Cartel sustainability and the overcharge as a measure of harm

(1)

Cartel Sustainability and the Overcharge as a

Measure of Harm

Ruben Bieze

Abstract

Antitrust laws enable cartel victims to retrieve their losses in court; however these damages that the court awards them are generally based on the direct purchaser overcharge. In this paper I argue that the quality of this measure of cartel harm turns out to deteriorate if only sustainable cartels are evaluated. Moreover, the damages can often only be claimed by the direct purchasers; further downstream in the production chain, other victims of cartel behaviour might not have standing to claim damages. I find that the overcharge may be over four times as big as the change of profits for the direct purchasers and that this effect increases when only sustainable cartels are taken into account.

Bachelor’s thesis BSc Econometrics

Faculty of Economics and Business University of Amsterdam

Date: June 22, 2015

Supervisor: prof. dr. Jan Tuinstra

(2)

1 Introduction

In the field of economics it can be challenging to find unanimous agreement about almost anything; strikingly, the theory that total welfare is greatest in a perfectly competitive market is such a rare thing most economists can agree on. However, perfect competition is unfortunately not very common in the real world. Among the reasons why perfect competition is so rare are the imperfect substitutability of products, the limited avail-ability of many resources and many more. Furthermore, firms may strive to reduce the amount of competition in a market in order to obtain more profit. They might do so through a legal merger of firms or by making illegal cartel agreements. In their pursuit to retain competitive markets, antitrust institutions both prevent mergers that reduce competition and punish discovered cartels.

Cartels are known to cause severe harm in their collusion aimed at reducing compe-tition. They agree to reduce production in order to raise prices and to make more profit. However, in doing so they severely harm other firms and consumers. Instead of just transferring wealth from one actor to another, anticompetitive acts reduce total social welfare. Therefore laws that forbid these collusive activities enable parties that suffer from a cartel to take the cartel to court and reclaim the damage done to them.

The Sherman Antitrust Act (1890) and the Clayton Act (1914) in the USA were the first such laws to forbid this anticompetitive behaviour; they dealt with fining the offenders and the Clayton Act enabled the directly affected parties to reclaim three times the damage they suffered from the cartel, treble damages. Much later, other Western countries followed suit, but not in all countries anticompetitive behaviour is illegal.

Generally, cartels are very hard to discover due to many fluctuations in demand, supply, interest, exchanges rates and practically every other economic variable that is relevant for the respective market; thus reduced production and the resulting higher prices can be attributed to a grand scope of external changes apart from anticompetitive agreements. Hence, cartels are discovered only after a substantial period of anticompeti-tive behaviour; as a consequence the damages have been passed on to multiple parties for quite some time and the real harm is particularly difficult to determine (Han et al. 2009).

Han et al. (2009) explain that in order to determine the true harm from a cartel the ‘but-for world’, the situation as it would have been without a cartel, needs to be calcu-lated. As more parties are involved this becomes increasingly arduous. For this reason the harm is usually approximated in a simplified way. In determining the antitrust harm courts generally take into account only the harm done to the next actor in the production

(4)

chain, termed the ‘direct purchaser overcharge’ by Han et al. (2009). This direct pur-chaser overcharge, that I use henceforth, is the amount the direct purpur-chasers overpay for the goods they purchase in the cartel situation; thus the price difference times the cartel quantity gives the total overcharge. In order to obtain this direct purchaser overcharge the ‘but-for’ price and quantity are still required, but apart from this it is obtained more easily than the actual harm.

However, as both Basso and Ross (2010) and Han et al. (2009) point out, the over-charge can be a very poor measure for the true harm that results from anticompetitive behaviour. This is due to the fact that this measure does not take into account the pass-on effect downstream from the direct purchasers, nor the lost profits and losses in consumer surplus resulting from the reduced equilibrium quantity and price, nor the effect on prices and quantities upstream from the colluding firms.

Han et al. (2009) investigate a supply chain with an arbitrary amount of layers but with perfectly substitutable products; in contrast Basso and Ross (2010) investigate a situation with a set number of layers but variable substitutability of products. Despite these differences in model specifications, they consistently find that the overcharge may be a rather poor measure of cartel harm.

Nevertheless, neither set of authors takes into account the sustainability of the cartel in their evaluation. It might follow that for situations where the overcharge is an espe-cially bad measure of the true harm, cartels are very unsustainable and therefore do not even occur in the first place. On the other hand, the case may also be the exact opposite, meaning that only where the overcharge underestimates true harm a cartel is sustainable and can exist. This distinction has serious implications on the research on the quality of the direct purchaser overcharge as a measure of the harm from cartel behaviour.

In general, cartels are sustainable if no firm within the cartel has an incentive to deviate from the cartel agreements in a repeated game. The focus here lies on covert and complete cartels. A firm can be considered incentivised to leave the cartel if leaving leads to a larger profit in the repeated game. Since the cartel agreements are illegal, no legal steps can be taken to enforce them. This leaves differences in profit as the only determinant of cartel sustainability (Tirole 1988).

The purpose of this paper is to investigate the relationship between cartel sustain-ability and the direct purchaser overcharge as a measure of true cartel harm; thus the suitability of the overcharge is being evaluated when only sustainable cartels are taken into account. Because there are different opinions about what should be defined as the

(5)

true cartel harm, these different options are being assessed. This is being done by way of a mathematical model. In the next section, the existing theory on the overcharge and cartel sustainability is discussed. Next, section 3 is directed at explaining the model used in this paper. Section 4 ensues with the results and in section 5 these are analysed; section 6 forms the conclusion.

2 Theoretical framework

This section discusses some of the main findings on the quality of the overcharge as a measure of harm and on cartel sustainability. As it is the purpose of this paper to bring these two aspects of cartel research together, it is crucial to provide a general understanding of the recent research on both topics.

It is beyond the scope of this paper to discuss in its entirety the incentives that lead to cartel behaviour, but it is necessary to understand at least the very basics. For a firm, more competition means less profit; hence, firms may strive to diminish the level of competition in order to obtain more profit. A common way of doing this is to make cartel agreements with competitors to keep prices high and/or production low, in order to make a higher total profit. The government, on the other hand, tries to prevent this behaviour by fining the offenders of antitrust laws. Firms take into account this possibility of a fine in their assessment of the profitability of potential collusion.

The way in which these factors are generally investigated is with the help of deductive models that are based on simplifications of the economic environment. These models are necessarily based on certain assumptions that are not automatically realistic. For instance the rationality of economic actors, the perfect substitutability of products and linear pricing may be questioned. However, these models can provide some insight in cartel behaviour and its consequences under certain conditions. Results are in many cases at least qualitatively applicable to real world situations.

2.1 The overcharge as a measure of cartel harm

In the evaluation of collusion cases courts in the US, and increasingly in Europe, take into account only the direct purchaser overcharge. This is the amount the supply chain layer that directly follows the cartel has overpaid due to the collusion. This does not allow for the effect on layers further downstream or upstream from the cartel and may therefore be an understatement of the total harm (Han et al. 2009). On the other hand, due to the same pass-on effect the damages paid to the direct purchasers may actually be an

(6)

overstatement of the harm that they suffered (Basso and Ross 2010). Firms in a certain layer that is not the colluding layer may pass on some of the harm they suffer from the cartel by overcharging the next layer; this is being referred to as the pass-on effect.

The quality of the overcharge as a measure of the true harm has been researched

in different model contexts. However, across different articles with different models,

consistency is found in the dismissal of the overcharge as a good measure of total harm. Most articles on cartel harm use a model with one or two layers in the supply chain next to a final layer of consumers. However, Han et al. (2009) set out to investigate the quality of the direct purchaser overcharge if firms in layer g collude in a K-layered production

chain; each layer k consists of nk firms and is homogeneous in terms of input and output.

π is the profit and CS is the consumer surplus.

∆W = g−1 X k=1 ∆πk+ ∆πg+ K X k=g+1 ∆πk+ ∆CS

In their search for the equilibria they use the following inverse demand function

P (Q) = a − bQγ

with a, b and γ constants. They determine the equilibrium quantity Qi for both the

Cournot-Nash and the cartel situations. In the Cournot-Nash situation, firms in each layer choose to produce the quantity that maximises their profits for a given demand function. If these firms collude into a cartel the quantity is determined to maximise the total cartel profit. This means in practice that the firms will produce less in order to drive up the prices.

Han et al. (2009) separate the effect of antitrust on total wealth W into the overcharge

effect ξk, the pass-on effect ωk and the output effect σk. The overcharge is the amount

that a certain layer is overcharged by the previous layer and the pass-on effect is the amount that a layer overcharges the next layer. Evidently the overcharge effect on layer k + 1 is equal to the pass-on effect of layer k. Lastly, the output effect is the loss of profit in a certain layer due to the reduced production level that is caused by the cartel. As the model assumes that the cartel remains a price-taker, the upstream effect is limited to fluctuations in the demand and can be both positive and negative. The effect on the consumer surplus on the other hand is strictly negative.

The total effect on welfare is the sum of the downstream effect, the upstream effect and the change in the cartel layer’s profits; this is equivalent to the total output effect, because the pass-on effects cancel out to the overcharge of the next layer. In this context downstream refers to all actors further down the production chain than the cartel and

(7)

upstream refers to all parties that are located higher up in the production chain than the cartel. ∆W = K X k=1 ξk− ωk+ σk+ ξC + σC ξk+1 = ωk Thus, ∆W = K X k=1 σk+ σC given that ξ1 = 0

Next, they look at the ratio of the antitrust effects to the direct purchaser overcharge in order to test the suitability of the direct purchaser overcharge as a measure of antitrust damage. Looking at the downstream effect, they find that this ratio is highly dependent on the measure of competition in both the colluding layer and the first layer downstream; it increases in the number of competitors in the former and it decreases in the number of competitors in the latter. Moreover, the ratio increases in the number of layers down-stream from the colluding one, due to the extra mark-up with each layer. Thus Han et al. (2009) show that the ratio of downstream effects to direct purchaser overcharge can vary with the market structure and that it is unbounded. However, this is the case for a situation of nonlinear demand. In case of linear demand this ratio is bounded between 1 and 3.

Han et al. (2009) find that the upstream damage increases in the number of colluding firms but because the direct purchaser overcharge does so too, the ratio remains constant. On the other hand, the ratio decreases in the number of firms in the layer upstream from the cartel. Moreover, the ratio increases in the number of upstream layers. Overall, the location of the cartel in the supply chain does not have a significant effect on the total harm as the upstream and downstream damage develop in opposite directions and the effect on the consumer surplus grows as the cartel moves toward the end of the chain (Han et al. 2009).

Han et al. (2009) conclude that in most cases the direct purchaser overcharge under-estimates the harmful effect of cartels on the total welfare. However, they do not come up with a practical measure that is both feasible and easily calculated. The overall effect of cartels on welfare proves to be too complex for this measure to be possible.

Basso and Ross (2010) too observe a trend of increasing numbers of antitrust class ac-tions, where plaintiffs seek to obtain damages from antitrust violating parties to make up for the economic harm that the collusion inflicted. However, in accordance with Han et al. (2009), they observe a disparity between the actual economic harm and the damages

(8)

awarded by the courts. They too set out to investigate the relationship between the harm done by price-fixing and the damages awarded to the victims. The authors distinguish between the direct effect and the indirect effect. Basso and Ross (2010) use the use the following inverse demand function, with a, b and e constants

pi(q) = a − bqi− e

X

j6=i

qj.

They determine the equilibrium quantity qi _{The constant e represents the measure of}

substitutability of the good.

Basso and Ross (2010) proclaim that the overcharge is a poor measure of the total harm done by the cartel for two reasons; firstly, because it does not take into account the lost consumer surplus that is caused by a reduction of the production; secondly, because it assumes that there are no layers in the supply chain apart from the cartel. As they show the harm increases with the number of layers in the supply chain between the colluding parties and the consumers. Apart from looking just at the amount of harm, Basso and Ross (2010) also research the distribution of harm among the direct and indirect purchasers. Moreover, they investigate to which extent taking into account the indirect as well as the direct harm improves the estimation of the total harm.

In looking at these effects Basso and Ross (2010) use a model of a double-layered supply chain; they include a parameter v that enables them to simulate Cournot, Bertrand or cartel situations. Instead of comparing the overcharge effect to the total harm they first compare the overcharge plus the associated loss of surplus with the total harm; their motivation in doing so is that the lost consumer surplus is generally considered part of the harm, although it is not always included in the calculations. Therefore, they compare the total actual harm to what is commonly perceived as harm from price-fixing.

Basso and Ross (2010), like Han et al. (2009), find that the harm done to direct purchasers gets smaller and the error decreases as this layer becomes more competitive. This implies that the share of the harm that befalls the direct purchasers decreases too. In comparing only the overcharge to the total harm they find that the loss of surplus increases with the cartel price, which is hardly a surprise; moreover, this loss depends on the elasticity of the product for both the downstream firms and the consumers. The ac-cepted measure of harm, the sum of the direct overcharge and the lost consumer surplus, always underestimates the total harm from price-fixing. However it can both underesti-mate and overestiunderesti-mate the harm inflicted on the direct purchasers; the ratio increases as the layer of direct purchasers becomes less competitive (Basso and Ross 2010).

(9)

Furthermore, Verboven and Van Dijk (2009) research the pass-on defence in case a cartel is discovered. This is a claim for damages by firms or consumers further downstream than the direct purchasers. They argue that direct purchasers have no right to claim the entire overcharge due to the pass-on effect. The direct purchasers allegedly pass on part of the overcharge to the next production layer or to the consumers. According to the current laws pass-on damages are not awarded. Interestingly, discovered cartels actually use the pass-on effect in order to lower the damages, as the overcharge is partially passed on by the direct purchasers. Although this may be true, they argue that it is hardly a good argument for a cartel on the defence to make, because the true harm exceeds the direct purchaser overcharge.

Verboven and Van Dijk (2009), in line with Basso and Ross (2010), find that the overcharge is an overstatement of harm done to the direct purchasers if these function in a highly competitive market; in these situations, the pass-on defence can make a good case for a fairer distribution of the damage payments.

Similarly, Boone and M¨uller (2008) argue in favour of a better way to calculate

dam-ages than the current overcharge measure. They show with their calculation of the

consumer harm share that it is not as insurmountable as it is thought to be to come up with a better measure for cartel harm. Moreover, they reason that the current system is unfair and that it puts direct purchasers in a too powerful position.

Most models that assess the overcharge are three layered, only Han et al. (2009) use a model with a variable amount of layers. However, the results are very similar throughout the economic literature; the direct purchaser overcharge is not a sufficiently good measure for total cartel harm. The next subsection deals with the economic literature on cartel sustainability and thereafter the article discusses what happens to the suitability of the overcharge as a measure of cartel harm if this cartel sustainability is taken into account.

2.2 Cartel sustainability

Economists have studied cartel sustainability for a long time and it goes beyond the scope of this paper to discuss all of this. However, it is most important to understand at least the game theory used to study cartel behaviour in the context of a model. Many case studies have been conducted in addition to this, but that will not be the focus of this paper.

Although in some countries anticompetitive behaviour is not against the law, this study is limited to cartels that are illegal. As their agreements are illegal, they cannot be enforced in court. Thus, cartels agreements are only sustainable if none of the colluding

(10)

players has an incentive to leave the cartel. This incentive may exist when cartels agree to produce less than in the original competitive equilibrium and one firm unilaterally decides to produce more, despite the agreements. Due to the others producing less, the total production goes down, which causes prices to rise; the deviating firm can maximise its production given the reduced production by the others to maximize its profits.

Such oligopoly situations are often studied in a supergame context. This means that the oligopoly firms are in a game situation together and that they know that they will be in similar situations in the future, an infinitely repeated game (Friendman 1971). In this supergame firms will not only take into account their competitors’ behaviour in the current period, but also their competitors’ reactions to their own behaviour. Thus, a firm that unilaterally deviates from the cartel agreements may be punished by the others in future periods (Friedman 1971).

Supergame settings require a market without deterministic finiteness in order to have any substantial meaning, because if the game were to end at a known moment in the future, firms will determine through backward induction that it is always wise to deviate. This implies that infinity or at least probabilistic finiteness is required, either the game keeps repeating forever, or it ends at a certain unknown moment in time with a cer-tain probability (Tirole 1988). However, this infinite setting is not altogether unrealistic, because although market situations will certainly end at some point, it is rarely known exactly when.

Within these supergames it is possible to evaluate firm strategies with regard to each other. Tirole (1988) introduces the term trigger strategies for the other firms’ reactions to a deviating firm. Among other possible strategies there is the grim strategy that implies that all firms will always play the competitive price or quantity from the moment that one firm deviates onwards. The grim strategy implies that actors compare the value of

deviating VD _{to the value of sticking to cartel agreements V}G_{, given a certain discount}

factor δ. πD _{is the profit of the firm if it deviates, π}G _{if it complies with the cartel and}

π∗ is the Cournot profit.

VD = πD+ ∞ X i=1 δiπ∗ and VG = ∞ X i=0 δiπG with πD > πG> π∗

(11)

Donsimoni et al. (1986) state that cartels are sustainable only if the colluding firms can recognise deviations from the agreements. As it is always better for a firm to be the only colluding firm on the outside of the cartel, all colluding firms must be able to control for this deviating behaviour. This implies that in situations with a very volatile market, or a market with many firms, the influence of one firm’s deviation is hardly noticeable and thus a cartel is less sustainable.

Escrihuela-Villar (2005) finds that within a certain market, a cartel becomes more sustainable if more firms partake. That is, a partial cartel is less sustainable than a cartel that includes the entire market. Intuitively, this seems reasonable enough as in this case there is no fringe firm that can free ride off the cartel agreements while producing more than the competitive equilibrium quantity.

2.3 Cartel sustainbility and the overcharge

Bringing together cartel sustainability and the overcharge is the purpose of this research. The relationship between the quality of the overcharge as a measure of cartel harm and the number of firms in both the colluding layer and the layer of direct purchasers is being investigated. In addition, there is a relationship between the sustainability of cartels and these numbers of firms. By combining both relationships dependent on the same external variables, conclusions can be formed about the interactions. In doing so a modification of the model introduced by Han et al. (2009) is being used. In addition, cartel sustainability is being evaluated with the help of Tirole’s (1988) equation of the value of both strategies.

3 The model

The model that I use in this paper is based for a substantial part on Han et al. (2009). For practical purposes some parameters are being excluded. The model consists of two production layers, followed by a layer of consumers. The first production layer consists of n firms and the second layer consists of m firms. The case that is being studied is a collusion in the first layer of producers and the effect it has on the subsequent layer of producers, as well as the consumers. A linear inverse consumer demand function P (Q) = 1 − Q, a modification of Han et al. (2009), is being assumed.

In this Cournot game the first layer starts of determining their production quan-tity, subsequently the firms in the second layer buy these goods and determine their own production levels; lastly the consumers buy the end products. However, the con-sumer demand is known and the firms base their maximisation on it. Using the inverse

(12)

consumer demand function the indirect demand function W (Q) is derived by way of

backward induction. With this function WQ, the firms in the first layer can determine

their production quantities by maximising their profits. Also, the equilibria for different cases are being derived. The Cournot-Nash equilibrium, the cartel equilibrium as well as

the unilateral deviation equilibrium. This leads to equilibrium quantities q_j∗, qG

j and qDj respectively Q∗ = n X j=1 q∗_1,j, QG = n X j=1 qG_1,j, QD = q₁D+ (n − 1)qG₁.

Consequently, the profits π_1,j∗ , πG

1,j and π1,jD are being compared in the context of grim

trigger strategies in the way Tirole (1988) suggests; this leaves me with a sustainability condition for the discount factor δ that depends on n and m,

VD = πD + ∞ X i=1 δiπ∗ = VG= ∞ X i=0 δiπG.

This equality is considered for both the situation without a fine and the situation with a fine equal to the overcharge. The chance that the cartel is discovered is φ ∈ [0, 1] in this case the sustainability equation becomes

VD = πD− φOC + ∞ X i=1 δiπ∗ = VG= ∞ X i=0 δi(πG− φOC).

In addition the overcharge OC is compared to different measures of the total harm. The overcharge is defined as the excess amount that the second layer firms paid for the goods they bought from the colluding firms due to said collusion. The change in total welfare is the cumulative influence of the collusion on consumer surplus CS, direct purchaser profit π₂G as well as cartel profit πG₁

∆W = n X j=1 ∆π1,j+ m X j=1 ∆π2,j + ∆CS, OC = m X j=1 (W (QG) − W (Q∗))qG_1,j.

The overcharge and the change of total welfare are represented graphically in figure 1. It is clear from the figure that the change of total welfare is equal to the total output effect, the loss of production due to the cartel. The overcharge and the pass-on only influence the division of the available total welfare. The pass-on effect for the second

(13)

P (Q)

W (Q)

OC

∆W

Q∗ QG Q PG P∗ WG W∗ P

Figure 1: The overcharge and the change of total welfare

In the evaluation of the overcharge as a measure of total harm four ratios are being

taken into account. ψ1 = _∆WOC is the ratio of the overcharge and the change in total

welfare, ψ2 = _{∆W +OC}OC is the ratio of the overcharge and the change in total welfare plus

the overcharge and ψ3 = _{∆W −}POCn

j=1∆π1,j is the ratio of the overcharge and the change

in total welfare minus the change in cartel profit. A case can be made for either these measures for total cartel harm; the first measure includes only the welfare change and is therefore amoral, the second measure takes into account the division of the new total welfare too; the third measure of harm corrects the welfare change for the increase in cartel profit.

Lastly, ψ4 = PmOC

i=1∆π2,i is the ratio of the overcharge and the cumulative change in

direct purchaser profits. This last ratio is used to investigate the relation between the amount by which the direct purchasers are overcharged and the actual damage that it does to them, taking into account the pass-on effects and the loss of sales.

The expressions for the total harm, the overcharge and the sustainability depend on m and n. Hence, a comparison between these can be made. In this comparison I evaluate the influence of taking into account only sustainable cartels in the assessment of the overcharge as a measure of total cartel harm.

(14)

4 Results

Using the model that is explained in the previous section I can derive the equilibrium situation before the cartel comes into existence, the cartel equilibrium and the equilibrium if a single firm deviates from the cartel agreements. For these situations the quantities, prices, profits and surplusses are being derived and compared. Next I look at the cartel sustainability, using these equilibrium values. Lastly, several ratios are being defined in order to evaluate the suitability of the overcharge as a measure of cartel harm.

4.1 Equilibria

I start of with the linear inverse demand function P (Q) = 1 − Q, with Q =Pm

i=1q2,i and

without any other costs than w, the price that the first layer charges the second layer for the . I derive the equilibrium values for a market with a two-layered production chain by

way of backward induction. By maximising over q2,i the following function of the profit

of a firm in the second layer

q2,i(1 − Q−i− q2,i− w), with Q−i=

X

k6=i

q2,i

and solving for the symmetric Nash-equilibrium, I obtain q2,i =

1

m + 1(1 − w) and Q =

m

m + 1(1 − w).

Solving for W (Q) this leaves the inverse demand function W (Q) = 1 −m+1_m Q for the first

layer in the production chain. With this out of the way the next step is to maximise over

q1,j the profit of a firm j in the first layer

q1,j(1 −

m + 1

m Q−j −

m + 1 m q1,j). The symmetric equilibrium quantities are

q1,j = 1 n + 1 m m + 1 and Q = n n + 1 m m + 1.

Thus the firms’ profit in both layers and the consumer surplus in this Cournot situation are obtained. Because there are no other costs than w taken into account the sum of these is equivalent to the total welfare.

π_1,j∗ = m (m + 1)(n + 1)2, π ∗ 2,i = n2 (m + 1)2_{(n + 1)}2 and CS = n2m2 2(m + 1)2_{(n + 1)}2

(15)

Subsequently, the equilibrium is being calculated in a similar way for the situation that the firms in the first layer of the production chain collude. It is assumed that all firms join the cartel and therefore the cartel equilibrium quantity can be found by maximising the cartel profit as though the colluding firms were a single company. The inverse demand

function W (Q) = 1 − m+1_m Q remains the same as nothing changes in the second layer.

Hence I maximize over Q

Q(1 − m + 1 m Q) and find QG = m 2(m + 1), q G 1,j = m 2n(m + 1) and q G 2,i = 1 2(m + 1).

In the cartel case the firms’ profits and the Consumer surplus are

π_1,j∗ = m 4n(m + 1), π ∗ 2,i = 1 4(m + 1)2 and CS = m2 8(m + 1)2.

In comparing both equilibria the overcharge OC and the change in total welfare ∆W can be found. The change in total welfare can be achieved in two ways, by taking the

integral over the inverse demand function from QG _{to Q}∗ _{or by adding the changes in}

profits of all firms to the change in the consumer surplus. This particular equality is the case here because there are no costs other than w included in this model.

OC = QG(W (QG) − W (Q∗)) = m(n − 1) 4(m + 1)(n + 1) ∆W = Z Q∗ QG (1 − Q)dQ = m(n − 1)(m(n + 3) + 4(n + 1)) 8(m + 1)2_{(n + 1)}2 or ∆W = n X j=1 ∆π1,j+ m X i=1 ∆π2,i+ ∆CS = m(n − 1)(m(n + 3) + 4(n + 1)) 8(m + 1)2_{(n + 1)}2

The overcharge increases in both n and m, which can be derived from the fact that the derivatives are both positive. The total welfare change too, increases in n but not monotonously in m. Intuitively, it seems to make sense that both increase in n, because the larger n, the more competitive the starting point and therefore the bigger the change if this layer colludes. The reason why the overcharge increases in m is less clear intuitively but it might be due to the weaker position of the firms in the purchaser layer if they are very competitive. If m is small, the few firms in this layer might have a stronger position and are therefore overcharged a smaller amount.

(16)

4.2 Sustainability

Successively the sustainability of the cartel is evaluated. In doing so, the value a firm attributes to the cartel strategy is being compared to the value of the strategy that involves deviating from the the agreements. The profit for the case that a single firm deviates is taken into account as well as a certain fine. The fine is equivalent to the overcharge OC and in their weighing of the fine firms take into account the probability of detection φ. The two options of either deviating from the cartel or sticking to it are being compared using Tirole’s (1988) grim trigger strategy.

Firstly, I maximise the profit firm j obtains if it deviates. W (Q) remains the same

and it is assumed that the other firms in the first layer stick to the cartel quantity qG_1,j. I

maximise over q1,j the profit of the deviating firm j.

q1,j(1 − m + 1 m Q G −j− m + 1 m q1,j)

This results in the following quantity and profit for the deviating firm j.

qD_1,j = m(n + 1) 4n(m + 1) and π D 1,j = m(n + 1)2 8n2_{(m + 1)}

The cartel is sustainable if, given the grim strategies, the value of the strategy of deviating is less than the value of the strategy of sticking to the cartel agreements. By equating these, a critical value for the discount factor δ can be found. The cartel is sustainable for

each δ ≥ δ+_{; δ}+ _{is obtained by equating V}D _{and V}G_.

VD = πD+ ∞ X i=1 δiπ∗ = VG = ∞ X i=0 δiπG δ+ = π G_{− π}D π∗_{− π}D = (n2_{+ 1)(n + 1)}2 n4_{+ 4n}3_{− 2n}2_{+ 4n + 1}

The critical value δ+_{is lowest for n = 5 and thus the cartel is most sustainable in this case.}

As n becomes either larger or smaller, the cartel becomes less sustainable. Strikingly, the cartel sustainability does not depend on m in the case of this particular model. In the case where a fine equal to the overcharge is taken into account the equation changes

slightly but still does not depend on m. δ± is the critical value for δ if a fine is taken into

account. δ± = π G_{− φOC − π}D_{+ φOC} π∗_{− π}D_{− φOC} = (n2_{+ 1)(n + 1)}2 n4_{+ 4n}3_{− 2n}2_{+ 4n + 1 − 2n}2_{φ(n − 1)(n + 1)}

(17)

4.3 Ratios

This sustainability is being compared to the overcharge as a measure of harm. There are multiple ways to define the harm of a cartel. Firstly, the change in total welfare due

to the cartel. The ratio ψ1 of the direct purchaser overcharge and the change in total

welfare is ψ1 = OC ∆W = 2(m + 1)(n + 1) m(n + 3) + 4(n + 1).

This does not take into account any other judgements than the observation that the total welfare has decreased. However, cartels also take a larger part of the new total welfare at the cost of others. Thus one might also look at the division of total welfare and define cartel harm as the change in total welfare plus the segment of the new welfare that the

cartel has appropriated. The ratio ψ2 of the direct purchaser overcharge and the change

in total welfare plus the direct purchaser overcharge is

ψ2 =

OC

∆W + OC =

2(m + 1)(n + 1)

m(3n + 5) + 6(n + 1).

Another way to regard cartel harm is to exclude the extra profits that the cartel obtains

from the change of total welfare. The ratio ψ3 of the direct purchaser overcharge and the

change in total welfare minus the change of cartel profits is

ψ3 = OC ∆W −Pn j=1∆π1,j = 2(m + 1)(n + 1) (m + 2)(3n + 1).

Lastly, it is interesting to look at the ratio of the direct purchaser overcharge and the direct purchaser harm. Thus cumulative change of profits in the second layer. The direct purchasers are overcharged by the cartel, but they pass part of the effect on to the consumers; on the other hand, they lose out on production due to the decrease in the

total quantity of goods that is produced.The ratio ψ3 of the direct purchaser overcharge

and the change second layer profits is

ψ4 = OC Pm j=1∆π2,j = (n − 1)(m + 1)(n + 1) 3n2_{− 2n − 1} .

These three ratios are being compared to the indicator of cartel sustainability δ+ _{in the}

subsequent section.

5 Analysis

Having established some results in the previous section I analyse the relationship between the sustainability of the cartel and the harm it does. In doing so I look for explanations

(18)

for the behaviour of these functions of m and n. I start off with an interpretation of the results and finish with some reflections thereafter.

5.1 Intrepretation

The overcharge increases in both m and n; the larger n, the more competitive the first layer before it colludes. After the collusion the situation is similar to n = 1; thus the change, and therefore the overcharge, is biggest for large n. This logic also applies to the change of total welfare, which also increases in n.

0.86 0.86 0.86 0.86 0.87 0.87 0.87 0.87 0.88 0.88 0.88 0.88 0.89 0.89 0.89 0.89 0.9 0.9 0.9 0.9 0.91 0.91 0.91 0.91 0.92 0.92 Delta m 2 4 6 8 10 12 14 16 18 n 2 4 6 8 10 12 14 16 18 0.77 0.77 0.77 0.77 0.78 0.78 0.78 0.78 0.79 0.79 0.79 0.79 0.8 0.8 0.81 0.81 0.82 0.82 0.83 0.83 0.84 0.84 0.85 0.85 0.86 0.86 0.87 0.87

Delta with fine

m 2 4 6 8 10 12 14 16 18 n 2 4 6 8 10 12 14 16 18

Figure 2: contour plots of δ+ _{and δ}± _{with φ = 0.1}

The critical discount factor δ+, which is a measurement for the sustainability as well

as the ratios ψ1, ψ2 and ψ4 are plotted in a contour plot of m and n. Both ψ1 and ψ2

increase in m and n because apparently the overcharge increases faster than the change of total welfare. The change in total welfare increases in n slower than does the overcharge because total welfare includes the profit of the first layer. Thus the effect of n works both ways in this case.

The top left plot shows the critical value δ+ _{for which the cartel is still sustainable for}

each m and n. The cartel is more sustainable as this critical value becomes lower. Hence it follows that the cartel is most sustainable for n = 5 and does not depend on m. The cartel sustainability decreases as n moves away from n = 5 in either direction but more so if n increases. Intuitively, it seems to make sense that the more firms are part of the cartel, the less sustainable the cartel is. This explains the decreasing sustainability if n

(19)

0.9 1 1 1.1 1.1 1.1 1.2 1.2 1.2 1.3 1.3 1.3 1.4 1.4 1.4 1.5 1.5 1.6

Overcharge over change of total welfare

m 2 4 6 8 10 12 14 16 18 n 2 4 6 8 10 12 14 16 18 0.5 0.5 0.52 0.52 0.52 0.54 0.54 0.54 0.56 0.56 0.56 0.58 0.58 0.58 0.6 0.6

Overcharge over change of total welfare plus overcharge

m 2 4 6 8 10 12 14 16 18 n 2 4 6 8 10 12 14 16 18

Figure 3: contour plots of ψ1 and ψ2

increases beyond 5. However, the decreased sustainability if n = 2 is somewhat contrary to my intuition. I believe this is due to the particular model specifications in this case.

The top right plot shows δ± for the chance of discovery φ = 0.1. Apart from the

values being lower, the structure of the function is very similar to the case without a fine. Therefore I compare the different ratios given the sustainability condition in the case without the fine. In cases for φ > 0 the effects are very similar.

In order to compare the cartel sustainability to the different measures of cartel harm I determine the average value of these ratios for a given value of δ, for sustainable cartels only. These ratios differ for separate values of δ because the latter influences the subset

of cartels that are in the sustainable zone. The ratio ψ2 gets smaller for smaller values of

the discount factor, that is, as the cartel becomes more sustainable. Thus the inclusion of the sustainability makes this ratio diverge even further from 1. Hence if one considers the change of total welfare plus the overcharge a good measure of cartel harm this implies that the overcharge becomes a worse measure after taking sustainability into account.

Likewise the average value of ψ1 decreases as the cartel becomes more sustainable.

However, in this case the ratio approaches rather than diverges from 1. In this case, the more sustainable the cartel, the better the overcharge approaches the change in total welfare.Therefore, the case for or against the overcharge boils down to one’s definition of cartel harm.

Accordingly ψ3 decreases in the sustainability. This ratio is larger than 1 and thus

approaches 1 as the level of substitutability goes up. Viewed this way, the overcharge becomes a slightly better indication of cartel harm. However, the ratio lies between 4.5

(20)

and 5.7 so it does not get particularly close to 1. Thus if one was to regard this measure of harm the correct one, the overcharge is not a very good proxy.

The ratio ψ4 of the overcharge and the change in profit for the second layer increases

with the sustainability. This is because the change of second layer profit increases in n faster than the overcharge. This ratio increases in m because the more firms are in the purchaser layer, the smaller the change in their profits due to the cartel because they can pass on a larger part of the overcharge.

2 2 3 3 4 4 4 5 5 5 6 6 6 7 7 8 8 9 10

Overcharge over change of CS and second layer profit

m 2 4 6 8 10 12 14 16 18 n 2 4 6 8 10 12 14 16 18 2 2 3 3 4 4 5 5 6 6 7 7 8

Overcharge over change of second layer profit

m 2 4 6 8 10 12 14 16 18 n 2 4 6 8 10 12 14 16 18

Figure 4: contour plots of ψ3 and ψ4

This ratio ψ4 shows that on average the overcharge is more than four times as big as

the change in direct purchaser profit. This indicates that the damages that these direct purchasers obtain in the case of a discovery of a cartel are far larger than the harm that has been inflicted upon them. This effect increases slightly if only sustainable cartels are considered. What is more, they receive a significant damage payment but other parties that are harmed too receive nothing. Thus the current overcharge-based fine, and the subsequent damage payments are insufficient in making up for the harm inflicted upon different parties.

I argue that the ratio ψ2 of the overcharge and the total change in welfare plus the

overcharge is the best indication of the suitability of the overcharge as a measure of harm. This measure takes into account not only the loss of welfare that can be attributed to the cartel but also the change in allocation of the remaining welfare, adding the unrightfully obtained overcharge to the total harm. I recognise that this measure of harm is by definition normative because it takes into account the distribution of wealth in addition

(21)

δ 0.86 0.88 0.9 0.92 1

ψ1 1.2358 1.2577 1.2997 1.3174 1.3293

ψ2 0.5510 0.5549 0.5627 0.5655 0.5677

ψ3 4.8409 5.0408 5.4088 5.6095 5.7220

ψ4 4.5121 4.4723 4.3832 4.3636 4.3395

Table 1: Average ratios ψ1, ψ2, ψ3 and ψ4 for given δ

to the evaluation of the total wealth.

At any rate, the overcharge appears to become a slightly worse measure for cartel

harm because ψ2 decreases slightly as the sustainability increases. Therefore, this finding

strengthens the conclusion that both Han et al. (2009) and Basso and Ross (2010)

put forward, that the direct purchaser overcharge is an insufficient measure of cartel harm. Thus, the fine, which is often based on the overcharge, becomes an even worse approximation when cartel sustainability is taken into account, but this effect is rather

small. On the whole, I would say that the ratios ψ2 and ψ3 indicate that the overcharge

is not a very good measure for cartel harm regardless if the sustainability is taken into account.

5.2 Reflections

The results that I have obtained are very much dependent on the restricted model that I have used. This model is by definition a simplification of the economic reality, which is not a problem as long as this is properly recognised. In this model demand is assumed to be linear, no costs are being included and perfect symmetry between firms is being presumed, to name just a few. Also, my definition of total harm as the sum of the overcharge and the change of the total welfare is not entirely amoral because it takes into account the distribution of wealth apart from the total amount of wealth.

However, given these restrictions, these results can prove to be useful in their indi-cation of certain effects. Some of these restrictions presumably have little influence on the validity of the results. When considered with appropriate reservations they might provide some useful insights on the unsuitability of the overcharge as a measure of cartel harm.

(22)

6 Conclusion

I have assessed the effect of taking into account only sustainable cartels, for a given discount factor, on the suitability of the overcharge as a measure of cartel harm. In the context of the model that I used the effect that I found is that the overcharge is even a worse measure of cartel harm if only sustainable cartels are taken into account. This serves as an enhancement of the point made by Han et al. (2009) and Basso and Ross (2010) that the overcharge is unsuitable as a measure of cartel harm. This point is relevant because the overcharge is often used in the determination of damage payments when a cartel has been discovered. Hence, this finding that this measure is even less appropriate than it was thought to be, indicates that other measures for determining cartel damages need to be used.

In the obtention of these results I first studied the existing literature on the overcharge and sustainability. I started off addressing the reasons why a cartel comes into existence in the first place. Firms may strive to reduce competition in order to obtain a larger profit. They agree to produce a smaller amount of goods or they set higher prices; either way these agreements lead to higher prices and lower quantities. This means that the direct purchasers, that buy from the cartel, get overcharged a certain amount. Moreover, due to lower the lower production quantities, not only the division of welfare, but also the total welfare changes.

Generally the direct purchaser overcharge (Han et al. 2009) is used to determine the harm done by cartels. However, both Han et al. (2009) and Basso and Ross (2010) find that this overcharge is a rather bad approximation of the actual harm done by the cartel. The problem in this case is threefold. Firstly, the overcharge is not a good representation of the cartel harm. Secondly, the damages are awarded to the direct purchasers, but the harm is done to more parties. Thus the damages that are paid to the direct purchasers overestimate the harm inflicted upon them and leave other victims entirely without damage payments. Thirdly, a better approximation of real cartel harm requires a lot of knowledge about the ‘but-for world’ (Han et al. 2009).

I addressed these first two issues in this research, the third one is more of a practical issue. The first issue was assessed using three different measures of harm. Which in this case lead to opposite conclusions about the influence of sustainability’s inclusion. However, I argued that the case where the overcharge is included in the total harm is the relevant measure. In this situation limiting the subset of cartels to the sustainable ones deteriorates the quality of the overcharge as a measure of harm.

(23)

to the direct purchasers only. I have shown that in the case of my model, the overcharge is an overestimation of direct purchaser harm. The damages that are awarded to the direct purchasers overestimate their losses and, more problematically, disregard any losses suffered by other parties. This is consistent with Basso and Ross (2010), who find that the direct purchaser loss is generally lower than the overcharge due to a passing on of part of the harm.

As was stated in the last section my research is based on a simplification of reality. Both the model and the game theoretic strategies could be expanded greatly. Future research might involve leaving some of the parameters unrestricted. Furthermore, other strategies for reacting to deviation behaviour than the grim trigger strategy might be considered.

In all, the inclusion of cartel sustainability in the assessment of the overcharge makes for an even greater disparity between the latter and total cartel harm. Therefore the overcharge might be an even worse measure for cartel harm.

Literature

Basso, L.J. and Ross, T.W. (2010): “Measuring the true harm from price-fixing to both direct and indirect purchasers”, Journal of Industrial Economics 58(4), 895-927.

Boone, J. and M¨uller, W. (2008): “The distribution of harm in price-fixing cases”,

mimeo, Tilburg University, Tilburg.

Donsimoni M., Economides, N. and Polemarchakis, H. (1986): “Stable cartels”, International Economic Review, 27(2), 317-327.

Escrihuela-Villar, M. (2005): “Cartel sustainability and cartel stability”, Working Paper. Friedman, J. (1971): “A non-cooperative equilibrium for supergames”, Review of

Economic Studies, 38 (1), 1-12.

Han, M.A., M.P. Schinkel en J. Tuinstra (2009): “The overcharge as a measure for antitrust damages”, ACLE Working Paper.

Motchenkova, E. (2004): “Effects of Leniency Programs on Cartel Stability”, CentER Discussion Papers Series 2004-98, Tilburg University, Tilburg.

Tirole, J. (1988): The Theory of Industrial Organization, MIT Press, 1988.

Verboven, F and Dijk, T. van (2009): “Cartel Damages and the passing-on defense”, Journal of Industrial Economics, 57(3), 457-491.

Cartel sustainability and the overcharge as a measure of harm