Product Differentiation with Multiple Qualities

Ba igozzi, F ancesca; Ma, Ching- o Albe Wo king Pape P oduc Di e en ia ion wi h Mul iple Quali ies Quade ni - Wo king Pape DSE, No. 1075 P o ided in Coope a ion wi h: Uni e si y o Bologna, Depa men o Economics Sugges ed Ci a ion: Ba igozzi, F ancesca; Ma, Ching- o Albe (2016) : P oduc Di e en ia ion wi h Mul iple Quali ies, Quade ni - Wo king Pape DSE, No. 1075, Alma Ma e S udio um - Uni e si à di Bologna, Dipa imen o di Scienze Economiche (DSE), Bologna, h ps://doi.o g/10.6092/unibo/amsac a/5407 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/159913 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. 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I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by-nc/3.0/ ISSN 2282-6483 P oduc Di e en ia ion wi h Mul iple Quali ies F ancesca Ba igozzi Ching- o Albe Ma Quade ni - Wo king Pape DSE N°1075 P oduc Di¤e en ia ion wi h Mul iple Quali ies F ancesca Ba igozzi Ching- o Albe Ma Depa men o Economics Depa men o Economics Uni e si y o Bologna Bos on Uni e si y ancesca.ba [email protected] [email protected] Augus 30, 2016 Abs ac We s udy subgame-pe ec equilib ia o he classical quali y-p ice, mul is age game o e ical p oduc di¤e en ia ion. Each … m can choose he le els o an a bi a y numbe o quali ies. Consume s’ alua ions a e d awn om independen and gene al dis ibu ions. The uni cos o p oduc ion is inc easing and con ex in quali ies. We cha ac e ize equilib ium p ices, and he equilib ium e¤ec s o quali ies on he i al’s p ice in he gene al model. We p esen necessa y and su¢ cien condi ions o equilib ium di¤e en ia ion in any o he quali ies. Keywo ds: mul idimensional p oduc di¤e en ia ion, quali y and p ice compe i ion JEL: D43, L13 Acknowledgemen : Fo hei commen s, we hank So onis Cle ides, Ba d Ha s ad, Hen y Mak, Kje il S o esle en and semina pa icipan s a Bos on Uni e si y, he Uni e si y o Oslo, and he 14 h In e na ional Indus ial O ganiza ion Con e ence in Philadelphia. 1 In oduc ion Fi ms using di¤e en ia ed p oduc s o so en in ense Be and p ice compe i ion is a basic p inciple in indus ial o ganiza ion. Following Ho elling (1929), D’Asp emon , Gabszewicz and Thisse (1979) cla i y heo e ical issues and sol e he basic ho izon al di¤e en ia ion model. Gabszewicz and Thisse (1979), and Shaked and Su on (1982, 1983) wo k ou equilib ia o he basic e ical di¤e en ia ion model. The s anda d model o ho izon al- e ical p oduc di¤e en ia ion is he ollowing mul is age game be ween wo … ms: in S age 1, … ms choose p oduc a ibu es, in S age 2, … ms choose p ices, and hen consume s pick a … m o pu chase om. In he li e a u e, models ha e seldom gone beyond wo possible quali ies, ha e assumed ha consume s’quali y alua ions a e uni o mly dis ibu ed, and ha e le p oduc ion o misma ch cos s be nonexis en , linea , o quad a ic. We make none o hese assump ions. In his pape , each o wo … ms p oduces goods wi h an a bi a y numbe o quali y a ibu es. Consume s’ alua ions on each quali y ollow a gene al dis ibu ion. A … m’s uni p oduc ion cos is an inc easing and con ex unc ion o quali ies. In his gene al en i onmen , we ully cha ac e ize subgame-pe ec equilib ia o he s anda d di¤e en ia ion model. Using he uni o m quali y- alua ion dis ibu ion and he sepa able cos assump ions, esea che s ha e managed o sol e o equilib ium p ices explici ly as unc ions o quali ies. Equilib ium quali ies hen can be cha ac e ized. Wha has eme ged in he li e a u e a e a ew classes o equilib ia wi h la ges o smalles di¤e ences in equilib ium quali ies (see he li e a u e e iew in he ollowing subsec ion). Indeed, in equi- lib ium … ms may success ully di¤e en ia e hei p oduc s in some quali ies, bu may ail o do so in o he s. Ob iously, ac abili y e sus gene ali y is he challenge ha has been posed by he li e a u e. How obus a e maximal o minimal di¤e en ia ion esul s? To wha ex en a e hey d i en by hese assump ions? In his pape we sol e he ac abili y-gene ali y dilemma. Fo he quali y-p ice, mul is age game, we comple ely cha ac e ize subgame-pe ec equilib ia. Fi s , we …nd ou how quali ies change equilib ium p ices— wi hou sol ing o he equilib ium p ices explici ly in e ms o quali ies. In o he wo ds, we do no need he usual assump ions in o de o compu e equilib ium p ices explici ly. Second, we iden i y wo sepa a e e¤ec s o he cha ac e iza ion o equilib ium quali ies. The … s is wha we call he p ice- eac ion 1 e¤ec , which is how a … m’s quali y in S age 1 a¤ec s he i al … m’s p ice in S age 2. The second is wha we call he Spence e¤ec (because i is o iginally exposi ed in Spence (1975); see oo no e 7). Fo maximum p o… , a … m chooses a quali y which is e¢ cien o he consume who is jus indi¤e en be ween buying om he … m and i s i al. Consume s mus buy om one o he wo … ms, so … ms sha e he same se o indi¤e en consume s. The Spence e¤ec says ha each … m should choose hose quali ies ha a e e¢ cien o he equilib ium se o indi¤e en consume s. The Spence e¤ec alone is a mo i a ion o minimal p oduc di¤e en ia ion: o each quali y, … ms should choose he same le el. Bu he e is a second e¤ec a wo k. In he s anda d mul is age game, p ices a e s a egic complemen s. In S age 1, each … m would like o use i s quali ies o induce he i al o choose a highe p ice in S age 2. The i al … m’s s a egic conside a ion, howe e , is exac ly he same. Hence, he wo … ms engage in a ace. Fo each quali y, he ela i e s eng h o he p ice- eac ion e¤ec de e mines which … m will, in equilib ium, choose a highe quali y han i s i al. Fo gene al dis ibu ions o quali y alua ions and a gene al cos unc ion, i is en i ely possible o he ela i e s eng h o a y ac oss di¤e en quali ies. The p ice- eac ion e¤ec implies ha … ms may di¤e en ia e hei p oduc s in many ways. La ges o smalles di¤e ences in equilib ium quali ies a e no obus . Ins ead, we p o ide a ull cha ac e iza ion o how equilib ium quali ies may di¤e be ween … ms. Ou esul s say ha … ms choosing he same le el o a quali y holds unde e y es ic i e condi ions: cos is sepa able ac oss quali y a ibu es oge he wi h consume s’quali y alua ions being uni o mly dis ibu ed (see Co olla y 3 below). Howe e , hese ha e been he exac assump ions used in he li e a u e. One casually obse es ha quali y a ibu es ac oss p oduc s a e ne e exac ly he same. Thus, … ms ha p oduce “high- end”p oduc s will s ill di¤e en ia e— i only in small de ails o hei p oduc quali ies. Fo example, BMW and Lexus a e companies ha di¤e en ia e e en in he high quali ies o hei ca s. All BMW and Lexus ca s a e high-quali y au omobiles, bu he common consensus is ha BMW has a highe “pe o mance” quali y han Lexus, bu he opposi e is ue when i comes o he ”com o ”quali y. Howe e , any ca by BMW o Lexus will be a be e pe o me and mo e com o able han any ca by Yugo. In ac , in he au omobile (and mos o he ) ma ke s, i is impossible o …nd p oduc s ha ha e iden ical quali y a ibu es. 2 These obse a ions a e consis en wi h he gene al ene o p oduc di¤e en ia ion, bu inconsis en wi h assump ions o uni o m quali y- alua ion dis ibu ion and sepa able cos unc ions. Wha is behind ou solu ion o he ac abili y-gene ali y dilemma? Ou key inno a ion is o show ha sol ing o he equilib ium p ices as unc ions o quali ies is equi alen o sol ing a single in eg al equa ion. The solu ion o he in eg al equa ion yields he equilib ium se o indi¤e en consume s, and hence he … ms’ demands. We ob ain his solu ion as an implici unc ion o he model p imi i es. Then a … m’s equilib ium p ice can be exp essed explici ly in e ms o quali ies, h ough he solu ion o he in eg al equa ion. In o he wo ds, we dispense wi h he need o explici compu a ion o equilib ium p ices, which would equi e explici speci…ca ion o he model p imi i es, a common esea ch s a egy in he li e a u e. Judicious analysis o mu ual quali y bes esponses, gi en quali ies’impac on equilib ium p ices, yields he undamen al p ice- eac ion and Spence e¤ec s. Fo many examples, he in eg al equa ion does admi an explici solu ion ( ha will be demons a ed la e ). We use a e ical di¤e en ia ion model, bu C eme and Thisse (1991) show ha o he usual model speci…ca ion, he Ho elling, ho izon al di¤e en ia ion model is a special case o he e ical di¤e en ia ion model. The in ui ion is simply ha … ms’demand unc ions in a Ho elling model can be di ec ly ansla ed o he demand unc ions in a e ical, quali y model. C eme and Thisse (1991) s a e he esul o a single loca ion o quali y dimension, bu hei esul ex ends s aigh o wa dly o an a bi a y numbe o such dimensions. (A model wi h a combina ion o ho izon al and e ical dimensions can also be ansla ed o a model wi h only e ical dimensions.) Hence ou esul s in his pape apply o ho izon al-di¤e en ia ion models. In pa icula , ou me hod o sol ing o equilib ium p ices is alid o Ho elling models.1 We con inue wi h a subsec ion on he li e a u e. In Sec ion 2, we de…ne consume s’p e e ences and … ms’ echnology. Then we se up he quali y-p ice, mul is age game. Sec ion 3 is di ided in o ou subsec ions. In Subsec ion 3.1, we cha ac e ize subgame-pe ec equilib ium p ices. Lemma 1 p esen s he solu ion o he in eg al equa ion, he key s ep in exp essing equilib ium p ices as unc ions o quali ies. In Subsec ion 3.2, 1Di¤e ences be ween ho izon al and e ial models may also be due o speci…ca ion o he s a egy se s. He e, we allow quali ies o ake any posi i e alues; in loca ion models, … ms’posi ions may no a y as much. See also he discussions ollowing Co olla y 3. 3 we cha ac e ize how p ices change wi h quali ies. In Subsec ion 3.3, we cha ac e ize equilib ium quali ies, and es ablish he p ice- eac ion and Spence e¤ec s. Subsec ion 3.4 p esen s a numbe o implica ions. We specialize ou model by adop ing common assump ions (uni o m quali y- alua ion dis ibu ion and sepa able cos unc ion), and d aw connec ions be ween ea lie esul s and ou s. A numbe o examples a e s udied in Sec ion 4. These examples illus a e how ou gene al esul s can be used. The las sec ion con ains some concluding ema ks. P oo s o esul s a e in Appendix A, and some s eps o compu a ion o Subsec ion 4.3 a e in Appendix B. 1.1 Li e a u e e iew The mode n li e a u e on p oduc di¤e en ia ion and compe i ion begins wi h D’Asp emon , Gabszewicz and Thisse (1979), Gabszewicz and Thisse (1979) and Shaked and Su on (1982, 1983). In he pas ew decades, he p inciple o p oduc di¤e en ia ion elaxing p ice compe i ion has been s a ed in ex s o indus ial o ganiza ion a all le els: Ti ole (1988), Ande son, De Palma, and Thisse (1992), and Belle‡amme and Pei z (2010) o g adua e le el, as well as Cab al (2000), Ca l on and Pe lo¤ (2005), and Pepall, Richa ds, and No man (2014). Many esea che s use he basic ho izon al and e ical di¤e en ia ion models as hei in es iga ion wo kho se. The esea ch he e ocuses on equilib ium di¤e en ia ion. In bo h ho izon al and e ical models, a com- mon heme has been o sol e o subgame-pe ec equilib ia in he quali y-p ice, mul is age game in a ious en i onmen s.2Fi s , ea lie pape s ha e looked a single o mul iple ho izon al and e ical dimensions o consume p e e ences. Second, mos pape s ha e adop ed he assump ion ha hese p e e ences a e uni- o mly dis ibu ed. Thi d, mos pape s in he ho izon al model ha e used a quad a ic consume misma ch disu ili y unc ion, whe eas hose in he e ical model ha e assumed ha he uni p oduc ion cos is ei he independen o , o linea in quali y. Single dimension o p e e ences Ande son, Goe ee and Rame (1997) s udy equilib ium exis ence and cha ac e iza ion in a single-dimension 2As we ha e al eady men ioned, C eme and Thisse (1991) ( ollowing on a sugges ion by Champsau and Roche (1989)) show ha ho izon al-loca ion models a e special cases o e ical models. 4 ho izon al model. They use a gene al consume p e e ence dis ibu ion bu quad a ic misma ch disu ili y. Ou mul idimensional e ical model can be ecas in o he single-dimension model in Ande son, Goe ee and Rame , and we will demons a e ha in Subsec ion 4.2. A ew o he pape s ha e adop ed nonuni o m dis ibu ions on consume loca ions. Ne en (1986) shows ha … ms end o loca e inside he ma ke when consume s’densi ies a e highe nea he cen e . Tabuchi and Thisse (1995) assume a iangula dis ibu ion and …nd ha he e is no symme ic loca ion equilib ia bu ha asymme ic loca ion equilib ia exis . Yu ko (2010) uses a e ical model o s udying en y decisions, bu he esul s a e based on nume ical simula- ions. Benassi, Chi co, and Colombo (2006) allow consume s he nonpu chase op ion. They ela e a ious apezoidal alua ion dis ibu ions o deg ees o equilib ium quali y di¤e en ia ion. Finally, Loe sche and Muehlheusse (2011) conside en y in loca ion games wi hou p ice compe i ion. They s udy equilib ia unde he uni o m and some nonuni o m consume -loca ion dis ibu ions. Mul iple dimensions o e ical p e e ences A ew pape s ha e s udied e ical models wi h wo dimensions. These a e Vandenbosch and Weinbe g (1995), Lauga and O ek (2011), and Ga ella and Lambe ini (2014). All h ee pape s use he uni o m alua ion dis ibu ion. In Subsec ion 3.4, we will p esen he ela ionship be ween ou esul s he e o hose in hese pape s. He e, we no e ha hese pape s ha e assumed ze o p oduc ion cos , o uni cos ha is linea o discon inuous in quali y. By con as we use a s ic ly con ex quali y cos unc ion. Mul iple dimensions o ho izon al p e e ences Fo ho izon al models wi h mul iple dimensions, he key pape is I men and Thisse (1998), who se up an Ndimensional model o de i e wha hey call “Max-Min-...Min”equilib ia. We will ela e ou esul s o hose in I men and Thisse in Subsec ion 3.4, igh a e Co olla y 4. Tabuchi (1994) and Vendo p and Majeed (1995) a e special cases o I men and Thisse (1998) a N= 2. Ansa i, Economides, and S eckel (1998) s udy wo and h ee dimensional Ho elling models, and de i e simila esul s as in I men and Thisse (1998). All assume ha consume s’loca ions a e uni o mly dis ibu ed, and ha he misma ch disu ili y is Euclidean and he e o e sepa able. We a e unawa e o any pape in he mul idimensional ho izon al li e a u e ha adop s gene al consume p e e ences dis ibu ions, o gene al, nonsepa able misma ch disu ili y. 5 Finally, Deg yse and I men (2001) use a model wi h bo h ho izon al and e ical di¤e en ia ion. Fo he ho izon al dimension, consume loca ions a e uni o mly dis ibu ed. Fo he e ical dimension, consume s ha e he same alua ion (as in he model in Ga ella and Lambe ini (2014)). Howe e , he misma ch disu ili y depends also on quali y, which co esponds o he case o a nonsepa able misma ch disu ili y o quali y cos unc ion. This can be hough o as a special case o he model he e; see Subsec ion 4.2. 2 The Model We begin wi h consume s and hei p e e ences. Then we p esen wo iden ical … ms. Finally, we de…ne demands, p o… s, and he ex ensi e o m o quali y-p ice compe i ion. 2.1 Consume s and p e e ences o quali ies The e is a se o consume s, wi h o al mass no malized a 1. Each consume would like o buy one uni o a good, which has N2quali y a ibu es. A good is de…ned by a ec o o quali ies (q1; q2:::; qN)2 <N +, whe e qiis he le el o he i h quali y, i= 1;2; :::; N. Some imes, we use he e m quali y qi o mean he le el o quali y a ibu e i. A consume ’s p e e ences on goods a e desc ibed by his quali y alua ions, ep esen ed by he ec o ( 1; :::; i; :::; N)2QN i=1[ i; i] <N ++. The alua ion on quali y qiis i, which a ies in a bounded, and s ic ly posi i e in e al. I a consume wi h alua ion ec o ( 1; :::; i; :::; N) buys a good wi h quali ies (q1; q2:::; qN)qa p ice p, his u ili y is 1q1+ 2q2+:::+ NqNp. (We may some imes call his consume ( 1; :::; N)o simply consume .) The quasi-linea u ili y unc ion is commonly adop ed in he li e a u e (see such s anda d ex s as Ti ole (1988) and Belle‡amme and Pei z (2010)) Consume s’he e ogeneous p e e ences on quali ies a e modeled by le ing he alua ion ec o be andom. We use he s anda d independence dis ibu ion assump ion: he alua ion i ollows he dis ibu ion unc ion Fiwi h he co esponding densi y i,i= 1; :::; N, and hese dis ibu ions a e all independen .Each densi y is assumed o be di¤e en iable (almos e e ywhe e) and logconca e. The logconca i y o iimplies ha he 6 P oposi ion 1 In subgame (q; ), equilib ium p ices a e he solu ion o p Ain (12) and p Bin (13): p AC(q) = Z 1 F1((q; ) 11(q; ))dF1 Z 1 1((q; ) 11(q; ))dF1 ( 1q1)(12) p BC( ) = Z 11F1((q; ) 11(q; ))dF1 Z 1 1((q; ) 11(q; ))dF1 ( 1q1);(13) wi h (q; )implici ly de…ned by (10), and k(q; ) = kqk 1q1 ; k = 2; :::; N: The impo ance o P oposi ion 1 is his. The equilib ium p ice p Ais gi en by (12), an explici unc ion o quali ies. Thus, a di ec di¤e en ia ion o p Awi h espec o quali ies yields all he ele an in o ma ion o how any o Fi m B’s quali y choice changes Fi m A’s equilib ium p ice. The same applies o p Band (13). The common link be ween p Ain (12) and p Bin (13) is he implici unc ion (10), he explici unc ions (11), and he dis ibu ions o quali y alua ions. How do quali ies change p ices? 3.2 Quali ies and equilib ium p ices We begin wi h w i ing equilib ium p ices p Ain (12) and p Bin (13) as p AC(q) 1q1 =G; 1and p BC( ) 1q1 =H; 1, whe e he unc ions: G; 1:<N! <, and H; 1:<N! < a e de…ned by G; 1Z 1 F1( 11)dF1 Z 1 1( 11)dF1 (14) H; 1Z 11F1( 11)dF1 Z 1 1( 11)dF1 :(15) The unc ions Gand Ha e he … ms’p ice-cos ma kups pe uni o quali y di¤e ence. Wo king di ec ly wi h hese p ice-cos ma kups has a¤o ded us ac abili y.6Howe e , ou …nal esul s will be exp essed in 6Fo a model wi h only one quali y a ibu e, he ma kups a e he e e se haza d a e, F= , and he haza d a e (1 F)= , whe e Fand a e he dis ibu ion and densi y unc ions o he quali y alua ion. 13 e ms o he p imi i es o he model: he densi y and dis ibu ion unc ions, iand Fi,i= 1;2:::; N and he cos unc ion C. We ha e omi ed he quali y a gumen s in and 1(2; :::; N) o a sho e se o no a ion he e. The nume a o s o G; 1and H; 1a e, espec i ely, Fi m A’s and Fi m B’s demands. The common denomina o is he o al densi y o he se o indi¤e en consume s. Indi¤e en consume s a e hose wi h alua ions sa is ying qpA=  pB. Hence, he o al densi y o his se is R qpA=  pBQN i=1dFi( i) = R 1 1(e  1)dF1. Recall ha (q; )and k(q; )a e gi en by Lemma 1. Hence, we can di ec ly di¤e en ia e p Awi h espec o Fi m B’s quali ies, and di¤e en ia e p Bwi h espec Fi m A’s quali ies, and we call hese he p ice- eac ion e¤ec s: @p A @ 1 = ( 1q1)@G ; 1 @ 1 =G; 1+ ( 1q1)"@G @ @ @ 1 + N X k=2 @G @k @k @ 1#(16) @p B @q1 = ( 1q1)@H ; 1 @q1 =H; 1+ ( 1q1)"@H @ @ @q1 + N X k=2 @H @k @k @q1#;(17) and @p A @ j = ( 1q1)@G ; 1 @ j = ( 1q1)@G @ @ @ j +@G @j @j @ j; j = 2; :::; N (18) @p B @qj = ( 1q1)@H ; 1 @qj = ( 1q1)@H @ @ @qj +@H @j @j @qj; j = 2; :::; N: (19) Because we label a di¤e en ia ed quali y a ibu e as he … s a ibu e (q1< 1), he e is a sligh di¤e ence be ween he o m o p ice- eac ion e¤ec s o he … s quali y and he o he quali ies. (Wi h all he a gumen s displayed, he unc ion Gis G(; 2; :::; N) = G; 2q2 1q1; :::; NqN 1q1. Simila ly, o he unc ion H. The a gumen s 2; :::; Nin Gand Hinclude 1and q1, so i we pa ially di¤e en ia e Go Hwi h espec o 1and q1, we ge he summa ion e ms in (16) and (17). Howe e , o j= 2; :::; N, jand qja e only in he a gumen jin Gand H. When we pa ially di¤e en ia e Go Hwi h espec o jo qj, we only ge he simple exp essions in (18) and (19).) The exp ession in (18) displays a composi ion o wo e¤ec s. The … s is how quali ies change he in e cep and slopes 1o he equa ion ha de e mines he equilib ium se o indi¤e en consume s 14 e  1( 1;q; ) = (q; ) 11(q; ); hese a e he e ms @ @ j ,@j @ j ,@ @qj , and @j @qj . The second is how he equilib ium se o consume s change he p ice-cos ma kups G; 1and H; 1; hese a e he e ms @G @ ,@G @j ,@H @ and @H @j ,j= 2; :::; N. We … s p esen how he sum in ma kups a ies wi h he equilib ium se o indi¤e en consume s. Then we can ob ain how he sum a ies wi h he in e cep and slope o he equa ions o he se o indi¤e en consume s. Lemma 2 In any subgame (q; ), he sum o he p opo ional changes in he … ms’ equilib ium p ice-cos ma kups and he p opo ional change in he o al densi y o he equilib ium se o indi¤e en consume s mus anish: dln[G(; 1) + H(; 1)] + dln Z 1 1( 11)dF1= 0.(20) I ollows ha he sum o he pa ial de i a i es o G(; 1)and H(; 1)wi h espec o and j, j= 2; :::; N a e @G @ +@H @ =Z 1 0 1( 11)dF1 Z 1 1( 11)dF1!2(21) @G @j +@H @j =Z 1 0 1( 11) jdF1 Z 1 1( 11)dF1!2:(22) Lemma 2 ollows om he de…ni ions o Gand Hand om s aigh o wa d di¤e en ia ion. F om he de…ni ions (14) and (15), we ha e: G+H=1 Z 1 1( 11)dF1 :(23) A each subgame (q; ) he ma ke is always co e ed, he o al p ice-cos ma kup is always equal o he ecip ocal o he o al densi y o equilib ium se o indi¤e en consumes. Hence, equa ion (20) ollows. I somehow he in e cep o he equilib ium se o consume s () inc eases, he sum o … ms’p ice-cos ma kups 15 will be in (21). Simila ly, i he j h slope o he equilib ium se o indi¤e en consume s (j) inc eases, he sum o hei ma kups will be in (22). Nex , we p esen how quali ies change he in e cep and slopes o he equa ion o he equilib ium se o indi¤e en consume s. Lemma 3 In any subgame (q; ), in equilib ium, o i= 1; :::; N and j= 2; :::; N, @(q; ) @qi +@(q; ) @ i =Cj( )Cj(q) ( 1q1)1 + @G @ @H @ and @j(q; ) @qi +@j(q; ) @ i = 0:(24) The con inua ion p ice equilib ium o subgame (q; )gene a es he equilib ium se o indi¤e ence con- sume s, e  1( 1;q; ) = (q; ) 11(q; ). Fi ms changing hei quali ies will impac he equilib ium p ices o bo h … ms, and he equilib ium se o indi¤e en consume s. In e cep (q; )and slopes k(q; ) will be changed by he change in qi. Lemma 3 desc ibes some ela ions o how quali ies change he in e cep and slopes. The sum o he e¤ec s o … ms’quali y changes on he in e cep is p opo ional o he di¤e ence in ma ginal cos . The sum o he e¤ec s o … ms’quali y changes on he slope is ze o. Lemmas 3 can be used o cha ac e ize … ms’ ela i e s eng h o s a egic p ice e¤ec s. P oposi ion 2 In subgame (q; ), o quali y j,j= 2; :::; N, he di¤e ence in he s a egic p ice e¤ec s, 16 @p B(q; ) @qj @p A(q; ) @ j , can be w i en in h ee equi alen o ms: ( 1q1)2 6 6 6 6 4 Z 1 0 1( 11)dF1 @ @ j Z 1 0 1( 11) kdF1 @j @ j 3 7 7 7 7 5 Z 1 1( 11)dF1!2+Z[Cj(q)Cj( )] (25) = ( 1q1)@ @ j"Z 1 1( 11)dF1# Z 1 1( 11)dF1!2+Z[Cj(q)Cj( )] (26) = ( 1q1)@ @qj"Z 1 1( 11)dF1# Z 1 1( 11)dF1!2+Z[Cj(q)Cj( )] ;(27) whe e Z= 8 > > > > > > > > > > > < > > > > > > > > > > > : 1 + H(; 1)Z 1 0 1( 11)dF1 Z 1 1( 11)dF1 3G(; 1)Z 1 0 1( 11)dF1 Z 1 1( 11)dF1 9 > > > > > > > > > > > = > > > > > > > > > > > ; : The p oposi ion says ha he di¤e ence in … ms’s a egic p ice e¤ec s o a quali y is de e mined by how he quali y changes he o al densi y o he equilib ium se o indi¤e en consume s, Z 1 1( 11)dF1, and by he di¤e ence in he ma ginal cos o quali y [Cj(q)Cj( )]. The sum o he unc ions Gand H was w i en in (23). The … s e m in each o he equi alen exp essions in he P oposi ion is he de i a i e o he sum o Gand Hwi h espec o a quali y j,j= 2; :::; N; mul iplied by ( 1q1). In ac , he e¤ec o Fi m A’s quali y on he o al densi y o he se o indi¤e en consume s is equal and opposi e o ha o Fi m B’s. The second e m is he di¤e ence in … ms’ma ginal cos s o a quali y adjus ed by Z. Finally, we no e ha when 1is a s ep unc ion, 0 1( 11) = 0, exp essions om (25) o (27) 17 simpli y o only he e m ela ed o he di¤e ence in ma ginal cos s: @p B(q; ) @qj @p A(q; ) @ j =1 3[Cj(q)Cj( )] : In o he wo ds, when he di¤e en ia ed dimension has a uni o m-dis ibu ion alua ion, each … m’s quali y aises he ma kup by he same amoun , so he s a egic p ice eac ion e¤ec s a e all in he ma ginal-cos di¤e ence. 3.3 Equilib ium quali ies Gi en he equilib ium p ices p A(q; )and p B(q; )in S age 2, we w i e he p o… unc ions in S age 1 in e ms o quali ies: A(p A(q; ); p B(q; ); q; ) = Z 1 F1(e 1( 1;p; q; ))dF1[p A(q; )C(q)] (28) B(p A(q; ); p B(q; ); q; ) = Z 1 [1 F1(e 1( 1;p; q; ))] dF1[p B(q; )C( )] ;(29) whe e e 1( 1;p; q; )p B(q; )p A(q; ) 1q1 PN k=2 k kqk 1q1 . Fo subgame-pe ec equilib ium p ices, p, equilib ium quali ies a e qand  ha a e mu ual bes esponses: q(q 1; :::; q N) = a gmax qZ 1 F1(e 1( 1;p(q; ); q; ))dF1[p A(q; )C(q)] (  1; :::;  N) = a gmax Z 1 [1 F1(e 1( 1;p(q; ); q; ))] dF1[p B(q; )C( )] ; whe e p(q; )(p A(q; ); p B(q; )). Quali ies qi,i= 1; :::; N a¤ec Fi m A’s p o… (28) in h ee ways. Fi s , hey ha e a di ec e¤ec h ough he cos s, C(q), as well as he demand. Second, hey a¤ec he p o… h ough Fi m A’s own equilib ium p ice p A(q; ). Thi d, hey a¤ec he p o… h ough Fi m B’s equilib ium p ice p B(q; ), cap u ed by @p B=@qi. Because he equilib ium p ices p A(q; )and p B(q; )a e mu ual bes esponses in he p ice subgame in S age 2, he en elope heo em applies. Tha is, Fi m A’s quali ies qi,i= 1; :::; N ha e second-o de e¤ec s on i s own p o… (28) h ough i s equilib ium p ice; he second e¤ec can be igno ed. 18 The … s -o de de i a i e o (28) wi h espec o qiis "Z 1 F1(e 1( 1;p(q; ); q; ))dF1#Ci(q) +@ @qiZ 1 F1(e 1( 1;p(q; ); q; ))dF1[p A(q; )C(q)] | {z } e¤ec s o quali y qion cos and demand (30) +@ @p B(Z 1 F1(e 1( 1;p(q; ); q; ))dF1)@p B @qi [p A(q; )C(q)] | {z } e¤ec o quali y on Fi m B’s p ice ; i = 1; :::; N; (31) whe e he (pa ial) de i a i e o p o… wi h espec o pAhas been igno ed. The e ms in (30) desc ibe how a quali y a¤ec s cos and demand, whe eas he e m in (31) desc ibes he s a egic e¤ec o a quali y on he i al’s p ice. Nex , we use he same s eps o ob ain he … s -o de de i a i es o Fi m B’s p o… (29) wi h espec o i, and hese a e "Z 1 [1 F1(e 1( 1;p(q; ); q; ))] dF1#Ci( ) +@ @ i(Z 1 [1 F1(e 1( 1;p(q; ); q; ))] dF1)[p B(q; )C( )] | {z } e¤ec s o quali y on cos and demand (32) +@ @p A(Z 1 [1 F1(e 1( 1;p(q; ); q; ))] dF1)@p A @ i [p A(q; )C( )] | {z } e¤ec o quali y ion Fi m A’s p ice ; i = 1; :::; N: (33) These exp essions ca y he same in e p e a ions as hose in he … s -o de de i a i es o Fi m A. We now s a e he main esul on equilib ium quali ies. We ob ain he se o equa ions in he nex p oposi ion by … s simpli ying he … s -o de de i a i es and hen se ing hem o ze o. Fo simpli…ca ion, we use he basic demand unc ion (3) and equilib ium p ices (12) and (13) in P oposi ion 1, and …nally d op common ac o s in he … s -o de de i a i es. (De ails a e in he p oo .) P oposi ion 3 Fo he quali y-p ice, mul is age game in Subsec ion 2.2, equilib ium quali ies (q; )(unde 19 he con en ion ha q 1<  1) mus sa is y he ollowing 2Nequa ions: @p B @q1 +Z 1 1( 11)e  1dF1 Z 1 1( 11))dF1 C1(q)=0 (34) @p A @ 1 +Z 1 1( 11)e  1dF1 Z 1 1( 11))dF1 C1( )=0;(35) and o j= 2; :::; N, @p B @qj +Z 1 1( 11) jdF1 Z 1 1( 11))dF1 Cj(q)=0 (36) @p A @ j +Z 1 1( 11) jdF1 Z 1 1( 11))dF1 Cj( )=0:(37) whe e and ja e he unc ions in (10) and (11), espec i ely, and e  1is e  1( 1;q; ), he solu ion o he in eg al equa ion in Lemma 1. The p ope ies o equilib ium quali ies in (34) and (36) can be explained as ollows. The e a e wo e¤ec s. The … s e m in each exp ession is he p ice- eac ion e¤ec : i desc ibes how Fi m A’s quali ies q1and qj, j= 2; :::; N a¤ec he i al’s p ice in he con inua ion subgame. In ac , all he p ice- eac ion e¤ec s a e w i en ou in equa ions (16) o (19) abo e. The second e¤ec conce ns he a e age alua ion o he j h quali y among he equilib ium se o indi¤e en consume s— he in eg als in (34) and (36)— and he j h quali y’s ma ginal con ibu ion o he pe -uni cos — Cj@C(q) @qj . These wo e ms oge he o m he Spence e¤ec . Indeed, Spence (1975) shows ha a p o… -maximizing … m chooses he e¢ cien quali y o he ma ginal consume (and hen aises he p ice o ex ac he ma ginal consume ’s su plus).7The same p ice- eac ion and Spence e¤ec s apply o Fi m B’s equilib ium quali y choices desc ibed by (35) and (37). 7Le P(D; q)be he p ice a … m can cha ge when i sells Duni s o i s good a quali y q= (q1; :::; qN). Le C(D; q) be he cos when he … m p oduces Duni s a quali y q. P o… is DP (D; q)C(D; q). The p o… -maximizing quali y qiis gi en by D@P @qi =@C @qi . Hence he quali y alua ion o he ma ginal consume @P @qi is equal o he ma ginal con ibu ion o quali y i o pe -uni cos @C=@qi D. See Spence (1975, p419; equa ion (8)). 20 Because he wo … ms ace he same equilib ium se o indi¤e en consume s, he Spence e¤ec pushes hem o choose he same quali ies. The p ice- eac ion e¤ec s gene ally pu he … ms in a ace si ua ion. P ices a e s a egic complemen s, so each … m wan s o use i s quali ies o aise he i al’s p ice. The p ice- eac ion e¤ec dic a es how much a … m’s equilib ium quali y de ia es om he e¢ cien quali y o he equilib ium se o indi¤e en consume s. The … m ha has a s onge s a egic p ice- eac ion e¤ec de ia es mo e. I he wo … ms we e playing ano he game in which p ices and quali ies we e chosen concu en ly (one wi h me ged S ages 1 and 2 in he ex ensi e o m in Subsec ion 2.2), he p ice- eac ion e¤ec would anish. Then he Spence e¤ec would dic a e equilib ium s a egies. Each … m would choose he quali ies op imal o he a e age alua ions o he common se o ma ginal consume s, so … ms choose he same le el o each quali y a ibu e. Fi ms mus hen se hei p ices a ma ginal cos . (Fo an illus a ion o a game wi h … ms choosing p ices and quali ies concu en ly, see Ma and Bu gess (1993)). P oposi ion 3 holds he key o he unde s anding o equilib ium p oduc di¤e en ia ion wi h mul iple quali ies, o which we now u n. 3.4 Quali y di¤e en ia ion P oposi ion 3 d aws a connec ion be ween he p ice- eac ion e¤ec s and quali ies’ma ginal con ibu ions o uni p oduc ion cos . We s a e his o mally: Co olla y 1 A he equilib ium (q; ), a … m’s j h quali y con ibu es mo e o i s own uni p oduc ion cos han a i al’s j h quali y con ibu es o he i al’s uni p oduc ion cos i and only i he … m’s p ice- eac ion e¤ec o ha quali y is s onge han he i al’s. Tha is, o each j= 2; :::; N, he ollowing a e equi alen : i) Cj(q)< Cj( ), ii) @p B(q; ) @qj <@p A(q; ) @ j , iii) he h ee equi alen exp essions (25), (26), and (27) in P oposi ion 2 a e nega i e a equilib ium (q; ). Co olla y 1 ollows om P oposi ion 3 s aigh owa dly; we simply ake he di¤e ence be ween … s -o de condi ions. On he cos side, he undamen al issue is how much a quali y con ibu es o he uni p oduc ion 21 cos . On he s a egic side, he undamen al issue is how much a quali y a¤ec s he i al’s p ice. The quali y ace conce ns only hese wo issues. Howe e , he co olla y does no di ec ly add ess he equilib ium quali y le els. We ha e used a gene al cos unc ion, so i is qui e possible ha Cj(q)< Cj( )bu q j>  j. (Fo an illus a ion, see he example a e Co olla y 3.) Sha pe esul s can be ob ained om he ollowing (wi h p oo omi ed): Co olla y 2 Suppose ha he cos unc ion Cis sepa able: C(q) = C(q1; q2; :::; qN) = 1(q1) + 2(q2); :::N(qN); whe e iis an inc easing and con ex unc ion, so Ci(q) = 0 i(qi),i= 1;2; :::; N. In an equilib ium (q; ), o j= 2; :::; N, q j<  j() @p B(q; ) @qj <@p A(q; ) @ j : Wi h sepa able cos , a quali y’s con ibu ion o he uni p oduc ion cos is independen o o he quali ies. A … m ha ing a s onge p ice- eac ion e¤ec a a quali y han i s i al’s mus choose a highe quali y han i s i al’s quali y. In he li e a u e, he sepa able cos unc ion has been adop ed. Co olla ies 1 and 2 can be ead as a cau ion. The undamen al issue is how a quali y con ibu es o he p oduc ion cos . Fo a model wi h many quali ies, a quali y’s con ibu ion o p oduc ion cos depends on he en i e ec o o quali ies. Co olla ies 1 and 2 oge he say ha such in e dependence is impo an ; see also he example ollowing he nex esul . Nex , we conside speci…c quali y alua ion densi y unc ions commonly used in he li e a u e. Recall ha , wi h a s ep unc ion he di¤e ence o he p ice e¤ec s educes o 1 3[Cj(q)Cj( )] :Hence: Co olla y 3 Suppose ha 1is a s ep unc ion, so 0 1= 0 almos e e ywhe e. In an equilib ium (q; ), Cj(q) = Cj( )and @p B(q; ) @qj =@p A(q; ) @ j ,j= 2; :::; N. Fu he mo e, i Cis sepa able, hen q j=  j, j= 2; :::; N; in o he wo ds, quali ies 2 h ough Na e nondi¤e en ia ed. Co olla y 3 p esen s a s iking esul . Fi s , he uni o m dis ibu ion is he mos common quali y- alua ion assump ion in he p oduc -di¤e en ia ion li e a u e, and i is a s ep unc ion. Unde he uni o m- 22 Fo gi en quali ies (q1; q2) = q om Fi m Aand quali ies ( 1; 2) = om Fi m B, i consume s alue he second quali y a b 2, he se o indi¤e en consume s in (3) becomes a single poin . The equilib ium p ices in (6) and (7) simpli y o p AC(q) = F1(e 1) 1(e 1)( 1q1)G1(e 1)( 1q1)(44) p BC( ) = 1F1(e 1) 1(e 1)( 1q1)H1(e 1)( 1q1);(45) whe e e 1(b 2;p; q; ) = p Bp A 1q1 b 2 2q2 1q1 :(46) He e, we ha e added he subsc ip 1on he p ice-cos ma gins Gand H o dis inguish hem om he gene al ones abo e.9The logconca i y o 1implies ha G1is s ic ly inc easing, and ha H1is s ic ly inc easing (Ande son, Goe ee and Rame 1997, p106). Equa ions (44) and (45), oge he wi h he indi¤e en consume (46), de…ne a p ice equilib ium. Le e  1(b 2;q; ) = e 1(b 2;p; q; ), whe e pis he ec o o equilib ium p ices which a e unc ions o qand . These h ee equa ions co espond o equa ions (2.1), (2.2) and (2.3) in Ande son, Goe ee and Rame (1997, p107). We use (44) and (45) o subs i u e o equilib ium p ices in (46). A e some simpli…ca ion we ob ain he implici de…ni ion o e  1: e  1+G1(e  1)H1(e  1) = [b 2q2C(q)] [b 2 2C( )] 1q1 :(47) F om (47) we compu e he de i a i es o e  1wi h espec o he quali ies; hese co espond o he de i a i es o (q; )in he p oo o Lemma 3. Then we use hose de i a i es o simpli y he p ice- eac ion e¤ec s ob ained om he explici di¤e en ia ion o (44) and (45) wi h espec o p ices. The a ious s eps a e spelled ou in he … s wo pa s o Appendix B. 9The unc ions G1and H1a e commonly called he e e se haza d a e and haza d a e, espec i ely. 29 The … s -o de condi ions om (34) o (37) in P oposi ion 3 can be simpli…ed o: @p B @q1 +e  1=C1(q)(48) @p B @q2 +b 2=C2(q)(49) @p A @ 1 +e  1=C1( )(50) @p A @ 2 +b 2=C2( ):(51) In he hi d pa o Appendix B, we p o e ha i he cos unc ion C(q)is sepa able, C(q) = 1(q1)+2(q2), hen b 2=0 2(q 2) = 0 2(  2), so q 2=  2. This is a na u al esul : … ms canno di¤e en ia e a quali y when consume s ha e homogenous alua ions on ha quali y when quali y cos s a e sepa able. Finally, a e pu ing he p ice- eac ion e¤ec s in o (48) and (50) (see he las pa o Appendix B o he compu a ion o he p ice- eac ion e¤ec s), we show ha equilib ium quali ies q 1and  1mus sa is y: e  1=C1(q) + H1(e  1)H0 1(e  1) 1 + G0 1(e  1)H0 1(e  1)C( )C(q)  1q 1 C1(q)(52) e  1=C1( )G1(e  1)G0 1(e  1) 1 + G0 1(e  1)H0 1(e  1)C1( )C( )C(q)  1q 1;(53) whe e e  1H1(e  1) + G1(e  1) = [b 2q 2C(q)] [b 2  2C( )]  1q 1 :(54) When he cos unc ion is sepa able, we ha e q 2=  2, so (54) becomes e  1+G1(e  1)H1(e  1) = 1(  1)1(q 1)  1q 1 : Fu he mo e, equa ions (52) and (53) become e  1=0 1(q 1) + H1(e  1)H0 1(e  1) 1 + G0 1(e  1)H0 1(e  1)1(  1)1(q 1)  1q 1 0 1(q 1) e  1=0 1(  1)G1(e  1)G0 1(e  1) 1 + G0 1(e  1)H0 1(e  1)0 1(  1)1(  1)1(q 1)  1q 1: These las h ee equa ions oge he wi h he equilib ium p ices co espond o equa ions (2.8) and (2.10) on pages 108 and 109 o Ande son, Goe ee, and Rame (1997). 30 Ga ella and Lambe ini (2014) use a discon inuous cos unc ion: a … m p oducing zuni s o he good a quali y (q1; q2)has a o al cos o cz +T(q1; q2)i q1> q1, bu only T(q1; q2)i q1=q1, whe e q1>0 and c > 0a e …xed pa ame e s, and Tis inc easing when q1> q1. Consume s ha e homogenous p e e ences on he second quali y, bu hei alua ions on he … s quali y ollow a uni o m dis ibu ion. They de i e equilib ia in which … ms choose di¤e en le els in bo h quali ies. We use a con inuous cos unc ion, bu i cos s a e nonsepa able, equilib ia in ou model will also exhibi di¤e en ia ions in bo h quali ies. Because o he uni o m dis ibu ion, @p B(q; ) @q2 =@p A(q; ) @ 2 , and C2(q) = C2( ), which gene ally implies ha q 26=  2. The example p esen ed jus be o e Co olla y 3 and he simula ions in Subsec ion 4.2 can be used again o his illus a ion. 5 Conclusion We eexamine he p inciple o p oduc di¤e en ia ion elaxing p ice compe i ion in he classical quali y-p ice game. The en i onmen o analysis in ou model is mo e gene al han exis ing wo ks. Ye , we a e able o cha ac e ize equilib ia wi hou comp omise, he eby con… ming he p inciple o di¤e en ia ion. The ou come o minimum di¤e en ia ion in ea lie wo ks can be a ibu ed o consume s’quali y alua ions (o loca ion) being uni o mly dis ibu ed and quali y cos (o misma ch disu ili y) being sepa able. The p inciple o p oduc di¤e en ia ion is obus : … ms end o choose di¤e en quali y a ibu es. Va ious open ques ions emain. Fi s , unlike p e ious wo ks ha use explici unc ional o ms, we ha e used a gene al se up. Whe eas di ec compu a ion o equilib ia has been he main echnique, we ha e used only necessa y condi ions o equilib ia. Al hough we do no p o ide exis ence esul s, ou cha ac e iza ion applies o each equilib ium when i does exis . The exis ence and uniqueness issues a e beyond he scope o he cu en esea ch, bu i may u n ou o be a ewa ding endea o . Second, ou inno a ion is he ep esen a ion o he p ice equilib ium by he solu ion o an in eg al equa ion. This echnique is uncommon. In ac , economis s seldom ha e o sol e in eg al equa ions. I may well be ha ou echnique can shed ligh on o he models wi h mul iple p e e ences o cos a ibu es. Thi d, we assume linea p e e ences: each quali y bene… s a consume a a cons an a e. The linea i y assump ion is so ubiqui ous in mode n mic oeconomics ha elaxing his is bo h challenging and consequen ial. Finally, i seems ha ex ending 31 ou en i onmen o allow o consume s ha ing co ela ed alua ions be ween quali ies may be wo hwhile. Ou amewo k may jus be ich enough o his ex ension. One may wo k wi h condi ional join densi ies o de i e demand unc ions. O cou se, he exac o mula ion will be u u e esea ch. 32 Appendix A: P oo s o Lemmas, P oposi ions, and Co olla ies P oo o Lemma 1: Equilib ium p ices p Aand p Bdepend on quali ies (q; ), so he igh -hand side o (8) is (a¢ ne) linea in 2; :::; N. The solu ion e  1( 1;q; )o (9) also sa is…es (8), so i mus also be linea in 2; :::; N. The e o e, we w i e e  1( 1;q; ) = (q; )PN k=2 kk(q; ), k2[ k; k]; k = 2; :::; N o some unc ions , and k,k= 2; :::; N. Then we subs i u e e  1( 1;q; )by (q; )PN k=2 kk(q; )in (9) o ge (q; ) N P k=2 kk(q; ) = Z 1h12F1((q; )PN k=2 kk(q; ))idF1 Z 1 1((q; )PN k=2 kk(q; ))dF1 +C( )C(q) 1q1  N P k=2 k kqk 1q1 ; o k2[ k; k]; k = 2; :::; N: Because his is ue o e e y 2; :::; N, he equa ions (10) and (11) in he lemma ollow. P oo o Lemma 2: F om he de…ni ions (14) and (15), a each (q; ), we ha e: G(; 1) + H(; 1) = 1 Z 1 1( 11)dF1 : Hence dln(G+H) + dln Z 1 1( 11)dF1= 0; so he … s s a emen o he lemma ollows. Because (20) holds o each (q; ), we can pa ially di¤e en ia e i wi h espec o and j,j= 2; :::; N, o ob ain (21) and (22). P oo o Lemma 3: F om (11), he unc ions j(q; )a e j= jqj 1q1 ,j= 2; :::; N. Hence, @j @q1 = jqj ( 1q1)2=@j @ 1 ;and @j @qj =1 1q1 =@j @ j ; j = 2; :::; N; (55) and all pa ial de i a i es o jwi h espec o qko k,k6=j, anish. These p o e he second equali y in (24). F om de…ni ions o Gand Hin (14) and (15), we w i e (10) as +G; 1H; 1=C( )C(q) 1q1 :(56) 33 We o ally di¤e en ia e (56) o ob ain 1 + @G @ @H @ d+ N X k=2 @G @k @H @kdj=dC( )C(q) 1q1; j = 2; :::; N: Using (55), we ob ain he pa ial de i a i es o wi h espec o each quali y, and hen simpli y hem o: 1 + @G @ @H @ @ @q1 =@G @1 @H @1 jqj ( 1q1)2C1(q) ( 1q1)+C( )C(q) ( 1q1)2 1 + @G @ @H @ @ @ 1 =@G @1 @H @1 jqj ( 1q1)2+C1( ) ( 1q1)C( )C(q) ( 1q1)2 (57) 1 + @G @ @H @ @ @qj =@G @j @H @j1 ( 1q1)Cj(q) ( 1q1)j= 2; :::; N 1 + @G @ @H @ @ @ j =@G @j @H @j1 ( 1q1)+Cj( ) ( 1q1)j= 2; :::; N; whe e Ci(q)@C(q) @qi deno es he i h pa ial de i a i e o he cos unc ion C. The … s pa o (24) ollows by summing up he … s wo equa ions and he las wo equa ions in (57) o compu e @(q; ) @qi +@(q; ) @ i , i= 1; :::; N. P oo o P oposi ion 2: F om (18) and (19), we ha e @p B(q; ) @qj @p A(q; ) @ j = ( 1q1)@H @ @ @qj +@H @j @j @qj@G @ @ @ j +@G @j @j @ j; j = 2; :::; N: Using Lemma 3, we ha e @ @qj =Cj( )Cj(q) ( 1q1)1 + @G @ @H @ @ @ j and @j @qj =@j @ j , and subs i u e hem in o he abo e o ob ain: @p B(q; ) @qj @p A(q; ) @ j = ( 1q1)8 > > < > > : 2 6 6 4 @H @ 0 B B @ Cj( )Cj(q) ( 1q1)1 + @G @ @H @ @ @ j1 C C A@H @j @j @ j3 7 7 5@G @ @ @ j +@G @j @j @ j9 > > = > > ; = ( 1q1)@H @ @ @ j @H @j @j @ j@G @ @ @ j +@G @j @j @ j+@H @ [Cj( )Cj(q)] 1 + @G @ @H @  =( 1q1)@G @ +@H @ @ @ j +@G @j +@H @j@j @ j+@H @ [Cj( )Cj(q)] 1 + @G @ @H @ : 34 Nex , we de…ne Z@H @ 1 1 + @G @ @H @ : We use (21) and (22) in Lemma 2 o ob ain @p B(q; ) @qj @p A(q; ) @ j = ( 1q1)Z 1 0 1( 11)dF1 Z 1 1( 11)dF1!2 @ @ j  ( 1q1)Z 1 0 1( 11) jdF1 Z 1 1( 11)dF1!2 @j @ j +Z[Cj(q)Cj( )] = ( 1q1) Z 1 0 1( 11)dF1 @ @ j Z 1 0 1( 11) jdF1 @j @ j! Z 1 1( 11)dF1!2+Z[Cj(q)Cj( )] ;(58) which gi es (25). Then we u he w i e he … s e m in (58) as ( 1q1)@ @ j"Z 1 1( 11)dF1# Z 1 1( 11)dF1!2;(59) which gi es he … s e m in (26). Finally, om Lemma 3, we ha e @ @ j =Cj( )Cj(q) ( 1q1)1 + @G @ @H @ @ @qj and @j @ j =@j @qj , so (59) also equals  ( 1q1)@ @qj"Z 1 1( 11)dF1# Z 1 1( 11)dF1!2; which gi es he … s e m in (27). Finally, om he de…ni ion o G; 1and H; 1in (14) and (15), we ha e: @ @G; 1= 1 Z 1 0 1( 11)dF1Z 1 F1( 11)dF1 Z 1 1( 11)dF1!2 @ @H; 1=1Z 1 0 1( 11)dF1Z 11F1( 11)dF1 Z 1 1( 11)dF1!2: 35 A e we subs i u e hese in o he de…ni ion o Z, we ob ain he same exp ession o Zin he P oposi ion. P oo o P oposi ion 3: We begin by simpli ying Fi m A’s … s -o de de i a i es wi h espec o quali ies. Fi s , o (30) we use (3) o ob ain @ @qjZ 1 F1(e 1( 1;p(q; ); q; ))dF1 =1 1q1Z 1 1(e 1( 1;p(q; ); q; )) jdF1j= 2; :::; N: Second, o (31), again we use (3) o ob ain @ @p BZ 1 F1(e 1( 1;p(q; ); q; ))dF1 =1 1q1Z 1 1(e 1( 1;p(q; ); q; ))dF1: We hen subs i u e hese exp essions in o (30) and (31), and he … s -o de de i a i e o Fi m A’s wi h espec o quali y qj,j= 2; :::; N, becomes "Z 1 F1(e 1( 1;p(q; ); q; ))dF1#Ci(q) +1 1q1Z 1 1(e 1( 1;p(q; ); q; )) jdF1[p A(q; )C(q)] (60) +1 1q1Z 1 1(e 1( 1;p(q; ); q; ))dF1 @p B @qi [p A(q; )C(q)] : We now e alua e (60) a he equilib ium quali ies, so eplace e 1( 1;p(q; ); q; )) as e  1( 1;q; ) = (q; ) 11(q; ). Using he equilib ium p ice (12) in P oposi ion 1 p A(q; )C(q)  1q 1 =Z 1 F1( 11))dF1 Z 1 1( 11))dF1 ; we simpli y he … s -o de de i a i e o Fi m A’s p o… wi h espec o qj o "Z 1 F1( 11)dF1#2 6 6 4 @p B @qj +Z 1 1( 11) jdF1 Z 1 1( 11))dF1 Cj(q)3 7 7 5; j = 2; :::; N; whe e we ha e omi ed he a gumen s in and 1. We se his o ze o o ob ain he … s -o de condi ion o q j: @p B @qj +Z 1 1( 11) jdF1 Z 1 1( 11))dF1 Cj(q) = 0 j= 2; :::; N: 36 Fo b e i y we do no lay ou all he s eps o ob aining he … s -o de condi ion o Fi m A’s equilib ium quali y q1, bu he key di¤e ence is ha (3) yields @e 1 @q1 =e 1 1q1 . The e¤ec o quali y q1on demand now becomes @ @q1Z 1 F1(e 1( 1;p(q; ); q; ))dF1 =1 1q1Z 1 1(e 1( 1;p(q; ); q; ))~ 1dF1: Following he same s eps, we ob ain he ollowing … s -o de condi ion o q 1 @p B @q1 +Z 1 1( 11)e  1dF1 Z 1 1( 11))dF1 C1(q) = 0: The … s -o de condi ions o Fi m B’s equilib ium quali ies a e de i ed analogously, so we do no epea he s eps he e. P oo o Co olla y 1: Fo each j= 2; :::; N, he e ms wi h he in eg als a e he same in he wo equa ions in (36) and (37). Taking hei di¤e ence, we ha e @p B @qj @p A @ j =Cj(q)Cj( ), and he equi alence o i) and ii) in he Co olla y ollows. Then we simply apply P oposi ion 2 on he equilib ium (q; ) o he equi alence o ii) and iii). P oo o Co olla y 3: We ac ually p o e a mo e gene al esul o he … s pa . Conside any subgame de…ned by equilib ium quali y ec o s (q; ). The di¤e ence in he … ms’p ice- eac ion e¤ec s is in P oposi ion 2. Ob iously, 0 1= 0 by assump ion, so (25) becomes @p B(q; ) @qj @p A(q; ) @ j =1 3[Cj(q)Cj( )]. Unde he s ep- unc ion assump ion, Co olla y 1 hen says ha @p B(q; ) @qj @p A(q; ) @ j =1 3[Cj(q)Cj( )] = Cj(q)Cj( ):Hence i mus be Cj(q) = Cj( ),j= 2; :::; N. Which also implies ha , in equilib ium, @p B(q; ) @qj @p A(q; ) @ j = 0:Finally, we apply Co olla y 2 o ob ain he nondi¤e en ia ion esul . P oo o Co olla y 4: Le he alua ion densi y o quali y jbe uni o m. Conside an equilib ium (q ). I q j=  j, hen he … s pa o he co olla y is i ially ue. Suppose ha q j6=  j. Wi hou loss o gene ali y we le q j<  j. Now we elabel he indexes so ha j= 1. Then Co olla y 3 applies, and he … ms choose iden ical quali ies o all quali y a ibu es k= 2; :::; N. 37 Appendix B: On collapsing one alua ion dis ibu ion in a model wi h wo quali ies De i a i es o e  1wi h espec o quali ies F om (47) we ob ain he de i a i es o e  1in (54) wi h espec o he ou quali ies: [1 + G0 1(e  1)H0 1(e  1)] @e  1 @q1 =[b 2q2C(q)] [b 2 2C( )] ( 1q1)2C1(q) 1q1 [1 + G0 1(e  1)H0 1(e  1)] @e  1 @q2 =b 2C2(q) 1q1 [1 + G0 1(e  1)H0 1(e  1)] @e  1 @ 1 =[b 2q2C(q)] [b 2 2C( )] ( 1q1)2+C1( ) 1q1 [1 + G0 1(e  1)H0 1(e  1)] @e  1 @ 2 =b 2C2( ) 1q1 : These a e he co esponding de i a i es o (q; )in he p oo o Lemma 3. P ice- eac ion e¤ec s Nex , om (44) and (45) we ob ain he ollowing: @p A @ 1 =G1(e  1)+( 1q1)G0 1(e  1)@e  1 @ 1 @p A @ 2 = ( 1q1)G0 1(e  1)@e  1 @ 2 (61) @p B @q1 =H1(e  1)+( 1q1)H0 1(e  1)@e  1 @q1 @p B @q2 = ( 1q1)H0 1(e  1)@e  1 @q2 :(62) The ou p ice- eac ion e¤ec s can be ob ained a e he de i a i es o e  1wi h espec o quali ies a e subs i u ed by hose abo e. Fi ms choosing he same second quali y unde sepa able cos unc ion We now claim ha q 2=  2so he wo … ms choose he same le el o he second quali y when cos is 38