## Abstract

We propose a new penalty regime for cartels in which the penalty *base* is the revenue of the cartel but the penalty *rate* increases in a systematic and transparent way with the cartel overcharge. The proposed regime formalises how revenue can be used as the base while taking into account the severity of the offence. We show that this regime has better welfare properties than the *simple* revenue-based regime under which the penalty rate is fixed, while having relatively low levels of implementation costs and uncertainty. We conclude that the proposed penalty regime deserves serious consideration by Competition Authorities.

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## Notes

- 1.
- 2.
We recognise that another dimension of severity is recidivism. While we have related work that takes this into account—see Katsoulacos et al. (2016)—including it in the context of penalty design introduces a degree of history dependence which makes the analysis intractable, so we leave analysis of this issues to future research.

- 3.
The issues of transparency, predictability of legal sanctions and legal certainty (or discretion) are discussed extensively in the ICN report (2008) that is based on the survey conducted by the ICN among the CAs. Discussion on pages 12 and 13 of this report suggests that transparency and legal certainty are preferred fundamental legal principles, which also help reduce litigation costs. In particular, it is mentioned that in jurisdictions, where sufficiently high sanctions are available (such as the US and the EU), the higher degree of certainty with respect to how fines are determined is preferred. The report also mentions that in some jurisdictions CAs take the view that risk-averse managers may be more deterred by a penalty regime that has more uncertainty. But at the same time it warns that CAs in these jurisdictions would have to incur higher costs to justify their ‘decisions in front of the bodies that approve the agency’s proposal or review the agency’s decision’. The report concludes that ‘the less discretion in determination of fines by the agency, the lower the degree of litigation on the amount of the fine by companies or individuals who have been fined. Enforcers in jurisdictions with uncertainty as to how fines are determined may also face public criticism of their fining system as subjective or arbitrary.’

- 4.
It should be clear that our paper is a simple piece of advocacy for replacing the existing penalty regime with a better one. We do not attempt to derive any sort of optimum penalty regime. This is because (1) we do not know what is the right objective function for CAs that encompasses welfare, implementation and transparency concerns; and (2) we do not know what is the appropriate distribution of cartel cases across different industries. As a matter of advocacy we think that it is better to present CAs with a single alternative that offers a refinement of the currently employed system: Our alternative allows a welfare improvement in terms of the reduction of consumer harm and at the same time remains transparent and easy to implement. As such our paper is consistent with a wide range of recent literature that recognises that penalty setting is an inherently second-best exercise. See for example: Bos and Schinkel (2006), Buccirossi and Spagnolo (2007), Schinkel (2007), Veljanovski (2007), Connor and Lande (2008), Allain et al. (2011), Bageri et al. (2013), Katsoulacos et al. (2015), Spagnolo and Marvão (2018), Dargaud et al. (2016), or Houba et al. (2018).

- 5.
See, e.g., Block et al. (1981), Tirole (1988, ch. 6), Chen and Rey (2013) or Katsoulacos and Ulph (2013). The homogeneous products/constant marginal costs framework has its limitations. However, what makes it such a powerful workhorse model is that it has the feature that in the absence of collusion competition is intense and the incentives to form a cartel are strongest. So it is important to test the effectiveness of penalty regimes in such an environment.

- 6.
- 7.
Note that in practice cartel duration will influence the size of penalties. It is difficult formally to introduce the impact of fines that depend on cartel duration in the stationary repeated-games framework. However, its influence will be exactly the same on the simple revenue-based and the sophisticated revenue-based penalties (as in both cases per period penalties will have to be adjusted by multiplying by the duration of the offence) and, hence, will not affect the comparison between them.

- 8.
A different assumption, where following a prosecution the cartel never forms again, has been made in a number of contributions, such as, e.g., Harrington (2004) or Bos et al. (2018). In Katsoulacos et al. (2016) we unify the two different assumptions by looking at the probability of re-emergence of collusive activity following successful prosecution. This generalization produces more complex formulae for cartel value

*V*(.) but does not affect the main qualitative results of the current paper, so we stick with the simpler assumption. - 9.
As will shortly become clear, even in the absence of a CA stable cartels can only exist if \( \Delta \le 1 \), so in order to understand the deterrence effects of a CA that operates under different penalty regimes it makes sense to restrict attention to the set [0,1]. The assumption of uniformity is made purely for convenience and can be replaced by a more general function without at all affecting the conclusions.

- 10.
While the qualitative nature of our results is largely unaffected by this assumption, we recognise that in practice this will not always be the case. However this assumption is made in many previous contributions by e.g. Motta and Polo (2003), while Spagnolo (2004) shows that not penalizing price deviating firms is the ideal policy. The reference to “any” future prosecution acknowledges that we are ignoring recidivism—see footnote 6.

- 11.
Note that concavity of cartel value function

*V(θ)*ensures the existence of the unique solution for overcharge that maximizes expression in (5). In what follows we assume that this condition is satisfied. - 12.
The maximum critical level of difficulty—Δ—is the direct analogue of the minimum critical discount rate that is used in much of the literature.

- 13.
One reason for repeating the proposition here is that we offer a new method of proof, which we exploit in our analysis of the pricing properties of a sophisticated revenue-based regime in Proposition 2 below.

- 14.
We thank the Editor for pointing this out.

- 15.
Equation (19) is derived by equating \( D_{SR} \) to \( D_{R} (\bar{\theta }^{M} ) \) and solving for \( \sigma_{SR} \), which achieves deterrence equivalence for the target industry type \( \bar{\theta }^{M} \).

- 16.
Nevertheless, e.g. Brander and Ross (2017) demonstrate reliable methods for calculation of overcharges in differentiated products setting, such as the methods that were employed in the

*Microsoft case*(*Pro–Sys Consultants Ltd. v. Microsoft Corporation*, 2013 SCC 57) and in the*Infineon case*(*Pro–Sys Consultants Ltd. v. Infineon Technologies AG*, 2009 BCCA 503). - 17.
Further such concerns apply also to those cases found in practice in which CAs use the overcharge as an “aggravating factor” in setting the penalty rate.

- 18.
See also Brander and Ross (2006).

- 19.
Extending the analysis to an asymmetric setting where not all firms are in the cartel is a non-trivial task, which requires the development of a very different model.

- 20.
To verify this see expressions (21) in Appendix 1 and (24) in Appendix 2, which are obtained by setting up the cartel value functions under simple revenue-based penalty in terms of overcharges or prices and taking the FOC with respect to the overcharge or price, respectively.

- 21.
At this level of generality without restricting analysis to a specific functional form for the demand structure, it is hard to obtain tractable results with respect to comparisons of the deterrence properties in the environment, where but-for prices are above marginal cost.

- 22.
- 23.

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## Acknowledgements

For helpful comments we are grateful to John Davies, Peter Dijkstra, Joe Harrington, Fabienne Ilzkowitz, Frederic Jenny, Tom Ross, Maarten-Pieter Schinkel, and participants at the 12th and 13th Annual CRESSE Conference (July 2017 and 2018 respectively) and at the Symposium on “Cartels: Insights on Fines and Enforcement” that was hosted by the Netherlands Authority for Consumers and Markets in the Hague on May 22nd 2018. We also thank the editor and two referees for their very useful comments and suggestions. We are grateful to acknowledge the financial support received through the Tinbergen Institute, Vrije Universiteit Amsterdam, Short-term Visitor Program.

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## Appendices

### Appendix 1: Proofs of Propositions

###
*Proof of Proposition*
*1*

It is a standard result that the monopoly overcharge is the solution to the equation

Insert (3) from the text into the maximand in (8), differentiate, set the derivative to zero, and re-arrange, and we find that \( \hat{\theta }_{R}^{C} \) is a solution to the equation:

It is readily verified that the RHS of the equation is a decreasing function of θ, while, from (1) in the text, the term on the LHS is a strictly increasing function of θ. Moreover since

Proposition 1 is established. The dashed line in Fig. 2 below illustrates the proof.

###
*Proof of Proposition*
*2*

Insert (4) into the maximand in (8), differentiate, set the derivative to zero and re-arrange and we find that \( \hat{\theta }_{SR}^{C} \) is the solution to the equation:

Then in order to derive the inequality in (11) we need to find the condition on the function \( \rho_{SR} (\theta ) \) such that \( \varphi_{SR} (\theta ) < \frac{1 + \theta }{\theta } \). This will reduce the overcharge \( \hat{\theta }_{SR}^{C} \) below the simple monopoly level \( \theta_{{}}^{M} \). Note that \( \varphi_{SR} (\theta ) < \frac{1 + \theta }{\theta } \) is equivalent to

The last inequality implies that (11) holds and Proposition 2 is established. Solid line in Fig. 2 illustrates the proof.

###
*Proof of Lemma*
*3*

Take a first-order Taylor approximation to \( Y\left( {\beta \rho_{R0} } \right) \) around 0. Then: (i) by the Envelope Theorem \( Y^{\prime}(z) = - R\left( {\hat{\theta }(z)} \right) \) where \( \hat{\theta }(z) \) is the overcharge that maximises \( Y\left( z \right) \); and (ii) when \( z = 0 \), \( \hat{\theta }(0) = \theta^{M} \) we have \( Y\left( {\beta \rho_{R} } \right) \approx Y(0) - \beta \rho_{R} R(\theta^{M} ) = c\theta^{M} Q\left( {c(1 + \theta^{M} )} \right) - \beta \rho_{R} c\left( {1 + \theta^{M} } \right)Q\left( {c(1 + \theta^{M} )} \right). \)

which proves the result.

### Appendix 2: Proof of Extension

This Appendix extends the result of Proposition 2 to industries in which the but-for prices in the absence of collusion are greater than unit cost. Because the but-for price \( p^{B} \ge c \) is now variable, it is useful to do the analysis directly in terms of price rather than overcharge. In such industries if a cartel forms and sets a price \( p > p^{B} \), then the percentage overcharge is \( \theta = {{\left( {p - p^{B} } \right)} \mathord{\left/ {\vphantom {{\left( {p - p^{B} } \right)} {p^{B} }}} \right. \kern-0pt} {p^{B} }} \). Proposition 6 stated in terms of prices shows that the result of Proposition 2 extends to this more general setting.

###
*Proof of Proposition*
*6*

First, it is easy to see that under a simple revenue-based penalty the unconstrained cartel price \( \hat{p}_{R}^{C} \) is independent of \( p^{B} \) and is given by the solution to:

Note that it is above the simple monopoly price \( p^{M} \), which is characterized by \( \eta (p) = \frac{p}{p - c} \). Under sophisticated revenue-based penalty regime the unconstrained cartel price is solution to:

After some manipulation, it is easy to see that

But if the function \( \rho (\theta ) \) satisfies our condition (11) \( \frac{{\rho^{\prime } (\theta )}}{\rho (\theta )} > \frac{1}{\theta (1 + \theta )} \), then it follows that

where the last inequality in (27) holds for all \( p^{B} \ge c \). The condition (11) that we imposed will guarantee that the unconstrained cartel price is below the monopoly price. Indeed, having a but-for price above marginal cost makes it even more likely to be true.

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Katsoulacos, Y., Motchenkova, E. & Ulph, D. Penalising on the Basis of the Severity of the Offence: A Sophisticated Revenue-Based Cartel Penalty.
*Rev Ind Organ* **57, **627–646 (2020). https://doi.org/10.1007/s11151-019-09738-x

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### Keywords

- Antitrust enforcement
- Antitrust penalties
- Antitrust law
- Cartels

### JEL Classification

- L4 antitrust policy
- K21 antitrust law
- D43 oligopoly and other forms of market imperfection