Market sharing agreements and collusive networks¤
Paul Belle‡ammey
Francis Blochz
March 18, 2003
Abstract
We analyze the formation of reciprocal market sharing agreements by
which …rms commit not to enter each other’s territory in oligopolistic
markets and procurement auctions. The set of market sharing agreements
de…nes a collusive network. We provide a complete characterization of
stable collusive networks when …rms and markets are symmetric. Stable
networks are formed of complete alliances, of di¤erent sizes, larger than
a minimal threshold. Typically, stable networks display fewer agreements
than the optimal network for the industry and more agreements than the
socially optimal network. When …rms or markets are asymmetric, stable
networks may involve incomplete alliances and be underconnected with
respect to the social optimum.
JEL Classi…cation Numbers: D43, D44
Keywords: market sharing, collusion, economic networks, oligopoly,
auctions.
¤
We are grateful to Olivier Compte, Sanjev Goyal, Philippe Jehiel, Mike Riordan and
Pierre Regibeau for helpful discussions on the paper. We have greatly bene…tted from the
comments of two anonymous referees. We also thank seminar participants at Barcelona,
Brown, Columbia, Essex, NYU, Penn, University College London, Warwick and the Vth
SCAET (Ischia, 2001) for their comments.
y
CORE and IAG, Université catholique de Louvain, Belgium, belle‡amme@core.ucl.ac.be.
z
GREQAM and Ecole Superieure de Mecanique de Marseille (France), bloch@ehess.cnrsmrs.fr
1
1
Introduction
Reciprocal market sharing agreements, whereby …rms refrain from entering each
other’s territory, have long been held under suspicion by antitrust authorities.
In one of the earliest cases litigated under the Sherman Act, the Addyston Pipes
Case of 1899, the Supreme Court struck down a group of iron pipe producers
which rigged prices on certain markets, and reserved some cities as exclusive
domains of one of the seller. (Scherer and Ross, 1990, pp. 318-319.) In recent
years, the globalization of markets and the deregulation of industries which used
to be regulated on a territorial basis (airlines, local telecommunication services
and utilities) have increased the scope for explicit or implicit market sharing
agreements.
The European Commission has been particularly aware of the potential
risk of market sharing, as …rms which used to enjoy monopoly power in some
territories seem reluctant to compete on the global European market. In a
landmark case against Solvay and ICI, in 1990, the European Commission has
established that the two companies had operated a market sharing agreement
for many years by con…ning their soda-ash activities to their traditional home
markets, namely continental western Europe for Solvay and the United Kingdom for ICI. It was also found that over many years, all the soda-ash producers
in Europe accepted and acted upon the ‘home market’ principle, under which
each producer limited its sales to the country or countries in which it had established production facilities.1 In the United States, the Telecommunications
Act of 1996 was speci…cally designed to encourage regional operators to enter
each other’s market. Five years later, it appears that both industries are still
dominated by a handful of dominant companies, each with highly clustered
regional monopolies.2
1
2
See O¢cial Journal L 152 , 15/06/1991, pp. 1-15.
The major providers of cable and local phone service seem to have chosen to merge rather
than compete. For example, thanks to a series of mergers, the Regional Bell Operating
Companies have shrunk to seven companies in 1996 into just four today. The last of these
mergers (between SBC and Ameritech) was subjected to a list of conditions requiring the
merged company to open its in-region local markets to competition and to enter local markets
outside its 13-state region. The conditions that the Federal Communications Commission
imposed made clear how dissatis…ed the agency was with the progress of local competition
2
Antitrust authorities have also reacted to this trend, issuing new guidelines
that emphasize market sharing agreements as an alternative form of collusion.
For example, in its 1999 merger guidelines, the Irish Competition Authority
states that:
“As an alternative to a price-…xing cartel, …rms [...] may divide
up the country between them and agree not to sell in each other’s
designated area. [...] At its simplest, a market-sharing cartel may
be no more than an agreement among …rms not to approach each
other’s customers or not to sell to those in a particular area. This
may involve secretly allocating speci…c territories to one another or
agreeing on lists of which customers are to be allocated to which
…rm.” (Irish Competition Authority, 1999).
In spite of this increasing concern from antitrust authorities, market sharing
agreements have not yet been studied in the theoretical literature. In this paper,
our objective is to propose a model of market sharing in order to study the
stability and welfare implications of market sharing agreements. We consider a
uni…ed model, which encompasses di¤erent market settings including oligopolies
and auctions. We suppose that each …rm is initially present on one market (its
home market) and can enter all foreign markets. We model market sharing
agreements as reciprocal agreements not to enter each other’s territory.3 The
set of bilateral agreements gives rise to a collusive network among …rms and
we draw on recent advances in the literature on strategic network formation
(Bala and Goyal (2000), Goyal (1993) and Jackson and Wolinsky (1996)) to
characterize stable and e¢cient networks of market sharing agreements.
Our …rst contribution is to identify a general condition on pro…t functions,
which guarantees that stable networks exist and can be characterized in a simple
way. This condition – the log-convexity of pro…ts in the number of competitors
on the markets – is satis…ed in most usual models of Cournot oligopoly and
under the Telecommunications Act of 1996 (Wilmer, Cutler & Pickering, 1999).
3
In this paper, our focus is on the stability of market sharing agreements, and we assume
that these agreements are enforceable. The issue of enforceability of market sharing agreements
is an important one, which cannot be answered in traditional models of repeated oligopoly
interaction. We leave it for further study.
3
in private value auctions. This condition is stronger than convexity. It states
that the rate at which pro…ts decline with the entry of a new competitor is
decreasing in the number of active …rms on the market.
Our second contribution is to provide a complete characterization of the
stable networks in a symmetric model where …rms and markets are identical.
We show that in a stable network, …rms form complete components, of di¤erent sizes, and we give an explicit formula for the minimal size of a component.
This formula can be applied to compute stable collusive networks in di¤erent
oligopoly and auction models. This characterization stems from two general
observations on the …rms’ incentives to form market sharing agreements. First,
we note that because pro…ts are log-convex, a …rm’s incentive to form an additional agreement is increasing in the number of agreements it has already
formed. This “convexity” e¤ect explains why …rms form market sharing agreements with all other …rms in an alliance. It also explains why alliances must
reach a minimal size to be stable. Second, we observe that …rms have an incentive to free-ride on the formation of market sharing agreements by other
…rms. Free-riding explains why alliances of di¤erent sizes may form, as …rms in
smaller groups free-ride on the alliances formed in larger groups and have no
incentive to form agreements with …rms in larger alliances.
It is instructive to contrast these results with the results which would be obtained if …rms formed multilateral cartel agreements. Because stable networks
are formed of complete components, the additional richness of the network
structure (enabling …rms to enter di¤erent agreements with di¤erent …rms)
is not exploited in our analysis. However, it is important to note that the
completeness of network components is an endogenous result of our analysis
– as opposed to an exogenous assumption in the traditional cartel literature.
Furthermore, our analysis indicates that it may be easier to sustain collusion
through bilateral agreements than through multilateral agreements, because a
…rm’s incentive to delete a link in the network is weaker than its incentive to
leave the cartel and delete all its links at once.
Our third contribution is to study the e¢ciency of stable collusive networks.
In our model, the e¢cient network from the industry point of view is the complete network, where all markets are local monopolies, while the socially e¢cient
4
network is the empty network, where all …rms participate on all markets. Not
surprisingly, because of the …rm’s free-riding incentives, stable collusive networks are under-connected with respect to the e¢cient network from the …rms’
point of view, but over-connected with respect to the socially e¢cient network.
This last result indicates that antitrust authorities should strike down market
sharing agreements in order to improve social welfare.
Our last contribution is to extend the model to asymmetric …rms and markets. While we are unable to obtain a complete characterization of stable networks in that case, we discuss several examples. The …rst example supposes
that markets are of di¤erent sizes, and shows that a stable network exists where
the …rm on a smaller market forms links to …rms on larger markets, but …rms on
larger markets compete with one another. The second example assumes that
…rms face an entry cost on foreign markets, and the …nal example considers
transportation costs to foreign markets. In both these examples, private incentives to enter may be excessive, and the formation of market sharing agreements
may improve welfare by discouraging entry on foreign markets.
We now discuss the relationship of our paper with the existing literature.
Starting with Stigler (1950)’s seminal contribution, the stability of price-…xing
cartels has been extensively studied in the literature. (See Selten (1973) and
d’Aspremont et al. (1983) in the case of Cournot oligopolies, Deneckere and
Davidson (1985) for a Bertrand model with di¤erentiated commodities and
Nocke (1999) for a recent contribution discussing the earlier literature.) In
auctions, the study of the stability of bidding rings is much more complex,
as it requires the analysis of auctions with asymmetric bidders. Mailath and
Zemsky (1991) study stability of bidding rings in the simpler context of secondprice auctions, and Mac Afee and Mac Millan (1992) provide an example in
the case of …rst-price auctions.4
The stability of collusive networks bears a
close resemblance to the stability of bidding rings and price-…xing cartels. The
“convexity e¤ect” in the formation of links is reminiscent of a similar e¤ect
found in some models of cartel stability.5 Moreover, as …rms bene…t from the
4
In general …rst-price auctions, Pesendorfer (2000) presents partial characterization results
on the equilibrium of an auction with a bidding ring and independent bidders. However, the
issue of stability of the bidding ring is not addressed in the model.
5
Mailath and Zemsky (1991) for instance show that in a second price auction, the game in
5
formation of collusive agreements by other …rms, free-riding incentives threaten
in the same way the stability of price-…xing cartels, bidding rings and collusive
networks. As in the case of cartels and bidding rings, our characterization of
stable collusive networks results from the balance between free-riding incentives
and the bene…ts of collusion.
Recent papers by Goyal and Joshi (2000a) and (2000b), Goyal and Moraga
(2001) and Konishi and Furusawa (2002) also apply the theory of economic networks to models of oligopoly. Goyal and Joshi (2000a) and Goyal and Moraga
(2001) study the formation of bilateral agreements by which …rms bene…t from
synergies and lower their production costs. Goyal and Joshi (2000b) and Furusawa and Konishi (2002) analyze the formation of bilateral trade agreements in
an international oligopolistic market. Trade agreements can be interpreted as
the converse of maket sharing agreements – trade agreements open up foreign
markets by abolishing tari¤s while market sharing agreements lead to the closure of foreign markets. The analyses in Goyal and Joshi (2000b) and Furusawa
and Konishi (2002) are not directly comparable to ours, as the objective functions – and hence the values of the network– are di¤erent in the two models. In
trade agreement models, a country’s objective function is the sum of consumer
and producer surplus and tari¤s, whereas the objective function in our model
is simply the …rm’s pro…t.
The rest of the paper is organized as follows. In the next section, we describe
the general model of market sharing and introduce some basic de…nitions. In
Section 3, we fully characterize the stable collusive networks in the symmetric
model with identical …rms and markets. In Section 4, we extend our analysis to
asymmetric markets and …rms. Section 5 discusses e¢ciency of stable collusive
networks, both from the industry and the social point of view. In the last
section, we conclude and discuss the limitations of our model.
2
The Model
We consider N …rms indexed by i = 1; 2; ::N. We associate to each …rm a
market. In the oligopolistic context, the market of …rm i is interpreted as its
characteristic function form generated by the formation of bidding rings is convex.
6
home market, and in the context of procurement auctions as a market to which
bidder i has priviledged access. For any market i, we denote by ni the number
of active …rms on the market. We consider a reduced form pro…t function on
each market, which could arise either from oligopolistic interaction or from
bidding competition, and which only depends on the number of active …rms on
the market. We let ¼ji (ni ) denote the pro…t of …rm j on market i.
We suppose that every …rm has an incentive to enter all foreign markets.
The only barrier to entry stems from reciprocal market sharing agreements,
whereby each …rm refrains from entering on the other …rm’s market. These
pairwise agreements are captured by binary variables, gij 2 f0; 1g. If gij = 1;
…rms i and j are linked by a market sharing agreement and are not active on
each other’s market, and if gij = 0, …rm i is present on market j and …rm j
on market i. Total pro…ts of …rm i are given by the sum of the pro…ts …rm
i collects on its home market and on all foreign markets for which it has not
formed agreements:
¦i = ¼ii (ni ) +
X
¼ij (nj ):
(1)
j;gij =0
2.1
Properties of Pro…t Functions
We impose three properties on the pro…t functions.
Property 1 Individual pro…ts are decreasing in the number of …rms active on
the market, ¼ij (nj ) · ¼ij (nj ¡ 1):
Property 2 Individual pro…ts are convex in the number of …rms active on the
market, ¼ij (nj ¡ 1) ¡ ¼ij (nj ) ¸ ¼ij (nj ) ¡ ¼ ij (nj + 1):
Property 3 Individual pro…ts are log-convex in the number of …rms active on
h
i
h
i
the market, ¼ij (nj ¡ 1) ¡ ¼ij (nj ) =¼ij (nj ) ¸ ¼ij (nj ) ¡ ¼ij (nj + 1) =¼ij (nj + 1):
Property 1 is a very intuitive condition, which guarantees that an increase in
the number of competitors reduces the pro…t of each …rm. Property 2 states that
this decline in pro…t (measured as a positive number) is a decreasing function of
the number of competitors. In other words, as the total number of active …rms
on a market increases, adding a new competitor leads to a smaller reduction in
7
pro…t. Property 3 is a stronger condition than Property 2: it indicates that the
rate of decline of pro…ts is decreasing in the number of …rms. The percentage
of pro…t lost after the addition of a new competitor gets smaller as the number
of active …rms on a market increases.
The three properties give a precise description of the e¤ects of entry on individual pro…ts: …rms su¤er from the entry of new competitors but competition
becomes less harmful, both in absolute and in relative terms, as the number of
…rms increases. Figure 1 depicts a pro…t function satisfying the three properties
and outlines the di¤erence between convexity and log-convexity.6
π(n)
45°
π(1)
π(2)
π(3)
π(1)/π(2)
π(n)
π(1)
π(2)
π(3)
1
2
3
n
π(2)/π(3)
Figure 1: Log-convex pro…t function
How restrictive are these three properties? We show below that Property 1
is satis…ed in most familiar models of oligopolies and symmetric auctions. We
6
It is apparent on the right panel of Figure 1 that pro…ts are decreasing and convex in the
number of …rms (clearly, ¼ (1) ¡ ¼ (2) > ¼ (2) ¡ ¼ (3) > 0). On the left panel, which maps the
function ¼ (n) against itself, we are able to measure the ratio between successive values of the
function; as ¼ (1) =¼ (2) > ¼ (2) =¼ (3), we observe that pro…ts are log-convex in the number
of …rms.
8
also provide su¢cient conditions on oligopoly models under which the convexity
properties are satis…ed, and show that pro…ts are always log-convex in private
value procurement auctions.
2.1.1
Cournot Oligopoly with Homogeneous Products
Consider a symmetric Cournot oligopoly with homogeneous products. Letting
P
qi denote the quantity produced by …rm i (i = 1; 2; :::; n) and Q = qi market
demand, the Cournot oligopoly is de…ned by an inverse market demand P (Q)
and individual cost functions c(qi ): Each …rm’s pro…t on the market is given by
¼i = qi P (Q) ¡ c(qi ):
We de…ne the elasticity of the slope of the inverse demand function, E(Q); as
E(Q) =
QP 00 (Q)
:
P 0 (Q)
Proposition 2.1 In a symmetric Cournot oligopoly with homogeneous products, (i) if costs are increasing and convex and E(Q) > ¡1, individual pro…ts
are decreasing in the number of active …rms on the market; (ii) if costs are lin-
ear, E(Q) > ¡1 and E 0 (Q) ¸ 0, individual pro…ts are log-convex in the number
of active …rms on the market.
Proof. See Appendix 7.1.
The …rst statement of the proposition is a classical comparative statics result
on Cournot oligopolies (termed “quasi-competitiveness” in Amir and Lambson,
2000). Vives (1999, pp. 105-107) gives early references to this result. Statement
(ii), providing su¢cient conditions for the log-convexity of individual pro…ts, is
a new result. While the conditions are clearly very restrictive, they are satis…ed
by speci…c families of inverse demand functions, as shown in Section 3.2.
2.1.2
Auctions
Consider now a set of …rms, i = 1; 2; :::; N; participating in procurement auctions. For each auction, …rm i draws a cost parameter ci distributed according to
the common distribution function F (ci ) with continuous density f (ci ) over the
support [0; C]: Furthermore, suppose that the function J(c) = c + (F (c)=f (c))
9
is increasing in c for any c 2 [0; C]: For the sake of simplicity, we assume that
the buyer sets a reservation price at C, and we consider any auction setting
which allocates the contract to the lowest bidder. By the revenue-equivalence
theorem (see, e.g., Riley and Samuelson, 1981, Proposition 1, p. 383), the ex
ante expected payo¤ of every …rm is independent of the auction rule, and is
equal to
¼(n) =
1
(E(c2n ) ¡ E(c1n ));
n
where cin denotes the i-th order statistic among n draws from the common
distribution F . By a simple application of the theory of order statistics (see
Mac Afee and Mac Millan, 1988, Lemma 1, p. 103), E(c2n ) = E(J(c1n )): Hence,
µ
¶ Z C
F (c1n )
1
=
(1 ¡ F (c))n¡1 F (c)dc:
¼(n) = E
n
f(c1n )
0
Proposition 2.2 In a private value procurement auction, ex ante individual
pro…ts are strictly decreasing and strictly log-convex in the number of active
…rms on the market.
Proof. See Appendix 7.2.
The comparative statics e¤ect of an increase in the number of bidders on
individual pro…ts is well-known in auction theory (see Mac Afee and Mac Millan
(1987, p. 711) and the references therein). The log-convexity of individual profits in the number of bidders is an original result, providing a strong structural
condition on the e¤ect of entry on individual pro…ts.
2.2
Incentives to Form Agreements
As a preliminary step, we consider …rms’ incentives to form reciprocal market
sharing agreements. Using expression (1), we compute …rm i’s incentive to form
an agreement with …rm j as
2
X
¢¦i = 4¼ ii (ni ¡ 1) +
k6=j;gik =0
3
2
¼ik (nk )5 ¡ 4¼ii (ni ) + ¼ij (nj ) +
£
¤
= ¼ii (ni ¡ 1) ¡ ¼ii (ni ) ¡ ¼ij (nj )
´ Fji (ni ; nj ) :
10
X
k6=j;gik =0
3
¼ik (nk )5
(2)
A market sharing agreement between …rms i and j leaves all markets k 6= i; j
una¤ected. Hence, …rm i’s incentive to form an agreement with …rm j only
depends on the characteristics of markets i and j. Two con‡icting e¤ects are at
work: by reducing competition on its home market, …rm i increases its pro…t by
¼ii (ni ¡1)¡¼ii (ni ) ; by losing access to foreign market j, it decreases its pro…t by
¼ij (nj ): The balance between these two e¤ects, noted Fji (ni ; nj ), depends on the
number of active …rms on the two markets, ni and nj : As the pro…t function ¼ii is
convex, Fji is a decreasing function of ni ; as the pro…t function ¼ij is decreasing,
Fji is an increasing function of nj . We conclude that …rm i’s incentive to form
an agreement with …rm j is increasing in the number of agreements …rm i has
already formed and decreasing in the number of agreements formed by …rm j:
The formation of a bilateral agreement requires the approval of both …rms.
Hence, an agreement between …rms i and j can only emerge if both incentives
are positive:
Fji (ni ; nj ) ¸ 0; Fij (nj ; ni ) ¸ 0:
These inequalities imply
¼ii (ni ¡ 1) > ¼ij (nj ) and ¼jj (nj ¡ 1) > ¼ji (ni ):
(3)
When …rms and markets are symmetric (¼ii = ¼ij = ¼jj = ¼ji = ¼), inequalities (3) yield a striking conclusion. If ¼(ni ¡ 1) > ¼(nj ) and ¼(nj ¡ 1) > ¼(ni ),
as ¼ is monotonically decreasing, we must have ni = nj : Hence, in a symmetric
model, a market sharing agreement can only be concluded among two …rms with
the same number of competitors on their home markets! This result is easily
understood. If one market had a smaller number of competitors, the …rm on
the other market would have no incentive to form an agreement, as the pro…t
it makes on the foreign market would already be larger than the pro…t it makes
on its home market.
If …rms i and j are symmetric ( ¼ii = ¼ji = ¼i and ¼ij = ¼jj = ¼j ) and one
market is more pro…table than the other (for example, ¼i (n) > ¼j (n) for all
n), inequalities (3) also result in a surprising conclusion. If ¼i (ni ¡ 1) > ¼j (nj )
and ¼j (nj ¡ 1) > ¼ i (ni ); as ¼i and ¼j are monotonically decreasing, the second
inequality can only be satis…ed if nj · ni : This result is easily interpreted. A
11
…rm on a less pro…table market only has an incentive to enter into an agreement
with a …rm on a more pro…table market if the pro…t it makes on the foreign
market is smaller than the pro…t it makes on the home market. Hence, the
number of competitors must be larger on the more pro…table market.
2.3
Stable Collusive Networks
We now consider the entire set of market sharing agreements among …rms.
Market sharing agreements can be interpreted as bilateral links, giving rise to
an undirected network g on the set of …rms.7 To study this network, we need
to introduce some notations and terminology from graph theory.
De…nition 2.1 (i) A network is complete if all …rms are linked (gij = 1 8 i; j;
i 6= j) and empty if no …rms are linked (gij = 0 8 i; j): (ii) A …rm i is isolated
if gij = 0 8 j 6= i: (iii) A network g is connected if there exists a path linking
any two …rms in g. (iv) A component g0 of g is a maximally connected subset
of g. We let m(g 0 ) denote the size of a component g 0 , i.e., the number of …rms
belonging to g 0 . A component is complete if all …rms inside the component are
linked.
We borrow our …rst concept of stability from Jackson and Wolinsky (1996)’s
general study of strategic networks. Formally, let g ¡ gij (respectively g + gij )
denote the network obtained from g by deleting (adding) the link between …rms
i and j.
De…nition 2.2 (Jackson and Wolinsky, 1996) A network g is (pairwise) stable
if and only if (i) 8i; j 2 N s.t. gij = 1; ¦i (g) ¸ ¦i (g ¡ gij ) and ¦j (g) ¸ ¦j (g ¡
gij ); and (ii) 8i; j 2 N s.t. gij = 0; if ¦i (g + gij ) > ¦i (g) then ¦j (g + gij ) <
¦j (g):
Jackson and Wolinsky (1996) do not speci…cally model the process of network formation but propose instead a test of stability in order to eliminate
unstable networks. Condition (i) states that whenever an agreement is formed,
both parties prefer to keep the agreement in place. Condition (ii) states that
there does not exist a pair of unlinked …rms which both have an incentive to
7
We require therefore gij = gji .
12
form an agreement. In our model, the two conditions for pairwise stability
rewrite as:
(i) 8i; j s.t. gij
(ii) 8i; j s.t. gij
8
< F i (n + 1; n + 1) ¸ 0
i
j
j
= 1;
j
: F (nj + 1; ni + 1) ¸ 0;
i
8
< if F i (n ; n ) > 0
i j
j
= 0;
: then F j (nj ; ni ) < 0:
(4)
(5)
i
The stability notion of Jackson and Wolinsky (1996) is a local criterion,
which considers pairwise links in isolation. It cannot be implemented as an
equilibrium of a well-de…ned game of network formation because …rms can only
create or sever links one by one. Furthermore, as …rms’ deviations are severely
constrained, this concept results in a very weak stability criterion.
In order to obtain a stronger stability concept, we allow …rms to form and
delete an arbitrary number of links. The concept we propose retains the intuition of Jackson and Wolinsky (1996) while resting on a simple noncooperative
model of network formation. We consider the simultaneous linking game introduced by Myerson (1991). Each …rm i chooses the set si of …rms with which it
wants to form a link. A link gij is formed if and only if i 2 sj and j 2 si . We let
g(s1 ; :::; sn ) denote the graph formed when every …rm i chooses si : The linking
game typically admits a large number of Nash equilibria, re‡ecting coordination
failures between agents who would bene…t from forming a link but do not form
it. In order to eliminate this coordination failure, we adopt a simple re…nement.
We say that an equilibrium is (pairwise) strong if it is immune to deviations by
coalitions of two …rms.
De…nition 2.3 A strategy pro…le fs¤1 ; :::; s¤n g is a pairwise strong Nash equilibrium of the linking game if and only if it is a Nash equilibrium of the game
and there does not exist a pair of …rms i; j and strategies si and sj such that
¦i (g(si ; sj ; s¤¡i;j )) ¸ ¦i (g(s¤i ; s¤j ; s¤¡i;j )) and ¦j (g(si ; sj ; s¤¡i;j )) ¸ ¦j (g(s¤i ; s¤j ; s¤¡i;j ))
with a strict inequality for one of the two …rms. A network g is strongly (pairwise) stable if and only if there exists a pairwise strong Nash equilibrium of the
linking game, fs¤1 ; :::; s¤n g, such that g = g(s¤1 ; :::; s¤n ):
Lemma 2.1 Any strongly pairwise stable network is pairwise stable.
13
Proof. The proof is immediate. Suppose that network g is not stable.
If gij = 1 and ¦i (g) < ¦i (g ¡ gij ) for some i; j, …rm i would bene…t from a
unilateral deviation, choosing si = s¤i nfjg: If gij = 0 , ¦i (g + gij ) > ¦i (g) and
¦j (g + gij ) ¸ ¦j (g), then g is not immune to a joint deviation by the two …rms,
si = s¤i [ fjg; sj = s¤j [ fig:
3
Symmetric Firms and Markets
In order to characterize stable collusive networks, we focus on the simplest
model of competition where …rms and markets are symmetric:
¼ ji (ni ) = ¼(ni ) 8i; j:
In this context, it is intuitive to require that monopoly pro…ts exceed total
duopoly pro…ts. We thus assume
Property 4 ¼ (1) ¸ 2¼ (2) :
We …rst derive a general characterization of stable and strongly stable networks. We then apply our results to speci…c examples of oligopoly and procurement auctions.
3.1
General Model
As we noted in the previous section, when two …rms have an incentive to form
a market sharing agreement, they must have the same number of competitors
on their home market. Hence, in a stable network, for any pair of linked …rms i
and j, ni = nj . Furthermore, if ni = nj = n, F (n + 1; n + 1) ¸ 0 implies that
¼ (n) ¸ 2¼ (n + 1). By log-convexity of pro…ts, ¼ (n) =¼ (n + 1) is a decreasing
function of n. Hence, if F (n + 1; n + 1) ¸ 0, then F (n0 + 1; n0 + 1) ¸ 0 for
all n0 < n. It follows that, for any pair of linked …rms, the incentive to form
additional agreements is increasing in the number of agreements already formed.
We conclude that, in a stable collusive network, if a set of …rms is linked through
market sharing agreements, they must all form bilateral agreements among
themselves. In other words, components in a stable collusive network have to
be complete.
14
Because ¼ (1) ¸ 2¼ (2) and pro…ts are log-convex, there exists a maximal
value n¤ for which ¼ (n¤ ) =¼ (n¤ + 1) ¸ 2. We associate to that number n¤ of
active …rms on the market the number m¤ of market sharing agreements generating a market of size n¤ : m¤ = N ¡n¤ . Clearly, …rms only have an incentive to
form an agreement if they have already formed at least m¤ agreements. Hence,
there exits a lower bound on the size of complete components in a stable network. Components in a stable network have to be of size greater than or equal
to m¤ + 1.
Finally, if di¤erent components exist in a stable network, they must involve di¤erent numbers of …rms. If two components had the same size, by
log-convexity of pro…ts, they would have an incentive to merge. Di¤erent components in a stable network must be of di¤erent sizes.
The preceding remarks provide necessary conditions on stable collusive networks. The next proposition shows that these conditions are also su¢cient.
Proposition 3.1 Let m¤ be the minimal integer such that ¼(N ¡ m¤ )=¼(N ¡
m¤ + 1) ¸ 2. A network g is stable if and only if it can be decomposed into
a set of isolated …rms and distinct complete components g1 ; g2 ; :::; gL such that
m(gl ) 6= m(gl0 ) 8l 6= l0 ; m(gl ) ¸ m¤ + 1 8l. Furthermore, if m¤ = 1, there is at
most one isolated …rm.
Proof. We only consider the su¢ciency part. Suppose that the network
g can be decomposed into isolated …rms and complete components of di¤erent
sizes, with m(gl ) ¸ m¤ + 1 8l. Clearly, as long as m¤ > 1, isolated players have
no incentive to create new links. We now consider a link between two players
i and j belonging to two components gl and gl0 with m(gl ) < m(gl0 ). Player
i, belonging to the smallest component, refuses to form a new link. Finally,
as m(gl ) ¸ m¤ + 1 8l, no player inside a component has an incentive to cut a
link, and the network is thus stable. If m¤ = 1, any two isolated …rms have
an incentive to form a link. Hence, there can be at most one isolated …rm in a
stable network.
Proposition 3.1 provides a full characterization of the pairwise stable networks when …rms and markets are symmetric. In this simple context, the complete network is always stable. Full collusion can always be sustained through
15
bilateral market sharing agreements. However, note that if m¤ > 1, the empty
network is also stable. Hence, while existence of a pairwise stable collusive network is always guaranteed, uniqueness will typically not be obtained. In fact, our
characterization puts no restrictions on the number of isolated …rms, free-riding
on the formation of agreements by other …rms.
Our characterization shows that stable collusive networks exhibit three properties: (i) in a collusive network, all alliances must reach a minimal size; (ii)
alliances must involve complete sets of bilateral agreements; and (iii) di¤erent
alliances must have di¤erent sizes.
We now turn to the analysis of strongly pairwise stable networks. The main
di¤erence between stability and strong stability stems from the …rms’ ability to
delete multiple links in the linking game underlying strongly stable networks.
In our setting, it turns out that a …rm prefers to renege on all its agreements
than on a single one. To see this, let h (k) denote a …rm’s incentive to delete k
links in a complete component of size m + 1:
h (k) = ¼ (N ¡ m + k) + k¼ (N ¡ m + 1) ¡ ¼ (N ¡ m) :
We compute
h (k + 1) ¡ h (k) = ¼ (N ¡ m + 1) ¡ ¼ (N ¡ m + k) + ¼ (N ¡ m + k + 1)
> 0 8k ¸ 1.
In a complete component, all …rms obtain the same pro…t. Hence, when deleting
one additional link, a …rm gains access to a foreign market which is at least as
pro…table as its home market. So, if a …rm initially has an incentive to renege
on one agreement, it necessarily has an incentive to renege on all agreements.
This implies that the criterion of strong stability typically selects a strict subset
of stable collusive networks.
Proposition 3.2 A network g is strongly stable if and only if it can be decomposed into a set of isolated …rms and distinct complete components g1 ; g2 ; :::; gL
such that (i) ¼(N ¡ m(gl ) + 1) ¸ ¼(N) + (m(gl ) ¡ 1)¼(N ¡ m(gl ) + 2) for all
l, and (ii) m(gl ) 6= m(gl0 ) for all l 6= l0 . Furthermore, if m¤ = 1, the network
contains at most one isolated …rm.
16
Proof. See Appendix 7.3.
The condition on component sizes in strongly stable networks is stronger
than the condition for stable networks. In fact, suppose that m satis…es
¼ (N ¡ m + 1) ¸ ¼ (N) + (m ¡ 1) ¼ (N ¡ m + 2) :
For m > 2, this implies that ¼ (N ¡ m + 1) > 2¼ (N ¡ m + 2) so that m ¸
m¤ + 1.
Because ¼(N ¡ m) ¡ m¼(N ¡ m + 1) is not a monotonic function of m, the
condition restricting component sizes in strongly stable networks does not determine an upper bound nor a lower bound on the sizes of components. However,
our examples below suggest that components in a strongly stable network cannot be too large. The incentive to free-ride and delete all links at once seems to
be higher in larger alliances. Note that if m¤ > 1, the empty network is always
strongly stable, which guarantees existence. However, as shown in Example 3.2,
a strongly stable network may fail to exist when m¤ = 1.
3.2
Examples
Example 3.1 Cournot oligopoly with iso-elastic inverse demand function
Let inverse demand be given by P (Q) = 1 ¡ Q® for ® > 0: (If ® = 1, demand
is linear; if 0 < ® < 1, demand is convex, and if ® > 1, demand is concave.)
Observe that E(Q) = ®¡1: As ® > 0; E(Q)+1 > 0 and furthermore, E 0 (Q) ¸ 0.
The su¢cient conditions of Proposition 2.1 are thus satis…ed. Straightforward
computations show that
¼(n) = ®n
1¡®
®
(n + ®)¡
1+®
®
:
In Appendix 7.4, we show that ¼ (n) =¼ (n + 1) ¸ 2 if and only if n = 1, so
the only stable networks are the empty and complete networks for a Cournot
oligopoly with isoelastic demand. The complete network is strongly stable if and
only if ¼(1) ¸ ¼(N) + (N ¡ 1)¼(2): We show in Appendix 7.4 that the latter
inequality can never be satis…ed. We conclude that the complete network is
never strongly stable, and hence the only strongly stable network is the empty
network.
17
In a Cournot market with iso-elastic inverse demand, incentives to share
markets only emerge when the number of competitors goes from two to one.
Hence, apart from the empty network, the only sustainable collusive network
is the complete network where every …rm is a monopolist on its home market.
This fully collusive network does not survive free-riding from individual …rms
deleting all their links at once.
Example 3.2 Cournot oligopoly with exponential inverse demand function
Let inverse demand be given by P (Q) = e¡Q . Here, E(Q) = ¡Q, and the
su¢cient conditions of Proposition 2.1 are not satis…ed. We can still show
directly that the equilibrium pro…t functions are decreasing and log-convex in
n. Each …rm’s pro…t function is given by ¼(q) = qe¡Q , which is strictly quasiconcave in q, and attains a maximum at q ¤ = 1: We compute the equilibrium
pro…t as ¼(n) = e¡n . Clearly, ¼(n) is a decreasing function of n and log ¼(n) =
¡n is a convex function.
Now note that ¼(n)=¼(n + 1) = e > 2; 8n: Hence, any two …rms have an
incentive to form a link, and the set of stable networks is very large: any
network with complete components of di¤erent sizes and at most one isolated
…rm is pairwise stable.
Consider now the condition characterizing strongly stable networks: ¼(N ¡
m + 1) ¡ ¼(N) ¡ (m ¡ 1)¼(N ¡ m + 2) ¸ 0: De…ne
f (N; m) = e¡N+m¡1 ¡ e¡N ¡ (m ¡ 1)e¡N+m¡2
= e¡N (em¡1 ¡ 1 ¡ (m ¡ 1)em¡2 ):
It is easy to check that f(N; m) ¸ 0 if and only if m = 2 or m = 3. We conclude
that in strongly stable networks, the sizes of components is either equal to 2 or
to 3: The following table characterizes strongly stable networks for N = 2; 3; 4; 5
and 6: For N ¸ 7; no network is strongly stable.
18
Number of …rms
Component sizes
2
f2g
3
f3g; f2; 1g
4
f3; 1g
5
f3; 2g
6
f3; 2; 1g
In a Cournot market with exponential inverse demand, the rate of decline
of pro…ts is very high. Any pair of …rms has an incentive to form an agreement,
resulting in an abundance of stable con…gurations. If …rms can renege on all
their agreements at once, large alliances become unstable. In a strongly stable collusive network, alliances cannot contain more than three members. This
implies that there does not exist a strongly stable network for N ¸ 7.
Example 3.3 Procurement auction with uniform distribution
Let F (c) = c for all c 2 [0; 1]: We obtain: ¼(n) = 1= [n(n + 1)]. It is easy to see
that ¼(2)=¼(3) = 2. Hence, there are three stable network architectures: the
complete network, the empty network and a network with one component of
size N ¡1 and one isolated bidder. Straightforward computations show that the
complete network is strongly stable if and only if N · 3 and that the network
with one component of size N ¡ 1 is never strongly stable. Hence, for N > 3,
the only strongly stable network is the empty network.
In a procurement auction with a uniform distribution, market sharing agreements become pro…table when the number of competitors is reduced from three
to two. This implies that apart from the empty network, there are two stable
con…gurations: full collusion and a network with one free-rider and N ¡ 1 colluding …rms. For N > 3, these alliances become subject to free-riding of …rms
deleting all their links at once.
Example 3.4 Procurement auction with exponential distribution
Let F (c) = 1 ¡ e¡c for c 2 [0; +1): We obtain
¼(n) =
1
for n > 1 and ¼(1) = +1:
n(n ¡ 1)
19
By analogy with the previous example, we see that ¼(3)=¼(4) = 2. Hence,
there are four stable network architectures: the complete network, the empty
network and networks with components of sizes N ¡ 1 or N ¡ 2. Notice that
the complete network is strongly stable, the network with a component of size
N ¡ 1 is strongly stable if and only if N · 4 and the network with a component
of size N ¡ 2 is never strongly stable.
Procurement auctions have a similar structure with an exponential and a
uniform distribution. In the exponential case, market sharing agreements become pro…table when the number of competitors is reduced from four to three.
This gives rise to a larger number of stable con…gurations, adding one where
N ¡ 2 …rms collude, facing two free-riders. Furthermore, when the distribution
is exponential, the fully collusive network is immune to deviations by individual
bidders reneging on all their agreements at once.
4
Asymmetric Markets and Firms
In this section, we depart from the assumption that …rms and markets are
symmetric. By extending the analysis beyond the simple symmetric case, we
are able to cover more realistic situations but lose the ability to provide a full
characterization of stable collusive networks. Instead, we discuss a collection of
examples pertaining to situations where (i) …rms are symmetric and markets are
not, and (ii) markets are symmetric and incumbent …rms enjoy an advantage.
4.1
Asymmetric Markets
Suppose that markets can be ranked according to pro…tability. We have already
observed in that setting that, whenever two …rms on asymmetric markets are
linked, the …rm with the less pro…table market enters into more agreements
than the other. This suggest that incomplete components may form in a stable
collusive network, as illustrated in the following example.
Example 4.1 Asymmetric markets and Cournot competition.
Consider Cournot competition with homogeneous products on three di¤erent
markets. Let inverse demand on market i be given by the exponential function
20
Pi (Q) = Ai e¡Q and assume A1 = 1 < A2 < A3 . The three markets can
thus be ranked according to pro…tability as ¼ 1 (n) = e¡n < ¼2 (n) = A2 e¡n <
¼3 (n) = A3 e¡n for all n. To emphasize the e¤ects of asymmetric markets,
let us …rst recall the results for the symmetric case described in Example 3.2.
When the three markets are symmetric, a stable network always exists, and
the two stable con…gurations are the complete network and the network with
two linked …rms and one isolated …rm. With asymmetric …rms, not only might
the latter con…gurations not be stable, but also other con…gurations might be
stable; moreover, a stable network might also fail to exist.
1. The complete network is stable only if markets are ‘not too asymmetric’.
The precise condition is A3 · e ¡ 1, which ensures that …rm 1 does not
want to renege on its agreement with …rm 3. Note that strong stability
calls for even less asymmetry: the complete network is strongly stable
¡
¢
provided that A3 · e2 ¡ 1 =e ¡ A2 < e ¡ 1. The three networks with
two linked …rms and one isolated …rm are stable for di¤erent combinations
of the parameters. For instance, the network with a single agreement
between the …rms on the larger markets (…rms 2 and 3) is stable if and
only if A3 · A2 (e ¡ 1).
2. The empty network is stable when markets are ‘su¢ciently asymmetric’.
More precisely, one needs (i) A2 > e ¡ 1, and (ii) A3 > A2 (e ¡ 1). These
conditions ensure that there is relatively enough pro…t to be made on
foreign markets, so that a …rm on a small market does not wish to form
an agreement with a …rm on a larger market.
3. An incomplete collusive network might be stable. In particular, if (i) A2 ¸
e= (e ¡ 1) and (ii) A2 (e ¡ 1) < A3 · e (e ¡ 1), the following network
is stable: the …rm on the smallest market (…rm 1) forms an agreement
with each of the other two …rms, while those two do not form a reciprocal
agreement. This network is strongly stable if, in addition, A2 +A3 · e2 ¡1.
4. A stable network might fail to exist. It can be checked that if (i) A2 < e¡1,
(ii) A3 > e (e ¡ 1), and (iii) A3 < A2 e (e ¡ 1), then no network is stable.
21
4.2
Incumbency Advantage
Firms involved in market sharing agreements often justify market sharing by
the presence of large entry costs into foreign markets. In order to analyze the
validity of this argument, we consider two models where incumbents bene…t
from an advantage in their home market. In the …rst model, …rms face an entry
cost in foreign markets. In the second model, …rms incur a unit transportation
cost when selling in a foreign market.
Entry Costs. Let K denote the entry cost in foreign markets. Suppose K <
¼(N), so that entry costs are not high enough to generate entry barriers. Firm
i’s pro…ts on home and foreign markets are given by ¼ii (n) = ¼ (n) and ¼ij (n) =
¼ (n) ¡ K. Hence, …rm i’s incentive to form an agreement with …rm j is equal
to
Fji (ni ; nj ) = [¼(ni ¡ 1) ¡ ¼(ni )] ¡ ¼(nj ) + K:
We …rst note that the presence of entry costs makes market sharing agreements
more attractive to …rms. We thus expect that agreements are easier to sustain when …rms bene…t from an incumbency advantage on their home market,
and that stable collusive networks exhibit a denser set of links. Second, it is
easy to see that if both …rms i and j have an incentive to form an agreement
(Fji (ni ; nj ) ¸ 0; Fij (nj ; ni ) ¸ 0), they must have the same number of competi-
tors on their home markets: ni = nj : (The presence of a symmetric cost of entry
does not modify our earlier observation in the symmetric case.) Finally, we note
that even if the gross pro…t function ¼(n) is log-convex, there is no guarantee
that net pro…ts on foreign markets be log-convex. The following example illustrates this point, and shows that, when entry costs on foreign markets are
high and log-convexity fails, stable collusive networks may have a very di¤erent
structure than in the symmetric case.
Example 4.2 Linear Cournot model with entry costs
Consider a Cournot oligopoly with zero marginal cost and a linear inverse demand P = 10 ¡ Q. In this example, pro…ts on foreign markets are given by
¼(n) ¡ K =
100
¡ K:
(n + 1)2
22
A direct computation shows that
@ 2 log[¼(n) ¡ K]
@n2
³
´
200 100 ¡ 3K (n + 1)2
=³
:
´2
100 ¡ K (n + 1)2 (n + 1)2
Hence, the pro…t function on foreign markets is not necessarily log-convex,
and we cannot characterize stable collusive networks generally. Instead, the
following table lists stable and strongly stable collusive networks for di¤erent
values of K and N = 6:
Fixed cost
Stable networks
Strongly stable networks
K=0
f6g; f1; 1; 1; 1; 1; 1g
f1; 1; 1; 1; 1; 1g
K = 0:5
f6g; f1; 1; 1; 1; 1; 1g
f1; 1; 1; 1; 1; 1g
K=1
f6g; f1; 1; 1; 1; 1; 1g
K = 1:5
f6g; f5; 1g; f2; 2; 2g
f1; 1; 1; 1; 1; 1g
K=2
f6g; f5; 1g; f4; 2g; f3; 2; 1g
f3; 2; 1g
f2; 2; 2g
This table re‡ects the fact that market sharing agreements are easier to
sustain when the entry cost increases. For low values of the entry cost, stable
networks are either complete or empty, and the only strongly stable network is
the empty network, as in the baseline model with no entry cost. By contrast,
for high values of the entry cost (K = 2), any market sharing agreement is
pro…table, and stable networks can be formed with any combination of components of di¤erent sizes. In that case, the strong stability criterion imposes an
upper bound on the sizes of components, and only one network survives this
criterion. Interestingly, there exists an intermediate case, K = 1:5, where, in a
stable collusive network, …rms may form di¤erent components of the same size.
This is due to the failure of log-convexity and stands in sharp contrast to the
baseline model with no entry cost.
Transportation costs. We assume now that …rms incur a unit transportation cost, denoted t > 0, when selling on a foreign market.
Example 4.3 Linear Cournot model with transportation costs
Consider a homogeneous Cournot market with inverse demand given by P (Q) =
1 ¡ Q, and assume t < 1=2 so that there is no exogenous barrier to entry: The
23
pro…t levels for domestic and foreign …rms are respectively
µ
¶
µ
¶
1 + (n ¡ 1) t 2
1 ¡ 2t 2
¼d (n) =
and ¼f (n) =
:
n+1
n+1
There exists no general method to compute pairwise stable collusive networks in this example. (In particular, two …rms may be linked but have di¤erent
numbers of competitors on their home markets.) We note that, as in the case
of …xed entry costs, collusion is easier to sustain when …rms face transportation
costs in foreign markets. In our example, this is re‡ected by the following two
facts.
² The complete network is always stable. Stability of the complete network
requires ¼d (1) ¸ ¼d (2) + ¼f (2) () (1=36) (10t + 1) (1 ¡ 2t) ¸ 0, which
is satis…ed since, by assumption, t < 1=2.
² The empty network is stable if and only if transportation costs are low
enough: ¼d (N ¡ 1) < ¼d (N) + ¼f (N) () t <
N 2 ¡2N¡1
2(2N 2 ¡N¡1)
< 1=2:
Moreover, in the presence of transportation costs, the sets of stable and
strongly stable collusive networks are likely to enlarge. This is illustrated in
the simple case with four …rms. In the absence of transportation costs, the
situation is as described in Example 3.1: the empty and complete networks are
the only stable networks, and the only strongly stable network is the empty
network. Simple computations establish two major di¤erences due to positive
transportations costs.
² The network with a complete component of three …rms and one isolated
…rm is stable if and only if (i) ¼d (2) ¸ ¼d (3) + ¼f (3) () t ¸ 1=14 '
0:07, and (ii) ¼d (4) + ¼f (2) > ¼d (3) () t < 319=998 ' 0:32; this
network is strongly stable if, in addition, ¼d (2) ¸ ¼d (4) + 2¼f (3) ()
t ¸ 97=674 ' 0:14.
² The complete network is stable for all t and is strongly stable if and only
if ¼d (1) ¸ ¼d (4) + 3¼f (2) () t ¸ 37=254 ' 0:15.
24
5
Industry Pro…ts and Social Welfare
In this section, we discuss e¢ciency of stable networks. We focus on the symmetric model, and later extend our analysis to the two cases where …rms enjoy
an incumbency advantage on their home market. De…ne total industry pro…ts
on market i as T (ni ) = ni ¼(ni ) and social surplus as W (ni ). We assume
Property 5 Total industry pro…ts are decreasing in the number of …rms active
on the market, T (ni ) · T (ni ¡ 1):
Property 6 Social surplus is increasing in the number of active …rms on the
market, W (ni ) ¸ W (ni ¡ 1):
We recall that these two properties are satis…ed in Cournot oligopolies with
homogeneous products and symmetric procurement auctions.
Proposition 5.1 In a Cournot oligopoly with homogeneous products, if costs
are linear and E(Q) > ¡1, total industry pro…ts are decreasing in the number
of active …rms on the market and social surplus is increasing in the number of
active …rms on the market. In a symmetric procurement auction, total pro…ts
are strictly decreasing in the number of active bidders and the expected social
surplus is increasing in the number of active bidders.
Proof. These results are well-known. References can be found in Vives
(1999, pp. 107-109) for Cournot oligopoly models and in Mac Afee and Mac
Millan (1987, p. 711) for symmetric auctions.
Total industry pro…ts are thus maximized in the complete network, where
all …rms are monopolies on their home market. Conversely, social surplus is
maximized in the empty network, where all …rms are present on all foreign
markets. As stable collusive networks always lie between these two extreme
con…gurations, we observe that stable collusive networks are under-connected
from the point of view of the industry and over-connected from the point of view
of the society.8
8
In many examples, both the complete network (maximizing industry pro…ts) and the
empty network (maximizing social surplus) are stable. Our point is that any stable collusive
network is either under-connected from the point of view of the industry, or over-connected
from the point of view of society, or maybe both.
25
Interestingly, this observation does not hold when …rms enjoy an incumbency advantage on their home market. In the presence of entry costs, it is
well known that private incentives to enter may be higher than social incentives, resulting in excessive entry on the market. In Example 4.2, the optimal
number of …rms on each market is equal to 6 for K = 0; 5 for K = 0; 5 and
K = 1 and 4 for K = 1:5 and K = 2: Hence, in stable (or even strongly stable)
collusive networks, the number of …rms on each market may be too large. A
similar conclusion holds in the presence of transportation costs: in Example 4.3
with four …rms, the optimal number of …rms on each market is equal to 4 for
0 · t · 9=58, to 3 for 9=58 · t · 7=38, to 2 for 7=38 · t · 5=22, and to 1 for
5=22 · t < 1=2. This suggests that, when …rms enjoy a signi…cant incumbency
advantage (through high entry or transportation costs), competition authorities
should actually favor the formation of market sharing agreements as a way to
discourage entry on foreign markets.
6
Conclusion
This paper analyzes the formation of market sharing agreements among …rms
in oligopolistic markets and procurement auctions. The set of agreements de…nes a collusive network, and the paper provides a complete characterization of
stable collusive networks when …rms and markets are symmetric. Stable networks are formed of complete alliances, of di¤erent sizes, larger than a minimal
threshold. Typically, stable networks display fewer agreements than the optimal network for the industry and more agreements than the socially optimal
network. When …rms or markets are asymmetric, incomplete alliances can form
in stable networks, and stable networks may be under-connected with respect
to the social optimum.
Our analysis sheds light on the recent wave of market sharing agreements in
a number of industries. We …nd that, in order to be stable, alliances must reach
a minimum size, and that an alliance grouping all …rms in the industry (as in
the chemical industry) is more likely to be stable. When di¤erent alliances form
(as in the airline industry), the market sharing agreements must be complete
– all …rms in the alliance are linked by bilateral agreements – and di¤erent
26
alliances must have di¤erent sizes, with …rms in smaller alliances free-riding on
the formation of agreements by other …rms. Our study also shows unambiguously that, in a symmetric setting, the formation of market sharing alliances
is harmful and should be corrected by an adequate antitrust policy. On the
other hand, when …rms face entry or transportation costs in foreign markets,
the formation of market sharing agreements may be bene…cial, as it helps to
correct excessive entry of …rms into foreign markets.
While we believe that our analysis provides a useful application of recent
developments in the theory of economic networks to a concrete problem in industrial organization, we are aware of two important shortcomings of our study.
First, in order to keep the problem tractable, we assume a very simple form
of market sharing, where …rms commit not to enter each other’s territory. In
reality, market sharing agreements may take much more complex forms, with
groups of …rms allocating a …xed set of markets among themselves. This form
of market sharing seems to be prevalent in procurement auctions, as exempli…ed by Pesendorfer(2001)’s recent study of Texas school milk contracts. While
we believe that our basic intuition on the trade-o¤ between reduction in the
number of competitors and access to markets will carry over to more general
models of market allocation, a complete analysis of market sharing agreements
in more complex settings remains to be done. Second, and most importantly,
we suppose that market sharing agreements are enforceable, without explicitly
modelling a dynamic framework of interaction. The analysis of the enforceability of market sharing agreements seems to us to be a particularly promising
area of research. By forming market sharing agreements, …rms can choose the
number of markets on which they will compete, and hence endogenously determine the level of multimarket contact. Hence, as in Bernheim and Whinston
(1990), the study of enforceability of market sharing agreements relies on the
interplay between collusion and multimarket contact. We plan to tackle this
issue in future research.
27
7
Appendix
7.1
Proof of Proposition 2.1
As the other result is well-known, we concentrate on the proof of statement (ii).
Di¤erentiating individual pro…ts with respect to qi , we obtain the …rst-order
condition: P 0 (Q)qi + P (Q) ¡ c0 (qi ) = 0: As all …rms are identical, we write
qi = qj = q 8i; j, so that
P 0 (Q)q + P (Q) ¡ c0 (q) = 0:
(6)
Treating n as a continuous variable and assuming linear costs, we obtain, by an
implicit di¤erentiation of Equation (6):
q
QP 00 (Q) + nP 0 (Q)
@q
=¡
;
@n
n QP 00 (Q) + (n + 1) P 0 (Q)
(7)
yielding
(n ¡ 1)
@q
q QP 00 (Q) + 2nP 0 (Q)
+q =
:
@n
n QP 00 (Q) + (n + 1) P 0 (Q)
(8)
Rearranging,
(n ¡ 1)
@q
Q (2n + E(Q)
+q = 2
@n
n (1 + E(Q) + n)
Hence,
Q2 P 0 (Q)(2n + E(Q))
d¼(n)
=
:
dn
n3 (1 + E(Q) + n)
Di¤erentiating this expression again with respect to n, we get:
d2 ¼(n)
A(n)
= 6
;
2
dn
n (1 + E(Q) + n)2
where
·
¸
dQ
dQ dQ2 P 0 (Q)
2 0
0
A(n) = [n (1 + E(Q) + n)] Q P (Q)(2 + E (Q)
) + (2n + E(Q))
dn
dn
dQ
·
µ
¶¸
dQ
¡[(2n + E(Q))Q2 P 0 (Q)] 3n2 (1 + E(Q) + n) + n3 1 + E 0 (Q)
:
dn
3
2
0
As costs are linear, c(q) = c0 (q)q. Hence ¼(n) = q(P (Q)¡c0 (q)) = ¡ Q
n2 P (Q).
We thus have:
d2 ¼(n)
¼(n)
¡
dn2
µ
d¼(n)
dn
¶2
=
¡Q2 P 0 (Q)[A(n) + n2 (2n + E(Q))2 Q2 P 0 (Q)]
:
n8 (1 + E(Q) + n)2
(9)
28
Using Equation (9), in order to show that ¼(n) is log-convex, it su¢ces to
establish A(n) + n2 (2n + E(Q))2 Q2 P 0 (Q) ¸ 0: Now,
dQ dQ2 P 0 (Q)
Q2 P 0 (Q)(2 + E(Q))
=
:
dn
dQ
n(1 + E(Q) + n)
Hence,
A(n) = Q2 P 0 (Q)[2n3 (1 + E(Q) + n) + n2 (2n + E(Q))(2 + E(Q))
¡3n2 (1 + E(Q) + n)(2n + E(Q)) ¡ n3 (2n + E(Q))]
dQ 3
[n (1 + E(Q) + n) ¡ n3 (2n + E(Q))]
+Q2 P 0 (Q)E 0 (Q)
dn
= Q2 P 0 (Q)[¡n2 (6n2 + 6nE(Q) + 2E(Q)2 + E(Q)]
dQ 3
¡Q2 P 0 (Q)E 0 (Q)
n (n ¡ 1):
dn
Rearranging,
dQ
A(n) + n2 (2n + E(Q))2 Q2 P 0 (Q) = ¡Q2 P 0 (Q)n2 [E 0 (Q)
n(n ¡ 1)
dn
¡
¢
+ E(Q)2 + (2n + 1) E(Q) + 2n2 ]:
By assumption, E 0 (Q) ¸ 0. Furthermore, E(Q)2 + (2n + 1) E(Q) + 2n2 > 0
8n ¸ 2 and 8E(Q): Hence, as P 0 (Q) < 0 we obtain
A(n) + n2 (2n + E(Q))2 Q2 P 0 (Q) > 0;
showing that the pro…t function ¼(n) is log-convex.
7.2
Proof of Proposition 2.2
We concentrate again on the log-convexity of pro…ts, as the other statement of
the proposition can be obtained using well-known arguments. A direct computation shows that
0
¼ (n) =
Z
C
0
log(1 ¡ F (c))(1 ¡ F (c))n¡1 F (c)dc < 0:
To show that pro…ts are log-convex, compute
Z C
00
¼ (n) =
(log(1 ¡ F (c))2 (1 ¡ F (c))n¡1 F (c)dc:
0
De…ne p(c) = (1 ¡ F (c))n¡1 F (c) and g(c) = ¡ log(1 ¡ F (c)): The function
p is …nite and strictly positive over (0; C) and the function g is …nite and
29
nonnegative over the same set. Hence, we may de…ne, as in Hardy, Littlewood
and Polya (1952, p.134), the weighted integral mean of the function g as
Mr (g; p) =
µR
p(c)gr (c)dc
R
p(c)dc
¶ 1r
:
By a generalization of Schwartz’s inequality (Theorem 192, p. 143 in Hardy,
Littlewood and Polya, 1952), if r < s, then Mr (g; p) < Ms (g; p): Applying this
result to the special case r = 1; s = 2, we obtain:
µZ
¶2
Z
Z
p(c)g(c) dc < p(c)dc p(c)g(c)2 dc:
Replacing p and g with their expressions:
µZ
<
Z
0
C
0
C
log(1 ¡ F (c))(1 ¡ F (c))
n¡1
(1 ¡ F (c))
F (c)dc
Z
0
C
n¡1
F (c)dc
¶2
(log(1 ¡ F (c))2 (1 ¡ F (c))n¡1 F (c)dc:
Hence, ¼0 (n)2 < ¼(n)¼00 (n); showing that individual pro…ts are strictly logconvex in n:
7.3
Proof of Proposition 3.2
Consider a pairwise strong Nash equilibrium s¤ . By Lemma 2.1, g(s¤ ) is a stable
network, and can be decomposed into complete components of sizes greater than
m¤ . Suppose, by contradiction, that some component gl does not satisfy the
condition: ¼(N ¡ m(gl ) + 1) ¸ ¼(N) + (m(gl ) ¡ 1)¼(N ¡ m(gl ) + 2): Then we
claim that s¤ cannot be a Nash equilibrium, as any …rm i in gl has a pro…table
deviation by choosing s0i = ;:
Conversely, suppose that the graph g can be decomposed into a set I of
isolated …rms and disjoint complete components g1 ; g2 ; :::; gL such that m(gl ) 6=
m(gl0 ) 8l 6= l0 ; and ¼(N ¡ m(gl ) + 1) ¸ ¼(N) + (m(gl ) ¡ 1)¼(N ¡ m(gl ) + 2) 8l.
Consider the following strategies for the …rms: If …rm i belongs to a component
gl , it announces s¤i = fjjj 2 gl ; j 6= ig: If i is isolated, it announces s¤i = ;. We
show that these strategies form a pairwise strong Nash equilibrium. Clearly,
no …rm i has an incentive to create a link to a …rm j in another component, as
i2
= s¤j . Furthermore, as m(gl ) 6= m(gl0 ) 8l 6= l0 , no pair of …rms has an incentive
to create an additional link. Now consider a …rm’s incentive to destroy some
30
links. As ¼(N ¡ m(gl ) + 1) ¸ ¼(N) + (m(gl ) ¡ 1)¼(N ¡ m(gl ) + 2); the …rm
cannot bene…t from destroying all its links. By the argument we give in the
text, this implies that it cannot bene…t from deleting any subset of links.
7.4
Iso-elastic Inverse Demand Function
We claim that in a Cournot model with iso-elastic demand:
¼(1)
¼(2)
>2>
:
¼(2)
¼(3)
The left inequality is immediately obtained as T (1) > T (2). The right inequality
is equivalent to log 2 + log ¼(3) ¡ log ¼(2) > 0; which can be rewritten as
¶
µ
3+®
> 0:
f(®) ´ (2® ¡ 1) log 2 + (1 ¡ ®) log 3 ¡ (1 + ®) log
2+®
Immediate computations show that f 00 (®) > 0 and f 0 (0) = 3 log 2 ¡ 2 log 3 +
1=6 > 0. Hence f(®) is a strictly increasing function and, as f(0) = 0, we
conclude that f (®) > 0 for all ® 2 (0; +1). Hence, the only stable networks
are the empty and complete networks for a Cournot oligopoly with isoelastic
demand.
Turning now to strongly stable networks, de…ne g(N) = ¼(N)+(N ¡1)¼(2).
The second derivative is given by g00 (N) = ¼00 (N). As ¼ is log-convex, it is
necessarily convex, so g 00 (N) > 0: Furthermore, evaluating g0 (N) at the lower
bound N = 2, we obtain
g0 (2) = ¼0 (2) + ¼(2) = ®(1 + ®)2
1¡2®
®
(2 + ®)¡
1+2®
®
> 0:
Hence g(N) is a strictly increasing function, and, if the complete network is
strongly stable for some N ¸ 3; ¼(1) ¸ ¼(3) + 2¼(2): Furthermore, as ¼(3) >
¼(2)=2, we obtain: ¼(1) > (5=2) ¼(2). We …nally show that the latter inequality
is never satis…ed. Rewriting it, we obtain:
h(®) = ® log 5 + (1 ¡ 2®) log 2 ¡ (1 + ®) log
µ
2+®
1+®
¶
< 0:
It is easy to see that h00 (®) > 0 and h0 (0) = log 5¡ 3 log 2 + 0:5 > 0: Hence, h(®)
is an increasing function and, as h(0) = 0, the inequality cannot be satis…ed
for any value of ®. We conclude that the complete network is never strongly
stable, and hence the only strongly stable network is the empty network.
31
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