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Economics Letters 99 (2008) 384 – 386
www.elsevier.com/locate/econbase
Sustainable collusion on separate markets
Paul Belleflamme a,⁎, Francis Bloch b
a
CORE and IAG-Louvain School of Management, Université catholique de Louvain, Belgium
b
Université de la Méditerranée and GREQAM, Marseille, France
Received 9 February 2006; received in revised form 24 July 2007; accepted 4 September 2007
Available online 14 September 2007
Abstract
In a Cournot duopoly where firms incur a fixed cost for serving each market, collusion is easier to sustain with production quotas if the fixed
cost is small enough, and with market sharing agreements if it is large enough.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Implicit collusion; Market sharing agreements; Production quotas; Optimal punishment
JEL classification: L11; L12
1. Introduction
2. The model
Market sharing agreements are commonly observed in many
industries.1 Bernheim and Whinston (1990) identified two
settings where market sharing agreements (in their terminology
“spheres of influence”) are likely to occur: when firms have
separate home markets and incur a transportation cost to serve
the other market,2 and when firms face a fixed cost of
production. In the latter case, they showed that, in a Bertrand
model with homogeneous products, collusion is easier to sustain
under market sharing agreements than when firms are present
on both markets and assign production quotas. In this paper, we
show that their conclusion is only partly true when firms
compete in quantities. While there always exists a threshold
value of the fixed cost over which firms always prefer to
specialize, for low values of the fixed costs, firms may prefer to
be present on the two markets.
We consider the same environment as Bernheim and
Whinston (1990) and Abreu (1986). Two identical firms produce
a homogeneous good that they can sell on two separate, identical
markets, with inverse demand P(Q) where P(·)is continuous,
nonincreasing, P(0) N c and limQ → ∞P(Q) = 0. Firms compete in
quantities, and a firm selling quantity qk on market k incurs a
cost C(qk) = F + cqk with C(0) = 0.
As in Abreu (1986), we assume that there is a unique
monopoly quantity defined by qm = argmaxqq(P(q) − c), and
that q(P(q) − c) is monotonically increasing until qm and
monotonically decreasing after qm . Monopoly profits are
denoted Πm. We also suppose that the one-shot duopoly
game has a unique symmetric pure strategy equilibrium, with
quantities qc ≠ qm/2 and profit Πc. For any q, let π(q) denote
the profit obtained by a firm which responds optimally to the
quantity q. Formally, π(q) = maxzz(P(q + z) − c). We also define
G(q) as the profit obtained by a firm when both firms produce
the same quantity q, G(q) = q(P(2q) − c). Notice that, at the
symmetric Cournot equilibrium, G(qc) = π(qc), and that by
assumption G(·)is monotonically increasing until qm/2 and
monotonically decreasing after qm /2. We finally assume
F b π(qm ), so that a firm has an incentive to enter the market
of another firm when it produces the monopoly quantity. As
qm N qc, this implies that F b Πc so that both firms serve both
⁎ Corresponding author. Mailing address: 34 Voie du Roman Pays, 1348
Louvain-la-Neuve, Belgium. Tel.: +32 10 47 82 91; fax: +32 10 47 43 01.
E-mail address: [email protected] (P. Belleflamme).
1
See Belleflamme and Bloch (2004) for a discussion of recent cases of market
sharing agreements.
2
See also Gross and Holahan (2003) for a model along these lines.
0165-1765/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.econlet.2007.09.020
Author's personal copy
P. Belleflamme, F. Bloch / Economics Letters 99 (2008) 384–386
markets in the noncooperative outcome. We recall the following
results in Abreu (1986).3
Lemma 1. (i) If q1 N q2 ≥ 0, either π(q1) b π(q1) or π(q1)= π(q1) =
0; (ii) qc N qm/2 N 0 and G(qm/2) NG(qc) N 0; (iii) the function π is
convex.
3. Optimal punishment schemes
Firms are engaged in an infinitely repeated duopoly game, in
which they play the “stick and carrot” strategies introduced by
Abreu (1986).
Definition 1. Stick and carrot strategy
(i) Start the game by abiding by the cooperative agreement.
(ii) Cooperate as long as the cooperative agreement has been
observed in all preceding periods. (iii) If one of the players
deviates from the cooperative agreement at period t, play q̄ at
period t + 1 and return to the cooperative agreement at period
t + 2 (Punishment phase). (iv) If one of the players chooses a
quantity q ≠ q̄ during the punishment phase, start the punishment phase again at the following period.
These strategies are optimal (in the set of stationary
symmetric strategies) when the quantity q̄ is chosen to minimize
the one-period payoff following the deviation. We let Π⁎ denote
the collusive profit of every firm and V the continuation value
following a deviation. Two different regimes appear according
to whether the most severe punishment, V = 0, can be sustained
or not. Let q̂ denote the quantity which induces a zero continuation value,
ð1 dÞð2GðqÞ̂ 2FÞ þ dP* ¼ 0
The most severe punishment can be sustained if and only if
π (q̂) − F ≤ 0.4 Because q̂ is increasing in δ (as shown below)
and π(·) is a decreasing function, there exists a lower bound δ
¯
(F ) such that the most severe punishment can be sustained for
5
δ ≥ δ (F). On the other hand, for δ b δ (F), firms are left with a
¯ continuation value after a deviation,
¯
positive
V N 0.
Given that firms may produce on two markets, we first show
that in an optimal punishment scheme, firms will always be
asked to produce on both markets.
385
Lemma 2. In the optimal punishment scheme, firms are asked
to produce q̄ on both markets.
Proof. If δ ≥ δ (F), the most severe punishment can be
sustained, with¯both firms making losses on the two markets,
as 2G (q̂) − 2F b 0. Hence, its is optimal to ask the firms to
produce on both markets. Suppose that δ b δ (F ). Let q¯1 and q̄2
¯
denote the optimal punishment strategies when
firms produce
on one and two markets respectively. For these punishment
strategies to be sustainable, we need
pð q̄1 Þ þ Pm 2FVð1 dÞV 1 ¼ ð1 dÞðGð q̄1 Þ FÞ þ dP⁎;
2pð q̄2 Þ 2FVð1 dÞV 2 ¼ ð1 dÞð2Gð q̄2 Þ 2FÞ þ dP⁎
It is clear that producing q̄ 2 = qc on both markets is a
sustainable punishment. Suppose that there also exists some
value q̄ 1 which is sustainable. Because δ b δ (F ), q̂ is not
¯ punishment
sustainable, and we can thus define the optimal
1
2
c
scheme as the maximal values q̄ and q̄ in [q ,q̂] for which:6
pð q̄1 Þ ð1 dÞGð q̄1 Þ ¼ dP⁎ Pm þ ð1 þ dÞF;
dP⁎
þ dF:
pð q̄2 Þ ð1 dÞGð q̄2 Þ ¼
2
Because dP⁎VPm F;
dP⁎
2
þ dFzdP⁎ Pm þ ð1 þ dÞ F:
Furthermore, as 2(π(q̂) −F)N 0 = (1 −δ)(2G(q̂)− 2F)+δΠ⁎,
̂
pðqÞ̂ ð1 dÞGðqÞN
dP⁎
þ dFzdP⁎ Pm þ ð1 þ dÞF:
2
This implies that at the maximal values q̄ 1and q̄ 2 in [qc,q̂]
for which the equalities are satisfied, the function π(q) − (1 − δ)
G(q) must be increasing. But then, we necessarily have q̄ 2 N q̄ 1.
Hence, π(q̄ 1) N π(q̄ 2) and
V1 ¼
pð q̄1 Þ þ Pm 2F 2pð q̄2 Þ 2F
N
¼ V 2;
1d
1d
completing the proof of the Lemma.
□
4. Market sharing vs production quotas
With the help of the preceding Lemma, we can compare the
continuation values under market sharing and production
quotas, V p and V s.7
Lemma 3. For any F N 0, V p ≥ V s. If F = 0, V p = V s.
3
These properties appear as Lemma 2 p. 198, Lemma 4 p. 200 and Lemma
21 p. 207 in Abreu (1986).
4
Sustainability requires that no firm have an incentive to produce a quantity
q ≠ q̂ during the first period of the punishment phase. Here, firms can deviate by
either abstaining from producing on both markets or by setting their optimal
response on both markets. In the first case, they obtain a zero profit for that
period. In the second case, they obtain a profit 2π(q̂) − 2F during that period.
Notice that, if π(q̂) − F b 0, the punishment scheme cannot be sustained.
5
When the most severe punishment can be sustained, Abreu (1986) shows
that the stick and carrot strategy is not only optimal in the set of symmetric
stationary strategies, but is globally optimal.
Proof. When F = 0, the collusive profits under market sharing
and production quotas coincide, so that the continuation values
after the punishment are identical. When F N 0, the collusive
6
The most severe punishment cannot be reached if the inequality is strict. In
that case, V can be decreased by raising q̄, while still satisfying the
sustainability condition. Note also that implicit differentiation shows that q̄ is
increasing in δ and decreasing in F.
7
We use superscripts p and s to denote values under production quotas and
market sharing respectively.
Author's personal copy
386
P. Belleflamme, F. Bloch / Economics Letters 99 (2008) 384–386
profits are higher under market sharing than production quotas:
Π s = Π m − F N Π p = Π m − 2F. We consider different cases. First,
suppose that δ ≥ δ¯p(F). Then,
ð1 dÞð2Gð b
q p Þ 2FÞ þ dPs zð1 dÞð2Gð b
qp Þ 2FÞ
þ dPp ¼ 0:
2F þ ð1 dÞð2pðqm =2Þ 2FÞ þ dV p zF þ ð1 dÞðpðqm Þ
As G is decreasing, this implies that qbs ≥ qbp and hence, as π
is also nonincreasing,
pð b
qs Þ FVpð b
qp Þ FV0:
Hence, one can also sustain the most severe punishment
under market sharing, so that δp(F) ≥ δs(F), meaning that the
region of parameters for which ¯the most¯ severe punishment can
be sustained for market sharing agreements always contains the
region of parameters for which the most severe punishment can
be sustained for production quotas.
Next, suppose that δ b δp(F). There are two possibilities.
¯ V s = 0 and V p N V s. Second, if
First, if δs(F) ≤ δ b δp(F), then
s
s
¯
¯
δ b δ (F), V N 0 By Lemma 2, the firms are asked to produce on
¯ markets during the punishment phase and the quantity q̄ is
both
chosen as the maximal quantity in [qc,q̂] for which:
pð q̄Þ ð1 dÞGð q̄Þ ¼
and collusion is easier to sustain with production quotas. By
continuity, there exists a lower bound F such that collusion is
¯
easier to sustain with production quotas
for F ≤ F. Next,
m
m
¯
suppose that F ≥ F̄= (1 − δ)(π(q )+Π ) − 2π(qm /2)). Then,
as
p
s
V ≥ V by Lemma 3,
dP⁎
þ dF:
2
By the same argument as in the proof of Lemma 2, at the
maximal quantity, π(q̄) − (1 − δ)G(q̄) is increasing, so that q̄ is
nondecreasing in Π⁎. Hence, q̄ s ≥ q̄ p and V p ≥ V s, completing
the proof of the Lemma.
□
Lemma 3 shows that the continuation value after a deviation
is always higher with production quotas, where the collusive
profits are lower. This is not surprising: if collusive profits are
lower, sustainability of the punishment requires that the
punishment quantity q̄ be smaller, so that the continuation
value is higher. We can now use the previous two lemmas to
prove our main result.
Theorem 1. Suppose that the two firms use the optimal stick and
carrot strategy. Then there exist values of the fixed cost, F and F̄,
such that collusion is easier to sustain with production¯ quotas
when F ≤ F and easier to sustain with market sharing
agreements¯when F ≥ F̄.
Proof. Consider the conditions under which collusion can be
sustained under market sharing and production quotas:
Pm Fzð1 dÞðpðqm Þ þ Pm 2FÞ þ dV s ;
Pm 2Fzð1 dÞð2pðqm =2Þ 2FÞ þ dV p :
When F = 0, by convexity of the best-response mapping,
π(qm)+ΠmN2π(qm /2) and Vs = Vp. Hence,
ð1 dÞ2pðqm =2Þ þ dV p bð1 dÞðpðqm Þ þ Pm Þ þ dV s
þ Pm 2FÞ þ dV m
and collusion is easier to sustain with market sharing
agreements.
□
Our analysis thus shows that when firms employ optimal
punishment strategies, collusion is easier to sustain with market
sharing agreements for sufficiently large values of the fixed
cost, but with production quotas for sufficiently small values of
the fixed cost. The intuition for the latter part comes from the
fact that for low values of the fixed cost, both forms of collusion
give rise approximately to the same profit. Yet, because of the
convexity of π(q), the instantaneous benefit from a deviation is
higher in the case of market sharing agreements (where the firm
still produces the monopoly quantity on one market, and
responds optimally to the monopoly quantity on the other
market) than under production quotas (where the firms responds
optimally to half the monopoly quantity on both markets). As
for the former part, it is intuitive that when the value of the fixed
cost increases, the incentives to deviate from market sharing
agreements are lower, as firms become more reluctant to enter
the other market. On the other hand, this effect is absent in the
case of production quotas (where firms always incur the fixed
cost on both markets). If punishment were constant, the whole
explanation for Theorem 1 would lie in these two arguments.
But, as we showed, punishment is not constant. This seriously
complicates the analysis and explains why Theorem 1 is silent
on what happens for ‘intermediary’ values of the fixed cost, i.e.,
for F ∈ [F, F̄]. The problem is that it is unclear whether V p − V s
¯ or decreasing in F.
is increasing
Acknowledgment
We thank an anonymous referee for the helpful suggestion.
References
Abreu, D., 1986. Extremal equilibria of oligopolistic supergames. Journal of
Economic Theory 39, 191–225.
Belleflamme, P., Bloch, F., 2004. Market sharing agreements and stable
collusive networks. International Economic Review 45, 387–411.
Bernheim, D., Whinston, M., 1990. Multimarket contact and collusive behavior.
Rand Journal of Economics 21, 1–26.
Gross, J., Holahan, W., 2003. Credible collusion in spatially separated markets.
International Economic Review 44, 299–312.