Collusion under product differentiation

Abstract

The present model analyses the possibility of stable cartels under vertical and horizontal product differentiation in the presence of cost asymmetry. This possibility is lesser for an agreement that allows the lower quality product to be produced when the quality difference (net of cost) increases or the level of horizontal product differentiation decreases. However, if side payments are allowed, and the cartel agreement does not allow the lower quality product to be produced, the result changes. In this second situation, the possibility of a stable cartel falls if the quality difference (net of cost) falls or the horizontal product differentiation increases. Welfare may increase after cartel formation if the lower quality good is not produced in the presence of side payments.

Appendices

A Appendix

1.1 A.1 Proof of Lemma 3

Lemma 3: $\delta ^{min}_{1}$ rises in $\theta$, whereas $\delta ^{min}_{2}$ falls in $\theta$. $\delta ^{min}_{2}>\delta ^{min}_{1}$ for $\theta \in (\bar{\theta },1)$.

Proof

From expression (19), we have $\delta ^{min}_{1}=\frac{\Big (\frac{(2-\gamma ^2)-\gamma \theta }{4(1-\gamma ^2)}\Big )^2-\Big (\frac{1-\gamma \theta }{4(1-\gamma ^2)}\Big )}{\Big (\frac{(2-\gamma ^2)-\gamma \theta }{4(1-\gamma ^2)}\Big )^2-\Big (\frac{2-\gamma \theta }{4-\gamma ^2}\Big )^2}$. Let us call the numerator of this expression as $Z_{1}$, where $Z_{1}(\alpha _{1}-c_{1})^{2}=(\pi ^{D}_{1}-\pi ^{C}_{1})$ and the denominator of this expression as $V_{1}$, where $V_{1}(\alpha _{1}-c_{1})^{2}=(\pi ^{D}_{1}-\pi ^{N}_{1})$. Partially differentiating $Z_{1}$ and $V_{1}$ w.r.t. $\theta$, we get $\frac{\partial Z_{1}}{\partial \theta }=\frac{{\gamma }^{2}(\theta -\gamma )}{4(1-{\gamma }^{2})^{2}}>0$ and $\frac{\partial V_{1}}{\partial \theta }=-2\gamma \Big [\frac{(2-\gamma ^2)-\gamma \theta }{16(1-{\gamma }^{2})^{2}}-\frac{2-\gamma \theta }{(4-{\gamma }^{2})^{2}}\Big ]<0$. This therefore implies that $\frac{\partial \delta ^{min}_{1}}{\partial \theta }>0$ for $\theta \in (\bar{\theta },1)$. From expression (19), we have $\delta ^{min}_{2}=\frac{\Big (\frac{(2-\gamma ^2)\theta -\gamma }{4(1-\gamma ^2)}\Big )^2-\Big (\frac{\theta (\theta -\gamma )}{4(1-\gamma ^2)}\Big )}{\Big (\frac{(2-\gamma ^2)\theta -\gamma }{4(1-\gamma ^2)}\Big )^2-\Big (\frac{2\theta -\gamma }{4-\gamma ^2}\Big )^2}$. Let us call the numerator of this expression as $Z_{2}$, where $Z_{2}(\alpha _{1}-c_{1})^{2}=(\pi ^{D}_{2}-\pi ^{C}_{2})$ and the denominator of this expression as $V_{2}$, where $V_{2}(\alpha _{1}-c_{1})^{2}=(\pi ^{D}_{2}-\pi ^{N}_{2})$. Similarly, partially differentiating $Z_{2}$ and $V_{2}$ w.r.t. $\theta$, we get $\frac{\partial Z_{2}}{\partial \theta }=\frac{1}{4(1-{\gamma }^{2})}\Big [\frac{(2\theta -\theta \gamma ^{2}-\gamma )(2-\gamma ^{2})}{2(1-{\gamma }^{2})}-(2\theta -\gamma )\Big ]=\frac{-\gamma ^{3}(1-\theta )}{8(1-\gamma ^{2})^{2}}<0$ and $\frac{\partial V_{2}}{\partial \theta }=2\Big [\frac{(2\theta -\theta \gamma ^2-\gamma )(2-\gamma ^{2})}{16(1-{\gamma }^{2})^{2}}-\frac{2(2\theta -\gamma )}{(4-{\gamma }^{2})^{2}}\Big ]>0$. This therefore implies that $\frac{\partial \delta ^{min}_{2}}{\partial \theta }<0$ for $\theta \in (\bar{\theta },1)$. Moreover, for $\theta =\bar{\theta }$, we know that $\pi ^{C}_{2}=\pi ^{N}_{2}$, therefore $\delta ^{min}_{2}=1$, and at $\theta =1$, $\delta ^{min}_{1}=\delta ^{min}_{2}$ as the firms become identical. Therefore, $\delta ^{min}_{2}>\delta ^{min}_{1}$ for $\theta \in (\bar{\theta },1)$, as $\delta ^{min}_{2}$ falls in $\theta$, whereas $\delta ^{min}_{1}$ increases in $\theta$. $\square$

1.2 A.2 Note on Lemma 5

Lemma 5 The possibility of the stable cartel without side payments decreases if $\gamma$ increases or $\frac{\partial \delta ^{min}_{2}}{\partial \gamma } >0$.

To understand the impact of $\gamma$ on the stability of the cartel, let us see how the increase in $\gamma$ affects $\delta ^{min}_{2}$. It is to be noted that $\bar{\theta }=\frac{\gamma }{2}+\frac{4-\gamma ^{2}}{2\sqrt{8+\gamma ^{2}}}~(>\gamma )$ (see Lemma 2 in the main text) is an increasing function of $\gamma$. It is also discussed in the proof of Lemma 2 in Section A.1. Therefore, if $\gamma$ increases from say $\gamma _{a}$ to $\gamma _{b}$, then $\bar{\theta }$ increases from $\bar{\theta _{a}}$ to $\bar{\theta _{b}}$ (such that $\gamma _{b}>\gamma _{a}$ and $\bar{\theta _{b}}>\bar{\theta _{a}}$).

Therefore, given $\gamma =\gamma _{a}$, at $\bar{\theta }=\bar{\theta _{a}}$, $\delta ^{min}_{2}=1$. This is because at $\theta =\bar{\theta }$, as stated in Section A.1, $\pi ^{C}_{2}=\pi ^{N}_{2}$, and therefore $\delta ^{min}_{2}=1$. Thus, without side payments the cartel formation is possible if $\theta \in (\bar{\theta _{a}},1)$ (See Case C of Proposition 1).

Similarly, given $\gamma =\gamma _{b}$, at $\bar{\theta }=\bar{\theta _{b}}$, $\delta ^{min}_{2}=1$ and without side payments the cartel formation is possible if $\theta \in (\bar{\theta _{b}},1)$.

Moreover, at $\theta =1$, $\delta ^{min}_{2}(1,\gamma )=\frac{(2+\gamma )^{2}}{(2+\gamma )^{2}+4(1+\gamma )} (<1)$ and it increases in $\gamma$. Therefore, at $\theta =1$, $\delta ^{min}_{2}(1,\gamma _{b})>\delta ^{min}_{2}(1,\gamma _{a})$ as $\gamma _{b}>\gamma _{a}$.

Thus, if $\gamma _{b}>\gamma _{a}$ then $\bar{\theta _{b}}>\bar{\theta _{a}}$ and the following is observed (as stated above):

i) For $\gamma =\gamma _{a}$ we have $\theta \in (\bar{\theta _{a}},1)$ as the cartel is possible without side payments and at $\theta =\bar{\theta _{a}}$, $\delta ^{min}_{2}=1$ and at $\theta =1$, $\delta ^{min}_{2}=\delta ^{min}_{2}(1,\gamma _{a})$.

ii) For $\gamma =\gamma _{b}~(>\gamma _{a})$ we have $\theta \in (\bar{\theta _{b}},1)$ as the cartel is possible without side payments and at $\theta =\bar{\theta _{b}}~(>\bar{\theta _{a}})$, $\delta ^{min}_{2}=1$ and at $\theta =1$, $\delta ^{min}_{2}=\delta ^{min}_{2}(1,\gamma _{b})~\big (>\delta ^{min}_{2}(1,\gamma _{a})\big )$.

Moreover, $\delta ^{min}_{2}(\theta ,\gamma )$ is a continuous function, and it also increases in $\gamma$ or $\frac{\partial \delta ^{min}_{2}}{\partial \gamma } >0$. We have checked it using Wolfram Mathematica that $\frac{\partial \delta ^{min}_{2}}{\partial \gamma } >0$ in zone C of Fig. 1. Therefore, the possibility of the stable cartel reduces if $\gamma$ increases.

This is also explained in Fig. 2 in the main text, where we measure $\theta$ on the x-axis and $\delta ^{min}_{2}$ on the y-axis. There the $\delta ^{min}_{2}$ curve shifts upward when $\gamma$ rises and thus we observe that $\frac{\partial \delta ^{min}_{2}}{\partial \gamma } >0$.

1.3 A.3 With side payments: critical discounting factors if good 2 is produced

Therefore, the critical discount factors in region $B_{1}$ for both the firms can be written as follows,

$$\begin{aligned} \delta ^{min}_{1}\equiv \frac{\pi ^{D}_{1}-\pi ^{C\prime }_{1}}{\pi ^{D}_{1}-\pi ^{N}_{1}}=\frac{2\pi ^{D}_{1}-\pi ^{C}_{1}-\pi ^{C}_{2}-\pi ^{N}_{1}+\pi ^{N}_{2}}{2\left( \pi ^{D}_{1}-\pi ^{N}_{1}\right) } (>0)~~and~~\delta ^{min}_{2}\equiv 0. \end{aligned}$$

Now, expressing the critical discount rates for both the firms in terms of $\gamma ~~and~~\theta$, we get,

$$\begin{aligned}{} & {} \delta ^{min}_{1}\equiv \frac{2\Big (\frac{(2-\gamma ^2)-\gamma \theta }{4(1-\gamma ^2)}\Big )^2-\Big (\frac{1-\gamma \theta }{4(1-\gamma ^2)}\Big )-\Big (\frac{\theta (\theta -\gamma )}{4(1-\gamma ^2)}\Big )-\Big (\frac{2-\gamma \theta }{4-\gamma ^2}\Big )^2+\Big (\frac{2\theta -\gamma }{4-\gamma ^2}\Big )^2}{2\Big [\Big (\frac{(2-\gamma ^2)-\gamma \theta }{4(1-\gamma ^2)}\Big )^2-\Big (\frac{2-\gamma \theta }{4-\gamma ^2}\Big )^2 \Big ]} ~~or, \\{} & {} \delta ^{min}_{1}\equiv \frac{\frac{(2-\gamma ^2-\gamma \theta )^{2}}{8(1-\gamma ^2)^{2}}-\frac{1-2\gamma \theta +\theta ^{2}}{4(1-\gamma ^2)}-\frac{1-\theta ^{2}}{4-\gamma ^2}}{\frac{(2-\gamma ^{2}-\gamma \theta )^{2}}{8(1-\gamma ^2)^{2}}-\frac{2(2-\gamma \theta )^{2}}{(4-\gamma ^2)^{2}} }. \end{aligned}$$

On the other hand, the critical discount factors in region $B_{2}~and~C$ for firm 1 will be same as above, whereas for the firm 2 it is

$$\begin{aligned}{} & {} \delta ^{min}_{2}\equiv \frac{\pi ^{D}_{2}-\pi ^{C\prime }_{2}}{\pi ^{D}_{2}-\pi ^{N}_{2}}=\frac{2\pi ^{D}_{2}-\pi ^{C}_{1}-\pi ^{C}_{2}-\pi ^{N}_{2}+\pi ^{N}_{1}}{2\left( \pi ^{D}_{2}-\pi ^{N}_{2}\right) }>0. \\{} & {} \delta ^{min}_{2}\equiv \frac{2\Big (\frac{(2-\gamma ^2)\theta -\gamma }{4(1-\gamma ^2)}\Big )^2-\Big (\frac{1-\gamma \theta }{4(1-\gamma ^2)}\Big )-\Big (\frac{\theta (\theta -\gamma )}{4(1-\gamma ^2)}\Big )+\Big (\frac{2-\gamma \theta }{4-\gamma ^2}\Big )^2-\Big (\frac{2\theta -\gamma }{4-\gamma ^2}\Big )^2}{2\Big [\Big (\frac{(2-\gamma ^2)\theta -\gamma }{4(1-\gamma ^2)}\Big )^2-\Big (\frac{2\theta -\gamma }{4-\gamma ^2}\Big )^2 \Big ]} ~~or, \\{} & {} \delta ^{min}_{2}\equiv \frac{\frac{(2\theta -\gamma ^2\theta -\gamma )^{2}}{8(1-\gamma ^2)^{2}}-\frac{1-2\gamma \theta +\theta ^{2}}{4(1-\gamma ^2)}+\frac{1-\theta ^{2}}{4-\gamma ^2}}{\frac{(2\theta -\gamma ^{2}\theta -\gamma )^{2}}{8(1-\gamma ^2)^{2}}-\frac{2(2\theta -\gamma )^{2}}{(4-\gamma ^2)^{2}} }. \end{aligned}$$

1.4 A.4 With side payments: cartel under the possibility of detection

1.4.1 A.4.1 Good 2 is produced

In contrast to Sect. 5.2 we assume here that cartels that involve side payment may be risky as side payment may make it much easier for the antitrust authority to detect the cartel (See Jaspers (2017)). We assume that in the presence of side payments, there is a probability s that the cartel will be detected by the antitrust authority in the period and if so, each firm pays a fine F. However, in the absence of side payments the cartel goes unnoticed. As in Allain et al. (2015), we assume that the antitrust authority may detect the side payment and thereby the cartel on the “spot” (if and when the cartel is active) but not retroactively. This implies that, once the side payment has ceased to exist without being discovered, no firm will be fined in the future.

Thus an optimal sharing agreement is determined by

$$\begin{aligned} \begin{aligned}&\max _{T} V = \left( \pi ^{C^{\prime }}_1-\pi ^N_1\right) \left( \pi _2^{C^{\prime }}-\pi _2^N\right) \\ \end{aligned} \end{aligned}$$

(42)

such that $\pi ^{C^{\prime }}_{i}>\pi ^{N}_{i}$, for $i=1,2$ where $\pi ^{C^{\prime }}_{1}=\pi ^{C}_{1}-T-sF~~and~~ \pi ^{C^{\prime }}_{2}=\pi ^{C}_{2}+T-sF$ are the expected collusive profits of firm 1 and firm 2 respectively. In expectation, this fine is assumed to be too small to deter the usage of side payments. Otherwise, firms will not use side payments for the formation of the cartel. After maximizing Eq. (21), we get the optimal side payment as,

$$\begin{aligned} T^{*}=\frac{\left( \pi ^{C}_{1}-\pi ^{C}_{2}\right) -\left( \pi ^{N}_{1}-\pi ^{N}_{2}\right) }{2}=\frac{3\gamma ^{2}[(\alpha _{1}-c_{1})^{2}-(\alpha _{2}-c_{2})^{2}]}{8(1-\gamma ^{2})(4-\gamma ^{2})} \end{aligned}$$

(43)

and assume that $T^{*}>sF$. Therefore, the profits post cartel formation are, $\pi ^{C^{\prime }}_{1}=\pi ^{C}_{1}-T^{*}-sF\left(>\pi ^{N}_{1}\right) ~~and~~ \pi ^{C^{\prime }}_{2}=\pi ^{C}_{2}+T^{*}-sF\left( >\pi ^{N}_{2}\right)$. We define $\overline{sF}=\pi ^{C}_{2}+T^{*}-\pi ^{N}_{2}$ and thereby assume $sF<\overline{sF}$.

As discussed before firm 1 may want to retain most of its realized profits and therefore, it would be optimal for firm 1 not to pay the side payment $T^{*}$ if it wants to break the cartel agreement. Consequently, the cartel will not be detected (as side payment is not paid) and the profits from deviation for both the firms i.e., $\pi ^{D}_{1}~~and~~\pi ^{D}_{2}$ will be same as before where there is no side payment (see the previous section, Eq. (18)). Hence, $\pi ^{D}_{1}>\pi ^{C^{\prime }}_{1}$, as after deviation firm 1 produces higher output, i.e. $q_{1}^{D}>q_{1}^{C}$, and refuses to pay the side payment ($T^{*}$) to firm 2. This is true because, $\pi ^{D}_{1}>\pi ^{C}_{1}>\pi ^{C^{\prime }}_{1}$. On the other hand, after deviation firm 2 produces higher output, i.e. $q_{2}^{D}>q_{2}^{C}$, but then it losses the side payment ($T^{*}$) from firm 1 from that period onward. After deviation from the cartel agreement firm 2 gets $\pi ^{D}_{2}$ as the cartel will not be detected (as side payment is not paid).

Moreover, $\pi ^{C^{\prime }}_{2}=\pi _{2}^{C}+T^{*}-sF$. Therefore, for $sF=0$, we observe that $\pi _{2}^{D}-\pi _{2}^{C}-T^{*}+sF>0$, if

$$\begin{aligned} \theta ^{2}\left( 6-2\gamma ^{2}-\gamma ^{4}\right) +\left( 5\gamma ^{2}-2\right) -2\gamma \theta \left( 4-\gamma ^{2}\right) >0. \end{aligned}$$

(44)

Thus, we can say with the help of Fig. 4 that when $sF>0$, the region $B_{2}$ (so that $\pi ^{C^{\prime }}_{2}<\pi ^{D}_{2}$) will expand and the region $B_{1}$ (so that $\pi ^{C^{\prime }}_{2}>\pi ^{D}_{2}$) will contract.

Therefore, the critical discount factors in region $B_{1}$ for both the firms are as follows:

$$\begin{aligned} \delta ^{min}_{1}\equiv \frac{\pi ^{D}_{1}-\pi ^{C\prime }_{1}}{\pi ^{D}_{1}-\pi ^{N}_{1}}=\frac{\frac{(2-\gamma ^2-\gamma \theta )^{2}}{8(1-\gamma ^2)^{2}}-\frac{1-2\gamma \theta +\theta ^{2}}{4(1-\gamma ^2)}-\frac{1-\theta ^{2}}{4-\gamma ^2}+2sF}{\frac{(2-\gamma ^{2}-\gamma \theta )^{2}}{8(1-\gamma ^2)^{2}}-\frac{2(2-\gamma \theta )^{2}}{(4-\gamma ^2)^{2}} } (>0)~~and~~\delta ^{min}_{2}\equiv 0, \end{aligned}$$

(45)

whereas the critical discount factors in the region $B_{2}~and~C$ for firm 1 will be the same as above (see Eq. (45)), but for firm 2 it is

$$\begin{aligned} \delta ^{min}_{2}\equiv \frac{\pi ^{D}_{2}-\pi ^{C\prime }_{2}}{\pi ^{D}_{2}-\pi ^{N}_{2}}=\frac{\frac{(2\theta -\gamma ^2\theta -\gamma )^{2}}{8(1-\gamma ^2)^{2}}-\frac{1-2\gamma \theta +\theta ^{2}}{4(1-\gamma ^2)}+\frac{1-\theta ^{2}}{4-\gamma ^2}+2sF}{\frac{(2\theta -\gamma ^{2}\theta -\gamma )^{2}}{8(1-\gamma ^2)^{2}}-\frac{2(2\theta -\gamma )^{2}}{(4-\gamma ^2)^{2}} }>0. \end{aligned}$$

(46)

The cartel will be stable in this case only if, $\delta >max\Big [\delta ^{min}_{1},\delta ^{min}_{2}\Big ]\nonumber$. Therefore, from Eq. (45), it can be said that in region $B_{1}$, the cartel is stable if $\delta >\delta ^{min}_{1}$. Moreover, in region $B_{2}~and~C$, after comparing we observe that $\delta ^{min}_{1}>\delta ^{min}_{2}$ when $sF=0$ as discussed in the main text. When $sF=\overline{sF}$, then $\delta ^{min}_{1}<\delta ^{min}_{2}=1$. Thus, we define $\widetilde{sF}$, such that at $sF=\widetilde{sF}$, $\delta ^{min}_{1}=\delta ^{min}_{2}$. Therefore, we can say that for in region $B_{2}~and~C$

(i) $sF\in (0,\widetilde{sF})$, $\delta ^{min}_{1}>\delta ^{min}_{2}$, and the cartel is stable if $\delta >\delta ^{min}_{1}$.

(ii) $sF=\widetilde{sF}$, $\delta ^{min}_{1}=\delta ^{min}_{2}$ and the cartel is stable if $\delta >\delta ^{min}_{1}=\delta ^{min}_{2}$.

(iii) $sF\in (\widetilde{sF},\overline{sF})$, $\delta ^{min}_{1}<\delta ^{min}_{2}$ and the cartel is stable if $\delta >\delta ^{min}_{2}$.

However, it is less clear cut to show the effect of the changes in $\theta$ and $\gamma$ on the possibility of the cartel as now the $\delta ^{min}_{i}$, $i=1,2$ depends of sF.

1.4.2 A.4.2 Good 2 is not produced

Let us now consider zone A of Fig. 1 where $\theta \in (0,\gamma$), such that $q^{C}_{2}=0$ and consequently $\pi ^{C}_{2}=0$. As in the previous section, an optimal sharing agreement is determined by

$$\begin{aligned} \begin{aligned}&\max _{T} \left( \pi ^{C^{\prime }}_1-\pi ^N_1\right) \left( \pi _2^{C^{\prime }}-\pi _2^N\right) \\ \end{aligned} \end{aligned}$$

(47)

such that $\pi ^{C^{\prime }}_{i}>\pi ^{N}_{i}$ for $i=1,2$ where $\pi ^{C^{\prime }}_{1}=\pi ^{C}_{1}-T-sF~~and~~ \pi ^{C^{\prime }}_{2}=T-sF$ ($>\pi ^{C}_{2}=0$) as in the previous section. After maximizing Eq. (27), we get the optimal side payment as,

$$\begin{aligned} T^{*}=\frac{\pi ^{M}_{1}-\left( \pi ^{N}_{1}-\pi ^{N}_{2}\right) }{2}=\Big [\frac{1}{8}-\frac{1-\theta ^{2}}{2(4-\gamma ^{2})}\Big ]({\alpha _{1}-c_{1}})^{2} \end{aligned}$$

(48)

and therefore the profits post cartel formation are, $\pi ^{C^{\prime }}_{1}=\pi ^{C}_{1}-T^{*}-sF\left( >\pi ^{N}_{1}\right)$ and $\pi ^{C^{\prime }}_{2}=T^{*}-sF\left( >\pi ^{N}_{2}\right)$. We define $\overline{sF}=T^{*}-\pi ^{N}_{2}$ and thereby assume $sF<\overline{sF}$.

Using $q^{C}_{1}=\frac{\alpha _{1}-c_{1}}{2}$, as the output produced by firm 1, in the response function for firm 2 in Eq. (4), we get the quantity produced by firm 2 when it deviates as, $q^{D}_{2}= \frac{[2(\alpha _{2}-c_{2})-\gamma (\alpha _{1}-c_{1})]}{4}.$ Consequently, the profit after deviation for firm 2 is

$$\begin{aligned} \pi ^{D}_{2}= \Big [\frac{2(\alpha _{2}-c_{2})-\gamma (\alpha _{1}-c_{1})}{4} \Big ]^2~~or~~\pi ^{D}_{2}= \left( q^{D}_{2}\right) ^2. \end{aligned}$$

(49)

However, $\pi ^{D}_{2}<\pi ^{C^{\prime }}_{2}$ when $sF=0$, but when sF tends to $T^{*}$ then $\pi ^{D}_{2}>\pi ^{C^{\prime }}_{2}$. Hence, firm 2 will never deviate from the cartel agreement if sF is low, i.e. $\delta ^{min}_{2}\equiv 0$. Otherwise, firm 2 will have the incentive to deviate and the critical discount factor for firm 2 is

$$\begin{aligned} \delta ^{min}_{2}\equiv \frac{\pi ^{D}_{2}-\pi ^{C\prime }_{2}}{\pi ^{D}_{2}-\pi ^{N}_{2}}=\frac{\frac{(2\theta - \gamma )^{2}}{16}-\frac{1}{8}+\frac{1-\theta ^{2}}{2(4-\gamma ^{2})} +sF}{\frac{(2\theta - \gamma )^{2}}{16}- \frac{(2\theta - \gamma )^{2}}{(4-\gamma ^{2})^{2}}} (>0). \end{aligned}$$

(50)

Firm 1 may have some incentives to deviate from the cartel agreement as $\pi ^{D}_{1}>\pi ^{C^{\prime }}_{1}$. Therefore, the critical discount factor for firm 1 is

$$\begin{aligned} \delta ^{min}_{1}\equiv \frac{\pi ^{D}_{1}-\pi ^{C\prime }_{1}}{\pi ^{D}_{1}-\pi ^{N}_{1}}=\frac{\pi ^{C}_{1}-\pi ^{N}_{1}+\pi ^{N}_{2}}{2\left( \pi ^{C}_{1}-\pi ^{N}_{1}\right) }=\frac{\frac{1}{4}-\frac{1-\theta ^{2}}{4-\gamma ^{2}}}{\frac{1}{2}-2\Big (\frac{2-\gamma \theta }{4-\gamma ^2}\Big )^2} (>0) \end{aligned}$$

(51)

The cartel will be stable in this case only if, $\delta >max\Big [\delta ^{min}_{1},\delta ^{min}_{2}\Big ]\nonumber$. Moreover, when $sF=\overline{sF}$, then $\delta ^{min}_{1}<\delta ^{min}_{2}=1$. Thus, we define $\widetilde{sF}$, such that at $sF=\widetilde{sF}$, $\delta ^{min}_{1}=\delta ^{min}_{2}$. Therefore, we can say that for

(i) $sF\in (0,\widetilde{sF})$, $\delta ^{min}_{1}>\delta ^{min}_{2}$, and the cartel is stable if $\delta >\delta ^{min}_{1}$.

(ii) $sF=\widetilde{sF}$, $\delta ^{min}_{1}=\delta ^{min}_{2}$ and the cartel is stable if $\delta >\delta ^{min}_{1}=\delta ^{min}_{2}$.

(iii) $sF\in (\widetilde{sF},\overline{sF})$, $\delta ^{min}_{1}<\delta ^{min}_{2}$ and the cartel is stable if $\delta >\delta ^{min}_{2}$.

However, it is less clear cut to show the effect of the changes in $\theta$ and $\gamma$ on the possibility of the cartel as now the $\delta ^{min}_{i}$, $i=1,2$ depends of sF.

B Welfare

What allocation, respectively, production volumes would a social planner choose to maximize welfare? We have done the following analysis to answer this question.

The industry profit is

$$\begin{aligned} IP= \pi _{1}+ \pi _{2}= \left[ \alpha _{1}-q_{1}-\gamma q_{2}-c_{1}\right] q_{1} + \left[ \alpha _{2}-q_{2}-\gamma q_{1}-c_{2}\right] q_{2}. \end{aligned}$$

(52)

and the consumer surplus is $CS=\frac{1}{2}\left[ {q_{1}^{N}}^{2}+{q_{2}^{N}}^{2}+2\gamma {q_{1}^{N}} {q_{2}^{N}}\right]$. Thus, the welfare is

$$\begin{aligned} W=CS+IP=(\alpha _{1}-c_{1})q_{1}-\frac{(q_{1})^{2}}{2}+(\alpha _{2}-c_{2})q_{2}-\frac{(q_{2})^{2}}{2}-\gamma q_{1}q_{2}. \end{aligned}$$

(53)

Therefore the F.O.Cs of the Welfare maximization problem are

$$\begin{aligned} \frac{\partial W}{\partial q_{1}}=\alpha _{1} - c_{1}-q_{1}-\gamma q_{2}\le 0 ~~and~~\frac{\partial W}{\partial q_{2}}=\alpha _{2} - c_{2}-q_{2}-\gamma q_{1}\le 0, \end{aligned}$$

(54)

with equality when optimal $q_{i}>0$. Thus, the welfare maximization problem has two types of solutions, as mentioned below in terms of the output produced by firm 1 and firm 2, respectively

$$\begin{aligned}{} & {} \mathrm{(i)}~~~ q^{E}_{1}= \frac{(\alpha _{1}- c_{1})-\gamma (\alpha _{2}-c_{2})}{(1-\gamma ^2)} (>0)~~and~~ q^{E}_{2} =\frac{(\alpha _{2}-c_{2})-\gamma (\alpha _{1}-c_{1})}{(1-\gamma ^{2})} (>0)~~or\nonumber \\{} & {} \mathrm{(ii)}~~~ q^{E}_{1}= (\alpha _{1}- c_{1})(>0)~~and~~ q^{E}_{2} =0. \end{aligned}$$

(55)

Thus, i) Case 1: $q^{E}_{2}=\frac{(\alpha _{2}- c_{2})-\gamma (\alpha _{1}-c_{1})}{(1-\gamma ^2)}>0$ if and only if $\theta >\gamma$; otherwise ii) Case 2: $q^{E}_{2}=0$ for $\theta \le \gamma$. We also assume here that $2\theta >\gamma$, as stated in Eq. (6), such that both firms earn positive profits if they compete.

Under case 1 (both goods are produced), after substituting the values of the outputs, we observe that the welfare is

$$\begin{aligned} W_{1}^{E}=\frac{(\alpha _{1}-c_{1})^{2}-2\gamma (\alpha _{1}-c_{1})(\alpha _{2}-c_{2})+(\alpha _{2}-c_{2})^{2}}{2(1-\gamma ^2)}. \end{aligned}$$

(56)

On the other hand, for case 2 (only good 1 is produced), after substituting the values of the outputs, we observe that the welfare is

$$\begin{aligned} W_{2}^{E}=\frac{(\alpha _{1}-c_{1})^{2}}{2}. \end{aligned}$$

(57)

After comparing, we observe that the outputs ($q^{E}_{1}$ and $q^{E}_{2}$), if the social planner maximizes the welfare, are twice the outputs ($q^{C}_{1}$ and $q^{C}_{2}$) if the firms collude; i.e. $q^{E}_{1}=2q^{C}_{1}$ and $q^{E}_{2}=2q^{C}_{2}$. Moreover, after comparing, we also observe that: i) $W_{1}^{E}>W^{N}$ when $\theta >\gamma$ and ii) $W_{2}^{E}>W^{N}$ when $\theta \le \gamma$.

We observe that it may be efficient for the welfare maximization that firm 2 is inactive (Case 2: $q^{E}_{2}=0$ for $\theta \le \gamma$). However, if the market is at play, and the firms are active, then competition will lead to an inefficient high activity of firm 2 as then $q^{N}_{2}>0$.

C A transfer maximizing the possibility of stable cartel

²⁷

We can also consider a transfer maximizing the possibility of stable cartel by minimizing the critical discounting factors w.r.t. T, by solving $\min _{T} \max \left\{ \delta ^{min}_{1}, \delta ^{min}_{2} \right\}$ or $\Leftrightarrow \delta ^{min}_{1}= \delta ^{min}_{2}$. Let us define $\bar{T}$ such that

$$\begin{aligned} \delta ^{min}_{1}=\frac{\pi ^{D}_{1}-\pi ^{C}_{1}+\bar{T}}{\pi ^{D}_{1}-\pi ^{N}_{1}}=\frac{\pi ^{D}_{2}-\pi ^{C}_{2}-\bar{T}}{\pi ^{D}_{2}-\pi ^{N}_{2}}= \delta ^{min}_{2}. \end{aligned}$$

(58)

From the above equation, we get

$$\begin{aligned} \bar{T}=\frac{\left( \pi ^{D}_{2}-\pi ^{C}_{2}\right) \left( \pi ^{D}_{1}-\pi ^{N}_{1}\right) -\left( \pi ^{D}_{2}-\pi ^{N}_{2}\right) \left( \pi ^{D}_{1}-\pi ^{C}_{1}\right) }{\left( \pi ^{D}_{2}-\pi ^{N}_{2}\right) +\left( \pi ^{D}_{1}-\pi ^{N}_{1}\right) }. \end{aligned}$$

(59)

Fig. 10

New diagram: Extension

It is important to note here that the denominator of the term “$\delta ^{min}_{2}$”, i.e. $\Big (\pi ^{D}_{2}-\pi ^{N}_{2}\Big )$ is independent of the side-payment (which is $\bar{T}$ here). The side-payment is paid if and only if none of the firms have deviated from the collusive agreement. Thus, if firm 2 deviates from the collusive agreement, it will get $\pi ^{D}_{2}$ in the present period only, i.e. it also loses the side-payment for that period.

Moreover, the numerator of the term “$\delta ^{min}_{2}$", i.e. the gain for firm 2 from deviation in one period $\Big (\pi ^{D}_{2}-\pi ^{C}_{2}-\bar{T}\Big )$, after substituting $\bar{T}$ (using Eq. (59)) is

$$\begin{aligned} \frac{\Big (\pi ^{D}_{2}-\pi ^{C}_{2}+\pi ^{D}_{1}-\pi ^{C}_{1}\Big )\Big (\pi ^{D}_{2}-\pi ^{N}_{2}\Big )}{\pi ^{D}_{2}-\pi ^{N}_{2}+\pi ^{D}_{1}-\pi ^{N}_{1}}. \end{aligned}$$

(60)

Thus, it is important to first check that whether the numerator as well as the denominator of the term “$\delta ^{min}_{2}$" is positive, i.e. $\pi ^{D}_{2}-\pi ^{N}_{2}>0$ where

$$\begin{aligned} \pi ^{D}_{2}-\pi ^{N}_{2}=(\alpha _{1}-c_{1})^{2}\bigg [\Big (\frac{(2-\gamma ^2)\theta -\gamma }{4(1-\gamma ^2)}\Big )^2-\Big (\frac{2\theta -\gamma }{4-\gamma ^2}\Big )^2\bigg ]. \end{aligned}$$

This is true ($\pi ^{D}_{2}-\pi ^{N}_{2}>0$) if

$$\begin{aligned} 2\theta -3\gamma +\theta \gamma ^{2}>0. \end{aligned}$$

(61)

This holds only in zone $B_{3}$ and zone C of Fig. 10.

Thus, in other zones $B_{4}$ and A as $\pi ^{D}_{2}-\pi ^{N}_{2}< 0$, $\delta ^{min}_{2}=0$ and there does not exist any $\bar{T}$ such that $\delta ^{min}_{1}=\delta ^{min}_{2}$ as $\delta ^{min}_{1}>0$. (Here zones $B_{3}$ and $B_{4}$ comprises zone B of Fig. 1 of the main text where zone A and zone C are also stated. Further as discussed before in Sect. 5.2, in zone $B_{3}$ and zone C, $q_{2}^{C}>0$.) Thus, this methodology, i.e. finding T such that $\delta ^{min}_{1}= \delta ^{min}_{2}$, is not applicable for the entire three zones (A,B and C). Therefore, this methodology is applicable only for zone $B_{3}$ and zone C of Fig. 10.

Thus, plugging in $\bar{T}$ in Eq. (58) for zone $B_{3}$ and zone C of Fig. 10, such that $\pi ^{D}_{2}-\pi ^{N}_{2}>0$, we get

$$\begin{aligned} \delta ^{min}_{i}=\frac{\pi ^{D}_{2}-\pi ^{C}_{2}+\pi ^{D}_{1}-\pi ^{C}_{1}}{\pi ^{D}_{2}-\pi ^{N}_{2}+\pi ^{D}_{1}-\pi ^{N}_{1}}~~i=1,2. \end{aligned}$$

(62)

Therefore, in zone $B_{3}$ and zone C of Fig. 10. ²⁸

$$\begin{aligned} \delta ^{min}_{i}\equiv \frac{\Big (\frac{(2-\gamma ^2)-\gamma \theta }{4(1-\gamma ^2)}\Big )^2-\Big (\frac{1-\gamma \theta }{4(1-\gamma ^2)}\Big )+\Big (\frac{(2-\gamma ^2)\theta -\gamma }{4(1-\gamma ^2)}\Big )^2-\Big (\frac{\theta (\theta -\gamma )}{4(1-\gamma ^2)}\Big )}{\Big (\frac{(2-\gamma ^2)-\gamma \theta }{4(1-\gamma ^2)}\Big )^2-\Big (\frac{2-\gamma \theta }{4-\gamma ^2}\Big )^2+\Big (\frac{(2-\gamma ^2)\theta -\gamma }{4(1-\gamma ^2)}\Big )^2-\Big (\frac{2\theta -\gamma }{4-\gamma ^2}\Big )^2}. \end{aligned}$$

(63)

Moreover, as in Lemmas 6 and 7 respectively mentioned in the main text in Sect. 5.2 here too we observe a similar result. We observe here using the new methodology that²⁹

Lemma 10

(i) The possibility of stable cartel with side payments increases or $\delta ^{min}_{i}$ decreases if $\theta$ increases.

(ii) The possibility of stable cartel with side payments decreases or $\delta ^{min}_{i}$ increases if $\gamma$ increases.

Thus, just as in Proposition 5 we observe that the possibility of the stable cartel decreases ($\delta ^{min}_{i}$ increases) if i) $\theta$ decreases (the quality difference net of cost increases) and/or ii) $\gamma$ increases (the horizontal product differentiation decreases).