Delivered versus Mill Nonlinear Pricing in Free Entry Markets

Transcription

1 Delivered versus Mill Nonlinear Pricing in Free Enry Markes Sílvia Ferreira Jorge DEGEI - Universidade de Aveiro Faculdade de Economia, Universidade Nova de Lisboa sjorge@egi.ua.p Cesalina Pacheco Pires Deparameno de Gesão de Empresas, Universidade de Évora cpires@uevora.p Sepember 004 Absrac This paper discusses a model where consumers simulaneously differ according o one unobservable (preference for qualiy) and one observable characerisic (locaion). In hese circumsances nonlinear prices arise in equilibrium. The main quesion addressed in his work is wheher firms should be allowed o pracise differen nonlinear prices a each locaion (delivered nonlinear pricing) or should be forced o se an unique nonlinear conrac (mill nonlinear pricing). Assuming ha firms can cosless relocae, we show ha he free enry long-run number of firms may be eiher smaller, equal, or higher under delivered nonlinear pricing. In addiion, we show ha delivered nonlinear pricing yields in he long-run higher welfare and, consequenly, our resuls suppor he view ha discriminaory nonlinear pricing should no be prohibied. Keywords: Delivered nonlinear pricing, Mill nonlinear pricing, Asymmeric informaion, Pricing regulaion. JEL classificaion: D43, L3, D8 This paper is no compeing for he Young Economis Award. Finanial suppor from Fundação para Ciência e Tecnologia and Fundo Social Europeu wihin he III Quadro Comuniário de Apoio - BD/857/000.

2 Inroducion Regulaion heory has been one of he more acive areas of research in he las decade. However, regulaion of firm s pricing policies has been almos negleced wihin his wave of research. This paper revisis he issue of wheher regulaory auhoriies should prohibi firms from pracicing price discriminaion among consumers who differ according o some observable characerisic. The currenly acceped view ha unregulaed markes are more compeiive has been jusified by economic heory using spaial pricing models. Mos of he exising sudies compare spaial discriminaory pricing wih he non-discriminaory mill pricing for a given marke srucure (see, for example, Norman (983) and Thisse and Vives (988)). Under spaial discriminaory pricing firms can price discriminae across locaions, hus firms compee in each of hem. On he oher hand, if firms have o pracice he same price in every locaion, compeiion occurs only a he boundary of each firm s marke. As a consequence, for a given marke srucure discriminaory pricing is more compeiive han mill pricing. However, as poined ou by Norman and Thisse (996), he previous analysis is incomplee if one does no consider he effec of pricing policies on marke srucure since he incenives for enry are no he same under boh pricing policies. Norman and Thisse (996) analyzed he economic jusificaion of firms pricing policies regulaion aking ino accoun he effec of pricing policies on welfare for a given marke srucure and he effec of pricing policies on marke srucure. They consider a circular model of horizonal produc differeniaion where he locaion of each consumer is observable, and firms may price discriminae (delivered pricing) or se a mill price (mill pricing). The free enry long-run equilibrium and is number of firms (number of produc varieies ) is compued for each pricing policy 3 and degree of spaial conesabiliy 4.Theirwork illusraes an imporan rade-off: as delivered pricing implies fiercer compeiion and lower profis, in he long-run i acs as an enry deerren and reduces he free enry number of firms. Therefore, mill pricing always leads o more variey han delivered pricing. However, his higher variey is no necessarily welfare improving. When relocaion is cosless, boh mill and delivered pricing have oo much produc variey, hus he equilibrium number of firms under discriminaory pricing is closer o he socially opimal one and discriminaory pricing leads o higher social surplus. Under spaial non-conesabiliy, he welfare comparisons are less clear. In his case, here is oo much produc variey under mill pricing, bu oo lile produc variey wih delivered pricing. The previously menioned works consider linear prices: mill linear price is compared wih discriminaory linear price, in a seup where consumers differ according o one observable characerisic (locaion). However, if consumers differ simulaneously according o some unobservable (e.g. qualiy preference) and Implici in his comparison is he idea ha wihou pricing regulaion firms will pracice discriminaory prices. Thisse and Vives (988) have shown ha if firms are free o choose heir pricing policy, in equilibrium, hey will in fac price discriminae even hough his implies lower profis for all firms. Thefreeenrylong-runnumberoffirms has been inerpreed as he number of produc variey (see MacLeod e al. (988) and Anderson and Palma (988)). When firms use delivered pricing, firms are seen o be redesigning he basic producs so as o offer he consumers opimal produc specificaion: firms cusomize heir basic/sandard producs. As firms locaion can be seen as he basic producs characerisic, he free enry long-run number of firms can give a measure of he degree of produc variey. 3 For simpliciy, hey assume ha consumers reservaion price is high enough wih respec o fixed coss such ha no firm has monopoly power in is marke area. 4 When firms is relocaion cosless we have spaial conesabiliy and when looking for he free enry long-run equilibrium we achieve he denses packing of firms. On he conrary, when locaion is once-for-all, we have spaial non-conesabiliy and, herefore, we achieve he looses packing of firms.

3 some observable characerisic (e.g. locaion - brand preference), we would expec nonlinear prices o arise in equilibrium and he issue of wheher firms should be allowed o pracise differen nonlinear prices a each locaion or should be forced o se he same nonlinear conrac in every locaion is relevan. In order o sudy his issue, which has no been explored before, our work adds verical differeniaion o he Norman and Thisse (996) se up. This is an imporan exension since horizonally differeniaed firms compeing in produc lines are noorious in various markes (e.g., car 5, elecommunicaions, airline ravel, ec.). Wihin his more realisic se up, we assume ha firms have uncerainy abou consumers qualiy preferences bu observe heir locaion (or brand preferences parameer). Under hese circumsances, firms se price/qualiy nonlinear conracs ha screen consumers according o heir preference for qualiy. The regulaory issue 6 is hen wheher firms should be allowed o pracice differen nonlinear price conracs a each locaion (delivered nonlinear pricing) or, on he oher hand, should be forced o se an unique nonlinear price conrac (mill nonlinear pricing). One ineresing example where our discussion applies and which has been he subjec of sizable debae wihin he European Union is he car marke. The recen policy sance 7 srongly acs upon is disribuion secor sopping he pracices upon which car manufacurers rely o preven arbirage (herefore acing upon a crucial feaure for price discriminaion). Alhough no acing upon price regulaion, he economic jusificaion of he European Commission is he need o creae more compeiion working o he advanage of European consumers: claiming ha he consumer should be in he driver s sea, o build a single marke may pu pressure on price differenials 8. By comparing delivered nonlinear pricing and mill nonlinear pricing, our analysis uncovers some feaures ha may be fel in he car marke afer is disribuion secor liberalizaion 9. Our work builds on previous resuls on nonlinear pricing under oligopoly. The framework we use is similar o he one firs presened by Sole (995), who named i verical uncerainy 0. Sole sudied delivered nonlinear price conracs considering a coninuous of consumer ypes. Delivered nonlinear pricing wih a discree number consumer ypes was sudied by Pires and Sarkar (000) and Vallei (00) (hey boh analyze he locaional equilibrium, he former considering price/quaniy nonlinear conracs and he laer price/qualiy nonlinear conracs). The discree case analysis showed ha when firms use delivered nonlinear pricing, in equilibrium, herecanbedifferen marke regions: monopoly, inermediae and compeiive regions. On he oher hand, Villas-Boas and Schmid-Mohr (999) sudied mill nonlinear conracs under discree verical uncerainy. They analyzed a credi-marke seing where banks offer one menu of credi conracs (which include a gross ineres rae and a collaeral requiremen), and showed ha hose conracs and heir screening degree depend upon he degree of 5 Ginsburgh and Weber (00) sudy his mulidimensional seing when firms use wo-par ariffs (price/qualiy schedules) wih an applicaion o he European car marke. 6 Regulaion always acs upon discriminaion in erms of some observable characerisic. In our model his characerisic is locaion. 7 For more deails see he new Block Exempion Regulaion 400/00-7 July Brenkers and Verboven (00) discuss he impac on compeiion from liberalizing he European car marke disribuion sysem and show empirically ha here may be a reducion in inernaional price discriminaion and price differenials. 9 Davidson e al.(989) sudy he welfare effecs of prohibiing hird degree price discriminaion in he European car marke and show ha in some cases price discriminaion is welfare-improving. 0 Sole also discusses anoher ype of uncerainy: he horizonal uncerainy case where firms are unable o observe he consumers brand preferences bu observe he qualiy preferences parameer. Spulber (989) sudies his case wih price/quaniy nonlinear conracs and concludes ha second-degree price discriminaion is he firms opimal sraegy. In his analysis second-degree price discriminaion leads o a greaer number of firms (more variey) han linear pricing. See also Spulber (984) and Hamilon and Thisse (997). They also sudy he coninuous ypes case - Villas-Boas and Schmid-Mohr (997). 3

4 horizonal and verical differeniaion sudying also welfare implicaions. Focusing also on he discree case, wih wo ypes of consumers a each locaion, he economic jusificaion for he pricing policies regulaion is discussed in a model of horizonal and verical differeniaion where firms se price/qualiy nonlinear conracs. Considering spaial conesabiliy, we sudy he impac on welfare and marke srucure of wo pricing policies: discriminaion in space (delivered nonlinear pricing - DNP) - firms offer differen nonlinear conracs a each locaion - or uniform nonlinear pricing (mill nonlinear pricing - MNP)- firms are no allowed o discriminae among consumers a differen locaions and hey mus simulaneously offer an unique nonlinear conrac. Our resuls show ha delivered nonlinear pricing may bring smaller, equal or higher produc variey (free enry long-run equilibrium number of firms) han mill nonlinear pricing and is always welfare-improving. Even when here is higher excess of enry wih delivered nonlinear pricing, he long-run welfare is higher han he one achieved wih mill nonlinear pricing. The paper will proceed as follows. Secion presens he model and sudies he linear ciy duopoly case for each pricing policy. Secion 3 exends i o a circular ciy model wih n idenical firms considering spaial conesabiliy. Secion 4 presens produc variey and welfare comparisons for each pricing policy in he long-run. Our main conclusions are presened in secion 5. 4

5 Linear ciy duopoly model. The model Consider wo idenical firms offering nonlinear conracs (price p i,qualiyu i ) o consumers disribued on a uni lengh line (brand preferences space). A each locaion d, d [0, ], here are wo ypes of consumers characerized by heir preference for qualiy parameer θ: θ > θ > 0. Boh ypes are uniformly disribued along he line, boh having mass equal o. We assume ha firms observe consumer s locaion d, while he qualiy preferences parameer θ is consumers privae informaion (i.e., here is verical uncerainy). We assume ha firms are locaed a d =0and d =and have idenical producion cos for a good of qualiy u i,givenbyc(u i )= u i. Each consumer ype buys a single uni of a cerain good and chooses he conrac (p i,u i ) ha yields he maximum ne surplus (purchasing a mos from one firm).ifaypeθ locaed a d buys a price p i a produc of qualiy u i produced by firm i, i =,, locaed a d i, d i [0, ], his ne surplus is given by 3 : V = U(θ,d,u i,d i ) p i = θu i d d i p i () where is a differeniaion parameer ( >0) and, as in oher spaial models, d d i measures he consumer s loss by no consuming her ideal brand. The differeniaion degree,, will be crucial for he compeiion analysis when firms use MNP and for he compeiion inensiy when firms use DNP. Also consider k o be he asymmeric informaion degree, i.e., he dispersion of ypes preference for qualiy parameers: k = θ 4 θ. Noice ha () implies ha, independenly of heir brand preferences, consumers wih higher θ will be willing o pay larger rises on price for a given increase in produc s qualiy.. Mill nonlinear pricing Le us firs consider he case where firms are no allowed o discriminae among consumers a differen locaions: hey mus simulaneously offer an unique nonlinear conrac ( p i, ū i and p i,u i ) for all consumers disribued in he line. All prices and qualiies mus be non-negaive. In general, under verical uncerainy, when firms se mill nonlinear conracs hey face a mulidimensional screening problem as hey may use boh θ and d as screening variables 5. However, under our uiliy specificaion, he soring condiion only holds wih respec o θ, since: [ ( V/ u) /( V/ p)] θ => 0 bu [ ( V/ u) /( V/ p)] d Therefore, in our model, mill nonlinear conracs only screen consumers according o heir qualiy preferences. Consumers differ in heir relaive preference for he wo firms bu hese brand preferences are When consumers are indifferen beween buying from firm or firm, we assume hey buy from he neares firm. 3 We considered a general specificaion given by V = θu i θ r d d i p i buresricouranalysisor =0for echnical simpliciy since for r 6= 0, brand preferences and qualiy preferences inerac and we may have changes in he consumer ranking since he consumer ype who is willing o pay more a given u may be ype θ consumers for locaions disan from he firm locaion. These cases are ineresing bu require addiional complex compuaions. 4 Noice ha θ can be reinerpre as he inverse of he marginal uiliy of income, in a seup where consumers have idenical ordinal preferences bu differ in heir incomes. Wealhier consumers have lower marginal uiliy of income and consequenly higher θ. Inhiscase,ahighk (k [; [) is relaed o a high asymmeric income disribuion degree. 5 Roche and Sole (000) discuss wih deail he mulidimensional screening scenario. =0. 5

6 neural wih respec o qualiy preferences 6. Under monopoly mill nonlinear pricing (see Appendix A), a monopolis locaed a d m =0always sops selling o ype θ a a locaion furher away han he one where he sops selling o ype θ and here are no local monopolies a leas for one ype (ype θ) aslongas< θ (henceforh we call his he non-exisence of local monopolies condiion - NLM) 7. Assuming ha belongs o he previous inerval, duopolisic firms may cover all he marke for boh ypes or cover all he marke only for ype θ. Alsofirms may choose o sell only o ype θ 8... Marke fully covered When firms cover all he marke for boh ypes, firms demand is deermined by he indifferen consumer locaion for each ype. From () if a consumer of ype θ locaed a d buys from firm she ges a ne surplus of θu d p while if she buys from firm she ges θu ( d) p. Hence, he locaion where a ype θ consumer is indifferen beween buying from firm or from firm is given by: d θ = θ(u u )p p Therefore, firmsolveshefollowingproblem: subjec o: µ max p ū ( p,ū )(p,u ) µ d θ p u d θ IRP θ : θū d θ p 0 IRP θ : θ u d θ p 0 IC θ : θū p θu p IC θ : θ u p θū p FC : 0 d θ for θ = θ, θ. The firs wo consrains are he individual raionaliy paricipaion consrains (IRP) he ne surplus of a ype θ consumer, locaed a he indifference poin, d θ, mus be non-negaive (noice ha his assures ha any ype θ consumer locaed o he lef of d θ ges a sricly posiive surplus). The hird and fourh consrains are he incenive compaibiliy consrains (IC) firm will se ( p, ū ) and (p,u ) so ha each ype prefers buying he conrac se for is own ype. I is ineresing o noice ha he incenive compaibiliy consrains do no depend on d. Thus, if hey are saisfied a one locaion, hey are saisfied for every locaion. The las consrains are he feasibiliy consrains (FC) for boh ypes, d θ mus belong o he uni line. 6 Vallei (00) considers ineracion beween horizonal (brand) and verical (qualiy) preferences, using he following uiliy funcion: V = U(θ,d,u i,d i ) p i = θ ( d d i ) u i p i. In his case screening is feasible boh on d and θ. However, his would increase considerably he complexiy of he mill nonlinear pricing problem and would require he use of recen echniques for solving mulidimensional screening. 7 Under monopoly delivered nonlinear pricing, he condiion for non-exisence of local monopolies would be less resricive. Therefore, we consider he limi given by he monopoly mill nonlinear pricing analysis. 8 For all hese cases we focus only on symmeric equilibrium soluions. 6

7 Obviously, firm faces a similar problem, wih demands ( d θ) and ( d θ ). Noice ha if he incenive compaibiliy consrains of firm are saisfied, hen ypes θ and θ locaed o he lef of he indifferen consumer do no wan o buy from firm a (p,u ) and ( p, ū ), respecively. Therefore, firm s demands are indeed d θ and d θ. From he Kuhn-Tucker (KT) condiions (see Appendix B), wo soluions are found: C θ θ C θ θ u = θ and p = θ ū = θ and p = θ u = θ and p = θ ū = θ and p = θ θ C θ corresponds o a siuaion where no consrain is binding, while IRP θ θ is binding a C θ. In θ oher words, a C θ boh ypes locaed a he middle of he marke (indifferen consumers locaion) ge θ posiive ne surplus, bu a C θ ype θ ges zero ne surplus. The price conracs offered a hese wo soluions are very similar. In boh cases he qualiy offered o each ype is he socially efficien one 9. The wo conracs only differ in erms of he price charged o θ ype θ consumers, p. A C θ boh prices (p and p) are increasing wih, hus marke power increases θ wih he differeniaion degree. However, a C θ p is decreasing wih. This las resul is due o he fac ha ype θ indifferen consumer has a zero ne surplus, hus when increases p has o decrease for he consumer o be willing o buy. As we will see aferwards in more deail, he marke is fully covered for boh ypes for low levels of θ. When is close o zero, all consumers ge posiive ne surplus (C θ ). However, as increases ype θ θ indifferen consumer sars geing zero ne surplus (C θ ). When increases even furher he marke is θ no longer compleely covered for ype θ. Thus,C θ may be inerpreed as an inermediae case beween full compeiion for boh ypes and local monopolies for ype θ... Marke fully covered only for ype θ When firms cover all he marke only for ype θ, demandfromypeθ consumers is deermined by he locaion where ype θ is indifferen beween buying from he firm and no buying a all. For firm, his consumer is locaed a ˆd θ where: ˆdθ i = θ u i p i. () In his case, firmsolveshefollowingproblem: µ µ max p ū d θ p ( p,ū )(p,u ) u ˆd θ subjec o: IRP θ IC θ : θū d θ p 0 : θū p θu p IC θ : θ u p θū p FC θ : 0 d θ PC θ : 0 ˆd θ < ˆd θ 9 Simple compuaions reveal ha u = θ is he socially opimal qualiy. 7

8 The las consrain is he Parially Covered consrain (PC). Again, firm will solve an analogue problem. From he KT condiions (see Appendix B), he following are some of he achieved soluions: S θ u = θ and p = θ S θ u = θ and p = θ LM θ LM θ u = θ and p = 3 4 θ ū = θ and p = θ u = θ and p = 3 4 θ ū = θ and p = θ A S θ and S θ firms sell only o ype θ, alm θ and LMθ firms sell o boh ypes bu have local monopolies for ype θ. When firms sell only o ype θ, IRP θ is binding a soluion S θ, bu no a S θ. Therefore, S θ and S θ have he same inuiions as he sandard Hoelling model: for S θ,whenrises, he price rises; for S θ, when rises, he price decreases. When he wo firms compee for ype θ consumers and have local monopolies for ype θ consumers, herearewopossiblesoluions: LM θ (when no consrain is binding) and LMθ (when IRP θ is he only binding consrain). The price conracs in hese wo soluions only differ for p. Inbohcases, firms offer he socially efficien qualiies and charge p = 3 4 θ, which is precisely he price ha would be charged by a monopolis. A LM θ he price charged o ype θ consumers, for whom firms are compeing, is increasing wih (marke power increases wih differeniaion). On he conrary, a LM θ p is decreasing wih. A boh soluions, firm sells o ype θ consumers o he lef of ˆd θ = θ 4 <.Theype θ s consumer locaed a d θ = ges a posiive surplus a LMθ,bunosurplusaLMθ. Unil now we have described some possible equilibrium soluion conracs under MNP, bu did no specify he se of parameers values o which each of hem holds. The nex secion analyzes he symmeric Nash equilibria as a funcion of he parameers values...3 Nash Equilibrium soluion For some parameers values he KT condiions for he previous problems have muliple soluions 0. In hese cases we consider he global maximum soluion. To achieve he maximum profis equilibrium soluion, we proceed compuing ³ each soluion profis covering all and k. We will only have equilibrium soluions for all < θ if k> 5 5. Figure shows he se of parameers values where each of he previously described soluion conracs hold. Figure illusraes several ineresing feaures of he equilibrium soluion conracs as a funcion of he horizonal and verical differeniaion parameers (, θ and θ). Firs, for a given level of k, as rises we move o less compeiive soluions. Second, he soluion conracs depend on k. 0 There are also some undeermined soluions. Bu since heir profis are equal o he profis of oher deermined soluions and heir inervals limis belong o he deermined soluions inervals limis, we can exclude hem from analysis. Since we only analyzed symmeric soluions of he Kuhn-Tucker condiions we canno guaranee ha he Nash equilibrium described is unique (here may be Nash equilibria corresponding o local maxima). However, even if he Nash equilibrium described is no unique, i cerainly is he mos reasonable symmeric equilibrium. Villas-Boas and Schmid-Mohr (999) also ge similar non exisence of (pure sraegies) equilibrium resul. 8

9 Figure : Duopoly Mill Nonlinear Pricing Conracs Discussion of resuls under mill nonlinear pricing Under MNP firms se a unique nonlinear conrac for all he marke. Curiously, some characerisics of he equilibrium nonlinear conracs are similar o he well known resuls of mill linear pricing: for low, firms face inense compeiion and sell unil he indifferen consumer d who ges posiive surplus; for inermediae, firms sell unil d bu he indifferen consumer ges no surplus; and for high, firms prefer o have local monopolies. When verical differeniaion is inroduced, his phenomenon happens for boh ypes bu as ype θ is characerized by a lower preference for qualiy parameer, he previous phenomenon resuls appear for lower levels of differeniaion for ype θ 3. As i is shown in Figure, for a given k and very low levels of, hemarke is fully covered for boh ypes, as rises firms sar having local monopolies for ype θ, andif rises above a cerain level here is no compeiion beween firms as hey have local monopolies for boh ypes of consumers 4. When is very low, here is efficien qualiy provision. We ge separaing equilibrium where firms screen consumers and cover all he marke for boh ypes. However, for low k, as rises firms end o use less screening since i becomes more difficul o se screening conracs and he ICs o hold. Therefore, for low k and high, firms prefer o sell only o θ (firs leaving surplus and hen sealing all he surplus of ype θ s indifferen consumer). For high k, which means ha he wo ypes of consumers are very differen and he separaion of hewoypesiseasy,if rises firms coninue o use screening and o sell o boh ypes (going from full compeiion, o local monopolies for ype θ and o local monopolies for boh ypes). Therefore, for very 3 For insance, for some degree of differeniaion, we may have ype θ s indifferen consumer wih posiive surplus bu ype θ s indifferen consumer wih no surplus or no buying a all. 4 Tha level of differeniaion depends on k. The higher is k, he higher is he level of differeniaion needed for local monopolies o hold for boh ypes. As menioned earlier, we will no discuss he case where firmshavelocalmonopoliesfor boh ypes. 9

10 high levels of k, we ge separaing equilibrium for all. For inermediae k levels, ha is no he case since we only ge separaing equilibrium for low and high levels of (for inermediae, firms sell only o ype θ, again, firs leaving surplus and hen sealing all he surplus of ype θ s indifferen consumer) and for low k, we only ge separaing equilibrium for low (when riseswemoveosoluionswherefirms do no sell o boh ypes) 5..3 Delivered nonlinear pricing Suppose now ha firms are able o offer differen nonlinear conracs a each locaion in he line ( p i (d), ū i (d) and p i (d),u i (d)) 6.Sincehewofirms are idenical, wih DNP, he marke will be spli in wo equal pars and each firm has is own marke area. Therefore, we will resric he analysis o firm s opimizaion problem wihin half of he spaial marke: d 0, - firm marke area 7. Also, for his pricing policy all prices and qualiies mus be non-negaive. When selling o boh ypes firm chooses, for each locaion d, he nonlinear conrac ha solve: max p u ( p,ū )(p,u ) p ū subjec o individual raionaliy and incenive compaibiliy consrains: : θū d p max {0, ne surplus of buying from firm } IR θ : θ u d p max {0, ne surplus of buying from firm } IR θ IC θ : θū p θu p IC θ : θ u p θū p The individual raionaliy consrains incorporae he fac he consumer has he ouside opion of buying from firm or no buying. Thus, in order o induce he consumer o buy from firm, his firm has o offer a ne surplus a leas as high as he one he consumer can ge wih he bes ouside opion. To deermine he consumer s bes ouside opion, we need o compue for each locaion d he ne surplus when buying from firm. Following an undercuing argumen, firm will offer o he consumers locaed a firm s marke area a price equal o is marginal cos, p = u (from he zero profi condiion). The maximum ne surplus ha a ype θ s consumer can ge when buying from firm is he soluion o max u θu ( d) u. The opimal qualiy is u = θ and he maximum ne surplus is hen given by θ ( d), which is posiive only when d θ > θ = d θ.consideringk>,for< θ 8 all ype θ s consumers locaed a firm s marke area always ge posiive ne surplus bu ha does no always occur for ype θ s consumers. More precisely, ype θ ne surplus of buying from firm behaves as follows: 5 The previous resuls have similariies o Roche and Sole (00) and Villas-Boas and Schmid-Mohr (999). The laer has an ineresing exposiion abou he hree effecs ha should be considered wih mill nonlinear pricing: business sealing or demand effec, he margin effec and he screening effec. 6 To simplify noaion we will drop he argumen d for he remaining of his subsecion. However, one should keep in mind ha a each locaion a differen nonlinear conrac is se. 7 We would have a symmerical siuaion for he res of he line bu wih reversed firms posiions. 8 On accoun of he menioned mill nonlinear pricing and local monopoly analysis we will only focus on hese limis. 0

11 ype θ s consumers locaed a d buying from firm 0 << θ d 0, have posiive ne surplus θ d h0, d i θ have negaive ne surplus <θ h i d dθ, have posiive ne surplus θ < θ d 0, have negaive ne surplus ): Therefore, when 0 << θ firm mus consider he following individual raionaliy consrains (case IR θ : θū d p θ ( d) IR θ : θ u d p θ ( d) when θ < θ (case ): IR θ : θū d p θ ( d) IR θ : θ u d p 0 and when θ <θ, firm considers case consrains for consumers locaed a d h0, d i θ and case h i consrains for d dθ,. Noice ha if IRs hold hen boh ypes θ and θ, will no buy from firm marginal cos pricing (p,u ) and ( p, ū ), respecively. When firm does no sell o ype θ a some locaion d (wihin firm marke area), firm migh consider o sell only o ype θ a ha locaion. Firm should maximize p ū subjec o: and θū p < 0. IR θ : θū d p θ ( d).3. Nash equilibrium soluion Since ype θ s individual raionaliy consrains change wih and d, we mus solve wo differen maximizaion problems when firm sells o boh ypes 9. Case Low When 0 << θ, he unique soluion is characerized by ICs no binding and boh ypesofconsumersindifferen beween buying from firm or firm. In is marke area, firm offers he following wo bundles a locaion d: u = θ and p = θ d ū = θ and p = θ d 9 The KT condiions of firm s problems are described in Appendix B. For more deails on similar mahemaical compuaions see Vallei (00).

12 Thus, when differeniaion is low, firm offers o boh ypes he socially opimal qualiies and maches he ne surplus offered by firm. This ype of nonlinear conrac is offered in all locaions of firm s marke area. We call his soluion conrac he full compeiion one (FC) since firms have o give consumers he posiive ne surplus hey would aain by buying from he compeior. Since FC conracs are se a all locaions of firm s marke area, his scenario is called Reg. Case High When θ < θ, we achieve a similar resul o Pires and Sarkar (000) and Vallei (00). Depending on and k, here may be hree differen regions a firm s marke area: monopolisic M, inermediae I and compeiive C regions (see Figure ). 0 M I C d * d^ Figure : Three regions a firm s marke area, where d = 6 θθ 3 θ 4θ and ˆd = θθ θ θ. These regions are characerized by differen equilibrium nonlinear conracs 30. In all of hem firm sells o boh ypes: Region M I C Conracs u =θ θ and p = θ(θ θ) d ū = θ and p = θ (θ θ)(θ θ) d u = θ d ( θ θ) and p = θ θ d(θ θ) θ ( θ θ) d ū = θ and p = θ u = θ and p = θ d ū = θ and p = θ d Region M is he neares region from firm locaion (unil d ). In his region, firm offers ypical monopolis asymmeric informaion conracs: only IR θ and IC θ are binding, ype θ consumes he socially efficien qualiy bu ype θ consumes a sub opimal qualiy (his region will be aained only for low k). Beween d and ˆd, wehaveaninermediaeregion,regioni, whereir θ is also binding and firm faces more compeiion (firm s conrac offer is very aracive for ype θ). In his region, firms do no disor u as much as hey do a region M, because heir power o exrac ren from ype θ consumers is limied. Finally, closer o he marke s cener region C, onlyir θ and IR θ are binding and firms face vigorous compeiion. A region C, boh ypes ge he socially opimal qualiies and ype θ s consumers ge he highes ne surplus of all he marke. When firm sells only o ype θ, he conrac offered o θ is equal o he conrac se a regions I and C for ype θ. Typeθ does no wan o buy he good from firm under his conrac, hus we denoe his region by C θ. The profis achieved are never higher han profis obained by selling o boh ypes 30 Compuaion deails are available from he auhors on reques.

13 seing conrac M. However, for high differeniaion degrees (closer o N LM limi) he firm is beer off selling only o ype θ han selling o boh ypes and seing conracs I and C. Considering he condiions for regions s exisence and comparing profis of selling only o ype θ or o boh ypes we can draw a plo ha reveals he se of parameers where differen equilibrium soluion conracs hold (see Figure 3). 3 Reg* Reg** 4 Reg* 3 Reg** Reg* Figure 3: Duopoly Delivered Nonlinear Pricing Conracs when is High where: Reg C ([0,d C [) and C θ ([d C, ]) 3Reg I ([0, ˆd[),C ([ ˆd, d C [) and C θ ([d C, ]) 4Reg M ([0,d [),I [d, ˆd[,C ([ ˆd, d C [) and C θ ([d C, ]) Reg I ([0,d I [) and C θ ([d I, ]) 3Reg M ([0,d [),I ([d,d I [) and C θ ([d I, ]) d C = θ and di = 6 θθ 3 θ 4θ p 4( θ θ) ( θ 4θ 4 θθ ). Figure 3 shows ha when is high, closer o he marke cener firms sell only o ype θ consumers. Moreover, i shows ha he number of regions in a firm s marke area depends on he parameers values. Noice ha locaions d, ˆd, d C and d I depend on and k, which means ha he way firm s marke area is divided depends on he parameer values. Figure 4 illusraes how firm s marke area is divided for differen ses of parameers values. For example, when k =.5 and = θ, firm offers monopolis nonlinear conracs o consumers locaed beween 0 and d =0.5, offers inermediae conracs for consumers locaed beween d = 0.5 and ˆd = 0.375, and, finally, for consumers locaed beween ˆd =0.375 and d C =0.5 he firm offer compeiive conracs. The firs plo in Figure 4 shows ha, for low k, here is a region M a firm s marke area and ha as rises his region becomes larger, i.e., here are more locaions where he monopolisic nonlinear conrac is se. This resul is quie inuiive, when differeniaion increases here are more locaions where maching he ne surplus offered by he compeior o ype θ is no a binding consrain and hus compeiion is no effecive. However, when k ges higher, region M becomes smaller and smaller ill i disappears (see he various plos in Figure 4). This happens because when k is high, he wo ypes are very differen and he ne 3

14 Figure 4: Case - Firm s Marke Area surplus offered by he compeior o ype θ is much higher han he one offered o ype θ, which implies ha IR θ sars binding for locaions closer o he firm. Noice ha he se of locaions closer o he cener where firms sell only o ype θ increase wih bu also wih k (see curves d C and d I in Figure 4). Case 3 Inermediae h levels of For θ <θ he ouside opion of ype θ depends on consumer s locaion. For d 0, d i he bes ouside opion is no o buy and we make he same reasoning as in he h i las case. For d d, he bes ouside opion for θ is o buy from he compeior and we ge he same Nash equilibrium conrac as in case. Figure 5 shows he Nash equilibrium for low and inermediae levels of (cases and 3 ). 4 Reg 3 Reg Reg Reg Figure 5: Duopoly Delivered Nonlinear Pricing Conracs for low and inermediae 4

15 where: Reg F C ([0, ]) Reg C ([0, d θ [) and FC ([ d θ, ]) 3Reg I ([0, ˆd[),C ([ ˆd, d θ [) and FC ([ d θ, ]) 4Reg M ([0,d [),I [d, ˆd[,C ([ ˆd, d θ [) and FC ([ d θ, ]) As expeced, for low and inermediae levels of, firms offer FC conracs closer o he marke cener. I is ineresing o analyze how firm s marke area is divided depending on he parameer values (see Figure 6), for inermediae levels of d d d Figure 6: Case 3 - Firm s Marke Area For low k, here may be up o four submarke regions (M, I, C and FC). However, he monopolisic region only appears for high. As k increases, he less compeiive conracs (regions M and I) are offered a a smaller se of locaions, and for k high enough only C and FC conracs are offered (see Figure 6). Noice ha he locaions where FC conrac is se do no change wih k bu decrease wih. Discussion of resuls under delivered nonlinear pricing Under DNP he equilibrium conrac soluion may change wih he locaion of he consumer. The marke served by a firm may be subdivided in more han one region, according o he nonlinear conrac offered. The nonlinear conracs may differ in erms of he degree of inefficiency in he qualiy offered o ype θ consumers and wih respec o he ne surplus offered o each ype of consumers. The qualiy chosen is always he social opimal for ype θ bu may be opimal or subopimal for ype θ. When is very low, he ouside opion of buying from he compeior is very aracive a every locaion in he firms marke. Consequenly, a large ne surplus has o be offered o boh consumer ypes. Moreover, he ne surplus ha has o be offered o ype θ consumers in order for hem no o buy from he compeior is so high ha his ype of consumers do no wan o buy he efficien qualiy bundle offered o ype θ. Therefore, he qualiy offered o boh ypes is he socially efficien one, and he ne surplus offered o each ype is exacly he same hey would receive if hey bough from he compeior. In oher words, for very low differeniaion we ge full compeiion in all locaions of he marke. For inermediae levels of, he ouside opion of buying from he compeior becomes less aracive and we may sar having locaions (closer o he firm, far away from he compeior) where for ype θ consumers, he relevan ouside opion is no o buy he good. In hese locaions he firm will be able o exrac more consumer ne surplus from ype θ consumers (and also from ype θ consumers, whenever he incenive compaibiliy consrain is binding). For locaions closer o he marke cener we coninue o 5

16 have inense compeiion. For hese inermediae levels of, he number of submarke regions (each wih differen conracs) depends on k. When k is very high, he wo ypes are very differen, implying ha he ne surplus offered by he compeior o ype θ is much higher han he ne surplus offered o ype θ. In his circumsances, i is very likely ha he relevan consrain wih respec o ype θ is o mee he compeior ne surplus offer (incenive compaibiliy is no an issue). Consequenly, for large k he efficien qualiy is offered o ype θ a every locaion in he firm s marke. However, he conrac se is no hesameinalllocaions,becauseheousideopionforypeθ depends on wheher he consumer is close o he marke cener or no. In oher words, for inermediae and high k, conracs are eiher C (closer o he firm) or FC (closer o he marke cener). As k decreases we may sar having more submarke regions. The inuiion is ha for smaller k he ne surplus offered by he compeior o ype θ is no so much higher han he one offered o ype θ, which implies ha he incenive compaibiliy consrain IC θ may sar o be binding, especially as we consider locaions furher way from he marke cener. As soon as his happens, one sars having conracs where he qualiy offered o ype θ consumers is subopimal. For inermediae k we may have a region where u is efficien and a region where u is subopimal, bu wih a smaller disorion han if here was a monopoly. And for low k we may have monopolisic, inermediae and compeiive regions (bu always wih FC conracs closer o he marke cener). Finally, when is high, he relevan ouside opion for ype θ is no o buy in every locaion of he firm s marke. In his case, for locaions very close o he marke cener he firm is beer off selling only o ype θ consumers. Again, he number of marke regions depends on k (for very high k here exiss only a compeiive region, for very low k here are monopolisic, inermediae and compeiive regions. Noice ha, similarly o MNP,foragivenk, as riseswemoveolesscompeiiveconracs. On one hand, when k is low as rises region M becomes larger and he region C smaller ( d and ˆd boh posiive). On he oher hand, for higher levels of k, as rises compeiion is only relevan for ype θ consumers and, closer o he marke cener, firms do no sell o ype θ consumers. are 6

17 3 Circular ciy oligopoly model Since one of our objecives is o sudy he impac of he pricing policies on he marke srucure i is convenien o exend he previous resuls o a circular ciy model wih n firms. Consider n idenical firms simulaneously offering nonlinear conracs (price p i,qualiyu i ) o consumers disribued on a uni lengh circle. All firms have a fixed cos F and firms relocaion is cosless (i.e., we focus on spaial conesabiliy). We assume a locaion paern where firms are equidisanly locaedaroundhecircle 3 : an arc disance of n separaes every wo neighbor firms in he marke. Each firm i will always compee direcly wih wo neighboring firms (he firs firm locaed a is lef and he firs a is righ) implying ha every firm deals wih wo independen symmerical problems. We will look for he equilibrium conracs beween wo of he n firms (le d =0and d = n )sincehe reasoning would be similar for all he oher pairs of firms Mill nonlinear pricing Les us look for he case beween wo of he n firms when hey simulaneously offer an unique nonlinear conrac similar o.. From he local monopoly analysis (see Appendix A) here are no local monopolies a leas for one ype (ype θ) aslongas0 < < n θ (Oligopoly NLM condiion). Following he same procedure as above and assuming ha belongs o he previous inerval, we have o consider wo siuaions. 3.. Marke fully covered Given he nonlinear conracs when n firms cover all he marke for boh ypes, he locaion, for each ype, where he consumers are indifferen beween buying from firm or firm is now given by: d θ n = n θ(u u )p p Following an idenical opimizaion procedure, he equilibrium soluions are similar o he linear ciy duopoly model. Subsiuing by n a he prices and soluions inervals limis33,weachievehen firms equilibrium soluions and he respecive individual profi funcions: 3 Noice ha his assumpion is appealing since his locaion paern is one of he possible locaional equilibrium configuraions. When firms pracice delivered nonlinear pricing, firms choose locaions a he median of heir sales disribuion (Pires and Sarkar (000), Pires (00) and Hamilon and Thisse (99)). If firms are equally spaced in he circle, each firm s equilibrium sales disribuion will be symmeric around is locaion. Bu hen he firm s equilibrium locaion is a he median of is sales disribuion, hus he equally spaced configuraion is an equilibrium of he wo sage locaion-price game. Kas (995) shows ha equally spaced locaion configuraion is one of he possible equilibrium when firms pracice mill linear pricing. When firms pracice mill nonlinear pricing and are equidisanly locaed around he circle, if a firm i ges closer o he firm j (locaed a he righ of firm i) i will rise he disance beween firm i and firm l (locaed a he lef of firm i). As firms have compeing firms a boh sides of heir locaion, he business sealing effec from geing closer o one of he neighboring firmswillcanceloubygeingfarfromheoherfirm. When equally spaced in he circle, firms will minimize ransporaion coss and seal he bigges share of he consumer surplus possible. For more discussion on symmeric locaional equilibrium see Eaon and Wooders (985), Lederer and Hurer (986), MacLeod e al. (988) and Novshek (980). 3 Firms profi funcion is achieved by duplicaing he profis aained a he marke region in analysis. 33 For insance, C θ θ inervals limis - see appendix B - we ge 0 < n < θ 3 0 < nθ 3. 7

18 C θ θ C θ θ u = θ and p = θ n ū = θ and p = θ n u = θ and p = θ n ū = θ and p = θ n θ n n F n F 3.. Marke fully covered only for ype θ When firms cover all he marke for ype θ bu no for ype θ and considering ˆd θ (as definedinequaion ()), firm will now solve: µ µ max p ū d θ ( p,ū )(p,u ) n p u ˆd θ Mainaining he same k inervals limis for each soluion case bu also subsiuing by n a he prices and soluions inervals limis, we achieve he n firms equilibrium soluions and respecive individual profi funcions 34 : S θ u = θ and p = θ S θ LM θ LM θ n u = θ and p = θ n u = θ and p = 3 4 θ ū = θ and p = θ n u = θ and p = 3 4 θ ū = θ and p = θ n n θ n θ n F n F θ 4 8 n F n θ4 8 F 3..3 Nash Equilibrium soluion ³ We will also only have equilibrium soluions for all < n θ if k> 5 5. Afer KT soluion profis comparison for a given n and for all and parameer ypes raio k, we achieve a plo represening he equilibrium as a funcion of he parameer values idenical o he linear ciy duopoly one (Figure ), bu wherehescaleof is expressed in erms of n θ. All he described profi funcions are decreasing wih n for he parameers values where hey are he θ equilibrium soluion. For a given n, he profi funcions of C θ, C θ θ, S θ and LM θ are increasing wih whereas S θ and LM θ profis are decreasing wih. There saprofis disconinuiy beween LMθ and θ S θ, C θ and S θ and beween S θ and LM θ. 3. Delivered nonlinear pricing Suppose firms mus simulaneously choose nonlinear conracs - similar o.3 - for all he locaions in he circle. The reasoning is similar o he linear ciy duopoly case bu here he maximum ne surplus aained when buying from he mos direc compeiors is equal o θ ( n d). This change lead us o he following ypes of conracs: 34 Noice ha he profis from he possible bu undeermined soluions are equal o he profis of S θ and S θ and heir inervals limis belong o hese soluions inervals limis. 8

19 Region M I C FC Conracs equal o he linear ciy duopoly case u = n θ dn ( θ θ)n and p = nθ θ dn(θ θ) θ ( θ θ)n ū = θ and p = θ d n u = θ and p = θ d ū = θ and p = θ d n u = θ and p = θ d n ū = θ and p = θ d n The conclusions are analogue o he linear ciy duopoly case bu wih n inervals limis. Figure 7 summarizes he resuls. replacing a he soluions 3 Reg* Reg** 4 Reg 3 Reg 4 Reg* Reg* 3 Reg** Reg Reg Figure 7: Oligopoly Delivered Nonlinear Pricing Conracs Firms profi funcions for each equilibrium soluion can be wrien as: 9

20 Reg Reg 3Reg 4Reg Reg 3Reg 4Reg Reg 3Reg n R h i o n θ 0 d n θ θ d n θ dx = n F n ( nθ ) 4n F n ( nθ ) 4n [n( θθ θ θ )] 3 F 4n 3 ( θ θ) n ( nθ ) 4n [n(6 θθ 3 θ 4θ )] 3 [n( θθ θ θ )] 3 F n θ4 4 F n θ4 4 [n( θθ θ θ )] 3 F 4n 3 ( θ θ) n θ4 4n 3 ( θ θ) 4 [n(6 θθ 3 θ 4θ )] 3 [n( θθ θ θ )] 3 F 4n 3 ( θ θ) 4θ )) 3 (n θ )( ( θ θ) n) (n(6 θθ 3 θ 4θ )) (n(6 θθ 3 θ 4( θ θ) n 3 6n( θ θ) (n θθ n θ nθ ) 4( θ θ) n(n( 4 θθ θ 4θ ) ) n( θ θ) 4( θ θ) n 3 F 4( θ θ) n 3 4( θ θ) n 3 4θ )) 3 (n θ )( ( θ θ) n)3 (n(6 θθ 3 θ 4θ )) (n(6 θθ 3 θ 4( θ θ) n 3 6n( θ θ) (n θθ n θ nθ ) 4( θ θ) n(n( 4 θθ θ 4θ ) ) n( θ θ) 4( θ θ) n 3 F 4( θ θ) n 3 4( θ θ) n 3 All he described profi funcions are decreasing wih n for he parameers where hey are he equilibrium soluion. For a given n, he profi funcions of Reg, Reg and 3Reg are always increasing wih whereas in all he oher soluions profi funcions here are some parameers k / where hey are decreasing / increasing wih. There is no profis disconinuiy. One imporan resul in he circular ciy model, which holds boh under MNP and DNP, isha for given k and, whenn rises we move oward more compeiive soluions. In addiion, when n rises he NLM limi also rises. Tha is, for given k, whenmorefirmsenerhemarkewemushavehigher differeniaion (higher ) in order o ge local monopolies for boh ypes of consumers. If we compare profis for a given se of he exogenous parameers,, θ, θ and F, for a given marke srucure (given n), we ge ambiguous resuls. For cerain se of parameers values he wo pricing policies yield he same profis. For oher parameers values MNP yields higher profis. Finally, here are some se of parameers values for which DNP has higher profis (his happens a soluion conracs where he monopolisic region exiss under DNP, i.e., for small levels of k and close o he NLM limi). This las resul is ineresing because i conradics he asserion ha delivered pricing is more enry deerren han mill pricing. Our resuls show ha his asserion is only valid when compeiion is effecive a all locaions of he marke. If firms are allowed o pracise differen nonlinear prices a each locaion and compeiion is no relevan a locaions closer o he firm, he firm will offer monopolisic nonlinear conracs a hese locaions and may ge higher profis han under MNP. Since, for a given marke srucure, profis under MNP may be higher, equal or smaller han profis under DNP, we also expec ambiguous resuls in he comparison of he free enry long-run equilibrium number of firms under he wo pricing policies. 3.3 Pricing policies and marke srucure Le us now urn o he sudy of he marke srucure under he wo pricing policies. The quesion we wan o answer is: for given values of he exogenous parameers,, θ, θ and F,whichishefreeenry long-run equilibrium number of firms for MNP (n MNP )anddnp (n DNP )? 0

21 Ignoring ineger consrains, he free enry long-run equilibrium number of firms is such ha all firms in he marke ge zero profi. Thus, we jus need o deermine n such ha each firm ges a zero profi. However, his urns ou o be quie complex since he profi funcion varies from soluion conrac o soluion conrac, and which soluion conrac is relevan, for a given se of exogenous parameers, also depends on n. Wha his implies is ha, for each se of exogenous parameers, we need o idenify simulaneously which is he relevan soluion conrac (and consequenly profi funcion o use) and he number of firms such ha his profi iszero 35. Due o he complexiy of some profi funcions 36 and he need o guaranee ha he free enry long-run equilibrium number of firms and he profi funcionusedareinernally consisen, we had o compue numerically he free enry long-run equilibrium number of firms for each pricing policy (n MNP and n DNP ). Using Gauss programs, we compued, for differen se of parameers, θ, θ and F,hefree enry long-run number of firms for each of he possible equilibrium soluion conracs (for MNP and for DNP, separaely). Then we checked which of hose free enry long-run number of firms is really he free enry long-run number of firms consisen wih he parameers considered, i.e., for each of hose free enry long-run number of firms we checked if for he parameers, θ, θ and F chosen we would be a se of parameers where ha equilibrium soluion conrac holds. In secion 4 we compare he free enry long-run equilibrium number of firms under he wo pricing policies, n MNP and n DNP. 3.4 Socially opimal marke srucure The social ne surplus for consumer θ a each locaion d is given by θu d u. Simple compuaions reveal ha u = θ is he socially opimal qualiy and he social ne surplus from a consumer θ locaed a d is given by θ subjec o: d. Therefore, he social planner problem is easily described by: max n S,d θ,d θ " Z d θ n S 0 ( θ d) Z d θ 0 # ( θ d) n S F d θ n d θ n and n S,d θ,d θ non-negaive. Noice ha if d θ < n he marke is no fully covered for ype θ, hus he previous formulaion allows for he possibiliy of no covering all he marke for one or for boh ypes. The socially opimal number of firms depends on he parameer values. When F θ4 i is socially opimal o cover all he marke for boh ypes and he opimal number of firmsisgivenby: r n S = F 35 For some se of parameers values, MNP may have wo free enry long-run equilibrium number of firms soluions since is profi funcions happen o be disconinuous. 36 Paricularly, DNP soluions Reg and 3Reg need nonlinear equaions sofware o solve i.

22 When θ4 F θ 4 θ 4 4 firms is: he marke should no be fully covered for ype θ and he opimal number of n S = p 4F θ 4 Finally, for F> θ 4 θ 4 4 i is socially opimal no o serve he marke (n S =0). In secion 4 we compare he socially opimal number of firms, n S, wih he free enry long-run equilibrium number of firms under he wo pricing policies, n MNP and n DNP. 3.5 Welfare comparisons Wheher discriminaory nonlinear pricing should be prohibied or no depends on which of he wo pricing policies is beer in welfare erms. Noice ha his involves comparing welfare under MNP and under DNP for differen values of n, since he long-run marke srucure under he wo pricing policies may be differen. Le us firs derive he welfare funcion for each pricing policy and hen compare welfare for he various parameers values 37. Wih n firms he aggregae oal surplus under mill nonlinear pricing, W MNP (n), isgivenby: Conrac θ θ C θ =C θ S θ =S θ LM θ =LMθ Welfare W MNP (n) θ θ θ θ 3nθ4 6 n nf 4n nf 4n nf On he oher hand, he welfare under delivered nonlinear pricing, W DNP (n), in each of he parameer regions is given by: 37 Noice ha under boh pricing policies he qualiy offered o ype θ is always he socially opimal qualiy (ū = θ). As a consequence, for a given n, welfaredifferences beween he wo pricing policies are due exclusively o differences wih respec o ype θ consumers ne surplus.