Partial coordination and mergers among
quantity-setting firms
Marc Escrihuela-Villar∗
Universidad de Guanajuato†
December, 2006
Abstract
This paper studies an infinitely repeated game where a subset of firms colludes
(cartel firms) while the rest maximize per period profits (fringe firms). For high
discount factors, cartel firms can sustain the fully collusive outcome while for low
discount factors, cartel firms only achieve partial coordination and their profits are
decreasing in the discount factor. The model provides a setting to study the effects
of horizontal mergers in a cartelized market. Although mergers have an ambiguous
effect on the possibility of sustaining full collusion, they definitely raise price. On
the other hand, cartel firms have fewer incentives to merge when they partially
coordinate their output. However, in this case, fringe firms have more incentives to
merge.
JEL Classification: L13,L40,L41
Keywords: Collusion, Partial Cartels, Trigger strategies, Mergers
∗
Mailing address: Department of Economics, University Carlos III Madrid. Calle Madrid 126, 28903
Getafe (Madrid), Spain. Phone: +34 916245737. Fax: +34 916249329. E-mail: [email protected]
†
Mailing address: Escuela de Economía, Universidad de Guanuajuato. UCEA-Campus Marfil,
Fracc.
I, El Establo.
Guanajuato GTO 36250 (México) Telf.
E-mail: [email protected] and [email protected]
1
+52 473 7352925 Ext.
2660.
1
Introduction
Faced with the multiplicity of equilibria arising in repeated games, most applied papers
have focused on studying the equilibrium where industry pro…ts are maximized assuming
that all …rms participate in the collusive agreement.1 However, at least to the best of our
knowledge, little attention has been paid to the situation where not all the …rms of the
industry join the cartel (an exception is Martin (1993) to be commented on later).2 In
practice, there are many cases where collusive agreements do not involve all …rms in the
industry. For instance, looking at recent measures of the European Commission against
collusion, we see that several cartels have been …ned hundreds of millions of euros by the
European Commission for price …xing and setting sales quotas. At the same time, it is
also veri…ed that in many of these industries not all …rms in the industry were …ned. For
example, in the citric acid industry three North-American and …ve European …rms were
convicted in United States, Canada and the European Union and …ned more than one
hundred million euros for …xing prices and allocating sales in the worldwide market, issuing
coordinated price announcements and monitoring one another’s prices and sales volumes
during the period 1991-1995. The joint market share of these eight …rms was only between
50% and 60% (see Levenstein, Suslow et al. (2003)).3 Despite this empirical evidence,
the Industrial Organization analysis of tacit collusion in quantity setting supergames has
1
The “Folk Theorems”show that any individually rational payo¤ vector of a one-shot game of complete
information can arise in a perfect equilibrium of the in…nitely-repeated game if players are su¢ ciently
patient (see for example Friedman (1971), Rubinstein, A. (1979) or Fudenberg et al. (1986)).
2
There is a well-known literature where the industry structure is characterized by a small group of
…rms plus a competitive fringe. However this literature investigates cartel stability in static models where
it is assumed that binding contracts could be signed. The seminal papers in this literature are Selten
(1973) and d’Aspremont et al. (1983); see also Sha¤er (1995), Donsimoni (1986) or Thoron (1998) for
example.
3
U.S. Department of Justice, Justice Department’s Ongoing Probe Into Food and Feed Additives
Yields Second Largest Fine Ever, Press Release, January 29, 1997.
2
generally focused on the symmetric subgame perfect equilibrium that maximizes industry
pro…ts and has established the conditions under which full collusion can be sustained.
However, the continuum of equilibria in supergames allows us to also select equilibria
such that only a subset of …rms colludes and only a partial coordination is achieved.
Indeed, there is a seminal paper on partial cartels (Martin 1993 p.111) from which this
paper gets part of its inspiration. He considers that only a subset of …rms belongs to the
cartel and the rest are called fringe …rms. Then, he studies conditions on the discount
factor such that the outcome where cartel …rms behave as a Stackelberg leader and fringe
…rms as followers can be sustained as equilibrium of the repeated game using “trigger”
and “stick-and-carrot”strategies.
In the present paper, Martin’s (1993) model is extended by identifying the best equilibrium for cartel …rms using “trigger strategies”. The usual “knife-edge” con…guration
(no collusion or full collusion if the discount factor is lower or higher than the critical
threshold) is questioned. For each possible level of the discount factor the output that
cartel …rms can produce to maximize their joint pro…ts is obtained. (This means that
for any value of the discount factor of cartel …rms, some degree of collusion is always
achieved.) It coincides with the Stackelberg-follower model for high discount factors and
converges to the Cournot outcome when the discount factor is close to zero. Thus, the
present model essentially represents two contributions; …rst a partial cartel is analyzed in
a repeated game, and second the way to approach collusion is from a new and perhaps
more realistic perspective as the framework does not necessarily consist of either no or
full collusion. In this paper, when …rms fail to sustain full collusion they can nevertheless
achieve some degree of collusion.
Although the speci…ed model with collusion can be used to study di¤erent subjects, we
specialize the model to deal with the issue of mergers. In contrast to noncollusion models,
collusion models have been unpopular in formal merger analysis. Curiously, there exists
3
substantial literature on the e¤ect of mergers when pre-and post-merger behavior is noncooperative, however, little attention has been paid by theorists to the consequences for
…rms of mergers when they take place in a collusive environment. In fact, despite its
in‡uence on matters of policy, there is almost no formal evidence from theory regarding
the relationship between mergers and collusion.4 Nevertheless, given the antitrust concern
with collusion, a formal analysis of mergers and collusive behavior is clearly called for.
To that extent, the second part of the paper considers the competitive e¤ects of horizontal mergers on pro…ts and welfare. In the sparse literature on the subject, the e¤ect
of mergers is analyzed to see what the e¤ect on the potential sustainability of the collusive agreements is. This is re‡ected in the e¤ect of mergers on the minimum discount
factor required for collusion to be sustainable and it is commonly believed that mergers, by reducing the number of competitors, facilitate collusion. Due to the particular
characteristics of our partial cartel model, the result of studying the e¤ect of mergers on
the threshold of the discount factor above which (full) collusion is sustained loses interest and is ambiguous. This follows naturally from the recognition that partially collusive
arrangements may exist before the change in market structure and that collusion does not
necessarily break down after a merger. Therefore, it would be more informative to study
the e¤ect of mergers on a variable that better re‡ects the degree of collusion achieved in
the industry. It seems that price is the most indicated candidate.
We obtain that mergers not creating synergies raise price. At one level, the result
may be interpreted as an extension of Farrell and Shapiro’s (1990) results when some (or
all) …rms can partially coordinate their output. At another level, this result raises the
question as to whether the traditional antitrust view that mergers facilitate collusion if
4
There are a few exceptions like Rotschild (1999). Another contribution of interest is that of Davidson
and Deneckere (1984) where they do not allow for a redistribution of output among …rms after the merger,
so it is assumed that each merged …rm conserves its pre-merger cartel quota after the merger. This implies
that horizontal mergers may make it more di¢ cult for a cartel to maintain its integrity.
4
they decrease the critical discount factor above which full or unconstrained collusion can
be sustained can be misleading.
On the other hand, studying the private incentives to merge represents a contrasting
result to the analysis of exogenous Cournot mergers in Salant et al. (1983) in a model
where …rms can partially coordinate their outputs. We observe that the incentives to
merge are sensitive to the actual discount factor of the …rms. In other words, they depend
on the degree of coordination that …rms achieve when they can partially coordinate their
outputs. It is obtained that cartel …rms have fewer incentives to merge when they are
already colluding. The basic intuition is that, in the absence of synergies, the motive for
mergers is purely anticompetitive. However, by colluding, …rms also restrict competition
without the need for any precommitment that tends to be irreversible in the case of a
merger. In contrast, coordination between cartel …rms increases fringe …rms’incentives
to merge. The intuition is clear: it is well known that in a linear oligopoly Cournot
market, …rms have few incentives to merge because non-merging …rms react to the merger
expanding their production. The last result, however, is explained by the fact that when
cartel …rms are coordinating their output, they react expanding their production less in
response to the merger of fringe …rms.
We can think of a possible interesting example of the last result. Looking at what
happened to the oil market in 1998-2000, we identify two di¤erent phenomena. On the
one hand, after being elected in 1998, the Venezuelan leader Chávez worked with the
OPEC president Alí Rodríguez to reinvigorate the cartel.5 On June 1998, in its 105th
meeting the cartel agreed to a cut of more than 2.5 million barrels a day. In March
1999, Mr. Rodriguez cut a deal with Saudi Arabia, the world’s biggest producer, to
reduce production by almost 2 million barrels a day and reverse the slide in prices. That
5
”And now, OPEC has arisen again,” President Chavez declared, adding that ”its resurrection will
rightly be celebrated in this land of the birth of OPEC” CNN, September 26, 2000.
5
move encouraged the rest of OPEC’s 11 member nations to follow suit. On the other
hand, during this period, many oil …rms merged. We have di¤erent examples: B.P. and
Amoco, Japan Oil and Mitsubishi Oil, Exxon and Mobil, B.P. and Arco or TotalFina and
Elf. Summarizing, at the end of the 90’s, a wave of fringe …rms mergers together with a
reinforcement of the collusive strategy of the cartel took place simultaneously. The present
model might illustrate the basic economic intuition about the relationship between these
two facts.
The structure of the paper is as follows. In the following section, the model of partial
coordination is set and the conditions under which a stable collusive arrangement can
occur are analyzed. In section 3, the e¤ect of mergers on the price and …rms’ private
incentives to merge are analyzed. In section 4 the equilibrium selection is based on the
assumption that …rms use a more severe punishment — Abreu’s (1986) stick-and-carrot
strategies— and it establishes that the general conclusions continue to hold. We conclude
in section 5. All proofs are relegated to the appendix.
2
Partial collusion
Consider an industry of n …rms, playing an in…nitely repeated oligopolistic game with
discounting. Firms are quantity setting and produce a homogeneous product at a constant
marginal cost c > 0. Firms discount the future at a common and known factor of 2 [0; 1).
Suppose they make output decisions in the beginning of each period. Let qi;t be the
quantity chosen by …rm i; i 2 f1; :::; ng in period t; t = 1; 2; :::
The stage game description is the following: we assume that the industry inverse
demand is piecewise linear:
p(Q) = max(0; a
Q)
(1)
where Q is the industry output, p is the price of the output and a > 0 with a > c. We
6
take n as …xed and exogenous. We shall explore the possibility that k of these …rms
behave cooperatively to maximize their joint pro…ts, while the n
k fringe …rms choose
their output levels noncooperatively. Then, let (K; F ) be a partition of the player set.
We assume that the subset K comprises k (1 < k
remaining (n
n) …rms of the industry, while the
k) …rms belong to the subset F . Hereafter, we will call “cartel …rms”
and “fringe …rms”, the …rms that belong respectively to the subsets K and F . It is also
assumed that only one cartel is formed. Assume that in the absence of collusion, all …rms
behave like Cournot competitors.
A basic insight from the supergame literature is that collusive agreements can bene…t
the players, but they also create incentives for some …rms to undermine the tacitly collusive
agreement by (secretly) expanding their individual output. Hence, in order for tacit
collusion to be possible, …rms have to use their ability to punish each other’s deviations
from any supposed equilibrium path by using a credible penal code. A standard example
of a credible penal code is the one proposed by Friedman (1971), which consists of a
Cournot-Nash reversal forever.
Following Friedman, in this section we restrict strategies to grim “trigger strategies”.
The essence of these strategies is as follows. Firms adhere to the cartel agreement for as
long as all cartelists …rms do so; in the event of deviation, the “loyal” members of the
coalition revert forever to the static (Cournot) noncooperative equilibrium. This forces
the deviant to do the same. Then, this is a deterrent to a prospective deviant whenever the
capitalized value of losses (due to retaliation) exceeds the one period gain from cheating.
The setting is one of perfect monitoring so that …rms’quantities over the preceding t
1
periods are common knowledge in period t. For each …rm let q, q f and qn denote the
output corresponding respectively to collusion, fringe …rms and Cournot non-cooperative
behavior.
More formally, consider an in…nite replication of a single stage game, in which there
7
exists for each …rm i 2 K a pure strategy denoted by the vector
t 2 [0; 1];
i;t
i;t .
At any stage t;
indicates an action (choice of output), which is a function of the actions
of all cartel …rms in all stages to t
1. Then, a trigger strategy for a cartel …rm can be
de…ned as follows:
i;1
i;t
=
q
8
< q if
=
: q
j;h
when t > 2
= q for any h < t and j 2 K,
Otherwise
n
Fringe …rms act as Cournot quantity-setting …rms. Then, the future play of cartel
…rms is independent of how fringe …rms play today. Therefore, the optimal response of
fringe …rms is to maximize their current period’s payo¤:
q f = maxf0; (
a c kq
)g
n k+1
(2)
The corresponding level of cartel …rms’pro…ts is:
c
(q; q f ) = (a
c
kq
k)q f )q
(n
(3)
We also denote the one period gain from deviation by:
d
(q; q f ; qid ) = (a
c
(k
1)q
qi
(n
k)q f )qid
(4)
where qid is the optimal cheat quantity produced by the cheating …rm:
qid = arg max
d
(q; q f ; qi )
qi
For the case at hand, cartel …rms producing q constitute a Subgame Perfect Nash
Equilibrium (SPNE) if and only if:
c
(q; q f )
1
c
d
(q; q f ; qid ) +
(q n ; q n )
1
(5)
However, we have multiplicity of equilibria as for each , (5) leaves a large set of SPNE
collusive outputs. To solve the multiplicity of equilibria we consider the best allocation
8
for the cartel i.e. maximizing cartel …rms’pro…ts, which is the solution to the following
program:6
max
c
(q; q f )
d
f
; qid )
q
(6)
st
(q; q f )
1
c
(q n ; q n )
1
c
(q; q
+
Observe that when the cartel is active ( > 0), it is implicitly assumed (as in all
cartel and fringe literature cited in the introduction) that in every period t the cartel
anticipates the fringe and behaves as a Stackelberg leader with respect to the fringe; the
reasonableness of this assumption will be explored in a subsection. Also note that due to
the monitoring setting we assume, cheating in period t
1 is detected by all …rms (fringe
or cartel) in period t.7
Solving (6) we obtain the unique maximizer of
q =
a c
).
2k
c
(q; q f ), which we denote by q (with
_
Therefore, if
is larger than a certain value (i.e. if
), the restriction
(5) evaluated at q is satis…ed and in this case, q is the solution to the program. It is a
standard exercise to verify that:
=
(n + 1)2
(n + 1)2 + 4k(n + 1
k)
_
However, when
<
…rms are not patient enough to sustain q, (5) is a binding
constraint and the distribution of output (q) is the solution to the equality constraint. In
this case, the main novelty of the present paper is that the formal structure of the model
6
This is also the widely extended way to solve the multiplicity of equilibria in in…nitely repeated games
with collusion at the industry level (k = n).
7
Simultaneous deviations are ignored since, as was pointed out by Fudenberg and Tirole (1991), “in
testing for Nash or subgame-perfect equilibria, we ask only if a player can gain by deviating when his
opponents play as originally speci…ed ” (p.281).
9
allows for coordination on output levels di¤erent from q. Then, the equilibrium quantities
(q; q f ) are connected to the magnitude of
and are given by:
q = qc
qf = (
where
qc =
and
8
(a
>
>
>
>
>
<
>
>
>
>
a
>
:
qf =
a c kq c
)
n k+1
c)((1 + n)2 + (1 2k + n)(3 2k + 3n) )
(1 + n)3 (1 + n)(1 2k + n)2
_
if
_
c
if
2k
8
(a c)(1 n(2 + n)( 1 + )
+ 4k 2 )
>
>
>
>
(1 + n)3 (1 + n)(1 2k + n)2
>
<
>
>
>
>
>
:
a
2(n
<
c
k + 1)
_
if
<
_
if
Observe that by solving (6), we chose the best equilibrium for cartel …rms. It is easy to
a c
). Note also that
see that then kq is the output of a unique Stackelberg leader (kq =
2
it does not depend on the number of …rms in the fringe and it amounts to the monopoly
output. This is because in the repeated game the best thing for cartel …rms is to sustain
the monopoly output. However, this is only possible when
. In this case, perfect or
full collusion is sustainable. Otherwise, perfect collusion is not sustainable because the
discount factor is not large enough and …rms are only partially coordinating their outputs.
In this case, an interesting question is how q c should be adjusted to maximize cartel pro…ts
_
when coordination is only partial ( < ). It turns out that it crucially depends on the
number of …rms in the cartel. Since the sign of
most …rms belong to the cartel (k >
n+1
),
2
@q c
@
and
@q f
@
is given by n
2k + 1, when
q c should be adjusted upwards in order to
reduce the incentives to deviate from the cartel agreement. This follows from the fact
10
that total output produced by all …rms (cartel and fringe …rms) except the deviant moves
in the same direction as the production quotas of the cartel. Then, the residual demand
left to the deviator is lower and therefore the potential deviant gains less by deviating.
This adjustment is greater the lower the discount factor. The standard Cournot output
is obtained when
= 0. However, when most …rms belong to the fringe (k <
n+1
),
2
quotas
must be adjusted downwards in order to reduce the incentives to deviate. The reason
is that now total output produced by all …rms except the deviant moves in the opposite
direction of the quotas because fringe …rms increase their output when the cartel quota
is reduced. This e¤ect now dominates because fringe …rms are the majority. Again, this
adjustment is greater the lower the discount factor and the standard Cournot output is
obtained when
= 0.
Figures 1 and 2 represent the output produced by cartel and fringe …rms as a function
of
for a given industry size n and a partition K when cartel …rms are a majority and a
minority respectively.
11
n +1
2
k>
0.2
q
Fringe
0.15
0.1
Cartel
0.05
0
0
0.25
0.5
0.75
1
delta
Figure 1
k<
n +1
2
q
Cartel
0.25
0.2
0.15
0.1
Fringe
0.05
0
0
0.25
0.5
0.75
1
delta
Figure 2
12
An implication is that market price decreases with when k <
k>
n+1
.
2
n+1
2
and increases when
Another basic insight from the present model concerns the comparison between
pro…ts for each …rm at the cartel and at the fringe respectively. Denote by
f
c f
n;k (q ; q )
the pro…t function of a fringe …rm:
f
c f
n;k (q ; q )
= (a
c
kq c
(n
k)q f )q f
(7)
Easy calculations show:
Proposition 1 For any , pro…ts of each cartel member are enhanced by cartelization
relative to symmetric Cournot oligopoly. Regarding fringe …rms, pro…ts are only enhanced
by cartelization when k >
n+1
.
2
Note (see Figures 1 and 2) that for k >
n+1
2
we have q f > q n > q c and for k <
have q c > q n > q f . Intuitively, this occurs because if k >
n+1
2
n+1
2
we
fringe …rms gain more than
cartel members from the high price induced by the cartel. Fringe …rms have incentives to
free ride from the output reduction agreed on by the cartel expanding their production
(see Figure 1). Hence, the pro…ts in the cartel are lower when k >
n+1
2
than when k <
By contrast, when fringe …rms are in a majority position (k <
n+1
2
see Figure 2) cartel
n+1
.
2
…rms use their leader’s role to expand production to achieve a higher market share. Then,
in this case the pro…t level for each fringe …rm is lower than the pro…ts of each cartel
member.
Once we examined the output levels and pro…ts of …rms, it is also interesting to ask
how the price varies with the number of cartel …rms for a given discount factor. Since
market price is given by the following expression:
8
2
< (1+n)2 (a+cn) (1 2k+n)(a(1+2k+n)+c(n+n
(1+n)3 (1+n)(1 2k+n)2
p(n; k; ) =
: a+c(1+2(n k))
2(n k+1)
Di¤erentiation with respect to k yields the following result:
13
2k(2+n)))
if
if
<
(8)
Proposition 2 For any , a relatively large cartel (k >
(n+1)
)
4
is a su¢ cient condition
for the market price to increase with the number of cartel …rms.
In other words, for a su¢ ciently large cartel when the group of collusive …rms is
bigger, the market price is also higher. Further, cartels cause prices to rise above Cournot
levels when k >
(n+1)
:However,
2
the reverse is true when k <
(n+1)
.
2
The intuition which
underlies this result is clear. When cartel …rms are a majority they cut production in
order to increase market price as they control most of the market. Then, a larger cartel
also implies a higher market price. Conversely, when cartel members are a minority they
increase production to achieve a higher market share and this lowers market price. The last
e¤ect makes prices fall even below Cournot levels. Moreover, in the case of a su¢ ciently
small cartel, market price could even decrease in the number of cartel members. Now we
are ready to study the conditions under which a stable collusive arrangement can occur.
2.1
Endogenous cartels
In this subsection, our aim is to analyze the stable coalition that would emerge if we
endogeneize cartel formation and the equilibrium sequence of moves between the cartel
and the fringe. Following d’Aspremont et al. (1983) we assume the standard notion of
stability: a cartel is stable when (i) …rms inside do not …nd it desirable to exit and (ii)
…rms outside do not …nd it desirable to enter.8
(i) Internal stability: apart from the degenerate case of k = 1, this property is equivalent to condition:
f
c f
n;k 1 (q ; q )
c
c f
n;k (q ; q )
(9)
(ii) External stability: likewise, apart from the case k = n this property is equivalent
8
It is assumed that …rms hypothesize that no other …rm will change its strategy concerning membership
in the cartel.
14
to condition:
f
c f
n;k (q ; q )
c
c f
n;k+1 (q ; q )
(10)
Following standard practice, we shall de…ne a cartel to be stable if it is both internally
stable and externally stable. Note that any combination of fringe and cartel satisfying
both (9) and (10) will constitute a SPNE outcome, in that no precommitment mechanism
would be needed to enforce the output levels implicit in each …rm’s initial choice of group
(fringe or cartel). To examine the issue of the stability of an industry equilibrium we focus
on a scheme where the timing of the game would be as follows. At the …rst stage (“participation stage”), …rms decide whether to join the cartel or the fringe. At the posterior
stages (“output stages”), …rms set quantities in an in…nitely repeated game. Then, in the
participation stage, …rms show their willingness to participate in a collusive agreement.
These decisions do not a¤ect the payo¤ of …rms, but can be used as a coordination device:
if k …rms decide to participate in a cartel agreement, only cartels of size k can be observed
in the repeated game. Hence, a stable cartel size it may be said is the stable industry
equilibrium that would emerge if …rms were able to collude. As proved in the appendix,
we have:
Proposition 3 In the game de…ned above, if
, when n
p
exists only for k 2 [f (n); f (n) + 1] where f (n) = 14 (1 + 3n 2 (n
4 stable partial cartels
2)n
7). When n < 4
the unique stable cartel is joint monopoly.
The result parallels earlier results for stable cartels with static games. Unfortunately,
stability in the model with partial coordination cannot be further simpli…ed. Nevertheless,
it is still possible to investigate whether stable cartel exists when
when n < 4 no partial cartel is stable either. When n
when k 2 [f (n; ); f (n; ) + 1], where
@f (n; )
@
< . In particular,
4, stable partial cartels exist only
< 0, lim_ f (n; ) = f (n) and lim f (n; ) =
!0
!
f (n)+ 2 with 2< 1. It follows then that the di¤erence in the relative size of stable cartels
_
when coordination is perfect (
_
) or only partial ( < ) is not striking.
15
We have thus established the intrinsic instability of cartels, as only cartels containing
just over half the …rms in the industry are stable.9 When cartel …rms are a majority, there
is a free rider problem in which fringe …rms bene…t from the actions of the cartel (cartels
are not internally stable). In contrast, when cartel …rms are a minority it is pro…table for
a …rm in the competitive fringe to join the cartel to bene…t from a higher market share
(cartels are not externally stable).
We next examine and discuss the equilibrium sequence of moves between the cartel
and the fringe. For instance, when
k …rms and n
with simultaneous play between the cartel of
k …rms, each fringe …rm earns
(a c)2
(n k+2)2
>
f
c f
n;k (q ; q )
for all 1 < k
n.
Thus, a fringe …rm prefers simultaneous play to the Stackleberg equilibrium in which the
cartel leads. However, this is not the end of the story because we must also consider
cartel members. It is also a standard exercise to verify that total pro…ts for a cartel
member under simultaneous play are smaller than
c
c f
n;k (q ; q ).
The cartel thus prefers the
leader’s role to simultaneous play, and this result suggests a motivation for the Stackleberg
sequence if the cartel is able to impose its will on the fringe …rms. One possible way for
imposing this sequence could be a threat of the cartel not to collude (but instead to behave
as a Cournot oligopolist) unless permitted by the fringe to lead. This threat would be
e¤ective because, by Proposition 1, each fringe …rm strictly prefers the endogenously
cartelized outcome of a leader cartel to symmetric Cournot.10
On the other hand, note that Salant et al. (1983) assuming likewise linear demand
and cost in a homogeneous Cournot oligopoly showed that the minimum pro…table merger
under simultaneous play involves at least 80 percent of the …rms in the industry. Then,
the threat would also be credible because comparing Proposition 3 with the results of
Salant et al. (1983) we see that …rms comprising the endogenous Stackelberg cartel are
9
10
Note that generally stable cartels contain slightly more than 50% of the …rms ( f (n) n2
1).
See Proposition 1 and observe that in the stable cartels, fringe …rms are always a minority (f (n) >
n+1
2 ).
16
too few to bene…t from collusion under simultaneous play and will therefore prefere to
compete as Cournot oligopolists if not allowed to lead.11 Thus, an endogenous sequence
of play between a stable cartel and a Cournot fringe will assign a leader’s role to the cartel
and a follower’s role to the fringe.
3
Mergers and partial coordination
Although the framework with partial coordination we set can be used to study di¤erent
subjects, in this section we specialize the model to deal with the issue of mergers. Curiously, while there exists substantial literature on the e¤ect of mergers when pre-and
post-merger behavior is non-cooperative, little attention has been paid by theorists to the
consequences for …rms of mergers when they take place in a collusive environment. In
fact, despite its in‡uence in matters of policy, there is almost no formal evidence from
theory regarding the relationship between mergers and collusion. Then, the purpose of
this section is to analyze both the competitive e¤ects of horizontal mergers and the private
incentives to merge in a model with partial coordination between …rms.
3.1
The e¤ect of mergers on collusion
To the best of our knowledge, little attention has been given by the literature to the
consequences of mergers when post-merger behavior is collusive. However, it is a popular
belief amongst industrial organization theorists, as well as policy makers, that horizontal
industrial merger increases the propensity of …rms to collude. In the sparse literature on
the subject, mergers are analyzed seeing what their e¤ect is on the potential sustainability of the collusive agreements, or in other words, the e¤ect on the minimum discount
11
Salant et al. (1983) is a Cournot merger model with simultaneous play that does not distinguish
between a cartel and a merged …rm except for the precommitment issue. Note that the endogenous
Stackelberg cartel comprises generally less than 80 % of the …rms of the industry (f (n) < 45 n).
17
factor required for full collusion to be sustainable. However, the present model allows for
partial coordination or coordination on output levels which maximize cartel …rms’pro…ts
conditional on being sustainable in equilibrium if …rms are not patient enough to sustain
the unconstrained collusive agreement (full or perfect collusion). Then, the purpose of
this subsection is to study how mergers among …rms change the picture when …rms can
partially coordinate. To that extent, we only consider mergers among …rms of the same
subset of …rms (either K or F ).12 Suppose that these mergers create no “synergies” so
that the marginal costs of the …rms remain unchanged after the merger.13 When k …rms
collude and n
k …rms belong to the competitive fringe, after the merger of m + 1 cartel
…rms there will be k
m cartel members and n
(k
m) fringe …rms. Similarly, after the
merger of m + 1 fringe …rms there will be k cartel members and n
k
m fringe …rms.
To focus the discussion we recall the literature most closely related to our setting. In
the repeated symmetric Cournot game with linear demand and costs (Vives (1999)), the
(n+1)2
(n+1)2 +4n
monopoly outcome can be sustained if
…rms. Then, as
@ (n)
@n
= (n), where n is the number of
> 0, we have that it is harder to collude with more …rms. Thus,
mergers, by reducing the number of …rms, facilitate tacit collusion.14
Observe that when k = n the present model encompasses the case previously analyzed
in the literature where all …rms participate in the collusive agreement. Following the logic
of the aforementioned model we could also study the e¤ect of mergers on : Surprisingly,
12
Mergers between cartel and fringe …rms are not considered. If we did so, it seems reasonable to
follow the reasoning of Huck et al. (2001): “If a leader merges with a follower in a market with quantity
competition the new …rm will stay a leader because the old …rm can still use the old commitment
technology of the former leader to commit itself on high output”. However, in the present model of tacit
collusion, this is not obvious as for example, if k >
n+1
2
fringe …rms pro…ts are larger than cartel …rms
pro…ts. Hence, it is not clear which should be the status of the new entity.
13
See Farrell and Shapiro (1990) for a formal de…nition.
14
See for example Osborne (1976). In the same setting, Davidson and Deneckere (1984) obtain the
opposite result by assuming that …rms are not allowed to redistribute output after a merger.
18
the result is partially ambiguous. Mergers can make either easier or more di¢ cult for
cartel …rms to sustain (full) collusion.
On the one hand, as
= (n; k) =
(n + 1)2
(n + 1)2 + 4k(n + 1
k)
the merger of m + 1 cartel …rms helps full collusion if
m < (n + 1)
because then (n
m; k
2k
n
k
1
(11)
m) < (n; k). It hinders full collusion if the inequality in (11)
is reversed. Condition (11) is only possible if k >
n+1
.
2
Therefore, the merger of m + 1
cartel …rms hinders full collusion for small cartels. On the other hand, the merger of m+1
fringe …rms helps full collusion if:
m < (n + 1)
because then (n
n + 1 2k
n k+1
(12)
m; k) < (n; k). It hinders full collusion if the inequality in (12) is
reversed. Condition (12) is only possible if k <
n+1
.
2
Therefore, the merger of m + 1 fringe
…rms hinders full collusion when the cartel is su¢ ciently large.
Then, there are some potential problems with the above analysis. However, these
disappointing results may not be so important in the present case where when …rms fail
to sustain full collusion they can nevertheless achieve some degree of collusion. First of
all, the present model di¤ers from the standard model of colluding …rms on two accounts.
On the one hand, we do not focus only on collusion at the industry level but also on the
ability of a subset of …rms to reach a collusive agreement (the …rst case is obtained in the
limit case when k = n). On the other hand, the present model is richer because for any
some degree of coordination is always achieved (Cournot competition occurs only when
= 0) and collusion is exploited at its maximum possible level.
As an alternative approach, in our case it would be more informative to study the
e¤ect of mergers on a variable that better re‡ects the degree of collusion achieved in the
19
industry. It seems that market price is the most indicated candidate for that purpose.
And then a robust conclusion is obtained: mergers always increase price. This result is
developed in the following Propositions that deal respectively with the case of mergers of
cartel …rms and mergers of fringe …rms.
Proposition 4 Any merger of cartel members does not decrease price.
The increase is always strict except for high values of the discount factor such that full
collusion is sustained both pre-merger and post-merger. In this case, when m + 1 cartel
…rms merge price does not change with the merger if:
maxf (n
m; k
m); (n; k)g
When both pre-merger and post-merger full collusion is sustained, the aggregate output of
a c
. Then, fringe …rms do not change
cartel …rms remains unchanged and amounts to
2
their output either in response to a merger, and price is una¤ected by the merger. Hence,
the most intriguing part of Proposition 4 is to understand why even when full collusion
is more di¢ cult to sustain after the merger price increases. The situation we consider is
the following:
(n; k) < (n
and then for
2 ( (n; k); (n
m; k
m; k
(13)
m)
m)) full collusion was sustainable before merger
but it is not sustainable after the merger. Analyzing (13) we see that it holds if and only
if:
m > (n + 1)
Note that (14) is only possible when k >
…rms after a merger of at least (n+1) 2k
n 1
k
2k
n+1
.
2
n
k
n+1
.
2
(14)
At the same time, the number of cartel
cartel …rms would always represent a minority
in the post merger industry. This is true because k
if k >
1
(n + 1) 2k
n 1
k
<(
n ((n+1) 2k
2
n 1
)+1
k
)
Then, the key point to solving the puzzle is to realize that (14) implies
20
m+1
i.e. cartel …rms are in a minority position after such a merger. Figure 2
2
shows that when full collusion is not sustainable cartel …rms adjust their quotas downwards
k m<
n
which drives the price upwards. So the puzzle created because we may have simultaneously
that full collusion becomes more di¢ cult with the merger and price increases is explained
by the peculiar feature of our model that price can decrease with the discount factor when
cartel …rms …nd themselves in a minority position.
The following picture illustrates the issue. It depicts the evolution of price pre-merger
and post-merger for the speci…c case of n = 10, k = 7 and m = 5, with a = 1 and c = 0.
Observe that in this case (14) is satis…ed and post-merger price is never below the price
prior to the merger.
Price
post - merger
pre - merger
delta
Figure 3
Proposition 5 identi…es the impact of mergers among fringe …rms on market price.
Proposition 5 Any merger of fringe …rms strictly increases price.
21
Now price strictly increases even when full collusion is sustained both pre-merger and
post-merger. In this case, the joint output of cartel …rms does not change after the merger.
However, fringe …rms will contract their aggregate output when they merge and this yields
the increase in the market price.
Observe that even when full collusion is more di¢ cult to sustain after the merger,
market price also increases. In this case, the situation we consider is the following:
_
_
(n; k) < (n
_
(15)
m; k)
_
Then, the most intriguing case would be if 2 ( (n; k); (n m; k)). That is, full collusion
was sustainable before the merger but it is not sustainable after the merger. The intuition
is that although the minimum discount factor required for full collusion to be sustainable
_
increases, more collusion can be achieved. In other words, as
> (n; k), there was no
_
room for additional collusion before merger. However, as
increases, …rms’present ability
to collude can be further exploited and this raises the market price.
Propositions 4 and 5 con…rm the results obtained in previous models that mergers
help to sustain higher prices.15 In this sense, assuming that …rms set quantities allows
a comparison with Farrell and Shapiro’s results. Thus, we can observe that mergers not
creating synergies raise price and not only in a Cournot oligopoly but also when some (or
all) …rms can partially coordinate their output.
Furthermore, within the limited context of the present model these results seem to invalidate the approach taken so far to focus on the ability of …rms to achieve full collusion.16
For instance, Compte, Jenny and Rey (2002) study collusion in a setting with …rms with
asymmetric capacities. They show that the main problem for collusion is to preventing the
largest …rm from deviating. Then, the e¤ect of mergers is ambiguous whenever it involves
15
16
See for example Farrell and Shapiro (1990) or Spector (2003).
To analyze real-world cases of mergers, …rms’capacities, cost asymmetries or synergies should also
be considered. The inclusion of these issues has been left for future research.
22
the largest …rm. On the one hand, it reduces the number of competitors which tends
to facilitate collusion. But, on the other hand, it increases the size of the largest …rm,
which increases its incentives to deviate. In particular, mergers increasing market share
asymmetry may hinder collusion. However, in Compte, Jenny and Rey (2002) restricting
attention to the sustainability of full collusion is validated by the fact that full collusion is
sustainable whenever some collusion is sustainable. By contrast, their model di¤ers from
the present in at least in two aspects. First, in our model …rms are symmetric in terms of
productions costs and second, here full collusion is not always achieved and when …rms are
not patient enough, cartel …rms achieve only the maximum degree of collusion which they
are able given their discount factor. Observe that the minimum discount factor required
for full collusion to be sustainable ( ) can either increase or decrease with both mergers of
fringe …rms or mergers of cartel …rms. Then, the results of the present paper indicate that
a policy concern based on the view of whether that increased concentration from mergers
facilitates collusion should not be focused on …rms’ability to sustain full collusion. This
is because whenever …rms can partially coordinate, a merger could facilitate or hinder full
collusion but the e¤ect on market price would remain unambiguously clear.
3.2
The e¤ect of collusion on mergers
In the last subsection we studied the e¤ect of mergers on market price. Here, we focus on
the private incentives to merge in a model of partial coordination. It is well-known that
the price increase by itself does not guarantee that a merger is pro…table. For example,
in a Cournot setting with …rms selling the same product (see the seminal paper by Salant
et al. (1983)), although mergers increase price they are (generally) not pro…table because
non merging …rms react to the merger expanding their production.17 Salant et al., which
17
This paradoxical result is valid only in Cournot environments, and generally fails to hold in di¤er-
entiated Bertrand models. This is an issue not raised here and left for future research. It would also
23
likewise assume linear demand and cost, show that the minimum pro…table merger under
Cournot oligopoly involves at least 80 percent of the …rms in the industry.18 To understand
this, recall that the output sold in equilibrium in the standard linear Cournot model is
given by: qi =
a c
:
n+1
When n decreases (because of a merger), qi increases. Thus, it is
the output expansion of the non-merging …rms that causes a reduction in pro…ts for the
merging …rms.
If as in Salant et al. we refer to the subset of …rms that do not participate in the
proposed mergers as “outsiders”, we have that the present model (which encompasses
the Cournot case when
= 0) shares the same characteristic that outsiders expand their
output and therefore merger pro…tability cannot be taken for granted. Observing the
equilibrium quantities (q c ; q f ) for cartel and fringe …rms respectively as a function of n; k
and , the following is true:
@q c (n
m; k
@m
m; )
> 0 and
@q f (n
m; k
@m
m; )
>0
and
@q c (n
m; k; )
@q f (n m; k; )
> 0 and
>0
@m
@m
Hence, a merger of m …rms of either cartel or fringe …rms causes the equilibrium output
of the outsiders to expand. Then, it should be noted that as in Salant et al., as the outputs
of the outsiders increase, the pro…ts of the merging …rms (insiders) decrease. Observe that
since the response of the non-merging …rms or outsiders depend on the discount factor,
the incentives for …rms to merge depend also on the coordination that …rms achieve. It
turns out then that the results of Salant et al. are in fact sensitive to the assumption of
no collusive coordination between …rms. This question is addressed in what follows.
be interesting to see an extension to a wider range of demand functions (see Cheung (1992), Faulí-Oller
(1997) and Hennessy (2000)).
18
“The break-even value for all industry sizes exceeds 80 percent” (p.193)
24
Using the pro…t functions de…ned in (3) and (7), it is assumed that the incentive for
m + 1 cartel …rms to merge is given by the following expression:
c
c f
n m;k m (q ; q )
c
c f
n;k (q ; q )
(m + 1)
(16)
and the incentive for m + 1 fringe …rms to merge by:
f
c f
n m;k (q ; q )
(m + 1)
f
c f
n;k (q ; q )
(17)
They are simply the di¤erence between the pro…ts obtained by merging …rms after and
before the merger. The (unique) root of (16) and (17) in m is respectively the minimum
number of cartel and fringe …rms that are involved in the minimum pro…table merger
that we denote by m .19 Equivalently, the fraction of …rms required for a merger between
either cartel or fringe …rms to be pro…table would be
m
k
and
m
n k
respectively. Restricting
attention to the case where full collusion is not sustained either before or after the merger,
we obtain the following:
Proposition 6 For any n and any k (1 < k
n), when k …rms are partially coordinating
their output the fraction of cartel …rms required for a merger to be pro…table increases
relative to symmetric Cournot oligopoly.
To appreciate Proposition 6, we can interpret the result as a direct comparison with
the results of Salant et al. when …rms can partially coordinate their outputs. For instance,
when
= 0, the present model encompasses the Cournot model and in this case the frac-
tion of …rms required for a pro…table merger ( mn ) is always above 80 percent. Proposition
6 states that if …rms are partially coordinating their output before the merger, this fraction of …rms increases and it is always above the break-even fraction mentioned in Salant
et al. Thus, in the present model mergers (among colluding …rms) become less pro…table
than in the model of Salant et al. Note that obviously if k is small compared to n mergers
19
16 and 17 are increasing functions of m in m .
25
between cartel …rms are never pro…table (for instance if k < 45 n). However, Proposition 6
indicates that the existence of output coordination decreases the incentives to merge for
all cartel sizes (including the limit case of k = n). To illustrate, note that in the present
model if n = 5 this fraction also reaches the value of 0.8 when
= 0. However, if
>0
for any k this fraction is above 0.8.20 Intuitively, in absence of any cost saving derived
from a merger, if
> 0 competition is already low because …rms are sustaining a collusive
agreement and a merger loses attractiveness as an anticompetitive device.
On the other hand, note also from section 2 (see Proposition 3) that in the endogenously determined cartels, mergers among cartel …rms would almost never be pro…table:
easy calculations show that when n
4 only a complete monopolization consisting of
all …rms merging is pro…table. It is also easy to see that when n
5 the stable cartels
obtained in subsection 2.1 always represent a proportion of …rms in the industry equal
to or smaller than 0.8. Therefore, from Proposition 6 it follows that in this case mergers
between …rms of a stable cartel would never be pro…table. Summarizing, within the limited context of the present model we can say that even a partially cooperative pre-merger
behavior may make it more di¢ cult or in most cases impossible for a merger between
colluding …rms to be pro…table.
We turn now our attention to the private incentives of fringe …rms to merge. From
_
section 2 it is clear that when
perfect or total collusion is sustainable. In this
case, it is also clear that the fraction of fringe …rms required for a merger to be pro…table
follows from the results of Salant et al. as fringe …rms act as Cournot quantity-setting
…rms facing a residual demand. To illustrate, note that for instance for any n and k if we
_
have 5 fringe …rms (n
20
k = 5) when
Note that Proposition 6 implies that when
pro…table is always larger than when
, the fraction of merging fringe …rms needed
> 0 the fraction of …rms required for the merger to be
= 0. However, the increase in the number of cartel …rms involved
in the minimum pro…table merger (m ) may not be strict because this number should be represented as
an integer number and not by a real number. This could be the case when k is large and
26
is low.
also reaches its minimum value of 0.8 (4 …rms). However, if
is very close to zero, the
number of merging …rms needed should be at least 45 n, which obviously in this case is
greater than 4, being 1 < k < n. Hence, it turns out again that merger incentives are
also sensitive to the partial coordination between cartel …rms. This observation yields the
following result:
Proposition 7 For any n and any k (1 < k < n), when k …rms are partially coordinating
their output the fraction of fringe …rms required for a merger to be pro…table increases
relative to full collusion.
Regarding the incentives for fringe …rms to merge, we get the opposite result from
Proposition 6. The key point to understanding Proposition 7 is that as the discount
factor increases, cartel …rms expand their output less in response to the merger of fringe
…rms. This is equivalent to
@ 2 q c (n; k; )
>0
@n@
(18)
Observe that in the limit case when full collusion is sustained both pre-merger and
post-merger, cartel …rms do not modify their output in response to the merger of fringe
…rms (q c =
a c
2k
if
). However, when cartel …rms can only partially coordinate
their output, mergers between fringe …rms are generally not pro…table. To illustrate,
consider again n
k = 5. Hence, a merger of at least 80% of the …rms (4 fringe …rms) is
pro…table if
but from Proposition 7 we know that for lower
this merger becomes
unpro…table.21 In other words, when partial coordination between cartel …rms is very
small ( close to zero), if m + 1 fringe …rms were to merge only whenever the merging
…rms represent at least 80 percent of total …rms in the industry (m + 1
21
For
4
n)
5
this merger
close to zero there is an industry size (n) large enough such that a merger of any number of
fringe …rms is unpro…table. For instance when n
k = 5 if n
7 then n
k < 54 n and for
any merger between fringe …rms would always be unpro…table. However, for larger
fringe …rms might become pro…table.
27
close to zero
a merger between
could be pro…table. In this case, for
close to zero, the present model reproduces the
standard Cournot model and merger pro…tability parallels Salant et al. results with n
…rms. Conversely, when partial coordination between cartel …rms is large ( close to ),
fringe …rms face a residual demand. Therefore, merger pro…tability of fringe …rms also
follows from the results of Salant et al. but with n
k …rms. In this case only when at
least 80 percent of fringe …rms merge, the merger may become pro…table. Then, for
<
the fraction of fringe …rms required for the merger to be pro…table is always larger than
0.8.
Analyzing the endogenous cartels obtained in section 2.1, we see that mergers between
fringe …rms would not be pro…table if partial coordination between cartel …rms is low
because the largest possible size of the fringe does not represent 80 percent of total …rms
in the industry (n f (n) < 45 n). However, Proposition 7 implies that in a less competitive
market, the incentives to merge for a fringe …rm are higher. Then, for larger
mergers
between fringe …rms may become pro…table if the number of insiders is large enough (for
instance when m + 1 > 45 (n
4
f (n)) and
).
Optimal punishment
In sections 2 and 3 we have assumed that a collusive output may be supported by the
threat of reverting to single period Cournot-Nash equilibrium. Notice, however, that
such an in…nite unforgiving punishment might seem rather extreme. Abreu (1986, 1988)
examined a type of more sophisticated punishment than reversions to the one-shot Nash
equilibrium. He pointed out that, without loss of generality, attention can be restricted to
what he de…ned as a simple penal code. A simple penal code has a very simple structure.
If …rms conform with the strategies of the prescribed equilibrium, then …rms will earn
the value of the best continuation equilibrium. If, instead, a single deviation occurs, all
28
…rms (including the deviant one) revert to a punishment which gives the deviant its worse
possible continuation equilibrium (Minimax).
Abreu (1986) outlines a symmetric, two-phase output path that sustains collusive
outcomes for an oligopoly of quantity setting …rms. The output path considered by Abreu
has a “stick-and-carrot”pattern. The path begins with a collusive output by cartel …rms
(q). The strategy calls for all cartel members to continue to produce q, unless an episode
of defection occurs. If a …rm cheats on the agreement, all cartel …rms expand output
p
for one period (qm
) (stick stage) and return to the most collusive sustainable output in
the following periods, provided that every player of the cartel goes along with the …rst
phase of the strategies (carrot stage). Unfortunately, the optimality of the simple penal
code of Abreu occurs in a model where …rms simultaneously choose quantities. Therefore,
the present model as formulated in section 2 is not suitable for using such a penal code.
However, to analyze the robustness of our results in a model with partial coordination
using an optimal penal code it will be su¢ cient to pay attention to the limit case of k = n.
In this case, the “stick-and-carrot”strategies ( ) can be formulated in the following way,
where qt;i denotes an action (choice of output) played by …rm i in period t:
Firm i, i = 1; :::; n plays:
8
>
>
q1;i = q
>
>
>
>
>
< qt;i = q if qt 1;j = q for j = 1; :::; n 8t = 2; 3; :::;
( )
>
p
>
qt;i = q if qt 1;j = qm
for j = 1; :::; n 8t = 2; 3; :::;
>
>
>
>
>
:
p
qt;i = qm
otherwise:
Following Abreu (1986), the strategies described above are considered optimal, or
in other words, they sustain the highest range of collusive outcomes among all possible
punishment phases, if continuation pro…ts after unilateral deviation in any period equal
the Minimax value of …rms. In the present model it is equal to 0 as …rms can always
decide not to be active and get 0 pro…ts. Therefore,
29
are optimal when the following
condition holds:
c
c
where
p
(qm
) and
c
c
p
(qm
)+
1
(q) = 0
(19)
(q) are cartel …rms’pro…ts if …rms are in a punishment phase (stick
stage) and a collusive stage (carrot stage) respectively with:
c
p
(qm
) = (a
p
nqm
p
c)qm
We need the conditions for the strategies
to constitute a SPNE. On the one hand, if
…rms are in a collusive phase (carrot stage), pro…ts that a …rm would obtain by deviating
from the collusive agreement should be smaller (given that the rest of the …rms adhere to
the strategies described) than collusion pro…ts. This is given by the next condition:
d
where
d
(q; qid )
c
(q)
(
c
(q)
c
p
(qm
)) no deviation in the carrot stage
(20)
(q; qid ) (already de…ned in section 2 ) are the one period gain from deviation
when cartel …rms collude and when k = n.
On the other hand, in a punishment phase (stick stage) …rms should obtain higher
pro…ts in the punishment phase than they do by deviating from it. Therefore, …rms do
not unilaterally deviate in the stick stage if the following condition holds:
d
where
d
p
(qm
; qid )
c
p
(qm
)
(
c
c
(q)
p
(qm
)) no deviation in the stick stage
(21)
p
(qm
; qid ) denote the pro…ts that a cartel …rm obtains by unilaterally deviating in
the stick stage:
d
p
(qm
; qid ) = max (a
qi
(n
p
1)qm
qi )
c)qi
p
We determine the optimal stick-and-carrot punishment seeing which qm
satis…es (19)
for a collusive output q. From (21) and (19) no deviation in the stick stage is only possible
if
d
p
(qm
; qid )
0, since otherwise a …rm can deviate in the …rst period and keep doing so
every time the punishment is reimposed. Hence, the total output produced by (n
30
1)
p
1)qm
)
p((n
0.22 It yields
p
)
1)qm
…rms must be large enough so that p((n
p
0 () qm
x
(22)
and this sets a lower bound on the quantity produced by cartel …rms in the stick stage.
We denote by
a
the lower bound of the discount factor such that (21) and (19) are
p
that satis…es (22):
satis…ed. We compute (19) for the lowest value of qm
c
(x) +
a
1
c
(q) = 0
a
It leads us to:
c
a
Hence, for a given q when
>
a
=
(x)
c (x)
(23)
c (q)
a harsher punishment is needed such that (19) is
satis…ed. As far as deviation in the carrot stage is concerned, from (20) and (19), …rms
do not deviate if
1
c
1
We denote by
b
(q)
d
(q; qid )
(24)
the lower bound of the discount factor such that (20) and (19) are
satis…ed:
d
b
=
c
(q; qid )
(q)
d (q; q d )
i
(25)
Summarizing, (19), (20) and (21) are satis…ed if and only if
maxf a ; b g
(26)
It is a standard exercise to verify that in this case for n large enough, (26) is satis…ed if
p
(n 1)2
.
Straightforward
calculations
show
that
if
n
>
3+2
2 ' 5:83 then b > a and
2
(n+1)
(n 1)2
.
(n+1)2
the cuto¤ of the discount factor becomes
However, it should be emphasized
again that the formal structure of the present model allows for coordination on output
levels which maximize cartel …rms’pro…ts conditional on being sustainable in equilibrium
22
The price p(Q) is interpreted as net price of marginal cost at zero.
31
if …rms are not patient enough (
< maxf a ; b g) to sustain in equilibrium the output
level that maximizes joint cartel pro…ts of the one stage game. In this case, it turns out
to be the case that assuming n large enough if
<
(n 1)2
,
(n+1)2
(25) is a binding constraint
and the distribution of output (q c ) is the solution to the equality constraint.
In this case, each cartel …rm produces a quantity given by the following expression:
8
p
>
(a c)(1 + + n(1
) 2 )
(n 1)2
>
>
if
<
>
(n+1)2
>
(1 + n)2 (1 n)2
<
qc =
>
>
>
>
>
(n 1)2
: a c
if
(n+1)2
2n
Hence, the market price is given by (a
nq c ). This yields the following result that
establishes the fact that Proposition 4 can also be extended to a model where …rms
sustain collusion using an optimal punishment.
Proposition 8 When …rms are partially coordinating their output using an optimal punishment, mergers raise the market price.
The increase is not strict for high values of the discount factor such that (full) collusion
is already sustained before the merger.23 In this case the market price remains unchanged
in the amount of p =
a+c
.
2
Once we study the e¤ect of mergers on market price, we focus on the private incentives
to merge in a model of partial coordination using an optimal punishment. In this case
the di¤erence between the pro…ts obtained by merging …rms after and before the merger
which is the incentive for m + 1 …rms to merge is given by the following expression:
c
n m (q)
23
(m + 1)
c
n (q)
(27)
Observe that mergers here always raise the minimum discount factor required for full collusion to be
(n
sustainable (
1)2
@( (n+1)2 )
@n
> 0).
32
As in the last section, the (unique) root of (27) is the minimum number of cartel …rms
that are involved in the minimum pro…table merger which divided by n is the fraction of
…rms required for a merger to be pro…table. We obtain the following:
Proposition 9 When …rms are partially coordinating their output using an optimal punishment, the fraction of …rms required for a merger to be pro…table increases relative to
symmetric Cournot oligopoly.
Again, we can interpret the result as a direct comparison with the results of Salant
et al. Therefore, Proposition 9 parallels Proposition 6 when …rms sustain collusion using
an optimal punishment. Again, the incentive for cartel …rms to merge decreases with
the partial coordination: Then, the fraction of merging …rms required for the merger
to be pro…table is always above the fraction mentioned in Salant et al. when
> 0.
In conclusion, we have studied a more severe punishment and established that general
conclusions of section 3 continue to hold.
5
Concluding comments
The main aim of this paper has been to develop a theoretical foundation on two di¤erent
issues that have been frequently not considered. First, most applied papers have focused
on studying the equilibrium where industry pro…ts are maximized assuming that all …rms
participate in the implicit collusive agreement. In practice, however, there are many cases
where collusive agreements do not involve all …rms in the industry, and surprisingly, little
attention has been paid to the situation where not all the …rms of the industry join the
cartel. A second contribution of this paper is that, using the trigger strategies, the usual
con…guration where the sustainability of collusion depends on whether the discount factor
is lower or higher than a critical threshold is questioned. It has also found that pro…ts of
each cartel member are enhanced by cartelization relative to symmetric Cournot oligopoly.
33
However, pro…ts of fringe …rms are only enhanced by cartelization when the cartel is large
enough. In addition, the intrinsic instability of cartels is also shown. When cartel …rms
are a majority, there is a free rider e¤ect in which …rms have incentives to leave the cartel
to sell at the same price but at the much larger output. In contrast, when cartel …rms
are a minority they have incentives to join the cartel to increase their market share and
the cartel is also unstable.
The model is specialized to deal with the issue of mergers. The assumption that …rms
set quantities allows a comparison with Farrell and Shapiro’s results. Mergers not creating
synergies raise price not only in Cournot oligopoly but also when some (or all) …rms can
partially coordinate their output. It is also shown that although the critical discount
factor above which joint pro…t maximization could be sustained may increase (due to a
merger), the e¤ect of a merger on market price is unambiguous; price increases. Within its
limited context, this result shows that the traditional antitrust view that mergers facilitate
collusion if they decrease the critical discount factor above which full or unconstrained
collusion can be sustained may not be adequate.
On the other hand, the paper has also analyzed the private incentives to merge. There
exists a traditional view in Industrial Organization, following Salant et al. according to
which there is little scope for merger pro…tability when mergers do not involve any cost
saving and …rms are in a Cournot environment. However, this paper proves that in a
repeated game, the results of Salant et al. are sensitive to the degree of coordination that
…rms achieve when they can partially coordinate their outputs. In other words, merger
incentives may depend on the actual level of the discount factor. Then, the present
model — that encompasses the Salant et al. case when
= 0— predicts that cartel …rms
have even fewer incentives to merge when …rms are partially coordinating their outputs.
However, the reverse is true for fringe …rms. Then, this can reinforce the tendency of
some groups of …rms to merge in a non-competitive environment. Therefore, it should be
34
taken into account that collusion may not only directly increase price but also indirectly
by rendering pro…table mergers among fringe …rms. The closest real example comes from
the oil market. At the end of the 90’s, mergers among fringe …rms and production cuts
by the OPEC cartel took place more or less simultaneously.
Finally, in section 4 we observed that using a more severe punishment — Abreu’s (1986)
stick-and-carrot strategies— the general conclusions of the paper continue to hold.
35
Appendix
Proof of Proposition 1. If
c
c f
n;k (q ; q )
=
< ,
(a c)2 (1 n(2+n)( 1+ ) +4k2 )((1+n)2 +(1 2k+n)(3 2k+3n) )
.
(1+n)6 2(1+n)4 (1 2k+n)2 +(1+n)2 (1 2k+n)4 2
(a c)2
.
(n+1)2
metric Cournot oligopoly), pro…ts become
each ; k and n,
c
c f
n;k (q ; q )
(a c)2
:
(n+1)2
(a c)2 ( 1+n(2+n)( 1+ )+ 4k2 )2
(1+n)6 2(1+n)4 (1 2k+n)2 +(1+n)2 (1 2k+n)4
,
c f
c
n;k (q ; q )
only when k >
=
(a c)2
4k(n k+1)
>
2
(a c)2
(n+1)2
= 0 (sym-
It is tedious but easy to see that for
On the other hand, if
which is above
and
When
(a c)2
(n+1)2
f
c f
n;k (q ; q )
=
f
c f
n;k (q ; q )
< , then
only when k >
n+1
.
2
(a c)2
4(n2 +k2 +1)+8(n k(n+1))
>
=
When
(a c)2
(n+1)2
n+1
.
2
Proof of Proposition 2. The market price is given by (8): If
,the derivative of
p(n; k; ) with respect to k is always positive. However, if < . this derivative becomes
p
( 1)(n+1)+ (1 )(n+1)2
positive only if k >
. Since the last expression is decreasing with ,
2
a su¢ cient condition for the market price to increase with the number of cartel …rms will
p
( 1)(n+1)+ (1 )(n+1)2
be found when is the smallest possible. Then, lim
= n+1
:
2
4
!0
Proof of Proposition 3. When
(a c)2
.
4(n k+1)2
,
c
c f
n;k (q ; q )
=
(a c)2
4k(n k+1)
and
f
c f
n;k (q ; q )
=
Then, internal and external stability ((9) and (10)) become
(a c)2 (2k2 +(2+n)2 k(5+3n))
4k(1+n k)(2 k+n)2
0 and
(a c)2 (1+k(2k 1)+n(n 3k+1))
4(1+k)(k n)(1 k+n)2
0 respectively. If n < 4
external stability is never met unless k = n. In the later case, internal stability holds
p
3
1
if n
4: When n
4 internal stability holds if k
n
(n 2)n 7 + 54 and
4
4
p
external stability holds if k > 41 (1 + 3n
(n 2)n 7). Hence, it follows that as
p
p
1
1
( 34 n
(n 2)n 7 + 45 )
(1 + 3n
(n 2)n 7) = 1; stable partial cartels
4
4
exists only when both conditions hold and that is for k 2 [f (n); f (n) + 1] where f (n) =
p
1
(n 2)n 7):
(1
+
3n
4
Proof of Proposition 4. From section 2, market price is given by the following expression:
36
8
< f (n; k; ) =
p(n; k; ) =
:
(1+n)2 (a+cn) (1 2k+n)(a(1+2k+n)+c(n+n2 2k(2+n)))
(1+n)3 (1+n)(1 2k+n)2
a+c(1+2(n k))
2(n k+1)
if
<
if
Observe that
p(n; k; ) = f (n; k; )
1
De…ne
2
and
as the pre-merger and post-merger cuto¤s respectively.
First we will prove that the following is true:
f (n
First, we see that f (n
there exists only one
0
1
m; ) > f (n; k; ) if
m; k
< minf ;
2
g
(28)
m; 0) > f (n; k; 0) On the other hand, we can see that
m; k
2 (0; 1) that solves f (n
m; ) = f (n; k; ). We will call it
m; k
.
We can check that if
then
0
2
=
=
1
1
<
2
, then
0
1
>
, but if
1
>
2
, then
0
>
2
. If
1
=
2
,
. Therefore, (28) is true.
If
< minf ;
1
2
If
maxf ;
1
2
g, (28) ensures that a merger (strictly) increases market price.
g the market price pre and post-merger is the same, as
a+c(1+2(n k))
2(n k+1)
=
a+c(1+2(n m (k m)))
.
2(n m (k m)+1)
For the remaining values of , we have two di¤erent relevant cases to consider: the …rst
one is when m > (n + 1) 2k
and
1
>
2
n 1
k
and
1
<
2
. The second case is when m < (n + 1) 2k
n 1
k
.
We know that p(n; k;
1
) = p(n
m; k
m;
2
)=
In the …rst case, it is enough to check that f (n
with . This is true if m > 2k
n
a+c(1+2(n k))
.
2(n k+1)
m; k
m; ) is strictly decreasing
1, and this holds as m > (n + 1) 2k
n 1
.
k
In the second case it is enough to check that f (n; k; ) is strictly increasing with ,
and this is true if 2k
n
1 > 0, and this holds as otherwise m < (n + 1) 2k
n 1
k
could
never hold.
Proof of Proposition 5. The market price is given by (8): We can easily see that the
37
following holds:
@p(n; k; )
<0
@n
1
De…ne
If
If
and
2
1
2
< minf ;
like pre-merger and post-merger cuto¤ respectively
1
maxf ;
a+c(1+2(n m k))
,
2(n m k+1)
(29)
g;(29) is enough to prove that price strictly increases.
2
g, we can see that as p(n; k;
1
)=
a+c(1+2(n k))
2(n k+1)
< p(n
m; k;
2
)=
price strictly increases.
For the remaining situations, we have two relevant cases: the …rst is if m > (n +
1
2k
1) n+1
, then we have that
n k+1
have then that
1
2
>
2
<
2k
, then we
. The second case is if m < (n + 1) n+1
n k+1
.
1
In the …rst case, as p(n m; k;
) > p(n; k;
is increasing with , which is true if m > n
In the second case, as p(n
m; k;
2
1
), it is enough to check that f (n m; k; )
2k
2k + 1, and this holds as m > (n + 1) n+1
.
n k+1
2
) > p(n; k;
) (remember (29)), it is enough to
check that f (n; k; ) is decreasing with , which is true if n + 1 > 2k, and this holds as
2k
otherwise m < (n + 1) n+1
could never be true.
n k+1
Proof of Proposition 6. Cartel …rms will have incentives to merge when (16) equal or
greater than zero. Since this equation has only one root in m,(with m 2 (1; n)) basically,
what we have to do is proving that
c
c f
n;k (q ; q )
=
@(
c
c f
n m;k m (q ;q )
(m+1)
c (q c ;q f ))
n;k
@
(a c)2 (1 n(2+n)( 1+ ) +4k2 )((1+n)2 +(1 2k+n)(3 2k+3n) )
.
(1+n)6 2(1+n)4 (1 2k+n)2 +(1+n)2 (1 2k+n)4 2
but straightforward to show that as
decreases. Therefore, if
< 0. If
< , then
Therefore, it is tedious
increases, the incentives for cartel …rms to merge
> 0 the fraction of cartel …rms required for a merger to be
pro…table is larger than when
= 0.
Proof of Proposition 7. As in the proof of Proposition 6, we just have to take the following expression: if
< , then
f
c f
n;k (q ; q )
=
(a c)2 ( 1+n(2+n)( 1+ )+ 4k2 )2
(1+n)6 2(1+n)4 (1 2k+n)2 +(1+n)2 (1 2k+n)4
2
and this function is continuous in . Again it is also easy but tedious to see that
@(
f
c f
n m;k (q ;q )
(m+1)
@
f
c f
n;k (q ;q ))
> 0.
38
Proof of Proposition 8. For large enough n, when …rms are partially coordinating their
(n 1)2
(n+1)2
the market price is given by the folp
a(1+n+( 1+n) )+n(c(1+n) c(n 1) +2 (a c)2 )
(n 1)2
lowing expression: p =
and p = a+c
if
:
(1+n)2 (n 1)2
2
(n+1)2
p
(a c)n((a c)((1+n)2 (n 1)2 )+2a((n 1)2 )+2(n2 1) (a c)2 )
@p
p
=
>
Then, it is easy to observe that @n
2
2
2 2
output using an optimal punishment, if
<
(a c)
((1+n)
(n 1)
)
0. Thus, mergers increase price if (full) collusion is not sustained either before or after the
merger. It is also easy to see that
(n 1)2
(n+1)2
increases with n. Then, a merger diminishes the
cuto¤ of the discount factor. Thus, the increase in market price is always strict except for
high values of the discount factor such that (full) collusion is sustained before the merger.
Proof of Proposition 9. The minimum number of cartel …rms that are involved in
the minimum pro…table merger
is the (unique) root of 27. It is given by the following
q p
p
3 p
3 + (
1) ( (4n 3) 4n 5) 1 2
. Moreover 27 is increasing in
expression: m = n
p
2
2(
1)
q p
p
3 p
1+ (
1) ( (4n 3) 4n 5)+2
@m
q p
m around m . The result follows since @ = p
p > 0.
2
3 p
( 1) ( 1) ( (4n 3) 4n 5)
39
Acknowledgements
This paper is based on chapter 2 of my PhD thesis defended at the Department of
Economics, University of Alicante. I am grateful to my supervisor Ramon Faulí-Oller for
his advice and encouragement, and Joel Sandonís, Luís Corchón, Pedro Barros, Javier
López Cunyat and Inés Macho for their helpful comments. I am also grateful to two
anonymous referees and the editor Bernard Caillaud for helpful comments which led to
substantial improvements. Earlier versions of this paper were presented at the University of Alicante (Spain), Jornadas de Economía Industrial 2003 (Castellón de la Plana,
Spain) and SMYE 2004 (Warsaw, Poland). Financial support by the I.V.I.E, (Instituto
Valenciano de Investigaciones Económicas) , the European Commission through its RTN
programme (Research Training Network), the Center for Financial Studies (Frankfurt)
and the Department of Economics, University of Alicante is gratefully acknowledged.
This paper was the winner of the 2004 Spring Meeting of Young Economists’second best
paper award. All errors are mine.
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