Equilibria for circular spatial Cournot markets
Daisuke Shimizu

Toshihiro Matsumura

Graduate School of Economics, University of Tokyo

Institute of Social Science, University of Tokyo

Abstract
This paper investigates a location−quantity model in a circular city. Pal (1998) investigates a
duopoly model and finds that an equidistant location pattern appears in equilibrium.
Matsushima (2001a) investigates an n−firm oligopoly model and shows that, if the number offirms is even, another equilibrium exists where half of the firms agglomerate at one point andthe other firms agglomerate at the opposite point. We find that there exist many otherequilibrium patterns that include the above two patterns as special cases.

We are grateful to participants of the seminars at Kyoto University, Kobe University of Commerce, and the University of Tokyo
for their helpful comments and suggestions. Needless to say, we are responsible for any remaining errors. We also gratefully
acknowledge financial supports from Grant−in−Aid of the Japanese Ministry of Education, Science and Culture.
Citation: Shimizu, Daisuke and Toshihiro Matsumura, (2003) "Equilibria for circular spatial Cournot markets." Economics
Bulletin, Vol. 18, No. 1 pp. 1−9
Submitted: April 4, 2003. Accepted: August 7, 2003.
URL: http://www.economicsbulletin.com/2003/volume18/EB−03R30001A.pdf

1. Introduction
Since the seminal work of Hotelling (1929), a rich and diverse literature on spatialcompetition has emerged. Location models fall into two categories: those in which ﬁrms
bear the transport costs are shipping or spatial price discrimination models; those inwhich consumers pay for transport are shopping or mill pricing models. In each type, one
can have either Bertrand-type price setting or Cournot-type quantity-setting.
Most papers on location theory use shopping (mill pricing) models with Bertrand
competition. Although Cournot-type and Bertrand-type non-spatial models are equally
popular, the body of literature on spatial competition that uses Cournot-type models
is relatively small. Economists have recently considered shipping models with Cournot
competition. Hamilton, Thisse, and Weskamp (1989) and Anderson and Neven (1991)
carry out pioneering works on these models. They use linear-city models and show that
all ﬁrms agglomerate at the central point. Pal (1998) shows that their result is cruciallydependent on the assumption of the linear-city. He investigates a circular city duopoly
model and ﬁnds that an equidistant location pattern appears in equilibrium, that is,
locational dispersion appears.1 Matsushima (2001a) extends Pal’s model to an n-ﬁrm
oligopoly and shows that another equilibrium exists where half of the ﬁrms locate at
one point and the other half locate at the opposite side. However, Matsushima (2001a)does not discuss the possibility of other types of equilibria, and says nothing when thenumber of ﬁrms is odd. We take a close look at the shipping spatial Cournot model withcircular market discussed by them. We ﬁnd that many other location patterns appear inequilibria, including those of Pal (1998) and Matsushima (2001a) as special cases. We
show that the location-quantity model with a circular city contains very rich locational
implications.
The paper is organized as follows. Section 2 formulates the model. Section 3 presents
equilibrium outcomes. Section 4 concludes the paper. All proofs of Lemmas and Propositions are presented in Appendix.
2. ModelThe model is from Pal (1998) except for the number of the ﬁrms. We extend a duopoly
model of Pal to an n-ﬁrm oligopoly model.
There is a circular market of length 1 where inﬁnitely many consumers lie uniformly.
The ﬁrms engage in the following location-quantity competition. In the ﬁrst stage, each1

For other circular Cournot models, see, e.g., Shimizu (2002). For applications of this model, see,e.g., Matsushima (2001b).

1

ﬁrm simultaneously decides where on the perimeter to locate. After observing the rivals’
locations, in the second stage each ﬁrm simultaneously chooses its output level at every
point (market) in the continuum [0, 1] as to maximize its proﬁt. Let x be a point on
the circle located at a distance from 0, measured clockwise. Assume that the demand
function at each market is linear, i.e., p(x) = A − B Q(x), where A and B are positive
constants and p(x) and Q(x) are the price and the total quantity supplied at market x,
respectively. Each ﬁrm incurs a symmetric constant marginal cost of production, which
we normalize to zero without loss of generality. The ﬁrms must pay transport costs. To
ship a unit of the product from its plant xi to a market at point x, ﬁrm i pays a transport
cost t|x − xi |, where t is a positive constant and |x − xi | is the distance between x andxi . The norm signiﬁes the shorter distance of the two possible ways to transfer the goods
along the perimeter. The consumers are assumed to have a prohibitively costly transportcost, preventing arbitrage.2 Finally, A > nt is assumed in order to ensure that every ﬁrm
serves the whole market. All the above assumptions are standard in the literature.
The equilibrium concept used is subgame perfect Nash equilibrium. Thus, we solve thegame by backward induction. Because constant marginal cost of production is assumed,
each local market can be analyzed independently. Thus the second-stage subgames andthe local Cournot competition are examined ﬁrst. Let the locations of the ﬁrms be denoted
x1 , x2 , . . . , xn respectively, with xi ∈ [0, 1], i = {1, . . . , n}. Also, let qi(x) be outputs
for ﬁrm i at market x and q−i(x) be the total output for ﬁrms other than i. Under theabove assumptions, at each point x ∈ [0, 1] ﬁrm i makes proﬁt given by
πi (qi (x), q−i (x), x) = [A − Bqi (x) −Bqj (x) − t|x − xi |]qi (x).

j=iTaking ﬁrst order conditions to solve for the unique Cournot equilibrium yields,qi∗ (x) =




1
A+
t|x − xj | − nt|x − xi | ,B(n + 1)
j=i

(1)

n



1
nA −
t|x − xj | ,B(n + 1)j=1

(2)

Q∗ (x) =and the proﬁt function can be rewritten asπi∗ (x) =

2

1
[A
+
t|x
−
x
|
−
nt|x
−
x
|
= Bqi∗ (x)2 .
ji
B(n + 1)2
j=i

2

(3)

This assumption is not essential. Unless transport costs for consumers are strictly smaller than thoseof firms, consumer arbitrage plays no role in our model. For this discussion, see Hamilton, Thisse, andWeskamp (1989).

2Each ﬁrm’s total proﬁt function Πi is,Πi (xi , x−i ) =x∈[0,1]πi∗ (x; xi , x−i )dx,

(4)and each ﬁrm chooses its location to maximize it. Let Xi denote the set of markets suchthat the derivative of |x − xi | with respect to xi is non-positive. If xi ∈ [0, 1/2], thenXi = [xi , xi + 1/2]. From (1), (3), and (4) we obtain





1

2nt
∂q ∗
∂Πi
= 2B
qi∗ i dx =(A +t|x − xj | − nt|x − xi |)dx
2
∂xi
∂xi
B(n + 1) x∈Xi
0
j=i

−



x∈X
/ i



(A +



t|x − xj | − nt|x − xi |)dx

j=i



2

=

2nt
B(n + 1)2where we usex∈Xi j=ix∈Xi|x − xj |dx −|x − xi |dx =







x∈X
/ i j=i|x − xj |dx ,(5)1|x − xi |dx = .
8
x∈X
/ i

3. Equilibrium analysis
We now look at the ﬁrst stage using the solutions derived above. First, we show thatthe outcome where all ﬁrms agglomerate at one point is not an equilibrium.
Proposition 1: x1 = x2 =, . . . , = xn is not an equilibrium outcome.
As is emphasized by Pal (1998), agglomeration at one point never appears in equilibriumin the circular-city model.
We then show that various types of equilibria exist in the game above. Before theactual analysis, let us introduce two related terms. The ﬁrst is “opposite”, which identiﬁes
a ﬁrm that are located exactly 1/2 away from the original ﬁrm. Hence, a ﬁrm is locatedas far away as possible from its opposite. The second is a “pair”, which signify two ﬁrms
that are opposites to each other. Therefore, the result in Pal (1998) shows that a pair isthe unique equilibrium of the two ﬁrm game. Note that even if there are multiple ﬁrms
located opposite to a ﬁrm, only one of them can be considered as a part of a pair at any
one time. Therefore, if ﬁrm 1 locates at 0 and ﬁrms 2 and 3 locate at 1/2, even thoughboth ﬁrms 2 and 3 can form a pair with ﬁrm 1, only one of them can be a part of a pairat a time.
3

Before discussing equilibria, we present one important lemma, from which two of ourmain results (Propositions 2 and 3) are derived straightforwardly.
Lemma 1: Suppose that ﬁrm g and ﬁrm h constitute a pair (i.e., |xg − xh | = 1/2), thenremoving this pair will not alter the location incentives for the remaining ﬁrms, regarding
where to locate, from that before the removal. Similarly, adding a pair will not aﬀect thelocation incentives for the original ﬁrms.
The following Propositions 2 and 3 are derived straightforwardly from Lemma 1.
Proposition 2: If a situation is such that all ﬁrms can be paired at one time, then thesituation is an equilibrium.
Proposition 3: Suppose that the number of ﬁrms is 2m+1, where m is a positive integer.
Then the situation (and others diﬀering by symmetry) where m + 1 ﬁrms locate at 0 andm ﬁrms locate at 1/2 is an equilibrium.
Note that the equilibrium location patterns described in Proposition 2 include both of
the results of Pal (1998) and Matsushima (2001a) as special cases.
We now discuss another type of equilibria where agglomeration appears at several
points. Proposition 2 implies that the following location pattern constitutes an equilibrium: There are even number of locations that splits the whole market into equidistantparts; at each location, some ﬁrms are located, where the number of ﬁrms at each location
is equal. Then a natural question arises: is it true when the number of locations is odd?
The answer is yes if and only if the number of the location is no larger than the numberof ﬁrms at each location.
Proposition 4: Let k be the number of potential locations, where the circular market isdivided up equidistantly apart by the locations. Let m be the number of ﬁrms locating at
each of these above locations. (i) Let k be even. Then this situation is an equilibrium forany k ≥ 2 and m ≥ 1; (ii) Let k be odd. Then this situation is an equilibrium if and only
if k ≥ m.
Note that this result implies that, for any number of ﬁrms (even or odd), the situation
where ﬁrms separate themselves equidistantly apart to the closest neighbors is an equilibrium outcome. Thus, the conjecture of the existence of this type of equilibrium in the
n-ﬁrm model of Pal (1998) is correct. Propositions 2–4 also indicate that many otherequilibrium location patterns exist.
We explain the intuition behind Proposition 4 (ii). Consider the situation where ﬁrms
agglomerate at three points. The situation where m ﬁrms locate at 0, m ﬁrms locate at
1/3, and m ﬁrms locate at 2/3 constitutes an equilibrium only if m ≤ 3. Suppose that4

initially ﬁrm 1 locates at 0, and it relocates from 0 to 1/2. The relocation increases theﬁrm’s distance from 0 by 1/2, and it mitigates competition between ﬁrms locating at 0.
At the same time the relocation decreases the distance from 1/3 and 2/3 by 1/6, and it
accelerates competition between ﬁrms locating at 1/3 and 2/3. After the relocation thenumber of ﬁrms locating at 0 is m − 1 and the number of ﬁrms locating at 1/3 or 2/3 is
2m. The former competition restricting eﬀect becomes relatively strong when m is large.
Thus, each ﬁrm has an incentive for relocating if m is large.
4. Concluding remarks
In this paper we reexamine the claim in Pal (1998) that the equilibrium outcome of
the circular Cournot game is when the ﬁrms are located symmetrically apart (therefore
no agglomeration). We show that such an equilibrium always exists regardless of thenumber of ﬁrms. We also ﬁnd that many other equilibria exist. We show that three typesof equilibrium structures exist, where any of them can have agglomeration at multiple
locations.
Finally, we make a remark on the applicability of the shipping spatial model. The
most natural interpretation of the model is that each ﬁrm chooses where it builds a plantin the model. There is another important interpretation. We can interpret “space” asproduct varieties. Each ﬁrm’s location indicates the product or sector in which it has
an advantage. Distant locations are the products the ﬁrm is in a disadvantage and to
produce them it incurs additional costs. In short, the location choice corresponds tothe technology choice and transportation costs correspond to the additional production
costs. Hence, the shipping model is a suitable analytical tool for both spatial and nonspatial competitions.3 Following this interpretation, our results indicate the existenceof various equilibrium patterns of technological choice. In other words, our model can
explain various partial ‘herd behavior’ of ﬁrms. It is never the case that all ﬁrms choose
the technology, while it is possible that some of them choose the same one. Such partial‘herd behavior’ is widely observed in many industries. Our model can explain such asituation without assuming any informational externality or network externality, which isassumed in standard models of herd behavior.

3

See Eaton and Schmitt (1994), and Norman and Thisse (1999).

5

Appendix
Proof of Proposition 1: We prove it by contradiction. Suppose that all ﬁrms agglomerate at one point in an equilibrium. Without loss of generality we assume that all ﬁrms
locate at zero. We now show that, if ﬁrm i deviates from the strategy above and choosesxi ∈ (0, 1/2), then its proﬁt increases. Substituting xj = 0 ∀ j = i into (5) we obtainthat, for all xi ∈ (0, 12 ),







2nt2 
2nt∂Πi=|x−0|dx−|x−0|dx=(xi −x2i ) > 0. (6)2∂xiB(n + 1) x∈Xi j=iB(n + 1)2
x∈X
/ i j=i(6) implies the deviation increases its proﬁt, a contradiction.
Proof of Lemma 1: Consider an m-ﬁrm game. From (5) we obtain







2mt2 
∂Πi
=
|x
−
x
|dx
−
|x
−
x
|dx
.
j
j
∂xi
B(m + 1)2 x∈Xi j=ix∈X
/ i j=i(7)

We then add two ﬁrms (ﬁrm m + 1 and ﬁrm m + 2) to the m-ﬁrm game above. Supposethat |xm+1 − xm+2 | = 1/2 and x1 , x2 , ..., xm are the same as the m-ﬁrm game. From (5)we obtain







∂Πi
2(m + 2)t2 
=
|x
−
x
|dx
−
|x
−
x
|dx
.
j
j
∂xi
B(m + 3)2 x∈Xi j=ix∈X
/ i j=i(8)

Obviously the sign of (7) is equal to that of (8). It implies that adding a pair does not
aﬀect the derivative of each ﬁrm’s proﬁt with respect to its own location; thus adding apair does not aﬀect the optimal location for ﬁrm i (i = 1, 2, ..., m). A similar principle
can be applied when considering removing a pair.
Proof of Proposition 2: A ﬁrm looking to optimize faces one ﬁrm to its opposite andother ﬁrms being paired. However, using Lemma 1 repeatedly, the situation after removing
all pairs must have the same solution as before as to whether it is an equilibrium or not.
Thus the ﬁrm needs to worry only about itself and the opposite ﬁrm. As shown in Pal(1998), the unique best response of a ﬁrm in a two ﬁrm game is to locate at the oppositeof the other ﬁrm. Therefore the original location is optimal for the ﬁrm. Since every ﬁrm
in this situation can apply the same process, no ﬁrm has an incentive to deviate and thisis an equilibrium. To reiterate, the “pairs” equilibria include as special cases situations
with even number of ﬁrms locating equidistantly apart.
Proof of Proposition 3: From Lemma 1, removing pairs has no eﬀect on the location
6

incentives for the remaining ﬁrms. Since removing all pairs create a situation where there
is only one ﬁrm, this ﬁrm can locate anywhere. Thus the ﬁrm can choose to locate at 0,
as every point is a best response. The same idea can be used for all m + 1 ﬁrms locatedat 0 to justify their locating at 0.
As for the ﬁrms at 1/2, we consider a situation after removing all but one pair fromthe original situation. Thus there are two ﬁrms at 0, and one at 1/2. The best response
for a ﬁrm given two ﬁrms at 0 is indeed to locate at 1/2. Thus the ﬁrms at 1/2 arewilling to locate at 1/2 in the original situation, also. Therefore the given situation is anequilibrium.
Proof of Proposition 4: Suppose that k is even. Then Proposition 2 implies Proposition
4 (i). We now suppose that k is odd.
Suppose that ﬁrms are located so that there are m ﬁrms each at 0, 1/k, 2/k, . . . ,and (k − 1)/k. Then we discuss whether or not x1 = 0 is optimal for ﬁrm 1 givenx2 =, ..., xm = 0, xm+1 = xm+2 =, ..., = x2m = 1/k, ..., x(k−1)m+1 =, ..., = xn = (k − 1)/k.
Without loss of generality we assume that x1 ≤ 1/2. Substituting x1 = 0 (i.e., substituting
Xi = [0, 1/2]) into (5), we obtain that the derivative is zero. Thus the ﬁrst order conditionis satisﬁed when x1 = 0. We then discuss the second order condition.4 In (5), only Xi
depends on xi . Diﬀerentiating (5) with respect to xi yields

4nt2 
1∂ 2 Πi
−
x
=
|x
+
|
−
|x
−
x
|
i
j
i
j .
∂x2i
B(n + 1)2 j=i
2
j=i

(9)

We now substitute i = 1, x1 = x2 =, ..., = xm = 0, xm+1 = xm+2 =, ..., = x2m = 1/k, ...,and x(k−1)m+1 =, ..., = xn = (k − 1)/k into (9). Since |0 − h/k| = |0 − (k − h)/k| = h/kfor h ≤ (k − 1)/2, we have

j=i

Since

|0 − xj | =k −1
2m(1 + 2 + 3+, ...., +
).
k
2

1 h
1 k−h
1
k−1
1
| − |=| −
|=
+(
− h)
2 k
2
k
2k
2
k

4

We can prove Proposition 4 without using local maximization conditions (the first order and thesecond order conditions) by showing that given other locations of other firms the profit of firm 1 is nonincreasing in x1 for x1 ∈ [0, 1/2] and strictly decreasing in x1 for x1 ∈ (0, 1/2). We avoid this alternative
proof because it requires tedious calculations and takes much space. This proof is available from theauthors upon request.

7and |1/2 − 0| = 1/2, we havek − 2 m − 1
2m  1 3+ +, ..., +.
| − xj | =
+
2
k 2 2
2
2
j=i

 1

Substituting these equations into (9), we have that the second order condition issatisﬁed if and only if−

2m k − 1 m − 1
m
m−1+≤ 0 ⇐⇒
≥
,
k
4
2
k
k−1

and it is satisﬁed if and only if k ≥ m.

8

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