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.
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Notes
The beer cartel is one example of a collusive agreement in a market where firms offer a differentiated product. Four Dutch beer brewers formed a cartel on the beer market in the Netherlands, and in 2007 the European Commission fined the leading Dutch brewers a total of around EUR 274 million for operating this cartel between at least 1996 and 1999. Thomadsen and Rhee (2007) mentions that collusion was found in studies of coffee roasting (as shown by Gollop and Roberts (1979)) and banking (Spiller and Favaro (1984)) to name a few.
Motta (2004) also argues that product homogeneity does not unambiguously raise the scope for collusion.
Cramton and Palfrey (1990) says “there are many industries, even within a free-market economy such as in the United States, where explicit collusion and side payments are not only legal but actually encouraged by the cost-less enforcement of cartel agreements by the government".
Among these are the cast iron pipe cartel (1894–98) which used a mechanism of side payments to allocate sales; a contract was awarded to the firm which offered the largest bonus payment for division among other members.
Gabszewicz et al. (2017) analyses whether pruning emerges and if so, a fighting brand is marketed under collusion when the firms produce vertically differentiated products.
Johnson and Myatt (2003) also studies the possibilities of pruning when firms sell multiple quality-differentiated products.
Xu and Coatney (2015) shows that output collusion alone is easier to sustain than collusive product market segmentation when the products are horizontally differentiated.
If \(\theta =1\), then the firms are symmetric and this implies that at the equilibrium their outputs as well as their profit should be the same.
It is true if \((8+\gamma ^{2})(1-\gamma \theta )-4\theta ^{2}(1-\gamma ^{2})>0\). The L.H.S. decreases in \(\theta\) and at \(\theta =1\), L.H.S.\(=0\). Therefore, \((8+\gamma ^{2})(1-\gamma \theta )-4\theta ^{2}(1-\gamma ^{2})>0\) for \(\theta \in (0,1)\). If \(\theta =0\) then \(\alpha _{2}-c_{2}=0\), thus \(q_{2}=0\) not only when the firms compete but also when they collude. Thus we have excluded this case from the discussion.
Bos and Pot (2012) also discusses about the side payments in the context of the cartel formation. They call the cartel without the requirement of the side payments as the hard core cartels, whereas the soft cartels involve side payments.
In this paper, we have used Wolfram Mathematica and Desmos for drawing the diagrams and thereby plotting the regions.
Putting \(\theta =1~~and~~\gamma =1\), we get \(\delta ^{min}_{1}=\delta ^{min}_{2}= \frac{9}{17}\). This case is relevant when we want to determine the value of the critical discount factors for both firms when there is neither horizontal nor vertical product differentiation.
We have checked it using Wolfram Mathematica that \(\frac{\partial \delta ^{min}_{2}}{\partial \gamma } >0\) in zone C of Fig. 1.
We are aware of the other types of models of bargaining that consider the dynamics of the repetition, say Rubinstein (1982). Interestingly, in Rubinstein (1982), if the game is repeated infinitely and the agents are perfectly patient (i.e. their discount factor is one) when they bargain over the division of the total gain, then it also results in equal gains, as stated in the previous paragraph. Thus, we have used Nash-bargaining in our model as in Sen et al. (2023), which studies collusion among heterogeneous firms in the presence of licensing opportunities. Mota et al. (2023) studies collusion between a public firm and a private firm facing linear demand and quadratic costs. They focus on the collusive agreement that results from Nash bargaining between the public and the private firm when bargaining power is equally distributed. The adoption of Nash bargaining for determining the collusive outcome between heterogeneous firms was proposed by Osborne and Pitchik (1983), and was later adopted by Schmalensee (1987) and Harrington (1991).
Symeonidis (1999) says “Unfortunately, other collusive technologies, such as the Nash bargaining solution, have proved analytically difficult to examine. Side payments are not allowed."
We get \(\pi _{2}^{D}-\pi _{2}^{C}-T^{*}=\frac{\gamma ^{2}}{16(1-\gamma ^{2})^{2}(4-\gamma ^{2})}\Big [(\alpha _{2}-c_{2})^{2}(6-2\gamma ^{2}-\gamma ^{4})+(\alpha _{1}-c_{1})^{2}(5\gamma ^{2}-2)-2\gamma (4-\gamma ^{2})(\alpha _{1}-c_{1})(\alpha _{2}-c_{2})\Big ]\)
Verified in Wolfram Mathematica and Desmos.
Here we have divided region B of Fig. 1 in two-segments as \(B_{1}\) and \(B_{2}\) respectively.
We have checked it in Wolfram Mathematica, as manually comparing it is near to impossible.
We have tried to analyze whether firm 2 has the incentive to forgo some (tiny) part of its transfer T to ensure that firm 1 saves enough to “stay on board" when \(\theta \rightarrow \gamma =\frac{1}{2}\). This would make it less attractive for firm 2 to collude. However, this is not true when \(\theta \rightarrow \gamma =\frac{1}{2}\). We thank one anonymous referee for pointing out this.
We have checked it in Wolfram Mathematica.
Similar pattern has been observed for other values of \(\gamma\).
We have checked it in Wolfram Mathematica.
We thank one anonymous referee for pointing out this.
We have checked it in Desmos.
We have checked it in Desmos.
We thank an anonymous referee for this suggestion.
It is also to be noted that both the numerator and the denominator of the following expression is positive. We have checked it in Wolfram Mathematica.
We have checked it in Wolfram Mathematica, as manually comparing it is near to impossible. See Supplementary material.
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The authors thank Arghya Dutta for hel** them in computing using Wolfram Mathematica. We are grateful to the anonymous referees of this journal for providing some insightful comments which helped us to improve our paper significantly. The authors thank Uday Bhanu Sinha, Amrita Chatterjee and Drishti Narula for their valuable suggestions and comments. The usual disclaimer applies.
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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,
Now, expressing the critical discount rates for both the firms in terms of \(\gamma ~~and~~\theta\), we get,
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
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
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,
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
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:
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
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
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,
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
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
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
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
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
Therefore the F.O.Cs of the Welfare maximization problem are
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
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
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
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
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
From the above equation, we get
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
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
This is true (\(\pi ^{D}_{2}-\pi ^{N}_{2}>0\)) if
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
Therefore, in zone \(B_{3}\) and zone C of Fig. 10. Footnote 28
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 thatFootnote 29
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).
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Sen, N., Tandon, U. & Biswas, R. Collusion under product differentiation. J Econ 142, 1–43 (2024). https://doi.org/10.1007/s00712-023-00852-9
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DOI: https://doi.org/10.1007/s00712-023-00852-9