1

Regional Science and Urban Economics 36 (2006) 790 – 802
www.elsevier.com/locate/regec

Spatial discrimination: Bertrand vs. Cournot with
asymmetric demands ☆
Wen-Jung Liang a , Hong Hwang b,c,⁎, Chao-Cheng Mai a,c
a
Department of Industrial Economics, Tamkang University, Taiwan
b
Department of Economics, National Taiwan University, Taiwan
c
Research Center for Humanities and Social Sciences, Academia Sinica, Taiwan
Received in revised form 24 March 2006; accepted 26 April 2006
Available online 8 August 2006

Abstract

This paper develops a barbell model a la Hwang and Mai [Hwang, H., and C.C. Mai, 1990, Effects of
spatial price discrimination on output, welfare, and location, American Economic Review 80, 567–575.] with
homogeneous product and asymmetric demands to compare prices, aggregate profits and social welfare
between Cournot and Bertrand competition, and to analyze the firms' equilibrium locations. It focuses on the
impacts of the spatial barrier generated from transport costs, and the market size effect resulting from
asymmetric demands. It shows that the market-size effect is crucial in determining firms' locations under
Cournot competition, but insignificant under Bertrand competition. Moreover, the equilibrium price of the
large market and the aggregate profits are lower but the social welfare is higher under Cournot competition
than under Bertrand competition if one of the markets is sufficiently large and the transport cost is high.
© 2006 Elsevier B.V. All rights reserved.

JEL classification: R32; L22
Keywords: Spatial discrimination; Bertrand competition; Cournot competition; Asymmetric demand structure

☆
We are indebted to an associate editor and anonymous referees for inducing us to improve our exposition and for
offering several suggestions leading to improvements in the substance of the paper. The first author would like to thank for
the financial support from National Science Counsil, Taiwan. Part of the work was completed while the second author was
a visiting research professor at the Department of Economics and Finance, the city University of Hong kong in January
2005. He would like to thank the department for the hospitality.
⁎ Corresponding author. Department of Economics, National Taiwan University, 21 Hsu Chow Road, Taipei 10020,
Taiwan. Tel.: +886 2 2351 9641x444; fax: +886 2 2358 4147.
E-mail address: echong@ntu.edu.tw (H. Hwang).

0166-0462/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.regsciurbeco.2006.04.001

W.-J. Liang et al. / Regional Science and Urban Economics 36 (2006) 790–802 791

1. Introduction

Since Bertrand's (1883) criticism of Cournot (1838), economists often debate the relative
merits of Bertrand and Cournot competition. They argue that competition in prices (i.e., Bertrand
competition) is more severe than in quantity (i.e., Cournot competition), and generally leads to
lower prices and profits but higher welfare.1 However, these results may not hold in a spatial
framework. As a specific example, D'Aspremont and Motta (2000), henceforth DM, use a model
a la Hotelling but with fixed locations of firms to analyze whether tougher price competition
might have an overall negative effect on welfare. More specifically, DM show that excessive
competition may not necessarily benefit consumers, as it may cause involuntary exit or
bankruptcies and provoke an undesirable increase in concentration. This increase in concentration
might be very damaging to consumers, and has a detrimental effect on welfare.2 Indeed, such a
danger has been officially recognized by National Recovery Administrations Code of Fair
Competition, as it was for some industries in the United States during the 1930s.
In addition, Anderson and Thisse (1988), henceforth AT, raise an intuitive idea that space acts
as a barrier to competition, allowing firms to build up higher profits. They indicate that this idea is
borne out in Bertrand competition but not in Cournot competition, and can be supported by the
analysis of Hamilton et al. (1989), henceforth HTW, which shows that Bertrand profits increase
whereas Cournot profits decrease with the transport cost when locations are endogenously
determined. Thus, both AT and HTW conclude that tougher competition might result in higher
profits under a high spatial barrier, but could not find striking results on prices and welfare.
Rowthorn (1992), and Haufler and Wooton (1999) emphasize the role of market size on
competition with a trade model. They point out that firms prefer to locate in a large market where
they will be able to charge a higher producer price. As a result, higher trade barriers reduce
competition by reducing the volume of imports in the small market, but the competition in the
large market becomes more severe due to the agglomeration of the firms in this market.
The main purpose of this paper is to use a barbell model a la Hwang and Mai (1990) to show
that prices and profits may become higher but welfare lower in Bertrand than in Cournot
competition if the demand in the two end markets is very asymmetric.
A barbell model can be described as two cities, such as Seattle and Vancouver, connected with
a highway (i.e., I-5). Anticipating either Cournot or Bertrand competition in the commodity
markets, each firm chooses an optimal location to set up its plant, which can be at one of the two
cities or a point on the highway. The barbell model fits the reality well, and can be used to
examine the trade between two countries as well.
It is worth pointing out that our model differs from the models in DM, AT, and HTW in the
following aspects: (i) The DM model takes firms' locations as exogenously given, but the number
of firms in each competition structure is variant. (ii) The AT and HTW models endogenously
1
For example, Singh and Vives (1984) show that in a differentiated duopoly, profits are larger (smaller) in Cournot
than in Bertrand competition if the products are substitutes (complements), and that Bertrand equilibrium is more efficient
than Cournot equilibrium because the consumer surplus and total surplus are higher at the former than at the latter
equilibrium. Vives (1985) also derives that given symmetric demands and unique equilibrium, the prices and profits at
equilibrium are larger and quantity is smaller in Cournot than in Bertrand competition. Cheng (1985) provides some
insights into the results obtained by Singh and Vives (1984) with a geometric analysis. In particular, he concludes that
profits are higher under Cournot than under Bertrand competition only if the demand structure is not sufficiently
asymmetric. Otherwise, one of the firms may earn a lower profit under Cournot than under Bertrand competition.
2
The idea that less intense competition allows for a higher number of firms in the industry, having possible desirable
effects on welfare, has been proposed by Motta (1992). The idea that collusion may lead to lower concentration and lower
profits is treated in Selten (1984). These ideas can be applied to the study if anti-trust laws are suitable.

792 W.-J. Liang et al. / Regional Science and Urban Economics 36 (2006) 790–802

determine the firms' locations, but the demands are symmetric. The spatial barrier which causes
the unusual outcome in the comparison of the Cournot and Bertrand equilibria occurs due to
firms' location choices and transport costs. Nevertheless, the present paper goes beyond the above
papers by showing that the demand asymmetry in the barbell model may generate a market size
effect which contributes to the unconventional outcome when comparing the Cournot and
Bertrand equilibria. Intuitively, when demands are asymmetric, the price and profits in the large
market are higher, creating a market-size effect, thereby attracting firms to move closer to the
large market. If the demand asymmetry is significant so as to make the market size effect striking,
under Cournot the two firms will agglomerate at the large market, serving the two markets
simultaneously. Under Bertrand, each firm will locate at its near market and be active in this
market only. Thus, space acts as a barrier to Bertrand but not to Cournot in our model.3 More
specifically, owing to price undercutting under Bertrand competition, each firm will charge a price
slightly lower than the rival's cost (transport cost plus marginal production cost). Thus, each
firm's excess profits are positively related to the transport cost. The higher the transport cost, the
larger will be the barrier and hence the excess profits. Consequently, the equilibrium price in the
large market will be higher, and the aggregate profits will be larger under Bertrand than under
Cournot competition, if the transport cost is high. Moreover, since the barrier causes a welfare
loss, the social welfare will be higher under Cournot than under Bertrand competition.
Another purpose of this paper is to analyze the firms' location choices under Cournot and Bertrand
types of competition when demands are asymmetric. HTW, and Anderson and Neven (1991) assume
that firms play Cournot quantity competition in Hotelling's linear city model with symmetric demands
and conclude that there is a unique location equilibrium in which the two firms agglomerate at the
center of the line market. Economists studying spatial competition have tried to overturn the central
agglomeration result. Research in this direction includes, Gupta et al. (1997), Sarkar et al. (1997), Pal
(1998), Mayer (2000), Chamorro-Rivas (2000), Matsushima (2001), Yu and Lai (2003), and Gupta et
al. (2004). In particular, Gupta et al. (1997) employ a linear city model in which consumers are
distributed non-uniformly within the line segment, and conclude that a dispersed equilibrium occurs if
the density of consumers around the market center is thin enough. It is noteworthy that the barbell
model used in this paper can be referred to as a non-uniformly distributed model in which the densities
of consumers at points between the two endpoints of the line segment are zero.
The focus of this paper is therefore on the impacts of the market size effect and the spatial barrier. A
two-stage game is employed in which firms choose locations to maximize their profits in the first stage
and compete in either Cournot or Bertrand fashion in the second stage. We shall solve these two cases,
respectively, and then compare the results with those derived by Gupta et al. (1997), DM, AT, and HTW.
The remainder of the paper is organized as follows. Section 2 explores the firms' location
choices under Cournot competition while Section 3 analyzes those under Bertrand competition.
Section 4 compares the equilibrium prices, aggregate profits, and social welfare derived under the
two types of competition. The final section concludes the paper.

2. Cournot competition

Consider a simple framework. There are two firms, denoted as firm A and firm B, who can
locate at any point along a line segment, such as a highway, with unit length. There are two
distinct markets, denoted as markets 1 and 2, located at the two endpoints of the line segment as

3
This means that the traditional barrier caused by cross hauling does not occur under Cournot competition, if the firms
agglomerate at the same point.


Fig. 1. A location line.

shown in Fig. 1.4 The firms sell a homogeneous product with zero production cost to consumers
who reside only in either of the two markets. The locations of firms A and B are assumed to be xA
and xB apart from the left endpoint of the line segment, respectively. Moreover, firm A is assumed
to locate to the left of firm B, i.e., xA ≤ xB.
In order to exhibit the feature of asymmetric demand structure, we assume that the demand
curves in markets 1 and 2 take the following forms:5
q1 ¼ gað1−p1 Þ;
ð1Þ
q2 ¼ að1−p2 Þ:
where qi, and pi are the quantity demanded and delivered price in market i (i = 1, 2); γ and a are
positive constants and γ is a measure of relative market size. Without loss of generality, we
assume throughout the paper that market 1 is larger than market 2 or equivalently γ is greater than
unity. Note that if γ = 1, the two demands become symmetric.
The game in this paper consists of two stages. In the first stage, the two firms simultaneously
select their locations. In the second stage, given the location decisions, the firms simultaneously
choose their quantities (prices) if they play Cournot (Bertrand) competition. The sub-game perfect
equilibrium of the model is solved by backward induction, beginning with the final stage.
Assume in this section that firms play Cournot competition in both markets—each firm
chooses a quantity schedule to maximize its profits given locations xA and xB, and also the rival's
quantity schedule. Profits of each firm are defined as the sum of its profits from markets 1 and 2.
They can be expressed as:

pA ¼ fqA −ð1=gaÞ½ðqA Þ2 þ qA qB Š−txA qA g
1 1 1 1 1
þ fqA −ð1=aÞ½ðqA Þ2 þ qA qB Š−tð1−xA ÞqA g;
2 2 2 2 2 ð2:1Þ

pB ¼ fqB −ð1=gaÞ½ðqB Þ2 þ qA qB Š−txB qB g
1 1 1 1 1
þ fqB −ð1=aÞ½ðqB Þ2 þ qA qB Š−tð1−xB ÞqB g;
2 2 2 2 2 ð2:2Þ

where qij (i = 1, 2, j = A, B) are firm j's sales in market i; and t is constant transport rate.

4
The two end-point markets setting, which we call it a barbell model, was first proposed by Hwang and Mai (1990)
and subsequently employed by Gross and Holahan (2003) etc.
5
Motta and Norman (1996) and Haufler and Wooton (1999) also use these demand curves to exhibit asymmetric
market size.


Maximizing each firm's profits with respect to its sales for the two markets and solving these
first-order conditions, we obtain the following Cournot quantity schedules:

qAC ¼ ðga=3Þð1−2txA þ txB Þ;
1
qAC ¼ ða=3Þ½1−2tð1−xA Þ þ tð1−xB ÞŠ;
2 ð3Þ
qBC ¼ ðga=3Þð1 þ txA −2txB Þ;
1
qBC ¼ ða=3Þ½1 þ tð1−xA Þ−2tð1−xB ÞŠ;
2

where variables with superscript “C” denote that their values are associated with the Cournot
equilibrium. Eq. (3) shows, as expected, that firm j's sales in market i increase as its location
becomes closer to the market.
The resulting delivered prices under Cournot competition can be obtained by substituting Eq.
(3) into Eq. (1):

pC ¼ 1−ð1=3Þð2−txA −txB Þ;
1 ð4Þ
pC ¼ 1−ð1=3Þ½2−tð1−xA Þ−tð1−xB ÞŠ:
2

We now turn to the first-stage problem. Anticipating the Cournot quantity schedules and
delivered prices in the second stage, each firm chooses a location to maximize its profits given the
rival's location. Substituting Eqs. (3) and (4) into Eqs. (2.1) and (2.2), we obtain the first- and
second-order conditions for profit-maximization with respect to each firm's location as follows:

ApA =AxA ¼ ð4t=3Þð−qAC þ qAC Þ;
1 2
A2 pA =Ax2 ¼ ð8a=9Þðg þ 1Þt 2 > 0;
A ð5Þ
ApB =AxB ¼ ð4t=3Þð−qBC þ qBC Þ;
1 2
A2 pB =Ax2
B ¼ ð8a=9Þðg þ 1Þt 2 > 0:

Eq. (5) shows that the profit functions for firms A and B are all strictly convex with respect to
location xj (j = A, B), respectively, which implies a corner solution. The Nash equilibrium location
can be derived by comparing the profits at the two corners, i.e., markets 1 and 2.
Note that given the assumptions of xB ≥ xA, there exist three possible solutions: (xA, xB) = (0, 0),
(0, 1) and (1, 1). However, the last can be ruled out as firm A's profits are necessarily higher to
locate in market 1 than in market 2, should firm B choose to locate in market 2. This is derivable by
the following condition:

pAC ð1; 1Þ−kAC ð0; 1Þ ¼ ð4gat=9Þf½ð1=gÞ−1Š−ðt=gÞg < 0; ð6:1Þ

The solution can therefore be derived by comparing the profits of firm B at the two markets
given firm A to stay in market 1:

pBC ð0; 1Þ−pBC ð0; 0Þ ¼ ð4gat=9Þft−½1−ð1=gÞŠg < ð>Þ0; if 1−ð1−gÞ > ð<Þt: ð6:2Þ

Eq. (6.2) shows that the solution can be either (0, 0) or (0, 1) as it depends on parameter
combinations. The two firms would agglomerate in market 1 (i.e., (xA, xB) = (0, 0)) if 1 − (1 /
γ) > t; they would locate separately at the two opposite endpoints (i.e., (xA, xB) = (0, 1)) if 1 −
(1 / γ) < t.


Intuitively, the optimal locations are determined by two effects—the market size effect and the
competition effect. The former moves firms to the large market while the latter pulls the firms away
from each other. When one market is sufficiently larger than the other market (i.e., a large γ), the
market-size effect dominates the competition effect; both firms locate in the large market.
Contrarily, if the size differential is not so great then firms locate at opposite end points so as to
mitigate competition in both markets. In sum, the asymmetry of demand does have an impact on
firms' location choices when they compete on quantity. We can thus establish:
Proposition 1. If market 1 is sufficiently larger than market 2, the two firms will agglomerate in
market 1 under Cournot competition. Otherwise, firms will locate separately at the opposite
endpoints of the line market.

This result is significantly different from the one derived by Gupta et al. (1997), which
indicates that a dispersed equilibrium occurs if the density of consumers around the market center
is thin enough. In contrast, the barbell model used in our paper can be regarded as a simplified
non-uniformly distributed model in which the densities at the interval between the two endpoints
are nil. Using the barbell model, we have found that the two firms agglomerate at the large market
if one of the markets is sufficiently larger than the other, and located separately at the two
endpoints, otherwise.6

3. Bertrand competition

In this section, we use the framework specified in the previous section to examine the case of
Bertrand competition. As before, the game in question consists of two stages. The two firms
choose locations in the first stage and then select their price schedules in the second stage.
In the second stage, the two firms simultaneously determine their prices piA and piB for market
i. Sales for firm j in markets 1 and 2 are then given by:
8
>0
<
j
if p1 > pk ;
1
j
q1 ¼ ðga=2Þð1−p1 Þ if p1 ¼ pk ; j; k ¼ A; B; jp k;
j j
ð7:1Þ
>
: j j
1
gað1−p1 Þ if p1 < p1 ;
k

8
>0
<
j
if p2 > pk ;
2
j
q2 ¼ ða=2Þð1−p2 Þ
j j
if p2 ¼ pk ; j; k ¼ A; B; jp k; ð7:2Þ
>
: j j
2
að1−p2 Þ if p2 < pk ;
2

Note that the Bertrand game with a homogeneous product is a game with the property of
winner-takes-all due to excessive competition. That is, the firm with a lower cost (marginal
production cost plus transport cost) will undercut the rival's price and takes the whole market.
Given the assumptions of xB ≥ xA and zero marginal production cost, firm A is closer to market 1,
and can use its location advantage to force out its rival from the market. Similarly, firm B has an
edge over firm A in market 2 and will use this edge to price firm A out of the market. Furthermore,
the winner's delivered price in its advantageous market must be slightly lower than the transport

6
As pointed out by a referee, what agglomeration means in this model is different from what it means in the traditional
spatial model where products are differentiated a la Hotelling. In our barbell model, consumers are located at one point
and are served a product at that point.


cost.7 Therefore, for any given xA and xB, the delivered prices of markets 1 and 2 charged by
firms A and B, respectively, under Bertrand competition are:

pA ¼ mA þ txA ¼ txB ;
1 1 ð8Þ
pB ¼ mB þ tð1−xB Þ ¼ tð1−xA Þ;
2 2

where mij denotes firm j's mill price in market i.
We now turn to the first-stage problem with respect to the firms' optimal locations. Making use
of the equilibrium in the second stage, the profit functions of firms A and B can be specified as
follows:

pA ¼ ð pA −txA ÞqA ¼ gatðxB −xA Þð1−txB Þ;
1 1 ð9:1Þ

pB ¼ ½ pB −tð1−xB ÞŠqB ¼ atðxB −xA Þ½1−tð1−xA ÞŠ:
2 2 ð9:2Þ

Totally differentiating Eqs. (9.1) and (9.2) with respect to xj respectively, we can derive the
profit-maximizing conditions for the two firms as follows:

ApA =AxA ¼ −tqA ¼ −gatð1−txB Þ < 0;
1 ð10Þ
ApB =AxB ¼ tqB ¼ at½1−tð1−xA ÞŠ > 0:
2

Eq. (10) shows that firm A will locate in market 1 in which it enjoys a competitive advantage
over firm B, while firm B will locate in market 2 for a similar reason. This outcome is regardless
of market sizes and is in sharp contrast to the one we derived under the Cournot case. The intuition
of the result is straightforward. As discussed before, under Bertrand competition, firm A always
captures the entire market 1 and firm B the entire market 2. Under such circumstances, any
increase in xA will lower firm A's mill price, and therefore resulting in a loss. The best location is
thus market 1. Similarly, increasing xB will move firm B closer to market 2, thereby raising its
profits from market 2. Firm B will thus locate in market 2. Accordingly, each firm will exit from
its distant (and also disadvantageous) market and act as a monopolist in its near (and also
advantageous) market. Therefore, we can establish:
Proposition 2. The two firms always locate at the opposite endpoints of the line segment under
Bertrand competition, regardless of market size.

Not surprisingly, under price competition, the firms never locate in the same market. The
products are identical and so to soften price competition, the Nash equilibrium is for firms to locate
at the opposite endpoints and this holds regardless of difference in market size. Hence, unlike the
Cournot case, demand asymmetry plays no role in plant locations under Bertrand competition. It
also implies that the principle of maximum differentiation applies only to the Bertrand case.

4. Comparison of the equilibria

Since the main purpose of this paper is to explore the role of asymmetric demand structure, we
will focus upon the case where market 1 is sufficiently larger than market 2 (i.e., 1 − (1 / γ) > t)

7
Without loss of generality, we assume that the winners' equilibrium prices are set equal to the transport cost.


throughout the rest of the paper.8 If so, the equilibrium location pair is (xA, xB ) = (0, 0) under
C C

Cournot competition and (xA, xB) = (0, 1) under Bertrand competition, where superscripts “C” and
T T

“T” denote Cournot and Bertrand equilibrium, respectively. Given these location equilibria, we
can then compare the differences in price, profit, and social welfare under the two types of
competition.

4.1. Delivered prices
C C
Substituting the equilibrium location pair (xA, xB ) = (0, 0) into Eq. (4), we can derive the
delivered prices of the two markets under the Cournot case as follows:

pC ð0; 0Þ ¼ 1=3;
1 ð11:1Þ

pC ð0; 0Þ ¼ ð1=3Þð1 þ 2tÞ:
2 ð11:2Þ

Alternatively, substituting (xT , xB) = (0, 1) into Eq. (8), we obtain the optimal delivered prices
A
T

of the two markets under the Bertrand case as follows:

pT ð0; 1Þ ¼ t;
1 ð12:1Þ

pT ð0; 1Þ ¼ t:
2 ð12:2Þ

It is noteworthy that Eqs. (12.1) and (12.2) is effective only if the value of t is no higher than 1/
2, which given (xT , xB) = (0, 1), is the highest (and also the monopoly) price a firm would charge.
A
T

If the transport cost t goes above 1/2, the firm would still set the price at 1/2 to secure the highest
profits. As the equilibrium for t being higher than 1/2 is the same as t equal to 1/2, we shall
assume t to be no higher than 1/2 for the rest of the paper.
Manipulating Eqs. (11.1), (11.2), (12.1) and (12.2), we obtain:

pC −pT ¼ ð1=3Þð1−3tÞ > ð<Þ0 if t < ð>Þ1=3;
1 1 ð13:1Þ

pC −pT ¼ ð1=3Þð1−tÞ > 0:
2 2 ð13:2Þ

Eq. (13.1) and (13.2) indicates that the price at the large market (i.e., market 1) can be higher or
lower, as it depends on the value of t, but the price at the small market (i.e., market 2) is definitely
higher under the Cournot case than the Bertrand case. The economic explanations of the results
are as follows. Under Cournot both firms agglomerate in the large market (i.e., market 1) and are
active in both markets, whereas under Bertrand each firm locates at its near market and is active in
only one market. Under Bertrand each firm sets a price so that the other firm would not find it
profitable to sell in its market—this has a contestable market flavor. This is not the case under
Cournot with large asymmetric demand. Thus, for market 1, the Cournot price is 1/3, regardless of
the transport cost9, while the Bertrand price is slightly below t. It implies that if the transport cost

8
We have proved that the rankings of delivered prices, aggregate profits, and social welfare under the two types of
competition where neither market is sufficiently large (i.e., 1 − (1 / γ) < t), are the same as those derived in AT and HTW.
To save space, we shall not go any further on this case.
9
Since the two firms agglomerate at market 1 and pay no transport cost for the sales to market 1, the Cournot price at
the market is 1/3, regardless of the transport cost.


is larger (smaller) than 1/3, the price at market 1 will be higher (lower) under Bertrand than under
Cournot competition. Moreover, the two firms face less competition (less than Bertrand) and
incur the maximum transport costs for their sales to the small market under the Cournot case.
Thus, the price at market 2 is necessarily higher under Cournot than under Bertrand competition.
Based on the above discussions, we can establish the following proposition.
Proposition 3. The equilibrium price of the large market is lower (higher) under Bertrand than
under Cournot competition, if t < (>) 1/3. However, the equilibrium price of the small market is
definitely lower under Bertrand competition than under Cournot competition.

In a spatial model with homogeneous goods as well as symmetric demand structure, AT and
HTW conclude that the equilibrium prices under Cournot competition are necessarily higher than
those under Bertrand competition. Given firms' locations, DM obtain that the prices might be
higher under Bertrand competition due to entry barrier caused by the reduction of the number of
firms. In contrast, the present paper has shown that Cournot price could be lower than Bertrand
price at the large market owing to spatial barrier as well as demand asymmetry.

4.2. Aggregate profits
C C
Substituting Eq. (3) into Eqs. (2.1) and (2.2) and noting that (xA, xB ) = (0, 0), we can derive the
aggregate Cournot profits from markets 1 and 2, respectively, as follows:

pC ð0; 0Þ ¼ 2ga=9;
1
ð14:1Þ
pC ð0; 0Þ ¼ ð2a=9Þð1−tÞ2 :
2

From Eqs. (12.1) and (12.2) and noting that (xT , xB) = (0, 1), the aggregate Bertrand profits of
A
T

markets 1 and 2 are derivable, respectively, as follows:

pT ð0; 1Þ ¼ gatð1−tÞ;
1
pT ð0; 1Þ ¼ atð1−tÞ:
2 ð14:2Þ

The profit differences between Cournot and Bertrand equilibrium for markets 1 and 2 are then
obtainable by comparing Eqs. (14.1) and (14.2):

pC −pT ¼ ðga=9Þð2−3tÞð1−3tÞ > ð<Þ0 if t < ð>Þ1=3;
1 1 ð15:1Þ

pC −pT ¼ ða=9Þð2−11tÞð1−tÞ > ð<Þ0 if t < ð>Þ2=11:
2 2 ð15:2Þ

We see from Eq. (15.1) that the profits of market 1 is higher under Cournot than under Bertrand
competition, if the transport cost between the two markets is lower than 1/3 (this is the Cournot
price if both firms pay no transport costs). The opposite holds if the condition is reversed. The
aggregate Cournot and Bertrand profits of the two firms from market 1 are depicted in Fig. 2. As
noted in Proposition 3, the price of market 1 is equal to 1/3 when t = 1/3; this holds for either
Cournot or Bertrand competition. Thus, the aggregate profits from market 1 must be the same
under the two types of competition when t = 1/3. Furthermore, a higher t has no effect on the price


Fig. 2. The total profits from market 1 under Cournot and Bertrand competition.

of and also the profits from the market under the Cournot case, but it necessarily raises the price of
and the profits from the market under the Bertrand case. This explains why the aggregate profits
from market 1 are higher under Bertrand than under Cournot competition for t > 1/3. In contrast,
the aggregate profits from market 2 are higher under the Cournot than under the Bertrand
competition, if the transport cost is lower than 2/11, and the ranking is reversed otherwise. This
result can be described by Fig. 3 in terms of profit curves. As shown in the figure, the Bertrand
profits of market 2 increase when the transport cost rises, reaching their maximum level when
t = 1/2. As for the corresponding Cournot profits, they fall when the transport cost rises.
The intuition of the above results is simple. We have shown that there exists a spatial barrier
under Bertrand competition, which generates excess profits. The higher the transport cost is, the
stronger the barrier and therefore the higher the profits. As a result, when the transport cost is
high, say higher than 1/3 for market 1 and 2/11 for market 2, the aggregate profits are larger
under the Bertrand than under the Cournot competition. We can thus establish the following
proposition:

Fig. 3. The total profits from market 2 under Cournot and Bertrand competition.


Proposition 4. The aggregate profits are higher under Cournot than under Bertrand competition
if the transport cost is low. The opposite holds if the transport cost is high.

4.3. Social welfare

The social welfare is defined as the sum of the consumer and producer surplus from markets 1
and 2. The welfare difference between the Cournot and Bertrand equilibria is derivable as follows:

SWC ð0; 0Þ−SWT ð0; 1Þ ¼ ða=18Þfg½ð−1 þ 3tÞð1 þ 3tÞŠ−½ð1−tÞð1 þ 17tÞŠg: ð16Þ

We see from Eq. (16) that the welfare difference is negative if the transport cost is less than 1/3,
but is positive if the transport cost is greater than 1/3 and the relative market size ratio γ is
sufficiently large.10 This result can be explained as follows. First of all, the equilibrium price is
always higher under Cournot than under Bertrand competition in market 2; hence the welfare in
market 2 is higher under Bertrand competition. Moreover, the equilibrium price in market 1 is
higher (lower) and hence the welfare is lower (higher) under Cournot than under Bertrand
competition, if the transport cost is less (greater) than 1/3. Consequently, when the transport cost
is greater than 1/3 and the market size of market 1 is sufficiently large, the welfare gain from
market 1 can outweigh the welfare loss from market 2 under the Cournot competition and make it
more socially desirable than the Bertrand competition.11
Intuitively, the spatial barrier, which leads to a welfare loss, arises under Bertrand competition
only, and the larger the transport cost, the higher the barrier. As the transport cost becomes large,
say t > 1/3, the welfare in market 1 is lower under Bertrand than under Cournot competition.
Consequently, when market 1 is sufficiently large, the welfare loss in market 1 prevails. As a
result, the social welfare is higher under Cournot than under Bertrand competition. Accordingly,
we have:
Proposition 5. The social welfare is higher under Cournot than under Bertrand competition, if
market 1 is sufficiently large and t > 1/3.

This result is significantly different from the one obtained by AT and HTW in which the social
welfare is higher under Bertrand than under Cournot competition. It is also interesting to note that
DM obtain a similar result to ours, but with a quite different rationale. In their paper, the
unconventional welfare ranking between the two types of competition is caused by entry barrier
through the reduction of the number of firms, which is different from ours.
Proposition 5 has the following policy implication. According to the proposition, making
markets more competitive, such as from Cournot to Bertrand competition, will not necessa-
rily benefit consumers once we allow for the impact of competition on location choices and
take into account the role of market size. Thus, tougher anti-trust law and anti-trust enforce-
ment might provoke an undesirable increase in spatial barrier, causing a detrimental effect on
welfare.

10
Since (1 − t)(1 + 17t) > 0 and γ [(− 1 + 3t)(1 + 3t)] < (>) 0 if t < (>) 1/3, the welfare difference is positive if t > 1/3 and γ
is sufficiently large.
11
For purpose of illustration, two numerical cases are provided. Let us assume γ = 5 for one case and γ = 10 for the other
and a = 1 for both cases. We then figure out that for the case of γ = 5, the welfare difference as defined in Eq. (16) is
positive if t ≥ 0.472. For the other case of γ = 10, it is positive if t ≥ 0.4074.


5. Concluding remarks

This paper has developed a barbell model with homogeneous products and asymmetric
demands to compare the delivered prices, aggregate profits, and social welfare between Bertrand
and Cournot equilibrium, and to analyze the firms' equilibrium locations. It focuses on the
impacts of the spatial barrier generated from transport costs and the market size effect of
asymmetric demands.
We have obtained several striking results. First of all, we have found that market asymmetry is
crucial in determining the equilibrium locations under Cournot competition, but insignificant
under Bertrand competition. In the case of Cournot competition, we have shown that the two
firms separately locate at the opposite endpoints of the line market if the demand is asymmetric
with neither market sufficiently large, but they agglomerate at the large market otherwise. This
result is different from the one derived by Gupta et al. (1997). They assume that consumers are
non-uniformly distributed along a line market and find that the two firms locate separately if the
density of consumers is thin enough around the center of the line market under Cournot
competition. Secondly, AT and HTW conclude that the equilibrium price is always lower under
Bertrand than under Cournot competition, while we have shown that if one market is sufficiently
larger than the other and the transport cost is high, the Bertrand equilibrium price in the large
market may be higher than the Cournot equilibrium price. Thirdly, AT and HTW conclude that
Bertrand equilibrium is more efficient than Cournot equilibrium. However, we have found that
the reverse holds if the demand structure is sufficiently asymmetric and the transport cost is high.
Lastly, although the effects on the delivered prices, aggregate profits, and social welfare derived
by DM are similar to those in our paper, their causes are quite different. Specifically, DM attribute
it to the entry barrier generated from the reduction of the number of firms, while we attribute it to
the spatial barriers caused by transport costs and asymmetric demands.
What underlies the main results in this paper, and specifically the comparison of the Cournot
and Bertrand outcomes, is that for cases considered, we have demonstrated that under Cournot
both firms are active in both markets, whereas under Bertrand each firm is active in only one
market. Under Bertrand each firm sets a price so that the other firm would not find it profitable to
sell in its market—this has a contestable market flavor and may lead to a low welfare level if the
other firm is relatively inefficient due to, for example, a high transport cost. This is not the case
under Cournot with large asymmetric demand. But in this case, having two firms located in one
market and ship their goods to the other may not be the most efficient way to organize production.
In sum, this paper has shown that, first, it is not necessarily the case that the more competitive
Bertrand environment will generate lower prices than the more accommodating Cournot
environment and that, second, making markets more competitive will not necessarily benefit
consumers once we allow for the impact of competition on location choices and take into account
the role of market size. These findings may be useful for anti-trust authorities when making
related policy recommendations.

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