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.

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