On the price effects of the intensity of competition and the number of competitors.

Transcription

1 On the price effects of the intensity of competition and the number of competitors. Marc Escrihuela-Villar Universitat de les Illes Balears April, 2015 Abstract This paper considers a theoretical model where firms are able to coordinate on distinct output levels than the unrestricted joint profit maximization outcome. We show that, in our model, the degree of collusion (captured by the common discount factor) and the number of firms are only imperfect substitutes in order to maximize consumer surplus. The main implication of this finding is that policy measures devoted to increase the number of competitors are more effective when the degree of collusion is small whereas the efforts to discourage tacit collusion should be applied especially in markets with a large number of firms. JEL Classification: L11; L13; L41 Keywords: Degree of collusion; Number of competitors; Trigger strategies I am grateful to Ramon Faulí-Oller for his advice and encouragement. Financial support by the Ministerio de Educación y Ciencia through its project Análisis de Economía Pública y Bienestar (Ref: ECO ) is gratefully acknowledged. The usual disclaimer applies. Mailing address: Departamento Economia Aplicada. Edificio Jovellanos Ctra. Valldemossa km Palma de Mallorca, Baleares. Spain. Tel Fax: marc.escrihuela@uib.es. 1

2 1 Introduction The analysis of the degree of competition in a market is of major concern for economists and policy makers. Non-competitive behavior includes, but is not limited to, collusive agreements and the market structure given for instance by a small number of firms in an industry. 1 A common problem for antitrust authorities is thus the trade-off in welfare terms between the number of competitors and the degree of competition, a typical example being the measures designed to promote more competitive behavior in a market that may lead to the exit of some firms. This problem may very well be a component of a phenomenon that has been referred to as the antitrust paradox by Bork (1978) where he basically warns against the potential effects of competition law in discouraging competitive behavior. This is also reflected in the fact that regulators with limited resources must often decide whether to allocate them to measures that facilitate entry (or that rigorously scrutinize mergers) or to measures designed to promote more competitive behavior by existing firms. Unfortunately, the standard one-shot quantity-setting and homogeneous product model does not provide a strong answer in this respect. For instance, in the Cournot model the equilibrium price is a decreasing function of the number of firms, but the concept of degree of collusion is often not considered as a determinant of the welfare consequences of the number of firms. 2 The Industrial Organization analysis of tacit collusion in quantity-setting supergames, on its side, has often considered the possibility of a collusive agreement to implement the joint monopoly outcome. Then, the potential sustainability of the collusive agreements is reflected in the minimum discount factor above which joint profit maximization could be sustained and it is obtained that reducing the number of competitors facilitates collusion (see for instance Vives (2001)). This exercise, however, does not allow us to consider the aforementioned trade-off since a knife-edge equilibrium is obtained where collusion (namely, the monopoly output) only exists if the 1 Other sources of imperfect competition arise for example when considering models where products are differentiated, such as in Tirole (1988, Chapters 2 and 7) and Anderson et al. (1992). 2 In this setting Salant, Switzer and Reynolds (1983) show that collusive agreements are (generally) not profitable because non participating firms react to collusion by expanding their production. 2

3 discount factor is large enough. Some other papers, though, have considered the degree of competition in linear supply schedules characterizing for example the competitiveness of the market in terms of the slope of the supply curves (see for instance among others Klemperer and Meyer (1989) or Vives (2011)). These papers, however, do not analyze the welfare effects of the trade-off mentioned above. On the other hand, in a recent paper with which our work is most closely related to, Menezes and Quiggin (2012) use the perceived slope of competitors supply functions as the intensity of competition to show that total welfare increases monotonically with the intensity of competition and the number of competitors. The present work differs from their, however, in some important ways. To begin with, Menezes and Quiggin consider the slope of the supply curves that make up their strategy space and the Cournot and Bertrand solutions arise as special cases. Consequently, a collusive market where firms might maximize joint profits is not considered. In contrast, we consider a model where firms maximize joint profits but where coordination on distinct output levels than the joint profit maximization outcome may also arise. Second, Menezes and Quiggin examine the relationship between the intensity of competition and the number of firms in a numerical analysis around the Cournot equilibrium, whereas the main goal of the present work is to shed some light on the price (as a welfare measure) implications of the aforementioned trade-off for the whole possible range of both market characteristics. To that extent, we develop a multi-period oligopoly model with homogeneous and quantity-setting firms using subgame perfect Nash equilibria henceforth, SPNE as solution concept. It is well known that our repeated game setting exhibits multiple SPNE collusive agreements. Therefore, to select among those equilibria and following Verboven (1997) and Escrihuela-Villar (2008a), we adopt the particular criterion of (i) restricting strategies to grim trigger strategies, and (ii) choosing the firm s profit maximizing allocation for each possible value of the discount factor, which means that firms coordinate their output even when the joint profit maximization agreement does not correspond to a SPNE of the repeated game. As a consequence, the discount factor can be interpreted as the degree of collusion of the market. In this manner, our model allows us to treat the 3

4 degree of collusion as a continuous variable in such a way that the existence of collusion is not dependent on the number of firms involved in the agreement. 3 After checking that market price monotonically decreases with the intensity of competition and the number of competitors, our main result shows that the market price sensitivity (i) with respect to the degree of collusion increases with the number of firms, and (ii) with respect to the number of firms decreases with the degree of collusion which implies, for example, that price increases due to horizontal mergers, and consequently most probably also their profitability, depend negatively on the pre-merger degree of collusion. Another direct implication of this result is also that the trade-off between the number of competitors and the degree of competition should be taken into account since both market characteristics are not perfect substitutes in welfare terms. Finally, and consistent with the previous result, it is also shown that the market price is less sensitive to changes in the degree of collusion than in the number of firms only when the degree of collusion is small enough since, in this case, most of the welfare gains from extra competition arecapturedwithanincreaseinthenumberoffirms. Although we focus on a collusive oligopoly model from a theoretical viewpoint, our paper is also motivated by real cases. In this sense, it is well known that one motivation to merge may be associated with the possibility to raise prices. 4 Empirical evidence is copious suggesting that merger incentives are consequently related to the pre-merger degree of competition, showing that in industries characterized by cartelist activities firms resort to mergers when cartels become a less viable alternative. In this sense, Bittlingmayer (1985) shows that during the Great Merger Wave between 1898 and 1902, mergers in several 3 We implicitly assume thus that the entrance of a new firm or a merger between firms does not modify the degree of collusion. Even though this might not be accurate, we have particularly in mind situations in which the competitive intensity is exogenous at the firm level since firms coordination is based, for example, on historical reasons, local legislation or a common corporate culture. Anyway, future research should address this limitation. 4 Accordingly, the literature on horizontal merger assessment has mainly focused on coordinated effects (if a merger induces rivals to alter their strategies, resulting in some form of price-increasing coordination) or unilateral effects (the merged firm charges a higher price) of horizontal mergers. 4

5 (cartelized) industries in the United States namely cotton oil, sugar, cast iron pipe, oil, meat packaging or steel and railroading, appear to have been the result of antitrust actions taken against cartels in those industries after the Sherman Act was passed. It is also well known that the Restrictive Trade Practices Act (1956), which outlawed cartels, triggered awaveofmergersintheunitedkingdom. Beforethat,mergerswerefewandfarbetween (see Symeonidis, 2002). In the same line, Evenett et al. (2001) examine the samples of international cartels during the 1990s, and observe that mergers are among the different measures adopted by firms for survival in collusive industries where cartel formation is restricted. A similar trend regarding the substitutability between merging and colluding has also been noticed in Germany where Neumann (2001) argues that German industries like cement, food processing machine building etc. adopted cartelist activities in order to attain monopolistic power only when mergers were not possible. One can argue thus that a stricter legal enforcement against a cartel increases the relative cost of collusion as compared to merger and consequently firms resort to mergers. On the other hand, the empirical relationship between firms capacity to raise prices within a collusive agreement and the market size has also been considered. In this regard, the 2011 Federal Sentencing Guidelines Manual (Chapter 2, Part R) says Another consideration in setting the fine is that the average level of mark-up due to price-fixing may tend to decline with the volume of commerce involved. However, as Connor and Lande (2005) emphasize, the commentary presents no conceptual or empirical justification. In fact, it is precisely the opposite what Bolotova (2009) obtains estimating the impact of cartel characteristics and the market environment on the magnitude of overcharges attained by cartels in different geographic markets. In her work it is shown that even though the majority of cartels operated in an environment characterized by low buyer concentration (i.e. many buyers), a positive relationship between the level of overcharge and the size of cartelized markets (i.e. volume of affected sales) is suggested. The rest of the paper is structured as follows. In Section 2, we present the model and analyze the sensitivity of the market price with respect to the degree of collusion and the number of firms. A numerical example is also provided to illustrate the results. 5

6 Section 3 tests the robustness of our results using an optimal punishment the stickand-carrot strategies proposed by Abreu (1986, 1988) and establishes that the main results continue to hold. We conclude in Section 4. All proofs are grouped together in the appendix. 2 The model and results We consider an industry with N 2 firms playing an infinitely repeated game at dates t =1,..., with a common discount factor δ [0, 1). Eachfirm produces a quantity of a homogeneous product with production costs that we normalize to zero. The industry inverse demand is given by the piecewise linear function, p(q) =max(0, 1 Q) where Q is the industry output, p is the output price. We assume that firms behave cooperatively so as to maximize their joint profits. Following Friedman (1971), we restrict attention to the well known grim trigger strategies. We assume perfect monitoring so that cheating in period t 1 is detected by each firm in period t in such a way that if a firm defects, the other firms revert from there on to the static Nash-Cournot quantity, and obtain a profit that we denote by Π N.Foreachfirm, let q denote the output corresponding to collusion. Then, the profit ofafirm is given by the function Π c =(1 Q)q. The fact that each firm produces q corresponds to a SPNE if and only if the following condition is satisfied for each firm: Π c 1 δ Πd + δπn 1 δ, (1) where Π d denotes the one period profit from an optimal deviation. There are many (SPNE) collusive output vectors that satisfy the system of inequalities in condition (1) above. As in Verboven (1997) and Escrihuela-Villar (2008a), we select an equilibrium from this large set assuming that if δ exceeds a certain critical level, the set of SPNE vectors is not a binding constraint, and the distribution of output is the symmetric distribution of the output of a monopolist. If δ is below that critical level, then the set of SPNE vectors is a binding constraint, and the distribution of output is the solution to any equality constraint in (1). Let us denote the above mentioned critical level of the discount 6

7 factor by δ. It can be verified that δ = (1+N)2. The model that we have described 1+N(6+N) thus exhibits equilibria where coordination on distinct output levels exists. Then, the equilibrium quantity of each firm q depends on δ. Note that coordination might not be complete according to whether firms agree or not on the monopoly outcome. Therefore, we consider that collusion is partial (respectively, full) whenever δ (0, δ) (respectively, δ δ). It can be easily checked that the quantity produced by each firm in the collusive equilibrium, that we denote by q(n,δ), isgivenby q(n,δ) = δ(n 1)(3+N) (1+N) 2 (1+N)(δ(N 1) 2 (1+N) 2 ) 1 2N if δ< δ if δ δ. Consequently, the market price is given by the following expression: δ N(2+N 2δ+3Nδ) 1 if δ< δ (N 1) p(n,δ) = 2 (1+N)δ (1+N) 3 1 if δ δ 2 (2) Remark 1 The present model encompasses the Cournot case if δ =0,andasδ δ we obtain the monopoly equilibrium. Hence, a variation in the discount factor when collusion is partial can be interpreted as a variation in the degree of collusion of the market. Admittedly, discount factors are primitives and although we build a direct link to collusive behavior, the reader might feel more comfortable when such link is made explicit and based on behavioral assumptions. However, we use the discount factor as a direct metric to measure collusion as long as conceptually the present model can be directly linked, for instance, to a static model where firms strategy set is q [ 1, 1 ] (a quantity in the interval between full collusion, 2N N+1 δ δ and the standard Cournot equilibrium, δ =0respectively). The present formulation thus allows us to parameterize the ability to collude presenting additionally a justification of the collusive actions chosen by firms. 5 5 Note also that in a collusive equilibrium firms are willing to cut production compared to the Cournot equilibrium ( q(n,δ) δ ( Πc δ > 0). < 0) and one can also check that profits of each firm are enhanced by cartelization 7

8 To study the welfare effects of the degree of collusion and the number of firms, we use total consumer surplus (namely, the market price) as a welfare measure. In this sense, several recent papers call for antitrust agencies to use a consumer surplus standard rather than a total welfare standard (see for instance Pittman (2007)). Often, the argument against a consumer welfare standard is that it implies a tolerance for monopsony. However, it is well known that this is more likely to occur in markets for intermediate goods and we focus here in a final good collusive market. Moreover, enforcement practice in most legislations (for instance in the guidelines on the assessment of horizontal mergers of the EU) is closer to a consumer welfare standard. 6 of the market price by E N (N,δ) p(n,δ) N the market price by E δ (N,δ) p(n,δ) δ δ p(n,δ). N p(n,δ) Lemma 1 N 2 and 0 <δ< δ, E N (N,δ) < 0 and E δ (N,δ) > 0. We denote the number of firms elasticity and the degree of collusion elasticity of Lemma 1 states that both E N (N,δ) and E δ (N,δ) have the intuitive sign since the market price decreases with N and increases with δ. In order to examine how the degree of collusion affects the gains from increasing the number of competitors and vice versa, that is how N affects the price effects of a change in δ, we use the derivatives of E N (N,δ) and E δ (N,δ) with respect to δ and N respectively. The following result identifies these relationships. Proposition 1 E δ (N,δ) increases with N and E N (N,δ) decreases with δ. Proposition 1 shows that when firms have the possibility to partially coordinate their outputs: (i) the effect on the market price of a change in the degree of collusion is greater when there are many firms than when there are few, and that (ii) the effect that mergers or the entry of new firms have on the market price is higher when the degree of collusion is 6 Additionally numerical simulations show that the results of the present paper also extend to the case where total welfare is used instead of consumer surplus. Unfortunately, the expression for total welfare cannot be further simplified in order to obtain explicit results. More details are available from the author upon request. 8

9 smaller. The intuition of the first part of Proposition 1 comes from the fact that when the number of firms is small, the intensity of competition is mostly captured by the number of firms and therefore the market price is proportionally less affected by a change in the degree of collusion. Analogously, the intuition of the second part is that with a high degree of collusion, most of the gains from extra competition are captured by a decrease in δ and as a consequence with a large δ the market price is less sensitive to changes in N. In other words and roughly speaking, the number of firms in the market is not so important in welfare terms when the market is already very collusive. Corollary 1 The increase in the market price due to an horizontal merger depends negatively on the pre-merger degree of collusion. Our results thus corroborate the empirical evidence provided in the introduction suggesting that in industries characterized by cartelist activities firms resort to mergers when cartels become a less viable alternative. 7 Corollary 1 thus states that, when the market is (partially) collusive, the effect of mergers on the market price is proportionally smaller than in absence of collusion, decreasing therefore merger profitability. Horizontal mergers thus would appear most likely in relatively competitive markets, since where firms are already relatively cooperative their private profit gain (derived from a price increase) is no longer likely to be high. Summarizing, we have analyzed the price sensitivity to two different dimensions of non-competitive behavior and we obtained that, ceteris paribus, when the market is less competitive in one dimension the market price is less sensitive to a change in the other dimension. A natural question arises at this point. Is the market price always more sensitive to one dimension than to the other? The comparison of both elasticities confirms that the answer to this question is no and reaffirms the intuition behind Proposition 1: The substitutability between N and δ in order to increase welfare (namely, to reduce the market price) is not perfect and depends crucially on the degree of collusion. 7 Note that this is also coherent with the results in Rodrigues (2001) and Escrihuela-Villar (2008a) showing that merger private profitability is increasing (among other factors) in the expected competitive intensity. 9

10 Proposition 2 N 2 there exists δ (0, δ) such that E N (N,δ) >E δ (N,δ) if δ<δ and E N (N,δ) <E δ (N,δ) if δ>δ. Proposition 2 states that when the degree of collusion is small enough, the market priceismoresensitivetochangesinn than to changes in δ. The reverse is true when the degree of collusion is high enough. Consequently, both market characteristics considered (N and δ) affect welfare and might be substitutes in policy terms since either a large N or a small δ would be sufficient to ensure very competitive markets. The present analysis shows, however, that their degree of substitutability is not perfect and depends on the respective value of the market characteristics. Therefore, our results also suggest for instance that in order to reduce the market price, efforts should be especially focused on regulatory measures designed to increase the number of competitors when the degree of collusion is small. In the same line, it seems plausible to recommend that more priority should be given to fight tacit collusion in markets where the number of firmsisrelatively large. 8 Numerical simulation A numerical simulation is provided to better illustrate our results. Table 1 gives the market price for different values of δ and N whereas Tables 2 and 3 show the percentage 8 As mentioned in the introduction, it could also be argued that if the number of firms increases, it becomes more difficult to sustain collusion. Escrihuela-Villar (2008b) shows, however, that this is not always true, for instance when not all the firms participate in the collusive agreement. Note furthermore that we deal here with the case where the existence of collusion depends only on the value of δ regardless of N. 10

11 in which the market price changes due to an increase in δ and N respectively. Market price p(n,δ) δ\n δ Table 1 Percentage rise in p(n,δ) if δ increases δ\n to to to Table 2 Percentage drop in p(n,δ) if N increases δ\n 2 to 3 3 to 4 4 to 5 5 to to Table 3 We can now thus check numerically the results in Proposition 1. If we compare the cells in each row of Table 2 we can observe that when δ increases, the percentage in which the market price increases is larger as the number of firms increases, or in other words, the price effects of a change in the degree of tacit collusion are larger in a market with a large 11

12 number of firms. On the other hand, comparing the cells in each column in Table 3, we note that the drop in the market price (due to an increase in the number of firms) decreases with δ or,whatisthesame,thepriceeffects of a change in N are larger as the degree of collusion decreases. Equivalently, for a given initial N, theriseinthemarketprice due to a merger decreases with the degree of collusion. 9 A concrete example may help to highlight the importance of our results: To evaluate competitive concerns, the Antitrust Agencies apply a wide range of analytical tools. For instance, market concentration is often one useful indicator of possible anticompetitive effects of a merger. Then, the lessening of competition resulting from a merger is more likely to be substantial the larger the post-merger market concentration and its change due to the merger. More precisely, the Horizontal Merger Guidelines of the U.S. Department of Justice (revised in 2010) state that the mergers resulting in highly concentrated markets (Herfindahl-Hirschman Index, hereafter HHI, above 2500) that involve an increase in the HHI of between 100 points and 200 points potentially raise significant competitive concerns and often warrant scrutiny. Consequently for example, in our symmetric collusive model and assuming away any cost synergies, if N =4and two firms merge the HHI changes from 2500 to Therefore, this merger would be likely to be challenged. From (2) it is immediate to check that the price increase due to the merger though is quite moderate (around 5%) if δ 0.5. Conversely, imagine that N =8and two firmsmerge. Inthiscase, thehhichanges from 1250 to 1428 and therefore, according to the U.S. Guidelines, the merger is unlikely to have adverse competitive effects and ordinarily requires no further analysis. The price increase due to the merger, however, might be above 12% if δ is close to zero. Admittedly, there are also other alternative methodologies for measuring the magnitude of potential unilateral effects of a merger. Merger simulation models have been used for years to estimate potential unilateral effects of mergers, see for instance Werden (1996), O Brien and Salop (2000) or Farrell and Shapiro (2010) that have proposed the use of different price pressure indices to aid this analysis. Interestingly enough, the U.S. Merger Guidelines of 9 Note also that the comparison between rows in Table 1 and columns in Table 2 would tell us about the convexity of p(n,δ) which is not the purpose of the present work. 12

13 2010 state that mergers should not be permitted to create or enhance market power or to facilitate its exercise. (p. 5). This statement thus appears to represent a policy choice where pre-merger market power is not considered either when price indices are considered in order to assess merger effects. Therefore, the only relevant policy concern seems to be to ensure that mergers will not enhance market power. This is explicitly observed for example in the use of the prevailing price as the starting point for the hypothetical monopolist pricing tests with the risk for instance of falling victim to the famous Cellophane Fallacy Extensions 3.1 Optimal Punishment We test whether our results hinge on the assumption of an oligopoly restricted to trigger strategies. It is then natural to consider a N firm oligopoly playing a set of strategies that are less grim than the trigger strategies. We consider here the two-phase output path (with a stick-and-carrot pattern) presented by Abreu (1986, 1988) in an industry of N > 2 collusive firms, indexed by i =1,..., N. As in Section 2, the strategy space consists of a sequence of decisions rules, describing each player s action as a function of the past history of the play. Then, a pure strategy for firm i is an infinite sequence of functions {Si} t t=1 with St i : Pt 1 Q where Pt 1 is the set of all possible histories of actions of all firms up to t 1, with typical element σ τ j, j =1,..., N, τ =1,..., t 1, and Q is the set of output choices available to each firm. Following Abreu (1986, 1988), we restrict our attention to the case where each firm is only allowed to follow a stickand-carrot strategy. In other words, we assume that if a deviation from the collusive 10 In the U.S. v. E.I. du Pont de Nemours & Co. (the cellophane case, 1957), du Pont produced 85 per cent of all cellophane. However, the Court ruled that other packaging materials were substitutes at prevailing market prices, and thus the market was not limited to cellophane. Such an analysis was flawed since profit-maximizing monopolists generally raise price to the point where other products become close substitutes. 13

14 agreement occurs, then all firms expand their output for one period so as to drive price below cost and return to the most collusive sustainable output in the remaining periods, provided that every player went along with the first phase of the punishment. Let q and q p denote the output produced by each firm in a collusive and in a punishment phase respectively. {Si} t t=1 can be specified as follows. At t =1, S1 i = q, whileatt =2, 3,... q if σ τ j = q for all j =1,..., N and τ =1,..., t 1 Si(σ t τ j )= q if σ τ j = q p for all j =1,..., N and τ = t 1 (3) q p otherwise. Under the conditions specified in Abreu (1986), all firms producing q in each period can be sustained as a SPNE of the repeated game with the strategy profile (3). As in Section 2, a multiplicity of equilibria is obtained. Then, for given values of N and δ, wechoose the profit maximizing allocation for the firms. Straightforward calculations show that if N>3+2 2 ' 5.83 the cutoff ofthediscountfactoris δ = (N 1)2. Let us concentrate on (N+1) 2 this case. As in the previous section we can easily calculate the optimal level of quantities produced in equilibrium by each firm and the market price for the case of partial collusion. 1 They are respectively: q(n,δ) = N 2 δ 1 1 and p(n,δ) = N( δ+1 1). Proposition 3 When N firms play the strategies described in (3), E δ (N,δ) increases with N and E N (N,δ) decreases with δ. We can also compare both elasticities to obtain the following result. Proposition 4 When N firms play the strategies described in (3), there exists δ 0 (0, δ) such that E N (N,δ) >E δ (N,δ) if δ<δ 0 and E N (N,δ) <E δ (N,δ) if δ>δ 0. Propositions 3 and 4 parallel Propositions 1 and 2 assuming an optimal punishment instead of trigger strategies. In conclusion, we have studied a more severe punishment and established that the results of Section 2 continue to hold. 14

15 3.2 Coefficient of cooperation An alternative attempt to allow for imperfect collusion involves assuming that each firm maximizes the sum of its own profit plustheaverageprofits of the remaining firms in such np awaythatagenericfirm i maximizes Π i (q i,q i )+ α Π n 1 j (q j,q j ) where no conjecture j6=i about the reaction of the rivals is expected. In this model, α it is assumed to be constant and symmetric and interpreted as the severity of competition in such a way that α =0 indicates the standard Cournot case and α = n 1 the monopoly case. Following Cyert and DeGroot (1973), we can denote this formulation by coefficient of cooperation model (CC model). 11 It is a standard exercise to obtain the quantities produced in equilibrium by each firm in the CC model with the linear demand function given in the previous section: 1 q(n,α) = 1+α(N 1) + N Consequently, the market price is given by the following expression: p(n,α) = 1+α(N 1) 1+α(N 1) + N Consequently, we can denote the number of firms elasticity of the market price by E N (N,α) p(n,α) N p(n,α) α N p(n,α) α p(n,α) and the degree of collusion elasticity of the market price by E α(n,α). Then, in a one-shot game where firmscareabouttheprofits of their rivals, Proposition 5 E α (N,α) increases with N and E N (N,α) decreases with α. In addition, N 2 there exists α such that E N (N,α) >E α (N,α) if α<α and E N (N,α) < E δ (N,α) if α>α. Proposition 5 confirms that the results obtained in the previous section also carry over to the CC model. 11 Admittedly α<0 might seem counterintuitive at first sight. We follow here however the reasoning in Matsumura and Matsushima (2012) stating that objective functions are often based on relative performance since, for instance, evaluations of management activities are also based on their relative performances as well as the absolute performances. Furthermore, some experimental works already pointed out reciprocal or altruistic behavior (see for instance Cason et al., 2002) that is closely related to firms objective functions based on relative performance (leading to either positive or negative α). 15

16 4 Concluding comments The purpose of the present paper is to analyze the effect of the number of firms and the degree of collusion on the market price as well as how both market characteristics might reinforce each other s effect on welfare measured by consumer surplus. The current work adds to the existing literature by examining these questions in an already collusive environment, a problem that, to the best of our knowledge, has not been extensively considered. We show that policy interventions regarding the number of firms or the degree of collusion should take into account that both market characteristics lose its degree of substitutability in welfare terms when they reach extreme values. A possible consequence of ignoring these findings is the opportunity cost of implementing ineffective policies. This paper thus highlights that the market structure or the intensity of collusion selected by firms can drastically change the potential trade-off between the number of competitors and the degree of competition. Finally, the limited context of the present model is acknowledged: to analyze real-world cases of collusion, cost asymmetries, the existence of non collusive firms or a wider range of demand functions should also be considered. These issues have been left for future research. Appendix ProofofLemma1.From(2)weobtainthat E N (N,δ) = N((1+N)4 +2(1+N) 2 (1+(N 6)N)δ (N 1) 2 (3+N(2+3N))δ 2 ) which is negative if δ =0. (1+N) 5 2(N 1)(1+N) 4 δ+(n 1) 3 (1+N)(1+3N)δ 2 Atthesametimewecancheckthattheonlyrootonδ of E N (N,δ) = 0 in the interval 0 < δ < 1 is δ = other hand, E δ (N,δ) = q (1+N)2 (1+(N 6)N) +2 1+N 3 (16+N(30+16N+N 4 )) > δ. On the (N 1) 2 (3+N(2+3N)) (N 1) 4 (3+N(2+3N)) 2 4(N 1)N(1+N) 2 δ (1+N) 4 2(N 1)(1+N) 3 δ+(n 1) 3 (1+3N)δ 2 which is obviously equal to 0ifδ =0. Then, in order to prove that the degree of collusion elasticity of the market price is positive we only have to check that it increases with δ, thatis E δ(n,δ) = δ 4δ((1+N) 5 (3N 1)+2(N 1) 2 (1+N) 4 δ+(n 1) 3 (1+N)(1+N(4+11N))δ 2 ) ((1+N) 4 +2(N 1)(1+N) 3 δ (N 1) 3 (1+3N)δ 2 ) 2 > 0. ProofofProposition1.FromLemma1weknowthat 16

17 4(N 1)N(1+N) 2 δ (1+N) 4 2(N 1)(1+N) 3 δ+(n 1) 3 (1+3N)δ 2 E δ (N,δ) = 4δ((1+N)5 (3N 1)+2(N 1) 2 (1+N) 4 δ+(n 1) 3 (1+N)(1+N(4+11N))δ 2 ) N ((1+N) 4 +2(N 1)(1+N) 3 δ (N 1) 3 (1+3N)δ 2 ) 2 E δ (N,δ) = > 0. We only have to check that > 0. On the other hand, E N (N,δ) = N((1+N)4 +2(1+N) 2 (1+(N 6)N)δ (N 1) 2 (3+N(2+3N))δ 2 ). FromLemma1we (1+N) 5 2(N 1)(1+N) 4 δ+(n 1) 3 (1+N)(1+3N)δ 2 know that E N (N,δ) < 0. Then, the second part of the result holds since E N (N,δ) = δ N 1 2N 4N(1 + N)(N 1)( + ) > 0. ((1+N) 2 (N 1) 2 δ) 2 ((1+N) 2 +(N 1)(1+3N)δ) 2 Proof of Proposition 2. It immediate to verify that E N (N, 0) >E δ (N,0) = 0. Also, we can check that the equation E δ (N,δ) ( E N (N,δ)) = 0 has only one root in δ given by δ δ =2 q (1+N) 4 (3 10N+8N 2 +2N 3 +N 4 ) (1+N)2 (6N+N 2 3) (N 1) 4 (3+2N+3N 2 ) 2 Proof of Proposition 3. From p(n,δ) = N δ N+(1+ andweonlyhavetocheckthat E δ(n,δ) = δ) 2 Nδ N hand, E N (N,δ) = where (N 1) 2 (3+2N+3N 2 ) δ < δ N > N( δ+1 1) we obtain that E δ(n,δ) = δ (1+N (N 1) δ) 2 > 0. On the other N(1 δ) 1 N (N 1) < 0 and E N (N,δ) N = δ δ (1+N (N 1) δ) 2 > 0. δ Proof of Proposition 4. As in the proof of Proposition 2, we can verify that E N (N,0) > E δ (N, 0) = 0. Then, the equation E δ (N,δ) ( E N (N,δ)) = 0 has only one root in δ given by δ δ 0 = 1(3 5) < δ = (N 1)2 if N>4.23 whichistruesincetheassumptionmade 2 (N+1) 2 regarding the number of firms ensures that N>5.83. Proof of Proposition 5. It is immediate to obtain that E N (N,α) = and E α (N,α) = 2 ) > 0 and also (1+α(N 1)+N) 2 E N (N,α) α (α 1)N (1+α(N 1))(1+a(N 1)+N) α(n 1)N Eα(N,α) 1. Then, for the first part = α( (1+α(N 1))(1+α(N 1)+N) N (1+α(N 1)) 2 = N( 1+2α(N 1)2 α 2 (N 1) 2 +N(2+N)) (1+α(N 1)) 2 (1+α(N 1)+N) 2 > 0. For the second part, we just have to solve the equation E α (N,α) = E N (N,α) for α to obtain that E N (N,α) >E α (N,α) if α> 1 N. References -Abreu, D. (1986). Extremal Equilibria of Oligopolistic Supergames. Journal of Economic Theory, 39(1), Abreu, D. (1988). On the Theory of Infinitely Repeated Games with Discounting. Econometrica, 56(2),

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