Building Stochastic Illinois Walls

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Building Stochastic Illinois Walls

Author:

Frank van der Ploeg

Supervisor: Prof. dr. J. Tuinstra Second Reader: Prof. dr. M.P. Schinkel

University of Amsterdam

Department of Quantitative Economics

Master’s Thesis

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Acknowledgements

This is my second master’s thesis, and hopefully a worthy end to 8 years of studying. In these 8 years, I have doubted my future plans on many occasions, and this time is no different. With a mind full of fears and doubt, these first words of my thesis are the last ones I will write. But eventually, even good things must come to an end, so here we are.

First and foremost, I would like to sincerely thank my two supervisors, Jan Tuinstra and Maarten Pieter Schinkel. I imagine that no thesis is a “textbook excercise”, but the freedom and flexibility I enjoyed are exceptional as far as I know. They let me be as stubborn, narrow-minded, headstrong, and at times tired as I wanted to be, and still had the patience to keep (or bring me back) on the right track. I also want to express my gratitude to Axel Augustinus. We have never met (yet), but his thesis was love at first sight for me, and reading it was almost as enjoyable as reading a well written book. I can only hope that some day yet another student will further expand our building, for which they laid the foundations, and to which I merely added yet another brick.

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7.3 Expected profits . . . 81 7.4 Summary . . . 81 8 Influence of F and µ 84 8.1 One purchaser . . . 85 8.1.1 (F, µ) = (0, 3) . . . 85 8.1.2 (F, µ) = (4, 3) . . . 86 8.1.3 (F, µ) = (4, 1) . . . 86 8.2 Two purchasers . . . 87 8.2.1 (F, µ) = (0, 3) . . . 87 8.2.2 (F, µ) = (4, 3) . . . 88 8.2.3 (F, µ) = (4, 1) . . . 89 9 Generalizations 91 9.1 Present values . . . 91

9.2 General discovery somewhere in the chain . . . 94

10 Conclusion and Outlook 98 A Proofs 100 B Cartel Lifetimes 109 B.1 Lifetime of the cartel for two purchasers, stable cartel . . . 109

B.2 Lifetime of the cartel for two purchasers, possibly unstable cartel . . . 115

C Mathematica notebooks 119 C.1 One purchaser, optimizing γ . . . 119

C.2 Two purchasers, both models . . . 120

C.3 Three purchasers . . . 121

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Chapter 1

Introduction

In the past century, in both the United States and later the European Union, efforts to find and end illegal collusive constructions have intensified. In 1914, the United States government passed the Clayton Antitrust Act, expanding on the already existing Sherman Antitrust Act from 1890. The main purpose of the Sherman Act was to protect the market from not being able to function as a market, by prohibiting collusion between entities that were supposed to be competing. Monopolies, however, were admissible, as long as the monopolist gained his position based on merit, and not by illegal means. So the intention of the Sherman Act was to prevent artificial raising of prices or other practices, but not the rise of prices due to bonafide superior players. Or, as interpreted by the US Supreme Court in the case Spectrum Sports Inc. v. McQuillan in 1993 [1]:

“The purpose of the [Sherman] Act is not to protect businesses from the working of the market; it is to protect the public from the failure of the market.”

This lack of restrictions on the emergence or existence of monopolies, however, meant that more and more companies realized that, instead of forming an illegal cartel, it would be wiser to merge companies, thereby creating a monopoly with the same corresponding payoffs. The Clayton Act put more restrictions on this kind of “disruptions of the market”, by forbidding or at least restricting mergers and acquisitions that decrease the level of competition, and also to prevent price discrimination and other practices even further. One important additional feature of the Clayton Act is the fact that parties that have been harmed as a consequence of illegal practices are able to claim treble damages in court [2]. In the second half of the 20th century, the European Union formed. Although initially constructed as a way of making trading and travelling between member countries easier, it has grown to operate as a governing body, whose laws and treaties overrule their national counterparts. With respect to cartels, the Treaty on the Functioning of the European Union, article 101, explicitly forbids practices that, among others “fix purchase or selling prices”, “limit or control production”, and “apply dissimilar con-ditions to equivalent transactions with other trading parties” [3]. However, there are marked differences between both approaches. In the US, an often seen phenomenon is the class-action lawsuit, where a private legal firm represents a large group of plaintiffs in a court case against one or several defendants. These legal firms are eager to take on these cases and have the incentive to strive for the highest possible monetary value awarded by the court or in a pre-trial settlement, because they get a certain share. However, in a recommendation from June 2013, the European Commission states that Member States should designate representative entities to bring representative actions, and one of the conditions for eligibility is that this entity has a non-profit character [4]. This might be an obstruction on the side of plaintiffs in receiving their

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just compensation, as these firms often do not have the means or incentives to get the most out of a trial. Moreover, in the same recommendation, the EC explicitly prohibits the use of punitive damages, also seemingly undermining the plaintiffs’ position.

On the other hand, damage compensation is not the only way to deter firms to form cartels. Governments can also sue companies to impose fines that are not meant to compensate any of the victims. In the period 1990-2010, the EU clearly fined higher than the US [5, 6]. This raises the question which tactic is most efficient in deterring firms from forming cartels.

Although in the US cartel victims can claim treble damages, there have been two landmark rulings by the Supreme Court that have seriously limited the position of consumers against cartels. First, in 1968, in Hanover Shoe v. United Shoe Machinery Co., the defendant argued that the purchaser of a cartel or monopoly could simply “pass-on” damages it suffered to its own purchasers. The Supreme Court ruled against this defensive argument, stating that it would result in unnecessarily long and complicated court cases. Then, in 1977, there was the case Illinois v. Illinois Brick Co.. The state of Illinois claimed that Illinois Brick had been involved in a conspiracy, fixing prices of concrete blocks and thereby overcharging contractors to raise the price of bricks. The contractor passed this overcharge on to its customer, one of whom was the state of Illinois. The Supreme Court ruled against this offensive argument, effectively denying indirect purchasers of a cartel to claim any damages. A more detailed treatment of these cases and their positive and negative implications is presented by R¨uggeberg and Schinkel [7]. In 2008, Schinkel, Tuinstra and R¨uggeberg published a paper on the implication of the combina-tion of these two court cases [8]. They showed that “Hanover Shoe” and ”Illinois Brick” could be combined to build what they call an “Illinois Wall”. An Illinois Wall is a construction where all direct purchasers of a cartel are aware of the existence of the cartel, but are better off in the current situation than in ordinary competition, and therefore remain silent. The cartel rations its output and charges high prices for the limited output it offers, and its purchasers charge correspondingly high prices, taking some of the excess in profits for themselves. Because of the combination of the aforementioned court cases, the customers at the end of the production chain have no standing in court against the cartel. Only the firms purchasing directly from the cartel could sue it, but these firms benefit from the situation, and if they benefit enough from it compared to the situation of “ordinary” competition, they will not use their knowledge to sue the cartel. It could also be that their silence and profiting from the existence of the cartel actually makes them part of that same cartel [9]. This is a tricky question, and not straightfor-ward to answer, but it would motivate purchasers to demand a scheme where it can in no way be proven that they were cooperating with their suppliers, except perhaps for the testimony of those suppliers, who would then immediately admit their own crimes as well.

Another option would be that a cartel is initially hidden from its purchasers, but then gets discovered at some point by one or more of its purchasers. These purchasers, having suffered economic harm as a consequence of the cartel, will be inclined to sue the cartel. However, instead of a court case, there is also the possibility of an out-of-court settlement, where both parties agree on some form of compensation for suffered damages. Such a settlement can be either disclosed or not. If it is, it will be filed in the record of some court, but if it is not, the contents or even the existence will not become public, and all sorts of arrangement could occur. Axel Augustinus developed a basic model in his Master’s thesis, where a cartel has two pur-chasers and gets discovered at deterministic moments in time [10]. He makes four assumptions about the market structure that seem reasonable. First, the direct purchasers of the cartel have the best opportunities to discover the cartel, because they have the most information.

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There-fore, the model is only concerned with interactions between the cartel and its direct purchasers, and does not contain provisions dealing with discoveries by others. Second, there should be an opportunity for bargaining about the form and value of settlements. Third, the form of the set-tlements must be such that it cannot be identified for what it really is, a bribe or “hush money”, and it must be legally enforceable, so that both parties are to some extent certain about the actions of others with respect to keeping their word. Finally, the discovering purchaser and the cartel must bear in mind that there is a possibility of other purchasers discovering the cartel, meaning that they have to take into account that the cartel may want to bribe others at later times. The virtue of a settlement that meets the third requirement is that it frees the purchaser from worrying about being part of the cartel or not. The mere fact that it has made some perfectly legal arrangement with a firm that happens to be part of the cartel, obviously does not imply that it is part of that cartel. This allows us to use the first assumption, that we are only dealing with interactions between the cartel and its purchasers, because customers or firm that buy from the purchasers of the cartel have no standing in court whatsoever.

Augustinus’ model is the first building block towards the model constructed in this thesis. One major limitation in the model developed there is the deterministic nature of the discovery pat-tern, something that simplifies the model greatly but is of course highly unrealistic. Another shortcoming is the fact that it is not clear how the model can be extended to accommodate more purchasers than the two it deals with. Aside from that, the basic mechanism of the model bears close resemblance to the US antitrust system, with treble damages, and the choice between an out-of-court closed settlement or a class-action lawsuit.

Motivated by the work of Schinkel, Tuinstra and R¨uggeberg on Illinois Walls [8] and Augustinus on private discoveries and subsequent settlements [10], we will try to answer the following questions:

1. Is it possible for a cartel to hide itself behind some or all of its own purchasers by including them in the scheme through some settlement?

2. How much of the surplus in profits must the cartel give to these purchasers in return for their silence?

3. Under what circumstances is it wise to start a cartel, given that the opportunity of these settlements exists?

4. Do antitrust laws and the vehicles of goverment fines and/or punitive damages contribute in deterring firms from forming cartels? Do they contribute in preventing the hiding of a cartel through settlements?

The first two questions are logical, and are also answered by Augustinus in his model. We are interested in seeing whether these answers change when we make the model more realistic by trying a range of parameter values in a stochastic discovery setting. This transition from a deterministic to a stochastic setting means a significant increase in complexity, because al-most nothing about the consequences of strategic decisions is certain, meaning we must take all possible outcomes into account. However, this means that the results we will obtain are much more robust and realistic, and offer a serious improvement. The third question, whether forming a cartel is a good decision, given the legislation and settlement possibilities, is a new one, but might be very interesting. Combined with the first two questions, we should be able to get answers to the final question, on what forms of punishment are effective in preventing the

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formation of cartels and why. Even though the model developed in this thesis is an artificial one, the results could have implications that are potentially useful in improving antitrust legislation.

Daughety and Reinganum [11, 12, 13, 14], as well as Che and Spier [15] have produced exten-sive material on the game-theoretic approach of signalling games that have all the properties of negotiations concerning a settlement that entails a lump-sum payment to one or more plaintiffs. However, in these games, the cartel collapses after the negotiations, and the settlement was merely meant to compensate the plaintiffs for the harm they suffered, without having a public trial. We would like to allow for the possibility to have several negotiations, after which the cartel and its purchaser keep interacting, making these models unsuitable for our goals. Instead, what we will use in this thesis is the concept of Nash bargaining. The idea of the bar-gain was developed by Nash in 1953 [16], and later expanded on by, among others, Binmore et al. [17] and Friedman and Wittman [18]. The idea is related to the Folk Theorem, which states loosely that any Nash equilibrium in an infinitely repeated game must weakly dominate the out-come where all players get the minmax payoff. This minmax payoff can be seen as a “threat” of other players, and is the worst-case scenario for all players, following from other players trying to minimize (min) their opponents’ payoff, and then choosing a strategy resulting in the highest (max) of these minimized values. In the Nash bargain, two players jointly maximize the product of the difference between their agreed upon payoff and this threat, which is the outcome that will emerge if they cannot meet each other’s demands. Of course, in a stochastic framework like the one we will use, payoffs are not always certain, and a lot of subtleties must be taken into account. However, at each moment in time, both players can calculate their expected payoffs, even when there is a good chance that the cooperation between the cartel and a purchaser will end soon due to new discoveries.

The outline of this thesis will be as follows. First, we will explain the fundamental model in which the cartel and its purchaser operate in chapter 2. Then, the mechanism of discovery, damage claims, settlements and the stochastic extension of the existing model will be worked out in detail in chapter 3. When the idea of the model is worked out, we will apply it to case-studies of increasing complexity in chapters 4 through 7. We will start with what seems like a trivial case, a cartel with one purchaser. Then, we move on to a scenario with two purchasers, but without concerns about the stability of the cartel, to keep the model relatively tractable, and then construct a model that covers all possible issues of stability and instability. Our last case study concerns a model with three purchasers, but due to the complexity of that model, we only present a small number of results. After that, we look back at the model with one purchaser and the full model with two purchasers to see how some of the results change when we change the legislative structure. Finallly, we take a brief look at what a true generalization to more complex models would look like, present necessary proofs and explain the Mathematica notebooks used to obtain the results.

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Chapter 2

The Market Environment:

Competition and Collusion

In this chapter, we will explain in detail what the setup of our model is. We will define the environment the cartel and its purchasers operate in, derive demand, price and profit functions for both, and point out the subtleties that will become important later on. At the end of this chapter, we will have all the ingredients to build the desired model for our research objectives, and we can begin studying specific cases.

First, we must define how the cartel interacts internally and with its purchasers, and also how its purchasers compete with each other, if there are more of them. For this, we adopt the same basic model as Augustinus, as illustrated in figure 2.1.

Here, we see an upstream layer of m companies, that produce goods at a marginal cost equal to c. These upstream firms produce homogeneous goods, i.e. they all make the same prod-uct. They sell this product to one or several of n downstream firms. Each downstream firm i is charged some price pi, at which it purchases Qi units. A different way to say this is that

the downstream firms face marginal costs equal to pi, assuming they have no other

produc-tion costs. Finally, the downstream firms produce new commodities from the material they purchased from the cartel, which they sell at a price Pi, and of which they sell a quantity

Qi. The prices charged and quantities sold by each layer of firms determine its profits. The

total profit of the firms in the upstream layer is denoted by π, and of those in the downstream layer it is denoted by Π. Note that this is a returning feature: we use lowercase symbols for up-stream prices, quantities and profits, and uppercase symbols for their downup-stream counterparts. The form of competition between firms in the same layer is important. We will use Bertrand competition, i.e. competition on prices. The reason for this is first of all that this occurs more frequently in everyday markets than Cournot competition, i.e. competition on quantities. A second reason is that Betrand competition in the case of homogeneous goods has the property that firms compete with each other to the point where they charge input prices for their prod-uct, forfeiting all potential profits. This simplifies computations later on, although the same computations could be performed with nonzero competitive profits.

Now, the very basis of the model is a utility function for the customers at the end of the chain. To this end, we reproduce some results from a prerpint of the Illinois Walls paper from Schinkel, Tuinstra and R¨uggeberg [19] and an extension to that paper by Schinkel and Tuinstra [20]. The proof of this and subsequent results can be found in the appendix.

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Figure 2.1: The market used in this model in a competitive situation. Figure from [10], page 12

Lemma 1. Suppose the consumers’ utility function is given by

U (Q0, Q1, Q2, ..., Qn) = Q0+ n X i=1 Qi− 1 2   n X i=1 Q2_i + e n X i=1 X k6=i QiQk  

where Qi, i = 1, ..., n are differentiated varieties of a commodity, e ∈ [0, 1) is a measure of

substitutability between these varieties, and Q0 is a composite commodity with price equal to

one, representing expenditures on all other available commodities. Then, denoting the price of variety i by Pi, the demand for variety i is given by

Qi(P1, ..., Pn) =

(1 − e)(1 − Pi) − eP_k6=i(Pi− Pk)

(1 + (n − 1)e)(1 − e) (2.0.1) In contrast to the upstream firms, the downstream firms do not necessarily produce homoge-neous goods. Instead, they produce heterogehomoge-neous goods, with the level of heterogeneity being specified in the form of a single parameter e. We see that if we use e = 0, the cross terms in the utility and demand functions vanish, corresponding to perfect differentiation. Using e = 1 im-plies treating products from firm i and j 6= i the same, i.e. perfect substitutability. An example of a situation where this market structure could emerge is where the upstream firms sell some raw material, like wood, and the downstream firms use this wood to manufacture furniture, like chairs and tables. If e = 1, they all manufacture exactly the same product (identical chairs, for example). If e ∈ (0, 1), they do not create exactly the same product, but something similar, like different kinds of chairs, or tables of different size. If e = 0, one firm produces chairs, another produces tables, and yet another might produce wooden doors.

With these demand functions for the customers at the end of the chain, we can derive two sets of prices, produced quantities and profits: for the competitive situation, and for the situation where the upstream firms collude in a cartel. Both these results can be found by backward induction. This means we first solve the maximization problem for downstream firms, find their prices and quantities, and from these solve the maximization problem for upstream firms.

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2.1 Competition

In the ordinary competition model, all firms maximize their own profits, given their competi-tors’ choices. Each downstream firm therefore maximizes (Pi− pi)Qi(P1, ..., Pn) with respect to

Pi, without constraints. The next lemma describes the charged prices as a function of a general

input price vector (p1, ..., pn):

Lemma 2. Given the input price vector (p1, ..., pn), the output price and demand for firm i are

Pi(p1, ..., pn) = A0e,n



B0_e,n+ C_e,n0 pi+ D0e,n

X k6=i pk   (2.1.1) where A0_e,n = 1 4 + 6(n − 2)e + (2n2_{− 9(n − 1))e}2 B_e,n0 = (2 + (2n − 3)e)(1 − e) C_e,n0 = (2 + (n − 2)e)(1 + (n − 2)e) D_e,n0 = e(1 + (n − 2)e)

and

Qi(p1, ..., pn) = Ae,n− Be,npi+ Ce,n

X k6=i pk (2.1.2) where Ae,n = 1 1 + (n − 1)e 2 + (4n − 7)e + (2n − 3)(n − 2)e2 4 + 6(n − 2)e + (2n2_{− 9(n − 1))e}2 Be,n = 1 + (n − 2)e (1 + (n − 1)e)(1 − e) 2 + 3(n − 2)e + (5 − 5n + n2)e2 4 + 6(n − 2)e + (2n2_{− 9(n − 1))e}2 Ce,n = 1 (1 + (n − 1)e)(1 − e) e(1 + (n − 2)e)2 4 + 6(n − 2)e + (2n2_{− 9(n − 1))e}2

Now we must maximize the profits of each upstream firm. They independently maximize (pi− c)Qi(p1, ..., pn), where we can now use Qi(p1, ..., pn) from equation 2.1.2. This is easy,

however, as in a Bertrand competition all firms charge their marginal costs for their products, i.e. pi = ci. Combining these results for upstream and downstream firms results in the following

proposition:

Proposition 1. The competitive equilibrium with both downstream and upstream firms com-peting in Bertrand competition, is given by

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Figure 2.2: The market with collusion between the upstream firms. Figure from [10], page 16 pc = c Pc = 1 − e + (1 + (n − 2)e)c (n − 3)e + 2 Qc = (1 + (n − 2)e)(1 − c) ((n − 3)e + 2)(1 + (n − 1)e) qc = nQc πc = 0 Πc = (1 − e)(1 + (n − 2)e)(1 − c) 2 ((n − 1)e + 1)((n − 3)e + 2)2

The superscripts “c” refer to the fact that this is the competitive setting. We will refer to the collusive equilibrium with superscripts “a”, for anti-competitive, or “m” for monopoly.

2.2 Upstream collusion

Now, imagine that the upstream firms collude to operate as one firm. This means that they will simply maximize the total upstream profit, instead of just their own, given the actions of competitors.Mathematically, they will maximizePn

i=1(pi− c)Qi(p1, ..., pn). The situation is

illustrated in figure 2.2. Solving this new maximization problem results in a new equilibrium: Proposition 2. The monopolistic equilibrium, resulting from a horizontal cartel in the upstream market and a competitive equilibrium in the downstream market, is given by

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pm = (1 + c) 2 Pm = 3 + (n − 4)e + (1 + (n − 2)e)c 4 + 2(n − 3)e Qm = (1 + (n − 2)e)(1 − c) 2(2 + (n − 3)e)(1 + (n − 1)e) qm = nQa πm = n(1 + (n − 2)e)(1 − c) 2 4(2 + (n − 3)e)(1 + (n − 1)e) Πm = (1 − e)(1 + (n − 2)e)(1 − c) 2 4(2 + (n − 3)e)2_{(1 + (n − 1)e)}

This equilibrium can be compared to the competitive equilibrium. First, the price charged by the upstream firm, pm exceeds the price charged in the competitive equilibrium, pcfor all c < 1. Furthermore, obviously πm> πcunless c = 1, which is exactly the limting case for the upstream prices. Finally, Πm = 1₄Πc, given that Πc 6= 0, which only happens if the downstream firms produce homogeneous goods, e = 1. This means that, whatever the number of downstream firms or the degree of heterogeneity e 6= 1, they will always lose 75% of their profits because of the cartel.

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Chapter 3

Discovery

In this chapter, we discuss the details of a discovery of the cartel by its purchasers. Before adjusting the existing model, it seems natural to explain in some detail what the actual “game” it describes looks like. Once we know that, we will see its limitations, and be able to overcome some of them.

Imagine that at some point in time, in the distant past, the upstream firms have decided to come together and increase their profits by colluding. From that point on, the downstream firms have paid the monopolistic price for their input commodities, and have adjusted their demands accordingly. There are no special treatments, and since downstream demand functions are symmetric, this implies that all downstream firms pay the same price for their input goods, i.e. pi is the same for all i ∈ {1, ..., n}. Now, suppose that long after the cartel has been formed,

and by long we mean long enough for the damage claims that downstream firms can make to accumulate to their maximum, for more details see section 3.3, one of its purchasers discovers the cartel. An important aspect of this event is the timing. Because it is so important we will go through it point by point:

1. At the beginning of every period, there is a probability that one of the downstream firms discovers the cartel. It is determined beforehand what the time between two discoveries is. So in terms of probabilities, in most periods there is a zero probability of discovery, and in some the probability of discovery is 1. If there is a discovery, negotiations take place with the purchaser that just discovered the cartel. If these negotiations result in a settlement between the cartel and this purchaser, this phase is over. If there is no settlement, the purchaser goes to trial, the cartel collapses and must pay damages to all its purchasers and might have to pay a government fine as well. Afterwards, the competitive equilibrium is played in every period.

2. If no discovery is made, or a settlement has been established with a discovering purchaser, the cartel members decide for themselves which prices they will charge. Charging p = pm

means they will comply with the cartel, charging p = pm− for some infinitessimal > 0 means defection by undercutting the other cartel members. If any cartel member defects, the cartel is revealed after the market clears, because the downstream firms have noticed the decrease in price (they certainly noticed, otherwise they would not all go to the same purchaser, and they also know that price cuts should be impossible in ordinary Bertrand competition). The cartel is sued in the next period by all its purchasers, and all upstream firms charge p = pcfrom then on.

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to customers. After the market clears without defection, go back to step 1. In case of defection, see step 2.

In the model described above, it seems peculiar that the cartel members know the exact moment of a discovery, but do not decide to defect one period earlier. However, the right combination of discount factors and payment obligations can persuade them not to do so. It is crucial that only one purchaser can make a discovery at some point in time, so negotiations with several purchasers can be ruled out. This assumption seems reasonable, but is not strictly necessary. The discovery is likely made because of some mistake by a cartel member, like sending an e-mail to the wrong address, or meeting with someone unexpected in public. These things are usually noticed by one person only, but it is possible that the cartel has to deal with multi-ple purchasers at the same time. The scenario where all purchasers collude (and in doing so, negotiate) with the cartel simultaneously has been explored in [8], where the “Illinois Wall” is erected at the beginning of the game. The complications arising from a defendant, the cartel, negotiating with multiple plaintiffs simultaneously, is discussed by Daughety and Reinganum in e.g. [12, 11, 13, 14] and Che and Spier [15]. In this thesis, however, we will restrict ourselves to discoveries by one purchaser at a time, but with the threat of future discoveries as a possibility in some scenarios.

In order to clarify the timing of lawsuits and market clearances, we adopt what we call the “C/M-model”, where “C” stands for court and “M” stands for markets. In this model, each period of time is divided in what we could refer to as a morning session in which the court is open and trials can take place, and an afternoon session in which markets are open. This means that, in the timing described above, if the cartel is discovered in some way but not because of a defecting cartel member, this discovery occurs in the morning, before the markets open. The purchaser discovering the cartel can then immediately sue the cartel if no settlement can be arranged, but it is important that everything such a trial is dealing with happened yesterday or further in the past. On the other hand, if a cartel member decides to defect, he will do so in the afternoon. We assume that such a defection leads to a trial at the first opportunity, i.e. the next morning. We argue that the sudden, slight change in price, perhaps even infinitessimal, causes all downstream firms to buy goods from that purchaser, because we assumed that there is Bertrand competition between upstream firms on homogeneous goods. This implies that, however small the decrease in price was, all purchasers have noticed it, and could reason that such a price decrease and consequential overtaking of the whole market should have generated negative profits if there were true Bertrand competition, where p = pc = c. Therefore, they reason that the original price was too high, and have reason to have some suspicion at the very least, suing the upstream firms in the next period.1 The mathematical implications of this timing will become clear later on. This important division of each time interval into two periods is illustrated in figure 3.1.2

1

One could also argue that the fact that one upstream firm takes the whole market, ignoring the restrictions on output and prices the cartel members have agreed upon, is only noticed that period by the other members of the cartel. If no purchaser has discovered the cartel until then, and the decrease in price was very small, it could be that nothing happens the next morning, but the next afternoon all cartel members charge p = pc= c the next afternoon. This serious decrease in prices is certainly noticed by the downstream firms, who sue the cartel the morning after, meaning this is actually two periods later. Both scenarios seem reasonable, but we opted for the one presented above, because it carries over more intuitively to the scenario where at least one of the purchasers has already discovered the cartel, and will then use the evidence he has about the collusion of the upstream firms as soon as he can in court.

2

Changing the order of C and M would change the model in some ways. It would imply that defection by a cartel member results in a trial on the same day, meaning cash flows are less heavily discounted, and also that damage claims are less depreciated (for details on depreciation, see section 3.3), since the last damage has

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Figure 3.1: The division of a single time interval into two periods.

Upon the discovery of the cartel, the purchaser in question can choose between some options. First, he could immediately sue the cartel. The advantage is that the outcome is relatively pre-dictable, and in our calculations we assume that “justice prevails”, so we do not allow the cartel to win such a trial. Another option is to use the leverage of a potential trial to try and make some deal with the cartel about sharing the profits. Technically, this could make the purchaser part of the cartel, since it is part of activities that also harm his competitors, but we assume that it is possible to come up with a legal construction that allows the cartel to give some of its profits to this purchaser. This arrangement compensates for the damages the downstream firm suffered and might even be more profitable in the long run than going to court right now. Finally, the purchaser could share the information with other victims of the cartel, and start or threaten to start a class action lawsuit, in which all purchasers that suffered damages sue the cartel at the same time. This last option, however, is never more profitable than the first two. The reason is that, in order to make a deal with the cartel and all, or at least several, of its purchasers, all those purchasers must be better of than they would be if they went to trial. But if one purchaser sues the cartel, everybody will join this lawsuit anyway, because trials take place in public, so nothing will change there. However, in order to prevent that court case, the cartel must present an offer that makes all the purchasers it is negotiating with better off than in the case of a trial, and this is obviously easier if only one purchaser is negotiating than when all purchasers are present. So the threat to the cartel is always a class action lawsuit, but preventing this lawsuit is easier if the cartel is only dealing with one purchaser, making this the better option for that single purchaser.

This model is easy to implement, and makes intuitive sense. Once we have defined this frame-work of discoveries, we can frame-work out in more detail what these damage claims would actually be, construct a rigorous mathematical framework to order the possible outcomes of negotia-tions based on the profits of upstream and downstream firms, and investigate what happens for several combinations of parameters. This is what Augustinus did in his thesis. However, there are some clear shortcomings to his model. The most obvious of these are the fact that he only

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used a model with two purchasers, and the fact that there is a fixed interval between discov-eries of the cartel. This latter assumption greatly simplifies the calculations, but is of course very unrealistic. The extension that we will make to that model is to allow for a stochastic discovery pattern, in which the only thing known to cartel members and purchasers who have discovered the cartel is the probability distribution the discoveries follow. This substantially complicates the analysis, and we will have to adjust some of the fundamental properties of the model. Furthermore, to present a detailed analysis of the case with two purchasers, we must first develop the scenario for one purchaser, and then extend that model, in order to deal with all the complications that come with the stochastic discovery pattern. In the next section, we will describe the implications this will have for basics of the model, adjust or augment the timing described above where appropriate, and also introduce new strategies for the cartel members. After we have set up this new, improved model, we will define the damage claims and settlement outcomes. Later, when we will investigate a number of case studies, we will also describe models with different numbers of purchasers.

3.1 Uncertainty

In the model proposed by Augustinus, the cartel is formed at some point, exists for quite some time, and is then discovered by its two purchasers with intervals between discoveries that are known and fixed. This means that there is no uncertainty, and because the cartel exists for a long time, charging the full monopoly price seems to make sense. The main extension that we are going to propose compared to that model is uncertainty about the moment of discovery. This has two implications. First, the cartel is formed at t = 0 and has a nonzero probability of being discovered almost immediately, i.e. its lifetime can be very short. Second, the exact lifetime can be any integer value, making it impossible to predict exactly what is going to happen once the cartel starts operating. The timing procedure described in the previous section must therefore be slightly altered, and now goes as follows:

1. At the beginning of every interval, after the cartel members decide on their pricing strat-egy, there is a probability p ∈ (0, 1) that the cartel is discovered by one (and only one) of its purchasers.3 Note that we use p for both the upstream price and the probability of discovery, but it will be clear from the context which of the two we are discussing. Nego-tiations with that purchaser start. If these fail, the cartel is revealed, fines and damages must be paid immediately, and competitive profits are earned from that moment onwards. This also means that settlements that were made with purchasers that discoverd the car-tel earlier are now obsolete. If these negotiations succeed, there is again a probability p ∈ (0, 1) that the cartel is discovered by another random purchaser (granted there are still purchasers left that are unaware of the existence of the cartel), and the negotiations with that purchaser start. Again, if they fail, the cartel is revealed, fines and damages are paid and competitive prices earned etcetera. If no discovery is made, this step is over. 2. If no further negotiations take place, but the cartel still exists, the cartel members decide

for themselves what prices they will charge. Charging p = pameans they will comply with the cartel, charging p = pa− for some infinitessimal > 0 means defection by

undercut-3_{For simplicity, we have chosen a constant p, that does not depend on the number of purchasers. An}

inter-pretation of such a scenario would be that a discovery takes place when the cartel makes an error in the form of an incorrectly addressed e-mail, i.e. a passive discovery by a purchaser. If purchaser actively investigate the existence of an upstream cartel, the probability would increase with the number of purchasers that are unaware of the existence of the cartel, but on their guard.

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Figure 3.2: Diagram explaining the timing of choices about prices, discovery and clearing of the market.

ting the other cartel members.4 If any cartel member defects, the cartel is revealed after the market clears. This means that all members will be sued by the downstream firms as soon as possible, and damage claims and government fines must be paid the next period (remember the C/M model).

3. The market opens, the downstream firms buy from the cartel and sell their products to costumers. After the market clears without defection, go back to step 1. In case of defection, see step 2.

The timing of the model is visualized in figure 3.2. In terms of the C/M-model, the morning session takes place up to and including the third node on the horizontal line, i.e. the loop where discoveries and negotiations take place. The last two nodes on the horizontal line and the bottom right portion where competitive prices are charged forever, all take place in the afternoon of this time interval. The order of events is basically the same as before, but there is uncertainty about the discovery. Now, this uncertainty might of course have an impact on the strategies that cartel members, and also their purchasers, adopt. We introduce a so-called “caution parameter” γ. As we will see later on, the consequences of a discovery depend partly on the harm the cartel caused its purchasers. If the cartel thinks it might be discovered shortly after its formation, it could therefore be wise not to charge the full monopoly price, but something between the ordinary competitive price pc = c and the monopoly price pm = 1+c₂ . At t = 0, the cartel can be allowed to choose a value γ ∈ [0, 1] to select any convex combination between these two. The new anti-competitive price then becomes

pa= γ(1 + c)

2 + (1 − γ)c = c + γ (p

m_{− c)} _(3.1.1)

The cartel could make higher periodic profits by charging any price that lies between this value and the monopoly price. However, if we look at the expected total discounted profit, πtot = P∞_t=0δtπ(t), the fact that the cartel will lose more money upon a discovery when it

charges higher prices might persuade them to be cautious. In some scenarios we do not allow the cartel to choose any value, but merely choose between γ = 0, so no cartel, or γ = 1, meaning an “ordinary” cartel. In other scenarios, it has the opportunity to choose this value once, at

4

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t = 0, and from that moment on it will remain the same until the cartel falls.

This new anti-competitive upstream price will of course influence the other parts of the model too. We will use Lemma 2 to rederive the anti-competitive model, because for the downstream firms it does not matter exactly how their input prices came to be, they simply have to deal with them. So, by substituting pa in formula 2.1.1, we find the following proposition:

Proposition 3. The anti-competitive equilibrium, where the upstream firms commit to a caution parameter γ ∈ [0, 1] and the downstream firms maximize their profits independently, is given by

pa = γ1 + c 2 + (1 − γ)c Pa = 2 − c(γ − 2)(e(n − 2) + 1) + γ + e(γ(n − 2) − 2) 2e(n − 3) + 4 Qa = (c − 1)(1 + (n − 2)e)(γ − 2) 2(2 + e(n − 3))(1 + (n − 1)e) qa = nQa πa = nγ(1 + (n − 2)e)(2 − γ)(1 − c) 2 4(2 + (n − 3)e)(1 + (n − 1)e) Πa = (1 + (n − 2)e)(γ − 2) 2_{(1 − c)}2_{(1 − e)} 4(2 + (n − 3)e)2_{(1 + (n − 1)e)}

It can easily be checked that if we put γ = 1, we get the previous formulas for the monopolistic equilibrium, as we should, of course. These are the formulas we will use for the remainder of this thesis when we refer to anti-competitive prices, demand and profit. We will regularly compare these to the competitive quantities, and not use the “monopoly” quantities from now on.

3.2 Geometric distribution and expected cartel profits

In this section, we briefly discuss the probability distribution we use for discoveries, the geo-metric distribution, and the expected cartel profits.

The geometric distribution is defined as follows: assume we perform independent Bernoulli experiments, with the probability of success equal to p. Then the probability that the first t experiments fail, and the (t + 1)-th experiments is succesful, follows a geometric distribution, and can be calculated as

P (T = t) = (1 − p)tp

In terms of cartel discoveries, this means that the probability of discovery is p in every period, and the time of discovery follows a geometric distribution. This seems like a reasonable and tractable distribution, and as long as we do not know whether it becomes easier or more difficult to discover the cartel for outsiders, it seems to make sense. Using the probability density and the profit function, we can derive the expected profit function:

E(πa(γ)) =

∞

X

t=0

P (T = t)πa(t, γ)

This is a complex function, mainly because for each γ ∈ [0, 1] we need to calculate for every value of T whether a settlement exists and if so, what its value is. Therefore, this sum will be

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computed numerically.5

Here we have simply chosen a probability distribution without a clear motivation, but when the formalism has been set, the exact probability distribution is merely a matter of choice that can be adjusted at will in the calculations. The reason why we chose the geometric distribution is that 1) it seems reasonable that the probability of discovery is the same in every period, because a discovery is probably made due to some mistake of a cartel member, like sending an e-mail to the wrong person, and 2) this constant probability means that it is a memoryless distribution, meaning that arguments like “the cartel has existed for some time now and is bound to fail soon” are not relevant. Also, computations sometimes become easier when using memoryless distributions.

The strategy used by the cartel is described by the value of γ, so the expected profit of the cartel, in the easiest scenario with just one purchaser depends on the value of that parameter as follows: πtot(T, γ) = (_P_T t=0δtπa(γ) + P∞ t=T +1δt(πa(γ) − Φ1(T, γ)) if a settlement is possible PT t=0δtπa(γ) − δT(D(T, γ) + F ) + P∞

t=T +1δtπc if a settlement is not possible

(3.2.1) γ is chosen by the cartel, and the moment of discovery T is unknown, but is assumed to fol-low the aforementioned geometric distribution, with respect to which we can take expectations. D(T ) denotes a damage claim that must be paid to the purchaser in case of trial, see section 3.3, and F is a government fine. Φ1 denotes a periodic payment that is to be made to the purchaser,

and will be explained in section 3.4. For multiple puchasers, this expectation obviously becomes more complicated. It then also contains the option that all purchasers except the last will settle, or all except the last two, etcetera. This extension is tedious, but in principle straightforward, and we will explain it later, when it becomes relevant.

3.3 Damage claims and harm

Now that we have both the expressions for the competitive and the collusive scenario, we can find expressions for the harm the downstream firms experience. A distinction must be made here between “harm” and ”damages”. By harm we mean the actual negative economic effects a downstream firm experiences as a consequence of the upstream collusion. Quantitatively, this is the difference in profits that firm would make in the competitive and the anti-competitive scenarios. This difference can be written as

5

What we will do for variable γ is perform a grid search. We know that for γ = 0 the expected profits are exactly zero, because you simply earn the competitive profit forever. We then calculate the expected profit for values of γ ∈ [0, 1] with a spacing of ∆γ = 0.01. Then, from all these profits, the highest one is of course the desired one, and from this we get the initial value of γ than will be chosen. The gap between the values of γ is chosen to find a good balance between computation time and accuracy. Also, replacing the upper limit of the summation, infinity, by 10

p reduces calculation times drastically, without sacrificing a serious amount of accuracy.

We can deduce that the probability of N >10 p is P (N > 10 p) = 1 − 10/p X t=0 (1 − p)tp = 1 − p1 − (1 − p) 10/p+1 1 − (1 − p) = (1 − p) 10/p+1

For example, if p = 0.1, we find that P (N > 100) = 0.0000239053, and for p = 0.00625, P (N > 1600) = 0.0000437224, so we are quite safe in ignoring the rest of the distribution’s tail.

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Ωγ = (Pc− pc) Qc− (Pa− pa) Qa (3.3.1)

Expression (3.3.1) can be decomposed intuitively into three seperate economic effects: the overcharge, the pass-on and the output effects. The total periodic harm is then expressed as

Ωt= ξt− ωt+ σt (3.3.2)

where the separate terms are given by

ξt = (pa− pc)Qa

ωt = (Pa− Pc)Qa

σt = (Pc− pc)(Qc− Qa)

In words, the overcharge effect ξt measures to what extent a downstream firm paid too much

for the goods it has actually purchased. This is the easiest of the three to compute, because two of the ingredients, pa and Qa, are well known, and only one of them, pc, needs to be estimated. In a Bertrand market, this is also fairly easy to find, since it should equal the marginal costs of production. The pass-on effect ωt indicates how much of this overcharge the downstream firm

was able to pass on to its own customers by raising its own prices. This is therefore a negative contribution to the total harm. Here, we are again sure about two factors determining this effect, but the third, Pc, is complicated. The reason is that it is supposed to follow from the pc and the input prices other firms face, and in practice this becomes already quite involved to compute. Finally, the output effect σt is the effect of the decrease in downstream sales due

to the fact that the downstream firm had to raise its price. Because it is difficult to estimate especially Qc_{, the amount a downstream firm would have sold if the cartel had never existed,}

this is very difficult to find.

The measure that is typically used for damage assessment is the overcharge ξt, and we will do

so as well. This means that, whereas the actual harm a downstream firm experiences in one period is given by expression (3.3.1), what we refer to as the damage suffered in a single period is simply Dt = ξt. Other factors in determining the damage a company can actually claim in

court are

• A depreciation factor β, indicating the factor by which the value of the damage or the evidence decreases with the passing of time. This factor is often equal to or very close to 1.

• A factor µ, by which suffered damages are to be multiplied to get the amount the cartel owes its victim. In the United States, this is equal to 3 under the Clayton Act [2]. It is equal to 1 in the European Union [4].

• A statute of limitations, Ts_{, which limits the number of periods over which damages can}

be claimed. This limitation is usually placed to keep things relatively simple. For example, it would not serve society if people were able to place damage claims on disputes that occured 100 years ago.

The most general expression we can therefore write down for the damage claim that can be made by a single downstream firm (so this is NOT the actual harm it experienced from collusion of the upstream firms) at time t, if the cartel started operating at t = 0 is

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D(t) = t−1 X k=τ βt−kµξk (3.3.3) where τ = max{0, t − Ts} = ( 0 if t < Ts t − Ts if t ≥ Ts

This expression for τ ensures that the cartel cannot be sued for an indefinite time interval. Note that this expression for the damages again takes into account that the damages are claimed up until yesterday, which means that even the most recent damage gets depreciated by a factor of β. We mentioned earlier that the timing of a trial and activity on the markets was important, and this is one of the reasons: if trials take place after the markets have closed, but on the same day, the sum in expression (3.3.3) would run until k = T , and τ would be max{0, t + 1 − Ts}.

When we will deal with more than one purchaser, we will need to make some small adjust-ment to this expression, to account for the consequences of settleadjust-ments that exceed the periodic overcharge of downstream firms, resulting in the possibility of negative damage claims. In that case, the cartel would be able to sue its own customers, which seems unnatural. The subscript k in the overcharge term is important for two reasons: first, it denotes the fact that the pur-chasers are overcharged in every period. Second, the amount by which they are overcharged is not constant. As soon as a purchaser discovers the cartel and makes a deal, he is effectively overcharged a lower amount, ξ = ξ0− Φi, where ξ0 is the “original” overcharge. This variable

behaviour is the main reason why we abstain from making the expression of the damages more explicit here.6

3.4 Settlements

As the last building block for our model, we must define what settlements between the cartel and a purchaser look like. There are two kinds of settlements: renegotiable and non-renegotiable settlements. The former, as one can guess from the name, is a settlement that may be adjusted in the future due to changing circumstances. In our case, such a change would be a discovery by another purchaser, necessitating the cartel to reconsider its previous settlements. We will focus on the latter form, the non-renegotiable settlements. As the name suggests, this is the kind of settlement you cannot get out of legally. There could be all sorts of legal issues arising from abandoning the construction. Whatever these are, we will simply assume that the only way for a cartel member to get out of a settlement is to defect from the cartel, thereby causing it to collapse, and we assume that purchasers will turn down a settlement if they’d rather go to court. In both cases, the resulting trial automatically includes all purchasers, and immediately ends settlements that were reached with purchasers that discovered the cartel at some earlier time.

6

If the overcharge is constant over the last Tsperiods, an assumption Augustinus makes by taking the interval between discoveries larger than Ts_{, then we can write an easy, explicit formula for the damages. If the effective}

overcharge changed a short while ago, however, we cannot do that, and because we will eventually try to make our models as general as possible, we will leave it as it is. Finally, it is worth stressing again the importance of timing. The damage claim at time t stands for the claim a purchaser is entitled to in a trial. Because of the C/M-model, this is always up until the previous period. So, for example, D(3) stands for the damage claim a purchaser can make at t = 3, and at that moment he can claim damages over t = 0, 1, 2.

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The first question we need to ask is what a settlement would actually consist of. When it comes down to money, the cartel can choose from lump-sum payments and periodic payments, or a combination of both. Let us say we have a settlement consisting of a lump-sum payment Ψ at the time of discovery, and subsequent discounts ψ, so that the new price paid becomes p = (1 − ψ)pa, expressed as a pair S = (Ψ, ψ). There are three ways to incorporate this discount in a scenario with multiple purchasers:

• Unrestricted: the cartel and all its purchasers reoptimize quantities and prices (for the cartel) after the discount has been given. This means purchasers get a new input price p0a, and order new quantities Q0a.

• Price Freeze: the cartel keeps it catalogue price at the old pa_{, but lets its purchasers}

reoptimize their quantities. This will lead to an increase in the quantity ordered by the purchaser that discovered the cartel, since it pays a lower price, and lower quantities for the other purchasers, since the purchasers with a discount steals some of their customers. • Rationing: the cartel keeps its catalogue price at the old pa_{, but also restricts the quantities}

sold to the old Qa_{. This has the downside of selling less units (the purchasers with discount}

will want to purchase more), but the benefit of raising as little suspicion as possible, because the purchasers that are unaware of the cartel notice no difference at all.

Augustinus has investigated for which parameter values the rationing scheme is feasible in terms of profits, and finds that for quite a wide range of parameters, it is. Because of its virtue of raising the least possible suspicion, we will adopt it from now on. Now, to raise even less suspicion, we would like to transform the lump-sum payment Ψ into a fixed payment every period. We write the after-settlement profit of the purchaser that has settled with the cartel as

Πs_t = [Pa− (1 − ψ)pa] Qa = [Pa− pa] Qa+ ψpaQa = Πa+ η

Also, we can write the lump-sum payment Ψ as the present value of an finite or infinite series of future payments Ψ = ∞ X t=T +1 δt−Tζ = Tend X t=T +1 δt−Tθ = δ1 − δ Tend−T 1 − δ θ

where the last equality is an application of the general formula for the sum of a geometric series,

b X k=a ck−(a−1)= b−a X k=0 ck+a−a+1= c b−a X k=0 ck= c1 − c b−a+1 1 − c

which gives the desired expression if we use a = T + 1, b = Tend and c = δ. We will choose

the infinite sum, since using a finite cash flow might give the purchaser the incentive to sue the cartel as soon as the periodic payments stop. So we amortize the lump-sum payment as an infinite stream of payments. Using ζ to indicate these periodic payments, we can write the settlement as a periodic payment

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where the subscript t denotes the fact that this payment is due every period. This construction is often used in all sorts of industries in the form of cashback arrangements, where purchasers get part of their money back when they meet certain sales targets or other requirements. So in principal, there is nothing illegal or suspicious about this deal, no illegal money changes hands and the other purchasers of the cartel will in principle never notice anything due to the rationing scheme. So, we have changed the combination of a lump-sum payment and a discount to a fixed cashback arrangement, and the question of whether a settlement can or cannot be designed depends solely on whether both the cartel and the purchaser can be satisfied by a value of Φt. From now on, because we will invariably deal with periodic payments, we will omit

the subscript t again, and simply refer to the settlement with purchaser i as Φi.

Now, the last thing to do is find out how the cartel and a purchaser would actually decide on a value of this cashback. Nash proposed an answer to this question in [16], in the form of a Nash bargaining game. In a Nash bargain, the players aim to jointly maximize the so-called Nash bargaining score function. This score function consists of the product of two factors, both of which consist of the difference between the values of some “agreement-outcome” and some “disagreement-outcome”. In this case, agreement means that there is going to be some periodic payment from the cartel to its purchaser, and disagreement means there will be a trial right now. The score function is given by

S(Φi) = V_i,Ts_i− V_i,Tl i α V_K,Ts _i − V_K,Tl i 1−α (3.4.1) Here Φistands for the periodic cashback, and will from now on be referred to as “the settlement

value”. V_i,Ts

i is the present value of the settlement for purchaser i if he settles for that specific

value at time t = Ti. V_i,Tl _i represents the present value of a trial for that purchaser. V_K,Ts _i and

V_K,Tl

i stand for these same present values, but then for the cartel. α ∈ [0, 1] is called the

bar-gaining power of the purchaser. α = 1 would mean that we simply maximize the payoffs for the purchaser, and do not worry about the cartel at all, whereas α = 0 implies the opposite, ignoring the interests of the purchaser and only maximizing the cartel payoffs. Given that all parameters are known to both parties, we assume they will jointly maximize this function. In doing so, there are of course constraints, which we will go into in much more detail in subsequent chapters. Now that the model and the context are clear, we are almost ready to deal with concrete examples. What we will do is examine a series of case-studies. We will start out with what seems to be an almost trivial case, namely a cartel that has only one purchaser. This will already turn out to be quite complicated when the time of discovery becomes stochastic, but almost all computations can still be performed analytically. When this basic model is worked out, we will move to two purchasers in two models: one where we do not concern ourselves with the possibility of defection from the cartel, and one where we do. In both models, we make γ binomial, meaning it can only take on values γ ∈ {0, 1}. The difficulty with two purchasers is that the outcome of the first settlement has to be calculated numerically, making this a time consuming effort. Finally, we will extend the model to three purchasers. Including the third purchaser means the computational effort increases dramatically, so this model will only be worked out for a small number of parameter combinations, and we also simplify it slightly by removing some defection possibilities again. This model is therefore to be viewed as a first exploration and comparison to the case of two purchasers, rather than a thorough investigation. Finally, in order to find some implications of antitrust policies, we investigate some scenarios with one and two purchasers for other values of (F, µ), the parameters that are connected to the damage claims and the government fine.

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Chapter 4

Case-study: One Purchaser

Now that the basics are laid out, we can turn to our first working model, namely one with one purchaser. We will do so by first deriving analytic espressions for the restrictions on the value of the settlement with the sole purchaser. After that, we can examine under what circumstances this settlement will exist, and if so, what its value will be. Then, we can investigate the optimal strategy for the cartel, looking for the value of γ that maximizes their expected profits.

4.1 Limits on Φ1

There are four values we can construct that give us more insight on the value of Φ1, and therefore

on the existence and hypothetical value of a settlement. These are:

• ¯Φ1: This is the maximum value of the settlement so that the cartel will remain stable in

the future. There must be such a limit, because the damage claims will generally become less after the settlement (due to the lower overcharge), and at some point reach a constant value. If at that point, it is still preferable for cartel members to honor their agreements, it will remain so indefinitely. If they prefer a lawsuit at that moment and deviate from the cartel strategy, backward induction implies that the settlement will never be made in the first place.

• Φ1,U: This is the upper bound above which a cartel member would rather go to court

than pay the periodic cashback. Augustinus argues that this upper limit is always higher than the previous one, because deviating in the future, after the damages have reduced somewhat due to the cashbacks, is always more profitable. That means that the amount a cartel member is willing to pay right now is always higher than in the future. In the more general case, however, it could very well be that the damages will rise at first, because the cartel was discovered before t = Ts.

• Φ_1,L: This is the lower limit below which it is more profitable for the purchaser to file a lawsuit.

• ˜Φ1: This is the optimal settlement value, following from the unrestricted maximization of

the Nash score function.

4.1.1 Cartel stability limit ¯Φ1

To find this “stability limit”, we must find the value of defecting or complying with the cartel, at the moment when the damages become constant at a lower value than before, because of the decreased overcharge. These values are:

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V_tC = 1 m ∞ X k=0 δk(πa− Φ₁) (4.1.1) V_tD = πD−Φ1 m − δ m(D(t + 1) + F ) + 1 m ∞ X k=1 δkπc (4.1.2)

The first expression is simply the discounted sum of the infinite stream of cartel profits, minus the periodic payments. The second expression contains first πD_{, the profit obtained when}

some-one decides to deviate from the cartel. Usually, we can set πD = πa, since a company could theoretically undercut the cartel price by an infinitessimal amount, take the whole market and obtain all the profits. From this profit we must still subtract the contribution of that cartel member to the cashback of the purchaser, Φ1

m. We could choose not to do so, but it is possible

that there is some legal contract forcing cartel members to do so. And if we did not include this term here, the damage term would have to be modified, because the damages the purchaser can claim are then slightly higher. After the defection, the cartel collapses, a damage claim D(t + 1), depending on the time t, and a fixed government fine F must be paid the next morning. We assume that the damages and fine are still distributed uniformly among the cartel members.1 These payments are therefore discounted. From the next period on, competitive profits will be earned.2

We are concerned with stability of the cartel at the moment the damage claims have reached a constant value that is lower than the value of the damage claims at the moment of discovery. Under no circumstances will the damages assume a lowest value at T1 < t < T1+ Ts, where T1

is the moment of the first and only discovery, making that moment in time the most preferrable to defect. The reason is that for this to happen, the damage claim must first decrease and then increase. But a decrease in the damage claim is caused by any settlement value if T1 ≥ Ts or a

settlement value that is greater than the overcharge, Φ1 > ξ0, if T1< Ts. But that would mean

that the damage claim is non-increasing until t = T1+ Ts in both cases, and strictly decreasing

as long as D(t) > 0. So we can safely ignore that situation. The constant value the damage claim assumes is D(t) = Ts₋₁ X k=0 βTs−kµ(ξ0− Φ1) = Dmax− Ts−1 X k=0 βTs−kµΦ1 = Dmax− φΦ1 (4.1.3) where

1_{As J. Tuinstra pointed out, we could refine the model by letting the defecting cartel member pay the full}

damages associated with the last period before the trial, because he obviously caused all damage in that period. We have not done so here, but it would make the arguments in this thesis possibly even stronger, as it makes defection less preferable and thus the cartel more stable.

2

Note that the C/M model is important here. The consequences of a trial are postponed one period and therefore discounted, and the damages caused today will carry a factor β in the final damage claim. If we had used an M/C model, where a trial takes place in the same period, we would miss a factor of β and a factor of δ. This would result in a lower value for VD

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φ = Ts₋₁ X k=0 βTs−kµ (4.1.4) and Dmax= Ts₋₁ X k=0 βTs−kµξ0= β − βTs+1 1 − β µξ0 (4.1.5)

and ξ0 indicates the overcharge before the discovery, i.e. ξ0 = (pa− pc)Qa. For the cartel to be

stable, we need a Φ1, such that

V_tD ≤ VC t

which results in the following proposition:

Proposition 4: The upper bound on Φ1, deriving from cartel stability, is

Φ1 ≤ ¯Φ1 = 1 δ + δ(1 − δ)φπ a_{− δπ}c_{+ (1 − δ) δD} max+ δF − mπD (4.1.6) It is interesting to see in what ways this upper bound is influenced by some of the model pa-rameters. Most dependencies can be found fairly easily by taking derivatives, which results in the following proposition:

Proposition 5: The upper bound on Φ1, deriving from cartel stability, ¯Φ1, depends on model

parameters as follows:

• ¯Φ1 increases with increasing government fine F.

• ¯Φ1 decreases with the number of cartel members m.

• ¯Φ1 increases with the damage multiplicative factor µ.

• ¯Φ1 increases with the depreciation factor of the damages β.

The dependence on m is intuitive, since the more cartel members there are, the less surplus profit there is for each, so defection becomes more attractive. The other three dependencies reflect the fact that a trial becomes less attractive for the cartel if F , µ or β increases, meaning defection becomes less attractive. It is difficult to make statements about the dependence of this upper limit on δ, because it depends on the discount factor in a complicated way. However, we can look at the extreme values δ = 0 (only present day cash flows matter) and δ = 1 (all future cash flows are weighed equally). As δ → 0, we see that ¯Φ1 → −∞ as long as m ≥ 2,

meaning the cartel will not settle under any circumstances. This might seem odd, but this limit comes from stability considerations, and δ = 0 means that all cartel members will defect from the cartel on the first opportunity, because they do not care about the damages and fine they must pay in the next period. This implies that collapse is inevitable in all cases, and settling does not change this. For δ → 1, we get ¯Φ1 → πa− πc, i.e. the cartel members can give up

exactly their surplus in profits, and nothing more. This makes sense, because if they would have a periodic loss, no matter how small, the infinite sum of these losses would have a present value of −∞, which is obviously not preferred. Also, if they must pay an arbitrary amount of money at the moment of discovery, this amount can be distributed over all future periods, making the effective periodic loss zero, so this is no longer a threat to the cartel.

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4.1.2 Cartel present value limit Φ1,U

We will now look at the present value of a settlement or a lawsuit at the moment of discovery, from the perspective of the cartel (or, equivalently, from the perspective of a single cartel member. In that case, we must divide all expressions by the number of cartel members m, but this will of course give the same results).

V_ts = ∞ X k=0 δk(πa− Φ1) (4.1.7) V_tl = − (D(t) + F ) + ∞ X k=0 δkπc (4.1.8)

Since we are dealing with only one purchaser, we can even write an explicit form of the damage claim here, in order to find the dependence on all model parameters more easily.

D(t) = t−1 X k=τ βt−kµξk= β − β1+t−τ 1 − β µξ0

where, again, τ = max{0, t − Ts} and ξ₀ is the full overcharge, before the settlement. Using this expression, we can state:

Proposition 6: The upper bound Φ1,U, deriving from the present values of a trial and a

settlement for the cartel is

Φ1 ≤ Φ1,U = (πa− πc) + (1 − δ)F +

1 − δ

1 − β β − β

1+t−τ_µξ

0 (4.1.9)

From this expression, we can again derive the dependence on several model parameters: Proposition 7: The upper bound on Φ1, deriving from the present values of a trial and a

settlement for the cartel depends on model parameters as follows: • Φ_1,U increases with increasing government fine F.

• Φ1,U increases with the damage multiplicative factor µ.

• Φ1,U increases with the depreciation factor of the damages β.

• Φ1,U decreases with the discount factor δ.

The first three dependencies in proposition 7 have the same reason as in the case of ¯Φ1. The

last one, with δ, can be explained from the fact that for higher values of δ, lower profits or even losses in the future have more weight in present day considerations. This makes a settlement less attractive, and therefore lowers this upper bound on Φ1. Looking at extreme cases again,

δ → 0 gives Φ1,U → πa− πc+ F + β−β 1+t−τ

1−β µξ0. δ = 0 corresponds to only counting profits

and losses of the current period, and we see that that this gives an upper limit where the cartel would give up their surplus in periodic profits, πa_{− π}c_{, and also the extra costs of a trial, the}

fine and damage claim. For δ → 1, we get the same limit as for ¯Φ1, namely Φ1,U → πa− πc, for

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