multiproduct quality competition: fighting brands and

Multiproduct Quality Competition

Fighting Brands and Product Line Pruning

By JUSTIN P JOHNSON AND DAVID P MYATT

Firms selling multiple quality-differentiated products frequently alter their productlines when a competitor enters the market We present a model of multiproductmonopoly and duopoly using a general ldquoupgradesrdquo approach that yields a powerfulanalytical framework We provide an explanation for the common strategies ofusing ldquo ghting brandsrdquo and of product line ldquopruningrdquo The optimal strategydepends on whether entry prompts an incumbent to expand or contract its totaloutput We also present a general condition that guarantees that a monopolist willsell but a single product Our model addresses other issues including intertemporalprice discrimination and ldquodamaged goodsrdquo (JEL D40 L10 M31)

Incumbent rms often adjust their productlines in response to competition Sometimesthey remove products from the market therebyldquopruningrdquo their product lines This was one re-sponse of Procter amp Gamble to private labelbrands in the early 1990rsquos1 At other times anincumbent responds to competition by expand-ing its product line often to include a lower-quality good called a ldquo ghting brandrdquo2 Thishappened following AMDrsquos entry into the mar-ket for 386DX microprocessors when Intel re-

leased the 486SX as a companion to its higher-quality 486DX processor3

Fighting brands are extremely widespreadFor instance ATampT launched Lucky Dog Tele-phone to help compete against lower-pricedldquodial aroundrdquo phone carriers4 Brian Adamik ofYankee Group5 commented ldquoTheyrsquove intro-duced a ghting brand in the market that goesafter price-sensitive consumers while allowingATampT to be their premier brand in the mar-ketrdquo6 BPL announced that it would introduce a ghting brand in the color television market totake on competition from Chinese and localbrands The head of corporate brand manage-ment at BPL noted that its ghting brand wouldbe ldquoa separate brand for the price-sensitive low-end CTV marketrdquo7 And in the Indian marketfor electric fans the growth in fringe competi-

Johnson Johnson Graduate School of ManagementCornell University 353 Sage Hall Ithaca NY 14853 (e-mail jpj25cornelledu) Myatt Department of Econom-ics University of Oxford and St Catherinersquos CollegeOxford OX1 3UJ United Kingdom (e-mail davidmyatteconomicsoxacuk) We thank Jon Dworak MichaelWaldman and two anonymous referees for helpful com-ments and suggestions

1 ldquoAnd in what amounts to virtual apostasy at a companythat never gave up on struggling brands PampG is consoli-dating some weak products with stronger siblings whiledumping othersrdquo Business Week July 19 1993

2 The ldquo ghting brandrdquo terminology is used in the man-agement literature (Michael E Porter 1980 Kevin LKeller 1998) and by industry participants In response tonew entrants in the credit card business American Expressintroduced the Optima card in 1991 Chairman James DRobinson said ldquoExpect to see Optima as a ghtingbrand I think that wersquove got initiatives going in all theareas where there is competitive pressurerdquo (The Wall StreetJournal July 18 1991)

3 We assess the AMD-Intel case ( rst highlighted byRaymond J Deneckere and R Preston McAfee 1996) inSection V

4 Dial-around carriers allow consumers to bypass theirhome carriers by dialing a special access code

5 Yankee Group provides industry research and consult-ing services

6 ldquoATampT Launches Lucky Dog Telephone in Responseto lsquoDial-Aroundrsquo Carriersrdquo The Wall Street Journal Eu-rope October 8 1998

7 ldquoBPL Adopts Multi-Brand Strategy for CTVsrdquo TheEconomic Times May 9 2000

748

tion led Usha to introduce a discount fan calledthe Racer8

Product line pruning is also common9 Timexrecently announced that it would be removing anumber of its lower-priced watches from theIndian market in response to growing competi-tion in the low end from Titan10 while Mitsu-bishi announced the phasing out of low-endversions of its Trium mobile phones in responseto a supply glut in that segment11 And aftermore than a century in the piano business Kim-ball International in 1996 discontinued all butits prestigious Bosendorfer model following thecapture of the low and middle markets by Jap-anese and Korean rms12

One feature common to these (and manyother) examples is that the increase in compe-tition manifested itself in the low end of themarket Since the incumbentrsquos decision to eitherexpand or contract its product line amounts to adecision about how to segment the marketthese examples suggest that understanding howprice discrimination opportunities change withthe presence of low-end competition might ex-plain the prevalence of both ghting brands andproduct line pruning Our main results implythat this is indeed the case

Consider for instance the IBM Laser-Printer13 A single version was initially soldcapable of printing ten pages per minute Theabsence of a lower-quality version suggests thatIBMrsquos gains from serving the low end of themarket were not large enough to justify intro-ducing a substitute product for its high-qualityunit (which would have limited IBMrsquos ability toextract surplus from high-value users) How-ever following Hewlett-Packardrsquos entry intothe market with its LaserJet IIP a lower-quality

substitute for IBMrsquos LaserPrinter IBM neededto reevaluate its product line strategy On onehand Hewlett-Packardrsquos entry meant that a sub-stitute for IBMrsquos LaserPrinter was already onthe market Inasmuch as a desire to avoid offer-ing such a substitute restrained IBM in the rstplace it might have been sensible to introduceone following Hewlett-Packardrsquos move On theother hand even though a lower-quality substi-tute would be on the market regardless of IBMrsquosdecision introducing its own such productwould have exacerbated (from IBMrsquos perspec-tive) the substitution possibilities for consum-ers In fact IBM decided to introduce a ghtingbrand the LaserPrinter E which was identicalto its original LaserPrinter except for the factthat its software limited its printing to ve ratherthan ten pages per minute This suggests that thedesire to compete in the newly opened low-endmarket was strong enough that IBM was willingto bear the additional erosion on its high-endpro ts resulting from its own entry into thatmarket

However as noted above not all rms mimicIBMrsquos strategy of expanding its product line inresponse to low-end competition some rmschoose to prune their product lines when facingsuch competition Therefore the full explana-tion of the in uence of competition on productline choices is more complicated than that sug-gested by the IBM example Consider DECwhich until 1996 sold a range of PCs to bothhome and business customers By offeringlower-quality PCs targeted at home users DECprovided a potential substitute to its businessclients but clearly felt this downside was worthbearing for the opportunity to serve the low-endmarket In 1996 however DEC faced increas-ing competition in the home computer marketand responded by exiting that market to focuson its high-end desktops and servers14 Despitethe contrast with IBMrsquos decision DECrsquos reac-tion also can be understood in terms of changingopportunities for market segmentation In par-ticular DECrsquos response suggests that competi-tion reduced the available pro ts in the low-endmarket enough that it became more importantto attempt to preserve pro ts on its high-end

8 ldquoSummer of Discontentrdquo The Business Standard July3 2001

9 Philip Kotler (1965) and John A Quelch and DavidKenny (1994) emphasize the bene ts of regular pruning ofproduct lines Kotler writes ldquoAs the pace of competitionquickens and as consumer tastes become surfeited the needfor pruning company product lines of casualties becomes asgreat as that for nding replacementsrdquo

10 ldquoTimex to Exit Mass Market for Watchesrdquo The Eco-nomic Times April 30 2001

11 ldquoMitsubishi to Phase Out Low-End Mobile HandSetsrdquo The Economic Times November 24 2001

12 ldquoPiano Industry Off-Keyrdquo Baltimore Sun February18 1996

13 For a more detailed review of this case see Section V

14 ldquoDigital Equipment to Exit Home-Computing SectorrdquoThe Asian Wall Street Journal January 31 1996

749VOL 93 NO 3 JOHNSON AND MYATT MULTIPRODUCT QUALITY COMPETITION

products by not contributing to a mass of low-priced substitutes Pruning its product line wasthe means to accomplish this

We argue in this paper that it is indeed truethat competitionrsquos in uence on market segmen-tation opportunities is crucial to understandingthe changing product line decisions of multi-product rms Moreover we show that it isstraightforward to understand when either ex-pansion or contraction of a product line willoccur More precisely we identify demand andcompetitor characteristics that make either ghting brands or pruning optimal responses toheightened competition This is important sincewhile much is known about the optimal productline choice of multiproduct monopolists rela-tively little is known about competition betweenmultiproduct rms and virtually nothing isknown about the decision of a rm to alter itsvertically differentiated product line (by prun-ing it or introducing a ghting brand) as aresponse to the arrival of new competition (Werelate our work to the existing literature in Sec-tion I)

In our analysis we presume that entry by asingle rm has occurred in a market originallydominated by a monopolist The duopolistscompete in quantities each potentially offeringa range of quality-differentiated productsWhether the incumbent will choose to extend orcontract its product line depends closely on theshape of the marginal revenue curves in themarket When marginal revenue is everywheredecreasing the incumbent never responds to theentrant by expanding its product line Ratherentry tends to be associated with the pruning oflower-quality products from the incumbentrsquosmenu meaning that it chooses to ldquofocus onqualityrdquo However when marginal revenue isincreasing in some regions an incumbent may nd it optimal to respond to entry through theintroduction of a lower-quality product

It might seem that marginal revenue that isincreasing in some regions represents an unim-portant case However there are two importantpoints to keep in mind when evaluating thisassumption First rms with constant marginalcost certainly do not choose output in regionswhere marginal revenue is increasing ie inequilibrium rms operate in regions where mar-ginal revenue is decreasing We are not suggest-ing therefore that marginal revenue will be

increasing in a region local to that observed inequilibrium Second while everywhere de-creasing marginal revenue is a convenient tech-nical assumption it is in fact incompatible withsome very plausible demand structures For ex-ample as we show in Section II subsection Bthe existence of a bimodal distribution of con-sumer preferences corresponding perhaps tosegments of home and business users readilygenerates marginal revenue that is increasing insome regions

The intuition for our results is straightfor-ward Suppose that marginal revenue is decreas-ing so that rms face the typical Cournotincentives to reduce their own output as theoutput of a rival increases Consider a simpleexample in which the incumbent originally mar-keted both a low- and a high-quality productand suppose that the entrant is able to offer onlya low-quality good When confronted with pos-itive output by the entrant the incumbent re-stricts output in the low-quality marketFurthermore since the total production in thelow-quality market also adversely effects theprice of the high-quality good the incumbentfaces additional pressure to lower its own outputin the low-quality market If the entrant nds itoptimal to produce beyond a certain level thebest course of action for the incumbent is tocede that market to the entrant in an effort topreserve margins on the high-quality goodmdashitwill prune its product line in order to focus onquality

As noted above marginal revenue that isincreasing in some regions can be consistentwith plausible demand structures such as theexistence of distinct ldquomarket segmentsrdquo Thisleads to the possibility that a suf ciently largeintrusion by a competitor may lead an incum-bent to expand its supply The intuition for thisis simply that as a monopolist the rm mightchoose not to serve the low-end market segmentin order to maintain high prices Once a com-petitor enters on a large enough scale howeverthe possible increase in marginal revenue mayencourage the incumbent to expand into thatsegment along with the entrant

Strikingly such an incentive for the incum-bent to increase its total supply is what drivesthe introduction of a ghting brand To see thissuppose once again that the entrant is able tooffer only a low-quality good and also that the

750 THE AMERICAN ECONOMIC REVIEW JUNE 2003

incumbent would choose to sell only the high-quality good as a monopolist Following entrythe incumbent may wish to expand its outputBut such pressure only applies at the qualitylevel offered by the entrant since that is theonly market in which the entrant is active Al-though the incumbent faces competition in thesupply of a complete high-quality product it(conceptually) maintains monopoly power inldquoupgradesrdquo from low to high quality and hencewishes to restrict their supply The desires of theincumbent are manifested by the introduction ofa low-quality ghting brand that allows its totaloutput to increase while still exercising marketpower through the restriction of supply of thehigh-quality good

Our analysis generates other predictions aswell which are potentially testable First ght-ing brands tend to emerge when the entrantoffers only low-end productsmdashthat is whenthere is some asymmetry between the techno-logical capabilities of the incumbent and theentrant Second if the incumbent introduces a ghting brand that brand will be of qualitycomparable to the lowest quality good of theentrantrsquos In other words any price-discriminat-ing behavior by the incumbent takes place atquality levels (weakly) above that of the en-trantrsquos worst product This result is consistentwith much observed behavior For instance theIBM LaserPrinter E was slightly faster than theHewlett-Packard IIP15 Third assuming thatmarginal revenue is decreasing an increase inthe maximum quality that an entrant can offer(whether due to the termination of a patentreverse engineering or some other factor)makes it more likely that the incumbent willchoose to exit the lower markets

Beyond our results on changing productlines we also provide a more specialized tech-nical result about price discrimination We dem-onstrate in a very general model of monopolynonlinear pricing that there exists a weak andplausible condition that is suf cient to ensurethat a rm never offers more than a singleproduct In the special case of multiplicativelyseparable preferences this condition reduces to

increasing returns in the provision of qualityThis condition might hold when there are pro-duction costs that must be incurred regardless ofthe nal quality choice as when an automobilemanufacturer chooses the nal trimline of avehicle only after rst building the basic plat-form We believe this result is of some interestbecause to our knowledge such a straightfor-ward condition for extreme ldquobunchingrdquo of con-sumers has been underexplored

Our paper is organized as follows In SectionI we relate our work to earlier literature Ourmodel is speci ed in Section II We present thegeneral monopoly results in Section III andalso construct the framework for the case ofduopoly when consumersrsquo preferences are sep-arable Duopolistic competition is our focus inSection IV In Section V we discuss a numberof illustrative case studies prior to offering con-cluding remarks

I Related Literature

Our work is related to that on product designdecisions and price discrimination by rms inimperfectly competitive markets There arethree main branches of interest monopolyprice-setting competition and quantity-settingcompetition We discuss each of these below

The incentive to engage in second-degreediscriminatory behavior has long been recog-nized In a classic contribution Michael Mussaand Sherwin Rosen (1978) consider the productline decisions of a price-discriminating monop-olist able to offer a range of products of differ-ent qualities An important insight of this workis that a monopolist may offer inef ciently lowqualities in the sense that the quality levelsupplied to lower-value customers is distorteddownward Inducing such a distortion is optimalas it reduces the substitution possibilities ofhigher-value customers

One situation in which this downward distor-tion does not occur is when a rm offers but asingle productmdashthe monopolist simply does notsupply lower-value customers Nancy Stokey(1979) shows that when a monopolist is able todiscriminate on the delivery date of a singleproduct so that products delivered in the futureare essentially of lower quality it may choose tooffer only products for sale today Multipledates of delivery are chosen only when costs fall

15 Some reviews claim that the IBM printer had slightlyinferior print quality and hence we might conclude that theoverall quality of the two products was approximatelyequal See Section V for more details


suf ciently quickly over time Stephen WSalant (1989) considers the conditions underwhich price discrimination occurs The mar-ginal production cost must be suf ciently con-vex in a productrsquos qualitymdashin a sense theremust be decreasing returns to quality In con-trast when there are increasing returns to qual-ity the monopolist will sell a single product atthe highest feasible quality level We obtainsimilar results with our own analysis (Proposi-tion 1)16

While the literature on monopoly price dis-crimination is obviously important such worknecessarily cannot address the matter of how rms with market power adjust their productlines in response to competition There are sev-eral papers that explore multiproduct competi-tion between rms Perhaps the most importantof these is that of Paul Champsaur and Jean-Charles Rochet (1989) who consider productline competition in prices between two rms ina general model They allow rms to commit toproducing in chosen intervals of quality beforecompeting on prices17 They nd that rmschoose to offer nonoverlapping product lines asthis reduces the intensity of price competitionHence the product line offered by a given rmneed not match the product line offered by amonopolist capable of offering the entire rangeof goods In particular the product line of the rm offering high-quality goods can containfewer products than a monopolist would offerAt rst this would appear to address the issueof product line pruning However we wish toask how an incumbent rm with a xed tech-nology adjusts its line following entry In theChampsaur and Rochet (1989) analysis thiscorresponds to comparing the product line of-

fering of the high-quality rm in monopolyversus duopoly in each case for a xed feasiblequality interval Importantly they show there is nodifference in the optimal product line in these twocases ie no pruning occurs As such for xedproduction opportunities product line pruningdoes not occur and ghting brands never arise

There is nonetheless a sense in which theequilibrium quality gap between the two rms isrelated to our analysis in Section IV subsectionB We show that when marginal revenue isdecreasing the incumbent never introduces a ghting brand Introducing a ghting brand isnot optimal because the best response to entry isfor an incumbent to reduce its total supplywhich leads to fewer distinct products Expand-ing into lower-quality markets only negativelyeffects all prices for the incumbent This issimilar in spirit to the desire of rms that canprecommit to qualities in order to avoid head-to-head competition

In contrast to these price-setting analyses wepresent a quantity-setting model as others doEsther Gal-Or (1983) assumes a symmetricCournot equilibrium where each rm offers arange of qualities (and states appropriate suf -cient conditions for a particular example) ob-taining comparative statics as the number of rms increases In the equilibria she considers rms do not change their product lines as thelevel of competition changes18 Giovanni DeFraja (1996) offers a quantity-setting modelwith the income-effect utility functions of Gab-szewicz and Thisse (1979) His main result isthat any equilibrium is symmetric when rmshave identical technologiesmdashwe offer a similarresult as part of Proposition 6 In contrast tothese contributions we offer a more completeanalysis of equilibrium product lines and con-sider a more general speci cation where one rmpotentially is limited in the qualities it can offerMoreover none of these papers considers the is-sue of ghting brands or product line pruning

Our analysis and that of many of the authorsmentioned above considers a single dimensionof quality where all consumers agree on the

16 Other authors offer similar results based upon slightlydifferent speci cations For instance Jean Jaskold Gabsze-wicz et al (1986) describe a model in which consumers aredistinguished by their income levels They nd that as theincome distribution narrows a monopolist focuses its pro-duction on a single quality level

17 A number of other authors offer price-setting modelsof competition in which rms precommit to quality levelsGabszewicz and Thisse (1979 1980) and Avner Shaked andJohn Sutton (1982 1983) allow rms to precommit to theirquality levels prior to the simultaneous choice of prices Inall of these papers rms are restricted to a single productand hence they cannot address the issue of product rangeswe consider

18 She moves on to combine her analysis with that ofGabszewicz and Thisse (1980) and Shaked and Sutton(1982) in Gal-Or (1985) In this later paper however sherestricts to single product provision and decreasing returnsto quality


ranking of products Alternative models com-bine quality provision with horizontal differen-tiation elements (Richard J Gilbert and CarmenMatutes 1993 Lars A Stole 1995 Frank Ver-boven 1999) Others allow preferences to differonly along a horizontal dimension (B CurtisEaton and Richard G Lipsey 1979 Kenneth LJudd 1985) and consider the ability of rms todeter entry by ldquocoveringrdquo the markets for cer-tain brands James A Brander and JonathonEaton (1984) address product line choices bytwo rms assuming that rms are able to com-mit to product lines prior to competing onprices In particular there are four productssplit into ldquogroupsrdquo of two Two products in onegroup are close substitutes for one another andless close substitutes for the other two goodswhich are in turn close substitutes for one an-other They show that an ldquointerlacerdquo equilib-rium can exist in which each rm offers oneproduct in each group The reason is that if one rm believes the other will occupy one productspace in each group it is certainly wise to avoidcompeting in those exact product spots as pricecompetition will drive pro ts to zero Howeverit might be pro table to occupy the ldquoopenrdquoproduct spots within the two groups if the re-sulting price competition is not too erce (as itmay not be because distinct products are some-what differentiated) Note that if there is nocommitment stage in their model each rmproduces all the products and interlace is not anequilibrium We show that absent such com-mitment there is no interlace equilibrium in avertically differentiated industry In particularthe entrant offers any product the incumbentdoes subject only to being physically capableof producing it (we actually prove the contra-positive in Section IV) Heuristically the en-trant has less to lose by offering a product sinceit has fewer high-quality products and is henceless concerned about negative price conse-quences associated with increasing the overalllevel of goods Hence if the incumbent wishesto offer a positive supply of some good theentrant must also wish to do so

In short the existing literature has made greatprogress in understanding monopoly productdesign decisions However less progress hasbeen made in understanding multiproduct com-petition and no one has addressed the matter ofchanging product lines that we consider

II A Market for Quality-DifferentiatedProducts

In this section we lay out the structure ofdemand and costs In Section II subsection Aand Section II subsection B we describe con-sumer preferences and the demand side of themarket in general In Section II subsection Cwe describe different cost structures that we willmake use of later on

A The Demand Side

Demand is generated from a unit mass ofconsumers indexed by a type parameter u Thecumulative distribution function is F(u ) whichhas support on [0 u] On its support F(u ) isstrictly increasing and continuously differentia-ble with density f(u ) A consumer of type uwho purchases a good of quality q at price penjoys utility uq 2 p19 She faces a selection ofn products indexed by i where product i is ofquality qi and price pi We order the goods sothat qn qn2 1 q1 0 Note thatgoods differ only with respect to quality andthat all consumers prefer goods of higher qual-ity Each consumer purchases a single unit ofthe good i that maximizes uqi 2 pi unless thisyields negative utility in which case she pur-chases nothing

Before deriving the entire system of inversedemand functions under this multiplicativespeci cation rst suppose that only a singlegood of quality q 0 is being sold at pricep 0 This good would be purchased by allconsumers satisfying u $ pq yielding demandof z 5 1 2 F( pq) Similarly the inversedemand curve satis es p 5 qH( z) where

(1) H~z 5 5 F21~1 2 z z 10 z $ 1

When z 1 u 5 H( z) is the type with amass of z consumers above her At a price of

19 This is equivalent to the more general formulationu(u q) 5 v(u )w(q) where v and w are both increas-ing We simply rede ne a consumerrsquos type to be v (u) andquality to be w(q) When we refer to quality q thereforewe are really considering the (scaled) monetary value ofquality since a consumer u is willing to pay uq for qualityq


p 5 uq it is exactly these z consumers who arewilling to buy with consumer u being just in-different between buying and not Hence p 5qH( z) equilibrates supply and demand For z 1 supply exceeds the number of willing pur-chasers and so the only possible equilibriumprice is p 5 0

As quantitieswill be the strategic choice vari-ables of rms in our later analysis and as prod-ucts are differentiated only with respect toquality in the eyes of consumers we can com-pute the market-clearing prices for the n prod-ucts knowing only that industry supply of eachproduct i is given by zi It turns out that theinverse demand function for a single goodbased on H( z) from equation (1) is central tothe general analysis with n products

Naturally we must consider the possibilitythat zi 5 0 for some products However it isconceptually easier to rst derive the inversedemands assuming that z i 0 for each i Wewill then explain how to incorporate the pos-sibility that some products are not supplied atall

When i51n zi 1 there is partial market

coverage Not all consumers are able to pur-chase a good Thus given a set of supplies z iwe require a set of positive prices pi suchthat exactly zi consumers wish to purchase goodi If a lower-quality good were priced no lowerthan a higher-quality good then it would attractno demand There must therefore be a pricepremium for higher quality Such higher-qualityproducts must be purchased by consumers withhigher types If a consumer u is willing to pay apremium for higher quality then higher typeswill strictly wish to do so Thus the highest zn

consumers purchase product qn and the nextzn2 1 purchase quality qn2 1 and so on Theconsumer with j51

n zj others above her must bejust indifferent between purchasing quality q1and not purchasing at all so that p1 5q1H( j51

n zj) Similarly the consumer withj5 in z j consumers above her must be just in-

different between products i and i 2 1 and sopi 5 pi2 1 1 (qi 2 qi2 1) H( j5 i

n z j) De ningq0 5 p0 5 0 for convenience for i [ 12 n we obtain

(2) p i 2 p i 2 1 5 ~q i 2 q i 2 1 HX Oj 5 i

n

z jD

Notice that pi 2 pi2 1 represents the price of anupgrade from quality qi2 1 to quality qi Thisobservation leads us to consider the cumulativevariables de ned by Zi 5 j5 i

n zj Z i is the totalsupply at quality qi and above We offer thefollowing interpretation We may suppose thatthe industry supplies Z1 units of a ldquobaselinerdquoproduct of quality q1 There are then supplies ofsuccessive upgrades to the baseline product inorder to achieve qualities above this For in-stance a product of quality q2 consists of abaseline product at price p1 plus an upgradeq2 2 q1 priced at p2 2 p1 Continuing in thismanner equation (2) becomes

(3) p i 2 p i 2 1 5 ~q i 2 q i 2 1 H~Z i

Hence the price of upgrade i depends only on itsown supply and not on the supply of any otherupgrades In contrast the complete product iwith quality qi has a price pi that depends on thesupplies of all n products

Equation (3) also applies when there is com-plete market coverage so that Z1 $ 120 IfZ1 5 1 then p1 must equal zero given ourassumptions on F(u ) If Z1 1 then p1 5 0as well because there is strictly excess supplySimilarly pi 2 pi2 1 5 0 for any upgrade isatisfying Zi $ 1 But of course if this holdsthen H(Zi) 5 0 by de nition [see equation (1)]and hence equation (3) continues to hold21

Note that if Zj1 1 $ 1 then there can be nodemand for product j The reason is that theprice of the upgrade to product j 1 1 is zero sothat all consumers will purchase a product atleast of quality j 1 1

It turns out that de ning prices in the mannerjust described easily allows us to address thepossibility that some products are in zero sup-ply To see how suppose that z i satis es onlythat zi $ 0 and de ne the prices pi andcumulative variables Zi exactly as above

20 Given our assumption that the lower bound of thesupport of F(u ) is zero there will only be partial coveragein equilibrium Nevertheless we must consider the possi-bility of complete coverage

21 Our assumption that the lower bound of F(u ) is zeroensures that all prices will be positive in equilibrium so thatthe market is not fully covered We still must address thepossibility that some prices are zero as we do here See alsofootnote 20 above


Note that a product i is in positive supply ifZ i 2 Zi1 1 0 at least if we de ne Zn1 1 50 while a product is in zero supply if Z i 2Z i1 1 5 0 If j is the rst product in positivesupply then there are a total of Zj 5 i5 j

n zi

products on the market If the prices de nedabove are correct in this case it must be thatconsumer u 5 H(Zj) is just indifferent betweenbuying good j and not The price of good j canbe obtained by adding up the incremental pricesgiven by the right-hand side of equation (3)Making use of the fact that for k j we haveZk 5 Zj we obtain

p j 5 p j 2 p0 5 Oi 5 1

j

~ p i 2 p i 2 1

5 Oi 5 1

j

~q i 2 qi 2 1H~Z i

5 H~Z jOi 5 1

j

~qi 2 q i 2 1

5 qjH~Z j

Hence a consumer with type u 5 H(Zj) isindeed indifferent between buying good j andnot buying at all The only subtlety is that she isalso indifferent between buying goods k jand not at the prices de ned but these are not inpositive supply Hence we adopt the conven-tion that when a consumer is indifferent be-tween several products she purchases the one ofhighest quality We have only dealt with the rst good in positive supply but a similar pro-cess can be applied recursively With somemore work it can be shown that the process justdescribed can be extended to consistently de nedemand for all possible supply con gurationsHence given the convention that a consumerpurchases the highest quality good to which sheis indifferent the prices de ned above are cor-rect for all circumstances

B Demand Segments and the Shape ofMarginal Revenue

Equation (3) reveals that the shape of inversedemand for all n products is tied to the function

H( z) which is itself the inverse demand curvefor a single good of quality q 5 1 A standardldquotextbookrdquo assumption would be to supposethat H( z) exhibits decreasing marginalrevenue22

De nition 1 (Decreasing marginal revenue)H( x 1 y) 1 xH9( x 1 y) is decreasing in y forall x

This states that if a rm is producing x unitsthen the marginal revenue of its nal unit isdecreasing in the output y of the rest of theindustry If marginal revenue satis es this def-inition then the marginal revenue of a rm alsois decreasing in its own output

Decreasing marginal revenue is a propertyexhibited by many demand curves Nonethe-less there are some very plausible speci ca-tions that are inconsistent with marginalrevenue being everywhere decreasing We illus-trate this with an example

Example 1 u is drawn from a mixture of twonormal distributions with means uH uL andvariances sH

2 and sL2 The former is chosen with

probability a Types u 0 do not purchase

Example 1 is a stylized representation of amarket where demand is drawn from two sep-arate sources as shown in Figure 1(a)23 Forinstance when consumers are drawn from homeand business sectors the speci cation of Exam-ple 1 may be appropriate24 Note that in thisexample marginal revenue is not decreasingeverywhere [Figure 1(b)]

22 The assumption of decreasing marginal revenue isalso a convenient suf cient condition for many existenceand uniqueness results in oligopoly For instance combin-ing decreasing marginal revenue with weakly convex mar-ginal costs ensures that a single-product Cournot game hasa unique and symmetric pure strategy Nash equilibriummdashsee for instance Xavier Vives (1999 Ch 4)

23 In our speci cation of the model above we assumedthat there was a nite upper bound to the support of uHowever allowing the support to be unbounded above asin this example does not affect our results

24 Many rms explicitly direct these categories of con-sumers toward different product lines The personal com-puter manufacturer Dell divides its web site into Consumerand Business products In the former division it offersDimension desktops and Inspiron laptops whereas in thelatter it offers Optiplex desktops and Latitude laptops


Our opening discussion of product lines andcompetition suggests that rms are able to iden-tify different ldquosegmentsrdquo of market demandUpon the introduction of Lucky Dog Tele-phone ATampT vice-president Howard E Mc-Nally commented that ldquo[w]hat we want to dowith this brand is attract a different group ofpeoplerdquo In the context of a formal model thiscomment suggests that the density f(u ) maywell be multimodal with each mode corre-sponding to a different segment of consumers

In Section IV we show that the presence orabsence of decreasing marginal revenue criti-cally affects the product ranges offered by com-peting duopolists As we wish to take bothpossibilities seriously we make two morepoints here First even if marginal revenue isincreasing in some regions rms will alwayschoose output in the region where it is decreas-ing in the textbook manner25 Thus incorpo-rating marginal revenue that is increasing insome regions is simply a way to admit otherdemand structures such as Example 1

To build to our second point note that the

monotonicity of marginal revenue is also re-lated to the price sensitivity of consumers For asingle good of quality q 5 1 the price elasticityof demand laquo( z) satis es

1laquo~z

5 2d log H~z

d log z5 2

H9~zz

H~z

yielding marginal revenue

H~z 1 zH9~z 5 H~zX 1 21

laquo~zD

Since H( z) is an inverse demand function itis automatically decreasing in z This meansthat if marginal revenue is both positive andincreasing in a region (as in Example 1) then itmust be the case that laquo( z) is increasing in z inthat region In other words an expansion in ztoward the ldquolow endrdquo of the market naturallyresults in an increase in the price elasticity ofdemand This is consistent with our example ofBPL where an expansion to the low end of themarket resulted in greater price sensitivity26

25 We assume constant marginal costs of productionwithin a quality level

26 Both marginal revenue and the price elasticity of de-mand are closely related to the hazard rate of F(u )

FIGURE 1 MULTIPLE MARKET SEGMENTS AND NONMONOTONIC MARGINAL REVENUE

Notes Figure 1 illustrates the speci cation of Example 1 The parameters chosen are uH 5 3 and uL 5 1 for the centers oftwo ldquomarket segmentsrdquo sH

2 5 sL2 5 03 for the variance in each segment and a 5 04 for the relative weight on the higher

segment The sharp drop in inverse demand (and corresponding negative marginal revenue) near z 5 04 corresponds to thedivision between the two segments


C The Cost Side Increasing and DecreasingReturns to Quality

We assume that any rm capable of produc-ing a product of quality qi has access to thesame production technologyPrecisely within aparticular quality level qi there are constantmarginal costs of production ci 0 There areno xed costs of production27 Costs may be re-lated to quality in a number of different waysFor instance we say that the production cost ci

exhibits ldquodecreasing returnsrdquo if both the averagecost of quality ciqi and the marginal cost ofquality (ci1 1 2 ci)(qi1 1 2 qi) are increasingfor all i Of course if the marginal cost ofquality exceeds the average cost then the aver-age cost of quality is clearly increasing Hencede ning c0 5 0

c1

q15

c1 2 c 0

q1 2 q0

c2 2 c1

q2 2 q1

cn 2 cn 2 1

qn 2 qn 2 1

3c1

q1

c2

q2

cn

qn

This simply says that if the marginal cost ofquality is increasing for all i it is necessarilytrue that the average cost of quality isincreasing

Likewise the average cost of quality is de-creasing when the marginal cost of quality isless than the average cost that is when (ci1 1 2ci)(qi1 1 2 qi) ciqi When the average costof quality is decreasing we say there are ldquoin-creasing returnsrdquo to quality

A production technology may easily exhibitboth increasing and decreasing returns For in-stance suppose that any product irrespective ofquality has an unavoidable marginal ldquobuildcostrdquo of c 0 In addition the production costincreases with quality according to the strictlyincreasing and convex function c(q) satisfyingc(0) 5 0 Hence

c i 5 c 1 c~q i 3c i

q i5

c 1 c~q i

q i

For i 1 the convexity of c(q) ensures thatthe marginal cost of quality is increasing Forsmall q however the average cost of qualityciqi is decreasing and exhibits the classic ldquoU-shaperdquo familiar from undergraduate textbooksThere are rst increasing and then decreasingreturns to quality Motivated by this examplewe categorize different cases of interest asfollows

De nition 2 There are increasing returns toquality when ciqi is decreasing for all i There aredecreasing returns to quality when both ciqi and(ci1 1 2 ci)(qi1 1 2 qi) are increasing for all i Theproduction technology is U-shaped if for some kthe average cost of quality is decreasing for i kand the marginal and average costs of quality areincreasing for i k

c1

q1

c2

q2

ck 2 1

q k 2 1

ck

qk

ck 1 1 2 ck

q k 1 1 2 qk

c n 2 1 2 cn 2 2

qn 2 1 2 qn 2 2

cn 2 cn 2 1

qn 2 qn 2 1

H~z 1 zH9~z 5 u 21 2 F~u

f~u

and

laquo~z 5uf~u

1 2 F~u

where u 5 H( z) If the hazard rate f(u )(1 2 F(u )) isincreasing in u then both marginal revenue and the priceelasticity of demand are decreasing in z This is inconsistentwith Example 1 and with greater price sensitivity upon anexpansion to serve the low end of a market Whereas themonotonicity of the hazard rate is a convenient assumptionand holds for many common unimodal distributions (includ-ing the uniform normal etc) it cannot hold if we wish tomodel the presence of multiple demand segments as inExample 1

27 Thus the only asymmetry between rms will be in therange of qualities that they are able to produce Allowing themarginal cost of production to vary across rms does notsubstantially change our analysis With some straightfor-ward modi cations the bulk of our results continue to holdin particular those regarding product line pruning and ght-ing brands


In this case k is the product that minimizes theaverage cost of quality

Notice that U-shaped costs of quality incor-porate increasing and decreasing returns as spe-cial cases The former occurs when k 5 n andthe latter occurs when k 5 1

Inspection of the production technology inobjective terms typically is not enough to de-termine the returns to quality consumer prefer-ences are part of this de nition For examplesuppose that the product in question is a micro-processor and that increases in clockspeed cor-respond to increased quality A consumer withtype u is willing to pay up to uq for a processorof quality q Thus q indexes the monetary valueof the microprocessor and not necessarily itsphysical clockspeed28 In some cases howeverwe can immediately identify the structure ofcosts Such is the case with ldquodamaged goodsrdquo(Deneckere and McAfee 1996) where a rmobtains a low-quality product by intentionallyldquocrimpingrdquo a higher-quality variant Such alow-quality product costs no less to produceand hence the returns to quality are automati-cally increasing

III Monopoly

In this section we derive the optimal productlines for a monopolist We rst consider thecase of multiplicative preferences which servesas a point of comparison when we introducecompetition in Section IV Following that webrie y consider the monopolistrsquos product linechoices with more general preferences

A Monopoly with Multiplicative Preferences

With n different goods available the monop-olistrsquos pro t on product i is simply z i( pi 2 ci)

The monopolist then chooses z i $ 0 to maxi-mize total pro ts i51

n z i( pi 2 ci) across allproducts It is equivalent and much easier in theend for the monopolist instead to choose arange of upgrade supplies Zi These mustrespect the constraint Z i Z i2 1mdashfor instancean upgrade to quality q2 may only be sold toconsumers who purchase the baseline productq1 Using this formulation the monopolistrsquosproblem becomes

max Oi 5 1

n

Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1

subject to Zi Zi2 1 for each i 1 Observethat the ith element of the summation involvesonly the term Z i In fact

(4) Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1

Z i H~Z i 2ci 2 ci 2 1

q i 2 q i 2 1

The last term is equivalent to the pro t fromselling a single good of quality q 5 1 with amarginal production cost of (ci 2 ci2 1)(qi 2qi2 1) If it were not for the constraint Z i Z i2 1 then the monopolist could maximize theobjective function termwise Neglecting thismonotonicity constraint the solution Ziwould satisfy

(5) H~Zi 1 Zi H9~Zi 5c i 2 c i 2 1

q i 2 q i 2 1

The ldquoupgraderdquo reformulation reveals the simpleeconomics of second-degree price discrimina-tion The monopolist sets marginal cost (of in-creased quality) equal to marginal revenue inthe market for each upgrade

Alas this approach ignores the monotonicityconstraint Zi Zi2 1 Imposing this constraintsheds light on the product range offered by themonopolist The simplest case is one of decreas-ing returns to qi so that (ci 2 ci2 1)(qi 2qi2 1) is strictly increasing in i for all i $ 1The unconstrained solutions to equation (5) nat-urally satisfy Zi Zi2 1 and hence zi 5 Zi 2Z i2 1 0 Simply put when there are decreas-ing returns to quality the monopolist offers thefull range of potential product qualities

28 With multiplicatively separable preferences changesin q represent common changes in the proportional willing-ness to pay of all consumers Under increasing returns toquality (when ciqi is decreasing in i) an increase in qualityraises a consumerrsquos willingness to pay proportionally morethan it raises the physical cost of producing the good Thisnotion of increasing returns to quality (and similarly fordecreasing and constant returns) is invariant to the labelingof quality levels since it deals solely with the willingness topay for and the production cost of particular physicalproducts


In contrast when there are increasing returnsto quality provision the monotonicity con-straints bind For simplicity consider the caseof n 5 2 Increasing returns to quality impliesthat c2q2 c1q1 or equivalently (c2 2 c1)(q2 2 q1) c1q1 Suppose that the monop-olist nds it optimal to offer two distinctproducts so that Z1 Z2 Then it must be thecase that

Z1 H~Z1 2 Z2 H~Z2 $c 1

q1~Z1 2 Z2

c2 2 c1

q2 2 q1~Z1 2 Z2

The rst inequality makes use of equation(4) and says that the monopolist does not wishto lower supplies of the baseline product to Z2which it may do without violation of the mono-tonicity constraint The second inequality fol-lows from increasing returns to quality keepingin mind that c0 5 q0 5 0 Combining theresulting strict inequality says that the monop-olist would strictly bene t by raising the supplyof the upgrade q2 2 q1 from Z2 to Z1 Thus theoriginal supplies cannot have been optimal andwe have a contradiction Thus the rm mustoptimally sell a single quality level

This argument naturally extends to the caseof general n In fact if there are increasingreturns to quality everywhere then a monopo-list will wish to set Z1 5 Z2 5 Zn andhence will offer only the highest quality productqn We summarize our results in the followingproposition29

PROPOSITION 1 If the production technol-ogy is U-shaped (De nition 2) with minimumaverage cost for product k then a monopolistoffers in positive supply exactly the n 2 k 1 1products of highest quality In terms of up-grades Z1 5 Z2 5 5 Zk Zk1 1 Zn

Proposition 1 demonstrates the crucial rolethat the shape of average and marginal costs of

quality play in a monopolistrsquos product selectiondecision In the region where average cost isdecreasing it sells a single product It is optimalto segment the market with multiple productsexactly in the region where average cost andmarginal cost are increasing

B Monopoly with General Payoffs

Here we brie y consider more general pref-erences before returning to the multiplicativespeci cation for the remainder of the paper Weask what properties of preferences and costs inthis more general setting lead to price discrim-ination using multiple products To answer thisquestion we suppose that a consumer of type upurchasing a good of quality q at price p enjoysutility u(u q) 2 p where u(u q) is strictlyincreasing in its arguments continuously differ-entiable and satis es u(u 0) 5 u(0 q) 5 0Consumers purchase a single unit of the utility-maximizing product so long as this yields non-negative utility A rst property we need is forthis function to satisfy the familiar sorting con-dition For q2 q1 u(u q2) 2 u(u q1) mustbe increasing in u Equivalently the functionu(u q) is supermodular in u and q30 Thisensures that higher types will purchase higherqualities This in turn implies the upgrade for-mulation of the inverse demand functions con-tinues to hold For n 5 2

p1 5 u~H~Z1 q1

p2 5 p1 1 u~H~Z2 q2 2 u~H~Z2 q1

The sorting condition ensures that price dis-crimination is feasible It does not howeverimply that such discrimination is optimal Toelicit an appropriate condition for optimalitysuppose that the monopolist optimally suppliestwo distinct products so that Z1 Z2 Forsimplicity of exposition we suppose that c1 5c2 5 0 so that there are increasing returns toquality31 It must be the case that

29 The proofs to this and subsequent results are given inthe Appendix

30 A differentiable function u(u q) is supermodular ifshy2ushyushyq $ 0 and log supermodular (when positive) ifshy2 log ushyushyq $ 0

31 In the formal speci cation of our model we assumethat marginal production costs are strictly positive Whenmarginal production costs are zero all of our monopoly


Z1 u~H~Z1 q1 $ Z2 u~H~Z2 q1

and

Z1 u~H~Z1 q2 2 u~H~Z1 q1

Z2 u~H~Z2 q2 2 u~H~Z2 q1

The rst inequality says that the monopolistdoes not wish to reduce supplies of the baselineproduct The second inequality says that themonopolist does not wish to expand supplies ofthe upgrade Dividing the second inequality bythe rst we obtain

u~H~Z1 q2

u~H~Z1 q1

u~H~Z2 q2

u~H~Z2 q1

Since H(Z2) H(Z1) this says that u(uq2)u(u q1) must be increasing in u so thathigher types prefer higher qualities proportion-ally more than lower types Mathematically theutility function u(u q) must be log supermodu-lar in the u and q at least over the range of pricediscrimination Of course this means that logsubmodularity [which means that u(u q2)u(uq1) is decreasing in u] is suf cient to ensure thatno price discrimination takes place Incorporat-ing costs and allowing for n products we ob-tain the following

PROPOSITION 2 Suppose that a monopolist nds it optimal to offer distinct products Thenthe surplus u(u q) 2 c(q) must be log super-modular over some range

To apply this proposition suppose that allqualities may be produced at an identical con-stant marginal cost of c 0 This seems to bean appropriate speci cation for the case of theIBM LaserPrinter since the two versions dif-fered only in their controller card For multipli-cative utility the surplus function satis es u(uq) 2 c(q) 5 uq 2 c By inspection (uq2 2c)(uq1 2 c) is strictly decreasing in u andhence the surplus is log submodular The mo-

nopolist will offer only the higher-quality prod-uct Moving back to the general speci cationu(u q) must be strictly log supermodular toovercome this

IV Duopoly

Here we consider a simple quantity-settingduopoly model Broadly our goal is to charac-terize the equilibrium product lines under avariety of conditions We allow for both de-creasing and nonmonotone marginal revenueand also consider equilibria for environmentsin which the entrant and incumbent haveeither symmetric or asymmetric technologicalcapabilities

Our leading results are as follows Whenmarginal revenue is decreasing ghting brandsnever emerge as optimal weapons in response toentry To the contrary we show that increases inthe entrantrsquos technological prowess tend to in-duce product line pruning of the lower-qualityproducts from the incumbentrsquos line

However moving away from the assumptionof decreasing marginal revenue reveals that ghting brands can emerge as effective tools tomaintain pro tability for the incumbent Thistends to happen when the total output of theincumbent expands following entry Moreoverthe incumbent never offers goods of lower qual-ity than the entrant Thus when competitiondoes lead the incumbent to expand its productline it chooses to maintain at least a weakquality advantage

Our analysis is arranged as follows First weset up the framework for analysis and presentseveral results that hold irrespective of theshape of marginal revenue Next we address thecase in which marginal revenue is everywheredecreasing Finally we expand our analysis toincorporate the possibility of nonmonotonicmarginal revenue and consider again the bi-modal type distribution described in Example 1

A Duopoly Framework and General Results

The n products are supplied by two rmsOur interpretation is that one is the incumbentand the other an entrant They supply xi and yi

units of good i respectively yielding a totalsupply of xi 1 yi The two rms simultaneouslychoose quantities The incumbent is able to pro-

results continue to hold Furthermore all of our duopolyresults continue to hold when we exclude equilibria where a rm offers a supply that oods an entire upgrade marketforcing the price of that upgrade to zero


duce the entire range of qualities The entranthowever is limited to products of quality qmand below for some m n and so yi 5 0 fori satisfying m i n32 As in Section II wede ne the upgrade quantities as follows

X i 5 Oj 5 i

n

x j Y i 5 Oj 5 i

m

y j

where of course Y i 5 0 for i m The upgradesupplies satisfy Xi Xi2 1 and Y i Yi2 1Employing equation (3) these yield pro ts forthe incumbent and entrant of

pI 5 Oi 5 1

n

~q i 2 q i 2 1

3 X iX H~X i 1 Y i 2c i 2 ci 2 1

q i 2 q i 2 1D

pE 5 Oi 5 1

m

~q i 2 q i 2 1

3 Y iX H~X i 1 Y i 2c i 2 c i 2 1

q i 2 q i 2 1D

As in the monopoly case these are conve-nient forms The ith term of each objectivefunction depends only on Xi and Yi Each rmchooses its supplies to maximize its objectivefunction subject to the monotonicity constraintson Xi and Yi We seek pure strategy Nashequilibria in these quantities and use Xi andYi to denote the equilibrium supplies Noticethat absent the monotonicity constraints wewould be able to seek separate Cournot equilib-ria in the supply of each upgrade The monoto-nicity constraints have to be satis ed howeverand they allow us to ascertain the relationshipsbetween the product lines of the two rms

Since the incumbent is active in all the up-

grade markets while the entrant may not be wemight expect that the total production of theincumbent exceeds that of the entrant at andabove any quality level This is indeed the case

We only consider pure strategy Nash equilib-ria throughout our entire paper

PROPOSITION 3 Any (pure strategy) equi-librium entails Xi $ Yi for each i If m 5 nthen Xi 5 Yi for each i

Note that the incumbent need not producemore of any single quality level Rather thetotal supply at and above any particular qualitylevel is greater for the incumbent Furthermorein the absence of any strict quality advantageany (pure strategy) equilibrium must be sym-metric We can push this analysis slightly fur-ther using the techniques of Proposition 3 toobtain the following

PROPOSITION 4 A product i m is sup-plied by the incumbent only if it is supplied bythe entrant

This says that the incumbent will not be ac-tive in any market that the entrant is not activein save potentially for those that the entrant isnot technologically capable of serving One im-plication of this is that the incumbent will neverchoose to offer products that are of qualityinferior to that of the entrantrsquos lowest-qualityproduct

The results so far are useful but do not suc-ceed in characterizing the precise nature of theequilibrium product lines To proceed furtherwe consider separately the cases of decreasing(Section IV subsection B) and nonmonotonicmarginal revenue (Section IV subsection C)

B Decreasing Marginal Revenue andProduct Line Pruning

When marginal revenue is decreasing andreturns to quality are increasing a monopolistsupplies only the highest feasible quality (Prop-osition 1) The entry of a competitor does notalter this

PROPOSITION 5 With increasing returns toquality and decreasing marginal revenue (Def-initions 1ndash2) both rms offer a single highest

32 We could allow for a more general speci cation oftechnological capabilities so that for example the entrantis able to produce some goods that the incumbent cannotWith some straightforward modi cations the bulk of ourresults continue to hold in particular those regarding prod-uct line pruning and ghting brands


available quality product In terms of the up-grade supplies X1 5 5 Xn 5 X and Y1 5 5 Ym 5 Y

Proposition 5 says that the incumbent offersonly product n and the entrant offers only prod-uct m If m 5 n we have a pair of symmetricduopolists selling a single product In contrastif m n the incumbent has a technologicaladvantage The pro t equations reduce to

(6) pI 5 X~qm H~X 1 Y 2 cm

1 X~~qn 2 qm H~X 2 ~cn 2 cm

(7) pE 5 Y~qm H~X 1 Y 2 cm

Heuristically the incumbent has stronger in-centives to produce because it is selling a higher-quality product that is produced at loweraverage cost The equilibrium therefore shouldexhibit X Y

PROPOSITION 6 Suppose marginal revenueis decreasing and there are increasing returnsto quality If m 5 n the rms operate assymmetric duopolists If m n the output levelof the incumbent exceeds that of the entrant sothat X Y In this case X is increasing inqn and decreasing in qm whereas Y is de-creasing in qn and increasing in qm

Proposition 5 implies that when marginalrevenue is decreasing and returns to quality areincreasing the emergence of competition can-not provide an explanation for the introductionof an additional product The reason is that weknow from Proposition 1 that a monopolist fac-ing increasing returns to quality also would sella single product of quality qn The optimalstrategy for the incumbent is to accept the de-cline in prices and continue selling only thehigh-quality good We might suspect that hadthe monopolist been selling multiple productsentry would induce the incumbent to removecertain products in an attempt to maintain mar-gins on the higher-quality goods

To see that pruning can occur in this casesuppose that the average cost of quality is U-shaped (De nition 2) We know from Proposi-tion 1 that the monopolist would offer distinct

products in the range where average cost isincreasing If entry occurs in this case the in-cumbent will tend to (weakly) reduce the num-ber of products offered The following lemma isa rst step in proving this

LEMMA 1 If the production technology isU-shaped (De nition 2) with minimum averagecost for product k and marginal revenue isdecreasing then the incumbent produces nomore distinct goods than it did as a monopolistIf m k then (i ) the entrant offers only productm and (ii) for some h n 2 k 1 1 theincumbent sells the h highest quality productsthat it can produce

There are no ghting brands even with ageneral U-shaped cost structure This resultplaces an upper bound on how many productsan incumbent will offer It is possible to saymore however and characterize precisely theproduct lines of rms in equilibrium at leastwhen the entrant is constrained to offer productsof relatively low quality (ie products in therange where the average cost of quality is de-creasing)33 As will be shown it is suf cient todetermine the lowest quality product offered bythe incumbent This is straightforward since inequilibrium it is as if the incumbent were com-peting with a single product against the en-trantrsquos single product and also selling certainupgrades in independentmarkets Conceptuallythe only question is which product the incum-bent wields against the entrant

Suppose that the incumbentrsquos lowest qualityproduct is r in equilibrium and that m kFrom Lemma 1 we know that r $ k and thatthe entrant supplies only product m To satisfythe appropriate monotonicity constraint it mustbe the case that the monopoly supply of upgrader 1 1 is below the incumbentrsquos duopoly supplyof product r De ne Xrm

dagger and Yrmdagger to be the

equilibrium outputs of the incumbent and en-

33 The entrant is constrained to offer products of rela-tively low quality when m k Johnson and Myatt (2003)consider a related model of quantity competition betweenmultiproduct oligopolists allowing for more general prefer-ences and arbitrary technological limitations They focus onthe equilibrium structure of rmsrsquo product lines as opposedto the product line response of an incumbent rm to entry


trant when they are forced to offer single prod-ucts of quality qr and qm respectively Theobjective functions for this restricted game aregiven by equations (6) and (7) changing the nto r for the incumbent Given that marginalrevenue is decreasing it turns out that there is aunique equilibrium to such a game (Vives1999 p 47) Now we can also consider theunrestricted monopoly supplies of upgradeproducts Zi

dagger for i $ k Increasing marginalcosts of quality for i k ensure that thissequence is decreasing and satis es Zi

dagger 5 Zi where Zi is the output of upgrade i that amonopolist would choose for i k (from theProof of Proposition 1) The following lemmaindicates that the sequences Zr

daggerr5 kn and

Xrmdagger r5k

n satisfy a single crossing property

LEMMA 2 Suppose that marginal revenue isdecreasing Fixing m dene R 5 r Zr

dagger $Xrm

dagger Zr11dagger r $ k If there is no such r then

de ne R 5 n The set R contains a singleelement r for each m k

We will now show that there is a real sensein which r determines the state of competitionin the industry In equilibrium r is the onlyproduct of the incumbent that is in directcompetition with the entrant and beyond thisquality level the incumbent exercises unre-stricted monopoly power over upgrades toquality34 The following result demonstratesthis and shows how r determines the equilib-rium outputs for both rms for all qualitylevels

PROPOSITION 7 Suppose that the produc-tion technology is U-shaped (De nition 2) withminimum average cost for product k and thatmarginal revenue is decreasing Also supposethat the entrantrsquos maximal quality is m k Inequilibrium the incumbent produces distinctproducts r r 1 1 n while the entrantproduces only product m Furthermore Xr m

dagger 5X1 5 5 Xr and for i r the incumbentproduces the same quantity of upgrades as if itwere a monopolist so that Xi 5 Zi 5 Z i

daggerFinally the output of the entrant is Yr m

dagger

When unrestricted duopoly levels of low-quality upgrades fall beneath monopoly levelsof higher-quality upgrades the equilibrium re-action is for the incumbent to remove its lower-quality goods from the market The equilibriumin the industry can therefore be determined byidentifying the product r such that if the incum-bent wields only this product against the en-trant the resulting duopoly supply for theincumbent is just large enough for the monop-oly supply of upgrade r 1 1 to be strictlyfeasible

Even though the incumbent is a monopolistin all markets for quality upgrades greater thanqm (the entrantrsquos highest quality) the monoto-nicity constraint on upgrade outputs must stillbe obeyed Hence in equilibrium the monopolypower of the incumbent is potentially more re-stricted In particular it is restricted to upgradesto quality levels greater than that of product r

This result also suggests that entrants who areable to produce goods of higher quality willcapture more of the market for low-qualitygoods forcing the incumbent out of those mar-kets As we now show this is indeed the caseSo long as the entrant is constrained to offergoods of quality less than qk increases in thequality of the entrantrsquos product tend to lead tothe further elimination of lower-quality prod-ucts by the incumbent

PROPOSITION 8 When marginal revenue isdecreasing the equilibrium number of productsoffered by the incumbent is decreasing in m solong as m k More precisely r is a (weakly)increasing function of m in the region m k

When m k the entrant will offer only asingle product In equilibrium it is as if theincumbent were competing only with a singleproduct itself This result says that the incum-bent nds it optimal to use a higher-qualityproduct in response to a higher-quality entrant

Note that in the region m k the logic ofProposition 8 does not hold There are tworeasons First even if the entrant continued tosell but a single good the effective marginalcost of that product would be increasing withm k This would tend to lower the entrantrsquosoutput and raise the incumbentrsquos which mightwell lead to the reintroduction of a previouslyremoved brand Second the entrant itself will

34 Notice that the incumbent does not hold monopolypower over the complete high-quality product


likely not choose to offer only a single productIncreasing average cost means that choosing tosell a second product for example raises thecost of that good and therefore further lowersthe entrantrsquos output

Thus while entry itself cannot lead the in-cumbent to expand its product line under theassumption of decreasing marginal revenuepronounced increases in the entrantrsquos techno-logical possibilities can have this effect In par-ticular only when the maximal quality of theentrant passes the threshold k where averagecost begins increasing can the incumbent pos-sibly introduce new products However theseare always goods that were sold as amonopolist

In contrast when marginal revenue is non-monotonic the incumbent might choose to sellproducts in the face of competition that it wouldnot sell as a monopolist We now examine thiscase

C Nonmonotonic Marginal Revenue andFighting Brands

In Section II subsection B we argued thatthe presence of multiple sources of demandmight generate a multimodal distribution for uand nonmonotonicmarginal revenue This leadsto the possibility that a suf ciently large incur-sion by an entrant may raise marginal revenueand hence lead an incumbent rm to expand itsown output

The existence of such expansionary pressureon an incumbentrsquos output leads to a novel ex-planation for the introductionof ghting brandsWhen an incumbent rm holds a quality advan-tage an entrant is unable to offer upgrades tohigher-quality levels and so the expansionarypressure is only felt at lower-quality levels Theincumbent expands output at such lower levelswhile continuing to restrict its output of higher-quality products This output expansion mani-fests itself as a ghting brand whereby theincumbent can compete for new customers(whom it would not have served as a monopo-list) while protecting the markup on its high-quality products

To explore this intuition in more depth sup-pose that the incumbent holds a strict qualityadvantage over the entrant so that n m andthat as a monopolist it would not sell product m

In terms of upgrades this means that the incum-bentrsquos output satis es Xm 5 Xm1 1 absentcompetition

Following entry the incumbent faces compe-tition in the upgrade market to quality m Thatis the incumbent will choose some Ym 0which in equilibrium will lead the incumbent toadjust its own upgrade level in that market Asalready noted and as we show by examplebelow it is possible that the equilibrium re-sponse is for the incumbent to increase its out-put in that market relative to pre-entry levels sothat its nal choice is some Xm Xm

It is important to note however that whilethe incumbent faces competition in upgrademarket m it faces no competition for the supplyof upgrades from qm to qm1 1 More preciselythe optimal choice of upgrade m 1 1 for theincumbent neglecting the monotonicity con-straint Xm $ Xm1 1 is unaffected by the com-petitorrsquos presence If this optimal choice liesbelow the duopoly supply of the product ofquality qm then Xm Xm1 1 in equilibriumThat is a ghting brand of quality qm emergesfollowing entry

It may be helpful to contrast this logic withthat from Section IV subsection B For Propo-sitions 7 and 8 we argued that even when theincumbent faces no direct competition in up-grade markets above m its optimal choices ofsuch upgrades are still potentially in uenced bythe entrant through the need to obey the mono-tonicity constraints on the upgrades This isparticularly important when marginal revenue isdecreasing since the presence of competitionwill lead the incumbent to restrict Xm therebypotentially causing more constraints to bindwhich is equivalent to the elimination of lower-quality goods Abandoning the decreasing mar-ginal revenue assumption the key conceptualdifference is that entry may lead to an expansionin Xm by the incumbent thereby causing fewerupgrade constraints to bind This leads naturallyto the introduction of new lower-qualityproducts

We now present two concrete examples of ghting brands In both cases the intuition thatwe wish to convey is as described above Whenentry leads an incumbent to expand total outputit may be optimal for it to introduce a newproduct of lower quality

We begin by exhibiting a nonmonotonicity in


the Cournot reaction functions for a single prod-uct of quality q 5 1 and marginal cost c 5 0generated by Example 1 [see Figure 2(a)] As amonopolist a rm restricts output to serve onlythe upper market segment centered at uH Thatis the monopoly output is Z 5 0377 and wecan see from Figure 1(b) that this corresponds toserving only the higher segment of the under-lying distribution of consumers A small incur-sion by a competitor results in a contraction ofoutput in the standard waymdashthe reaction func-tion is initially decreasing If the incursion issuf ciently large however then the incumbentwill wish to expand output and serve the lowermarket segment (the one around uL) This re-sults in a single jump upward in the reactionfunction which decreases smoothly again there-after Inspecting Figure 2(a) notice that thesymmetric duopoly quantity satis es X 5 Y5 0437 In other words the incursion of acompetitor prompts the incumbent to expand itssupply beyond the monopoly quantity

We now show how a ghting brand can ariseunder this demand structure To this end as-sume that the incumbent has two products both

of which can be produced at zero cost withqualities q1 5 1 and q2 5 2 From Figure2(a) note that unconstrained monopoly supplyof each upgrade is given by the best response ofthe incumbent to a zero supply by the entrantand is equal to 0377 for each upgrade Hencea monopolist would set Z1 5 Z2 5 0377 ineffect selling only the high-quality good

Now suppose that entry occurs by an entrantonly capable of producing good 1 Figure2(a) reveals that if each rm were to only sellproduct 1 there would be an equilibrium inwhich each rm produces 0437 When the in-cumbent can offer both products and sinceZ2 5 0377 0437 there is an equilibriumin which the incumbent responds to entry byraising its output of upgrade 1 resulting in anincrease in total output In particular the outputof product 2 remains constant but the incum-bent raises its output of product 1 from 0 to006 5 0437 2 0377 Equivalently the incum-bent serves 6 percent of consumers with a low-quality ghting brand

Our second example is given by a discreteversion of the bimodal Example 1 which

FIGURE 2 COURNOT REACTION FUNCTIONS FOR EXAMPLE 1

Notes These are reaction functions for the Cournot duopoly selling a single good manufactured at zero marginal cost Bothcon gurations involve uL 5 1 The demand speci cation is taken from Example 1 and for Figure 2(a) is identical to thatused in Figures 1(a) and 1(b)


corresponds to letting the standard deviations ofthe two normal distributions fall to zero so thatsH 5 sL 5 0

Example 2 There is a mass a of types uH anda mass 1 2 a of types uL with uH uL Twoqualities q2 q1 are both produced at zerocost

Proposition 1 is valid in this discrete modelas inspection of its proof reveals Since theproduction cost for both quality levels is zerothere are increasing returns to quality35 and soa monopolist will wish to offer only the high-quality product It is optimal to restrict supplyand sell only to the high types when a uLuH

Consider next a pair of duopolists each offer-ing a single product q1 In a symmetric equilib-rium they serve either the entire market at aprice of uLq1 or restrict to the fraction a of hightypes at a price uHq1 When a 12 there is asymmetric equilibrium in which the entire mar-ket is served36 The reason is that if one rm isselling 05 units the other rm cannot raise theprice even it were to drop its output to zeroHence it would be best off also selling 05 units

Therefore when uLuH a 1 2 is satis- ed we can nd an equilibrium in which theincumbent releases a ghting brand In particu-lar entry leads the incumbent to expand itsoutput of upgrade X1 from a to 05 whilekeeping its output of upgrade X2 constant at aAs a result the incumbent introduces the low-quality product as a ghting brand in supply05 2 a

The two examples of ghting brands just pre-sented are representative of a more general resultFormally we have the following proposition

PROPOSITION 9 Consider the restricted duo-poly game in which each rm can offer only upto quality level qm for some m n Fix anequilibrium of this game in which the incum-bentrsquos production of upgrade m is Xm Supposethat

Xm maxm i n

5 arg maxX

XX H~X 2ci 2 c i 2 1

q i 2 q i 2 1D 6

Then there is an equilibrium of the unre-stricted duopoly game in which the incum-bent offers multiple products In particu-lar the incumbent sells product m so thatXm1 1 Xm

To understand how this proposition relates to ghting brands suppose that in the absence ofcompetition the incumbent would not sell prod-uct m We know from Proposition 1 for exam-ple that this would be the case if the averagecost of quality were decreasing for product mas in De nition 2 When the hypotheses ofProposition 9 are satis ed however product memerges as a ghting brand37

An alternative viewpoint is as follows Wecould expand our model to allow for the possi-bility that a rm could innovate allowing it toproduce goods of higher qualities We mightask whether the innovating rm would continueto sell its old product To address this possibil-ity suppose that each rm is initially able tooffer only quality levels at or below qmmdashthat iseven the incumbent cannot produce the highestquality products Proposition 6 ensures that anypure strategy equilibrium is symmetric andhence Xm 5 Ym Imagine now that the incum-bent rm bene ts from a successful productinnovation which permits it to offer all prod-ucts up to quality qn If the condition of Prop-osition 9 is satis ed then the unconstrainedsupplies of upgrades m 1 1 n lie belowXm In this case the incumbent will sell notonly products that it has newly developed butalso its older lower-quality products In con-trast if marginal revenue is everywhere de-creasing then the lower-quality products wouldbe removed following the innovation

We close with a technical remark regarding ghting brands Proposition 9 assumes that the

35 We de ne increasing returns to quality as decreasingc iqi which clearly is constant if ci 5 0 for all i Howeverinspecting the proof of Proposition 1 reveals that in fact amonopolist would also sell only the highest quality goodwhen all costs are zero

36 There are multiple equilibria as a result of the discretetypes We focus on the symmetric one since in the continuous-type version of this example equilibrium outputs would besymmetric We are also ignoring equilibria in which there isstrictly excess supply even though zero marginal costsimply that equilibria in which each rm is ooding themarket with its own output may exist (albeit in weaklydominated strategies)

37 In fact it is possible that other products of qualitiesexceeding qm also emerge as ghting brands


incumbent has a quality advantage over theentrant In fact however ghting brands mayarise when the incumbent and entrant have iden-tical technological capabilities When marginalrevenue is nonmonotonic a Cournot quantity-setting game may well have multiple equilibriaIt is possible therefore for two rms to coor-dinate on one equilibrium in the market for abaseline product and a second equilibrium in themarket for an upgrade This idea is illustrated inFigure 2(b) When costs are zero the payofffunctions in the baseline and upgrade market fortwo quality levels are proportional to eachother Hence it is possible for the duopolists toplay the low-output equilibrium in the upgrademarket and the high-output equilibrium in thebaseline market However note that this multi-plicity of equilibria is not required for ghtingbrands to emerge In fact neither example pre-sented above leans upon the presence of multi-ple equilibria

V Discussion

We point to price-discrimination-based rea-sons for a rm to expand or contract its productline in the face of entry Thus we nd noveltheoretical support for both the use of ghtingbrands and the practice of pruning product linesin order to focus on quality two commonly ob-served strategies Moreover we show how theshape of marginal revenue plays a critical role

We now turn to a number of empirical exam-ples that help to illustrate our ideas

A Computer Hardware

The Intel 80486SX microprocessor and theIBM LaserPrinter 4019E are examples of ght-ing brands in the computer hardware industry38

Both of these examples feature prominently in

the work of Deneckere and McAfee (1996)They analyze the phenomenon of damagedgoods where a monopolist intentionally (andperhaps at some cost) damages a high-qualitygood hence enabling price discriminationThey offer two models The rst employs mul-tiple type dimensions and generates relativelyweak conditions for the damaged goods phe-nomenon The second considers a single typedimension and the authors show that a condi-tion that is equivalent to log supermodularity ofpreferences is needed to generate the damagingof goods They comment on the strict nature oftheir condition

Reviewing the examples in more detail isworthwhile In early 1991 Intel released the80486SX microprocessor This chip was a mod-i ed version of the earlier 80486 subsequentlyrenamed 80486DX The sole difference was theomission of an internal oating point mathemat-ics coprocessor yielding an initial pricing pointof $258 relative to the 486DX price of $588Interestingly the industry literature recognizedthat the 486SX was a damaged version of the486DX

In a move aimed at replacing the 386DXas the midrange processor of choice Intelhas launched the 486SX the 486SX issimply a 486 without the oating-pointunit In fact the initial silicon includes theFPU but it has been disabled (MichaelSlater 1991 p 1)

Interestingly the release of the 486SX fol-lowed the entry of Advanced Micro Devices(AMD) into the 386DX market3940 The 486SXwas therefore a ghting brand Slater (1991)agreed predicting that ldquoAMD may be forcedto lower 386DX prices signi cantly to maintainmomentum in that marketrdquo It would appear thatthe presence of competition may have in u-enced Intelrsquos decision to expand its product line

38 There is a growing empirical marketing literature onthe product line decisions of rms For example William PPutsis Jr and Barry L Bayus (2001) investigate productline expansions and contractions in the personal computerindustry in the years 1981ndash1992 They jointly consider theempirical determinants both of the decision to change aproduct line and the magnitudes of such changes Amongother results they nd that a rm is more likely to prune itsproduct line when the number of new products in theindustry is higher

39 ldquoAMDrsquos 486 Clone Ready But Will It Sellrdquo Com-puter World March 1991

40 The 80386DXSX series was an earlier generation ofx86 architecture microprocessors that omitted the internalcache feature of the 486 series AMD previously manufac-tured 386 generation processors under licence Followinglengthy litigation they were able to continue to produce thedesign independently (ldquoThe Chips Are Downrdquo PC UserJanuary 1991)


The release of the IBM LaserPrinter 4019Eprovides a similar example This device was aslower version of the IBM 4019 (5 ppm versus10 ppm) The only reason it was slower was thatthe controller card made it so Deneckere andMcAfee (1996 p 170) view this as a secondexample of a relationship between product dam-aging and the presence of competition

At least two of the damaged goods theIntel 486SX and the IBM LaserPrinter Eappear to have been introduced in re-sponse to competition by another pro-ducer This is dif cult to explainparticularly in the LaserPrinter case

In both of these cases entry by a lower-quality competitor induced the introduction (bythe incumbent) of a new product that was infe-rior to the incumbentrsquos but (at least weakly)superior to the entrantrsquos This exactly re ectsour results in Section IV We arrive at thefollowing explanation for say the IBM caseInitially IBM was content to serve only high-value business customers with the LaserPrinterThe entry of the HP LaserJet IIP forced it toserve home and low-value business users Nev-ertheless it retained monopoly power on a qual-ity increase from 5 ppm to 10 ppm and wishedto deliver this quality premium only to high-value customers Hence it introduced thecrimped LaserPrinter E to execute the strategyof serving a broader market while still maintain-ing a markup on its premium product

B Airlines

Air travel is a canonical example of second-degree price discrimination The UK carrierBritish Airways (BA) provides examples of ex-pansion and contraction in its product lineBArsquos former CEO Robert Ayling engaged inan explicit strategy of reducing the economyclass capacity on long-haul routes and ex-panding the business class and rst class pro-vision In other words BA chose to focus onquality

Now the focus has moved to presentingproducts that Mr Ayling hopes will at-tract higher numbers of premium custom-ers the executives and travellers willing

to pay higher prices for premium servicesand facilities If the plan works there willbe little space left for the passengerswanting to travel on deeply discountedeconomy class tickets41

Interestingly however BA adopted a differ-ent strategy in the European short-haul marketincluding the market for UK domestic ightsIt introduced a ghting brand the ldquono-frillsrdquosubsidiary Go4243

A possible explanation for this stems fromMarch 1995 with the incorporation of the easy-Jet airline by Stelios Haji-Ioannou the son of aGreek shipping magnate easyJet operates fromairports in the United Kingdom on a no-frillsbasis similar to that of Southwest Airlines inthe United States In common with other oper-ators of this kind it offers a ticketless servicedevoid of complimentary in- ight meals andother nonessential features44

Media reports at the time suggested that BAhad predatory intentions45 Was the entry of Goa predatory move by British Airways If so ithas failed as easyJet has continued to expand inthe no-frills market Our analysis offers an al-ternative explanation for this event It is possi-ble that BA was initially reluctant to enter thismarket segment due to the anticipated negativeeffect on its core operations However follow-ing the creation of easyJet this segment wasopened up and BA thus found it pro table toenter

41 ldquoBA Aims at Business Traveller with Revamp ofAircraft Fleetrdquo Financial Times (London) February 12000

42 ldquoBArsquos New Cut-Price Airline Hots Up Sky WarsrdquoThe Independent (London) October 1997

43 This behavior has also been seen in other marketsDavid Haugh and Tim Hazledine (1999) describe events inthe Trans-Tasman air travel market Following the entry ofa former charter airline Kiwi International Air New Zea-land launched a no-frills subsidiary Freedom Air as a ght-ing brand in this market

44 The no-frills market has also seen the success ofRyanair These carriers tend to use secondary airports suchas Luton and Stansted for London ights rather than themainstream Heathrow and Gatwick Arguably the betterconnection opportunities at these latter airports are the maincomponent of increased quality for full-service carriers

45 ldquoBA Sets Up No-Frills Airline Budget Flight RivalsFear Service Aims to Wipe Out Competitionrdquo FinancialTimes (London) November 18 1997


C The Market for Watches

In 1998 Timex and Titan ended a joint ven-ture in the Indian market for watches The orig-inal purpose of the joint venture was to captureeconomies of scale in retail and distributionand also to split the market for watches Titanwas to serve the high end of the market andTimex the low end46

Timex and Titan together were able to dom-inate much of the market in India47 Howeverthe alliance proved unstable in part becauseeach felt the other was cannibalizing sales byoffering products not in line with the originaldemarcation of the market With the terminationof the joint venture each rm moved aggres-sively Timex launched the high-end Vistabrand while Titan introduced the Sonata brandin the under Rs 1000 price range

The Sonata was immensely successfulquickly becoming the second best-selling watchbrand in India following the parent brand Ti-tan48 In 2001 Timex announced that it wouldexit the low-end watch market in India by phas-ing out most of its under Rs 1000 watches Itsnew strategy would be to focus on becoming atrendier ldquosports and technology brandrdquo target-ing its watches at the Rs 1000ndash5000 range49

Since both Timex and Titan are capable ofproducing watches across a very broad qualityrange it might seem that the exit of Timex fromthe low-end market is inconsistent with Propo-sition 3 which states that rms with identicaltechnologies should offer products in the samerange Even if we imagine that Titan has anadvantage in the high end our Proposition 4seemingly implies that Titan should not sellproducts of lower quality than Timex Howeverin this situation it seems likely that in factTitan was at a variable cost advantage in thelow-end segment Under the terms of the orig-

inal joint venture Titan (which is based in In-dia) was in charge of all distribution for bothTimex and Titan watches Following the termi-nation of the relationship Timex had to estab-lish its own channels It initially fought tomaintain its position in the low-end market butthis was complicated by the fact that the low-end market is concentrated in many smallerurban and rural areas which might have made itmore costly for Timex to distribute there com-pared to Titan which already had an establishednetwork

The possibility that Titan has an edge indistribution is attested to by Titan vice-chairmanand managing director Xerxes Desai who saidthat ldquoa strong grip on distributionrdquo has contrib-uted to the success of Titan50 If true this po-tentially reconciles the asymmetric productofferings of Timex and Titan since in the mar-ket for basic low-end watches Titan would pro-duce more baseline upgrades in equilibriumthan Timex Hence it could well be the casethat Timex would prune its product line follow-ing the entry of Titan in that segment and thatTitan would maintain a presence there follow-ing the end of the joint venture and exit ofTimex from the segment

D Concluding Remarks

Fighting brands and product line pruning arepervasive and widespread responses to intensi- ed competition We have proposed a model ofmultiproduct quality competition analysis ofwhich provides an integrated explanation forboth phenomena In particular we consideredthe product line response of an incumbent toentry by another multiproduct rm

The equilibrium product lines of rms werederived under a variety of conditions Whetheran incumbent will choose to expand or contractits product line depends on whether entryprompts it to expand or contract its total outputThis in turn is connected closely to whethermarginal revenue is everywhere decreasing ornot

When marginal revenue is everywheredecreasing entry induces a restriction in the

46 ldquoStitch in Timexrdquo The Economic Times June 282000

47 ldquoThe joint venture did extremely well leveraging thestrengths of both partners While Titan chalked up a marketshare of 60 percent in the premium segment Timex literallymonopolised the low end of the marketrdquo The EconomicTimes June 28 2000

48 ldquoThe Changing Face of Titanrdquo The Strategist August24 1999

49 ldquoTimex Corp to Hike Stake in Indian Armrdquo TheEconomic Times January 14 2002

50 ldquoClash of the Titansrdquo The Times of India December13 2000


output of the incumbentrsquos low-quality productsThus entry can lead naturally to the incum-bentrsquos exit from the lower markets therebypruning its line of products In this case there isa real sense in which the incumbent uses but asingle product to compete against the entrantwhile enjoying unrestricted monopoly power inthe markets for upgrades to higher quality Iden-tifying which product the incumbent will useagainst the entrant in equilibrium essentiallydetermines the entire equilibrium structure ofthe industry

When marginal revenue is increasing in someregions it may be optimal for the incumbent toexpand output in response to entry If this is thecase the incumbent may decide to expand intoa lower-market segment by using a low-quality ghting brand thereby allowing it to be com-petitive in that segment while preserving mar-gins on its high-quality good This is possiblewhen the incumbent maintains a technologicaladvantage over the entrant since it thereforecontinues to hold market power over higher-quality upgrades We showed that if the incum-bent does offer new products it will neveroperate at quality levels inferior to those of the

entrant We also argued in Section II that de-mand structures exhibiting increasing marginalrevenue in some regions are not pathological Inparticular simple bimodal consumer-type dis-tributions naturally can generate nonmonotonic-ity in marginal revenue

Introducing ghting brands may therefore bethe optimal strategy when the incumbent wishesto expand its total output following entry Prod-uct line pruning may be the optimal strategywhen the incumbent wishes to restrict its totaloutput following entry

From a technical standpoint a general up-grades approach has been presented that yieldsa powerful analytical framework We alsoprovided a new condition on costs and pref-erences that determines when a monopolistwould choose to pursue market segmentationopportunities

While we have performed no substantive em-pirical analysis we have presented examplesfrom numerous industries that seem to resonatewell with our theory Our theory of ghtingbrands and product line pruning provides oneexplanation for the observed product line ex-pansions and contractions in these industries

APPENDIX

Elements of the proofs to Lemma 1 and Propositions 2 4 5 and 9 are contained in a furthertechnical Appendix available from the AER web site (httpwwwacaweborgaercontents) or fromthe authors on request

PROOF OF PROPOSITION 1Suppose that the lowest quality product i supplied by the monopolist satis es i k It must be

the case that Z1 5 Z2 5 5 Zi Zi1 1 Product i is therefore the effective ldquobaselinerdquo productFollowing the argument given in the text

Zi H~Zi 2 Zi 1 1 H~Zi 1 1 $c i

q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1

The rst inequality says that the monopolist does not wish to lower supplies of the baseline productto Zi1 1 The second inequality follows from the fact that i k But this means that it would bebetter to raise the supply of the upgrade i 1 1 to Zi and hence the original con guration was notoptimal We conclude that Z1 5 Z2 5 5 Zk Consider the monopolistrsquos problem with the relaxedconstraint that Z1 5 Z2 5 5 Zk Increasing average and marginal costs imply that thequality-normalized costs [ie (ci1 1 2 ci)(qi1 1 2 qi)] are increasing for i $ k Hence ignoring


the monotonicity condition we immediately see that Zk Zk1 1 Zn occurs naturally Sincethis relaxed solution satis es the monotonicity constraint it solves the maximization problem

PROOF OF PROPOSITION 2Suppose that the monopolist offers distinct products and that product i is the lowest-quality

product offered so that Z1 5 Z2 5 5 Zi Zi1 1 As in the proof of Proposition 1 product iis effectively the baseline product and hence we may relabel it as i 5 1 The argument given in thetext upon the inclusion of costs yields the required proof

PROOF OF PROPOSITION 3If n m then this proposition is true by assumption for all i m For i m such that Xi 5

Xi1 1 and Yi 5 Yi1 1 amalgamate neighboring upgrades by viewing upgrades i and i 1 1 as acombined upgrade Following this suppose that the proposition does not hold for some i m Takethe lowest such i so that Yi Xi Either i 5 1 or i 1 and Xi2 1 $ Yi2 1 $ Yi Xi In eithercase the upward monotonicity constraint on the incumbent is not locally binding The incumbentcannot have a strict incentive to locally raise Xi and thus has a weak incentive to lower it Thisimplies that the entrant must have a strict incentive to lower Y i

H~Xi 1 Yi 1 Yi H9~Xi 1 Yi H~Xi 1 Yi 1 Xi H9~Xi 1 Yi

ci 2 ci 2 1

qi 2 qi 2 1

The rst inequality follows from Yi Xi and H9(Xi 1 Yi) 0 and the second from the weakincentive for the incumbent to lower Xi Since there is an incentive for the entrant to lower Yi themonotonicity constraint must bind and hence Yi 5 Yi1 1 Xi1 1 We have amalgamated allidentically supplied neighboring upgrades and therefore we know that Xi Xi1 1 Turning toupgrade i 1 1 the upward monotonicity constraint on the incumbent is not locally binding and wemay repeat our argument until we conclude that the entrant has a strict incentive to lower Ym Butthere is no constraint on downward movement of Ym and thus we have a contradiction

PROOF OF PROPOSITION 4Use an identical approach to Proposition 3


PROOF OF PROPOSITION 6When m 5 n the result is obvious For m n and when both marginal revenue and returns to

quality are decreasing the reaction functions are continuously decreasing with absolute slope of lessthan 1 Under these standard conditions the incumbentrsquos reaction function intersects the entrantrsquosfrom above and there is a unique equilibrium (Vives 1999 p 47) The incumbentrsquos reaction functionis an outward shift of the inverse of the entrantrsquos reaction function This ensures that X YFinally note that an increase in qn pushes the incumbentrsquos reaction function outward To see thisnote that we may replace pI from equation (6) by pI where

pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn

Hence a marginal expansion in quantity yields


shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn

This is increasing in qn (due to decreasing marginal revenue and returns to quality) pushing theincumbentrsquos reaction function outwards Similar operations using qm yield the desired results

PROOF OF LEMMA 1Use an identical approach to Propositions 1 and 5

PROOF OF LEMMA 2Suppose not so that both s and r are members of the set R and satisfy s r $ k $ m It follows

that Xrmdagger Zr1 1

dagger $ Zsdagger $ Xsm

dagger so that Xrmdagger Xsm

dagger In other words an increase in the incumbentrsquosquality level results in a drop in its output Write pIr(X Y) for the incumbentrsquos pro t with qualityr and similarly for quality s Observe that

shyp Is ~X smdagger Y sm

dagger

shyX5

shypIr ~X smdagger Y sm

dagger

shyX1 ~qs 2 qr H~X sm

dagger 2c s 2 cr

q s 2 q r1 X sm

dagger H9~X smdagger

$shypIr~Xsm

dagger Ysmdagger

shyX1 ~qs 2 qr H~Xsm

dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

The rst inequality follows from the assumption that marginal cost is increasing beyond k Thesecond follows from the fact that Zs

dagger $ Xsmdagger and hence Xsm

dagger is below the pro t-maximizing supplyof upgrade s This means that the reaction function with quality qr must lie weakly below that forquality qs evaluated at Xsm

dagger But this of course means that Xrmdagger Xsm

dagger and we have reached acontradiction

PROOF OF PROPOSITION 7By assumption m k and hence Y1 5 5 Ym and X1 5 5 Xk from Lemma 1 Suppose

that for r $ k we have X1 5 5 Xr Xr1 1 A local increase in Xr1 1 must (weakly) lower pro tson upgrade r 1 1 The upgrade pro t functions are concave and hence Xr1 1 $ Zr11

dagger where Zr11dagger

is the unrestricted monopoly supply of this upgrade Of course if this inequality holds then it mustbe optimal to set Xi 5 Zi

dagger for all i r We have X1 5 5 Xr and hence it must be the case thatthe incumbent produces the duopoly supply of product r and hence Xr 5 Xrm

dagger We need to checkthat it cannot do better by varying Xr downwards For this we need Xr Zr

dagger Hence r must satisfyZr $ Xrm

dagger Zr1 1 or equivalently r 5 r

PROOF OF PROPOSITION 8Take r and observe that by de nition Xim

dagger 2 Zi1 1dagger 0 for all i such that k i r 2 1 Now

suppose the entrant can offer good m 1 1 k The effective marginal cost of the entrant is lowerin the hypothetical game where it sells only product m 1 1 and the incumbent sells only some goodi Now decreasing marginal revenue together with constant marginal costs is suf cient to imply thatthe resulting hypothetical outputs of the incumbent are lower when the entrant sells good m 1 1given that the incumbentrsquos reaction curves intersect the entrantrsquos curve from above That is Xi(m11)

dagger

Ximdagger for all i This obviously implies that Xi(m1 1)

dagger 2 Zi1 1dagger 0 for all i such that k i r 2

1 which proves the result


PROOF OF PROPOSITION 9Throughout we assume that all of our maximization problems have a unique solution Generically

this will be the case but the proof can be easily modi ed to allow multiple maxima For m i n suppose that the set of upgrade supplies Xj solve the problem

max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1

q i 2 qi 2 1D subject to Xm 1 1 $ $ Xn

If Xm1 1 Xm then this yields an equilibrium of the unrestricted games The incumbent achievesthe unconstrained maximum pro ts on upgrades above product m Its pro ts on upgrades m andbelow are maximized since we began with an equilibrium of the original restricted duopoly gameSimilarly the entrant is maximizing given the supplies of the incumbent To verify the propositionwe need only check

Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6

The remainder of the proof follows a similar approach to our other results

REFERENCES

Brander James A and Eaton Jonathan ldquoProd-uct Line Rivalryrdquo American Economic Re-view June 1984 74(3) pp 323ndash34

Champsaur Paul and Rochet Jean-CharlesldquoMultiproduct Duopolistsrdquo EconometricaMay 1989 57(3) pp 533ndash57

De Fraja Giovanni ldquoProduct Line Competitionin Vertically Differentiated Marketsrdquo Inter-national Journal of Industrial OrganizationMay 1996 14(3) pp 389ndash414

Deneckere Raymond J and McAfee R PrestonldquoDamaged Goodsrdquo Journal of Economicsand Management Strategy Summer 19965(2) pp 149ndash74

Eaton B Curtis and Lipsey Richard G ldquoTheTheory of Market Preemption The Persis-tence of Excess Capacity and Monopoly inGrowing Spatial Marketsrdquo Economica May1979 46(182) pp 149ndash58

Gabszewicz Jean Jaskold Shaked Avner Sut-ton John and Thisse Jacques-Francois ldquoSeg-menting the Market The MonopolistrsquosOptimal Product Mixrdquo Journal of EconomicTheory April 1986 39(2) pp 273ndash89

Gabszewicz Jean Jaskold and Thisse Jacques-Francois ldquoPrice Competition Quality and In-come Disparitiesrdquo Journal of EconomicTheory June 1979 20(3) pp 340ndash59

ldquoEntry (and Exit) in a DifferentiatedIndustryrdquo Journal of Economic TheoryApril 1980 22(2) pp 327ndash38

Gal-Or Esther ldquoQuality and Quantity Compe-titionrdquo Bell Journal of Economics Autumn1983 14(2) pp 590ndash600

ldquoDifferentiated Industries without En-try Barriersrdquo Journal of Economic TheoryDecember 1985 37(2) pp 310ndash39

Gilbert Richard J and Matutes Carmen ldquoProd-uct Line Rivalry with Brand DifferentiationrdquoJournal of Industrial Economics September1993 41(3) pp 223ndash40

Haugh David and Hazledine Tim ldquoOligopolyBehaviour in the Trans-Tasman Air TravelMarket The Case of Kiwi InternationalrdquoNew Zealand Economic Papers June 199933(1) pp 1ndash25

Johnson Justin P and Myatt David P ldquoMulti-product Cournot Oligopolyrdquo University ofOxford Department of Economics DiscussionPaper No 145 February 2003

Judd Kenneth L ldquoCredible Spatial Preemp-tionrdquo RAND Journal of Economics Summer1985 16(2) pp 153ndash66

Keller Kevin L Strategic brand managementUpper Saddle River NJ Prentice Hall 1998

Kotler Philip ldquoPhasing Out Weak ProductsrdquoHarvard Business Review MarchndashApril1965 43(2) pp 108ndash18


Mussa Michael and Rosen Sherwin ldquoMonopolyand Product Qualityrdquo Journal of EconomicTheory August 1978 18(2) pp 301ndash17

Porter Michael E Competitive strategy Tech-niques for analyzing industries and competi-tors New York Free Press 1980

Putsis William P Jr and Bayus Barry L ldquoAnEmpirical Analysis of Firmsrsquo Product LineDecisionsrdquo Journal of Marketing ResearchFebruary 2001 38(1) pp 110ndash18

Quelch John A and Kenny David ldquoExtendPro ts Not Product Linesrdquo Harvard Busi-ness Review SeptemberndashOctober 199472(5) pp 153ndash60

Salant Stephen W ldquoWhen Is Inducing Self-Se-lection Suboptimal for a Monopolistrdquo Quar-terly Journal of Economics May 1989104(2) pp 391ndash97

Shaked Avner and Sutton John ldquoRelaxing PriceCompetition Through Product Differentia-

tionrdquo Review of Economic Studies January1982 49(1) pp 3ndash13

ldquoNatural Oligopoliesrdquo EconometricaSeptember 1983 51(5) pp 1469ndash83

Slater Michael ldquoIntelrsquos 486SX Aims to Dis-place 386DXrdquo Microprocessor Report May1991 5(8) p 1

Stokey Nancy ldquoIntertemporal Price Discrimina-tionrdquo Quarterly Journal of Economics Au-gust 1979 93(3) pp 355ndash71

Stole Lars A ldquoNonlinear Pricing and Oligop-olyrdquo Journal of Economics and Manage-ment Strategy Winter 1995 4(4) pp 529ndash62

Verboven Frank ldquoProduct Line Rivalry andMarket Segmentationrdquo Journal of IndustrialEconomics December 1999 47(4) pp 399ndash425

Vives Xavier Oligopoly pricing CambridgeMA MIT Press 1999


tion led Usha to introduce a discount fan calledthe Racer8

Product line pruning is also common9 Timexrecently announced that it would be removing anumber of its lower-priced watches from theIndian market in response to growing competi-tion in the low end from Titan10 while Mitsu-bishi announced the phasing out of low-endversions of its Trium mobile phones in responseto a supply glut in that segment11 And aftermore than a century in the piano business Kim-ball International in 1996 discontinued all butits prestigious Bosendorfer model following thecapture of the low and middle markets by Jap-anese and Korean rms12

One feature common to these (and manyother) examples is that the increase in compe-tition manifested itself in the low end of themarket Since the incumbentrsquos decision to eitherexpand or contract its product line amounts to adecision about how to segment the marketthese examples suggest that understanding howprice discrimination opportunities change withthe presence of low-end competition might ex-plain the prevalence of both ghting brands andproduct line pruning Our main results implythat this is indeed the case

Consider for instance the IBM Laser-Printer13 A single version was initially soldcapable of printing ten pages per minute Theabsence of a lower-quality version suggests thatIBMrsquos gains from serving the low end of themarket were not large enough to justify intro-ducing a substitute product for its high-qualityunit (which would have limited IBMrsquos ability toextract surplus from high-value users) How-ever following Hewlett-Packardrsquos entry intothe market with its LaserJet IIP a lower-quality

substitute for IBMrsquos LaserPrinter IBM neededto reevaluate its product line strategy On onehand Hewlett-Packardrsquos entry meant that a sub-stitute for IBMrsquos LaserPrinter was already onthe market Inasmuch as a desire to avoid offer-ing such a substitute restrained IBM in the rstplace it might have been sensible to introduceone following Hewlett-Packardrsquos move On theother hand even though a lower-quality substi-tute would be on the market regardless of IBMrsquosdecision introducing its own such productwould have exacerbated (from IBMrsquos perspec-tive) the substitution possibilities for consum-ers In fact IBM decided to introduce a ghtingbrand the LaserPrinter E which was identicalto its original LaserPrinter except for the factthat its software limited its printing to ve ratherthan ten pages per minute This suggests that thedesire to compete in the newly opened low-endmarket was strong enough that IBM was willingto bear the additional erosion on its high-endpro ts resulting from its own entry into thatmarket

However as noted above not all rms mimicIBMrsquos strategy of expanding its product line inresponse to low-end competition some rmschoose to prune their product lines when facingsuch competition Therefore the full explana-tion of the in uence of competition on productline choices is more complicated than that sug-gested by the IBM example Consider DECwhich until 1996 sold a range of PCs to bothhome and business customers By offeringlower-quality PCs targeted at home users DECprovided a potential substitute to its businessclients but clearly felt this downside was worthbearing for the opportunity to serve the low-endmarket In 1996 however DEC faced increas-ing competition in the home computer marketand responded by exiting that market to focuson its high-end desktops and servers14 Despitethe contrast with IBMrsquos decision DECrsquos reac-tion also can be understood in terms of changingopportunities for market segmentation In par-ticular DECrsquos response suggests that competi-tion reduced the available pro ts in the low-endmarket enough that it became more importantto attempt to preserve pro ts on its high-end

8 ldquoSummer of Discontentrdquo The Business Standard July3 2001

9 Philip Kotler (1965) and John A Quelch and DavidKenny (1994) emphasize the bene ts of regular pruning ofproduct lines Kotler writes ldquoAs the pace of competitionquickens and as consumer tastes become surfeited the needfor pruning company product lines of casualties becomes asgreat as that for nding replacementsrdquo

10 ldquoTimex to Exit Mass Market for Watchesrdquo The Eco-nomic Times April 30 2001

11 ldquoMitsubishi to Phase Out Low-End Mobile HandSetsrdquo The Economic Times November 24 2001

12 ldquoPiano Industry Off-Keyrdquo Baltimore Sun February18 1996

13 For a more detailed review of this case see Section V

14 ldquoDigital Equipment to Exit Home-Computing SectorrdquoThe Asian Wall Street Journal January 31 1996




































A The Demand Side



(1) H~z 5 5 F21~1 2 z z 10 z $ 1














(2) p i 2 p i 2 1 5 ~q i 2 q i 2 1 HX Oj 5 i

n

z jD



(3) p i 2 p i 2 1 5 ~q i 2 q i 2 1 H~Z i









n zi


p j 5 p j 2 p0 5 Oi 5 1

j

~ p i 2 p i 2 1

5 Oi 5 1

j

~q i 2 qi 2 1H~Z i

5 H~Z jOi 5 1

j

~qi 2 q i 2 1

5 qjH~Z j




















1laquo~z

5 2d log H~z

d log z5 2

H9~zz

H~z


H~z 1 zH9~z 5 H~zX 1 21

laquo~zD












c1

q15

c1 2 c 0

q1 2 q0

c2 2 c1

q2 2 q1

cn 2 cn 2 1

qn 2 qn 2 1

3c1

q1

c2

q2

cn

qn





q i5

c 1 c~q i

q i



c1

q1

c2

q2

ck 2 1

q k 2 1

ck

qk

ck 1 1 2 ck

q k 1 1 2 qk

c n 2 1 2 cn 2 2

qn 2 1 2 qn 2 2

cn 2 cn 2 1

qn 2 qn 2 1

H~z 1 zH9~z 5 u 21 2 F~u

f~u

and

laquo~z 5uf~u

1 2 F~u







III Monopoly






max Oi 5 1

n

Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1


(4) Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1


q i 2 q i 2 1


(5) H~Zi 1 Zi H9~Zi 5c i 2 c i 2 1

q i 2 q i 2 1






Z1 H~Z1 2 Z2 H~Z2 $c 1

q1~Z1 2 Z2

c2 2 c1

q2 2 q1~Z1 2 Z2








p1 5 u~H~Z1 q1

p2 5 p1 1 u~H~Z2 q2 2 u~H~Z2 q1






Z1 u~H~Z1 q1 $ Z2 u~H~Z2 q1

and

Z1 u~H~Z1 q2 2 u~H~Z1 q1

Z2 u~H~Z2 q2 2 u~H~Z2 q1


u~H~Z1 q2

u~H~Z1 q1

u~H~Z2 q2

u~H~Z2 q1





IV Duopoly











X i 5 Oj 5 i

n

x j Y i 5 Oj 5 i

m

y j


pI 5 Oi 5 1

n

~q i 2 q i 2 1

3 X iX H~X i 1 Y i 2c i 2 ci 2 1

q i 2 q i 2 1D

pE 5 Oi 5 1

m

~q i 2 q i 2 1

3 Y iX H~X i 1 Y i 2c i 2 c i 2 1

q i 2 q i 2 1D

















(6) pI 5 X~qm H~X 1 Y 2 cm


(7) pE 5 Y~qm H~X 1 Y 2 cm
















daggerr5 kn and

Xrmdagger r5k



dagger $Xrm







dagger






































Xm maxm i n

5 arg maxX


q i 2 q i 2 1D 6










V Discussion



A Computer Hardware














B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES


































































A The Demand Side



(1) H~z 5 5 F21~1 2 z z 10 z $ 1














(2) p i 2 p i 2 1 5 ~q i 2 q i 2 1 HX Oj 5 i

n

z jD



(3) p i 2 p i 2 1 5 ~q i 2 q i 2 1 H~Z i









n zi


p j 5 p j 2 p0 5 Oi 5 1

j

~ p i 2 p i 2 1

5 Oi 5 1

j

~q i 2 qi 2 1H~Z i

5 H~Z jOi 5 1

j

~qi 2 q i 2 1

5 qjH~Z j




















1laquo~z

5 2d log H~z

d log z5 2

H9~zz

H~z


H~z 1 zH9~z 5 H~zX 1 21

laquo~zD












c1

q15

c1 2 c 0

q1 2 q0

c2 2 c1

q2 2 q1

cn 2 cn 2 1

qn 2 qn 2 1

3c1

q1

c2

q2

cn

qn





q i5

c 1 c~q i

q i



c1

q1

c2

q2

ck 2 1

q k 2 1

ck

qk

ck 1 1 2 ck

q k 1 1 2 qk

c n 2 1 2 cn 2 2

qn 2 1 2 qn 2 2

cn 2 cn 2 1

qn 2 qn 2 1

H~z 1 zH9~z 5 u 21 2 F~u

f~u

and

laquo~z 5uf~u

1 2 F~u







III Monopoly






max Oi 5 1

n

Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1


(4) Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1


q i 2 q i 2 1


(5) H~Zi 1 Zi H9~Zi 5c i 2 c i 2 1

q i 2 q i 2 1






Z1 H~Z1 2 Z2 H~Z2 $c 1

q1~Z1 2 Z2

c2 2 c1

q2 2 q1~Z1 2 Z2








p1 5 u~H~Z1 q1

p2 5 p1 1 u~H~Z2 q2 2 u~H~Z2 q1






Z1 u~H~Z1 q1 $ Z2 u~H~Z2 q1

and

Z1 u~H~Z1 q2 2 u~H~Z1 q1

Z2 u~H~Z2 q2 2 u~H~Z2 q1


u~H~Z1 q2

u~H~Z1 q1

u~H~Z2 q2

u~H~Z2 q1





IV Duopoly











X i 5 Oj 5 i

n

x j Y i 5 Oj 5 i

m

y j


pI 5 Oi 5 1

n

~q i 2 q i 2 1

3 X iX H~X i 1 Y i 2c i 2 ci 2 1

q i 2 q i 2 1D

pE 5 Oi 5 1

m

~q i 2 q i 2 1

3 Y iX H~X i 1 Y i 2c i 2 c i 2 1

q i 2 q i 2 1D

















(6) pI 5 X~qm H~X 1 Y 2 cm


(7) pE 5 Y~qm H~X 1 Y 2 cm
















daggerr5 kn and

Xrmdagger r5k



dagger $Xrm







dagger






































Xm maxm i n

5 arg maxX


q i 2 q i 2 1D 6










V Discussion



A Computer Hardware














B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES










































(2) p i 2 p i 2 1 5 ~q i 2 q i 2 1 HX Oj 5 i

n

z jD



(3) p i 2 p i 2 1 5 ~q i 2 q i 2 1 H~Z i









n zi


p j 5 p j 2 p0 5 Oi 5 1

j

~ p i 2 p i 2 1

5 Oi 5 1

j

~q i 2 qi 2 1H~Z i

5 H~Z jOi 5 1

j

~qi 2 q i 2 1

5 qjH~Z j




















1laquo~z

5 2d log H~z

d log z5 2

H9~zz

H~z


H~z 1 zH9~z 5 H~zX 1 21

laquo~zD












c1

q15

c1 2 c 0

q1 2 q0

c2 2 c1

q2 2 q1

cn 2 cn 2 1

qn 2 qn 2 1

3c1

q1

c2

q2

cn

qn





q i5

c 1 c~q i

q i



c1

q1

c2

q2

ck 2 1

q k 2 1

ck

qk

ck 1 1 2 ck

q k 1 1 2 qk

c n 2 1 2 cn 2 2

qn 2 1 2 qn 2 2

cn 2 cn 2 1

qn 2 qn 2 1

H~z 1 zH9~z 5 u 21 2 F~u

f~u

and

laquo~z 5uf~u

1 2 F~u







III Monopoly






max Oi 5 1

n

Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1


(4) Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1


q i 2 q i 2 1


(5) H~Zi 1 Zi H9~Zi 5c i 2 c i 2 1

q i 2 q i 2 1






Z1 H~Z1 2 Z2 H~Z2 $c 1

q1~Z1 2 Z2

c2 2 c1

q2 2 q1~Z1 2 Z2








p1 5 u~H~Z1 q1

p2 5 p1 1 u~H~Z2 q2 2 u~H~Z2 q1






Z1 u~H~Z1 q1 $ Z2 u~H~Z2 q1

and

Z1 u~H~Z1 q2 2 u~H~Z1 q1

Z2 u~H~Z2 q2 2 u~H~Z2 q1


u~H~Z1 q2

u~H~Z1 q1

u~H~Z2 q2

u~H~Z2 q1





IV Duopoly











X i 5 Oj 5 i

n

x j Y i 5 Oj 5 i

m

y j


pI 5 Oi 5 1

n

~q i 2 q i 2 1

3 X iX H~X i 1 Y i 2c i 2 ci 2 1

q i 2 q i 2 1D

pE 5 Oi 5 1

m

~q i 2 q i 2 1

3 Y iX H~X i 1 Y i 2c i 2 c i 2 1

q i 2 q i 2 1D

















(6) pI 5 X~qm H~X 1 Y 2 cm


(7) pE 5 Y~qm H~X 1 Y 2 cm
















daggerr5 kn and

Xrmdagger r5k



dagger $Xrm







dagger






































Xm maxm i n

5 arg maxX


q i 2 q i 2 1D 6










V Discussion



A Computer Hardware














B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES

































n zi


p j 5 p j 2 p0 5 Oi 5 1

j

~ p i 2 p i 2 1

5 Oi 5 1

j

~q i 2 qi 2 1H~Z i

5 H~Z jOi 5 1

j

~qi 2 q i 2 1

5 qjH~Z j




















1laquo~z

5 2d log H~z

d log z5 2

H9~zz

H~z


H~z 1 zH9~z 5 H~zX 1 21

laquo~zD












c1

q15

c1 2 c 0

q1 2 q0

c2 2 c1

q2 2 q1

cn 2 cn 2 1

qn 2 qn 2 1

3c1

q1

c2

q2

cn

qn





q i5

c 1 c~q i

q i



c1

q1

c2

q2

ck 2 1

q k 2 1

ck

qk

ck 1 1 2 ck

q k 1 1 2 qk

c n 2 1 2 cn 2 2

qn 2 1 2 qn 2 2

cn 2 cn 2 1

qn 2 qn 2 1

H~z 1 zH9~z 5 u 21 2 F~u

f~u

and

laquo~z 5uf~u

1 2 F~u







III Monopoly






max Oi 5 1

n

Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1


(4) Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1


q i 2 q i 2 1


(5) H~Zi 1 Zi H9~Zi 5c i 2 c i 2 1

q i 2 q i 2 1






Z1 H~Z1 2 Z2 H~Z2 $c 1

q1~Z1 2 Z2

c2 2 c1

q2 2 q1~Z1 2 Z2








p1 5 u~H~Z1 q1

p2 5 p1 1 u~H~Z2 q2 2 u~H~Z2 q1






Z1 u~H~Z1 q1 $ Z2 u~H~Z2 q1

and

Z1 u~H~Z1 q2 2 u~H~Z1 q1

Z2 u~H~Z2 q2 2 u~H~Z2 q1


u~H~Z1 q2

u~H~Z1 q1

u~H~Z2 q2

u~H~Z2 q1





IV Duopoly











X i 5 Oj 5 i

n

x j Y i 5 Oj 5 i

m

y j


pI 5 Oi 5 1

n

~q i 2 q i 2 1

3 X iX H~X i 1 Y i 2c i 2 ci 2 1

q i 2 q i 2 1D

pE 5 Oi 5 1

m

~q i 2 q i 2 1

3 Y iX H~X i 1 Y i 2c i 2 c i 2 1

q i 2 q i 2 1D

















(6) pI 5 X~qm H~X 1 Y 2 cm


(7) pE 5 Y~qm H~X 1 Y 2 cm
















daggerr5 kn and

Xrmdagger r5k



dagger $Xrm







dagger






































Xm maxm i n

5 arg maxX


q i 2 q i 2 1D 6










V Discussion



A Computer Hardware














B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES




































1laquo~z

5 2d log H~z

d log z5 2

H9~zz

H~z


H~z 1 zH9~z 5 H~zX 1 21

laquo~zD












c1

q15

c1 2 c 0

q1 2 q0

c2 2 c1

q2 2 q1

cn 2 cn 2 1

qn 2 qn 2 1

3c1

q1

c2

q2

cn

qn





q i5

c 1 c~q i

q i



c1

q1

c2

q2

ck 2 1

q k 2 1

ck

qk

ck 1 1 2 ck

q k 1 1 2 qk

c n 2 1 2 cn 2 2

qn 2 1 2 qn 2 2

cn 2 cn 2 1

qn 2 qn 2 1

H~z 1 zH9~z 5 u 21 2 F~u

f~u

and

laquo~z 5uf~u

1 2 F~u







III Monopoly






max Oi 5 1

n

Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1


(4) Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1


q i 2 q i 2 1


(5) H~Zi 1 Zi H9~Zi 5c i 2 c i 2 1

q i 2 q i 2 1






Z1 H~Z1 2 Z2 H~Z2 $c 1

q1~Z1 2 Z2

c2 2 c1

q2 2 q1~Z1 2 Z2








p1 5 u~H~Z1 q1

p2 5 p1 1 u~H~Z2 q2 2 u~H~Z2 q1






Z1 u~H~Z1 q1 $ Z2 u~H~Z2 q1

and

Z1 u~H~Z1 q2 2 u~H~Z1 q1

Z2 u~H~Z2 q2 2 u~H~Z2 q1


u~H~Z1 q2

u~H~Z1 q1

u~H~Z2 q2

u~H~Z2 q1





IV Duopoly











X i 5 Oj 5 i

n

x j Y i 5 Oj 5 i

m

y j


pI 5 Oi 5 1

n

~q i 2 q i 2 1

3 X iX H~X i 1 Y i 2c i 2 ci 2 1

q i 2 q i 2 1D

pE 5 Oi 5 1

m

~q i 2 q i 2 1

3 Y iX H~X i 1 Y i 2c i 2 c i 2 1

q i 2 q i 2 1D

















(6) pI 5 X~qm H~X 1 Y 2 cm


(7) pE 5 Y~qm H~X 1 Y 2 cm
















daggerr5 kn and

Xrmdagger r5k



dagger $Xrm







dagger






































Xm maxm i n

5 arg maxX


q i 2 q i 2 1D 6










V Discussion



A Computer Hardware














B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES



































c1

q15

c1 2 c 0

q1 2 q0

c2 2 c1

q2 2 q1

cn 2 cn 2 1

qn 2 qn 2 1

3c1

q1

c2

q2

cn

qn





q i5

c 1 c~q i

q i



c1

q1

c2

q2

ck 2 1

q k 2 1

ck

qk

ck 1 1 2 ck

q k 1 1 2 qk

c n 2 1 2 cn 2 2

qn 2 1 2 qn 2 2

cn 2 cn 2 1

qn 2 qn 2 1

H~z 1 zH9~z 5 u 21 2 F~u

f~u

and

laquo~z 5uf~u

1 2 F~u







III Monopoly






max Oi 5 1

n

Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1


(4) Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1


q i 2 q i 2 1


(5) H~Zi 1 Zi H9~Zi 5c i 2 c i 2 1

q i 2 q i 2 1






Z1 H~Z1 2 Z2 H~Z2 $c 1

q1~Z1 2 Z2

c2 2 c1

q2 2 q1~Z1 2 Z2








p1 5 u~H~Z1 q1

p2 5 p1 1 u~H~Z2 q2 2 u~H~Z2 q1






Z1 u~H~Z1 q1 $ Z2 u~H~Z2 q1

and

Z1 u~H~Z1 q2 2 u~H~Z1 q1

Z2 u~H~Z2 q2 2 u~H~Z2 q1


u~H~Z1 q2

u~H~Z1 q1

u~H~Z2 q2

u~H~Z2 q1





IV Duopoly











X i 5 Oj 5 i

n

x j Y i 5 Oj 5 i

m

y j


pI 5 Oi 5 1

n

~q i 2 q i 2 1

3 X iX H~X i 1 Y i 2c i 2 ci 2 1

q i 2 q i 2 1D

pE 5 Oi 5 1

m

~q i 2 q i 2 1

3 Y iX H~X i 1 Y i 2c i 2 c i 2 1

q i 2 q i 2 1D

















(6) pI 5 X~qm H~X 1 Y 2 cm


(7) pE 5 Y~qm H~X 1 Y 2 cm
















daggerr5 kn and

Xrmdagger r5k



dagger $Xrm







dagger






































Xm maxm i n

5 arg maxX


q i 2 q i 2 1D 6










V Discussion



A Computer Hardware














B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES



































III Monopoly






max Oi 5 1

n

Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1


(4) Z i ~q i 2 q i 2 1 H~Z i 2 ~c i 2 c i 2 1


q i 2 q i 2 1


(5) H~Zi 1 Zi H9~Zi 5c i 2 c i 2 1

q i 2 q i 2 1






Z1 H~Z1 2 Z2 H~Z2 $c 1

q1~Z1 2 Z2

c2 2 c1

q2 2 q1~Z1 2 Z2








p1 5 u~H~Z1 q1

p2 5 p1 1 u~H~Z2 q2 2 u~H~Z2 q1






Z1 u~H~Z1 q1 $ Z2 u~H~Z2 q1

and

Z1 u~H~Z1 q2 2 u~H~Z1 q1

Z2 u~H~Z2 q2 2 u~H~Z2 q1


u~H~Z1 q2

u~H~Z1 q1

u~H~Z2 q2

u~H~Z2 q1





IV Duopoly











X i 5 Oj 5 i

n

x j Y i 5 Oj 5 i

m

y j


pI 5 Oi 5 1

n

~q i 2 q i 2 1

3 X iX H~X i 1 Y i 2c i 2 ci 2 1

q i 2 q i 2 1D

pE 5 Oi 5 1

m

~q i 2 q i 2 1

3 Y iX H~X i 1 Y i 2c i 2 c i 2 1

q i 2 q i 2 1D

















(6) pI 5 X~qm H~X 1 Y 2 cm


(7) pE 5 Y~qm H~X 1 Y 2 cm
















daggerr5 kn and

Xrmdagger r5k



dagger $Xrm







dagger






































Xm maxm i n

5 arg maxX


q i 2 q i 2 1D 6










V Discussion



A Computer Hardware














B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES

































Z1 H~Z1 2 Z2 H~Z2 $c 1

q1~Z1 2 Z2

c2 2 c1

q2 2 q1~Z1 2 Z2








p1 5 u~H~Z1 q1

p2 5 p1 1 u~H~Z2 q2 2 u~H~Z2 q1






Z1 u~H~Z1 q1 $ Z2 u~H~Z2 q1

and

Z1 u~H~Z1 q2 2 u~H~Z1 q1

Z2 u~H~Z2 q2 2 u~H~Z2 q1


u~H~Z1 q2

u~H~Z1 q1

u~H~Z2 q2

u~H~Z2 q1





IV Duopoly











X i 5 Oj 5 i

n

x j Y i 5 Oj 5 i

m

y j


pI 5 Oi 5 1

n

~q i 2 q i 2 1

3 X iX H~X i 1 Y i 2c i 2 ci 2 1

q i 2 q i 2 1D

pE 5 Oi 5 1

m

~q i 2 q i 2 1

3 Y iX H~X i 1 Y i 2c i 2 c i 2 1

q i 2 q i 2 1D

















(6) pI 5 X~qm H~X 1 Y 2 cm


(7) pE 5 Y~qm H~X 1 Y 2 cm
















daggerr5 kn and

Xrmdagger r5k



dagger $Xrm







dagger






































Xm maxm i n

5 arg maxX


q i 2 q i 2 1D 6










V Discussion



A Computer Hardware














B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES
































Z1 u~H~Z1 q1 $ Z2 u~H~Z2 q1

and

Z1 u~H~Z1 q2 2 u~H~Z1 q1

Z2 u~H~Z2 q2 2 u~H~Z2 q1


u~H~Z1 q2

u~H~Z1 q1

u~H~Z2 q2

u~H~Z2 q1





IV Duopoly











X i 5 Oj 5 i

n

x j Y i 5 Oj 5 i

m

y j


pI 5 Oi 5 1

n

~q i 2 q i 2 1

3 X iX H~X i 1 Y i 2c i 2 ci 2 1

q i 2 q i 2 1D

pE 5 Oi 5 1

m

~q i 2 q i 2 1

3 Y iX H~X i 1 Y i 2c i 2 c i 2 1

q i 2 q i 2 1D

















(6) pI 5 X~qm H~X 1 Y 2 cm


(7) pE 5 Y~qm H~X 1 Y 2 cm
















daggerr5 kn and

Xrmdagger r5k



dagger $Xrm







dagger






































Xm maxm i n

5 arg maxX


q i 2 q i 2 1D 6










V Discussion



A Computer Hardware














B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES

































X i 5 Oj 5 i

n

x j Y i 5 Oj 5 i

m

y j


pI 5 Oi 5 1

n

~q i 2 q i 2 1

3 X iX H~X i 1 Y i 2c i 2 ci 2 1

q i 2 q i 2 1D

pE 5 Oi 5 1

m

~q i 2 q i 2 1

3 Y iX H~X i 1 Y i 2c i 2 c i 2 1

q i 2 q i 2 1D

















(6) pI 5 X~qm H~X 1 Y 2 cm


(7) pE 5 Y~qm H~X 1 Y 2 cm
















daggerr5 kn and

Xrmdagger r5k



dagger $Xrm







dagger






































Xm maxm i n

5 arg maxX


q i 2 q i 2 1D 6










V Discussion



A Computer Hardware














B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES


































(6) pI 5 X~qm H~X 1 Y 2 cm


(7) pE 5 Y~qm H~X 1 Y 2 cm
















daggerr5 kn and

Xrmdagger r5k



dagger $Xrm







dagger






































Xm maxm i n

5 arg maxX


q i 2 q i 2 1D 6










V Discussion



A Computer Hardware














B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES



































daggerr5 kn and

Xrmdagger r5k



dagger $Xrm







dagger






































Xm maxm i n

5 arg maxX


q i 2 q i 2 1D 6










V Discussion



A Computer Hardware














B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES





























































Xm maxm i n

5 arg maxX


q i 2 q i 2 1D 6










V Discussion



A Computer Hardware














B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES

































V Discussion



A Computer Hardware














B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES



































B Airlines




































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES























































APPENDIX





q i~Zi 2 Zi 1 1

ci 1 1 2 ci

q i 1 1 2 qi~Zi 2 Zi 1 1









ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES






































ci 2 ci 2 1

qi 2 qi 2 1






pI 5 X X 1 2qm

qnD H~X 1

qm

qnH~X 1 Y 2

cn

qn



shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES
































shypI

shyX5 X 1 2

qm

qnD ~H~X 1 XH9~X 1

qm

qn~H~X 1 Y 1 XH9~X 1 Y 2

c n

qn









dagger

shyX5


dagger


dagger 2c s 2 cr

q s 2 q r1 X sm


$shypIr~Xsm

dagger Ysmdagger


dagger 2cs 2 cs 2 1

qs 2 qs 2 11 Xsm

dagger H9~Xsmdagger

$shypIr~Xsm

dagger Ysmdagger

shyX

















dagger







max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES


































max Oi 5 m11

n

~q i 2 q i 2 1 X iX H~X i 2c i 2 ci 2 1



Xm 1 1 X 5 maxm i n

5 arg maxX

XX H~X 2c i 2 c i 2 1

qi 2 q i 2 1D 6


REFERENCES
































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