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
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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