Cartel Penalties Under Endogenous Detection

Abstract

We compare the performance of cartel penalties that are proportional to a cartel’s revenue and cartel penalties that are proportional to the difference between the cartel price and the competitive price: the overcharge. Prior literature has shown that when the probability of cartel detection does not depend on the cartel price, penalties that are based on a cartel’s overcharge generate greater total surplus and consumer surplus than do penalties that are based on a cartel’s revenue. In contrast, we find that when the probability of detection depends on the cartel price, penalties that are based on revenue can generate greater total surplus and consumer surplus.

1 Introduction

The recent discovery of a number of large and harmful cartels demonstrates that illegal price fixing remains a significant concern for antitrust authorities.¹ To deter cartels, antitrust authorities penalize firms that are proven guilty of illegal collusion. The structure of the penalties that are imposed by an antitrust authority affects both cartel formation and pricing.²

Revenue-based and aggregate overcharge-based penalties are two often-proposed types of cartel penalties. A revenue-based penalty is a multiple of cartel revenue. An aggregate overcharge-based penalty is a multiple of the difference between the collusive price and the competitive price (the overcharge) multiplied by some base volume of sales. Revenue-based penalties are the current practice in all major jurisdictions.³

Recent research (Bageri et al., 2013; Katsoulacos et al., 2015) suggests that cartel penalties that are based on revenue generate a lower level of social surplus than do penalties that are based on the cartel aggregate overcharge.⁴ This is the case because revenue-based penalties induce cartels to price above the monopoly level in order to reduce revenue and thereby reduce the potential penalty. Intuitively, because a cartel that is maximizing profits already prices on the elastic part of its demand curve, the cartel increases its price so as to reduce revenue. Conversely, aggregate overcharge-based penalties induce cartels to price below the monopoly level so as to reduce the measured overcharge. These studies assume a constant and exogenous probability of cartel detection.

In this study, we compare revenue-based and overcharge-based cartel penalties while allowing the probability of detection to depend on the cartel price.⁵ We consider two specifications for the probability of detecting a cartel: In the first specification, the probability of detection is increasing in the cartel (per-unit) overcharge.⁶ In the second, the probability of detection is increasing in the rate at which the cartel price increases, as in Harrington (2004, 2005). Intuitively, relatively high prices or large price increases are more likely to attract the attention of an antitrust authority and lead to the cartel’s detection. For each specification, we analyze and compare cartel pricing, deterrence, total surplus and consumer surplus under revenue-based and overcharge-based penalties.

We find that cartel penalties that are based on revenue generate a higher level of total and consumer surplus when the probability of detection is sufficiently sensitive to price under both specifications. Cartels do not charge prices above the monopoly price because such a high price is likely to raise suspicions of collusion and may cause the cartel’s detection. Thus, the pricing result of Katsoulacos et al. (2015) does not occur. Instead, cartels set low prices to avoid detection. However, a revenue-based penalty is larger than an aggregate overcharge-based penalty for low cartel prices. As a result, collusion is more difficult to sustain under revenue-based penalties. In addition, cartels that face revenue-based penalties have a stronger incentive to reduce price further so as to avoid detection and the payment of larger revenue-based penalties.

The rest of the article is organized as follows: Sect. 2 presents the model. Section 3 compares the two types of penalties when the probability of detection depends on the cartel overcharge. Section 4 compares the two types of penalties when the probability of detection depends on the rate at which the cartel price increases. Section 5 concludes. All proofs can be found in “Appendix 1”.

2 Base Model

The model closely follows Katsoulacos et al. (2015, 2020a, 2020b).⁷$N\ge 2$ symmetric firms compete in prices in each of infinitely many periods and have a common discount factor $\delta <1$. Firms produce an homogeneous product at a constant marginal cost $c>0$. Market demand when firm i charges price $p_{i}$ is $D(p_{1},\dots p_{N})=N\left[ 1-{\underline{p}}\right]$ where ${\underline{p}}$ is the lowest price that is charged by firms. Market demand is split evenly among all firms that charge the lowest price. All other firms receive zero demand. Let $D(p)=1-p$ denote per-firm demand when all firms charge a common price p. When all firms charge price p, the profit of each firm is $\pi (p)=D(p)\left[ p-c\right]$. Demand and cost conditions do not change over time.

Industry suppliers have the opportunity to form an illegal cartel at the beginning of the initial period. If a cartel forms, the firms agree to set a common price. An antitrust authority may detect and penalize a cartel. Cartels can be detected in a variety of ways. A cartel could be detected because of: a report from an internal whistleblower; a complaint from a buyer; or the discovery of evidence during a merger review or separate antitrust investigation. Let $\phi (p_{t-1},p_{t})$ denote the probability of detection when the price in the current period is $p_{t}$ and the price in the preceding period is $p_{t-1}$. We consider two different specifications for $\phi$ in the following sections. If a cartel is detected, we assume that each firm is prosecuted, convicted and penalized with probability 1.⁸

We compare two penalty structures: revenue-based penalties and overcharge-based penalties. Revenue-based penalties are of the form $x_{R}(p)=\gamma _{R}D(p)p$ where $\gamma _{R}$ is a positive constant and p is the cartel price during the period of detection.⁹ Overcharge-based cartel penalties are of the form $x_{O}(p)=\gamma _{O}D(p^{BF})\left[ p-p^{BF}\right]$ where $p^{BF}$ is the but-for price and $\gamma _{O}$ is a positive constant. The but-for price is the price that would prevail in the absence of collusion. Following Katsoulacos et al. (2015), we assume that the but-for price is the Nash equilibrium price c which implies $x_{O}(p)=\gamma _{O}Q_{N}\left[ p-c\right]$ where $Q_{N}=1-c$. Following Motta and Polo (2003), Katsoulacos et al. (2015), Chen and Rey (2013) and Katsoulacos et al. (2020b), we assume that a cartel has a renewed opportunity to form after detection.

A firm will defect from the cartel if the expected present discounted value of the payoff from collusion is less than the payoff from defection. After defection, the market reverts to Nash competition for all future periods. A defecting firm undercuts the cartel price and serves all demand. Defection profit for a firm when all rivals charge price p is¹⁰

$$\begin{aligned} \pi ^{D}(p)={\left\{ \begin{array}{ll} ND(p)\left[p-c\right] &{} {\text{if}}\, p\le p^{m}\\ ND(p^{m})\left[p^{m}-c\right] &{} {\text{if}}\, p>p^{m} \end{array}\right.},\end{aligned}$$

where $p^{m}$ is the monopoly price. We assume that cartels can be detected only when all firms charge the collusive price: The cartel cannot be detected in a period where any firm defects or after the breakdown of a cartel.¹¹ Once a firm has defected, we assume the cartel cannot be detected.¹²

If a cartel forms, the firms set a common price to maximize the expected present discounted value of per-firm profit less penalties, subject to the constraint that no cartel member wishes to defect. Cartel prices satisfy the Bellman equation

$$\begin{aligned} V_{i}(p_{t-1})&=\max _{p\in [c,1]}\pi (p)-\phi (p_{t-1},p)x_{i}(p)+\delta \left[ 1-\phi (p_{t-1},p)\right] V_{i}\left( p\right) +\delta \phi (p_{t-1},p)V_{i}\left( c\right) \nonumber \\&\text {s.t.}\quad \pi (p)-\phi (p_{t-1},p)x_{i}(p)+\delta \left[ 1-\phi (p_{t-1},p)\right] V_{i}\left( p\right) +\delta \phi (p_{t-1},p)V_{i}\left( c\right) \nonumber \\&\quad \quad \ge \pi ^{D}(p) \end{aligned}$$

(1)

where $V_{i}(p_{t-1})$ denotes the expected present discounted value of the payoff from collusion when the cartel price in the prior period was $p_{t-1}$ and the penalty type is $i\in \left\{ R,O\right\}$ where R denotes revenue-based penalties and O denotes overcharge-based penalties. We assume that the market is competitive prior to cartel formation so the initial price is the Nash equilibrium price c.¹³ We refer to the inequality in Eq. (1) as an incentive compatibility constraint.

Consumers have discount factor $\delta <1$. The expected present discounted value of consumer surplus (total surplus) under penalty type $i\in \{O,R\}$ when a cartel forms is denoted $CS_{i}$ ($TS_{i}$). $CS_{i}$ and $TS_{i}$ are defined and derived in “Appendix 2”.

A cartel forms if and only if collusion is sustainable and profitable. Collusion is profitable if $V_{i}(c)>0$. Collusion is sustainable if there exists a sequence of prices that satisfy the constraint in (1) in every period. As $\delta$ increases, the constraint in (1) is more easily satisfied. The critical discount factor $\delta _{i}$ is the discount factor such that a cartel forms under penalty type $i\in \left\{ O,R\right\}$ for all $\delta >\delta _{i}$ and does not form under penalty type $i\in \left\{ O,R\right\}$ if $\delta <\delta _{i}$.¹⁴ If a cartel does not form, firms engage in Nash competition in all future periods and earn zero profit.

3 Overcharge (Price-Level) Detection Specification

In this section, the probability of detection depends on the per-unit cartel overcharge: $[p_{t}-c]$. Since c is exogenous, the probability of detection effectively depends on the price: $p_{t}$. A large overcharge—a relatively high price—is more likely to attract the attention of an antitrust authority—either directly, or through the complaints of buyers or other industry observers – and cause the detection of the cartel.

Let

$$\begin{aligned} \phi \left( \cdot ,{p_{t}}\right) =\min \left\{ {\alpha _{0}}+{\alpha_{1}}\left[ {p_{t}}-c\right]^{2},1\right\} \end{aligned}$$

(2)

where $\alpha _{1}>0$ measures the sensitivity of the probability of detection to the cartel overcharge.¹⁵ This specification is not a function of $p_{t-1}$—the cartel price in the prior period—and depends only on the level of the cartel price in the current period. For example, in industries where a high overcharge (a relatively high price) is considered anomalous and indicative of collusion, $\alpha _{1}$ is high. On the other hand, in industries where a high overcharge is unlikely to attract the attention of an antitrust authority and lead to detection, $\alpha _{1}$ is low. $\alpha _{0}>0$ represents the probability of detection that does not depend on the cartel price.¹⁶

Houba et al. (2010, 2012, 2015) also analyze a specification where the probability of detection is increasing in the cartel price, but they do not compare revenue and overcharge-based penalties.¹⁷ We assume $\gamma _{R}>1$ and $\gamma _{O}>1$ to ensure penalties are large enough that cartels do not choose a price that would cause immediate detection.¹⁸ We also assume $\alpha _{0}\gamma _{R}<1-c$ and $\alpha _{0}\gamma _{O}<1$ to ensure expected penalties are not so large as to preclude all collusion.¹⁹

3.1 Cartel Formation Under Price-Level Specification

In this section, we analyze the critical discount factor under revenue and aggregate overcharge-based cartel penalties when the probability of detection is given by Eq. (2). A cartel forms if collusion is sustainable—no cartel member wishes to defect in any period – and profitable.

Theorem 1

$\delta _{O}=\frac{N-1}{N}+\frac{\alpha _{0}\gamma _{O}}{N}$.

$\delta _{O}<1$ because $\alpha _{0}\gamma _{O}<1$ by assumption. A cartel forms for any $\delta >\delta _{O}$. The critical discount factor under overcharge-based penalties does not depend on $\alpha _{1}$.

As $\alpha _{1}$ increases, cartels must reduce the cartel price to avoid detection (i.e., $p\rightarrow c$). However, a reduction in the cartel price does not make collusion unprofitable. This is the case because while a reduction in the cartel price diminishes profit, it also causes a commensurate drop in the aggregate overcharge-based penalty. Cartel profit approaches 0 as $p\rightarrow c$ but—as is illustrated in Fig. 1—aggregate overcharge based penalties also approach 0 as $p\rightarrow c$. This is the case because both a cartel’s profit and the aggregate overcharge-based penalty are tied to the same outcome—the markup $p-c$. In essence, the benefit (cartel profit) and cost (the expected penalty) of collusion decrease in tandem as the cartel price declines. Thus, although the payoff from collusion declines as $\alpha _{1}\rightarrow \infty$, it remains positive if the cartel charges a sufficiently low price.

When we turn to sustainability, a similar intuition holds: As $\alpha _{1}\rightarrow \infty$, the cartel price falls which reduces profits. However, the payoff from defection is also reduced proportionately. Thus, the incentives to defect are limited as $\alpha _{1}\rightarrow \infty$, and collusion remains sustainable—provided firms are sufficiently patient: $\delta >\delta _{O}$. And from Theorem 1, the critical discount factor above which a cartel forms—$\delta _{O}$—does not depend on $\alpha _{1}$.

Fig. 1

Revenue-based penalty ($\gamma _{R}p(1-p)$) and aggregate overcharge-based penalty ($\gamma _{O}Q_{N}(p-c)$)

Theorem 2

$\delta _{R}\rightarrow 1$ monotonically as $\alpha _{1}\rightarrow \infty$.²⁰

Under revenue-based penalties, the critical discount factor increases as the sensitivity of the probability of detection to the cartel price increases. When the probability of detection is highly sensitive to price, cartels must set low prices to avoid detection (as was the case under overcharge-based penalties). However, in contrast to the setting with overcharge-based penalties, collusion is unprofitable under revenue-based penalties when the cartel price is low.

To understand why, observe that the per-period profit from collusion—$\pi (p)$—approaches 0 as $p\rightarrow c$. However, the penalty to be paid if detected—$\gamma _{R}pD(p)$—approaches $\gamma _{R}cD(c)>0$ as $p\rightarrow c$ (see Fig. 1) because, even when the cartel price is low, revenue is strictly positive. Thus, the expected penalty is greater than the per-period profit from collusion when the cartel price is sufficiently low (which occurs when $\alpha _{1}$ is sufficiently large)—which implies that collusion is unprofitable and a cartel does not form.²¹

Intuitively, revenue-based penalties more effectively penalize cartels that set low prices because, as is shown in Fig. 1, revenue-based penalties are greater than overcharge-based penalties for low cartel prices. However, overcharge-based penalties are typically more effective at penalizing cartels that set high prices because, as is shown in Fig. 1, overcharge-based penalties exceed revenue-based penalties for high cartel prices. Crucially, when the probability of detection is highly sensitive to the cartel price, cartels must set low prices to avoid detection. Therefore, revenue-based penalties are superior and deter the formation of a greater number of cartels.

3.2 Cartel Pricing Under Price-Level Specification

In this section, we assume that a cartel forms under both penalty types, and we compare cartel prices under revenue and overcharge-based penalties. Note that the cartel’s problem under the $\phi$ specification in Eq. (2) is identical in every period because the probability of detection does not depend on the cartel price in the prior period $p_{t-1}$. Therefore, the cartel sets the same price in each period.

Let $p_{O}$ denote the price under overcharge-based penalties and let $p_{R}$ denote the price under revenue-based penalties.²² Katsoulacos et al. (2015) show that $p_{O}<p^{m}<p_{R}$ if $\alpha _{1}=0$. Intuitively, cartels increase price above the monopoly price in order to reduce revenue under revenue-based penalties and set a price below the monopoly price in order to reduce the overcharge under overcharge-based penalties. However, we show in this section that cartel prices may be lower under revenue-based penalties when $\alpha _{1}>0$.

Theorem 3

There exists an ${\bar{\alpha }}_{1}^{L}$ such that $p_{R}<p_{O}$ if $\alpha _{1}>{\bar{\alpha }}_{1}^{L}$ and a cartel forms under both penalty types.

Theorem 3 implies that prices are lower under revenue-based penalties than under overcharge-based penalties when a cartel forms under both penalty types and the probability of detection is sufficiently sensitive to the cartel overcharge.

As Katsoulacos et al. (2015) show, cartels set a high price under revenue-based penalties—$p_{R}>p^{m}$—when $\alpha _{1}=0$. However, when $\alpha _{1}$ is sufficiently large, such a high price is likely to raise suspicions of collusion and increase the likelihood of the cartel’s detection. Instead, cartels set low prices when $\alpha _{1}$ is high so as to reduce the likelihood of detection. When cartel prices are relatively low—close to marginal cost—revenue-based penalties are strictly larger than overcharge-based penalties (see Fig. 1). Faced with the possibility of a relatively large penalty if detected, cartels under revenue-based penalties further reduce price so as to reduce the likelihood of detection and penalization.

Intuitively, the sensitivity of the probability of detection to the price level encourages cartels to set prices in a region where prices are relatively close to marginal cost—but this is also a region where revenue-based penalties are large. Recognizing this, cartels that face revenue-based penalties are more exposed to detection and, as a result, reduce their price to a level that is below the price that is set under overcharge-based penalties so as to disguise collusion and reduce the likelihood of detection.

3.3 Surplus Under Price-Level Specification

In this section, we compare total surplus and consumer surplus under overcharge-based penalties and revenue-based penalties. There are two cases to consider: $\delta \le \delta _{O}$ and $\delta >\delta _{O}$. First, we consider the case of $\delta \le \delta _{O}$. Theorem 2 implies that $\delta _{R}>\delta _{O}$ if $\alpha _{1}$ is sufficiently high. Therefore, no cartel forms under either penalty type and both penalties generate the same level of total and consumer surplus when $\delta \le \delta _{O}$ and the probability of detection is sufficiently sensitive to the cartel price.

Next, we consider the case of $\delta >\delta _{O}$. When $\delta >\delta _{O}$, a cartel forms under overcharge-based penalties. However, a cartel does not form under revenue-based penalties when $\alpha _{1}$ is sufficiently high, by Theorem 2. Therefore, both total and consumer surplus are greater under revenue-based penalties than under overcharge-based penalties when $\delta >\delta _{O}$ and the probability of detection is sufficiently sensitive to the cartel price.

Theorem 4

When $\delta >\delta _{O}$, there exists an ${\tilde{\alpha }}_{1}^{L}$ such that $CS_{R}>CS_{O}$ and $TS_{R}>TS_{O}$ if $\alpha _{1}>{\tilde{\alpha }}_{1}^{L}$. When $\delta \le \delta _{O}$, there exists an ${\tilde{\alpha }}_{1}^{L}$ such that $CS_{R}=CS_{O}$ and $TS_{R}=TS_{O}$ if $\alpha _{1}>{\tilde{\alpha }}_{1}^{L}$.

In summary, the optimal penalty type depends on $\alpha _{1}$. As Katsoulacos et al. (2015) show, overcharge-based penalties generate a greater level of surplus when the probability of detection does not depend on the cartel overcharge ($\alpha _{1}=0$). Conversely, surplus is greater under revenue-based penalties when the probability of detection is sufficiently sensitive to the cartel overcharge.

4 Price-Change Detection Specification

In this section, the probability of detection depends on the difference between the current period price and the price in the preceding period. We assume that large price increases are more likely to lead to detection than is true for smaller increases. As noted above, a large increase in price might create suspicions of collusion among buyers or other industry observers which may be reported to a competition authority.²³ Alternatively, a large and anomalous price increase could directly attract the attention of competition authorities.

Let

$$\begin{aligned} \phi (p_{t-1},p_{t})={\left\{ \begin{array}{ll} \min \left\{ \alpha _{0}+\alpha _{1}\left[ p_{t}-p_{t-1}\right] ^{2},1\right\} &{} \text { if }p_{t}>p_{t-1}\\ \alpha _{0} &{} \text { if }p_{t}\le p_{t-1} \end{array}\right. } \end{aligned}$$

(3)

where $\alpha _{1}\ge 0$ represents the sensitivity of the probability of detection to the price increase.²⁴$\alpha _{0}$ represents the probability of detection when the cartel price is constant or decreasing. Harrington (2004, 2005) first introduced and analyzed a similar specification, but those studies do not consider revenue-based penalties. As in the previous section, we assume $\gamma _{R}>1$ and $\gamma _{O}>1$ to ensure that a cartel does not choose a price that would cause detection with probability 1. In addition, we assume that $\alpha _{0}\gamma _{R}<1-c$ and $\alpha _{0}\gamma _{O}<1$ so as to ensure that the expected penalties are not so large as to preclude all collusion.

4.1 Cartel Formation Under Price-Change Specification

In this section, we analyze the critical discount factor under revenue and overcharge-based cartel penalties when the probability of detection is given by Eq. (3). A cartel forms if collusion is sustainable—no cartel member wishes to defect in any period – and profitable.

Theorem 5

$\delta _{O}=\frac{N-1}{N}+\frac{\alpha _{0}\gamma _{O}}{N}$.

$\delta _{O}<1$ because $\alpha _{0}\gamma _{O}<1$ by assumption. A cartel forms for any $\delta >\delta _{O}$. As was the case under the price-level specification, the critical discount factor under overcharge-based penalties does not depend on $\alpha _{1}$.

As $\alpha _{1}$ increases, cartels must increase the cartel price more gradually to avoid detection: $p_{t}\rightarrow p_{t-1}$. Raising the cartel price more gradually effectively reduces the cartel price in each period. However, the more gradual price increase does not render collusion unprofitable. This is the case because while increasing the cartel price more gradually diminishes profit, it also causes a commensurate drop in the overcharge-based penalty that the cartel faces in each period: The benefit (cartel profit) and cost (the expected penalty) of collusion decrease in tandem as the cartel price rises more slowly. Thus, while the payoff from collusion declines as $\alpha _{1}\rightarrow \infty$, the payoff from collusion remains positive if the cartel increases price at a sufficiently slow rate.

When we turn to sustainability, a similar intuition holds: As $\alpha _{1}\rightarrow \infty$, the cartel price falls in each period, which reduces profit. However, the payoff from defection is also reduced proportionately. Thus, the incentives to defect are limited as $\alpha _{1}\rightarrow \infty$, and collusion remains sustainable if firms are sufficiently patient: $\delta >\delta _{O}$. The critical discount factor above which a cartel forms—$\delta _{O}$—does not depend on $\alpha _{1}$.

Theorem 6

$\delta _{R}\rightarrow 1$ monotonically as $\alpha _{1}\rightarrow \infty$.²⁵

Under revenue-based penalties, the critical discount factor is increasing in the sensitivity of the probability of detection to changes in the cartel price. When the probability of detection is highly sensitive to changes in the cartel price, cartels must increase price slowly in order to avoid detection (as was the case under overcharge-based penalties). However, unlike under overcharge-based penalties, collusion becomes unprofitable under revenue-based penalties when the cartel is forced to increase price slowly.

To see this, first note that gradually increasing the cartel price effectively reduces the price in each period. Next, recall that the per-period profit from collusion—$\pi (p_{t})$—approaches 0 as $p_{t}\rightarrow c$. However the penalty—$\gamma _{R}p_{t}D(p_{t})$ – approaches $\gamma _{R}cD(c)>0$ as $p_{t}\rightarrow c$ (see Fig. 1) because, even when the cartel price is low, revenue is strictly positive. Thus, the expected penalty is greater than the per-period profit from collusion when the cartel price is sufficiently low.²⁶

As $\alpha _{1}\rightarrow \infty$, the cartel is induced to price in a region (low prices) where the expected penalty exceeds cartel profit in an increasingly large number of periods. Rapidly increasing the price to a level where collusion is profitable would cause detection. Thus, as $\alpha _{1}$ increases, the cartel incurs a per-period loss in increasingly more and more periods; and, for sufficiently large $\alpha _{1}$, collusion is altogether unprofitable, and a cartel does not form.

4.2 Cartel Pricing Under Price-Change Specification

In this section, we assume that a cartel forms under both penalty types and compare cartel prices under revenue and overcharge-based penalties. Under the $\phi$ specification in (3), the Bellman equation in (1) does not readily generate tractable closed-form solutions or analytical results.

However, cartel prices under each penalty type can be determined through numerical solutions. Specifically, we solve the Bellman equation in (1) with the use of value function iteration.²⁷ We first analyze this issue in an illustrative baseline setting. Next, we consider modifications of the baseline setting to assess the robustness of our findings.

Baseline Setting $\delta = 0.9$, $c=0.1$, $N=2$, $\gamma _R = 5$, $\gamma _O = 3.05$, $\alpha _0 = 0.05$

The baseline setting represents a duopoly. $\gamma _{O}$ and $\gamma _{R}$ are chosen to ensure that the cartel penalty is the same under the two penalty types when each member of the cartel charges the monopoly price.²⁸ We consider a range of $\alpha _{1}$ values. Let $p_{i}^{t}$ denote the cartel price under penalty type $i\in \left\{ O,R\right\}$ in period t. Figure 2 shows that the cartel price is constant and higher under revenue-based penalties than for aggregate overcharge-based penalties when $\alpha _{1}=0$, as is shown in Katsoulacos et al. (2015). However, Fig. 2 also shows that cartel prices are lower under revenue-based penalties in early periods and lower under aggregate overcharge-based penalties in later periods when $\alpha _{1}=5$, $\alpha _{1}=10$ or $\alpha _{1}=15$.

To understand these results, first note that cartels increase their price gradually as bigger price movements are likely to attract attention and lead to the cartel’s detection. As Fig. 1 shows, revenue-based penalties exceed aggregate overcharge-based penalties when the cartel price is low. Thus, a cartel that is detected in one of the first few periods of collusion pays a large penalty under revenue-based penalties and a smaller penalty under overcharge-based penalties. Recognizing this, cartels that face revenue-based penalties increase the cartel price more slowly so as to disguise collusion and avoid the payment of relatively high penalties. Conversely, cartels that face aggregate overcharge-based penalties are less wary of rapidly increasing their price and raising suspicions of collusion because they face a smaller penalty. These considerations cause the cartel price to be lower under revenue-based penalties during the early periods of collusion.

In the later periods of collusion, $p_{t}^{R}>p_{t}^{O}$. This occurs because the cartel price converges to the price that is derived in Katsoulacos et al. (2015)—the cartel price when $\alpha _{1}=0$—as $t\rightarrow \infty$.²⁹ Increasing the cartel price any further would reduce the payoff from collusion. Thus, $p_{t}^{R}>p_{t}^{O}$ in later periods when the cartel price has converged to its limiting value. We demonstrate the robustness of our results to alternative parameter configurations in the online appendix.³⁰

Fig. 2

Cartel prices in the baseline setting

In summary, the following conclusion arises in every numerical solution we conducted:

Conclusion 1

If $\alpha _{1}$ is sufficiently large and a cartel forms under both penalty types, then $p_{t}^{R}<p_{t}^{O}$ in early periods and $p_{t}^{R}>p_{t}^{O}$ in later periods.

4.3 Surplus Under Price-Change Specification

In this section, we compare total and consumer surplus under overcharge-based penalties and revenue-based penalties. There are two cases to consider: $\delta \le \delta _{O}$ and $\delta >\delta _{O}$. First, consider the case of $\delta \le \delta _{O}$. Theorem 6 implies that $\delta _{R}>\delta _{O}$ when $\alpha _{1}$ is sufficiently high. Therefore, no cartel forms under either penalty type and both penalties generate the same level of total and consumer surplus when $\delta \le \delta _{O}$ and the probability of detection is sufficiently sensitive to the change in cartel price.

If $\delta >\delta _{O}$, a cartel forms under overcharge-based penalties. However, a cartel does not form under revenue-based penalties when $\alpha _{1}$ is sufficiently high, by Theorem 6. Therefore, both total and consumer surplus are greater under revenue-based penalties than under overcharge-based penalties when $\delta >\delta _{O}$ and the probability of detection is sufficiently sensitive to the change in cartel price.³¹

Theorem 7

When $\delta >\delta _{O}$, there exists an ${\tilde{\alpha }}_{1}^{C}$ such that $CS_{R}>CS_{O}$ and $TS_{R}>TS_{O}$ if $\alpha _{1}>{\tilde{\alpha }}_{1}^{C}$. When $\delta \le \delta _{O}$, there exists an ${\tilde{\alpha }}_{1}^{C}$ such that $CS_{R}=CS_{O}$ and $TS_{R}=TS_{O}$ if $\alpha _{1}>{\tilde{\alpha }}_{1}^{C}$.

As under the price-level specification, the optimal penalty type depends on $\alpha _{1}$. As Katsoulacos et al. (2015) show, overcharge-based penalties induce a greater level of surplus when the probability of detection does not depend on the cartel price ($\alpha _{1}=0$). Conversely, surplus is greater under revenue-based penalties when the probability of detection is sufficiently sensitive to price increases.

5 Conclusion

We have examined revenue- and aggregate overcharge-based cartel penalties when the probability of detection depends on the cartel price. We have compared the two penalty structures on the basis of both consumer surplus and total surplus. First, we consider a model in which the probability of detection depends on the cartel overcharge. Next, we analyze a model in which the probability of detection depends on the rate at which the cartel price increases. Under both models, we obtain a similar result: When the probability of detection is sufficiently sensitive to the cartel price, both consumer surplus and total surplus are higher under revenue-based penalties.

In summary, which penalty structure secures a higher level of surplus depends crucially on the detection process. In industries where a large price increase or a high cartel overcharge is considered anomalous, indicative of collusion, and likely to lead to detection, revenue-based cartel penalties generate a higher level of surplus. For example, in a stable industry which does not experience significant price variation when competitive, a large price increase is likely to attract the attention of an antitrust authority and lead to detection. In this case, revenue-based penalties are better.³²

In industries where a large price increase or high overcharge is not likely to lead to detection, overcharge-based penalties generate a higher level of surplus. For instance, in a turbulent industry which experiences regular price variation even when competitive, a large price increase is unlikely to raise suspicions of collusion. In this case, aggregate overcharge-based penalties are better.

Appendices

Appendix 1: Proofs

1.1 Price-Level Specification

Let $W_{R}(p)$ denote the present discounted value of profit less expected penalties under revenue-based penalties when the cartel price is p. Let $W_{O}(p)$ denote the present discounted value of profit less expected penalties under overcharge-based penalties when the cartel price is p. $p_{R}=\text {argmax}_{p\in [c,1]}W_{R}(p)$ subject to $W_{R}(p)\ge \pi ^{D}(p)$ is the cartel price under revenue-based penalties, if a cartel forms. $p_{O}=\text {argmax}_{p\in [c,1]}W_{O}(p)$ subject to $W_{O}(p)\ge \pi ^{D}(p)$ is the cartel price under overcharge-based penalties, if a cartel forms.

Lemma 1 provides bounds on the cartel price, which necessarily hold if a cartel forms.

Lemma 1

$p_{O}\in \left( c,c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}\right)$ and $p_{R}\in \left( c,c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}\right)$ if a cartel forms.

Proof

If $p_{R}=c$, the payoff from collusion is

$$\begin{aligned} W_{R}(c)=\frac{(1-c)(c-c)-\alpha _{0}\gamma _{R}c(1-c)}{1-\delta }=-\frac{\alpha _{0}\gamma _{R}c(1-c)}{1-\delta }<0, \end{aligned}$$

which implies that collusion is not profitable.

If $p_{R}\ge c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}$, then the probability of detection is 1. Therefore, the payoff from collusion, when the cartel price is $p_{R}$, is

$$\begin{aligned} W_{R}(p_{R})= & {} \frac{(1-p_{R})(p_{R}-c)-\gamma _{R}p_{R}(1-p_{R})}{1-\delta }<\frac{(1-p_{R})(p_{R}-c)-p_{R}(1-p_{R})}{1-\delta }\\= & {} \frac{-(1-p_{R})c}{1-\delta }\le 0, \end{aligned}$$

which implies that collusion is not profitable (the first inequality follows from $\gamma _{R}>1$).

If $p_{O}=c$, the payoff from collusion is

$$\begin{aligned} W_{O}(c)=\frac{(1-c)(c-c)-\alpha _{0}\gamma _{O}Q_{N}(c-c)}{1-\delta }=0, \end{aligned}$$

which implies that collusion is not profitable.

If $p_{O}\ge c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}$, then the probability of detection is 1. Therefore, the payoff from collusion, when the cartel price is $p_{O}$, is

$$\begin{aligned} W_{O}(p_{O})= & {} \frac{(1-p_{O})(p_{O}-c)-\gamma _{O}Q_{N}(p_{O}-c)}{1-\delta }<\frac{(1-p_{O})(p_{O}-c)-(1-c)(p_{O}-c)}{1-\delta }\\= & {} \frac{-(p_{O}-c)^{2}}{1-\delta }<0, \end{aligned}$$

which implies that collusion is not profitable (the first inequality follows from $\gamma _{O}>1$). $\square$

Lemma 2

$W_{R}(p)$ and $W_{O}(p)$ are strictly concave on $\left[ c,c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}\right]$ if $\alpha _{1}>\max \left\{ \left( \frac{5}{H(c)}\right) ^{2}(1-\alpha _{0}),\frac{4\left( 1-\alpha _{0}\right) }{(1-c)^{2}}\right\}$ where $H(c)=\min \left\{ c(1-c),\left( \frac{1+c}{2}\right) \left( 1-\frac{1+c}{2}\right) \right\}$.

Proof

First, we consider revenue-based penalties. Note that $W_{R}(p)=\frac{1}{1-\delta }\left( (1-p)(p-c)-\left( \alpha _{0}+\alpha _{1}(p-c)^{2}\right) \gamma _{R}p(1-p)\right)$ when $p\in \left[ c,c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}\right]$. The second derivative of $W_{R}(p)$ when $p\in \left[ c,c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}\right]$ is

$$\begin{aligned} \frac{\partial ^{2}W_{R}(p)}{\partial p^{2}}&=\frac{\partial ^{2}}{\partial p^{2}}\frac{1}{1-\delta }\left( (1-p)(p-c)-\left( \alpha _{0}+\alpha _{1}(p-c)^{2}\right) \gamma _{R}p(1-p)\right) \\&=\frac{\partial }{\partial p}\frac{1}{1-\delta }\left( 1-2p+c-\alpha _{1}\gamma _{R}\left( (p-c)^{2}\left( 1-2p\right) +2p(1-p)(p-c)\right) \right. \\&\quad \left. -\alpha _{0}\gamma _{R}\left( 1-2p\right) \right) \\&=\frac{\partial }{\partial p}\frac{1}{1-\delta }\left( 1-2p+c-\alpha _{1}\gamma _{R}\right. \\&\quad \left. \left( (p-c)^{2}-2p(p-c)^{2}+2p(p-c)-2p^{2}(p-c)\right) -\alpha _{0}\gamma _{R}\left( 1-2p\right) \right) \\&=\frac{1}{1-\delta }\left( -2-\alpha _{1}\gamma _{R}\left( 2(p-c)-2(p-c)^{2}\right. \right. \\&\quad \left. \left. -4p(p-c)+2(p-c)+2p-4p(p-c)-2p^{2}\right) +2\alpha _{0}\gamma _{R}\right) \\&=\frac{1}{1-\delta }\left( -2\left( 1-\alpha _{0}\gamma _{R}\right) -\alpha _{1}\gamma _{R}\left( 4(p-c)-2(p-c)^{2}-8p(p-c)+2p-2p^{2}\right) \right) \end{aligned}$$

which is less than 0 if $4(p-c)-2(p-c)^{2}-8p(p-c)+2p-2p^{2}\ge 0$. Note that $p(1-p)\ge H(c)$ because $c\le p\le c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}<\frac{1+c}{2}$ where the last inequality follows from $\alpha _{1}>\frac{4\left( 1-\alpha _{0}\right) }{(1-c)^{2}}$.³³ Therefore,

$$\begin{aligned}&4(p-c)-2(p-c)^{2}-8p(p-c)+2p-2p^{2}\\ \ge&-2(p-c)^{2}-8p(p-c)+2p(1-p)\\ \ge&-2(p-c)^{2}-8p(p-c)+2H(c)&\text {by }&p(1-p)\ge H(c)\\ \ge&-2(p-c)-8(p-c)+2H(c)\\ \ge&-10(p-c)+2H(c)\\ \ge&-10\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}+2H(c)\\ \ge&-10\sqrt{\frac{1-\alpha _{0}}{\left( \frac{5}{H(c)}\right) ^{2}(1-\alpha _{0})}}+2H(c)&\text {by }&\alpha _{1}>\left( \frac{5}{H(c)}\right) ^{2}(1-\alpha _{0})\\ \ge&-10\left( \frac{H(c)}{5}\right) +2H(c)\\ \ge&-2H(c)+2H(c)=0. \end{aligned}$$

Therefore, $W_{R}(p)$ is strictly concave on $\left[ c,c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}\right]$.

Next, we show $W_{O}(p)$ is strictly concave. Note that $W_{O}(p)=\frac{1}{1-\delta }\left( (1-p)(p-c)-\left( \alpha _{0}+\alpha _{1}(p-c)^{2}\right) \gamma _{O}Q_{N}(p-c)\right)$ when $p\in \left[ c,c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}\right]$. The second derivative is

$$\begin{aligned} \frac{\partial ^{2}W_{O}(p)}{\partial p^{2}}&=\frac{\partial ^{2}}{\partial p^{2}}\frac{1}{1-\delta }\left( (1-p)(p-c)-\alpha _{1}(p-c)^{2}\gamma _{O}Q_{N}(p-c)-\alpha _{0}\gamma _{O}Q_{N}(p-c)\right) \\&=\frac{\partial }{\partial p}\frac{1}{1-\delta }\left( 1-2p+c-3\alpha _{1}(p-c)^{2}\gamma _{O}Q_{N}-\alpha _{0}\gamma _{O}Q_{N}\right) \\&=\frac{1}{1-\delta }\left( -2-6\alpha _{1}(p-c)Q_{N}\gamma _{O}\right) <0. \end{aligned}$$

Therefore, $W_{O}(p)$ is strictly concave when $p\in \left[ c,c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}\right]$. $\square$

Proof of Theorem 1

We show that a cartel forms when $\delta >\delta _{O}$ and does not form when $\delta \le \delta _{O}$. A cartel forms if collusion is sustainable and profitable. For collusion to be sustainable, collusion must be incentive-compatible: The constraint in Eq. (1) must hold. Note that the cartel price p satisfies both $c<p<c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}$, by Lemma 1, and $p\le p^{m}$.³⁴ Collusion is sustainable at a price $p=c+\epsilon$ where $p\le p^{m}$ and $p\in \left( c,c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}\right)$ if

$$\begin{aligned} N\pi (p)&\le \frac{1}{1-\delta }\pi (p)-\frac{\alpha _{0}x_{O}(p)}{1-\delta }-\frac{\alpha _{1}(p-c)^{2}x_{O}(p)}{1-\delta }\nonumber \\ 1-\delta&\le \frac{1}{N}-\frac{\alpha _{0}x_{O}(p)}{N\pi (p)}-\frac{\alpha _{1}(p-c)^{2}x_{O}(p)}{N\pi (p)}\nonumber \\ \delta&\ge \frac{N-1}{N}+\frac{\alpha _{0}x_{O}(p)}{N\pi (p)}+\frac{\alpha _{1}(p-c)^{2}x_{O}(p)}{N\pi (p)}\nonumber \\ \delta&\ge \frac{N-1}{N}+\frac{\alpha _{0}\gamma _{O}Q_{N}\epsilon }{N\epsilon D(c+\epsilon )}+\frac{\alpha _{1}\epsilon ^{2}\gamma _{O}Q_{N}\epsilon }{N\epsilon D(c+\epsilon )}\nonumber \\ \delta&\ge \frac{N-1}{N}+\frac{\alpha _{0}\gamma _{O}Q_{N}}{ND(c+\epsilon )}+\frac{\alpha _{1}\epsilon ^{2}\gamma _{O}Q_{N}}{ND(c+\epsilon )}\nonumber \\ \delta&\ge \frac{N-1}{N}+\frac{\alpha _{0}\gamma _{O}\left( 1-c-\epsilon +\epsilon \right) }{ND(c+\epsilon )}+\frac{\alpha _{1}\epsilon ^{2}\gamma _{O}Q_{N}}{ND(c+\epsilon )}\nonumber \\ \delta&\ge \frac{N-1}{N}+\frac{\alpha _{0}\gamma _{O}D(c+\epsilon )}{ND(c+\epsilon )}+\epsilon \frac{\alpha _{0}\gamma _{O}}{ND(c+\epsilon )}+\frac{\alpha _{1}\epsilon ^{2}\gamma _{O}Q_{N}}{ND(c+\epsilon )}\nonumber \\ \delta&\ge \frac{N-1}{N}+\frac{\alpha _{0}\gamma _{O}}{N}+\epsilon \frac{\alpha _{0}\gamma _{O}}{ND(c+\epsilon )}+\epsilon ^{2}\frac{\alpha _{1}\gamma _{O}Q_{N}}{ND(c+\epsilon )}. \end{aligned}$$

(4)

If $\delta >\delta _{O}=\frac{N-1}{N}+\frac{\alpha _{0}\gamma _{O}}{N}$, then there exists a sufficiently small $\epsilon >0$ such that inequality (4) holds and collusion is sustainable. Collusion is also profitable at price $p=c+\epsilon$ because

$$\begin{aligned} 0<N\pi (p)\le \frac{1}{1-\delta }\pi (p)-\frac{\alpha _{0}x_{O}(p)}{1-\delta }-\frac{\alpha _{1}(p-c)^{2}x_{O}(p)}{1-\delta }, \end{aligned}$$

where the first inequality follows from $c<p\le p^{m}$ and the second inequality follows from the sustainability of collusion at price $p=c+\epsilon$. Therefore, collusion is profitable and sustainable if $\delta >\delta _{O}$. If $\delta \le \frac{N-1}{N}+\frac{\alpha _{0}\gamma _{O}}{N}$, collusion is unsustainable for all $\epsilon \in \left( 0,\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}\right)$ (see inequality (4)) and unprofitable for $\epsilon =0$ and $\epsilon \ge \sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}$ (see Lemma 1). Thus, a cartel does not form. $\square$

Proof of Theorem 2

First, we show $\delta _{R}\rightarrow 1$ as $\alpha _{1}\rightarrow \infty$. A cartel forms if collusion is sustainable and profitable. By Lemma 1, $p\in \left( c,c+\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}\right)$. We show collusion is unprofitable for all $\delta <1$ if $\alpha _{1}>\frac{(1-\alpha _{0})\left( 1-\alpha _{0}\gamma _{R}\right) ^{2}}{\left( \alpha _{0}\gamma _{R}c\right) ^{2}}$. Therefore, a cartel does not form and $\delta _{R}=1$ if $\alpha _{1}>\frac{(1-\alpha _{0})\left( 1-\alpha _{0}\gamma _{R}\right) ^{2}}{\left( \alpha _{0}\gamma _{R}c\right) ^{2}}$. For collusion to be profitable, $\pi (p)-\alpha _{0}x_{R}(p)>0$ must hold for some price $p\in \left( c,c+\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}\right)$.³⁵ Note that

$$\begin{aligned} \pi (p)-\alpha _{0}x_{R}(p)&=D(p)(p-c)-\alpha _{0}\gamma _{R}D(p)p\\&=D(p)\left[ p\left[ 1-\alpha _{0}\gamma _{R}\right] -c\right] \\&\le D(p)\left[ \left[ c+\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}\right] \left[ 1-\alpha _{0}\gamma _{R}\right] -c\right] \\&=D(p)\left[ c-\alpha _{0}\gamma _{R}c+\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}\left[ 1-\alpha _{0}\gamma _{R}\right] -c\right] \\&=D(p)\left[ -\alpha _{0}\gamma _{R}c+\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}\left[ 1-\alpha _{0}\gamma _{R}\right] \right] \\&\le D(p)\left[ -\alpha _{0}\gamma _{R}c+\frac{\alpha _{0}\gamma _{R}c}{1-\alpha _{0}\gamma _{R}}\left[ 1-\alpha _{0}\gamma _{R}\right] \right] \\&=0 \end{aligned}$$

for all $p\in \left( c,c+\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}\right)$ where the last inequality follows from $\alpha _{1}>\frac{(1-\alpha _{0})\left( 1-\alpha _{0}\gamma _{R}\right) ^{2}}{\left( \alpha _{0}\gamma _{R}c\right) ^{2}}$. Thus, collusion is unprofitable and a cartel does not form – $\delta _{R}=1$—if $\alpha _{1}>\frac{(1-\alpha _{0})\left( 1-\alpha _{0}\gamma _{R}\right) ^{2}}{\left( \alpha _{0}\gamma _{R}c\right) ^{2}}$, which implies $\delta _{R}\rightarrow 1$ as $\alpha _{1}\rightarrow \infty$.

Next, we show $\delta _{R}\rightarrow 1$ monotonically as $\alpha _{1}\rightarrow \infty$. Let $\delta _{R}(\alpha _{1})$ denote the critical discount factor when the sensitivity of the probability of detection to the overcharge is $\alpha _{1}$. Suppose that $\delta _{R}(\alpha '_{1})>\delta _{R}(\alpha ''_{1})$ for some $\alpha '_{1}<\alpha ''_{1}$. Let $\delta$ be such that $\delta _{R}(\alpha ''_{1})<\delta <\delta _{R}(\alpha '_{1})$. A cartel forms when the discount factor is $\delta$ if $\alpha _{1}=\alpha ''_{1}$ and does not form if $\alpha _{1}=\alpha '_{1}$. Let $p''$ denote the cartel price when $\alpha _{1}=\alpha ''_{1}$ and the discount factor is $\delta$. Let $W_{R}(p;\alpha _{1},\delta )$ denote the payoff from collusion when the cartel price is p, the sensitivity of the probability of detection to the overcharge is $\alpha _{1}$ and the discount factor is $\delta$. Note that $\pi ^{D}(p'')\le W_{R}(p'';\alpha ''_{1},\delta )$ because collusion is sustainable when the cartel price is $p''$. Also, note that $0<W_{R}(p'';\alpha ''_{1},\delta )$ because collusion is profitable when the cartel price is $p''$. However, $\pi ^{D}(p'')\le W_{R}(p'';\alpha ''_{1},\delta )\le W_{R}(p'';\alpha '_{1},\delta )$ and $0<W_{R}(p'';\alpha ''_{1},\delta )\le W_{R}(p'';\alpha '_{1},\delta )$ because $\alpha '_{1}<\alpha ''_{1}$.³⁶ Thus, collusion is sustainable and profitable at a price of $p''$ when $\alpha _{1}=\alpha '_{1}$. This implies that a cartel forms when $\delta <\delta _{R}(\alpha '_{1})$, which is a contradiction. Therefore, we conclude that $\delta _{R}\rightarrow 1$ monotonically as $\alpha _{1}\rightarrow \infty$. $\square$

Proof of Theorem 3

Let

$$\begin{aligned} {\bar{\alpha }}_{1}^{L}=\max \left\{ \frac{9\left( 1-\alpha _{0}\right) }{(1-c)^{2}},\frac{1}{3\gamma _{O}Q_{N}^{2}\left( 1-\alpha _{0}\gamma _{O}\right) }\left( \left( \frac{3\gamma _{O}Q_{N}}{2\gamma _{R}H(c)}\left( 1-c+\gamma _{R}\right) +1\right) ^{2}-1\right) ,\right. \nonumber \\ \left. \left( \frac{5}{H(c)}\right) ^{2}(1-\alpha _{0}),\left( \frac{\gamma _{O}Q_{N}}{\gamma _{R}H(c)}\right) ^{2}\left( 1-\alpha _{0}\right) \right\} \end{aligned}$$

(5)

where $H(c)=\min \left\{ c(1-c),\left( \frac{1+c}{2}\right) \left( 1-\frac{1+c}{2}\right) \right\}$. Suppose that $\alpha _{1}>{\bar{\alpha }}_{1}^{L}$.

The proof involves three steps: In step 1, we establish that the cartel price under overcharge-based penalties exceeds the cartel price under revenue-based penalties when $\alpha _{1}>{\bar{\alpha }}_{1}^{L}$ and incentive compatibility constraints do not bind. This is proven by establishing that the derivative of the payoff from collusion under revenue-based penalties is negative at the unconstrained cartel price under overcharge-based penalties. In step 2, we establish that, when $\alpha _{1}>{\bar{\alpha }}_{1}^{L}$, revenue-based penalties exceed overcharge-based cartel penalties. In step 3, we show, using the results from step 1 and step 2, that $p_{R}<p_{O}$ generally.

Step 1: Let $p_{i}^{NB}$ denote the cartel price when the incentive compatibility constraint does not bind under penalty type $i\in \left\{ R,O\right\}$. First, we show $p_{R}^{NB}<p_{O}^{NB}$. Note that $p_{O}^{NB},p_{R}^{NB}\in \left( c,c+\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}\right)$ by Lemma 1. By the concavity of $W_{O}(p)$ (Lemma 2), the first order condition which determines $p_{O}^{NB}$ is

$$\begin{aligned} \frac{\partial W_{O}(p)}{\partial p}(1-\delta )&=1-2p+c-3\alpha _{1}\gamma _{O}Q_{N}(p-c)^{2}-\alpha _{0}\gamma _{O}Q_{N}=0. \end{aligned}$$

Solving for p yields

$$\begin{aligned} p_{O}^{NB}&=\frac{\sqrt{1+3\alpha _{1}\gamma _{O}Q_{N}^{2}\left( 1-\alpha _{0}\gamma _{O}\right) }-1}{3\alpha _{1}\gamma _{O}Q_{N}}+c. \end{aligned}$$

(6)

By the strict concavity of $W_{R}(p)$ (Lemma 2), the first-order condition that determines $p_{R}^{NB}$ is

$$\begin{aligned} \frac{\partial W_{R}(p)}{\partial p}(1-\delta )&=1-2p+c-\left( \alpha _{0}+\alpha _{1}(p-c)^{2}\right) \gamma _{R}(1-2p)\nonumber \\&\quad -2\alpha _{1}(p-c)\gamma _{R}p(1-p)=0. \end{aligned}$$

(7)

Note that

$$\begin{aligned} \frac{\partial W_{R}(p)}{\partial p}(1-\delta )&\le 1-2p+c+\gamma _{R}-2\alpha _{1}(p-c)\gamma _{R}p(1-p)\\&\le 1-c+\gamma _{R}-2\alpha _{1}(p-c)\gamma _{R}p(1-p), \end{aligned}$$

where the first inequality follows from $-\left( \alpha _{0}+\alpha _{1}(p-c)^{2}\right) \gamma _{R}(1-2p)<\gamma _{R}$ and the second inequality follows from $p>c$. By the strict concavity of $W_{R}(p)$, $p_{R}^{NB}<p_{O}^{NB}$ if $\frac{\partial W_{R}(p_{O}^{NB})}{\partial p}<0$.

$$\begin{aligned} \frac{\partial W_{R}(p_{O}^{NB})}{\partial p}(1-\delta )&\le 1-c+\gamma _{R}-2\alpha _{1}(p_{O}^{NB}-c)\gamma _{R}p_{O}^{NB}(1-p_{O}^{NB})\\&=1-c+\gamma _{R}-2\alpha _{1}\frac{\sqrt{1+3\alpha _{1}\gamma _{O}Q_{N}^{2}\left( 1-\alpha _{0}\gamma _{O}\right) }-1}{3\alpha _{1}\gamma _{O}Q_{N}}\gamma _{R}p_{O}^{NB}(1-p_{O}^{NB})\\&=1-c+\gamma _{R}-2\frac{\sqrt{1+3\alpha _{1}\gamma _{O}Q_{N}^{2}\left( 1-\alpha _{0}\gamma _{O}\right) }-1}{3\gamma _{O}Q_{N}}\gamma _{R}p_{O}^{NB}(1-p_{O}^{NB})\\&\le 1-c+\gamma _{R}-2\frac{\sqrt{1+3\alpha _{1}\gamma _{O}Q_{N}^{2}\left( 1-\alpha _{0}\gamma _{O}\right) }-1}{3\gamma _{O}Q_{N}}\gamma _{R}H(c), \end{aligned}$$

where the last inequality follows from $p_{O}^{NB}(1-p_{O}^{NB})\ge H(c)$, which follows from $c<p_{O}^{NB}<\frac{1+c}{2}$.³⁷ Next, note that

$$\begin{aligned}&1-c+\gamma _{R}-2\frac{\sqrt{1+3\alpha _{1}\gamma _{O}Q_{N}^{2}\left( 1-\alpha _{0}\gamma _{O}\right) }-1}{3\gamma _{O}Q_{N}}\gamma _{R}H(c)< 0\\&\quad \iff 1-c+\gamma _{R}<2 \frac{\sqrt{1+3\alpha _{1}\gamma _{O}Q_{N}^{2}\left( 1-\alpha _{0}\gamma _{O}\right) }-1}{3\gamma _{O}Q_{N}}\gamma _{R}H(c)\\&\quad \iff \frac{3\gamma _{O}Q_{N}}{2\gamma _{R}H(c)}\left( 1-c+\gamma _{R}\right)< \sqrt{1+3\alpha _{1}\gamma _{O}Q_{N}^{2}\left( 1-\alpha _{0}\gamma _{O}\right) }-1\\&\quad \iff \frac{3\gamma _{O}Q_{N}}{2\gamma _{R}H(c)}\left( 1-c+\gamma _{R}\right) +1< \sqrt{1+3\alpha _{1}\gamma _{O}Q_{N}^{2}\left( 1-\alpha _{0}\gamma _{O}\right) }\\&\quad \iff \left( \frac{3\gamma _{O}Q_{N}}{2\gamma _{R}H(c)}\left( 1-c+\gamma _{R}\right) +1\right) ^{2}< 1+3\alpha _{1}\gamma _{O}Q_{N}^{2}\left( 1-\alpha _{0}\gamma _{O}\right) \\&\quad \iff \frac{1}{3\gamma _{O}Q_{N}^{2}\left( 1-\alpha _{0}\gamma _{O}\right) }\left( \left( \frac{3\gamma _{O}Q_{N}}{2\gamma _{R}H(c)}\left( 1-c+\gamma _{R}\right) +1\right) ^{2}-1\right) < \alpha _{1} \end{aligned}$$

which holds by (5). Therefore, $\frac{\partial W_{R}(p_{O}^{NB})}{\partial p}<0$ and $p_{R}^{NB}<p_{O}^{NB}$.

Step 2: Next, we show that revenue-based penalties are greater than overcharge-based penalties for all $p\in \left[ c,c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}\right)$. Note that

$$\begin{aligned} x_{R}\left( p\right) -x_{O}\left( p\right)&=\gamma _{R}p(1-p)-\gamma _{O}Q_{N}(p-c)\\&\ge \gamma _{R}H(c)-\gamma _{O}Q_{N}(p-c)\\&\ge \gamma _{R}H(c)-\gamma _{O}Q_{N}\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}. \end{aligned}$$

$\gamma _{R}H(c)-\gamma _{O}Q_{N}\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}>0$ holds if

$$\begin{aligned} \gamma _{R}H(c)&>\gamma _{O}Q_{N}\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}\\ \sqrt{\alpha _{1}}&>\frac{\gamma _{O}Q_{N}}{\gamma _{R}H(c)}\sqrt{1-\alpha _{0}}\\ \alpha _{1}&>\left( \frac{\gamma _{O}Q_{N}}{\gamma _{R}H(c)}\right) ^{2}\left( 1-\alpha _{0}\right) , \end{aligned}$$

which holds by (5). Thus, $x_{R}\left( p\right) >x_{O}\left( p\right)$ and $W_{O}(p)>W_{R}(p)$ for all $p\in \left[ c,c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}\right)$.

Step 3: Suppose $p_{O}<p_{R}$. We show that $p_{R}+\epsilon$, for sufficiently small $\epsilon >0$, is incentive-compatible under aggregate overcharge-based penalties and achieves a greater payoff than $p_{O}$. Therefore, $p_{O}$ is not optimal. Note that

$$\begin{aligned} W_{O}(p_{R}+\epsilon )>W_{R}(p_{R})\ge \pi ^{D}(p_{R}), \end{aligned}$$

where the first inequality follows from $W_{O}(p)>W_{R}(p)$ and the continuity of $W_{O}(p)$. The second inequality follows from the incentive-compatibility of $p_{R}$.³⁸ Thus, $p_{R}+\epsilon$ is incentive compatible under overcharge-based penalties.

Next, note that $p_{O}<p_{R}\le p_{R}^{NB}<p_{O}^{NB}$ which implies that $p_{O}<p_{R}+\epsilon <p_{O}^{NB}$ for sufficiently small $\epsilon >0$.³⁹ Therefore, $W_{O}(p_{O})<W_{O}(p_{R}+\epsilon )$ by the strict concavity of $W_{O}(p)$. We have shown $p_{R}+\epsilon$ is incentive-compatible under overcharge-based penalties and results in a greater collusive payoff. Therefore, $p_{O}$ is not the optimal cartel price, which is a contradiction. $\square$

Proof of Theorem 4

First, consider the case of $\delta >\delta _{O}$. The proof follows from the observation that, by Theorems 1 and 2, there exists an ${\tilde{\alpha }}_{1}^{L}$ such that $\delta _{R}>\delta >\delta _{O}$ if $\alpha _{1}>{\tilde{\alpha }}_{1}^{L}$. Next, consider the case of $\delta \le \delta _{O}$. When $\delta \le \delta _{O}$, a cartel does not form under overcharge-based penalties (see the proof of Theorem 1). By Theorem 2, there exists an ${\tilde{\alpha }}_{1}^{L}$ such that $\delta _{R}>\delta$ if $\alpha _{1}>{\tilde{\alpha }}_{1}^{L}$. Thus, a cartel does not form under either penalty type when $\alpha _{1}>{\tilde{\alpha }}_{1}^{L}$ which implies $CS_{R}=CS_{O}$ and $TS_{R}=TS_{O}$. $\square$

1.2 Changes Specification

Proof of Theorem 5

The proof involves two steps: First, we show a cartel forms when $\delta >\delta _{O}$. Second, we show that a cartel does not form when $\delta \le \delta _{O}$.

Suppose that $\delta >\delta _{O}$. Let $W_{O}^{j}(p)$ denote the payoff from collusion under specification $j\in \left\{ L,C\right\}$ when the cartel price is p in all periods.⁴⁰ A cartel forms under specification L when $\delta >\delta _{O}$ (see Theorem 1). Let $p_{O}^{L}$ denote the cartel price under specification L. $p_{O}^{L}>c$, and therefore $\pi ^{D}(p_{O}^{L})>0$, by Lemma 1. In addition, $W_{O}^{L}(p_{O}^{L})\ge \pi ^{D}(p_{O}^{L})$ because collusion must be sustainable for a cartel to form. Lastly, note that $W_{O}^{C}(p)\ge W_{O}^{L}(p)$ for all p. Thus,

$$\begin{aligned} W_{O}^{C}(p_{O}^{L})\ge W_{O}^{L}(p_{O}^{L})\ge \pi ^{D}(p_{O}^{L})>0, \end{aligned}$$

which implies that collusion is sustainable and profitable under specification C when $\delta >\delta _{O}$ at a constant price of $p_{O}^{L}$. Thus, a cartel forms under specification C when $\delta >\delta _{O}$.

Next, suppose $\delta \le \delta _{O}$ and a price path of $\left\{ p_{t}\right\} _{t=1}^{\infty }$ is sustainable and profitable under specification C. Let $W_{O}^{j}\left( p_{t-1},\left\{ p_{t}\right\} _{t=1}^{\infty };\alpha _{1}\right)$ denote the present discounted value of collusion when: the prior period’s price is $p_{t-1}$; the cartel price path is $\left\{ p_{t}\right\} _{t=1}^{\infty }$; the sensitivity of the probability of detection to price is $\alpha _{1}$ and the $\phi$ specification is $j\in \left\{ L,C\right\}$. First, note that $W_{O}^{C}\left( p_{t-1},\left\{ p_{t}\right\} _{t=1}^{\infty };\alpha _{1}\right) \le W_{O}^{C}\left( p_{t-1},\left\{ p_{t}\right\} _{t=1}^{\infty };0\right) =W_{O}^{L}\left( p_{t-1},\left\{ p_{t}\right\} _{t=1}^{\infty };0\right)$ for all t.⁴¹ As $\left\{ p_{t}\right\} _{t=1}^{\infty }$ is sustainable under specification C,

$$\begin{aligned} \pi ^{D}(p_{t})\le W_{O}^{C}\left( p_{t-1},\left\{ p_{t}\right\} _{t=1}^{\infty };\alpha _{1}\right) \le W_{O}^{C}\left( p_{t-1},\left\{ p_{t}\right\} _{t=1}^{\infty };0\right) =W_{O}^{L}\left( p_{t-1},\left\{ p_{t}\right\} _{t=1}^{\infty };0\right) \end{aligned}$$

for all t. This implies that collusion is sustainable when $\alpha _{1}=0$ and $\delta \le \delta _{O}$ under specification L. Because the price path $\left\{ p_{t}\right\} _{t=1}^{\infty }$ is profitable under specification C,

$$\begin{aligned} 0<W_{O}^{C}\left( c,\left\{ p_{t}\right\} _{t=1}^{\infty };\alpha _{1}\right) \le W_{O}^{C}\left( c,\left\{ p_{t}\right\} _{t=1}^{\infty };0\right) =W_{O}^{L}\left( c,\left\{ p_{t}\right\} _{t=1}^{\infty };0\right) \end{aligned}$$

which implies collusion is profitable when $\alpha _{1}=0$ and $\delta \le \delta _{O}$ under specification L. We have shown the price path $\left\{ p_{t}\right\} _{t=1}^{\infty }$ is profitable and sustainable when $\alpha _{1}=0$ under specification L. Therefore, a cartel forms when $\alpha _{1}=0$ and $\delta \le \delta _{O}$ under specification L, which is a contradiction. $\square$

Lemma 3

$p_{t}<p_{t-1}+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}$ for all $t\ge 1$ under both overcharge and revenue-based penalties.

Proof

Let $\left\{ p_{t}\right\} _{t=1}^{\infty }$ be an optimal price path where $p_{t'}\ge p_{t'-1}+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}$ for some $t'$ (and let $t'$ denote the first such t). Note that $\phi (p_{t'-1},p_{t'})=1$. The optimality of $p_{t'},p_{t'+1},p_{t'+2}\dots$ implies that it yields a payoff that is greater than or equal to the payoff from the alternative price sequence of $p_{1},p_{2},p_{3}\dots$ from $t'$ onwards.⁴² Thus,

$$\begin{aligned} V(p_{t'-1})&=\pi (p_{t'})-x(p_{t'})+\delta V(c)\nonumber \\&=\pi (p_{t'})-x(p_{t'})+\delta \left[ \pi (p_{1})-\phi (c,p_{1})x(p_{1})\right. \nonumber \\&\quad \left. +\delta \left[ 1-\phi (c,p_{1})\right] V(p_{1})+\delta \phi (c,p_{1})V(c)\right] \nonumber \\&\ge \pi (p_{1})-\phi (p_{t'-1},p_{1})x(p_{1})+\delta \left[ 1-\phi (p_{t'-1},p_{1})\right] V(p_{1})\nonumber \\&\quad +\delta \phi (p_{t'-1},p_{1})V(c). \end{aligned}$$

(8)

Note that $\pi (p_{t'})-x(p_{t'})\le 0$ because $\gamma _{O}>1$ and $\gamma _{R}>1$.⁴³ Thus, inequality (8) implies

$$\begin{aligned} \delta V(c)\ge \pi (p_{1})-\phi (p_{t'-1},p_{1})x(p_{1})+\delta \left[ 1-\phi (p_{t'-1},p_{1})\right] V(p_{1})+\delta \phi (p_{t'-1},p_{1})V(c), \end{aligned}$$

or

$$\begin{aligned}&\pi (p_{1})-\phi (p_{t'-1},p_{1})x(p_{1})+\delta \left[ 1-\phi (p_{t'-1},p_{1})\right] V(p_{1})+\delta \phi (p_{t'-1},p_{1})V(c)\nonumber \\ \le \quad&\delta \left[ \pi (p_{1})-\phi (c,p_{1})x(p_{1})+\delta \left[ 1-\phi (c,p_{1})\right] V(p_{1})+\delta \phi (c,p_{1})V(c)\right] . \end{aligned}$$

(9)

Note that $\pi (p_{1})-\phi (p_{t'-1},p_{1})x(p_{1})\ge \pi (p_{1})-\phi (c,p_{1})x(p_{1})$ because $\phi (p_{t'-1},p_{1})\le \phi (c,p_{1})$. Also, note that $\delta <1$. Thus, inequality (9) implies

$$\begin{aligned} \delta \left[ 1-\phi (p_{t'-1},p_{1})\right] V(p_{1})+\delta \phi (p_{t'-1},p_{1})V(c)<&\delta \left[ 1-\phi (c,p_{1})\right] V(p_{1})+\delta \phi (c,p_{1})V(c). \end{aligned}$$

(10)

Given that $V(p_{1})\ge V(c)$,⁴⁴ inequality (10) holds only if $\phi (p_{t'-1},p_{1})>\phi (c,p_{1})$ which is false. Therefore, $p_{t'}$ is not optimal and the proof is complete. $\square$

Proof of Theorem 6

Note that, because the initial price is c, the cartel price in period t satisfies

$$\begin{aligned} p_{t}<p_{t-1}+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}<p_{t-2}+2\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}<...<c+t\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}, \end{aligned}$$

where the inequalities follow from Lemma 3. Let $T^{*}$ be such that $T^{*}>2$ and $\frac{\delta ^{T^{*}}}{1-\delta }<\frac{\alpha _{0}\gamma _{R}c}{2}D\left( \frac{1+c}{2}\right)$. Let

$$\begin{aligned} \alpha _{1}>{\hat{\alpha }}_{1}^{C}=\max \left\{ \left[ T^{*}\frac{\sqrt{1-\alpha _{0}}}{\alpha _{0}\gamma _{R}c}\right] ^{2},\frac{4\left( 1-\alpha _{0}\right) }{(1-c)^{2}}\right\} . \end{aligned}$$

(11)

We show that collusion is unprofitable when $\alpha _{1}>{\hat{\alpha }}_{1}^{C}$, and therefore a cartel does not form for all $\delta <1$.

Suppose a cartel forms for some $\alpha _{1}>{\hat{\alpha }}_{1}^{C}$. Let $\left\{ p_{t}\right\} _{t=1}^{\infty }$ denote the cartel price path. First, observe that

$$\begin{aligned} p_{1}&<c+\sqrt{\frac{1-\alpha _{0}}{\alpha _{1}}}\\&\quad <c+\sqrt{\frac{1-\alpha _{0}}{\frac{4\left( 1-\alpha _{0}\right) }{(1-c)^{2}}}}\\&\quad =c+\sqrt{\frac{(1-c)^{2}}{4}}\\&\quad =c+\frac{1-c}{2}=\frac{1+c}{2}, \end{aligned}$$

where the first inequality follows from Lemma 3 and the second inequality follows from inequality (11).

Next, observe that

$$\begin{aligned} V\left( c\right)&=\pi (p_{1})-\phi (c,p_{1})x_{R}(p_{1})+\phi (c,p_{1})\delta V(c)+\left[ 1-\phi (c,p_{1})\right] \delta V(p_{1})\\&\le \pi (p_{1})-\phi (c,p_{1})x_{R}(p_{1})+\delta V(p_{1})\\&=\pi (p_{1})-\phi (c,p_{1})x_{R}(p_{1})\\&\quad +\delta \left[ \pi (p_{2})-\phi (p_{1},p_{2})x_{R}(p_{2})+\phi (p_{1},p_{2})\delta V(c)+\left[ 1-\phi (p_{1},p_{2})\right] \delta V(p_{2})\right] \\&\le \pi (p_{1})-\phi (c,p_{1})x_{R}(p_{1})\\&\quad +\delta \left[ \pi (p_{2})-\phi (p_{1},p_{2})x_{R}(p_{2})+\delta V(p_{2})\right] \\&\le \sum _{t=1}^{\infty }\delta ^{t-1}\left[ \pi (p_{t})-\phi (p_{t-1},p_{t})x_{R}(p_{t})\right] \\&\le \sum _{t=1}^{\infty }\delta ^{t-1}\left[ \pi (p_{t})-\alpha _{0}x_{R}(p_{t})\right] , \end{aligned}$$

where the first, second and third inequalities follow from the observation that $V(p_{t})\ge V(c)$ for all t.⁴⁵ The last inequality follows from $\phi \ge \alpha _{0}$. Therefore,

$$\begin{aligned} V\left( c\right)&\le \sum _{t=1}^{\infty }\delta ^{t-1}\left[ D(p_{t})(p_{t}-c)-\alpha _{0}\gamma _{R}D(p_{t})p_{t}\right] \\&=\sum _{t=1}^{\infty }\delta ^{t-1}\left[ D(p_{t})\left( p_{t}\left[ 1-\alpha _{0}\gamma _{R}\right] -c\right) \right] \\&=\underbrace{D(p_{1})\left( p_{1}\left[ 1-\alpha _{0}\gamma _{R}\right] -c\right) }_{A}\\&\quad +\underbrace{\sum _{t=2}^{T^{*}}\delta ^{t-1}\left[ D(p_{t})\left( p_{t}\left[ 1-\alpha _{0}\gamma _{R}\right] -c\right) \right] }_{B}\\&\quad +\underbrace{\sum _{t=T^{*}+1}^{\infty }\delta ^{t-1}\left[ D(p_{t})\left( p_{t}\left[ 1-\alpha _{0}\gamma _{R}\right] -c\right) \right] }_{C}. \end{aligned}$$

We show $A+C<0$ and $B\le 0$which implies $V\left( c\right) <0$ and, therefore, a cartel does not form. First, note that

$$\begin{aligned} A&=D(p_{1})\left( p_{1}\left[ 1-\alpha _{0}\gamma _{R}\right] -c\right) \\&\le D(p_{1})\left[ \left( c+\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}\right) \left[ 1-\alpha _{0}\gamma _{R}\right] -c\right] \\&=D(p_{1})\left[ c+\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}-c\alpha _{0}\gamma _{R}-\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}\alpha _{0}\gamma _{R}-c\right] \\&=D(p_{1})\left[ \frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}-c\alpha _{0}\gamma _{R}-\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}\alpha _{0}\gamma _{R}\right] \\&\le D(p_{1})\left[ \frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}-c\alpha _{0}\gamma _{R}\right] \\&<-\frac{c\alpha _{0}\gamma _{R}}{2}D(p_{1})\\&<-\frac{c\alpha _{0}\gamma _{R}}{2}D\left( \frac{1+c}{2}\right) \end{aligned}$$

where the last inequality follows from $p_{1}<\frac{1+c}{2}$ and the second to last inequality follows from Eq. (11):

$$\begin{aligned}&\left[ \frac{T^{*}\sqrt{1-\alpha _{0}}}{c\alpha _{0}\gamma _{R}}\right] ^{2}<\alpha _{1}\\&\quad \implies \left[ \frac{2\sqrt{1-\alpha _{0}}}{c\alpha _{0}\gamma _{R}}\right] ^{2}<\alpha _{1}\\&\quad \implies \frac{2\sqrt{1-\alpha _{0}}}{c\alpha _{0}\gamma _{R}}<\sqrt{\alpha _{1}}\\&\quad \implies \frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}<\frac{c\alpha _{0}\gamma _{R}}{2}\\&\quad \implies \frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}-c\alpha _{0}\gamma _{R} <-\frac{c\alpha _{0}\gamma _{R}}{2}. \end{aligned}$$

Thus, we have shown $A<-\frac{c\alpha _{0}\gamma _{R}}{2}D\left( \frac{1+c}{2}\right)$. $B\le 0$ because

$$\begin{aligned} B&=\sum _{t=2}^{T^{*}}\delta ^{t-1}\left[ D(p_{t})\left( p_{t}\left[ 1-\alpha _{0}\gamma _{R}\right] -c\right) \right] \\&\le \sum _{t=2}^{T^{*}}\delta ^{t-1}D(p_{t})\left[ \left( c+t\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}\right) \left[ 1-\alpha _{0}\gamma _{R}\right] -c\right] \\&=\sum _{t=2}^{T^{*}}\delta ^{t-1}D(p_{t})\left[ t\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}-\alpha _{0}\gamma _{R}c-\alpha _{0}\gamma _{R}t\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}\right] \\&\le \sum _{t=2}^{T^{*}}\delta ^{t-1}D(p_{t})\left[ t\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}-\alpha _{0}\gamma _{R}c\right] \\&\le \sum _{t=2}^{T^{*}}\delta ^{t-1}D(p_{t})\left[ T^{*}\frac{\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}-\alpha _{0}\gamma _{R}c\right] \\&\le 0, \end{aligned}$$

where the last inequality follows from Eq. (11):

$$\begin{aligned} \alpha _{1}&>\left[ T^{*}\frac{\sqrt{1-\alpha _{0}}}{\alpha _{0}\gamma _{R}c}\right] ^{2}\\ \sqrt{\alpha _{1}}&>T^{*}\frac{\sqrt{1-\alpha _{0}}}{\alpha _{0}\gamma _{R}c}\\ \alpha _{0}\gamma _{R}c&>\frac{T^{*}\sqrt{1-\alpha _{0}}}{\sqrt{\alpha _{1}}}. \end{aligned}$$

Lastly, note that

$$\begin{aligned} C&=\sum _{t=T^{*}+1}^{\infty }\delta ^{t-1}\left[ D(p_{t})\left( p_{t}\left[ 1-\alpha _{0}\gamma _{R}\right] -c\right) \right] \\&\le \sum _{t=T^{*}+1}^{\infty }\delta ^{t-1}\\&=\frac{\delta ^{T^{*}}}{1-\delta }<\frac{\alpha _{0}\gamma _{R}c}{2}D\left( \frac{1+c}{2}\right) , \end{aligned}$$

where the first inequality follows from the observation that per-period profit is bounded above by 1. The second inequality follows from the definition of $T^{*}$. Therefore, $A+C<-\frac{c\alpha _{0}\gamma _{R}}{2}D\left( \frac{1+c}{2}\right) +\frac{c\alpha _{0}\gamma _{R}}{2}D\left( \frac{1+c}{2}\right) =0$. Thus, collusion is not profitable, and no cartel forms when $\alpha _{1}>{\hat{\alpha }}_{1}^{C}$. This implies $\delta _{R}\rightarrow 1$ as $\alpha _{1}\rightarrow \infty$.

Next, we show $\delta _{R}\rightarrow 1$ monotonically as $\alpha _{1}\rightarrow \infty$. Let $\delta _{R}(\alpha _{1})$ denote the critical discount factor when the sensitivity of the probability of detection to the change in price is $\alpha _{1}$. Suppose $\delta _{R}(\alpha '_{1})>\delta _{R}(\alpha ''_{1})$ for $\alpha '_{1}<\alpha ''_{1}$. Let $\delta$ be such that $\delta _{R}(\alpha ''_{1})<\delta <\delta _{R}(\alpha '_{1})$. A cartel forms when the discount factor is $\delta$ if $\alpha _{1}=\alpha ''_{1}$ and does not form if $\alpha _{1}=\alpha '_{1}$. Let $\{p''_{t}\}_{t=1}^{\infty }$ denote the cartel price path when $\alpha _{1}=\alpha ''_{1}$ and the discount factor is $\delta$. Let $W_{R}(p_{t-1},\{p_{t}\}_{t=1}^{\infty };\alpha _{1},\delta )$ denote the payoff from collusion when the cartel price path is $\{p_{t}\}_{t=1}^{\infty }$, the prior period price is $p_{t-1}$, the sensitivity of the probability of detection to the change in price is $\alpha _{1}$ and the discount factor is $\delta$.

Note that

$$\begin{aligned} \pi ^{D}(p''_{t})\le W_{R}(p''_{t-1},\{p''_{t}\}_{t=1}^{\infty };\alpha ''_{1},\delta )\le W_{R}(p''_{t-1},\{p''_{t}\}_{t=1}^{\infty };\alpha '_{1},\delta ) \end{aligned}$$

for all t where the first inequality follows from the sustainability of collusion when $\alpha _{1}=\alpha ''_{1}$. The second inequality follows from $\alpha '_{1}<\alpha ''_{1}$. Also, note that

$$\begin{aligned} 0<W_{R}(c,\{p''_{t}\}_{t=1}^{\infty };\alpha ''_{1},\delta )\le W_{R}(c,\{p''_{t}\}_{t=1}^{\infty };\alpha '{}_{1},\delta ), \end{aligned}$$

where the first inequality follows from the profitability of collusion when $\alpha _{1}=\alpha ''_{1}$. The second inequality follows from $\alpha '_{1}<\alpha ''_{1}$. Thus, collusion is sustainable and profitable with a price path of $\{p''_{t}\}_{t=1}^{\infty }$ when $\alpha _{1}=\alpha '_{1}$ and the discount factor is $\delta$. Therefore, a cartel forms when $\alpha _{1}=\alpha '_{1}$ which implies $\delta >\delta _{R}(\alpha '_{1})$, a contradiction. We therefore conclude that $\delta _{R}\rightarrow 1$ monotonically as $\alpha _{1}\rightarrow \infty$. $\square$

Proof of Theorem 7

First, consider the case of $\delta >\delta _{O}$. The proof follows from the observation that, by Theorems 5 and 6, there exists an ${\tilde{\alpha }}_{1}^{C}$ such that $\delta _{R}>\delta >\delta _{O}$ if $\alpha _{1}>{\tilde{\alpha }}_{1}^{C}$. Next, consider the case of $\delta \le \delta _{O}$. When $\delta \le \delta _{O}$, a cartel does not form under overcharge-based penalties (see the proof of Theorem 5). By Theorem 6, there exists an ${\tilde{\alpha }}_{1}^{C}$ such that $\delta _{R}>\delta$ if $\alpha _{1}>{\tilde{\alpha }}_{1}^{C}$. Thus, a cartel does not form under either penalty type when $\alpha _{1}>{\tilde{\alpha }}_{1}^{C}$ which implies $CS_{R}=CS_{O}$ and $TS_{R}=TS_{O}$. $\square$

Appendix 2: Additional Derivations

In this section, we derive expressions for consumer surplus and total surplus.

Let the cartel’s price path be denoted $\left\{ p_{t}\right\} _{t=1}^{\infty }$. Let $CS_{t}$ denote the expected present discounted value of consumer surplus when the price in the prior period was $p_{t-1}$. $CS_{1}$ is the expected present discounted value of consumer surplus when the price in the prior period was c: the initial period or any period immediately after detection. Let $CS(p_{t})$ denote per-period consumer surplus when the cartel price is $p_{t}$ and let $\phi _{t}=\phi (p_{t-1},p_{t})$.

$$\begin{aligned} CS_{t}=CS(p_{t})+\delta (1-\phi _{t})CS_{t+1}+\delta \phi _{t}CS_{1}. \end{aligned}$$

If undetected in period t (which happens with probability $(1-\phi _{t})$), the cartel continues into the next period (period $t+1$) with an initial price of $p_{t}$. If detected (which happens with probability $\phi _{t}$), the cartel reforms with an initial price of c. Note that

$$\begin{aligned} CS_{1}&=CS(p_{1})+\delta (1-\phi _{1})CS_{2}+\delta \phi _{1}CS_{1}\\&=CS(p_{1})+\delta (1-\phi _{1})\left( CS(p_{2})+\delta (1-\phi _{2})CS_{3}+\delta \phi _{2}CS_{1}\right) +\delta \phi _{1}CS_{1}\\&=CS(p_{1})+\delta (1-\phi _{1})CS(p_{2})+\delta ^{2}(1-\phi _{2})(1-\phi _{1})CS_{3}+\delta ^{2}\phi _{2}(1-\phi _{1})CS_{1}+\delta \phi _{1}CS_{1}. \end{aligned}$$

Continuing this pattern yields

$$\begin{aligned} CS_{1}=\sum _{t=1}^{\infty }\delta ^{t-1}\left( \prod _{\tau =1}^{t-1}(1-\phi _{\tau })\right) CS(p_{t})+\sum _{t=1}^{\infty }\delta ^{t}\phi _{t}\left( \prod _{\tau =1}^{t-1}(1-\phi _{\tau })\right) CS_{1}. \end{aligned}$$

Solving for $CS_{1}$ yields

$$\begin{aligned} CS_{1}=\frac{\sum _{t=1}^{\infty }\delta ^{t-1}\left( \prod _{\tau =1}^{t-1}(1-\phi _{\tau })\right) CS(p_{t})}{1-\sum _{t=1}^{\infty }\delta ^{t}\phi _{t}\left( \prod _{\tau =1}^{t-1}(1-\phi _{\tau })\right) }, \end{aligned}$$

where we let $\prod _{\tau =1}^{0}\left[ 1-\phi _{\tau }\right] =1$ as a convention.⁴⁶

Computations analogous to those above show that the expected present discounted value of total surplus—the sum of consumer surplus and aggregate cartel profit—is

$$\begin{aligned} TS_{1}=\frac{\sum _{t=1}^{\infty }\delta ^{t-1}\left( \prod _{\tau =1}^{t-1}(1-\phi _{\tau })\right) TS(p_{t})}{1-\sum _{t=1}^{\infty }\delta ^{t}\phi _{t}\left( \prod _{\tau =1}^{t-1}(1-\phi _{\tau })\right) } \end{aligned}$$

(12)

where $\phi _{t}=\phi (p_{t-1},p_{t})$ and $TS(p_{t})$ is total surplus when the cartel price is $p_{t}$.⁴⁷