DISCUSSION PAPER SERIES
2008-01
Cournot competition in a two-dimensional city
Takanori Ago

October 10, 2008
Revised: October 31, 2008

Discussion Papers can be downloaded:
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Cournot competition in a two-dimensional city
Takanori Ago
October 31, 2008

Abstract
This paper analyzes a spatial Cournot competition model in a twodimensional rectangular city. Consequently, there exists a unique spatial
equilibrium such that both …rms agglomerate in the center of the city withsu¢ ciently low transport costs.

Keywords: spatial Cournot competition, two-dimensional city, central agglomeration
JEL Classi…cation: L13; L15; R39

1Introduction

Hotelling’s (1929) seminal work showed that duopolistic …rms agglomerate inthe center of a one-dimensional space (a linear city). Hamilton et al. (1989)
and Anderson and Neven (1991) developed location-then-quantity (Cournot)
Faculty of Regional Policy, Takasaki City University of Economics, 1300 Kaminamie,
Takasaki, Gumma 370-0801, Japan. E-mail: ago@tcue.ac.jp

1

competition models rather than Bertrand ones. They then showed the agglomeration of …rms in the center, while spatial Bertrand competition shows noagglomeration (see, e.g., d’Aspremont et al. (1979)).
This paper extends such a spatial Cournot competition model to a twodimensional rectangular city.1 Maldonado et al. (2005) have already shownthat both …rms agglomerate in the center when the space is a disk (a circularcity). However, such a circular city is di¢ cult to interpret when we consider
the space to be a characteristic one instead of a geographical one. Further,because the ratio of the length to the breadth of our rectangle can vary, ourmodel contains more types of spaces. That is why the analysis of a rectangularcity has merit.
We consider a location-then-quantity competition game involving duopolists;
then, we show the same result of central agglomeration with su¢ ciently lowtransport costs. In other words, the central agglomeration is robust in a spatial
Cournot competition for a wide range of spaces.

2The model

We consider a city expressed by a rectangle, L = f(x; y) 2 R2 :lx =2; ly =2yly =2g, where lx and ly are constants (lxlx =2xly ) and consumers

are uniformly and continuously distributed with a density of one at each location
on L. The total mass of the consumers is normalized to one; thus, we can rewrite
1 With regard to spatial Bertrand competition models in a multiple dimensional space, see,e.g., Tabuchi (1994) and Irmen and Thisse (1998).

2lx = l and ly = 1=l (l1). There are two …rms (…rm 1 and …rm 2) that supply ahomogeneous good with zero marginal costs and engage in Cournot competition.
Each consumer has the same inverse demand function as follows:

P =abQ;Q = q1 + q2 ;(1)

where P is the price, qi (i = 1; 2) is …rm i’s supply amount and a; b are parameters.
Based on the literature, subgame perfection is adopted as the equilibrium
concept, and we consider a two-stage location-then-quantity game. We assume
that the …rms bear transport costs and they can set a supply amount for eachlocation independently because arbitrage between consumers is assumed to beprohibitively costly. Further, the transport costs are the same for the …rmsand are linear to the supply amount. The unit transport cost for …rm i is only
dependent on the Euclidean distance di to a consumer and is quadratic withregard to the distance. Hence, the cost function is given by td2i , where t is atransport cost parameter and is assumed to be su¢ ciently low such that

1a > 2t lx2 + ly2 = 2t l2 + 2l;(2)which ensures that both …rms serve the entire city, irrespective of the locationsof the …rms.2 Let …rm i’s location be (xi ; yi ) 2 L; then, the distance between a
2 When the …rms are diagonally located at the vertexes (e. g., (x ; y ) = (l =2; l =2) andxy1 1(x2 ; y2 ) = ( lx =2; ly =2)), in which the distance between …rms is maximized, a …rm must
serve at a location of the rival …rm. This condition ensures such a positive supply with the3consumer at (x; y) 2 L and …rm i is
2di (x; y) = [(xix) + (yi2y) ]1=2 :First, we analyze the second-stage game by backward induction. From (1),the local pro…t for …rm i at point (x; y) isi (x; y) = qi (x; y)P (x; y)tdi (x; y)2(3)By solving the …rst-order conditions: @ i (x; y)=@qi (x; y) = 0, we have the equilibrium quantity for …rm i at (x; y) as follows:qi (x; y) =a2tdi (x; y)2 + tdj (x; y)23b

for i; j 2 f1; 2g; i 6= j:(4)Then, the equilibrium local pro…t is

2i (x; y) = b [qi (x; y)]for i; j 2 f1; 2g; i 6= j;where qi (x; y) is de…ned by (4). Hence, the total pro…t for …rm i is given byi (xi ; yi ) =ZZ

equilibrium quantity in (4).

4i (x; y)dxdy:
L(5)

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The result: location equilibrium

We analyze the …rst-stage game given the results in the second stage. We
propose the three lemmas below.
Lemma 1 The central agglomeration (x1 = y1 = x2 = y2 = 0) is a Nashequilibrium.
Proof. We assume that …rm 2 is located at the center (x2 = y2 = 0). Then,
we have27bl2[t1 (0; 0)

1 (x1 ; y1 )]=12al2 x21 + y12t 12l2 x4 + x21 1 + 5l4 + 24l2 y12 + y12 5 + l4 + 12l2 y12>tx21 23 + 19l412l2 x21

tx21 23 + 19l43l424l2 y12 + ty12 19 + 23l4

6 + ty12 19 + 23l4

12l2 y12

3 >0

for all x1 6= 0; y1 6= 0, where the …rst inequality is due to (2) and the …rst
inequality is due to 0 < x21l2 =4 and 0 < y12

1=4l2 . Hence, x1 = y1 = 0

is the unique best response to x2 = y2 = 0. Because of the symmetry withrespect to the …rms, it is clear that x2 = y2 = 0 is the unique best response tox1 = y1 = 0.

Lemma 2 It cannot be a Nash equilibrium unless at least one …rm locates onan axis.

5Proof. Without loss of generality, we assume that 0 < x2
1=2l and x21 + y121(x1 ; y1 ) >x22 + y22 . First, when x1 > 0 or y1 > 0, we have either
1 (x1 ; y1 ) or1 (x1 ;y1 ) >when x1 < 0 and y1 < 0, we can show thatif y1

x1 =l2 andl=2, 0 < y2

1 (x1 ; 0) >1 (x1 ; y1 ) for all x2 ; y2 . Second,1 (0; y1 ) >1 (x1 ; y1 ) for all x2 ; y2

1 (x1 ; y1 ) for all x2 ; y2 if otherwise.

Hence, thecase of x1 6= 0; y1 6= 0; x2 6= 0; y2 6= 0 cannot be an equilibrium.

Lemma 3 It cannot be an equilibrium in which a …rm locates on an axis except
for the central agglomeration.
Proof. We assume (without loss of generality) that …rm 2 is on an axis (x2 =0 or y2 = 0). First, when x2 = 0, we have

1 (0; y1 )>

1 (x1 ; y1 ) for allx1 6= 0; y1 ; y2 . Second, when y2 = 0, we obtain

1 (x1 ; 0) >1 (x1 ; y1 ) for all

x1 ; y1 6= 0; x2 . These facts have shown that it cannot be an equilibrium unless
both …rms are located on the same axis. Next, we consider such cases.
Consider that x1 = x2 = 0. We assume (without loss of generality) thatjy1 j

jy2 j. Then, we have

1 (0; 0) >1 (0; y1 ) for all y1 6= 0, y2 . Thus, there isan incentive for a …rm that is nearer to the center to move into the center, whichshows the case that no …rm is located at the center cannot be an equilibrium.
Further, when a …rm (without loss of generality, …rm 2) is located at the center,
we readily have

1 (0; 0) >1 (0; y1 ) for all y1

6= 0. Hence, it cannot be anequilibrium unless both …rms are located at the center when x1 = x2 = 0.
When y1 = y2 = 0, a similar calculation shows that it cannot be an equilibrium unless both …rms are located at the center. Thus, we have the lemma.
6Clearly, Lemma 1, Lemma 2 and Lemma 3 have established our main result
as follows.
Proposition 1 The central agglomeration is the unique Nash location equilibrium.

4Conclusion

We have shown that the central agglomeration result is as robust in a rectangularspace as in a linear one or a circular one. This suggests that …rms have astronger incentive to reduce transport costs by establishing a central locationthat provides better access to consumers than to relax competition by locational
dispersion in a spatial Cournot competition.

References[1] Anderson, S. P. and D. J. Neven, 1991, Cournot competition yields spatialagglomeration, International Economic Review 32, 793-808.
[2] D’Aspremont, C., J. Jaskold-Gabszewicz and J.-F. Thisse, 1979, On
Hotelling’s Stability In Competition, Econometrica 47, 1045-1050.
[3] Hamilton, J. H., J.-F. Thisse and A. Weskamp, 1989, Spatial discrimination:
Bertrand vs. Cournot in a model of location choice, Regional Science andUrban Economics 19, 87-102.
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[4] Hotelling, H., 1929, Stability in competition, Economic Journal 39, 41-57.
[5] Irmen, A. and J.-F. Thisse, 1998, Competition in Multi-characteristics
Spaces: Hotelling Was Almost Right, Journal of Economic Theory 78, 76102.
[6] Maldonado, M. I. B., S. C. Valverde and M. Á. Escalona, 2005, Cournot
competition in a two dimensional circular city, The Manchester School 73,
40-49.
[7] Tabuchi, T., 1994, Two-stage two-dimensional spatial competition between
two …rms, Regional Science and Urban Economics 24, 207–227.

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