Collusion in Quality-Segmented Markets

This paper analyzes price collusion in a repeated game with two submarkets; a standard and a premium quality segment. Within this setting, we study four types of price-…xing agreement: (i) a segment-wide cartel in the premium submarket only, (ii) a segment-wide cartel in the standard submarket only, (iii) two segment-wide cartels, and (iv) an industry-wide cartel. We present a complete characterization of the collusive pricing equilibrium and examine the corresponding effect on market shares and welfare. Partial cartels operating in a sufficiently large segment lose market share and the industry-wide cartel prefers to maintain market shares at pre-collusive levels. The impact on consumer and social welfare critically depends on the cost of producing quality. Moreover, given that there is a cartel, more collusion can be bene…cial for society as a whole.


Introduction
Virtually all markets can be segmented by product variety or customer group. In principle there are many segmentation dimensions, but perhaps the most common one is (perceived) quality. Quality submarkets emerge due to factors on both the demand and the supply side. On the demand side, di¤erences exist in the ability and willingness to pay for product quality. On the supply side, there are typically larger costs associated with the production of higher quality goods and services. The presence of such quality segments has a nontrivial e¤ect on …rms'strategies and on pricing in particular.
The purpose of this paper is to examine price collusion in quality-segmented markets. We do so by studying a price-setting supergame with two quality submarkets; a standard and a premium segment.
It is assumed that products within each segment are perfect substitutes and that they are di¤erentiated across segments. In the spirit of Mussa and Rosen (1978), Gabszewicz and Thisse (1979) and Shaked and Sutton (1982), consumers have heterogeneous valuations for quality and buy from the …rm with the best value proposition. Within this setting, we consider four types of price-…xing agreement: (i) a segmentwide cartel in the premium submarket only, (ii) a segment-wide cartel in the standard submarket only, (iii) two segment-wide cartels, one in each submarket, and (iv) an industry-wide cartel. It is worth noting that these scenarios have been observed in (antitrust) practice. For example, a recent German co¤ee roaster cartel consisted exclusively of premium brands and faced competition from lower quality private labels (e.g., store brands of Aldi and Lidl). 1 During the interwar period, an incomplete Swiss dyestu¤ export cartel competed with inferior quality rivals from other countries. 2 By contrast, a French yogurt cartel …xed prices of supermarket own-brand yogurt, but premium producers like Danone did not participate in the collusive agreement. 3 As yet another example, a German sausage cartel spanned a large part of the market and included both private label and premium brands. 4 With this in mind, we employ our model to address questions such as: what does optimal segmentwide collusion look like? How does price-…xing a¤ect the market share of the quality-segments? What are the welfare implications of the di¤erent types of price conspiracies? Is a cartel in a premium segment more or less detrimental to consumers and society at large than a cartel in a standard quality segment?
We begin our analysis by providing a complete characterization of the collusive price equilibrium for all four scenarios. Each partial coalition is shown to be capable of sustaining the joint pro…t maximum, whereas the industry-wide cartel charges monopoly prices only when its members are su¢ ciently patient.
Cartel prices are increasing with the inclusiveness of the coalition. That is, prices are higher when collusion spans the whole market and the industry-wide cartel sets (weakly) higher prices compared to when there are two segment-wide partial cartels. The latter is driven by the fact that the all-inclusive cartel internalizes the pricing externality across segments.
These collusive equilibria have an impact on the market share distribution. If there is a single segment-wide cartel, then there is a net movement of customers to the non-collusive segment. With two partial cartels, the change in segment size e¤ectively depends on the competitive market share. If a segment is su¢ ciently small absent collusion, then cartel prices are set such that its sales share increases (and vice versa). The all-inclusive cartel prefers to maintain market shares at pre-collusive levels, which requires prices to rise more in the premium segment. It is noteworthy that such a market division rule has been frequently observed in antitrust practice. 5 Also, we have assumed such an allocation scheme in Bos and Marini (2019), whereas here it emerges endogenously.
As to welfare, we delineate two distinct e¤ects: a market size e¤ect and a market share e¤ect. The market size e¤ect is the change in welfare that comes from buyers leaving the market. Part of the consumers in the standard segment derive little value from quality and they may no longer buy the product when its price increases. This market size e¤ect is therefore absent when there is a partial cartel in the premium segment only since in that case the price of the standard quality product remains unaltered. In the other three scenarios, the price of the standard product rises so that the market size e¤ect is strictly negative. The market share e¤ect describes the movement of customers from one segment to the other. Since the value created is larger in the premium segment, this e¤ect is positive when more consumers buy the high quality product. By contrast, it is negative when customers switch from the premium to the standard quality segment. These distinct e¤ects, and in particular the fact that they might work in opposite directions, make the welfare impact of the di¤erent types of price conspiracy far from trivial.
We …nd that customer damages increase with the inclusiveness of the cartel, which is in line with expectations. Yet, whether a single partial cartel in the premium or in the standard submarket is more harmful e¤ectively depends on the relation between quality and costs. Speci…cally, a single cartel in the standard segment is more detrimental to buyers when unit costs rise su¢ ciently with quality. In that case, sellers in the standard segment have a relatively strong market position and this enables them to sustain a relatively high cartel price. This induces many buyers to leave the market, i.e., there is a strong negative market size e¤ect which signi…cantly harms consumer welfare. In a similar vein, the impact on societal welfare also depends on the degree to which costs increase with quality. Among others, we …nd that the (positive) market share e¤ect may outweigh the (negative) market size e¤ect.
This has the implication that a single segment-wide cartel may be more detrimental to society than two segment-wide partial cartels. Therefore, given that there is collusion in one segment, a social planner 5 See, for example, Harrington (2006).
3 might well prefer collusion over competition in the adjacent segment.
Our research is naturally related to studies on collusion in markets with vertically di¤erentiated products. 6 Häckner (1994) and Symeonides (1999), for instance, both analyze how the incentive to chisel on a collusive agreement depends on quality and reach opposite conclusions. In Bos Bos and Harrington (2010) …nd that the incentive to collude is positively related to …rm size and that su¢ ciently small sellers may prefer not to take part in the anti-competitive coalition.
More recently, de Roos and Smirnov (2020) study pricing strategies of a partial cartel assuming that consumers are imperfectly attentive. Among other things, they characterize the optimal collusive price path and show it may involve intertemporal price dispersion. Finally, Bos, Marini and Saulle (2020) establish that many di¤erent cartel sizes may emerge when products are vertically di¤erentiated. None of these works considers partial collusion in quality-segmented markets, however.
This paper proceeds as follows. The next section introduces the model. In Section 3, we use this setting to study partial and full price collusion. Section 4 builds on this by presenting a welfare analysis. Section 5 concludes with a summary and suggestions for further work. All proofs are relegated to Appendix A.

Model
Consider an industry with two quality segments; a low (standard) and a high (premium) quality segment.
Let the set of low and high quality sellers be respectively given by L = f1; : : : ; lg and H = f1; : : : ; hg, with l; h 2. Each …rm supplies a single quality variant of the product so that l + h is the total number of …rms. 8 Within each segment, sellers are identical and their quality is indicated by v i , where i = l; h 6 For a detailed overview of this literature, see Marini (2018). 7 Hasnas and Wey (2015) study partial and full collusion in a three-…rm spatial setting. They e¤ectively consider a high quality segment in the sense that two horizontally di¤erentiated premium producers are assumed to o¤er the same quality and face a lower quality private label …rm. In our model, each quality variant is supplied by two or more sellers and …rms within a particular segment solely compete on price. 8 We elaborate on the possibility of multi-product …rms in Appendix B.

4
and v h > v l > 0. Corresponding marginal costs of production are constant per unit of output and given by c h c l 0. It is therefore weakly more expensive to manufacture the high quality product.
Interaction takes place for an in…nite number of discrete periods and in each period t 2 N producers simultaneously pick prices to maximize their pro…ts. The discount factor is 2 (0; 1) and all prices set up until t 1 are assumed common knowledge.
The demand side comprises consumers who are uniformly distributed over [0; 1] with a mass normalized to unity. The parameter describes the degree to which buyers value quality and a higher corresponds to a higher gross utility when consuming variant v i . Consumers either purchase one unit of the product or choose an outside option for which the valuation is normalized to zero. Someone 'located'at therefore obtains the following utility: where p i 2 R + is the price set by a …rm o¤ering quality v i , i = l; h. Observe that this speci…cation implicitly assumes …rms within one segment to set the same price; a property that we verify below.
To further economize on notation, we write p l and p h to indicate the prices set in the standard and premium segment, respectively. Finally, it is worth noting that some consumers prefer not to buy the product when prices are strictly positive in which case the market is not covered.
Let us now describe the demand for each …rm. To begin, a consumer 'located'at 0 is indi¤erent between buying from a …rm i 2 L and the outside option when: In a similar vein, a consumer at 1 is indi¤erent between buying from a …rm i 2 L and a …rm i 2 H when: Under the assumption that consumers spread evenly across equally-priced …rms, …rm demand is then given by: with concomitant pro…t functions: and In the ensuing analysis, we concentrate on situations in which each …rm is active (i.e., has strictly positive sales). The following two assumptions provide su¢ cient conditions for an interior solution: Assumption 2. Each …rm chooses 'to be productive' over 'not to be productive' when both yield the same pro…t.
The …rst assumption ensures that production costs are not 'too high'in relation to the quality of the products and the valuation thereof. The second assumption states that …rms have a weak preference for positive sales. An implication of these assumptions is that there is a unique static Nash equilibrium in which each seller sets price at marginal costs. Thus, none of them makes an economic pro…t absent collusion. Below, we indicate this noncollusive outcome with a superscript '*'.

Collusion
We now proceed by considering the possibility of …rms colluding on supra-competitive prices. In the following, four scenarios are studied: (1) segment-wide collusion in the low-quality submarket only, (2) segment-wide collusion in the high-quality submarket only, (3) segment-wide collusion in both submarkets separately and simultaneously, and (4) industry-wide collusion. For all these cases, we characterize the optimal collusive contract under the assumption that conspirators'aim is to maximize joint pro…ts and a deviation from the agreement by any of them results in in…nite reversion to the static Nash equilibrium.

Partial collusion
To begin, suppose that collusion is not industry-wide so that conspirators receive competition from …rms not taking part in the agreement. Since our focus is on segment-wide collusion, a partial cartel faces one of the following two constrained maximization problems: where the superscript 'd'denotes the optimal deviating price. 9 Let us now discuss the deviating strategy in more detail. Notice …rst that, given the pro…tability of the cartel, it never pays to cheat by lifting the price since this would leave the deviant …rm with no demand. Next, shaving price slightly below the collusive segment price would yield one hundred percent of the sales in this particular submarket. However, whether this is the optimal deviating strategy depends on the best-reply toward the adjacent segment. If that price is lower than the collusive price, then a deviating …rm …nds it in its interest to follow this best-response. Consequently, where p c i indicates the collusive price and e p i is the best-reply price toward the adjacent quality segment. 10 The next proposition speci…es optimal prices in the event of a standard segment partial cartel, a premium segment partial cartel and a partial cartel in both segments simultaneously. The superscript indicates the type of price-…xing conspiracy under consideration. Proposition 1. If there is a price cartel in the standard segment only, then If there is a price cartel in the premium segment only, then If there is a price cartel in the standard segment and in the premium segment, then The critical discount factor h can be derived in a similar fashion. Notice that this speci…cation gives one hundred percent market segment share to a deviant …rm; something that will be veri…ed in the ensuing analysis. 10 Note that when e p i > p c i , p d i should, strictly speaking, be the maximum price below p c i . Yet, this is problematic since action sets are continuous in our model. We therefore follow the convention by simply writing p d i = p c i in this case, which e¤ectively means that the deviant seller prices arbitrarily close to p c i and obtains all demand in the respective market segment(s). See, amongst many others, Tirole (1988), Bos and Harrington (2010), and de Roos and Smirnov (2020).
As a …rst observation, note that there is a single optimum in all three scenarios. Speci…cally, each partial cartel is capable of sustaining the joint pro…t maximum. The reason is that the pro…t-maximizing price is a best-response to the price set in the adjacent quality segment. This means that the optimal deviating strategy is to undercut the cartel price by the smallest possible amount, i.e., p d i = p c i , i = l; h. In turn, this has the implication that the incentive compatibility constraints reduce to: Thus, the unconstrained solution is feasible whenever some collusion is sustainable. Given a high enough discount factor, the latter requires the number of …rms in the segment to be su¢ ciently small.
Next, it can be easily veri…ed that cartel prices are higher when …rms in the adjacent segment are also colluding, which is due to the strategic complementarity of the choice variables. In terms of comparative statics, cartel prices are rising with unit production costs in both segments. Moreover, in case of a single segment-wide partial cartel, prices are increasing with the own quality level and decreasing with the quality o¤ered in the noncollusive segment.
Yet, the e¤ect of changes in quality is more subtle when there are two partial cartels. Similar to the case of a single segment-wide cartel, the collusive price of the premium product is increasing in v h and decreasing in v l . This is less straightforward in the standard segment, however. To see this, note that: Hence, dp lh l =dv l < 0 and dp lh l =dv h > 0 when unit costs are (approximately) zero and the quality di¤erence, v h v l , is small enough. In both cases, the sign of the direct e¤ect is as expected (@p lh l =@v l > 0 and @p lh l =@v h < 0). However, the indirect (or strategic) e¤ect may work in the opposite direction (@p lh l =@p lh h dp lh h =dv l < 0 and @p lh l =@p lh h dp lh h =dv h > 0) and dominates the direct e¤ect when the situation is su¢ ciently symmetric.
We conclude this subsection by considering the shifts in market shares resulting from partial collusion.
Let s l and s h indicate the sales share of the standard and premium segment, respectively. The next result shows how the size of the respective segments changes under the di¤erent (partial) collusive scenarios.

Corollary 1.
If there is a price cartel in the standard segment only, then If there is a price cartel in the premium segment only, then If there is a price cartel in the standard segment and in the premium segment, then The change in market shares in case of a single segment-wide cartel is in line with the existing literature.
Theoretical work on incomplete cartels robustly predict a decrease in demand for the colluding …rms' products and an increase in demand for non-cartel suppliers. 11 The Corollary reveals that the standard and premium segment partial cartels not only lose buyers in absolute terms, but also in relative terms, i.e., their share of total sales decreases.
In case of two partial cartels, the change in market share depends on the segment sizes absent collusion as well as the value created. If, say, the premium segment is relatively small and it is relatively costly to provide additional quality (i.e., both s h and v h c h are 'small'and s l and v l c l are 'large'), then the price increase in the premium segment is smaller than in the standard segment. The strong market position of the sellers in the standard segment allows them to raise their price signi…cantly. By contrast, the weak market position of the premium suppliers limits the scope for a price increase. Since in this case it holds that p lh l c l > p lh h c h , some buyers move from the standard to the premium segment so that high quality suppliers gain back some market share.

Full collusion
We now direct our attention to the possibility of an all-inclusive cartel. In comparison to the partial cartel cases that we have analyzed above, such a collusive contract is more sophisticated in the sense that the anti-competitive combination has to simultaneously select a price for both segments. Formally, the industry-wide cartel faces the following constrained optimization problem: This a priori allows for a plethora of cartel contracts in which none, some or all of the incentive constraints are binding. The next result restricts the set of collusive outcomes, however, by showing that all incentive constraints bind when the unconstrained solution cannot be sustained. In stating this result, let e max 1 1 l ; 1 1 h . Lemma 1. Assume > e . If the industry-wide cartel cannot sustain the joint pro…t maximum, then This …nding basically leaves two types of collusive outcome. Either all incentive constraints bind or the industry-wide cartel sets the 'unconstrained' monopoly prices (p m Note that the all-inclusive cartel could always mimic the two partial cartels case of Proposition 1 whenever some collusion is sustainable. Yet, it is willing and able to raise prices further provided that …rms are su¢ ciently patient. In fact, and as the next proposition reveals, there is a whole range of prices that may result from industry-wide collusion. We use the superscript 'a'to indicate the all-inclusive price-…xing agreement. What prices emerge under industry-wide collusion e¤ectively depends on the level of the discount factor. The all-inclusive cartel can sustain the joint pro…t maximum when the discount factor is su¢ ciently high. Speci…cally, the most pro…table outcome is feasible when: Observe that l > 1 1 l and h > 1 1 h so that the all-encompassing cartel sets prices below monopoly levels when 2 The previous result presents the pro…t-maximizing prices. Assuming the joint pro…t maximum, the following proposition speci…es the pro…t-maximizing production level for each cartel participant.
Proposition 3. Assume an industry-wide cartel and suppose that b . Each participant produces precisely half its Nash demand: The next result follows almost immediately.
Corollary 2. Assume an industry-wide cartel and suppose that b . All market shares are maintained at pre-collusive levels.
This is a remarkable result for at least two reasons. First, we have studied full collusion under quality di¤erentiation elsewhere assuming that the industry-wide cartel maintains market shares at pre-collusive levels. 12 This …nding shows that the cartel may …nd it in its interest to select this market division scheme when collusion spans several quality-segments simultaneously. Second, the literature on collusion in vertically di¤erentiated markets has repeatedly shown that it may well be optimal to price lower-quality products out of the market. 13 Such a strategy is suboptimal within our setting due to the substantial di¤erence in production costs (as speci…ed by Assumption 1). Indeed, and in line with the existing literature, it would be optimal to exclusively sell the premium product when the cost di¤erence is su¢ ciently small.

Welfare
The previous section provides a complete characterization of collusive pricing for the di¤erent types of cartel agreement. Let us now turn to the welfare consequences. We start with analyzing the impact on consumer welfare and then proceed by examining the e¤ect on societal surplus.

Consumer Surplus
How are consumers a¤ected by the di¤erent types of price-…xing agreement? To answer this question, we begin by presenting the benchmark of no collusion. Recall that the static Nash equilibrium has all …rms price at marginal cost. In this case, therefore, a buyer who is indi¤erent between staying at home and purchasing the low quality item is 'located'at 0 = c l =v l . A consumer who is indi¤erent between the standard and the premium product is at 1 = (c h c l ) = (v h v l ). Combining gives the (net) consumer surplus (henceforth CS): which is decreasing in price (c l and c h ) and increasing in quality (v l and v h ).
Now consider the four collusive scenarios studied above. For any of these cases, consumer surplus is generally given by: By Proposition 1 and Proposition 2, we know that prices di¤er for the di¤erent cartel regimes. The next result shows how the various price-…xing agreements rank in terms of their impact on consumer welfare. To facilitate comparison between the di¤erent scenarios, we assume without loss of generality that c l = 0 and c = c h c l .

Proposition 4.
There is a threshold k 2 (0; 1) such that: Not surprisingly, collusion is bad for buyers. The industry-wide cartel is the most harmful and, also in line with expectations, the scenario of two segment-wide partial cartels is second worst. The ranking of single segment-wide cartels is sensitive to di¤erences in costs and quality, however. Speci…cally, if a substantial quality improvement can be obtained for little additional cost, then a partial cartel in the premium segment is more detrimental to consumers. By contrast, a single partial cartel in the standard segment is more harmful when unit costs rise su¢ ciently with quality.
To see the intuition behind this latter result consider two extreme scenarios. If c= (v h v l ) ! 0, then low quality …rms are unable to e¤ectively compete with their high quality rivals, either in competition or under collusion. In this case, premium suppliers o¤er a superior value proposition even when p l = c l = 0. A cost di¤erence is thus necessary for a standard segment-wide cartel to be viable and raise price to supra-competitive levels. More generally, the impact of such a cartel on consumer welfare remains limited when low quality suppliers have a relatively weak market position as re ‡ected by a low c or a high v h v l .
By contrast, if c= (v h v l ) ! 1, then a partial cartel in the standard segment increases its price substantially above costs. This not only induces some customers to switch from the standard to the premium segment, but it also makes some buyers leave the market. Note that the latter e¤ect is not present when there is a single partial cartel in the premium segment. As part (ii) of Proposition 4 reveals, this reduction in market size may be su¢ ciently strong to make consumers worse o¤ with a cartel in the standard segment.

Social Welfare
What about the welfare of society as a whole? How do the various types of cartel agreement a¤ect societal surplus? To address this issue, we shall evaluate the following general social welfare function: As established in Proposition 1 and Proposition 2, prices will di¤er for the di¤erent scenarios, and consequently so will societal surplus.
We start the comparison with the next proposition, which reveals the preference of a social planner. Since prices are set at marginal costs in both segments absent collusion and each cartel raises price above costs, the implication of this result is that social welfare is highest when there is no cartel. Not surprisingly, therefore, a social planner prefers a competitive world. Let us now turn to the other extreme; an industry-wide cartel. As before, we assume without loss of generality that c l = 0 and c = c h c l to facilitate comparison between the di¤erent scenarios.

Proposition 6.
Social welfare under all forms of partial collusion is (weakly) higher than with an industry-wide cartel.
This result con…rms the intuition of an all-inclusive price-…xing cartel being most detrimental to society as a whole. As the proof in the Appendix reveals, this holds both when the incentive constraints are binding and when the industry-wide cartel is capable of sustaining the monopoly solution. In fact, societal surplus monotonically declines gradually when both segment prices continuously increase along the intervals as speci…ed in Proposition 2.
As the next proposition shows, the same does not hold for partial cartels.
(i) If v h =v l = , then there exists a unique threshold k 2 (0; 1) such that: (ii) If v h =v l > , then there exist three thresholds k 1 ; k 2 ; k 3 2 (0; 1), with k 3 > k 2 > k 1 , such that: (iii) If v h =v l < , then there exist three thresholds k 1 ; k 2 ; k 3 2 (0; 1), with k 1 > k 2 > k 3 , such that: This result reveals that the impact of partial cartels on societal surplus is everything but trivial. In fact, each partial collusion scenario can be most or least detrimental to society and two cartels can be better than one.
To explain the intuition underlying this …nding it is useful to distinguish between two distinct e¤ects: a market size e¤ect and a market share e¤ect. The …rst captures the impact on social welfare that comes from a reduction in market size due to a raise in the standard segment price. It can be easily veri…ed that the magnitude is increasing in the price of the low quality product. This implies that it is lowest for a single premium segment-wide cartel (in fact, it is zero because in that case p l = c l like in competition) and highest in case of two partial cartels. Note further that this e¤ect is weakly negative so that absent any other e¤ect we would have SW h > SW l > SW lh .
Yet, there is a second e¤ect that is driven by a shift in market shares. This market share e¤ect captures the extent to which buyers are induced to switch to the segment generating most value. Following Assumption 1, the value created is higher in the premium segment. Hence, collusion may positively a¤ect societal surplus by incentivizing consumers of standard products to switch to the premium segment.
With these two concepts in mind, let us now compare social welfare in case of a single partial cartel: SW h and SW l . To start, note that the market size e¤ect is zero in case of a single premium cartel. Moreover, the price rise in the premium segment induces some buyers to move to the standard segment so that the market share e¤ect is negative. A single standard segment-wide cartel is worse in terms of the market size e¤ect since it is strictly negative. At the same time, however, it incentivizes buyers to move to the premium segment so that the market share e¤ect is positive. Proposition 7 shows that this positive market share e¤ect su¢ ciently mitigates the negative market size e¤ect when the cost di¤erence is su¢ ciently low and the quality gap is su¢ ciently high. In other words, a single cartel in the premium segment is more detrimental to society than a single cartel in the standard segment when the value created in the premium segment, v h c h , is signi…cantly higher than the value created in the standard segment, v l c l . With regards to the two partial cartels scenario, note that compared to a single premium cartel the negative market size e¤ect is stronger since the standard segment price is higher (p lh l > p h l ). The market share e¤ect is less negative, however. The above result reveals that the market size e¤ect dominates the market share e¤ect when the marginal production costs of quality are high. In that case, the di¤erence in value created between segments is limited and SW h > SW lh . By contrast, the market share e¤ect may dominate the market size e¤ect when marginal production costs of quality are low in which case it holds that SW lh > SW h .
Finally, comparing the two partial cartels scenario with a single segment-wide cartel in the standard segment, the former has a stronger market size e¤ect (since p lh l > p l l ). As to the market share e¤ect, this e¤ectively depends on the price di¤erence between the standard and the premium quality segment.
Speci…cally, the di¤erence is smaller in the two partial cartel case when the marginal production costs of quality is su¢ ciently high: Proposition 7 shows that the market share e¤ect may be su¢ ciently strong to make a single standard segment cartel more detrimental than two segment-wide cartels.
In sum, what partial cartels are most harmful for society critically depends on the di¤erences in cost and quality. A partial cartel in the premium (standard) segment is more detrimental to society when the di¤erence in value created between both segments is su¢ ciently high (low), all else unchanged.

Concluding Remarks
In many industries, …rms can be grouped into more homogeneous classes based on quality. We have addressed the question of what collusion may look like in such quality-segmented markets through studying an in…nitely repeated price-setting game with two submarkets; a standard and a premium quality segment. We used this framework to analyze four types of price-…xing agreement: a segmentwide cartel in the premium submarket only, a segment-wide cartel in the standard submarket only, two segment-wide cartels, one in each submarket, and an industry-wide cartel. For all these cases, we provided a complete characterization of the collusive equilibrium and examined the impact on market shares and welfare.
Let us summarize some of our main …ndings. Partial cartels are capable of sustaining the joint pro…t maximum, whereas the all-inclusive cartel can sustain the unconstrained optimum only when its members are su¢ ciently patient. Incomplete cartels loose market share when the segment in which they operate is su¢ ciently large and the industry-wide cartel prefers to set its prices such to maintain market shares at pre-collusive levels. In terms of welfare, we show that a partial cartel in the premium segment is more detrimental to consumers than a partial cartel in the standard segment when the marginal costs of quality are su¢ ciently small. The impact on societal welfare also critically depends on the relation between quality and costs. Among other things, we …nd that a single segment-wide partial cartel may be more harmful for society than two segment-wide partial cartels.
There are three natural avenues for future research. First, it may be interesting to also allow for within-segment heterogeneity, i.e., include some degree of horizontal di¤erentiation. Second, one might explore the potential impact of additional quality segments. Both extensions will likely prove to be computationally challenging, however. Finally, we have studied price collusion given qualities. Though it seems natural to assume the available qualities to be exogenous in the short run, we can imagine situations in which the quality o¤ered is (partly) endogenous. It is worth exploring how our main …ndings would be a¤ected when allowing …rms to reposition themselves along the quality spectrum. We leave this issue for future research.

Appendix A: Proofs
Proof of Proposition 1. To start, consider a single segment-wide partial cartel in the standard segment. Such a cartel faces the following constrained maximization problem: Since there is more than one …rm in the adjacent premium segment, it follows from Assumption 1 and Assumption 2 that p h = c h . This means that the joint pro…t-maximizing price is a best-reply to c h , which implies p d l is (approximately) equal to this collusive solution. The incentive compatibility constraint therefore reduces to: Consequently, the joint pro…t-maximizing price can be sustained whenever some collusion is sustainable.
Taking the …rst-order condition and rearranging gives: Next, consider a single segment-wide partial cartel in the premium segment. Such a cartel faces the following constrained maximization problem: Since there is more than one …rm in the standard segment, it follows from Assumption 1 and Assumption 2 that p l = c l . Similar to the previous case, the joint pro…t-maximizing price is a best-reply to c l , which implies p d h is (approximately) equal to this collusive solution. The incentive compatibility constraint therefore reduces to: The joint pro…t-maximizing price can thus be sustained whenever some collusion is sustainable. Taking the …rst-order condition and rearranging gives: Finally, consider the possibility of a partial cartel in each segment. Also in this case, the joint pro…t-maximizing price is a best-response to the price set by …rms in the adjacent segment. Thus, p d i , i = l; h, is again (approximately) equal to this collusive solution and the incentive constraints reduce to: Hence, the joint pro…t-maximizing prices can be sustained whenever some segment-wide collusion is feasible. The collusive optimum is therefore given by the respective …rst-order conditions. Rearranging gives: Proof of Corrolary 1. To begin, note that the market shares of the two segments absent collusion are given by: Using the collusive prices as speci…ed in Proposition 1, when there is a partial cartel in the standard segment only its sales share is: which is smaller than the Nash equilibrium share when: Rearranging gives: which holds.
Using the collusive prices as speci…ed in Proposition 1, when there is a partial cartel in the premium segment only its sales share is: which is smaller than the Nash equilibrium share when: which holds.
Finally, consider the case in which there is a partial cartel in both segments. We show that s lh h > s h when (v l c l ) s l > (v h c h ) s h . By the prices as speci…ed in Proposition 1, s lh h is given by: Comparing with the Nash equilibrium share: Rearranging and using the above speci…cations for s l and s h gives: Hence, we conclude that if (v l c l ) s l > (v h c h ) s h , then s lh l < s l and s lh h > s h (and vice versa). Proof of Lemma 1.
To begin, note that the all-inclusive cartel could mimic the situation with two segment-wide partial cartels as speci…ed in Proposition 1. Yet, since > e max 1 1 l ; 1 1 h , the industry-wide cartel can improve upon this outcome. Next, recall that the joint pro…t-maximizing prices in the two partial cartels case are in fact best-responses to the adjacent segment. This has the implication that when the cartel increases prices further it holds that p d i = e p i , i = l; h. That is, a deviant …rm …nds it optimal to not just cut its price slightly below the collusive segment price, but to lower it further to the best-reply level.
Given this deviating strategy, let us now proceed by writing down the Lagrangian of the constrained maximization problem: The corresponding Karush-Kuhn-Tucker (KKT) conditions are: To show that all incentive constraints must be binding when the joint pro…t maximum cannot be sustained, we consider two cases: (i) the incentive constraints of all high-quality …rms are binding, whereas the incentive constraints of the low-quality …rms are not and (ii) the incentive constraints of all low-quality …rms are binding, whereas the incentive constraints of the high-quality …rms are not. For both cases, we derive a contradiction. Case (i): 1 > 0 and 2 = 0. In this case, the KKT conditions reduce to: As the price of the basic good is unconstrained, we know that: Hence, by the second KKT condition it must hold that: whereas by the last KKT condition, we know: Combining gives: Combining with the …rst KKT condition, this implies: which cannot hold as p l > c l under collusion; a contradiction.
Case (ii): 1 = 0 and 2 > 0. In this case, the KKT conditions reduce to: As the price of the premium product is unconstrained, we know that: v h v l 2p h + 2p l + c h c l = 0: Hence, by the …rst KKT condition it must hold that: whereas by the last KKT condition, we know that: Combining gives: Combining with the second KKT condition, this implies: which cannot hold as p h > c h under collusion, a contradiction. We thus conclude that when a pro…tmaximizing industry-wide cartel sets prices below the joint pro…t maximum all incentive constraints must be binding.
Proof of Proposition 2. Industry-wide collusion is sustainable when: To begin, the all-inclusive cartel can mimic the two partial cartels scenario as described in Proposition 1. In that case, p a l = p d l and p a h = p d h so that the incentive constraints reduce to: Thus, the industry-wide cartel can sustain the following prices whenever some collusion is sustainable: This would be the outcome when = e .
If > e , then the industry-wide cartel is able and willing to raise prices further. In that case, deviating prices are a best-response to the adjacent quality segment and strictly below the collusive 22 level. Using , the critical discount factors can be written as: Note that, in both cases, the last term between brackets is strictly below 1 so that the incentive constraints are tighter than in the scenario with two partial cartels. By Lemma 1, when collusion is constrained, the industry-wide cartel sets its prices such that = l = h .
Finally, if ! 1, then none of the incentive constraints bind in which case collusive prices are given by the …rst-order conditions. Rearranging gives: Proof of Proposition 3. Using the full collusive unconstrained prices as speci…ed in Proposition 2, the pro…t-maximizing production levels are given by: which is precisely half the Nash demand: Proof of Corollary 2. Absent collusion, total market demand is given by Under full collusion at the joint pro…t maximum, it is: Combining with the results from the previous proposition, it can be easily veri…ed that Proof of Proposition 4. To begin, note that consumer surplus can be written as: which is monotonically decreasing in both prices. Hence, all types of cartel are bad for buyers in the sense that at least one consumer pays more for the same quality and no consumer pays less. It follows that consumer surplus is highest absent collusion. Moreover, since consumer welfare is strictly decreasing in segment prices, it holds that CS l ; CS h > CS lh (by Proposition 1) and CS lh CS a (by Proposition 2). We conclude that CS > CS l ; CS h > CS lh CS a .
Finally, let us compare CS l and CS h . Combining the consumer surplus formula with the prices given in Proposition 1, one obtains: Taking the di¤erence yields: Next, CS l CS h is convex in c and negative at the upper bound, c = v h v l (Assumption 1): Also note that: which has e c = v h p v h v l as the smaller root. Combining these …ndings then leads to the following conclusion. Comparing the societal surplus under a single segment-wide cartel regime with the social welfare with a industry-wide cartel yields: Next, let us compare social welfare under the two partial cartels scenario with social welfare under unconstrained full collusion: Notice that SW lh SW a > 0 for c ! 0 and that the di¤erence SW lh SW a decreases monotonically in c. Substituting the upper bound c = v h v l (Assumption 1) in SW lh SW a gives: We conclude that social welfare under partial collusion is strictly higher than under an industry-wide cartel capable of sustaining the joint pro…t maximum.
Finally, let us consider the case where 2 h e ; b so that the industry-wide cartel sets prices below the monopoly level. We show that social welfare declines when the incentive constraints are binding and the prices set by the …rms continuously increase along the interval ranging from the two partial cartels to the industry-wide cartel prices as speci…ed in Proposition 2.
Now suppose that both incentive constraints are binding (Lemma 1) and consider a marginal increase in the discount factor. The binding incentive constraints are given by: Since a marginal increase in the discount factor relaxes both incentive constraints this yields an increase in collusive prices. Speci…cally, we can distinguish a direct e¤ect and an indirect e¤ect. The latter comes from the fact that a price increase in one segment enables a price increase in the adjacent segment. Using the binding incentive constraints and solving for p h and p l yields: These expressions show that the two prices p h and p l are positively related. Starting with an increase in p h , the impact on social welfare is The …rst term captures the market size e¤ect. Since an increase in p h has a positive indirect e¤ect on p l this e¤ect is negative. The second term captures the market share e¤ect. If p h increases, then some customers move to the standard segment. This e¤ect is, however, countered by the indirect positive e¤ect on p l . Nevertheless, a su¢ cient condition for the market share e¤ect to be negative is that dp l (p h ) dp h < 1, which holds because 0 < (l ( 1) + 1) < 1 and: Let us now turn to the increase of p l . The impact on social welfare is given by: As before, the …rst term captures the market size e¤ect and the second term captures the market share e¤ect. Since dp h dp l 0 and D h h < 1, the combined e¤ect is negative. Taken together, societal surplus therefore gradually decreases when prices increase marginally along the interval as speci…ed in Proposition 2. We conclude that SW lh SW a .
Proof of Proposition 7. Social welfare under the di¤erent partial cartel scenarios is provided in the proof of Proposition 6. Comparing them yields the following thresholds in reference to k = c=(v h v l ): It can be readily veri…ed that there is a quality-ratio: v h =v l = p 2=2 + 3=4; for which all thresholds are equal, i.e., k 1 = k 2 = k 3 . Moreover, if v h =v l > , then k 1 > k 2 > k 3 and if v h =v l < , then k 3 > k 2 > k 1 . The welfare ranking under the di¤erent partial cartel regimes as speci…ed in the proposition then follows accordingly.
di¤erence with the two segment-wide partial cartels case is, however, that the price in the noncollusive segment equals marginal costs when the number of outsiders exceeds two. Therefore, independent of the number of multi-product …rms, prices remain as in Proposition 1. Yet, this situation changes when there are only two …rms active in the adjacent segment. To see this, suppose there is a single partial cartel in the premium submarket. Suppose further that k = 1 and l = 2. The multi-product …rm could now contemplate the following strategy. Rather than keeping price at costs in the standard segment it may increase this price. This implies that the multi-product …rm sacri…ces all its market share in the low quality submarket, because the best-reply by the outsider is to price below it. Such a strategy is nevertheless pro…table since pro…ts remain zero in the low quality submarket and increase in the premium segment. The latter is due to the strategic complementarity of the choice variable. To see that this strategy is also feasible, consider the incentive constraint of the multi-product …rm when it would indeed set price above costs in the standard segment: First note that in this case the multi-product …rm faces a tighter incentive constraint than singleproduct …rms because its deviating pro…ts are (weakly) higher, whereas collusive pro…ts are the same. Second, since price exceeds costs in the standard segment, the optimal deviating strategy for the multiproduct …rm is to cut price in both submarkets simultaneously. Finally, the outsider charges a price weakly below the best-response to the collusive premium price and that collusive price itself is weakly below the best-response to the price in the low quality segment. Optimal deviating prices are therefore (approximately) equal to the segment prices, i.e., p d h = p c h and p d l = p l . This reduces the incentive constraint of the multi-product …rm to: Provided that some collusion is sustainable, the collusive price p c h is the same as in Proposition 1 when p l = c l . However, the above strategy is also feasible for a high enough discount factor. In fact, if ! 1, then the multi-product …rm could set and sustain the best-responses: p l = (v l p c h + v h c l ) =2v h and p c h = (v h v l + p l + c h ) =2. Notice that these are precisely the prices that would result in case of two segment-wide partial cartels. In a similar vein, multi-product …rms may lead to higher collusive prices when there is a single segment-wide cartel in the standard segment and there are two …rms active in the adjacent premium segment.
In sum, in most cases the presence of multi-product …rms does not a¤ect the collusive outcome, but when it does, cartel prices are higher. This can occur when there are two partial cartels or when there is a single partial cartel and there are two …rms active in the adjacent segment.