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Mixed oligopolies and collusion

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Abstract

We introduce a firm partially owned by the government in a dynamic model of collusion between a subset of private firms. We show that increasing the public ownership of the non-colluding firm may help collusion between the private firms.

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Notes

  1. For example, in a context of mixed oligopoly, Cremer et al. (1991) and Matsushima and Matsumura (2003) consider the locational choices of firms, Cato and Matsumura (2013) consider taxation, Matsumura (1998) considers the optimal degree of privatization, Ishibashi and Matsumura (2006), Heywood and Ye (2009) and Gil-Moltò et al. (2011) study R&D competition between public and private firms, whereas Matsumura and Ogawa (2012) and Scrimitore (2013) analyse the choice of the strategic variable. For a survey of theoretical issues about mixed oligopolies, see for example De Fraja and Delbono (1990) or Ishibashi and Matsumura (2006), while empirical investigations can be found in Megginson and Netter (2001).

  2. One exception is a recent work by Delbono and Lambertini (2014), where it is shown that the threat of nationalizing one private firm may act as device to reduce the incentive of the remaining private firms to constitute a cartel.

  3. We refer to semi-public firm, as we allow for the possibility that the government holds a positive proportion of the shares of Firm X (see later). As a limit case, the government entirely owns the firm. In this case Firm X is a pure public firm.

  4. All the results in this section can be replicated for the case of quadratic costs, which are commonly adopted in the mixed oligopoly literature (see for example De Fraja and Delbono 1989; Fjell and Pal 1996). Details are available by the author upon request. In general, constant marginal costs and increasing marginal costs yield similar results. However, Matsumura and Okamura (2015) show an example where the results depend on the type of the cost function adopted. Therefore, robustness check for quadratic costs is important when considering mixed oligopolies.

  5. Therefore, we assume a monopolistic sector with three firms producing differentiated goods, and a competitive numeraire sector.

  6. When \(\gamma =1\), the demand functions cannot be defined. For details, see Singh and Vives (1984).

  7. We are assuming that Firm X does not participate to the cartel, even if it may be profitable for it to participate to the agreement. This can be rationalized as follows. As noted for example by Escrihuela-Villar (2008), collusion is much easier between similar firms. As Firm X’s objective function is different from the objective function of Firm 1 and Firm 2, including Firm X into the cartel might imply excessive coordination costs (Thomadsen and Rhee 2007; Colombo 2013) and the agreement may not be reached. Note that our model extends Escrihuela-Villar (2008), by assuming that the non-colluding firm may be a public or semi-public firm.

  8. Intuitively, collusion involves negotiation costs and transaction costs (Thomadsen and Rhee 2007; Colombo 2013), that make the collusive quantity a less flexible decision than the quantity set autonomously by the non-colluding firm.

  9. It is well known that other more sophisticated punishment mechanisms may exist. In particular, simple penal codes as in Abreu (1986, 1988) with stick-and-carrot phases may be preferred (in the sense that they allow expanding the discount factor set that supports collusion in equilibrium). However, the simple two-phase penal code derived in Abreu (1986) for the case of quantity firms cannot be easily transferred to the present model for two reasons. First, the optimality of the penal code in Abreu (1986) is shown under the assumption of simultaneous moves by the players, while here, during the collusive phase, a sequential play occurs. Second, using the optimal penal code at least for the limit case where all firms collude (as done for example in Escrihuela-Villar 2008), is of limited interest here, as it is possible only in the case where the semi-public firm is not active. Furthermore, the grim-trigger strategy is the most popular formulation, and it has been adopted in several papers, as for example Gupta and Venkatu (2002), Matsumura and Matsushima (2005) and Ishibashi and Shimizu (2010).

  10. In particular, when \(\gamma \le 1/2\), \(\delta ^{*} \) strictly increases with \(\theta \); instead, \(\delta ^{*} \) strictly decreases with \(\theta \) only when \(\gamma =1\).

  11. We are assuming that the “switching” private firm does not collude.

  12. The explicit expressions of the profits and the critical discount factor are not reported here and they are available upon request.

  13. Note that the best-reply function of both the semi-public firm and the non-colluding private firms takes into account the fact that one of the cartel’s members has deviated.

  14. The following numerical example has been repeated with different values of N. Details are available upon request.

  15. The complete expression of the critical discount factor is omitted from the text as it is quite long.

  16. In the mixed oligopoly literature, the case of multiple public firms is considered for example by Matsumura and Okamura (2013), Bose and Gupta (2013) and Haraguchi and Matsumura (2015).

  17. We assume that the government owns the same share in both the semi-public firms.

  18. Note that \(\partial {\pi _k^D }/ {\partial \theta }-\partial {\pi _k^C }/{\partial \theta }<0\).

  19. Note that the government chooses \(\theta \) in order to maximize W rather than \(\Gamma \).

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Authors and Affiliations

  1. Department of Economics and Finance, Università Cattolica del Sacro Cuore, Largo A. Gemelli 1, 20123, Milan, Italy

    Stefano Colombo

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  1. Stefano Colombo
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Correspondence to Stefano Colombo.

Additional information

I would like to thank the Editor Giacomo Corneo and three anonymous reviewers whose suggestions greatly improved this work. I wish to thank Toshihiro Matsumura, Luigi Filippini, Noriaki Matsushima, Akira Ogawa, Nicola Doni, Federico Boffa, Lucia Visconti Parisio, Bruno Bosco, Vittoria Cerasi, Johannes Meya and participants to XVI Centre for competition and regulatory policy workshop in Milan (Italy), and SIDE-ISLE 9\(\mathrm{th}\) Annual Conference, Lugano. All remaining errors are my own.

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Colombo, S. Mixed oligopolies and collusion. J Econ 118, 167–184 (2016). https://doi.org/10.1007/s00712-015-0467-z

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