Penalising on the Basis of the Severity of the Offence: A Sophisticated Revenue-Based Cartel Penalty

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.

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

$$ \eta \left[ {c\left( {1 + \theta } \right)} \right] = \frac{1 + \theta }{\theta }. $$

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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:

$$ \eta \left[ {c\left( {1 + \theta } \right)} \right] = \frac{{1 - \beta \rho_{R} }}{{\frac{\theta }{1 + \theta } - \beta \rho_{R} }} \equiv \varphi_{R} (\theta ). $$

(21)

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

$$ \frac{{1 - \beta \rho_{R} }}{{\frac{\theta }{1 + \theta } - \beta \rho_{R} }} > \frac{1 + \theta }{\theta }, $$

(22)

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

Fig. 2

Unconstrained cartel overcharges for a simple revenue-based penalty regime and for a sophisticated revenue-based penalty regime

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:

$$ \eta \left( {c(1 + \theta )} \right) = \frac{{1 - \beta \rho_{SR} (\theta ) - \beta (1 + \theta )\rho_{SR}^{\prime } (\theta )}}{{\frac{\theta }{1 + \theta } - \beta \rho_{SR} (\theta )}} \equiv \varphi_{SR} (\theta ). $$

(23)

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

$$ \frac{{1 - \beta \rho_{SR} (\theta ) - \beta (1 + \theta )\rho_{SR}^{\prime } (\theta )}}{{\frac{\theta }{1 + \theta } - \beta \rho_{SR} (\theta )}} < \frac{1 + \theta }{\theta } \Leftrightarrow \frac{{\rho_{SR}^{\prime } (\theta )}}{{\rho_{SR} (\theta )}} > \frac{1}{\theta (1 + \theta )}$$

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 $Y\left( {\beta \rho_{R} } \right) \approx Y(0) - \beta \rho_{R} R(\theta^{M} ) = c\theta^{M} Q\left( {c(1 + \theta^{M} )} \right) - \beta \rho_{R} c\left( {1 + \theta^{M} } \right)Q\left( {c(1 + \theta^{M} )} \right).$

$$ \text{So}\,D_{R} = 1 - \frac{{Y\left( {\beta \rho_{R} } \right)}}{Y(0)} = 1 - \left\{ {1 - \beta \rho_{R} \left[ {\frac{{c(1 + \theta^{M} )Q\left[ {c\left( {1 + \theta^{M} } \right)} \right]}}{{c\theta^{M} Q\left[ {c\left( {1 + \theta^{M} } \right)} \right]}}} \right]} \right\} = \beta \rho_{R} \frac{{(1 + \theta^{M} )}}{{\theta^{M} }}, $$

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:

$$ \eta (p) = \frac{p}{{p - \frac{c}{{1 - \beta \rho_{R} }}}}. $$

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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:

$$ \eta (p) = \frac{{p\left[ {1 - \beta \rho \left( {\frac{{p - p^{B} }}{{p^{B} }}} \right) - \beta \rho^{\prime}\left( {\frac{{p - p^{B} }}{{p^{B} }}} \right)\frac{p}{{p^{B} }}} \right]}}{{p - c - \beta p\rho \left( {\frac{{p - p^{B} }}{{p^{B} }}} \right)}}. $$

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After some manipulation, it is easy to see that

$$ \frac{{p\left[ {1 - \beta \rho \left( {\frac{{p - p^{B} }}{{p^{B} }}} \right) - \beta \rho^{\prime}\left( {\frac{{p - p^{B} }}{{p^{B} }}} \right)\frac{p}{{p^{B} }}} \right]}}{{p - c - \beta p\rho \left( {\frac{{p - p^{B} }}{{p^{B} }}} \right)}} < \frac{p}{p - c}\; \Leftrightarrow \;\frac{c}{p - c} < \frac{{\rho^{\prime}\left( {\frac{{p - p^{B} }}{{p^{B} }}} \right)\frac{p}{{p^{B} }}}}{{\rho \left( {\frac{{p - p^{B} }}{{p^{B} }}} \right)}}. $$

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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

$$ \frac{{\rho^{\prime } \left( {\frac{{p - p^{B} }}{{p^{B} }}} \right)\frac{p}{{p^{B} }}}}{{\rho \left( {\frac{{p - p^{B} }}{{p^{B} }}} \right)}} > \frac{{p^{B} }}{{p - p^{B} }} \ge \frac{c}{p - c}, $$

(27)

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.