Collusion incentives in a mixed oligopoly. (Work in progress)

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1 Collusion incentives in a mixed oligopoly Marc Escrihuela-Villar Universitat de les Illes Balears Carlos Gutiérrez-Hita Universitat d Alacant (Work in progress) Abstract We investigate the extent to which private firms are willing imperfectly collude in a mixed quantity-setting oligopoly where an inefficient public firm fully or partially maximizes total welfare, depending on the grade of privatization. We obtain that perfect collusion might not enhance profits of private firms due to the aggressive output behavior of the public firm. The scope for collusion, however, increases with the inefficiency of the public firm. We also obtain that generally only a partial privatization scheme is advisable since the existence of a nationalized firm might discourage collusion, specially if the regulator is able to correctly anticipate that private firms cooperation depends on the degree of privatization. Finally, we also show that a production subsidy might also increase firms incentives to collude. JEL Classification: L11; L13; L41; D43 Keywords: Imperfect collusion; Mixed oligopoly; Partial privatization; Subsidies. Departamento de Economía Aplicada. Edificio Jovellanos Ctra. Valldemossa km Palma de Mallorca Baleares - España. Tel Fax: marc.escrihuela@uib.es Corresponding author. Mailing address: Departamento de Fundamentos del Análisis Económico. Campus San Vicente del Raspeig, Alicante (Spain). Tel carlosghita@ua.es. Financial support by the Ministerio de Economía y Competitividad and the Fondo Europeo de Desarrollo Regional through the project Conflicto e influencia estratégica: Los efectos de la actividades de captación de rentas en las decisiones colectivas (Ref: ECO P MINECO/FEDER) and "Strategic management on complex systems with multiplicity of agents" (Ref: MTM P) are gratefully acknowledged. The usual disclaimer applies. 1

2 1 Introduction Public or at least to some extent state-owned firms are very common in many markets such as airlines, telecommunications, railways, electricity, banking, broadcasting and education. Consequently, the literature on mixed oligopolies where a public or semi-public firm competes with one or more private profit-maximizing firms is also extensive. Different market structures like Cournot, leader-follower or price competition canbefoundinthisliterature. Manypapers have often examined these mixed oligopolies following two different strands; whether privatization or subsidization is optimal and which is the endogenous order of moves between public and private firms. A good example from the first strand is De Fraja and Delbono (1989) and subsequent literature showing that in some cases a public firm should be privatized and should maximize profits rather than welfare. The intuition is that privatization might increase welfare since with a large number of private firms the public firm must produce a very high level of output, driving private profits to a very low level. Regarding the second strand, some studies like Pal (1998) argue that an observable delay game like in Hamilton and Slutsky (1990) will create an endogenous equilibrium where the public and private firms behave like a Stackelberg leader and followers respectively. A good review of various mixed oligopoly models, including different move orders in oligopolistic games, can be found in De Fraja and Delbono (1990) and Matsumura and Shimizu (2010). The general trend in the literature on mixed oligopolies, however, has been to ignore the possibility that private firms achieve a collusive agreement. Therefore, in our opinion a proper analysis of collusive behavior in industries with nationalized firms deserves attention. To that extent, we consider a situation in which private firms are able to imperfectly collude 1 in a mixed oligopoly market were the government has two different policy measures; privatization and a production subsidy. In this setting, our focus is twofold: Firstly, to analyze the effect of the existence of a relatively more inefficient public firm in private firms incentives to achieve a collusive agreement. Secondly, to evaluate the optimal government intervention contrasting two different regulators; a myopic Social Planner (hereafter, SP) that does not consider that 1 Imperfect collusion is usually related to the degree of cooperation among firms and is different to partial collusion which refers to a market where only a subset of firms colludes. 2

3 a policy measure applied might enhance or hinder collusion, and a forward-looking SP that is able to correctly anticipate that private firms cooperation depends on the policy measure implemented. To study the above mentioned issues, we firstly develop as a benchmark model a mixed oligopoly where private firms simultaneously compete in quantities with a relatively more inefficient and partially welfare-maximizing public firm. 2 Additionally, we assume that private firms maximize the summation of its own profits plus a proportion of the profits of the other private firms. As a consequence, this proportion, that we endogenize, may be considered as the degree of coordination which implies that firms can agree on a distribution of the output quotas different to that arising from a perfect joint profit maximization agreement. This way of modeling the intensity of competition, which has received growing attention of scholars (see for instance Symeonidis, 2008), is closely related to the coefficient of cooperation defined by CyertandDeGroot(1973)andcapturestherelative performance approach that is evolutionary stable (see Vega-Redondo, 1997). It is also in line with the growing and more recent behavioral economics literature, as well as with experimental games that test that subjects are concerned with reciprocity (see for instance Fehr and Schmidt, 1999 and Charness and Rabin, 2002 respectively). Furthermore, since any combination of individually rational profits is sustainable as equilibrium of an infinitely repeated game if firms are sufficiently patient, following the reasoning of Symeonidis (2008) one can assume that firms always achieve the highest level of collusion that is sustainable given a number of exogenous parameters. As a consequence, a fall in the importance given to rivals profits might correspond to a lower critical discount factor in an infinitely repeated game and the payoff function based on relative performance might be used to parameterize the intensity of competition. Intermediate values of this parameter might represent imperfect collusion and may be justifiedbyreferencetosomeimplicitdynamicmodel of collusion, a reduced-form representation of which being the quantity competition subgame of the model we present. Secondly, we study the extent to which the above mentioned policy measures may enhance welfare. Previous literature has also considered privatization in a mixed oligopoly. For example 2 This is in line with several empirical studies that show that private firms produce at lower costs. See for instance Bös (1991). 3

4 Matsumura (1998) has shown that neither full privatization nor full nationalization is optimal in a symmetric cost duopoly. Another example is Matsumura and Kanda (2005) that show that partial privatization is the optimal policy in the short-run, whereas full nationalization becomes optimal in the long-run with free entry among private firms. With respect to subsidization, White (1996) shows that in a Cournot setting when a production subsidy is used only before privatization, it always lowers welfare regardless of the number of firms in the industry. On the contrary, there are no welfare consequences when a subsidy is implemented before and after privatization. However, partial privatization is not considered in White s paper. In addition some existing works, like Poyago-Theotoky (2001) and Myles (2002), have demonstrated that there are no consequences from privatization of a public firm whenever a subsidy ensures the first-best allocation. In Fjell and Heywood (2004) the model is extended to assume that after privatization the leader s role changes. When the public firm remains a leader after privatization welfare is reduced. Interestingly, Kato and Tomaru (2007) show that in a Cournot setting in which each firm s objective function is to maximize the weighted average of its own profits plus another factor such as revenue, the irrelevance results concerning the effect of subsidization in welfare still hold. In all these studies though, collusion among private firms is not considered. An exception is the recent paper by Delbono and Lambertini (2016) where they show that, with symmetric linear cost functions, tacit collusion may be prevented by the threat of nationalizing a private firm. Our papers differ, however, in some important ways. To begin with, in their paper only perfect collusion is considered. In contrast, our focus is more general in that we consider all possible degrees of collusion. Second, Delbono and Lambertini consider neither the effect of different strategic profiles of the regulator nor subsidization. In addition, to the best of our knowledge privatization rather than nationalization seems to be in line with the processes experienced in many mixed oligopolies, especially whenever nationalization would imply the acquisition of firms that are successfully colluding. 3 Our first result is that collusion profitability is limited by the output expansion of the public firm intended to maximize welfare. However, the scope for cooperation of private firms is enhanced by public firm s cost inefficiency that reduces its output expansion reaction to private 3 This difference is important, since perfect collusion profits also depend on the number of participants in the agreement. It implies that their paper actually compares different cartel sizes. 4

5 firms collusion. We also obtain that welfare may be enhanced by using an optimal privatization scheme or by subsidizing production contingent on the cost inefficiency of the public firm. However, the success of the policy implemented and consequently the level of welfare depend on whether the regulator is myopic or forward-looking. We show that privatization incentives of a forward-looking SP are generally smaller compared to those of the myopic SP since the former is aware of the fact that by keeping a larger amount of shares in a partially privatized firm, private firms voluntarily reduce their degree of imperfect collusion. On the other hand, when a myopic government incorporates a production subsidy, the private sector also benefits from a high subsidy and achieves perfect collusion. In this case thus the government incurs a higher expenditure. On the contrary, a forward-looking SP offers a lower subsidy reducing therefore firms incentives to collude and yielding higher welfare. The results of the present paper thus prove useful to highlight that in a plausible scenario of colluding private firms, the welfare effects of a policy measure depend on the capacity of the regulator to assess not only the current but also the subsequent level of market competition. Although we focus on a mixed oligopoly from a theoretical viewpoint, our paper is also motivated by some real cases. Mixed public-private firms are increasingly used in several European countries and regulators make use of mixed firms following cost considerations or financial constraints. Some empirical studies have also analyzed how weak competition makes it difficult for local governments to obtain benefits from contracting out. For instance, in the Netherlands, Dijkgraaf and Gradus (2007) estimate whether collusion exists and what the impact is on tariffs for waste collection. Their results indicate the existence of collusion between private firms and that the presence of competing public firms might be essential to ensure more and fair competition. In the market for water and waste production, and based on calculation of the Herfindahl index for some Spanish municipalities, Bel and Fageda (2011) obtain that a tendency toward concentration may diminish the advantages of partial privatization. The rest of the paper is structured as follows. Section 2 describes the imperfectly collusive market in the presence of a public firm. Section 3 presents two different policy measures in order to enhance welfare. Section 4 presents some extensions of the model showing, for instance, that the effect of the existence of a public firm on private firms collusion incentives is also 5

6 robust to other ways to parameterize the product-market competition. Section 5 concludes. All proofs are grouped together in the Appendix. 2 The benchmark model: Imperfect collusion in a mixed oligopoly We consider an industry with N +1firms simultaneously producing a homogeneous product. N of these firms, indexed by i = 2, 3,..., N +1, are profit-maximizing private firms that produce a quantity q i of the product with a quadratic cost function given by c i (q i )= 1 2 q2 i.on the contrary, the firm indexed by 1 is a welfare-maximizing public firm that produces with a quadratic cost function given by c 1 (q 1 )= c 2 q2 1,withc 1. 4 Therefore, c accounts for the cost asymmetry between public and private firms. Welfare is given by the following expression: W = P N+1 i=1 Π i + CS where Π i accounts for firm s i profits and CS for the consumer surplus. The industry inverse demand is given by the piecewise linear function p(q) =max(0,a Q) where Q = P N+1 i=1 q i is the industry output, p is the output price and a>0. Through the paper we focus on the short-run equilibrium in which neither entry nor exit in the market are possible. In many cases, though, a mixture of private and public ownership where the government still holds a non-negligible proportion of shares in privatized firms can be observed. These semi privatized firms thus must respect the interests of private shareholders and they cannot be pure welfare-maximizers. We also allow for the possibility of partial privatization where the unique semi public firm maximizes the weighted sum of own profit andwelfare:β( P N+1 i=1 Π i + CS)+(1 β)π 1 where β [0, 1]. Therefore, the present formulation encompasses the model with one pure public firm and N private firms if β =1and the quantity competition among N +1private firms if β =0, or to put it differently, if the shares owned by the government 4 As stated in De Fraja and Delbono (1990), if each firm s marginal cost is constant the public firm will imposetheruleofpricingatmarginalcost.thisistrueindependentlyoftherelativeefficiency of private and public firms. Then, if there were any fixed costs, the public firm would be unable to cover the losses which would then need to be funded by the taxpayer. We abstract from this issue by considering increasing marginal costs. 6

7 increase, then β also increases. As mentioned in the introduction, we characterize imperfect collusion considering a particular model where output is determined in such a way that private firms maximize the sum of its own profit and a fraction of the profits of the rest of the private firms. Explicitly, each private firm i =2,..., N +1maximizes Π i + α( P N+1 j6=i Π j ) where α [0, 1]. Since private firms are symmetric, we also assume α to be symmetric and constant. This parameter can be interpreted as representing the degree of reciprocal loyalty to a collusive agreement and therefore may represent the degree of collusion. 5 This model exhibits equilibria where private firms coordination on distinct output levels exists and therefore, the equilibrium quantity of each firm depends on α. Admittedly, although we build a direct link between α and the degree of collusion, the reader might feel more comfortable when such link is made explicit and based on a direct behavioral assumption like the output produced. Arguably, we could also interpret our model as one in which private firms strategy set is a quantity in the interval between private firms joint-profit maximizing allocation and the Cournot equilibrium in such a way that α merely parameterizes the most collusive output achievable. Definition 1 Collusionissaidtobeimperfectifα (0, 1). On the contrary, collusion is said to be perfect if α =1. It is a straightforward exercise to obtain the (partially) cooperative equilibrium for given values of α and β, a ϕ(n,α) a (1+c β) q 1 (α, β) =,q (2+c β)ϕ(n,α)+n(1+c βα) i(α, β) =, (1) (2+c β)ϕ(n,α)+n(1+c βα) where ϕ(n,α) 2+(N 1)α. The associated level of profits for the (semi) public and private firms are Π 1 (α, β) = (2(1 β)+c) q1(α, 2 β) and Π i (α, β) =(3+2(N 1)α) qi 2 (α, β), respectively where throughout the paper and abusing notation, we assume exogenously given values of c and N. It can be easily checked that public firm s output decreases with c and increases with α and β while the reverse is true for private firms. Therefore, cooperation among private firms implies an output expansion by the public firm that could constrain collusion profitability. 5 Escrihuela-Villar (2015) shows that using conjectural variations and the coefficient of cooperation leads to equivalent closed-form solutions. 7

8 A natural question arises as to what extent private firms are willing to collude or, what is the same thing, how α could be made endogenous. We search thus for the private firms profitmaximizing level of collusion for any level of privatization of the public firm. To that extent, we present here a simple two-stage game where at the first stage private firms simultaneously decide an individually profit-maximizing symmetric level of collusion, which we denote by bα, for any exogenously given level of partial privatization β. At the second stage private firms (possibly cooperatively) decide their outputs according to the collusive level previously decided and therefore produce the output levels given in (1). 6 We also assume that at the second stage firms cannot deviate from the degree of collusion non-cooperatively decided in the first stage. 7 Using subgame perfect Nash equilibrium as solution concept henceforth, SPNE, we are ready to present our first result. Proposition 1 The symmetric profit-maximizing degree of imperfect collusion by private firms is bα = N(1+c β) (2+c β) < 1 where bα increases with c and N and decreases with β. In addition, (N 1)(2+c β) lim bα =1. c By replacing bα in (1) we obtain the following equilibrium quantities and profits (where 1+c β): α(1+(1+n) ) q 1 (bα, β) =, q α (1+ )(1+(1+2N) ) i(bα, β) = (1+(1+2N) ), Π 1 (bα, β) = (1+ β) 2 q 1 (bα, β), Π i (bα, β) = 1 2(1+ ) q i(bα, β), (2) where 1+c β. The intuition behind Proposition 1 is as follows. Private firms cooperation 6 Our model presents also a similarity with partial ownership arrangements. Even though cross ownership also takes into account rivals profits in the Cournot game, in our model the above mentioned reciprocity only implies cooperation among firms if firms collude whereas firms reciprocal concern disappears in a competitive environment. 7 We believe that several reasons justify this assumption. Firstly, assuming cooperation when setting α is more complex as long as possible subsequent deviations should also be considered markedly extending the set of strategies available to firms. Secondly, we have in mind a non-cooperative game where the type of collusion considered is merely tacit. One could think that the extent of collusion to be sustained should also be decided non-cooperatively. Relaxing this assumption has been left for future research. 8

9 consists of a reduction in their output in order to increase the market price. However, the output expansion of the public firm in response to the collusive behavior of private firms limits the scope of collusion, and as a consequence, only (some) imperfect collusion is profitable. If the public firmbecomes more inefficient, the aforementioned output expansion is also reduced since q 1 (bα, β)/ c < 0. As a consequence, the scope for cooperation of private firms is enhanced by public firm s cost inefficiency. In fact, private firms have only incentives to perfectly collude in the limit case of c. To put it differently, by enhancing collusion, a potential additional harmful effect associated to cost inefficiency of the public firm beyond the direct effect of c on the market price is also suggested. On the other hand, an increase in the number of private firms also increases the scope for collusion. Intuitively, since Q(bα, β)/ N < 0, an increase in the number of private firmsmitigatestheoutputexpansionofthepublicfirm in response to private firms cooperation increasing therefore collusion profitability. Interestingly enough this is contrary to the classic view where collusion is less likely to benefit firms when the number of colluding firms increases (see for instance Vives, 2001). Finally, with a smaller β where the public firm cares less about welfare but more about its individual profits the output expansion effect of the public firm is also mitigated. 8 Remark 1 In a mixed oligopoly, private firms always prefer imperfect rather than perfect collusion. To conclude this section it is worth noting that, as mentioned in the introduction, the literature on quantity-setting mixed oligopolies has often also assumed that public firms act as Stackelberg leaders. In the present analysis, however, the role of the public firm as a Stackelberg leader would just increase the output expansion effect described above. In fact, it can be easily checked that the equivalent bα with a public firmendowedwithastackelberg leadership is always smaller than the one obtained in Proposition It is also worth noting though that even for β =0we have bα <1. As mentioned above, private firms incentives to collude are undermined by the output expansive behavior of public firms. In the present model, however, even with β =0the former public (now private) firm does not collude and consequently also expands its output if α increases in order to benefit fromotherfirms collusive agreement. As long as this former public firm is less efficient than the rest of the firms, we only obtain bα =1in the limit case where c. 9 We opted thus for the most favorable timing for the collusive private firms in our mixed oligopoly, and even 9

10 3 Enhancing Welfare In what follows we investigate the extent to which a welfare maximizing SP can mitigate the negative effect that imperfect collusion exerted by private firms has in welfare. Since the effect of the number of private firms on the optimal privatization has already been stated among others by De Fraja and Delbono (1989), we assume hereafter to simplify that N =2. We discuss two possible different policy measures to enhance welfare in our setting: (i) privatization of the public firm that consists of choosing the value of β that maximizes welfare and (ii) the implementation of an optimal production subsidy for all firmsthatwedenotebys. In the latter case, welfare must also incorporate the cost of the subsidy: W = Π 1 + P 3 i=2 Π i + CS s(q 1 + P 3 i=2 q i). In both cases we consider two different scenarios. First, welfare is maximized by a myopic SP that treats α as an exogenous parameter; that is, the regulator does not consider that a change in the policy measure applied might enhance or hinder collusion. Second, the SP is forward-looking and it is able to correctly anticipate that private firms cooperation depends on the policy measure implemented; that is, the forward-looking SP knows that the policy measure will be followed by private firms choosing a profit-maximizing degree of collusion contingent on the policy measure implemented. 3.1 Optimal privatization AmyopicSPthatignorestheeffect of β on α can be modeled as the following three-stage game: at the first stage private firms simultaneously decide the value of α that maximizes individual profits. In a second stage, the SP decides how many shares of the public firm must be sold for the value of α previously decided without taking into account a possible collusive reaction of private firms once β has been determined. In a third stage, quantity competition takes place for the previously determined values of α and β. As a consequence, at the first stage private firms maximize individual profits with respect to α considering thus the effect of α on the value of β that will be determined at the second stage. At the third stage firms will produce the outputs described in (1) for the α and β previously determined. in this case bα <1. Consequently, Proposition 1 does not rely on this assumption. Further details are available from the authors upon request. 10

11 A forward-looking SP that correctly anticipates that private firms cooperation depends on β and that therefore, the degree of market competition will be adjusted after β is determined, can also be modeled as a three-stage game: at the first stage the SP decides how many shares of the public firm must be sold (namely, the value of β). In a second stage, private firms observe β and decide the value of α that maximizes individual profits. The point here, and unlike the case of a myopic SP, is that regardless of the level of market competition that could be previously observed by the forward-looking SP before setting β, the regulator anticipates that adifferent α might follow the level of β determined. From Proposition 1 we know that private firms profits are maximized at bα which is, therefore, the Nash equilibrium of the second stage of the game with a forward-looking SP. At the third stage, quantity competition takes place inducing firms to produce the outputs given in (2) for the value of β previously determined. Regarding the myopic SP, the following lemma describes the equilibrium choice of β in the second stage of the game. Lemma 1 The optimal privatization policy of a myopic SP is β(α) =1 2c(1+α) 6+α(4+α) whereas if c 3, β(α) =0. if 1 c<3 The intuition behind Lemma 1 is crucial to understand the solution of the three-stage game proposed when the SP is myopic. Since β(α) > 0 decreases with α, the optimal privatization policy always decreases if cooperation between private firms is larger. The intuition is that private firms profits, as a proportion of total welfare, are larger if α increases and, as a consequence, a lower β is called for in order to maximize welfare. We are now ready to present the solution of the games described above. We denote by β M,β FL,α M, and α FL the optimal privatization policies and the profit-maximizing degrees of imperfect collusion by private firms with a myopic and with a forward-looking SP respectively. Proposition 2 In our mixed oligopoly model with imperfect collusion, (a) If 1 c<3, then 1 >β FL >β M > 0, and1 >α M >α FL;(b)if3 c<4+2 7,then1 >β FL >β M =0, and α M = c 2+c >α FL;and(c)whenc 4+2 7, β FL = β M =0and α M = α FL = c. 2+c Furthermore, α M and α FL increase with c whereas β FL and β M decrease with c whenever they are positive. 11

12 Proposition 2 establishes that full privatization is advisable only if the public firm is inefficient enough compared to private firms. The intuition is fairly simple. Profits of both private and public firm are maximized when β =0.However,CS is maximized when β =1. Then, if c increases beyond a certain level, the effect of β on CS is too small to boost total output as long as the inefficient public firm s market share becomes also very small. In this case, a positive β barely increases CS but decreases private and public firm s profits compared to β =0. In general, an inverted-u relationship between welfare and β is obtained for a given α and partial privatization seems to be the best policy unless the public firm is markedly inefficient. If β exceeds the critical positive value (β M or β FL respectively) a further nationalization would imply that the decline in private and public firms profits would offset the increase in CS. On the other hand, β M is lower than β FL becausefromlemma1weknowthattheformer privatization policy always decreases with α. Then, inaspneprivatefirms are aware of this effect which allows them to further expand collusion. Regarding the forward-looking SP, privatization is usually less desirable than with a myopic SP. Intuitively, we can distinguish two different effects. On the one hand, if c is not too large, a (up to some extent) larger β might increase welfare due to the output expansion of a not markedly inefficient public firm. On the other hand, there is an additional effect that the forward-looking SP also takes into account which is that, as Proposition 1 states; a larger β also implies that the scope for cooperation of private firms is reduced. Proposition 2 shows that by incorporating the second effect, privatization incentives of a forward-looking SP are reduced compared to the myopic SP. The comparison between both optimal privatization policies thus also indicates that in some cases under a forward-looking SP only a partial privatization scheme is advisable whereas a myopic SP, that ignores the effects of privatization on collusion incentives, would fully privatize. In other words, Proposition 2 confirms that privatization in order to obtain a less collusive market is generally less desirable with a forward-looking SP since when private firms cut production to raise the price, a further output expansion from the public firm (namely a larger β) is called for in order to maximize welfare. 10 In both scenarios though, a more inefficient public firm 10 We note also that this result is in line with the results of De Fraja and Delbono (1989) even though in the present paper the degree of competition in a mixed oligopoly is captured by α instead of by the number of private firms. 12

13 should be further privatized. Interestingly enough, it can also be easily checked that welfare is always larger with a forward-looking SP than with a myopic SP. The intuition is that, comparedwiththelatter,theformerregulator provides a positive effect on CS due to less collusion and less privatization that compensate the reduction in private and public firms profits. This highlights the importance for the success of regulation that a SP also takes into account the market reaction to the regulation. Finally, it is also worth to note that a pure public firm is never optimal (namely, β FL and β M are always smaller than 1) since the restraint of the collusive practices that a too large β might represent is more than offset by the excessive output expansion of an inefficient public firm. 3.2 Optimal Subsidization In this subsection we study the effect on welfare of an optimal subsidy assuming that the degree of privatization β is exogenously given. The subsidy is included in the total welfare as part of the public and private firms profits but also as an equivalent expenditure. It is interesting to note though that the subsidy also affects public firm s profits through its effect on the output of private firms even if β =1. As in the previous subsection we also consider the case of a myopic and a forward-looking SP. In the latter case, the game at hand can be modeled as follows: at the first stage the SP decides the level of the subsidy s, at the second stage private firms observe the value of s and simultaneously decide the value of α. Finally, at the third stage, quantity competition takes place. However, when the SP is myopic, first and second stages are reversed. Then, firstly private firms decide on α and afterwards the SP chooses the value of the subsidy taking as given the level of collusion α. By backward induction we can obtain the common (partially) cooperative equilibrium at the third stage, regardless of whether the SP is myopic or not: q 1 (α, β, s) = (a+s)(2+α) βs(4+α),q (2+α)+ i (α, β, s) = a +s(1+c). (3) (2+α)+ In (3) we can easily check that the subsidy positively affects private firms output whereas it has a negative effect in the output of the public firm. Lemma 2 presents the solution of the second stage of the game for both scenarios: 13

14 Lemma 2 With a forward-looking SP, private firms choose α =(c β)/(1 + ) which does not depend on the subsidy s, whereas when the SP is myopic the optimal subsidy is given by s(α) = a[(c(2+α)+ )(β(4+α) (2+α)) 4 (1+c)] where s(α) increases with α. (1+c)[18+6c β(2+β)(4+α) 2 +α(8+α)] The first part of this lemma points out that the level of collusion chosen by private firms does not depend on the subsidy when the SP is forward-looking. The intuition behind is that private firms are symmetrically affected by the subsidy in such a way that the output reduction of firm i due to an increase in α only depends on the relative inefficiency of the public firm for a given value of s and consequently, the subsidy does not affect the profit-maximizing level of collusion. The second part states that, with a myopic SP, the level of the subsidy increases if α increases as long as the subsidy is intended to mitigate the negative effect that collusion has in total output that decreases CS and, consequently, welfare. Now we are ready to present the main result of this subsection. We denote by s M,s FL, α M and α FL the optimal privatization policies and the profit-maximizing degrees of imperfect collusion with a myopic and with a forward-looking SP respectively. Proposition 3 For given values of c and β, s M > s FL > 0. Furthermore, private firms perfectly collude with a myopic SP (namely, α M = 1) whereas with a forward-looking SP α FL =(c β)/(1 + ) < 1 where α FL increases with c. Proposition 3 states that when the SP is myopic private firms achieve perfect collusion because private firms know that, for all α < 1, an additional level of collusion is always rewarded with a larger subsidy intended to encourage production. It can be also checked that, in both scenarios, the subsidy generally increases if the public firm is more inefficient. The intuition is that even though subsidization is symmetric (s is the same for public and private firms) 11 if the public firm becomes more inefficient, a higher subsidy is intended to offset the negative effect of the associated increase in the level of collusion. Consequently, a higher 11 We have also checked the asymmetric case where the SP only subsidizes private firms. In this case, the results do not importantly change. The main difference is that profits for public firm are lower than in the symmetric case and, consequently, private firms profits increase. 14

15 subsidy induced by a more inefficient public firm also increases private firms collusion with a forward-looking SP. Analogously to the previous subsection, it can also be easily checked that welfare is larger with a forward-looking SP than with a myopic SP. Intuitively, when comparing both scenarios in welfare terms we can observe that there is a trade-off between the level of the subsidy (and hence, collusion) and the level of firms profits: the larger the subsidy (which is an expenditure for the government) the higher the level of firms profits, including the public firm. In other words, compared to the forward-looking SP, a myopic SP basically subsidizes collusion at the cost of a higher expenditure and lower CS. 4 Extensions 4.1 Multiple public firms Some previous studies, such as Bose and Gupta (2013), Matsumura and Shimizu (2010) or Matsumura and Okumura (2013), have allowed for multiple public firms. We check here the effect of an increase in the number of public firms on collusion profitability by assuming that, in the model presented above, K>1symmetric welfare-maximizing public firms simultaneously set outputs competing with N private firms. To simplify, we assume also that β =1. As Proposition 4 shows, a sufficiently high number of public firms prevent collusion since private firms find it unprofitable to reduce output in order to increase the price. Proposition 4 Private firms profitsareonlyenhancedbycooperationifα<ˆα c(n 1) K (K+c)(N 1) where ˆα decreases with K. Therefore, when N =2, ˆα 0 if c<2 and when N>2, ˆα 0 if K N(c 1). Proposition 4 shows that the output expansion of public firms increases with K. Consequently, if the number of public firmsislargeenough, privatefirms voluntarily do not collude since an output contraction of private firms would be followed by an output expansion of public firms turning private firms profits below the ones obtained in the Cournot allocation. This effect though is alleviated when public firms are relatively inefficient compared to private firms. 15

16 4.2 Discount Factor We test here whether Proposition 1 hinges on the assumption made in Section 2 regarding the way to measure the intensity of competition assuming that the degree of cooperation is captured by the discount factor. We consider an industry like the one described in our benchmark model with one public and N private firms but for the case where α =0.However, we assume now that firms play an infinitely repeated game at dates t =1,..., with a common discount factor δ [0, 1). Following Friedman (1971), we restrict attention to the well-known grim trigger strategies for private firms. Hence, if a private firm defects, the other firms revert from there on to the static Nash-Cournot output obtaining a profit denotedbyπ N.Regarding the public firm, we assume that its optimal response consists of maximizing welfare at each period. In such a way, if each private firm produces q the output produced by the private firms, denoted by q p,isq p =(a Nq)/(1 + c). Eachprivatefirm producing q corresponds to a SPNE if and only if the following condition is satisfied for each private firm: Π c 1 δ Πd + δπn 1 δ, (4) where Π d denotes the one period profit from deviation and Π c the profits obtained by each private firm in the collusive equilibrium. There are many SPNE collusive output vectors that satisfy the system of inequalities in condition (4) above. As in Verboven (1997) and Escrihuela- Villar (2008), 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 under perfect collusion. 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 (4). Let us denote the above mentioned critical level of the discount factor by δ. It can be verified that δ = (N 1)(2+c(2+N)) 2 (c(n 1) 1)(7+5N+c(7+N(10+N))).12 Then, it can be easily checked that the quantity produced by each private firm in the collusive 12 We also note that perfect collusion might only be sustainable (namely, δ (0, 1)) ifeitherc is large enough since public firm s inefficiency decreases its output expansion behavior or if N is also large enough whenever c is small. In the latter case, the (relatively efficient) public firm expanding output behavior is diluted by the existence of a large number of private collusive firms. 16

17 equilibrium depends on δ. Wedenoteitbyq(N,c,δ) and it is given by ac(4c(1+c)(δ 1)N (c+1) 2 (4+5δ)+c 2 (δ 1)N 2 ) 2(1+c) q(n,δ,c) = 3 (δ 4) 3c(1+c) 2 (4+δ)N 6c 2 (1+c)N 2 +c 3 (δ 1)N 3 ac N(2c+1)+c+1 if δ δ if δ< δ (5) where δ =0represents the non-collusive equilibrium, and as δ δ we obtain the perfectly collusive equilibrium. Consequently, a variation in the discount factor when δ (0, δ) can be interpreted as a variation in the degree of collusion of the market. The collusive profits of private firms, that we denote by Π c (N,δ,c), alsodependonδ. 13 Proposition 5 Private firms profits are only enhanced by cooperation if δ< δ (2+c(2+N)) 2 7+2c(7+5N)+c 2 (7+N(10+N)) < δ where δ increases with c and N. In addition, lim c δ = lim c δ. Proposition 5 shows that Proposition 1 also carries over to the case where δ captures the degree of collusion. 5 Concluding comments We have developed a theoretical framework to study how the existence of a (possibly semi) public firm affects private firmsincentives tocollude. Weobtainthatprivatefirms voluntarily collude less than what they would do in absence of a public firm as long as the public firm reacts to the effort of collusive private firms to cut production by expanding its output. Interestingly enough, perfect collusion is never the most preferred outcome for private firms. We also obtain that the degree of collusion that maximizes private firms profits increases if the public firm becomes more inefficient because, in this case, its output expanding response to collusion is smaller. This result proves useful to present an additional and possibly ignored consequence of the existence of welfare maximizing firms providing a strong disciplining mechanism that reduces collusion profitability. We also obtain that, if we allow for multiple public firms, a sufficiently high number of public firms implies that collusion is completely unprofitable. We 13 Note also that in a collusive equilibrium firms are willing to cut production compared to the non-collusive equilibrium ( q(n,δ,c) δ ( Πc (N,δ,c) δ > 0). < 0) andonecanalsocheckthatprofits of each private firm are enhanced by collusion 17

18 also test the robustness of our first result to other ways to parameterize the product-market competition. Additionally, two different policy measures are considered in our welfare analysis. Regarding privatization we obtain that, especially when its deterring effects on collusion are considered, the existence of a (at least to some extent) public firm seems a more appropriate policy unless a public firm is very inefficient compared to private firms. Full privatization is anyway only advisable if the public firm is markedly inefficient. On the other hand, we also showed that, through a larger production subsidy, the degree of collusion is also larger when the regulator is myopic. Our results also prove that the welfare effects of a policy measure crucially depend on the capacity of the regulator to assess the competitive reaction of private firms to the policy implemented. The framework we have worked with is only a particular approach of a more general issue. To analyze real-world mixed oligopolies with collusive private firms, additional research is required, and for instance a wider range of demand and cost functions should also be considered. Webelievethatthosearesubjectsforfutureresearch. Appendix Proof of Proposition 1. Π i (α, β) defined in Section 2 that the following is true: Π i (α,β) α We just have to check in the profit function for private firms = a2 (1 β+c) 2 (1 N)(β c+n(1 β)+cn 2+(β c 2)(N 1)α) (2(β c)+n(β 1) cn 4+(β 2 c)(n 1)α) 3 2+c+β(N 1) (1+c)N (2 β+c)(1 N) ᾱ = 1 N (2 β+c)(n 1) 2 > 0 only if α<ᾱ whereas if α > ᾱ then Π i(α,β) α < 0. We can also easily check that > 0, ᾱ c = Proof of Lemma 1. expression that we denote by N ᾱ > 0 and (2 β+c) 2 (N 1) = N < 0. β (2 β+c) 2 (N 1) By adding firms profits and consumer surplus we obtain the welfare W (α, β) = a2 ((2(1 β)+c)(2+α) 2 +(4+2(c β)+α) 2 +2(1 β+c) 2 (3+2α)). Then, we just have to check that 2(2(3+α) β(4+α)+c(4+α)) 2 W(α,β) β = a2 (2+α)(2c(1+α) 6 α(4+α)+β(6+α(4+α))) (β(4+α) 2(3+α) c(4+α)) 3 < 0 if c>3. On the contrary, W (α, β) is maximized with respect to β whenever W(α,β) =0. Thisistrueifβ(α) =1 2c(1+α). β 6+α(4+α) Finally, the second order condition also holds since 2 W (α,β) = a2 (2+α)(2β(4+α)(6+α(4+α)) (6+α)(6+α(4+α))+c(4+α)(12+α(10+α))) < 0. 2 β (2(3+α) β(4+α)+c(4+α)) 4 18

19 Proof of Proposition 2. The outputs expressed in (1) represent the equilibrium at the third stage. On the other hand, the bα provided in Proposition 1 (for the case N =2)andthe β(α) provided in Lemma 1 are the solution of the second stage for the forward-looking and the myopic SP cases respectively. Therefore, regarding the first stage, in the myopic SP case we just have to maximize the private firms profit function provided in Section 2, Π i (α, β), when β = β(α) if 1 c<3, and when β(α) =0if c 3. Then, if 1 c<3, α M is the solution to this equation Π i(α,β(α)) = a2 c 2 (4+α)(6+c(α 1)(4+α) 2 +α(22+α(11+α))) =0for α and β α (6+α(4+α)+c(4+α) 2 ) 3 M = β(α M ). Unfortunately the explicit expression for α M cannot be simplified to be included in the paper. We used the program Wolfram Mathematica 7.0 to simplify the expressions and tedious but straightforward details for this and subsequent proofs are available at or from the authors upon request. If c 3, Π i(α,0) = a2 (1+c) 2 (c(α 1)+2α) =0gives α α (2(3+α)+c(4+α)) 3 M = c. 2+c Regarding the forward-looking SP, the solution of the first stage of the game is given by the solution for β to the equation W( α,β) = a2 (41β 4 β 3 ( c) ((c 8)c 12)(8+c(11+4c))+3β 2 (124+c(141+37c)) β(308+c(500+c(243+29c)))) =0 β (5β 5c 6) 3 (β c 2) 3 and leads us to β FL where α FL corresponds to bα evaluated at β FL.Itcanalsobechecked that W( α,β) β < 0 if c. Finally, as long as a just appears as a single multiplicative parameter in β M,β FL,α M, and α FL,wejusthavetoplottheminordertoobservehowthey change with c. ProofofLemma2. First, with a forward-looking SP, at the second stage private firms optimally choose α contingent to any value of s. The associated level of profits for public and private firms at the second stage of the game can be denoted by Π 1 (α, β, s) and Π i (α, β, s) respectively where Π i (α, β, s) =[(3+2α)(a(1 β)+ac+s(1+c)) 2 ]/[(2(2(3+α)+(c β)(4+α)) 2 )]. Hence, by taking the first order condition Π i (α, β, s)/ α =0,itisobtainedα =(c β)/(2 + c β) which does not depend on s. Second, with a myopic SP, at the second stage the public firm chooses s taken as given the level of α. By using (3) welfare at the second stage, that we denote by W (α, β, s), can be easily obtained. Hence, by taking the first order condition W(α, β, s)/ s =0,weobtains(α) = a(6+(2c+c2 )2(1+α)+β 2 (2+α)(4+α)+α(6+α) 2β(7+c+(6+c)α+α 2 )). (1+c)(18+6c+(β 2)b(4+α) 2 +α(8+α)) Proof of Proposition 3. First, the optimal subsidy under the forward-looking case is obtained at the first stage by solving W(α,β,s) =0,whenα =(c β)/(2 + c β), (which s 19

20 is the α FL ) and yields s FL. It is also immediate to check that α FL 2 = > 0. In the c (2 β+c) 2 myopic case at the second stage (Lemma 2) we have to obtained the solution for s: s(α). At the first stage, private firms decide the degree of collusion by maximizing individual profits with respect to α when s = s(α). Then, we can check that Π i(α,β,s(α)) > 0 is always true which α yields α M =1and the optimal subsidy is the s(α) provided in Lemma 2 evaluated at α =1. Then, s M = a(13+15β2 +4c(2+c) 4β(7+c)) (1+c)(27+25(β 2)β+6c) >s FL. See the proof of Proposition 2 for further details. ProofofProposition4. are given by the expression Π i (K) = Π i (K) α = a2 c 2 (N 1)(c cn+k+(n 1)(c+K)α) (c(2+n)+2k+(n 1)(c+K)α) 3 It can be easily checked that collusive profits for private firms a 2 c 2 (3+2(N 1)α) 2(c(2+N)+2K+(N 1)(c+K)α) 2. Then, < 0 if α<ˆα c(n 1) K. Therefore, ˆα 0 if (K+c)(N 1) either 1 <c<2 and N =2or K N(c 1).when N>2. Finally,wejusthavetocheckthat ᾱ = cn < 0. K (K+c) 2 (N 1) ProofofProposition5. Private firms profits Π c (N,δ,c) can be easily obtained with the outputs given in (4). See the proof of Proposition 2 for further details. References -Bel, G., & Fageda, X. (2011). Big guys eat big cakes: Firm size and contracting in urban and rural areas, International Public Management Journal, Vol. 14(1), pp Bös, D. (1991). Privatization: a theoretical treatment, Clarendon. -Bose, A., & Gupta, B. (2013). Mixed markets in bilateral monopoly, Journal of Economics, Vol. 110(2), pp Cyert, R. M., & DeGroot, M. H. (1973). An Analysis of Cooperation and Learning in a Duopoly Context, The American Economic Review, Vol. 63(1), pp Charness, G., & Rabin, M. (2002). Understanding Social Preferences with Simple Tests, Quarterly Journal of Economics, Vol. 17(3), pp De Fraja, G., & Delbono, F. (1989). Alternative strategies of a public enterprise in oligopoly, Oxford Economic Papers, Vol. 41(2), pp De Fraja, G., & Delbono, F. (1990). Game theoretic models of mixed oligopoly, Journal of Economic Surveys, Vol. 4(1), pp

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