Endogenous Cartel Formation with Differentiated Products and Price Competition

Transcription

1 Endogenous Cartel Formation with Differentiated Products and Price Competition Tyra Merker * February 2018 Abstract Cartels may cause great harm to consumers and economic efficiency. However, literature on endogenous cartel formation in a dynamic setting is scarce. Furthermore, models always assume quantity competition with homogeneous products, despite the fact firms generally compete in prices and that products rarely are homogeneous. This paper is therefore the first to endogenize cartel formation in a model using price competition with differentiated products. The model may yield key insights into what characteristics colluding firms are likely to have, and thus has important policy implications. For instance, I find that closeness in competition between firms is of high importance for cartel size. This contrasts findings from models of quantity competition, where firm size is the most important factor. Thus, the type of competition assumed leads to very different conclusions for what markets should be scrutinized by competition authorities. For a model with symmetric firms, linear costs and quadratic profit functions, I find that closeness in competition measured by cross-price derivatives or so-called diversion ratios is the most important parameter for defining the optimal size of a stable cartel. Furthermore, I find that cartel size rarely exceeds five to six firms, as the relative pay-off from free-riding increases faster in cartel size than the pay-off from collusion irrespective of industry size. Finally, I find that if firms are sufficiently patient, the stable cartel is incentive compatible in the sense that cartel members do not have an incentive to deviate from the collusive agreement. Keywords Collusion, Cartel formation, Cartel stability, Differentiated products JEL Classification D43, L11, L41 *University of Oslo, Tel.: , tyra.merker@econ.uio.no 1

2 1 Introduction Real-life cartels - more often than not - include only some of the firms in a market. Hay & Kelley 1974 report statistics such as market shares and industry size number of firms for over 60 US cartels in the period Their overview shows that 20 of 27 cartels where market shares are reported are partial, in the sense that they do not include all firms in the market. The overview also shows that these cartels on average have a market share of 92 percent share of sales and make up 74 percent of firms in an industry. 53 of 54 cartels studied in Griffin 1989 were partial. More recent examples from European Commission decisions between include the 1990s citric cartel that encompassed 60 percent of global production, cartels in vitamins B1, B2 and C that excluded Chinese suppliers and the European industrial tubes cartel that lasted 13 years, excluded two significant suppliers and controlled percent of total production Harrington Observations of actual cartels therefore suggest that an assumption of all-inclusive cartels is unrealistic. What firm or product characteristics determine cartel participation and size therefore poses an important theoretical question. For instance, the empirical examples above may suggest that larger firms are more likely to part-take in a cartel than smaller firms. The literature on endogenous cartel formation originates from static models, and early works include Selten 1973, d Aspremont et al. 1983, Donsimoni 1985, and Donsimoni et al In the literature on collusion, it is common to assume quantity competition. See for example Yi 1997, Compte et al. 2002, Escrihuela-Villar 2008, 2009 and Bos & Harrington These coalition models generally consist of a twostage coalition game, where firms first choose coalition membership and subsequently either play a game of quantity competition or a capacity constrained price game with homogeneous products. Thus, these models assume that products are homogeneous and therefore perfectly substitutable. In reality, markets rarely involve perfect substitutes, but rather consist of a limited number of differentiated firms or brands. Furthermore, consumers are likely to prefer one brand or product to another, and may differ in their preferences. Intuitively, firms gain more by colluding with a close competitor than they do by colluding with weaker substitutes. Collusive agreements including all firms in a market may therefore be less desirable for the firms. The question of how firms compete may therefore have important implications for antitrust work. Bos & Harrington 2010 show that mergers between intermediate-size firms may cause the most severe coordinated effects when firms are capacity constrained. This may not hold true, however, if firms do not engage in a capacity game, but instead engage in price competition with differentiated products. 1 1 Note that there exist models of coalitions in markets with differentiated price competition in the 2

3 So far, little work has been done on optimal cartel size and partial cartelization of industries in a dynamic setting. Escrihuela-Villar 2008, 2009, and Bos & Harrington 2010 do however present dynamic models infinitely repeated games of endogenous cartel formation. All three articles assume homogeneous products and capacityconstrained price competition or quantity competition. For markets with differentiated products, Selten 1973 shows that mergers are increasingly profitable in the exogenous number of merging firms, and that non-merging firms gain more from the merger than the merging firms. A merger in the context of Selten 1973 is equivalent to a cartel that maximizes joint profit. The article shows that, contrary to quantity competition, mergers with differentiated price competition are advantageous for the merging firms. However, the article treats merger size as exogenous and does not assess stability of the agreement. This article will, to my knowledge, be the first to examine endogenous cartel formation under the assumption of price competition with differentiated products. As in a capacity-constrained price game, a model of price competition between firms with differentiated products introduces a free-riding incentive: the existence of a partial cartel in a market raises demand for non-colluding firms products. Therefore, firms may have incentives to free-ride rather than part-take in collusion and an incentive compatible cartel may only include a subset of firms in a given market. Thus, modelling collusion in markets with differentiated products may be important to obtain a more realistic model of real world markets - and explain why cartels do not necessarily include all agents. This raises several interesting research questions: i Under what industry conditions can we expect a cartel to be industry-wide? ii What market characteristics matter the most for the degree of cartelization? iii Which firms are most likely to form a cartel? and iv How does a merger affect the degree of cartelization? In this paper, I introduce a simple model with no renegotiation and a grim-trigger punishment strategy. The main purpose of the model is to serve as a baseline for endogenous cartel formation with differentiated price competition. In section 2, I will present the basic set-up of the model. Section 3 presents general equilibrium pricing strategies for firms when there exists a cartel in the industry, and characterizes the solution under the assumption of linear demand quadratic profit functions. Section 4 characterizes the equilibrium when product differentiation is symmetric and equal between firms, and illustrates the optimal cartel size of the industry. I show that full cartelization rarely is optimal, as the condition for internal cartel stability is not satisfied. Furthermore, I find that closeness in competition between firms is the most important parameter to determine merger literature. For example Deneckere & Davidson 1985 present a model of coalitions with Bertrand competition. The article shows that profits are increasing in the size of a coalition, and that free-riders benefit more than coalition members. However, the article does not assess coalition stability and treats coalition sizes as exogenously given. 3

4 optimal cartel size. Finally, I assess whether the incentive compatibility constraint holds. I find that if firms are sufficiently patient, collusion can be sustained for any cartel share, and that the incentive compatibility constraint rarely binds. Section 5 concludes and summarizes important extensions of the model. 2 The model The figure in table 1 illustrates the timing of the game. In the first period, firms simultaneously choose whether or not to become a cartel member. The firms only make the choice of cartel-membership once. This implies that renegotiation and recartelization is not possible. Clearly, no renegotiation may be an unrealistic assumption. However, the assumption simplifies the model and eases the analysis of comparative statics, allowing me to focus on optimal cartel size. After the cartel-membership choice, firms play an infinitely repeated price-game with differentiated products. In each stage-game of the price-game, firms simultaneously choose prices. In this price game, cartel members will choose prices differently than non-members. I will assume that cartel break-down will lead to a reversion to the Nash equilibrium of the stage game for all future periods the punishment strategy is a so-called trigger strategy. Distribution of cartel size Table 1: Timing of the game Let N = {1,2,...,n} denote a set of firms in a market, that compete in an infinitely repeated price game with differentiated products. Let Q i p i, p i, for i N, denote the demand for firm i s product, where p i denotes firm i s price and p i denotes the vector of prices set by i s competitors. Assumption 1 Properties of the demand function Assume that demand functions are twice and continuously differentiable, and that for all i, j N : 4

5 i Q ip i,p i Q p i 0, where lim i p i,p i p i p i = 0 ii Q ip i,p i p j 0, for i j iii p i c i 2 Q i 2 Q p 2 i p i i Thus, I assume that the demand for a firm s product is decreasing in its own price, but that demand increases when a competitor raises its price. Part iii ensures concavity of the firms profit functions. Firms set prices simultaneously in each stage-game. Assuming constant marginal costs, 2 firm i N chooses p i to maximize its profit Π i p i, p i = p i c i Q i p i, p i, where c i denotes the marginal cost of production. Note that competition with differentiated products implies that, in choosing price p i, firm i exerts an externality on the other firms in the market. Industry profit can be characterized as Π = p i c i Q i p i, p i. Thus, a change in one firm s price changes industry profit by: Π = p i c i Q i + Q i + p i p }{{ i p j c j Q j } j N, j i p i }{{} firm i s optimality condition FOC sum of externalities on other firms in market An individual firm only considers how a price change affects its own profit, given by its first order condition FOC. As Q i p i 0 and Q j p i 0, firms set lower prices - earning lower profits - when they set prices individually than they would if they maximized joint profit. Therefore, firms have an incentive to collude by setting prices collectively. As Q i p j 0, there also exists a free-riding incentive: firm i benefits if its competitors collude, as their increased prices lead to higher demand for firm i. 3 Equilibrium pricing strategies the general case In this section, I assess the firms optimal pricing strategies in the repeated pricing-game, thus taking cartel size from the first stage of the game as given. Firstly, I present the optimal pricing behavior of non-cartel members, given by Nash-equilibrium strategies in each stage game. Subsequently, I will characterize optimal pricing behavior when there 2 Note that the assumption of linear costs may be unrealistic. However, it simplifies the first order conditions of profit maximization, and allows me to illustrate some of the models dynamics. Extending the analysis to other specifications of the cost function is left for future research. i N 5

6 exists a cartel in the industry. In the final subsection, I will present conditions for cartel break-down or lack thereof, given by the incentive compatibility constraint for deviation. These equilibrium strategies serve as the basis for evaluating cartel stability and optimal cartel size in section 4. In each subsection, I also characterize equilibrium strategies given linear demand functions. Linear demand 3 may be a reasonable approximation for small price changes. However, a cartel price may be significantly larger than a firm s competitive price. As a firm increases its price, it is likely that price-conscious consumers will be the first to stop purchasing the product. Consequently, the firm s remaining customers are less price sensitive, and the slope of the demand function is likely to become less steep. Linear demand may therefore be an unrealistic assumption. Despite being a limiting assumption, linear demand enables us to solve analytically for equilibrium solutions. 4 Furthermore, linear demand may yield similar dynamics and insights as more advanced demand functions. Linear demand therefore serves as a useful benchmark for my analysis. 3.1 The static and subgame perfect Nash-equilibria Non-cartel members maximize their profit, given the other firms prices. The FOCs then implicitly define static Nash-equilibrium prices for non-cartel members: Q i p i, p i = Q ip i, p i p i p i c i 1 If demand is linear, the first-order derivatives are constant. assumptions about the linear demand function: I make the following Assumption 2 Linear demand Assume that linear demand for the product of firm i N takes the form Furthermore, assume that Q i p i, p i = A + d i j p j. j N i a firm s demand increases in its competitors prices: d i j 0 if j i 3 Note that linear demand implies quadratic and weakly concave profit functions for the firms. This assumption mirrors an assumption of quadratic benefit functions frequently used in coordination models in the field of environmental economics. See for example Barrett The quadratic shape of the profit functions leads to a set of linear first-order conditions for the firms, that can be solved analytically for optimal prices. 6

7 ii a firm s demand decreases in its own price: d i j 0 if i = j iii the total increase in demand experienced by a firm s competitors cannot be greater than the direct demand decline by that firm: d ji d ii. 5 j i Non-members FOCs therefore become: 2d ii p i + d i j p j = d ii c i A 2 j i, j N With linear demand, we can thus characterize the NE by the price vector p NE : where p NE = D NE 1 B NE, 3 2d 11 d d 1n d 11 c 1 A D NE d 21 2d 22 =....., and d 22 c 2 A BNE =.. d n1... 2d nn d nn c n A Finally, the subgame perfect Nash-equilibrium SPNE is an equilibrium where firms price according to p NE in each stage game. 3.2 Equilibrium prices with collusion Let κ N denote the set of cartel members. Cartel members set prices to maximize their joint profit Π C = p j c j Q j, yielding the following FOC for cartel member j: j κ Q i p i, p i Q j p j, p j = p i c i,for i, j κ 4 i κ p j The FOCs for non-cartel members in equation 1 and the FOCs for cartel members in equation 4 together implicitly define the optimal prices in the industry. 5 Note that if we assume that there is no exit by consumers following a price increase, the condition holds with equality. 7

8 If demand is linear, the optimality condition for cartel member j yields: d i j + d ji p i + i κ l / κ,l N d jl p l = d i j c i A 5 i κ By combining the linear FOCs in 2 and 5, we can solve for optimal prices p : p = D 1 B, 6 where and D D 1n D =..... D n1... D nn B 1 B 2 B n B =.,D i j = { d i j + d ji if i, j κ d i j otherwise d mi c m A if i κ,b i = m κ. d ii c i A otherwise 3.3 Incentive compatibility and cartel break-down For a cartel to exist in equilibrium, deviation from the collusive agreement and subsequent deviation to the Nash equilibrium must be weakly less lucrative than continuing in the collusive agreement for all cartel members i κ. Let δ denote the firms common discount factor. Furthermore let Π C i, ΠD i and Π NE i denote firm i s profits when it part-takes in collusion C, deviates from the collusive agreement D and plays the Nash equilibrium NE, respectively. The incentive compatibility constraint ICC for i κ is given by where 1 1 δ ΠC i }{{} NPV of C Π D i + }{{} One-period profit from D Π C i = p i c i Q i p i,p i δ 1 δ ΠNE i }{{} NPV of NE for all subsequent periods is firm i s profit when all firms play the collusive prices p given in equation 6,, 7 Π D i = p D i c i Q i p D i,p i 8

9 is firm i s profit when it maximizes its profit, given that all other firms play p, 6 and Π NE i = p NE i c i Q i p NE i,p NE i is firm i s profit when all firms play the Nash-equilibrium prices p NE given in equation 3. If each cartel member is sufficiently patient, namely if δ ΠD i ΠC i for all i κ, then the Π D i ΠNE i cartel can indeed be an equilibrium. However, if the ICC is violated for at least one cartel member, the cartel cannot be an equilibrium. 4 Cartel formation with symmetric firms Throughout the first parts of this section, I will assume that a cartel once formed can be sustained forever. In other words, I assume that the incentive to deviate is sufficiently small according to equation 7. I will return to the incentive constraint at the end of the section. The concepts of internal and external cartel stability was first introduced by d Aspremont et al A cartel is considered internally stable if no participant prefers to leave the cartel: Π j κ Π j / κ for all j κ. Similarly, it is considered externally stable if no free-rider prefers to join the cartel: Π i/ κ Π i κ for all i / κ. Of particular interest is how these stability conditions are affected by product differentiation between the cartel members and free-riders. Intuitively, if the cartel members products are closer substitutes to a free-riding firm, the free-rider can gain more by joining the cartel. However, she also benefits more from free-riding than she would with larger product differentiation, as the cartel members increased prices lead to a larger increase in the free-rider s demand. It is therefore not clear whether increased substitutionality makes the cartel more or less inclusive. Symmetry between firms implies that d ii = d j j d and d i j = d ji = d ik for all i, j,k N. Let η denote the share of sales that exit the market following a price increase, such that: 7 N 1 1 ηd = i=1 d ji = N 1d i j d i j = 6 With linear demand, firm i s optimal deviation price is given by its FOC: p D i c i d }{{} ii +A + d ii p D i + d i j p j = 0 Q i p D j i, j N i,p i }{{} p D i = 1 2 c i + Q i p D i,p i p D i d1 η N 1 j i, j N d i j p j d ii 7 Note that some exit η > 0 is necessary to ensure that demand for an all-inclusive cartel varies with the 9

10 Furthermore, let N denote the number of firms in the industry, and K the number of cartelmembers. The profit functions for the cartel and non-member i are, respectively: Π C N K = p j c A + d p j j κ N 1 1 ηd p d1 η NM + N 1 p l l κ,l j Π i = p i c A + d p i N K 1 1 ηd p NM K N 1 N 1 1 ηd p κ, f or j κ, f ori / κ, where p κ is the price set by cartel-members and p NM is the price set by non-members. Calculating first-order conditions and using that, in equilibrium, p κ = p j and p NM = p i, I find the optimal prices 8 N 1 [ c + d A ] [ ] 2N 1 K 21 η + KK 11 η 2 c p NM K = 2 [ 2N 1 N K 11 η ][ N 1 K 11 η ] [ KN K1 η 2] 8 p κ K = N 1[ c + d A ] ck 11 η N K1 η + 2N 1 K 11 η 2N 1 K 11 η p NM 9 and profits Π κ K = p κ K c A + Π NM K = p NM K c A + N K + ηk 1 N K d p κ K N 1 N 1 1 ηd p NMK K + ηn K 1 d p NM K N 1 K N 1 1 ηd p κk, f or j κ 10, f ori / κ, 11 cartel price: In equilibrium, p j = p κ, such that Q j κ = A + d p j + N 1d i j p κ Q j κ = A + d p κ 1 ηd p κ = A + ηd p κ 8 Note that when K = 1, both the cartel member s and the non-members prices are equal to the Nash equilibrium price: p κ 1 = p NM 1 = c+ A d 1+η 10

11 Both prices and profits are increasing in K, as illustrated in table 2. The model also illustrates the intuitive result that the damage caused by the cartel is increasing in the cartel share. Furthermore, the marginal damage of an increase in cartel share is also increasing as the profit functions are convex in the cartel share. Prices Profits Table 2: Prices and profits for cartel members and non-members by degree of cartelization for parameter values: A = 1000, N = 100, d = 0.2, c = 10, and η = 0.1 K N Given the profit functions in 10 and 11, I can now define cartel stability as follows: Definition 1 A cartel in a market with symmetric firms, linear demand and differentiated price competition, such that d ii = d j j d and d i j = d ji = d ik for all i, j,k N, is: Internally stable iff Π κ K Π NM K 1 Externally stable iff Π NM K Π κ K + 1 Define the net benefit from collusion as BK = Π κ K Π NM K 1. As K is an integer, optimal cartel size is the maximum K such that BK is positive. Table 3 illustrates how the net benefit of collusion as a function of K evolves for different parameter values. The figures in the top-right and bottom-left panels show that the level of marginal costs and price sensitivity of consumers have little impact on the shape of the benefit-curve. Furthermore, they have little effects on the optimal cartel size, that remains relatively constant at approximately six firms. As discussed above, as market exit becomes small, the profit of an all-inclusive cartel becomes nearly independent of the cartel price p κ. Therefore, the benefit curve may increase for a sufficiently large K, making full cartelization optimal. This is shown in the bottom-right panel of the table. Perhaps most interesting is the effect of changing the industry size N, shown in the topleft panel of table 3. An important result of the model is that full cartelization is always optimal when the industry is sufficiently concentrated. Note that, for a given market exit 11

12 Industry size Marginal cost Price sensitivity Market exit Table 3: Net benefit from collusion BK sensitivities for parameters N, c, d and η. Standard parameter values: A = 1000, N = 8, d = 0.2, c = 10, and η = 0.1 η, a smaller industry size implies that the cross derivatives d i j, and consequently diversion ratios, 9 increase. In other words, when the industry becomes more concentrated, firms mechanically become closer substitutes in this model. Consequently, it may be closeness in competition, rather than industry size that determines the optimal cartel share of firms. The figure also shows that, given that the industry is sufficiently large, the optimal cartel size remains relatively stable at only 5-6 firms. Table 4 illustrates how the optimal cartel share and diversion ratios, as well as the optimal cartel size relates to different 9 Another measure of closeness in competition is the the diversion ratio. The diversion ratio between firms i and j is defined as D i j Q j p i Q i p i = d ji d. Intuitively, firm i looses customers when it raises its price. The diversion ratio can hence be thought of as the share i s lost customers that buy from firm j instead. 12

13 Diversion ratios and optimal cartel share Optimal cartel size Table 4: Diversion ratio, optimal cartel share and size of stable cartel K for different industry sizes N with symmetric product differentiation. Parameter values: A = 1000, d = 0.2, c = 10, and η = 0.1 industry sizes. As the figure shows, the diversion ratio and optimal cartel share are closely related this model. However, the optimal cartel size remains fairly constant at five to six firms as industry size increases. Using the data collected on US cartels operating between by Hay & Kelley 13

14 1974, I find an indication of whether the theoretical model fits reality. This comparison is illustrated in table 5, where I compare the 33 observations of cartel share and industry size with the theoretical predictions. Note that the theoretical predictions only are an illustration for one set of parameters. Despite being a stylized example, the model seems coincide relatively well with real-life cartel shares. Even though this observation hardly proves the dynamics of the model, it may give an indication of the accuracy of its predictions. Indeed, Hay & Kelley 1974 only observe full cartelization for small industries. Furthermore, the cartel share appears to be decreasing with industry size, as do the predicted values. Cartel share and industry size Table 5: Distributions of cartel share and industry size, based on data from Hay & Kelley 1974 and theoretical model with parameter values: A = 1000, d = 0.2, c = 10, and η = 0.1 So far in this section, I have assumed that a cartel, once formed, remains for an infinite number of future periods. For an optimal cartel size K, it is however also necessary that the cartel members have no incentive to deviate from the collusive arrangement. If the incentive compatibility constraint does not hold for the cartel member, the cartel of size K cannot be a feasible equilibrium. Let δ min K ΠD K Π κ K Π D K Π κ 1, 12 where Π D is the profit a cartel member earns when deviating and using that Π κ 1 = Π NM 1. δ min is therefore the smallest level of patience that cartel members may have, 14

15 such that the incentive compatibility constraint is satisfied. If δ min δ, the cartel of size K is not incentive compatible, and consequently cannot be an equilibrium. Note that δ min 1, as Π κ K Π κ 1 for all K N. 10 Thus, a cartel is always incentive compatible if firms are sufficiently patient, i.e. when δ δ min K. It remains to be shown how δ min is affected by the parameter values of the model, and perhaps most importantly how it is affected by K. The figure in Table 6 illustrates the required level of patience for different values of K, and shows how these results are sensitive to differing parameter values for industry size and market exit. 11 The results suggest that δ mink K 0, such that δ min is largest with full cartelization. Furthermore, as long as market exit η is sufficiently large, δ min remains relatively low, as shown in the bottom panel of the figure. Thus, even with full cartelization, the ICC is likely to hold. Consequently, the conditions for cartel stability will determine optimal cartel size. 10 See table 2 for a numerical example. The proof of Π κk 0 for all K is analogous to the proof in Selten 1973, and is therefore omitted here. 11 Changing the other parameter values has little impact on δ min. I have therefore omitted the figures. 15

16 Industry size Market exit Table 6: Values for δ min sensitivities for parameters N and η. Standard parameter values: A = 1000, d = 0.2, c = 10, N = 100 and η =

17 5 Conclusions I have presented a simple model of endogenous cartel formation, with symmetric firms with linear costs and quadratic profit functions. I show that closeness in competition is the most important parameter for defining the optimal size of a stable cartel. Furthermore, I find that cartel size remains relatively stable at five to six firms, as the relative pay-off from free-riding increases faster in cartel size than the pay-off from collusion irrespective of industry size. Finally, I find that if firms are sufficiently patient, the stable cartel is incentive compatible in the sense that cartel members do not have an incentive to deviate from the collusive agreement. The model also illustrates the intuitive result that the damage caused by the cartel is increasing in the cartel share. Furthermore, the marginal damage of an increase in cartel share is also increasing. This model serves as a useful benchmark for collusion with price differentiation. However, much work remains to be done to make the model more realistic. The following list summarizes work yet to be done and important extensions of the current model, that I may wish to explore in future versions of the model: Separate the effect of closeness in competition from industry size. This may require a model with heterogeneous firms. Introduce renegotiation/re-cartelization. Does this change cartel formation and the main results of the simple model? Coordinated effects of mergers: What kinds of mergers are most harmful in terms of merging and non-merging parties characteristics? 12 Include competition policy in model, e.g. through risk of detection. Introduce other membership rules for the first stage of the game. Allow for multiple coalitions. This is trivial for the symmetric case, but may be important for a model with heterogeneous firms. 12 An advantage of not assuming full cartelization is that there is no need to define a market and which firms should be included in order to determine coordinated effects. This would correspond well with current models for measuring unilateral effects. 17

18 References Barrett, S. 1994, Self-enforcing international environmental agreements, Oxford Economic Papers pp Bos, I. & Harrington, J. E. 2010, Endogenous cartel formation with heterogeneous firms, The RAND Journal of Economics 411, Compte, O., Jenny, F. & Rey, P. 2002, Capacity constraints, mergers and collusion, European Economic Review 461, d Aspremont, C., Jacquemin, A., Gabszewicz, J. J. & Weymark, J. A. 1983, On the stability of collusive price leadership, Canadian Journal of economics pp Deneckere, R. & Davidson, C. 1985, Incentives to form coalitions with bertrand competition, The RAND Journal of economics pp Donsimoni, M.-P. 1985, Stable heterogeneous cartels, International Journal of Industrial Organization 34, Donsimoni, M.-P., Economides, N. S. & Polemarchakis, H. M. 1986, Stable cartels, International economic review pp Escrihuela-Villar, M. 2008, On endogenous cartel size under tacit collusion, Investigaciones económicas 323. Escrihuela-Villar, M. 2009, A note on cartel stability and endogenous sequencing with tacit collusion, Journal of Economics 962, Griffin, J. M. 1989, Previous cartel experience: Any lessons for opec?, in Economics in theory and practice: An eclectic approach, Springer, pp Harrington, J. E. 2006, How do cartels operate?, Foundations and Trends in Microeconomics 21, Hay, G. A. & Kelley, D. 1974, An empirical survey of price fixing conspiracies, The Journal of Law and Economics 171, Selten, R. 1973, A simple model of imperfect competition, where 4 are few and 6 are many, International Journal of Game Theory 21, Yi, S.-S. 1997, Stable coalition structures with externalities, Games and economic behavior 202,

19 A Finding prices for symmetric firms A.1 First-order conditions for non-cartel members Firm i maximizes its own profit: N K 1 Π i = p i c A + d p i 1 ηd p NM K N 1 N 1 1 ηd p κ, f ori / κ, subject to it s price p i, and using that p i = p NM in equilibrium. The FOC becomes: N K 1 p NM c }{{} d + A + d p NM 1 ηd p NM K N 1 N 1 1 ηd p κ = 0 Q i }{{} p i Q i 2d p NM N K 1 1 ηd p NM = cd + A + K N 1 N 1 1 ηd p κ 2N 1 N K 11 η p NM = N 1c + A d + K1 ηp κ 13 A.2 First-order conditions for cartel members The cartel maximizes joint profit: Π C N K = p j c A + d p j j κ N 1 1 ηd p d1 η NM + N 1 p l l κ,l j, f or j κ subject to each of the cartel members prices. The FOC for cartel member j is: N K p κ c }{{} d + A + d p κ N 1 1 ηd p K 1 NM N 1 1 ηd p d1 η κ +p κ ck 1 = 0 N 1 Q j }{{}}{{} p j Q j Q l p l κ,l j j K 1 2d p κ N 1 1 ηd p K 1 N K κ p κ c 1 ηd = cd + A + N 1 N 1 1 ηd p NM [ 2N 1 K 11 η p κ = N 1 c + A ] + N K1 ηp NM ck 11 η d p κ = N 1[ c + d A ] ck 11 η N K1 η + 2N 1 K 11 η 2N 1 K 11 η p NM 14 19

20 A.3 Solving for optimal prices By substituting for p κ from 14 into 13, I find the optimal price for non-members as a function of K: 2N 1 N K 11 η pnm = N 1c + A d + K1 η N 1[ c + A d ] ck 11 η 2 [ N 1 K 11 η ] N K1 η +K1 η 2 [ N 1 K 11 η ] p NM [ [ ] [ ] 2N 1 N K 11 η 2 N 1 K 11 η pnm = N 1 c + A ] 2 [ N 1 K 11 η ] d [ [ +K1 η N 1 c + A ] ] ck 11 η d +KN K1 η 2 p NM p NM = 2 [ 2N 1 N K 11 η ][ N 1 K 11 η ] [ KN K1 η 2] p NM = [ N 1 c + A ] [ 2 [ N 1 K 11 η ] ] + K1 η + KK 11 η 2 c d N 1 [ c + d A ] [ ] 2N 1 K 21 η + KK 11 η 2 c 2 [ 2N 1 N K 11 η ][ N 1 K 11 η ] [ 15 KN K1 η 2] Thus, equations 14 and 15 characterize the optimal prices for cartel members and non-members, respectively. 20