Monopolistic Competition with Outside Goods
Author(s): Steven C. Salop
Reviewed work(s):
Source: The Bell Journal of Economics, Vol. 10, No. 1 (Spring, 1979), pp. 141-156
Published by: RAND Corporation
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Monopolistic
competition
withoutsidegoods
Steven C. Salop
BureauofEconomics
FederalTradeCommission
The Chamberlinianmonopolisticallycompetitiveequilibriumhas been explored
and extendedin a numberof recentpapers. These analyses have paid only
cursoryattentionto the existence of an industryoutside the Chamberlinian
group.In thisarticleI analyze a model ofspatial competitionin whicha second
commodityis explicitlytreated. In this two-industry
economy, a zero-profit
with
located
exhibit
ratherstrangepropequilibrium
symmetrically
firmsmay
erties.First,demand curves are kinked,althoughfirmsmake "Nash" conjectures. If equilibriumlies at the kink,the effectsof parameter changes are
perverse. In the shortrun,prices are rigidin theface of small cost changes.
In thelong run,increases in costs lowerequilibriumprices. Increases in market
size raiseprices. The welfarepropertiesare also perverseat a kinkedequilibrium.
1. Introduction
* The Chamberlinian
(1931)zero-profit
monopolistically
competitive
equilibriumhas been exploredand extendedin a numberof recentpapers.These
and have
analyseshavefocusedon themonopolistically
competitive
industry
attention
totheexistenceofan industry
outsidetheChamberpaidonlycursory
liniangroup.In thispaper,a modelofspatialcompetition
is analyzedinwhich
is explicitly
a secondcommodity
treated.
withsymmetrically
In thistwo-industry
economy,a zero-profit
equilibrium
locatedfirms
First,demandcurvesare
mayexhibitratherstrangeproperties.
Ifequilibrium
firms
make"Nash" conjectures.
is a tangency
eventhough
kinked,
toparameter
andlong-run
thekink,
theshortsolution
responses
awayfrom
changes
ofparamliesat thekink,theeffects
areconventional.
However,ifequilibrium
eterchangesareperverse.In theshortrun,pricesare rigidinthefaceofsmall
cost changes.In the long run,increasesin costs lowerequilibrium
prices.
thecostincreaseas an excisetax,thisresultstatesthattheincidence
Interpreting
ofthetaxis negative.Increasesin marketsize raiseprices.The welfarepropThisisa revisedversion
ofa survey
Reconstituted"
paperentitled
"Monopolistic
Competition
to Don Hester,Lew Johnson,
Bob Mackay,PerryQuick,SteveSalant,Joe
(1976).I amindebted
and to MaryAnnHenryfor
AndyWeiss,therefereeand editor,forhelpfulcomments
Stiglitz,
in thispaperrepresent
The remarks
editingand typing.
onlymypersonalviews.Theyare not
intended
tobe,andshouldnotbe construed
oftheviewsofanyothermember
of
as, representative
theFederalTradeCommission
staff
or theindividual
Commissioners.
141
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142 / THE BELL JOURNALOF ECONOMICS
ertiesarealso perverseat a kinkedequilibrium.
Decreasesincostandincreases
in marketsize lowerbothconsumerand aggregate
welfare.
In thenextsectiontheformalmodelis presented
andthesymmetric
zero(SZPE) is defined.Conditionsforexistenceof the
profitNash equilibrium
SZPE are derivedin Section3. Comparative
staticsand welfareproperties
are
in
Sections
and
4
The
concludes
with
a
short
5,
explored
respectively. paper
discussionofthedeterrence
equilibrium
concept.
2. The basic model
* In thissectionwe analyzea variantofthetraditional
(1929)model
Hotelling
of spatialcompetition
whichis derivedfromLernerand Singer(1937).In this
variantthe economythatis envisionedconsistsof two industries.
The one
withdifferentiated
brands
uponwhichwe focusis monopolistically
competitive
and decreasingaveragecosts; the otheris a competitive
industry
producing
a homogeneous
Each ofL consumers
commodity.
purchaseseitherone unitor
none1ofthedifferentiated
topreferences,
commodity
according
prices,andthe
distribution
of brandsin productspace. Remainingincomeis spenton the
homogeneous
commodity.
Eachconsumer
hasa most-preferred
brandspecification
1*.A brandI differentfromthemostpreferred
is valuedloweraccordingto preferspecification
encesinproductspace U(l,l*). Theproductspaceoftheindustry
is takentobe
an infinite
lineor theunit-circumference
ofa circle.Whileneither
assumption
is realistic,
bothallowthe"corner"difficulties
oftheoriginalHotellingmodel
tobe ignored
andan industry
withidentical
equilibrium
pricesbyequally-spaced
firms
toobtain.Eliminating
thetechnical
difficulties
makesitsimpler
toanalyze
thequalitative
ofthemodel.Thus,themodelis a benchequilibrium
properties
markforsubsequentanalyseswithnonuniform
acrossempirically
preferences
validatedproductspaces. By eliminating
technicalproblems,
thismodelallows
a focuson theessentialinteractions
offirms
in an industry.
Iftherearen brandsofthedifferentiated
availableat pricespi
commodity
andlocations
whosemostpreferred
is l* willpurchase
li,a consumer
specification
one unitfromsomebrandifthemaximum
less priceacross
surplusof utility
brandsoutweighs
thesurplusfromthehomogeneous
othergood.Denotingthat
surplus
bys, wehavethedecisionrule:Purchaseoneunitofthebrandsatisfying
max [U(li,l*) - pi] - .
(1)
The traditional
modelofconstanttransport
costsis capturedwithpreferences
givenby
(2)
U(li,l*) = u - c l - 1*i,
wherethe"distance" I li - l* I refersto theshortestarc lengthbetweenli andl*.
In thiscase, equation(1) maybe rewritten
as follows:
max [v - c |li - /*I - Pi]
i
0,
(3)
wherethe effectivereservationprice is given by
v= u -
> 0.
The model easily generalizesto elastic demands.
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(4)
SALOP / 143
FIGURE 1
TYPICAL DEMAND CURVE
PRICEA
v
MONOPOLY
COMPETITIVE
-_.--.....
-
.--SUPERCOMPETITIVE
,,
,QUANTITY
We now explore the existence and propertiesof a symmetriczero-profit
Nash equilibrium(SZPE). By symmetricwe mean an equilibriumin whichthe
brandsare equallyspaced aroundthecircularproductspace and chargeidentical
prices. By zero profits,we mean an equilibriumin whichfreeentryleads to a
situationofeach brandearningzero profits.The equilibriumis Nash in thateach
brandchooses a best price, given a perceptionthatall otherbrands hold their
prices constant.The conclusion discusses the requirementthatthe numberof
brandsbe integer-valued.
The methodologyhereconsistsof derivingtheperceiveddemandcurvefor
a singlerepresentativebrandas a functionof otherbrands' prices and locations
and thenfindinga tangencybetween that demand curve and the average cost
curve. Three regionsof the representativebrand's demand curve may be disthe"monopoly,""competitive,"and "supercompetitive"
tinguished:
regions.The
"monopoly" regionconsists of those prices in whichthebrand's entiremarket
consistsofconsumersforwhomthesurplusof no otherbrandexceeds thesurplus
of the homogeneousoutside good. The "competitive" regionis composed of
those prices in which customersare attractedwho would otherwisepurchase
some other differentiated
brand. The "supercompetitive" region consists of
those prices in which all the customersof the closest neighboringbrand are
captured. These three regions of a typical demand curve are illustratedin
Figure 1.
brandchargespricep and its nearestcompetitors
Suppose therepresentative
located at distance lln charge/, as shown in Figure 2. We derive the regions
of the demand curve as follows. In the absence of competitionfromother
differentiated
brands, the representativebrand captures all consumers living
withina distance where the net surplusgiven in (3) in nonnegative.Denoting
the maximumdistance by x and substituting
into (3) we have,
v
v-p=
.
x =
=(5)
c
If thereare L consumersaroundthe circle, since the brandcapturescustomers
withina distancex on each side, its monopolydemand qmis given by
qm =
2L
L(vc
p).
(6)
This definesthe potentialmonopoly marketof the representativebrand.
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144 / THE BELL JOURNALOF ECONOMICS
FIGURE 2
THE CIRCULAR MARKET
P
r
/n
1
(c
1/n
I
p
t
Those consumersresidingin thepotentialmonopolymarketof two brandspurchase fromthe one offeringhighernet surplus. If the brands are located a
distance apart of 1/nand the neighboringbrand on one side charges a pricep,
then from(3) the representativebrand captures all those consumers withina
distancex given by
-
v - ex - p
v - c
(n
- x
)
-(7)
Denotingby x, the value forwhich (7) holds withequality,we have
x =-
2c
(P + cin - p)
(8)
and hence a firm'scompetitivedemand qC = 2Lx is given by
qc = _ (p + c/n - p).
c
(9)
(6) and (9), the slopes sl(D) of the demand curve in these
Differentiating
two regionsare given by
= -c/2L
(10)
sl(Dc) = -c/L.
(11)
sl(Dn)
Thus, we have the unusual resultthatdemand is more elastic in the monopoly
region than in the competitiveregion. Moreover, as illustratedin Figure 1,
the monopolyregioncomes at higherprices.
The two regionsfittogetheras follows.Suppose theright-sideneighborhas
a potentialmonopoly marketillustratedin Figure 3. At prices above v, the
brandobtainsno customers.As itbeginsloweringpricesbelow v,
representative
it captures demand fromthe homogeneous good according to the monopoly
FIGURE 3
MARKET SEGMENTS
/
p
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SALOP / 145
FIGURE 4
MARKET OVERLAP
P
slope c/2L. Eventuallyits price becomes low enoughthatits monopolymarket
overlapsthemonopolymarketofitsneighboras illustratedin Figure4. Now as it
it begins to capture customersfromits neighboraccordlowers price further,
ingto the steepercompetitiveslope cIL. At the kinkin Figure2, the monopoly
regionsjust touch.Note thatthekinkarises herefromtheexistenceoftheother
industry,not fromthe non-Nash perceptionsdiscussed by Sweezy (1939). It
generalizesto higherdimensionalspaces.2
At some lowerprice, even those customersresidingat the neighborare indifferent
betweenthe representativefirmat p and the neighborat p (and additional surplusc(1ln)). This pricepz is given by
Pz = P - c/n.
(12)
At prices below pz, the representativefirmcaptures the entiremarketof its
neighbor,fornot only are those consumersresidingat the neighborwillingto
incur the surplus loss cln for the price differential
p - p, but so are all the
customersof the neighbor.Thus the representativebrand's demand has a discontinuityat pz fromthis "predatory" pricing.
The demand curve in Figure 2 displays the typical shape of these three
brands.
regions.It shiftsaccordingto thepricesand locationsoftheneighboring
Since demandcan neverexceed the monopolydemand,the kinkalways lies on
thatmonopolycurve, as illustratedin Figure 5 (withsupercompetitiveregions
deleted). Note that the marketmay be so competitiveas to make the kink
nonexistent.This occurs when the neighbors'potentialmonopolymarketincludes the location of the representativebrand.
3. Existenceofa symmetric
zero profit
(SZPE)
equilibrium
* A SZPE is definedas a pricep and a numberof brandsn such thatevery
equally spaced3 Nash price setter's maximumprofitprice choice earns zero
profits.We ignoretheadditionalrequirementthatthenumberofbrandsmustbe
an integerand discuss itlater.In addition,thepotentialnonexistenceof equilibin demandis also postponed.Ifan equilibrium
riumarisingfromthediscontinuity
exists,therepresentativebrand's demand curve and average cost curve willbe
tangent,forthen the zero-profitpoint is surelyalso one of maximumprofits.
are possible, as illustratedin Figure6, where
Three equilibriumconfigurations
the monopoly,kinked,and competitiveequilibriumprices are denotedby subscripts(m,k,c), respectively.
At themonopolyequilibrium,some consumerslyingbetweentwo neighborcommodity.Thus, the markets
ing brandsmay not purchase the differentiated
2
This may be confirmedin a two-dimensionalproduct space. The monopoly market is
circular,while competitivemarketsare polygonal.
3 This
equilibriumconcept is static. In a dynamiccontext,it assumes that firmsmay costlessly relocate in response to entryand, in fact, do relocate. Thus, equal spacing is maintained.
For a discussionof an alternativeequilibriumconcept, see the conclusions.
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146 / THE BELL JOURNALOF ECONOMICS
FIGURE 5
FAMILY OF DEMAND CURVES
Pt
vL
L/n
of neighborsmay not overlap and each can act as a monopolist,constrained
onlyby the outside commodity.Monopoly equilibriawithand withoutoverlap
are picturedin Figure6. At a kinkedequilibrium,marketsjust touch. As illustratedin Figure6, since theextensionofthemonopolydemandcurvelies above
theaverage cost curve,themonopolypricepmlies below thekinkedequilibrium
pricePk. At the competitiveequilibriumconfiguration,
monopolymarketscompletelyoverlap. However,Pc maybe above or below Pm, dependingon demand
and technologies.
It is easy to show graphicallywhich equilibriumconfiguration
obtains for
any set of technologyand demand parameters{F,m,v,c,L}. Simplydrawing
the average cost curve and the entirefamilyof demand curves, existenceof an
equilibriumconfigurationrequires maximumprofitsequal to zero-pointE in
Figure7. A zero-profit
pointlike G does not satisfymaximumprofitsbecause
it is dominatedby a point like F.
FIGURE 6
EQUILIBRIUM CONFIGURATIONS
MONOPOLY EQUILIBRIUM
KINKED EQUILIBRIUM
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COMPETITIVE
EQUILIBRIUM
SALOP / 147
FIGURE 7
EXISTENCE OF EQUILIBRIUM
I\
L
L/n*
The SZPE satisfiestwo conditions:marginalrevenue(less thanor) equal to
marginalcost and price equal to average cost. For constantmarginalcost m
and fixedcost F, the SZPE is given by
p +q
dp
= m
dq
p = m + Flq,
(13)
(14)
and fromsymmetry,
ifthe equilibriumhas no gaps,
q = L/n.
(15)
At themonopolyequilibriumdpldq is givenbysl(Dm) in (10); at thecompetitive
equilibriumby sl(Dc) in (11); and at the kinkedequilibriumby a slope between
sl(Dm) and sl(Dc). Substituting(15) and (10) into (13) and (14), the monopoly
price and numberof brands4are given by
Pm = m + c/2nm
(16)
/cL/F.
(17)
nm=\--
Using (11) instead of (10), the competitiveequilibriumis given by5
Pc = m + clnc
(18)
n =
(19)
VcL/F.
4 There
Thiscalculation
number
maybe gapsata monopoly
equilibrium.
yieldsthemaximum
ofbrandsat a monopoly
equilibrium.
5 See Grubel
ofthecompetitive
(1963)fora shortderivation
equilibrium.
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148 / THE BELL JOURNALOF ECONOMICS
The values (pk,nk) fora kinkedequilibriumlie between the values given
in (16)-(19). Since thereis no tangencyat a kinkedequilibrium,(13) holds as
an inequality.Instead of being given by the equalityin (13), price is given by
the monopolisticdemand function,or
pk - v - (c/L)q = v - c/n.
(20)
Solving (20) and the price equal to average cost conditiongiven by equation (14) forequilibriumvarietynk, we have
F
-nk
L
+ clnk = v - m.
(21)
The monopolyequilibriumconfiguration
requirestheveryrestrictiveconditionthattheexogenouslygivenaveragecost curvebe tangentto theexogenously
givendemandcurve or v - m = V2cF/L.6 We ignorethislimitingcase forthe
remainderoftheanalysis. The competitiveequilibriumconfiguration
occurs for
all parametervalues such that v - m _ 3/2cF/.7 The kinked equilibrium
occurs for values of v - m in the interval[V'2cF/L,3/2V'cF/L],
configuration
whichis small relativeto the range of values v - m can assume.8
The demand discontinuitycan imply the nonexistence of any SZPE.9
Recallingfrom(12) thattherepresentativefirmcan captureits neighbor'sentire
marketat pricesbelow p - c/n,an additionalconditionforexistenceof a SZPE
is that such pricingbehavior is unprofitable.A sufficient
conditionforthis is
thatthepredatorypricep - c/ndoes notexceed marginalcost m, forpriceequal
to or below marginalcost necessarily is a losing strategyin the presence of
fixedcosts. Referring
to (16) and (18), supercompetitive
behavioris notprofitable,
since the equilibriumprice is no greaterthan m + c/n. Similarly,if marginal
costs are increasing,as with U-shaped AC curves, thenthe market-capturing
AC price.However,ifmarginalcosts are decreasing,
pricelies belowtheminimum
thensuch price cuts may be profitableand cause nonexistenceof a SZPE.
4. Comparativestatics
U As the exogenous technologicalor demandparameters{F,m, v,c,L} vary,
theequilibriumprice-variety
pairalso changes.These changesmaybe calculated
fromthe equilibriumvalues in equations (16)-(19).
O Competitive
equilibria.The comparativestaticsat competitiveequilibriaare
and
straightforward traditional.Substituting(19) into (18) we have
6
Pc = m +
cL
(22)
n, =
(23)
/- .
Forthederivation,
see thederivation
ofequation(31) belowwhenprofits
are zero.
to Figure6, theequilibrium
follows.Referring
Pc derivedin (18) and
7 The derivation
is as
(19) must lie below the monopoly portion of the demand curve, or v - V(c/L)(L/nc) > Pc.
forp, andnc,thestatedcondition
obtains.
Substituting
8 Thisinterval
anddemandspecifications.
maybe largerforalternative
technological
9 See Roberts-Sonnenschein
(1977) forexamples of discontinuousreactionfunctionsleading
to nonexistenceof equilibrium.
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SALOP / 149
As fixedcosts(F) rise,pricerisesandequilibrium
falls.Changesin
variety
costs(m) arefullyshifted
ontoconsumers;
remains
marginal
variety
equilibrium
thesame.Surprisingly
on
perhaps,changesinthenetvaluationv haveno effect
theequilibrium.
is competitive
Themarket
demandis unenoughthataggregate
affected
bychangesin thisdemandprice.
As thevalueofproductdifferentiation
falls.
(c) falls,pricesfallandvariety
As market
size(L) rises,pricesfallandvariety
rises.As c/Ldecreases,demand
cost.Thus,theratioc/L
becomesmoreelasticandpricemovestowardmarginal
is therelevantmeasureof monopolistic
in themodel.
productdifferentiation
obtainswhenc/L = 0,forthen
ForU-shapedaveragecosts,perfect
competition
elasticdemandfunction.
everybrandfacesa perfectly
O Kinkedequilibria.The comparativestaticsat kinkedequilibriaare all
costslowersprices.Thisis illusAnincreaseineitherfixedormarginal
perverse.
fromE to E'. Intuitively,
cost
belowas a movement
trateddiagrammatically
numberof brands,allowingthe remaining
increasesreducethe equilibrium
result.Ifthe
brandsto further
exploitscale economies.Thisis a verystriking
thenthe
as an excisetaxleviedon theindustry,
increaseincostsis interpreted
In termsof
incidenceofthatexcisetax is negativeat thekinkedequilibrium.
ofcourse.
thelowerpriceis offset
consumer
welfare,
bythedeclineinvariety,
However,itis showninthenextsectionthatconsumerwelfaredoes risefrom
thetax,eveniftheproceedsofthetax are ignored.
It shouldbe emphasizedthatthisperversereactionto a costincreaseis a
In theshortrun,
firms.
long-run
responsethatresultsfromtheexitofmarginal
at
revenuecurveis discontinuous
thereis no reactionat all. Sincethemarginal
a smallchangein marginalcosts inducesno price
the kinkedequilibrium,
respondsto a smallmarginalcost increaseas
response.Thus, the industry
do notchange,though
fall
follows.In theshortrun,pricesandquantities
profits
inhigher
toexit,resulting
belownormal(zero). Theselossesinducesomefirms
firms
demandforthosethatremain.Thisincreaseddemandallowstheremaining
in decreasedlong-run
to betterexploitscale economies,resulting
prices.
as illustrated
Anincreaseinthevaluation
bythe
(v) raisespriceandvariety,
E toE". As maybe seenfromFigure8,pricerisesbymorethan
movement
from
as scaleeconomiesarelost.As withcostincreases,the
theincreaseinvaluation,
welfareeffect
ofthisincreaseinvaluationis also perverse;itmaybe shownthat
theincreasein valuationas arisingfrom
consumerwelfarefalls.Interpreting
and the cost increaseas the cost of thatinformative
informative
advertising
the valuationeffectlowerswelfare,whilethe cost effectraises
advertising,
welfare.
Decreasesin c/L,arisingfromeitheran increasein marketsize (L) or a
decreaseinthevalueofproductdifferentiation
(c), raisepricesinequilibrium,
in Figure9 as a movementfromE to E'. As withthe other
as illustrated
staticsdiscussed,thisresultis thereverseof whatoccursin the
comparative
configuration.
competitive
equilibrium
5. Welfareanalysis
It has beenpointedoutby Spence(1976)andothersthat
* Productselection.
andtheproduccommodities
ofsomeunprofitable
theproduction
maybe optimal
tion of some profitable
commoditiesmay be nonoptimal.For the circular
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150 / THE BELL JOURNAL OF ECONOMICS
FIGURE 8
COMPARATIVESTATICS OF (v,m,F): KINKED EQUILIBRIUM
$4
AC'
L/n
thismaybetestedbycomparing
thecondition
underwhicha segment
ofthe
market,
marketwillbe servedby a monopolist
condition)withthe
(the profitability
conditionunderwhichserviceyields positivenet surplus(the optimality
condition).
to the
Supposeeach brandproducesup to thepointwherethenetbenefit
consumer,who is at a distancex* fromthe brandservinghim,is
marginal
zero. Then, the net social benefitper brandof servingthe entirecircular
marketis:
B B=2L L \
Jo
- F,
cx -- m)dx
m
(v -- cx
FIGURE 9
STATICSOF c/L:KINKEDEQUILIBRIUM
COMPARATIVE
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(24)
SALOP / 151
wherethe marginalconsumer's benefitis given by
x* - m = 0.
v-
(25)
we have
Substituting
(25) into (24) and integrating,
L
B = -(v
- m)2 - F,
c
(26)
and this surplusis nonnegative(B _ 0) if and only if the followingoptimality
conditionis satisfied:
v -m
(27)
LF.
On theotherhand, a monopolisticfirmwillchoose to serve any segmentof
whereprofitsforthe monopoly
the marketonlyifits profitsare nonnegative,10
portionof the demand functionare given by
Im
=
(v
q - m
_
-F.
(28)
Maximizing(28) withrespectto q, we have
L
qm = - ( - m).
c
(29)
Substituting(29) into (28), profitsare given by
=
- (v - m)2 - F.
2c
(30)
Profitsare nonnegative (IHm- 0) if and only if the followingprofitability
conditionis satisfied:
/2cF
v- m
.(31)
L
Comparing the optimality and profitabilityconditions, we see that
is sufficient
but not necessary foroptimality.All marketsserved
profitability
should be served, but not vice versa.
O Optimalvs. equilibriumvariety.Giventhattheentirecircularmarketshould
be served,theoptimalprice-variety
pair maybe comparedwiththeequilibrium
price-varietypair. A tradeoffbetween price and varietyexists because of the
scale economies presentin production.
If n firmsoperate and serve theentireunit-circumference
market,thenthe
1
consumer
a
2½n
a
travels
distance
and
consumer
located
at x - ½n
marginal
obtains a surplusin excess of marginalcost of v - m - cx. Since thereare L
consumers per unit distance and 2n intervalsof length½2n, total surplus is
given by
W = 2n
rl/2n
Jo
(v - m - cx)Ldx - nF.
(32)
10Itshouldbe notedthatifonemonopolist
doesnotwishtoserveonesegment,
nomonopolist
willserveanysegment
sincethecircularmarket
is symmetric.
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152 / THE BELL JOURNALOF ECONOMICS
we have:
Integrating,
W= (v - m _- cL
4 n
- nF.
(33)
of (33) is the following.Since (i) the marginalconsumer
The interpretation
travels a distance of ½2nin product space, (ii) the consumer who travels the
shortestdistance(zero) obtainshis mostpreferredbrand,and (iii) L consumers
in productspace, theaverage distancetravelledis 4¼n
are distributeduniformly
at an imputedcost of c per unit.Then the average net surplusper consumeris
v - m - c/4n.Total fixedcosts are nF. Maximizing(33) with respect to the
numberof brandsn to findthe optimalprice-varietymix, we have
*1 cL
2 F
(34)
Comparingthisoptimumto thepossible equilibriagivenby(17), (19), and (21) we
have
n* <nm
< nk < nc.
(35)
Thatis, optimalvarietyis less thanequilibriumvarietyforthiscircularmarket,if
the marketshould be served.
This resultoftoo manybrandsis notrobust,butratherdependscruciallyon
the distributionof consumersand preferences.As Spence (1976) and others"1
have pointedout, the optimumdepends on the differencebetweenthe average
surplusand the surplus of the marginalconsumer relativeto fixedcosts; the
value of adding an extra brand (and respacing the others) effectivelyconvertsmarginalconsumersto average ones, at fixedcost F.12
Graphically,the comparisonof the equilibriumwiththe optimummay be
made as follows. The planningproblem in (33) is equivalent to maximizing
average consumer welfare minus price W(n,p) subject to the price equal to
average cost breakeven constraint.13Since the average consumer travels a
distance V¼nin productspace, we have
max W(n,p) = v - p --
1c
4n
F
subject to p = m + - n.
L
(36)
(37)
curves in (p,L/n) space withslope of
Equation (36) defineslinearindifference
-V4(c/L) while (37) expresses the constraint.As illustratedin Figure 10, a
smallervalue of S expresses a highersurplusv - S. Then, the optimumlies at
the point where the average cost has slope equal to - ¼(c/L), whereas
equilibriumlies at the point where the average cost curve has slope between
-½2(c/L)(formonopolyequilibrium)and -cIL (forcompetitiveequilibrium).
A graphicalrepresentationof the optimumvs. equilibriumprice-variety
pair may be used to show the welfare effectsof the comparative statics at
n Forexample,Dixitand Stiglitz
(1977)and Lancaster(1975).
12
thatis concaveindistancewillyieldexcessvariety,
Itcanbe shownthatanyutility
function
fora uniform
distribution.
Convexfunctions
are necessaryfordeficient
consumer
variety.
13
demandsinthisexample,thatpricedoesnotequalmarginal
Sinceconsumers
haveinelastic
no distortion.
costintroduces
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SALOP
/ 153
FIGURE 10
EQUILIBRIUMVS. OPTIMAL VARIETY
-SLOPE = -1/4 c/L
\~-
SLOPE = -
/n)2
(L/n)2
thekinkedequilibrium.(See Figure 11.) For example, an increase in costs from
AC to AC' thatlowers prices will improvewelfare,since the slope of the indifferencecurve (- 4c/L) is flatterthan the slope of the monopoly demand
curve (-½2c/L). Thus, movementsdown the demand curve representhigher
welfareas illustratedbelow by comparingS to S'. Similaranalysiswillshow that
FIGURE 11
DERIVATION OF OPTIMAL VARIETY
E
S'-
--^
SLOPE = -1/4 c/L
---~~s
\
\%-
-SLOPE
= 1/2 c/L
AC
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154 / THE BELL JOURNALOF ECONOMICS
FIGURE 12
DERIVATION OF THE p(L/n) FUNCTION
v
\\
_-p
)
p(L/n
M
I \
I\
AC
\
I\\
_/,.,....
..-
nc
L/n
an increase in valuation(v) and an increase in marketsize (L) lowers consumer
welfare.
Finally, we can determinethe optimal numberof brands given monopolistically competitive pricing. Spence (1976) shows that for his partial
equilibriummodel,themarketsolutionis optimal.For thecircularmodelstudied
here,the optimumis eitherthe marketequilibriumor completemonopoly.We
may prove this as follows. First we derive the Nash equilibriumprice for
different(exogenously given) numbersof symmetricallyspaced brands. We
denotethisrelationship
is illustrated
in Figure12
byp(Lln).14 Thep(L/n) function
as EE'M, where we assume the SZPE (pointE) is competitive.The complete
monopoly equilibriumis labeled M, and E'M is a portionof the monopoly
demand curve.
The consumerwelfaremaximumcould thenbe foundby placing indifference curves, which have slope -/4(clL), in Figure 12. It is clear that the
optimummustlie at E or M. If the SZPE were kinked,say at E', thenp(Lln)
wouldbe theportionE'M, and theoptimumwould lie at thecompletemonopoly
point. Thus, forthe circularindustry,optimalentrypolicy is eitherfreeentry
or entryrestrictedto the point of each brand having a complete monopoly
market.
6. Conclusions
* In the example studiedhere, explicitattentionhas been paid to the role a
thepropertiesofmonopolistisecondindustry
(outsidegoods) playsin determining
This
focus
us
cally competitiveequilibrium.
required to ignorethe possibilityof
14
Thederivation
Foranynumber
ofbrandsn,
ofp(Lln) relieson thefollowing
observation.
is notserved).
thereexistsa leveloffixedcostsF' suchthatann-brand
SZPE obtains(orthemarket
whenfixedcostsareF, yieldsequilibrium
Nash equilibrium,
Thus,an n-brand
pricep(Lln) and
excess profitsper brandof H = F' - F.
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SALOP / 155
nonuniform
preferencesacross more complicatedand realisticproductspaces
or different
technologiesacross brands.
The majorcontributionof the approach taken here, whichwas firstnoted
by Lernerand Singer(1937), is to providea rationalizationofthekinkeddemand
curve in termsof symmetric"Nash" conjecturalvariations.Previous rationalizations by Sweezy (1939) and others were based upon asymmetriccompetitiveresponses to price increases and decreases. This paper goes beyond
Lerner and Singer by derivingthe industryequilibriumand analyzing its
properties.
The industryequilibriummodel displays conventional propertieswhen
equilibriumoccurs at a Chamberliniantangencyaway fromthe kink. On the
other hand, the propertiesof equilibria occurringat the kink are perverse.
In the shortrun,industryprices do not adjust to small cost changes, as noted
by Sweezy in his model. It is only throughthe process of entryand exit that
the industryadjusts to cost changes. Moreover,at the kinkedequilibrium,the
long-runresponse to a cost increase is exit by some brands followed by a
decrease inindustry
prices,as remainingbrandsbetterexploitscale economies.
The short-run
price rigidityof the kinkedequilibriumaccords withcasual
to confirmor
empiricism.However, the long-runpropertiesare more difficult
run
since
on
they depend
longer
entryadjustments. Moreover, the
reject,
the
abstractand unrealisticassumpactual symmetric
entails
example analyzed
tionsof uniformpreferencesarounda circularproductspace and identicalcost
functionsand valuations among competingbrands. While these assumptions
thetheoreticalanalysis,theymakeempiricalconfirmation
considerablysimplify
more difficult.
Other shortcomingsof the approach taken here are the zero-profitand
costless relocation assumptions we have made. Because the technologyis
characterized by an indivisible fixed cost, the number of brands must be
integer-valued.Therefore,freeentryneed notlead to a zero-profit
equilibrium,
as originallypointed out by Kaldor (1935) and analyzed by Eaton (1976).
An interesting"deterrence" equilibriumconcept built on these foundations
has been exploredfora circularmarketby Hay (1976) and Schmalensee (1977)
and forotherspatial marketsby Prescottand Visscher (1977). In a deterrence
equilibriumsequentialentrantslocate in such a way thatno new entrantwishes
to locate in the interval between two firms. As a result, the deterrence
equilibriumhas halfthe numberof brands as the SZPE.
In the model analyzed here, the deterrenceequilibriumconfigurationis
generallythatpoint in thep(L/n) curve withthe numberof brandsn equal to
/2 the number at the SZPE.15 If the SZPE is competitive,the deterrence
equilibriummay be competitive,kinked, or at the monopoly point; which
equilibriumoccurs depends on the particularparametervalues. Hay showed
thatif the deterrenceequilibriumis competitive,prices are higherthan at the
SZPE. However, kinkedor monopolydeterrenceequilibriamay resultin lower
pricesthanthecompetitiveSZPE. Finally,iftheSZPE is kinked,thedeterrence
equilibriumlies at the monopolypoint and entails lower prices and unserved
segmentsof the circularmarket.
15The
exceptionariseswhenthispointon p(Lln) lies belowthe monopolypointon the
ofthedemandcurve.In thiscase, whilethenumber
ofbrandsis stillhalved,the
monopoly
portion
brandsraisepriceand loweroutputto the monopolypoint.Hence, the deterrence
remaining
entailsunservedsegments
ofthecircularmarket.
equilibrium
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156 / THE BELL JOURNAL OF ECONOMICS
In closing,it shouldbe reemphasized
thattheexactresultsderivedhere
is
theexampleused. Thatthereis excess varietyat equilibrium
merelyreflect
ofproduct
notrobust.The kinkappearsrobustas thenumberof dimensions
(1975)and Weiss(1977)
space increases.However,as Archibald-Rosenbluth
maynotexistwithhigherdimensional
productspaces.
pointout,equilibrium
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