# Empirical methods in the analysis of collusion

## Abstract

Regression methods are commonly used in competition lawsuits for, e.g., determining overcharges in price-fixing cases. Technical evaluations of these methods’ pros and cons are not necessarily intuitive. Appraisals that are based on case studies are descriptive but need not be universally valid. This paper opens up the black box called econometrics for competition cases. This is done by complementing theoretical arguments with estimation results. These results are obtained for data that is generated by a simulation-model of a collusive industry. Using such data leaves little room for debate about the quality of these methods because estimates of, e.g., overcharges can be compared to their true underlying values. This analysis provides arguments for demonstrating that thoroughly conducted econometric analyses yield better results than simple techniques such as before-and-after comparisons.

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## Notes

1. The mentioned cases were investigated by the European Commission under the following case numbers: Bathroom fittings and fixtures 39.091, Belgian beer market 37.614.

2. For μ = ∞ goods are perfect substitutes. For μ = 0 goods are independent. As this paper is interested in analyzing (imperfect) substitutes, μ is set at values greater than 0.

3. For readers interested in reproducing the below regressions and calculations, I will be happy to provide the data in EViews-format on request. Please send an email to johannes.paha@wirtschaft.uni-giessen.de.

4. It can be shown that firms' reaction function is $$p_{i} = {\frac{{c_{i} }}{2}} + {\frac{{n\nu + \mu \sum\nolimits_{(j = 1)\backslash i}^{n} {p_{j} } }}{2 \cdot (n + n\mu - \mu )}}$$ (see, e.g., Paha (2010)).

5. To see this even more clearly, one may choose arbitrary values for n and μ, and consider an arbitrary composition of the cartel. Then, writing Eq. (9) in non-matrix notation for collusive periods, competitive periods, and transition-periods supports the above statements.

6. The calculation is done as follows: One observes that firm 1 sets a collusive price p 1,42 = 31.994 in period 42 and a collusive price of p 1,43 = 31.173 in period 43. The common cost-shock in period 43 is s 43 = −1.354. Applying the above regression, one predicts a hypothetical competitive price of $$\hat{p}_{c,1,43} = - 11.070 + (1 + 0.194) \cdot p_{1,42} + 0.717 \cdot s_{43} = 26.16$$. From the industry-simulation, one can infer the true competitive price p c,1,43 = 25.956 that would be unknown in a real case. Therefore, the predicted price is found to be a good predictor for its true underlying.

7. Please note that the variables used for the coefficients of the regression have a different meaning in Eq. (12) than in Eq. (10) above. On the one hand, this choice is made for reasons of parameter parsimony and, on the other hand, for being more in line with Baker's (1999) notation.

8. In particular, I started with a specification that closely resembled the final specification in Table 3 but included firm-specific interaction effects for all firms. By successively applying F-tests it was possible to show that the fringe firm is the only one for whom the coefficients of p i,t−1 D t−1 and D t differ from those of the other firms in a statistically significant sense. Including the prices of competitors and the Herfindahl–Hirschman-Index supports the above proposition of biased estimates as these variables are correlated with the error term. The consequence can be seen particularly well in the coefficient of p i,t−1 that is biased away from its economically plausible value of one.

9. Suppose that the above cartel was found to have been active in periods 21–24, 39–73, and 82–100. This specification only neglects one cartel-period and three price war periods in times, where short price wars interrupt longer cartel-periods. Estimating Eq. (10) with this faint misspecification results in a downwards biased overcharge estimate of 4.55%.

10. Please note that the last column presents the regression-results of Eq. (13) for firm 2 in periods t = 1–20. The last four rows summarize the results of tests for structural breaks. These results are not interpreted here, as they are used further below.

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## Acknowledgments

I would like to thank Oliver Budzinski, Henning Fischer, Georg Götz, Maarten Janssen, Peter Winker, my colleagues in Giessen, and the participants of the 35th Hohenheimer Oberseminar for their valuable comments on this paper.

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Correspondence to Johannes Paha.

## Appendix 1: Generating a collusive industry

### Appendix 1: Generating a collusive industry

The dataset that is used in the above regressions is generated by running the simulation model with the parameter values given in Table 5. Except for the number of cartelists m, these parameter values are randomly determined from uniform distributions within the bounds provided by Table 5.

Choosing n ∈ [7,15] is reasonable because, first, all firms would join the cartel with probability 1 if the number of firms was too small. Second, the time for calculating the participation-equilibrium rises exponentially in the number of firms and, thus, would be undesirably long for n > 15. From the viewpoint of economic theory, the size of ν is irrelevant as it only affects the size of prices and quantities but does not have an impact on the ratio of profit-measures. Using μ upper  = 100 as an upper bound is reasonable as it suffices to produce rather homogeneous goods. Choosing a 1 ∈ [0.05, 1.0] is reasonable because values below 0.05 would indicate that marginal costs are quite negligible. The production of such firms may be supposed to rather generate fixed costs that, however, are beyond the scope of this model. Choosing a 4 ∈ [0.05, 0.15] gives economically meaningful yet not unrealistically large cost shocks. Drawing P from the wide interval [0.05, 0.4] reflects our lack of knowledge about the effectiveness of competition authorities. This is because one knows the number of discovered cartels but can hardly determine the number of undiscovered ones. The interval encloses the 15–20% detection probability that some studies suggest. Choosing r ∈ [0.05, 0.25] suggests that firms’ discount rate is somewhere between the return of government bonds and some (ambitious) firms’ target value of their return on equity. The number of cartelists m is determined endogenously as the mixed-strategy Nash-equilibrium of the cartel-formation game.

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Paha, J. Empirical methods in the analysis of collusion. Empirica 38, 389–415 (2011). https://doi.org/10.1007/s10663-010-9160-1

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• DOI: https://doi.org/10.1007/s10663-010-9160-1

### Keywords

• Collusion
• Empirical methods
• Industry simulation

• C51
• D43
• K21
• L41