Equilibria in a Capacity-Constrained Di erentiated Duopoly

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1 Equilibria in a Capacity-Constraind Di rntiatd Duopoly Maxim Sinitsyn McGill Univrsity Prliminary and incomplt. Abstract In this papr I analyz th modl of pric-stting duopoly with capacity-constraind rms producing di rntiatd products. I show that a pur stratgy quilibrium xists if th capacitis of th rms ar both small or both larg. For th othr capacity valus, mixd stratgy quilibria xist, with th support of th pric distribution ithr bing an intrval or consisting of a nit numbr of points. I show that if bfor th pric comptition th rms simultanously choos thir capacity lvls, thir optimal choic of capacitis would lad to a pur stratgy quilibrium in prics. Th optimal capacity lvls dcras with an incras in consumr htrognity. Introduction Rcnt advancs in stimations of dmands for di rntiatd products ncssitat rsarch in th supply sid of th markt how th rms act whn facing such dmands. Whil this qustion has bn studid thoroughly for th rms that ar not constraind in thir production, th litratur on th bhavior of th capacity-constraind rms producing

2 di rntiatd products is scarc. In this work, I charactriz and comput th quilibria that occur in th situations, whn th rms ar capacity-constraind. Th litratur on pric comptition with capacity constraints initially considrd only homognous products (Brtrand-Edgworth comptition). Du to discontinuitis in dmand and pro t functions, pur stratgy quilibria do not always xist, thus, it was ncssary to work with mixd stratgy quilibria. First, th quilibria in symmtric Brtrand- Edgworth modls wr computd for th proportional (Bckmann (965)) and surplusmaximizing (Lvitan and Shubik (972)) rationing ruls for unsatis d dmand. rps and Schinkman (983) charactrizd th mixd stratgy quilibrium for asymmtric capacitis using th surplus-maximizing rul. This allowd thm to solv a two-stag gam, in which th rms rst choos capacitis and thn compt in prics. Thy found that th rms choic of capacitis in th rst stag coincids with th Cournot quilibrium th production lvls th rms would choos if thy wr to compt in quantitis. Davidson and Dnckr (986) argud that this rsult crucially dpnds on th assumption of th surplus-maximizing rationing rul, and it dos not hold for almost all othr rationing ruls. Thy showd that for th proportional rationing rul, th rms ncssarily will choos capacitis that xcd th Cournot lvl, and th quilibria ar asymmtric if th capacity cost is small. Furthrmor, th mixd stratgy quilibria for th gnral form of th dmand function wr charactrizd by Osborn and Pitchik (986) for th surplus-maximizing rationing rul and Alln and Hllwig (993) for th proportional rationing rul. Only rcntly, modls of pric comptition with capacity constraints wr xamind for di rntiatd products (Brtrand-Edgworth-Chambrlin comptition). Bnassy (989) provd that a pur stratgy quilibrium in ths modls fails to xist if th dgr of substitutability of th products is larg nough. In a rlatd papr, Canoy (996) providd a paramtrizd duopoly xampl in which a pur stratgy quilibrium dos not xist if Bnassy (989) also shows that for a givn dgr of product substitutability a pur stratgy quilibrium xists if th numbr of comptitors is larg nough. 2

3 th products ar su cintly similar. Thus, it was stablishd that whn th products ar su cintly homognous only th mixd stratgy quilibrium xists. Howvr, no attmpt has bn mad to charactriz and study th quilibrium in mixd stratgis for th Brtrand-Edgworth-Chambrlin comptition. Undrtaking this study is th goal of this papr. I tak th dmand to hav a logit spci cation, which is a form of dmand for di rntiatd products, widly usd in mpirical litratur. Th rms produc at a constant marginal cost, but ar limitd in production by thir capacity constraints. I show that a pur stratgy pric quilibrium xists whnvr th rms capacitis ar both small or both larg. For th othr valus of capacitis, a pur stratgy pric quilibrium dos not xist, and I comput th mixd stratgy quilibrium. For th cas of symmtric capacitis in th intrmdiat rang, this quilibrium involvs a nit support for th optimal pric distributions. For th cas of asymmtric capacitis, thr ar also mixd quilibria, for which th support of th pric distributions is an intrval. Th knowldg of th rms pro ts in th pricing gam allows m to nd th quilibrium in a two-stag gam, whr th rms rst choos th capacity lvls. Th optimal capacity lvls lad to pur stratgy pricing in th scond stag. Th capacitis incras with a dclin in consumr htrognity. 2 Basic Modl Considr a markt whr a particular good is producd by 2 rms at a zro marginal cost. Firms compt in prics p. Th st of consumrs has masur : Consumrs hav htrognous tasts for th goods producd by th rms. Each consumr rcivs utility U i = p i +" i from purchasing a product from rm i. " i is iid standard doubl xponntial. is th masur of consumr htrognity. An outsid option givs th consumrs a utility of U = + ". Thus, an outsid option could b considrd as anothr product which is sold at a xd pric of. Consumrs purchas th product that givs thm th 3

4 highst utility. Thn, th dmand facd by rm i is a standard logit dmand: G i (p i ; p j ) = p i = p i = + p j = + =. On th supply sid, assum that both rms hav zro marginal cost, but can produc only up to capacity i. Givn th natur of th consumrs utility functions th rationing rul for unsatis d dmand is proportional. If th rst rm s dmand G (p ; p 2 ) is gratr than, th rmaining ( ) consumrs hav iid draws for " and " 2, thus, th rsidual dmand is ( ) p 2 = p 2 = + =. In summary, th contingnt dmand of rm i is Gi (p i ; p j ) if G j (p j ; p i ) 6 j G i (p i ; p j ) = ( j ) p i = if G p i = + = j (p j ; p i ) > j This sction dals with th symmtric cas whn both rms hav th sam capacity constraint = = 2, and th nxt sction will addrss th cas 6= 2. First, I will xamin for what valus of and a pur stratgy quilibrium xists. Thr ar two di rnt cass: (a) both rms produc at th capacity constraint; and (b) both rms produc at th lvl blow th capacity constraint. Production at th capacity constraint. A singl-pric quilibrium with both rms oprating at capacity constraints xists whn th paramtrs satisfy th following conditions 2 : < + ln, if 2 2 >, if 2 + ln 2 + ln > () Th prics chargd by th rms ar 2 All drivations ar in th Appndix. p = ln 2 (2) 4

5 ln + ln 2 2 = for = a : For th smallr valus of ; is lss than zro. Thrfor, a singl pric quilibrium xists for all lvls of consumr htrognity whn th capacity constraints ar small ( is lss than a ). For th largr valus of, whn th consumr htrognity is larg nough, a singl pric quilibrium with both rms raching thir capacity lvls fails to xist. Whn is qual to zro or is ngativ and > a, this typ of a singl pric quilibrium nvr xists. Rgion a in gur shows th paramtr valus for which () holds. Figur : Typs of quilibrium in th symmtric cas for = a b µ Production at th lvl blow th capacity constraint. 5

6 If both rms ar producing at th lvl blow thir capacity constraint, thy charg th prics p that satisfy th following quation: Th rm arns a pro t of = p G i (p ; p ). = p p + 2 p (3) + For ths prics to b a global maximum, must b highr than th monopoly pro t on rsidual dmand. Anothr potntial local maximum could occur at a highr pric bp, whn th rival is capacity-constraind. This pric solvs th quation = bp and givs a pro t b = bp( ) bp bp + bp + ; (4). p is a global maximum only if > b or if th rival is not capacity-constraind whn th pric is bp. That is, th following inqualitis must hold: > b or G i (p ; bp) 6 ; (5) whr p is th solution to (3), and bp is th solution to (4). Th rgion whr both rms ar producing at th lvl blow thir capacity constraint is dpictd in Figur as b. As gos to in nity, th boundary of th rgion b convrgs to th valu b. is possibl to nd th valu of b. First, (3) could b rwrittn as = p Whn!,!. So, as = bp bp +, from whr bp dscribd by th quation bp = b or bp( ) bp + bp bp = = p 2 p p +! b :858 as!. bp + It p + 2 p. + p! t :2268. Similarly, (4) could b rwrittn! q :2785. Th boundary of th rgion b is 2 = p p p +, from whr. Using th valus for t and q found prviously, 6

7 Th following statmnts summariz th ndings about th xistnc of a pur stratgy quilibrium: ) For th low lvls of th capacity constraint ( 6 a :78855) a pur stratgy quilibrium always xist with both rms producing at th capacity constraint lvl and charging pric p from (2). 2) For th vry small rgion of low capacitis ( a < 6 b :858) a pur stratgy quilibrium xists for th low nough lvls of consumr htrognity (rgion a in Figur ). For th high lvls of a pur stratgy quilibrium dos not xist. 2) For th intrmdiat lvls of th capacity constraint ( b < < :5) a pur stratgy quilibrium xists for th high lvls of consumr htrognity (rgion b in Figur ) and for th low lvls of consumr htrognity (rgion a in Figur ). Thr always xist intrmdiat valus of, for which a pur stratgy quilibrium dos not xist. 3) For th high lvls of th capacity constraint ( > :5) a pur stratgy quilibrium xists only whn th consumr htrognity is high nough (rgion b in Figur ). 4) For any lvl of th consumr htrognity a pur stratgy quilibrium xists for th low lvls of capacity constraint (rgion a in Figur ) and for th high lvls of capacity constraint (rgion b in Figur ), but not for th intrmdiat valus. As dcrass, th rgion, whr a pur stratgy quilibrium dos not xist, incrass. Ths conclusions ar robust to th changs in. If is incrasing, rgion a incrass in siz with its boundary approaching th vrtical lin at :5 whil rgion b dcrass. If is approaching zro, rgion a dcrass in siz with its boundary approaching zro for th valus of > a, whil rgion b incrass. In th ara, whr a pur stratgy quilibrium dos not xist, a mixd stratgy quilibrium xists. 3 Thorm in Sinitsyn (26) shows that if th dmands ar analytic, a mixd stratgy pric quilibrium has to hav a nit support. In our cas, th dmands ar not analytic thy hav a kink at th pric which maks th rival capacity-constraind. 3 Th xistnc rsult is du to Glicksbrg (952). 7

8 Nvrthlss, I found that for th symmtric cas a mixd stratgy quilibrium also has a nit support. Figur 2 illustrats th pricing stratgis of th rms for th small capacity lvls. Figur 2: Equilibrium pricing stratgis and pro ts for = :4. p 2 Prics p.9 p p 3 p 4 p µ Probabilitis γ γ.5 γ 2 γ 3 γ 4 γ µ Prof it π µ.2. I x to b :4 and chang th lvl of consumr htrognity. I start with a rlativly high lvl of ( = :6) and thn dcras it. For th high valus of a pur stratgy pric quilibrium xists (th paramtrs fall in rgion b from Figur ) with both rms oprating undr thir capacity constraints. As dclins, th dmands bcom mor lastic, th quilibrium mor comptitiv, and th optimal prics dcras. Finally, th prics rach th lvl p, whr it bcoms qually pro tabl for th rms to dviat to th high pric p as (p; p) = (p; p) ( is approximatly qual to :5). This 8

9 point is th boundary of th rgion b from Figur. If kps dcrasing, a pur stratgy quilibrium dos not xist anymor as th rms prfr dviation to th high pric. Howvr, an quilibrium with two prics appars. Each rm chargs two prics p and p 2 with corrsponding probabilitis and 2. From gur 2 it is vidnt that whn th two prics ar chargd th lowr of two prics p dos not dcras as stply as in th rgion with on pric only. This mans that if p wr th only pric chargd, not only thr would b a pro tabl dviation to th highr pric p 2, but also thr would b an incntiv to undrcut p. This dos not happn bcaus th prsnc of a highr pric p 2 chargd with som probability 2 rducs th undrcutting incntivs. Similarly, a pric p 2 chargd with probability will also b undrcut, but th fact that thr is a lowr pric p chargd with som probability kps th high pric at p 2. This happns bcaus in th nighborhood of p 2 th rival charging a lowr pric is always capacity-constraind, so thr is an incntiv to charg a pric highr than p 2. In quilibrium, this incntiv is xactly o st by th undrcutting incntiv and th optimal pric is p 2. As kps dcrasing, th incntiv to undrcut also incras. This puts a gratr wight on p 2 ( 2 incrass) and dcrass both prics. Surprisingly, th pro t incrass slightly as th ct of putting a gratr wight on a highr pric p 2 outwighs th ct of dclining prics. Finally, dclins to such a lvl (approximatly :345 in Figur 2) that again a pro tabl dviation appars. Th rms start charging thr prics and th cycl kps rpating. In Figur 2 I calculatd th mixd stratgy quilibria with up to v prics chargd by th rms. It could b obsrvd that th rang of th prics chargd dcrass and nally collapss to a point whn rachs approximatly :27. This is a boundary point of th rgion a from Figur so both rms oprat at th capacity constraint and th singl pric quilibrium rappars. As could b sn from (2), th optimal pric p will incras with dclin of only if ln 2 is positiv, which holds for > =3. Sinc in Figur 2 = :4, th optimal pric and th pro t incras as dclins. 9

10 Figur 3 illustrats th optimal pricing stratgis for th larg capacity lvls ( is takn to b :75 for this gur). Figur 3: Equilibrium pricing stratgis and pro ts for = :75: Prics p p 2 p 3 p 4 p γ µ Probabilitis π µ Profit.2 γ 2 γ3 γ4 γ π µ Th movmnt of prics and th mrgnc of nw quilibria with multipl prics is similar to th procss dscribd for th small capacity lvls and illustratd in Figur 2. Th only major di rnc btwn th cass with small ( < :5) and larg capacitis is that for th larg capacitis, th quilibria with multipl prics do not convrg to a singl pric quilibrium as dcrass. Instad thy approach a mixd stratgy quilibrium with an intrval for a pric support. In Figur 3 th pric rang of th mixd stratgy quilibrium for = is illustratd by th sgmnt [p; p] and th corrsponding pro t is labld. Figur shows th rgions, whr mixd stratgy quilibria with th rms charging multipl prics xist. Each rgion is numbrd in accordanc with th numbr of prics th

11 rms us. In th unshadd rgion in Figur both rms us mor than v prics in th quilibrium. 3 Asymmtric Capacitis Now, I will considr th cas 6= 2. This will srv as a building block for solving th two-stag gam, in which th rms rst choos capacitis and thn compt in prics. Pur stratgy quilibria Th cas whr both rms produc at thir capacity lvl or both rms produc at th lvl blow thir capacity is handld xactly th sam way as in th symmtric cas. Figur 4 shows th rgions whr such quilibria xist as a and b, corrspondingly. Figur 4: Typs of quilibria in th cas of asymmtric capacitis for = and = : (2; ) b.7.6 (3; 2) (2; 2).5.4 (3; 3) (2; 3) (; 2).3 a

12 It sms intuitivly plausibl that thr could xist pur stratgy quilibria with on (largr) rm producing blow its capacity lvl and anothr (smallr) rm producing at its capacity lvl. Howvr, th following proposition provs that such quilibria do not xist. Proposition A pur stratgy quilibrium with on rm producing at th lvl blow its capacity constraint, and anothr rm producing at its capacity constraint dos not xist. Proof. Without loss of gnrality assum that th rst rm producs at th lvl blow its capacity constraint ( G (p ; p 2 ) < ) and th scond rm producs at its capacity constraint ( G 2 (p 2 ; p ) = 2 ). I will xamin th bhavior of th pro t function of th rst rm at p. If th rst rm chargs a pric lowr than p th scond rm oprats at th lvl blow its capacity constraint, othrwis th scond rm could pro tably incras its pric. Thrfor, to th lft of p th pro t function of th rst rm is (p ; p 2 ) = p G (p ; p 2 ). Whn th rst rm chargs a pric abov p th scond rm is capacityconstraind, thus th pro t to th right of p is + (p ; p 2 ) = ( 2 )p p = p = + = = ( 2 )p b G (p ), whr b G (p ) = p = p = + =. Th lft hand sid drivativ of th pro t function at p is qual (p ;p 2 ) = G (p ; p 2 ) p (p ;p 2 )( G (p ;p 2 )). Th right hand sid drivativ of th pro t function at p 2 is qual +(p ;p 2 = ( 2 ) b G (p ) p ( 2 ) b G (p )( b G (p )). Using th fact that ( 2 ) b G (p ) = G (p ; p 2 ), w + (p ;p 2 (p ;p 2 = p (p ;p 2 )(G (p ;p 2 ) G b (p )) >. Thus, th right hand sid drivativ of th rst rm s pro t function at p is gratr than th lft hand sid drivativ. Th lft hand sid drivativ has to b gratr or qual to zro at p (othrwis, thr xists a maximum to th lft of p ), thus, th right hand sid drivativ has to b strictly gratr than zro. This mans that th pro t function of th rst rm incrass to th right of p, so p can not b th maximum. Thrfor, pur stratgy quilibria xist only in rgions a and b in Figur 4. For all othr valus of and 2 it is ncssary to sarch for th mixd stratgy quilibria. Mixd Stratgy Equilibria 2

13 Figur 4 shows svral rgions, in which mixd quilibria xist. Th numbrs in brackts show th numbr of prics chargd by ach rm, i.. (3; 2) indicats that th rst rm chargs 3 prics, and th scond rm chargs 2 prics. Th larg spacs in th uppr-lft and bottom-right corrspond to th mixd stratgy quilibrium, in which th support of th pric distributions is an intrval. A complt charactrization of ths quilibria and prcis boundaris of th rgion, whr thy xist, ar to b addd. 4 Choic of Capacitis (prliminary) Now, considr a two-stag gam, in which th rms rst costlssly accumulat capacitis, and thn compt in prics. Figur 5 illustrats how th choic of th capacity of th scond rm a cts its pro ts givn that th rst rm s capacity is xd. It could b sn from Figur 5 that th optimal choic of capacity for th scond rm rst dclins as dcrass, but always stays smallr than. It kps dclining until = 2 = that solvs + ln 2 2 =. Aftrwards, th optimal valu of 2 incrass. Thus, thr is a uniqu symmtric quilibrium in a two-stag gam. First, both rms choos th capacity lvl that solvs + ln 2 2 =. Thn, thy both charg prics p from (2). To b addd. 5 Conclusion To b addd. 3

14 Figur 5: Pro ts of th scond rm for th di rnt capacity lvls of th rst rm..5 =.5.4 =.3.3 =.45 =.6.2 = Appndix Production at th capacity constraint. First, I will nd pric p, at which th capacity constraint is rachd. G i (p ; p ) =, so p = =. Thn, p = ( 2) = V=, from whr p = ln, 2 p = + = 2 and (2) follows. Now I nd to stablish conditions undr which p is th global maximum. If th rm chargs any pric p blow p it will b capacity-constraind, so its pro t at p will b blow th on at p. In ordr to chck that no prics abov p rsult in a highr pro t it is nough to chck that th right-hand sid drivativ of th pro t function i (p i ; p ) = p i Gi (p i ; p ) 4

15 is ngativ at p. i (p ; p ) = G i (p ; p ) + G i (p ; p ) (6) Whn p i is gratr than p, Gi (p i ; p ) = ( ) p i = p i = + =. xampl, Andrson and d Palma (992), Lmma ) that if i = p i = It is known (s, for p i = + = is th probability givn by th multinomial logit, i = i( i ). In our cas, = G i (p ; p ) = ( ) p i = p i = + =, so G i (p ;p ) pi= =. Thus, p i = + G i (p ; p ) = ( ) from (7) and p from (2) into (6), w i (p ; p ) = + i (p ;p ) is lss than whn + 2 ln 2 This is quivalnt to (). 2! = ( 2) ( ) Production at th lvl blow th capacity constraint. ( 2) ( ) (7) (8) ( 2) < or ( ) > 2 +ln 2. If both rms oprat blow thir capacity constraints, thn G i (p i ; p j ) = G i (p i ; p j i (p i ;p j ) = G i (p i ; p j ) p i (p i ;p j ) = G i (p i ; p j ) = p ( G i (p ; p )), which is quivalnt to (3). p i G i (p i ;p j )( G i (p i ;p j i (p ;p ) =, so Firm i can choos to charg a highr pric p i for which its rival is capacity-constraind (G j (p ; p i ) > ). For such pric th pro t is i (p i ) = ( ) p i p i + sam argumnt as in th paragraph abov, th optimal pric bp should solv = bp : Following th p i + 4 i (p i ; p j ) = p i Gi (p i ; p j ), whr G i (p i ; p j ) is logit has a uniqu maximum, thus onc it starts dcrasing, it dcrass forvr.. 5

16 Rfrncs [] Alln, B. and M. Hllwig (993). Brtrand-Edgworth Duopoly with Proportional Rsidual Dmand, Intrnational Economic Rviw, 34(), [2] Andrson, S. P. and A. d Palma (992). "Th Logit as a Modl of Product Di rntiation," Oxford Economic Paprs, 44(), [3] Bckmann, M. (965). Edgworth-Brtrand Duopoly Rvisitd, in R. Hnn, d., Oprations Rsarch Vrfahrn III, Misnhim: Vrlag Anton Hin, [4] Bnassy, Jan-Pascal, "Markt Siz and Substitutability in Imprfct Comptition: A Brtrand-Edgworth-Chambrlin Modl," Th Rviw of Economic Studis, 56(2), 989, [5] Canoy, Marcl (996). "Product Di rntiation in a Brtrand-Edgworth Duopoly," Journal of Economic Thory, 7, [6] Davidson, C. and R. Dnckr (986). Long-Run Comptition in Capacity, Short- Run Comptition in Pric, and th Cournot Modl, Th RAND Journal of Economics, 7(3), [7] Glicksbrg, I. L. (952), "A Furthr Gnralization of th akutani Fixd Point Thorm with Application to Nash Equilibrium Points," Procdings of th Amrican Mathmatical Socity, 38, [8] rps, D. M. and J. A. Schinkman (983). Quantity Prcommitmnt and Brtrand Comptition Yild Cournot Outcoms, Bll Journal of Economics, 4(2), [9] Lvitan, R., and M. Shubik (972). Pric Duopoly and Capacity Constraints, Intrnational Economic Rviw, 3(), -22. [] Osborn, M. J. and C. Pitchik (986). Pric Comptition in a Capacity-Constraind Duopoly, Journal of Economic Thory, 38(2),

17 [] Sinitsyn, Maxim (26). "Charactrization of th Mixd Stratgy Pric Equilibria in Oligopolis with Htrognous Consumrs," McGill Univrsity Working Papr. [2] Vivs, Xavir. "Oligopoly Pricing: Old Idas and Nw Tools," Cambridg, Mass.: MIT Prss

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