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Richa rd Schmalen s ee
FE B 24
Economet r ic D ia gno s i s o f Comp e t i t ive Lo ca l i za t ion
Richa rd Schmalen s e e9’
1390—8 3 Janua ry 198 3
Ec onome t r i c D i agno s i s o f Compe t i t i ve Local i za t i on
R i chard S chmalensee *
S loan S choo l o f Manag emen tM as sachus e t t s Ins t i tut e o f Technology
ABS TRACT
Comp e t i t ive loc al i za t i on i s p re sent i n an ex t reme form i n
one-dimens i onal s pa t i al mod el s ; e ach b rand f ac e s only two r ival s
no mat t e r how many brand s ar e on t he ma rke t . A t e s t tha t do e s
n o t re qu i re unreal i s t ic ally long da t a s e ri e s i s p ropo se d f o r
de t ec t ing the p re s enc e o f locali z ed c omp e t i t i on i n marke t s wi t h
d i ffe rent i at ed p r od uc t s,and t e chni que s f o r a s se s s ing t he na t ure
and impor tanc e o f suc h loc al i za t i on a r e pre s ent e d . Appl i c at i o n
o f t he s e me thod s t o da ta on t he U . S . r eady- t o— ea t b reakfas t
c e rea l indus try sugge s t s t he i r po t ent i a l value i n o the r c ont ex t s
b u t , b e c ause o f a pparent da t a p roblems,y i e lds no d e f ini t e
conc lus i on s abou t t ha t indu s t ry .
January 1983
0 745195
1 . Int roduc t i o n
Two p olar—ca s e model s o f marke t demand and s elle r int erac t i on emerg e d
r oughly s imul tane ously a s e conomi s t s b egan t o s t udy p roduc t d i f fe rent i a t i on
sys t emat i c ally" t he spa t i al mode l o f Ho t elling ( 1929) and t he symme t ri c mode l
u s ually a s so c i a t ed wi t h Chambe rl in The s pa t i al f ramework s t re s se s
buye r he t e rogene i t y ; add i t i ona l brand s make i t mor e l ike ly tha t any pa rt i c ula r
b uye r wi ll f ind a s ingle one we ll-s ui t ed t o h i s o r he r t a s t e s . S ymme t r i c
model s usually involve a re pre sent a t i ve buye r who i s imag ined t o buy al l
b rand s and t o bene f i t d i re c tly f rom i nc reas ed var i e t y . B o th polar c as e s
appea r f req uent ly i n t he mode rn l i t era t ure " c ompar e t he spat i a l analy s e s o f
S alop ( 1979) and S chmalens e e ( 1978 ) wi th t he symme t r i c model s o f S penc e (1976 )
and Dix i t and S t ig l i t z
In t he o r ig inal Ho t el ling mod el,wi th s e lle rs a rrayed along a l ine in
ge ographi c o r p roduc t spac e , ri val ry i s local i z e d i n t he s ens e tha t e ach brand
f ac e s only a small numbe r o f d i re c t c ompe t i t or s,no mat t e r how many s e lle r s
a re p re s en t i n t he ma rke t a s a whole . Thi s s or t o f local i za t i on can c onve r t
a n apparent ly large-numbe rs s i t ua t i on i nt o one o f ove rlap p ing o l igopo l i e s , s o
t ha t ove ral l marke t c onc ent ra t i on may be a s er i ous ly mi s lead ing i nd ic at o r o f
t he l ike l i hood o f non— compe t i t i ve b e havi o r . In symme t r i c model s , on t he o t he r
hand , each f i rm affec t s and i s af fec t e d by all o t he r s i n a symme t ri c fas hi on ,
s o t hat t he f ac t o f p roduc t d i f fe rent i a t i on has no e f fe c t on t he pa t t e rn o f
rivalry,and c onc ent ra t i on may be measured on a marke t—wide ba s i s .
Whi le one—dimens i onal mod el s o f t he Ho t e ll ing var i e t y a re u se ful f o r
i llus t ra t ing the impl ica t i on s o f loc al i za t i on [ a s i n S chmalens e e the y
b ui ld i n an e xt reme f o rm o f t hi s e f fec t . As Ar chi bald and Ro senblut h ( 1975 )
have shown , loc al i za t i on ne ed no t be impo rtant o r even p re s ent i n s pa t i a l
m ode ls o f highe r d imens i onal i ty . Thus,e ven i f one ac c ep t s t he pe r sua s ive
argument s o f Archi bald , Eat on , and Li p sey (1982) tha t addre s s mode ls , i n
whi ch brand s a re i den t i f i ed by t he i r l oca t i ons i n t he s pac e o f po s s i ble
produc t s , gene rally p rovi d e t he ri ght framework fo r t he s t udy o f di ffe rent i a t e d
ma rke t s , one i s l ed t o t he c onc lu s i on t ha t t he impor tanc e o f l ocal i za t i on i n
any par t ic ula r d i ffe ren t i a t e d marke t i s an emp i ri ca l que s t i on . Thi s e s say i s
c onc e rned wi t h deve lop ing t echni que s f or answe ring such q ue s t i ons . I t i s t hus
int ended a s a c ont ri but i on t o t he g rowing l i t e ra t ur e on ec onome t ri c analy s i s
o f individual marke t s . l
W i t h enough da t a , o f c ours e , one c ould a s s e s s t he impo r tanc e o f
local i za t i on by examining t he c oe ff i c i ent s of e s t ima t e d unre s t r i c t e d
b rand— s pec i f i c d emand f unc t i ons . In s uch f unc t i ons,e ach brand ' s s ale s would
depend on suc h t hing s a s t he p ri c e , adve rt i s ing , and lagged sale s of al l
b rands on t he marke t,along wi t h var i able s s uch a s i nc ome t ha t would i nf luenc e
t o t a l marke t demand . Bu t t hi s app roach enc ount e r s a se ri ou s problem o f
d imens i onal i t y" i f t he re a re a s ub s t ant i al numbe r o f b rand s , e ach brand ' s
e qua t i ons wi ll have many unknown parame t e r s , and unreal i s t i cally long da t a
s e r i e s would be r equi red t o o b t a in r e l i able e s t ima t e s . The e xi s t ing
l i t e ra tur e i n ec onomi c s p rovide s no al t e rna t ive app roache s tha t are not i c eabl y
l e s s da ta-int ens i ve .
Mo s t o f t he re levant e c onome t r i c wo rk i n marke t ing s imp ly a s sume s a
2s ymme t ri c mode l . On t he o the r hand
,marke t e rs f re quent ly c ons t ruc t
pe rc ep t ual map s , ba s e d on que s t i onnai re and o the r non-marke t da ta , tha t
3dep i c t ind ividua l brand s ' l oca t i on s i n produc t spac e . Fo r a vari e t y o f
r easons , i nc luding t he d i f f i cul ty o f i nt e rpre t ing d i s tanc e s i n s uch map s ,
the s e and rela t e d t echni que s ar e no t g ene rally us eable by an ec onomi s t
i nt ere s t ed i n t he o ve rall impor tanc e o f local i za t i on i n any p ar t i cular marke t .
The approac h t aken he re s ide s t e p s t he d imens i onal i t y problem tha t plague s
unre s t ri c t ed b rand— s pec i f ic d emand s ys t ems by f ocus ing on t he re s i dual
cova rianc e ma t ri x o bta ine d unde r a fa i rly gene ra l symme t r i c s p ec i f i ca t i on .
4
Tha t s pe c i f i ca t i on i s p re sent ed i n S e c t i on 2, and a large-s ample l ikel i hood
ra t i o t e s t f o r de par tu re s f rom symme t ry i s de ri ve d . S ec t i on 3 r epor t s t h e
r e s ul t s o f applying t ha t t e s t t o da t a on t he U . S . r eady-t o-ea t b reakfa s t
c e rea l indus t ry , whi c h i s re fe r re d t o s imply a t t he RTE ce re al indu s t ry i n
wha t f ol lows . The p re senc e o f l o c al i za t i on was an impor tant e lement o f t he
gove rnment ' s ca s e i n a rec ent ant i t ru s t p roc e ed ing involving t ha t i ndu s t r y ;
s ee S chmalense e ( 1978 ) f o r a s ummary o f t he a rgument . Whi le t he ini t i al
de c i s i on i n tha t proc e ed ing f ound tha t local i za t i on exi s t s t o a s uff i c i en t
d egr e e t o have t he t yp e o f i mpac t t he o ri z ed by c omp la int c ounse l , t he nat ure
o f t he avai lable evidenc e c omp e lle d t he f ind ing tha t t he ext ent t o whi c h
l ocal i za t i on p reva i l s i n t he RTE c e real i ndus t ry i s an unknown"
( B e rman 1981 ,
p . Te chniq ue s f o r a s se s s ing the pa t t e rn and impor tanc e o f loc a li za t i on
a r e p re s ent ed i n S e c t i on 4, and t he i r a ppl i ca t i on t o t he RTE c e real s da t a i s
d i s cus sed . Our c onc lus i on s a re br i e f ly s ummari z e d i n S ec t i on 5.
2 . Te s t ing f or Local i za t i on
Th e o b j e c t ive o f t hi s s e c t i on i s t o d e r ive re fut able imp li ca t i ons o f t he
null hypo the s i s o f no local i za t i on , t ha t i s , o f pe rf ec t symme try i n
c omp e t i t i ve i nt e rac t i ons . O pe ra t i onally,I i nt e rpre t t hi s hypo t he s i s a s
f o llows . Cons ide r a marke t i n whi ch N brand s o f some p roduc t ar e s old .
Cons i de r a c hange i n an o b s e rvable o r unob se rvable var iable t ha t ha s t he
di rec t e f fe c t o f maki ng brand 1 mo r e a t t rac t i ve t o buye rs . Example s would b e
an i nc rea s e i n i ' s adve r t i s ing,a r educ t i on i n i t s p r i c e
,a s hi f t i n buye r
pre fe renc e s t oward i , and ( f o r normal g ood s ) an i nc rea s e i n c onsume r i nc ome .
I int e rpre t p e r fe c t s ymme t ry t o r e qui re t ha t s uch c hange s no t af fe c t t he ra t i o
5o f brand j
's sale s t o b rand k
's sale s
,fo r all j
, k. # i . Chang e s i n
r ela t ive s ale s would r e f lec t s ub s t i t ut i on p at t e rns d e ri ved f rom depar t ure s
f r om symme t ry i n t he p at t e rn o f b rand s ' l oca t i ons i n p roduc t s pac e , and o ur
nul l hyp o the s i s rule s ou t suc h depar t ure s .
Unde r p e rfe c t s ymme t ry , c hange s i n marke t-wi de var iable s ( such a s
cons ume r i nc ome ) tha t aff ec t t he leve l o f t o ta l marke t demand mus t leave al l
ma rke t share s unchanged . The impl i ed mul t i p li ca t i ve s ep arabi l i ty o f
brand- sp ec i f i c demand func t i ons t hen allows u s t o neg lec t suc h vari able s when
working wi th marke t s hare s and l ead s na turally t o t he u se o f what ha s been
calle d an"
a t t rac t i on mode l t o de t e rmine tho s e share s
N
( 1 ) M . oic xi ,
1
J 1J J J
whe re M1
i s brand i ' s marke t share,®i
i s i t s non—nega t ive at t rac t i on
( t he par t o f i t s demand func t i on no t involving "marke t-wid e var i able s ) , Xi
i s a vec t o r o f obse rvabl e p rede t e rmined variable s t ha t d i rec t ly affe c t th e
i s ad emand f or b rand i ( such a s i t s own p r i c e and adve rt i s ing ) , and ui
di s turbanc e t e rm . The di s tu rbanc e s a re mo s t na turally i nt erp re t e d a s
s ummar i z ing t he e f fec t s o f unob s e rvable var iable s t ha t d i re c t ly af fec t e ach
brand ' s a t t rac t ivene s s and tha t chang e during t he sampl e pe ri od . Thi s
i nt erpre t a t i on i s c ent ral t o o ur a pproach,a s we re s t r i c t t he u
it o a ffec t
marke t share s only through change s i n t he oi
Func t i ons l ike ( 1 ) re la t e marke t share s t o p r i c e s i n t he symme t r i c
r epre sent a t ive i nd ividual mode l s o f S p enc e ( 1976 , S ec t . 5 ) and Dix i t and
S t ig l i t z ( 1977) t ha t emp loy gene rali z e d CES ut i l i t y func t i ons . (No t al l
s ymme t ri c u t i li t y f unc t i ons a re c ons i s t ent wi th ( l ) , o f c our se . ) S inc e
c onsumers ' t as t e s appear t o di ff e r,and all c onsume r s do no t buy all brand s i n
m o s t marke t s,a more p laus i ble i nt e rpre ta t i on o f ( l ) and o f t he null
hypo t he s i s o f p erfec t symme t ry i s t he followi ng . As wa s no t e d above , i n
addre s s mode ls"
o f h igh d imens i onal i ty,t he p at t e rn o f buye r and brand
addre s se s nee d no t imp ly t he ex i s t enc e o r impor tanc e o f compe t i t iv e
l oc ali za t i on . Pre c i s e c ond i t i ons on t he p a t t e rns o f buye rs ' t a s t e s and
brands ' a t t r i but e s suf f i c i en t t o rule ou t local i za t i on i n suc h mode l s ar e no t
known , howeve r . Mo re ove r,d i rec t ve r i f i ca t i on o f any s uch c ond i t i ons would
almo s t c e r t ainly requ i re bo t h larg e quant i t i e s o f da t a on ind i vi dual brand s
and buye rs and t he u se o f s t rong mai nt a ined hypo t he se s . The a rgument he re i s
tha t i f t he p a t t e rn s o f t a s t e s and a t t ri but e s i n any part i cular marke t ar e
s uch a s t o rende r local i za t i on a bs ent o r unimpor tant,an e quat i on l ike ( 1 )
mus t p rovid e a good app roximat i on t o agg rega t e behavi o r .
B e ll , Keeney , and L i t t le ( 1975) and Barne t t ( 1976 ) p rovide al t e rna t i ve
axi oma t i c j us t i f i ca t i ons f o r a t t rac t i on model s , and the surve y of Par son s and
S chul t z ( 1976 , c h . 7) a t t e s t s t o t he i r p op ulari t y i n app l i ed work on marke t
share s . In add i t i on t o p e rf ec t symme try , suc h model s have t he imp or t an t
p rope r ty t ha t p red i c t ed s hare s a re a lway s be twe en z e ro and one and a lways sum
6t o one .
In o rde r t o g o f ar the r,i t i s nec e s sary t o make a s sump t i ons abou t t he
d i s tu rbanc e t e rms . I t i s c onveni ent , s t andard , and no t unu sually re s t r i c t i ve
t o a s sume t ha t t he ui
a re j o int ly no rmally d i s t ri but ed wi t h z e ro means and
t hat t he Oimay be wr i t t en a s Unde r thi s las t
a s sump t i on , i f we us e brand N a s t he ba s e brand , t ake logar i t hm s of ( 1 ) and
s ub t rac t , we ob t a in
( 2 ) 1n [Mi/MN] (u
i-l .L1
N
I f t he ®ihave a f ini t e numbe r o f unknown pa rame t e rs , t hi s sy s t em o f
e quat i on s can be j o int ly e s t ima t e d by s tandard me thod s .
7I f t he u
ifo llow
aut o regre s s i ve proc e s s e s wi t h the s ame c oe f f i c i ent s , qua s i-d i f fe renc ing c an b e
u s ed t o e limina t e s e ri al c o rre la t i on,and t he c o e f f i c i ent s o f t he
8aut o regre s s ive p roc e s s can be e s t ima t e d along wi t h the parame t e r s o f the m
i.
We make the usua l a s sump t i on t ha t all di s turbanc e va rianc e s and c ovar i anc e s
a re c ons t ant o ve r t ime .
S up p ose t ha t t he null hyp o the s i s i s c o rrec t and t ha t a sys t em l ike ( 2) ha s
be en prope rly spec i f i e d , s o t ha t t he X1inc lud e ( a t lea s t ) var ia ble s t ha t
provide a good de sc ri p t i on o f se lle rs ' marke t ing ac t ivi ty . I now wan t t o
a rgue t ha t unde r tho s e a s sump t i ons,
mus t e qual s ome c ons t ant ,
2 , fo r al l i 7 j . The argument i s j u s t an app l i cat i on o f t he de f ini t i on o f
p e r fec t s ymme t ry g iven above t o t he unob se rvable d e t e rm inant s o f b rands '
a t t rac t i ons . I f t he null hypo the s i s i s c o rrec t and t he demand sys t em i s
p rope rly s pe c i f i ed , t he unob s e rvable d e t e rm inant s mus t r e f le c t e i t he r i ndus t ry
wid e o r brand- sp ec i f i c fac t o r s . ( I f t he sy s t em i s prope rly sp ec i f i e d , al l
i mpor tant marke t ing va r iable s a re i nc luded i n t he Xi ’ s o t ha t s i gni f i c ant
unobse rvable f i rm—wid e marke t ing change s by mult i-brand f i rm s are ruled ou t .
Ta s t e change s t ha t a f fe c t s ubse t s o f t he s e t o f b rand s a re ruled out by t he
null hyp o the s i s o f no loc al i za t i on . ) Unde r our def ini t i on o f pe rf ec t symme try ,
an inc reas e i n ui
c aus ed by a change i n e i t he r t ype o f f ac to r c anno t b e
a s soc ia t ed wi t h a c hange i n ( uj
uk) i n ( 2 ) f o r any j i k , s inc e
o the rwi s e t he r ela t ive s ale s o f b rand s j and k would be a ffe c t ed . But t hi s i s
eas i ly s e en t o re qu i r e tha t t he cova rianc e o f uiand u
jb e id ent i c a l f o r
al l i and j , a s as s e rt e d .
The p re ce edi ng a rgument i mp li e s t e s t able r e s t r i c t i ons on t he c ovar i anc e
ma t r ix o f the reduc ed- f orm d i s turbanc e s i n I f 2 i s t ha t ma t r ix , wi t h
t yp i cal e lement 0 and vi
i s t he va rianc e o f ui, t he re s t r ic t i ons
a re the following
( Vi
2 ) (vNz ) ,
( 3 ) —l o
( v z ) ,
By cons ide r ing the c ovar ianc e ma t r i x o f t he ui ,
i t i s ea sy t o se e t ha t 2 c an
exc e e d a t mo s t one o f the vii f t hey ar e al l d i s t inc t . Thi s c ons t ra i nt i s
d i f f icul t t o i mpo se , howeve r . We lo s e l i t t le g ene ral i ty by r equ i ring t ha t a ll
t he viexc ee d 2 , s o tha t we can s impl i f y no ta t i on i n wha t fo llows by s e t t ing
Z t o ze ro .
We c an t e s t f o r d ep ar ture s f rom p er fe c t s ymme t ry by e s t ima t ing a sys t em
l ike (2) wi t h an unc ons tra ined c ont emporaneou s d i s turbanc e c ovar ianc e ma t r i x
and t hen u s ing t he re s i dual c ova ri anc e ma t r ix t o t e s t f o r t he val i di t y o f t he
re s t r i c t i ons g ive n by We employ the a symp t o t i c di s t r i but i on o f a
l ike l i hood ra t i o s t a t i s t i c,s o t ha t t he t e s t has only t he u sual large— sample
prope rt i e s . We los e none o f t ho s e p rope r t i e s by t rea t ing t he parame t e r s o f
t he ¢ia s known i n d e r iving t he t e s t . The re s i dual c ovar i ance ma t rix , S ,
w i t h t yp i cal e l ement 8 i s t hu s t rea t ed i n wha t f ollows a s havi ng be eni j
’
g ene ra t ed by o b s e rvat i ons f rom a mul t i var ia t e no rmal d i s t r i but i on wi t h
o 9c ovarianc e ma t r i x 2 and z e ro mean vec t o r .
Unde r t hi s a s sump t i on,t he ( N l ) x ( N 1 ) mat r ix S ha s t he Wi shar t
di s t r i but i on , and t he log-li kel i hood func t i on i s t he f ollowing
(4) L c (T/2) 1n lz [ (T1s) ,
whe re C i s a c ons t ant and T i s t he numbe r o f o b se rva t i ons . ( S ee , fo r
i ns tanc e , Rao ( 1973 , p . Rao us e s S t o deno t e t he d i s p e rs i on ma t r i x ,
whi ch e qual s T t ime s t he samp le c ovar ianc e mat r ix . ) G iven S , t he unre s t r i c t ed
max imum o f L i s o bta ined by s e t t ing 2 S , and t he max imum value i s
Lu
c ( N /2) ln I S I T (N
In o rde r t o maximi z e L s ub j e c t t o t he r e s t r i c t i ons g i ven by ( 3 ) wi t h z 0,
we make u s e o f the fac t t ha t t he f ollowing ar e val i d i f tho s e re s t r ic t i on s
hold (Rao 1973 , pp . 32-3 )
N -l
'Z | H vND ) , whe r e
i=11
N —l
D Z an d
l( 6 c ) [ 2 l
i jdi j/ vi
vN/ [ v
iv
j( l
whe re si j
1 i f i j and i s z e r o o the rwi s e . Us ing i t i s s t ra igh t
f orward bu t t ed i ous t o show t ha t t he f i rs t- orde r c ondi t i ons f o r max imi z ing L
wi t h re sp ec t t o t he e s t ima t e s V i a re t he f o llowing
( 7a ) vi
si i
vN[ 1 Z (DF — l ,
and
( 7b ) V ( F r 5) /E whe reN
AN -l
( 7C ) Pi
i=l, o o o ,
N -l ,
AN -l
( 7d ) F X F / v, ,
1 11
— 1
and D i s gi ven by ( 6b ) a s a func t i on of the V i ' The s e e qua t i on s may b e
solve d i t e ra t ively fo r t he maximum like l ihood e s t ima t e s . The c on s t ra i ne d
maximum o f L c an be s hown t o b e e qual t o
( 8 ) Lr
c [ m T (N
whe re I Z I i s g i ven a s a func t i on o f t he V i by ( 6a ) and ( 6 b ) .
The t e s t s ta t i s t i c i s then j u s t
A -2(L L Tr u
Th e unre s t r i c t ed max imum i nvo lve s N ( N l ) /2 p arame t e rs , whi le t he re s t r i c t ed
maximum involve s N pa rame t e rs , s o t ha t X i s a symp t o t ic ally di s t r i bu t e d a s
x2w i t h N ( N d egre e s o f f re edom .
In o rd e r t o a pply t hi s t e s t , one mus t c omple t e t he s pe c i f i ca t i on o f ( 2) by
making a s sump t i ons abou t t he X1and t he ¢
i° Thi s of c ours e g i ve s r i s e
t o a c las s i c p roblem" i f t he t e s t de s c ri bed above re j ec t s t he nul l
hypo the s i s , i t may be s i gnall ing m i s-s pec i f ic a t i on o f t he p ar t i cular symme t r ic
mode l employed , rat he r than a t rue vi ola t i on of symme t ry . Examina t i on o f
a l t e rnat ive s pe c i f i ca t i ons and u s e o f t he d iagno s t i c t ec hn i que s p re s ent ed i n
Se c t i on 4 should he lp one choo s e be tween t he s e al t e rnat ive s , a s we i llus t ra t e
b e low .
3 . An Appl i ca t i on t o t he RTE Ce real s Indus t ry
Th e re a re two reas ons f or a t t emp t ing t o t e s t f o r l ocal i za t i on i n t he RTE
ce real s indus t ry . F i r s t , a s wa s ment i oned i n t he i nt roduc t i on , t he exi s t enc e
and imp or tanc e o f l o cal i za t i on i n t ha t i ndus t ry we re i s sue s i n a r e c ent
ant i t rus t p roc ee ding . S ec ond , i n t he c our s e o f tha t p roc eed ing , t he U . S .
F ede ral Trade Commi s s i on ( FTC ) o b t a ined f rom t he l ead ing s e lle rs mont hly
brand-sp e c i f i c da t a o f t he s or t t ha t would appea r su i table f o r t he appl i c a t i o n
o f the p roc edure de sc r i bed i n S e c t i on 2 . Tho se da t a a re u sed he re ; t hey have
al s o been employed by Aake r,Ca rman
,and Jac ob son who p rovi d e
a dd i t i onal de sc r i p t ive i nfo rma t i on .
Al t hough s ome mont hly s e ri e s f o r s ome f i rms run f rom 1957 t hrough 1973 ,
only f o r t he 40— month pe r i od from Sep t embe r , 1969 through De c embe r , 1972 d o
t he da ta c ont a in s ale s ( f ac t o ry s hi pment s ) , whole sale l i s t p r i c e s , and
adve r t i s ing out lay s f o r t he four larg e s t p roduc e r s , along wi t h s ome
inf o rma t i on on t he o the r two na t i onal se lle r s .
1 0 The s e s i x f i rms , wh i ch
a cc ount ed f o r o ve r 97 p e r c ent o f R TE c e real s ale s i n 1970 ( Be rman 1981 , p .
had ove r 70 brand s i n na t i onal d i s t r i bu t i on dur ing t hi s p e ri od . I t would
c learly no t be po s s i ble t o examine local i za t i on by e s t ima t ing an unre s t ri c t e d
d emand s y s t em i nvolving a ll brand s on t he marke t . Even sy s t em e s t imat i on o f
our symme t ri c mode l (2) us ing all brand s i s c omput a t i onally infe as i ble .
Ac co rdingly , we cho se t o wo rk e xc lu s i vely wi th t he 11 l ead ing brands l i s t ed i n
Tabl e 1 . The s e we r e t he only brand s tha t ind i vidual ly account ed f o r a t lea s t
p e r c ent o f i ndu s t ry d ollar s ale s in e ach o f t he f our year s 1969-1972. As
Tabl e 1 indic a t e s , t he s e brand s i n aggrega t e acc ount e d f o r j u s t unde r half the
s i x l ead ing f i rms ' t o t al RTE c e real r evenue s d ur ing t hi s p e r i od . Our s ample
inc lude s a t lea s t one brand o ffe re d by each o f t he four large s t p roduce r s .
I f e qua t i on ( 1 ) holds i n s ome marke t,and t he add i t i onal a s sump t i ons t ha t
le d t o equa t i on s ( 2) a re c or rec t , one c an employ (2) us ing any subs e t of t he
b rand s on t he marke t . F or p urpo se s o f t e s t ing f o r l ocal i za t i on , we can t hu s
t rea t t he s e 11 brand s a s i f t he y we r e t he whol e ma rke t . Do ing t hi s naturally
l imi t s o ur a bi l i ty t o make d e f ini t e s t a t ement s a bou t p a t t e rns o f l ocal i za t i on
i f t he null hypo the s i s o f pe rf ec t symme t ry i s re j ec t e d , howeve r . Our cho i c e
o f 11 brand s r e f lec t s a t rade of f be tween c omprehens i ve c ove rage and
c omput a t i on c os t .
G iven t he f o rm o f e qua t i ons i t i s no s urpr i se t ha t i t i s s t andard in
t he relevan t l i t e ra t ur e t o as sume tha t t he ¢ i are Cobb-Dougla s func t i ons ,
and we adop t t ha t a s sump t i on . As Ca s e ( 1981 ) demons t ra t e s , i f sha re s ar e t o
b e invar i ant t o e qui p ropo r t i onal c hange s i n e lement s o f t he X1
a c ro s s b rand s
( a doubling o f al l pr i c e s , f o r ins tance ) , t he ¢imu s t be Cobb-Dougla s wi t h
e qual exponent s . Thi s form emerg e s i n t he symme t r i c CES mode l s of S p e nc e
( 1976 , S e c t . 5 ) and D ixi t-S t ig li t z N ae rt and We v e rbergh ( 1981 ) no t e
tha t relaxing a s sump t i on o f equa l exp onent s o f t e n c aus e s nume ri c a l p ro blem s
akin t o t ho se c aused by mul t i c ollinear i ty,and we enc ount e red s uch p roblems
12
wi t h the s e da t a . (We we r e i n fac t unabl e t o o bt ai n c onve rg enc e f o r
s p ec i f i c a t i ons t ha t r e laxed t hi s a s sump t i on . ) Acc o rdingly , we als o make t he
as sump t i on o f e qual exponent s ac ro s s brand s and spec i fy t he oia s f ollows
( 10 ) ai
B y 6 1nMi[ ( t
whe r e A i s adver t i s ing sp end ing , P i s whole sale p ri c e , and M i s marke t shar e
i n p ound s o f c e real s old . The lagged s hare t e rm re pre sent s an a t t emp t t o
c ap tur e the e ff ec t s o f buye r ine r t i a . ( Thi s i s alge bra ic ally e qu ivalent her e
t o t he u se o f lagged p ound sale s . ) In addi t i on,we a s sume t ha t t he u
i( t ) i n
(2 ) f ollow a f i r s t-o rde r aut o regre s s ive proce s s wi t h aut oc or re lat i o n
l lc oe f f i c i ent p . F inally
,g i ven f i rms ' info rmat i on and d ec i s i on—making
lags , i t s eems p lau s i bl e t o t r ea t bo th pr i c e and adve r t i s ing a s prede t e rm ined
i n monthly dat a . ( S e e S chmalens e e ( 1972) on f i rms ' adve r t i s ing dec i s i ons . )
D i f fe renc e s i n t he aiac ro s s b rand s may be i nt e rpre t ed a s re flec t ing
d i fferenc e s i n unob se rvable d e t e rminant s o f b rands ' a t t rac t i ons t ha t d o no t
vary during t he s ample pe r i od . As we ar e i nt e re s t e d i n change s i n quant i t i e s
d emanded , i t i s a ppropr i at e t o wo rk wi th s hare s o f sale s measured in pound s .
Change s i n share s t hen re f lec t movement s i n quant i t i e s sold , e ve n though one
m ight a rgue t ha t leve ls o f sha re s s o mea sured have d oubt ful s igni f i c anc e .
( Pound sale s f o r Gene ral Mi ll s we r e c ons t ruc t e d u s ing p ri c e and dolla r sale s
s e ri e s ; s ale s in p ound s we re re por t ed f or t he o t he r f i rms . ) We u se d t he
repor t e d whole sal e l i s t p r ic e s f o r t he Pi( t ) ‘ (A pound—we ight e d ave rag e o f
l i s t p r ic e s f o r seve ra l sub-brand s wa s us ed f o r Cap ' n Crunch . ) Al tho ugh t he s e
f i rms g ene rally d id no t d epar t f rom l i s t p r ic e dur ing mo s t o f t he l96o 's ,
t rade deal s , shor t- t e rm di scount s below li s t p r ic e , we re us e d wi t h
i nc reas ing f requency d uring our s ample p eri od ( Be rman 1981 , p p . 106 Thi s
i s a po t ent i ally impo rt ant s ourc e o f sp ec i f ic a t i on e r ro r , a s suc h deal s
c anno t r eal i s t i cally be t r ea t ed a s exogenous , random e vent s , but da t a t o
c ons t ruc t a s e r i e s o f t ransac t i on s p r i c e s do no t exi s t .
The r ep o r t ed adve r t i s ing s e r i e s e xhi b i t ed large mont h-t o-mont h
varia t i ons . ( The med ian c oe ff i c i ent o f va ria t i on i n t he 11 adve r t i s ing s e r i e s
wa s t he se c oe f f i c i ent s we re a bove f o r a ll brand s and above f o r
al l bu t one . In cont ra s t , only one o f t he 11 c oe ff i c i ent s o f va r iat i on f o r
p ound sale s e xc e eded Mo re su rpr i s ingly,t he re we re two nega t i ve
adver t i s ing numbe r s i n ou r da t a . ( The s e we r e f o r brand s 9 and As Aake r ,
Ca rman , and Jac ob son ( 1982) d i scu s s,t he se f ea ture s o f t he adve r t i s ing s e ri e s
undoubt edly re f lec t bi lling and acc ount ing p rac t i c e s tha t cau s e shor t- t e r m
t iming d i f fe renc e s be twe en t he a ppearanc e o f ad s and t he manufac ture r ' s
paymen t fo r t hem . In orde r t o smoo t h t he s e r i e s and , i t wa s hop ed , e l imina t e
t he e f fec t s o f random t iming d i f fe renc e s,we appl i ed a moving ave rage f i lt e r ,
wi t h we ight s of on the nex t month and t he la s t mont h and on t he
c urrent mont h . The se we i gh t s we re cho sen i n p ar t s o a s t o e l imina t e t he
nega t i ve ob se rva t i ons . Us e o f t hi s f i l t e r , along wi t h t he pre senc e o f s e ri a l
c o rre la t i on and lagged dep endent va ri abl e s i n o ur s pe c i f i ca t i on , r e duced t he
numbe r o f obse rva t i on s t o 37.
As we no t ed a bove ( f oo tno t e t he cho i c e o f a base b rand [ brand N i n
eq uat i on s i s arbi t rary . Tabl e 2 p re sent s c o e ff ic i ent s and s tandar d
e rro rs o bt a ined by j o int nonl inear g ene ral i ze d l eas t s quare s u s ing t he large s t
and smalle s t b rand s ( i n t e rm s o f ave rag e pound sale s ) a s t he ba se . (We
e mp loyed t he Z ELLNER o p t i on i n t he GREMLIN p ackage on t he TROLL sys t em a t
MIT . ) The c oeff i c i ent s us ing t he larg e s t brand (Kellogg ' s Corn Flake s ) a s a
bas e s e em reasonable , bu t f i ve o f t he 1 0 ind ividua l R2s ta t i s t i c s ar e
2neg at ive . Only one R i s neg at ive when t he smalle s t b rand (Kellogg ' s Fr o o t
Loop s ) i s us ed a s t he ba s e , bu t t he c oe ff i c i ent e s t ima t e s ar e much le s s
sa t i s fac t o ry . (As the dependent va r iable s d i f fe r , one canno t c ompar e R
value s be twe en t he two s e t s o f e s t ima t e s . ) The s ens i t i vi ty o f c oe ff i c i ent
e s t imat e s t o t he cho i c e o f t he ba s e brand i s t roubl ing . I t s t rongly sugge s t s
t he p os s i b i li ty o f m i s-s pec i f i c a t i on . ( S e e f oo tno t e 7, above . )
Table 2 als o repo r t s t he re sul t s o f t he l ike l i hood rat i o t e s t f o r
1 2local i zat i on pre sent e d i n Sec t i on 2 . I t i s i nt e re s t ing t o no t e tha t t he
e s t ima t ed d i s t urbanc e var ianc e s d o no t d i f fe r much be twe en t he two s e t s o f
e s t ima t e s . Al s o , t he e s t ima t e d va rianc e s f o r t he s ix Kellogg ' s b rand s
( numbe rs 1 , 3 , 4, 6
,7, and 11 ) a re much smalle r t han f o r t he o t he r f i ve
brand s . The t e s t s ta t i s t i c,X, de f ined by equa t i on (9) i s d i s t ri but e d a s
X2w i t h 44 degree s o f f re edom und e r t he null hypo the s i s . The value s s hown
i n Table 2 a re t hu s bo t h h ighly s igni f i cant . ( The p e r c ent c ri t i cal l eve l
i s
Taking t he se t e s t r e sult s a t f ac e value,one would b e c ompe lled t o re j ec t
t he hyp o t he s i s of p erf ec t symme t ry and acc ep t t he p re senc e o f loc al i zat i on .
B ut t he q ual i ty o f our e s t imat e s and t he da t a p robl ems d i sc us s ed above
s t rongly sugge s t t he po s s i b i l i t y o f mi s—sp e c i f i ca t i on,s o t ha t cau t i on i s
c alle d f o r a t thi s s tage .
1 3
4. As s e s s ing Pa t t e rns and Impo r tanc e o f Local i za t i on
As wa s no ted in t he i nt r oduc t i on,t he l ea s t re s t r i c t ive approac h t o
as se s s ing depar t ure s from symme t ry involve s s imply e s t ima t ing unre s t r i c t e d
b rand— s pe c i f i c d emand f unc t i ons . Bu t t hi s u sually r e qui re s unreal i s t i cally
long da t a se ri e s , and i t i s no t fea s i ble wi t h t he par t i cula r da t a s e t use d
h ere . We c an make s ome headway i f we a re will ing t o re t a in t he a s s ump t i on
t ha t marke t share s dep end only on t he ®1(X1 ’
ui) even i f ( 1 ) doe s no t
hold . In such a general i z e d a t t rac t i on mode l , t he pa t t e rn of c ompe t i t ive
Table 2
Es t imat i on and Te s t Re sul t s fo r RT E Ce re al s Da t a
No t e s
1 . F igure s i n p arent he se s a re s t andard e rro rs .
_ l6
e f f ec t s i s a s sumed t o b e t he s ame f or a ll b rand-s pec i f i c marke t ing var iable s ,
like pr ic e and adve r t i s ing . Whi l e t hi s i s o f c our s e s omewha t re s t r ic t i ve , i t
d oe s no t s eem g ene rally implau s i ble . An a symme t r i c g ene rali z ed a t t rac t i on
mode l s e em s a reasonable reduc e d-fo rm rep re s enta t i on of an addre s s mode l
w i th lo cal i za t i on,i n whi ch t he e f fec t s o f c hange s i n t he 9
1on t he
rela t ive sale s o f r ival brand s re flec t and summar i z e t he pa t t e rns o f brand and
b uye r loca t i ons i n p roduc t s pac e .
In s uch a m odel , t he deve lopment i n S chmalense e ( 1982 , S ec t . 2 ) s ugge s t s
t ha t t he fo llowing can be us e d t o analyz e t he na ture and impor tanc e o f
l ocal i za t i on"
( 1 1 ) Ki j
i=j ;
In t he symme t r i c mode l , equa t i on i t i s ea sy t o s e e t ha t t he Ki j
al l
e qua l uni ty . In gene ral , t he add ing-up re s t r i c t i on on the M1re qui re s
share-we ight e d ave rage s o f t he Ki j
t o e qual uni t y ;
(12) X Ki
,Mj/ Z M . 1 ,
m Jj
‘
7‘iJ
I f one o f t he Ki j
e xce eds uni ty,i t impl i e s t ha t i nc rease s i n b rand i ' s
s ha re c ome mo re a t t he exp ens e o f b rand j t han a symme t r i c model would
p redi c t . Di rec t examinat i on o f t he Ki j
can thu s yi e ld inf o rmat i on on
pa t t e rns o f loc al i za t i on .
To summar i z e t he impor tanc e o f lo cal i za t i on ac ro s s t he marke t as a who le ,
S chmalens e e ( 1982 , S ec t . 2) p rop o s e s us ing t he fo llowing general i zat i on o f t he
s tandard H— index o f c onc ent ra t i on "
I
I
MZ
( 1 3a ) m (My / G1,
whe rei l
2( 13 b ) G M
1( 1 M
i) 1
_ l7 _
I t i s ea sy t o s e e t ha t H* i s b ounded be tween t he H-index and uni t y and t ha t i t
a t t a ins i t s lowe r bound only i n t he symme t r i c ca s e i n whi ch t he Ki j
al l
e qua l one . The ra t i o o f H t o H* t hu s p rovide s a mea sur e of t he imp or t anc e o f
l ocal i za t i on t ha t i s b ounded be twe en z e ro and one"
N2
(14) c * Z (Mi=1
1
Low value s o f 0 * s i gna l imp or t ant local i za t i on of c ompe t i t i ve i nt erac t i on ; G *
i s e qual t o one only i n t he p er fe c t ly symme t r i c c a s e .
Le t u s now c ons ide r thre e app roache s t o e s t ima t ing t he Ki j
and G* tha t
ec onomi z e on dat a . Al l t hre e involve u s ing s t a t i s t i c s de ri ved f r om e s t imat i o n
o f e qua t i ons whi ch a re m i s—s pec i f i ed i f l oc al i za t i on i s pr e sent . The s e
approache s can thu s y i e ld di agno s t i c inf o rmat i on bu t no t rig o rous ly de f ens i bl e
e s t ima t e s . (All t hi s i s i n t he s p i r i t o f t he d i agno s t i c t echniq ue s p r e s ent ed
by Phlip s ( 1974, pp . 212 The f i r s t two t e chni que s emp loy re s i dua l
c ovari anc e s f rom a symme t r i c mode l t o p rovide i nfo rmat i on on t he l ocal i za t i on
o f t he eff e c t s o f t he unob se rvabl e de t e rminant s o f b rand s ' a t t rac t i ons . Unde r
t he g ene rali z ed a t t rac t i on a s sump t i on,t he same p at t e rn o f l ocal i za t i on
appl i e s t o bo t h obse rvable and unob se rvable va ria ble s .
Th e f i rs t and s imple s t t e chni que wo rks wi t h t he ( s ingular ) c ovar i anc e
ma t r ix o f t he eit "i t "i t , t he d i f fe renc e s b e twe en t he a c t ual s hare s
a nd t ho s e p redi c t ed by t he s ymme t r i c model . Le t t hi s mat r ix be S'
, wi t h e lement s
sgj . Then u se "B [ ( ej t ei ) whe re c i s any
c on s tant,a s an e s t imat e o f t he ra t i o o f de ri va t i ve s i n The ra t i onale
i s t ha t t hi s e s t imat o r provide s a rough answe r t o t he f ollowing que s t i on
G i ven an i nc reas e i n b rand i ' s s hare c au sed by uno bserv ables , how much o f t ha t
i ncrea s e i s exp e c t e d t o c ome a t t he exp ens e o f brand j's shar e" A pro ble m
w i t h t hi s non-s t ruc tural a pproach i s t ha t we c anno t a s soc i a t e inc rea s e s
18
i n ei t
di rec t ly wi t h i nc rea s e s i n brand i ' s own a t t rac t i on . Und e r
n o rmali t y,and t rea t ing t he ei a s known , our e s t imat or i s j us t
and t he e s t ima t ed Kij a re o b t a ined by s ubs t i t ut i on i n
We a ppli ed t hi s t e c hn i que t o t he RTE c e reals da ta , u s ing t he e s t ima t e s
repo r t e d i n Sec t i on 3 and t he s ampl e mean value s o f t he M1. Fo r bo t h se t s
o f e s t ima t e s , we o bt a ined a surp r i s i ng ly larg e numbe r of nega t ive e s t ima t e d
Ki j
' One m ight expec t t o o bt a in negat i ve e s t imat e s o f t he se p arame t e r s i f
e xogenous , unmea sured s hocks a f fe c t ed t he a t t rac t ivene s s o f g roup s o f c lo s e
subs t i t ut e b rand s a s agai ns t al l o t he r s . Fo r example , larg e s hi f t s i n buyer s '
p re fe renc e s f o r ra i s ins m ight be e xp ec t ed t o c ause t he s ale s o f brand s 7 and 9
( the two ra i s i n brans ) i n ou r s amp l e t o move t oge the r . I f one c ould i dent i f y
s uch i nf luenc e s d uring t he s amp le p er i od,and i f t he p a t t e rn o f nega t i ve K
i j
made sens e i n l i gh t o f tho s e inf luenc e s , one migh t be le d t o c onc lud e tha t
Ki j
0 i mpl i e s an unusually c lo se s ub s t i t ut e re la t i on b e twe en brands i
and j . Thi s would be i nfo rmat ive,e ven t hough one c ould no t s ens i bly emp loy
the ove ral l local i za t i on measur e G* i n the p re s enc e of nega t ive e s t ima t e s o f
t he K . . o
1 3
Unfor tuna t e ly,no s uch e xogenou s g roup-s p ec i f i c d i s t urbanc e s a re a pparent
he re . Mo re ove r , t he pa t t e rn o f negat i ve value s ac tually o bs e rved s e ems t o
p o int c learly t o s pec i f i cat i on and da t a p roblems . F o r bo t h s e t s o f e s t ima t e s ,
i f brand s i and j ar e e i t he r bo t h Kellogg brand s o r bo t h non-Ke llogg brands ,
Ki j
i s a lways negat ive,whi le i f i i s a Kellogg brand and j i s no t ,
Ki j
i s
a lways p o s i t i ve . ( Thi s s ame p at t e rn g ene rally holds when s imple r
spec i f i ca t i on s ar e us e d t o g ene ra t e p redi c t ed share s . ) I t doe s no t make muc h
s ense t o t hink o f t hi s a s r e f le c t ing loc al i za t i on,g i ven t ha t b o t h Kellogg and
non-Kellogg s e t s cont a i n brand s i n mo s t o f the segment s di scus s e d b y
i ndus try o bse rve rs . ( Brand s 3 , 8 , and 11 a re p re-swe e t ened , brands 7 and 9
mode l ( 15) then sugge s t s t he f ollowing log-l inea r sp ec i f i cat i o n
N
( 1 7) si t
ri j
u
j t, t=l , o o o , T o
J- l
whe re ui t
s ummar i z e s t he e f fec t s o f unob se rvable d e t e rminant s o f brand i ' s
a t t rac t i on i n p e ri od t , a s b e f o re . F ollowing t he d i sc u s s i on i n S ec t i on 2 , i f
t he symme t ri c mode l i s p rope rly sp ec i f i e d , i t i s rea sonable t o a s sume tha t t he
ui t
a re unc orre la ted ac ro s s b rand s . Be c au se o f t he homogene i t y o f t hi s
e xp re s s i on,and be c au se we a re only c onc e rned wi t h ra t i o s o f t he f o rm
we can s e t t he var ianc e s o f t he ui t
t o uni t y wi t h no lo s s o f
gene ral i t y . Le t R be t he ma t ri x o f t he ri J
I t i s na tural t o c hoos e t he
s igns o f t he e lement s o f R s o tha t p o s i t i ve value s o f uia re a s soc i a t ed wi t h
po s i t i ve value s o f Ei' Thi s i s mo s t eas i ly done by re qu i ring R t o b e
non—nega t ive de f ini t e . Unde r the s e a s sump t i ons,t he c ova r ianc e ma t r ix o f t he
61i s j us t RR '
, and one can e s t ima t e R a s the uni que non-nega t ive def ini t e
s quare r o o t o f t he s amp le c ovar i anc e mat ri x o f t he ei. Eq uat i ons
wi t h ri j
replac i ng ei j
eve rywhe r e t hen y i e ld e s t ima t e s o f t he Ki j
’
The re sul t s repo r t ed j u s t above se em s o s t rong a s t o make sy s tema t i c
a ppli ca t i on o f t hi s a pproach t o t he RTE c e rea ls d a ta ra the r p o int le s s ,
howeve r .
15
F inally , i f e s t ima t i on o f ( 2) y i e ld s p laus i ble e s t ima t e s o f t he p arame t e rs
o f t he mi , i t may be ins t ruc t ive t o c ons t ruc t and employ a se ri e s o f
e s t imat ed a t t rac t ions, ¢ i t ’ f or e ach b rand . Apply ing t he t rans format i on
d i s cus s ed by Bulte z and N aert ( 1975) t o one o bt a ins
N N
( 18 ) ln (Mi) "M
kln (M.
k) "o
whe r e the Gi j
a re de f ined by ( l6 b ) . The s e la s t quant i t i e s ar e no t i n
g ene ral c ons t ant o ve r t ime , o f c ours e . But i f t he var ia t i on i n share s i s no t
gr ea t , subs t i t ut ing ¢if o r O
ii n add ing an i nt e rc ep t t erm , an d
runn ing a s e t o f N l inea r r egre s s i ons s hould y i e ld serv i c able e s t ima t e s o f t he
1 6ave rag e value s of t he O
, ,t ha t c an be us e d i n ( l 6a ) . The qual i t y an d
1 J
i ns t abi l i ty o f t he e s t ima t e s r ep o r ted i n S e c t i on 3 would a lso s eem t o rule out
s e ri ou s us e o f t hi s t e chni que wi t h t he ava i lable RTE ce real s da ta .
5 . Conclus i ons
S e c t i on 2 p re sent ed an a pproach t o t e s t ing f o r t he p re senc e o f l ocal i z ed
comp e t i t i on t ha t c an b e implement e d wi t hou t enormou s quant i t i e s of da ta . In
o rde r t o p roduc e s uch an a pp roach,i t was nec e s sary t o make a s sump t i ons b o t h
abou t func t i ona l fo rm and abou t t he rela t i ve unimpor tanc e o f ind ivi dua l
unob s erved de t e rminant s o f b rand s ale s . App li ca t i on o f t hi s approach t o da t a
o n t he RTE ce real s indus t ry i n S ec t i on 3 s t rong ly re j ec t e d t he nul l hypo t he s i s
o f p erfec t s ymme t ry,but t he e s t ima t e s on whi ch t hat r e j ec t i on wa s bas e d we re
unsa t i s fac t ory i n seve ra l re sp ec t s . S ec t i on 4 p re sent e d t e chni que s f o r
a s s e s s ing t he na t ure and i mpo r tanc e o f lo cali za t i on t ha t al so e c onomi z ed on
da t a re qui rement s . App li c a t i on o f t he s imple s t of t ho s e t e chni q ue s t o t he RTE
c e re als da ta c learly d emons t ra t ed t he impor tanc e o f s pe c i f i ca t i on o r
measuremen t p roblems tha t appa rent ly make i t inapp rop ri a t e t o i nt e rp re t
S e c t i on 3 's t e s t re sul t s a s p roviding a r e futa t i on o f t he null hypo t he s i s o f
no local i za t i on . Tho s e re sul t s s e em ra the r t o be dri ven by inadequac i e s o f
t he ava i lable da t a .
O ur t e chni que s f or d i agno s i s o f lo cal i za t i on i n marke t analys i s t hu s f ai l
t o produc e po s i t ive re sul t s when appl i e d t o t he FTC's RTE ce re al s da t a . Bu t
t he re i s e ve ry r eason t o s u s pe c t t ha t t hey o r r e f inement s o f t hem wi ll be
use ful i n t he d iagno s i s o f loc al i za t i on i n o the r c ont ex t s . (Af t e r all , t he
_ 22 _
RTE c e rea l f i rms re sp onde d i n a p re sumably unc oordinat ed fa shi on t o a broad ly
worded r e que s t f o r da t a f rom t he FTC , whi ch wa s i n t he proc e s s o f b ring ing a
maj o r ant i t ru s t cas e agai ns t t hem . O the r app roache s t o da t a c o llec t i on shoul d
y i eld sy s t emat i cally b e t t e r r e sult s . ) I t i s al so enc ourag ing t ha t a
s trai ght f orwar d app l i c at i on o f t he d iagno s t i c t e c hni que s p re sent e d i n Sec t i on
4 was a ble t o d e t ec t s e ri ous unde rly ing weakne s se s i n t he p ubl i cly-avai lable
da t a on t hi s int e re s t ing indus try , we akne s s e s t ha t migh t o the rwi s e have g on e
unno t i c ed . The se t echni que s s hould b e o f value e ven i f t hey a re only u se d t o
a s s e s s t he val id i t y o f symme t ri c sp e c i f ica t i ons o f t he s or t t ha t ar e wide l y
u s ed i n marke t s ha re analys i s .
Re fe renc e s
Aake r , D . A . , Ca rman , J .M . and Jac ob son , R . Mode ling Adve rt i s ing-Sa le s
R e la t i onshi p s Involving Fee dback" A T ime S e ri e s Analy s i s o f S ix Ce real
Ar chi bald,C . C . and Ro senblu th
,G . The ' New ' The ory o f Cons ume r
Demand and Monop oli s t i c Compe t i t i on, "uar t e rly Journal o f Ec onomi c s ,
( February ) , 569-90 .
Ar chi bald,C.C.
,Eat on
,B .C. , and L i p sey ,
R . G .
"
Addre s s Model s o f
Value Theory,
P re sent e d a t t he Int e rnat i ona l Ec onomi c s As soc ia t i o n
Co nfe renc e on New Deve lopment s i n Ma rke t S truc t ure,O t tawa , May 1982.
Re s ea rch , 1 3 ( Fe bruary ) , 1 04-09.
B e l l , D . E . ,Ke eney
,R . E . and L i t t le
,J . D .C. A Marke t S har e The orem ,
Journal o f Marke t ing Re search ,1 2 (May ) , 1 3 6—41 .
B e rman,A . L .
"
Ini t i al D ec i s i on i n t he Ma t t e r o f Ke llogg ,e t al . ,
Docke t No . 888 3 , Was hi ng t on " U . S . Fede ra l Trad e Commi s s i on ,
m ime og raphed,S e p t embe r 1 .
B e rnd t ,E .R . and S avin
,N . E . E s t imat i on and Hyp o the s i s Tes t ing i n
S ingula r Equat i on Sy s t ems wi t h Au t o regre s s ive D i s t urbance s , Ec onome t ri ca ,
43 ( S ep t embe r—Novembe r ) , 937-57.
B r e snahan ,T . F . I dent i f i ca t i on o f Ma rke t Powe r , S tanfo rd Uni ve rs i t y ,
Workshop on Fac t o r Ma rke t s , Re s ear c h Pape r No . 15 , Oc t o be r .
Bulte z ,A .V . and N aert ,
P .A. Cons i s t ent S um— Cons t ra ined Mode l s ,
Journal o f t he Ame ri c an S t at i s t i cal As soc i a t i on ,70 ( Se p t embe r ) , 529-3 5.
Cas e , J .
"
Ad —Re s pons e Mode l s and Ea rly Entry ,
"
i n Ke on , J .W . ( ed . )
Analys i s ( Providenc e " The Ins t i t ut e o f Management S c i enc e ) .
Chamberlin ,E . R . ( 1933 ) The Theory o f Monopol i s t i c Compe t i t i on , l s t Ed .
( Cambri dge " Ha rvard ) .
D i x i t ,A . R . and S t ig l i t z
,J . E .
"
Monopol i s t i c Compe t i t i on and O p t imum
P roduc t D ive r s i ty,
Ame ri can Ec onomi c Revi ew , 67 ( June ) , 297-308 .
Ha use r , J .R . and Koppelman,F . S .
"
Alt e rna t i ve Pe rc ep t ual Mapp ing
R esea rch , 1 6 ( Novembe r ) , 495— 506 .
Ho t elli ng , H .
"S t abi l i t y i n Compe t i t i on
,Economi c Journal , 39
( March ) , 41—57.
Hu be r , J . and Ho lbro ok,M . E . Us ing At t r i but e Ra t ing s f o r Produc t
Po s i t i oning " S ome D i s t i nc t i on s Among Comp o s i t i onal App roache s , Journa l
o f Ma rke t ing Re s earch , 16 ( Novembe r ) , 507-1 6 .
Jeuland,A . P .
"
Emp i ri ca l Inve s t igat i on o f P ri c e and Adver t i s ing
Co mpe t i t i on U s ing a Marke t S ha re Model,
"
i n Montgome ry ,D . E . and Wi t tink ,
In s t i t ut e ) .
N ae r t,P . and Wev e rbe rgh ,
M . On t he Pre di c t i on Powe r o f Marke t Share
Pa rsons,L . J . and S chul t z
,R . L . ( 1976 ) Ma rke t ing Model s and Econome t ri c
Re s ear c h (Ams t e rdam" Nor th-Holland ) .
Phlip s ,L . ( 1974) Appl i ed Consump t i on Analys i s ( New Y o rk" Elsevi er ) .
Rao ,C.R . ( 1973 ) Linear S ta t i s t i ca l Infe renc e and I t s Appli c a t i ons , 2nd Ed .
( New Y o rk" W i ley ) .
S alo p , S .C.
"
Monop oli s t i c Compe t i t i on wi th Ou t s i de Goods ,
"
B ell
Journal o f Economic s , 10 ( S pr ing ) , 141-56 .
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N o rth~Ho lland ) .
_ 25
S c hmalens e e,R . Entry De t e r renc e in t he Ready-t o-Ea t Breakfa s t Ce re a l
Indus t ry,
Be ll Journal o f Ec onomi c s , 9 (Aut umn ) , 3 05—27.
S chmalens e e,R .
"
The New Indu s t r i al Organi za t i on and t he Economi c
Analys i s o f Mode rn Marke t s , p p . 253-85 i n H i ldenbrand,W . ( e d . ) Advanc e s
S pence,A .M . Produc t S e lec t i on , F ixe d Co s t s , and Monopoli s t i c
Compe t i t i on,
Revi ew of Ec onomi c S t udi e s , 43 ( June ) , 217-3 5.
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_ 26
Foo tno t e s
I am inde bt e d t o t he Nat i onal S c i enc e Founda t i on f o r f inanc ia l s uppor t , t o
W i ll i am Burne t t f or c ons i de rable he lp wi th da t a p roblems , t o Je rry Hausman
f o r many he lp ful c onve rsa t i ons , t o Thoma s S t oke r f o r valuable c omment s on an
e arl i e r draf t,and t o M i chae l S al inge r and , e s pe c ially , S eve rin Borens t e in
f o r supe rb resera c h as s i s t anc e . The usual wave r of l iabi l i t y appli e s , o f
C Curs e 0
S e e B re snahan ( 1981 ) f o r a valuable s urvey and b i bl i og raphy .
Par sons and S chul t z ( 1976 , ch . 7) p rovide a surve y ; s e e N aert an d
We v erbergh ( 1981 ) f o r a r ec ent e xamp le .
Fo r rec ent ove rvi ews o f t hi s work , s e e Haus e r and Koppelman ( 1979) and
Hu be r and Ho lbrook
O ur a pproach i s i n t he s p i r i t o f t he se cond moment model o f The i l
e sp ec ially a s regard s t e s t ing f o r devia t i on s f rom symme try , and o f
t he d iagno s t i c t echni que s p re s ent ed by Phlips ( 1974, ch .
5 . Whi l e t hi s requi rement ha s a s t rong f o rmal re s emblanc e t o th e
"
indep endenc e o f i rre levant a l t e rna t ive s " a s sump t i on i n cho i c e t he o ry , i t
should bec ome c le ar be low tha t I am no t a s suming t ha t ind i vidual behavi o r
c onforms t o t hi s a s sump t i on .
On the imp li ca t ions o f t he se r e s t r i c t i ons,s e e B e rnd t and S avin
Bulte z and N aer t N aer t and We v erberg h and the re fe renc e s
t hey c i t e .
I t s e ems s omewha t more c ommon i n a ppl i ed wo rk t o s ubt rac t t he ave rage o f
all log-share s f rom each o f the N logg ed equat i on s and t hen dele t e one o f
t he re sul t ing e qua t i ons " s e e Bulte z and N aert N aert and
We v erberg h and e qua t i on be low . (Maximum like l i hoo d
e s t imat e s a re i nvari ant t o t he e qua t i on de le t ed . ) Whi le t hi s approach
pe rfo rm l ocali za t i on t e s t s u s ing raw c ova rianc e mat ri c e s of d i ff e renc e s i n
l og-s hare s and s imple a ut o regre s s ive mode l s ; p e rfec t symme t ry was always
dec i s i ve ly re j ec t ed . ) Wi t hou t t ho s e re sul t s , i t would have be en na t ura l
t o e xp lore t he u se o f d ollar s hare s ( as i n t he t rans —log l i t e ra t ure ) and
t o expe riment wi t h the al t e rna t ive t iming a s sump t i on us ed by Aake r ,
Ca rman , and Jac ob son ( They f ound s urpr i s ing weak brand-s pe c i f i c
adve r t i s ing-sale s re la t i ons hi p s f o r a s e t of Ke llogg b rands . )
The s e re sul t s s ugge s t t ha t da t a p roblems may g o a l ong way t oward
expla ining t he surp ri s ing ly weak bivaria t e a s soc ia t i ons be twee n
a dve r t i s ing and sale s f ound by Aake r,Ca rman
,and Jac ob son ( 1982) f o r
Kellogg brands .
S i nc e we lacked c onf i dence i n t he c oe f f i c i ent e s t ima t e s r epor t ed i n
S ec t i on 3 , we appl i ed t hi s t e chni que wi th devi at i on s from means replac ing
p red i c t i on e rro rs . All but 3 o f t he 50 K1
e lement s above t he ma inj
d iagonal c onformed t o t he s i gn pa t t e rn de sc r i be d above . ( The mat r i x o f
Ki j
i s s ign-symme t r i c . )
In c ons t ruc t ing the ¢ i unde r s pe c i f i ca t i ons l ike an arbi t rary
p o s i t ive value of aNmus t be cho sen . The add i t i on o f an i nt erce p t
t e rm t o ( 18 ) s e rve s t o c ompensa t e f o r e rro rs i n t hi s c ho i c e . S inc e a ll N
e qua t i ons have the same ri ght-hand s id e variable s , the r e i s no e ff i c i enc y
g a in f rom sy s t em e s t ima t i on he re . W i t h b e t t e r da t a , o f c our s e , one c ould
a t t emp t t o e s t ima t e al l o f t he parame t e r s of mode l s l ike ( 1 5) d i rec t ly .
