Predation enforcement options: an evaluation in a Cournot framework

Abstract

The paper characterises the building blocks of a framework to enforce anti-predation rules and subsequently evaluates selected enforcement options in a Cournot-type duopoly predation model. Differentiating between a no rule approach, an ex ante approach and two ex post approaches, it is shown that an ex post approach typically maximises overall welfare. However, an ex ante approach can be the preferred option in cases where the entrant has a large cost advantage over the incumbent.

Appendix

1.1 Proof of inequality (1)

As discussed above, an initial welfare assessment has to compare the welfare situation of a successful predation strategy against the welfare realised if the monopoly situation in the pre-predation period would have continued. Based on the setup shown in Fig. 1, the welfare if predation is successful,

$$ \begin{gathered} {\text{W}}_{{{\text{Predation}}\;{\text{successful}}}} = \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {{\text{CS}}^{\text{Pred}} } \right)} \right] + \left[ {\left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {{\text{CS}}^{\text{Mono}} } \right)} \right] \hfill \\ + \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{E}}^{\text{Pred}} } \right)} \right] + \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{I}}^{\text{Pred}} } \right) + \left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {\pi_{\text{I}}^{\text{Mono}} } \right)} \right], \hfill \\ \end{gathered} $$

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has to be larger than the welfare realised in the case of continuous monopoly,

$$ \begin{aligned} {\text{W}}_{{{\text{Continuous}}\;{\text{monopoly}}}} = & \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {{\text{CS}}^{\text{Mono}} } \right)} \right] + \left[ {\left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {{\text{CS}}^{\text{Mono}} } \right)} \right] \\ + \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{I}}^{\text{Mono}} } \right)} \right] + \left[ {\left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {\pi_{\text{I}}^{\text{Mono}} } \right)} \right] \\ \end{aligned} $$

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Substituting $\left( {t^{exit} - t^{enrty} } \right) = \alpha$ and $\left( {t^{end} - t^{exit} } \right) = \beta$ and simplifying both expressions leads to

$$ {\text{W}}_{{{\text{Continuous}}\;{\text{monopoly}}}} = \alpha \left( {{\text{CS}}^{\text{Mono}} + \pi_{\text{I}}^{\text{Mono}} } \right) $$

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$$ {\text{W}}_{{{\text{Predation}}\;{\text{successful}}}} = \alpha \left( {{\text{CS}}^{\text{Pred}} + \pi_{\text{E}}^{\text{Pred}} + \pi_{\text{I}}^{\text{Pred}} } \right). $$

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Further simplifying and rearranging leads to

$$ {\text{CS}}^{\text{Pred}} - {\text{CS}}^{\text{Mono}} > \pi_{\text{I}}^{\text{Mono}} - \left( {\pi_{\text{E}}^{\text{Pred}} + \pi_{\text{I}}^{\text{Pred}} } \right). $$

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1.2 Proof of inequality (2)

As discussed above, antitrust rules and interventions increase welfare as long as the overall welfare realised with such interventions,

$$ \begin{aligned} {\text{W}}_{\text{Antitrust}} = & \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {{\text{CS}}^{\text{Pred}} } \right) + ({\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} )({\text{CS}}^{\text{Duo}} )} \right] \\ + \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{E}}^{\text{Pred}} } \right) + \left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {\pi_{\text{E}}^{\text{Duo}} } \right)} \right] \\ + \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{I}}^{\text{Pred}} } \right) + \left( {t^{\text{end}} - t^{\text{exit}} } \right)\left( {\pi_{\text{I}}^{\text{Duo}} } \right)} \right], \\ \end{aligned} $$

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is larger than the welfare realised when the incumbent can successfully apply a predation strategy

$$ \begin{aligned} {\text{W}}_{{{\text{No}}\,{\text{Antitrust}}}} = & \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {{\text{CS}}^{\text{Pred}} } \right)} \right] + \left[ {\left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {{\text{CS}}^{\text{Mono}} } \right)} \right] \\ + \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{E}}^{\text{Pred}} } \right)} \right] + \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{I}}^{\text{Pred}} } \right) + \left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {\pi_{\text{I}}^{\text{Mono}} } \right)} \right]. \\ \end{aligned} $$

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Substituting $\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{enrty}} } \right) = \alpha$ and $\left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right) = \beta$ and simplifying both expressions leads to

$$ {\text{W}}_{{{\text{No}}\,{\text{Antitrust}}}} = \beta \left( {{\text{CS}}^{\text{Mono}} + \pi_{\text{I}}^{\text{Mono}} } \right) $$

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$$ W_{\text{Antitrust}} = \beta \left( {{\text{CS}}^{\text{Duo}} + \pi_{\text{E}}^{\text{Duo}} + \pi_{\text{I}}^{\text{Duo}} } \right) $$

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Further simplifying and rearranging leads to

$$ {\text{CS}}^{\text{Duo}} - {\text{CS}}^{\text{Mono}} > \pi_{\text{I}}^{\text{Mono}} - \left( {\pi_{\text{E}}^{\text{Duo}} + \pi_{\text{I}}^{\text{Duo}} } \right). $$

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1.3 Proof of equalities (3) and (4)

As discussed above, optimal fines can be calculated on a gain-basis and on a harm-basis. In the following, proofs for both fines are provided.

1.3.1 Optimal gain-based fine

The optimal gain-based fine for an antitrust violation is equal to the additional gain the offender realises due to its misbehaviour. In the setup of Fig. 1, the optimal fine is therefore defined as the difference between the incumbent’s overall profits realised under successful predation

$$ \pi_{\text{I}}^{\text{Pred}} = \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{I}}^{\text{Pred}} } \right)} \right] + \left[ {\left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {\pi_{\text{I}}^{\text{Mono}} } \right)} \right] $$

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and the incumbent’s profit if it accommodates the entrant

$$ \pi_{\text{I}}^{\text{Duo}} = \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{I}}^{\text{Duo}} } \right)} \right] + \left[ {\left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {\pi_{\text{I}}^{\text{Duo}} } \right)} \right]. $$

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Substituting $\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{enrty}} } \right) = \alpha$ and $\left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right) = \beta$ and simplifying both expressions leads to

$$ \pi_{\text{I}}^{\text{Pred}} = \beta \pi_{\text{I}}^{\text{Mono}} + \alpha \pi_{\text{I}}^{\text{Pred}} $$

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$$ \pi_{\text{I}}^{\text{Duo}} = \beta \pi_{\text{I}}^{\text{Duo}} + \alpha \pi_{\text{I}}^{\text{Duo}} $$

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Generally, the optimal gain-based fine is ${\text{F}}_{{{\text{Gain}}\;{\text{based}}}} = \pi_{\text{I}}^{\text{Pred}} - \pi_{\text{I}}^{\text{Duo}} .$ Using the expressions above leads to the following optimal gain-based fine:

$$ {\text{F}}_{{{\text{Gain}}\;{\text{based}}}} = \beta \left( {\pi_{\text{I}}^{\text{Mono}} - \pi_{\text{I}}^{\text{Duo}} } \right) + \alpha \left( {\pi_{\text{I}}^{\text{Pred}} - \pi_{\text{I}}^{\text{Duo}} } \right). $$

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1.3.2 Optimal harm-based fine

As explained in the text, the optimal harm-based fine refers to the ‘net harm to others’ caused by the violation. In the predation period, harm is therefore given by the sum of the difference between the duopoly and the predation consumer surpluses and the difference between the entrant’s duopoly and predation profits

$$ {\text{Harm}}_{\alpha } = \alpha \left[ {\left( {{\text{CS}}^{\text{Duo}} - {\text{CS}}^{\text{Pred}} } \right) + \left( {\pi_{\text{E}}^{\text{Duo}} - \pi_{\text{E}}^{\text{Pred}} } \right)} \right]. $$

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If predation is successful, the net harm to others is given by the difference between the duopoly and the monopoly consumer surpluses and the entrant’s duopoly profits (it would have earned without a successful predation strategy)

$$ {\text{Harm}}_{\beta } = \beta \left[ {\left( {{\text{CS}}^{\text{Duo}} - {\text{CS}}^{\text{Mono}} } \right) + \pi_{\text{E}}^{\text{Duo}} } \right] $$

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The optimal harm-based fine is therefore given by

$$ {\text{F}}_{{{\text{Harm}}\;{\text{based}}}} = \alpha \left[ {\left( {{\text{CS}}^{\text{Duo}} - {\text{CS}}^{\text{Pred}} } \right) + \left( {\pi_{\text{E}}^{\text{Duo}} - \pi_{\text{E}}^{\text{Pred}} } \right)} \right] + \beta \left[ {\left( {{\text{CS}}^{\text{Duo}} - {\text{CS}}^{\text{Mono}} } \right) + \pi_{\text{E}}^{\text{Duo}} } \right] $$

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1.4 Proof of equality (5) and inequality (6)

The choice between an ex post I approach and an ex post II approach with an optimal fine can be expressed as follows. In an ex post I approach the overall welfare is given by

$$ \begin{aligned} {\text{W}}_{{{\text{Ex}}\,{\text{post}}\,{\text{I}}}} = & \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {{\text{CS}}^{\text{Pred}} } \right) + ({\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} )({\text{CS}}^{\text{Duo}} )} \right] \\ + \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{E}}^{\text{Pred}} } \right) + \left( {t^{\text{end}} - t^{\text{exit}} } \right)\left( {\pi_{\text{E}}^{\text{Duo}} } \right)} \right] \\ + \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{I}}^{\text{Pred}} } \right) + \left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {\pi_{\text{I}}^{\text{Duo}} } \right)} \right]. \\ \end{aligned} $$

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As shown above, the optimal fine is 0 in an ex post I approach, as the predator did not cause any harm. The overall welfare in an ex post II approach with an optimal fine is given by

$$ \begin{aligned} {\text{W}}_{{{\text{Ex}}\,{\text{post}}\,{\text{II}}}} = & \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {{\text{CS}}^{\text{Pred}} } \right)} \right] + \left[ {\left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {{\text{CS}}^{\text{Mono}} } \right)} \right] \\ + \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{E}}^{\text{Pred}} } \right)} \right] + \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{I}}^{\text{Pred}} } \right) + \left( {t^{\text{end}} - t^{\text{exit}} } \right)\left( {\pi_{\text{I}}^{\text{Mono}} } \right)} \right] \\ + \left\{ {\left. {\alpha \left[ {\left( {{\text{CS}}^{\text{Duo}} - {\text{CS}}^{\text{Pred}} } \right) + \left( {\pi_{\text{E}}^{\text{Duo}} - \pi_{\text{E}}^{\text{Pred}} } \right)} \right]} \right\}} \right. + \varepsilon \left[ {\left( {{\text{CS}}^{\text{Duo}} - {\text{CS}}^{\text{Mono}} } \right) + \pi_{\text{E}}^{\text{Duo}} } \right]. \\ \end{aligned} $$

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The welfare realised is just the welfare in an approach where the antitrust authority does not intervene and the welfare of collecting the optimal fine after ε periods. As the predator successfully reached the exit of the entrant, he can still charge monopoly prices for the remaining β-ε periods.

Substituting $\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right) = \alpha$ and $\left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right) = \beta$ and simplifying both expressions leads to

$$ {\text{W}}_{{{\text{Ex}}\,{\text{post}}\,{\text{I}}}} = \beta \left( {{\text{CS}}^{\text{Duo}} + \pi_{\text{E}}^{\text{Duo}} + \pi_{\text{I}}^{\text{Duo}} } \right) $$

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$$ \begin{aligned} {\text{W}}_{{{\text{Ex}}\,{\text{post}}\,{\text{II}}}} = & \beta \left( {{\text{CS}}^{\text{Mono}} + \pi_{\text{I}}^{\text{Mono}} } \right) \\ + \left\{ {\left. {\alpha \left[ {\left( {{\text{CS}}^{\text{Duo}} - {\text{CS}}^{\text{Pred}} } \right) + \left( {\pi_{\text{E}}^{\text{Duo}} - \pi_{\text{E}}^{\text{Pred}} } \right)} \right]} \right\} + \left\{ {\varepsilon \left[ {\left( {{\text{CS}}^{\text{Duo}} - {\text{CS}}^{\text{Mono}} } \right) + \pi_{\text{E}}^{\text{Duo}} } \right]} \right\}.} \right. \\ \end{aligned} $$

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The welfare differential can be calculated by subtracting ${\text{W}}_{{{\text{Ex}}\,{\text{post}}\,{\text{II}}}}$ from ${\text{W}}_{{{\text{Ex}}\,{\text{post}}\,{\text{I}}}}$. The value of the positive differential shows how much the antitrust authority should invest at the maximum in the quicker but more expensive ex post I approach to increase overall welfare compared to an ex post II approach.

1.5 Proof of inequality (8)

As discussed above, an alternative to ex post antitrust rules is ex ante antitrust rules. If such rules work frictionless they turn predation into an unprofitable strategy before it is actually played by the incumbent. Consequently, the entrant will be accommodated under such a regime.

Ex ante rules are superior to ex post rules if the welfare realised under the former regime,

$$ \begin{aligned} {\text{W}}_{{{\text{Ex - ante}}\;{\text{rule}}}} = & \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {{\text{CS}}^{\text{Duo}} } \right) + \left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {{\text{CS}}^{\text{Duo}} } \right)} \right] \\ + \left[ {\left( {{\text{t}}^{\text{exit}} - t^{\text{entry}} } \right)\left( {\pi_{\text{E}}^{\text{Duo}} } \right) + \left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {\pi_{\text{E}}^{\text{Duo}} } \right)} \right] \\ + \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{I}}^{\text{Duo}} } \right) + \left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {\pi_{\text{I}}^{\text{Duo}} } \right)} \right] \\ \end{aligned} $$

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is larger than the welfare realised under an ex post regime,

$$ \begin{aligned} {\text{W}}_{{{\text{Ex - post}}\;{\text{rule}}}} = & \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {{\text{CS}}^{\text{Pred}} } \right) + ({\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} )({\text{CS}}^{\text{Duo}} )} \right] \\ + \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{E}}^{\text{Pred}} } \right) + \left( {{\text{t}}^{\text{end}} - t^{\text{exit}} } \right)\left( {\pi_{\text{E}}^{\text{Duo}} } \right)} \right] \\ + \left[ {\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{entry}} } \right)\left( {\pi_{\text{I}}^{\text{Pred}} } \right) + \left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right)\left( {\pi_{\text{I}}^{\text{Duo}} } \right)} \right] \\ \end{aligned} $$

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Substituting $\left( {{\text{t}}^{\text{exit}} - {\text{t}}^{\text{enrty}} } \right) = \alpha$ and $\left( {{\text{t}}^{\text{end}} - {\text{t}}^{\text{exit}} } \right) = \beta$ and simplifying both expressions leads to

$$ {\text{W}}_{{{\text{Ex - ante}}\,{\text{rule}}}} = \alpha \left( {{\text{CS}}^{\text{Duo}} + \pi_{\text{E}}^{\text{Duo}} + \pi_{\text{I}}^{\text{Duo}} } \right), $$

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$$ {\text{W}}_{{{\text{Ex - post}}\,{\text{rule}}}} = \alpha \left( {{\text{CS}}^{\text{Pred}} + \pi_{\text{E}}^{\text{Pred}} + \pi_{\text{I}}^{\text{Pred}} } \right) $$

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Further simplifying and rearranging leads to

$$ {\text{CS}}^{\text{Duo}} - {\text{CS}}^{\text{Pred}} > \pi_{\text{E}}^{\text{Pred}} + \pi_{\text{I}}^{\text{Pred}} - \pi_{\text{E}}^{\text{Duo}} - \pi_{\text{I}}^{\text{Duo}} . $$

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