DISCUSSION PAPER SERIES
2009-02
Mixed oligopoly in a two-dimensional city
Takanori Ago

November 6, 2009

Discussion Papers can be downloaded:
http://www1.tcue.ac.jp/home1/c-gakkai/dp/dp09-02

Mixed oligopoly in a two-dimensional cityy
Takanori Agoz
November 6, 2009

Abstract
This paper analyzes a mixed oligopoly model with one public …rmand two private …rms in a two-dimensional square city. Three types ofspatial equilibria are presented: (i, central symmetry) the public …rm
locates at the center, the private …rms locate equidistantly away fromthe center on a diagonal in the opposite direction each other; (ii, full
agglomeration) the public …rm locates slightly away from the center ona diagonal and two private …rms agglomerate on the opposite half ofthe diagonal; (iii, partial agglomeration) in one dimension the locationsof private …rms coincide but di¤er from that of the public …rm, andwith regard to the other dimension the public …rm locates at the centerand the private …rms locate equidistantly away from the center in the
opposite direction each other.

Keywords: mixed oligopoly, two-dimensional city, spatial competition
JEL Classi…cation: L13; L15; R39
I am grateful to Yoshitsuge Kanemoto, Takatoshi Tabuchi, and participants of Urban
Economics workshop at University of Tokyo for their helpful comments. All remaining
errors are of my own.
yVery preliminary. Please do not cite this version without my approval.
zFaculty of Regional Policy, Takasaki City University of Economics, 1300 Kaminamie,
Takasaki, Gumma 370-0801, Japan. E-mail: ago@tcue.ac.jp1

1

Introduction

Hotelling’s (1929) seminal work showed that duopolistic …rms agglomerate in
the center of a one-dimensional space (a linear city) although d’Aspremont
et al. (1979) revised the result as maximum di¤ erentiation under spatial
Bertrand competition. On the other hand, Hamilton et al. (1989) and Anderson and Neven (1991) developed location-then-quantity (Cournot) competition models. By contrast, they consequently showed the agglomerationof …rms in the center (minimum di¤ erentiation).
Matsushima and Matsumura (2003, henceforth “MM”) focused on an
observation that in a mixed market private …rms often provide a di¤erent
good or service from public sector but it is similar to other private …rms’(herd behavior of private …rms). For this to be explained, MM incorporated
a welfare-maximizing public …rm into a spatial Cournot model. As a result,in their circular city all private …rms agglomerate at a point that is thefarthest from the public …rm. And as an extension, in their linear city there
are two types of equilibrium: (i) All private …rms agglomerate (Proposition
1, p. 71). (ii) Firms agglomerate at two points if and only if the number ofthe private …rms is even (Proposition 4, p. 73). These results indicate thatpublic …rm has a strong repelling-e¤ ect against private …rms because thepublic …rm sets the price of each market at its transport cost (marginal cost2

pricing), which means markets near the public …rm are very competitive forthe private …rms. At the same time, private …rms share the same objective
to rival the public …rm; they consequently choose the same location.
This paper extends MM into a two-dimensional square city. When it
comes to two-dimensional Cournot competition without a public …rm, both
Berenguer-Maldonado et al. (2005) and Ago (2008) have shown that …rms
agglomerate in the center when space is a disk or a rectangle. Because this isan essentially identical result as in linear space, one may think that we do notneed any change in dimensionality. However, with regard to spatial Bertrand
competition, dimensionality matters. In their location-then-price competition in a multi-dimensional city, Tabuchi (1994) and Irmen and Thisse (1998)
showed that maximum di¤erentiation occurs in only one dimension, while
minimum di¤erentiation is achieved in the rest of the dimensions. Hence,with dispersion force or repelling e¤ect, it is important to have a formal
result of how …rms run away from their rivals, a problem that is more complex and non-trivial because there are many directions for dispersion. That
is why this paper tackles on the dimensionality problem.
Another reason for multi-dimension is relevance to the real world. MM
suggested “Television programs supplied by the Japan Broadcasting Corporation (NHK) are quite di¤ erent from those of private broadcasting companies,3

which are quite similar to each other (p. 63).” It seems true that a publicbroadcasting company prefers a more serious program like news to an entertainment or a shopping one, which rather concerns private broadcasting
companies. However, with regard to a political aspect, the public company(is e¤ectively required to) chooses a moderate course, while positions of theprivate companies di¤er: conservative or liberal stance. Such a complex position can be analyzed by a multi-dimensional model. That is another meritof the paper.
The remainder of the paper is organized as follows. In Section 2, thetwo-stage location-then-quantity game is presented. In Section 3, quantity
choice is analyzed in the second stage by backward induction. Then, Section
4 yields location equilibria in the …rst stage. Section 5 summarizes theresults.

2

The model

Our model is based on a one-dimensional model developed by MM (Matsushima and Matsumura, 2003), who analyzed a location-then-quantity competition model with a welfare-maximizing public …rm among private …rms. I
extend it to a two-dimensional model with simpli…cation in two ways: First,
the functional form of transport costs is assumed quadratic in Euclidean4distance. Second, the number of the private …rms is two. Except for those
aspects, MM and this model are identical in order to be better compared.
We consider a two-dimensional city expressed by a square on x-y coordinates, L = f(x; y) 2 R2 :

1

x

1; 1

y

1g, and consumers

are uniformly and continuously distributed on L with a density of one ateach location. There are a welfare-maximizing public …rm (…rm 0) and two
pro…t-maximizing …rms (…rm 1 and …rm 2) that supply a homogeneous goodwith zero marginal cost. Let …rm i’s location be (xi ; yi ) 2 L.
We analyze a two-stage location-then-quantity game with subgame perfection being the equilibrium concept. In the …rst stage, each …rm simultaneously chooses its location. In the second stage, each …rm simultaneously
chooses its quantity given their locations. Based on the literature, …rms beartransport costs and they can set a supply amount for each location independently because arbitrage between consumers is assumed to be prohibitively
costly.
Each consumer at (x; y) has the same inverse demand function as follows:

P (x; y) = abQ(x; y);Q(x; y) =2Xqi (x; y);

(1)

i=0

where P (x; y) is the price at (x; y), qi (x; y) (i = 0; 1; 2) and Q(x; y) is supply

5amount of each …rm and the total supply amount there. a and b are positive
constants.
The transport costs are the same for the …rms and are linear to supply
amount. The unit transport cost is quadratic with regard to Euclidean
distance between a …rm and a consumer. Let di (x; y) denote the distance
between a consumer at (x; y) and …rm i. Then, the transport cost is givenbyx)2 + (yitdi (x; y)2 = t[(xiy)2 ];

where t is the transport cost parameter and is assumed to be su¢ ciently low
such thata > 24t;(2)

which ensures that the public …rm always serves the entire city, irrespectiveof the locations of the …rms.1
For private …rm i, the local pro…t at (x; y) and the total pro…t are given
1More generally, the condition is a > (n + 1)t, where n is the number of private…rms and t is the maximal transport cost for the public …rm. In our model, n = 2 andp
2t = t (max d0 (x; y))2 = t 22 + 22 yield the threshold of (2).

6byi (x; y)i

= qi (x; y) P (x; y) tdi (x; y)2 ;ZZ=
i (x; y)dxdy;
Lwhere P (x; y) is given by (1). Let w(x; y) denote the social surplus (consumer surplus plus the sum of the pro…ts) at (x; y):w(x; y) =Z Q(x;y)(abm) dm0

2
X

tdi (x; y)2 qi (x; y);i=0

and the (total) social surplus is

ss =

ZZ

w(x; y)dxdy:

L

3

Quantity choiceFirst, we analyze the second-stage game. In this stage, we can apply thesame analysis in MM because only distance matters, irrespective of dimensionality. Therefore, following important results in MM are clearly valid
here as well:7Remark 1 (Lemma 1 in MM, p.69) In equilibrium,

P (x; y) = td0 (x; y)2 ; Q(x; y) =a

td0 (x; y)2
:
b

(3)

Remark 2 (Lemma 2 in MM, p.69) Consumer surplus, cs, does not dependon (xi ; yi ) (i = 1; 2) and is given by

cs =

ZZ

cs(x; y)dxdy

L

=

2
[45a2
45b

30at 2 + 3x20 + 3y02

+t2 28 + 45x40 + 45y04 + 120x20 + 120y02 + 90x20 y02 ];where cs(x; y) is consumer surplus at (x; y), which iscs(x; y) =td0 (x; y)2
2b

a

2

:

Note that if we rewrite (x0 ; y0 ) as (r cos ; r sin ), then we have

cs =

2
45a2
45b

30at 2 + 3r2 + t2 28 + 120r2 + 45r4

and
@cs
@cs
= 0;
= 60tr@@r

3a + 4 + 3r2 t < 0;

8

;

(4)

where the inequality is due to (2). Therefore, the nearer to the center the
public …rm is, the greater cs becomes.

Remark 3 (Lemmas 3 and 4 in MM, p.70) Firm i (i = 1; 2) supply positive
amount of qi (x; y) at (x; y) if and only if d0 (x; y) > di (x; y), whereqi (x; y) =t d0 (x; y)2di (x; y)2b;

and its pro…t is2i (x; y) = b [qi (x; y)] :Remark 4 (Lemma 5 in MM, p.71) The pro…t of each private …rm does
not depend on the location of the other private …rm:i (xi ; yi ; x0 ; y0 ) =ZZ

i (x; y)dxdy;

(5)

Li

where Li (i = 1; 2) denotes the domain in which …rm i serves:

Li = f(x; y) : d0 (x; y) > di (x; y)g \ L:

Note that the market boundary of d0 (x; y) = di (x; y) is the perpendicularbisector of the line segment jointing the locations of …rm 0 and …rm i.

9

4

Location equilibrium

We analyze the …rst-stage game given the results in the second stage. Without loss of generality, we henceforth assume that 0

x0 ; y0

1. Fur-thermore, due to symmetry and for notational convenience, we will omit
any equilibria that is symmetric with another equilibrium: For example,
if f(x0 ; y0 ); (x1 ; y1 ); (x2 ; y2 )g = f(0; 0); (1=2; 1=2); ( 1; 1)g were an equilibrium; f(0; 0); ( 1=2; 1=2); (1; 1)g, which were clearly another equilibrium,
would be omitted because it is regarded as essentially the same equilibrium.
First of all, symmetry yields the following intuitive result.

Lemma 1 The public …rm locates on x-axis (y-axis) if and only if the private …rms locate equidistantly away from x-axis (y-axis) in opposite direction
each other. In other words, in equilibrium, x0 = 0 () x1 + x2 = 0 and
y0 = 0 () y1 + y2 = 0.
Proof. See the appendix.
This lemma classi…es the types of location equilibria. For the public…rm to choose the center, (x0 ; y0 ) = (0; 0), the private …rms must locate
symmetrically with regard to the center, (x2 ; y2 ) = ( x1 ; y1 ). Let thiscase be named central symmetry, which will be analyzed in Subsection 4.1
below.
10

On the other hand, unless the public …rm locates on either of two axes,
symmetry breaks down. From Remark 4, the pro…t functions of both private…rms are only dependent on locations of public …rm and of its own; hence,both …rms would choose the same location, (x1 ; y1 ) = (x2 ; y2 ). We namethis case full agglomeration, which we will deal with in Subsection 4.2.
At last, there is another possibility: the public …rm locate on one axis but
not on the other axis. For example, suppose that x0 = 0 and y0 6= 0. In thiscase, x2 =x1 must hold, but the symmetry with regard to y-coordinate

has broken down. Again from Remark 4, the private …rms would locate thesame position with regard to y-coordinate. We call it partial agglomeration.
Subsection 4.3 will see the case.
Among three cases above, central symmetry and agglomeration werealready presented in the linear-city case of MM. Meanwhile, the last case is
new.

4.1

Central symmetryFirst, we focus on the possibility that the public …rm is located at the center,(x0 ; y0 ) = (0; 0) and the private …rms are symmetrically located with regard
to the public …rm in equilibrium.

Proposition 1 Under the scheme of central symmetry, there exists a unique11

location equilibrium2 such that(x0 ; y0 ; x1 ; y1 ; x2 ; y2 ) = (0; 0; 2=3; 2=3; 2=3; 2=3):

Proof. See the appendix.
This equilibrium corresponds to the equilibrium in MM (Proposition 4(ii) in MM), where the public …rm locates at 1=2, half of the private …rms
locate at 1=10 and the others locate at 9=10 in a linear city of a line segment[0; 1]3 .
Unlike Tabuchi (1994) and Irmen and Thisse (1998), private …rms differentiate with regard to all dimensions here. Irmen and Thisse (1998) give
us a clue how to interpret an equilibrium in multi-dimensional models:In a symmetric equilibrium with …xed market size pro…ts arehighest when prices are highest, that is, when the elasticity ofdemand is lowest. The lower the density of marginal consumers,
the lower is the elasticity. Accordingly, as the consumer distribution is uniform, demand has minimal elasticity when thecorresponding hyperplane has minimal surface area. Since the2

Remember the sentence at the top of this section. We omit other symmetric equilibria
like (x0 ; y0 ; x1 ; y1 ; x2 ; y2 ) = (0; 0; 2=3; 2=3; 2=3; 2=3) because this is essentially the sameequilibrium in this proposition.
3
If we change the space into a rectangle, f(x; y) 2 R2 : cxc; 1=cy1=cg(c > 1); we can …nd when c gets large, the equilibrium approaches MM’s.

12strategy space is a hypercube this hyperplane is the one thatis parallel to one of the facets, which in turn implies that the
products di¤er only in one dimension.

In this paper, when (x0 ; y0 ; x1 ; y1 ) = (0; 0; 2=3; 2=3) in Proposition
1, the length of market boundary between …rm 0 and …rm 1 is 1:88562 =p
4 2=3. On the other hand, if (x0 ; y0 ; x1 ; y1 ) = (0; 0; 1; 0), which gives differentiation in one dimension; the length of market boundary is 2. Therefore,
the elasticity of demand is lower in di¤erentiation in both dimensions.
Needless to say, locating at (x1 ; y1 ) = ( 1; 1) minimizes the boundary when (x0 ; y0 ) = (0; 0). However, Cournot competition has a relatively
stronger centripetal force (Hamilton et al., 1989; Anderson and Neven,1991). Therefore, the tradeo¤ between these e¤ects yields our interior location equilibrium.

4.2

Full agglomeration

Next, we analyze the possibility that the private …rms agglomerate. Unfortunately, due to complexity (high order of equations), it is impossible toget the solutions analytically. Hence, we will investigate some properties of
equilibrium and conduct a numerical analysis below.
At …rst, remember that cs depends on a (@cs=@a > 0) but the pro…ts
13

of the private …rms do not (Remark 2 and Remark 4). Hence, when a issu¢ ciently large, the public …rm cares the consumer surplus much more.
Clearly, when a ! 1, the public …rm chooses (x0 ; y0 ) = (0; 0). When(x0 ; y0 ) = (0; 0), as Subsection 4.1 shows, a private …rm would choose(x1 ; y1 ) = ( 2=3; 2=3). Because of independence from the other …rm’s
location on its pro…t, we have the following.

Proposition 2 Under the scheme of full agglomeration, we havelim (x0 ; y0 ; x1 ; y1 ; x2 ; y2 ) = (0; 0; 2=3; 2=3; 2=3; 2=3):

a!1

However, from Lemma 1, (x0 ; y0 ) = (0; 0) is consistent only when thecase of central symmetry. Therefore, when a is very large but not in…nity,
the public …rm would locate not at the center but near the center due tocontinuity. Then, we proceed to an analysis of the …rst-order conditions,@ss=@x0 = 0, @ss=@y0 = 0, @i =@xi = 0 and @

0<

t=a <

1
;
24

i =@yi = 0 (i = 1; 2). Let(6)then the solutions depend only on . In the vicinity of (x0 ; y0 ; x1 ; y1 ; x2 ; y2 ) =14

(0; 0; 2=3; 2=3; 2=3; 2=3), a comparative statics yields

dx0
dy0
dxi
dyi
512
=
=3
=3
=
> 0 (i = 1; 2);dddd729 2252

where the inequality is due to (6). This result indicates that the public …rm
locates slightly away from the center to a point on a diagonal of L while theprivate …rms slightly approach the public …rm on the same diagonal whenis su¢ ciently small (a is very large).
Unfortunately, all we can do is to proceed to a numerical analysis as afurther investigation because of analytical unsolvability. The analysis yields
the following numerical result.

Solution 3 Under the scheme of full agglomeration, the public …rm locatesnear the center on a diagonal while the private …rms agglomerate at a pointon the opposite half of the diagonal with regard to the public …rm. Whenincreases from zero, the public …rm goes away from the center; while theprivate …rms approach the center from the point that divides internally thehalf of the diagonal in the ratio of 2:1.

Figure 1 and Figure 2 around here.

15

Figure 1 and Figure 2 show this result, which corresponds to an equilibrium of Proposition 4 (i) in MM. This result is consistent with our intuition.
The greater

becomes, the more the public …rm cares the pro…ts of theprivate …rms. Suppose that all …rms are on a diagonal, then the movement
along the diagonal is the best way for separating from the private …rms withthe loss of the consumer surplus being minimized because the consumer
surplus is dependent only on the distance between the public …rm and thecenter. Then, once the public …rm goes away from the center, the private…rms have an incentive to approach the center due to relaxed competition.

4.3

Partial agglomeration

We can guess from Lemma 1 that if the public …rm locates on either x-axis
or y-axis, then there is an equilibrium where the private …rms are symmetricwith regard to that axis. Unfortunately, because the analytical unsolvability
continues, we will repeat a similar analysis as in the case of full agglomeration. Without loss of generality, we only deal with the case where the public…rm locates on y-axis (x0 = 0).
First of all, the limit case of a ! 1 yields the same outcome as in thecase of full agglomeration (Proposition 2).

16

Proposition 4 Under the scheme of partial agglomeration, we havelim (x0 ; y0 ; x1 ; y1 ; x2 ; y2 ) = (0; 0; 2=3; 2=3; 2=3; 2=3):

a!1

Next, we proceed to an analysis in the vicinity of the limit case above.
Letting x0 = 0, a comparative statics that is evaluated at (y0 ; x1 ; y1 ; x2 ; y2 ) =(0; 2=3; 2=3; 2=3; 2=3) yields

dy0
d
dx1
d
dy1
d

=
=
=

512
> 0;
729 1996
dx2
1024
=
> 0;
d
2187 5998
dy2
512
=
< 0;
d
2187 5998

where the inequalities are due to (6). Note thatdy0 dx1 dy1
2
:
:
=1: :
d
d
d
3

1
:
3

Hence, when the public …rm moves slightly away from the center in the
direction of north, …rm 1 moves in the direction of southeast from (x1 ; y1 ) =( 2=3; 2=3). A numerical analysis con…rms such a successive movement
as follows.

Solution 5 Suppose that the public …rm is on y-axis (x0 = 0). When17

increases, the public …rm goes away from the center on y-axis while …rm 1
goes southeastern away from ( 2=3; 2=3) with …rm 2 being kept symmetricwith regard to y-axis.

Figures 3-5 around here.

Figures 3-5 show this result. This solution seems somewhat di¤erent
from that in the case of full agglomeration, where the public …rm exhibitsa strong repelling e¤ect: When the public …rm approaches the center, theprivate …rms move away from it. In the case of partial agglomeration, theprivate …rms do move away in one dimension although they approach the
public …rm in the other dimension. The market boundary is a key to understand.
Suppose that (x1 ; y1 ) = ( 2=3; 2=3). If the public …rm slightly moves
northern from the center (x0 ; y0 ) = (0; "), the boundary rotates counterclockwise a little bit. Hence, the northwestern markets become more competitive while the southeastern markets become less competitive. Therefore,…rm 1 has an incentive to move southeastern, by which the …rm can avoid
keen competition.

18

4.4

Welfare implications

We compute the consumer surplus, the pro…ts, and the social surplus inequilibrium. Note that only consumer surplus depends on a; the greater a
becomes, the greater the consumer surplus. Meanwhile, b is just a multiplierfor each value. Let CSj , P Sj and SSj (j = sym; agg; par) denote consumer
surplus, producer surplus (pro…ts) and social surplus, respectively; wheresym; agg; par denote the three cases (central symmetry, full agglomeration,
partial agglomeration in order). Then, we have the following result.

Solution 6 CSagg < CSpar < CSsym , P Sagg > P Spar > P Ssym for each
t=a. The di¤ erence between each pair is increasing in t. SSagg 7 SSpar 7
SSsym when a ? a~(t), where a~(t) is a threshold of a function of t.
These orders are quite intuitive. Because the consumer surplus is maximized when the public …rm locates at the center, central symmetry makes
the consumer surplus maximal. On the other hand, roughly speaking, because the private …rms’pro…ts increase when the public …rm goes away fromthe center, central symmetry minimizes the producer surplus. Hence, social
surplus depends on which surplus is more important. If a is su¢ ciently large,
consumer surplus is dominant. Therefore, central symmetry maxmizes thesocial surplus when a is su¢ ciently large.

19

5Conclusion

We have analyzed an extended model of spatial mixed oligopoly. We have
had some similar results as in a linear city of MM (Proposition 1 and Solution3). As a new result, we have Solution 5, where the private …rms di¤erentiate
only in one dimension. Further, the reactions of the …rms to a change oftransport costs are analyzed. In our two-dimensional model, when the public…rm approaches a private …rm, the private …rm can more di¤erentiate in adimension while it less di¤erentiates in the other.

References[1] Ago, T., 2008, Cournot competition in a two-dimensional city, TCUE
Discussion Paper Series, No. 08-01.
[2] Anderson, S. P. and D. J. Neven, 1991, Cournot competition yields spatial agglomeration, International Economic Review 32, 793-808.
[3] D’Aspremont, C., J. Jaskold-Gabszewicz and J.-F. Thisse, 1979, On
Hotelling’s Stability In Competition, Econometrica 47, 1045-1050.
[4] Hamilton, J. H., J.-F. Thisse and A. Weskamp, 1989, Spatial discrimination: Bertrand vs. Cournot in a model of location choice, Regional
Science and Urban Economics 19, 87-102.
20

[5] Hotelling, H., 1929, Stability in competition, Economic Journal 39, 4157.
[6] Irmen, A. and J.-F. Thisse, 1998, Competition in Multi-characteristics
Spaces: Hotelling Was Almost Right, Journal of Economic Theory 78,
76-102.
[7] Berenguer-Maldonado, M. I., S. Carbó-Valverde and M. Á. FortesEscalona, 2005, Cournot competition in a two dimensional circular city,
The Manchester School 73, 40-49.
[8] Matsushima, N. and T. Matsumura, 2003, Mixed oligopoly and spatial
agglomeration, Canadian Journal of Economics 36, 62-87.
[9] Tabuchi, T., 1994, Two-stage two-dimensional spatial competition between two …rms, Regional Science and Urban Economics 24, 207–227.

21

APPENDIX
Proof of Lemma 1
Proof. Di¤erentiating the social surplus with regard to x0 and evaluating
it at x0 = 0, we have@ss=
@x0 x0 =032t2(x1 + x2 ) :3b

The …rst-order condition requires x1 + x2 = 0. Furthermore, using (2),we can readily con…rm @ 2 ss=@x20 < 0. Therefore, for x0 = 0 to be an
equilibrium, x1 + x2 = 0 is required.
Conversely, assume that x1 + x2 = 0. Then, we have@ss=
@x0 x2 = x1

8t2
a=t + 4x21 + 2y12 + 2y22
b

5x20

5y02

4 x0 :

Because a=t > 24 from (2), the value of the parenthesis is positive. Therefore, ss is maximized at x0 = 0.
Hence, the public …rm locates at x0 = 0 if and only if x1 + x2 = 0. Dueto symmetry, we can apply identical analysis with regard to y-coodinate.

22

Proof of Proposition 1
Proof. Without loss of generality we assume that1y11)2 + (y1 + 1)2(x2 ; y2 ) = ( x1 ; y1 ). First, when (x1of d0 (x; y) = d1 (x; y) intersects the lines of x =1 and y =x1

0 with2, the boundary
1. Hence, …rm

1’s pro…t is given by1 =ht2 (x0 + 1)2 + (y0 + 1)2
48b (x0

y1 )i4( x1 + 1)2( y1 + 1)2(x1 + 1)2

x1 ) (y0(y1 + 1)2

and due to symmetry, …rm 2’s pro…t is2 =ht2 (x0 + 1)2 + (y0 + 1)2

48b (x0 + x1 ) (y0 + y1 )i4:Then, the social surplus is given byss = cs +1+2;where cs is given by (4). We can readily show that ss is maximized at(x0 ; y0 ) = (0; 0) from the …rst-order and the second-order conditions forthe public …rm. Next, the …rst-order conditions for …rm 1 evaluating at

23

(x0 ; y0 ) = (0; 0) and (x2 ; y2 ) = ( x1 ; y1 ) yield the following equations:

6x1 + 7x21

2y1

y12 = 0; 6y1 + 7y12

2x1

x21 = 0:

The admissible solution is only (x1 ; y1 ) = (2=3; 2=3), which satis…es thesecond-order condition.
Second, when (x11)2 + (y1 + 1)2 < 2, the boundary of d0 (x; y) =d1 (x; y) intersects the lines of x = 1 and x =1. Similar calculation yields

the social surplus and the pro…ts, and we can show that no interior solutions
exist for pro…t maximization and the pro…t functions are di¤erentiable ateah point on d0 (x; y) = d1 (x; y). Therefore, the location pair of (x0 ; y0 ) =(0; 0); (x1 ; y1 ) = ( 2=3; 2=3); (x2 ; y2 ) = (2=3; 2=3) is a unique equilibrium
when

1

y1

x1

0. Clearly, due to symmetry, we have the result.

24

0. 015
0. 0125
0. 01
0. 0075
0. 0050. 0025
0. 01

0. 02

0. 03

0. 04

Figure 1: The location of the public firm. (The horizontal axis showst/a and the vertical axis shows x0(= y0).)- 0. 661- 0. 662- 0. 663- 0. 664- 0. 665

0. 01

0. 02

0. 03

0. 04

Figure 2: The location of the private firm. (The horizontal axis
shows t/a and the vertical axis shows x1(= y1).)

0. 015
0. 0125
0. 01
0. 0075
0. 0050. 0025
0. 01

0. 02

0. 03

0. 04

Figure 3: The location of the public firm given x0 = 0. (The horizontal axis
shows t/a and the vertical axis shows y0.)- 0. 668- 0. 669

0. 01

0. 02

0. 03

0. 04

- 0. 671- 0. 672

Figure 4: The location of the private firm. (The horizontal axis shows t/aand the vertical axis shows y1.)

- 0. 656- 0. 658

0. 01

0. 02

0. 03

0. 04

- 0. 662- 0. 664

Figure 5: The location of the private firm. (The horizontal axis shows t/aand the vertical axis shows x1.)

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