1).18Proof. See Appendix 1A.11Given the two firms' supports and equilibrium profits, Proposition 1.3 establishes the mixedstrategy equilibrium.Proposition 1.3 Given Assumptions 1.1 and 1.2, a unique mixed-strategy equilibrium forthe price subgame is given by the following distribution functions:where{ ac(p) if 1) < P < PcH,1^if P > Pcil ,a — bp (a — K, — bac)2 vIC,^4b1Ce(p — ac) p E [p, p].Proof It is straightforward to show that Gc(.) and G,(•) are increasing functions, withG(pi) = 0, Gi(pc-11) = 1, i^c,e. Further, both functions are right continuous on [g,Thus, (G,,Ge) are distribution functions. Next, given Ge, the conference's profit from charg-ing a price p islic(p) = Ge(p)[(p — ac)(a — Ke — bp)] + (1 — Ge(p))(p— ac)(a — bp),which, after substituting in Ge, simplifies to the equilibrium profit of the conference. Sim-ilarly, given Gc, noting that the entrant is capacity constrained, i.e., Ke < (a — bp) for allp E ], the profit of the entrant is11,(p) G(p) (0) + (1 — Gc(p))pKe,which simplifies to the equilibrium profit of the entrant. Thus (Gc, Ge) are indeed a pair ofequilibrium strategies. Finally, the equilibrium is unique since (G c,Ge) uniquely solve the11Deneckere and Kovenock (1990) give a complete characterization of the capacity-constrained price gamewhen firms' marginal costs differ. They show that the supports of firms' strategies are not necessarily thesame and that each firm's support is not necessarily connected. These cases occur when the high-cost firmhas a small capacity. Kreps and Scheikman (1983) and Osborne and Pitchik (1986) give results pertaining tocases in which firms' marginal costs are the same.19Note that if pc < pc, then qe Ke for all pc Eif Pc > Pcif Pe = Pc = Pif Pe < Pc.:PcP] I!0q, =^min(Ke, A(a — bp))I. Kfollowing equations= G c(p)[(p — )(a —IT — bp)] + (1 — G e(p))(p — a c)(a — bp),Le(p) = (1 — s(OPIG.Note that the conference's equilibrium strategy, G c(p), has a mass point at^. That is, theconference charges the price p-P. with a strictly positive probability, which implies that theconference is more likely to be undercut in equilibrium. This result is hardly surprising sincethe conference can still earn a strictly positive profit Hc* if it is undercut, whereas the entrantgets nothing if it is the high-price firm. Thus, the entrant will tend to be more aggressive insetting its price.Given the solution of the price subgame in the second stage, it is straightforward to solvethe full game. Note that for any given pair of prices (pc, pc), the average price (weighted byquantities) the shipper pays isA = qcp, q,p,^(a — — bpc)pc q,p,p 7qC + qe a — bpswhereThus, given the mixed-strategy equilibrium of the price subgame, the expected price theshipper pays is obtained by integrating over all possible pairs of prices, i.e.,PH P Hc f cEpJP-1 ApG/e(pc)G(pe) dpc dpepIPr[ pc =^] Gie(pe)KePe + (a ^ bec )Plic  dpe.a — bpti(1.7)Now, if the conference can find a price at which it can make more profit, and yet is lower thanthe price that the shipper who does not sign a contract expects to pay, then the conference20can exclude the lower cost entrant. That is, suppose there exists a price 25 such thatfi < Ep and II,(P) >Then the conference can profitably exclude a lower-cost entrant by offering a loyalty contractthat charges 25. Proposition 1.4 shows that this is always possible.Proposition 1.4 Given Assumptions 1.1 and 1.2, there always exists a price fi such that theconference can profitably exclude the potential entrant.Proof: Note that the lowest price the shipper pays in the mixed strategy equilibrium is p!-,which occurs with a probability of strictly less than one. Thus, from the definition of Ep in(1.7) above, it is obvious that Ep > g. This implies that there exists a price p ^.73 < E.Further, 11,(5)^(23 — ac)(a — b25), and note that by definition, pc1 is the price which solvesthe equation (p — etc)(a — bp) = H. Since p < < p, where Km is the monopoly pricefor the conference, it follows then 11() > _Tic*. Thus, the conference earns a higher profit byoffering the shipper a loyalty contract with a price 25. It is to the advantage of the shipper toaccept the contract. Therefore, entry is deterred.Intuitively, if the shipper rejects the contract, the conference and entrant play their respectivemixed strategies, and prices will likely be high since the conference's equilibrium strategy callsfor charging pH, with a strictly positive probability. Hence, the shipper is willing to acceptthe contract. The conference, on the other hand, is willing to offer such a contract becauseit would otherwise face a residual demand and charge a high price p supply a smallquantity. If it offers a contract, it captures the whole market, although at a lower price.Proposition 1.4 shows that the latter is always more profitable.211.5 DiscussionThe equilibrium for the full game involves the conference offering a loyalty contract witha price j5 and the shipper accepting the contract. By signing a loyalty contract, both theconference and the shipper gain, at the expense of the entrant. This result may appear torely heavily on the assumption of a single shipper. However, if one regards capacities offirms as the frequency of services over a period of time, then the result holds for n identicalshippers. Each shipper will prefer to sign the contract with the conference.This result does not depend on the type of competition assumed in the second stage. In fact,the same result holds if competition in the second stage is Cournot rather than Bertrand. Thiscase is analysed in Appendix 1B. The intuition is as follows. Under the Cournot assumption,firms supply different quantities (due to different marginal costs), but charge the same price.Competition is less intense, and both firms earn strictly positive profit at this price. Theconference can therefore offer a loyalty contract supplying all the shipper's demand at aslightly lower price. In effect the conference achieves a discrete jump in revenue by charginga slightly lower price. This means that the conference earns a higher profit by offering thecontract. The entrant, although having a lower cost, is again deterred due to its limitedcapacity.The equilibrium outcome, however, is not socially efficient, since the shipping service isprovided at a total cost of C ac(a — V), not the lowest possible. A more efficient outcomeis qe = IC, and qc = a — —V, with a total cost of C* = cte(a — Ke— < C. Nonetheless,the outcome is socially more efficient than that under monopoly, since the contract price islower than the monopoly price. The presence of a potential entrant, as predicted by thecontestable market hypothesis, exerts a downward pressure on price. The entrant's capacityconstraint, however, prevents it from completely disciplining the conference. Instead, theconference exploits this 'weakness' of the entrant by introducing a loyalty contract system to22exclude the entrant. Therefore, the market fails to be contestable because of the entrant'scapacity constraint.In part, the entrant is unable to reap the efficiency gain because, according to the rules ofthe game, the entrant does not get to move if the shipper accepts the contract. This can beinterpreted as saying that there is no way in which the entrant can credibly commit to charginga low price before it enters the market. The no-entry result is weakened if precommitment bythe entrant is possible. To see this, consider a slightly modified version of the game. Supposethat the shipper, having accepted the contract, is allowed to breach it at no cost; and thatthe entrant can commit to charging its marginal cost, i.e., p, = 0. Whether the entrant willdefault obviously depends on the contract price /3.To decide what contract price to offer, the conference has to anticipate what will happen ifthe shipper breaches the contract. Since in that event the entrant is committed to chargingp, = 0, the best response of the conference is to charge p , which gives a profit of H. The(weighted) average price the shipper pays is thenpc11(a — K, — bpi!)11-P^a — be (1.8)(The superscript "1" is used to indicate that this is the lowest possible average price theshipper pays.)If on the other hand, the conference and the shipper sign a loyalty contract at an agreed priceof , then the conference's profit is II,(73) = (fi — ac)(a — 473). In order to induce the shippernot to default, the contract price must be less than the average price in (1.8). Assume, forthe moment, that ct, < A, so that the conference can find a contract price /3 such thatcc, < < Ap1.12 However, for the conference to offer such a contract, it must be able to earna higher profit than H. The highest price the conference may charge is /3 = Apl , which gives12Note that, a priori, Alp need not be greater than a,. It will be shown later that Alp > a, whenever theconference can profitably deter entry.23a profit of ,(Api ).13 Therefore, a contract will be offered if= (A ac)(a —^> H:^ (L9)If this inequality holds, it also ensures that Api > a,. To see this, note that by definition,< p , thus a — bApl > a —^> 0. Since Hc* > 0, this implies that if the inequality in(1.9) holds, then Api > ac.After substitutions and some algebraic manipulations, the entry deterrence condition in (1.9)simplifies to144a/Ce(a — ba,) < (a — Ke — bac)(a+ Ke bac)2 (1.10)The inequality in (1.10) is depicted in Figure 1.2 in (as, KO space, with a = 2 and b = 1.The shaded region represents combinations of a, and K where a contract is offered (andthus entry is deterred). Note that entry is likely to be deterred if the cost disadvantage of theconference is small, or the entrant's capacity is low, or both. Entry deterrence is thereforeonly optimal for the conference for some values of K e and a,.It is worth noting that the above entry deterrence condition is derived under the extremeassumptions that the shipper is allowed to breach the contract without paying any damages,and that the entrant is committed to charging the lowest possible price (i.e., its own marginalcost). Both assumptions tend to reduce the feasibility and profitability of offering an exclu-sive dealing contract by the conference. Hence, entry deterrence is least likely under thesecircumstances.Central to the entry deterrence results is the assumption that the entrant is capacity-cons-trained. Absent this capacity constraint, the standard Bertrand result applies. That is, thelow-cost entrant simply displaces the conference, and loyalty contracts cannot be an effectiveentry barrier. This is in essence the argument of the Chicago School. However, an entrant13Since 0 < 11,1 < p,H < pr, the conference's profit function tic is thus increasing over the interval [0, A,].-"See Appendix 1C for the derivation.24Figure 1.2: Graphical representation of entry deterrence condition, with a 2, b = 1who has a limited capacity is not able to displace the conference. The entrant's small capacitymeans that the shipper cannot satisfy all her demand through the entrant (at a relatively lowprice), and must also use the conference's services (at a relatively high price). This resultsin a high average price, hence a low surplus for the shipper. As a result, the conference isable to compensate the shipper for her loss of a more efficient but low-capacity supplier. Itis at this point that the Chicago School argument breaks down. Loyalty contracts indeedrepresent an effective entry barrier against low-capacity entrants.Why, then, does the entrant have a limited capacity? Is there not an incentive for the entrantto increase its capacity? Two responses can be offered. First, this capacity assumption is arealistic one for the shipping industry if capacity is interpreted as the frequency of servicesover, say, a year. Note that a conference consists of several members who are usually thewell-established lines in the business. Not surprisingly, on most trade routes independentlines and tramps are capable of providing far less frequent services than conferences due tolimited fleet sizes. It is therefore unreasonable, from a practical viewpoint, to expect an25independent line or a tramp ship to be able to match the capacity of the conference.Second, on a theoretical note, the imperfection of capital markets may justify the capacityassumption. Fudenberg and Tirole (1986), Poitevin (1989), and Bolton and Scharfstein (1990)show that it is more difficult for an entrant to obtain financing if the capital market ischaracterized by asymmetric information. Because of this imperfection, an incumbent firmis able to prey on a financially constrained rival by inflicting losses on the rival.' The sameargument can be applied to show why the entrant has a limited capacity and is unable toincrease that capacity.1.6 ConclusionThis paper shows that loyalty contracts can be effective barriers to entry when the entrant hasa limited capacity. This result does not depend on the assumption of Bertrand competition(and hence the nonexistence of pure-strategy equilibrium). A weaker result is obtained ifprecommitment by the entrant is possible. The resulting equilibrium is not socially efficient,since a lower-cost entrant is unable to enter the market. Central to this result is the capacityconstraint of the entrant, which enables the conference to stay in the market. The ensuingprice (or quantity) competition results in a high average price, hence a lower surplus for theshipper. This in turn allows the conference to compensate the shipper for her loss of analternative supplier.Without the capacity constraint, the standard Bertrand result applies: a more efficient entrantsimply displaces the conference. Exclusive dealing contract is therefore not an effective entrybarrier. The Chicago School argument is thus valid in this case. The use of exclusive dealingcontracts cannot possibly cause any efficiency loss. However, this argument breaks downwhen the entrant is capacity-constrained. Casual observations about the shipping industry15 This is often referred to as the 'long-purse' story of predation in industrial organization.26suggest that, as compared to the Chicago School argument, the present model seems moreconvincing. In fact, conferences openly admitted that the main purpose of loyalty contractswas to exclude entry of smaller entrants. Mr. Sutherland, an executive of a liner company,testified before the Royal Commission on Shipping Rings (1909) that,To put the matter quite clearly and openly, the conference system and the rebatesexclude fairly what you might call the casual competition. Without the rebatesystem, you are liable to have a state of chaos; with the rebate system, that casualcompetition which would throw things into a state of chaos is excluded. (Citedin McGee, 1960, p.243; emphasis added.)Presumably, by 'chaos,' Mr. Sutherland referred to the volatility of freight rates when therewas entry. Clearly, the practice of rebate system (a form of exclusive dealing) has an adverseeffect on the competitiveness of 'casual competitor' such as tramps. Therefore, we concludethat the Chicago School's position of letting all exclusive dealing contracts be per se legal inliner shipping is questionable, if not unwarranted.27Chapter 2Vertical Contracts and Entry Deterrence2.1 IntroductionFirms in a vertical relationship often enter into various forms of contractual arrangementsbroadly known as vertical control or vertical restraints. These arrangements range from asimple supply contract to a complex vertical merger (i.e., vertical integration) between twoor more firms. Broadly, there are two strands of literature on vertical control in industrialorganization. The first focuses on the control problem of a monopoly or monopsony, whowishes to influence the actions of firms at other stages of production. For example, a manu-facturer may like to impose restrictions on retailers' choices of price, output, location and soon (see, for example, Mathewson and Winter, 1984). The welfare effect of these restrictionsis generally ambiguous. (See the surveys by Katz, 1989; and Perry, 1989.)More recently, many studies have examined whether vertical integration can lead to theforeclosure of competition in upstream or downstream markets when these markets are char-acterized by oligopolistic competition. (See Salinger, 1988; Hart and Tirole, 1990, Ordover,et al., 1990; and Bolton and Whinston, 1991, among others.) These studies show that it ispossible for vertical integration to lead to anticompetitive foreclosure, and social welfare isgenerally lower.28Little, however, has been said in the literature regarding the possibility of an incumbentfirm deterring more efficient entrants through some forms of vertical control. This paperexamines whether an upstream monopolist is able to deter the entry of more efficient entrantsinto the upstream market through establishing contractual relationships with its downstreamcustomers. We show that entry deterrence is possible if the downstream firms are competingagainst each other in an imperfectly competitive product market. This is, however, not trueif the downstream firms are not in direct competition against each other (e.g., they operatein unrelated product markets).The paper is organized as follows. Section 2.2 reviews the relevant literature. Section 2.3outlines the basic setting of the model and considers the simple case where the downstreamfirms are not in direct competition. The case of Cournot competition between downstreamfirms is considered in Section 2.4. The results are discussed in Section 2.5. Some concludingremarks are contained in Section 2.6.2.2 Related LiteratureIn the antitrust literature, it is widely believed that vertical contractual arrangements aremade mainly for efficiency reasons (see, for example, Marvel, 1982). In particular, a school ofthought that is often referred to as the Chicago School holds that these practices cannot haveany anti-competitive effects. Among the chief proponents are Bork (1978) and Posner (1976,1981). These authors argue that a downstream firm, for example, is unlikely to want toparticipate in schemes that will limit entry in the upstream market. This is because by makingthe upstream market less competitive, the downstream firm loses alternative and perhapsless costly sources of supply. Thus, for an upstream producer to successfully implement suchschemes, it must compensate the downstream firm for its loss. However, this cannot beprofitable for the upstream producer if it is less efficient than potential entrants whom it29wishes to exclude. Based on these arguments, Posner (1981) calls for the courts to treat allvertical restraints as per se legal.In industrial organization, however, the anticompetitive effect of vertical contractual arrange-ments in oligopolistic markets remain a concern. For examples, Comanor and Frech (1985),Kranttenmaker and Salop (1986), and Mathewson and Winter (1987), among others, haveshown that firms in imperfectly competitive markets can use exclusive dealing contracts toreduce the competitiveness of rivals or even eliminate rivals altogether.On the question of entry deterrence, recent contributions by Aghion and Bolton (1987) andRasmusen et al. (1991) show that, under certain institutional settings, a monopolist seller isable to deter entry through signing exclusive dealing contracts with customers. Aghion andBolton consider a particular type of contract that contains two elements: a contract priceand liquidated damages to be levied on the customer if a breach of contract occurs. In themodel, entry is uncertain, with the probability of entry depending on the liquidated damagesagreed upon between the seller and buyer. A marginally more efficient entrant may be unableto set a price attractive enough for the buyer to breach the contract. On the other hand, avery efficient entrant may find it profitable to offer the buyer a sufficiently low price that thebuyer is fully compensated in breaching the contract. In this way, liquidated damages actas an entry fee, which is divided between the seller and buyer. The authors show that, bysetting the liquidated damages optimally, the seller can exclude the entry of some, thoughnot all, lower-cost entrants.On the other hand, Rasmusen et al. assume the existence of a minimum efficient scale ofproduction such that an entrant will need a minimum number of customers, say x, in orderfor entry to be worthwhile. To deter entry, the incumbent therefore need only sign up enoughcustomers through contracts so that there are less than x customers for the entrant. In fact,Rasmusen et al. show that all customers signing the contract with the incumbent is a Nash30equilibrium.This paper considers the use of supply contracts as a device for entry deterrence. Thesecontracts take a simple form: they commit downstream firms to acquire all their requirementsof a certain input from the upstream supplier.1 Unlike the contracts considered by Aghionand Bolton, no liquidated damages are specified. Further, the use of two-part franchisecontracts is permitted, i.e., the contract may contain a fixed fee as well as a per-unit charge.It is worth noting that, in the simple environment considered below, there is no substantivedifference between signing a supply contract and outright vertical merger. The analysis belowrequires little modification if the latter possibility is considered.This paper differs from Aghion and Bolton, and Rasmusen et al. in one important aspect:downstream firms are in direct competition against each other. It is this downstream com-petition which enables the upstream incumbent to use contracts as barriers to entry.2.3 Basic SettingAn upstream monopolist, denoted U, supplies an input z to two downstream firms, 1 and 2.For concreteness, we refer to all upstream firms as producers and all downstream firms asretailers. Producer U holds an exclusive right (e.g., a patent) to a technology that is necessaryto produce the input. Assume, for simplicity, that this is the only factor of production forthe two retailers, who share a production technology which turns one unit of z into one unitof final product. The upstream monopolist faces a potential entrant, denoted E, who has todecide whether to invest in research and development. If producer E invests and the RD is successful, it will be able to supply the input to retailers 1 and 2 at a lower cost. Ineffect, the entrant is in a position to displace the incumbent as the sole upstream supplier.In the antitrust literature, these contracts are known as requirements contracts. They do not differ fromexclusive dealing contracts in the present context. Under the antitrust laws, these contracts are not illegal,rather they are subject to the rule of reason. That is, these practices will be decided on a case-by-case basisby the courts. For greater detail, see Blair and Kaserman (1983), Chapter 9.31The probability that producer E succeeds in the R & D is exogenously given as 0 < a < 1,which is known to all parties.Consider the following hypothetical example. The upstream monopolist is a utility company,that supplies the energy requirements of two nearby factories. There exists an alternativesource of energy which, if successfully developed, will be able to meet the energy requirementsof the two factories at a lower cost. There is, however, a risk involved in developing thisalternative source. In other words, success is not assured when the R D investment ismade. The questions of interest are: Is there any incentive for the upstream monopolist toestablish a contractual relationship with firms 1 and 2 so that the entrant is discouraged frominvesting in R & D? Will the downstream firms accept such a contract?It is assumed that the upstream monopolist is prohibited from discriminating between down-stream retailers. In other words, if a contract is offered, it must be made available to bothretailers. This assumption rules out the possibility that the upstream producer can create amonopoly in the downstream market by contracting with only one retailer. It is also assumedthat all firms are risk neutral.Given that the industry structure consists of successive oligopolies, the problem of doublemarginalization may arise if only linear pricing is considered (Spengler, 1950).2 We assumethat firms are sophisticated enough to recognize and avoid this problem through nonlinearpricing. For the present purpose, it suffices to allow the use of two-part tariffs, so thatelimination of double marginalization is not a motive for signing contracts. That is, firmsmay negotiate contracts that contain a fixed fee and a constant per-unit charge. In theevent that no contract is signed, firms are also free to negotiate two-part tariffs on spottransactions.3 It is easy to show that such a pricing scheme maximizes the combined profits2 Unless, of course, the downstream market is characterized by Bertrand competition.3It goes without saying that linear contracts are a special case of two-part tariffs, with the fixed fee beingset at zero.32of the upstream and downstream firms.4Consider a two-stage game. In stage 1, producer U offers retailer i (i=1,2) a supply contract,which specifies a wholesale price w and a fixed fee f. If both retailers accept the contract, itis implemented in stage 2, and no entry will occur, regardless of whether producer E investsin R & D. If neither retailers accept the contract, two possibilities arise in stage 2. Eitherentry occurs or it does not. Entry does not occur if producer E fails in its R D effort,which occurs with probability 1 — a. There are then three firms in the industry: producerU upstream, and retailers 1 and 2 downstream, and they bargain over a two-part tariff onspot transactions. We do not specify the bargaining process, although we assume that thebargaining outcome is efficient; that is, firms will not agree on an inferior outcome when anoutcome is available in which they can all be made better off. The joint profit to be dividedis therefore the amount an integrated monopoly would earn. Note that each firm's share ofthe joint profit is unknown, since it depends on the firms' bargaining power, which is leftunspecified. However, the possibility that the upstream producer appropriates all of the jointprofit through the fixed fee is not ruled out.On the other hand, entry occurs if producer E invests in R & D and succeeds, which occurswith probability a. In this event, there are four firms in the industry: producers U andE upstream, and retailers 1 and 2 downstream. We assume throughout that retailers 1and 2, being in the same downstream market, do not engage in direct bargaining betweenthemselves. The same assumption also applies for producers U and E.' There are fourpossibilities in this case: either producer U or E sells to both retailers 1 and 2, or eachproducer sells to a different retailer.' The bargaining issues involved are complex, since4 For a clear exposition of the problem of double marginalization and ways to overcome it, see Tirole (1988),pp.174-177.5This assumption is made for practical reason, since it is illegal in most cases for firms in the same marketto attempt to divide the market among themselves.6We rule out the possibility that both producers selling to a single retailer. Since producer E is moreefficient, it is unlikely that a retailer would find it advantageous to acquire its input from both producers.33not only are the producers competing for customers downstream, but also the retailers arechoosing suppliers upstream. To avoid these bargaining issues, we proceed by letting the tworetailers choose, simultaneously, their respective upstream supplier. That is, retailers 1 and2 play a one-shot game in choosing, non-cooperatively whether to acquire their inputs fromproducer U or E. Henceforth we will refer to this game as the 'Choose a Supplier' game. Thedetail of this game can be found in Section 2.4.For simplicity, we rule out the possibility that one of the retailers, say retailer i, agrees to,while retailer j (j i), rejects, the contract. In effect, we assume that there is no incentivefor a retailer to hold out by not signing if the other retailer has agreed to sign the contract.Ideally, this should constitute part of the equilibrium outcome of the model. However, morestructure is needed to analyse this possibility; for example, one must specify what happensto the retailer who holds out if no entry occurs in stage 2. Obviously, different specificationscould lead to vastly different results. We therefore assume that there are ways by which theparties to the contract can avoid such opportunistic behavior. For example, the parties maybe able to agree that the contract is void unless both retailers enter the agreement. Furtherdiscussion of this assumption is deferred until Section 2.5.The timing of the game is illustrated in Figure 2.1.Consider first the simple case where retailers 1 and 2 are not in direct competition against eachother in the downstream product market. They may, for example, be located in geographicallyseparated regions or operate in unrelated markets. Assuming that they are identical in everyother aspect, there is then no loss of generality in considering a representative retailer, sayretailer k. Proposition 2.1 shows that signing a contract in stage 1 cannot be part of asubgame perfect equilibrium.Proposition 2.1 Suppose that the retailers do not compete against each other in the down-34three-playerbargainingretailers 1 and 2 play'Choose a Supplier' gameProduction takes placeand profit dividedProduction takes placeand profit dividedFirm Uoffers a long-term contractVFirm i, i=1,2accepts^rejectsContracts implemented,game endsentry does not occur(with prob. 1-a)entry occurs(with prob. a)Four firms in industry:U, 1 and 2^U, E, 1 and 2V^ I'C/):SooR1■.)Three firms in industry:Entrant invests in R & DFigure 2.1: Timing of game35stream product market. Then, there does not exist a subgame perfect equilibrium in which theupstream monopolist and the retailers sign a contract.Proof: Consider first stage 2. Suppose no contract is signed in stage 1. Then, in the eventthat no entry occurs, only firms U and k are in the industry. This is the case of successivemonopolies, and total profit is maximized through the use of two-part tariffs. Let IV denotethe total profit, and Rk > 0 be retailer k's share of the total profit. Thus, the payoff toproducer U is II* — Rk. Next, suppose entry occurs in the upstream market. Then, there aretwo producers competing for a single buyer in the downstream market. This competition forbuyer causes the more efficient entrant to offer a wholesale price and fixed fee combinationthat gives a profit of IV+ c to firm k, where c > 0 is some arbitrarily small amount, and theincumbent makes no sales. Let II denotes the total profit to be shared between firms E andk. Note that by supposition, fl > II*. The payoffs to firms k and E are, respectively, II* + cand .11 — (II* + c).7In stage 1, retailer k's expected profit for not signing the contract is a weighted sum of itsgains in stage 2 under the two cases considered above, i.e.,(1 — a)Rk a(11* c).Similarly, if a contract is not signed, the expected profit for producer U in stage 2 is(1 — a)(II* — Rk)d- a • (0) = (1 — a)(11* — Rk)•Suppose, contrary to the hypothesis, that a contract is signed. Let Rck be retailer k's shareof the total profit, 11*. Then, the payoffs to producer U is H* — R. Hence, for the contractto be offered by producer U, it must be true that— Irk > (1 — ct)(fl* — Rk),7Alternatively, suppose that firms E and k engage in a noncooperative bargaining game, in which retailer kholds the outside option of dealing with producer U instead. Assuming that players have a common discountfactor which approaches unity, then the payoffs for firms k and E are, respectively, II* and 1-1 — II*. SeeOsborne and Rubinstein, (1990), pp.54-63.36which, after rearranging terms, simplifies toR7, < (1 — tx)Rk + all*.^ (2.1)However, for the same contract to be acceptable to retailer k, it must also be true thatR> (1 — oz)Rk a(1-1* e).which contradicts (2.1).The intuition behind Proposition 2.1 is as follows. Suppose no contract is signed in stage 1.Since the entrant has a lower cost, the retailer is assured of at least II* + a in the event thatentry occurs in stage 2. If no entry occurs, the joint profit between the monopolist incumbentand retailer is II*. Hence, the sum of the incumbent and retailer's expected profits is at leastH* plus some small amount in stage 2. However, in stage 1, the total profit to be dividedbetween the incumbent and retailer through a contract is at most II*. Clearly, by not signingthe contract, either the incumbent or the retailer can be made better off without making theother worse off. Therefore, a contract cannot be part of an equilibrium.This result is in sharp contrast to that of Aghion and Bolton, who show that it is possible forthe upstream producer to deter lower-cost entrants through contracts. Their model, however,contains an important feature. The contract they consider contains two components: acontract price and liquidated damages, the latter are to be paid by the retailer if a breachof contract occurs. Thus, it is possible for the retailer to breach the contract by payingthe agreed liquidated damages. By setting the liquidated damages, the incumbent in effectimposes an entry fee that the entrant must pay in order to trade with the retailer. This entryfee is set in the same way a monopoly would set its price. Thus, a very efficient entrant(as compared to the incumbent) is willing to pay the liquidated damages on behalf of theretailer to induce the retailer to breach the contract. On the other hand, a marginally moreefficient entrant is not willing to do so. Hence, the incumbent is able to exclude some, but37not all entrants. For comparison, we may interpret the contract in this paper as one which,once signed, is prohibitively costly to breach. In effect, the 'liquidated damages' in this caseare large as compared to the benefit of the contract, and do not accrue to the incumbent.8It can be shown that, given this modification, the result of Aghion and Bolton no longerholds. However, while this interpretation is possible, we emphasize that the present modeldoes not focus on an alternative specification of the liquidated damages assumption in theAghion-Bolton model.2.4 Cournot Competition in the Downstream MarketIn this section, we consider Cournot (quantity) competition between retailers 1 and 2 inthe downstream product market. The case of Bertrand (price) competition when the finalproducts are differentiated is considered in Appendix 2B. It is shown there that the qualitativenature of the results remains unchanged. We do not consider the case of homogeneous-goodBertrand competition because, by definition, the retailers earn zero profit in all cases. Theretailers are therefore indifferent between signing and not signing a contract.For simplicity, we assume that the final product is homogeneous and the retailers face lineardemand of the formP = 1 — (qi^q2), (2.2)where p denotes the price and q, denotes the quantity of retailer i, i = 1, 2.Let wz and A be, respectively, the wholesale price and fixed fee facing retailer i, i = 1, 2.Note that, by the no-discrimination assumption, wi = w and A = f if both retailers acquiretheir inputs from the same producer.8For example, one can think of liquidated damages as litigation costs, most of which go to the lawyers.38The decision problem of retailer i is to maximize its share of the total profit,R=max (p—w)q— f.^ (2.3)qiThe first-order condition gives rise to the reaction function for retailer i, i = 1,2:1qi(qj) = —2(1 qi wi)' jSolving the two reaction functions gives the optimal quantity supplied by each retailer,= —1(1— 2wi wj), i = 1,2, j^ (2.4)3Substituting (2.4) into (2.3) gives the Cournot profit for retailer i, i = 1,2:1Ri = —(1 — 2wi wi)2 —9Consider next the upstream market, where we assume that the production is characterizedby constant marginal cost. Let en and ce denote, respectively, the marginal costs of producersU and E, if producer E succeeds in the R & D. Assumption 2.1 states that the entrant is amore efficient producer, and that the marginal costs are not too high in relation to demand.Assumption 2.1: ce < en < 1.We now proceed to analyse various possible market configurations in stage 2 of the game.Recall that if entry occurs in stage 2, there are four firms in the industry: producers U and Eupstream, and retailers 1 and 2 downstream. There are four possible market configurations:either both retailers buy from the same producer, or each retailer buys from a different pro-ducer. In order to derive the equilibrium market configuration(s), we examine each possibleconfiguration in turn.Case I: Both retailers acquire their inputs from producer U.We denote this case as { U-1-2}. The decision problem facing producer U isSu = max (wi —^f2).wt (2.6)(2.5)39Since we do not allow producer U to discriminate between retailers, we have wi = w and= f (i = 1,2), which imply that, from (2.4),1= —3(1 — w), q*, i = 1,2,and from (2.5),1= —9(1 — w)2 — f = R, i = 1,2.Hence (2.6) reduces toSu = max (w — ci,)(2q*)+ 2f,which, after substituting q* and f, becomesSt, = max —2 (w —c)(1 — w) 2[-1(1 — w)2 — R].w 3^9Note that the objective function in (2.7) is strictly concave, thus a global maximum existsand the unique solution is given by,1w* = —4 (1 + 3c,i).Substituting (2.8) into (2.7) gives the maximum total profit, denoted 11:(-L=. St, +2R), to bedivided among the three firms,ll^1= 74(1 — cu)2. (2.9)It is worth noting that 11„* is the same as the monopoly profit of an integrated monopolist.Because the use of two-part tariffs is allowed there is no double marginalization. ProducerU, being the sole upstream supplier, is able to set the wholesale price w* such that the retailprice and quantity are identical to those under an integrated monopolist.Case 2: Both retailers acquire their inputs from producer E.This case is denoted V-1-21. The above results apply, with appropriate changes in labels,to this case. In particular, the maximum total profit to be divided among firms E, 1, and 2is given by,II: = -14(1— c02. (2.10)(2.7)(2.8)40Case 3: Each retailer acquires input from a different producer.We denote this case as {U-i, E-j} (i, j = 1,2, i^j). Note that the retailers' decisionproblems remain the same as before. Thus, retailer i's optimal output and profit are given,respectively, by (2.4) and (2.5). However, the decision problems facing producers U and Eare quite different. In particular, each has an incentive to expand the market share of itsdownstream ally at the expense of the other downstream firm. To this end, producers Uand E may find it desirable to charge a negative price, i.e., give a per-unit subsidy to theirrespective downstream allies. Formally, for a given wi E R, firm E's decision problem isgiven by,Se = max (wi — ce)(q;)+Wi(2.11)1subject to w < —2 (wi + 1),where the constraint ensures that q , the quantity supplied to and sold by retailer j, is non-negative. The decision problem of producer U is similarly given, with the subscripts i and jinterchanged and ce replaced by cu. DefineHu_i^+ Ri andlle-j^Se + R3*In words, H1 and He are the total profits for the alliances U-i and E-j, respectively.Presumably, these profits are divided according to some bargaining process, which is leftunspecified. Proposition 2.2 states the equilibrium prices charged by the two producers andthe respective total profits for the two alliances.9Proposition 2.2 (i) If cu < (2c, + 1), the equilibrium upstream prices are1= —1(8c,, — 2c, —1) and tv* = —(8c e — 2cu — 1)5^ 3^5(2.12) 9The analysis in Proposition 2.2 assumes that each retailer sets its output price by taking its own as wellas its rival's wholesale prices as given. Implicitly, it is assumed that each retailer believes that the wholesaleprices alter the rival's marginal costs in a credible manner.41and the equilibrium total profits are2= —25(1 – 3cti 2c,)2 and 11*e_i = 2 25(1 – 3c, 2c„)2.(ii) If Cu > -A- (2c, + 1), the equilibrium upstream prices are1^ 1= –3(2c, + 1) and w3 = –3 (4c, – 1),and the equilibrium total profits are211° • = 0 and lre = –9(1 – ce)2.u-t(2.13)(2.14)(2.15)Proof: See Appendix 2A.It should be noted that in case (i), producer U's cost is not too high, thus its downstreamally is able to compete against its rival in the product market. This is, however, no longertrue in case (ii), where producer U's cost is so high that the optimal choice is to charge aprice which results in no sales and zero profit for its downstream ally.1° To illustrate, werewrite the total profit function for the alliance U-i as (after substituting ql and L),llu_i(wi) = –1(1 – 2wi w3)(1 wi w – 3ci,).9Hence, for a given w3, H(wi) <0 if and only if wi < 3c„ – w3 – 1, or the constraint binds,i.e., w = 1(1 + w3), or both. In case (i) of Proposition 2.2, we have 3cii– w3 – 1 < -1(1 + w3)(which, after rearranging terms, simplifies to Cu < + 1)), while the reverse holds in case(ii). These two cases are illustrated in Figure 2.2.Note also that in both cases (i) and (ii), total industry profit (i.e., 11,u_i^lle_j) is lessthan that under an integrated monopoly. This is due to the Cournot competition betweenretailers, which results in a partial dissipation of profits.We are now in a position to analyse the game. Consider first stage 2. Suppose no contractis signed between firms U and i (i = 1,2) in stage 1. Then, two possibilities exist in stage 2:'If exit is costless, this can be interpreted as both firms U and i exiting the market.423cu—w.-1in case GO3C—w1-1uin case (i)Case (i)Figure 2.2: Firm U's objective function under cases (i) and (ii)either entry occurs or it does not. If entry does not occur, there are three firms in theindustry: producer U upstream and retailers 1 and 2 downstream. Since the use of two-parttariffs is allowed, producer U simply charges a wholesale price which results in maximumtotal profit, 11:, for all three firms. This profit is then divided through the use of a fixedfee, which is presumably set through a bargaining process between the three firms. Let /-='t beretailer i's (i = 1,2) share of the total profit. Thus, producer U's share is II: — 271.Next, consider the event that entry occurs. There are then four firms in the industry: produc-ers U and E upstream, and retailers 1 and 2 downstream, and there are four possible marketconfigurations: { U-1-2}, f U-1, E-21, { U-2, E-1}, and {E-1-2}. As mentioned earlier, toavoid the bargaining issues involving four firms, we let retailers 1 and 2 play a one-shot gamein choosing, non-cooperatively, whether to acquire their inputs from producers U or E. Werefer to this game as the 'Choose a Supplier' game. Since producer E is more efficient, itappears that the only reasonable equilibrium market configuration is {E-1-2}. It will be43shown later that this is indeed the case.In addition, we seek to establish an upper bound on the two retailers' equilibrium payoffs,while leaving the bargaining process unspecified. For this purpose, we let all bargaining powerreside with the retailers in the other three market configurations. That is, we assume that allprofits go to the two retailers in each of the market configurations { U-1-2} and { U-i, E-j}(i j). We show in Section 2.5 that this assumption is not crucial to the result that follows.Specifically, the result remains valid as long as the bargaining position of the two retailers isnot weaker in the case { U-i, E-j} (i j) than in the case {U-1-2}.Given the above assumptions, suppose producer E offers the payoffs (C, C) to retailers 1 and2. The question, then, is how large must G be in order for both retailers to choose E as theirupstream supplier. To answer this, we examine the normal form of the 'Choose a Supplier'game, which is depicted in Figure 2.3.Retailer 2-H *, ', , I 1 ', i ', flu-1,^Ile-211e-1, 11u-2 G, GFigure 2.3: Normal form of the 'Choose a Supplier' gameIn Figure 2.3, the first entry to each cell is the payoff to retailer 1, while the second entry is44a11the payoff to retailer 2. The matrix is constructed as follows. The top left-hand corner of thematrix applies if both retailers choose producer U as their upstream supplier—the marketconfiguration is {U-1-2}. From the previous analysis, the joint profit is Hu*, as given in (2.9).Since, by supposition, all bargaining power resides with the two retailers, they thus dividethe joint profit equally among themselves, leaving zero profit for producer U. On the otherhand, if retailer 1 chooses producer E while retailer 2 chooses producer U as their respectiveupstream suppliers, then the lower left-hand corner of the matrix is relevant. The marketconfiguration is {U-2, E-1}, and from the previous analysis, the joint profits for the alliancesU-2 and E-1 are, respectively, liu-2 and ll , which are given in Proposition 2.2. The upperright-hand corner of the matrix is constructed in a similar manner. However, when bothretailers choose producer E as their upstream supplier, their share of total profit is G each.In what follows, we show that 2G is strictly less than the total profit, H:, despite the factthat the retailers get all the profits in each of the other market configurations.Referring to Figure 2.3, from the perspective of retailer 1, if retailer 2 chooses U, retailer 1is better off by choosing E if and only if1He-i> 2 uOn the other hand, if firm 2 chooses E, retailer 1 is better off by choosing E if and only ifG>Similarly, from the perspective of retailer 2, choosing E is always optimal regardless of whatretailer 1 does if and only if the following inequalities hold:He-2 > 1 , andG > ilu-2.However, since flu-i = Hu-2 and 14_1 = H,-2, the four inequalities above can be summarized45as1lle"^-2-i = 1, 2, and^ (2.16)G >^i = 1,2. (2.17)Proposition 2.3 states the condition under which 'both retailers 1 and 2 choosing E' is aunique Nash equilibrium outcome.Proposition 2.3 For the 'Choose a Supplier' game in Figure 2.3, producer E can ensurethat the outcome (E, E) is a unique Nash equilibrium outcome by setting G >^i.e.,G^3cu 2c,)2 c if cu < (2c, + 1),otherwise,where c > 0 is arbitrarily small, if the following condition holds:1cu >^+ 1).(2.18)(2.19)Proof: See Appendix 2A.The condition in (2.19) is depicted in Figure 2.4. It will be shown later that the samecondition also ensures that signing a contract is a unique subgame perfect equilibrium forthe whole game.The intuition behind Proposition 2.3 is as follows. If cu is large as compared to cc, i.e., (2.19)holds, suppose retailer j chooses U, retailer i's payoff for choosing U, AII„*, is low. Retailer iis better off by choosing E. This is no longer the case if c„, is close to ce. In particular, imaginethat cu = cc, then Ilu_i 11,-3, and we have Diu* > 113 since the monopoly profit is alwaysgreater than the sum of duopoly profits. Thus, condition (2.16) is never satisfied. However,when c, is lower than c„ such that (2.19) holds, a retailer who associates itself with the low-cost producer E gains a substantial advantage in the downstream product market. Giventhat its wholesale price is lower, its downstream market share is larger, and consequently its46Figure 2.4: Condition for (E, E) as a Nash equilibrium outcomeduopoly profit is greater than half of the monopoly profit. For this reason, if (2.19) holds, the'Choose a Supplier' game is simply a standard Prisoner's Dilemma. Note that the maximumthat G need attain is2—2(1 – 3c„ 2c6)2 E < -2(1 – 3cu 2c.)", = y5-(1 – ci,)2 < 125^ 25Thus, the payoff under (E,E) is strictly less than that under ( U, U) for each retailer. Hence,we have1G < –II*.2 (2.20)In other words, both retailers would like to 'cooperate' by choosing U, however, becauseof (2.20), each is better off by 'defecting' to E if the other retailer chooses U. Hence, byboth choosing E, the two retailers actually end up with the worst possible outcome. Thisresult also makes signing a contract in stage 1 attractive for retailers 1 and 2, as stated inProposition 2.4.47Proposition 2.4 Suppose cu > 11-(12c, + 1). Then, in the unique subgame perfect equilib-rium, producer U and retailer i (i =1,2) sign a contract.Proof: Suppose no contract is signed in stage 1. Then, given cu >^+ 1), retailer i's(i = 1,2) expected profit for not signing the contract is,(1— a).1?-F aG.Similarly, producer U's expected profit is(1 — a)(11: — 2h) + a (0) = (1 — a)(11: — 2R).To show that there exists a contract which can make firms U, 1 and 2 better off, it sufficesto show that the sum of their expected profits when no contract is signed is strictly less thanthe total profit when a contract is signed. Note that if a contract is signed, the total profitis 11u*, as given in (2.9). The sum of firms U, 1 and 2's expected profits when no contract issigned is,2[(1 — a)h ceG] + (1 —^— 2h) = 2aG (1 —1=^— 2a(-2-1T:, — G)< I17„where the last inequality follows from (2.20). Hence, by signing a contract, firms U, 1 and2 can divide the total profit in such a way that all parties to the contract are made betteroff.The result is a direct consequence of Proposition 2.3, which says that, in the event that entryoccurs, competition between the two retailers results in the worst possible outcome for eachfirm. Therefore, both retailers are willing to sign the contract in stage 1 to avoid the 'choosea supplier' game in stage 2. As a consequence, entry is deterred. Central to the result is theassumption that the two retailers are not able to collude in choosing a supplier in the event48that entry occurs. If the retailers are able to cooperate in choosing a supplier, they effectivelyact as a single firm in the input market. Hence the result in Proposition 2.1 applies, that is,a contract in stage 1 cannot be an equilibrium outcome, and entry cannot be deterred. Itshould, however, be noted that any such arrangements which permit the retailers to colludein the input market are likely to run afoul of the antitrust laws.It should be noted that if the condition depicted in Figure 2.4 is violated, i.e., if ct, <1A(12c, + 1), there are then two Nash equilibria, (U, U) and (E, E), for the 'Choose aSupplier' game (if G remains as in Proposition 2.3). It is unclear, a priori, which is a morelikely equilibrium outcome, neither is it clear what is an appropriate value for G. However,suppose it is possible for the retailers to communicate their choices before the 'Choose aSupplier' game begins. For example, suppose retailer 1 is able to communicate its choice toretailer 2 before the game begins. This then enables the retailers to coordinate their choicesby choosing the outcome which gives the highest payoffs to both firms. In this case, producerE has an incentive to set G > 111,2* to ensure that (E, E) is the equilibrium outcome. It isthen straightforward to show that Proposition 2.4 no longer obtains; that is, no contract willbe signed in stage 1. Thus, the condition depicted in Figure 2.4 represents the necessary aswell as the sufficient condition for Proposition 2.4.112.5 DiscussionWe show that a contract is likely to be an equilibrium outcome of the game if the retailersare competing against each other in the product market. This result is contingent on theassumption that the entrant has no means of making a credible offer to the downstreamfirms in stage 1. Suppose, instead, that such a possibility exists. Specifically, we allow the"Note that as long as the condition depicted in Figure 2.4 is satisfied, Proposition 2.4 remains valid evenif communication between players is allowed. This is because the incentive for each player to 'defect' remainsunchanged.49entrant to offer a cash payment M to each downstream firm it it refuses to sign the contractin stage 1. Proposition 2.5 states that the retailers then no longer have any incentive to signthe contracts.Proposition 2.5 Given that cu > -g(12c, + 1), suppose that the entrant offers each retailera payment M in stage 1. Then, in the unique subgame perfect equilibrium, no contract issigned in stage 1, instead the entrant's offer is accepted by both retailers.Proof: Consider stage 2. If entry occurs, retailers 1 and 2 then play the 'Choose a Supplier'game in Figure 2.3, and the resulting payoff for each firm is G. On the other hand, if noentry occurs, each retailer gets, as before, 1-k. Let M a(11-1„* — G). The expected profit foreach retailer for not signing the contract is thusM + aG + (1 — a-211I: +(1 —The expected profit for producer U remains as before, which is given by(1 — a)(H: — 2Th.Suppose, contrary to the hypothesis, producer U and retailer i ( = 1,2) sign a contract. LetRC be retailer i's share of the total profit. Hence producer U's share is 1I*„ — 2Rc. For thiscontract to be acceptable to retailer i, it must be the case that1> all + (1 —2 n (2.21)Further, for the contract to be acceptable to producer U, it must also be the case that— 2/r > (1 — a)(1I: — 21-k),which, after rearranging terms, reduces to1< a—II* + (1 — a)f?.— 250The contradicts (2.21). To verify that the entrant is willing to pay retailer i the amount M,we note that the entrant's expected profit is a(II: — 2G) — 2M > 0 if neither retailer signsthe contract in stage 1. Hence the entrant is strictly better off by inducing the retailers notto sign the contracts.Proposition 2.5 reflects the fact that the entrant has much to gain due to its lower cost if itsuccessfully enters the market. Hence it is willing to pay the necessary amount to induce theretailers not to sign the contracts.Note that in constructing the 'Choose a Supplier' game, we assume that all profits go tothe two retailers in each of the market configurations {U-1-2} and { U-i, E-j} (i j). Weshow now that this assumption can be relaxed without affecting the results. Proposition 2.4continues to hold as long as the retailers' bargaining power (measured in terms of profitshares) under { U-i, E-j} is greater than or equal to that under f U-1-21. Note that eachretailer is bargaining with a different producer in the former, while the two retailers arebargaining against a single upstream firm in the latter. Hence, each retailer is likely to be ina stronger bargaining position in the former case.To see that the result remains unchanged, suppose that the retailers' share of profits isA E (0,1] in each of the market configurations {U-1-2} and {U-i, E-j} (i^j). Then inthe 'Choose a Supplier' game in Figure 2.3, the payoffs under ( U, U) become OFP:i,and the payoffs under (E, U) are^All„2), and similarly for the case (U, E). Withan appropriate choice of G, and if condition (2.19) holds, then the unique Nash equilibriumremains (E, E), which means that Proposition 2.4 remains valid. It is easy to see thatcondition (2.19) can be further relaxed if the retailers' share of profits is less than A under( U, U). In particular, suppose the retailers' share of profit is zero under ( U, U), (Se_1, S„2)under (E, U), and (S,1, Se_2) under ( U, E), where Sk-2> 0, k = e,u, i = 1,2. Then, bysetting G = Su-, + E, conditions (2.16) and (2.17) is satisfied for all values of ci, and ce. That51is, producer E can always ensure that (E, E) is the only Nash equilibrium.In Section 2.3, we ruled out the possibility that one of the retailers, say retailer i, rejects thecontract while retailer j, j i, signs the contract. We justify this assumption by assumingthat firms U, 1 and 2 are able to agree that the contracts are void unless both retailersenter the agreements. Alternatively, we may assume that if the entrant decides to invest inR & D, it has to invest an amount I. This amount is large enough so that the entrant'sexpected return net of the investment cost is positive if and only if there are two customersdownstream, i.e., in the event that both retailers reject the contracts. Therefore, if retaileri is the only customer downstream, the entrant will not invest in the R Sz D. There is thenno incentive for retailer i to hold out if retailer j has signed the contract since no entry willoccur anyway. Note that the investment cost does not affect the entrant's decision in stage2 of the game, since the amount is already sunk by then. Hence all results in Section 2.3remain valid.2.6 ConclusionThis paper shows that it is possible for an upstream monopolist to use supply contracts as abarrier to entry if retailers are in competition against each other in the downstream productmarket. Social welfare in this case is lower, since the inefficient incumbent is allowed tocontinue to operate instead of being displaced, with probability a, by more efficient entrants.In view of this, we believe that the proposal by the 'Chicago School' of making verticalcontracts per se legal is unwarranted. However, the other extreme of making these contractsper se illegal is also indefensible, since there can be substantial efficiency gains. For example,these contracts may reduce supply uncertainties for retailers or protect specific assets fromopportunistic behavior, as in Marvel (1982). Therefore, rule of reason is the only sensibleposition on the legal status of vertical contracts seems to be the rule of reason.52Chapter 3Quality Differentiation and StrategicProduct Line Expansion3.1 IntroductionMost firms produce several rather than a single product. In many industries, it is not un-common to find a small number of firms supplying a large number of products.' In manycases these multiproduct firms do not begin with a full spectrum; instead, some products areadded while others are dropped over time. IBM, for example, first established itself in themainframe computer market, later expanded into the mini and personal computer markets.There are, however, few studies in industrial organization that focus on the product linedecisions of firms. In most product differentiation models, for example, it is often assumedthat each firm produces only a single product.The conventional explanation for multiproduct firms is that there are economies of scope inproducing several products jointly. This is, however, not the only reason that firms producemultiple products. For example, some firms introduce lower-quality products that are morecostly to produce. For example, Intel, the maker of microprocessors for personal computers,unveiled the 486SX processor shortly after it introduced the 486DX processor. The 486SX is'Indeed, this is listed as one of the awkward facts of product differentiation in the real world by Eaton andLipsey (1989), p.726.53identical to the 486DX except that the floating point processor in the 486SX has been turnedoff (PC Magazine, December 31, 1991). In other words, the 486SX is a crippled version of the486DX, and it is more costly to produce. The same phenomenon is observed in the computersoftware industry, where it is common to find software publishers introduce 'light' versionsof their main products. Clearly economies of scope do not provide a satisfactory account forsuch phenomena.This paper examines the strategic motive behind firms' product line decisions in the contextof a vertically differentiatied product market. Unlike some previous studies (e.g., Brander andEaton, 1984; and Bhatt, 1987), which fix the number of potential products and the degrees ofsubstitutability among products, the number of potential products is infinite in the presentframework, and firms are free to choose the degrees of differentiation among products. Thequestions we seek to answer are: Can an incumbent firm expand its product line strategicallyso as to pre-empt its rivals? What are the optimal number of products for firms to bringto the market? What are the equilibrium market structures under different assumptions ofthe model? Can an incumbent firm deter a more efficient entrant? How do firms adjusttheir product lines when a change in technology makes introducing higher-quality productspossible?This paper is organized as follows. A review of the related literature is found in Section 3.2.Section 3.3 outlines the basic model, while Section 3.4 considers the question of entry de-terrence by an incumbent monopolist. Section 3.5 extends the analysis to consider firms'product line decisions when technological advances make the introduction of higher-qualityproducts possible.543.2 Related LiteraturePrevious studies of multiproduct firms tend to fall into one of two categories. The first consistsof studies which focus on cost factors (i.e., economies of scope) as the main reason that firmsproduce several products. A recent survey of the literature can be found in Panzar (1989).Studies in the second category consider whether a firm can gain a strategic advantage inthe market by supplying several products. Examples are Schmalensee (1978), Brander andEaton (1984), Bhatt (1987), Bonanno (1987), and Shaked and Sutton (1990).It has been suggested in the literature that firms competing in oligopolistic markets tend toproduce close substitutes (Spence, 1976; Schmalensee, 1978). This phenomenon is termedproduct segmentation by Brander and Eaton (1984). In contrast to a segmented market, aninterlaced market is one in which each firm produces products that are less closely related. Ina model with two firms, two markets and four possible products, Brander and Eaton find thatif firms enter the market sequentially, market segmentation (as opposed to market interlacing)is indeed the likely outcome. Not surprisingly, a segmented market structure gives rise tohigher prices and profits for both firms than an interlaced one because competition is lessintense. In contrast, Bhatt (1987), in a model with two firms, two markets and six possibleproducts, finds that an interlaced market structure is the likely outcome, although firmswould prefer a segmented market. A feature of Bhatt's model is that each firm occupies anestablished market to begin with. Thus regardless of what the other firm does, each firmis strictly better off by invading the other firm's market. In other words, producing distantsubstitutes is a dominant strategy for both firms.Schmalensee (1978), Eaton and Lipsey (1979), and Bonanno (1987) consider the issue ofentry deterrence using a Hotelling-type spatial model. The central result is that monopolypersists. The incumbent firm maintains its monopoly position by crowding the productspace. Bonanno (1987) further shows that in some cases, entry deterrence need not be55achieved through product proliferation; rather, it is sometimes possible for the incumbentfirm to deter entry by strategically locating its products.In a more general setting, Shaked and Sutton (1990) examine oligopolistic firms' decisionsregarding the number of products to produce. They argue that there are two opposing effectswhen a new product is introduced. The expansion effect measures the degree to which totaldemand is increased by introducing a new product; and the competition effect measures theeffect on profit due to more intense competition because of the new product. Using thesemeasures, they are able to completely characterize the two-good case. The result is thenapplied to a linear demand schedule model. They find that, contrary to previous studies,entry deterrence is not necessarily the optimal strategy for the incumbent. For example,for the three-good case, there exists an equilibrium in which two firms each offers a singleproduct. As they point out, the model is considerably less tractable as the number of productsgoes beyond three.More recently, Donnenfeld and Weber (1992) consider a model of non-simultaneous entryinto an industry where firms compete in a vertically differentiated product market. Thereare three firms in the model, each producing a single product. The authors show that the firsttwo firms to enter the market supply the highest- and the lowest-quality product, whereasthe third firm chooses an intermediate quality and earns a higher profit than the incumbentwho supplies the lowest-quality product. A notable feature of the model is that firms do notincur any sunk cost in choosing a particular quality to produce. This assumption enablesthe authors to obtain comparative static results through differentiations. Thus, for example,the authors show that firms' equilibrium profits increase as the quality spectrum is increasedexogenously. This result is obtained through taking the first derivatives of firms' profitfunctions with respect to the range of quality spectrum. Clearly, this exercise is meaningfulonly if firms are able to adjust their quality choices costlessly in response to any exogenous56(and infinitesimally small) changes in the environment. However, as is pointed out later, thisraises the question of whether quality choices represent a credible commitment by firms tominimize price competition.3.3 Basic ModelConsider a one-dimensional market where goods differ in quality. Each consumer buys at mostone unit of the good with the most preferred quality, given prices. There are N consumers,each is assumed to have an indirect utility function of the form2= Os — p if the consumer buys a good of quality s at price p,V 0^if the consumer buys nothing, (3.1)where 0 is a taste parameter, it represents consumers' willingness to tradeoff quality againstprice (i.e., preference intensity); and is assumed distributed uniformly on the unit interval[00-1, 00], where 00 > 1. Without loss of generality, normalize N = 1 (e.g., one million). LetF(0) denote the cumulative distribution function, i.e., F(0) = 0 — (00 — 1). In what follows,0 will be referred to as the consumer's characteristic. The quality of goods, s, is assumedto lie in a bounded interval, s E [1,], where > 1. Let 1 = — 1 denote the length ofthe quality spectrum. Note that the representation of consumer preference (3.1) differ fromthat of Shaked and Sutton (1982, 1983), who assume that consumers differ in incomes ratherthan tastes. However, as pointed out by Tirole (1988, p.96), a simple transformation of (3.1)yields= s — (110) p if the consumer buys a good of quality s at price p,0 if the consumer buys nothing,which says that all consumers derive the same utility from consuming a good of quality sbut have different marginal rates of substitution between income and quality (1/0), due todifferent income levels. Thus, a higher 0 corresponds to a lower marginal utility of income,2This particular form of utility function is used in previous studies of vertical differentiation by Tirole (1988),Donnenfeld and Weber (1992), and Motta (1993), among others.57and hence a higher income level. Under this interpretation, the preference representationhere is analogous to that of Shaked and Sutton.Two features of the preference representation in (3.1) are worth emphasizing. First, goodswith higher quality are preferred by all consumers at the same price; and second, a consumerwith higher 0 is willing to pay more for a higher quality good. Thus, this preference rep-resentation is different from horizontal differentiation, where some consumers strictly preferproduct x while others strictly prefer product y even if both products are sold at the sameprice.There are two potential firms in the market: firms A and B. A firm may choose to produceone or several products of different qualities in the interval [1,3]. A sunk cost K > 0 isincurred for each quality firm i plans to produce. It is worth noting that firm i incursthe same sunk cost K, for each quality it chooses to produce. There are no other costsof production.3 Note that there are neither economies nor diseconomies of scope in jointlyproducing several products of different qualities. It is assumed that firms cannot alter thequality level of their products once a choice is made (i.e., after the sunk cost K, is incurred).This is the case, for example, if it is prohibitively costly to retool plants and equipment toproduce products of a different quality.'Consider first the case of a monopolist who faces no threat of entry. In what follows it will bereferred to as a protected monopolist. The monopolist has to decide the number of productsto produce, the quality of each product, and then set a price for each product.Proposition 3.1 A protected monopolist produces only one product; and its optimal qualitychoice is 3 , the highest quality level.3We deliberately keep the cost structure as simple as possible so as to focus on the strategic effects ofproduct line expansion.4This assumption receives much emphasis in Prescott and Visscher (1977), who argue that in many instancesit is more reasonable to model firms as making once-and-for-all location decisions (quality choices in the presentcontext).58Proof: See Appendix 3A.The highest-quality product is supplied at the monopoly price pm = -1003. The monopolistwill not produce more than one product even if the market is not covered, i.e., there aresome consumers who do not buy any product at all. Formally, this occurs when 00 < 2,i.e., when 00 is small. Note that we restrict the distribution of 0 on a unit interval, thus asmall 00 corresponds to the case where consumers' valuations of the products differ greatlyin relative terms. It is evident from the proof that the result is true even if the sunk cost iszero. This result is due primarily to the particular form of consumer preference, which doesnot permit the monopolist to extract surplus by offering a low-end and a high-end product.If a low-end product is introduced, it attracts customers away from the high-end product.Given the preference representation in (3.1) as well as the distribution assumption of 0, theloss from the high-end product strictly dominates the gain from the low-end product and asa result, the monopolist's overall profit is lower.This result is similar to that obtained by Stokey (1979), who considers the intertemporalpricing problem of a durable goods monopolist. In her model, consumers differ in theirvaluations of a single infinitely durable good, which all consumers prefer to consume earlierrather than later. The monopolist has an opportunity to price discriminate by graduallylowering its price over time, thus inducing high-valuation consumers to buy earlier, and ata higher price, than low-valuation consumers. However, Stokey proves that, for a particularclass of utility function (to which the specification in (3.1) belongs), the optimal amount ofintertemporal price discrimination is no price discrimination at all. That is, the monopolistsimply sets a price in the first period and does not lower the price over time. The reasonis simple. Given this particular form of consumer preference, the price cuts necessary toattract a wider market induce too many buyers to postpone their purchases, thus makingprice discrimination unprofitable. By renaming the durable good at different dates as goods59of different qualities, it can be seen that the present model closely resembles that of Stokey.Proposition 3.1 also states that the monopolist will always choose to produce the highest-quality product. This is in fact true for any firm which enters the market first. By producingthe highest quality, a firm can charge a higher price since consumers with higher 0 are willingto pay more. For this reason, products of different qualities enter firms' profit function in anonsymmetric fashion in this model. This is in contrast to Shaked and Sutton (1990), whoassume that all products are symmetric as far as firms' profits are concerned.Note that the first-best outcome involves the provision of the highest-quality good at zeroprice (the marginal cost), and every consumer buys one unit of the product. Thus thenumber and quality of products under monopoly coincide with the social optimum. Theprice, however, is greater than the marginal cost; furthermore, for some values of 00, notevery consumer buys the good.Suppose now the monopolist faces a potential entrant. Does the monopolist have an incentiveto deter entry by producing more than one product? To answer this question, it is necessaryto consider the duopoly case: Two firms, A and B, play a three-stage game. Suppose, fornow, that each firm produces only one product. Firm A chooses a quality level, sA in stage1; in stage 2, firm B decides on a product quality, 5B, given firm A's choice. The two firmsthen compete in prices in stage 3.5Three assumptions are made.Assumption 3.1: <0 < 2.Assumption 3.2: 3 <Assumption 3.3: KB > 19(2-00)24(2-00)2+1.Assumptions 3.1 and 3.2 are made so that every consumer buys a unit of one of the products.°It will be shown later that, if given a chance to introduce another product in stage 3, neither firm has anincentive to do so.60The first inequality in Assumption 3.1 places a lower bound on consumers' preference intensity00 to avoid cases where some consumers do not buy either of the products; the secondinequality ensures that consumers' preference intensity is not so high that all consumersdemand the highest-quality product. Assumption 3.2 imposes an upper bound on the highestpossible quality to ensure that the market is sufficiently competitive so that equilibrium pricesare low enough for every consumer to buy one of the products. Assumption 3.3 puts a lowerbound on the sunk cost of firm B so that it is possible (though it may not be profitable) forfirm A to deter entry by introducing a finite number of products in the market.'Proposition 3.2 states a well-known maximum differentiation result.7Proposition 3.2 Given Assumptions 3.1-3.3, and if KB < ( 2_0)2, firms A and B chooseto supply, respectively, the highest- and the lowest-quality products in a unique subgame perfectequilibrium; that is, s*A = and 513 = 1.Proof: Consider the third-stage price subgame, given firms' quality choices (SA, sB). Thereare two possibilities: either sA > sB or sA < sB.8 Assuming, for now, that sA > 5B• Let 0Abe the consumer who is indifferent between buying from firms A and B. That is,PA — PB OA =^-sA SBThe demand for 5A isQA =1— F(04= 00 — OA,and the demand for sB isQB = F(0A) = 0A — (00 — 1).6Without a lower bound on the entrant's sunk cost KB, entry deterrence through product proliferation isnot possible when Kg approaches zero.7See Shaked and Sutton (1982). The formulation here follows that of Tirole (1988, pp.296-7). Note thatthe result does not hold if the market is not covered, see Choi and Shin (1992).8The third possibility, namely sA = sB is ignored since it is never optimal for firms to choose the samequality.61Each firm maximizes revenue piQi — Ki, i = A, B. The first-order conditions give rise to thefollowing reaction functions:PA(PB) = —21[00(sA — sB)-f-PB]^ (3.2)PB(PA) =^[PA - (00 - 1)(SA 813)}^ (3.3)Solving (3.2) and (3.3) yields the following optimal prices, expressed as functions of s[SA SB]:1PA(s) = —3(1+ go)(8A — sB)^ (3.4)pB(s) = —31 (2 — 00)(8A — sB) (3.5)Substituting (3.4) and (3.5) into firms' revenue functions yields the following 'reduced-form'profit functions:1HA(s) = —9(1 + go)2(sA — sB)— KA^ (3.6)IIB(s) = —9 (2 — 00 )2 (sA — sB ) — KB (3.7)Note that Assumption 3.1 ensures that firm B's price and market share are strictly positive.Thus, in stage 2, firm B chooses the optimal quality to maximize (3.7), which yields s 1.Similarly, in stage 1, firm A's optimal quality choice is sTo show that every consumer buys one of the products, it suffices to show that the consumerwith the lowest preference intensity, (Go — 1), buys a unit from firm B. The utility of thisconsumer is (Go — 1) if she buys from firm B, the price she pays is= —3 (2 — Go) < 00 — 1where the latter inequality follows from Assumption 3.2.The maximum gross profits of firms A and B are, respectively,llf = —/ (1 + 00)2, and911,t = —9(2 — 00 )2 .(3.8)(3.9)62Note thatIIA > ll > KB. (3.10)The first inequality follows from the fact that 00 > 1, and the second inequality is true bysupposition. Since firm A is to move first, it can choose to produce the low- or high-qualityproduct. That is, it effectively faces a choice of the gross profits in (3.8) and (3.9). Thus, bythe inequality in (3.10), firm A's choice of SA= .3 is indeed optimal. This establishes thatsA = 3 and sB = 1 constitute a unique subgame perfect equilibrium. ifNote that the inequality KB < t • (2 — 00)2 is necessary so that firm B finds it worthwhileto enter the market, otherwise, firm B's entry is blockaded and firm A becomes a naturalmonopolist.This maximum differentiation result comes from the fact that, when setting prices in stage 3,firms are not able to costlessly alter their quality choices given that the sunk costs have alreadybeen incurred. This allows firms to credibly commit themselves to the quality choices whichresult in the least competition in prices. In other words, firms choose the two extreme qualitylevels to minimize price competition in stage 3. Note that the same result is obtained byDonnenfeld and Weber (1992, Proposition 1). In their model, however, firms incur no sunkcost when making product quality choices. It is unclear, then, how firms could crediblycommit to their quality choices to minimize price competition.We next show that, in the duopoly equilibrium stated in Proposition 3.2, neither firm A norB has any incentive to introduce another product. That is, suppose we add another stageto the game: in stage 4, each firm is given an opportunity to introduce another product.Proposition 3.3 shows that neither firm has an incentive to do so.Proposition 3.3 Given that firms A and B supply, respectively, the highest- and the lowest-quality products in stage 3, in a Nash equilibrium of the subgame in stage 4, neither firm has63an incentive to expand its product line.Proof: There are two existing products in the market, QA and QB, with quality levels sA =and sB = 1. Note that both firms A and B have incurred the necessary sunk cost for theproduction of these products.Consider first the incentive of firm A to introduce another product. Suppose firm A decides tointroduce a second product, denoted QA, with quality s° < .3. There are now three productsin the market: QA, Q, and Q B. Refer to Figure 3.1. Define 01 as the characteristic of theconsumer who is indifferent between buying quality QA and Q. Let 02 be similarly defined.Let PA, p°A, and pB denote the prices of products SA, s°A, and sB, respectively. Then,QB^QA°^QA00- 102 01 00Figure 3.1: Firm A introduces a second productPA — PA and 02 = PA — PB 1=8A — sA^sA — 8BThe demands for the three products areQA = 1 - F(01) = 00 - 81,QA^F(01)- F(02)^-02 and,Q B = F(02) = 02 - (00 - 1).The decision problems of firms A and B are, respectively,max PA Q A + PA Q°A- KA,pA,pAandmax pB Q B.PB64Solving these two maximization problems yields the following optimal prices:P*A = -00(sA — s°A)+ i(00 + 1)(8°A — sB),+ 1)(sA sB),PA* =^ and11Y:9^— 00)(sA — sB).Substituting these prices into the profit function of firm A, and noting that sA^.3 andsB = 1, we have^1^1IIA(sA) = —404(3 _ soA)+ —9(0o + 1)(s_ 1) — KA.^(3.11)However, by differentiating the profit function with respect to sA, we get^allA^1= -9 (00 + 1)2 —0oOsA 4=6(2 — 00)(2 + 500) > 0.Hence, firm A's profit is strictly increasing in sA, which means that it is maximized by settingsA = SA = Hence, there is no incentive for firm A to introduce a second product, sinceKA> 0.Consider next the incentive of firm B to introduce another product. Suppose firm B decidesto introduce a second product, denoted Q. The quality of this product is s 8B• Thereare now three products in the market: QA,Q°B, and QB. Refer to Figure 3.2. Define 01 asthe characteristic of the consumer who is indifferent between buying QA and Q. Let 02 besimilarly defined. Let pA,pcb, and pB be, respectively, the prices of QA, Q, and QB. Then, 02^0100- 1 00Figure 3.2: Firm B introduces a second productPA — P°B^P°B PB 01 =^and 02 =SA — S°B SB65The demands for the three products areQA = 1— F(01) = 00— 01,Q°B = F(0i) — F(02) =^02, andQ B = F(02) = 02 — (Go — 1)The decision problem of firms A and B are, respectively,maxpA QA,PAandmax^P^+ PB QB — KB^ (3.12)1373 P Bsubject to Q B > 0,where the constraint in (3.12) requires firm B to supply non-negative quantity of Q B. It willbe shown later that this constraint binds at the solution to firm B's problem. We solve thesetwo maximization problems by first ignoring firm B's constraint; and the resulting optimalprices are1P*A = i(eo +1)(sA — 43),PB^1=-3-(2 — 9o)(8A — 43), and1^1P*B =^(2 00)(s A — 43) — (00 1)(.5°B SB)•It is straightforward to show that, at these prices, Q B < 0, which violates the constraint in(3.12). Thus, the optimal quantity of Q B supplied by firm B is constrained at Q*B = 0. The'reduced-form' profit function of firm B is then111/3 (4) = —9 (2 — 002(sA — 8°B) — K B.Hence, firm B maximizes its profit by setting s°B = sB = 1. Therefore, firm B has no incentiveto expand its product line either.^411-66The intuition behind this result is as follows. Given that each firm is occupying one endof the quality spectrum, a firm introducing a second product can only choose a qualitylevel in between the two existing qualities. As such, although the new product expands thefirm's overall market share, it also attracts some consumers away from its existing product.Proposition 3.3 shows that the overall effect on the firm's profit is negative. In the languageof Shaked and Sutton (1990), the competition effect dominates the expansion effect, hencethe firm's profitability is adversely affected. Thus, unlike the result obtained by Bhatt (1987),neither firms find it profitable to introduce a second product. This result also implies that ifthe duopolists were to simultaneously choose the number of products to bring to the market,each would choose to supply a single product.9 Note that this equilibrium is not necessarilyunique. In particular, there may exist an equilibrium in which each firm simultaneouslychooses to introduce another product. Regrettably, the algebra in this case is intractable,and we are not able to obtain any result. Note, however, that if each firm is restricted tosupplying a single product, then the equilibrium stated in Proposition 3.2 is unique.3.4 Entry DeterrenceSuppose now an incumbent monopolist, firm A, faces a potential entrant, firm B. The questionis: Does the incumbent have any incentive to deter entry by expanding its product line? Putdifferently, is the resulting market structure a two-good monopoly or a duopoly with eachfirm producing a single good?Consider again a three-stage game similar to the one above, except now the incumbent hasalready established a product, Q Ai, with quality 8A1 = 3 in the market.1° In stage 1, theincumbent decides whether to introduce a second product, denoted by Q A2. In stage 2, the9It is assumed that both duopolists have decided to enter the market, thus the option of not producing,i.e., choosing zero product, is ruled out.10It will become clear later that this specification is equivalent to letting the incumbent choose the twoproducts' quality levels in stage 1. The analysis below is greatly simplified by fixing 3A1.67potential entrant decides whether to enter the market and if it does enter, makes a qualitychoice, taking into account the incumbent's choice in stage 1. If the entrant enters the market,the two firms compete in prices in stage 3. The game ends if the entrant does not enter.There are two issues that need to be addressed. First, is it feasible to deter entry by intro-ducing two products? Second, can the incumbent increase its net profit by deterring entry?That is, does the incumbent has an incentive to deter entry? Proposition 3.4 states that themonopolist can always deter entry by introducing two products in the market.Proposition 3.4 Given Assumptions 3.1-3.3, a monopolist incumbent can always deter en-try by introducing two products, with quality choices 8A1 = -s- and(a) 8 A 2 =^if K B ?_ 316 , or(b) — 36KB < sA2 < 1 + (29_11), if K B <Further, the monopolist maximizes profit by setting QA2 = 0, i.e., the monopolist incurs thesunk cost for the second product but does not carry out any production.Proof. See Appendix 3A.From Propositions 3.1 and 3.2, and in view of Proposition 3.4, it is clear that regardlessof whether the entrant enters or not, the incumbent always chooses to produce the highestquality, 3. There is therefore no loss of generality in setting 5A1 = 3 and only letting firm Achoose 3A2 in stage 1.Judd (1985) argues that, if exit costs are low, it may not be credible for the incumbent todeter entry through product proliferation. The reason being that, if the entrant enters oneof the product markets, intense post-entry competition in this market will adversely affectthe profitability of the incumbent's other products. Therefore, the incumbent is better off byexiting this market if exit costs are not prohibitive. Unlike Judd, we rule out the possibilityof exit by either firm. Specifically, we assume that exit costs are so high that exit is not an68option for the incumbent. Besides the obvious costs of shutting down factories and layingoff or relocating workers, there are many other reasons why this may be so. For example,by exiting the market, the incumbent may suffer a loss of reputation if building a reputationof toughness is desirable. (See Kreps and Wilson, 1982; and Milgrom and Roberts, 1982b).Alternatively, if product quality is not observable by consumers, by withdrawing the productin question, the incumbent may signal a low quality for all its other products. See Choi andScarpa (1992).Consider next the incentive of the incumbent to deter entry. If entry is deterred, the incum-bent becomes a two-good monopolist, its profit is maximized by setting QA2 = 0. In otherwords, the monopolist maintains some form of "excess capacity" to deter entry—it incurs thesunk cost for QA2 but does not carry out any production." The intuition behind this resultis exactly the same as that for Proposition 3.1. The resulting net profit for the monopolist is1117,1  = —439?) — 2KA.On the other hand, if entry is allowed, the incumbent earns the duopoly profitll = —9 (1 + 00)2 — KA.Thus, entry deterrence is profitable if and only if rir:,1  > IVA. That is, if1—4301) — 4(1 + 90)2 > KA(3.13)(3.14)(3.15)The right-hand side is the cost of entry deterrence, and the left-hand side represents thebenefit in terms of an increase in profit. Thus, the condition in (3.15) simply states thatfor entry deterrence to be profitable, the benefit must outweight the cost. The inequality in(3.15) is illustrated in Figure 3.3 below, for three different values of 3, namely,,3 = 1.5, 3 = 3and 3 = 5. The shaded regions represent combinations of Bo and KA such that the incumbenthas an incentive to deter entry."If, however, there exists a minimum efficient scale of production, the optimal quantity of QA2 may wellbe strictly positive. This is the case, for example, if the average cost curve is U-shaped and the cost is verylarge when output is small.69901^1.14^1.4^1 57 23•=3KA> " d iA n shaded regionsFigure 3.3: Incentive of incumbent to deter entryIt is immediately clear from Figure 3.3 that entry deterrence is not necessarily optimal forthe incumbent. In particular, when its sunk cost K A is high and the preference intensity 00is low, the incumbent is usually better off by allowing entry. This is because a high sunkcost makes entry deterrence costly, while a low preference intensity makes monopolizing themarket less attractive. However, the condition under which the incumbent has no incentiveto deter entry does not imply that entry will necessarily occur. DefineK*^- -/(1 + 9)2.4^9Proposition 3.5 states the conditions for different equilibrium market outcomes.Proposition 3.5 Given Assumptions 3.1-3.3, the following market structures are possible:(i) If KB > (2 - 00)2, entry is blockaded; the incumbent is a natural monopoly.(ii) If KB < 4(2 - 00)2, two possibilities exist:(a) If K A < K*, the incumbent is a two-good monopoly, i.e., entry is deterred.(b) If K A > K*, the incumbent and the entrant each produce a single product, i.e.,entry is accommodated.70Proof: (i) Note that from the proof of Proposition 3.2, firm B's gross profit isIll^19 (2 00)2.Thus, entry is blockaded if 4(2 — 002 < KB•(ii) Given that KB < 00)2, firm B enters if firm A does not pursue any entry deterrencestrategy. However, from previous discussion, entry deterrence is profitable if and only if(3.15) holds; i.e., K* > KA. This proves (ii)(a). On the other hand, entry is accommodatedif K* < KA, which is (ii)(b).Note that the equilibrium market structure is a duopoly only if the incumbent's sunk cost ishigh while that of the entrant is low. Specifically, the entrant's sunk cost must be strictlylower than that of the incumbent. To see this, we first establish Lemma 3.1.Lemma 3.1: K* > ( 2 — 00)2.Proof. See Appendix 3A.By Proposition 3.5 and Lemma 3.1, it can be easily shown that the equilibrium marketstructure is a duopoly if the following inequalities hold:KB < -9 (2 — 00)2 < K* <That is, entry can only occur in the event that firm B has a strictly lower sunk cost. Propo-sitions 3.5 and Lemma 3.1 can also be used to partition the (KA, KB) space into three zoneswhich correspond to the different market structures. This is illustrated in Figure 3.4.It is worth noting that if KA = KB, entry deterrence is always profitable, and hence monopolypersists. This result is consistent with that obtained under horizontal differentiation models,e.g., Schmalensee (1978), Eaton and Lipsey (1979), and Bonanno (1987), among others.Intuitively, if the entrant can earn a positive net profit by introducing a new product, theincumbent can always mimic the entrant and do strictly better. This is because when a new71KB—/ (2 -00299^0 K,Figure 3.4: Equilibrium market structuresproduct is introduced, it affects the demand and hence the profitability of the incumbent'sproduct. The entrant will not take this negative 'external effect' into account whereas theincumbent will. Hence the incumbent can do strictly better than the entrant. In other words,the net profit of a two-good monopoly is always greater than the sum of the duopoly profits,i.e., H > HdA Ht. The incumbent therefore has more of an incentive to deter entry thanthe entrant has to enter. This also explains why the entrant needs to have a strictly lowersunk cost in order to enter the market. The welfare implication is clear. Social welfare sufferssince exclusion of a more efficient entrant, i.e., one with a lower sunk cost, is possible.3.5 Technological AdvancesSuppose, due to an (exogenous) improvement in technology, the range of quality increasesto, say, [1, -I- a], where a > 0 is some constant. The questions of interest are: Under whatcondition will a protected monopolist introduce a new product? What if the monopolist is72unprotected, i.e., facing a potential entrant who has the same access to the new technology?Next, in the context of a single-good duopoly, how does the firm which produces the high-endproduct before the technological advance reposition its product line if it has the exclusiveright to the new technology? What if this right belongs to the firm that produces the low-endproduct before the technological advance?These questions are of interest because quality spectra never remain the same in the realworld. Technological advances frequently redefine the best available product quality in manyindustries. For example, a 24-pin dot-matrix printer was regarded as the high end productin the early 1980s. Along came laser technology, which pushed all dot-matrix printers to thelow end of the product spectrum. Similar examples abound in other industries.Consider first the case of a monopoly. Suppose a protected monopolist is producing an ex-isting product of quality . Given the technological advance, the monopolist has to decidewhether to introduce a new product by incurring the sunk cost K. It follows from Proposi-tion 3.1 that, if a new product is introduced, its quality level is set optimally at 3 + a; andthe output of its existing product is set to zero. The monopolist's net earning in the nextperiod isIIN = :41 (Ts +c)0(1 — K. (3.16)On the other hand, if the monopolist does nothing with the new technology, it continues toearn 14-04 in the next period.12 Hence, the protected monopolist will introduce a new productif and only ifRN > 02— 4 0. (3.17)Consider next the unprotected monopolist, who is facing a potential entrant who has thesame access to the new technology. We assume that the incumbent enjoys an advantage in'Note that the sunk cost has already been incurred, thus it does not appear in the monopolist's next-periodprofit.73that the new technology is available to the incumbent first. In other words, the entrant is ableto enter the market only if the incumbent decides not to introduce a new product. As before,if the incumbent introduces a new product, it incurs the sunk cost K, and earns the net profitRN in (3.16). On the other hand, if the incumbent does not introduce a new product, thenew technology will then be employed by the entrant. The market then becomes a duopolywith each firm producing a single product. To ensure that every consumer buys a unit ofone of the products in the duopoly case, we assume that Assumptions 3.1 and 3.2 continueto hold with the new technology. In particular, we re-state Assumption 3.2 as follows.Assumption 3.4: a <The product quality of the incumbent is fixed at ;sr, while that of the entrant is optimally setat - a . The latter follows directly from Proposition 3.2, which states that firms in a duopolymaximize profits through maximum quality differentiation. Here the incumbent becomes asupplier of low-end products, and from Proposition 3.2, its profit is— e0)2.^ (3.18)Hence, the unprotected monopolist will introduce a new product if and only ifITN^-190 2.^ (3.19)Proposition 3.6 states that a protected monopolist has less incentive to introduce a newproduct.Proposition 3.6 Given the technological advance described above, a protected monopolisthas less incentive to introduce a new product than a monopolist who faces the threat of entry.Proof We show that, for a new product to be introduced, it needs to generate a higher profit(i.e., a higher ITN) for the protected monopolist. That is, by comparing (3.17) and (3.19),744^9^36 [(130,1 _ 1600 + 16)+ (80.4 _ 2000 + 8)].1 1se4 a (2 00)2 > (3.21)we need to show that1—.70g > (1(2 — 00)2.4 9To derive this inequality, we note that(3.20)11—302 — (1(2 — 00)2 = —36 [9* — 4a(2 — 00)2].4 ° 9By making use of Assumption 3.4, and after rearranging terms, we haveBut the term (1304 _ 1600 + 16) is strictly positive,13 and since 7s- > 1, by setting 7s- = 1 at theright-hand side of (3.21), we have1—1:04 — —a(2 — 0)2> -i-2 (704 _ 1200 + 8) > 0 V Oo•4^9Hence the inequality in (3.20) obtains.^IfThe result is hardly surprising. A protected monopolist gains less from introducing a newproduct since it merely displaces the monopolist's existing product. On the other hand,an unprotected monopolist not only has to take into account the benefits of introducing anew product, it must also consider what will happen if it does not introduce a new productwhile the entrant does. It is therefore not surprising that the unprotected monopolist hasmore incentive to introduce a new product. This result seems to be supported by somecasual empirical observations. Intel, for example, has recently decided to quicken the paceof product developments through developing two rather than one microprocessor generationat once. This is primarily in response to the challenge posed by recent entrants such asAdvanced Micro Devices Inc. and Cyrix Corp. (Business Week, June 1, 1992).We consider next the case of a single-good duopoly. Assuming that, before the technologicaladvance, firms A and B are producing, respectively, the high- and low-end products, i.e.,'3It can be easily shown that mine(130(3 — 1600 + 16) > 0.75sA = -.9- and sB = 1. For the purpose of comparison, let RA = KB = K. We further assumethat the new technology is proprietary in nature, that is, one of the two firms has an exclusiveright to the technology. Proposition 3.7 states the conditions under which each firm is willingto introduce a new product if it holds the exclusive right.Proposition 3.7 Given the technological advance described above,(i) suppose firm A has the exclusive right to the new technology, it will introduce a new productif and only if4K>^ (3.22)002 •(ii) Suppose firm B has the exclusive right to the new technology, it will introduce a newproduct if9K a > (3.23)(Bo + 1)2.Proof: See Appendix 3A.Note that (3.22) represents the necessary and sufficient condition for firm A to introduce anew product, whereas (3.23) is only a necessary condition for firm B. The latter is derivedby establishing a lower bound on firm B's profit when a new product is introduced. Thesufficient condition is more complex and not needed for the result that follows. The nextproposition states that firm A has less incentive to introduce a new product.Proposition 3.8 The firm producing the high-end product (firm A) before the technologi-cal advance has less incentive to introduce a new product as compared to the firm which isproducing the low-end product (firm B).Proof: Referring to (3.22) and (3.23), we need to show that (3.22) is a more stringent condi-tion. In other words, there exists some values of a for which firm B would introduce a new76product whereas firm A would not, i.e.,4K^9K (00 + 1)2.To show this, we note that4K ^91(^K 1911^(0 0 + 1)2^04(1 + 00)2 (2 00)(2 + 500),(3.24)which is strictly positive for all relevant values of 00. Hence the inequality in (3.24) ob-tains.The intuition is simple. For firm A, the high-quality producer before the technological ad-vance, introducing a new product draws customers away from its existing product. On thecontrary, for firm B, introducing a new product primarily draws customers away from its ri-val. Clearly, then, firm B has more incentive to introduce a new product. The result suggeststhat low-quality producers are willing to invest more in bringing new products to the mar-ket. In contrast, high-quality producers are more inclined to maintain their existing productlines. Consequently, there is a tendency for firms to 'leap-frog' each other in introducing newproducts. This is indeed a common phenomenon in the real world. For example, Epson, oncea prominent dot-matrix printer manufacturer, has been relegated to also-run status since theintroduction of laser printer technology. Another example is the emergence of Japanese auto-makers, who were for years regarded as low-quality producers, into high-quality luxury-carmarkets.3.6 ConclusionIn the context of a vertically differentiation model, we show that a monopolist produces nomore than one product. The monopolist's choice the number and quality of product coincidewith the social optimum. In the case of a duopoly, if product quality represents a crediblecommitment, firms choose the maximum degree of product differentiation to minimize price77competition. Furthermore, neither firm has any incentive to expand its product line in aNash equilibrium. However, if a firm enters the market first, it may wish to expand itsproduct line solely for the purpose of deterring later entrants. This result is consistent withthat obtained under the horizontal differentiation literature, e.g., Schmalensee (1978), Eatonand Lipsey (1979) and Bonanno (1987). We also show that entry deterrence is possible eventhough the entrant is more efficient (i.e., has a lower sunk cost).In the event of a technological advance, we show that a protected monopolist is less willingto expand its product line. This implies that, if firms are to bid for the right to the newtechnology, a protected monopolist is not willing to offer more than an unprotected monop-olist. 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(1988), The Theory of Industrial Organization, Camb. Mass.: MIT Press.[64] Varian, H.R. (1980), "A Model of Sales," American Economic Review, 70, 651-9.82Appendix 1Appendix 1A Proof of Proposition 1.2This appendix contains six lemmata which establish Proposition 1.2. A more general ver-sion of the capacity-constrained price game when firms' marginal costs differ is discussed inDeneckere and Kovenock (1990). They show that in a mixed strategy equilibrium, firms'supports do not necessarily be the same and they may not be connected. These possibilities,however, are ruled out by Assumptions 1.1 and 1.2 in this paper. These assumptions alsogreatly simplify the proof.Throughout this Appendix, (11,, He) is used to denote, respectively, the equilibrium profitsof the conference and entrant. The support of each firm's strategy is denoted [p,, pi], i = c, e.It is worth noting that in a mixed-strategy equilibrium each firm randomizes in such a waythat keeps the other firm indifferent among its strategies. Formally, if P', P" E pi], then11,(P', G3) = 11,(P", G3). This fact is used repeatedly in the proofs below.Lemma 1.1 Suppose pc = pc = p. Then in equilibrium at most one firm has a mass pointat p.Proof: By Proposition 1.1 there does not exist a pure strategy equilibrium. Thus, only mixed83strategy equilibria need to be considered. By assumption,^p. > a,.^ (A1.1)Further, the upper bound of a mixed strategy equilibrium is strictly greater than the lowerbound, i.e.,^> Pi•^ (A1.2)Combining (A1.1) and (A1.2) gives^p > a,.^ (A1.3)Suppose, contrary to the hypothesis, both firms have a mass point at p. Then, by (A1.3),both firms make positive profit at However, by reducing the price slightly, one of thefirm would get a strictly higher profit by avoiding the probability of a tie. This contradictsequilibrium. ifLemma 1.2 There exists a firm i E {c, el such that Hi = H.Proof: Suppose, without loss of generality, pi > p. Consider first pi > p3 . Since Gj(p.i) = 1,by charging pi, firm i gets for certain Hi = Hi(pi) < H7, by definition of H. But (Gi, Gi) is apair of equilibrium strategy, so lli(pi) > H. Thus = H. Next, consider pi = = p.By Lemma 1.1, at most one firm has a mass point at p. Suppose firm j has no mass point atp. Then, lli(p) = H(p) <H. But, again, iii(p) > H7, which implies Hi =Lemma 1.3 ll = II*, Ile> H.Proof: Since a firm is at worst undercut by its rival in playing its mixed strategy, in equilibriumll > H7, i = c, e. Suppose a strict inequality holds for the conference, i.e., He > II*, thenby Lemma 1.2, He = H = 0. But note that pc > ac by assumption. By charging aprice p E (0, (lc), the entrant earns a strictly higher profit. Thus, He = H,* cannot be theequilibrium profit. Therefore He > H in equilibrium, and by Lemma 1.2, He = H*.84Lemma 1.4 (i) pc > pe, (ii) pc = p, pe < pH, .Proof: (i) Suppose not, i.e., pc < pe. Then, by charging pe, the entrant gets for certainHe = 0. But this contradicts Lemma 1.3, which states that 11, > H: = 0.(ii) Suppose pc > . Then II, = H. But from Lemma 1.3, II, = 11, which implies thatPc = p . Suppose pc = pe = p. By Lemma 1.1, at most one firm has a mass point at p.Thus, H = H'," again implies that p = p.Lemma 1.5 pc= pc= pi,.Proof: From Lemma 1.3, II, = H. This impliesp > pi.c —.c1since any price below p' gives a profit strictly below H. Further,(A1.4)P <PI— c (A1.5)Suppose not, then the conference can set a price p E (p, p) such that it earns strictly morethan H , which contradicts Lemma 1.3. Finally,Pc> Pc'^ (A1.6)Suppose not, then the entrant's equilibrium profit is 11,(p). But by charging a price p Eit earns strictly more than^a contradiction. Combining the inequalities in(A1.4), (A1.5) and (A1.6) yields pc = pc =Lemma 1.6 Ile = Le(p).Proof: From Lemma 1.5, pc = pc = p!.. Thus, by charging pci, the entrant gets either L(p)or Te(p,i). Since Le(p,I)> Te(g), this implies that Le(p!')> He. Suppose a strict inequality85holds. Then, noting that Le(p) = pK e, which is continuous on [0, pH, ], there exists an c > 0such that Le(g — €) > 11,, but (p!) [g, ple-1], a contradiction. Thus, L(p) = Ile.Lemmata 1.1 to 1.6 establish that in a mixed strategy equilibrium, both the conference andthe entrant's strategies share a common support [pci,p,11], and their respective equilibriumprofits are .11,* and Le(g). This completes the proof of Proposition 1.2.86Appendix 1B Cournot CompetitionSuppose the conference and the entrant engage in Cournot (quantity) rather than Bertrand(price) competition in the event that the contract is rejected by the shipper. All other aspectsof the game remain unchanged. Again, the game can be solved by the logic of backwardinduction. Consider the second-stage quantity game. Recall that the inverse demand functionisp= (a— Q)lbwhere Q = q, q6. The decision problem of firm i, i=c,e, ismax (p — ai)qi subject to qi < Kiqiwhere a, = 0.(A1.7)Since (A1.7) is a well-defined maximization problem, there exists a pure-strategy equilib-rium for the second-stage quantity game. Note that the capacity constraint of the entrantmay or may not bind since the equilibrium quantities are typically smaller under Cournotcompetition.Proposition A1.1 Under Assumptions 1.1 and 1.2, the pure-strategy equilibrium for thesecond-stage quantity subgame is given by(i) if 3K, a + bac,q = (a — 2ba,) and q:^(a + bac),(ii) if 31C, < a + bac, q:* =^— ba,— 1(6) and q:* = Ke.Proof Solving the firms' maximization problem in (A1.7), ignoring the constraints, gives riseto the following optimal-response functions:qc(qe) ==1—(a — ba, — q,),21(a^qc).(A1.8)(A1.9)87_ Solving these optimal-response functions yieldsqc* =qc* =1—(a — 2bac),31—(a bac).3(A1.10)(A1.11)Notice that the capacity constraint of the conference does not bind but the same does nothold for the entrant. Consider two cases. (i) Suppose the entrant is not capacity-constrained,i.e., If e> A (a + bac). Then the expressions in (A1.10) and (A1.11) constitute the equilibriumquantities. (ii) Suppose the entrant is capacity-constrained, i.e., lie < -A-(a bac). Then, thecapacity constraint of the entrant binds, and the optimal quantity for the entrant is qe** = K e.The optimal response of the conference is, from (A1.8), given byq'c"* = —1 (a — bac —2 (A1.12)Given the solution for the second-stage quantity subgame, the equilibrium for the full gamecan be readily derived. Proposition A1.2 states that entry exclusion is still possible andprofitable for the conference.Proposition A1.2 Under Assumptions 1.1 and 1.2, and that the post-entry competition isCournot in nature, the conference can always profitably exclude the potential entrant.Proof. Consider two cases.(i) Suppose 3K, > a bac. Then the entrant is not capacity constrained and the equilibriumquantities are given by (A1.10) and (A1.11). The total quantity demanded by the shipper is1Q* = q: + q = — (2a — bac),3at the equilibrium pricep* = (a — Q*)lb =^(a + bac).88To show that p* > ac, note that by supposition, 3K, > a + bac, which is equivalent toKe > (a — 2K,) bac, but the expression in bracket is strictly positive due to Assumption1.1. Thus,Re > ba,^ (A1.13)Next, Assumption 1.2 states that a > Ke + bac. Combining this with the inequality in(A1.13) gives a > 2bac. This inequality implies that a + ba, > 3bac, which in turn impliesthat p* > ac.Since p* > ac, there exists a price fi such thatSuppose the conference offers a loyalty contract supplying the quantity Q* at a price 5. Sincethis is the same quantity demanded but at a lower price, the shipper will surely accept thecontract. Note that the entrant is not able to offer such a contract since Ke < Q*. To see this,note that by Assumption 1.1, 2a > 4K,, but 4K, > 3K, -I- ba, due to (A1.13). Combiningthe two inequalities yields 2a > 3K, + bac, which after rearranging terms, gives IC, < Q*.It remains to show that the conference is strictly better off by offering the contract. Byallowing entry, the conference earnsIrc‘ = (P* ac)q:,while by offering the loyalty contract, the conference earns= (73 ctc)Q*•Since Q* is strictly greater than qc* while fi can be set arbitrarily close to p*, it follows then>(ii) Suppose 3K, < a + bac. Then the entrant is capacity-constrained, and the equilibriumquantities are q:* = Ke and qc** as given by (A1.12). The total quantity demanded by the89shipper isQ**K e q'e" =^— bac + Ke),and the equilibrium price isp** = (a — Q**)lb = —21b(a + bac — Ke).It follows immediately from Assumption 1.2 that p** > ac. Thus, the same reasoning incase (i) above applies, the conference is better off by offering a loyalty contract supplyingthe quantity Q** at a price /5 < p**, and it is to the advantage of the shipper to accept thecontract.90Appendix 1C Derivation of the Entry Deterrence ConditionThis appendix contains the algebraic derivation of the entry deterrence condition given in(1.10), which is reproduced as follow.4a Ke(a—K, + bac) < (a — K — bac)(a + K — bac)2^(A1.14)Recall that if the entrant can commit to charging a price equals to its marginal cost, then theaverage price the shipper pays is A, as given by (1.8). The conference, being the high-pricefirm, earns H,* if it allows entry. On the other hand, if the conference offers the shipper aloyalty contract, the highest price that is acceptable to the shipper is A, which gives theconference a profit of11,(Alp) = (A ^ac)(a — bAlp).^ (A1.15)After substituting in A, (A1.15) simplifies toñ(A) =^— acKe)(1^bpH K )(a — be)2 (A1.16)A contract will be offered (and hence entry is deterred) if and only if fic(Ap/ ) > H. Substitute(A1.16) into this expression yields(a — bp,11)2ac > bP,H(1/: — acKe).^(A1.17)Recall that pell = 1(a — K + bac)2 and H,* = -41t(a — K — bac)2. Substituting these into(A1.17) gives2bac(a Ke — bac)2 > (a —^+ bac)Ra — Ke — bctc)2 — 4bacKe].After adding and subtracting 2aKe in the square-bracket term on the right-hand side, thedesired expression in (A1.14) obtains.91Appendix 2Appendix 2A Proofs of Results in Chapter 2Proof of Proposition 2.2(i) After substituting qI and f3 into (2.11), the maximization problem of producer E becomes1^ 1Se^max — (wj — cu)(1 —^wi) + —9(1 — 2wi wi)2 — Siw, 31subject to wj < —2(w, + 1).Note that, given w„ the objective function is strictly concave, hence a global maximum exists.We first ignore the constraint, and solve for the first-order condition, which gives the following'reaction function' for producer E,1= —4 (6c —^— 1).Similar exercise gives the following 'reaction function' for producer U,1w, = —4 (6c„ — wj — 1).(A2.1)(A2.2)Solving (A2.1) and (A2.2) yields (w:`, WI) in (2.12). It remains to verify that these pricessatisfy the constraints in the respective maximization problems. First,1—2 (1 + wi ) = —3 (3c, — 2c„, — 1) < 0,592where the inequality follows from Assumption 2.1. Hence, producer E's constraint is satisfied.Next, note that by supposition, c < (2, + 1), hence1—2 (1 + w) =^— 2c, — 1) < 0.r^5Thus, producer U's constraint is also satisfied. The equilibrium profits in (2.13) are obtainedby substituting (w:', w;) into the respective objective functions of producers U and E.(ii) Given that c > A(2c, + 1), the derivation above suggests that producer U's constraintis binding. That is, producer U's 'reaction function' is now given by the constraint,1wi = —2 (1 + w.i ). (A2.3)As before, producer E's 'reaction function' is obtained by first ignoring the constraint. This isgiven in (A2.1). Solving (A2.1) and (A2.3) results in (w°, wy) in (2.14). This case is illustratedin Figure A2.1. The shaded region represents the inequality constraint, wi < 1(wj + 1).However, the two reaction functions intersect at point A, which violates the constraint. Hencethe equilibrium is at point B, which is the intersection of wi = 1(wi + 1) and retailer j'sreaction function.To show that producer E's constraint is satisfied, note that1+ 1) = c, — 1 <0,where the inequality again follows from Assumption 1. Hence firm E's constraint is satisfied.The equilibrium profits in (2.15) are obtained by substituting (w°, w) into the respectiveobjective functions of producers U and E. 1.Proof of Proposition 2.3To prove that (E, E) is a unique Nash equilibrium, it suffices to show that the conditions in(2.16) and (2.17) in Chapter 2 are satisfied. First, suppose cu < (2c, + 1). Then,211,_,^—25(1 — 3c, + 2ci,)2.93Figure A2.1: Reaction functions of firms U and E under case (ii)94Given cu > —113(12c, + 1), which is equivalent to ce < 1÷-2(13c„ — 1), we have2^ 1^111,_, = —25(1 — 3c, 2c„)2 > -2 {1 — —3 (13c„ — 1) + 2c„}2 = —8(1 — cur = —2^2 ^12Hence condition (2.16) is satisfied. Since in this case producer E sets2G =^—^-I- 2ce)2 E,condition (2.17) is also satisfied. It remains to verify that producer E is willing to make suchan offer, i.e., fle* > 2G. Note that by Assumption 2.1,^TT ^1^2^1 1= -4 ( 1 — ce)- = 71(1 — 3c, + 2c)2> —4 (1 — 3c„ 2c,)2which is clearly greater than 2G. Hence (E, E) is a unique Nash equilibrium outcome.^Next, suppose Cu >^+ 1). Then,2rie_i^-6(1 — c,)2.Thus, condition (2.16) is clearly satisfied. Given that producer E in this case setsG = c > 0,condition (2.17) is also satisfied. Hence (E, E) is a unique Nash equilibrium outcome.95Appendix 2B Bertrand Competition in a Differentiated Prod-uct Market: An ExampleThis Appendix considers an example of Bertrand competition in the downstream productmarket, where goods are substitutes. The purpose is to examine whether the central resultsin Chapter 2 remain valid in this case. Note that, as before, the two retailers, 1 and 2, use ahomogeneous input, supplied either by the upstream incumbent, U, or the entrant, E.We consider a particularly simple demand function facing firm i, i = 1,2:q1(P) = 1 – pi + –12pj,where qi and pi denote, respectively, the quantity and price of firm i, and P [pi, pi].Consider first the decision problems of the two retailers. Let w, and A be the wholesale priceand fixed fee faced by firm i. Note that if both retailers acquire their inputs from the sameproducer, then w, = w and f, = f, i = 1,2. Retailer i's decision problem isRi = max (pi – wi)qi(P) –PiThe first-order condition gives rise to retailer i's reaction function:pi = –1(1 + wi –1 pi),i , j= 1,2, j i.2^2 (A2.4)Solving the reaction functions of retailers 1 and 2 yields the following optimal retail prices,expressed as a function of the wholesale prices W E [w„ w3]:pi(W) = —2(5 + 4wi wi), i,j = 1,2, j i.^(A2.5)15Substituting these prices into the demand function of firm i yields1q( W) = —15 (10 – 7w + 2w).Hence, retailer i's 'reduced-form' profit function is1R(W) = —(10 – 7wi 2wi)2 –225(A2.6)(A2.7)96We next consider the two upstream producers' decision problems. Suppose each producersupplies to a different downstream firm. That is, we consider the case { U-i, E-j}, i j.Producer U's decision problem ismax (w — cu)qi(W) -I- L.^(A2.8)Noting that fi = (pi(W)— wi)qi(W)— Ri, we can rewrite (A2.8) asmax (pi — cu)qi(W)— R.^ (A2.9)w,The first-order condition gives producer U's reaction function:1w(w) = 112 (10 + 2w + 105cu).A similar exercise gives the reaction function of producer E,11-v3( /D) = 112 (10 + 2wi + 105c, ).Solving (A2.10) and (A2.11) gives the optimal wholesale prices,1418 (38 -I- 7c, + 392cu),1418 (38 + 7cu 392ce).(A2.10)(A2.11)Substituting these wholesale prices into producer U's profit function in (A2.9) yields theoptimal joint profit of the affiance {U-i}:ll^ 14 = (209)2 (38 + 7c, — 26cu)2, i = 1, 2.^(A2.12)The profit of the affiance {E-j} is derived in a similar manner, and is given by141-4_3 = (209)2(38 + 7cu-26ce)2, j = 2, 1. (A2.13)Next, suppose both retailers acquire their inputs from the same upstream producer. Thatis, we consider the two cases { U- i-j} and {E- i-j}. By the no-discrimination assumption, we97have wi = w, and fi^f, i = 1,2. Hence, from (A2.5), (A2.6) and (A2.7), retailer i's price,quantity and profit are, respectively,p(w) =^(1 w),q(w) = -3-1 (2 — w), andR(w) =^(2 — w)2 — f.Thus, the decision problem of producer k, (k = U, E) ismax(w — ck) q(w) + 2f = max —2 (2 + 2w — 3ck )(2 — w) — 2R.w 9Solving this maximization problem yields the optimal wholesale price1w* = —4 (2 + 3ck),which, after substituting into (A2.14), gives the optimal profit for firm k asirk = -1 ( 2 — ck)2, k = U,E.4(A2.14)(A2.15)Given the profits in (A2.12), (A2.13) and (A2.15), we can construct the normal form ofthe 'Choose a Supplier' game as in Figure 2.3. As before, for this game to be a standardPrisoner's Dilemma game, we require conditions (2.16) and (2.17) to hold. These conditionsare reproduced as follows:1lle_i >^i = 1, 2, andG > 112, i = 1,2.In the present context, condition (A2.16) reduces to, approximately,(A2.16)(A2.17)ct, > 0.056 + 0.972ce. (A2.18)Compared to condition (2.19) in Chapter 2, which is illustrated in Figure 2.4, (A2.18) rep-resents a weaker requirement, i.e., the shaded region associated with (A2.18) is larger thanthat of (2.19).98Next, condition (A2.17) is satisfied by setting^G =^c,14(209)2(38 + 7c, — 26c,i)2^E.It is straightforward to verify that 2G < II:, thus producer E is willing to pay the amountG to each retailer. The payoff under (E,E) is strictly lower than that under ( U, U). To seethis, note that by Assumption 2.1, c > c,, hence^= ^14 G (209)2 (38 + 7c,— 26cu)214 (209)2 (38 + 7c,, — 26cu)214 (209)2 (2 cu)21This inequality implies that both retailers would like to coordinate their actions by choosingU, however, each is tempted to cheat by choosing E. As such, both retailers are willing tosign the long-term contract with the upstream incumbent in stage 1 to avoid this Prisoner'sDilemma. The proof of this is identical to that of Proposition 2.4.99Appendix 3Appendix 3A Proofs of Results in Chapter 3Proof of Proposition 3.1:Suppose the monopolist produces n > 2 products of different qualities: 1 < sn < • • < s2 <si < The monopolist incurs a sunk cost K for each quality it chooses. Define the (n+l)th'product' to be the consumers' option of buying nothing, with pn+i = 0 = 8n+1. Withoutloss of generality, suppose prices are set such that there exists a 9k,k = 1,- • • , n where theconsumer with characteristic Ok is indifferent between buying product k and k + 1. Then, itfollows from (3.1) thatPk Pk+1 Ok =^•Sk Sk+1The demand for product si isQi = 1— F(01) = —and the demand for product sk, k = 2,3, • • , n is^Qk = F(Ok—i) F(0k) =^OkThe monopolist's decision problem ismax piQi + • • • +PnQn - nK(P1 ,•••,Pn)100The first-order conditions give rise to the following n equations:1 „ ,P1 = —2 uo0i — 82) +P21Pk-1(Sk — Sk+1) + Pk-Fl(Sk-1 — SO Pk =^ , k = 2,• ••,n,8k-1 — 8k+18nPn =^Pn-1Sn-1Note that these equations are recursive in nature. By substituting the nth equation into the(n-1)th equation, we obtain8n-1Pn-1 =^Pn-2,sn-2(A3.1)and a similar expression is obtained if (A3.1) is substituted into the (n-2)th equation. Thefollowing solutions are obtained by successive substitutions:f,Pk = —2 uosk ' k = 1,2,• • •,n.Substituting these prices into the demand equations gives1Q = —2 00 and Q k = 0, k = 2, 3, • • , n.Thus the monopolist does not produce more than one product. The 'reduced-form' profitfunction of the monopolist is-1802 - K4 °which is maximized by choosing si =^411Proof of Proposition 3.4:(a) Suppose firm A supplies two products with quality 8A1 =.3 and 8A2 = 1. If the entrantenters, it could only choose a quality level between 8A1 and 8A2. Let 91 be the characteristicof the consumer who is indifferent between qualities sAi and 8B. Let 02 be similarly defined.That is,PA1 PBPB PA201 =^and 02 =sAi — SB^8B 8A2101The demands for firm A's two products areQ Al = 1 — F(01) = 00 — 01,Q A2 = F(02) = 02 — (00 — 1).and firm B's demand isQ B = F(01) — F(02) = 01 — 02.The entrant maximizes its profit pBQB — K B. The reaction function derived from the first-order condition is1 PB (PA1 PA2)^ [PA1 (313 — 5A2) P A2(3 Al — SB)1•5A2)(A3.2)Similarly, the incumbent maximizes its profit PA1Q Al+ P A2QA2 —2KA. The resulting reactionfunctions are:Define the vectorPAi (PB )^[611::1(sAi — sB)d- PB],1PA2(PB) = —2 PB^— 1)(5B s A2)1•s^[sAi, sA2, si3]. Solving (A3.2), (A3.3) and (A3.4) yields(SA1^SB)  r,,nLou0(SA1^8212)^(SB^SA2)116(8A1 — SA2)(SB SA2) [(8/41 SB) — 3(00 — 1)(8A1 — 5A2)b6(SAl 8A2)1^ [(8A1^8B)(8/3^8A2)1.3(8A1 8A2)(A3.3)(A3.4)(A3.5)(A3.6)(A3.7)Substituting (A3.5), (A3.6) and (A3.7) into Q B yields the equilibrium demand for the en-trant's product, which is Vig = 1/3. The 'reduced-form' profit function of the entrant istherefore1HB(S) = n(^ (S Al — SB)(SB 5A2)^B •u0 Al — 8A2)(A3.8)Since (A3.8) is concave in 5B, the optimal quality choice is s'13^8A2), which gives amaximum profit of1 ,1-1*B = —36 (sin — 5A2) — K B • (A3.9)102It is straightforward to verify that, by virtue of Assumption 3.2, every consumer buys a unitof one of the products.Since 5A1 = 3 and 5A2 = 1, it follows then ll =^— KB. Thus, it does not pay for theentrant to enter the market if KB>(b) Note that 8A2 > 3 – 36KB > 1, since KB < 1. Thus the entrant can either choose aquality level that is between 5A2 and 3, or one that is lower than sA2. Consider the former,i.e.,SA2 < SB SAl.Then, from the proof of (a) above, the entrant's optimal quality choice iss*B –2 0A1 sA2)with a net profit ofll^1 ,= —36 – sA2) – KB•But 5A2 > – 36KB, which implies that 1113 < KB. Thus, choosing a quality level between3 and sA2 is not profitable for the entrant.Next, suppose the entrant chooses a quality level that is lower than 5A2, that is,sB < SA2 < SA1-The entrant effectively competes only with the lower-quality product of the incumbent.1From Proposition 3.2, the entrant's optimal quality choice is 3B = 1 with a profit offl*B = —91 (2 — 002 (S A2 — 1) — KB.But 5A2 < 1 + (29R0.'302 implies that 11*B < KB. Thus, choosing a quality level that is lowerthan 5A2 is not profitable for the entrant either.'Again, by virtue of Assumption 3.2, every consumer buys a unit of one of the products.103Therefore, whether the entrant chooses a quality lower than sA2 or one between 3 and 8A2, itcannot make a positive profit net of its sunk cost KB. It remains to show that there alwaysexists such a quality choice 5A2• To show this, note that by Assumption 3.3,1^(2 – 00)2KB > 9 1 + 4(2 – 00)2which, after rearranging terms, and noting that by definition, 1 = – 1, gives1 + (29KB> 3 — 36KB.— 002(A3.10)The inequality in (A3.10) ensures that the incumbent can always find a quality level sA2 suchthat entry is never profitable if KB <Further, from Proposition 3.1, the monopolist's profit is maximized by setting QA2:= 0.Proof of Lemma 3.1:Note that by definition of K*, we have1K* – –1 (2 – 00)2 = –430g – –/ [(1 + 00)2 + (2 – 0o)2]•9^9After some algebraic manipulations, and noting that 1^– 1, (A3.11) reduces toK* – –1(2 – 00)2 –1 – 1-(10 + 00)(2 – 00).9^4^36(A3.11)(A3.12)Since the last term on the right-hand side of (A3.12) is positive, and from Assumption 3.2,0 < 1 < 3(90 – 1)/(2 – 00), by substituting 1 = 3(00 – 1)/(2 – 00) into the right-hand side of(A3.12), we get1^ 1K* – –9(2 – 00)2 > - 1406 – —12 (10 + 00)(00 – 1).After simplifying terms, (A3.13) reduces to1K* – –9 (2 – 00)2 > —112 (0o – 2)(200 – 5).(A3.13)(A3.14)It is straightforward to show that for < 00 < 2, the expression on the right-hand side of(A3.14) is strictly positive.104Proof of Proposition 3.7:(i) There are two existing products in the market, QA and QB, with quality levels sA = 3 andsB = 1. Given that firm A holds the exclusive right to the new technology, it has to decidewhether to make use of the new technology by introducing a new product. Suppose firm Aintroduces a new product, denoted by Q, with quality 811. Define 01 as the characteristicof the consumer who is indifferent between buying Q % and QA. Similarly, define 02 as thecharacteristic of the consumer who is indifferent between buying QA and QB. Let p-Ni , PA, andpB be the prices of products Q, QA, and QB, respectively. Then, the problem is equivalentto the one considered in Proposition 3.3. The profit function of firm A is, from (3.11), andnoting that sA = ts- and sB =1,1lfiv4 (si) = —4 cs/IT _ .3) + (0o + 1)2_ K.Hence, firm A maximizes its profit by setting siltr = + a; and the resulting profit is.c,11611) + —91(00 + 1)2 — K. (A3.15)On the other hand, if firm A decides not to introduce a new product, it continues to earn itsduopoly profit HdA, which is given in (3.8). Therefore, it is profitable for firm A to introducesa new product if and only if ail > WI, which, after rearranging terms, gives the requiredcondition in (3.22).(ii) Since firm B holds the exclusive right to the new technology, if it introduces a newproduct, denoted by QIE`31', with quality 4, there are then three products in the market.Firm B cannot do worse than producing only the new, high-end product Q by abandoningits low-end product QB, i.e., set QB = 0. By doing so, firm B's optimal quality choice issNB = + a, and it earns the duopoly profitN 1 + =^(0o + 1)2 — K,9which represents the lower bound of firm B's profit.105On the other hand, if firm B decides not to introduce a new product, it continues to earnthe duopoly profit III given in (3.9). Thus, it is profitable for firm B to introduce the newproduct if 11/1 > 1113, which simplifies to the desired inequality in (3.23). li106

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