Collusion, Firm Numbers and Asymmetries Revisited∗
Luke Garrod† and Matthew Olczak‡
June 10, 2016
Abstract
In an infinitely repeated game where market demand is uncertain and where firms with
(possibly asymmetric) capacity constraints must monitor the agreement through their privately observed sales and prices, we analyse the firms’ incentives to form a cartel when
they could alternatively collude tacitly. In this private monitoring setting, tacit collusion
involves price wars on the equilibrium path if a firm cannot infer from its low sales whether
the realisation of market demand was unluckily low or whether at least one rival has undercut the collusive price. In contrast, explicit collusion involves firms secretly forming an
illegal cartel to share their private information to avoid such price wars, but this runs the
risk of sanctions. We show, in contrast to the conventional wisdom and consistent with the
empirical evidence, that the incentives to form an illegal cartel can be smallest in markets
with a few symmetric firms, because tacit collusion is most successful in such markets.
JEL classification: D43, D82, K21, L12, L41
Key words: cartel, tacit collusion, imperfect monitoring, capacity constraints
∗ We have benefitted from the comments of seminar participants at the ESRC Centre for Competition Policy
and the Bergen Center for Competition Law and Economics (BECCLE) conference 2016. The usual disclaimer
applies.
† School of Business and Economics, Loughborough University, LE11 3TU, UK, email: [email protected]
‡ Aston Business School, Aston University, Birmingham, B4 7ET, UK, email: [email protected]
1
1
Introduction
One of the main aims of competition policy is to prevent explicit collusion, where members
of a cartel reach an agreement over their conduct by directly communicating with each other.
Competition authorities impose sanctions on detected cartels in an attempt to deter such conduct,
and leniency programmes that grant immunity to the first cartel member to come forward create
a race to the authorities to avoid such sanctions. To ensure that such policies are designed in
a way to deter and detect cartels as effectively as possible, it is important to understand the
circumstances in which cartels are most likely to form. Regarding firm numbers and asymmetries,
the conventional wisdom is that cartels are most likely to form in markets where there are a few
relatively symmetric firms, often considered to be no more than three or four. One reason for this
is that reaching an agreement is easier when firms are similar or when there are fewer of them.
A second reason is that economic theory has consistently shown that it is difficult to incentivise
some members from deviating on the collusive agreement when there are a large number of firms
or when there are asymmetries among the firms.1
Surprisingly, empirical evidence from prosecuted cartels in Europe and the US provides little
evidence in favour of this conventional wisdom. Instead, research shows that most prosecuted
cartels tend to include a large number of generally very asymmetric firms. For instance, Levenstein and Suslow (2006) reviewed the literature for prosecuted cartels in the US throughout
the 20th century (see Posner, 1970; Hay and Kelley, 1974; Fraas and Greer, 1977; and Dick,
1996) and showed that the median cartel commonly had between six to ten members, depending
upon the sample. More recently, Davies and Olczak (2008) analysed the prosecuted cartels in
Europe between 1990 and 2006. They found that the median cartel had five members, but the
asymmetries among the firms were so large that it “calls into question whether symmetry of
market shares is a pervasive feature of real world cartels” (p.198).
One potential explanation of this puzzle is that it is a result of a sample selection bias:
cartels with few relatively symmetric firms may be harder to detect and are consequently underrepresented in any sample of prosecuted cartels. However, this pattern is still observed for samples
of legal cartels where no such selection bias exists (see Dick, 1996). An alternative explanation
is that firms may tacitly collude in markets with few symmetric firms rather than colluding
1 This result is robust to a large number of possible forms of heterogeneity. For example, collusion is hindered
by asymmetries whether they are in terms of firms’ capacity constraints (see Compte et al., 2002; Bos and
Harrington, 2010 and 2014; Garrod and Olczak, 2016), the number of differentiated products that each firm sells
(see Kühn, 2004), the cost structures of firms (Vasconcelos, 2005; and Miklós-Thal, 2011), or the rate at which
firms discount the future (see Harrington, 1989).
2
explicitly by forming an illegal cartel. This may arise because tacit collusion is not usually
considered to be illegal, despite causing similar effects as explicit collusion.2 Interestingly, the
evidence on tacit collusion is more consistent with the convential wisdom. For example, Davies
et al. (2011) show that the European Commision’s intervenions in merger due to the increased
likelihood of tacit collusion have almost always been confined to cases where there would have
been only two relatively symmetric players post-merger.3
In this paper, we explore the previously under-researched idea that this puzzle can be due
to firms tacitly colluding in markets with few symmetric firms rather than colluding explicitly
by forming an illegal cartel. We achieve this by extending Garrod and Olczak (2016), where we
modelled tacit collusion in a setting where market demand is uncertain and where firms with
(possibly asymmetric) capacity constraints never directly observe their rivals’ prices or sales.
In this private monitoring setting, similar to that first discussed by Stigler (1964) and recently
analysed by Harrington and Skrzypacz (2011), the firms monitor a collusive agreement through
their privately observed sales, and when they receive low sales they may be unsure whether it
is due to low demand or due to a rival undercutting the collusive price. Consequently, to solve
this non-trivial signal extraction problem, the firms initiate price wars on the equilibrium path,
similar to other models of imperfect monitoring (see Green and Porter, 1984; Tirole, 1988). We
extend this model of tacit collusion to allow the firms to instead form an illegal cartel that allows
them to share their private information with each other to avoid the non-trivial signal extraction
problem. Thus, firms face a tradeoff between these two forms of collusion: on the one hand, a
cartel is likely to generate higher profits than tacit collusion, because the cartel can avoid costly
price wars; but, on the other hand, forming a cartel is illegal and runs the risk of sanctions.
We use this model to investigate under which market structures firms have incentives to collude
explicitly by forming a cartel when they also have the ability to collude tacitly, and consider the
policy implications.
Our paper is related to the literature that analyses explicit collusion in the presence of a
competition agency, where collusion requires explicit communication and where cartel members
face the threat of sanctions. This literature has focussed mainly on the effects of leniency
programmes in markets with symmetric firms (see Motta and Polo, 2003, and Chen and Rey,
2 Hard evidence of a cartel is usually required to prove guilt (e.g., recorded conversations, minutes of meeting, or
emails). This approach to the law is arguably desirable because it ensures legal certainty over illegal conduct, and
it prevents competitive behaviour from being punished erroneously, which could undermine the market mechanism
across the economy (Motta, 2004: 185–190).
3 It is also supported by experimental evidence in Fonseca and Normann (2012) where the gains from communication are much large with 4 rather than 2 players.
3
2013).4 However, in a Bertrand-Edgeworth setting similar to ours, Bos and Harrington (2014)
also consider the effect of sanctions on the number and type of firms that join the cartel when
firms are asymmetrically capacity constrained. All of these papers model collusion under perfect
observability yet assume that tacit collusion is not possible. If tacit collusion were possible,
then any sanction would be sufficient to prevent explicit collusion. It is often argued that
communication is required in these models to coordinate on price, so these models are most
appropriate for price-fixing cartels. However, the need to coordinate is often not modelled as
usually the collusive price is the same in each period. An implication of this assumption is that
the benefit of communication compared to tacit collusion is exogneous. In contrast, we model
collusion in a private monitoring setting, where informational problems can mean that tacitly
colluding firms are unable to extract the full monopoly profit. Thus, in our model firms can
exchange their private information on sales and prices to increase the collusive profits, so that
the benefits of communication are endogenous. Furthermore, this exchange of private information
captures one of the main functions of a large number of cartels (see, for example, Harrington’s
(2006b) discussion of the following cartels: carbonless paper, choline chloride, copper plumbing
tubes, graphite electrodes, plasterboard, vitamins, and zinc phosphate).
In relation to explicit collusion, we show first that, if firms are sufficiently patient, then firms
are able to set the monopoly price in each period. However, the sum of their profits is less than
the monopoly profit due to the expected fine, and the profits are independent of the capacity
distribution. In contrast, the static Nash equilibrium profits are increasing in the size of the
largest firm, because competition is less intense. Thus, the difference between each firm’s static
Nash equilibrium profits and the best equilibrium profits under explicit collusion decreases as the
largest firm gets larger. Consequently, consistent with the previous literature, if we do not take
account of tacit collusion, then we would also conclude that firms have the greatest incentive to
form a cartel when they are symmetric.5
When tacit collusion is taken into account, however, we show that firms’ incentives to form
a cartel can be very different. For instance, we show that if monitoring is perfect under tacit
collusion, then there is no incentive to form a cartel, because firms can extract the monopoly profit
tacitly without the need to form a cartel. This happens when fluctuations in market demand
are small. The reason is that, in the absence of communication, all firms can always infer when
4 One other exception is Martin (2006) who considers tacit and explicit collusion in a Cournot framework.
However, our model allows for a much broader range of comparative statics on the chosen form of collusion,
including allowing for asymmetries between firms.
5 For the same reasons as just described, the critical discount factor is higher when the largest firm is larger.
4
at least one firm’s sales are below some firm-specific “trigger level”, which is determined by the
largest possible sales consistent with them or a rival being undercut on price. Thus, if all firms
set a common price, then all firms’ sales will exceed their respective trigger levels when the
realisation of market demand is high, otherwise they can all fall below the trigger levels. Yet, if
all firms do not set a common price, then at least one firm will receive sales below their trigger
level. Thus, when fluctuations in market demand are small, firms will only ever receive sales
below their trigger levels if they are undercut, so monitoring is perfect.
In contrast, if fluctuations in market demand are large, such that there is imperfect monitoring
because sales can also fall below the trigger levels when firms set a common price, then firms
have the least incentive to form a cartel when they are symmetric. This is due to the fact that
the best equilibrium profits under tacit collusion decrease as the smaller firm gets smaller. The
reason is that deviations by the smallest firm are most difficult for rivals to detect, because
each rival’s resultant sales are most similar to its collusive sales. Thus, monitoring is more
difficult when the smallest firm has less capacity, so punishment phases occur more often on the
equilibrium path. Thus, firms have a greater incentive to form a cartel when they are sufficiently
asymmetric, because they prefer to tacitly collude when they are symmetric. Finally, given both
equilibrium profits are independent of the largest firm, it actually follows that firms have the
smallest incentive to form a cartel in a symmetric duopoly. This is in stark contrast to the
conventional wisdom that suggests that cartels have the greatest incentive to form in such a
case.
More broadly, our paper is also related to a literature that is interested in explaining how firms
could when they share non-verifable sales reports. For instance, Compte (1998) and Kandori
and Matsushima (1998), charactise equilibria under private monitoring in which firms truthfully
report and this enables them to acheive joint profit maximisation. Harrington and Skrzypacz
(2011) model a collusive agreement in which firms truthfully reveal non-verifiable sales information to each other and make inter-firm transfers based on this. In a related paper, Awaya and
Krishna (2014) show that when firms’ sales at similar prices are sufficiently correlated such communication can facilitate collusion without the need for transfers between firms. Furthermore,
Spector (2015) shows that such information exchange can facilitate collusion even if sales data
will eventually be publicly verified. This is because communication allows firms to immediately
put in place temporary market share reallocations and removes the need for price wars.6 While
6 This
literature focuses on how communication helps to monitor compliance. In a related literature (Athey
and Bagwell, 2001 and 2008) sharing of cost information increases cartel profits by ensuring that the firm with
the low costs produces more of the industry output.
5
we assume in the main body of paper that sales are verifable, we show in a robustness section
that our analysis still holds if sales are non-verifable.
The rest of the paper is organised as follows. Section 2 sets out the assumptions of the model
and solves for the static Nash equilibrium. In section 3, we analyse the two forms of collusion. In
section 4, we analyse firms’ incentives to form a cartel and analyse how such incentives changes
as the capacity distribution changes. Section 5 explores the robustness of our results, and section
6 concludes. All proofs are relegated to appendix A. In appendix B, we analyse an alternative
strategy profile to show when our main analysis is the most profitable. This appendix is best
read after section 3.
2
The Model
To model the difference between explicit and tacit collusion for asymmetric firms, we extend the
capacity-constrained private monitoring repeated game in Garrod and Olczak (2016). We now
allow firms to form a cartel to exchange their private information at the risk of being detected
and penalised by a competition agency.
2.1
Basic assumptions
Consider a market in which a fixed number of n ≥ 2 capacity-constrained firms compete on price
to supply a homogeneous product over an infinite number of periods. Firm i = {1, . . . , n} can
produce a unit of the product at a constant marginal cost but the maximum it can produce in
P
any period is ki . We denote the total industry capacity as K ≡ i ki , the sum of firm i’s rivals’
P
capacities as K−i ≡ j6=i kj , and we let kn ≥ kn−1 ≥ . . . ≥ k1 > 0, without loss of generality. In
any period t, firms set prices simultaneously, where pt = {pit , p−it } is the vector of prices, pit is
the price of firm i and p−it is the vector of prices of all of firm i’s rivals. Firms have a common
discount factor, δ ∈ (0, 1), and we normalised their marginal costs to zero.
Market demand consists of a mass of mt (infintesimally small) buyers, each of whom are
willing to buy one unit provided the price does not exceed their reservation price, which we
normalise to 1. We assume that firms do not observe mτ , for all τ ∈ {0, . . . , t}, but they know
that mt is independently drawn from a distribution G(m), with mean m
b and density g(m) > 0
on the interval [m, m]. Furthermore, firm i never observes firm j’s prices, pjτ , or sales, sjτ , j 6= i,
for all τ ∈ {0, . . . , t}. In contrast, buyers are informed of prices, so they will want to buy from
the cheapest firm. Thus, this setting is consistent with a market in which all buyers are willing
6
to check the prices of every firm in each period to find discounts from posted prices, but actual
transaction prices are never public information.
2.2
Demand allocation and sales
Consistent with much of the previous literature, we assume demand is allocated according to the
following rule:
The proportional allocation rule
Unsupplied buyers want to buy from the firm(s) with the lowest price among those with spare
capacity.
• If the joint capacity of such firms is insufficient to supply all of the unsupplied buyers, then
such capacity is exhausted, and the remaining unsupplied buyers now want to purchase
from the firm(s) with the next lowest price among those with spare capacity, and so on.
• If the joint capacity of such firms suffices to supply all of the unsupplied buyers, then each
firm supplies an amount of buyers equal to its proportion of the joint capacity.
Following Garrod and Olczak (2016), we also place the following plausible yet potentially
restrictive assumption on the capacity distribution:
Assumption 1. K−1 ≤ m.
This says that the joint capacity of the smallest firm’s rivals should not exceed the minimum
market demand. This is a necessary condition that ensures firm i’s sales in period t are strictly
positive, for all i and all mt > m, even if it is the highest-priced firm. An implication of
Assumption 1 is that if m < K, then there is a restriction on the size of the smallest firm in that
it cannot be too small. Given the smallest firm’s capacity can be no larger than for a symmetric
duopoly, a necessary condition for Assumption 1 to hold is that the minimum market demand
must be greater than 50% of the total capacity, m ≥ 0.5K. We believe that Assumption 1 is not
very restrictive for the capacity distributions where tacit and explicit collusion are likely to be
substitutes, The reason is that tacit collusion is most likely to occur in markets with two or three
relatively symmetric firms (see Davies et al., 2011), so the smallest firm is likely to be relatively
large. We place no restriction on the level of the maximum market demand, m.
Thus, denoting Ω(pit ) as the set of firms that price strictly below pit and pmax
≡ max{pt },
t
Assumption 1 and the proportional allocation rule together imply that firm i’s sales in period t,
7
sit (pit , p−it ; mt ), for any pit ≤ 1, are:
sit (pit , p−it ; mt ) =



k
if pit < pmax
t
i


min
P
K−
ki
j∈Ω(pit )
kj
mt −
P
(1)
j∈Ω(pit ) kj , ki
≥0
if pit = pmax
t
This says that a firm will supply its proportion of the residual demand if it is the highest-priced
firm in the market and if capacity is not exhausted, otherwise it will supply its full capacity. This
´m
implies that firm i’s expected per-period profit is πit (pit , p−it ) = pit m sit (pit , p−it ; m) g(m)dm,
where we drop time subscripts if there is no ambiguity. We write πi (p) = ki pS (p) if pj = p for
all j, where S (p) is the expect
 sales per unit of capacity, such


1
if

 ´
´
K m
m
S (p) =
g(m)dm + K g(m)dm if
m K



b
 m
if
K
that:
K≤m
m<K<m
(2)
m ≤ K.
So, such profits are maximised for pm ≡ 1.
2.3
Static Nash equilibrium
Lemma 1 states the static Nash equilibrium profits, which can result from pure or mixed strategies. An important part of the analysis is firm i’s minimax payoff, which is:



m
b − K−i
if

 ´
´m
K
πi ≡
(m − K−i ) g(m)dm + ki K g(m)dm if
 m


 k
if
i
m ≤ K,
m<K<m
(3)
K≤m<m
for all i. The intuition is that if the realisation of market demand is below total capacity, then
a firm that sets the monopoly price expects to supply the residual demand, otherwise it expects
to supply its full capacity. The proof is the same as Lemma 1 in Garrod and Olczak (2016), so
we only provide the intuition below.
Lemma 1. For any given n ≥ 2 and K−1 ≤ m:
i) if m ≥ K, the unique pure strategy Nash equilibrium profits are πiN = ki ∀ i;
π
ii) if m < K, the mixed strategy Nash equilibrium profits are πiN (kn ) = ki knn ∀ i.
Competition is not effective if the minimum market demand is above total capacity, m ≥ K,
so firms set pi = 1 in equilibrium and receive πiN = ki for all i. In contrast, if market demand
8
can be below total capacity, firms are not guaranteed to supply their full capacity for every level
of demand, so they have incentives to undercut each other. However, by charging pi = 1, firm
i can obtain its minimax payoff, π i . Assumption 1 is sufficient to ensure that such profits are
nonnegative for all i, so competition does not imply price equals marginal cost. Instead, the
largest firm will never set a price below p ≡ π n /kn in an attempt to be the lowest-priced firm.
This implies that the smaller firms i < n can sell their full capacity with certainty by charging
a price slightly below p to obtain a profit of ki p ≥ π i . Consequently, the mixed strategy Nash
equilibrium profits are given by πiN (kn ) = ki p. The lower bound of the support is p, where it
follows from (3) that limm→K p = 1.
3
Two Forms of Collusion
We now move on to analyse the repeated game. In any period, firm i’s prices and sales are
initially private information to it, but it may have an incentive to share this information with
its rivals to facilitate collusion. In this section, we first set out the assumptions regarding the
exchange of this information. Then we solve for the collusive equilibria when firms exchange
their private information with each other, and we refer to this as explicit collusion. We then
briefly restate the collusive equilibria in the absence of this information exchange, which was the
focus of Garrod and Olczak (2016), and we refer to this as tacit collusion. Henceforth, we impose
m < K, as collusion is unnecessary otherwise from Lemma 1.
We use the term cartel to refer to a group of firms that exchange their private information
with each other. We say that a cartel is active in period t, if at the start of the period there is a
chance that the firms will exchange their private information. Otherwise the cartel is inactive.
An active cartel is subject to enforcement. If such a cartel is detected, then each of the cartel
members i are convicted with probability 1 and are fined ki F , where F ≥ 0 is the fine per unit of
the industry’s capacity, such that larger firms receive larger fines. This is consistent with most
jurisdictions, including Europe and the US, where the fine for each cartel member is initially
linked to the size of its sales (see International Competition Network, 2008). We assume that in
each period an active cartel may be detected by the authorities with some probability, θ ∈ (0, 1].
In addition, we also allow for the possibility that each firm can inform the authorities of the
cartel in return for leniency, in which case the informant is not fined. Consistent with leniency
programs in Europe and in the US, we assume that applying for leniency is publically observable.
The timing of the game in any given period t > 1 is as follows:
9
Stage 0 (pricing stage). The firms set prices simultaneously. The game continues to stage 1.
Stage 1 (communication stage). The firms realise their sales and profits privately:
• If there is not an active cartel, then period t ends and period t + 1 begins.
• If there is an active cartel, then each firm chooses whether to share its private information
with its rivals secretly and whether to inform the authorities of the cartel publically in
return for leniency. After this, enforcement is realised:
– If no firm has informed the authorities of the cartel, then the cartel is detected and
convicted with a probability θ ∈ (0, 1], and all firms are each fined ki F . Otherwise,
the cartel is not detected and no firm is fined.
– If at least one firm has informed the authorities of the cartel, then the cartel is
detected and convicted with probability 1. Leniency is given to only one informant
and the competition agency selects the informant with the lowest price (or randomly
selects among these informants with equal probability if there is more than one). This
selected informant is not fined and all other firms are each fined ki F .
Finally, period t ends and period t + 1 begins.
There are two assumptions that are worth discussing here. First, for simplicity we initially
assume that the sales reports that firms exchange are verifiable, so that firms just choose whether
to exchange their private information with their rivals. This implies that firms do not need to
question whether the reported information is accurate or not. However, this assumption is not
important, because our results are robust to when these sales reports are nonverifiable, as we
explain in section 5.2. Second, we focus on full leniency in the main analysis, where an informant
is not fined at all. This is consistent with the current leniency programs that operate in Europe
and in the US. However, the European leniency program has evolved over time, so we analyse
the effects of partial leniency on the firms’ incentives to form a cartel in section 5.1 to understand
the effects of this evolution.
3.1
Explicit collusion
In this subsection, we analyse explicit collusion where there is an active cartel in period 0. We
consider the following strategy profile which we refer to as explicit trigger strategies. There are
‘active’ phases, where there is an active cartel, and ‘inactive’ phases where there is not. In the
10
pricing stage of a period during an active phase, a firm sets the collusive price pc > p. Then
in the communication stage, the firm realises its sales and each firm secretly shares its private
information with its rivals and does not apply for leniency. The active phase continues into
period t + 1 if pjt = pc and sjt = ki S (pc ) for all j and if all firms did not apply for leniency.
Otherwise, firms enter an inactive phase. Once in the inactive phase, each firm prices according
to the static Nash equilibrium forever and never exchanges its private information.
There are two comments to make regarding this profile of strategies. First, reversion to
the static Nash equilibrium is the harshest possible punishment under our assumptions. The
reason is that, as showed by Lambson (1994), the harshest punishments under the proportional
allocation rule are such that the largest firm receives the stream of profits from its minimax
strategy. In our setting, the per-period minimax payoff of the largest firm is equivalent to its
static Nash equilibrium profits, so it is not possible to implement a harsher punishment given the
proportional allocation rule. Second, the fact that the cartel may still be active after detection
and conviction implies that, under certain conditions, the cartel will want to exploit the leniency
programme by applying for leniency in every period in an attempt to reduce its expected fines.
We focus on the above strategy profile in the main body of the paper and show that our main
results are robust to when firms want to exploit the leniency programme in Appendix B. In
Appendix B, we show that θ <
1
2
is a sufficient condition to ensure that cartel members cannot
exploit the leniency programme. Given the low detection rates of cartels, this seems likely to
hold in most jurisdictions, hence the reason why we focus on explicit trigger strategies in the
main analysis.7
We now solve for the equilibria. Given firms can observe whether or not they share information
or apply for leniency, the game is one of obserable actions. Thus, it follows from the one-stage
deviation principle (see Fudenberg and Tirole, 1991, p.108-110) that the profile of explicit trigger
strategies is a subgame perfect Nash equilibrium (SPNE) if there is no history that leads to a
subgame in which a deviant will chose an action that differs to that prescribed by the strategy,
then conforms to the strategy thereafter (assuming the deviant believes others will also conform
to the strategy). We say that collusion under explicit trigger strategies is not sustainable if no
such equilibrium strategies exist.
Denoting firm i’s expected (normalised) profit in an active phase as ki V e , if all firms follow
7 An important determinant for the analysis in Appendix B is which informant is selected for leniency when
there is more than one. We have assumed that the deviant with the lowest price is given leniency, which is
consistent with Spagnolo (2005), who assumes that a deviant informant will be given leniency over a colluding
informant. This assumption has little effect on the main analysis, however, because firms do not apply for leniency
on the equilibrium path, so there is only ever one deviating informant in this case, who will have the lowest price.
11
explicit trigger strategies, then:
ki V e = (1 − δ) (πi (pc ) − θki F ) + δki V e ,
where solving yields:
ki V e = πi (pc ) − θki F.
This says that in a period during an active cartel phase firm i expected profits are the expected
per-period profit from setting pc minus its expected fine. We must find the conditions under
which no firm will deviate from its prescribed strategy. Firms play the static Nash equilibrium
during each period of an inactive phase, so it is clear that they have no incentive to deviate
in any such periods. Consequently, we need only consider deviations during active phases. We
begin by considering deviations in the communication stage of a given period t, and then we
move back to the pricing stage. Recall that during an active phase, firm i believes that its rivals
will set pc in the pricing stage, and that they will share their private information and not apply
for leniency in the communication stage.
Communication Stage
We first suppose firm i has abided by its strategy in the pricing stage of period t by setting
pi = pc and consider deviations in the communication stage. Firm i can deviate at this point
by applying for leniency and/or by not sharing its private information. If firm i applies for
leniency, then firms will enter an inactive phase and firm i’s expected (normalised) discounted
profit from this stage on is δπiN (kn ), regardless of whether firm i shares its information. This is
firm i’s optimal deviation at this stage, because if it deviates by not sharing its private information
without applying for leniency, then firm i’s profits would be:
− (1 − δ) θki F + δπiN (kn ) ≤ δπiN (kn ) .
Thus, firm i has no incentive to deviate in the communication stage if:
− (1 − δ) θki F + δki V e ≥ δπiN (kn ) ,
(4)
where the left-hand side is firm i’s expected (normalised) discounted profit at this stage onwards
from abiding by its strategy, and the right-hand side is the profit from firm i’s optimal deviation.
We refer to this as the “communication” ICC.
12
Now suppose firm i had deviated from its prescribed strategy in stage 0 of period t by setting
pi 6= pc . This implies that firms will enter a punishment phase, regardless of whether firm i’s
shares its private information or not. Thus, firm i has a dominant strategy to apply for leniency
to raise its expected profits by eliminating its fine.
Pricing Stage
Next consider firm i’s incentive to deviate from its prescribed strategy in the pricing stage of
period t by setting pi 6= pc , in which case it follows from the above that in the communication
stage it will apply for leniency and firms will enter an inactive phase. In contrast, if a firm
does not deviate in the pricing stage, then it will also not deviate in the communication stage,
provided the communication ICC is satisfied. Thus, assuming the communication ICC is satisfied,
it follows that firm i has no incentive to deviate in the pricing stage if:
ki V e ≥ (1 − δ) ki pc + δπiN (kn ) ,
(5)
where the left-hand side is firm i’s profit from abiding by its strategy, and the first term on
the right-hand side is firm i’s (normalised) profit from undercuting pc marginally to supply its
full capacity, ki , which is firm i’s optimal deviation from pc > V e > p. We refer to this as the
“pricing” ICC.
Note that if the pricing ICC is satisfied, then the communication ICC is also satisfied. This
point can be seen by adding (1 − δ) πi (pc ) to both sides of (4) such that the communication ICC
becomes:
ki V e ≥ (1 − δ) πi (pc ) + δπiN (kn ) ,
Then it is easy to see that the right-hand side of (5) is greater than or equal to the right-hand
side of the above, and the left-hand sides of the two ICCs are the same. Thus, the pricing ICC
is more stringent than the communication ICC. Rearranging (5) in terms of the discount factor
yields:
δ ≥1−
Ve−p
≡ δe (kn , F, pc ) .
pc − p
(6)
It follows from the fact that the critical discount factor δe (kn , F, pc ) is independent of ki that,
despite potential asymmetries, each firm has the same incentive to deviate as its rivals. Furthermore, δe (kn , F, pc ) < 1 for any V e > p, such that collusion under explicit trigger strategies is
only sustainable if it is strictly more profitable than the static Nash equilibrium. Finally, note
13
that δe (kn , F, pc ) is strictly increasing in the fine per unit of capacity, F . This implies that there
is a greater incentive for each firm to deviate from its prescribed strategy in the pricing stage as
the fine per unit of capacity becomes larger.
It follows from the above that explicit trigger strategies are SPNE strategies if pricing ICC
is satisfied. Proposition 1 now finds the optimal SPNE profits.
Proposition 1. For any given n ≥ 2, K−1 ≤ m < K and m ≥ m, there exists a unique
fine per unit of capacity, F (kn ) = θ1 S (pm ) − p > 0, such that if F ∈ 0, F (kn ) and if
m
)+θF
δ ≥ δe∗ (kn , F ) ≡ 1−S(p
∈ kKn , 1 , then firm i’s optimal SPNE profits under explicit trigger
1−p
strategies are:
ki Ve∗ = πi (pm ) − θki F ∈ πiN (kn ) , πi (pm ) ∀ i.
(7)
Otherwise, collusion under explicit trigger strategies is not sustainable.
This says that, if firms are sufficiently patient, then the optimal equilibrium profits have the
firms set the monopoly price. As is common in collusion models with capacity constraints, any
collusive price below the monopoly price not only lowers profits but it also raises the critical
discount factor. Thus, either the profile of explicit trigger strategies, with firms setting the
monopoly price, is a SPNE or collusion under explicit trigger strategies is not sustainable at any
collusive price.
3.2
Tacit collusion
In this subsection, we analyse tacit collusion where there is never an active cartel in any period.
This analysis is the same as in Garrod and Olczak (2016). However, since it is central to our
story, we briefly restate the analysis here.
First, consider the information available to each firm when firms do not exchange their private
information. As we showed in Garrod and Olczak (2016), firm i not only has a private history of
its past prices and sales, denoted zit ≡ (pi0 , si0 ; . . . ; pit−1 , sit−1 ), but there there is also a public
history that firms can condition their play on in the absence of communication. The reason is
that all firms can always infer from their own sales when at least one firm’s sales are below some
firm-specific “trigger level” given by s∗i ≡ min kKi m∗ (k1 , m) , ki for all i, where m∗ (k1 , m) ≡
14
K (m−k1 )
.
K−1
The public history is ht = (y0 , y1 , . . . , yt−1 ) where, for all τ = {0, 1, . . . , t − 1}:
yτ =


y
if siτ (piτ , p−iτ ; mτ ) > s∗i ∀ i

y
otherwise.
This says that yτ = y if all firms’ sales in period τ exceed their trigger levels, but yτ = y if at
least one firm’s sales does not.
Notice that ht is a public history, because yτ is common knowledge for all τ , for any zit .
To see this, first consider the case where maximum market demand is above the total capacity,
m ≥ K. In this case, the trigger levels are so high that all firms’ sales can never exceed them
for any prices, that is, s∗i = ki so yτ = y for all τ . Next, consider the case of m < K, where it
is possible for firms to receive sales above their trigger levels, since s∗i < ki . In this case, if all
firms set a common price p ≤ 1, then the sales of all firms will exceed their respective trigger
levels if the realisation of market demand is high, so yt = y, otherwise they can all fall below the
trigger levels, so yt = y. Yet, if all firms do not set such a common price, then the sales of the
firm(s) that set the highest price will not exceed their trigger levels, so yt = y, and their rivals
can infer this. For instance, it follows from (1) that, for any nonempty set of rivals with a price
below pmax , Ω (pmax ), the sales of firm i with pi = pmax ≤ 1 are:
si =
ki
K−
P
j∈Ω(pmax )
kj
mt −
P
j∈Ω(pmax )
k (m − k )
1
i
= s∗i < ki ,
kj ≤
K−1
(8)
and all of firm i’s low-priced rivals will supply their full capacities. Any firm that supplies its full
capacity knows, from (1), that it will do so only if its price is below the highest in the market.
Thus, it can infer from this that at least one of its rivals’ sales is below its trigger level.
The above implies that each firm knows that all firms’ sales will exceed their trigger levels,
such that y = y, only if pj = p ≤ 1 for all j and if m > m∗ (k1 , m); otherwise, at least one firm’s
sales will not exceed its trigger level, so y = y. Thus, if m > m∗ (k1 , m), then there is perfect
monitoring of a strategy in which all firms set a common collusive price, even in the absence
of more information. The reason is that each firm would only receive sales below its trigger
level, if it has been undercut. In contrast, there is imperfect monitoring of such an agreement, if
m ≤ m∗ (k1 , m), where the probability of observing y if firm i sets pi and its rivals price according
15
to p−i is:
Pr y|pi , p−i =
 ´
∗
 min{m (k1 ,m),m} g (m) dm ∈ [0, 1]
if pj = p ∀j
 1
otherwise.
m
(9)
This says that a firm’s sales can be below their trigger level if the realisation of market demand is
sufficiently low, even when firms set a common price. Thus, without more information, colluding
firms face a non-trivial signal extraction problem: each firm does not know whether the realisation
of market demand was unluckily low or whether at least one rival has undercut them.
Lemma 2 states the conditions for perfect and imperfect monitoring in terms of the maximum
market demand, holding the minimum market demand constant.
Lemma 2. For any given n ≥ 2, K−1 ≤ m < K, and δ ∈ (0, 1), there exists a unique level
of market demand, x (k1 ) ∈ (m, K), such that if m ∈ (m, x (k1 )), then monitoring is perfect.
Otherwise, there is imperfect monitoring.
Deviations by the smallest firm are most difficult to detect, so it follows that detecting
deviations is less difficult when the smallest firm is larger. As a result, the critical level x (k1 ) is
strictly increasing in the capacity of the smallest firm, k1 .
To solve the game, we restrict attention to sequential equilibria in public strategies, known
as perfect public equilibria (PPE). In what follows we assume firms follow a strategy profile in
which, similar to Green and Porter (1984) and Tirole (1988), firms punish each other by reverting
to the static Nash equilibrium for a fixed number of periods, if they receive a bad signal during
a collusive phase. We formally describe the strategy profile below and we henceforth refer to it
as tacit trigger-sales strategies. In Garrod and Olczak (2016), we used the techniques of Abreu
et al. (1986, 1990) to show that these equilibrium strategies generate the maximal PPE payoffs.
Finally, note that in the case of perfect monitoring, any PPE is also a subgame perfect Nash
equilibrium (SPNE).
Tacit trigger-sales strategies are formally defined as follows. There are ‘collusive phases’ and
‘punishment phases’. Suppose period t is in a collusive phase. In any such period, a firm should
set the collusive price, pc > p. If yt = y, such that all firms received sales above their trigger
levels, then the collusive phase continues into the next period t + 1. If yt = y, such that at least
one firm received sales below its trigger level, then firms enter a punishment phase in the next
period t + 1. In the punishment phase, each firm should play the static Nash equilibrium for T
16
periods, after which a new collusive phase begins. This sequence repeats in any future collusive
phase.
The profile of trigger-sales strategies is a PPE if, for each date t and any history ht , the
strategies yield a Nash equilibrium from that date on (see Fudenberg and Tirole, 1994, p.187191). We say that collusion under tacit trigger-sales strategies is not sustainable if no such
equilibrium strategies exist. Given firms play the static Nash equilibrium during each period
of the punishment phase, it is clear that they have no incentive to deviate in any such periods.
Thus, we need only consider deviations during collusive phases. The ICC for firm i is:
ki V c ≥ (1 − δ) ki pc + δki V p , ∀ i,
(10)
where ki V c is firm i’s expected (normalised) profit in a collusive phase and ki V p is its expected
(normalised) profit at the start of a punishment phase, such that:
ki V c = (1 − δ) πi (pc ) + δ 1 − Pr y|pc ki V c + Pr y|pc ki V p
PT −1
ki V p = (1 − δ) t=0 δ t πiN (kn ) + δ T ki V c .
The ICC says that firm i will not deviate in any period in a collusive phase if it cannot gain by
marginally undercutting pc > p to supply its full capacity ki , in which case Pr y|pi , pc = 1 for
any pi 6= pc from (9). Note that (10) can never satisfied when the maximum market demand
is greater than total capacity, m ≥ K, as then Pr y|pc = 1, from (9). Thus, collusion under
trigger-sales strategies is not sustainable if m ≥ K, so we can henceforth focus on the case where
m < K.
Lemma 3 solves for the optimal PPE profits under trigger-sales strategies.
Lemma 3. For any given n ≥ 2 and K−1 ≤ m < K, there exists a unique level of market
demand, x (k1 , kn ) ∈ (x (k1 ) , K), that solves G (m∗ (k1 , x (k1 , kn ))) = KK−n < 1, such that, if
δ ≥ δc∗ (k1 , kn ) ≡ 1−G(m∗1(k1 ,m)) kKn ∈ kKn , 1 , then firm i’s optimal equilibrium profits under tacit
trigger-sales strategies are, ∀ i:

 πi (pm )
ki Vc∗ =
∗
b
(k1 ,m))K
 ki m−G(m
∈ πiN (kn ) , πi (pm )
∗
K
1−G(m (k1 ,m))
if m < x (k1 )
(11)
if x (k1 ) ≤ m < x (k1 , kn ) .
Otherwise, collusion under tacit trigger-sales strategies is not sustainable.
When there is perfect monitoring, such that m < x (k1 ), punishment phases do not occur on
17
the equilibrium path. Thus, firms are able share the monopoly profits between them, if the threat
of reverting to the static Nash equilibrium forever is sufficiently harsh. In contrast, when there is
imperfect monitoring, such that x (k1 ) ≤ m, if firms are sufficiently patient and if the probability
of receiving a bad signal when they set a common price is sufficiently low, m < x (k1 , kn ), then
the optimal equilibrium profits under trigger-sales strategies have the firms set the monopoly
price during a collusive phase, and the optimal punishment phase duration is finite and set at
a level such that the ICC (10) is binding with no slack. Given punishment phases occur on the
equilibrium path, the sum of the optimal equilibrium profits is below the monopoly profit.
4
Cartels, Firm Numbers and Asymmetries
We now use our equilibrium analysis to investigate the incentives of firms to form a cartel when
firms can alternatively collude tacitly. We are particularly interested in how the distribution of
capacity affects such incentives. The conventional wisdom is that cartels are most likely to arise
in markets with a few relatively symmetric firms. The reason is that this is where collusion is
the least difficult to sustain. However, such theories have not considered that instead of forming
a cartel firms could alternatively collude tacitly.
In what follows, we say that it is privately optimal for a firm to be a cartel member if its
profits from being a member are strictly greater than its profits from not being a member.
Furthermore, we define a cartel phase as a sequence of periods that begins the period after
detection and ends the period of the next detection. Thus, the expected duration of a cartel
P∞
t−1
phase is t=1 t (1 − θ)
= 1/θ, which implies that the lower the probability of detection, the
longer the expected duration of a cartel phase.
4.1
The incentives to form a cartel
In this subsection, we analyse the firms’ incentives to form a cartel. We begin by considering this
when the most profitable alternative is the static Nash equilibrium. This provides a benchmark
for our analysis, since it is the comparison that is commonly considered by the previous literature
that exogenously assumes that firms cannot collude without forming a cartel. It is also the
appropriate comparison in our framework if collusion under tacit trigger-sales strategies is not
sustainable, such that m ≥ x (k1 , kn ) or δ < δe∗ (k1 , kn ) for any m < x (k1 , kn ). We then also
consider the firms’ incentives to form a cartel when collusion under tacit trigger-sales strategies
is sustainable.
18
4.1.1
Comparing cartel against competition
If tacit collusion is not sustainable, or indeed if it is not taken into account, then it follows from
Proposition 1 that joining a cartel is privately optimal for each firm i if collusion under explicit
trigger strategies is sustainable, such that δ ≥ δe∗ (kn , F ) for any F ∈ 0, F (kn ) . This is due to
the fact that ki Ve∗ > πiN (kn ) for such conditions, since rewriting this inequality yields:
F <
1
S (pm ) − p = F (kn ) .
θ
(12)
Note that the left-hand side of (12) is, from the cartel’s perspective, the expected cost of a cartel
phase per unit of capacity and the right-hand side is the expected benefit of a cartel phase per
unit of capacity. The expected benefit is the multiplication of the expected duration of a cartel
phase, 1/θ, and the per-period difference between the profits per unit of capacity of explicit
collusion and of the static Nash equilibrium, S (pm ) − p.
4.1.2
Comparing cartel against tacit collusion
We now move on to investigate the incentives for firms to form a cartel when they could alternatively collude tacitly.
Proposition 2. For any given n ≥ 2 and K−1 ≤ m ≤ m < x (k1 , kn ), there exists a unique fine
per unit of capacity given by:

 0
F ∗ (k1 ) ≡
 1
θ
if m < x (k1 )
m
b
K
−
Vc∗
∈ 0, F (kn )
if x (k1 ) ≤ m < x (k1 , kn ) ,
such that if and only if F ∈ [0, F ∗ (k1 )), then the optimal profits that could be sustained as an
equilibrium under explicit trigger strategies are greater than under tacit trigger-sales strategies,
Ve∗ > Vc∗ .
This says that collusion under explicit trigger strategies is more profitable than tacit triggersales strategies if the fine per unit of capacity is sufficiently low, such that:
F <
1
θ
m
b
− Vc∗
K
= F ∗ (k1 ) .
(13)
Similar to (12), the above inequality compares, from the cartel’s perspective, the expected cost
of a cartel phase per unit of capacity on the left-hand side against the expected benefit per unit
19
of capacity on the right-hand side. The only difference compared with (12) is that the term in
brackets on the right-hand side of (13) is the per-period difference between the profits per unit
of capacity of explicit collusion and of tacit collusion, where S (pm ) =
m
b
K
for all m < K from
(2). If the maximum market demand is sufficiently low, such that there is perfect monitoring,
m < x (k1 ), then the critical fine F ∗ (k1 ) equals zero. The reason is that firms are able to extract
the monopoly profit by colluding tacitly, so Vc∗ =
m
b
K
such that being a cartel member is never
privately optimal for any firm. For larger fluctuations in market demand where there is imperfect
monitoring, such that m ∈ [x (k1 ) , x (k1 , kn )), the critical fine F ∗ (k1 ) is positive but it less than
F (kn ) for all m < x (k1 , kn ). This is due to the fact that the profits from collusion under tacit
trigger-sales strategies are below the monopoly level but are above the profits of the static Nash
equilibrium, Vc∗ > p, from Lemma 3. Finally, F ∗ (k1 ) converges to F (kn ) as m → x (k1 , kn ),
where Ve∗ = Vc∗ = p.
It follows from the above that if there is imperfect monitoring such that colluding tacitly
requires punishments on the equilibrium path, then forming a cartel is not privately optimal if
the expected fine is sufficiently high. However, this is only true if firms are sufficiently patient
such that both tacit and explicit collusion are sustainable. Proposition 3 shows that if explicit
collusion is more profitable than tacit collusion, then it is sustainable at lower discount factors
than tacit collusion, and vice versa.
Proposition 3. For any given n ≥ 2 and K−1 ≤ m ≤ m < x (k1 , kn ), if and only if F ∈
[0, F ∗ (k1 )), then the critical discount factor under explicit trigger strategies is less than under
tacit trigger-sales strategies, δe∗ (kn , F ) < δc∗ (k1 , kn ).
Figure 1 depicts the critical discount factors under tacit trigger-sales strategies and under
explicit trigger strategies for some m ∈ (x (k1 ) , x (k1 , kn )) for any given fine per unit of capacity.
Note that δc∗ (k1 , kn ) is a horizontal line because it is independent of F , and it lies is between
kn /K and 1 since it equals the former at m = x (k1 ) and it tends to the latter as m → x (k1 , kn ) .
In contrast, δe∗ (kn , F ) is linear and strictly increasing in F . It follows from Proposition 1 that
if we were to ignore tacit collusion, then forming a cartel is privately optimal for all firms if
δ ≥ δe∗ (kn , F ) for any F < F (kn ). However, when tacit collusion is taken into account, the
parameter space for which forming a cartel is privately optimal contracts. The reason is that if
collusion under tacit trigger-sales strategies is sustainable, such that δ ≥ δc∗ (k1 , kn ), then forming
a cartel is privately optimal only if the fine is sufficiently low, such that F < F ∗ (k1 ). We refer to
this as the parameter space of explicit collusion. Similarly, the parameter space of tacit collusion
20
Figure 1: The parameter space of explicit collusion
is the space where tacit collusion is most profitable.
4.2
Comparative statics
In this subsection we analyse the comparative statics of our model when the total capacity
is held constant to investigate the effects of asymmetries on the incentives for firms to form a
cartel. Only changes to the capacity of the smallest firm or the largest firm affect the equilibrium
analysis so we restrict attention to these. We begin by considering the benchmark case where
the alternative to explicit collusion is the static Nash equilibrium.
Proposition 4 analyses the effects of reallocating capacity on δe∗ (kn , F ) and F (kn ).
Proposition 4. For any given n ≥ 2, K−1 ≤ m < K and m > m, raising the capacity of the
largest firm, kn , strictly increases the critical discount factor δe∗ (kn , F ) and strictly decreases the
critical fine F (kn ).
The fact that increasing the capacity of the largest firm, kn , increases the critical discount
factor implies that if tacit collusion is not sustainable or indeed if it is not taken into account,
then the scope for explicit collusion reduces as the largest firm gets larger. The reason is that
the static Nash equilibrium profits are higher when the largest firm has more capacity, because
21
competition is less intense. Consequently, the punishment is weaker than before and the pricing
ICC is tighter, so the critical discount factor rises. Moverover, the critical fine F (kn ) falls as
the largest firm gets larger, because the expected benefit of a cartel phase is reduced. This is
due to the fact that the per-period difference between the profits per unit of capacity of explicit
collusion and of the static Nash equilibrium is smaller.
In summary, the above implies, consistent with the conventional wisdom, that if tacit collusion
is not sustainable or indeed if it is not taken into account, then the incentives to form a cartel
are greatest in markets with symmetric firms, k1 = kn = K/n, and these incentives reduce as the
largest firm gets larger, especially for high fines. In contrast, given both δe∗ (kn , F ) and F (kn )
are independent of the size of the smallest firm, it follows that reducing the size of the smallest
firm has no effect on the incentives to form a cartel, when the size of the largest firm is held
constant.
We now move on to consider the case where the alternative to explicit collusion is tacit
collusion. Henceforth we restrict attention to m < x (k1 , kn ), because tacit collusion is not
sustainable otherwise from Lemma 3. In Garrod and Olczak (2016), we showed the the critical
discount factor δc∗ (k1 , kn ) is strictly increasing in the capacity of the largest firm and strictly
decreasing in the capacity of the smallest firm under imperfect monitoring. The reason for the
former is that an increase in the capacity of the largest firm weakens the punishment that lasts
an infinite number of periods, because the static Nash equilibrium profits are higher, so the
critical discount factor rises. The latter is due to the fact that an increase in the smallest firm
ensures that monitoring is easier for the firms, because it is less likely that firms will receive sales
below their trigger levels when they set a common price. Consequently, it is less likely that a
collusive phase will switch to a punishment phase on the equilibrium path than before and the
expected future profits from collusion are higher. As a result, a punishment phase that lasts an
infinite number of periods is relatively harsher, so the critical discount factor falls. Although the
critical discount factor under perfect monitoring is independent of the smallest firm, monitoring
is perfect for a wider range of fluctuations in market demand as the smallest firm gets larger.
Next we analyse the effects of reallocating capacity among the firms on the critical fine,
F ∗ (k1 ). This fine equals zero under perfect monitoring and hence it is independent of the
capacity distribution. Given that increasing the size of the smallest firm implies that monitoring
is perfect for a wider range of fluctuations in market demand, it follows that F ∗ (k1 ) = 0 for a
wider range. Proposition 5 now investigates the effect of reallocating capacity on this critical
fine under imperfect monitoring.
22
Proposition 5. For any given n ≥ 2 and K−1 ≤ m < x (k1 ) ≤ m < x (k1 , kn ), the critical fine
F ∗ (k1 ) is strictly decreasing in the capacity of the smallest firm, k1 .
The fact that the critical fine F ∗ (k1 ) is strictly decreasing in the capacity of the smallest firm,
k1 , implies that if firms are sufficiently patient, then they have less incentive to form a cartel as
the smallest firm gets larger. The reason is that the expected benefit of a cartel phase decreases,
because the difference between the per-period profits of colluding explicitly and tacitly is smaller.
This is due to the fact that the optimal equilibrium profits of tacit trigger-sales strategies under
imperfect monitoring are strictly increasing in the capacity of the smallest firm. This is due to two
effects. First, it is less likely that firms will receive sales sales below their trigger levels when they
set a common price. Consequently, a collusive phase is less likely to switch to a punishment phase
than before, so profits rise on the equilibrium path, other things equal. Second, this increase in
profits also introduces slack into the ICC for tacit collusion, so the optimal punishment phase
duration shortens to such an extent that the ICC binds with no slack. Both effects imply that
firms expect there to be fewer and shorter punishment phases on the equilibrium path when the
smallest firm has more capacity than before. Thus, collusion under tacit trigger-sales strategies
is more profitable, so there is less incentive to form a cartel.
The critical fine F ∗ (k1 ) is independent of the capacity of the largest firm. The is due to
the fact that the optimal equilbrim profits of tacit trigger-sales strategies does not depend on
the capacity of the largest firm. This results from the fact that changing the capacity of the
largest firm causes two effects on profits that perfectly offset each other. The first effect is that
an increase in the capacity of the largest firm raises profits on the equilibrium path, other things
equal, because the static Nash equilibrium profits of each firm are greater than before. However,
this also tightens the ICC for tacit collusion, so the second effect is that the optimal punishment
phase duration lengthens to ensure that the ICC is binding with no slack. This second effect
cancels out the first, so the size of the largest firm has no effect on the critical fine F ∗ (k1 ).
Figure 2 builds on the illustration in Figure 1 to depict the effects of asymmetries on the
parameter space of explicit collusion. It shows that increasing the size of the largest firm contracts
the parameter space of explicit collusion, when the capacity of the smallest firm is held constant.
Note that if firms are sufficiently patient, such that the comparision is between explicit and tacit
collusion, then increasing the size of the largest firm does not affect the incentives to form a
cartel, because F ∗ (k1 ) is independent of the size of the largest firm. Instead, the reason for the
contraction is that the critical discount factors of both explicit and tacit collusion are higher,
23
Figure 2: Changes to the capacity distribution
because collusion is more difficult to sustain. Similarly, increasing the size of the smallest firm
contracts the parameter space of explicit collusion, when the capacity of the largest firm is held
constant. This is due to two effects. First, the critical fine F ∗ (k1 ) falls because tacit collusion
is more profitable than before. Second, tacit collusion is easier to sustain than before, so the
critical discount factor falls. Consequently, the parameter space of explicit collusion contracts to
the left, because the parameter space of tacit collusion expands.
It follows from these results that, in contrast with the conventional wisdom but not inconsistent with the evidence, the parameter space of explicit collusion is limited in markets with a few
symmetric firms, but it expands as the number of symmetric firms rises or as the smallest firm
gets smaller, other things equal. Consider the reasons, starting with the number of symmetric
firms. Such firms share the total capacity evenly between them, such that k1 = kn = K/n, so
an increase in the number of symmetric firms from n to n + 1 reduces both the smallest and the
largest firms’ capacities by
1
n(n+1) .
This expands the parameter space of explicit collusion for two
reasons. First, explicit collusion becomes easier to sustain from Proposition 4, because the largest
firm is smaller. Second, tacit collusion becomes more difficult to sustain and is less profitable
from Proposition 5, because the smallest firm is smaller. These two effects are the reverse of the
effects illustrated in Figure 2(a) and (b), respectively, so the parameter space of explicit collusion
expands with the number of firms. Next consider introducing asymmetries by only decreasing
the size of the smallest firm. This expands the parameter space of explicit collusion further due
24
to the second effect just discussed. Thus, it follows from this that, for example, the parameter
space of explicit collusion under a symmetric triopoly is larger than under a symmetric duopoly
but is smaller than under an asymmetric distribution with k1 < K/3 and k3 = K/3. Both of
these results are in contrast with the conventional wisdom.
Finally, while the effect of decreasing the capacity of the smallest firm on the parameter space
of explicit collusion is only present if tacit collusion is sustainable, the effect of decreasing the
largest firm is present even when monitoring is perfect. This implies that explicit collusion is
still more difficult to sustain as the number of symmetric firms increases, because the ‘largest’
firm is smaller.
4.3
Example
We now complement our general results by analysing an example. We do this for three reasons. First, we wish to show that the fine required to deter firms from forming a cartel can
be substantially lower for markets with a symmetric capacity distribution than for market with
an asymmetric capacity distribution where only the smallest firm is smaller. Second, we want
to show that such a fine can be substantially lower for markets with few symmetric firms than
with more firms. Third, we wish to consider how the critical fine changes with the capacity
distribution under duopoly, where a decrease in the size of the smallest firm increases the size of
the largest firm.
With these objectives in mind, we consider an example where total capacity is K = 100 and
suppose that this is divisible into 6 equal sized parts. We consider four capacity distributions: the
first two distributions are a symmetric duopoly and a symmetric triopoly, denoted (3/6, 3/6) and
(2/6, 2/6, 2/6), respectively, where the first element of the vector relates to firm 1’s proportion
of the total capacity, the second is firm 2’s proportion, and so on. The other two distributions
are an asymmetric duopoly, (2/6, 4/6), and an asymmetric triopoly, (1/6, 2/6, 3/6). Note that
these distributions allow us to analyse the effects of asymmetries when the size of the largest firm
is held constant by comparing (3/6, 3/6) with (1/6, 2/6, 3/6). Furthermore, we can investigate
the effect of a change in the number of symmetric firms by comparing (3/6, 3/6) with (2/6, 2/6,
2/6). Finally, we can show the effects under duopoly by comparing (3/6, 3/6) with (2/6, 4/6).
We want to analyse how the capacity distribution affects the level of the fine that is necessary
and sufficient to deter firms from forming a cartel, and we do so for δ → 1 such that collusion
under trigger-sales strategies is not sustainable only if m ≥ x (k1 , kn ). Thus, it follows from
this that this fine level is F ∗ (k1 ) if collusion under tacit trigger-sales strategies is sustainable,
25
Figure 3: G(m) =
m−m
m−m ,
m
b = 92, K−1 ≤ 56 (100) < 100 = K, and δ → 1
otherwise it is F (kn ). For expositional purposes, we transform these fines into the fine per unit
sold per period of a cartel phase, which is given by f ≡ θ K
m
b F for any given fine per unit of
capacity F if m < K. Denoting the level of f at the critical fine levels as f ∗ , it follows from (12)
and (13) that:



0


∗
f =
1 − pbc (k1 , m)



 1 − pbN (k )
n
where pbc (k1 , m) ≡
K ∗
m
b Vc
if m < x (k1 )
if x (k1 ) ≤ m < x (k1 , kn )
if x (k1 , kn ) ≤ m,
is the optimal average price under tacit collusion and pbN (kn ) ≡
K
m
bp
is
∗
the average price of the static Nash equilibrium. Note that f could also be interpreted as the
expected consumer surplus per unit from the most profitable alternative to explicit collusion.
The intuition is that f ∗ is equivalent to the expected benefit of the cartel per unit sold per
period of a cartel phase, and this is the difference between setting pm and the average price of
the most profitable alternative, pb, which is the same as the consumer surplus per unit in the
absence of a cartel, CS (b
p) = 1 − pb, given demand is perfectly inelastic. Figure 3 plots f ∗ as a
function of ∆m ≡
m−m
m
b
for the various capacity distributions, assuming demand is drawn from
a uniform distribution. Parameter values are chosen such that m
b = 92 for all ∆m and that
K−1 ≤ 56 (100) ≤ m ≤ m ≤ K = 100, so Assumption 1 holds.
Each of the plotted lines in Figure 3 has a similar shape. If f ∗ equals zero, then monitoring is
26
perfect and forming a cartel has no benefit, since pbc (k1 , m) = pm . If f ∗ is upward-sloping, then
there is imperfect monitoring and pbc (k1 , m) is strictly decreasing in ∆m. Thus, the expected
benefit per unit sold per period of a cartel phase strictly increases with ∆m, and the critical
fine to deter the cartel rises as a result. If f ∗ is positive and constant, then collusion under
tacit trigger-sales strategies is not sustainable, such that the expected benefit per unit sold per
period of a cartel phase is the difference between pm and pN (kn ). Finally, note that comparing
(2/6, 2/6, 2/6) with (2/6, 4/6) for any given ∆m is consistent with moving horizontally from
left to right on Figure 2(a) as δ → 1, because only the capacity of the largest firm increases.
Similarly, comparing (1/6, 2/6, 3/6) with (3/6, 3/6) for any given ∆m is consistent with moving
horizontally from left to right on Figure 2(b) as δ → 1, because only the capacity of the smallest
firm changes.8
First, consider the difference between the symmetric duopoly (3/6, 3/6) and the asymmetric
triopoly (1/6, 2/6, 3/6). Despite the change in the number of firms, the only difference that
affects the equilibrium analysis for these distributions is that the size of the smallest firm has
changed. The fact that the smallest firm is larger under the symmetric distribuition than under
the asymmetric distribution implies that monitoring is perfect for a larger range of fluctuations
and tacit collusion is more profitable when it is sustainable. Thus, in contrast to the conventional
wisdom, the fine require to deter the cartel in the symmetric duopoly can be lower than the
asymmetric triopoly. This difference between these two fines is largest over the range 0.02 <
∆m < 0.06, where the asymmetric triopoly requires a fine equivalent to 9% of total welfare per
unit and the symmetric duopoly requires 0%. However, deterrence of the cartel in the symmetric
duopoly over this range does not affect total welfare, because the firms extract all of the consumer
surplus through tacit collusion. This difference becomes smaller as ∆m increases towards 0.09
as monitoring becomes increasingly difficult and tacit collusion becomes less profitable.
Next, consider the difference between the two symmetric distributions (3/6, 3/6) and (2/6,
2/6, 2/6). Here there is a difference in the number of the symmetric firms, and this affects
the equilibrium analysis by reducing the size of the smallest and the largest firm. The effect of
changing the size of the smallest firm are the same as those just illustrated above. The decrease
in the size of the largest firm reduces the static Nash equilibrium profits, which makes explicit
collusion more appealing when tacit collusion is not sustainable, and it also implies that collusion
under imperfect monitoring is sustainable under a wider range of fluctuations in market demand.
8 If
δ < 1, then there would be a discontinuity in each of the f lines at the threshold of ∆m where the outcome
is noncollusive. At this thresholds, a line would jump up to the level of the horizontal line, such that this level of
fine extends for lower levels of ∆m than in Figure 3.
27
Thus, again in contrast to the conventional wisdom, the fine required to deter the cartel in the
symmetric duopoly can be lower than the symmetric triopoly. The difference begins to emerge
at around ∆m = 0.04 and after approximately ∆m > 0.06 the difference is in excess of 7% of
total welfare.
Finally, consider the difference between the two duopolies (3/6, 3/6) with (2/6, 4/6). Here the
size of the smallest firm decreases but the size of the largest firm increases. Figure 1 shows that,
in contrast to the conventional wisdom, the fine required to deter the cartel in the symmetric
duopoly is lower than the asymmetric duopoly when fluctuations in market demand are ∆m <
0.04, such that monitoring is easier and the optimal profits under tacit collusion are close to the
monopoly level. At its largest, this distance in 4% of total welfare. For larger fluctuations in
market demand, the conventional wisdom holds.
5
Extensions
In this section, we extend our results in two ways. First, we consider the effects of the leniency
program on the parameter space of explicit collusion. Second, we show that our results are robust
if firms’ private information is nonverifable.
5.1
The effects of leniency
Up to this point, we have assumed that an informant receives full leniency. While this is consistent
with contemporary leniency programs in Europe and the US, this has not always been the case.
For example, leniency was introduced to Europe in 1996, but full leniency was not guaranteed
to the first firm to come forward until the program was adjusted in 2002. In this subsection we
consider our results when an informant receives less than full leniency to analyse the effect of
such programs. More specifically, we consider a leniency program which reduces the informant’s
fine from ki F to ki F (1 − λ), such that the informant receives less than full leniency if λ < 1.
For brevity, we assume that leniency is sufficiently generous that λ > 1 − θ. This implies that
the expected fine from informing is less than the expected fine if no firm informs the authorities,
ki F (1 − λ) < θki F . Nevertheless, to obtain the results for the case of 1 − λ ≥ θ, one simply
needs to substitute λ = 1 − θ into the below, because then no firm will apply for leniency on or
off the equilibrium path. All other assumptions are unchanged.
This change to the extent of leniency only affects explicit collusion, because tacit collusion
is not illegal. Furthermore, it also does not change the profits under explict collusion, because
28
firms do not apply for leniency on the equilibrium path. Hence, ki V e is the same as in (7). Thus,
it will only change the critical discount factor as it will affect the deviation profits. Following
similar arguments as in section 3.1, firm i has no incentive to deviate in the communcation stage
if:
− (1 − δ) θki F + δki V e ≥ − (1 − δ) ki F (1 − λ) + δπiN (kn ) ,
(14)
and firm i has no incentive to deviate in the pricing stage if
ki V e ≥ (1 − δ) (ki pc − ki F (1 − λ)) + δπiN (kn ) .
(15)
These ICCs are the same as (4) and (5), resepectively, except that deviating by applying for
leniency only reduces the fine by ki F λ < ki F . Furthermore, consistent with the main analysis,
by adding (1 − δ) πi (pc ) to both sides of (14) it is easy to check that the pricing ICC (5) is more
stringent and hence determines the critical discount factor. Thus, rearranging (15) in terms of
the discount factor yields:
δ ≥ δe (kn , F, λ, pc ) ≡ 1 −
Ve−p
pc − p − (1 − λ) F
The critical discount factor δe (kn , F, λ, pc ) is strictly increasing in fine per unit of capacity, F .
The difference with the main analysis is that δe (kn , F, λ, pc ) is convex in F for any λ < 1, which
implies that increasing the fine has a greater impact for higher fines. Furthermore, the critical
discount δe (kn , F, λ, pc ) is strictly decreasing in pc such that the critical discount factor is lowest
and the equilibrium profits are highest when pc = 1.
Consider the effect of leniency in the presence of tacit collusion. Figure 4 illustrates the
parameter space for which forming a cartel is privately optimal for the firms when there is less
than full leniency. It shows that the critical discount factor δe∗ (kn , F, λ) no longer intersects with
δc∗ (k1 , kn ) at F ∗ (k1 ) for any λ < 1. Instead, it does so at Fe (k1 , kn , λ) > F ∗ (k1 ) and this implies
there exist some values of the discount factor such that tacit collusion is more profitable than
explicit collusion but tacit collusion is not sustainable. Recall that for the case of full leniency,
where λ = 1, δe∗ (kn , F, 1) is linear and given by the upward sloping dashed-line in Figure 4. Thus,
it follows from Figure 4 that raising the degree of leniency from λ < 1 to full leniency would
only contract the parameter space of explicit collusion for a range of discount factors less than
δc∗ (k1 , kn ), where collusion under trigger-sales strategies is not sustainable. Furthermore, this
parameter space contracts as the smallest firm gets larger due to the fact that tacit collusion is
easier to sustain and is more profitable. Consequently, it follows that leniency programs will be
29
Figure 4: The parameter space of explicit collusion
more successful at reducing the parameter space of explicit collusion in markets with asymmetric
firms.
5.2
Non-verifable sales
Up to this point, we have assumed that the private information that firms exchange in a cartel
is verifiable such that firms do not need to consider the truthfulness of each other’s information.
This simplifies the analysis because we have not needed to consider whether a firm could deviate
in an active phase by undercutting the collusive price and underreporting the level of their sales.
This could be a profitable deviation from explicit trigger strategies if an active phase continued
into the next period rather than switching to an inactive phase. In this subection, we wish to
establish that this assumption is not important for our main analysis, because the results are
the same if sales reports are non-verifiable.
We assume that in the communication stage of each period firms must simultaneously submit
a sales report, rit , which then becomes common knowledge among the firms. Denote the total
P
reported sales in period t as Rt ≡ i rit and the sum of firm i’s rivals’ reported sales in period
P
t as R−it ≡ j6=i rjt . We assume that firms follow explicit trigger strategies with the addition
that the strategy prescribes each firm must submit a truthful sales report in the communication
30
stage, rjt = sjt , and that an active phase continues into the next period if and only if:
rjt =
kj
Rt ∀ j
K
and
Rt ∈ [m, m] .
(16)
Thus, the difference between this section and the main analysis is that in the main anlysis we
effectively restricted attention to rit = sit , but here we allow rit 6= sit . Following the main
analysis, we maintain assumption 1 and impose m < K to restrict attention to where tacit
collusion can be sustainable.
Proposition 6 shows that an active phase will only continue into the next period if pi = pc
and rit = sit .
Proposition 6. For any given n ≥ 2 and K−1 ≤ m ≤ m < K, if firm i’s rivals set the collusive
price and report their sales truthfully, such that pj = pc and rjt = sjt for all j 6= i, then an active
phase will continue into the next period if and only if firm i set the collusive price and reports
its sales truthfully, pi = pc and rit = sit .
The intuition is that a non-deviant firm can infer from its own sales what other firms will
report if they have set the collusive price. In contrast, since a deviant with a lower price sells ki ,
it has no information on the level of the market demand. Consequently, it must guess at the level
of sales that will ensure that the active phase continues into the next period. However, given
market demand is drawn from a continuous distribution, the probability of guessing correctly is
infintesmially small. Thus, firms cannot gain by deviating and submitting an untruthful report.
6
Concluding remarks
The conventional wisdom is that cartels should form in symmetric market structures, so competition agencies should actively seek them there. Our model suggests that once tacit collusion
is take into account, firms have a greater incentive to form cartels in more asymmetric market
structures. This fits well with the previously puzzling empirical evidence from detected cartels
which, as described in the introduction, shows that many tend to include a large number of very
asymmetric firms.
Our paper emphaises the need to recognise tacit collusion as a potential substitute for expicit
collusion. This raises a number of further policy implications. First, it strengthens concerns that
have previously been raised about the use of structural screens based on market characteristics to
31
detect cartel activity. For example, Harrington (2006a) argues that, whilst structural indicators
may suggest that cartel activity is more likely, there is still a high chance of false positives
because, based on the available evidence, cartels are relatively rare and a collusive outcome is
just one of the possible equilibria. In addition, our paper demonstrates that if the sturctural
indicators are identified according to the conventional wisdom, it will not even be the case that
they indicate that cartel activity is more likely. Instead, the screen may simply pick-up markets
where explicit collusion is unnecessary as tacit collusion is sustainable and more profitable.
Second, as has been recognised by Harrington (2011), tacit collusion following cartel detected,
has implications for using post-detection prices to estimate the but-for price had the cartel not
been in place for damage calculations. Furthermore, once tacit collusion is recognised as an
alternative to explicit collusion, further questions are raised about the approriate counterfactual
that should be used and whether the appropriate but-for price is at the competitive level. More
generally, our analysis has taken a normative stance by considering when, for a given level of
fines, cartel behaviour occurs. However, a related body of theoretical literature considers how
alternative penalty regimes affect cartel formation and pricing (see for example Katsoulacos et
al. 2014). As yet, to the best of our knowledge, this literature has not taken into account that
tacit collusion may be a susbsitute for cartel formation.
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Appendix A
(p(1−S(pc ))+θki F )
< 0
2
(1−p)
that raising the price raises profits and lowers the critical discount factor. This implies that the
Proof of Proposition 1. It follows from
∂V e
∂pc
= S (pc ) > 0 and
∂δe
∂pc
= −
most profitable equilibrium has pc = 1. Substituting this into k i V e and δe (kn , F, pc ) shows that
k i Ve∗ and δe∗ (kn , F ) are as claimed. Finally, δe∗ (kn , F ) < 1 if F ∈ 0, F (kn ) where F (kn ) ≡
1
m
θ S (p ) − p > 0. 35
Proof of Lemma 2. Substituting in for m∗ (k1 , m) and rearranging yields m = k1
K−m
K
+m ≡
x (k1 ), where x (k1 ) ∈ (m, K) for any m < K. Thus, it follows that monitoring is perfect if
m < x (k1 ), as this implies m > m∗ (k1 , m). Otherwise, there is imperfect monitoring. Proof of Lemma 3. Substituting ki V p into ki V c and solving yields:
(1 − δ)
πi (pc ) − πiN (kn ) .
1 − δ + G (m∗ (k1 , m)) δ (1 − δ T )
ki V c = πiN (kn ) +
(17)
Then substituting ki V p and ki V c into (10) and rearranging yields:
(K − m)
b pc
(1 − G (m∗ (k1 , m))) K pc − p −
≥ δ T (1 − G (m∗ (k1 , m))) K pc − p − (K − m)
b pc .
δ
(18)
Note that the left-hand side of (18) is less than the expression in square brackets on the right-hand
side, such that (18) can only hold if both are positive, since δ T ∈ (0, δ] for all T ∈ [0, ∞).
Given ki V c is strictly decreasing in T , the optimal equilibrium profits for firm i can be found
by evaluating it at the level where . Thus, it follows from (18) that:
∗
1 − δT =
δ
(1 − δ) (K − m)
b pc
.
1 − G (m∗ (k1 , m)) K pc − p − (K − m)
b pc
(19)
Then, substituting this into (17) yields:
ki
ki V =
K
c
m
b − G (m∗ (k1 , m)) K
1 − G (m∗ (k1 , m))
pc , ∀ i.
This is strictly increasing in pc , so pc = 1 and k i V ∗ is as claimed.
kn
1
1−G(m∗ (k1 ,m)) K ≡
G (m∗ (k1 , m)) < KK−n .
Finally, note that the left-hand side of (18), with pc = 1, is positive if δ ≥
δ ∗ (k1 , kn ) and the right-hand side of (18), with pc = 1, is positive if
∂G(m∗ )
> 0 that
∂m
= KK−n < 1, where
It follows from
there is a unique level of m, denoted x (k1 , kn ), that sets
G (m∗ (k1 , m))
x (k1 , kn ) < K and where G (m∗ (k1 , m)) ∈ 0, KK−n for all
m
b
for all
m ∈ [x (k1 ) , x (k1 , kn )). This implies δ ∗ (k1 , kn ) ∈ kKn , 1 and ki V ∗ ∈ πiN (kn ) , ki K
m ∈ [x (k1 ) , x (k1 , kn )). Proof of Proposition 2. Differentiating δe∗ (kn , F, λ) =
1−S(pm )+θF
1−p
respect to kn yields:
∂δe∗
∂kn
=
=
1−S(pm )+θF ∂p
2
∂kn
(1−p)
∂p
1
∗
δe 1−p ∂kn
( )
36
and F (kn ) =
S(pm )−p
θ
with
and
∂F
1 ∂p
=−
,
∂kn
θ ∂kn
respectively. Thus, it follows from from
and
∂F
∂kn
> 0 and S (pm ) < 1 for all m < K that
∂δe∗
∂kn
>0
< 0. Proof of Proposition 3. Differentiating
∂p
∂δe∗
(1−p) ∂kn
∂F
1
∂p
∂kn
. Thus,
∂ 2 δe∗
∂kn ∂F
> 0 from
∂p
∂kn
∂δe∗
∂kn
∂p
1
δ ∗ with respect to F yields
(1−p) ∂kn e
∂δ ∗
θ
> 0 and ∂Fe = 1−p
> 0. =
∂ 2 δe∗
∂kn ∂F
=
m
b
− Vc∗ ≡ F ∗ (k1 ) ,
Proof of Proposition 4. Note that ki Ve∗ > ki Vc∗ if and only if F < θ1 K
∗
m−G(m
b
)K
1
where Vc∗ = K
from Lemma 3. For any m < x (k1 ), such that G (m∗ ) = 0, then
1−G(m∗ )
F ∗ = 0. It also follows from limm→x(k1 ,kn ) Vc∗ = p that limm→x(k1 ,kn ) F ∗ (k1 ) = F (kn ). Finally,
m
b
if x (k1 ) ≤ m < x (k1 , kn ), then F ∗ ∈ 0, F (kn ) from Vc∗ ∈ p, K
. Proof of Proposition 5. For any m < K, δe∗ < δc∗ if and only if:
m
b
+ θF
1− K
1
kn
<
.
∗
1−p
1 − G (m (k1 , m)) K
Rearranging the above yields:
F
<
1
θ
h
<
1
θ
1−p
kn
1−G(m∗ (k1 ,m)) K
m
b
K
−
1
K
where the right-hand side can be rewritten as
− 1−
∗
m−G(m
b
)K
,
∗
1−G(m )
1
θ
m
b
K
m
b
K
i
− Vc∗ = F ∗ (k1 ). m
b
Proof of Proposition 5. Differentiating F ∗ = θ1 K
− Vc∗ with respect to k1 yields:
∂F ∗
1 ∂Vc∗
=−
.
∂k1
θ ∂k1
where:
∂Vc∗
(K − m)
b
∂m∗
=−
g (m∗ ) ,
2
∂k1
K (1 − G(m∗ )) ∂k1
such that
∂Vc∗
∂k1
> 0 from
∂m∗
∂k1
< 0 and K > m.
b Thus,
∂F ∗
∂k1
< 0. Proof of Proposition 6. We wish to show that if firm i’s rivals set pj = pcj and report rjt = sjt
for all j 6= i, then firm i cannot gain from deviating from its prescribed report of rit = sit , for
any arbitraty price for firm i, and for any feasible level of sales for firm i, sit .
37
To begin, consider the case where pi = pc . If all firm i’s rivals set pc and report truthful sales,
t
then rjt = sjt = kj m
K for all j 6= i such that R−it =
i reports truthfully such that rit = sit =
t
ki m
K ,
K−i
K mt .
It follows from this that if firm
then the active phase continues into the next
t
period, since rjt = kj m
K ∀ j and Rt = mt ∈ [m, m]; otherwise, firms enter an inactive phase.
Thus, firm i will report its sales truthfully such that the active phase continues if:
− (1 − δ) θki F + δki V e ≥ − (1 − δ) θki F + δπiN (kn ) ,
which is true for all F < F (kn ).
Next, suppose pi < pc . If all firm i’s rivals set pc and report sales truthfully, then rjt = sjt =
i
for j 6= i such that R−it = mt − ki . Thus, the active phase continues into the next
kj mKt −k
−i
period if firm i reports its sales untruthfully, such that:
rit = ki
m t − ki
K−i
< sit = ki ,
and if Rt ∈ [m, m]; otherwise firms enter an inactive phase. Given firm i does not know mt ,
i
suppose it reports rit = ki m−k
for some m ∈ [m, m]. Then its present discounted value of
K−i
i
submiting an untruthful report of rit = ki m−k
is:
K−i
− (1 − δ) θki F +δ Pr (m = mt ) ki V e + (1 − Pr (m = mt )) πiN (kn ) = − (1 − δ) θki F +δπiN (kn ) ,
for all m ∈ [m, m], since Pr (m = mt ) = 0 due to the continuous distribution. These profits are
the same as it would get if it submitted a truthful report. Therefore, firm i, for all i, cannot gain
by submitting an untruthful report, rit 6= ki for any pi < pc .
Finally, suppose pi > pc . If all firm i’s rivals set pc and report sales truthfully, then rjt =
sjt = kj for j 6= i such that R−it = K−i . Thus, the active phase continues into the next period
if firm i reports its sales untruthfully, such that:
rit = ki > sit = mt − K−i ,
and if Rt = K ∈ [m, m]; otherwise firms enter an inactive phase. Therefore, m < K suffices to
ensure that firm i cannot gain by submitting an untruthful report, rit = ki > sit . 38
Appendix B
In this subsection, we analyse the alternative strategy described in section 3.1 in which the firms
in the cartel apply for leniency in every period in an attempt to reduce their expected fines. We
refer to this as strategic leniency strategies. Formally, under this strategy profile, in the pricing
stage of a period during an active cartel phase, a firm sets the collusive price pc . Then, in the
communication stage the firm realises its sales and always applies for leniency. If yt = y, such
that all firms’ sales are above their trigger levels, then the active phase continues into the next
period. However, if yt = y, such that at least one firm’s sales are below its trigger level, then
each firm secretly shares its private information with its rivals. The active phase continues into
the next period if pjt = pc and sjt = ki S (pc ) for all j. Otherwise, firms enter an inactive phase
and each firm prices according to the static Nash equilibrium forever.
Since under this strategy profile all firms apply for leniency and the cartel is detected every
period, the per-period probability that firm i’s leniency application is successful and it avoids
paying a fine is
1
n.
Therefore, given conviction does not lead to the breakdown of the cartel, firm
i’s expected (normalised) discounted profit if each firm abides by its prescribed strategy is:
ki V s = πi (pc ) −
n−1
ki F,
n
(20)
As before, this says that in a period during an active cartel phase that firm i expects to receive
the expected per-period profit from setting pc minus the expected fine. The only difference is
that the expected fine now depends upon the probability that a strategic leniency application is
successful rather than the probability the cartel is detected.
We next consider whether firm i has an incentive to deviate from its prescribed strategy. We
first suppose firm i has abided by its strategy in the pricing stage of period t by setting pi = pc
and consider deviations in the communication stage. Firm i can deviate here by not applying for
leniency and/or not sharing its private information in the case of yt = y. However, not applying
for leniency increases the expected fine a firm faces and not sharing its private information means
that firms will enter an inactive phase. Therefore, for any ki V s ≥ πiN (kn ) there is no gain from
deviating in the communication stage. This means that we only need to consider the pricing
ICC.
Now, consider firm i’s incentive to deviate from its prescribed strategy in the pricing stage
of period t. It follows from above that if firm i deviates in the pricing stage, then in the
communication stage it will apply for leniency and firms will enter a punishment phase. Note
39
that, as explained in section 3.1, we assume that the deviant with the lowest price is given
leniency. This follow the approach of Spagnolo (2005) because the competition agency has an
incentive to favour a deviant informant over colluding informants. This makes collusion more
difficult (in that the critical discount is higher) than if there is no such preference. In reality, the
first informant through the door is selected for leniency. Our approach is consistent with this if
the first firm through the door is determined in discrete time periods as opposed to a continuous
period. In our setting, firms apply for leniency in the same discrete period (day, week, month)
and the competition authority gets to select which informant within this discrete time period.
In contrast, Chen and Rey (2014) assume that any informant is randomly selected with equal
probabilities, so preference is not given to a deviant informant over a colluding informant. In
contrast, if a firm does not deviate in the pricing stage, then it will share its private information
and apply for leniency. Thus, firm i has no incentive to deviate in the pricing stage if:
ki V s ≥ (1 − δ) ki pc + δπiN (kn )
(21)
where the left-hand side is firm i’s profit from abiding by its strategy, and the right-hand side
is the profit from firm i’s optimal deviation. The first term on the right-hand side is firm i’s
profit from undercutting pc marginally to supply its full capacity, ki , which is firm i’s optimal
deviation from pc > V s > p. Rearranging (15) yields:
δ ≥1−
Vs−p
≡ δs (kn , n, pc , F ) .
pc − p
This critical discount factor is independent of ki , so if it holds for firm i it holds for j 6= i.
Furthermore, it follows that δs (kn , n, pc , F ) < 1 for any V s > p, where δs (kn , n, pc , F ) is strictly
increasing in the fine per unit of capacity, F .
It follows from the above that any given V s ∈ p, pc are SPNE profits if the pricing ICC is
satisfied. It is straight forward to check that raising the price raises profits and lowers the critical
discount factor. Therefore, either the profile of strategic leniency strategies, with firms setting
the monopoly price, is a SPNE or collusion under strategic leniency strategies is not sustainable
at any collusive price. Therefore, substituting pc = 1 into k i V s and δs (kn , n, F, pc ) shows that
1−S(pm )+ n−1 F
n
collusion is sustainable under strategic leniency strategies if δ ≥ δs∗ (kn , n, F ) ≡
1−p
n
where δs∗ (kn , n, F ) < 1 if F ∈ 0, F s (kn , n) where F (kn , n) ≡ n−1
S (pm ) − p > 0 and in
N
m
this case firm i’s optimal SPNE profits are ki Vs∗ = πi (pm ) − n−1
n ki F ∈ πi (kn ) , πi (p ) ∀ i.
40
Otherwise, collusion under strategic leniency strategies is not sustainable.
Comparing the optimal SPNE profits under strategic leniency strategies with those under
explicit trigger-sales strategies in Proposition 1, shows that firms will not want to inform the
competition agency about the cartel as part of the collusive strategy if θ <
n−1
n .
This is because
in this case informing the competition agency raises the expected fine firms face. The collusive
strategy profile is then exactly as analysed in the main text. Whereas if θ >
n−1
n ,
all firms will
inform the competition agency of the cartel in every period as part of the collusive strategy.
Furthermore, comparing the critical discount factor under stratwgic leneicny strategies with the
equivalent under explicit trigger-sales strategies as given in Proposition 1, shows that δe∗ (kn , F ) ≤
δs∗ (kn , n, F ) if and only if θ ≤
n−1
n .
This implies that if collusion under strategic leniency
strategies is more profitable than under explicit trigger-sales strategies, then it is sustainable at
lower discount factors than collusion under explicit trigger-sales strategies, and vice versa. It
follows that θ <
1
2
is a sufficient condition for firms to never inform the competition agency
about the cartel as part of the collusive strategy. Given the low detection rates of cartels, this
seems likely to hold in most jurisdictions. Hence, the focus on explicit trigger-sales strategies in
our main analysis.
41