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American Economic Association
A Generalized Model of Spatial Competition
Author(s): Dennis R. Capozza and Robert Van Order
Source: The American Economic Review, Vol. 68, No. 5 (Dec., 1978), pp. 896-908
Published by: American Economic Association
Stable URL: http://www.jstor.org/stable/1811321 .
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A GeneralizedModel of Spatial Competition
By
DENNIS
R.
CAPOZZA
AND
Recentresearchon the theoryof the firm
in economic space (for example, see M. L.
Greenhut,M. Hwang, and H. Ohta, 1975;
Capozza and Kazem Attaran) has argued
that in a spatial contextmany of the conclusions of classical price theory are reversed.Since mostfirmsoperate in a spatial
ratherthan a nonspatial environment,the
resulthas farreachingimplications.For example, Greenhut, Hwang, and Ohta have
shown that some formsof (spatial) competitionwill resultin higherpricesthan those
realizedunder(spatial) monopoly. Capozza
and Attaranhave shown that spatial comparativestatics can be different
fromnonspatial. In particular,cost increases (fixed,
marginal, or transport) may cause lower
marketequilibriumprices.
Many of the spatial resultsappear to be
sensitiveto the assumptionmade concerningcompetitionamong firms(see Greenhut,
Hwang, and Ohta, 1975; the authors,
1977a,b). The purposeof thispaper is to investigatethe extentto which spatial price
theory replicates the results of classical
pricetheorywhenthereis freeentry.To this
end we analyze in detailtwo models of location under freeentrywhichdifferprimarily
in their assumption about price competition. An importantaspect of the analysis is
the use of the second model as a basis for
analyzing monopolistic competition. Indeed, we believe that monopolistic competitioncan bestbe viewedas a resultof the
spatial dimension and that the results of
monopolisticcompetitionshould be derived
accordingly.
The traditionalmodel of spatial competition follows the assumption in location
theory(see, for example, Edwin Mills and
Michael Lav; Martin Beckmann; Greenhut
* University
of SouthernCaliforniaand Department
of Housing and Urban Development,respectively.We
are indebtedto anonymous readers for helpfulcomments.
ROBERT
VAN
ORDER*
and Ohta; John Hartwick; Nicholas Stern
among many others)that firmsset price as
iftheyare monopolistswithintheirmarket
area. In this paper a second model is developed froman assumption about pricing
that is similar to that assumed by Harold
Hotelling and followed by Arthur N.
Smithiesin the analysisof spatial duopoly.
Under the Hotelling assumptioneach firm
conjecturesthat all other firmswill leave
theirpriceunchanged.' We believe that this
assumption is closer to the way firmsbehave in most real world situations.A third
assumption developed by Greenhut and
Ohta is also analyzed.
To previewthe conclusions,we findthat
manyof the perverseresultsfoundin spatial
price theoryusing the traditional assumption are also found in the second model
using the Hotelling monopolistic competition assumption,but usually as special extremecases. Thus while we are not able to
rule out all perversespatial results,we are
able to showthattheconditionsunderwhich
theyare likelyto be realized are much more
restrictive
than earlierbelieved. The second
model also leads to a generalized class of
models encompassinga wide range of competitivereactions.
In the next section we consider the relationshipbetweenspace and the theoryof
the firm.In addition we list the characteristics that we think reasonable spatial
models should possess; these are subsequently used in discussing alternative
models. The second section outlines the
standard assumptions of location models
and discusses the possible price reactions.
The thirdand fourthsections analyze the
two models. Finally, the last two sections
presentand evaluate a generalizationof the
earlierresults.
I The term"zero conjecturalvariation" is sometimes
used to describe this assumption, see, for example,
B. CurtisEaton and Richard Lipsey.
896
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VOL. 68 NO. 5
CAPOZZA AND VAN ORDER: SPATIAL COMPETITION
1. Spatial Microeconomics
Beforedevelopingany spatial model, it is
worthwhileto inquire into the relationship
between spatial and nonspatial microeconomics. There are two essential distinguishing
featuresof spatial competition.
The firstis transportationcost. If transportationwere costless, firmswould have
no protection from spatially separated
rivals. A firm1,000 miles away would be
just as formidablea competitoras one next
door. As a result,space would be of no consequence; firmsmightas well all be located
on the head of a pin, perfectcompetition
would prevail. Transport cost gives the
spatial firmitsmonopolypower over customersclose to it.
The second essentialfeatureis that average cost curvesmust be downward sloping
over some range. The curve may have the
negativeslope, for instance,eitherbecause
thereare some fixedcosts or because there
are economies of scale in production. If
averagecostcurvesare nonnegatively
sloped
throughoutthe range, there will be no advantage to concentratingproduction at
specific locations. Each consumer could
produce his required consumptionjust as
cheaplyas any concentratedfirm,and again
spatial competitionwould disappear.
If either of the above features fails to
hold in practice, there is no need for a
spatialmodel. In fact,both are characteristic
of mostreal worldindustries;but thatis not
the interestingpoint for model building.
The issue that must be raised here is that
nonspatial perfectcompetition would appear to be essentiallya special case of imperfectspatial competition that arises if
eitherof the above featuresis absent.Therefore,we should expect a reasonable spatial
model to behave like the perfectcompetition model in the limit. In particular,we
should expectthe followingcharacteristics:
1) As transportcosts approach zero,
perfectcompetitionshould be approached
and priceshould approach marginalcost.
2) As fixedcosts approach zero, concentratedproductionis less essential;spatial
monopoly power is diminished;and again
897
priceshould approach marginalcost.
The above would have to hold fora model
to be valid over a wide range of values. In
additionto 1) and 2), we mightalso expect
that,at least forsome values of the parameters(i.e., low transportand low fixedcosts),
the model should obey the intuitivelyappealingpropertiesof nonspatialcompetitive
theorythat:
3) As costs (fixed,marginal,or transportation)rise,priceshould rise.
4) As demand density rises, firms
should be able to take advantage of economies of scale. Price should fall in the long
run.
5) As more firmsenter the industry,
thereshould be increased competitionand
priceshould fall.
The most widelyused Loschian model of
location violates all of the above. The inconsistenciesof the Loschian model can be
attributedto the factthat it is a special extremecase of a more general class of location models. In the next section we begin
to develop themodels.
ofLocationModels
II. TheAssumptions
underFreeEntry
Following the writingsof August Losch
a somewhat standard set of assumptions
has evolved for partial equilibriummodels
of location withfreeentry.We accept most
of these,including:
ASSUMPTION 1: Thereis a singlecommoditythat can be produced in two-dimensionalspace withthesame costfunction(i.e.,
ubiquitousresourcesand technology).
ASSUMPTION 2: The cost functionhas
constantmarginalandfixedcosts
C =f + cX
(1)
where X = output; C = productioncost;
f = fixedcost; c = marginalcost.
3: Transport cost per
ASSUMPTION
mileis identicalbetweenany two points and
equal to t unitsper mile.
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898
THE AMERICAN ECONOMIC
ASSUMPTION 4: Potential consumers
occupy a homogeneousunboundedplain at
unijormdensityD.
ASSUMPTION
5: All consumers are
identicaland have a demand curve that is
linearindelivered
price(p + tu).
REVIEW
DECEMBER
1978
(G-O)
ASSUMPTION 8c: Greenhut-Ohta
competition Each firm assumes the price
at theedge of themarketarea (borderprice)
to befixed.
All threeassumptionscan be interpretedin
termsof how the firmexpectsrivalsto react
to a change in its price. The differencesare
(2)
x = a - b(p + tu)
crucial to the results that follow and rewhere p = the mill price; x = demand per
quire some elaboration.
consumer; u = distance to the firm; and
Note that the buyerhas two decisions to
a, b > 0. There is no price discrimination. make: whereto buy and how much to buy.
Firmsset the mili price, and transportation The firstchoice is made by buying where
the delivered price (including transport
costsare paid by theconsumer.
costs) is smallest.If we considertwo neighASSUMPTION 6: Firms continueto enboringfirms,the boundaryof theirmarket
ter untilprofitsfor all firmsare drivento
areas will be at a point where both firms'
zero.
productssell for the same delivered price.
Hence, ifwe have, as in Figure 1, two firms
To theabove we add:
at a distance U apart, and draw each firm's
deliveredprice(mill price plus transport)as
ASSUMPTION 7: All marketareas are
a functionof distance,then the boundary
circular.
betweenfirmswill occur at the intersection
of the two deliveredprice lines. At this inAs is well known,circles will not cover a
tersection,consumers will be indifferent
plane while hexagons will; however, the betweenthetwo sellers.
analysis of hexagonal market areas is not
This intersectionoccurs at that radius R,
materiallydifferent.
Extensionto hexagonal satisfying
areas is straightforward
(see fn.4). Assumpp + tR = p + t(U - R)
tion 7 narrowsand simplifiesthe investigation. Since we are not concernedwith the where p = themillpriceof thefirstfirm
p = themillpriceof the second firm
shape of marketareas, thereis no loss of
= the distancebetweenfirms
U
generalityby includingthis assumption as
= the radius of the first firm's
R
long as we restrictthe discussion to twomarketarea
dimensionalmarketareas.
The finalassumption,upon whichour re- SolvingforR we have
sultsdepend,concernsthereactionof a firm
I
R =
to a change in a competitor's price. The
+ tU)
(3)
(P-p
usual assumptionwhich has been analyzed
p+Ap +t(U-U)
extensivelyin thiscontextis:
ASSUMPTION 8A: Loschian competition-Eachfirm assumesits marketarea to
befixed, and sets prices like a monopolist
withinitsmarketarea.
However, other assumptions are possible
including:
ASSUMPTION
8B: Hotelling-Smithies
(H-S) competition Each firmassumes the
pricesof competitors
to befixed.
p+Ap+tU
p+t (U-U
B
~-Ap+t(o-U)
ptU
+Ap
p+Ap
p
__
_ __
_
0
FIGURE
_I
RD
1.
THREE
RC
PRICE
-
. 71U
RA
REACTION
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ASSUMPTIONS
VOL. 68 NO. 5
CAPOZZA AND VAN ORDER: SPATIAL COMPETITION
That is, theradiusof the firm'smarketarea
dependson transportcosts,distanceto other
firms,and itspricerelativeto others'prices.
Since we assume that all firmsare identical,
thisformulaapplies in the same way to all
itscompetitors,so thatR is the "radius" of
themarketin all directions.
ConsiderAssumption8. The representative firmlocated at U = 0 in Figure 1 is
chargingprice p; its competitorat U = U
chargesprice p = p initially.The market
will be split betweenthem, each having a
marketof radiusRA = U/2. If the firstfirm
should raise itsprice to p + Ap, then under
the Lbschian assumptionthe marketradius
must remain unchanged at RA. However,
theonly way this resultcould be obtained
would be if the rival raises its price at the
same timeto p + /p. Thus, one implication
of the Lbschian assumptionis that the price
reactionof a competitoris unitary,that is,
dp/dp= 1.
From (3) we have
I dp
1
2t dp
dR
dp
(4)
and since dp/dp = 1, we have dR/dp = 0,
whichis the Lbschian case.
Under Assumption8B of Hotelling and
Smithies,the competingfirmsare assumed
not to changetheirprice(dp/dp = 0). If so,
the intersectionof the price lines would
move to pointC in Figure 1 and the market
radiusof thefirstfirmwould fallto
Rc=
2t (p
-
-
p - Ap + tU)
< RA
In this case, from(4) above, since dp/dp =
0, we have dR/dp = - 1/2t.
Under Greenhut-Ohtacompetition (8c)
the borderprice is fixed.2In the contextof
our analysisthe border price could remain
constantwhen the firstfirmraises its price
if rivals lower theirprice to p - Ap (i.e.,
dp/dp = - 1). The intersection would then
be at point D in Figure 1. Market radius
2It should be noted that G-O competitionwas developed in thecontextof zonal pricingby competitors
ratherthan by mill pricing.Thus the application in
this context is somewhat strained. We retain the
stylizedversionhere because it helps to highlightthe
generalizationin Section V.
TABLE
1-EXPECTED
899
RADIUS
Loschian competition
H-S competition
G-O competition
Spatial Monopoly
AND PRICE
RESPONSE
Radius
Change
dR/dp
Conjectural
VariationforPrice
dp/dp
0
- 1/2t
- l/t
- l/t
1
0
-1
would fallto
RD
D=
-
2t
(P
-
p
2Ap + tV) < Rc
and since dp/dp = -1, we have from (4),
dR/dp = - I /tforthe G-O case.
For completenesswe consider also the
spatial monopolist. The maximum price a
consumerwill pay is given by the price interceptof the demand curve, which is a/b.
The monopolist'smarketwill extendto the
point where delivered price equals this
maximumprice or p + tR = a/b, whence
R = (1/t)((a/b) - p), and dR/dp = - 1/t.
Thus each assumptioncan be converted
to an equivalent assumption concerning
eitherthe change in market radius or the
pricereactionof competitor'sprice. Table 1
summarizesthesecases.
The classificationaccording to expected
pricereactionis particularlyuseful,because
it offerscomparison with the nonspatial
market structureliterature.In this light,
Lbschian competition is seen to be the
spatial equivalent of noncompetitiveoligopoly. Firms raise and lower price in unison
eitherbecause of collusion or price leadership. H-S competition is comparable to
monopolistic competition or competitive
oligopoly.3Firms assume no reaction from
3The terminologyemployed in this paper is somewhat differentfrom standard usage in the spatial
economics literature.The ULschian model described
hereas spatial noncompetitiveor collusive oligopoly
is oftenreferredto elsewhereas spatial monopolistic
competition.We preferto reservethe term "monopolisticcompetition"forthe zero price reactioncase to
retain comparabilitywith the nonspatial microeconomics literature.The term "spatial monopolistic
competition" is used advisedly since many authors
argue thatmonopolisticcompetitionis not possible in
a spatial contextwhere there are only six immediate
competitors.
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900
THE AMERICAN ECONOMIC
competitors.G-O competitiondoes not appear to have a nonspatialanalog.
Whileit is clearlypossible forany of these
marketstructuresto be realized in practice,
our suspicion is that the most common
marketstructureis somethingclose to H-S
competition. We have two reasons for
favoringthe H-S assumption: first,it incorporatesthecontestformarketarea which
Lbschian competitionignores by assuming
marketareas to be fixed;and secondly, it
allows us to analyze spatial monopolistic
competition,whichmany believe to be the
most common industry structure. This
analysiscan be done in a relativelyrigorous
mannerwhich gives a precise way of distinguishingindividual consumer demand
fromdemand facedby thefirm.In addition,
the model degeneratesto perfectcompetition under intuitively
appealing conditions.
We turnnow to the actual models.
X = DjR (a - b(p + tu))2ll udu
= HDR2(a - bp - (2/3)btR)
Profitsare
Y = pX-
C
Substituting
(1) and (5) into(6) we have
(7) Y = HDR2(a - bp - (2/3)btR)
*(p-
1978
P
P=
b -tR
b
L
L
~~~~Z
PI
p2
R2
Rmax
D
R
FIGURE 2. THE L6SCHIAN MODEL
Making use of Assumptions 1-8, the
equilibriumpriceand radiuscan be derived.
Total demand in a marketof radius R will
be4
(6)
DECEMBER
N
RI
III. TheLoschianModel
(5)
REVIEW
c) -f
From Assumption6, profitswill be zero in
long-runequilibrium.This impliesthat
(8) HDR2(a - bp - 2/3btR)
(p - c) -f
= 0
Equation (8) definesa curve in the (R,p)
4Notice that equation (5) can be writtenX = DAxT
whereA is marketarea and x is demand of a representativeconsumer.Writtenin this form,the equation
is applicable to all marketareas that are regularpolygons. For example, if market areas are hexagonal
A = 3.45R2 and x = a - bp - .76tR.
plane for each level of fixed costs f. This
curveis shownin Figure2 forthreelevelsof
fixedcost and is designatedthe zero profit
locus (ZPL). Two items are worthnoting:
first,this curve is invariantto the type of
competitionprevailingin the industry,so
thatit will be repeatedin our discussion of
H-S and G-O competition;second,the curve
collapses around the point M as fixedcosts
rise.This is significantforthreereasons: 1)
M is theZPL forf = frax; forf> fnax nonnegativeprofitsare not possible, the ZPL
is not defined,and no firmswill survive.
2) The point M gives the price-radiuscombination a regional monopolist would
choose' (i.e., the mill price at each production point and the optimal spacing of production points). 3) M also defines the
5Since the monopolistis unconstrainedby competition,profitsneed not be zero. The monopolist's price
is found by substitutingdR/dp = - I/t into equation
(9) and solvingforp which gives p = 1/4 (a/b + 3c).
The monopolist'spriceis independentof density,fixed
costs,and transportcosts. It should be noted that this
resultholds only in a marketof uniformdensity.See
Dennis Heffleyfor a discussion of the spatial monopolist in a marketwithvariabledensity.
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VOL. 68 NO. 5
CAPOZZA AND VAN ORDER: SPATIAL COMPETITION
maximum possible extent of the market
given by Rmax= (3/4)t((a/b) - c). Radii
beyondRmaxare ruled out because beyond
thatpointpriceis greaterthan ((a/b) - tR)
and some customerswould have negative
demands.
A second condition is derived from the
profit-maximizing
behavior of firms.Maximizing(7) withrespectto pricegives
(9)
ddY=
dp
= 2(p-
c)(a-
bp- btR)
dR _ R(a - 2bp - (2/3)btR + bc)
dp
Under Lbschian competitiondR/dp = 0 so
that(9) becomes
91)
tR
a + c
p
and givestheprofit-maximizing
price that a
Loschian firmwill charge in the short run
givena marketof radius R. It is shown as
LL' in Figure2.
Notice that the profit-maximization
condition (9') and the zero profitsconditions
are satisfiedtwice fora given level of fixed
costs. However, for demand to be nonnegativeat theedge of the marketarea, it is
requiredthat(a/b) - p > tR > 0. Therefore
all pointsto therightof thelinep = (a/b) tR whichis shown as ND (nonnegativedemand) in Figure 2, are not admissible. If
fixedcosts are fl, long-runequilibriumoccurs at (R,, p1) wherethe profit-maximization line (LL') cuts the ZPL in the admissibleregion.
TABLE
2-EFFECT
ON PRICE
901
For marketswitha radius largerthan R1,
positiveprofitswill be earned and firmswill
enterthe industry.Entrywill cause market
areas to shrinkand the industrywill move
towards long-run equilibrium. For radii
smaller than R, profitsare negative, firms
will leave the industry,marketradii will increase, and again the industry moves
towardslong-runequilibrium.6
It is straightforward
to show that this
modelis inconsistent
withthe characteristics
1)-5) outlinedabove. We note the following
behaviorof the Loschian model.
IA) As transportcosts approach zero,
the profit-maximization
line LL' becomes
flat (note equation (9')), and price approaches the nonspatial monopoly price
((a/2b) + (c/2)) rather than the perfectly
competitivemarginalcost price.
2A) As fixedcosts approach zero, the
ZPL expands, the equilibrium price rises
and again approaches p = (a/2b) + (c/2),
the nonspatial monopoly price,ratherthan
themarginalcost price.
3A) As fixedcosts rise fromf1 to f2 in
Figure2, theZPL shrinks,and price fallsin
thelong run to P2. An increasein marginal
costs shiftsboth the ZPL and the profitmaximizationline (LL') so that the price
6Readers may be more familiarwith the tangency
solutionthatoccurs in the(X,p) plane. The problem is
actually three-dimensionalin (X,p, R) space. The
tangencysolution arises graphicallyin the projection
onto the(X,p) plane. Our graphicalsolution(Figure 2)
is the projection onto the (R,p) plane. (See the authors, 1976.) The point (RI, P1) corresponds to the
pointof tangency.
IN THE LOSCHIAN
AND SMC
MODELS
SMC
Loschian
Noncompetitive
Normal
Parameter
Spatial
Change
p <a/b - 4/3 tR
Oligopoly
a
+
D
f
t
+
c
i
-
-
4
+
-
+
+
Perverse
p > a/b - 4/3tR
Spatial
Monopolist
4
+
+
0
0
i
-
-
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+
0
902
THE AMERICAN ECONOMIC
REVIEW
DECEMBER
1978
K
change is ambiguous. The comparative
statics are developed more fully in ApatR
b 4~
3
pendix A. The signs are summarized in
Table 2.
4A) Appendix A shows that price inZPL,~
creaseswhenthereis an increasein density.
Intuitivelythis occurs because the higher
densitypermitssmaller market areas. The
spatial demand curve is more inelastic for
smallermarketradii;7 that is, the firmhas
M
moremonopolypowerand raisesprice.
fi m
imizotion (SMC)
5A) As firms
enter,marketarea shrinks;
p13
the firmmoves back along the profit-maximizationline,and pricesincrease.
The above might be acceptable for a
model of organized (price-leader) spatial
Cl
oligopolybut is counterto what one might
expectto hold in spatial monopolisticcomI ~K
_
~
R
petitionor unorganized oligopolistic comR3
Rmox
petition.It is particularlyimportantto note
FIGURE
3. A MODEL
OF SPATIAL
MONOPOLISTIC
thatin the limitas transportcosts and fixed
COMPETITION:
NORMAL CASE
costs approach zero, the Loschian model
approaches the nonspatial monopoly price
ratherthan the perfectlycompetitiveprice. (9") p = c + tR(a - bp - 2/3btR)
The model of the next section does ap(a - bp)
proach perfectcompetitionin thelimit.
This can also be writtenas
IV. A ModelofSpatialMonopolistic
p =
t+R
(9"')
Competition
(SMC)
xo
The model to be developed is one with
Hotelling-Smithiescompetition (Assumption 8B) instead of Loschian competition.
With this zero conjectural variation assumptionwe have shown above that dR/
dp = - 1/2t. Substitutingin (9) gives the
profit-maximization
line forthe SMC firm.8
7The elasticityof the spatial demand curve in the
Loschian model is
17=
-bp
a - bp - 2/3 btR
and becomes moreinelasticas R decreases.
8To the individualfirmR is endogenous. The firm's
price equation is obtained from substituting(3) into
(9"). In equilibrium if firmsare identical they will
move untiltheyare equidistant.We can thereforeinterpret(9") as an equilibriumcondition with R set
equal to one-halfthedistancebetweenfirms(assuming
firmsare not regional monopolists). For a briefdiscussionof short-rundynamics,see AppendixB.
where x = averagedemand per person
xo = demand perpersonat R = 0
That is, profit-maximizing
price is an additive markup over marginalcost where the
markupis given by the second termon the
rightin (9"').
Equation (9") has an invertedU-shape
in the admissible region and is shown in
Figure3. This profit-maximizing
pricecurve
startsat marginalcost (p = c, whereR = 0)
and rises to a maximum at p = (a/b) (4/3)tR, thendeclines.This conditionfora
maximumis satisfiedif average demand is
unitaryelastic with respect to transportation cost.9 The SMC curve is downward
9This elasticityis givenby
775FI
= dx-T
--
and is unitaryifp
=
3- btR
3
=2
a - bp - - btR
a/b - 4/3 tR.
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3
VOL. 68 NO. 5
CAPOZZA AND VAN ORDER. SPATIAL COMPETITION
sloping,however,ifaveragedemand is elasticwithrespectto transportcost.
The zero profitlocus (ZPL) is as before
and is superimposedon Figure 3. Long-run
equilibriumoccurs at (R3, p3) if fixedcosts
are fl. For market radii larger (smaller)
thanR3 firmsearnpositiveprofits;new firms
enter(exit) and move the industrytowards
equilibrium.
In thismodel two forcesdetermineprices.
Firstis the demand effect:the more elastic
is averagedemand,the lower the price. The
second is the competitiveeffect:the closer
competingfirmsare, the greater is competitionand theloweris price.When market
radius is small (i.e., when firmsare very
close together)price is close to marginal
cost. Competition forces price down to
marginalcost in this model whereas in the
Loschian model price approaches (a/2b) +
(c/2) > c as radiusapproacheszero.
There is, however,a point (given by p =
(a/b) - (4/3)tR, thelineKK' in Figure3)
wherefurtherincreasesin marketarea correspondto lower prices. For small market
areas with densely packed firms,competitionis important.However,as marketareas
approach the maximal size Rmax, competition over the size of the market becomes
less and less importantsince the consumer
at theedge of the marketis buyingless and
less (note that at Rmax he is buyingnothing
and is not worthfighting
over). Indeed, the
SMC profit-maximization
line approaches
the ULschianline and intersectsit at Rmax.
Hence, the Loschian-type effects noted
above wheremonopoly power decreases as
marketarea increases may tend to be importante.venin the SMC model. Thus, we
cannotruleout a priorithe "perverse" case
in which enteringfirmsraise rather than
lower the equilibriumprice. This perverse
case occurs when firmsare so spread out
that individual firms are almost regional
monopolists.
Mathematicallywe solve the model by
substitutingthe profit-maximization
equation (9") into the zero profitsequation (8)
to obtain
(10) (a -
bp)2(p-
C)3=
(
f) x
[(a-bp-(?
3
()(
(
HID
903
I
-1)2
(a -bp)(p -C)2]
Explicitsolution forprice is difficult,
but
we can outlinesome propertiesof the longrunequilibrium.Note the following:
1) The zero profitslocus is the same as
thatin theLbschian model (equation (8)).
2) Changes in f or D affectthe zeroprofitslocus but not the profit-maximization curve(equations (8) and (9")).
3) Since t,f, and D enterequation (10)
as t2f/D, solvingforthe impact of changes
in one impliestheeffectsof theothers.
There are threepossiblecases:
CASE
1: Degenerate case nonnegative
profitsare notpossible in the admissibleregionand no firmsexist. This mightoccur if
fixedcostswereverylarge or populationdensityverysmall.
CASE 2: Normalcase (p < (a/b) - (4/3)tR,
i.e., to the leftof KK' in Figure3)-zeroprofits curve cuts the profit-maximization
curveon the upward-sloping
portion as in
Figure3. If additionalfirms
enter,pricefalls.
CASE 3: Perverse case (p > (a/b) (4/3)tR, i.e., to the rightof KK' in Figure
3) zero profitscurve cuts the profit-maximizationcurveon thedownward-sloping
portionas in Figure4. If additionalfirmsenter,
prices rise. This could occurfor large fixed
costs or small populationdensity.However
theimplicationis that transportcosts are a
largeproportionof deliveredprice (p + tR)
at theedge of themarketarea. Greenhutand
Greenhut,Hwang, and Ohta (1976) present
empiricalevidenceindicatingthat in general
thisis unusual.It wouldcertainlybe unusual
in urbanareas wherethepacking of firmsis
relatively
dense.
Only the lattertwo cases are of interest.
In Table 2 the comparative statics have
been summarized.
To derive the results consider firstthe
effectsof increasing fixed costs f Rising
fixedcosts affectonly the zero profitslocus,
shrinkingit. Price rises if the profit-maximization line is upward sloping (normal
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904
P
THE AMERICAN ECONOMIC
K
a -4
P= b --4
tR
ZPLf.f
M
c
>
~
DECEMBER
1978
approaches marginalcost.
3B, 4B) The comparative statics are
normaliftheintersection
takes place on the
rising portion of the profit-maximization
line as discussed above. That is, increases
in costs will raise price; increasesin density
will lowerprice.
SB) Entryforcespricedown if the profit-maximization
line is upward sloping,i.e.,
in the normalcase.
t
P4
REVIEW
V. SpatialCompetition-A
Generalization
profit
maximization
(SMC)
We have not developed a model with
G-O competition,but it would be straightforward to do so. Since dR/dp = - 1/t
underG-O competition,(9) would become
K
(9"")
I
c + tR(a
p
2(a
R
FIGURE 4. THE PERVERSE CASE OF THE SMC
case) and fallsif the line is downwardsloping. From property3) the same holds fora
risein transportcosts t, or a fall in density
D. Increasing marginal cost c raises the
profit-maximization
line and shrinks the
ZPL, raisingprice in the normal case, but
withambiguous effectin the perversecase.
Similarly,a rise in the demand intercepta
expandsthe zero profitlocus and raises the
profit-maximization
line, leading to an ambiguous effecton price in the normal case
and an increasein theperversecase.
The effectson price are quite different
in
thetwo cases. In theabsence of a restriction
on the sign of (a - bp - 4/3btR) nothing
can be said about the sign of the price
changes. If we compare the signs for the
SMC model with those for the Loschian
model,we findthat the Loschian signs are
identicalto the perversecase of the SMC
model.
Note that characteristics1)-5) hold for
theSMC model.
I B) From (9 ") as t approacheszero the
profit-maximizationline becomes horizontal at p = c, themarginalcost price.
2B) As fixed costs approach zero the
intersectionof the two curves moves back
line and price
along theprofit-maximization
-
bp - 2/3btR)
bp -
1/2btR)
This equation definesa curve in the (R,p)
plane that starts at p = c and rises to a
maximumat M, the monopoly price, and
falls afterward.The comparativestatics in
this model will be unambiguous (since the
profit-maximization
curve is monotonically
increasing in the admissible region) and
similarto the normalcase of theSMC.
The threeformsof competitionsuggesta
generalization.Equation (9) definesa family
of curvesin the(R,p) plane which is shown
in Figure 5. The two extremecases in the
p
a
c Loschion
b2
\dp
dp
M
dp
dp
Rmax
FIGURE
5.
THE PROFIT-MAXIMIZATION
CURVE
DIFFERENT
FORMS OF COMPETITION
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UNDER
VOL. 68 NO. 5
CAPOZZA AND VAN ORDER: SPATIAL COMPETITION
figureare the Loschian profit-maximization
curvewheretheprice reactiondpj/dpis plus
one and the G-O curve where the reaction
is minus one. Hotelling-Smithiescompetition is one intermediatecase; but we can
conceive of any number of other intermediate cases as the price reaction varies
from1 to -1. Notice that as the price reaction increases,themaximumpoint on the
curvemovesto the left.
profit-maximization
This impliesthat the perverseresultsnoted
fortheLoschian model and forsome values
oftheSMC model are more likelythe larger
the price reaction,that is, the greater the
tendencyforfirmsto collude.
A relevantquestion to raise at this point
is which price reaction is most likely. We
feelthat in practice the value probably is
fairlyclose to zero. In two-dimensional
space, thereare six competitorssurrounding each firm.Each surroundingfirmis a
small proportionof the total competition
faced by the firm.Therefore,it seems reasonable to argue that unless thereis collusion,firmswould not react dollar fordollar
to price changes of just one of their competitors,and would probably have a comuch closer to zero than unity.In
efficient
any case, the qualitativeresultsof theS MC
model would also hold for any value of
pricereactionbetween 1 and - 1. We view
the Loschian model as extreme,largelybecause it fails to meet the requirementsset
out earlier in the paper. In particular: it
does not allow price to approach marginal
cost as fixed or transportationcosts approachzero. It unambiguouslyrulesout the
plausible cases where fixed and marginal
cost increasesraise price. It rules out the
possibility'that a densityincrease will decrease price.The argumentthatfirmsmatch
price increases of rivals dollar for dollar
seemsimplausible.As a model of collusive
oligopoly,however,it is moreacceptable.
The Greenhut-Ohtamodel has the advantagethat it does give plausible and unambiguouscomparativestatics results,and
handles the extremecases properly. However, in the contextof spatial competition
withmillpricing,the impliednegativeprice
reactionseems unlikely.In another context
or zonal
(forexample,underdiscriminatory
905
pricingwhere a negative price reaction is
not necessarilyimplied) it may be more appropriate.10
competitiondoes
The Hotelling-Smithies
satisfythecriteriawe have set up. While we
should not expect price reactions to be
exactlyzero, we should (for reasons mentionedabove) not expectit to be especially
large, so that in the absence of collusion
the zero conjectural variation assumption
would appear to be a suitable approximation.
VI. Conclusion
We have shown that it is indeed the case
that many perverse results are possible
whenthefirmis placed in a spatial context.
We have developed an imperfectcompetition model that shows that the perverseresultsappear under relatively"unusual" circumstancesand that provides a convenient
way of interpretingimperfectcompetition
models as an outgrowthof spatial considerations.
In particular we have shown that the
widely applied Loschian model, which exhibits the perverse results, can be interpreted as an extreme case of a class of
models defined by the price reaction of
firms.The perversecomparativestatics results appear when the profit-maximization
curve is negatively sloped in the (R,p)
plane. The negativeslope is more likelyto
be obtained if the price reaction(dp/dp) is
close to one; if transportcosts are a large
proportionof deliveredprice or, more specifically,if average demand is elastic with
respect to transportationcost; or if fixed
costs are so large that firms are barely
viable."
All of these are clearlypossible; but we
have arguedthatin urban areas each would
be an exception. Thus, the results of the
theoryof the nonspatial firmare usually
?0Seefn.2 above.
I I There is a fourthfactorthatinfluencesthe slope of
curve that has not been anathe profit-maximization
lyzedin thispaper. Ohta has shown that the convexity
of the individual demand curve also influencesthe
slope. The negativeslope is less likelythe more convex
individualdemand is.
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906
THE AMERICAN ECONOMIC
qualitativelysimilarto those forthe spatial
firm;however, there are conditions under
whichthetraditionaltheoryis violated.
We have not addressed the welfareissue
raised by the lack of perfectcompetition.
We have shown in our 1977a paper that
total costs (production and transport)are
not minimized.There are too manyfirmsso
that not enough advantage of scale economies is taken. This is a standardmonopolistic competition result which we conjecturewould carryover to the model here.
We do not however,offera proof.
Also, we have deliberatelykept withina
partialequilibriumframework.This framework is in the spiritof most work in monopolisticcompetition,but it is not entirely
satisfactory.In particular,the distribution
of population is assumed to be given; but
one mightexpectthe distributionto depend
upon the price structure and transport
costs; that is, if transportcosts are high,
consumerswould concentrateat production
sites and producers would cluster. We do
not attempt to incorporate this aspect,
viewingour analysis as a kind of market
equilibriumcurve,whichshould eventually
be imbeddedin a moregeneralsystem.
APPENDIX A-THE
COMPARATIVE STATICS
OF LONG-RUN EQUILIBRIUM PRICES
IN THE LOSCHIAN MODEL
In theshortrun,withfixedmarketareas,
priceis given by equation (9') and is an increasingfunctionof demand densitya and
marginalcost c but a decreasing function
of transportcost t.
To determinethe long-runprice we combine equations (8) and (9') which gives the
quarticinp
(A1)
a
(p-b2b
2
)2(p
-
c)2 =
ft2
9171Db
The solutionis
a
(A2) p = 4 + 3c
-
2
I a
[ 4b~
(
4t(
3
-
C)2
c)1/2]1/2
HDb
The relevantroot, after taking economic
REVIEW
DECEMBER
1978
admissibilityintoconsideration,is the positiveone, so that
(A3)
p =! (' + 3c + 2Z)
4 b
whereZ
=
_ c)2 -4t 3
-4(a
fb)1/2]1/2
Differentiating
with respect to the parameterswe have
(A4) aP
aa
I
8 ( - c + 6bZ) > 0
8~b_Z
b
(A5) ap =-8b1[2a(q
8bZ
ab
b
- c + Z)
) 1/2] t<
-4 t (
3 11Db
(A6) a1
Oc
p
(A7) a=
aD
(A8) ap Of
(A9) aP
At
0
O
(a -c-6Z)
8Z b
(
6DZ
t
6DZ
I
3Z
D
)1/2
11Db
(f)1/2
11Db
11)/2
H1Db
>
0
< 0
< 0
The sign of ap/la follows directlyfrom
the requirementthata/b > p > c for firms
not to shut down. The others follow from
the requirementthat Z be real forthe solutionto exist.
Intuitivelythe results come from the
workingof two effects.In Figure 2, changes
in some parameters shift the short-run
profit-maximization
lineLL', butat thesame
time the zero profitslocus may expand or
contract.An increase in a, which can be
as an increase in income, shifts
interpreted
the profit-maximizationline up and expands the zero profitslocus. Thus we have
theunambiguouspositivesign. Increases in
fixedcosts shrinkthe ZPL and raise prices.
Increasesin transportcost lower the profitmaximizationline and shrinkthe ZPL so
thatpricesfall.
The sign of marginal costs is ambiguous
because increasesin c shiftthe profit-maximizationline up, but shrinkthe ZPL. The
resultingchange in price depends on the
slope of the profit-maximization
line. If
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CAPOZZA AND VAN ORDER: SPA TIAL COMPETITION
VOL. 68 NO. 5
transportcosts are low, prices will rise; but
iftransportcosts are sufficiently
high,prices
can fall.
Some of the resultsare somewhat counterintuitive.
One would not expect pricesto
fall if fixed or marginal costs rose. Nor
would one expect population density to
have a positiveeffectand transportcosts a
negativeeffect.In general,prices are lower
in urban areas than in low density rural
areas because of highervolume and greater
competitionamong sellers. Similarly,high
transportcosts are oftencited as a cause of
high prices not low prices, because of the
lessenedcompetitionbetweenfirms.
APPENDIX
B-SHORT-RUN
DYNAMICS
The "true" short-runprice equation for
thefirmis (9") with
R =
U-
-t
(p-
p-)
substitutedforR in theequation. This gives
thefirm'spriceas a functionof the distance
betweenfirms(a proxyforthe "number" of
firms) and other firms' price. Short-run
equilibriumis givenin Figure6.
The firm'sprice line is AA'. For pricesto
theleftof Pe firmsthinktheyhave charged
more than theirrivals price p but discover
theyhave not and increase their estimates
ofp, increasingprices.Similarlyto the right
p
X
~~~A'
of Pe prices must fall, and Pe is a stable
equilibrium.It is possible to show that 1) if
p-is zero firmsshould set a positive price
and 2) a unitincreasein T leads to less than
a unitincreaseinp so thatAA' looks as it is
drawnin thefigure.
Then, in short-runequilibrium,p must
equal F and R mustequal 1/2U, so that(9")
can be intepretedas givingboth profitmaximization and (with R = 1/2U) short-run
equilibrium. We assume that short-run
equilibriumis obtained "instantaneously"
and suppressthe dynamicsof short-runadjustments.For a more completediscussion,
see our 1976paper.
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THE AMERICAN ECONOMIC
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