Optimal Cartel Pricing when
Anticipating Antitrust Enforcement
Wouter Antonius Griffioen
Bachelor’s Thesis to obtain the degree in Econometrics & Operations Research
Specialization: Econometrics & Actuarial Sciences University of Amsterdam
Faculty of Economics and Business
Author: Wouter Antonius Griffioen Student nr: 10017631
Date: June 22, 2015
Abstract
This study analyses the effect on the overcharge as an estimate of cartel harm when colluding firms account for the possibility of receiving fines. It does so under perfect and asymmetric information assuming linear demand. First the expected profit is modelled for both cases on which the analysis is based and subsequently ratios of the overcharge to total harm, welfare loss and consumer harm are defined. On the basis of these three ratios the approximation strength of the overcharge is analysed. The ratios show that ac-counting for fines in general improves the approximation of the overcharge. Nevertheless, the overcharge underestimates the total harm, the consumer harm and overestimates the loss in welfare as a consequence of cartel formation showing that courts and other institutions wrongfully use the overcharge to approximate the cartel damage. Allow-ing for information asymmetry imposes more uncertainty but in general improves the approximation.
Keywords Cartel damage, Antitrust, Overcharge, Asymmetric information
Acknowledgements
I am grateful for the support of my thesis supervisor Prof. Dr. Jan Tuinstra, as he gave me valuable insights and comments throughout the process of writing my thesis. Furthermore, I would like to thank Annemiek Griffioen and Jasmin Salik for revising. The views in this paper are those of the author and do not represent the views of any other isntitution.
Contents
1 Introduction 1
2 Theoretical framework 3
2.1 Static antitrust models . . . 5
2.2 Dynamic antitrust models . . . 7
2.3 The overcharge . . . 9
2.4 Combining antitrust models and the overcharge . . . 10
3 Modelling profit and overcharge ratios 11 3.1 Modelling expected profit . . . 11
3.2 Decomposing total harm . . . 13
4 Results 14 4.1 General case . . . 15
4.2 Linear demand, perfect information . . . 18
4.3 Linear demand, asymmetric information . . . 27
5 Conclusion 36
1
Introduction
In December 2013, the European Commission fined eight financial institutions a total of e1.71 billion for participating in illegal cartels in markets for financial derivatives covering the European Economic Area, also known as the Libor scandal (European Commission, 2013a). According to the European Commission (2013a), Barclays received full immunity for revealing the existence of the cartel and thereby avoided a fine of almost e700 million for its participation in the infringement.
The above example is a combination of a leniency program and damage dependable fines, used by an antitrust authority as instrument to fight the existence of illegal cartels. According to the European Commission (2013b), its leniency program is the main and most effective tool to detect illegal cartels. Cartels are illegal because they are generally perceived as bad for the economy since they create losses to welfare. The objective of any antitrust authority is the minimisation of two types of costs, those resulting from the harmful conduct by the colluding firms subjected to regulation and those incurred in the dismantlement of that conduct (Schwartz, 1979). An antitrust authority then tries to allocate its resources as efficiently as possible in order to detect and, subsequently, sanction colluding firms or to deter firms from collusive behavior.
In addition to leniency programs and fines associated to the harm on welfare done by cartels other instruments used by antitrust authorities are fixed fines or even crimi-nal sanctions on an individual basis (Aubert, 2007). Authorities, however, face practical constraints in their enforcement. As Souam (2001) states, an authority’s resources are limited, they cannot monitor all the markets and pursue every firm which is suspected of collusive behavior. Next to that authorities are unable to perfectly observe the char-acteristics and behavior of the firms because markets are rarely transparent. This is known as asymmetric information and enhances problems of moral hazard and adverse selection that reduce the efficiency and impact of public interventions (Souam, 2001).
The limited resources, constraints and possible instruments give the antitrust authorities the optimization problem of how to allocate their resources and what instruments to use in order to detect and deter collusive behavior. Fines in the form of fixed fees are found to have no effect on the cartels’ prices theoretically (Harrington, 2005). In addition, leniency programs work best when rewards are given for the first reporting firm, but this is often considered immoral (Spagnolo, 2005). If fines as multiple of the damage caused by the cartel are used there arises a problem of how to determine the damage as
is the case with damage claims from plaintiffs. For instance, in the United States direct purchasers can claim treble damages, but in order to do so the size of the damage needs to be determined (Clayton Antitrust Act, 1914).
The most common approach used is taking the difference by the cartel price and the price that would have existed but for the price fixing activity and multiplying this by the units purchased by the direct purchasers, which is called the overcharge (Basso and Ross, 2010). But as the studies of Han, Schinkel and Tuinstra (2009) and Basso and Ross (2010) show, the overcharge generally provides a poor estimate of the true harm. In addition, colluding firms might already anticipate on the probability of being fined a multiple of the overcharge. It could, for instance, be optimal for a cartel to lower the mark-up on the competitive price in order to maximise their expected pay-off from participating in the cartel since lowering the markup would lower the overcharge, and thus the fine imposed on the participating firms.
While there exists literature on both the optimal use of instruments for antitrust en-forcement and the overcharge as a measure of the harm caused by collusive activities the link between these is missing. If firms already take the probability of being fined depending on the overcharge into account when colluding this might alter the mark-up on the but for price and thus the overcharge. Therefore the main goal of this study is to analyse the effect of fines on the overcharge as an estimate of cartel harm when colluding firms account for the possibility of receiving fines. This study gives an anal-ysis for this effect by providing a model allowing for both symmetric and asymmetric information, from which the implications of fines for collusive behavior are deducted. Before studying the situation where there exists an informational asymmetry between the authority and the colluding firms this study derives the results in a setting assuming perfect information as an initial benchmark.
It is important to gather insight in the existing literature on the optimal application of the tools used by antitrust authorities and the overcharge as a measure of true harm caused by collusive activities. Therefore the next chapter gives a review of previous stud-ies on these subjects in combination with outlining the theoretical framework. The third chapter models the expected profit in case of symmetric and asymmetric information and defines ratios used to analyse how well the overcharge approximates cartel harm. In the fourth chapter the results are derived and analysed and chapter five summarises the main results, treats the limitations of this study and provides suggestions for further research.
2
Theoretical framework
This chapter discusses the main literature on the effects of antitrust enforcement on the collusive behavior of firms and on the overcharge as a measure for true cartel damage. But first it is essential to understand the basics of collusive activity and its implications on welfare as well as some basic understanding of the simplifications made in order to study implications in the economic environment.
The Treaty on the functioning of the European Union (TFEU) defines prohibited anti-competitive behavior as:
. . . all agreements between undertakings, decisions by associations of under-takings and concerted practices which may affect trade between Member States and which have as their object or effect the prevention, restriction or distortion of competition within the internal market . . . (Treaty on the Functioning of the European Union, 2012, article 101).
The prevention, restriction or distortion of competition is prohibited because it is gener-ally perceived to reduce welfare caused by artificial high prices or low output. Competing firms, however, are dictated by the incentive to raise prices or lower output by colluding in order to maximise their profits. In absence of any authority it is optimal to charge monopoly prices and divide the profits between the participating firms (Block, Nold & Sidak, 1981).
Figure 2.1 illustrates this in the simple case of constant marginal costs and a linear inverse demand function under perfect competition. The price is given on the vertical axis and the quantity on the horizontal axis, furthermore pm gives the monopoly price, pcthe competitive price and Qmand Qcthe industry’s monopoly and competitive quan-tities. In the competitive setting when competing in prices firms sell at marginal costs leaving them zero profit and therefore have an incentive to collude and artificially raise prices resulting in a positive profit. The overcharge is then given by (pm− pc)qm, char-acterised in figure 2.1 by the checkered square OC, this corresponds to the profit of the cartel and therefore is merely a transfer of surplus from the consumers to the producers which does not reduce welfare. Nevertheless, harm is done to the direct purchasers as a consequence of the collusive activity. The loss in welfare is given by the shaded dead-weight loss triangle DW L in figure 2.1. This triangle represents consumers that would have bought the good for a competitive price but do not do so at the monopoly price the cartel sets.
C0(Q) P (Q) OC DW L Qc Qm Q pm pc p
Figure 2.1: Overcharge and deadweight loss
In the presence of an antitrust authority that enforces sanctions dependable on the overcharge, prices different from the monopoly price might become optimal when the colluding firms anticipate the possibility of being fined. The cartel might have an incen-tive to set a lower cartel price to reduce the size of the overcharge (the OC square in figure 2.1) in order to maximise their expected profit.
In order to study the economic phenomena of the real world it is necessary to impose assumptions. The economic models resulting from this are a simplification of reality and thus do not perfectly describe these phenomena. Nevertheless, these models give useful insights in the behavior of economic agents or structure of equilibria. These insights on their turn can be consulted to explain the economic phenomena that are not suited to perfectly reconstruct.
Although there are multiple variations possible the two basic models on oligopolistic competition are given by Cournot (1838), where firms compete in quantities and by Bertrand (1883) where firms compete in prices. Next to their specific assumptions both models assume rational behavior, homogenous goods meaning that there is no product differentiation, market power which means firms take in consideration the prices or quantities set by others and that the number of firms is fixed and known by all firms. When these firms possess the same characteristics it is possible to arrive at a symmetric equilibrium.
This study is based on a Cournot setting with homogenous goods in order to be able to more conveniently analyse the effects on the overcharge since in a Betrand setting heterogeneous goods are needed in order to establish analogous results which complicates derivations. Chapter 3 provides a more detailed description of the model and assumptions this study uses.
The remainder of this chapter provides a review on the existing literature on antitrust policies and cartel interactions. Section 2.1 and 2.2 divide the past literature on antitrust policies in two categories, the literature that uses a static setting and the literature that
uses a dynamic setting. The static models are in a steady state or equilibrium which provides a basis for structural analysis, while the dynamic models are more focused on the series of sequential steps prior to a steady state forming a basis for behavioral analysis. Section 2.3 elaborates on the overcharge used as a measure of true cartel harm while section 2.4 provides the intuition in combining the discussed literature in this chapter and gives an introduction of the modelling choices which are elaborated in chapter 3.
2.1
Static antitrust models
This section discusses literature on the effects of antitrust enforcement on collusive activity and the consequences from a welfare point of few using static models. In these settings the model is in a steady state or equilibrium which provides a basis for structural analysis.
Block et al. (1981) attempt to estimate the impact of antitrust enforcement on horizontal price fixing using a static theoretical model. They specify a penalty function which relates to the price mark-up the cartel exhibits so that the sanctions given by the antitrust enforcement relates to the overcharge. Next to that, their model consists of a probability function which depends on the size of the mark-up the cartel uses and the level of enforcement efforts by the antitrust agency. Their model assumes that the probability of detection increases with the size of the mark-up. In general, their hypothesis states that the more efficient a cartel produces mark-ups, the more likely it will be detected and the cartel members convicted.
The findings of Block et al. (1981) are that antitrust penalties not necessarily elim-inate price fixing but are, however, likely to reduce the optimal mark-up the cartels use. In addition they find that an increase in either the probability of detection or the penalty for price fixing is expected to reduce the collusive mark-up. They conclude that deterring cartel members from setting the monopoly price and thus the full overcharge is rather easy to attain for an antitrust agency, but that to deter all collusive pricing is nearly impossible.
There have been multiple modifications to the model of Block et al. (1981). For instance, Salant (1987) argues that Block et al. (1981) fail to notice that the prospect of treble damages for direct purchasers of the cartel stimulates the demand at any given price. He attains a neutrality result which shows that buyers will still consume at the monopoly price as long as buyers and sellers share the same information. Therefore any enforcement by an antitrust agency is superfluous and if the enforcement involves any resource costs than no enforcement is strictly preferable (Salant, 1987).
Baker (1988) extends the model from Block et al. (1981) by allowing buyers and sellers to have private information concerning the likelihood of a successful cartel detec-tion and convicdetec-tion. He argues that the resource allocadetec-tion in the absence of antitrust law does not differ to the resource allocation with antitrust law. Without information
differences in Baker’s (1988) model the private damage does not deter cartel formation nor gives cartels an incentive to alter output from the monopoly level because the buyer and seller can contract around the possible damage when they already anticipate the possibility of harm. His conclusions are thus consistent with those of Salant (1987).
A shortcoming to the practical applicability of the above described models is that in reality there is an information asymmetry between the colluding firms, their direct purchasers and the relevant antitrust agency. Besanko and Spulber (1989) use a game of incomplete information to model firms and the antitrust enforcement. They consider the optimal design of antitrust policy when the cartel is unobservable and the production costs are private information. The marginal costs of the firms are either one of two values, high or low, for each of which the authority assigns a probability so that they do not have complete information. They show that asymmetric information can be a significant factor in the decision to allow for some degree of collusion even though price fixing is an infringement of antitrust law.
The approach of Besanko and Spulber (1989) to use asymmetric information is ex-tended in studies of LaCasse (1995), Polo (1997), Souam (2001) and Schinkel and Tuin-stra (2006). LaCasse (1995), however, uses a different setting by modelling a sealed-bid auction in which agents who rig their bids face the threat of government prosecution. In her model agents make their decision by weighing, on the one hand, the benefits that could be realized from conspiracy and, on the other hand, the costs associated with a potential prosecution. The legal authority, however, cannot ascertain the presence of a conspiracy, it can only try to detect whether agents at the auction are cooperating by examining their bids. LaCasse (1995) finds that agents avoid a policy of complete deter-rence by randomizing between rigging their bids and bidding as if they are competitive, such that a legal authority being active causes agents to not totally collude.
Closer to the study of Besanko and Spulber (1989) is the study of Polo (1997). He argues that the optimal allocations implemented by antitrust enforcement depend on the informational constraints, limited resources and bounded penalties the authorities face. The extension of the model used by Polo (1997) is that he accounts for the possibility of an unexpected fine for firms. He finds that the optimal price schedule entails prices higher than costs for all the types, which is consistent with the finding of Besanko and Spulber (1989).
Another possible extension is allowing marginal costs to be in a bounded continuum of values, which is used by Souam (2001). He studies whether different systems of antitrust enforcement are equally efficient in terms of deterring collusion. From the point of view of the authorities, fines are cost-less while investigation is costly. So in order to obtain a certain level of expected fines the authorities have to set the fine at the highest possible level while keeping the probability of investigation as low as possible (Souam, 2001). He concludes that a fine system based on a fixed proportion of the total sales can always dominate from a social welfare point of view under the right
circumstances. Furthermore, for markets where possibilities of collusion are small fines as a proportion of sales are more efficient whereas in industries where the possibilities of collusion are high it is more efficient to fine based on a multiple of the damage caused (Souam, 2001). From this study it can thus be interpreted that the treble damages as stated in the Clayton Antitrust Act (1914) are only most efficient when a market with a high possibility of collusion is considered.
Schinkel and Tuinstra (2006) go back from the continuous costs used by Souam (2001) to a model in which firms can be of three different cost types. They simplify further by using a discrete action space, where firms choose from a set of three different outputs. Their addition, however, is that they allow the antitrust authorities to make errors. The authorities may find industries that are competitive liable of anticompetitive behavior or the authorities might by acquitting firms that actually behaved anticompet-itive. Schinkel and Tuinstra (2006) find that the incidence of anticompetitive behavior increases in the antitrust authority’s enforcement error. On the one hand because the probability of getting caught goes down, and on the other hand because the risk of being fined while not colluding goes up causing firms to collude as a precautionary measure.
All of the studies discussed above use a static model and therefore give qualitative insight in the interaction between possibly colluding firms and the antitrust enforce-ment authorities. Where Block et al. (1981) provide a base model with the assumption of symmetric information between firms and authorities, Besanko and Spulber (1989) provide the reference model for incorporating the restriction of asymmetric information.
2.2
Dynamic antitrust models
This section discusses relevant literature on price-fixing and detection in a dynamic setting. Although this study uses a more static setting it is still very relevant to discuss the literature using dynamic models because the dynamic setting of these models provide more insight in the steps towards an equilibrium. This provides the information of how to arrive at a static situation. Therefore these models are better suited to, for instance, analyse the effects of leniency programs on the deterrence of cartels.
Friedman (1971) in his study on non-cooperative equilibria defines an infinite series of ordinary games over time as a supergame. He argues that an oligopoly can also be viewed as such a supergame since in each period the firms are in a game and they know they will be in similar games with the same other firms in subsequent periods as well. Friedman (1971) shows that an equilibrium exists in a non-cooperative supergame. The Cournot equilibrium seems unsatisfactory for firms since higher profits can be achieved simultaneously for each, therefore firms engage in something called tacit collusion under which firms act as if they collude (Friedman, 1971). An explanation offered by Fried-man (1971) for the existence of tacit collusion is that firms’ market movements are interpretable as messages working as some sort of cartel self-enforcement mechanism.
demand uncertainty is given by Green and Porter (1984). They present a model of a non-cooperatively supported cartel and discuss the characteristics of the industry structures that would make such a cartel viable. One of their main findings is that, given particular industry characteristics, collusive behavior may result in a pattern of recurrent periods in which price and profit levels sharply decrease. This shows that industry performance of this type not necessarily indicates an industry where firms are not able to form a cartel (Green and Porter, 1984).
The model of Green and Porter (1984) is used by Cyrenne (1999) to examine an an-titrust policy of using significant price changes in a certain industry for deciding where to allocate resources to policing collusion. He argues that this investigation strategy can reduce the expected profits generated by the cartel. However, a large enough pun-ishment is needed to reduce the frequency of collusion and next to that it can cause the colluding firms to reduce output or raise prices (Cyrenne, 1999). In addition, he finds that if the punishment mechanism of the antitrust authority is anticipated by the colluding firms they will alter their agreement to offset the authority’s policy.
Next to the decision where to allocate their resources another enforcing policy instru-ment may be the use of leniency programs to increase the efficiency of destabilizing cartels. Motta and Polo (2003) and Spagnolo (2005) investigate the use of leniency programs by the antitrust authorities.
In the model of Motta and Polo (2003) reduced fines are given to firms that reveal information to the antitrust authority. After observing the authority’s decisions firms decide whether they want to deviate or collude, and then whether they want to reveal information about the cartel or not. They find two possible effects of using a leniency program. It might lead to firms desisting from collusive behavior more frequently but it might also stimulate collusive behavior because it decreases the expected cost from anticompetitive behavior (Motta and Polo, 2003). They conclude that in the optimal policy the first effect dominates the second and consequently leniency programs should be used when the antitrust authority is restricted by limited resources.
Spagnolo (2005) first examines the optimal law enforcement policy in the absence of leniency programs. He concludes that antitrust authorities should commit not to target agents that defect from collusive behavior because if agents know they will not be punished for their past wrongdoing if they defect from the cartel, they are more prone to do so. When incorporating leniency programs into the policy, Spagnolo (2005) finds the existence of the same two effects as Motta and Polo (2003). Furthermore he argues no collusive agreement would be sustainable if sufficiently high rewards are offered for the first reporting agent. Finally Spagnolo (2005) finds that moderate leniency programs like protection from fines or protection from punishment may also increase deterrence from collusive behavior.
Although above discussed models consider collusive behavior in a dynamic setting, the sources of dynamics are excluded from the current analysis (Harrington, 2005).
According to Harrington (2005), these studies do not provide information about the transitional pricing dynamics associated with a newly formed cartel because they do not allow the detection and penalties to be sensitive to firms’ current and past pricing behavior. He models these transitional pricing dynamics and his main findings are that the cartel price is decreasing in the probability of detection, fines and the damage multiple. In addition he states that the steady-state price is decreasing in the damage multiple and the probability of detection but independent of a fixed fine (Harrington, 2005).
Summarising, Friedman (1971) provides the basic reference model in a dynamic setting for non-cooperative equilibria in supergames. The model of Green and Porter (1984) extends Friedman’s model this into a model with demand uncertainty. Cyrenne (1999) uses this model to establish how authorities should use their resources whereas Motta and Polo (2003) and Spagnolo (2005) examine the effect of leniency programs. Harrington (2005) models the transitional dynamics and finds that the cartel price is dependent on the damage multiple and the probability of detection.
2.3
The overcharge
The goal of this section is to review and discuss the overcharge as a measure of the true harm caused by cartels. To achieve optimal deterrence Landes (1983) showed that the damages from illegal collusive behavior should be equal to the expected net harm caused by this behavior divided by the probability of detection. Currently, the overcharge is the most common measure used by courts for the damage caused, but recent research shows that it generally gives a poor estimate.
In the United States, any person who is injured in his business or property by any infringement of the antitrust laws is able to recover three times the damages sustained (Clayton Antitrust Act, 1914). The damages sustained are, as in most western countries, based on the overcharge. This leaves the treble damages applied as an example of where the fine a colluding firm receives is some multiple of the overcharge. But even the American court recognises problems with the overcharge as an estimation of cartel damage stating that overcharged direct purchasers often sell in imperfectly competitive markets and thus compete with other sellers that have not been subject to the overcharge (Hanover Shoe, 1968).
Connor and Lande (2005) state that next to the overcharge the cartel causes there are other factors that influence the total damages. First, they argue, the market power produces allocative inefficiency which is expressed in the deadweight loss triangle in figure 2.1. Second, cartel formation can produce a so called umbrella for non-colluding firms to raise prices which constitutes harm to direct purchasers (Connor and Lande, 2005). In addition, they argue that cartel members sometimes have a smaller incentive to innovate or to offer as much non-price variety or quality as they would have in com-petitive circumstances. Some of these factors are found in the following studies on the
approximation of true cartel harm by the overcharge.
The study of Han et al. (2009) considers the effects of anti-competitively raised prices in an arbitrary layer in the production chain with homogeneous goods. They show that the formation of a cartel does not only have its implication on the direct purchasers but also causes damages upstream in the production chain and to indirect purchasers. Furthermore Han et al. (2009) argue that the direct purchaser overcharge is equal to the sum of all passed on overcharges but misses the output effects in every production layer, again corresponding to the deadweight loss triangle in figure 2.1. They derive the estimation power of the overcharge by defining ratios of change in welfare, cartels gains and downstream harm to the overcharge. With these ratios they argue that the overcharge generally underestimates the actual harm and that the closer the pricefix-ing to the consumers and the longer the supply chain the more problematic uspricefix-ing the overcharge as an estimation for true cartel harm becomes (Han et al., 2009).
Basso and Ross (2010) also aim to explore the relationship between the true harm caused by price-fixing and the overcharge. In contrast to Han et al. (2009) they fix the amount of production layers to two and assume a cartel is formed in the first layer but they allow for heterogeneous goods in their model. Nevertheless, Basso and Ross (2010) also find that the overcharge as a measure of true cartel harm is poor and grossly underestimates the true harm, again providing evidence for the other factors mentioned by Connor and Lande (2005).
Summarising, the literature finds evidence of the overcharge being a poor estimate of the cartel damage and argues that there are more factors to be considered. Nevertheless, the overcharge is the most common used estimate of the cartel damage by courts in order to determine the fines to impose.
2.4
Combining antitrust models and the overcharge
With the knowledge of the literature that approximates the size of the overcharge in relation to true harm and the knowledge of the literature that models the possible imperfections in the enforcement of an antitrust authority in the previous sections, an interesting combination arises.
If the overcharge is used by courts as a measure of the cartel damage in order to sanction firms that participate in the cartel, then these firms can already anticipate a fine that is dependable on the size of the overcharge their price mark-up creates. Participating firms then account for the possibility of a fine as multiple of the overcharge in the profit maximisation while colluding. A consequence is that colluding firms might have an incentive to lower their price mark-up and thus the overcharge in order to receive a smaller expected fine if this optimises their expected profit. It is exactly this effect, and how this effect changes the overcharge as an estimate of the cartel harm, that this study aims to analyse.
For the analysis of the approximation strength of the overcharge as a consequence of firms accounting for the possibility of a fine it is useful to first compose a model in a static setting assuming perfect information in line with the models of Block et al. (1981) and Salant (1987). This gives qualitative insight in the behavior of the colluding firms in their optimisation process. But since cartels are illegal, firms want to collude without being caught and thus conceal as much information as possible and, in addition, markets are generally non-transparent. This causes information asymmetry between the colluding firms and the antitrust authority. Therefore it is a vital generalisation to add information asymmetry to the basic model in line with the model of Besanko and Spulber (1989). If the antitrust authority assigns a probability to a firm either having high marginal costs or low marginal costs, the firm with low marginal costs might have an incentive to raise its price since on expectation this firm receives a lower fine than it actually deserves and vice versa for a high cost firm. Another interesting addition would be to incorporate leniency programs into the model in line with the models of Polo (2003) and Spagnolo (2005). However, this is beyond the scope of this study since it requires modelling a dynamic setting because the sequential decisions to report a cartel or continue colluding are needed.
The precise definitions and set up of the model in this study are given in the next chapter together with the assumptions imposed.
3
Modelling profit and overcharge ratios
This chapter describes the model for analysing the effect of colluding firms anticipating the possibility of being fined on the estimation power of the overcharge. The benchmark model assuming symmetric information is based on the model of Block et al. (1981) whereas the generalisation to a situation with asymmetric information is inspired on the additions made by Besanko and Spulber (1989). The model is set up in section 3.1 whereas section 3.2 decomposes the total harm to create the possibility of estimating different shifts in welfare and surplus by the overcharge.
3.1
Modelling expected profit
As stated in the theoretical framework this study’s model uses a Cournot setting so that a fixed and known amount of N firms compete in quantities qi, i ∈ 1, ..., N . An one layered vertical production chain is considered so that there is only one industry directly serving the consumers. The industry has an inverse demand function P (Q) which is twice continuous differentiable and strictly concave which ensures the existence
of a unique interior maximum that corresponds to the monopoly output. All firms engage in the collusive agreement so that the total cartel output is given by Q = q1+ ... + qN. Furthermore homogenous goods are assumed and firms act rational in the sense that they maximize expected profit π(Q). All N firms have the same linear cost function C(q) with constant marginal costs ∂C(q)∂q = c known to the antitrust authority. The symmetry of the cost functions implies a symmetric equilibrium in which all N firms produce the same output. The probability of detecting a cartel is given by d ∈ [0, 1), which depends on exogenous factors like the investigation efforts of the antitrust authority but also on the price markup relative to the marginal costs.
The final essential part of this model is the fine imposed by the antitrust authority if they detect a cartel. This fine is given by F = µ(P (Q) − p∗)Q, where (P (Q) − p∗)Q is the overcharge and µ the damage multiple. So, for instance, in the United States there would hold µ = 3 as treble damages may be claimed by the plaintiffs. For notational convenience ϕ = µd is defined as the effective damage multiplcator. If a cartel is detected it will always be successfully prosecuted so that there are no inefficiencies in the law enforcement. Furthermore, non-colluding firms are never charged implying that d = 0 for P (Q) = p∗ and there is always the possibility that a cartel is not detected so that indeed d < 1.
Combining these model specifications results in the following expected profit function of the cartel:
π(Q) = (P (Q) − c)Q − dF = (P (Q) − c)Q − ϕ(P (Q) − p∗)Q (3.1) Where the first part on the right gives the ordinary Cournot profits for the cartel and the second part gives the fine times the probability of being detected.
In order to model information asymmetry between the colluding firms and the antitrust authority this study adopts the strategy from Besanko and Spulber (1989). The marginal cost is assumed to take either one of two possible values ci ∈ c1, c2 with c2 > c1 for a given industry. The cartel knows which of the two cost types it is but this information is successfully disclosed from the antitrust authority. The antitrust authority assigns a probability α ∈ (0, 1) to the event c = c1 and subsequently (1 − α) to the event c = c2. The cases α = 1 or α = 0 correspond to a situation where the authority would know the cost type of the industry with certainty and are therefore excluded.
The case of information asymmetry results in the following expected profit function: π(Q, ci) = (P (Q) − ci)Q − φ(α(P (Q) − p∗1) + (1 − α)(P (Q) − p∗2))Q, i ∈ {1, 2}.
(3.2) As can been seen from this profit function, a cartel with low marginal costs expects to receive a lower fine than they deserve since c1 < c2. Therefore it might be raising its markup in order to maximise the expected profit. The reverse can be hypothesised for an industry with high marginal costs assuming that a cartel is unable to successfully
recover or object the undeserved part of the fine. Therefore information asymmetry is expected to have its implications on the size of the overcharge.
The next section provides a strategy to derive the estimation power of the overcharge for the total cartel harm using the results of the above described models.
3.2
Decomposing total harm
The goal of this section is to show how this study derives the results on the estimation power of the overcharge, this is done according to a graph of the situation described in the model specifications in section 3.1. Ratios of different parts of the total harm to the overcharge are used. Although the ratios itself are defined differently this strategy is very much alike to that of Han et al. (2009).
Recall figure 2.1 of the previous chapter. Rebuilding this figure to the setting of the model defined in the previous section gives insight in the changes in welfare and surpluses due to collusive activities. Figure 3.1 shows a rise in prices from the Cournot equilibrium p∗ to the price under cartel formation pF. There again arises an overcharge but now the welfare change is given the consumer welfare loss plus the producer welfare loss (CW L + P W L), the colluding firms thus also bear a part of the loss in welfare. The consumer surplus under cartel formation is given CSF whereas the consumer surplus in the competitive cournot equilibrium would be given by CSF + OC + CW L and P S is the part of the producer surplus attained by the firms in the market in both cases.
The quantities of interest for the overcharge to estimate then are the total harm, the change in welfare and the change in consumer surplus. Therefore three ratios are defined. The overcharge to the total harm given by OCH = OC+CW L+P W LOC = −∆W +OCOC , where ∆W = P S +OC +CSF−(P S +OC +CSF+CW L+P W L) = −(CW L+P W L). To correct for the fact that the firms attain more profit in the case of cartel formation it is economically more relevant to estimate the change in welfare caused by the cartel formation. Therefore this study also derives the ratio of the overcharge to the change in welfare given by −∆WOC . In addition the interest of the antitrust authority may lie solely with the consumer welfare. It then is of interest to derive how well the overcharge esti-mates the consumer harm or change in consumer surplus. This ratio is given by −∆CSOC where ∆CS = CSF − (CSF + OC + CW L) = −(OC + CW L).
C0(Q) P (Q) OC CW L P W L CSF P S Q∗ QF Q pF p∗ p c
Figure 3.1: Welfare and surplus changes
The next chapter derives and analyses the results from the model outlined in this chapter to establish the effect of the possible fines to the behaviour of colluding firms and the estimation power of the overcharge for the total harm caused by cartel formation. The power of approximation is analysed according to the ratios defined in this chapter.
4
Results
The model specified in the previous chapter provides the basis for addressing the ques-tion whether the expected fines alter the size of the overcharge and its estimaques-tion power of total cartel harm. This chapter provides the results based on this model through a structured derivation of the change in welfare and the ratio of the overcharge to total harm.
The structure of this chapter is as follows. First, section 4.1 provides an analysis for the general case, no further assumptions on the demand are made. The goal of section 4.1 is to establish for which effective damage multiplier ϕ it is optimal for firms to form a cartel and for which it is not. This provides restrictions on ϕ that are useful in the analysis of the subsequent sections. Next to that, section 4.1 derives the effect of the effective damage multiplier on the quantity produced in the market in general.
Section 4.2 then imposes a linear demand function and assumes perfect information for the antitrust authority in order to analytically derive the total harm caused by the colluding firms. This is done by first deriving the equilibria for Cournot competition, an unrestricted cartel and a cartel expecting fines. These equilibria provide the prices, quantities and profits to derive the overcharge, changes in consumer and producer sur-plus and with that the change in welfare as a consequence of cartel formation within
the market. Using the overcharge and the change in welfare the ratios of the overcharge to welfare change and total cartel harm are derived. Furthermore, the influence of the effective damage multiplicator on changes in surplus and welfare is of interest as it gives an implication of how an antitrust authority could control or influence the harm caused by collusive activity. Finally the dependence of the ratios on the effective damage mul-tiplicator is an important factor in the analysis since it shows how the estimation power of the overcharge for total harm which might have implications for the policy of an antitrust authority. Section 4.3 has the exact same build up as section 4.2 only that in this section it is not longer assumed that the antitrust authority is able to observe the industry’s characteristics, which is modelled as described in the previous chapter. To prevent mathematical clutter section 4.3 does not provide derivating for the ratios, but omitting these does not harm the reader’s insight in the analysis.
4.1
General case
This section gives analysis to the influence of the effective damage multiplicator on the total quantity produced by the industry and derives for which values of the multiplicator it is optimal for the firms in the industry to collude. These bounds are used in the anlysis of the subsequent sections.
The first step necessary for obtaining results is to establish the conditions for a symmetric Cournot equilibrium and a cartel without the existence of any antitrust authority. Finding the Cournot equilibrium starts with optimising the profit function of an individual firm.
max qi
πi(qi) = p(qi+ Q−i)qi− c(qi) Which gives the following first order condition:
∂πi(qi) ∂qi
= P0(qi+ Q−i)qi+ P (qi+ Q−i) − c = 0.
Using the assumption of a symmetric market gives qi = Qn, then the Cournot quantity Q∗ satisfies:
P0(Q∗)Q ∗
n + P (Q
∗) − c = 0 (4.1)
and the second order condition: ∂2πi ∂qi2 = P 00(q i+ Q−i)qi+ 2P0(qi+ Q−i) ≤ 0 ⇒P00(Q∗)Q ∗ n + 2P 0(Q∗) ≤ 0.
A full cartel without an antitrust authority would act like a monopolist which gives the following optimisation problem:
max
With first order condition ∂π(Q)
∂Q = P
0(Q)Q + P (Q) − c
⇒P0(Qk)Qk+ P (Qk) − c = 0, (4.2) and second order condition
∂2π(Q) ∂Q2 = P
00(Q)Q + 2P0(Q) ⇒P00(Qk)Qk+ 2P0(Qk) ≤ 0
It is useful to establish for which effective damage multiplicator ϕ a collusive quan-tity between Q∗ and Qk would be optimal because if producing more than Q∗ is optimal a cartel would not form in the first place and a quantity smaller than Qk is expected to always be suboptimal. The condition for which this holds is captured in the following proposition.
Proposition 4.1.1. For competing firms it is optimal to collude, if and only if 0 ≤ ϕ < ϕ∗ = n − 1
n .
Proof. Start by rewriting equation 3.1 to
π(Q) = (1 − ϕ)P (Q)Q + (ϕp∗− c)Q Optimising results in the following first order condition
∂π(Q)
∂Q = (1 − ϕ)P
0(Q)Q + (1 − ϕ)P (Q)Q + ϕp∗− c = 0
⇒ (1 − ϕ)(P0(Q)Q + P (Q)) + ϕp∗− c = 0. (4.3) Define g(ϕ) := ∂π(Q)∂Q , which makes it possible to distinguish between different cases for ϕ. The case ϕ = 0 gives g(0) = P0(Q)Q + P (Q) − c, so an effective damage multiplicator of 0 gives the unrestricted cartel quantity as expected since there is effectively no fine. The case ϕ = 1 gives g(1) = p∗− c > 0 showing that the first order condition is never met for any Q.
The are, however, more options for the case ϕ ∈ (0, 1). Substituting the unrestricted cartel quantity Qk for Q in equation 4.3 gives
∂π(Q) ∂Q Q=Qk = (1 − ϕ)(P (Qk) + QkP0(Qk)) + ϕp∗− c = ϕ(p∗− c) > 0,
where equation 4.2 is substituted in the first line. This result shows that if ϕ ∈ (0, 1) it is never optimal to produce the unrestricted cartel quantity since the first order condition would never hold. In policy terms this means that if the antitrust authority is able
to impose an effective damage multiplier that is greater than zero the cartel will be deterred from producing the monopoly quantity.
Substituting Q∗ for Q in equation 4.3 and using Q∗P0(∗) = −n(P (Q∗) − c) from equation 4.1 gives ∂π(Q) ∂Q Q=Q∗ = (1 − ϕ)(P (Q∗) + Q∗P0(Q∗)) + ϕP (Q∗) − c = (1 − ϕ)(P (Q∗) − n(P (Q∗) − c)) + ϕP (Q∗) − c = (1 − n(1 − ϕ))(P (Q∗) − c). If ∂π(Q)∂Q Q=Q∗
> 0 it is optimal for a cartel to produce Q∗ which is the competitive amount and thus a cartel will not be formed. If, however, ∂π(Q)∂Q
Q=Q∗ < 0 there exists a ¯Q ∈ (Qk, Q∗) which is the optimal cartel quantity.
Solving ∂π(Q)∂Q Q=Q∗ < 0 for ϕ gives ∂π(Q) ∂Q Q=Q∗ = (1 − n(1 − ϕ))(P (Q∗) − c) < 0 ⇒ 1 − n + nϕ ≤ 0 ⇒ ϕ < ϕ∗ = n − 1 n .
This shows that in order for an antitrust authority to totally deter cartel formation it must at least establish an effective damage multiplicator ϕ of 12 for a market with two firms where a ϕ approaching 1 is needed if the amount of firms in the market becomes large. Furthermore, this implies that in the further analysis of the effect of ϕ on the overcharge and welfare in the subsequent sections the largest ϕ that is still relevant is given by ϕ∗. The remainder of this study assumes that a cartel will form, so it that 0 ≤ ϕ < ϕ∗ holds.
A second result to obtain in the general case is how the quantity to be produced in the market reacts to the size of ϕ. This is given in the following proposition.
Proposition 4.1.2. There exists a positive relation between the effective damage multiplicator ϕ and the total output in the market.
Proof. To find this result, apply the implicit function theorem on the first order condi-tion given by equacondi-tion 4.3, and assume ∂2∂Qπ(Q)2 = (1 − ϕ)(P00(Q)Q + 2P (Q)) < 0. This
gives dQ dϕ ≈ − ∂2π(Q) ∂Q∂ϕ ∂2π(Q) ∂Q2 = − p ∗− (P0(Q)Q + P (Q)) (1 − ϕ)(P00(Q)Q + 2P0(Q) = − p ∗−c−ϕp∗ 1−ϕ (1 − ϕ)(P00(Q)Q + 2P0(Q) = − p ∗− c (1 − ϕ)2(P00(Q)Q + 2P0(Q)) > 0
Where on the second line the substitution P0(Q)Q + P (Q) = c−ϕp1−ϕ∗ is obtained from equation 4.3.
This establishes a positive relation between the effective damage multiplicator and the produced quantity. Thus, if ϕ is set at a higher level then the quantity produced goes up and if ϕ is set at a lower level the quantity produced goes down. This result corre-sponds to the hypothesis that if the expected fine to be paid by the cartel increases, the produced collusive quantity increases and thus the price mark-up decreases. The lower price mark-up means less damage done to the direct purchasers but also a lower cartel profit partly diminishing the welfare increase from the lower direct purchasers harm. This ensures the antitrust authority that as long as they are able to rise the effective damage multiplier by, for instance, stricter legislation or a high investigation effort they will incentivise a higher total output.
Having established that there exists a positive relationship between the effective damage multiplier and that for ϕ ∈ (0,n−1n ) it is optimal for a cartel to produce a quantity that is lower than the quantity in the Cournot equilibrium and thus for firms to participate in collusive activity, this can be used in further analysis since it gives the knowledge that an effective damage multiplier ϕ > n−1n is not relevant for the effects the overcharge as estimate of total cartel harm. The next section introduces a linear demand function and assumes perfect information for the antitrust authority.
4.2
Linear demand, perfect information
This section assumes linear demand with corresponding inverse demand function P (Q) = a − bQ with a > c and b > 0 arbitrary constants to enhance generality. Furthermore the antitrust authority is able to perfectly observe the characteristics of the firms in the industry.
The analysis starts with the symmetric Cournot and the unrestricted cartel equilib-ria, summarised in lemma 4.2.1.
Cournot equilibrium are given by Q∗= n n + 1 a − c b , p ∗ = a + nc n + 1 π∗ = (a − c) 2 b n (n + 1)2, for the case of an unrestricted cartel these are given by
Qk= a − c 2b , p k= a + c 2 πk= (a − c) 2 4b . Proof. max qi πi(qi) = (a − bqi− Q−i)qi− cqi ∂πi(qi) ∂qi = a − 2bqi− bQ−i− c ≡ 0 ⇒ qi = a − bQ−i− c 2b , using symmetry: q = Q n ⇒ q ∗ = a − c b(n + 1), Q∗= n n + 1 a − c b , p ∗ = a − bQ∗= a + nc n + 1 π∗= (p∗− c)Q∗ = (a − c) 2 b n (n + 1)2 give the competitive cartel quantity and price, and
max Q π(Q) = (a − bQ)Q − cQ ∂π(Q) ∂Q = a − 2bQ − c ≡ 0 ⇒ Qk= a − c 2b , p k = a + c 2 πk= (pk− c)Qk= (a − c) 2 4b give the unrestricted cartel quantity and price.
For the situation where the probability of receiving a fine is anticipated by the col-luding firms the linear demand function is substituted in equation 3.1. The results are captured in the following proposition.
Proposition 4.2.2. The price, aggregate quantity and aggregate profit when the colluding firms anticipate antitrust enforcement are given by
QF = ((1 − ϕ)n + 1)(a − c) 2b(1 − ϕ)(n + 1) , pF = (1 − ϕ)((a + c)n + 2a) − a + c 2(1 − ϕ)(n + 1) , πF = (a − c) 2(1 + n(1 − ϕ))2 4b(1 − ϕ)(n + 1)2 .
Proof. The following optimisation process derives the above results max Q π(Q) = (1 − ϕ)(a − bQ)Q + (ϕp ∗− c)Q ∂πi(qi) ∂qi = (1 − ϕ)(a − bQ) − b(1 − ϕ)Q + ϕp∗− c ≡ 0 ⇒ (1 − ϕ)a − 2b(1 − ϕ)Q + ϕp∗− c = 0 QF = (1 − ϕ)a + ϕp ∗− c 2b(1 − ϕ) substituting p∗ and rewriting gives
QF = ((1 − ϕ)n + 1)(a − c) 2b(1 − ϕ)(n + 1) pF = a − bQF = a − ((1 − ϕ)n + 1)(a − c) 2(1 − ϕ)(n + 1) = (1 − ϕ)((a + c)n + 2a) − a + c 2(1 − ϕ)(n + 1) πF = (1 − ϕ)pFQF + (ϕp∗− c)QF = (a − c) 2(1 + n(1 − ϕ))2 4b(1 − ϕ)(n + 1)2 .
Figure 4.1 shows the course of the profit, price and quantity under cartel formation. In figure 4.1a the effective damage multiplicator ϕ varies and n = 10 whereas in figure 4.1b n varies and ϕ = 12. From figure 4.1a it is observed that the price and profit are decreasing in ϕ whereas the total output rises in ϕ so that a higher effective damage multiplicator increases welfare. The Cournot profit is, as it should be, independent of the damage multiplicator and the intersection with the cartel profit shows that indeed for ϕ = n−1n competing becomes more profitable for firms than colluding. Figure 4.1b shows that prices increase when the number of colluding firms grows but that quantity and profit slightly decrease in n. Furthermore, the Cournot profit is at the same level as the cartel profit for n = 2 since ϕ = 12 but is always below the cartel profit for a market with more than two firms. The difference between the Cournot profit and the cartel profit increases when the number of firms increases because the competition becomes more fierce before cartel formation increasing the aggregate profit gains from colluding.
Using the results from lemma 4.2.1 and proposition 4.2.2 it is possible to calculate the overcharge and derive the effect of ϕ on the overcharge. The overcharge is given by
OC = (pF − p∗)QF = (a − c)
2((1 − ϕ)n − 1)((1 − ϕ)n + 1)
4b(1 − ϕ)2(n + 1)2 , (4.4)
taking the derivative with respect to ϕ gives ∂OC
∂ϕ = −
(a − c)2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 QF pF πF π∗ ϕ
(a) Cartel profit, quantity and prices for n = 10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 0.5 1 1.5 2 2.5 3 QF pF πF π∗ n
(b) Cartel profit, quantity and prices for ϕ = 0.5
Figure 4.1
This shows that the effective damage multiplicator has a negative effect on the over-charge. Meaning that if an antitrust authority is able to effectively increase the damage multiplicator the size of the overcharge will decline. It is thus optimal for colluding firms to lower the overcharge when the expected fine rises in order to maximize their profit.
To derive the effects on welfare as a consequence of the possible fining by an antitrust authority the change in consumer and producer surplus are needed so that the change in welfare can be defined as the sum of these two. Define the change in consumer surplus as ∆CS = CSF−CS∗and the change in producer surplus as ∆π = πF−π∗. The consumer surplus in the Cournot equilibrium is given by the triangle CSF+ OC + CW L in figure 3.1 where the maximum price is given by the intersection of the inverse demand function with the y-axis corresponding to the constant a.
CS∗ = 1 2(p max− p∗)Q∗ = 1 2 a −a + nc n + 1 (a − c)n (n + 1)b = (a − c) 2 2b n n + 1 2 , CSF = 1 2(p max− pF)QF = 1 2 a − a +(a − c)((1 − ϕ)n + 1) 2(1 − ϕ)(n + 1) ((1 − ϕ)n + 1)(a − c) 2b(1 − ϕ)(n + 1) = 1 8b (a − c)((1 − ϕ)n + 1) (1 − ϕ)(n + 1) 2 , ∆CS = CSF − CS∗= 1 8b (a − c)((1 − ϕ)n + 1) (1 − ϕ)(n + 1) 2 −(a − c) 2 2b n n + 1 2 = (a − c) 2 8b(n + 1)2(1 − ϕ)2(1 + 2n(1 − ϕ) − 3n 2(1 − ϕ)2) < 0. Taking the derivative with respect to ϕ gives
∂∆CS ∂ϕ = a − c n + 1 2(1 − ϕ)n + 1 4b(1 − ϕ)3 > 0
As expected the change in consumer surplus is always negative, so that collusive activ-ities always harm the direct purchasers. The strict positiveness of the derivative shows
that the change in consumer surplus depends positively on the change in the damage multiplicator. This result is intuitive since a higher damage multiplier then narrows the difference between the consumer surplus in the competitive equilibrium and the consumer surplus when firms are colluding.
The change in producer surplus is given by ∆π = πF − π∗ = (a − c) 2(1 + n(1 − ϕ))2 4b(1 − ϕ)(n + 1)2 − n(a − c)2 b(n + 1)2 = (a − c) 2(1 − n(1 − ϕ))2 4b(n + 1)2(1 − ϕ) > 0, with derivative ∂∆π ∂ϕ = (a − c)2(1 + n(1 − ϕ))(1 − n(1 − ϕ)) 4b(n + 1)2(1 − ϕ)2 < 0.
This shows that the profit when colluding is always higher than the competitive Cournot profit for ϕ ∈ (0,n−1n ). Furthermore, the negative derivative with respect to ϕ implies a negative effect of the effective damage multiplicator on the change in the profit. This result, again, is intuitive since it means that if the expected fine is larger the expected additional profit from colluding is lower and therefore the difference diminishes.
Now that the change in consumer and producer surplus is known, it is possible to define the change in welfare caused by cartel formation. Define ∆W = ∆π + ∆CS + ϕOC, this means that the expected fine paid by the cartel to the antitrust authority is redistributed to the consumers in some way. The fine is then just a shift in surplus between consumers and producers and has no influence on the welfare change itself. Using the former definition the change in welfare becomes
∆W = ∆CS + ∆π + ϕOC = 1 8b a − c (1 − ϕ)(n + 1) 2 (1 − n(1 − ϕ))(3 + n(1 − ϕ) − 4ϕ) < 0 (4.5) with derivative ∂∆W ∂ϕ = (a − c)2(1 + n(1 − ϕ) − 2ϕ) 4b(n + 1)2(1 − ϕ)3 > 0
Showing that the change in welfare is always negative corresponding to −(CW L+P W L) in 3.1. This shows that if the fine is a shift of surplus from producers to consumers a positive relation between ϕ and the welfare effect exists. It is thus always optimal to increase ϕ in order to decrease the loss in welfare.
Before analysing the approximation precision of overcharge it is of interest to analyse the harm, welfare and overcharge itself. In order to do so define the total harm as the area H = OC + CW L + P W L in figure 3.1. The course of the total harm together with
that of the overcharge and the welfare is shown in figure 4.2. Figure 4.2a shows that, as expected, the overcharge and total harm decrease when the effective damage mul-tiplicator rises whereas the welfare is increasing in the damage mulmul-tiplicator ϕ. When the damage multiplicator approaches n−1n both the overcharge and the total harm de-crease to zero because firms start competing at that point instead of colluding. When the damage multiplicator is held constant figure 4.2b shows that the overcharge and the total harm increase when the number of colluding firms increases whereas the reverse holds true for the welfare. This corresponds to the findings of figure 4.1b, since if there is a higher degree of competition before cartel formation the price mark-up that can be achieved by colluding is higher. This raises the overcharge and the total harm as a consequence of cartel formation when there are a lot of competing firms in the market pre-cartel formation. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 H W OC ϕ
(a) Total harm, welfare and the overcharge for n = 10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 0.5 1 1.5 2 2.5 3 H W OC n
(b) Total harm, welfare and the overcharge for ϕ = 0.5
Figure 4.2
To answer the question if the expectation of fines taken into account by colluding firms alters the degree of correctness of the overcharge as a measure of total cartel harm the ratio of the overcharge to total harm is of interest. This ratio is given by
OC
H =
2(1 + n(1 − ϕ)) 5 + 3n(1 − ϕ) − 4ϕ > 0
This ratio is drawn in figure 4.3 for fixed values of n. The graph shows that for an effective damage multiplier lower than 12 the overcharge underestimates the total harm more for a market with a low number of firms than for a market with a high number of firms. This effect reverses for a damage multiplicator higher than 12. It is clear that the overcharge is smaller for a small market since there is less competition prior to collusive agreements so the change in welfare is relatively large for small markets when the effective damage multiplicator is small and this reverses when the effective damage multiplicator is large. Remarkably for a multiplicator of exactly 12 the number of firms is not of influence on the ratio. The overcharge then underestimates the total harm by 33 percent.
To see what happens for the special case where the damage multiplicator is 12, substituting ϕ = 12 in equations 4.4 and 4.5 gives a useful insight shown in the equations below. OC|ϕ=1 2 = − (a − c)2 b (1 −12n)(1 +12n) (n + 1)2 ∆W |ϕ=1 2 = (a − c)2 2b (1 −12n)(1 + 12n) (n + 1)2 = − 1 2OC|ϕ=12
This shows that the loss in welfare is exactly half the overcharge independent of the amount of firms colluding when the damage multiplicator is fixed at 12. For damage multiplicators lower than 12 the loss in welfare is larger than half the overcharge and when the damage multiplicator is larger than 12 the loss in welfare is smaller than half the overcharge.
Takings limits for ϕ → 0 results in values ranging between 116 for n = 2 to 23 for n → ∞ and for ϕ → n−1n this ranges between 23 for n = 2 to 1 for n → ∞. In the worst possible case the overcharge thus underestimates the total harm by 45 percent. In addition, the ratio is dependent on the amount of firms in the industry. For instance, if the damage multiplicator approaches its limit for cartel formation to be optimal an industry with a small amount of firms still has a positive loss in consumer and producer welfare. The loss in welfare would decrease for damage multiplicators closer to unity, but in these cases firms will not collude in the first place so that the loss in welfare takes a jump from zero to the corresponding loss in welfare at the critical damage multiplicator ϕ∗ = n−1n . In general this shows that the overcharge underestimates the total harm caused by cartel formation.
To analytically determine the dependency of the ratio on the effective damage mul-tiplactor take derivative of the overcharge to the total harm with respect to ϕ, given by
∂OCH
∂ϕ =
4(n + 2)
4ϕ − 5 − 3n(1 − ϕ))2 > 0
The strict positiveness of the derivative proves that the ratio is increasing in ϕ, so that the higher the effective damage multiplicator becomes the better the overcharge approximates the total true cartel harm. This implies that as the damage multiplicator increases the loss in welfare decreases faster than the overcharge.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 n = 2 n = 10 n = 100 OC H ϕ
Figure 4.3: Ratio of overcharge to total harm
The total harm however, does not correct for the fact that colluding firms achieve higher profits than when they are competing. From an economic perspective this should be corrected since a rise in producer surplus is not lost welfare. The question arises how well the overcharge estimates the loss in welfare −∆W = −(CW L + P W L). Using the assumption of the fine being a shift of surplus between consumers and producers the ratio of the overcharge and loss in welfare is given by
OC
−∆W =
2(1 + n(1 − ϕ) 3 − 4ϕ + n(1 − ϕ) > 0
Figure 4.4a shows this ratio for fixed values of n. These curves show that the overcharge is always larger than the change in welfare. Furthermore the graph coincides with figure 4.3 in the sense that for ϕ = 12 the ratio is independent of the number of firms in the market. In contrast, however, figure 4.4a shows that for a low damage multiplicator the overcharge is a better estimate of the change in welfare for a market with a small number of firms than for a market with a large number of firms and this effect is reversed for a high damage multiplicator.
Taking the absolute value of the change in welfare the limits for ϕ → 0 give a ratio of 65 for n = 2 to 2 for n → ∞ and taking the limit for ϕ → n−1n gives a ratio of n which implies that for a high effective damage multiplicator the overcharge over change in welfare ratio can grow infinitely large if n → ∞, as is also implied by the curves in figure 4.4a. The overcharge thus overestimates the welfare loss by 20 to 100 per cent as long as the damage multiplicator is below 12 but the percentage rises sharply for a large damage multiplicator when the number of firms participating grows very large. So the welfare not only decreases faster when the damage multiplicator increases, as found in the harm to overcharge ratio, the loss in welfare is also always smaller than the overcharge. In addition, figure 4.4a shows consistency with the results found above for a damage multiplicator equal to 12 because for lower damage multiplicators the loss in
welfare is larger than half the overcharge and the reverse for high damage multiplicators. Again, the derivative is needed to show the effect of the damage multiplicator. The derivative of the overcharge to welfare ratio with respect to ϕ is given by
∂−∆WOC
∂ϕ =
4(n + 2)
(4ϕ − 3 − n(1 − ϕ))2 > 0
This proves that the ratio is strictly increasing in ϕ. Thus as the damage multiplica-tor rises the power of the overcharge in approximating pure welfare loss deteriorates. Concluding, using the overcharge to estimate the change in welfare can lead to very misleading results.
The antitrust authority might only be interested in pursuing minimisation of the harm to consumers, corresponding to −∆CS = −(OC + CW L) in figure 3.1. Therefore, next to using the overcharge to estimate total harm and welfare change it is of interest to know if the overcharge is a suited estimator for the consumer harm when the probability of being fined is taken into account by the cartel. In order to do so define the ratio of the overcharge to the consumer harm by
OC −∆CS =
2(1 + n(1 − ϕ)) 1 + 3n(1 − ϕ) > 0
Figure 4.4b graphs this ratio for fixed n. Observing from the curves, the lower the number of firms in the market the closer the ratio is to one for low values of ϕ. For ϕ approaching its upperbound n−1n all ratios are equal to one. This implies that if the effective damager multiplicator is large enough the overcharge almost perfectly estimates the consumer harm. The lower the amount of firms the faster a one to one ratio is reached.
Taking limits for ϕ → 0 the overcharge to consumer harm ratio ranges from 67 for n = 2 to 23 for n → ∞ and the limit for ϕ → n−1n is given by 1 for all n corresponding with figure 4.4b. This analysis therefore shows that if the antitrust authority aims to fine according to the lost consumer surplus the overcharge is only underestimating the consumer harm with a small amount for markets that consists of a small number of firms. For instance if the market consists of two firms the overcharge underestimates consumer harm by 0 to 15 per cent while with a large amount of firms the underestimation percentage can rise to 33 per cent.
The effective damage multiplicator has a positive effect on this ratio as the following derivative shows ∂ OC −∆CS ∂ϕ = 4n (1 + 3n(1 − ϕ))2 > 0
so a rise in the effective damage multiplicator results in a better estimation of the consumer harm by the overcharge. The implication of this effect is that the deadweight loss decreases faster than the overcharge and therefore the overcharge divided by the consumer harm gets closer to a one to one ratio.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10 11 n = 2 n = 10 n = 100 OC −∆W ϕ
(a) Ratio of overcharge to welfare change
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 n = 2 n = 10 n = 100 OC −∆CS ϕ
(b) Ratio of overcharge to consumer harm
Figure 4.4
This section has derived the effects of the effective damage multiplicator on the estimation power of the overcharge for total cartel harm, welfare change and consumer harm by calculating their ratios with the overcharge under perfect information for the antitrust authority. All ratios are increasing in ϕ which ensures a better estimate for the total harm and the consumer harm but a worse estimate for the welfare change when the damage multiplicator increases. The overcharge generally underestimates the total harm caused but cartel formation but the estimate becomes better when the number of firms in the market and the damage multiplicator rise. The next section performs the same derivation steps but loosens the assumptions of perfect information.
4.3
Linear demand, asymmetric information
This section expands the analysis of the previous section by loosening the assumption of perfect information. Now the antitrust authority is not able to observe the characteristics of the firms and therefore has to estimate the costs colluding firms make in order to determine the overcharge. The model describes this by introducing the probability α of firms being cost type 1 and 1 − α of firms being cost type 2, α is what is called a behavioural variable and represents the beliefs of the antitrust authority and firms know the size of α. Therefore the expected profit function of colluding firms changes, incorporating the costtype probability α.
Again the analysis begins with the symmetric Cournot and the unrestricted cartel equilibria. Recalling from lemma 4.2.1 in section 4.2 results in
qi∗= a − ci b(n + 1), Q ∗ i = n n + 1 a − ci b , p∗i = a − bQ∗ = a + nci n + 1 , π ∗ i = (p∗− ci)Q∗= (a − ci)2 b n (n + 1)2, Qki = a − ci 2b , p k i = a + ci 2 , πki = (pk− ci)Qk = (a − ci) 2 4b , for i = {1, 2},
representing prices, quantity and profits in Cournot competition and unrestricted cartel formation for a cartel with cost type i.
Proposition 4.3.1. below captures the equilibrium output, price and aggregate profit for cartel formation anticipating fines with asymmetric information for the antitrust authority.
Proposition 4.3.1. The output, prices and aggregate profits for cartel formation anticipating fines under asymmetric information for cost type 1 and cost type 2 are given by QF1 = (a − c1)(n + 1) − nϕ(a − αc1− (1 − α)c2) 2b(n + 1)(1 − ϕ) , QF2 = (a − c2)(n + 1) − nϕ(a − αc1− (1 − α)c2) 2b(n + 1)(1 − ϕ) , pF1 = a − bQF1, pF2 = a − bQF2, π1F = ((a − c1)(n + 1) − nϕ(a − αc1− (1 − α)c2)) 2 4b(n + 1)2(1 − ϕ) , π2F = ((a − c2)(n + 1) − nϕ(a − αc1− (1 − α)c2)) 2 4b(n + 1)2(1 − ϕ) .
Proof. The optimisation problem incorporating cost type assignment of the antitrust authority becomes max Q πi(Q) = P (Q)Q − ciQ − ϕ((P (Q) − p ∗ 1)Qα + (P (Q) − p ∗ 2)Q(1 − α)) = (1 − ϕ)(a − bQ)Q + (ϕαp∗q+ (1 − α)ϕp∗2− ci)Q,
with first order condition ∂πi(Q) ∂Q = (1 − ϕ)(a − 2bQ) + ϕαp ∗ 1+ (1α)φp∗2− ci ≡ 0 ⇒ QFi = (1 − ϕ)a + αϕp ∗ 1+ (1 − α)ϕp∗2− ci (1 − ϕ)2b .
Substituting p∗1 and p∗2 for cost types 1 and 2 and substituting the resulting quantities back in the profit function gives the results in lemma 4.3.1.
The profits of both cost types show that the profit is dependent of the prediction probability α. Furthermore, the profit functions show a familiar structure with the profit function under perfect information but that is partly dependable on the marginal costs of the other cartel. It is clear that the efficient cartel with low marginal costs always obtains profits at least as high as the inefficient cartel since c1 < c2.
Important to note however, is that the cartel price pF1 should always be at least as high as the Cournot equilibrium price p∗2 since if the low cost cartel sets a lower price than the competitive equilibrium price of the high cost type industry the antitrust au-thority knows the cartel has low marginal costs.
