Cartel Stability under Quality Differentiation
aa r X i v : . [ ec on . T H ] D ec CARTEL STABILITY UNDER QUALITY DIFFERENTIATION
IWAN BOS AND MARCO A. MARINI
Abstract.
This note considers cartel stability when the cartelized products are verticallydifferentiated. If market shares are maintained at pre-collusive levels, then the firm withthe lowest competitive price-cost margin has the strongest incentive to deviate from thecollusive agreement. The lowest-quality supplier has the tightest incentive constraint whenthe difference in unit production costs is sufficiently small.
Keywords:
Cartel Stability , Collusion, Vertical Differentiation . JEL Classification:
D43, L13, L41 . Introduction
In this note, we examine cartel stability when the cartelized products are vertically dif-ferentiated. Goods or services are differentiated vertically when there is consensus amongconsumers about how to rank them quality-wise; comparing products A and B, all agree Ato have a higher (perceived) value than B or vice versa . There might, however, still be ademand for lower-quality goods when buyers face budget constraints or differ in their will-ingness to pay for quality. This creates an incentive for suppliers to compete through offeringdifferent price-quality combinations.One implication of this price-quality dispersion is that firms that consider colluding typ-ically face heterogeneous incentive constraints. The fact that firms are induced to chargedifferent prices, for example, affects both collusive and noncollusive profits. From a supply-side perspective, there commonly exists a positive relationship between the quality of a goodand its production costs. This, too, impacts both sides of the constraint. It is therefore apriori unclear how quality differentiation impacts the sustainability of collusion.The scarce literature on this topic provides mixed results and, moreover, does not con-sider the potential impact of cost heterogeneity. Assuming identical costs, H¨ackner (1994)and Symeonidis (1999) both analyze an infinitely repeated vertically differentiated duopolygame. H¨ackner (1994) studies a variation of the setting in Gabszewicz and Thisse (1979)
Date : October 2018.We appreciate the comments of Maria Rosa Battaggion, Boris Ginzburg, Joseph E. Harrington Jr., RonaldPeeters, an anonymous referee, participants at the 2018 Oligo Workshop and the 2018 EARIE conference inAthens. All opinions and errors are ours alone.Iwan Bos, Department of Organization & Strategy, School of Business and Economics, Maastricht University.E-mail: [email protected] A. Marini, Department of Social and Economic Sciences, University of Rome La Sapienza. E-mail:[email protected]. An extensive and detailed overview of this literature is provided by Marini (2018). Apart from being analytically convenient, the identical cost assumption can be defended on the groundsthat the difference in quality may mainly come from upfront sunk investments in which case the impact onprices would be limited. and Shaked and Sutton (1982) and finds that it is the high-quality supplier who has thestrongest incentive to deviate. By contrast, Symeonidis (1999) considers a representativeconsumer model with horizontal and vertical product differentiation and establishes that itis the low-quality seller who is most eager to leave the cartel.In the following, we analyze an n -firm infinitely repeated game version of the classic verticaldifferentiation model of Mussa and Rosen (1978) where production costs are assumed to beincreasing in quality. Under the assumption that colluding firms maintain their pre-collusivemarket shares, we find that it is the competitive mark-up rather than the quality of theproduct that drives the incentive to deviate. Specifically, it is the supplier with the lowestnoncooperative price-cost margin who has the strongest incentive to chisel on the cartel.Moreover, our analysis confirms the above-mentioned conclusion by Symeonidis (1999) whenthe difference in unit costs is sufficiently small.The next section presents the model. Section 3 contains the main finding. Section 4concludes. 2.
Model
There is a given set of suppliers, denoted N = { , . . . , n } , who repeatedly interact overan infinite, discrete time horizon. In every period t ∈ N , they simultaneously make pricedecisions with the aim to maximize the expected discounted sum of their profit stream.Firms face a common discount factor δ ∈ (0 ,
1) and all prices set up until t − i ∈ N sells a single variant of the product with quality v i . We assume ∞ >v n > v n − > ... > v > n as the top firm , firm 1 as the bottom firm and all others as intermediate firms . Unit production costs of firm i ∈ N are given by theconstant c i and we suppose these costs to be positive and (weakly) increasing in quality, i.e. , c n ≥ c n − ≥ . . . ≥ c > θ , which is uniformly dis-tributed on [ θ, θ ] ⊂ R ++ with mass normalized to one. A higher θ corresponds to a highergross utility when consuming variant v i . Buyers purchase no more than one item so thatsomeone ‘located’ at θ obtains the following utility(2.1) U ( θ ) = (cid:26) θv i − p i when buying from firm i p i ∈ (cid:2) , θv n (cid:3) is the price set by firm i . Using (2.1), it can be easily verified that aconsumer at θ i ∈ [ θ, θ ] is indifferent between buying from, say, firm i + 1 and firm i when(2.2) θ i ( p i , p i +1 ) = p i +1 − p i v i +1 − v i , for every i = 1 , , ..., n −
1. In the ensuing analysis, we further assume that the market isand remains covered ( i.e. , all consumers buy a product). Like the model in H¨ackner (1994), this is a model with heterogeneous customers. Apart from the numberof firms and cost heterogeneity, it differs in terms of consumers’ utility specification. Note that none of the buyers would buy at prices in excess of θv n . This is a common assumption in contributions that employ this type of spatial setting. See, for example,Tirole (1988, pp.296-298) and Ecchia and Lambertini (1997). We discuss some implications of this assumption
ARTEL STABILITY UNDER QUALITY DIFFERENTIATION 3
Current profit of the bottom firm ( i = 1) is therefore given by(2.3) π ( p , p ) = ( p − c ) · ( θ − θ ) , where θ = θ ( p , p ) is as specified by (2.2). For each intermediate firm ( i = 2 , , ..., n − π i ( p i − , p i , p i +1 ) = ( p i − c i ) · ( θ i − θ i − ) , and for the top firm ( i = n ) it is(2.5) π n ( p n − , p n ) = ( p n − c n ) · (cid:0) θ − θ n − (cid:1) . Before analyzing the infinitely repeated version of the above game, let us first considerthe one-shot case in more detail. In this setting, each firm simultaneously picks a price tomaximize its profit as specified in (2.3)-(2.5). Following the first-order conditions, this yieldsthree types of best-response functions:(2.6) b p ( p ) = 12 ( p + c − θ ( v − v ))for the bottom firm ( i = 1). For each intermediate firm ( i = 2 , , ..., n − b p i ( p i − , p i +1 ) = 12 p i − ( v i +1 − v i ) + p i +1 ( v i − v i − )( v i +1 − v i − ) + 12 c i . The best-response function of the top firm ( i = n ) is(2.8) b p n ( p n − ) = 12 (cid:0) p n − + c n + θ ( v n − v n − ) (cid:1) . Since the action sets are compact and convex and the above best-reply functions are con-tractions , there exists a unique static Nash equilibrium price vector p ∗ for any finite numberof firms. Finally, we impose two more conditions to ensure that the equilibrium solution isinterior ( i.e. , all firms have a positive output at p ∗ ) and that the market is indeed coveredat the single-shot Nash equilibrium:(2.9) θ > θ ∗ n − > θ ∗ n − > . . . > θ ∗ i > ... > θ ∗ > θ > p ∗ v > , where θ ∗ i ≡ θ i (cid:0) p ∗ i , p ∗ i +1 (cid:1) and p ∗ i ≥ c i , for all i ∈ N . at the end of Section 3 and consider the possibility of an uncovered collusive market in an online appendixto this paper. See, for instance, Friedman (1991, p.84). A sufficient condition for the contraction property to hold is(see, for example, Vives 2000, p.47): ∂ π i ∂ ( p i ) + P j = i (cid:12)(cid:12)(cid:12)(cid:12) ∂ π i ∂p i ∂p j (cid:12)(cid:12)(cid:12)(cid:12) < , which, using (2.4) for all intermediate firms i = 2 , ..., n −
1, becomes v i − − v i +1 ( v i +1 − v i ) ( v i − v i − ) < , which holds. The same applies for the top and the bottom firm. IWAN BOS AND MARCO A. MARINI Sustainability of Collusion
Within the above framework, we now study the sustainability of collusion assuming astandard grim-trigger punishment strategy. The incentive compatibility constraint (ICC) ofa firm i ∈ N is then given by:(3.1) Ω i ≡ π ci − (1 − δ ) · π di − δ · π ∗ i ≥ , where π ci = π i (cid:0) p ci − , p ci , p ci +1 (cid:1) is its collusive payoff, π di = π i (cid:0) p ci − , p di , p ci +1 (cid:1) , with p di = b p i ( p ci − , p ci +1 ), is its deviation payoff and π ∗ i = π i (cid:0) p ∗ i − , p ∗ i , p ∗ i +1 (cid:1) is its Nash equilibrium payoff.A cartel comprising the entire industry is thus sustainable only when Ω i ≥ i ∈ N .In principle, this set-up allows for a plethora of sustainable collusive contracts. In the fol-lowing, we limit ourselves to what is perhaps the simplest possible agreement. Specifically,we consider the maximization of total cartel profits without side payments under the as-sumption that firms maintain their market shares at pre-collusive levels. Such an agreementis appealing for several reasons. First, it seems a natural focal point in the issue of howto divide the market. Second, there have been quite a few cartels that employed such (orsimilar) market-sharing scheme. Third, it is arguably one of the most subtle arrangementsin that firm behavior maintains a competitive appearance, thereby minimizing the possibilityof cartel detection.Let us now address the question of what collusive price vector such an all-inclusive cartelwould pick. As an initial observation, notice that the fixed market share assumption impliesthat the price ranking should stay the same ( i.e. , collusive prices are strictly increasing inquality). Moreover, as market size is given, the lowest-valuation buyer should still be willingto buy the product. This means that θv − p c ≥ . Next, note that the fixed market share rule in combination with the covered marketassumption implies that each ‘marginal consumer’s location’ remains unaffected. Specifically,the consumer who was indifferent between firm 1 and 2 absent collusion has now the followingutility when buying from firm 1: U ( θ ∗ ) = (cid:18) p ∗ − p ∗ v − v (cid:19) v − p c , which in turn determines the collusive price for the product of firm 2: (cid:18) p ∗ − p ∗ v − v (cid:19) v − p c = (cid:18) p ∗ − p ∗ v − v (cid:19) v − p c . Rearranging gives, p c = p ∗ + ( p c − p ∗ ) . A higher collusive price by firm 2 would mean that the customer on the boundary prefers firm1, which contradicts market shares being fixed. Likewise, a lower price implies a decreasein sales for firm 1 and therefore cannot occur either. The collusive prices for all other firms See, for example, Harrington (2006).
ARTEL STABILITY UNDER QUALITY DIFFERENTIATION 5 can be determined in a similar fashion. In general, the collusive price of firm i ∈ N \{ } isequal to its Nash price plus the price increase by the lowest-quality firm:(3.2) p ci = p ∗ i + ( p c − p ∗ ) . Both cartel profits and the incentive constraints are therefore effectively a function of p c alone.Since prices are strategic complements, it is clear that the cartel would like to set p c = θv .The collusive contract with p c = θv might not be sustainable, however, because one ormore ICC’s may be binding. The next result shows that it is the supplier with the lowestnoncollusive profit margin who has the tightest incentive constraint. Proposition 1.
For any i, j ∈ N and j = i , if p ∗ i − c i > p ∗ j − c j , then Ω i > Ω j .Proof. Consider the ICC of an intermediate firm i = 2 , , ..., n − i ≡ π ci − (1 − δ ) · π di − δ · π ∗ i ≥ ⇔ δ ≥ δ i ≡ π di − π ci π di − π ∗ i . To evaluate the critical discount factor δ i of every intermediate firm, let us focus on π di , π ∗ i and π ci in turn. Every firm’s i = 2 , , ..., n − π di = (cid:0) p di − c i (cid:1) (cid:0) θ di − θ di − (cid:1) , which, using best-replies (2.7), yields p di − c i = p ci − ( v i +1 − v i ) + p ci +1 ( v i − v i − ) − ( v i +1 − v i − ) c i v i +1 − v i − )or(3.3) 2( v i +1 − v i − ) (cid:0) p di − c i (cid:1) = p ci − ( v i +1 − v i ) + p ci +1 ( v i − v i − ) − ( v i +1 − v i − ) c i . Moreover, θ di − θ di − = ( v i − v i − ) p ci +1 + ( v i +1 − v i ) p ci − − ( v i +1 − v i − ) c i v i +1 − v i ) ( v i − v i − ) , from which(3.4)2 ( v i +1 − v i ) ( v i − v i − ) (cid:0) θ di − θ di − (cid:1) = ( v i − v i − ) p ci +1 + ( v i +1 − v i ) p ci − − ( v i +1 − v i − ) c i . Combining (3.3)-(3.4) yields2( v i +1 − v i − ) (cid:0) p di − c i (cid:1) = 2 ( v i +1 − v i ) ( v i − v i − ) (cid:0) θ di − θ di − (cid:1) and therefore (cid:0) θ di − θ di − (cid:1) = ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) (cid:0) p di − c i (cid:1) . Hence, deviating profits can be written as(3.5) π di = (cid:0) p di − c i (cid:1) (cid:18) ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) (cid:19) . In a similar vein, intermediate firms’ Nash profit can be written as
IWAN BOS AND MARCO A. MARINI (3.6) π ∗ i = ( p ∗ i − c i ) (cid:18) ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) (cid:19) . Following the covered market assumption, we know that θ ci − θ ci − = θ ∗ i − θ ∗ i − and p ci = p ∗ i + p c − p ∗ . Hence, every intermediate firm’s collusive profit can be written as(3.7) π ci = ( p ci − c i ) (cid:0) θ ci − θ ci − (cid:1) = ( p ∗ i + p c − p ∗ − c i ) (cid:18) ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) ( p ∗ i − c i ) (cid:19) . Note further that p di = ( v i − v i − ) p ci +1 + ( v i +1 − v i ) p ci − + ( v i +1 − v i − ) c i v i +1 − v i − )= ( v i − v i − ) (cid:0) p ∗ i +1 + p c − p ∗ (cid:1) + ( v i +1 − v i ) (cid:0) p ∗ i − + p c − p ∗ (cid:1) + ( v i +1 − v i − ) c i v i +1 − v i − )= p ∗ i + 12 ( p c − p ∗ ) . Combining (3.5)-(3.7) yields(3.8) π di − π ci = (cid:18) ( v i +1 − v i − )( v i +1 − v i ) · ( v i − v i − ) (cid:19) (cid:20)
14 ( p c − p ∗ ) (cid:21) , and(3.9) π di − π ∗ i = (cid:18) ( v i +1 − v i − )( v i +1 − v i ) · ( v i − v i − ) (cid:19) ( p c − p ∗ ) (cid:20)
14 ( p c − p ∗ ) + ( p ∗ i − c i ) (cid:21) . Thus, by (5.1) and (5.2), the critical discount factor of every intermediate firm i = 2 , , ..., n − δ i ≡ π di − π ci π di − π ∗ i = ( p c − p ∗ ) ( p c − p ∗ ) + ( p ∗ i − c i ) , which is decreasing in the noncollusive price-cost margin.Turning to the top-quality firm, by following the above steps, it can be verified that π dn = (cid:0) p dn − c n (cid:1) ( v n − v n − ) , π ∗ n = ( p ∗ n − c n ) ( v n − v n − ) and π cn = ( p c − p ∗ + p ∗ n − c n ) θ ( v n − v n − ) − (cid:0) p ∗ n − p ∗ n − (cid:1) v n − v n − ! , yielding δ n ≡ π dn − π cn π dn − π ∗ n = ( p c − p ∗ ) ( p c − p ∗ ) + ( p ∗ n − c n ) . ARTEL STABILITY UNDER QUALITY DIFFERENTIATION 7
Hence, it is the supplier with the smaller noncooperative price-cost margin (between firm n and firm n −
1) who has the tighter incentive constraint.Finally, in a similar fashion, it can be shown that π d = (cid:0) p d − c (cid:1) ( v − v ) , π ∗ = ( p ∗ − c ) ( v − v ) and π c = ( p c − c ) (cid:18) p c − p c − ( v − v ) θv − v (cid:19) = ( p c − c ) ( p ∗ − c )( v − v ) , yielding δ ≡ π d − π c π d − π ∗ = ( p c − p ∗ ) ( p c − p ∗ ) + ( p ∗ − c ) . Hence, p ∗ − c > p ∗ − c implies Ω > Ω , whereas p ∗ − c > p ∗ − c implies Ω > Ω . Wethus conclude that, if p ∗ i − c i > p ∗ j − c j , then Ω i > Ω j for all i, j ∈ N, j = i . (cid:3) The incentive to deviate from the collusive agreement is determined by the short-termgain of defection (cid:0) π di − π ci (cid:1) and the severity of the resulting punishment ( π di − π ∗ i = π di − π ci + π ci − π ∗ i ). The proof of Proposition 1 reveals that the ‘extra profit effect’ is the same acrossfirms and that the heterogeneity in incentive constraints is exclusively driven by differencesin the punishment impact ( π ci − π ∗ i ). Specifically, there is a positive relation between a firm’smarket share and its noncollusive price-cost margin. Since market shares are fixed at pre-collusive levels, members with a higher competitive mark-up are hit relatively harder by acartel breakdown and this creates a stronger incentive to abide by the agreement.The next result follows immediately. In stating this result, let △ c ij = c i − c j for any firm i, j ∈ N and j = i . Corollary 1.
For any firm i, j ∈ N and j = i , ∃ µ ∈ R ++ such that if △ c ij < µ and v i > v j ,then Ω i > Ω j . Hence, if the difference in unit production costs is sufficiently small, then it is the lowest-quality supplier who has the tightest incentive constraint. Our analysis thus confirms theabove-mentioned conclusion by Symeonidis (1999) in case quality heterogeneity is primarilydriven by (sunk) fixed costs rather than variable costs. Let us conclude this section with a remark on the covered market assumption. Notethat since each ICC is strictly concave in the own cartel price, a sufficient condition for thecartel to keep market size constant is that at p c = θv , Ω i ≤ ∂ Ω i ∂p ci < i ∈ N .Endogenizing the size of the market would therefore not affect the above findings when thediscount factor is sufficiently low. If its members are patient enough, however, then thecartel would like to uncover the market. This case is generally far less tractable analytically,but we show in an online appendix that this paper’s results also hold when the number oflow-valuation buyers leaving the market is sufficiently small. This result can also be shown to hold in an n -firm variation of the model in H¨ackner (1994). The analysisis available upon request. IWAN BOS AND MARCO A. MARINI Conclusion
Many markets are characterized by some degree of quality differentiation with correspond-ing firm heterogeneity in cost and demand. One implication of such differences is that collud-ing firms typically face non-identical incentive constraints. Existing literature on this topicfocuses on demand differences, while ignoring the potential impact of cost heterogeneity.In this note, we considered how cartel stability is affected when unit costs are increasing inproduct quality. Under the assumption that colluding firms maintain their pre-collusive mar-ket shares, we found that the incentive to deviate from the collusive agreement is monotonicin the noncollusive price-cost margin. Specifically, the supplier with the lowest competitivemark-up is ceteris paribus most inclined to leave the cartel. Moreover, it is the lowest-qualityseller who has the tightest incentive constraint when differences in unit costs are sufficientlysmall. 5.
Appendix 1: Uncovered Market Case
In the note we show that the incentive to deviate from a cartel agreement is monotonic inthe noncollusive price-cost margin and that the firm with the lowest profit margin has thetightest incentive constraint (Proposition 1). This result is derived under the assumption ofa fixed market size. In this appendix, we consider the possibility that the cartel ‘uncovers themarket’ by setting collusive prices at a level where some of the lowest-valuation customersprefer to no longer buy the product. In the following, we show that the result of Proposition1 also holds when the number of buyers leaving the market is sufficiently small.To begin, consider the incentive compatibility constraint (ICC) of an intermediate firm i ∈ N :Ω i ≡ ( p ci − c i ) (cid:0) θ ci − θ ci − (cid:1) − (1 − δ ) (cid:0) p di − c i (cid:1) (cid:0) θ di − θ di − (cid:1) − δ ( p ∗ i − c i ) (cid:0) θ ∗ i − θ ∗ i − (cid:1) ≥ . As the cartel uses a fixed market share rule, it holds that: θ ci − θ ci − θ − p c v = θ ∗ i − θ ∗ i − θ − θ . The ICC can thus be written as:Ω i ≡ ( p ci − c i ) (cid:0) θ ∗ i − θ ∗ i − (cid:1) (cid:16) θ − p c v (cid:17)(cid:0) θ − θ (cid:1) − (1 − δ ) (cid:0) p di − c i (cid:1) (cid:0) θ di − θ di − (cid:1) − δ ( p ∗ i − c i ) (cid:0) θ ∗ i − θ ∗ i − (cid:1) ≥ , which is equivalent to δ ≥ δ i ≡ π di − π ci π di − π ∗ i = (cid:0) p di − c i (cid:1) (cid:0) θ di − θ di − (cid:1) − ( p ci − c i ) (cid:0) θ ∗ i − θ ∗ i − (cid:1) (cid:18) θ − pc v (cid:19) ( θ − θ ) (cid:0) p di − c i (cid:1) (cid:0) θ di − θ di − (cid:1) − ( p ∗ i − c i ) (cid:0) θ ∗ i − θ ∗ i − (cid:1) . Let us now specify π di , π ∗ i and π ci . Following the proof of Proposition 1, deviating andNash profits are respectively given by π di = (cid:0) p di − c i (cid:1) (cid:18) ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) (cid:19) , ARTEL STABILITY UNDER QUALITY DIFFERENTIATION 9 and π ∗ i = ( p ∗ i − c i ) (cid:18) ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) (cid:19) . As to collusive profits, note that uncovering the market in combination with the fixed marketshare rule implies that the lowest quality firm has the smallest price increase and that theprice increase is rising in quality. In other words, cartel prices must be chosen such thateach marginal consumer’s location ‘shifts upwards’ in order to maintain market shares atpre-collusive levels. Rather than adding p c − p ∗ to its Nash price p ∗ i (as in the covered marketcase), intermediate firm i should therefore raise its price by more. Let this additional amountbe indicated by x i > p ci = p ∗ i + p c − p ∗ + x i . Collusiveprofits are then π ci = ( p ci − c i ) (cid:0) θ ci − θ ci − (cid:1) = ( p ∗ i + p c − p ∗ + x i − c i ) (cid:18) ( v i +1 − v i − ) ( p ∗ i − c i )( v i +1 − v i ) ( v i − v i − ) (cid:19) (cid:16) θ − p c v (cid:17)(cid:0) θ − θ (cid:1) . Focusing on the numerator of the critical discount factor first, we have π di − π ci = (cid:0) p di − c i (cid:1) ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) − ( p ∗ i + p c − p ∗ + x i − c i ) (cid:18) ( p ∗ i − c i ) ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) (cid:19) (cid:16) θ − p c v (cid:17)(cid:0) θ − θ (cid:1) = (cid:18) ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) (cid:19) (cid:0) p di − c i (cid:1) − ( p ∗ i + p c − p ∗ + x i − c i ) ( p ∗ i − c i ) (cid:16) θ − p c v (cid:17)(cid:0) θ − θ (cid:1) . In this case, the deviating price is given by p di = ( v i − v i − ) p ci +1 + ( v i +1 − v i ) p ci − + ( v i +1 − v i − ) c i v i +1 − v i − )= ( v i − v i − ) (cid:0) p ∗ i +1 + p c − p ∗ + x i +1 (cid:1) + ( v i +1 − v i ) (cid:0) p ∗ i − + p c − p ∗ + x i − (cid:1) + ( v i +1 − v i − ) c i v i +1 − v i − )= p ∗ i + 12 ( p c − p ∗ ) + ( v i − v i − ) x i +1 + ( v i +1 − v i ) x i − v i +1 − v i − ) . To facilitate the presentation of the analysis, let us denote y i = ( v i − v i − ) x i +1 + ( v i +1 − v i ) x i − v i +1 − v i − ) and s = θ − p c v θ − θ . Substituting in the above equation gives π di − π ci = (cid:18) ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) (cid:19) (cid:20) (cid:0) p ∗ i + ( p c − p ∗ ) + y i − c i (cid:1) − ( p ∗ i + p c − p ∗ + x i − c i ) ( p ∗ i − c i ) s (cid:21) = (cid:18) ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) (cid:19) (cid:20) ( p c − p ∗ ) + (1 − s ) ( p ∗ i − c i ) [( p ∗ i − c i ) + ( p c − p ∗ ) + x i ]+ ( p ∗ i − c i ) (2 y i − x i ) + y i (( p c − p ∗ ) + y i ) (cid:21) or(5.1) π di − π ci = (cid:18) ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) (cid:19) (cid:20)
14 ( p c − p ∗ ) + z (cid:21) , where z ≡ (1 − s ) ( p ∗ i − c i ) [( p ∗ i − c i ) + ( p c − p ∗ ) + x i ] + ( p ∗ i − c i ) (2 y i − x i ) + y i (( p c − p ∗ ) + y i ) . Note that with a covered market it holds that s = 1, x i = 0 and y i = 0, in which case z = 0and the numerator reduces to the corresponding value in the proof of Proposition 1. Turningto the denominator of the critical discount factor, we have π di − π ∗ i = (cid:0) p di − c i (cid:1) ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) − ( p ∗ i − c i ) ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − )= ( v i +1 − v i − ) h(cid:0) p di − c i (cid:1) − ( p ∗ i − c i ) i ( v i +1 − v i ) ( v i − v i − )= ( v i +1 − v i − ) h(cid:0) p ∗ i + ( p c − p ∗ ) + y i − c i (cid:1) − ( p ∗ i − c i ) i ( v i +1 − v i ) ( v i − v i − )= ( v i +1 − v i − ) (cid:2) ( p c − p ∗ ) + ( p ∗ i − c i ) ( p c − p ∗ + 2 y i ) + y i (( p c − p ∗ ) + y i ) (cid:3) ( v i +1 − v i ) ( v i − v i − )or(5.2) π di − π ∗ i = ( v i +1 − v i − ) (cid:2) ( p c − p ∗ ) + r (cid:3) ( v i +1 − v i ) ( v i − v i − ) , where r ≡ ( p ∗ i − c i ) ( p c − p ∗ + 2 y i ) + y i (( p c − p ∗ ) + y i ) . Note that in case of a covered market y i = 0 so that r = ( p c − p ∗ ) ( p ∗ i − c i ) as in the proofof Proposition 1. Combining both terms (5.1) and (5.2) gives the critical discount factor: δ ≥ δ i ≡ ( p c − p ∗ ) + (1 − s ) ( p ∗ i − c i ) [( p ∗ i − c i ) + ( p c − p ∗ ) + x i ]+ ( p ∗ i − c i ) (2 y i − x i ) + y i (( p c − p ∗ ) + y i ) ( p c − p ∗ ) + ( p ∗ i − c i ) ( p c − p ∗ + 2 y i ) + y i · (( p c − p ∗ ) + y i ) . As a final step, let us evaluate this critical discount factor with respect to the price-costmargin ( p ∗ i − c i ). Taking the first-derivative with respect to the noncollusive profit marginof firm i yields: ∂δ i ∂ (( p ∗ i − c i )) = (cid:26) ( p c − p ∗ ) + ( p ∗ i − c i ) ( p c − p ∗ + 2 y i )+ y i (( p c − p ∗ ) + y i ) (cid:27) (cid:20) (1 − s ) [2 ( p ∗ i − c i ) + ( p c − p ∗ ) + x i ]+ (2 y i − x i ) (cid:21)(cid:0) ( p c − p ∗ ) + ( p ∗ i − c i ) ( p c − p ∗ + 2 y i ) + y i (( p c − p ∗ ) + y i ) (cid:1) − (cid:26) ( p c − p ∗ ) + (1 − s ) ( p ∗ i − c i ) [( p ∗ i − c i ) + ( p c − p ∗ ) + x i ]+ ( p ∗ i − c i ) (2 y i − x i ) + y i (( p c − p ∗ ) + y i ) (cid:27) ( p c − p ∗ + 2 y i ) (cid:0) ( p c − p ∗ ) + ( p ∗ i − c i ) ( p c − p ∗ + 2 y i ) + y i (( p c − p ∗ ) + y i ) (cid:1) , ARTEL STABILITY UNDER QUALITY DIFFERENTIATION 11 which is negative when(1 − s ) ( p ∗ i − c i ) (cid:26) y i (( p c − p ∗ ) + y i ) + ( p ∗ i − c i ) ( p c − p ∗ + 2 y i ) + 12 ( p c − p ∗ ) (cid:27) − s { ( p c − p ∗ ) + x i } (cid:20) y i (( p c − p ∗ ) + y i ) + 14 ( p c − p ∗ ) (cid:21) < . Note that this condition holds for s →
1. In a similar fashion, this can be shown to be truefor the bottom and top quality firm cases. Thus, we conclude that the result of Proposition1 also holds when the cartel uncovers the market and the fraction of buyers no longer buyingthe product is sufficiently small.6.
Appendix 2: Example with Nonuniform Distribution
Let us present a simple two-firm example with a non-uniform distribution of θ . The resultsof the paper are also shown to hold in this case. Though of course far from being a proof,this suggests that our findings may apply for a wider class of customer distributions.Consider the model of the paper, but with two firms and a simple ‘two-step uniform’distribution function where a subset of consumers (of mass s ) is uniformly distributed overa given interval h θ, e θ i , with e θ ∈ (cid:0) θ, θ (cid:1) and another subset (of mass (1 − s ) = s ) is uniformlydistributed over (cid:16)e θ, θ i (higher willingness to pay consumers). The density function takesthe form: f ( θ ) = s ( e θ − θ ) for θ ∈ h θ, e θ i and − s ( θ − e θ ) for θ ∈ (cid:16)e θ, θ i . In the following, suppose that θ ( p , p ) = p − p v − v ≤ e θ so that the profit functions are givenby: π = ( p − c ) (cid:18) p − p v − v − θ (cid:19) s ( e θ − θ ) , and π = ( p − c ) (cid:18)e θ − p − p v − v (cid:19) s ( e θ − θ ) + ( p − c ) (cid:16) θ − e θ (cid:17) − s ( θ − e θ )= ( p − c ) e θ ( v − v ) s − sp + sp + ( v − v ) ( e θ − θ ) (1 − s )( v − v ) ( e θ − θ ) ! . The first-order conditions give the best response functions: b p = 12 ( p − ( v − v ) θ + c ) b p = ( v − v ) (cid:16)e θ − θ (1 − s ) (cid:17) + sp + sc s Combining gives the Nash prices and corresponding profits: p ∗ = ( v − v ) (cid:16)e θ − θ (1 + s ) (cid:17) + 2 sc + sc sp ∗ = ( v − v ) (cid:16) (cid:16)e θ − θ (cid:17) + θs (cid:17) + 2 sc + sc sπ ∗ = ( v − v ) (cid:16)e θ − θ (1 + s ) (cid:17) − sc + sc s ( v − v ) (cid:16)e θ − θ (1 + s ) (cid:17) − sc + sc v − v ) ( e θ − θ ) = ( p ∗ − c ) ( v − v ) s ( e θ − θ ) .π ∗ = ( v − v ) (cid:16) (cid:16)e θ − θ (cid:17) + θs (cid:17) − sc + sc s ( v − v ) h (cid:16)e θ − θ (cid:17) + θs i − sc + sc v − v ) ( e θ − θ ) = ( p ∗ − c ) ( v − v ) s ( e θ − θ ) . Under collusion, prices and profits are respectively given by: p c p c = p ∗ + ( p c − p ∗ ) = ( v − v ) (cid:16)e θ − θ + 2 θs (cid:17) + sc − sc + 3 sp c s and π c = ( p c − c ) (cid:18) p c − p c v − v − θ (cid:19) s ( e θ − θ ) = ( p c − c ) (cid:18) p ∗ − p ∗ v − v − θ (cid:19) s ( e θ − θ )= ( p c − c ) ( v − v ) (cid:16)e θ − θ (1 + s ) (cid:17) + sc − sc v − v ) ( e θ − θ ) = ( p c − c ) ( p ∗ − c )( v − v ) s ( e θ − θ ) .π c = ( p c − c ) e θ ( v − v ) s − s ( p c − p c ) + ( v − v ) ( e θ − θ ) (1 − s )( v − v ) ( e θ − θ ) ! = ( p c − c ) e θ ( v − v ) s − s ( p ∗ − p ∗ ) + ( v − v ) ( e θ − θ ) (1 − s )( v − v ) ( e θ − θ ) ! = ( p c − c ) e θ ( v − v ) s − ( v − v ) (cid:16)e θ − θ + 2 θs (cid:17) − sc + sc + 3 ( v − v ) ( e θ − θ ) (1 − s )3 ( v − v ) ( e θ − θ ) = ( p c − c ) ( v − v ) h (cid:16)e θ − θ (cid:17) + θs i − sc + sc v − v ) ( e θ − θ ) = ( p c − c ) ( p ∗ − c )( v − v ) s ( e θ − θ ) . ARTEL STABILITY UNDER QUALITY DIFFERENTIATION 13
Finally, deviating price and profits are: p d = 12 ( p c − ( v − v ) θ + c ) p d = ( v − v ) (cid:16)e θ − θ (1 − s ) (cid:17) + sp c + sc s and π d = (cid:0) p d − c (cid:1) (cid:18) p c − p d v − v − θ (cid:19) s ( e θ − θ ) == 12 ( p c − ( v − v ) θ − c ) (cid:18) p c − ( v − v ) θ − c v − v ) (cid:19) s ( e θ − θ )= (cid:0) p d − c (cid:1) ( v − v ) s ( e θ − θ ) , and, π d = (cid:0) p d − c (cid:1) e θ ( v − v ) s − s (cid:0) p d − p c (cid:1) + ( v − v ) ( e θ − θ ) (1 − s )( v − v ) ( e θ − θ ) ! = ( v − v ) (cid:16)e θ − θ (1 − s ) (cid:17) + sp c − sc s ( v − v ) he θ − θ (1 − s ) i + sp c − sc v − v ) ( e θ − θ ) = (cid:0) p d − c (cid:1) ( v − v ) s ( e θ − θ ) . Combining to obtain the critical discount factor of each firm gives: δ = π d − π c π d − π ∗ = (cid:0) p d − c (cid:1) ( v − v ) s ( e θ − θ ) − ( p c − c ) ( p ∗ − c )( v − v ) s ( e θ − θ ) (cid:0) p d − c (cid:1) ( v − v ) s ( e θ − θ ) − ( p ∗ − c ) ( v − v ) s ( e θ − θ )= (cid:0) p d − c (cid:1) − ( p c − c ) ( p ∗ − c ) (cid:0) p d − c (cid:1) − ( p ∗ − c ) = 14 ( p c − p ∗ ) ( p c − p ∗ ) (cid:0) ( p c − p ∗ ) + ( p ∗ − c ) (cid:1) = 14 ( p c − p ∗ )14 ( p c − p ∗ ) + ( p ∗ − c ) . and δ = π d − π c π d − π ∗ = (cid:0) p d − c (cid:1) ( v − v ) s ( e θ − θ ) − ( p c − c ) ( p ∗ − c )( v − v ) s ( e θ − θ ) (cid:0) p d − c (cid:1) ( v − v ) s ( e θ − θ ) − ( p ∗ − c ) ( v − v ) s ( e θ − θ )= (cid:0) p d − c (cid:1) − ( p c − c ) ( p ∗ − c ) (cid:0) p d − c (cid:1) − ( p ∗ − c ) = 14 ( p c − p ∗ ) ( p c − p ∗ ) (cid:0) ( p c − p ∗ ) + ( p ∗ − c ) (cid:1) = 14 ( p c − p ∗ )14 ( p c − p ∗ ) + ( p ∗ − c ) . Appendix 3: Comparisons with Other Models
Symeonidis (1999) analyzes a representative consumer model, which differs from ours inmany ways. Meaningful comparisons are therefore difficult to make. H¨ackner’s (1994) modelis also different, but in many ways comparable. Specifically, he is using a utility specificationof the form U ( θ ) = v i ( θ − p i ), whereas we follow Mussa and Rosen (1978) which uses U ( θ ) = v i θ − p i . To address this issue, we have tried to clarify the differences between thedifferent settings in the introduction. Moreover, we have performed the same analysis inH¨ackner’s (1994) setting. This analysis is presented below, but let us first summarize themain conclusion. The main finding of Proposition 1 may not generally hold in H¨ackner’s(1994) model. In particular, conclusions may be different when the highest quality firmshave the lowest profit margins and vice versa . Yet, the results of Proposition 1 do holdwhen the noncollusive profit margin is increasing in quality. Moreover, if differences in unitcosts are sufficiently small, then it is indeed the lowest-quality seller who is most inclined todeviate, all else unchanged. Consequently, the result of the Corollary also holds when takingH¨ackner’s (1994) approach. We have added a footnote highlighting this point (footnote 8).Consider an intermediate firm in an n -firm variant of H¨ackner’s (1994) model with costheterogeneity. A consumer located at θ i is indifferent between buying from firm i and i + 1when: θ i v i +1 − v i +1 p i +1 = θ i v i − v i p i or θ i = v i +1 p i +1 − v i p i v i +1 − v i . The profit function is then given by π i = ( p i − c i ) (cid:18) v i +1 ( v i − v i − ) p i +1 + v i − ( v i +1 − v i ) p i − − v i ( v i +1 − v i − ) p i ( v i +1 − v i ) ( v i − v i − ) (cid:19) , which yields the following best-response function: b p i ( p i − , p i +1 ) = v i +1 ( v i − v i − ) p i +1 + v i − ( v i +1 − v i ) p i − + v i ( v i +1 − v i − ) c i v i ( v i +1 − v i − ) . ARTEL STABILITY UNDER QUALITY DIFFERENTIATION 15
Following the same steps in the proof of Proposition 1, this gives demand: θ ∗ i − θ ∗ i − = v i ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) ( p ∗ i − c i )and Nash profits π ∗ i = ( p ∗ i − c i ) (cid:18) v i ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) (cid:19) . In a similar fashion, it can be shown that: π di = (cid:0) p di − c i (cid:1) (cid:18) v i ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) (cid:19) . As to collusive profits, we know that (due to the fixed market share rule): θ ∗ i v i +1 − v i +1 p ci +1 = θ ∗ i v i − v i p ci or p ci +1 = p ∗ i +1 + v i v i +1 ( p ci − p ∗ i )In general, the collusive price is then: p ci = p ∗ i + v v i ( p c − p ∗ ) . Following the covered market assumption, we know that θ ci − θ ci − = θ ∗ i − θ ∗ i − and therefore, π ci = ( p ci − c i ) (cid:0) θ ci − θ ci − (cid:1) = (cid:18) p ∗ i + v v i ( p c − p ∗ ) − c i (cid:19) (cid:18) v i ( v i +1 − v i − )( v i +1 − v i ) ( v i − v i − ) ( p ∗ i − c i ) (cid:19) . Moreover, p di = v i +1 ( v i − v i − ) p ci +1 + v i − ( v i +1 − v i ) p ci − + v i ( v i +1 − v i − ) c i v i ( v i +1 − v i − )= v i +1 ( v i − v i − ) (cid:18) p ∗ i +1 + v ( p c − p ∗ ) v i +1 (cid:19) + v i − ( v i +1 − v i ) (cid:18) p ∗ i − + v ( p c − p ∗ ) v i − (cid:19) + v i ( v i +1 − v i − ) c i v i ( v i +1 − v i − )= v i +1 ( v i − v i − ) p ∗ i +1 + v i − ( v i +1 − v i ) p ∗ i − + ( v i +1 − v i − ) v ( p c − p ∗ ) + v i ( v i +1 − v i − ) c i v i ( v i +1 − v i − )= p ∗ i + 12 v v i ( p c − p ∗ ) . Combining the above profit specifications gives π di − π ci = v i ( v i +1 − v i − ) (cid:0) p di − c i (cid:1) ( v i +1 − v i ) ( v i − v i − ) − v i ( v i +1 − v i − ) ( p ∗ i − c i ) (cid:16) p ∗ i + v v i ( p c − p ∗ ) − c i (cid:17) ( v i +1 − v i ) ( v i − v i − )= v i ( v i +1 − v i − ) (cid:20)(cid:16) p ∗ i + v v i ( p c − p ∗ ) − c i (cid:17) − ( p ∗ i − c i ) − v v i ( p c − p ∗ ) ( p ∗ i − c i ) (cid:21) ( v i +1 − v i ) ( v i − v i − )= v i ( v i +1 − v i − ) (cid:20) (cid:16) v v i (cid:17) ( p c − p ∗ ) (cid:21) ( v i +1 − v i ) ( v i − v i − ) . and π di − π ∗ i = v i ( v i +1 − v i − ) (cid:0) p di − c i (cid:1) ( v i +1 − v i ) ( v i − v i − ) − v i ( v i +1 − v i − ) ( p ∗ i − c i ) ( v i +1 − v i ) · ( v i − v i − )= v i ( v i +1 − v i − ) (cid:18)(cid:16) p ∗ i + v v i ( p c − p ∗ ) − c i (cid:17) − ( p ∗ i − c i ) (cid:19) ( v i +1 − v i ) · ( v i − v i − )= v i ( v i +1 − v i − ) (cid:18) (cid:16) v v i (cid:17) ( p c − p ∗ ) + v v i ( p c − p ∗ ) ( p ∗ i − c i ) (cid:19) ( v i +1 − v i ) · ( v i − v i − )= v i ( v i +1 − v i − ) (cid:16) v v i ( p c − p ∗ ) (cid:17) (cid:16) (cid:16) v v i (cid:17) ( p c − p ∗ ) + ( p ∗ i − c i ) (cid:17) ( v i +1 − v i ) · ( v i − v i − ) . Thus, the critical discount factor of an intermediate firm i in H¨ackner’s (1994) setting withcost heterogeneity and n firms is: δ ≥ δ i = π di − π ci π di − π ∗ i = (cid:16) v i ( v i +1 − v i − )( v i +1 − v i )( v i − v i − ) (cid:17) (cid:20) (cid:16) v v i (cid:17) ( p c − p ∗ ) (cid:21)(cid:16) v i ( v i +1 − v i − )( v i +1 − v i ) · ( v i − v i − ) (cid:17) (cid:16) v v i ( p c − p ∗ ) (cid:17) (cid:16) (cid:16) v v i (cid:17) ( p c − p ∗ ) + ( p ∗ i − c i ) (cid:17) = (cid:16) v v i (cid:17) ( p c − p ∗ ) (cid:16) v v i (cid:17) ( p c − p ∗ ) + ( p ∗ i − c i ) = v ( p c − p ∗ ) v ( p c − p ∗ ) + v i ( p ∗ i − c i ) . Results for the highest and lowest quality firm can be derived in a similar way (see the proofof Proposition 1).Recall that the corresponding critical discount factor in Proposition 1 is given by δ ≥ δ i = π di − π ci π di − π ∗ i = ( p c − p ∗ ) ( p c − p ∗ ) + ( p ∗ i − c i ) . Observe that, compared to the result of Proposition 1, it is now possible that a firm with ahigher competitive profit margin is more eager to leave the cartel when it is of sufficientlylow quality ( i.e. , a high p ∗ i − c i may be more than neutralized by a low v i ) and vice versa .The result of Proposition 1 does apply, however, when the noncollusive price-cost marginis increasing in quality. Moreover, and in contrast to H¨ackner, it is indeed the lowest-quality supplier who has the tightest incentive constraint when differences in unit costs aresufficiently small (in accordance with the result of the Corollary). As to the latter, note thatH¨ackner (1994) considers a duopoly without costs and does not impose a fixed market sharerule, which is an important driver of our finding. References [1] Ecchia, Giulio and Luca Lambertini (1997), “Minimum Quality Standards and Collusion,”
Journal ofIndustrial Economics , 45(1), 101-113.[2] Friedman, James W. (1991),
Game Theory with Applications to Economics . Oxford University Press,Oxford.[3] Gabszewicz, Jean J. and Jacques-Fran¸cois Thisse (1979), “Price Competition, Quality and IncomeDisparities,”
Journal of Economic Theory , 20(3), 340-359.
ARTEL STABILITY UNDER QUALITY DIFFERENTIATION 17 [4] H¨ackner, Jonas (1994), “Collusive pricing in markets for vertically differentiated products,”
Interna-tional Journal of Industrial Organization , 12(2), 155-177.[5] Harrington, Joseph E. Jr. (2006), “How do Cartels Operate?,”
Foundations and Trends in Microeco-nomics , 2(1), 1-105.[6] Marini, Marco A. (2018), ”Collusive Agreements in Vertically Differentiated Markets”, in
Handbook ofGame Theory and Industrial Organization , Volume 2: Applications , L. C. Corchon and M. A. Marini(eds.), Edward Elgar, Chelthenam, UK, Northampton, MA, USA.[7] Mussa, Michael and Sherwin Rosen (1978), “Monopoly and Product Quality,”
Journal of EconomicTheory , 18(2), 301-317.[8] Shaked, Avner and John Sutton (1982), “Relaxing Price Competition Through Product Differentiation,”
Review of Economic Studies , 49 (1), 3-13.[9] Symeonidis, George (1999), “Cartel stability in advertising-intensive and R&D-intensive industries,”
Economics Letters , 62, 121-129.[10] Tirole, Jean (1988), “
The Theory of Industrial Organization ,” MIT Press, Cambridge, Massachusetts.[11] Vives, Xavier (2000),