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Berlin, March 2004
Discussion Papers
Transport Cost Sharing and Spatial
Competition
Hrachya Kyureghian
Sudipta Sarangi
Opinions expressed in this paper are those of the author and do not necessarily reflect views
of the Institute.
DIW Berlin
German Institute
for Economic Research
Königin-Luise-Str. 5
14195 Berlin,
Germany
Phone +49-30-897 89-0
Fax +49-30-897 89-200
www.diw.de
ISSN 1619-4535
Transport Cost Sharing and Spatial Competition
Sudipta Sarangi† Hrachya Kyureghian‡
(Research Professor of DIW Berlin)
September, 2003; Revised February 2004
Abstract
We consider a linear city model where both firms and consumers have to
incur transport costs. Following a standard Hotelling (1929) type
framework we analyze a duopoly where firms facing a continuum of
consumers choose locations and prices, with the transportation rate being
linear in distance. From a theoretical point of view such a model is
interesting since mill pricing and uniform delivery pricing arise as special
cases. Given the complex nature of the profit function for the two-stage
transport cost sharing game, we invoke simplifying assumptions and solve
for two different games. We provide a complete characterization for the
equilibrium of the location game between the duopolists by removing the
price choice from the strategy space. We then find that if the two firms are
constrained to locate at the same spot, the resulting price competition leads
to a mixed strategy equilibrium with discriminatory rationing. In
equilibrium both firms always have positive expected profits. Finally, we
derive a pure strategy equilibrium for the two-stage game. Results are then
compared with the mill pricing and uniform delivery pricing models.
Keywords: Spatial Competition, Cost Sharing
JEL Classification: R1, L13, D43, C72
WWe are grateful to Hans Haller, Amoz Kats, Simon Anderson, two annonymous referees and
the editor Laura Razzolini for helpful suggestions. We thank Robert Gilles, Georgia
Kosmopoulou, Cathy Johnson, Tito Cordella, Rickard Wall and partcipants at Southeastern
Economic Theory Meetings 1999, Games 2000 and Southern Economics Association
Meetings 2000 for comments. Sudipta Sarangi gratefully acknowledges the hospitality of
the American University of Armenia and DIW Berlin where a part of this research was
carried out. The usual disclaimer applies.
†DIW Berlin and Department of Economics, 2107 CEBA, Louisiana State University,
Baton Rouge, LA 70803, USA. email: sarangi@lsu.edu
‡Department of Industrial Engineering, American University of Armenia, Yerevan, Ar-
menia 375019.
1
Transport Cost Sharing and Spatial Competition
September, 2003; Revised February 2004
Abstract
We consider a linear city model where both Þrms and consumers
have to incur transport costs. Following a standard Hotelling (1929)
type framework we analyze a duopoly where Þrms facing a continuum
of consumers choose locations and prices, with the transportation rate
being linear in distance. From a theoretical point of view such a model
is interesting since mill pricing and uniform delivery pricing arise as
special cases. Given the complex nature of the proÞtfunctionfor
the two-stage transport cost sharing game, we invoke simplifying as-
sumptions and solve for two different games. We provide a complete
characterization for the equilibrium of the location game between the
duopolists by removing the price choice from the strategy space. We
then Þnd that if the two Þrms are constrained to locate at the same
spot, the resulting price competition leads to a mixed strategy equilib-
rium with discriminatory rationing. In equilibrium both Þrms always
have positive expected proÞts. Finally, we derive a pure strategy equi-
librium for the two-stage game. Results are then compared with the
mill pricing and uniform delivery pricing models.
1Introduction
The spatial competition literature in the Hotelling tradition has two main
strands. One concerns itself with models of mill pricing in which Þrms
choose location and prices, while the spatially dispersed consumers pay the
cost of travelling to the Þrm to buy the product. The other strand of the
literature assumes that Þrms absorb the transport cost of shipping the item
to the consumers and is called uniform delivery pricing since all consumers
pay the same price.1In this paper we analyze a model of a linear city that
1A third concept, less frequently encountered is that of spatial price discrimination
(Hoover, (1937)). For an insightful exposition of this issue see Anderson, de Palma and
1
incorporates features of both mill pricing and uniform delivery pricing. We
assume that Þrms charge the same price to all consumers but have a cost of
delivering to all those who purchase from them as in the models of uniform
delivery pricing. Buyers on the other hand pay the price and also incur
a transport cost which, for instance, reßects the delivery time associated
with the good. This delivery time increases with the consumer’s distance
from the Þrm and is a source of disutility. It captures the opportunity cost of
being able to consume sooner than later.2The consumers’ share of transport
cost can be interpreted broadly to include time, effort and other transaction
costs, asides from the costs of travel. This feature is common to models
of mill pricing. Thus our model is a hybrid of the standard mill price and
uniform delivery price models.
Buyers and sellers in the real world are dispersed over geographical space.
It has been argued that the dispersed nature of market activities can be a
source of market power for Þrms. Each Þrm has only a few rivals in its
immediate neighborhood. Similarly, consumers who are at a considerable
distance from a Þrm will not buy from that Þrm since they have to pay
very high transport costs. The relative location of the Þrms with respect
to the consumers is a crucial determinant of the degree of competition.
Consequently, once one recognizes the importance of space, it is obvious
that competition in the real world occurs only among a few and is best
analyzed in a strategic game setting.
The economic relevance of location games does not stem exclusively from
their initial geographical set-up. The idea can be extended to competition
among Þrms selling differentiated products where each Þrm’s product is
viewed as a point in the characteristic space. This product differentiation
aspect of location theory dates back to Hotelling’s (1929) seminal work. He
recognized that while location was a source of market power in itself, it could
also be a proxy for other characteristics of the product. The following quote
serves to illustrate this point quite well: “... distance, as we have used it for
illustration, is only a Þgurative term for a great congeries of qualities. In-
stead of sellers of an identical commodity separated geographically we might
consider two competing cider merchants side by side, one selling a sweeter
liquid than the other.”
Aside from the purely theoretical aspects of the model, one encounters
many examples of this sort in the real world. Retailers bear the cost of
Thisse (1989).
2One need look no further than the wide array of shipping options provided to con-
sumers by FedEx, UPS and the United States Postal Service to be convinced of the value
of consuming earlier.
2
bringing the commodity over to the shopping center, while the buyers must
drive there to actually inspect and purchase the items. Buying furniture
usually involves a trip to the furniture store and selecting the desired items,
and the furniture store usually delivers the items to the consumer location
free of charge. An almost perfect example is the Chicago based video rental
Þrm Facets C i n´emath`eque or Facets Video Rent By Mail. Members can order
videos of their choice by mail. The Þrm pays shipping and handling one
way, while the consumer incurs the mailing expenses involved in returning
the video. The labor market also has similar features. The commute time to
work has to be borne by the employees. Hence, one consideration for Þrms
in choosing to locate in the suburbs is the desire to avoid traffic congestion
thereby making the job attractive to workers. The large numbers of hi-tech
Þrms located in sub-urban Washington D.C. provide ample testimony to
this fact.
A similar phenomenon can also be observed in certain types of differenti-
ated product markets. In particular, it is quite common in some segments of
the software industry. Often each Þrm produces its own standard product
and then customizes it to suit the needs of individual buyers, while buy-
ers have to learn the intricacies of the software. For example, software for
Supply Chain management in the food industry differs from that designed
for the apparel industry (which is equivalent to choosing location) and the
software Þrm has to tailor the package to suit the needs of individual clients
in each of those industries. The cost of learning new software or customizing
it to suit the individual client’s needs can be treated as transport cost in
our framework. Also, Þrms often attempt to reduce their buyers’ learning
costs by providing training classes, on site implementation and customer
service. This is deÞnitely true for the Electronic Resource Planning [ERP]
segment of the industry where Þrms like PeopleSoft and SAP constantly
provide training to their clients and have created a new professional class
called information technology consultants.
For the purpose of modelling these issues one might imagine that there
is a total cost for moving a commodity from the store to the consumer’s
location. We then assume that the total pecuniary burden of shipping a
commodity from the Þrm to a consumer is shared by both buyers and sell-
ers. So consumers in our model pay an exogenously set proportion of the
transport cost while Þrms pay the remainder. For most of the examples
listed above, assuming an exogenously given transport cost sharing rule is
reasonable since the consumers have their own transport cost, while Þrms
have to incur transport costs which are speciÞc to them. Notice that when
the consumers’ share of costs goes to zero we have the uniform delivery
3
price model and when they bear the entire cost we obtain the mill pricing
formulation.
In the subsequent sections we develop a model to analyze the two stage
game. We Þrst analyze for a pure location game with exogenous prices
and its counter-part where Þrms locate at the same spot (thereby removing
location choice from the strategy space) and compete in prices. The most
interesting Þnding for the location game is that when transport costs are
shared, a unique and symmetric equilibrium in locations will exist. We Þnd
that as the consumers’ share of the transport cost decreases, both Þrms move
further away to the center to reduce their transport cost bill. On the other
hand when the exogenously given price in the location game is raised, Þrms
move towards the center to minimize transport costs. We also Þnd that
when the two Þrms are constrained to locate at the same spot, the resulting
price competition leads to a mixed strategy equilibrium with discriminatory
rationing.BothÞrms always have positive expected proÞts in equilibrium.
Using the insights from these two games, we then model the two stage
location price game. These allow us to simplify the otherwise complex and
intricate expression obtained for the proÞt function of each Þrm. We identify
conditions under which the two stage game has a pure strategy equilibrium.
We also demonstrate that Hotelling’s claim about increasing transport costs
leading to greater proÞts is incomplete −properly stated, it must account
for consumer reservation prices.
The remainder of the paper is organized as follows. The next section
provides a brief overview of the related literature. Section 3 provides the
basic model setup. In Section 4 we solve for the location equilibria, assuming
Þxed prices. Section 5 analyzes a particular price game where both Þrms
are located at the same spot. The following section solves for the location-
price equilibrium of a two stage game where the two Þrms Þrst choose their
locations and then compete in prices. We look for subgame perfect equilibria
in this game. Section 7 contains concluding remarks.
2 Review of Literature
Given the plethora of work both on models of mill pricing and uniform de-
livery pricing, an exhaustive survey of all aspects of the literature would be
a considerable digression. We limit the scope of our review only to those re-
sults which are pertinent to the model under consideration. Graitson (1982)
is an early survey of the literature. A more up-to date and comprehensive
survey can be found in Anderson, de Palma and Thisse (1992). The litera-
4
ture on mill price is more abundant and we will start by discussing those.
The mill price models trace their heritage from the original Hotelling
(1929) model.3Typically in these models Þrms choose locations and then
prices and consumers incur the transportation cost. Hotelling ((1929), pg.
53) claimed that under mill pricing the two Þrms in the market would
“...crowd together as closely as possible.” , while he noted the possibility of
Bertrand competition for the extreme concentration case only.4Fifty years
later d’Aspremont, Gabszewicz, and Thisse (1979) (henceforth DGT) revis-
ited the model and formally characterized the ßawed nature of Hotelling’s
solution.5They found that the price equilibrium found by Hotelling holds
only if the two Þrms are sufficiently far apart. If the two Þrms were located
close to each other, undercutting the opponent is proÞtable. Higher proÞts
destroy the pure strategy equilibrium in prices. Consequently, Þnding the
location equilibrium for the two stage game is also jeopardized. Of course
as pointed out by Hotelling, when the two Þrms are exogenously located at
the same spot, the game reduces to pure Bertrand competition. What he
missed though (and was pointed out by DGT), was that price undercutting
or Bertrand competition would arise ‘earlier’ −long before the Þrms ‘arrived’
at the same location. The tendency to undercut which allows the success-
ful Þrm to capture the entire market would arise as soon as the Þrms are
sufficiently close. DGT suggest one way out of the nonexistence problem:
assume quadratic transport costs. The game now exhibits a ‘centrifugal’
location tendency rather than central location tendency. The Þrms would
like to locate outside the linear city, and hence in equilibrium the two Þrms
charge the same price and locate at the endpoints of the line segment.
There are also some other approaches to deal with the non-existence
problem. One of the more ingenious ones by de Palma et al. (1986) shows
the existence of Nash equilibrium in pure strategies by introducing suffi-
ciently heterogenous products. A different solution has been provided by
Kats (1995) where the linear city was replaced by a one dimensional bounded
space without boundary, i.e., a circle. In this case a pure strategy equilib-
3Note however that Ferreira and Thisse (1996) provide evidence of the fact Launhardt
had already proposed such a model of a spatial duopoly in 1885.
4In the industrial organization literature this result is also referred to as the principle
of minimum differentiation. The term was coined by Boulding (1955) who used it among
other things to explain the existence of similarities between Methodists, Quakers and
Baptists.
5As noted in Osborne and Pitchik (1987), Vickrey (1964) had already identiÞed the
problem with Hotelling’s analysis.
5
rium does exist −any pair of locations where the Þrms are at least a quarter
away from each other and p∗
1=1
2=p∗
2constitutes a pure strategy equilib-
rium. Another approach is to characterize the mixed strategy equilibrium.
This line of research stems from the two Dasgupta and Maskin (1986) pa-
pers on games with discontinuous payoffs guaranteeing the existence of an
equilibrium in Hotelling type models. Osborne and Pitchik (1987) under-
take the task of actually identifying the equilibrium mixed strategy price
distribution functions of Hotelling’s original model. They identify a support
for mixed strategies in prices when the Þrms locate close to each other. They
Þnd a unique pure strategy equilibrium in locations, in which the Þrms are
located at about 0.27 from the respective endpoints. For those locations, the
equilibrium support for prices consists of two distinct line segments. As an
intuitive explanation Osborne and Pitchik suggest a parallel with the phe-
nomenon of ‘sales’. It is worth emphasizing that during this analysis they
encounter highly nonlinear equations and resort to computational methods
to come up with approximate numbers.
For the uniform delivery pricing models, where each Þrm quotes a single
delivery price to all its customers, the non-existence problem is even more
severe. It arises because the rationing of some consumers by one Þrm allows
its rival to service this segment of the market at a high price. This gives
the Þrst Þrm an incentive to undercut, thereby destroying the equilibrium
(see Beckmann and Thisse (1986)).6A comprehensive analysis of the circu-
lar space scenario can be found in Kats and Thisse (1993). After showing
the nonexistence of a pure strategy equilibrium in prices, they invoke Das-
gupta and Maskin (1986) and characterize the mixed strategy equilibrium
in prices. The location equilibrium in the Þrst stage of the game is in pure
strategies. The second part of their paper is devoted to the endogenous
choice of the pricing policy by the Þrms. For the monopoly case, uniform
delivery pricing is the optimal policy, partly because it allows the monop-
olist to extract all the surplus from the consumers. In the duopoly case,
the consumer’s reservation price ris the crucial parameter. For low r<5
8,
both Þrms choosing uniform delivery pricing is the unique equilibrium of the
pricing policy game. For higher r, the competitive region (the overlapping
market area) for the two Þrms becomes larger, intensifying price competition
between the Þrms, making mill pricing quite attractive. Hence both price
policies can be sustained as equilibria for the duopoly with mill pricing re-
6For more on uniform delivery pricing models also see Greenhut and Greenhut (1975)
and de Palma, Portes and Thisse (1987).
6
sulting in higher proÞts. Another solution to the nonexistence problem also
using heterogenous products can be found in de Palma, Labb´eandThisse
(1986). The interested reader may also refer to Anderson, de Palma and
Thisse (1989) for an excellent comparison of the above two pricing poli-
cies, as well as spatial price discrimination using a heterogeneous product
formulation.
3 The Model
Consider a linear city of length lwith a continuum of consumers distributed
uniformly on this line. Each consumer derives a surplus from consumption
(gross of price and transportation costs) denoted by V. In keeping with
the terminology used in the spatial competition literature we will refer to
this as the consumer’s reservation price. Consumers are assumed to have
unit demands when their reservation value exceeds the price plus the trans-
port cost they incur. Otherwise, they do not purchase the commodity. The
transportation rate tis assumed to be linear in distance. Consumers pay
a proportion αand Þrms pay a proportion (1 −α)ofthetransportcost.
Consequently, a consumer who travels a distance of dpays αtd as transport
cost and the Þrm pays the remaining (1 −α)td of the cost. For notational
convenienceweset(1
−α)t=sand αt=t−s. Due to the sharing of
transport costs by consumers and Þrms, consumers face horizontal product
differentiation and Þrms engage in some price discrimination in our model.
There are two Þrms in the market called Aand B.TheÞrms are located
at respective distances aand bfromtheendsoftheline(
a+b≤l, a ≥0,
b≥0), and charge prices of p1and p2respectively. In order to focus on the
transport cost issue, we assume that there are zero marginal costs. Utility
maximizing consumers buy from the Þrm that quotes the smallest effective
price (mill price plus their share of the transport cost). The location of the
indifferent customer is denoted by z=p2−p1
2(t−s)+1
2(l−b+a). Firms in the
model Þrst choose a location and then quote a price. Based on the price and
the transport cost consumers make their purchase decision. Figure 1 (all
Þgures have been attached at the end) represents the most general situation,
i.e., the two stage location-price game and provides a graphical depiction of
the notation developed here.
We are now in a position to obtain the proÞt function of the two stage
game. Note that the expression below is derived for Þrm A. We re q u i r e t hat
a≤l−b,orthatÞrm Ais located to the left of Þrm B.FortheÞrm on the
7
right, a symmetric expression applies with only relevant change in notation.
Set
∆=max
{0,min{a, max{0,l−b−p2
s}} −max{0,a−V−p1
t−s,a−p1
s}},
Φ=min
{max{a, l −b−p2
s},a+V−p1
t−s}−a,
Γ=max
{0,min{l, a +p1
s,a+V−p1
t−s}−max{a, (b+p2
s)}},
H=min
·½p2−p1
2(t−s)+1
2(l−b+a)−a¾,p1
s,V−p1
t−s¸,
K=a−max{0,a−V−p1
t−s,a−p1
s},
M=min
{p1
s,V−p1
t−s,l−a},
P=(t−s)(l−b−a).
Then the general expression for the proÞt function is as follows:
Π1(p1,p
2,a,b)=
(i)∆·{p1−s·max[0,a−(b−p2
s)]}
−s
2∆2+Φ·p1−s
2Φ2+Γ·p2−s
2Γ2if p1>p
2+P;
(ii)H·p1−s
2H2+K·p1−s
2K2if |p1−p2|≤P;
(iii)M·p1−s
2M2+K·p1−s
2K2if p1<p
2−P.
Notice that the expression depends on the relationship between the price
difference p1−p2and P=(t−s)(l−b−a),thecosttoaconsumerto
go the extra way from ato b. Further, observe that if the Þrm serves an
adjacent market area (an interval immediately to its left or to its right) of
length N, then the revenue from these customers is N·p1and the cost of
serving them is s
2·N2. It remains to determine that ∆,Φ,... are the correct
market sizes. We will explain the Þrst component of the proÞt function in
detail and provide brief discussions of the last two cases.
(i)p1>p
2+(t−s)(l−b−a). This case occurs when Þrm Ais being
undercut by Þrm B.Here∆isthesizeofthemarketareatotheleftofits
location and Φis the size of the market area to the right of its location. We
now discuss each part in some detail.
Consider {p1−s·max[0,a−(b−p2
s)]}Þrst. This is part of the expression
for revenue. As noted above, if the Þrm were to serve an adjacent market
8
area, the revenue would be p1times the base, ∆.However,itispossible
that the low price Þrm, by undercutting, is serving part of the market to
theleftofÞrm A’s location. This would correspond to the case when in the
expression max[0,a−(b−p2
s)] the greater number is a−(b−p2
s). Here the
height of the rectangle representing revenues would be less than p1by the
amount s·max[0,a−(b−p2
s)].
Next consider ∆in more detail. The part max{0,l−b−p2
s}within ∆
represents situations where depending on parameters Þrm B’s left hand side
bound could be to the left of a. The expression min{a, max{0,l−b−p2
s,l−
b−V−p2
t−s}} allows for the fact that the effective right hand side boundary
will be determined by the minimum of aand the left hand side bound of
Þrm B’s market segment as determined by the preceding expression. Now
consider max{0,a−V−P1
t−s,a−p1
s}. This depicts the possibility that Þrm A
may not serve to all the customers to the left of a. It depends on whether
the Þrm A’s constraint through price and transport cost is binding or the
consumer’s reservation price constraint goes into effect Þrst. The difference
∆is the total market area available to Þrm Ato the left of a.
We now m ove o n t o Φ.Thetermmax
{a, l −b−p2
s}inside Φrepresents
the possible right-hand side bounds on Þrm A’s market area depending on
the price and transport cost that Þrm Bhas to incur. Intuitively, by under-
cutting, Þrm Bcaptures Þrm A’s territory and the parameters determine
the limits of Þrm A’s market area. The l−b−p2
sexpression here repre-
sents the case when the Þrm B’s transport cost determines the limits of its
territory. Taking the maximum of this expression and aensures that we
conÞne ourselves to considering market segments to the right of Þrm A’s
location. The next term a+V−p1
t−srepresents the case when given Þrm A’s
price, customers’ decision about whether to buy from this Þrm determines
its market area. Since both numbers, i.e., max{a, l −b−p2
s}and a+V−p1
t−s
represent possible bounds on the right hand side for Þrm A, the smallest of
them is the effective bound. This accounts for the “min” in the expression.
Finally,sinceweareconsideringonlytherighthandsidemarketarea,we
need to subtract afrom this expression.
The last two terms represent the possible case of ‘leapfrogging’ a far
away market: Γis a possible market area to the right of the opponent’s
territory. It occurs when the opponent loses market share from the right,
thus making it feasible for the left-hand side Þrm to serve that chunk of the
market by sufficiently raising its price. The min{l, a +p1
s,a+V−p1
t−s}part of
Γrepresents the possible right-hand side boundary of that market segment.
It could be either l(the right-hand side boundary of our linear city), or a
9
bound arising either because the seller is unwilling to sell or the buyers do
not wish to purchase on the far right hand side area of the linear city. The
term max{a, (b+p2
s)}represents the possible left-hand side boundary for
that market segment. The maximum of zero and the above has to be taken,
to account for the possibility that the right hand side border of Þrm B’s
market area is to the right of min{l, a +p1
s,a+V−p1
t−s}.
(ii)|p1−p2|≤(t−s)(l−b−a). In this case no Þrm is able to undercut its
rival. The Þrst two terms here signify proÞts from the right hand side and
thelasttwotermssignifyproÞtsfromthelefthandside. His the minimum
of the three following possibilities: either the line segment between aand
location of the indifferent consumer, or p1
swhich is total the market the
Þrm Awould like to serve (on its right hand side), or only the line segment
representing the locations of those consumers (located to the right of a)
who would like to buy from Þrm A. Since this is the no undercutting case,
the (left-hand side) extent of the market area to which Þrm Bis willing to
sell plays no role here. The expression Kjust depicts the Þrm A’s captive
market on the left hand side.
(iii)p1<p
2−(t−s)(l−b−a). This case occurs when Þrm Aundercuts
Þrm B.TheÞrst two terms are proÞts from the market area of size Mon
the right and the last two are proÞts from the market area of size Kon the
left. Consider M.FirmAcan sell to the market segment it wishes to, given
by p1
s, unless of course some of those customers themselves do not want to
purchase from it which is given by ( V−p1
t−s). Finally, it allows for the fact
that Þrm Asells to the entire line segment, from its own location aup to
the right hand side boundary of the linear space, l.Thelasttwotermsof
this part of the proÞt function represent the market area of the Þrm to the
left of its location.
The general expression for the proÞt function given below indicates a
host of possibilities from which one may surmise that multiple equilibria
can exist in our setting. Clearly, it will not be possible to analyze the model
without making some simplifying assumptions. Any equilibrium outcome of
the model will be determined by the interplay of the consumer’s reservation
val ue and the Þrm’s choice of location and prices. Since our model combines
elements from both the mill pricing and the uniform delivery pricing models,
absence of sales can occur for two reasons. First, consumers may not wish
to purchase the product at the price offered by the Þrm because of low
reservation utilities. Secondly, for certain location-price pairs, it is also
10
possible that a Þrm may not want to sell to some consumers who are willing
to buy from it. Keeping these in mind we analyze two different games to gain
some insight into the two-stage game. We Þrst study a pure location game.
Here Vdoes not play any role and we are able to focus on the interaction
between price and location choice. We then look at the situation where the
Þrms are located at the same spot. In this case choice location does not play
a role and allows us to concentrate on the interaction between Vand the
prices set by the Þrm. Finally, in the two stage game all three parameters
are allowed to vary.
4TheLocationGame
In this section we assume that price is exogenously given, as in a regulator’s
world. This can also happen if prices have been chosen earlier in the distri-
bution channel by manufacturers or wholesalers, and retailers are subject to
resale-price maintenance. In this section we assume that the price set by the
regulator is low enough to ensure that consumers can buy from either Þrm.7
In order to look for equilibria we deÞne three price ranges based the Þrms’
ability to recover the transport cost of shipping to consumers and identify
the equilibrium for each case. Our Þrst proposition concerns one of these
ranges.
Proposition 1. If1
4sl ≤p≤sl, there exists a unique equilibrium in
locations. The equilibrium locations are symmetric and are given by
a∗=b∗=l
6+p
3s
Equilibrium proÞts for each seller are identical and given by
Πi(a∗,b
∗)= 1
72s[40pls −8p2−5(sl)2].
Proof: See Appendix.
In contrast to the original Hotelling model, here transport cost consider-
ations in maximizing proÞts prevent the Þrms from locating at the center in
all instances. Next, in equilibrium the indifferent consumer is always located
at l
2irrespective of the location of the two Þrms. Further, a∗∈[l
4,l
2]with
7We analyze the role of reservation prices in subsequent sections of the paper.
11
the Þrmneverlocatingtotheleftof l
4to ensure that cost minimization.
The corresponding proÞts lie in the range [ 2
8pl, 3
8pl]withproÞts increasing
as the Þrm Amoves to the right. When p∈[1
4sl, sl]asassumedinthe
proposition, the optimal location varies inversely with sand directly with
p. Comparative statics results suggest that Þrms in our model also have
a central location tendency. Clearly, da∗
dp >0,suggesting that both Þrms
want to locate closer to the center as the exogenously given price grows. As
the regulator raises the price, each Þrm can sell to a larger segment of the
market and in order to minimize costs moves towards the centre. Since this
is true for both Þrms, equilibrium behavior ensures that the position of z
remains unchanged. Finally, using the fact that s=(1
−α)t,itisalsopossi-
ble to show that da∗
dα>0. Thus as the consumers bear a greater proportion
of the transport cost, we get an outcome closer to Hotelling’s mill pricing
model. This is intuitive since the closer the situation is to Hotelling’s case,
the stronger is the central location tendency.
Equilibrium location for the remaining two cases cannot be obtained
using standard Þrst order conditions and is discussed next.
Remark 1. Case (i)p< 1
4sl : Prices are so low in this range that the Þrms
are unable to serve the entire market. At best each Þrm can only sell to a
market size of l
2(see Figure 2 ). Toseethisweequatea∗+p
s=(l−b∗)−p
sand
solve for p. This equation enables us to Þnd the price at which the market
areas of the two Þrms are adjacent without overlapping which occurs at
p=1
4sl. For prices below this the Þrms can have isolated markets. In fact
the Þrms will locate such that their market segments do not overlap while
maximizing the market area served. They choose locations so as (1) not to
have overlapping market areas (p
s<l−b−a
2), and (2) to ensure that (p−as)
is nonpositive. Thus, it is possible to have a whole range of locations as
equilibria in this instance.
Remark 2.Case(iii)p>sl:HereeachÞrm can cover the entire market
by itself. Both Þrms locate at l
2. From the previous proposition we have
already seen that no seller wants to choose a location to the left of their
original location with increases in price. By symmetry of the proÞt functions
rightward movements are ruled out, as this amounts to relabelling the Þrms
and therefore both Þrms choose l
2.
Thus, with a slight modiÞcation of the location game we Þnd that Bould-
ing’s ubiquitous principle of minimum differentiation is no longer so perva-
sive. The implications of this for the regulator are also fairly obvious. If
the regulator decides to lower prices after the Þrms have chosen locations,
12
each Þrm’s location will be sub-optimal. Hence, in order to maximize proÞts
both Þrms will deliver to a smaller market than prior to the price reduction.
Similarly, if prices are raised locations will still be sub-optimal, but fewer
customerswillbeleftout. WenextsolveforapricegamewherebothÞrms
are at the same location by decree.
5ASpecialPriceGame
In this section we analyze a price game when both players are constrained
to be at the same location while retaining all other assumptions of the pre-
vious section. Admittedly, this is an extreme assumption, but it is also the
analogue of the problem in the previous section where the Þrms faced exoge-
nously given prices and were allowed to compete only in locations.8Given
that transport cost is shared between the Þrms and consumers some inter-
esting possibilities can arise in this game. First, for low Vsome consumers
will not purchase the product at the price offered by the Þrm. Another pos-
sibility is the existence of prices at which a Þrm does not sell to some willing
consumers. Then given a sufficiently high V, the other Þrm can alter (raise)
its price and sell to the excluded section of the market. Thus, the two Þrms
may sell the same product to different segments of the markets at different
prices. Clearly, rationing of certain consumers is a distinct possibility in
this model. Furthermore, this rationing will be of a discriminatory nature,
as each additional consumer will pay a higher effective price based on the
distance from the seller’s location.
Since both Þrms are located at the same place, there is no horizontal
product differentiation −we have a case of pure price competition. As shown
by DGT, in the Hotelling model there exists a pure strategy equilibrium for
such a price subgame where prices are equal to the marginal cost, i.e., zero.
Given positive transport costs for Þrms however, the standard Hotelling
result no longer holds. We Þnd that there is no pure strategy equilibrium
in the price subgame. Instead, we show the existence of a mixed strategy
equilibrium exists where prices always exceed the marginal cost. This is
similar to the results in models of Bertrand-Edgeworth competition (see for
example Allen and Hellwig (1986), Dasgupta and Maskin (1986), and Kats
and Thisse (1993) in the context of spatial models.
We will Þrst establish a result about the upper (pu)andlower(
pl) bounds
on prices. Without loss of generality, consider a realization of the mixed
8One might imagine a situation where fastidious city planners will only let Þrms set up
shop at a particular location!
13
strategy where Þrm Acharges a low price and Þrm Bsets a high price.9
We obtain puby computing the monopoly price that takes Vinto account.
Consequently, there are two possible upper bounds on price. When the
reservation price is below a certain threshold (say b
V), at the price upper
bound denoted by pu
rsome consumers in the market will not wish to purchase
from Þrm B.10 When V≥b
V, the upper bound is given by the highest price
at which Þrm Bcan sell to consumers located furthest from it, and is denoted
by pu
a(>p
u
r) since the Þrm sells to all residual consumers.
Lemma 1. The support of any equilibrium in the (same location)
price game is a strict subset of [0,V].
Proof: See Appendix.
The next proposition shows the precise support of this mixed strategy
equilibrium when the Þrms locate at the center. We then argue that this
can be generalized to asymmetric location choices of the Þrms.
Proposition 2. Fo r a=b=l
2, the price game has no pure strategy
equilibrium. A mixed strategy equilibrium does exist for this price game. For
V<b
V, the support is given by hsV ³t−√2ts−s2
(t−s)2´,sV
√2ts−s2iand for V≥b
V,
the support is given by hs(l−a)³1+ s(l−a)
2(V−(l−a)(t−s)) ´,V−(l−a)(t−s)i.
Proof: See Appendix.
We now argue that the same result is also true for any other location of
the two Þrms. The problem becomes asymmetric in this case and the critical
val ue of Von the right hand side segment of the Þrm’s location can differ
from the critical value on the left hand side. This alters the proÞts of the
high price Þrm and consequently the value of the bounds. Given that the
problem is computationally intensive and qualitatively no different from the
one shown above, we just provide the rationale for the argument without
explicitly computing the bounds.
Remark 3. For a+b=l, the price game has no pure strategy equilib-
rium. A mixed strategy equilibrium however, does exist in this price game.
9Although we often refer to Þrm Aas the low price Þrm and Þrm Bas the high price
Þrm, here we have in mind only a particular realization of the mixed strategy.
10Theprecisevalueofthethresholdreservation price is not relevant to the argument
here. Exact computations are shown for the next result.
14
To deal with the case of any location, we will consider situations where
a< l
2and a+b=l, using the rationale suggested above. While the method
for computing the upper and lower bound remains the same, three distinct
possibilities can arise in this situation. Two of these situations have already
been described above (see also Figure 3 ). The third possibility arises in the
asymmetric case because there is an intermediate value of the reservation
price at which some consumers to the right of awill not be able to purchase
at the monopoly price. This case will also yield different values of phand
pl. Thus we will have three different inequalities which have to be solved
using a technique similar to the one used for a=l
2. The main difference
with the previous case therefore stems from the fact that the computation
of monopoly proÞts changes. This affects the high price Þrm’s best response
and consequently the lower bound without altering the logic of the calcula-
tion. Since Theorem 5 of Dasgupta and Maskin (1986) holds in this case as
well, the mixed strategy equilibrium exists.
It is worth pointing out that while the upper bound can be the monopoly
price, the lower bound differs from zero and from sas well. Since smay
be thought of as the marginal cost to the Þrm of (delivering) an additional
unit, this is different from the usual lower bound of the support of the mixed
strategy in rationing models. These differences arise because the rationing
mechanism in our model can be described as discriminatory rationing.Con-
sumers not served by the low price Þrm are served by the high price Þrm, but
each additional consumer pays a higher effective price which is proportional
to the distance from the Þrm’s location. It is precisely this reason which
also prevents the price from going down to zero due to price undercutting,
as it becomes worthwhile for one Þrm to sell to the market segment that is
left out instead of lowering prices further.
The mixed strategy equilibrium in our formulation has another attractive
feature. Equilibrium proÞts in the DGT model are always zero when a+b=l
and involves the play of a pure strategy with both Þrms choosing zero prices.
In general, a mixed strategy equilibrium is often considered unattractive as
players are indifferent between all the pure strategies involved. Moreover,
it does not give any reason to select among these different strategies (see
Osborne and Rubinstein, (1994) for more on interpretations and criticisms
of mixed strategies, including points on which even the authors of the book
disagree). The redeeming feature of the equilibrium mixed strategy in our
model is the fact that expected proÞts are always positive, whereas in the
DGT framework they are always zero. In fact, proÞts are positive for any
realization of the mixed strategies since plis always positive in our model.
15
The insights from these two special games and their implications for the
two-stage game are discussed in the concluding section.
6TheLocation
−Price Game
In this section we will look for pure strategy equilibria of the two stage
game where Þrms Þrst choose their locations and then set prices. Figure 1
is a representative situation for this case. The subgame perfect equilibrium
obtained here is the central result of this paper and ties together all the
elements of the last two sections. Given the general setting of this section
multiple equilibria cannot be ruled out. However we focus on identifying
a pure strategy equilibrium.11 This is the most frequently sought after
equilibrium in the spatial competition literature, and is perhaps the most
interesting one as well. Besides being an interior equilibrium we Þnd that
it also has other intuitive properties. We compare our results with those of
DGT and the uniform delivery price models.
The strategy for constructing the proof is as follows. We identify three
conditions to simplify the proÞt function for the two stage game and give it
the relevant shape, i.e., one that enables us to Þnd a pure strategy equilib-
rium −the only one that is “strictly interior in all respects” for the two stage
game. Then we show that an interior pure strategy equilibrium cannot ex-
ist unless these conditions are satisÞed, thereby justifying these conditions.
Thus while the constructed equilibrium might seem to require endogenous
conditions, we show that if these conditions do not hold an equilibrium can-
not exist. These conditions essentially allow us to focus on a speciÞcpartof
range of the proÞt function to construct the desired equilibrium.
Condition 1:Buyer Participation condition. This condition consists of
two inequalities and is assumed by most models of spatial competition (see
for example DGT or Graitson’s (1982) survey).
(i)p1+max
{(t−s)(l−a),(t−s)a}<V
(ii)p2+max
{(t−s)(l−b),(t−s)b}<V
These inequalities require that the reservation price be so high that con-
sumers are not prevented from buying from either Þrm. The absence of this
condition will lead to the type of outcomes associated with Case (i) of the
location model described in Section 4.
11Following Dasgupta and Maskin (1986), it can be shown that the proÞt function
satisÞes conditions for the existence of a mixed strategy equilibrium and is available from
the authors on request.
16
Condition 2:Seller Participation condition. All of the four inequalities
given below must be satisÞed for no Þrm to lose any market share.
(i)p1−as ≥0andp1−(z−a)s≥0 and,
(ii)p2−(l−b)s≥0andp2−(l−b−z)s≥0,
where z=p2−p1+(t−s)(l−b+a)
2(t−s)denotes the location of the indifferent con-
sumer. It ensures that the total cost imposed on each Þrm by the market
segment both to its left and right hand side can be represented by the area
of a triangle. Essentially it says that the prices are so high that after ac-
counting for costs, the Þrmiswillingtoselltoallconsumerswhowishto
purchase from it, in its captive market on one side and up to the indifferent
consumer on the other side. Note that this condition has consequences for
both location and price choices and affects the possibility of ‘leapfrogging’
by Þrms.
Condition 3:Market Capture condition.
|p1−p2|<(t−s)(l−b−a)
This is the situation that frequently appears in mill pricing models. It
implies that no single Þrm can set its price to sell to the entire market by
completely undercutting its rival.12
Next, assuming that Conditions (1) −(3) hold, we have surprisingly well
behaved proÞt functions for both Þrms given by
Π1(p1,p
2)=p1z−s
2(2a2+z2−2az)and
Π2(p1,p
2)=p2(l−z)−s
2(b2+(l−b)2+z2−2(l−b)z).
This is easy to verify since Condition 3 eliminates a large portion of the
proÞt function and the other two conditions give us the desired expression.
Region 3 of (Figure 5 ) depicts a proÞt function of this sort. We now show
that a (interior) pure strategy equilibrium in prices and locations does not
exist when either Condition 2 or Condition 3 is not satisÞed.
12Although we call it the Market Capture condition, this is somewhat of a misnomer.
When the Þrm’s share of transport costs is high, the Þrm that undercuts may not choose to
sell to all the customers who wish to purchase from it. Condition 2 prevents the occurence
of this situation.
17
Lemma 2.A pure strategy equilibrium of the price-location game
does not exist when the Seller Participation condition does not hold.
Proof: See Appendix.
Lemma 3.A pure strategy equilibrium of price-location game does
not exist when the Market Capture condition is violated.
Proof: See Appendix.
Note that in the DGT framework Condition 3 is enough to guarantee
that the proÞt functions are well-behaved. However, in our framework this
condition does not suffice, as the Þrm that has undercut its rival may not
wish to sell to all the willing consumers. The other Þrm then will be able
to get the residual consumers. By raising the price Þrm Ais able to reach
the boundary where the customers wish to purchase from it. Note that the
proÞt function of Þrm Awill have a kink here. Consequently, the proÞt
function will not be well behaved, which renders it impossible to solve the
two stage game in the standard fashion. However, Conditions (1) −(3)
together whose violation ensures that a pure strategy equilibrium will not
exist enables us to circumvent those sorts of problems for the current anal-
ysis.
DeÞne
a∗=b∗=1
8lt 4t−3s
s(3t−2s)and p∗
1=p∗
2=1
8l20t2+8s2−25st
3t−2s
At these prices and location proÞts are given by
Π∗
1=Π∗
2=l2
64s−96s4+ 416s3t−609s2t2+312
st3−16t4
(3t−2s)2.
Proposition 3. Suppose Conditions (1) −(3) are satisÞed. Then
for a+b<l, the tuple (a∗,b
∗,p
∗
1,p
∗
2)as deÞned above is the unique sym-
metric equilibrium in pure strategies of the location-price game.
Proof: See Appendix.
We are now in a position to compare our results with those of the mill
pricing and uniform delivery pricing models. As demonstrated by DGT, the
18
Hotelling model is inherently unstable. Optimal prices require that Þrms
be far apart but ∂Π1(p∗
1,p
∗
2)/∂aand ∂Π2(p∗
1,p
∗
2)/∂bare positive and both
Þrms have a tendency to move to the center. Thus the Market Capture
condition is violated and the second stage equilibrium does not exist. We
employ a similar reasoning to investigate the existence of our pure strategy
equilibrium. Let Θ=(Π∗
i−Π0
i)>0whereΠ0
idenotes proÞts from capturing
the whole market. We Þnd that Θ>0whens>0.5034tsuggesting that
Þrms must share more than half of the total transport cost to ensure that
equilibrium proÞts are higher than undercutting proÞts. Thisisequivalent
to the DGT condition requiring Þrms to be sufficiently far enough. In other
words a low sfacilitates capturing the market. It becomes easier for the two
Þrms to locate closer to each other which facilitates undercutting.
Note that Θ=0ats=1,which coincides with the uniform delivery
pricing case. When s= 1, optimal proÞts are the same as those from
undercutting, simply because optimal prices are the same as undercutting
prices: p0
1=p∗
1−(l−b∗−a∗)(t−s), and at s=t, clearly p0
1=p∗
1.This can also
be explained intuitively. As s→1, p∗→3
8lt and a∗=b∗→1
8l. Therefore,
the right hand side bound of the desirable market area for Þrm Agiven by
a∗+p∗
scan be written as 1
8l+3
8tlt =l
2.Obviously, the desirable market
areas for two Þrms in this case do not overlap. They are contiguous meeting
at the center of the city. Also, as mentioned earlier, the optimizing price
here is the same as undercutting price. In this situation, each Þrm has an
incentive to increase its price, provided the buyer participation constraint
is satisÞed. By doing so the Þrm looses only an inÞnitesimal amount in
proÞts as a result of market loss to the competitor. On the other hand it
gains a larger amount in proÞts due to the higher prices. This is the usual
explanation for the non-existence of price equilibria in uniform delivery price
models (see de Palma, Labb´e and Thisse, 1986). Hence, for the case of s=1
or in the uniform delivery price model, there is no pure strategy equilibrium
in prices. Thus, while a mill pricing type situation does not arise unless
the Þrm share of transport cost is greater than half the total transport cost,
the uniform delivery pricing type situation arises only when Þrmspaysthe
entire cost.
Next we Þnd that ∂Θ/∂sis a concave function with the maximum at
s'0.7. The concavity of this function can be explained by the fact that
it affects both prices and locations differently and the value of Θclearly
depends on which of these two dominates the other. Our model suggests
that differential proÞts are maximized when both Þrms and consumers incur
a part of the transport cost. This echoes the Þndings of Kats and Thisse
(1993) on the issue endogenous pricing policy choice. They Þnd that for high
19
reservation values (as in our model) both mill pricing and uniform delivery
pricing can be sustained here as an equilibrium. It is also worth mentioning
that a special case of Proposition 3 arises when sellers are constrained to
choosing only symmetric locations around the center. In this case we Þnd
that a=b=l
4, and optimal prices are given by p=l(t−3
4s).
An intriguing feature of optimal proÞts seems to be the fact that proÞts
rise with an increasing t.13 This however is fairly intuitive. As the transport
technology becomes more expensive, it becomes easier for the Þrms to en-
gage in monopolistic behavior. The consumers’ reservation price must also
increase in order to enable the consumers to purchase the commodity at the
higher effective price. Since the equilibrium here assumes that Conditions
(1) −(3) must hold, it leads to higher proÞts for the two Þrms. On the
other hand, if Vis Þxed, then the proÞts shrink at tincreases. This is easily
veriÞable if we use the intuition from the problem where Þrms are located
at the same place and compete only in prices. Recall that the optimal proÞt
functions there are given by Π1=Π2=s³V
(t−s)2´2³t−√2ts −s2´2,where
Vis the Þxed reservation price. Thus proÞts tend to zero as tincreases.
7 Conclusion
This paper analyzes a model where both Þrms and consumers have trans-
portation costs. In the standard mill pricing model the pure strategy equi-
librium breaks down since Þrms have an incentive to move to the center
and this makes it easier for the rival to undercut. The Þrm that undercuts
successfully gains the entire market. In our model while choosing locations
the Þrms also have to ensure that they minimize their of transport cost bill.
So, while there is a central location tendency, in our model there is also a
countervailing force. The Þrm that undercuts its rival may not be able to
sell to the entire market. Similarly, note that in the uniform delivery price
models one Þrm may charge a high price and sell to customers who its rival
may be unwilling to service. In our model the ‘rationed’ consumers may
be not be willing to buy from a high price Þrm since the price inclusive of
transport costs may exceed their reservation price. However, by imposing
13Hotelling’s remark in this context is especially interesting. “These particular mer-
chants would do well, instead of organizing improvement clubs and booster associations
to better the roads, to make transportation as difficult as possible. Still better would be
their situation if they could obtain a protective tariffto hinder the transportation of their
commodity between them.” (pg. 51) Thus Hotelling’s intuition seems to be incomplete as
the effect of the reservation price of the consumers is not taken into account.
20
a set of simple requirements and invoking two lemmas, we are able to rule
out such possibilities in the location-price game. Under certain parameter
values on the transport cost share, we Þnd a unique symmetric pure strategy
equilibrium where Þrms do not locate at the city center. This result differs
from the earlier results on location games and is clearly a consequence of
requiring both types of players to incur transport costs.
We also solve two other games to develop some insights for the two-stage
game. In the Þrst of these we consider parametric prices, thereby restricting
Þrms to choosing a location only. We Þnd that when the share of transport
costs borne by the consumers increase Þrms move closer to the center. An
interesting feature of the model is that there is a symmetric location equi-
librium where the Þrms chose their location keeping their transport cost in
mind. It means that the Þrms would never locate at the end points. In fact
there is a threshold location ( l
4)whichtheÞrms will never cross. In the
second game Þrms are assumed to locate at the same spot. Thus location
choice is no longer an element of the strategy space. Here we Þnd the ex-
istence of a mixed strategy equilibrium whose support is identiÞed in the
paper. This is useful since the existence problem is resolved indicating that
the two stage game would at least have a mixed strategy equilibrium. This
game also revealed that when the Þrms locate too close to each other there
cannot be an equilibrium in pure strategies, i.e., the incentive to undercut is
too strong. The importance of the reservation price was also demonstrated
by the this game. Another interesting feature of this price game is the pos-
sibility of discriminatory rationing. These key insights were valuable for
Þnding the pure strategy equilibrium of the two-stage game.
Finally, the paper also raises another interesting question −the issue of
endogenizing the transport cost sharing decision. We believe that this will
provide an alternative approach to modeling the choice between uniform
delivery pricing or mill pricing for Þrms.
21
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23
Appendix:
ProofofProposition1: The location of the consumer who is indif-
ferent between buying from Aand buying from BsimpliÞes to z=l−b+a
2.
Then we can write the proÞt function as Π1(a, b)=pz−(1−α)t
2{a2+(z−a)2},
and Π2(a, b)=p(l−z)−(1−α)t
2{b2+(l−b−z)2}. IneachoftheseproÞt
functions the Þrst term denotes revenues and the second term is the share of
the transport cost. Geometrically, total costs are triangles whose areas the
Þrms try to minimize. The proof consists of taking the derivative of each
Þrm’s proÞt function with respect to its location and solving the following
system of two equations obtained from the Þrst order conditions.
a=4
5
1
2p+1
4ls −1
4sb
sand b=4
5
1
2p+1
4ls −1
4sa
s
We also verify that the second order conditions are satisÞed. Substituting
the optimal locations in the proÞt functions yields the equilibrium proÞts.
Furthermore, we can check that Þrm Adoes not wish to locate to the right
of Þrm B.Given
b∗, we know that l−b∗is less than l
2which denotes the
location of the indifferent consumer in the above equilibrium. Consequently,
in the above equilibrium Þrm Ahas half the market. If Þrm Alocates to
the right of Þrm Bthen the new indifferent consumer will lie in the interval
l−b∗and Þrm A’s market area will be strictly less than half the market area
l
2. Hence, Þrm Awill not gain by selecting a location to the right of Þrm
B.Bysymmetry,Þrm Bwill never locate to the left of Þrm A.So,(a∗,b∗)
constitutes an equilibrium.
ProofofLemma1: We demonstrate that the interval [pl,p
u]⊂[0,V]
for each of the two possible cases. We show that when Þrm Araises its price
starting from zero, Þrm Bwill not raise its price beyond pu. Similarly, Þrm
Awill not reduce its price below a lower bound pl.Case (I): For V<b
V,
pu
ris the candidate upper bound. By deÞnition, we know that pu
rdominates
all prices in the interval (pu,V]. When Þrm Araises its price beyond zero,
Þrm Bloses customers from the center of its market. The optimal response
for Þrm Bis to either lower its price and sell to the consumers previously
left out, or to undercut Þrm A. Thus prices do not exceed pu
r.Case (II):
For V≥b
V,the candidate upper bound is pu
a.WhenÞrm Araises its prices,
Þrm Bcan continue to sell to the residual market at pu
a,or undercut Þrm
A. Thus in either case there is an upper bound on prices. Now consider the
24
existence of the associated lower bounds for the two possible cases. Case
(III):LetV<b
V.SinceÞrm A’s proÞts are increasing in its price, it will
prefer to raise its price. The lower bound on prices can be obtained by
equating Π2(p1,p
0
2(p1)) (where p0
2(p1)isÞrm B’s best response) with Þrm
A’s proÞt when it raises price. Prices in the range [0,p
l
r) are dominated by
pl
rand for prices above pl
r,therivalÞrm will have an incentive to undercut.
Case (IV):ForV≥b
V,the candidate lower bound is pl
a.Asimilarargument
establishes the lower bound on prices for this case. Thus prices will not fall
below pl. This allows us to conclude that prices above puare dominated by
it. Also, choosing plyields higher proÞts than prices below it. So there can-
not be a pure strategy equilibrium outside this interval. Hence there cannot
be one in mixed strategies either as it would involve the play of dominated
strategies.
ProofofProposition2: From Lemma 1 we know that any equilibrium
must lie in the interval [pl,p
u]. Assuming a=l
2we will now compute
the critical Vand the two associated intervals. The threshold Vis found
by taking the derivative of Þrm B’s proÞt function at the price when the
furthest consumer is indifferent between buying and not buying from Þrm
Band occurs at p=V−(t−s)l
2.For V<l
2(2t−s)= b
V, the upper
bound is the monopoly solution. When Þrm Bbehaves as a monopolist on
the residual demand (area) left by Þrm A,itsproÞts are Π2=1
2(V−p2
t−s−
p1
s)(2p2−s(V−p2
t−s+p1
s)). Hence its best response is p0
2(p1)=stV −p1(t−s)2
(2t−s)s.
Using this we compute proÞts of Þrm Bin terms of p1which are given
by Π2(p1,p
0
2(p1)) = 1
2
(sV −p1t)2
(2t−s)s2. Clearly ∂Π2(p1,p0
2(p1))
∂p1<0.For th e l owe r
bound on prices we equate the proÞts of the two Þrms using the fact that
Π1(p1,p
2)=p2
1
2s.SinceÞrm Ais the low price Þrm, its proÞt expression does
not contain a p2term. Equating the two proÞtexpressionsgivesusp1=
sV ³t±√2ts−s2
(t−s)2´. The root with the positive discriminant yields a negative p2
and hence is eliminated. The optimal value of p1which is the lower bound
is then given by
pl
r=sV Ãt−√2ts −s2
(t−s)2!and pu
r=sV
√2ts −s2
This is intuitive: a high p1implies that the low-price Þrm is selling to a
large section of consumers leaving out very little for the other Þrm. To Þnd
the upper bound we use the root with the negative discriminant and it can
be checked that pu
r>p
l
r. Also, using these prices we Þnd that proÞts are
25
Π1=Π2=s³V
(t−s)2´2³t−√2ts −s2´2.
Let V≥l
2(2t−s)= b
V. A similar argument establishes the lower
bound on prices for this case. The only difference is that the monopoly
proÞtofÞrm Bis now different. So, pu
a=V−(l−a)(t−s)andpl
a=
s(l−a)³1+ s(l−a)
2(V−(l−a)(t−s)) ´. Now let us consider what happens when prices
are in the range [pl,p
u]. Suppose a Þrm is charging the price pu.Thenby
chargingapricepu−ε(where ε>0, and small) its rival can undercut the
Þrm completely. This phenomenon of successive undercutting will occur for
any price above pl. Once prices reach pl,one of the Þrms is better offsell-
ingtotheremainingconsumersatapriceofphinstead of undercutting its
rival further. However, the Þrm charging plwould now prefer to undercut
the high price Þrm. Hence there are no pure strategy equilibria. Finally,
it can be shown that the two stage location-price game satisÞes Theorem 5
of Dasgupta and Maskin (1986, pg. 14). Hence it is satisÞed for this price
subgame. Hence we assert that a mixed strategy equilibrium identiÞed here
exists.
ProofofLemma2: This is a non-existence result that proves the ne-
cessity of the Seller Participation condition. Case (i) p1−as < 0. Here
Þrm Ahas market loss on its left hand side. Also assume that α>1−α.
Then it is easy to check that by charging a price p1+ε(ε>0), Þrm A
can increase its proÞts. Suppose α≤1−αas shown in Figure 4.Then
thereexistsapricepair(p1,p
2) which is an equilibrium in pure strategies.
However, it is not robust to the Þrm’s location choice decision. It is easy
to check that Þrm Acan always do better by moving to its left. Since we
require conditions for an equilibrium in the two stage game, any candidate
equilibrium must survive the next stage of subgame perfection −the choice
of optimal locations. Note that the tendency to move to the left for Þrm
Ais present irrespective of the relationship between αand 1 −α. Hence it
is not possible for a pure strategy equilibrium to exist in this case. Case
(ii) p1−(z−a)s<0. Firm Anow has market loss from the right hand
side. One can check that by charging a price p1+ε,Þrm Ais better off.It
gains market share and sells at a higher price. Case (iii) p1−as < 0,and
p1−(z−a)s<0. Here Þrm Ais losing market areas on both sides. By
raising its price Þrm Awill increase market areas on both sides, and sell to
all customers at the higher price eventually leading to one of the two situa-
tions described above. The second part of Condition 2 consists of symmetric
conditions for Þrm B.
26
ProofofLemma3: Suppose that (p1,p
2) is an equilibrium but |p1−p2|>
(t−s)(l−b−a). Then there are two possibilities. First let p2−as ≥0.
Here the high price Þrm has no market area and will lower its price down to
(at least) the delivered price of the opponent at its own location, ensuring
apositiveproÞt. Hence this case cannot be sustained in equilibrium. Next
let p2−as < 0. Here the high price Þrm has some residual market area. If
this Þrm does not alter its location then a mixed strategy equilibrium (as in
Section 5) is the only possibility. Alternatively, the Þrm can move inwards
to make greater proÞts. Thus there is no pure strategy equilibrium. Finally,
let p2−p1=(
t−s)(l−b−a) where the modulus has been ignored for the
sake of simplicity. If p1=0,thenÞrm Acan gain by charging a price less
than p2+(t−s)(l−b−a). If p1>0, given the general form of our proÞt
function Þrm Acan sell to buyers up to the location of Þrm B.SoÞrm Acan
reduce its price slightly and increase its market share making larger proÞts.
Alternatively if at p1>0andÞrm Awas not selling to all the customers
up to Þrm B’s location, it can increase proÞts by raising prices. Hence for
price equilibrium we need that |p1−p2|<(t−s)(l−b−a).
ProofofProposition3: We know from Lemma 2 and 3 that if any
of the conditions are not satisÞed then a pure strategy equilibrium does not
exist. So assuming Conditions (1) −(3) are satisÞed, we have well behaved
proÞt functions in this range given by Π1=p1z−s
2(2a2+z2−2az)and
Π2=p2(l−z)−s
2(b2+(
l−b)2+z2−2(l−b)z). Assuming Þxed locations
we compute the optimal prices in terms of locations. These are given by
p1=1
2
4s2a+2
ls2+stb −7sat −7slt +2
at2+6
lt2−2t2b
3t−2s
p2=1
2
4s2b+2
ls2+sat −7stb +2
t2b−7slt +6
lt2−2at2
3t−2s
Substituting these in the proÞt function we obtain the optimal locations
stated in the proposition. Note that in order to satisfy a∗≤l
2we need
s>0.32195t. These are substituted back into the price equations to obtain
the equilibrium prices. Recall using optimal locations and prices proÞts are
given by Π∗
1=Π∗
2=l2
64s−96s4+416s3t−609s2t2+312st3−16t4
(3t−2s)2. While it is quite
cumbersome due to the higher order polynomials involved in the proÞtex-
pression, we verify that proÞts are almost always positive through graphical
analysis. The expression for optimal proÞts is positive for all s>0.0575t.
Hence (a∗,b
∗,p
∗
1,p
∗
2) is the unique equilibrium.
27
Figure 1: The Location-Price Game
Figure 2: The Low Prices Case
Figure 3: Mixed Stratgey Equilibrium for a+b=l
Figure 4: Seller Participation Condition violated for Firm A
Figure 5: Profit Function for the Location-Price Game
Figure 6
Figure 6: Profit Function for the Special Price Game
Figure 8
