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Biased Managers as Strategic Commitment in a Mixed Duopoly with Relative Profit-Maximizers


This study analyzes a mixed duopoly composed of a social welfare-maximizing public firm and a relative profit-maximizing private firm wherein the owners of both firms hire relative profit-maximizing managers biased with respect to the market size they face. In particular, we explore the quantity competition that gives a result different from that obtained in a standard mixed duopoly that differentiates between ownership and management, as considered by (Nakamura, Theor Econ Lett 4(3), 889–896, 2014a), which follows the approach of (Englmaier, and Reisinger, Manag Decis Econ 35(5), 350–356, 2013). More precisely, when we introduce the degree of importance of each firm’s relative performance into quantity competition, we see that the owner of the private firm reverts to a less aggressive manager when the degree of importance is relatively high, which is similar to how the classical strategic managerial delegation works. Furthermore, we consider the influence of an increase in degree of importance of each firm’s relative performance on the aggressiveness of managers hired by both the public and private firm. Moreover, we explore the case in which the owner hires his manager with salary based on social welfare.

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  1. 1.

    Hoernig (2012) provided a theoretical foundation for why firms’ owners hire aggressive managers even in price competition by introducing network effects à la Katz and Shapiro (1985). This differs from the classical studies on strategic managerial delegation such as Fershtman and Judd (1987), Sklivas (1987), and Vickers (1985).

  2. 2.

    The relative performance approach has frequently been used in the context of macroeconomics as well (for example, Corneo and Jeanne (1997), Corneo and Jeanne (1999), and Futagami and Shibata (1998)). Moreover, in laboratory and experimental works, Cason et al. (2002), Brandts et al. (2004), and Coats and Neilson (2005) considered spiteful behavior and reciprocal or altruistic behavior. Both behaviors are closely related to the firms’ objective functions based on relative performances; more precisely, the positive value of α corresponds to spiteful behavior whereas the negative value of α corresponds to reciprocal or altruistic behavior. On the basis of relative performance, Alchian (1950) and Vega-Redondo (1997) considered evolutionary stability.

  3. 3.

    The analysis of price competition is available upon request. In price competition, an increase in degree of importance of each firm’s relative performance is seen to make the owners of both the public firm and private firm more likely to hire a less aggressive manager.

  4. 4.

    More concretely, this is the inverse demand function the owners of firms face. Thus, although the objective functions of the owners are explained in detail in Section 3, the social welfare that the owner of firm 0 and the relative profit that the owner of firm 1 maximize in the first stage are calculated as the functions of k 0 and k 1 based on this “true” inverse demand function the owner of firm i faces, \({p^{o}_{i}} \left (q_{i}, q_{j} \right )\). \({q^{m}_{0}} \left (k_{0}, k_{1} \right )\), and \({q^{m}_{1}} \left (k_{0}, k_{1} \right )\), which are set by the managers of firms 0 and 1 in the second stage.

  5. 5.

    b∈(0,1)indicates that the goods produced by firms 0 and 1 are substitutable.

  6. 6.

    Firm i’s manager is not biased when k i = 1,(i = 0,1).

  7. 7.

    We assume that a>c≥0 and α∈[0,1) in order to ensure the non-negativity of all equilibrium outcomes.

  8. 8.

    As in Matsumura and Okamura (2010), Nakamura and Saito (2013a, 2013b), we regard social welfare as the sum of consumer surplus and the profit of firms 0 and 1, rather than the sum of consumer surplus and the relative profit of firms 0 and 1. Although the CEOs of firms emphasize their relative profit rather than original profit, since their relatively good performance would raise their current and future income, we consider this simply an income transfer. Therefore, in this paper, social welfare is the sum of consumer surplus and the profit of firms 0 and 1.

  9. 9.

    As indicated in Englmaier and Reisinger (2013), Nakamura (2014a, 2014b), the type of rival manager is observable to both firms 0 and 1. Since the identity of the manager of each firm is public information, the type of manager is also observable to the manager of the rival firm; this factor becomes important in the market competition stage.

  10. 10.

    In this paper, we deal with the degree of importance of each firm’s relative performance as exogenous variable, since we investigate the influence of the intensity of competition based on the type of the managers of firms 0 and 1, k 0 and k 1. In another strand of the research on the relative performance approach, Miller and Pazgal (2001) considered the situation in which each firm’s owner endogenously sets the degree of importance of his relative performance as the weight between his absolute profit and the absolute profit of his rival firm in his relative profit.

  11. 11.

    Furthermore, we have

    $$\begin{array}{@{}rcl@{}} \partial {p^{o}_{i}} \left( k_{i}, k_{j} \right) / \partial k_{i} = a \left[2 - b^{2} \left( 1 - \alpha\right) \right] / \left[4 - b^{2} \left( 1 - \alpha\right)^{2} \right] < 0. \end{array} $$

    Thus, as k i increases, p i decreases, implying that market competition becomes more intense as k i increases, (i = 0,1). In addition, we obtain

    $$\begin{array}{@{}rcl@{}} \partial {p^{m}_{i}} \left( k_{i}, k_{j} \right) / \partial k_{j} = - a b \left( 1 + \alpha\right) / \left[4 - b^{2} \left( 1 - \alpha\right)^{2} \right] < 0. \end{array} $$
  12. 12.

    In a private duopoly with network effects à la Katz and Shapiro (1985), Nakamura (2014b) presented the same result and intuition as those obtained in this paper.

  13. 13.

    We thank the anonymous referee for suggesting an extended analysis in this section.


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We are deeply grateful to anonymous referees for helpful comments and suggestions. This research is financially supported by KAKENHI (25870113). All remaining errors are our own.

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Correspondence to Yasuhiko Nakamura.

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Nakamura, Y. Biased Managers as Strategic Commitment in a Mixed Duopoly with Relative Profit-Maximizers. J Ind Compet Trade 15, 323–336 (2015).

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  • Strategic delegation
  • Biased expectations
  • Aggressiveness
  • Quantity competition

JEL Classifications

  • L20
  • L32
  • D43