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Combining the Endogenous Choice of the Timing of Setting the Levels of Strategic Contracts and Their Contents in a Managerial Mixed Duopoly

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Abstract

This study considers a game in which both, the timing of setting the levels of strategic contracts and their contents, are endogenized in a managerial mixed duopoly composed of a public firm with a welfare-maximizing owner and a private firm with a profit-maximizing owner. We suppose that the managers of both, the public firm and the private firm, adopt sales delegation contracts that are equal to the weighted sum of their profits and sales revenue with respect to the incentive parameters. We show that the market structure such that the manager of the public firm with a quantity contract is the follower and the manager of the private firm with a price contract is the leader can become the unique equilibrium market structure in the game in which both, the timing of setting the levels of strategic contracts and their contents, are endogenized in a managerial mixed duopoly. In addition, highest social welfare can be achieved in such a unique equilibrium market structure. Therefore, it is not necessary for the relevant authority, including the government, to regulate free decisions on the timing of the levels of the strategic contracts for the owners of both, the public firm and the private firm.

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Notes

  1. In the context of a private oligopoly, Singh and Vives (1984) and Klemperer and Meyer (1986) find that both firms choose quantity (price) contracts when the goods are substitutes (complements).

  2. For example, Maggi (1996) develops a model of capacity price competition, where the equilibrium outcomes range from Bertrand to Cournot outcomes as capacity constraints from the perspective of strategic trade policy theory. Thisse and Vives (1998) examine the implications of the strategic choice of spatial price policies, including a commitment to a uniform pricing policy before actual price competition takes place.

  3. This model was later extended by Scrimitore (2013), Haraguchi and Matsumura (2014, 2016). Note that Haraguchi and Matsumura (2016) also considered a situation in which more than one public firm exists. In addition, Nakamura (2015b) examines each firm’s endogenous choice of a price or quantity contract in a mixed duopoly with a social welfare-maximizing public firm and a profit-maximizing private firm.

  4. For example, Chirco and Scrimitore (2013) use FJSV delegation to endogenize the effect of a firm’s choice of strategic contract on the product market in the presence of network externalities, à la Katz and Shapiro (1985) and Hoernig (2012), in a managerial private differentiated-goods duopoly. Nakamura (2017a) studies the endogenous choices of strategic contracts in a managerial private duopoly, where the owner and manager bargain over the content of the managerial delegation contract. Using the same approach, Nakamura (2017b) investigates firms’ endogenous choices of strategic contracts in a managerial private duopoly with asymmetric demand functions. Here, the managerial contracts are again determined through bargaining between the owner and the manager.

  5. Bárcena-Ruiz (2007) investigates the endogenous timing of setting prices in a differentiated-goods mixed duopoly composed of one public firm and one private firm. Bárcena-Ruiz and Garzón (2010), Matsumura and Ogawa (2010), and Méndez-Naya (2015) investigate an endogenous timing game in a mixed oligopoly with a partially (semi-)privatized firm, as first introduced in Matsumura (1998). In another strand of timing games in the context of mixed oligopolies, Balogh and Tasnádi (2012) explore a homogeneous-good price-setting mixed duopoly with capacity constraints. They find that \(\left (1 \right )\) the order of moves does not matter, \(\left (2 \right )\) social welfare is higher in the mixed version of the Bertrand—Edgeworth game than it is in the standard version with private firms only, and \(\left (3 \right )\) the mixed version of the Bertrand—Edgeworth duopoly game has an equilibrium in pure strategies for any capacity level. Subsequently, Bakó and Tasnádi (2017) extended the results of Kreps and Scheinkman (1983) to mixed duopolies. In doing so, they show that a quantity pre-commitment and Bertrand competition yield Cournot outcomes, not only in the case of private firms, but also when a public firm is included in the same industry.

  6. Japan provides a recent example of the same managerial delegation contract being used for a private firm and a public firm as private finance initiative activities. This corresponded to the private delivery of business activities from a (local) public firm to a private firm in a water supply business in Tokyo, Yokohama, and Fukushima, and in the gas industry in Gunma. For managerial mixed duopolies, studies on managerial delegation contracts in public firms include the works of Barros (1995), White (2001), Heywood and Ye (2009), and Tomaru et al. (2011).

  7. Similarly, Nakamura (2018b) combines the endogenous choice of when firms set their incentive parameters and strategic contracts in a managerial private duopoly with FJSV delegation.

  8. Whereas we focus on the endogenous timing of setting the levels of the strategic contracts in a managerial mixed duopoly, Nakamura (2018c) endogenizes the timing in their FJSV incentive parameter. As such, Nakamura (2018c) employs an observable delay game, where the owners of the public and private firms announce the timing of setting their FJSV incentive parameters and the content of their strategic contracts in the pre-play stage.

  9. In the game in which there is no equilibrium market structure under the pure strategic contract class, we attempt to derive a mixed-strategy equilibrium market structure when the contents of the contracts are fixed, but the levels are determined endogenously in both firms.

  10. In the Supplementary Material, we provide the market outcomes, including the incentive parameters of firms 0 and 1 and the payoffs of their owners, as well as their rankings in the four games and the three cases. Moreover, we describe the process of deriving the market outcomes for the four games and the three cases. Then, we present the relation between the strategic contracts of the two firms in the third stage and between their FJSV incentive parameters in the second stage for the four games and three cases.

  11. Therefore, when we derive the payoff of the owner of firm i, we can ignore the part of Ai + BiMi corresponding to the wage of his manager. This is a standard assumption when assuming a mixed oligopoly with an FJSV delegation, in which the owners of firms delegate decisions to their managers (see Barros 1995; White 2001; Bárcena-Ruiz 2009, 2013).

  12. Throughout this paper, we denote α − ci as ai, \(\left (i = 0, 1 \right )\). Thus, ai represents the efficiency of firm i, where firm i becomes more efficient as ai increases. Therefore, the sign of the difference between ai and aj indicates the efficiency of firm i relative to that of firm j, \(\left (i, j = 0, 1; i \neq j \right )\). Moreover, we assume that \(\left |a_{i} - a_{j} \right |\) is sufficiently small in order for the equilibrium market outcomes to be positive in all games when the goods are substitutes or complements, \(\left (i, j = 0, 1; i \neq j \right )\). More precisely, if we assume a ≡ a0 = a1 by supposing α − c0 = α − c1 ⇔ c0 = c1, our analyses and results are adequate. Moreover, when δ is sufficiently close to zero, our analyses and results are appropriate, even if the difference between a1 and a0 is relatively large. Furthermore, when δ approaches one, a1 − δa0 must hold (e.g., the quantity of firm 1, \(q^{SSmm}_{1qq}\), is strictly positive) in order for the analyses and results to be persistent. Consequently, even though δ is sufficiently near one, \(\left |a_{i} - a_{j} \right |\) must be sufficiently near zero (implying almost no difference in product efficiency between the firms) in order for the market structures in the three cases in all four games to be strictly positive, \(\left (i, j = 0, 1; i \neq j \right )\).

  13. In the Supplementary Material, we give all the market outcomes, except for the FJSV incentive parameters of firms 0 and 1 and the payoffs of their owners in the q–q game.

  14. In the Supplementary Material, we give the ranking orders of the equilibrium market outcomes other than the payoffs of the owners of firms 0 and 1 for the three cases in the q–q game. Nakamura and Inoue (2007) consider the q–q game and the timing of setting the quantities for firms i and j when the managerial contract of firm i is \(M^{mm}_{iqq} = \pi ^{mm}_{iqq} + \theta ^{mm}_{iqq} q^{mm}_{i}\) in a managerial mixed duopoly, for \(\left (i, j = 0, 1; i \neq j \right )\). Thus, the ranking orders of the equilibrium market outcomes in the q–q game in this study are the same as those in Nakamura and Inoue (2007). Note that Nakamura and Inoue (2007) limit their analysis to the case when the goods are substitutes.

  15. More precisely, we obtain that \(CS^{LFmm}_{qq} > CS^{SSmm}_{qq} > CS^{FLmm}_{qq}\) if \(\delta \in \left (0, 1 \right )\), whereas \(CS^{FLmm}_{qq} > CS^{LFmm}_{qq} > CS^{SSmm}_{qq}\) if \(\delta \in \left (- 1, 0 \right )\). On the other hand, we have that \(PS^{FLmm}_{qq} > PS^{SSmm}_{qq} > PS^{LFmm}_{qq}\) if \(\delta \in \left (0, 1 \right )\), whereas \(PS^{LFmm}_{qq} > PS^{SSmm}_{qq} > PS^{FLmm}_{qq}\) if \(\delta \in \left (- 1, 0 \right )\).

  16. More precisely, we obtain \(p^{FLmm}_{1qq} > p^{SSmm}_{1qq} > p^{LFmm}_{1qq}\) if \(\delta \in \left (- 1, 0 \right ) \cup \left (0, 1 \right )\).

  17. Note that Nakamura and Inoue (2007) use \(M^{mm}_{iqq} \left (q_{i}, q_{j}; \theta _{i} \right ) = \pi ^{mm}_{iqq} \left (q_{i}, q_{j} \right ) + \theta _{i} q_{i}\) as the managerial delegation contracts adopted by firms 0 and 1. In addition, Nakamura and Inoue (2007) limit the degree of product differentiation (i.e., the area of b) to \(b \in \left [0, 1 \right ]\).

  18. In the Supplementary Material, we give all the market outcomes except for the FJSV incentive parameters of firms 0 and 1 and the payoffs of their owners in the p–p game.

  19. In the Supplementary Material, we give the ranking orders of the equilibrium market outcomes other than the payoffs of the owners of firms 0 and 1 for the three cases in the p–p game. In particular, Nakamura and Inoue (2009) consider the p–p game and the timing of setting the quantities of firms i and j when the managerial contract of firm i is \(M^{mm}_{ipp} \left (p_{i}, p_{j}; \theta _{i} \right ) = \pi ^{mm}_{ipp} \left (p_{i}, p_{j} \right ) + \theta _{i} q^{mm}_{ipp} \left (p_{i}, p_{j} \right )\), \(\left (i, j = 0, 1; i \neq j \right )\). Thus, the ranking orders of the equilibrium market outcomes in the p–p game in this study are the same as those in Nakamura and Inoue (2009). Note that Nakamura and Inoue (2009) limit their analysis to the case when the goods are substitutes, which is similar to the q–q game considered in Nakamura and Inoue (2007).

  20. More precisely, we obtain \(CS^{FLmm}_{pp} \geq CS^{SSmm}_{pp} > CS^{LFmm}_{pp}\) if \(\delta \in \left [- 0.897962, 0 \right )\), and \(CS^{LFmm}_{pp} > CS^{SSmm}_{pp} > CS^{FLmm}_{pp}\) otherwise. On the other hand, we have that \(PS^{LFmm}_{pp} > PS^{SSmm}_{pp} > PS^{FLmm}_{pp}\) if \(\delta \in \left (0, 1 \right )\), whereas \(PS^{FLmm}_{pp} > PS^{SSmm}_{pp} > PS^{LFmm}_{pp}\) if \(\delta \in \left (-1, 0 \right )\).

  21. In the Supplementary Material, we give all the market outcomes, except for the FJSV incentive parameters of firms 0 and 1 and the payoffs of their owners in the p–q game.

  22. In the Supplementary Material, we give the ranking orders of the equilibrium market outcomes other than the payoffs of the owners of firms 0 and 1 for the three cases in the p–q game.

  23. In the Supplementary Material, we give all the market outcomes other than the FJSV incentive parameters of firms 0 and 1 and the payoffs of their owners in the q–p game.

  24. In the Supplementary Material, we give the ranking orders of the equilibrium market outcomes other than the payoffs of the owners of firms 0 and 1 for the three cases in the q–p game.

  25. More precisely, we obtain \(q^{FLmm}_{1qp} > q^{SSmm}_{1qp} > q^{LFmm}_{1qp}\) if \(\delta \in \left (-1, 0\right ) \cup \left (0, 1 \right )\), whereas \(p^{LFmm}_{1qp} = p^{FLmm}_{1qp} > p^{SSmm}_{1qp}\) if \(\delta \in \left (- 1, 0 \right ) \cup \left (0, 1 \right )\).

  26. More precisely, we obtain \(PS^{FLmm}_{qp} > PS^{LFmm}_{qp} > PS^{SSmm}_{qp}\) if \(\delta \in \left (0, 1 \right )\), whereas \(CS^{FLmm}_{qp} > CS^{SSmm}_{qp} > CS^{LFmm}_{qp}\) if \(\delta \in \left (-1, 0 \right )\).

  27. Nakamura (2018c) consider the observable delay game in which the timing of setting the FJSV incentive parameters of firms 0 and 1 and the contents of their strategic contracts are announced endogenously by their owners in the pre-play game.

  28. Din and Sun (2016) deal with the same discussion as that in Matsumura and Ogawa (2012).

  29. Sun (2013) shows that a unique equilibrium market structure under the pure strategic contract class is composed of a pair of strategies of the private firms \(\left ({q^{1}_{0}}, {q^{1}_{1}} \right )\), which is equal to the q–q game with the simultaneous setting of the levels of the strategic contracts in the first period.

  30. This type of delegation contract can also be used for a socially responsible firm.

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Acknowledgments

We are grateful to the two anonymous referees for their feedback. All remaining errors are our own. This study was conducted with the financial support of the Seimeikai Foundation (16-002), KAKENHI (16H03624), and KAKENHI (16K03665).

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  1. College of Economics, Nihon University, 1-3-2 Misaki-cho, Chiyoda-ku, Tokyo, 101-8360, Japan

    Yasuhiko Nakamura

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Nakamura, Y. Combining the Endogenous Choice of the Timing of Setting the Levels of Strategic Contracts and Their Contents in a Managerial Mixed Duopoly. J Ind Compet Trade 19, 235–261 (2019). https://doi.org/10.1007/s10842-018-0286-3

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