2009-5
Collusion, Agglomeration, and Heterogeneity of Firms
Toshihiro Matsumura Noriaki Matsushima

Collusion, Agglomeration, and Heterogeneity of Firms

Toshihiro Matsumura
Institute of Social Science, University of Tokyo
and
Noriaki Matsushima∗
Graduate School of Business Administration, Kobe University
February 18, 2009

Abstract
Friedman and Thisse (RAND Journal of Economics, 1993) show that spatial agglomeration appears in a standard two-stage location price model if the symmetricfirms can collude in prices. We introduce a cost difference between two firms. We
show that agglomeration never appears in a collusive equilibrium even when the costdifference between the firms is sufficiently small.

JEL classification numbers: L13, R32, L41
Key words: cost asymmetry, spatial agglomeration, collusion∗

Corresponding author: Noriaki Matsushima, Graduate School of Business Administration, Kobe University, Rokkodai 2-1, Nada, Kobe, Hyogo 657-8501, Japan. Phone: +81-78-803-6981, Fax: +81-78-8036977, E-mail: nmatsush@kobe-u.ac.jp1

1

Introduction

Since the seminal work of Hotelling [6], the model of spatial competition has been seen
by many subsequent researchers as an attractive framework for analyzing product differentiation. The major advantage of this approach is that it allows an explicit analysis ofproduct selection. Of particular interest is the equilibrium pattern of product locations
and the degree of product differentiation. The original finding of Hotelling [6] is that firms
produce similar products (minimum differentiation). d’Aspremont et al. [2] consider twostage location–price games on the Hotelling line. They show that products are maximallydifferentiated when transport costs are quadratic. The firms in their study never choose
minimal differentiation so as to avoid cutthroat competition in the price competition stage.
Friedman and Thisse [4] investigate a collusion in the price-setting stage. They assume that firms choose their locations non-cooperatively and then divide collusive profits
according to their relative profits at status quo. They show that firms agglomerate at thecentral point, restoring minimal differentiation. The logic behind the result is as follows.
Suppose that firm 1 (firm 2) locates initially at the left (right) side in the city, and not at
the central point. Now, suppose that firm 1 moves to the central point. First, consider
the competitive phase. The move by firm 1 increases the demand for it (as is discussed byHotelling [6]) and, at the same time, accelerates competition, thereby resulting in lower
prices (as is discussed by d’Aspremont et al. [2]). The accelerated competition as a resultof the move by firm 1 reduces the profits of firm 2 more significantly: while both effects
discussed above diminish the profits of firm 2, only the latter effect reduces the profits
of firm 1. Next, consider the collusive phase. Since the accelerated competition reduces
the profits of firm 2 at status quo more significantly than those of firm 1, the move byfirm 1 improves its bargaining position (by increasing its relative profits). Thus, when
they can collude, firm 1 has a strong incentive to locate close to its rival; this results incentral agglomeration.1 Jehiel [7] also investigates this problem and shows that central
1

There are many other papers that discuss central agglomeration without collusion. See, de Palma etal. [3], Hamilton et al. [5], Anderson and Neven [1], and Mai and Peng [8].

2agglomeration appears under Nash bargaining without monetary transfer (side payment).
Rath and Zhao [10] extend these studies to more general bargaining games and show that
central agglomeration is always an equilibrium outcome, though it may not be the uniqueequilibrium outcome.2
We introduce a cost asymmetry between two firms. We find that even a slight differencein cost has a drastic impact on the result: under a general allocation rule of collusive profits,
firms never agglomerate.
The paper is organized as follows. Section 2 formulates a basic model where monetary transfer (side payment) is possible. Section 3 presents our non-agglomeration result.
Section 4 investigates the model without monetary transfer and shows a similar nonagglomeration result. Section 5 concludes the paper.

2

The Model

In sections 2–3, we assume that monetary transfer (side payment) between two firms ispossible and firms split monopoly profits directly. In section 4, we investigate a nontransferable case, where firms split their market shares through a coordination of theirprices.
We formulate a duopoly model with a linear city a la Hotelling. The linear city of length
1 lies on the abscissa of a line and consumers are uniformly distributed with density 1 along
this interval. Suppose that firm i (i = 1, 2) is located at point xi ∈ [0, 1]. A consumer
living at y ∈ [0, 1] incurs a transport cost of t(xi − y)2 when (s)he purchases the productfrom firm i. The consumers have unit demands, i.e., each consumes one or zero unit of the
product. Each consumer derives a surplus from consumption (gross of price and transport
costs) equal to s. We assume that s is so large that the market is fully covered.3 Firms
1 and 2 produce the same physical product. Each firm maximizes its own profits. The
unit cost of the product for each firm is ci , which is given exogenously. Without loss of2

They show that under the egalitarian solution and the Kalai-Smorodinsky solution, multiple equilibria,

including the case of central agglomeration, exist.
3

A sufficient condition is s > 3t + c.

3generality, we assume that firm 1’s marginal cost is zero and firm 2’s is c ≥ 0. We assume
that c ≤ t.
The game runs as follows. In the first stage, each firm i chooses its location xi ∈ [0, 1] simultaneously. In the second stage, each firm i chooses its price pi ∈ [ci , ∞) simultaneously.
The firms collude in the second stage, while they choose their locations non-cooperatively
in the first stage.4
The firms split the collusive profit through bargaining at the beginning of the second
M
C
C
M
B
stage. Firm 1’s profit is given by ΠB
1 (Π , Π1 , Π2 ) and firm 2’s profit is Π − Π1 , whereΠM is the collusive profit (maximized joint profit) and ΠC
i is the competitive profit (profit
when firms do not collude in prices) of firm i. We make the following assumption on the
payoff function through bargaining.
Assumption 1
1≥

∂ΠB
∂ΠB
∂ΠB
1
1
1
≥
0,
>
0,
< 0.
C
∂ΠM
∂ΠC
∂Π
1
2

The first part in Assumption 1 implies that an increase in joint profit benefits all firms;
the second and third inequalities imply that an increase in its own profits at the threat
point increases its profits through bargaining.

3

Results

Friedman and Thisse [4] present an attractive explanation of minimum differentiation:
firms dare to choose minimum differentiation to improve their bargaining position. The
following proposition indicates that this holds only if c = 0 (no cost differentiation).
Proposition 1 Under Assumption 1, minimal differentiation constitutes an equilibrium
only if c = 0.
Proof
4

Suppose that c > 0 and x1 = x2 in equilibrium. Without loss of generality we

We implicitly consider the following repeated price-setting game. The firms simultaneously and non-cooperatively select their locations at the beginning of time; once chosen, the locations are fixed, but thefirms will select their prices in each of the subsequent periods. In other words, the firms collusively arrange
a trigger strategy equilibrium in prices, and select their locations knowing that the trigger strategy priceequilibrium will ensue.

4assume that x1 ≤ 1/2. In the equilibrium, at the competitive stage (at the threat point),
firm 2 sets p2 = c, and firm 1 undercuts firm 2’s price and obtains the whole market. As
Ca result, ΠC
1 = c and Π2 = 0.

Suppose that firm 2 deviates from the equilibrium location x1 = x2 and chooses x2 =x1 + ε, where ε is positive and sufficiently small. This relocation never reduces ΠMbecause production efficiency is improved by the differentiation. After the deviation,5Π1C = c − (t(1 − x1 )2 − t(1 − x1 − ε)2 = c − tε(2 − 2x1 − ε) < c and ΠC
2 = 0. Thus, thedeviation improves firm 2’s payoff, a contradiction.

Q.E.D.

When two firms are symmetric, both firms’ profits at the threat point (competitive
stage) are minimized when x1 = x2 . Thus, both firms choose agglomeration in order tominimize the rival’s profits. When firm 2 is inferior to firm 1, however, firm 1’s profits are
locally maximized when x1 = x2 and firm 2’s profits are independent of its location as longas |x1 − x2 | is sufficiently small.6 Thus, firm 2 has an incentive to avoid agglomeration.
This yields our discontinuous result with respect to c (the cost difference between two
firms).

4

Nontransferable profits

In sections 2–3, we assume that monopoly profit is transferable between two firms. Supposethat x1 = x2 . Then, the joint profit is maximized by the monopoly of firm 1, and monetarytransfer from firm 1 to firm 2 is made. In this section, we consider a situation wheremonetary transfer is impossible (no side payment).
At the beginning of the second stage, firms coordinate their prices through bargaining.
CC
The profits of firm 1 and firm 2 at the collusive stage are given by ΠBN
1 (F, Π1 , Π2 ) andC
Π2BN (F, ΠC
2 , Π1 ), respectively, where F is a feasible set of their profits.

We make the following three assumptions on the payoff function through bargaining.
5

See Result 1 in Matsumura and Matsushima [9]. It shows that firm 1 obtains the whole market if andonly if c ≥ t(x2 − x1 )(4 − x1 − x2 ).
6This statement means that the profit of firm 2 is zero if x1 and x2 satisfy the following inequality,c ≥ t(x2 − x1 )(4 − x1 − x2 ). See Matsumura and Matsushima [9].

5Assumption 2 The bargaining yields an efficient outcome, i.e.,
C
C
BN
C
C
(ΠBN
1 (F, Π1 , Π2 ), Π2 (F, Π2 , Π1 )) lies on the frontier of the feasible set.

Assumption 3
∂ΠBN
∂ΠBN
i
i
>
0,
<0
C
∂ΠC
∂Π
i
j

(i = 1, 2, i 6= j).

0
C
C
BN
C
C
Assumption 4 For a positive constant δ, if (ΠBN
1 (F, Π1 , Π2 ), Π2 (F, Π2 , Π1 )+δ) ∈ F ,
C
C
BN
C
0
C
then ΠBN
2 (F , Π2 , Π1 ) ≥ Π2 (F, Π2 , Π1 ).

Assumption 4 implies that if the feasible set is enlarged and firm 2 has an opportunity
to increase its payoff without reducing firm 1’s profits, firm 2’s profits through bargaining
never decrease by the change of the feasible set.
Suppose that x1 < x2 . Firm 1 obtains the consumers living at x ∈ [0, x̃] and firm 2
obtains consumers living at x ∈ (x̃, 1], where x̃ is given by
½

½

¾

¾

x1 + x2
p2 − p1x̃ = max min+,0 ,1 .
22t(x2 − x1 )(1)Henceforth, at the collusive stage we assume that x̃ ∈ (0, 1), i.e., both firms obtain positive
profits at the collusive stage. Let pE
i be the equilibrium price at the collusive stage. Then,2
E2
because the market is fully covered, both pE
1 ≤ s − t(x1 ) and p2 ≤ s − t(1 − x2 )are satisfied.7 At least one of the two inequalities stated above must be satisfied withE
strict equality at the collusive stage; otherwise, firms could raise both pE
1 and p2 withoutchanging the demand for either firm. If the two firms agglomerate at a point (x1 = x2 ),they set the highest price that induces all consumers to buy (the market is fully covered),and then coordinate their market shares through bargaining (if their prices are different,
the firm setting the lower price appropriates the entire demand).
E
We then suppose that xE
1 = x2 in equilibrium and derive a contradiction. WhenExE
1 = x2 , we assume without loss of generality that firm 2 locates just to the right-sideof firm 1. Then, firm 2 obtains the consumers living at x ∈ (x̃, 1] at the collusive stage.
If firm 2 deviates from the equilibrium location and chooses x02 > xE
2 , then firm 2 obtains
the consumers living at x ∈ [x0 , 1] at the collusive stage. If firm 2 deviates from the7

Full coverage at the collusive stage holds since we assume that s is sufficiently large.

6equilibrium location and chooses x002 < xE
2 , then firm 2 obtains the consumers living at
x ∈ [0, x00 ] at the collusive stage (note that firm 2 now locates to the left-side of firm 1).
Given the locations of the two firms, x̃, x0 and x00 are determined indirectly through pricebargaining.
We now make the final assumption.
E
Assumption 5 If c > 0 and xE
1 = x2 , then x̃ > 1/2.
E
Suppose that Assumption 5 is not satisfied (x̃ ≤ 1/2 when c > 0 and xE
1 = x2 ). Then,both firms 1 and 2 obtain the same market share or firm 2 obtains a larger market share,
despite firm 2’s profits being smaller at the threat point. We believe that it is natural toassume that the superior firm (firm 1) obtains the larger market share. The Nash bargaining solution discussed by Jehiel [7] satisfies this assumptions.
Proposition 2 Under Assumptions 2–5, minimum differentiation constitutes an equilibrium only if c = 0.
Proof

E
Suppose that c > 0 and xE
1 = x2 . Before the deviation, firm 2 obtains the con-sumers living at x ∈ [x̃, 1] at the collusive stage. We then show that firm 2 can improveEE
its payoff by a deviation. We consider three cases: (i) xE
2 < x̃; (ii) x̃ < x2 ; (iii) x̃ = x2 .

In the proof of Proposition 1, we show that the deviation reduces ΠC
1 without changing
0
Π2C if firm 2 deviates from the equilibrium strategy and chooses x02 6= xE
2 where x2 is8
sufficiently close to xE
2.

Let F A and F B respectively be the feasible sets of the collusive payoffs of firms 1 and
2 after and before the deviation.
0
E
First, we suppose that xE
2 < x̃. Firm 2 now deviates and chooses x2 > x2 . After the2
02
deviation, at the collusive stage, firm 2 can increase p2 by t(xE
2 − x̃) − t(x2 − x̃) without
8

E
0
0
E
E
0
This statement means that xE
1 = x2 and x2 satisfy the following inequality, c ≥ t(x2 −x2 )(4−x1 −x2 ).

7

9
BN
B
BN
B
E
2
0
changing its demand given pE
1 . Thus, Π1 (F , ..), Π2 (F , ..) + t((x2 − x̃) − (x2 −x̃)2 )(1 − x̃) ∈ F A . This implies that the deviation improves firm 2’s payoff (Assumption4), a contradiction. Note that we have already shown that the deviation improves thethreat point of firm 2 (Proof of Proposition 1).
E
E
Second, we suppose that x̃ < xE
2 . This is possible only if x1 = x2 > 1/2, because

firm 1 has to secure more than half of the market demand (Assumption 5). Firm 2 now
deviates and chooses x02 < xE
2 . After the deviation, firm 2 obtains the consumers living at
x ∈ [0, x0 ] at the collusive stage. Let pE
i be the equilibrium price of firm i at the collusive
stage before the deviation. At the collusive stage, firm 2 can set x0 = 1 − x̃ by setting
E
0 2
0
0 2
E 10 This again implies that the deviation
p02 = pE
2 + t(x1 − x ) − t(x2 − x ) given p1 .

improves firm 2’s payoff (Assumption 4), a contradiction.
E
E
Finally, we suppose that x̃ = xE
2 . This is possible only if x1 = x2 > 1/2 because firm

1 has to secure more than half of the market (Assumption 5). Firm 2 now deviates and0chooses x02 < xE
2 . After the deviation, firm 2 obtains the consumers living at x ∈ [0, x ] at

the collusive stage. Before the deviation, since the market is fully covered, and xE
1 > 1/2,
E 2
E
E 2
E
pE
1 = s−t(x1 ) . Note that the constraint p2 ≤ s−t(1−x2 ) is not binding since x2 > 1/2.
2
0 2
After the deviation, at the collusive stage, both firms can raise prices by t(xE1 ) − t(x )
E 2
E
0 2
since the constraint pE
1 ≤ s − t(x1 ) is replaced with p2 ≤ s − t(x2 ) and hence relaxed.
E 2
E
E 2
On the other hand, the constraint pE
2 ≤ s − t(1 − x2 ) is replaced with p1 ≤ s − t(1 − x1 )E 2
Eand thereby strengthened. However, pE
1 ≤ s − t(1 − x1 ) is not binding since x1 > 1/2(See Figure 1). The deviation does not affect the collusive price pE
2 . Thus, the deviation
enlarges the feasible set. This again implies that the deviation improves firm 2’s payoff
9This is derived by the following equationµ E
1 − x̃ = 1 −x1 + x02
p02 − pE1+22t(x02 − xE
1 )

¶
.

E
As mentioned earlier, when xE
1 = x2 , firms coordinate their market shares through bargaining.
10

p02 is derived by the following equation
x0 =

0
0
pE
xE
1 − p2
1 + x2
+
.
0
2
2t(xE
1 − x2 )E
As mentioned earlier, when xE
1 = x2 , they coordinate their market shares through bargaining.

8

(Assumption 4), a contradiction.

Q.E.D.

Propositions 1 and 2 imply that under a broad class of bargaining situations, location
then collusive pricing models do not yield agglomeration when two firms are asymmetric.11
This is because a cost difference results in a conflict of interests between two firms at thecompetitive stage. The superior firm’s profit is locally maximized around an agglomeration
point, while the inferior firm’s profit is independent of locations around the agglomeration
point. This is why the inferior firm has a strong incentive to avoid minimum differentiation.

5

Concluding Remarks

In this paper we discuss how cost asymmetry between firms affects the possibility of central
agglomeration in location then collusion models. In contrast to the existing results undercost symmetry, firms never agglomerate under conditions of even slight cost asymmetry.
Cost asymmetry creates a conflict of interest between firms. The superior firm’s profits
are locally maximized at the agglomeration, while it is locally minimized without cost
asymmetry. The inferior firm, therefore, has a strong incentive to avoid agglomeration
so as to reduce the rival’s status quo profit under cost asymmetry. This yields our contrasting results. We believe that this result holds true for more than two firms and otherspecifications of space, such as a circular-city model.
Our result holds for cases where the superior firm chooses its location first and theinferior firm then selects its location on the basis of the former’s choice. If the location
choice is inversed, our result does not hold and agglomeration equilibrium appears. We
believe, however, that the superior firm naturally moves first. The superior firm, in all
likelihood, is an innovator, while the inferior firm is quite possibly an imitator. In such asituation, it is natural that the innovator chooses its product position first.
[2009.2.17, 909]
11

The Kalai-Smorodinsky solution does not satisfy Assumption 4. We can, however, show that under

this solution agglomeration does not appear when c 6= 0. The proof is almost the same as the current proofof proposition 2. When x1 = x2 ≤ 1/2, firm 2 can improve its payoff with the deviation x02 = x1 + ε; whenx1 = x2 > 1/2, firm 2 can improve its payoff with the deviation x02 = x1 − ε.

9

s
E 2
pE
1 = s − t(x1 )2E
p2 = s − t(xE2)

0

1/2 xE1xE
2

E 2
Figure 1: pE
1 ≤ s − t(1 − x1 ) is not binding.

101References[1] S. P. Anderson and D. J. Neven, Cournot competition yields spatial agglomeration.
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[4] J. W. Friedman and J.-F. Thisse, Partial collusion fosters minimum product differentiation. RAND Journal of Economics 24(4) (1993) 631–645.
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[9] T. Matsumura and N. Matsushima, Cost differentials and mixed strategy equilibria in a Hotelling model. Annals of Regional Science, forthcoming, available at
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11