Optimal Collusion under Cost
Asymmetry
Jeanine Thal∗
Gremaq, Toulouse
Abstract
Cost asymmetry is generally thought to hinder collusion because a more efficient firm has both less to gain from collusion
and less to fear from retaliation. This paper reexamines this
common wisdom and characterises optimal collusion between cost
heterogeneous firms without any prior restriction on the class of
strategies. We first show that even the most efficient firm can
be maximally punished by means of a simple stick-and-carrot
punishment. This implies that collusion is sustainable under cost
asymmetry whenever it is sustainable under cost symmetry. Only
efficient collusion is more difficult when costs are asymmetric. Finally, we show that, in the presence of side payments, cost asymmetry actually facilitates collusion.
1
Introduction
There is general agreement that collusion is less likely when firms are
heterogeneous.1 With regard to cost structures for example, Scherer
(1980) states that ”...the more cost functions differ from firm to firm,
the more trouble firms will have maintaining a common price policy...”.
One reason for this is that reaching a collusive agreement is more difficult under heterogeneity as firms have divergent preferences about the
collusive price. Yet even abstracting from the cartel coordination problem as we do in this paper, it is usually thought that asymmetry also
hinders the sustainability of collusion. Two main arguments support this
opinion: (i) it may be more difficult to retaliate against a low-cost firm if
∗
I am grateful, above all, to Patrick Rey. This paper has also been benefited
from comments by Bruno Jullien, Jean Tirole, and participants of the 2004 Enter
Jamboree in Barcelone.
1
See for example Scherer (1980) or Tirole (1988).
1
it deviates since it earns positive profits even under tough competition,
and (ii) the lower a firm’s costs the higher its potential short-term gain
from deviation at any given collusive market share. Finally, asymmetry
adds to the difficulty of reaching a collusive agreement not because of
coordination issues, but simply because it may be impossible to induce
the low-cost firm to participate and share the market if its competitive
profits are high.
This paper critically examines these arguments and their implications in a context of repeated price competition. We show that it is
possible to design tough punishments even for a low-cost firm: firms can
”collude” on punishments that leave the cheating firm with its minmax
profits, no matter whether the deviator has high or low costs. These
punishments have a stick-and carrot structure as in Abreu (1986, 1988).
Thus, cost asymmetry weakens retaliation only if there is some rationale why firms should use standard trigger strategies or other restricted
forms of punishments instead of these maximal punishments. It remains
nevertheless true that a low-cost firm’s short-term gain from deviation is
higher than that of a high-cost firm at any given market share. However,
this deviation concern can be accommodated by increasing the collusive
market share of a low-cost firm. Finally, we affirm that cost asymmetry
can impede the existence of collusion even if it were sustainable, provided
the low-cost firm’s competitive profits are large enough.
The existing literature on collusion under cost asymmetry is rather
small. In particular, we are not aware of any analysis of optimal punishments without any restriction on the class of strategies.
A large part of the early literature on collusion under cost asymmetry is concerned with the determination of price and output quotas when
firms have different cost functions. As Bain (1948) has argued more than
50 years ago, the objective of maximising joint profits makes sense only
when side payments are possible. In the absence of side payments, total
industry profits may have to be reduced so as to attain a division of the
gains from collusion which induces all firms to stick to the collusive outcome. Following Bain, several papers have analysed the question of how
cost heterogeneous firms should set price and allocate production in the
absence of side payments. Schmalensee (1987), for example, models the
selection of price and output quotas subject to the constraint that each
firm is at least as well off as without collusion applying a variety of selection criteria. However, he does not explicitly examine the circumstances
under which an outcome is a sustainable collusive equilibrium.
Bae (1987) as well as Harrington (1991), whose papers are closest to
our one, combine the analysis of collusive implementability with that of
selecting price and output quotas. Both papers first determine the set of
2
price and output quotas that can be sustained by grim trigger strategies
in a price competition setting. The papers then use different criteria to
select an allocation within the set of sustainable collusive outcomes; Bae
uses the balanced temptation requirement of Friedman (1971), while
Harrington applies the more general Nash Bargaining Solution. Both
authors find that cost asymmetry hinders collusion, but this result hinges
upon the use of grim trigger strategies.
Let us also mention some other recent papers on collusion between
heterogeneous firms which are less related to our work, but which sometimes use optimal punishment. Rothschild (1999) and Vasconcelos (2004)
both deal with collusion under cost asymmetry when firms compete à la
Cournot. Whereas Rothschild uses standard grim trigger strategies, Vasconcelos looks for optimal punishments. However, Vasconcelos restricts
attention to equilibria with proportional market shares on all equilibrium
paths. As he shows, optimal punishments that have a stick-and-carrot
structure as proposed by Abreu (1986, 1988) exist within this restricted
class of equilibria. For a limited range of parameters, these punishments
are also maximal and would thus be optimal even without any restrictions. In the related literature of collusion under asymmetric capacity
constraints, firms compete in prices, but no paper uses general optimal punishments either.2 Davidson & Deneckere (1990) employ trigger
strategies, while Compte, Jenny & Rey (2000) use optimal punishments
but restrict attention to a particular class of equilibria.
Our paper differs from Bae and Harrington mainly in the use of
optimal punishments, which in our model allow to punish both firms
maximally, that is down to zero continuation profits. This implies that,
in contrast to previous conclusions, some collusion is sustainable between heterogeneous firms at any discount factor for which collusion is
sustainable between homogeneous firms. Only efficient collusion, in the
sense that production is allocated in a Pareto optimal way under the
constraint that side payments are impossible, is rendered more difficult
by cost asymmetry. This result is driven by the increased deviation incentive of a low-cost firm whenever the collusive price exceeds its own
monopoly price.
Moreover, we also analyse the sustainability of collusion when side
payments are allowed. The analysis of collusion with side payments supported by optimal punishments is interesting because, somewhat surprisingly, cost asymmetry facilitates collusion in this setting. Collusion
allows a less efficient firm to profit from the other firm’s superior efficiency. As a result, a high-cost firm’s gain from deviation is lower the
2
Lambson (1987) shows that optimal punishments exist in a model of price competition with symmetric capcity constraints.
3
larger the cost advantage of the other firm. The low-cost firm’s incentive
to deviate, on the other hand, is the same as under cost symmetry. When
it comes to selecting a sustainable outcome, a high-cost firm prefers an
efficient outcome, whereas a low-cost firm prefers an inefficiently low
price to reduce the high-cost firm’s gain from deviation.
The analysis proceeds as follows. Section 2 sets out the framework.
Section 3 discusses the use of optimal punishments in models of repeated
price setting in which firms have asymmetric costs. In particular, we
show that subgame perfect maximal punishments, which do not involve
any weakly dominated strategies, exist for the low-cost firm. Section 4
deals with stationary collusion without side payments. We first derive
the set of sustainable collusive outcomes and then restrict attention to
efficient allocations. We also analyse the selection of collusive equilibrium in the set of sustainable allocations as a function of the relative
bargaining power of each of the two firms. Section 5 allows for side
payments. We derive the set of sustainable allocations and compare it
to the one obtained without side payments, as well as to the one under
cost symmetry. Finally, we analyse the selection of a sustainable outcome with side payments as a function of the firms’ respective bargaining
powers.
2
Framework
We consider a simple model of infinitely repeated Bertrand competition
between two firms. The firms produce perfect substitutes. However, one
firm is more efficient than the other one. The low-cost firm produces
at a constant marginal cost c, whereas the high-cost firm produces at
a constant marginal cost c > c. There are no other relevant costs of
production. Market demand at price p is D(p). The demand function
D(p) is assumed to be strictly downward sloping with limp−→−∞ D(p) =
∞.
The monopoly profit functions of the low-cost and the high-cost firm,
respectively, are:
π(p) = (p − c)D(p)
π(p) = (p − c)D(p)
These functions are assumed to be twice differentiable, and strictly
concave. Since c > c, π(p) > π(p) for all p. We denote the (unique)
monopoly prices by pm and pm . A standard revealed preferences argument ensures that pm < pm . Unless stated otherwise, it is assumed that
the difference in marginal costs is non-drastic so that
pm > c.
4
The firms compete in prices in the absence of any capacity constraints. Since the firms produce perfect substitutes, the market demand
goes entirely to the low price firm if the two firms set different prices.
In case of a price tie, the firms allocate the market amongst themselves
in some arbitrary way consistent with the equilibrium. We shall denote
the market share of the low-cost firm by s, and that of the high-cost firm
by (1 − s).
In this set-up, we analyse a noncooperative supergame, where in each
period, the two firms simultaneously set prices, and earn the resulting
profits from price competition. The firms have perfect memory and can
condition their actions on past prices. Each firm aims to maximise its
discounted profit stream. The common and constant discount rate is
δ ∈ (0, 1). We look for a subgame perfect equilibrium.
3
Optimal Punishments
The sustainability of collusive (i.e. supracompetitive) prices in a repeated game requires a credible punishment mechanism which deters
deviation from the collusive outcome. We will focus on optimal punishments, i.e. the harshest sustainable punishments. The use of optimal
punishments permits the characterisation of the maximum scope for collusion. If the firms employed non-optimal punishments, some outcomes
sustainable under optimal punishments would no longer be sustainable.
The lowest possible per period profit that can be imposed on a firm is
its minmax. In our model, the minmax is zero for both firms, as (in the
absence of any restrictions on prices) each firm can drive the other firm’s
profits down to zero by undercutting its price. We will therefore say that
a punishment is maximal whenever the firm’s continuation value along
the punishment path is equal to zero. Obviously, if any such maximal
punishment is sustainable, then it will constitute an optimal punishment,
since no other punishment can possibly be harsher.
Denote by v the continuation value of the low-cost firm’s punishment for the low-cost firm. Similarly, let v be the continuation value of
the high-cost firm’s punishment for the high-cost firm. Then, the punishment of the low-cost firm is maximal if and only if v = 0, and the
punishment of the high-cost firm is maximal if and only if v = 0.
3.1
Grim Trigger Strategies
In a Bertrand model with homogeneous firms, reversion to the (unique)
static equilibrium in which both firms charge their common marginal cost
constitutes a maximal and thus optimal punishment. Reversion drives
the deviating (and all other) firm’s profits down to zero, the minmax
value. It is also obvious that reversion to the static equilibrium forever
5
is a credible threat that constitutes an equilibrium of the subgame played
after a deviation.
Under cost asymmetry, the standard static Nash equilibrium is such
that both firms charge price c, and the low-cost firm makes all the sales
earning π(c) > 0. In the past literature on collusion on prices under
cost asymmetry (Bae (1987), Harrington (1991)), the collusive outcome
is supported by grim trigger strategies with reversion to this static equilibrium after deviation by any of the two firms. While such a trigger
strategy provides a maximal punishment for the high-cost firm, this is
not true for the low-cost firm that still makes positive profits. The lowcost firm is therefore less deterred from deviating than the high-cost firm
or one of the firms in a symmetric duopoly.
3.2
Credible Maximal Punishments
A punishment path for a firm consists of a sequence of prices and market
shares for both firms from time t = 1 until infinity. Let us denote
the punishment paths of the two firms by τ and τ , respectively. The
corresponding continuation values for the punished firm are v and v. A
punishment path is credible (and thus subgame-perfect) if neither of the
two firms has an incentive to deviate from the prescribed punishment
path at any stage.
Starting from the collusive outcome, the two firms start playing punishment τ i if firm i deviates from the collusive path.3 If a firm deviates
from the prescribed action at any stage of a punishment path, its own
punishment is started (or restarted, respectively).
3.2.1
High-Cost Firm
As argued above, reversion to the standard static equilibrium constitutes
a maximal punishment for the high-cost firm. Repetition of the static
equilibrium is evidently also subgame perfect. The following punishment
τ is therefore optimal:
τ : (p, p, s) = (c, c, 1) for all t.
3.2.2
Low-Cost Firm
We have seen that reversion to the standard static equilibrium does not
constitute a maximal punishment for the low-cost firm. In the following,
we will consider two possible types of maximal punishments for the lowcost firm: (i) reversion to a different static equilibrium, in which the
price is c and s = 1, and (ii) stick-and carrot punishments as in Abreu
(1986, 1988). Both are subgame perfect and therefore constitute optimal
3
If both firms deviate simultaneously, no punishment is played.
6
punishments. However, the first option involves a weakly dominated
strategy, whereas an appropriately designed stick-and carrot punishment
does not. Therefore, a stick-and carrot punishment appears preferable.
Reversion to a Static Equilibrium Consider the situation in which
both firms charge the same price c, and the low-cost firm makes all the
sales. This is a static equilibrium: none of the firms has a (strict) incentive to charge a different price given the other firm’s price. Reversion
to this static equilibrium forever constitutes a maximal punishment for
both firms since even the low-cost firm’s continuation payoff is equal to
zero. As usual, reversion to a static equilibrium is also subgame perfect.
However, this punishment path has an undesirable feature: the highcost firm plays a weakly dominated strategy. Given any strategy of the
low-cost firm, the high-cost firm could do equally well or strictly better
by charging a higher price such as c forever.4
Stick-and-Carrot Punishments It is rather straightforward to construct other punishment paths that leave the low-cost firm with a continuation payoff of zero and that do not involve any weakly dominated
strategies. These punishments have a stick-and-carrot structure as in
Abreu (1986, 1988). In the following, we will characterise a class of such
punishments, show that they are credible, and finally prove that they do
not involve any weakly dominated strategies.
Lemma 1 Suppose there exists a collusive path with price p∗ > c and
market shares s∗ and (1 − s∗ ) that can be supported by maximal punishments. Then, any punishment path with price pP < c and market shares
s = 1 and (1 − s) = 0 for a number T of periods and reversion to the
collusive path afterwards, where pP and T are such that
v=
T
X
t−1
δ
π(pP ) +
t=1
δT ∗ ∗
s π(p ) = 0,
1−δ
(1)
provides a maximal, subgame perfect punishment of the low-cost firm.
Moreover, such a punishment path does not involve any weakly dominated
strategy.
4
Note that the standard static equilibrium in which both firms charge c equally
involves a weakly dominated strategy for the high-cost firm. Given any strategy
of the low-cost firm, the high-cost firm could do equally well or strictly better by
charging a price above c forever. However, the weak domination of the standard static
equilibrium is less problematic, as the weakly dominated strategy of the high-cost firm
is the limit of a sequence of undominated strategies. Since we are considering a game
with continuous strategy spaces, this means that the standard static equilibrium is
nevertheless a perfect equilibrium (see Simon & Stinchcombe (1995)). Perfection
strictly excludes the play of weakly dominated strategies only in games with finite
strategy spaces.
7
Proof. Let the punishment path τ have the following structure:
τ : (p, p, s) =
(
(pP , pP , 1) for t ≤ T
,
(p∗ , p∗ , s∗ ) for t > T
where p∗ is the collusive price and s∗ the collusive market share of the
low-cost firm, and collusion can be supported by maximal punishments.
Thus, the firms return to the pre-deviation collusive equilibrium if no
firm has deviated during the first T stage of the punishment path. Let
the price pP and the number of ”stick” periods T such that condition (1)
of the lemma is satisfied, i.e. such that the continuation payoff v of the
low-cost firm is equal to zero. Since prices are assumed to be continuous,
an exact solution pP < c to (1) exists for any T . If prices are bound
to be non-negative, solutions to (1) with finite T only exist as long as
c > 0.
We will now show that this punishment is both maximal and sustainable, which implies that it is optimal.
Maximality:
As v = 0, the punishment is maximal by construction.
Subgame Perfection:
To prove subgame perfection, we need to show that neither firm has
a strict incentive to deviate at any stage of the punishment path. As
usual, it is sufficient to consider unilateral one-shot deviations.
It is also sufficient to only examine the first stage of the punishment
path. First, by construction a firm has no incentive to deviate for t > T
as the collusive equilibrium is played. Second, a firm has no incentive to
deviate in any of the periods t ∈ [2, T ] if it has no incentive to deviate
at t = 1. The short-term gains from a deviation at any stage t ∈ [1, T ]
are the same, whereas the cost of foregoing the future return to collusion
becomes more important in each period.
The low-cost firm’s best deviation at t = 1 is to charge a price above
P
p to earn zero instead of negative profits. After this deviation, τ would
be restarted yielding zero discounted profits to the low-cost firm. The
low-cost firm therefore has no strict incentive to deviate as
0 + δv = 0 ≤ v = 0.
Finally, the high-cost firm cannot benefit from deviating at t = 1:
it cannot make any one-period gains but a deviation would trigger the
punishment τ . Sticking to the punishment τ instead, the high-cost firm
will earn collusive profits in the future. Thus, the high-cost firm has a
strict incentive not to deviate:
0 + δv = 0 < 0 +
δ
(1 − s∗ )π(p∗ ).
1−δ
8
No Weak Domination:
It still remains to be shown that τ does not involve any weakly dominated strategies. We will use the following argument: a strategy is not
weakly dominated if it is the unique best reply to some strategy of the
other firm.
Consider the strategy of the low-cost firm on punishment path τ . At
t = 1, the firm is indifferent between compliance and an optimal upward
one-shot deviation. For t ∈ [2, T ], compliance is the unique best reply.
Suppose now that the high-cost firm (with some small chance) trembles
at and plays the following strategy where ε is some positive constant:













p=
pP − ε at t = 1
pP for t ∈ [2, T ]
p∗ for t > T







pP



 p∗
if p = pP at t = 1
and no further deviation
for t ∈ [2, T + 1]
if p 6= pP at t = 1
for t > T + 1 and no further deviation
Given this strategy of the high-cost firm, playing the strategy prescribed
by τ strictly dominates any deviation at t = 1 for the low-cost firm. This
result is driven by the other firm’s capacity to ”reward” the low-cost firm
for first-period compliance.
We still need to show that compliance is an undominated strategy
at later stages of the punishment path (t > T ), i.e. along the collusive
path. However, the possibility of weak domination in the case where the
low-cost firm is indifferent between collusive compliance and deviation
can be easily ruled out by the introduction of a tremble. If the highcost firm trembles upwards and charges a price above p∗ with some small
chance, then compliance is clearly the low-cost firm’s best reply.
The high cost firm’s strategy along τ is obviously undominated for
t ≤ T because it is the strict best reply the low-cost firm’s strategy along
τ . For t > T , the high-cost firm’s strategy is undominated for the same
reasons as just employed when analysing the low-cost firm.
Consequently, the low-cost firm’s punishment path does not involve
any weakly dominated strategy.
3.2.3
Extension to Three or More Firms
The analysis of optimal punishments under cost asymmetry can easily
be extended to situations in which three or more firms compete in prices.
Let the firms be indexed by i = 1, 2, ...n where n ≥ 3 is the number of
firms. The firms are ordered by their marginal costs c1 ≤ c2 ≤ c3 ≤
... ≤ cn with c1 < cn . The standard static equilibrium is such that the
9
price is equal to cbi = mini∈[2,n] {ci : ci > c1 }, i.e. the price is equal to
the lowest cost above c1 . Firms i ∈ [1, (bi − 1)] do all the production and
share profits (cbi − c1 )D(cbi ). All other firms charge prices equal to their
own costs, but have zero market shares and make no profits.
Then, reversion to the standard static equilibrium is a credible maximal and thus optimal punishment for firms i ∈ [bi, n]. For firms i ∈
[1, (bi − 1)], lemma 1 tells us that maximal credible punishments that
do not involve any weakly dominated strategies also exist. For a finite
number T of periods, all firms charge a price strictly below c1 and the
deviator has a market share of 1. Afterwards, the firms return to the
collusive outcome.
4
Collusion without Side Payments
Our analysis of collusion without side payments consists of four parts.
First, we derive the set of price-market share pairs that are sustainable
collusive outcomes under optimal punishments, as a function of the discount factor. Second, we analyse the sustainability of outcomes in which
production is allocated in a privately efficient way given the absence of
side payments. Third, we address the selection among collusive outcomes, as a function of the firms’ relative bargaining power. Finally, we
consider the low-cost firm’s decision to participate in collusion.
4.1
Sustainable Collusive Outcomes
A collusive outcome is defined by a pair (p, s), where p is the collusive
price and s the market share of the low-cost firm. We restrict p ∈
(c, pm ].5 We focus on stationary equilibria in which the collusive price
and collusive market shares are stationary across periods.
A collusive outcome is sustainable if it can be induced by punishment
strategies which form a subgame perfect equilibrium. We will use the
optimal punishments constructed in the previous section, so that our
analysis is not restricted to any particular class of equilibria. More
specifically, we will characterise the collusive outcomes that could be
sustained by maximal punishments (v = v = 0). Using lemma 1, we
then know that there exist an optimal punishment path yielding maximal
punishments so that (p, s) is indeed a sustainable collusive outcome.
For the high-cost firm, the optimal one-shot deviation from any collusive outcome is to slightly undercut the collusive price in order to
capture the whole market. Thus, the firm has no incentive to deviate
The firms will never find it optimal to price above pm , as higher prices render
collusion more difficult while a Pareto improvement could be achieved by moving to
pm .
5
10
from the collusive outcome if and only if
1
(1 − s)π(p)
1−δ
π(p) ≤
For the low-cost firm, the optimal deviation is to charge pm if the
collusive price lies above pm , or to undercut otherwise. The low-cost
firm has therefore no incentive to deviate if and only if
³
´
1
sπ(p).
1−δ
π min[p, pm ] ≤
These conditions are equivalent to
δ≥s
(CS)
and
s ≥ (1 − δ)
³
´
π min[p, pm ]
π(p)
π(p)
´,
⇔ δ ≥1−s ³
π min[p, pm ]
(CS)
respectively.
A collusive outcome (p, s) is sustainable by maximal punishments if
and only if it satisfies the collusive sustainability conditions CS and CS.
The set of such outcomes is represented by
where
and
e
∆(δ) = {(p, s) | p ∈ (c, min{p(δ),
pm }] , s ∈ [se(δ, p), δ] |} ,
e
p(δ)
≡π
−1
Ã
!
1 − δ ³ m´
π p
δ
se(δ, p) ≡ (1 − δ)
³
(2)
´
π min[p, pm ]
π(p)
.
(3)
Equation (2) defines the highest price at which collusion is sustaine
≤ pm . This highest price corresponds to an outcome
able, as long as p(δ)
in which both CS constraints are simultaneously binding. (2) only has
a solution for δ ≥ 12 , which as we will show below is the discount factor
threshold for collusion to be sustainable. Equation (3) defines the minimum market share se(δ, p) for which collusion at a price p is sustainable,
given the discount factor δ.
Collusion at price p is sustainable if and only if
e
δ ≥ δ(p),
11
(4)
e
can be found by adding up the two collusive sustainability
where δ(p)
conditions:
³
´
π min[p, pm ]
e
³
´.
δ(p) ≡
π(p) + π min[p, pm ]
m
For p ∈ (c, p ],
e
δ(p)
∈
·
π(pm )
1
,
2 π(pm )+π(pm )
¸
. Hence, the discount factor
threshold above which some collusion is sustainable is the same as under
e
e for p ≥ pm .
is the inverse of p(·)
cost symmetry, namely 12 . Note that δ(·)
Figure 1 gives a graphical representation of ∆(δ) for different discount
factors.
If iδ = 12 , othe set of sustainable allocations is ∆( 12 ) =
n
³
(p, s) | p ∈ c, pm , s = 12 , just as under cost symmetry between two
low-cost firms. Note that whenever the collusive price lies below or at
pm , both firms’ optimally deviate to the same price so that the CS constraints are symmetric.
Collusion on higher prices is only possible for higher discount factors
0
e
as δ (p) > 0. Consider a discount factor δ 1 > 12 . Now, the set ∆(δ 1 )
includes all outcomes which satisfy both CS (i.e. lie to the left or on
the vertical line s = δ 1 ) and CS (i.e. which lie to the right or on the
boundary which is defined by s = δ 1 for p ≤ pm , and by sπ(p) = (1 −
δ1 )π(pm ) for p > pm ). The maximum collusive price that can be attained
e 1 ) (assuming that p1 ≤ pm ). At (p1 , δ 1 ), both CS constraints
is p1 ≡ p(δ
are simultaneously binding. As the discount factor grows to δ 2 > δ 1 ,
the set of sustainable allocations becomes even larger: ∆(δ 2 ) ⊃ ∆(δ 1 ).
π(pm )
e m) =
Since δ 2 > δ(p
m )+π(pm ) , the monopoly price of the high-cost firm
π(p
is now sustainable.
Note already that any sustainable outcome with p < pm is Pareto
dominated by another sustainable outcome. As one can easily see in
figure 1, any outcome (p, s) with p < pm is sustainable if and only if
(pm , s) is sustainable. However, the firms achieve a Pareto improvement
by moving from (p, s) to (pm , s).
Drastic Cost Difference In the preceding analysis, we assumed that
the cost difference was sufficiently small so that c < pm . Here we consider
the implications of dropping this assumption and allowing for a drastic
cost advantage. In the static Bertrand equilibrium, the low-cost firm now
charges its monopoly price pm < c and earns profits π(pm ). Using grim
trigger strategies, there is therefore no sustainable collusive outcome.
Once we consider optimal punishments however, collusion is enforceable even in the presence of a drastic cost difference. First note that
lemma 1, which guarantees the existence of a subgame perfect undominated maximal punishment for the low-cost firm, holds whenever the
collusive price exceeds c, irrespective of the size of the cost difference.
12
p
p
~
~
p ( s ) = δ −1 ( s )
m
p1
CS
CS
p
m
∆(δ1 )
c
0
1 − δ 2 1 − δ1
1
2
δ1
Figure 1:
13
δ2
1
s
When the cost difference is drastic, the range of collusive prices (c, pm ] is
strictly included in [pm , pm ]. The analysis of the set of sustainable collusive outcomes remains unchanged. In particular, the incentive compatibility constraints remain IC and IC with min[p, pm ] = pm . Yet, since
p ∈ (c,µpm ] ⊂ [pm , pm ], collusion
is only possible for discount factors
¸
e
δ(p)
∈
π(pm )
π(pm )
,
π(c)+π(pm ) π(pm )+π(pm )
. As pm < c,
π(pm )
π(c)+π(pm )
> 12 .
Using optimal punishments which are maximal in our unrestricted
model, collusion is therefore sustainable even if the cost difference is
drastic. However, collusion is more difficult in the sense that the collusive
price must be relatively high (strictly above pm ), and that collusion at
such high prices is only sustainable for higher discount factors.
4.2
4.2.1
Efficient Collusion
Privately Efficient Production in the Absence of Side
Payments
Let us first analyse the efficient allocation of production between the two
firms in the absence of side payments neglecting the issue of collusive
sustainability. We solve the following problem:
max [sπ(p)]α [(1 − s)π(p)]1−α ,
{p,s}
(P 1)
where α ∈ [0, 1] represents the bargaining power of the low-cost firm.
The analysis of this problem6 yields the following results. The optimal market share of the low-cost firm is equal to its bargaining power:
s=α
(5)
The optimal price is determined by
−α
π 0 (p)
π0 (p)
= (1 − α)
.
π(p)
π(p)
(6)
As α ∈ [0, 1], the optimal market share s ∈ [0, 1], and the optimal price
p ∈ [pm , pm ]. It is straightforward that the solution of (P 1) indeed chracterises the whole set of privately efficient allocations. We will denote the
optimal price as a function of α by pO (α).
Combining expressions (5) and (6) yields the following one-to-one
relationship between the low-cost firm’s market share and the price:
sO (p) =
6
(c − c)D(p) + (p − c)π 0 (p)
, p ∈ [pm , pm ].
(c − c)D(p)
(7)
See exercise 6.1 in Tirole(1988) for a detailed treatment of an equivalent problem.
14
As can be checked easily, sO (·) is strictly downward-sloping.7 The
inverse of sO (·) is pO (·), where pO (s) as defined above gives the price at
which the two firms’ iso-profit lines are tangent.
Note that the Pareto profit frontier of this problem is convex. Thus,
allocations where firms take turns being the monopolist could improve
upon deterministic allocations where both firms produce and share the
market in each period.
4.2.2
Sustainability of Efficient Outcomes
Let us now analyse the collusive sustainability of efficient outcomes as
characterised by condition (7). We shall first approach the problem
graphically by means of figure 2 which displays the line of efficient outcomes pO (·). Given the shape of the sets ∆(δ) and the fact that pO (·)
e
is upward-sloping, it is evident that the
is downward-sloping while p(·)
minimum discount factor compatible with some efficient collusion is such
that the set of sustainable collusive outcomes touches pO (·) in just one
b and the correpoint. This minimum discount factor is denoted by δ,
b b
b b
s). The outcome (p,
s) is sustainable if and
sponding allocation by (p,
O
b
e
b
b = δ(p)
b = δ.
only if δ ≥ δ. Note that bs = s (p)
b the constraint which deFor efficient outcomes with prices above p,
termines the discount factor threshold is CS. For efficient outcomes
b the constraint which determines the discount factor
with prices below p,
threshold is CS.
These results are summarised in the following proposition (a formal
proof is delegated to the appendix):
³
Proposition 2 Collusion on an efficient market sharing allocation pO (α), α
with
α
[0, 1]
is
sustainable
if
and
only
if
∈
(i)
δ ≥ δb = α > 12 if α = δb ⇔ pO (α) = pb
(ii)
e p),
b ≡ δ(
e p).
b = δ(
b and δ
b
where pb ∈ (pm , pm ) is uniquely defined by sO (p)
(iii)
δ ≥ α > δb if α > δb ⇔ pO (α) < pb
O
(α))
δ ≥ 1 − α π(p
> δb if α < bδ ⇔ pO (α) > pb
π(pm )
Note that in all three subcases, collusion is only sustainable for discount factors which strictly exceed 12 , the threshold under cost symmetry.
7
It is also strictly concave if π000 (p) < 0 and Q00 (p) < 0 or these derivatives are
positive but not ”too” large. However, concavity of sOP (p) is not needed for the
results of the following analysis.
15
´
~p ( s ) = δ~ −1 ( s )
p
p
m
p O (s) =
s
O −1
(s)
p̂
p
m
s
c
0
1 − δˆ
Figure 2:
16
1
2
δˆ
1
4.3
Selection of a Collusive Outcome
The previous subsection dealt with the question which discount factor
was necessary to sustain any given privately efficient allocation as a
collusive outcome. In this section, we examine which allocation the firms
select within the set of sustainable collusive equilibria ∆(δ) as a function
of the bargaining power parameter α. This approach takes account of the
methodological point underlined by Harrington (1991) that an allocation
only provides a sensible collusive outcome if it is implementable by a
self-enforcing agreement. By restricting their choice of price and market
shares to the set of sustainable collusive equilibria a priori, the firms
automatically solve this implementation problem.
Formally, the firms maximise the same objective function as in (P 1) under
the additional constraints that collusion be sustainable given the discount factor:
max[sπ(p)]α [(1 − s)π(p)]1−α
{p,s}
(P2)
s.t. CS :
s ≤ δ³
´ ⇔ (p, s) ∈ ∆(δ)
CS : sπ(p) ≥ (1 − δ)π min[p, pm ]
Previous authors (Harrington and Bae) have also imposed that each
firm’s profits be at least as high as its competitive profits. We will postpone the analysis of such an additional restriction to the next section.
We already know that the price at any solution of (P 2) must be
larger equal to pm , since the firms could achieve a Pareto improvement
within ∆(δ) otherwise. We also know from the previous analysis that
δ ≥ 12 . Solving (P 2) then leads to the following results:
Proposition 3 Let α ∈ [0, 1], and δ ∈ [ 12 , 1]. Then, the solution of (P 2)
as a function of α and δ can be characterised as follows:
b
(i) Case 1: α ∈ [0, δ]
Define δ(α) by απ(pO (α)) = (1 − δ(α))π(pm ).
b then both CS constraints are binding. The solution is
• If δ ∈ [ 12 , δ],
e
(p(δ),
δ).
b δ(α)), then CS is the only binding constraint and the
• If δ ∈ (δ,
solution lies on pO (s). The optimal price and market share are
determined by
p = pO (s)
and
sπ(pO (s)) = (1 − δ)π(pm ).
17
• ³If δ ∈ [δ(α),
´ 1], then the solution is the unconstrained optimum
O
p (α), α .
b 1]
(ii) Case 2: α ∈ (δ(p),
e O (α))], then both CS constraints are binding. The
• If δ ∈ [ 12 , δ(p
e
solution is (p(δ),
δ).
e O (α)), α), then only CS is binding. The solution is
• ³If δ ∈ (δ(p
´
pO (α), δ .
³
´
• If δ ∈ [α, 1], then the solution is the unconstrained optimum pO (α), α .
Proof. see appendix.
The results of proposition 3 are illustrated graphically in figure 3.
b For discounts factors between 1 and δ,
b the
First, consider α2 ∈ [0, δ].
2
e
solution lies on p(·)
such that s and p are maximal given the discount
factor. For larger discount factors, the price keeps on increasing but the
market share
of the low-cost
firm is decreasing until the unconstrained
³
´
optimum pO (α2 ), α2 becomes attainable.
b 1], the solution lies on p(·)
e
only as long as the optimal
For α1 ∈ (δ,
O
price p (α1 ) is unsustainable. For discount factors larger or equal to
e
δ(pO (α1 )), the price is equal to pO (α1 ) and the market share is s =
min{δ, α1 }.
4.4
Participation
Cost asymmetry may also hinder collusion because of the difficulty of inducing the low-cost firm to participate. In our model, this translates into
an additional constraint on the low-cost firm’s collusive profits, which
would correspond to a threat point (π(c), 0) instead of (0, 0) in the bargaining problem.8
An additional ”participation constraint” reduces the set of sustainable outcomes to:
∆P C (δ) = ∆(δ) ∩ {(p, s) | sπ(p) ≥ π(c)}
8
If we introduced the additional constraint in our bargaining problem by means of
a threat point (π(c), 0), the solution of the unconstrained problem would still lie on
pO (·), but the low-cost firm’s optimal market share would exceed α. The qualitative
findings of proposition 2 would go through, although the market share α in case 1 and
case 2 would need to be replaced by the new higher market share. If the competitive
profits were very high, then the market share at the unconstrained optimum would
always exceed b
δ, so that only case 2 of the proposition would be relevant.
18
~
~
p ( s ) = δ −1 ( s )
p
p
m
p O (s) =
s
O −1
(s)
p̂
p O (α1 )
p
c
m
0
α2
1
2
δˆδˆ
Figure 3:
19
α1
1
s
For any δ ≥ 12 , the set ∆P C (δ) is non-empty if and only if
δπ(pm ) ≥ π(c).
(8)
This condition ensures that the profits in at least one point in ∆(δ),
namely the most profitable allocation for the low-cost firm, exceed the
firm’s competitive profits. Condition (8) is binding if the discount factor
π(c)
is equal to π(p
m ) ∈ (0, 1). Thus, some collusion with high enough profits
π(c)
for the low-cost firm is sustainable if and only if δ ≥ max[ 12 , π(p
m ) ].
This analysis confirms that the low-cost firm might not want to collude for low discount factors under cost asymmetry , simply because
none of the sustainable collusive outcomes is more attractive than competition. As cost asymmetry grows, this possibility obviously becomes
more likely.
π(c)
For discount factors above max[ 12 , π(p
m ) ], collusion is sustainable but
the additional constraint nevertheless excludes outcomes that are not
profitable enough for the low-cost firm from ∆(δ). This is particularly
important if the discount factor is very high, so that outcomes with a
high price p and a low market share s would be sustainable otherwise.
Thus, although collusion is sustainable under cost asymmetry whenever it is sustainable under cost symmetry, cost asymmetry may hinder
collusion. Firstly, privately efficient collusion is more difficult to sustain
under cost asymmetry than under cost symmetry. Secondly, even if collusion is sustainable, it might be impossible to induce the low-cost firm
to participate.
5
5.1
Collusion with Side Payments
Sustainable Collusive Outcomes
The firms now have the possibility to allocate production to the low-cost
firm without inhibiting collusion, as joint profits can be shared by means
of side transfers. We restrict attention to collusive allocations in which
the low-cost firm carries out all the production. This restriction does not
limit the scope of the analysis, because collusion with side payments is
easiest if production is indeed allocated efficiently. Letting the high-cost
firm produce a positive amount would imply lower collusive profits but
leave deviation profits unchanged.
A collusive outcome is then defined by a pair (p, S), where p is the
price, and S the share of its profits that the low-cost firm keeps. In each
period along the collusive path, the low-cost firm charges a price p, and
20
the high-cost firm a price (slightly above) p.9 Then, the low-cost firm
serves the whole demand at that price, and finally makes a side payment
(1 − S)π(p) to the high-cost firm.
As before, we restrict the price to p ∈ (c, pm ]. We will say that
collusion is efficient if p = pm , so that the firms cannot increase private
efficiency by either reallocating production or changing the price. In contrast to the previous analysis without side payments, collusion at prices
above pm turns out to be unattractive now. However, lower ”inefficient”
prices may occur in order to facilitate collusion.
We will use the same punishments as previously. Since the proposed
punishments are maximal, there is no point in extending the analysis to
allow for side payments during the punishment path.
The low-cost firm could optimally deviate from the collusive outcome
by refusing to make the side payment, and charging its monopoly price
if p ≥ pm or remaining at p otherwise. The low-cost firm’s no-deviation
constraint is
³
´
1
π min[p, pm ] ≤
Sπ(p),
1−δ
which is equivalent to
(1 − δ)
³
π min[p, pm ]
´
≤ S.
(ND)
π(p)
The high-cost firm could deviate from the collusive outcome by producing itself and undercutting the price. Therefore, the high-cost firm
has no incentive to deviate if and only if
1
π(p) ≤
(1 − S)π(p).
1−δ
This condition is equivalent to
π(p)
π(p)
⇔ 1 − (1 − S)
≤ δ.
(ND)
π(p)
π(p)
Adding up the two no-deviation constraints yields the following condition
on the discount factor:
π(p)
S
³
´.
δ ≥ δe (p) ≡ 1 −
(9)
π(p) + π min[p, pm ]
S ≤ 1 − (1 − δ)
9
This means that the high-cost firm is ready to produce in every period without
ever aking any sales along the collusive path. In an alternative scenario, the highcost firm is shut down and can only start production with a time lag of one period.
This would affect the deviation possibilities of the low-cost firm. In particular, the
low-cost firm’s optimal deviation price in N D would always be pm , even if p < pm .
consumers cannot by from the high-cost firm in the same period, so that the lowcost firm’s optimal deviation price would always be pm , even if p < pm .
21
Interestingly, condition (9) implies that collusion with side payments
is sustainable even for discount factors below 12 . Efficient collusion
at price pm for example is sustainable for any discount factor above
S
e
δ (pm )
=
π(pm )
π(pm )+π(pm )
< 12 . Figure 4 illustrates the sets of sustainable
S
S
outcomes with side payments for eδ (pm ) and for δ 1 > δe (pm ).
S
In fact, the function eδ (p) is increasing in p for p ∈ (c, pm ], so that
lowering the price below pm further facilitates collusion as it alleviates
S
the high-cost firm’s deviation concern. Since δe (p) → 0 as p → c, some
collusion is sustainable for any δ > 0.10
It is easy to see that any sustainable collusive outcome (p, S) with
p > pm is Pareto-dominated by the sustainable outcome (pm , S). We
will therefore concentrate on outcomes with prices smaller or equal to
pm in the following discussion.
5.1.1
Comparison to Previous Results
Cost Symmetry We find that collusion with side payments between
asymmetric firms is easier to sustain than collusion between symmetric firms.11 Under cost asymmetry, efficient collusion is sustainable for
discount factors below 12 , the threshold for efficient collusion under cost
symmetry. This result occurs because cost asymmetry lowers the highcost firm’s incentive to deviate. Thanks to the side payments, collusion
with a more efficient competitor is more ”valuable” than collusion with
an identical firm, although the deviation profits are identical in both
cases. The low-cost firm’s deviation incentives, on the other hand, are
unaffected by the cost asymmetry.
Cost Asymmetry in the Absence of Side Payments Under cost
asymmetry, it is easier to sustain collusion with side payments than
without side payments. Consider any collusive outcome (p0 , s0 ) ∈ ∆(δ0 ).
Then, the allocation in which the low-cost firm does all the production,
charges price p0 and keeps a share S = s0 of its profits, is a sustainable
collusive outcome with side payments. It is straightforward to see this
graphically in figure 4, as, for any δ, ∆(δ) is smaller than the set of
sustainable outcomes with side payments. The high-cost firm has higher
10
In the alternative scenario described in footnote 10, the discount factor threshold
S
does not tend towards zero, but lies somewhere strictly between 0 and e
δ (pm ) <
1
2.
Thus, collusion is still sustainable for discount factors below 12 . Moreover, the
discount factor threshold for efficient collusion to be sustainable remains unchanged.
11
The feasibility of side payments is irrelevant under cost symmetry. In the absence
of fixed costs, no advantage can be derived from allocating production to only one of
the firms.
22
p
p
m
ND
ND
p
m
c
0
1 − δ1
1
2
~ m
1− δ S ( p )
δ1
Figure 4:
23
1
S
collusive profits than without side payments (s0 π(p0 ) > s0 π(p0 )), whereas
its deviation profits are unchanged. The low-cost firm’s deviation concern, on the other hand, remain unchanged. Intuitively, side payments
permit greater ”flexibility”: production can be allocated efficiently without inhibiting collusion.
5.2
Efficient Collusion and Selection of a Collusive
Outcome
As argued earlier, any allocation is privately efficient for the two firms
as long as the
low-cost firm is the only producer and maximises profits. Formally,
solving the bargaining problem
max[Sπ(p)]α [(1 − S)π(p)]1−α ,
{p,S}
(P 3)
where α ∈ [0, 1] is the bargaining power of the low-cost firm, yields
p = pm , S = α.
As α varies between 0 and 1, the solution of (10) characterises the whole
set of efficient outcomes.
Interestingly, even if some efficient outcomes are sustainable, the
firms will not always want to select one of them. To see this, consider
S
any given discount factor δ > δe (pm ). Then, the preferred outcome of
the high-cost firm is (pm , 1 − δ). The low-cost firm’s preferred allocation
is not such that the price is pm and S as large as possible given the price.
In fact, the low-cost firm prefers to move to a price strictly below pm .
Such a move has a negative second-order effect on π(pm ), but this effect
is more than offset by a positive first-order effect on the low-cost firm’s
share S, since it alleviates the high-cost firm’s no-deviation constraint.
The general selection problem we address is
max[Sπ(p)]α [(1 − S)π(p)]1−α
{p,S}
³
(P4)
´
s.t. ND : (1 − δ) π min[p, pm ] ≤ Sπ(p)
ND : (1 − δ)π(p) ≤ (1 − S)π(p)
We already know that the firms will never choose any outcome with
p > pm , so that we can simplify the low-cost firm’s no-deviation constraint min[p, pm ] = p.
24
Proposition 4 Let α ∈ [0, 1], and δ ∈ (0, 1). Then, the solution of
(P4) as a function of
· α and δ¸has the following characteristics:
S
(i) Case 1: α ∈ 0, eδ (pm )
S
• If δ ∈ (0, δe (pm )], both no-deviation constraints are binding. The
price lies strictly below pm .
• If δ ∈
µ
¶
S m
e
δ (p ), (1 − α) ,
only binding constraint.
the solution is (pm , 1 − δ). ND is the
• If δ ∈ [(1 − α), 1), the solution is the unconstrained optimum
(pm , α).
(ii) Case 2: α ∈
µ
¸
S m
e
δ (p ), 1
Define S E (δ) as the market share at which the ND constraint is
binding at price pm :
S E (δ) ≡ 1 − (1 − δ)
µ
S
π(pm )
.
π(pm )
¶
There exists some δ T ∈ 1 − α, δe (pm ) , such that
• If δ ∈ (0, δ T ], both no-deviation constraints are binding. The price
lies strictly below pm .
³
−1
´
• If δ ∈ δ T , S E (α) , ND is the only binding constraint. The price
lies strictly below pm .
h
−1
´
• If δ ∈ S E (α), 1 , the solution is the unconstrained optimum
(pm , α).
Proof. see the appendix.
25
References
[1] Abreu, D. (1986), ”Extremal Equilibria of Oligopolistic Supergames”, Journal of Economic Theory, Vol.39: 191-225.
[2] Abreu, D. (1988), ”Towards a Theory of Discounted Repeated
Games”, Econometrica, Vol.56: 383-396.
[3] Bae, H. (1987), ”A Price-Setting Supergame between Two Heterogeneous Firms”, European Economic Review, Vol.31: 1159-1171.
[4] Bain, J. S. (1948), ”Output quotas in imperfect cartels”, Quarterly
Journal of Economics, Vol.62: 617-622.
[5] Compte, O., Jenny, F., and Rey, P. (2000), ”Capacity Constraints,
Mergers and Collusion”, European Economic Review, Vol.46: 1-29.
[6] Davidson, C. and Denerecke, R. (1990), ”Excess Capacity and Collusion”, International Economic Review, Vol.31(3): 521-541.
[7] Friedman, J. W. (1971), ”A Non-cooperative Equilibrium for Supergames”, Review of Economic Studies, Vol.38: 1-12.
[8] Harrington, J. E. Jr. (1991), ”The Determination of Price and Output Quotas in a Heterogeneous Cartel”, International Economic
Review, Vol.32(4): 767-792.
[9] Lambson, V. E. (1987), ”Optimal Penal Codes in Price-setting Supergames with Capacity Constraints”, Review of Economic Studies,
Vol.54: 385-297.
[10] Rothschild, R. (1999), ”Cartel Stability when Costs are Heterogeneous”, International Journal of Industrial Organization, Vol.17:
717-734.
[11] Scherer, F. M. (1980), Industrial Market Structure and Economic
Performance, Houghton Mifflin Company, Boston.
[12] Schmalensee, R. (1987), ”Competitive Advantage and Collusive
Equilibria”, International Journal of Industrial Organization, 5:
351-368.
[13] Simon, L. K., and Stinchcombe , M. B. (1995), ”Equilibrium Refinement for Infinite Normal Form Games,” Econometrica, Vol.63(6):
1421-1444.
[14] Tirole, J. (1988), The Theory of Industrial Organization, MIT Press,
Cambridge.
[15] Vasconcelos, H. (2004), ”Tacit Collusion, Cost Asymmetries, and
Mergers”, Rand Journal of Economics, forthcoming.
A
Appendix
Proof of proposition 2:
e p),
b the two
Define pb as the price at which, for a discount factor of δ(
IC constraints are simultaneously binding at an efficient market-sharing
26
allocation:
O
s
b
(p)
=
e p)
b
δ(
b + (p − c)π 0 (p)
b
π(pm )
(c − c)D(p)
=
.
⇔
b
b + π(pm )
(c − c)D(p)
π(p)
Such a pb exists and is unique because sO (p) is strictly decreasing with
e
sO (pm ) = 1 and sO (pm ) = 0, while δ(p)
is strictly increasing
over the
³
´
1
m
m
m
e
relevant range and δ(p ) = 2 . It also follows that pb ∈ p , p , which
implies that
1
e p)
b > .
δb ≡ δ(
2
b b
b is sustainable as long as
s), where bs ≡ sO (p),
The efficient outcome (p,
δ ≥ bδ.
This proves part (i) of the proposition.
Collusion at an efficient outcome (pO (α), α) with pO (α) 6= pb is not
e O (α)). This is because
sustainable when the discount factor is equal to δ(p
e O (α)), the only market share compatible with collusion at price
if δ = δ(p
e O (α)), which is different from α unless pO (α) = p.
b Hence,
pO (α) is δ(p
e O (α)) for pO (α) 6= p,
b which implies that (at least)
we must have δ > δ(p
one of the CS constraints is slack.
h
´
³
i
Consider first the case in which pO (α) ∈ pm , pb ⇔ α ∈ bδ, 1 . For
e O (α)). This implies that the r.h.s.
such prices, sO (pO (α)) = α > bδ > δ(p
of CS is larger than the r.h.s. of CS. Therefore, the relevant constraint
on the discount factor δ is CS:
1
δ ≥ sO (pO (α)) = α > δb > .
2
This proves part (ii) of the proposition.
b implying
b pm ] ⇔ α ∈ [0, δ),
Next, consider the case where pO (α) ∈ (p,
O O
O
b
e
that s (p (α)) = α < δ < δ(p (α)). Now, the relevant constraint on δ
is CS:
δ ≥ 1 − sO (pO (α))
π(pO (α)) b 1
π(pO (α))
=
1
α
>δ> .
−
π(pm )
π(pm )
2
This proves part (iii) of the proposition.
q.e.d.
Proof of proposition 3:
This proof proceeds in two steps. First, we exclude all Pareto dominated allocations from ∆(δ). Second, we characterise the exact solution
27
of the problem as a function of the bargaining power and the discount
factor, making use of the results found in the first step.
step 1:
Firstly, any allocation (p, s) in ∆(δ) such that p < pm is Pareto
dominated by the allocation (pm , s) which is also included in ∆(δ).
Secondly, any allocation in ∆(δ) that lies above pO (·) in the (s, p)space is Pareto dominated by an allocation in ∆(δ) that lies on pO (·).
To
³ see this,
´ recall that the solution of the unconstrained problem P 1 is
O
p (α), α . At any such Pareto optimal outcome, the isoprofit lines of
the two firms are tangent. In any point above pO (·), the isoprofit lines
of the high-cost firm are steeper than those of the low-cost firm. Note
also that for p ≥ pm the low-cost firm’s profits are constant along each
CS constraint, so that the CS constraints are in fact isoprofit curves.
Now consider any (p, s) ∈ ∆(δ) such that p > pO (s). The firms could
achieve a Pareto improvement for example by moving to a lower price
and a lower market share along the low-cost firm’s isoprofit line. Such
a move would leave the low-cost firm’s profits unchanged, but increase
the high-cost firm’s profits. The move would take place within the set
of sustainable outcomes, since the low-cost firm’s isoprofit line in (p, s)
coincides with the CS constraint passing through (p, s), and moving to
a lower market share relaxes the CS constraint.
Thirdly, any allocation (p, s) in ∆(δ) such that p < pO (s) and s < δ
is Pareto dominated by another allocation (p0 , s0 ) ∈ ∆(δ) such that p0 ≤
pO (s0 ) and s0 = δ. To see this, consider any (p, s) ∈ ∆(δ) that lies below
pO (·) at which CS is slack, i.e. s < δ. Since (p, s) lies below pO (·), the
isoprofit lines of the low-cost firm are steeper than those of the high-cost
firm. Thus, the firms could achieve a Pareto improvement by moving
to a higher price and a higher market share along the low-cost firm’s
isoprofit line. Such a Pareto improvement is possible within ∆(δ), as
CS is slack initially, and the low-cost firm’s isoprofit line coincides with
the CS constraint passing through (p, s).
To summarise, we define the set of Pareto-undominated sustainable
outcomes Ω(δ):
³n
Ω(δ) = ∆(δ)∩
h
´
o
n
o´
(p, s) | (p ∈ pm , pO (δ) , s = δ) ∪ (p, s) | p = pO (s), s ∈ [0, 1]
Hence, the problem simplifies to maximising the objective function subject to (p, s) ∈ Ω(δ).
step 2:
To characterise the solution, we form the Lagrangian of P 2 taking
into account that min[p, pm ] = pm since Ω(δ) does not include any p <
28
.
pm . The derivative with respect to price is:
Ã
(1 − s)π(p)
αsπ 0 (p)
sπ(p)
!1−α
Ã
(1 − s)π(p)
+(1−α)(1−s)π 0 (p)
sπ(p)
!−α
+λsπ 0 (p) = 0
(FOC1)
Solving (F OC1) for λ yields
µ
1−s
λ=−
s
¶1−α Ã
π(p)
π(p)
!−α "
#
π 0 (p)
π(p)
(1 − α) 0
.
+α
π (p)
π(p)
(10)
It must also hold that
λ[sπ(p) − (1 − δ)π(pm )] = 0, sπ(p) − (1 − δ)π(pm ) ≥ 0, λ ≥ 0.
Equation 10 implies that:
λ > 0 ⇔ p < pO (α)
since
π 0 (p)
π(p) h
λ = 0 ⇔ p = pO (α)
h
i
is decreasing in p for p ∈ pm , pm , while
i
(11)
(12)
π 0 (p)
π(p)
is decreasing in
p for p ∈ pm , pm .
Using the results of step 1 as well as (11) and (12), we can complete
the solution by distinguishing two cases:
Case 1: α ≤ δb
First, we define δ(α) as the discount factor at which the CS constraint is strictly binding at the unconstrained solution: απ(pO (α)) =
(1 − δ(α))π(pm ). Then, the unconstrained solution is attainable if and
only if δ ≥ δ(α).
b CS must be binding whenever the unconstrained soluα ≤ δ,
Since
³
´
O
tion p (α), α is not attainable. Assume that CS were slack. Then,
necessarily λ = 0 ⇔ p = pO (α). However, any outcome with price pO (α)
lies
³ outside
´ the set of Pareto-undominated sustainable outcomes Ω(δ) if
O
p (α), α ∈
/ Ω(δ). This contradiction proves the claim that CS must
be binding.
We still need to examine for which values of δ the CS constraint is
binding, too:
b all points in ∆(δ) lie below pO (·) so that CS must be
For δ ≤ δ,
binding for the solution to lie in Ω(δ).
For δb < δ ≤ δ(α), CS is the only binding constraint, since the solution would lie above the pO (·) curve outside of Ω(δ) if both constraints
were binding. Since CS is not binding, the solution must lie on pO (·) to
be part of Ω(δ).
To summarise:
29
b both CS constraints are binding at the solution of P 2.
• For δ ≤ δ,
• For δb < δ ≤ δ(α), the solution lies on pO (·) and only CS is binding.
• For δ > δ(α), the solution of the constrained
³
´ problem is the soluO
tion of the unconstrained problem p (α), α .
Case 2: α > δb
b CS must be binding whenever the unconstrained solution
> δ,
³ For α ´
pO (α), α is not attainable (i.e., whenever δ < α). We can prove this
result by contradiction. Suppose that CS is slack (i.e., s < δ). Then,
the solution must lie on pO (·) to be in Ω(δ). Consequently, p = pO (s) >
pO (δ) > pO (α). This violates p ≤ pO (α) ⇔ λ ≥ 0. Thus, CS must be
binding as long as δ does not exceed α.
We still need to establish when CS is the only binding constraint:
e O (α)), the CS constraint must be slack. If both conFor δ > δ(p
e
straints were binding, the price (i.e., p(δ))
would exceed pO (α), which
violates λ ≥ 0.
e O (α)) < δ
b
For δ < eδ(pO (α)), CS must be binding, too. Since δ(p
b only points below pO (·) are sustainable so that CS must be
for α > δ,
binding for the solution to be part of Ω(δ).
To summarise:
e O (α)), both Cs constraints are binding at the solution
• For δ ≤ δ(p
of P 2.
³
´
e O (α)), the solution is pO (α), δ .
• For α > δ > δ(p
• For δ > α, the solution of the ³constrained
´ problem is the solution
O
of the unconstrained problem p (α), α .
q.e.d.
Proof of proposition 4:
As in the previous proof, we first restrict the set of possible solutions
by excluding all Pareto dominated outcomes from the set of sustainable
outcomes.
step 1 :
Firstly, any sustainable outcome (p, S) such that p > pm is Pareto
dominated by the sustainable outcome (pm , S), which leaves higher profits for both firms.
Secondly, any sustainable outcome (p, S) such that p < pm and (1 −
δ)π(p) < (1 − S)π(p) is Pareto dominated by the sustainable outcome
(p0 , S) such that (1−δ)π(p0 ) = (1−S)π(p0 ). The move to (p0 , S) increases
30
profits for both firms because p < p0 ≤ pm . The inequality p < p0 follows
0)
0 −c)
0 −c)
> π(p
> (p
↔ (p
from π(p)
.
(p−c)
(p−c)
π(p)
π(p0 )
We thus know that any solution of (P 4) will involve a price p ≤ pm .
Moreover, if p < pm , then the ND constraint must be binding.
step 2:
To complete the solution, we form the Lagrangian of (P 4), taking
into account that min[p, pm ] = p. The multipliers of the two constraints
are denoted by µ and µ, respectively. The first-order conditions are:
" µ
1−S
α
S
¶1−α
µ
1−S
− (1 − α)
S
¶−α #
π(p) = µπ(p) − µ
(13)
S α (1 − S)1−α π 0 (p) = µ [(1 − δ)π 0 (p) − (1 − S)π 0 (p)]
(14)
µ [S − (1 − δ)] = 0, S ≥ 1 − δ, µ ≥ 0
(15)
µ [(1 − S)π(p) − (1 − δ)π(p)] ≥ 0, (1−S)π(p) ≥ (1−δ)π(p), µ ≥ 0 (16)
Distinguish two cases:
Case 1: α ≤ δ S (pm )
³
´
In this case, the unconstrained solution pm , α is sustainable if and
only if δ ≥ 1 − α.
Consider any δ < 1 − α. Then, S must be larger than α, since
outcomes with lower market shares are unsustainable. If S > α, firstorder condition (13) implies that µ > 0, so that S = 1 − δ must hold
by (14). Thus, we know that ND is binding whenever the solution is
constrained.
³
i
If δ ∈ 0, δS (pm ) , only outcomes with p ≤ pm are sustainable. We
therefore know
from step´ 1 that ND must binding, too.
³
S m
If δ ∈ δ (p ), 1 − α , both constraints are simultaneously binding
for p > pm . Since we have excluded prices above pm in step 1, we can
conclude that only ND is binding. Since ND is slack, µ = 0 by (16),
which implies that p = pm by (14).
To summarise:
1.
³
i
• For δ ∈ 0, δ S (pm ) , both ND constraints are binding at the
solution.
³
´
• For δ ∈ δS (pm ), 1 − α , the solution is (pm , 1 − δ).
• For δ ∈ [1 − α, 1), the solution of the constrained problem is
the unconstrained solution.
31
Case 2: α > δ S (pm )
Define S E (δ) as the highest market share at which efficient collusion
(p = pm ) is sustainable. Then, the unconstrained solution (pm , α) is
−1
sustainable whenever δ ≥ S E (δ).
−1
Consider any δ < S E (δ), and assume that p = pm . Then, only
market shares smaller than α are sustainable. If S < α, then by (13)
µ > 0, so that N D is binding. This contradicts p = pm , because by (14)
−1
p = pm if and only if µ = 0. Thus, p < pm for any δ < S E (δ). Step 1
then implies that ND is binding at any constrained solution.
It remains to determine for which values of δ the other no-deviation
constraint is binding, too. If δ ≤ (1 − α), so that only market shares
above α are sustainable, both constraints must be binding by (13) and
(16). For high discount factors δ ≥ δ S (pm ), on the other hand, it is
impossible that both constraints are binding at a price below pm .
Comparative statics on the solution of (P 4) ignoring the ND constraint show that in this case the optimal profit share S is strictly increasing in δ if and only if S < α. It is evident that S ≤ α for 1 − α < δ,
since otherwise both constraints would need to be binding by (13) and
this is ruled out by 1 − δ < α. Thus, either both constraints are binding,
or only ND is binding and S is increasing in δ. Since we know that for
large enough discount factors, only ND is binding, it must be that S
decreases first so as to ensure that S ≤ α even when S finally strictly increases. Moreover, once only ND is binding, it remains the only binding
constraint: since the market share is increasing in δ, N D is not binding
0
for δ 00 > δ 0 if ND is not
³ binding for δ´ . We can conclude that there exists
some threshold δ T ∈ 1 − α, δ S (pm ) such that both ND constraints are
−1
binding for δ ≤ δ T , but only ND is binding for δ T < δ ≤ S E (δ).
q.e.d.
32