cartelisation

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International Conference on Applied Economics ICOAE 2009 551

Cartelisation of Oligopoly

Jacek Prokop224

Abstract

This paper considers possibilities of cartelisation in the oligopolistic industries. We distinguish between two important issues. The firstone, traditionally discussed in the literature, is the problem of cartel stability. The other one, much less analysed, is the process of

cartel formation. Both aspects have important repercussions for the market structure and the role of regulatory agencies.

Even though the profit-maximizing firms have incentives to restrict competition by forming cartels, the collusion has often beenbroken due to the prisoners dilemma. Therefore, it could have been claimed that antitrust agencies do not need to intervene. Further

analysis shows, however, that there are markets where stable cartels may be created and persist. We present a model of collusive price-

leadership and show the existence of stable cartels. Next, we investigate the cartel-formation process in a game-theoretical framework.Our analysis allows to formulate general guidelines for the regulators intervention to prevent cartelisation.

JEL codes: D43, L13, L49

Key Words: antitrust policy, cartels, collusive behaviour

1. Introduction

The profit-maximizing firms in oligopolistic markets have strong incentives to restrict competition bycreating cartels. They recognize that coordination of output or pricing strategies could help them increase profits. In themost extreme case when all companies in the industry form a cartel, they can duplicate the monopoly outcome, andshare the largest pool of profits that can be generated.

The increased profit is achieved at the expense of consumers, and usually leads to a reduction in total welfare.Therefore, cartels are illegal and antitrust authorities actively engage in fighting them. Even though there have beenmany success stories of governmental activities leading to detection and punishment of firms forming cartels225, there isno doubt that it is not an easy task, and additional tools of analysis should be developed to help eliminate illegal

cooperation.

Following the universal rule that it is better to prevent than cure, antitrust agencies could improve theireffectiveness in eliminating cartels from the markets by identifying the circumstances in which cartelisation is most

likely to occur. It is the objective of this paper to determine conditions under which the antirust authorities should getactively involved in correcting the market, and to show the basic directions of the intervention.

The first indication of possible cartelisation in a given industry is the theoretical existence of a stable

cartel.226 Such existence is not guaranteed in every market structure. In some industries, the cartel stability is destroyedby the occurrence of the prisoners dilemma. In these circumstances, cartelisation does not constitute a major threat.

However, the markets, for which the theoretical existence of a stable cartel can be shown, should be the target of acloser analysis by the antitrust agency.

The theoretical existence of a stable collusion among firms is not an immediate argument for an active marketintervention. We should analyse the industry situation to determine whether there is a chance for the potentially viablecartel to form. [Selten: 1973] and [Prokop: 1999] have shown that the discussion about the process of cartel creation is

224Professor of Economics, Warsaw School of Economics, Al. Niepodleglosci 162, 02-554 Warsaw, POLAND, e-mail: [email protected]

Several interesting examples of successful cartel detection by the antitrust authorities are presented, e.g., in [Pepall - Richards - Norman: 2002, pp.386-400].226

For references on cartel stability see, e.g., [Martin: 1993].


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552 International Conference on Applied Ecomonics ICOAE 2009

at least as important as the research on cartel stability itself.227 Following that literature, we present a model of cartel-formation process, and formulate the key conditions that make the cartel creation highly probable. We demonstrate thatin an industry composed of more than five firms the chance of having a cartel is quite small due to the free-riding

problem among the market participants. In the case of industries with up to five firms, we should expect a cartel to becreated for sure; there are strong incentives for the firms to collude.

These results are used to provide the guidance for an intervention of the antitrust authority to prevent industry

cartelisation. When the dynamics of the market structure leads to collusion among firms, the antitrust agency shouldstep in; such situation takes place for industries with the number of competitors falling below six. Industries with a

larger number of firms (more than five) should not be of large concern by the antitrust authorities; the market itselfmakes the cartelisation difficult.

The rest of this paper is organized as follows. In section 2, the cartel stability is defined, and the classicalexample of prisoners dilemma is given to illustrate the lack of stable collusion in some oligopolies. Next, we present amodel of an industry characterized by collusive price-leadership where the existence of a stable cartel is shown. In

section 3, we offer the cartel-formation game and discuss the equilibrium behaviour of industry participants as well asthe role of the antitrust agency. The conclusions are in section 4.

2. Cartel Stability

2.1. The Definitions

We consider an oligopolistic industry composed of n > 1 identical firms producing a homogeneous output.Any number of firms in the industry may form a cartel, i.e. agree to maximize joint profit. Such collusive agreement is

usually aimed at restricting the joint output in order to raise the price, and thereby increase profits. For any cartel to stayin the market the collusion among firms must be stable.

In order to define cartel stability, we need to distinguish between internal and external stability. A cartel is

said to be internally stable if it is not profitable for a member firm to defect from the collusive agreement. A cartel isexternally stable if it is not profitable for any non-member firm to join the collusive agreement. A stable cartel is acartel which is both internally and externally stable.

It has been long known that cartels have a tendency to break due to the prisoners dilemma among thecooperating firms (lack of internal stability). The most typical behaviour of cartel members can be illustrated by thefollowing example of the two-firm game given in Table 1.

Table 1: Prisoners dilemma game of cartel members.Firm 2

Cooperate Defect

Firm 1 Cooperate (25, 25) (15, 35)

Defect (35, 15) (20, 20)

There is only one Nash equilibrium of this game; in the equilibrium both firms decide to deviate from the cartelagreement i.e. (Defect, Defect). Even though, it is Pareto dominant for the firms to cooperate, there is lack of internalstability of the cartel due to the strong incentives for the individual members to defect (to cheat).

It has been argued that by taking into account an infinite horizon of the competition the firms are encouraged tocooperate, and to avoid cheating.228 Unfortunately, the imperfections of information could destroy the collusion even inan infinite horizon interactions between companies. When cheating by cartel members is unobservable, the

participating firms may not be sure about the source of their poor performance. The low profit could result either fromnot obeying the cartel agreement, or from a reduction in the overall market demand.229 Hence, the existence of the

prisoners dilemma makes cartels not to function too smoothly. This result provides justification for many observers to

conclude that government intervention is not required because competitive forces keep destroying the cartels.

227Selten [1973], Prokop [1999], and Morasch [2000] model the cartel formation process and show that not all profitable and stablecartels will form in a subgame perfect equilibrium.228See, e.g., [Tirole: 1997, pp. 258-259].229For a discussion of the literature on secret price cuts see, e.g., [Tirole, 1997, pp. 251-253, 262-265].


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The prisoners dilemma game however does not arise in every oligopolistic market. Table 1 captures, forexample, an industry when companies with linear cost functions are engaged in a Cournot-type competition when thecartel agreement fails.230 Since the cooperative (cartel) output levels do not constitute a pair of best responses, each

firm has a stronger profit incentive to defect or to cheat upon the cooperative agreement than to stick with it. Underdifferent assumptions about the cost functions and the type of competition among firms, it can be shown that a stablecartel exists. An example of the existence is discussed below.

2.2. The Existence of Stable Cartels

Let us assume that the market demand for our industrys product is given as

(1) D(p) = 100p.

Each firm has the following cost function:

(2) 22

1)( ii qqC ,

where qi is the output of firm i. Further suppose that this market is characterized by price leadership of a dominant firmor cartel. The non-dominant firms are price takers; they constitute a competitive fringe. 231 Denote the number of cartel

members by k. Then, there are nkremaining firms in the competitive fringe. The latter firms take the cartel price p(k)as given and choose their production level by setting marginal cost equal to this price.

The cartel behaves as a monopolist with respect to the residual demand curve, i.e. the market demand at price p(k)reduced by the supply of the competitive fringe. Since all companies are identical, the production within the cartel will

be spread uniformly among its members. We do not allow any redistribution of profits among the firms. In this market,the profit of each firm in the cartel is

(3),

)1(

15000)(

22 knkd

and the profit of each firm in the competitive fringe is

(4).

])1[(

)1(5000)(

222

2

kn

nkc

From (3) and (4), it is easy to see the following:

Proposition: The profit of a cartel member is smaller than the profit of a competitive-fringe participant, i.e.,.0)()( kallforkk cd

An immediate implication of this proposition is that it is strictly preferred for any firm to be in the competitivefringe rather than in the cartel. This preference, however, should not be confused with the firms incentives to leave thecartel for the competitive fringe. As it will be demonstrated, such a departure is not necessarily profitable.

Now, let us turn to the cartel-stability conditions. A cartel will be internally stable ifk 1and

(5) ,)()1( kk dc External stability of a cartel takes place when k n1 and

(6) )()1( kk cd .

Thus for a cartel to be (internally and externally) stable both conditions (5) and (6) must hold.

Using the profit functions (3) and (4), a straightforward algebra shows that:232(a) for an industry composed ofn 5 firms,the stability conditions (5) and (6) are satisfied only fork=n,(b) for an industry composed ofn > 5 firms, the stability conditions (5) and (6) hold only fork=3.

Thus, in the oligopolistic industries discussed in this subsection there exists a unique stable cartel. 233 When thenumber of industry participants does not exceed five, the cartel consists of all of them. For industries with more thanfive companies, the size of the stable cartel does not depend on the number of firms, and is always equal to three.234

230 For a detai led numerical example see, e.g., [Pepall - Richards - Norman: 2002, pp. 353-357].

231 Other types of competition in the industry characterized by collusion may be found in [Martin: 1993]. For example, a cartel may

act as a Stackelberg leader, and the fringe firms follow the Cournot behaviour.

232 For comparison see [Daskin: 1989].

233 The existence of a stable cartel in the industries with collusive price leadership has been first shown in [ dAspremont - Jacquemin

- Gabszewicz - Weymark: 1983]. Further discussion of the existence of stable dominant cartels can be found in [Donsimoni: 1985],[Donsimoni - Economides - Polemarchakis: 1986], and [Nocke: 1999]. A different approach is offered by [Yong: 2004] who argues


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The existence of a stable cartel does not automatically mean that such cartel can be formed. The formationprocess may face some obstacles that prevent the creation of cartels. Some of the difficulties arise from the fact thatcartels are prohibited by the law of most of the democratic countries. Even in the case when these hurdles could be

overcome, however, there are also other reasons why independent firms in some industries may not be able to colludedespite the theoretical existence of a stable cartel in these markets. In the next section, we propose a model of cartel-formation process and analyse its potential outcomes.

3. The Process of Cartel Creation

3.1. The Cartel-Formation GameConsider again an industry with n > 1 identical firms. The market demand for the industrys product and the

cost functions of the firms are as described in the previous subsection. The cartel-formation game is played in twostages.235 In the first stage, the firms simultaneously and independently decide whether to participate in a cartel orremain in the competitive fringe. In the second stage, the members of the cartel choose the price that maximizes their

joint profit, and the firms in the competitive fringe take this price as given and choose their profit-maximizingproduction level.

Each firm iN={1, , n} has two pure strategies, 0 to stay in the competitive fringe, and 1 to join the cartel.

Denote the strategy space {0, 1} of firm i by Si. Then S1 Sn is the outcome space. When each firm i chooses its

strategy si and the dominant cartel is formed, the payoff to firm i is determined. Firm i that chooses to stay in the

competitive fringe obtains a profit of c(k), where k=jN sj. We assume this profit to be the payoff to firm i if itdecides to stay in the competitive fringe. The payoff to firm i from joining the dominant cartel is d(k). Thus the payofffunction of firm i is given as

(7) .1,0),...,()( 1 iNj jd iNj jcnii sifs sifssshsh In addition to pure strategies, we allow the firms to play mixed strategies. Denote the set of mixed strategies of firm

i by Ti, i.e. Ti = {pi: 0pi1} foriN, wherepi is the probability of joining the dominant cartel by firm i; we call it theparticipation probability. Define the expected payoff to firm i by

(8)

....)...,,(

|)(|)1()1(

)1|(|)1()(

11

}{

}{

}{}{

nn

Sj

iNj

Sjcjj

iNSi

iNS SjiNj

Sjdjjii

TTpppfor

Sppp

SppppH

The first bracket in (8) is the expected payoff to firm i from joining the dominant cartel, and the second bracket is the

expected payoff from staying in the competitive fringe.

We apply the Nash equilibrium concept to our game. A strategy n-tuple )...,,( 1 nppp is called a Nashequilibrium if for all iN,

,),()( iiijiii TpallforppHpH where )....,,,...,,( 111 njjj ppppp

Since it is expected that identical firms will behave in a similar way, we restrict our analysis to the symmetric Nash

equilibria. Denote the symmetric equilibrium strategy of a firm by0p . In the symmetric case, it follows from (8) that

the equilibrium payoff of firm i is

(9)

,)()1(1

)1(

)1()1(1

)(

1

00

1

0

0

1

0

1

000

kppk

np

kppk

nppH

cknk

n

k

d

n

k

knki

that the Nash notion of stability used in the literature is too weak, because it only permits deviations by individual firms,i.e. group deviations are not allowed.

234 It is interesting to observe that for many other specification of demand and cost functions the stable cartelis usually composed of 3 companies when the number of industry participants is greater than five. See, forexample,

comments by [Daskin: 1989, pp. 7-12].235 Compare [Prokop: 1999].


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where d() and c() are given by (3) and (4).In the next section, we use the above definitions to describe the equilibrium strategies of the market players.

3.2. The Equilibrium Behaviour of Firms

The following theorem characterises the symmetric equilibrium of our game:236

Theorem: The cartel formation game has exactly one symmetric equilibrium point. At this point:

.510

,51

0

0

nforp

nforp

According to theorem, in an industry composed of not more than five firms, the equilibrium participationprobability is equal one, i.e. the cartel-formation game leads to the creation of a stable cartel. The probability of joining

the cartel by an individual firm falls below one when the number of firms is greater than five.

The actual chance of forming a dominant cartel in the case ofn>5 can be seen only by calculating the exactlevels of the equilibrium participation probabilities. Table 1 shows the simulation results for various numbers of firms

in the industry. Observe that the individual probability of joining the cartel is declining with an increasing number offirms in the industry. The explanation of this outcome is not difficult. The stable cartel consists of three firms in theindustries with more than five competitors. Hence it is becoming a coordination problem to reach the cooperativeagreement. The firms have a tendency to free ride by waiting for others to form the cartel.

Table 1: Symmetric equilibrium of the cartel formation game.

Number of firms

in the industry

(n)

Participation probability)( 0p

2

3

4

5

6

7

8

9

1011

12

13

14

15

20

50

1,00001,0000

1,0000

1,00000,42820,31590,2582

0,2206

0,19320,17230,1555

0,14180,13030,12060,0879

0,0335

The above outcomes have very important consequences for the behaviour of firms in the market where a

cartel plays the role of the price leader. In the case of industry composed of not more than 5 firms we should expect allof the firms to create a cartel. The cartel will become a monopolist in this market. Thus, from the point of view of thetotal welfare, such market structure is inefficient, and there should be some involvement of the antitrust agency tocorrect it.

When the number of firms in the market exceeds five than the participation probability suddenly andsignificantly declines to the values much below 0,5. It means that the probability of creating a stable cartel becomessmall, and it is rather hard to expect that it will be formed.

Based on the presented results, in general, the antitrust agency should take measures to block mergers andacquisitions that lead to a reduction of the number of firms below six in any given industry. Moreover, it would be

beneficial to stimulate an increase of the number of firms in any industry. A relatively large number of companies(more than five) serves as the best guarantee of a low probability of cartel creation.

236The proof of theorem follows directly from the reasoning presented in [Prokop: 1999, pp. 250-251].


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Since the industries with a small number of firms (less than six) are expected to be cartelised, an active roleof the antitrust agency in this case is of great importance. One of the suggested actions that could be undertaken by theantitrust authorities is to break up large companies (e.g. Microsoft) into few smaller ones in order to increase the

number of independent industry participants. Another possibility is to enable other firms to enter the market. Otherwise,the industries comprised of up to five competitors should be the target of a close watch to detect signs of cartel

behaviour, because all the incentives work here for the market participants to collude.

In summary, only the industries composed of less than six firms should be of major concern to the antitrustauthorities. The markets with at least six firms may be viewed as quite competitive and require much less attention

from the regulators side.

4. Conclusions

Any regulatory measures imposed by antitrust authorities make sense only when the market mechanism leads

to some undesired results. Therefore, on the one hand, research should be conducted towards identification ofconditions under which undertaking any regulatory measures is redundant. On the other hand, it is important to describethe circumstances when some intervention of antitrust is indispensable. The current paper aimed at contributing to theliterature on cartelisation of industries in order to provide guidance on the needs for governmental regulation of some

markets.

The analysis of markets characterized by price leadership of one firm or a group of firms shows that there areindustries in which the theoretical existence of a stable cartel can be proved. However, such potential existence of

collusion among firms does not necessarily mean that a stable cartel can be actually formed. We illustrated the problemby offering the cartel-formation game. It is demonstrated that in the industries composed of less than six firms, theequilibrium outcome has all firms forming a dominant cartel. The process of cartel formation will not be successful in

the markets comprised of more than five firms. The reason for these results stems from the incentives of firms to freeride on the existence of a cartel in the case of a larger number of companies in the industry.

The analyses of both the cartel stability and the formation process are crucial in the discussion about theinvolvement of antitrust agency in fighting cartels. It has been shown that in the industries characterised by collusive

price leadership and quadratic cost functions of firms, a stable cartel always exists. Thus, such industries should be thetarget of a closer watch by the antitrust agency. Even though the existence of a stable cartel can be shown, the actualcreation of a cartel in this case is not automatic. We have demonstrated that only in the industries composed of up tofive firms, a cartel composed of all of them will be created for sure. Thus an active involvement on the part of an

antitrust agency is necessary to fight the collusive behaviour. When there are more than five firms in the industry, ourmodel predicts that the free riding among firms will prevent the creation of a stable cartel. Such industries do notrequire immediate regulatory measures by antitrust authorities.

Clearly, further research on the prevention and detection of cartels is necessary.

First, the case of possible cartelisation in the industries with asymmetric firms should be analysed.The stability of collusion by heterogeneous firms in some industries has been discussed, forexample, by [Donsimoni: 1985], but the process of cartel creation still remains to be considered.

Second, many other types of cost functions and forms of competition among firms should beconsidered from the viewpoint of both stability and formation process of a potential collusion.

Third, the role of asymmetric information in the cartelisation process should be investigated.

Fourth, other than the Nash notion of stability could be discussed. For example, [Yong: 2004]

suggested an inclusion of group decisions in the definition of stability. The existence of suchstability should be supplemented by the analysis of the collusion process, as well.

Fifth, the analysis of cartelisation should be extended to a dynamic framework. The discussion ofthe collusive behaviour has been conducted in the framework of one shot games. A multi-period

analysis would constitute a natural extension.


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Finally, the empirical studies237 will be the ultimate judge of the theoretical results of the cartelisation ofoligopolistic industries and the effectiveness of the antitrust authorities.

References

dAspremont, C. - Jacquemin, A. - Gabszewicz, J.J. - Weymark, J.A. (1983), On the Stability of Collusive Price

Leadership, Canadian Journal of Economics, 16:17-25.

Daskin, A.J. (1989), Cartel Stability in the Price Leadership Model: Three -Firm Cartels and the Role of Implicit

Collusion, Working Paper 89-17, Boston University School of Management.

Donsimoni, M.P. (1985), Stable heterogeneous cartels,International Journal of Industrial Organization, 3:451-467.

Donsimoni, M.P. - Economides, N.S. - Polemarchakis, H.M. (1986), International Economic Review, 27:317-327.

Martin, S. (1993),Advanced Industrial Economics, Blackwell: Basil.

Morasch, K. (2000), Strategic Alliances as Stackelberg Cartels-Concept and Equilibrium Alliance Structure,

International Journal of Industrial Organization, 18, 257-282.

Nocke, V. (1999), Cartel Stability Under Capacity Constraints: The Traditional View Restored, Working Paper,Nuffield College, Oxford.

Pepall, L. - Richards, D.J. - Norman, G. (2002), Industrial Organization: Contemporary Theory and Practice, SecondEdition, South-Western: Cincinnati, Ohio.

Prokop, J. (1999), Process of Dominant-Cartel Formation,International Journal of Industrial Organization, 17:241-257.

Selten, R. (1973), A Simple Model of Imperfect Competition, where 4 are Few and 6 are Many,International Journal

of Game Theory, 4:25-55.

Suslov, V. (2005), Cartel Contract Duration: Empirical Evidence from Inter-War International Cartels,Industrial andCorporate Change, 14:705744.

Tirole, J. (1997), The Theory of Industrial Organization, The MIT Press: Cambridge, Massachusetts.

Yong, J.-S. (2004), Horizontal Monopolization via Alliances, or Why a Conspiracy to Monopolize is Harder Than It

Appears, Working Paper, MIAESR, University of Melbourne.

237For some interesting empirical research on stability of international cartels see, e.g., [Suslov: 2005].


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