Endogenous Choice of the Timing of Setting Incentive Parameters and the Strategic Contracts in a Managerial Mixed Duopoly with a Welfare-Based Delegation Contract and a Sales Delegation Contract

Abstract

This study explores the game in which both the timing of setting the welfare-based incentive parameter of a public firm with a welfare-maximizing owner and the sales incentive parameter of a private firm and the contents of their strategic contracts in a managerial mixed duopoly with a welfare-based delegation à la (Nakamura in Int Rev Econ Financ 35:262–277, 2015) and a sales delegation are endogenously determined by their owners. We show two market structures: (1) the market structure in which the owner of the public (private) firm with a quantity contract is the leader (follower) in the determination of the incentive parameters and (2) the market structure in which the owner of the public (private) firm with a quantity (price) contract is the follower (leader) in the determination of the incentive parameters. The equilibrium market structures and highest social welfare are achieved in market structure (1). Therefore, in a managerial mixed duopoly with a welfare-based delegation and a sales delegation, it is not as necessary for the relevant authority including the government to regulate the free determination of both the timing of setting the welfare-based incentive parameter of the public firm and the sales incentive parameter of the private firm and the contents of their strategic contracts by their owners. The payoffs of the owners of both the public firm and the private firm stay relatively low in market structure (2), another equilibrium market structure. Finally, we confirm the robustness of the results on the equilibrium market structures against the change in the cost functions of both the public firm and the private firm from their constant marginal costs to their increasing marginal costs on the basis of the quadratic cost functions with their quantities.

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Notes

  1. 1.

    In the context of a managerial private oligopoly composed of private firms with profit-maximizing owners, Fershtman et al. (1991), Polo and Tedeschi (1992), Bárcena-Ruiz and Paz Espinoza (1999), Lambertini (2000a), and Lambertini (2000b) also considered the interplay between market competition structures and internal organizational structures à la Fershtman and Judd (1987), Sklivas (1987), and Vickers (1985).

  2. 2.

    The theoretical literature has also proposed numerous developments in the context of a private oligopoly. For instance, Amir (1995), Sadanand and Sadanand (1996), and Van Damme and Hurkens (1999) strengthened the statement in Hamilton and Slutsky (1990) by taking the monotone condition, demand uncertainty, and the notion of risk dominance into account. In addition, Amir and Grilo (1999) and Amir (2005) considered multiple Nash equilibria in a simultaneous game and used the theory of supermodular games, respectively.

  3. 3.

    Matsumura and Ogawa (2014) considered the importance of the change in the (a)symmetry of the objective function between public and private firms by introducing the effect of corporate social responsibility. Most recently, Lee and Xu (2017) examined an endogenous timing game in the contexts of both private and mixed duopolies under price competition with differentiated goods when each firm’s emission tax is imposed on the environmental externality.

  4. 4.

    Before, Nakamura (2018a, 2018b) considered the endogenous timing game of setting the incentive parameter by each firm’s owner on the basis of the observable delay game, focusing on admitting that the strategic contracts of the two firms are different from each other when the contents of their strategic contracts are fixed.

  5. 5.

    In this paper, we further confirm the robustness of the occurring equilibrium market structure(s) along with the change in the cost functions of the public firm and the private firm from their constant marginal costs to their increasing marginal costs on the basis of the quadratic cost functions with respect to their quantities. We find that the change in the cost functions from their constant marginal costs to their increasing marginal costs yields the different policy implications of the regulation policy for the endogenous determination of (1) the timing of setting the welfare-based incentive parameter of firm 0 and the FJSV incentive parameter and (2) the contents of their strategic contracts by their owners from the viewpoint of social welfare, since the figures of the equilibrium market structures strikingly depend on the homogeneity of the goods produced by both the public firm and the private firm when their technologies are represented as quadratic cost functions with respect to their quantities in the full observable delay game such that the owners endogenously determine the factors of (1) and (2).

  6. 6.

    In the game with respect to the strategic contracts of both the public firm and the private firm in which no equilibrium market structure exists under the pure strategic contract class, we attempt to derive a mixed strategy equilibrium market structure in the game in which only the timing of setting the welfare-based incentive parameter of the public firm and the FJSV incentive parameter of the private firm is endogenously determined by their owners in the case in which the contents of their strategic contracts are fixed.

  7. 7.

    For the market outcomes to be strictly positive in all the situations, we assume that the interval of δ, which means the degree of the substitutability of the goods of firms 0 and 1, is restricted to (0, 1).

  8. 8.

    In general, the consumer surplus is defined as \(CS = \alpha \left (q_{0} + q_{1} \right ) - \left [\left ({{q}^{2}_{0}} + 2 \delta q_{0} q_{1} + {{q}^{2}_{1}} \right ) / 2 \right ] - \left (p_{0} q_{0} + p_{1} q_{1} \right )\) and social welfare is defined as the sum of the consumer surplus and producer surplus, (i, j = q, p).

  9. 9.

    The value of the delegation parameter of firm 0, 𝜃0ij, emphasizes the consumer surplus relative to the producer surplus, (i, j = p, q). Thus, we can regard the presence of firm 0 as a consumer-friendly firm or a socially responsible firm, which implies that the welfare-based incentive parameter of 𝜃0ij within its managerial contract M0ij means the importance of the consumer surplus relative to the producer surplus within social welfare for its owner including the government, (i, j = p, q). Following this approach, in the fashion of Goering (2012), Kopel and Brand (2013), and Nakamura (2015, 2019), in this paper, in a managerial mixed duopoly with a consumer-friendly firm (firm 0) with a welfare-maximizing owner and a private firm (firm 1) with a profit-maximizing owner, we consider the equilibrium market structures in the full observable delay game such that their owners endogenously determine (1) the timing of setting the welfare-based incentive parameter of firm 0 and the FJSV incentive parameter of firm 1 and (2) the contents of their strategic contracts.

  10. 10.

    This is a standard assumption when assuming a mixed oligopoly with an FJSV delegation in which the owners of firms delegate decisions to managers. See Barros (1995), White (2001), and Barcéna-Ruiz (2009, 2013) for a detailed discussion on this.

  11. 11.

    w denotes the market outcomes in the three cases, namely the simultaneous determination of the welfare-based incentive parameter of firm 0 and the FJSV incentive parameter of firm 1, SSw, and the two types of sequential determination of their incentive parameters, LFw and FLw, in a managerial mixed duopoly with a welfare-based delegation and an FJSV delegation.

  12. 12.

    We recognize that ai represents the efficiency of firm i in this paper such that firm i becomes more efficient as ai rises. Thus, we can understand that the sign of the difference in ai, and aj indicates the efficiency of firm i relative to that of firm j, (i, j = 0, 1; ij). Moreover, we assume that |aiaj| is sufficiently small in order for the equilibrium market outcomes to be positive in all the games, (i, j = 0, 1; ij).

  13. 13.

    Thus, in the q-q game, the effect of the increases in the welfare-based incentive parameter of firm 0 and the FJSV incentive parameter of firm 1 on the respective quantities are the opposite of each other. More concretely, in the q-q game, the owner of firm 0 makes his manager more aggressive by increasing his welfare-based incentive parameter, while the owner of firm 0 makes his manager less aggressive by increasing his FJSV incentive parameter.

  14. 14.

    In the Supplementary Material of this paper, the derivation of the equilibrium market outcomes are given in detail in the q-q game for the three cases, namely the SSw case, LFw case, and FLw case.

  15. 15.

    We provide concrete values of the equilibrium market outcomes except for the welfare-based incentive parameter of firm 0 and the FJSV incentive parameter and the payoffs of their owners in the q-q game in the Appendix.

  16. 16.

    Thus, in the p-p game, similar to the q-q game, the effect of the increases in the welfare-based incentive parameter of firm 0 and the FJSV incentive parameter of firm 1 on the respective quantities are the opposite of each other. More concretely, similar to the q-q game, also in the p-p game, the owner of firm 0 makes his manager more aggressive by increasing his welfare-based incentive parameter, while the owner of firm 0 makes his manager less aggressive by increasing his FJSV incentive parameter.

  17. 17.

    In the Supplementary Material of this paper, the derivation of the equilibrium market outcomes are given in detail in the p-p game for the three cases, namely the SSw case, LFw case, and FLw case.

  18. 18.

    We provide concrete values of the equilibrium market outcomes except for the welfare-based incentive parameter of firm 0 and the FJSV incentive parameter and the payoffs of their owners in the p-p game in the Appendix.

  19. 19.

    Thus, in the p-q game, similar to both the q-q game and the p-p game, the effect of the increases in the welfare-based incentive parameter of firm 0 and the FJSV incentive parameter of firm 1 on its price and its quantity, respectively, are the opposite of each other. More concretely, similar to both the q-q game and the p-p game, also in the p-q game, the owner of firm 0 makes his manager more aggressive by increasing his welfare-based incentive parameter, while the owner of firm 0 makes his manager less aggressive by increasing his FJSV incentive parameter.

  20. 20.

    In the Supplementary Material of this paper, the derivation of the equilibrium market outcomes are given in detail in the p-q game for the three cases, namely the SSw case, LFw case, and FLw case.

  21. 21.

    In the p-q game, the FJSV incentive parameter of firm 1 does not depend on the welfare-based incentive parameter of firm 0 in \(\theta ^{SSw}_{1pq} \left (\theta _{0} \right ) = R^{SSw}_{1pq} \left (\theta _{0} \right )\), which is \(\theta ^{SSw}_{1pq} = 1\) for any δ ∈ (0, 1).

  22. 22.

    We provide concrete values of the equilibrium market outcomes except for the welfare-based incentive parameter of firm 0 and the FJSV incentive parameter and the payoffs of their owners in the p-q game in the Appendix.

  23. 23.

    Thus, in the q-p game, similar to the other three games, namely the q-q game, p-p game, and p-q game, the effect of the increases in the welfare-based incentive parameter of firm 0 and the FJSV incentive parameter of firm 1 on its quantity and its price, respectively, are the opposite of each other. More concretely, similar to the other three games, namely the q-q game, p-p game, and p-q game, in the q-p game, the owner of firm 0 makes his manager more aggressive by increasing his welfare-based incentive parameter, while the owner of firm 0 makes his manager less aggressive by increasing his FJSV incentive parameter.

  24. 24.

    In the Supplementary Material of this paper, the derivation of the equilibrium market outcomes are given in detail in the q-p game for the three cases, namely the SSw case, LFw case, and FLw case.

  25. 25.

    We provide concrete values of the equilibrium market outcomes except for the welfare-based incentive parameter of firm 0 and the FJSV incentive parameter and the payoffs of their owners in the q-p game in the Appendix.

  26. 26.

    In Nakamura (2015), in a mixed strategy equilibrium in the simultaneous determination of the welfare-based incentive of firm 0 and the FJSV incentive parameter of firm 1 by their owners, he considered the probability that the owner of firm i chooses his price contract.

  27. 27.

    Throughout this paper, the subscript wk denotes the situation in which the technologies of firms 0 and 1 are represented as quadratic cost functions with respect to their quantities.

  28. 28.

    The FJSV incentive parameter of firm 1 in the type of its managerial contract, namely \(M^{wk}_{1ij} \left (\pi _{1ij}, S_{1ij}; \theta _{1ij} \right ) = \theta _{1ij} {\pi }^{wk}_{1ij} + \left (1 - \theta _{1ij} \right ) S^{wk}_{1ij}\) where \(S^{wk}_{1ij} = p_{1ij} q_{1ij}\), which has been employed until Section 4, implies the less aggressiveness of the manager of firm 1 in the market, while the slightly different FJSV incentive parameter in the managerial contract of firm 1, which is employed in this section, implies the (more) aggressiveness of its manager in the market, (i, j = p, q).

  29. 29.

    The superscript ∗ denotes the possibilities that the owners of firms 0 and 1 choose the first period in the second stage as the timing of setting the welfare-based incentive parameter and FJSV incentive parameter, respectively, in the equilibrium mixed strategy market structures. In addition, the concrete values of \(z^{*}_{0pp}\) and \(z^{*}_{1pp}\) when δ ∈{1/6, 1/4, 1/3, 1/2, 2/3, 3/4, 5/6, 7/8, 9/10} are given in the Supplementary Material of this paper.

  30. 30.

    When δ = 1/4, the q-q game, one of the equilibrium market structures, can become the socially preferable market structure from the viewpoint of social welfare. Therefore, in the LFwk case, including δ = 1/4, it is not as necessary for the relevant authority including the government to regulate the endogenous determination of the contents of the strategic contracts of firms 0 and 1 by their respective owners from the viewpoint of social welfare.

  31. 31.

    When δ = 1/4, the q-p game, one of the equilibrium market structures, coincides with the socially preferable market structure from the viewpoint of social welfare, while the highest social welfare is not achieved in the q-p game, which is the other equilibrium market structure.

  32. 32.

    When the level of δ is around 7/8, \(\left ({{p}^{1}_{0}}, {{q}^{2}_{1}} \right )\) and \(\left ({{q}^{2}_{0}}, {{p}^{1}_{1}} \right )\) can become equilibrium market structures exceptionally.

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Acknowledgments

We thank an anonymous referee and the Editor-in-Chief Professor Kai Hueschelrath for helpful comments and suggestions. All remaining errors are our own.

Funding

We are grateful for the financial support of the Seimeikai Foundation (16-002), KAKENHI (16H03624) and KAKENHI (16K03665).

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Correspondence to Yasuhiko Nakamura.

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Appendix: Market Outcomes Except for the Welfare-Based Incentive Parameter of Firm 0 and the FJSV Incentive of Firm 1 in the Four Games

Appendix: Market Outcomes Except for the Welfare-Based Incentive Parameter of Firm 0 and the FJSV Incentive of Firm 1 in the Four Games

q-q Game in the Constant Marginal Cost Function Case

Prices of Firms 0 and 1 in the Three Cases in the q-q Game

We provide the prices of firms 0 and 1 in the three cases in the q-q game as follows:

$$\begin{array}{@{}rcl@{}} &&\text{SSw case} \left\{\begin{array}{l} {p}^{SSw}_{0qq} = c_{0} + \frac{\delta (a_{1} - a_{0} \delta)}{4 - \delta^{2}}, \\ {p}^{SSw}_{1qq} = c_{1} + \frac{2 (a_{1} - a_{0} \delta)}{4 - \delta^{2}}. \end{array}\right. \\ &&\text{LFw case} \left\{\begin{array}{l} {p}^{LFw}_{0qq} = c_{0} + \frac{\delta (a_{1} - a_{0} \delta)}{4 - 3 \delta^{2}}, \\ {p}^{LFw}_{1qq} = c_{1} + \frac{(a_{1} - a_{0} \delta) (2 - \delta^{2})}{4 - 3 \delta^{2}}. \end{array}\right. \\ &&\text{FLw case} \left\{\begin{array}{l} {p}^{FLw}_{0qq} = c_{0} + \frac{\delta (a_{1} - a_{0} \delta)}{2 (2 - \delta^{2})}, \\ {p}^{FLw}_{1qq} = c_{1} + \frac{a_{1} - a_{0} \delta}{2 - \delta^{2}}. \end{array}\right. \end{array} $$

Quantities of Firms 0 and 1 in the Three Cases in the q-q Game

We provide the quantities of firms 0 and 1 in the three cases in the q-q game as follows:

$$\begin{array}{@{}rcl@{}} &&\text{SSw case} \left\{\begin{array}{l} {q}^{SSw}_{0qq} = \frac{2 a_{0} (2 - \delta^{2}) - a_{1} \delta (3 - \delta^{2})}{\beta (4 - 5 \delta^{2} + \delta^{4})}, \\ {q}^{SSw}_{1qq} = \frac{2 (a_{1} - a_{0} \delta)}{\beta (4 - 5 \delta^{2} + \delta^{4})}. \end{array}\right. \\ &&\text{LFw case} \left\{\begin{array}{l} {q}^{LFw}_{0qq} = \frac{a_{0} (2 - \delta^{2})^{2} - a_{1} \delta (3 - 2 \delta^{2})}{\beta (1 - \delta^{2}) (4 - 3 \delta^{2})}, \\ {q}^{LFw}_{1qq} = \frac{(a_{1} - a_{0} \delta) (2 - \delta^{2})}{\beta (1 - \delta^{2}) (4 - 3 \delta^{2})}. \end{array}\right. \\ &&\text{FLw case} \left\{\begin{array}{l} {q}^{FLw}_{0qq} = \frac{a_{0} (4 - 3 \delta^{2}) - a_{1} \delta (3 - 2 \delta^{2})}{\beta(4 - 6 \delta^{2} + 2 \delta^{4})}, \\ {q}^{FLw}_{1qq} = \frac{a_{1} - a_{0} \delta}{2 \beta(1 - \delta^{2})}. \end{array}\right. \end{array} $$

Consumer Surplus in the Three Cases in the q-q Game

We provide the consumer surplus in the three cases in the q-q game as follows:

$$\left\{\begin{array}{ll} &[\text{SSw case}] {CS}^{SSw}_{qq} = \frac{4 {a_{0}^{2}} (4 - 3 \delta^{2}) - 8 a_{0} a_{1} \delta (2 - \delta^{2}) + {a_{1}^{2}} (4 + \delta^{2} - \delta^{4})}{2 \beta (4 - \delta^{2})^{2} (1 - \delta^{2})}, \\ &[\text{LFw case}] {CS}^{LFw}_{qq} = \frac{{a_{0}^{2}} (2 - \delta^{2})^{2} - 2 a_{0} a_{1} \delta (2 - \delta^{2}) + {a_{1}^{2}}}{2 \beta (1 - \delta^{2}) (4 - 3 \delta^{2})}, \\ &[\text{FLw case}] CS^{FLw}_{qq} = \frac{{a_{0}^{2}} (16 - 20 \delta^{2} + 5 \delta^{4}) - 2 a_{0} a_{1} \delta (8 - 9 \delta^{2} + 2 \delta^{4}) + {a_{1}^{2}} (4 - 3 \delta^{2})}{8 \beta (2 - \delta^{2})^{2} (1 - \delta^{2})}. \end{array}\right. $$

Producer Surplus in the Three Cases in the q-q Game

We provide the producer surplus in the three cases in the q-q game as follows:

$$\left\{\begin{array}{ll} &[\text{SSw case}] PS^{SSw}_{qq} = \frac{2 {a_{0}^{2}} \delta^{4} - a_{0} a_{1} \delta (4 - \delta^{2} + \delta^{4}) + {a_{1}^{2}} (4 - 3 \delta^{2} +\delta^{4})}{\beta (4 - \delta^{2})^{2} (1 - \delta^{2})}, \\ &[\text{LFw case}] PS^{LFw}_{qq} = \frac{a_{1} (a_{1} - a_{0} \delta)}{\beta (4 - 3 \delta^{2})}, \\ &[\text{FLw case}] PS^{FLw}_{qq} = \frac{(a_{1} - a_{0} \delta) \left[a_{1} (4 - 5 \delta^{2} + 2 \delta^{4}) - a_{0} \delta^{3} \right]}{4 \beta (2 - \delta^{2})^{2} (1 - \delta^{2})}. \end{array}\right. $$

Social Welfare in the Three Cases in the q-q Game

We provide social welfare in the three cases in the q-q game as follows:

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} &[\text{SSw case}] \!W^{SSw}_{qq} = \frac{4 {a_{0}^{2}} (4 - 3 \delta^{2} + \delta^{4}) - 2 a_{0} a_{1} \delta (12 - 5 \delta^{2} + \delta^{4}) + {a_{1}^{2}} (12 - 5 \delta^{2} + \delta^{4})}{2 \beta (4 - \delta^{2})^{2} (1 - \delta^{2})}, \\ &[\text{LFw case}] \!W^{LFw}_{qq} = \frac{+ {a_{0}^{2}} (2 - \delta^{2})^{2} - 2 a_{0} a_{1} \delta (3 - 2 \delta^{2}) + {a_{1}^{2}} (3 - 2 \delta^{2})}{2 \beta (1 - \delta^{2}) (4 - 3 \delta^{2})}, \\ &[\text{FLw case}] \!{W}^{FLw}_{qq} = \frac{{{a}_{0}^{2}} (16 - 20 \delta^{2} + 7\delta^{4}) - 2 {a}_{0} a_{1}\delta (12 - 13 \delta^{2} + 4 \delta^{4}) + {a_{1}^{2}} (12 - 13\delta^{2} + 4\delta^{4})}{8 \beta (2 - \delta^{2})^{2} (1 - \delta^{2})}. \end{array}\right. \end{array} $$

p-p Game in the Constant Marginal Cost Function Case

Prices of Firms 0 and 1 in the Three Cases in the p-p Game

We provide the prices of firms 0 and 1 in the three cases in the p-p game as follows:

$$\begin{array}{@{}rcl@{}} &&\text{SSw case} \left\{\begin{array}{l} p_{0pp}^{SSw} = c_{0} + \frac{\delta (a_{1} - a_{0} \delta)}{4 - 5 \delta^{2} + \delta^{4}}, \\ {p}_{1pp}^{SSw} = c_{1} + \frac{(a_{1} -a_{0} \delta) (2 - \delta^{2})}{4 - 5 \delta^{2} + \delta^{4}}. \end{array}\right.\\ &&\text{LFw case} \left\{\begin{array}{l} p_{0pp}^{LFw} = c_{0} + \frac{\delta (a_{1} - a_{0} \delta)}{4 - 3 \delta^{2}}, \\ {p}_{1pp}^{LFw} = c_{1} + \frac{2 (a_{1} - a_{0} \delta)}{4- 3 \delta^{2}}. \end{array}\right.\\ &&\text{FLw case} \left\{\begin{array}{l} {p}^{FLw}_{0pp} = c_{0} + \frac{\delta (a_{1} - a_{0} \delta)}{4 (1 - \delta^{2})}, \\ {p}^{FLw}_{1pp} = c_{1} + \frac{(a_{1} - a_{0} \delta) (2- \delta^{2})}{4 (1 - \delta^{2})}. \end{array}\right. \end{array} $$

Quantities of Firms 0 and 1 in the Three Cases in the p-p Game

We provide the quantities of firms 0 and 1 in the three cases in the p-p game as follows:

$$\begin{array}{@{}rcl@{}} &&\text{SSw case} \left\{\begin{array}{l} {q}^{SSw}_{0pp} = \frac{2 a_{0} (2 - \delta^{2}) - a_{1} \delta (3 - \delta^{2})} {\beta (4 - 5 \delta^{2} + \delta^{4})}, \\ {q}^{SSw}_{1pp} = \frac{(a_{1} - a_{0} \delta) (2 - \delta^{2})}{\beta (4 - \delta^{2}) (1 - \delta^{2})}. \end{array}\right. \\ &&\text{LFw case} \left\{\begin{array}{l} {q}^{LFw}_{0pp} = \frac{a_{0} (2 - \delta^{2})^{2} - a_{1} \delta (3 - 2 \delta^{2})}{\beta (1 - \delta^{2}) (4 - 3 \delta^{2})}, \\ {q}^{LFw}_{1pp} = \frac{(a_{1} - a_{0} \delta) (2 - \delta^{2})}{\beta (1 - \delta^{2}) (4 - 3 \delta^{2})}. \end{array}\right. \\ &&\text{FLw case} \left\{\begin{array}{l} {q}^{FLw}_{0pp} = - \frac{3 a_{1} \delta + a_{0} (-4 + \delta^{2})}{4 \beta (1 - \delta^{2})}, \\ {q}^{FLw}_{1pp} = \frac{a_{1} - a_{0} \delta}{2 \beta (1 - \delta^{2})}. \end{array}\right. \end{array} $$

Consumer Surplus in the Three Cases in the p-p Game

We provide the consumer surplus in the three cases in the p-p game as follows:

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} &[SSw \text{case}] \!CS^{SSw}_{pp} = \frac{{a_{0}^{2}} (2 - \delta^{2})^{2} (4 -\! 3 \delta^{2}) - 2 a_{0} a_{1} \delta (2 - \delta^{2})^{3} \!+ {a_{1}^{2}} (4 - 7 \delta^{2} + 5 \delta^{4} - \delta^{6})}{2 \beta (4 - 5 \delta^{2} + \delta^{4})^{2}}, \\ &[LFw~ \text{case}] \!CS^{LFw}_{pp} = \frac{4 {a_{0}^{2}} - 4 a_{0} a_{1} \delta + {a_{1}^{2}}}{2 \beta (4 - 3 \delta^{2})}, \\ &[FLw~ \text{case}] CS^{FLw}_{pp} = \frac{{a_{0}^{2}} (16 - 20 \delta^{2} + 5 \delta^{4}) - 2 a_{0} a_{1} \delta (8 - 7 \delta^{2}) + {a_{1}^{2}} (4 - 3 \delta^{2})}{32 \beta (1 - \delta^{2})^{2}}. \end{array}\right. \end{array} $$

Producer Surplus in the Three Cases in the p-p Game

We provide the producer surplus in the three cases in the p-p game as follows:

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} &[SSw~ \text{case}] PS^{SSw}_{pp} = \frac{{a_{1}^{2}} (4 - 7 \delta^{2} + 2 \delta^{4}) - a_{0} a_{1} \delta (4 - 9 \delta^{2} + 3 \delta^{4}) - {a_{0}^{2}}\delta^{4} (2 - \delta^{2})}{\beta (4 - 5 \delta^{2} + \delta^{4})^{2}}, \\ &[LFw~ \text{case}] PS^{LFw}_{pp} = \frac{a_{1} (a_{1} - a_{0} \delta)}{\beta (4 - 3 \delta^{2})}, \\ &[FLw~ \text{case}] PS^{FLw}_{pp} = \frac{(a_{1} - a_{0} \delta)\left[a_{1} (4 - 5 \delta^{2}) + a_{0} \delta^{3} \right]}{16 \beta (1 - \delta^{2})^{2})}. \end{array}\right. \end{array} $$

Social Welfare in the Three Cases in the p-p Game

We provide social welfare in the three cases in the p-p game as follows:

$$\left\{\begin{array}{ll} &[SSw \text{case}] {W}^{SSw}_{pp} = \frac{{a_{0}^{2}} (16 - 28 \delta^{2} + 12 \delta^{4} - \delta^{6}) - 2 a_{0} a_{1} \delta (12 - 21 \delta^{2} + 9\delta^{4} - \delta^{6}) + {a_{1}^{2}} (12 - 21 \delta^{2} + 9 \delta^{4} - \delta^{6})}{(2 \beta (4 - 5 \delta^{2} + \delta^{4})^{2}}, \\ &[LFw \text{case}] {W}^{LFw}_{pp} = \frac{4 {a_{0}^{2}} - 6 a_{0} a_{1} \delta + 3 {a_{1}^{2}}}{2 \beta (4 - 3 \delta^{2})}, \\ &[FLw \text{case}] {W}^{FLw}_{pp} = \frac{{a_{0}^{2}} (16 - 20 \delta^{2} + 3 \delta^{4}) - 2 a_{0} a_{1} \delta (12 - 13 \delta^{2}) + {a_{1}^{2}} (12 - 13 \delta^{2})}{32 \beta (1 - \delta^{2})^{2}}. \end{array}\right. $$

pq Game in the Constant Marginal Cost Function Case

Prices of Firms 0 and 1 in the Three Cases in the p-q Game

We provide the prices of firms 0 and 1 in the three cases in the p-q game as follows:

$$\begin{array}{@{}rcl@{}} &\text{SSw case} \left\{\begin{array}{l} {p}^{SSw}_{0pq} = c_{0} + \frac{\delta (a_{1} - a_{0} \delta)}{4 - 3 \delta^{2}}, \\ {p}^{SSw}_{1pq} = c_{1} + \frac{(a_{1} - a_{0} \delta) (2 -\delta^{2})}{4 - 3 \delta^{2}}. \end{array}\right. \\ &\text{LFw case} \left\{\begin{array}{l} {p}^{LFw}_{0pq} = c_{0} + \frac{\delta (a_{1} - a_{0} \delta)}{4 - 3 \delta^{2}}, \\ {p}^{LFw}_{1pq} = c_{1} + \frac{(a_{1} - a_{0} \delta) (2 -\delta^{2})}{4 - 3 \delta^{2}}. \end{array}\right. \\ &\text{FLw case} \left\{\begin{array}{l} {p}^{FLw}_{0pq} = c_{0} + \frac{\delta (a_{1} - a_{0} \delta)}{4 (1 - \delta^{2})}, \\ {p}_{1pq}^{FLw} = c_{1} + \frac{(a_{1} - a_{0} \delta) (2- \delta^{2})}{4 (1 - \delta^{2})}. \end{array}\right. \end{array} $$

Quantities of Firms 0 and 1 in the Three Cases in the p-q Game

We provide the quantities of firms 0 and 1 in the three cases in the p-q game as follows:

$$\begin{array}{@{}rcl@{}} &&\text{SSw case} \left\{\begin{array}{l} {q}^{SSw}_{0pq} = \frac{a_{0} (2 - \delta^{2})^{2} - a_{1} \delta (3 - 2 \delta^{2})}{\beta (1 - \delta^{2}) (4 - 3 \delta^{2})}, \\ {q}^{SSw}_{1pq} = \frac{(a_{1} - a_{0} \delta) (2 - \delta^{2})}{\beta (1 - \delta^{2}) (4 - 3 \delta^{2})}. \end{array}\right. \\ &&\text{LFw case} \left\{\begin{array}{l} {q}^{LFw}_{0pq} = \frac{a_{0} (2 - \delta^{2})^{2} - a_{1} \delta (3 - 2 \delta^{2})}{\beta (1 - \delta^{2}) (4 - 3 \delta^{2})}, \\ {q}^{LFw}_{1pq} = \frac{(a_{1} - a_{0} \delta) (2 - \delta^{2})}{\beta (1 - \delta^{2}) (4 - 3 \delta^{2})}. \end{array}\right. \\ &&\text{FLw case} \left\{\begin{array}{l} {q}^{FLw}_{0pq} = \frac{a_{0} (4 - \delta^{2}) - 3 a_{1} \delta}{4 \beta (1 - \delta^{2})}, \\ {q}^{FLw}_{1pq} = \frac{a_{1} - a_{0} \delta}{2 \beta (1 - \delta^{2})}. \end{array}\right. \end{array} $$

Consumer Surplus in the Three Cases in the p-q Game

We provide the consumer surplus in the three cases in the p-q game as follows:

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} &[SSw~ \text{case}] CS^{SSw}_{pq} = \frac{{a_{0}^{2}} (2 - \delta^{2})^{2} - 2 a_{0} a_{1} \delta (2 - \delta^{2}) + {a_{1}^{2}}}{2 \beta (1 - \delta^{2}) (4 - 3 \delta^{2})}, \\ &[LFw~ \text{case}] CS^{LFw}_{pq} = \frac{{a_{0}^{2}} (2 - \delta^{2})^{2} - 2 a_{0} a_{1} \delta (2 - \delta^{2}) + {a_{1}^{2}}}{2 \beta (1 - \delta^{2}) (4 - 3 \delta^{2})}, \\ &[FLw~ \text{case}] CS^{FLw}_{pq} = \frac{{a_{0}^{2}} (16 - 20 \delta^{2} + 5 \delta^{4}) - 2 a_{0} a_{1} \delta (8 - 7 \delta^{2}) + {a_{1}^{2}} (4 - 3 \delta^{2})}{32 \beta (1 - \delta^{2})^{2}}. \end{array}\right. \end{array} $$

Producer Surplus in the Three Cases in the p-q Game

We provide the producer surplus in the three cases in the p-q game as follows:

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} \left[SSw~ \text{case}\right] & PS^{SSw}_{pq} = \frac{a_{1} (a_{1} - a_{0} \delta)}{\beta (4 - 3 \delta^{2})}, \\ \left[LFw~ \text{case}\right] & PS^{LFw}_{pq} = \frac{a_{1} (a_{1} - a_{0} \delta)}{\beta (4 - 3 \delta^{2})}, \\ \left[FLw~ \text{case}\right] & PS^{FLw}_{pq} = \frac{(a_{1} - a_{0} \delta) \left[a_{1} (4 - 5 \delta^{2}) + a_{0} \delta^{3} \right]}{16 \beta (1 - \delta^{2})^{2}}. \end{array}\right. \end{array} $$

Social Welfare in the Three Cases in the p-q Game

We provide social welfare in the three cases in the p-q game as follows:

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} \left[SSw~ \text{case}\right] & W^{SSw}_{pq} = \frac{{a_{0}^{2}} (2 - \delta^{2})^{2} - 2 a_{0} a_{1} \delta (3 - 2 \delta^{2}) + {a_{1}^{2}} (3 - 2 \delta^{2})}{2 \beta (1 - \delta^{2}) (4 - 3 \delta^{2})}, \\ \left[LFw~ \text{case}\right] & W^{LFw}_{pq} = \frac{{a_{0}^{2}} (2 - \delta^{2})^{2} - 2 a_{0} a_{1} \delta (3 - 2 \delta^{2}) + {a_{1}^{2}} (3 - 2 \delta^{2})}{2 \beta (1 - \delta^{2}) (4 - 3 \delta^{2})}, \\ \left[FLw~ \text{case}\right] & W^{FLw}_{pq} = \frac{{a_{0}^{2}} (16 - 20 \delta^{2} + 3 \delta^{4}) - 2 a_{0} a_{1} \delta (12 - 13 \delta^{2}) + {a_{1}^{2}} (12 - 13 \delta^{2})}{32 \beta (1 - \delta^{2})^{2}}. \end{array}\right. \end{array} $$

q-p Game in the Constant Marginal Cost Function Case

Prices of Firms 0 and 1 in the Three Cases in the q-p Game

We provide the prices of firms 0 and 1 in the three cases in the q-p game as follows:

$$\begin{array}{@{}rcl@{}} &\text{SSw case} \left\{\begin{array}{l} {p}^{SSw}_{0qp} = c_{0} + \frac{\delta (a_{1} - a_{0} \delta)}{4 - 3 \delta^{2}}, \\ {p}^{SSw}_{1qp} = c_{1} + \frac{2 a_{1} - 2 a_{0} \delta}{4 - 3 \delta^{2}}. \end{array}\right. \\ &\text{LFw case} \left\{\begin{array}{l} {p}^{LFw}_{0qp} = c_{0} + \frac{\delta (a_{1} - a_{0} \delta)}{4 - 3 \delta^{2}}, \\ {p}^{LFw}_{1qp} = c_{1} + \frac{2 (a_{1} - a_{0} \delta}{4 - 3 \delta^{2}}. \end{array}\right. \\ &\text{FLw case} \left\{\begin{array}{l} {p}^{FLw}_{0qp} = c_{0} + \frac{\delta (a_{1} - a_{0} \delta)}{2 (2 - \delta^{2})}, \\ {p}^{FLw}_{1qp} = c_{1} + \frac{a_{1} - a_{0} \delta}{2 - \delta^{2}}. \end{array}\right. \end{array} $$

Quantities of Firms 0 and 1 in the Three Cases in the q-p Game

We provide the quantities of firms 0 and 1 in the three cases in the q-p game as follows:

$$\begin{array}{@{}rcl@{}} &\text{SSw case} \left\{\begin{array}{l} {q}^{SSw}_{0qp} = \frac{4 a_{0} - 3 a_{1} \delta}{\beta (4- 3 \delta^{2})}, \\ {q}^{SSw}_{1qp} = \frac{2 a_{1} - 2 a_{0} \delta}{\beta (4 - 3 \delta^{2})}. \end{array}\right. \\ &\text{LFw case} \left\{\begin{array}{l} {q}_{0qp}^{LFw} = \frac{4 a_{0} - 3 a_{1} \delta}{\beta (4 - 3 \delta^{2})}, \\ {q}^{LFw}_{1qp} = \frac{2 a_{1} - 2 a_{0} \delta}{\beta (4 - 3 \delta^{2})}. \end{array}\right. \\ &\text{FLw case} \left\{\begin{array}{l} {q}^{FLw}_{0qp} = \frac{a_{0} (4 - 3 \delta^{2}) - a_{1} \delta (3 - 2 \delta^{2})}{2 \beta(2 - 3 \beta \delta + \delta^{3})}, \\ {q}^{FLw}_{1qp} = \frac{a_{1} - a_{0} \delta}{2 \beta (1 - \delta^{2})}. \end{array}\right. \end{array} $$

Consumer Surplus in the Three cases in the q-p Game

We provide the consumer surplus in the three cases in the q-p game as follows:

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} &[SSw~ \text{case}] CS^{SSw}_{qp} = \frac{4 {a_{0}^{2}} - 4 a_{0} a_{1} \delta + {a_{1}^{2}}}{2 \beta (4 - 3 \delta^{2}}, \\ &[LFw~ \text{case}] CS^{LFw}_{qp} =\frac{4 {a_{0}^{2}} - 4 a_{0} a_{1} \delta + {a_{1}^{2}}}{2 \beta (4 - 3 \delta^{2})}, \\ &[FLw~ \text{case}] CS^{FLw}_{qp} = \frac{{a_{0}^{2}} (16 - 20 \delta^{2} + 5 \delta^{4}) -2 a_{0} a_{1} \delta (8 - 9 \delta^{2} + 2 \delta^{4}) + {a_{1}^{2}} (4 - 3 \delta^{2})}{8 \beta (2 - \delta^{2})^{2} (1 - \delta^{2})}. \end{array}\right. \end{array} $$

Producer Surplus in the Three Cases in the q-p Game

We provide the producer surplus in the three cases in the q-p game as follows:

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} &[SSw~ \text{case}] PS^{SSw}_{qp} = \frac{a_{1} (a_{1} - a_{0} \delta)}{\beta (4 - 3 \delta^{2})}, \\ &[LFw~ \text{case}] PS^{LFw}_{qp} = \frac{a_{1} (a_{1} - a_{0} \delta)}{\beta (4 - 3 \delta^{2})}, \\ &[FLw~\text{case}] PS^{FLw}_{qp} = \frac{(a_{1} - a_{0} \delta)\left[a_{1} (4 - 5 \delta^{2} + 2 \delta^{4}) - a_{0} \delta^{3} \right]}{4 \beta (2 - \delta^{2})^{2} (1 - \delta^{2})}. \end{array}\right. \end{array} $$

Social Welfare in the Three Cases in the q-p Game

We provide social welfare in the three cases in the q-p game as follows:

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} &[SSw~ \text{case}] W^{SSw}_{qp} = \frac{4 {a_{0}^{2}} - 6 a_{0} a_{1} \delta + 3 {a_{1}^{2}}}{2 \beta (4 - 3 \delta^{2})}, \\ &[LFw~ \text{case}] W^{LFw}_{qp} = \frac{4 {a_{0}^{2}} - 6 a_{0} a_{1} \delta + 3 {a_{1}^{2}}}{2\beta (4 - 3 \delta^{2})}, \\ &[FLw~ \text{case}] W^{FLw}_{qp} = \frac{{a_{0}^{2}} (16 - 20 \delta^{2} + 7 \delta^{4}) - 2 a_{0} a_{1} \delta (12 - 13 \delta^{2} + 4 \delta^{4}) + {a_{1}^{2}} (12 - 13 \delta^{2} + 4 \delta^{4})}{8 \beta (2 - \delta^{2})^{2} (1 - \delta^{2})}. \end{array}\right. \end{array} $$

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Nakamura, Y. Endogenous Choice of the Timing of Setting Incentive Parameters and the Strategic Contracts in a Managerial Mixed Duopoly with a Welfare-Based Delegation Contract and a Sales Delegation Contract. J Ind Compet Trade 19, 679–737 (2019). https://doi.org/10.1007/s10842-018-0291-6

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Keywords

  • Cournot competition
  • Bertrand competition
  • Endogenous timing of incentive parameters
  • Mixed duopoly
  • Welfare-based delegation
  • Sales delegation

JEL Classification

  • L13
  • D43
  • D21