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Mixed duopoly and the indirect effect in linear supply function competition

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Abstract

The 1980s market reforms in Europe led to far-reaching privatization. However, empirical data show that some countries experienced deterioration in market performance, such as in consumer prices and welfare, during a certain period. Although theorists have focused on the change in the stock holding rate and the degree of product differentiation between privatization, few paid attention to the change in competition mode or intensity. To grasp how the structural change of privatization can affect market performance, we investigate the infinitesimal change in competition intensity by parameterizing it, apart from using the traditional method of making a simple comparison between Cournot and Bertrand equilibria. Thus, we parameterize the intensity of competition, and consider a mixed duopoly market with product differentiation accompanying partial privatization using a linear supply function approach. We observe a non-monotonic change in prices, profits, and social welfare in the plane of the degree of product differentiation and privatization, when competition becomes aggressive. Intuitively, when the semi-public firm promotes privatization, the market is an ordinary private duopoly and the competition becomes weak, while the competition can be aggressive with nationalization, since the semi-public firm tries to lower its price to improve consumer surplus. Further, the thresholds’ roughly common property—the right-upwardness—depends on the degree of the cross-price effect. Analytically, these counterintuitive phenomena are caused by the indirect effect inherent in linear supply function competition.

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Notes

  1. See, for instance, Zhang et al. (2008) for the electricity case; see Vickers and Yarrow (1988) for a comprehensive survey and analysis.

  2. See Matsumura (1998).

  3. The mixed oligopoly theory dates back to Merrill and Schneider (1966). Since then, substantial literature, centered on De Fraja and Delbono (1989), has been produced on the different methods of analyses. For ordinary private oligopoly theory and policy, see Tirole (1988) and Motta (2004).

  4. Matsumura and Okamura (2015), who examined a mixed market using a relative profit approach, noted that evaluating the intensity of competition by the number of firms can be misleading because it changes the market structure. Supporting their claims, we refrain from constructing the oligopoly model.

  5. After the early work on SFE by Grossman (1981) and Hart (1985), Klemperer and Meyer (1989) constructed a basic model. Green and Newbery (1992) analyzed the British electricity industry using SFE. Baldick et al. (2004) studied electricity markets using the linear version of SFE. For a comprehensive survey of SFE, see Vives (2001) (section 7.2).

  6. Scrimitore (2014) argued that reversal phenomena, as in Ghosh and Mitra (2010), do not occur if a government adopts an optimal partial privatization strategy.

  7. There are two primary reasons for adopting the symmetric cost function. The first is for a theoretical comparison with other studies, such as Delbono and Lambertini (2015). The second is to avoid bankruptcy within the LSFE model. That is, the setting of asymmetric costs probably yields separate equilibrium prices between the two firms, whereas SFE is programmed to yield an identical price between the two.

  8. This discussion was developed earlier in Matsumura (1998).

  9. As in Delbono and Lambertini (2015), replacing the linear supply function as \(q_i=\alpha _i+\beta _ip_i\) leads to the same equilibria.

  10. Delbono and Lambertini (2015) showed that linear dependence holds in both linear cost and quadratic cost cases. Moreover, the authors note that the dependence holds even if the timing of determining \(\alpha _i\) and \(\beta _i\) differs.

  11. For a model in which firms select quantity competition or price competition endogenously (alternatively, dispersive \(\beta _i\)) in mixed markets as in Singh and Vives (1984), see Matsumura and Ogawa (2012) and Kopel (2015).

  12. The denominator of \(RF_0\) is minimized when \(\lambda =1\), which is positive. Further, the stability condition between \(RF_0\) and \(RF_1\) is satisfied.

  13. The main results are obtained; the existence of non-monotonicity on prices and profits is also supported on other points of \(\beta\). Note that the interval of \(\delta\) is limited as \(\beta\) increases because of \(\beta <1/\sqrt{1-\delta ^2}\). For example, if \(\beta =10\), \(\delta\) is limited to \(\delta >0.995\); to analyze the \(\delta -\lambda\) square, \(\beta \le 1\) must be assumed.

  14. For earlier work on the cross-price effect, see Dixit (1979).

  15. See Green and Newbery (1992) (Fig. 3).

  16. See, for instance, Megginson and Netter (2001), Zhang et al. (2001), and Fries and Taci (2005).

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Acknowledgements

The author is grateful to Tomomichi Mizuno (Kobe University), Takashi Yanagawa (Kobe University), and the anonymous referees for their helpful comments and suggestions.

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  1. Graduate School of Economics, Kobe University, 2-1 Rokkodai, Nada, Kobe, 657-8501, Japan

    Keita Yamane

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Yamane, K. Mixed duopoly and the indirect effect in linear supply function competition. Econ Polit Ind 45, 519–532 (2018). https://doi.org/10.1007/s40812-018-0103-3

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