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Price Versus Quantity in a Duopoly with a Unilateral Effect and with Bargaining over Managerial Contracts

Abstract

This study examines the endogenous choice of each firm’s strategic contract, that is, a price contract or a quantity contract, in a duopoly in which their demand functions are asymmetric when the content of their managerial contracts is determined through bargaining between the owner and the manager. The degree of asymmetry between their demand functions corresponds to the relation between the goods they produce. In contrast to the case wherein each firm’s delegation parameter is determined through profit maximization, we show that the quantity competition cannot become the equilibrium market structure when the bargaining power of the manager relative to that of the owner is sufficiently low. In particular, when the relation between the two goods is complementary, two asymmetric market structures can be observed in equilibrium. Furthermore, we consider the situation in which the relative bargaining power of the manager to that of the owner within each firm is different between the two firms.

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Notes

  1. As indicated in Choi and Lu (2012) and Nakamura (2013, 2015b), the degree of asymmetry between the demand functions of the two firms is positively associated with the price-raising effect of the goods of one firm on the goods of the other firm. When the degree of asymmetry between their demand functions is lower than 1, the goods produced by the two firms are substitutes. However, when the degree of asymmetry is greater than 1, the goods are complements.

  2. In particular, we focus on the changes in the results between Nakamura (2015b) and this study.

  3. Most recently, in the context of a mixed duopoly composed of a welfare-maximizing public firm and a profit-maximizing private firm, Matsumura and Ogawa (2012) showed that, regardless of the relation between the goods produced by the two firms, price competition can become the equilibrium market structure by the dominant strategies of their strategic contracts by both firms. Furthermore, Scrimitore (2013) considered the endogenous choice problem of the strategic contracts by both the public firm and the private firm in a mixed duopoly, with subsidization by the central government. Chirco and Scrimitore (2013) investigated the endogenous choice of strategic contracts by each firm in both entrepreneurial and managerial cases in a private duopoly with network externalities. Chirco et al. (2014) investigated the hiring problem of managers by the owners of both public and private firms in a differentiated goods mixed duopoly, in which the owners simultaneously select their strategic contracts.

  4. Moreover, by considering a standard oligopolistic market without asymmetry between the demand functions of the two firms, Bulow et al. (1985a, 1985b) discussed how the dominant firm in the industry can consider the products of the fringe firms as strategic complements. However, the fringe firms can consider the output level of the dominant firm as a strategic substitute.

  5. For example, Tombak (2006) discussed Boeing’s reaction to the entry of Airbus into the intercontinental commercial jumbo jet market as an illustration of strategic asymmetry.

  6. Nakamura (2008a) extended the analysis conducted in Van Witteloostuijn et al. (2007) to that of the differentiated goods case. Subsequently, Kamaga and Nakamura (2008) and Nakamura (2008b) extended the analyses in Van Witteloostuijn et al. (2007) and Nakamura (2008a) by considering sequential move competition and the situation in which each firm’s technology is represented as a quadratic cost function with respect to its quantity. Furthermore, Nakamura (2011a, 2011b) incorporated bargaining over the sales delegation contract in each firm, à la Van Witteloostuijn et al. (2007), in the context of the merger problem in a domestic oligopoly and an international oligopoly, respectively.

  7. As in Nakamura (2015b), where the bargaining power of the manager relative to that of the owner is relatively low, both quantity and price competition can be observed in the equilibrium when the degree of asymmetry between the firms’ demand functions is lower than 1. However, when the degree of asymmetry is greater than 1, the two types of asymmetric competition can be observed in the equilibrium.

  8. In this study, even when α ∈ (0, 1). is introduced to determine the bargaining over the delegation parameters between the owner and the manager in each firm, the two asymmetric market structures do not have a much wider area to become the equilibrium structure.

  9. Similarly to Choi and Lu (2012), we omit the case of 𝜃 = 1 because the price level of firm 0’s product is independent of firm 1’s output level; this is a trivial case. This assumption is adopted in Nakamura (2013, 2015b), who extended the analyses conducted in Choi and Lu (2012).

  10. Similarly to Choi and Lu (2012), firm 1’s fixed cost f 1 is assumed to be positive because it has the price-raising effect as a unilateral externality. Analogous to Choi and Lu (2012), this “unilateral externality” effect is also referred to in Nakamura (2013, 2015b). Thus, we also assume that this fixed cost is sufficiently small, implying that it has little influence on our results. Therefore, we suppose that f 1 is equal to zero, similarly to Choi and Lu (2012)and Nakamura (2013, 2015b).

  11. This fact is supported by the assumption that the payoff to the manager of firm i is defined as λ i + μ i V i , for some real number λ i and some positive real number μ i . Moreover, this type of delegation contract has been used by Lambertini (2000a, 2000b), Lambertini and Trombetta (2002) and Nakamura (2011a, 2011b).

  12. One may wonder if the convexity of the payoff possibility set, one of the analytical assumptions in Nash’s (1950) formulation of the bargaining problem, is ensured in our model. This difficulty is resolved by Kaneko (1980). Kaneko (1980) proposed a direct extension of the Nash bargaining solution for the bargaining problem with a compact, but not necessarily convex, payoff possibility set, and showed that his extension is the only solution (set-valued function) that satisfies his moderate conditions, which are similar to those in Nash (1950).

  13. Note that we consider the situation in which the bargaining power of the manager relative to that of the owner in each firm is different between firms 0 and 1 in Section 4. More precisely, we consider the case in which the bargaining power of the manager relative to that of the owner differs by exploring the following two situations: (1) the relative bargaining power of the manager and the owner within firm 1 is fixed at 1/2, and the relative bargaining power of the manager and the owner within firm 0, α, is taken at any α ∈ (0, 1); and (2) the relative bargaining power of the manager and the owner within firm 0 is fixed at 1/2, and the relative bargaining power of the manager and owner within firm 1, α, is taken at α ∈ (0, 1).

  14. This assumption follows that of Van Witteloostuijn et al. (2007) and Nakamura (2011a, 2011b), among others.

  15. Note that the superscript pp represents the equilibrium market outcomes with the quantity, price, and profit levels of firms 0 and 1 in the p-p game.

  16. Irrespective of each firm’s strategy, we consider \(\partial q^{ij}_{k} \left (\gamma _{0}, \gamma _{1}; \theta \right )/ \partial \gamma _{k}\), (i, j = p, q; k = 0, 1). This is because we would like to confirm the aggressiveness in the market of each firm’s manager among the four games by limiting the comparative statistics of each firm’s quantity with respect to γ k , (k = 0, 1). This is the case in both p-q and q-p games.

  17. More precisely, we have

    $$\begin{array}{@{}rcl@{}} d \gamma^{pp}_{0}\! (\theta ) / d \theta \,=\, \frac{a [-15 + 20 \theta + 38 \theta^{2} + 20 \theta^{3} + \theta^{4} + 4 \alpha (11 + 12 \theta + 16 \theta^{2} + 8 \theta^{3} + \theta^{4}) + 2 \alpha^{2} (1 + \theta)^{2} (7 - 2 \theta + 3 \theta^{2}) - 4 \alpha^{3} (9 + 4 \theta + 4 \theta^{2} - \theta^{4}) + \alpha^{4} (3 - 2 \theta - \theta^{2})^{2} ]} {[5 + 10 \theta + \theta^{2} + 2 \alpha (1 - \theta)^{2} - \alpha^{2} (3 - 2 \theta - \theta^{2}) ]^{2}}. \end{array} $$
  18. More precisely, we have

    $$d \gamma^{pp}_{1} \left(\theta \right)/ d \theta = \frac{2 a \left(- 1 + \alpha^{2}\right) \left[5 - 10 \theta - 11 \theta^{2} - 2 \alpha \left(9 + 2 \theta - 3 \theta^{2}\right) + \alpha^{2} \left(3 + \theta\right)^{2} \right] } {\left[5 + 10 \theta + \theta^{2} + 2 \alpha \left(1 - \theta\right)^{2} - \alpha^{2} \left(3 - 2 \theta - \theta^{2}\right) \right]^{2}} $$
  19. In the asymmetric demand case, \(d \gamma ^{pp}_{1} \left (\theta \right )/ d \theta > 0\).

  20. When 𝜃 ∈ (1, 2), in the asymmetric demand case, although p 1 is negatively associated with p 0 in the reaction function of p 1 with respect to p 0, this effect is weaker than in the positive reaction function of p 0 with respect to p 1.

  21. Note that the superscript qq is used to represent the equilibrium market outcomes with the quantity, price, and profit levels of firms 0 and 1 in the q-q game.

  22. More precisely, we have

    $$\begin{array}{@{}rcl@{}} d \gamma^{qq}_{0} \left(\theta \right)/ d \theta = \frac{32 a \left(1 - \alpha^{2}\right) \left(\alpha - \theta\right) \left(1 + \theta\right)} {\left[5 + 10 \theta + \theta^{2} - 2 \alpha \left(1 - \theta\right)^{2} - \alpha^{2} \left(3 - 2 \theta -\theta^{2}\right) \right]^{2}} > 0. \end{array} $$
  23. Note that the superscript pq is used to represent the equilibrium market outcomes with the quantity, price, and profit levels of firms 0 and 1 in the p-q game.

  24. More precisely, we have

    $$\begin{array}{@{}rcl@{}} d \gamma^{pq}_{0} \left(\theta \right) / d \theta = - 8 a \left(1 - \alpha\right)^{2} \left(1 + \alpha\right) \left(1 + 2 \theta + 5 \theta^{2}\right) / \left[1 + 10 \theta + 5 \theta^{2} - \alpha^{2} \left(1 + 2 \theta - 3 \theta^{2}\right) \right]^{2} > 0. \end{array} $$
  25. Note that the superscript qp is used to represent the equilibrium market outcomes for the quantity, price, and profit levels of firms 0 and 1 in the q-p game.

  26. More precisely, in the asymmetric demand case, we have

    $$\begin{array}{@{}rcl@{}} d \gamma^{qp}_{0} (\theta )/ d \theta = \frac{\{a [- 3 + 4 \theta + 38 \theta^{2} + 20 \theta^{3} + 5 \theta^{4} + 4 \alpha (1 + 4 \theta + 18 \theta^{2} + 20 \theta^{3} + 5 \theta^{4}) + 2 \alpha^{2} (1 - 4 \theta - 2 \theta^{2} + 28 \theta^{3} + 9 \theta^{4}) - 4 \alpha^{3} (1 + 4 \theta + 10 \theta^{2} + 4 \theta^{3} - 3 \theta^{4}) + \alpha^{4} (1 + 2 \theta - 3 \theta^{2})^{2} ] \} } {[1 + 10 \theta + 5 \theta^{2} - \alpha^{2} (1 + 2 \theta - 3 \theta^{2}) ]^{2}}. \end{array} $$
  27. In this case, we observe that the behavior of \(d \gamma ^{qp}_{0} \left (\theta \right ) / d \theta \) (positive or negative) is almost the same in the asymmetric and symmetric demand cases.

  28. More precisely, we have

    $$d \gamma^{qp}_{1} \left(\theta \right) / d \theta = \frac{2 a \left(-1 + \alpha^{2}\right) \left[\left(\alpha + 3 \alpha \theta\right)^{2} + 5 + 6 \theta + 5 \theta^{2} - 2 \alpha \left(1 + 2 \theta + 5 \theta^{2}\right) \right]} {\left[1 + 10 \theta + 5 \theta^{2} - \alpha^{2} \left(1 + 2 \theta - 3 \theta^{2}\right) \right]^{2}} $$
  29. We give the signs between \(\pi ^{pp}_{0}\) and \(\pi ^{qp}_{0}\) and between \(\pi ^{pq}_{0}\) and \(\pi ^{qq}_{0}\) when 𝜃∈(0, 1) in the Appendix.

  30. We give the signs between \(\pi ^{pp}_{1}\) and \(\pi ^{pq}_{1}\) when 𝜃∈(0, 1) in the Appendix.

  31. We give the signs between \(\pi ^{pp}_{0}\) and \(\pi ^{qp}_{0}\) and between \(\pi ^{pq}_{0}\) and \(\pi ^{qq}_{0}\) when 𝜃 ∈ (1, 2) in the Appendix.

  32. We give the signs between \(\pi ^{qp}_{1}\) and \(\pi ^{qq}_{1}\) when 𝜃 ∈ (1, 2) in the Appendix.

  33. Note that \(\pi ^{pq}_{0} > \pi ^{qp}_{0}\) is achieved only when α and 𝜃 are sufficiently high.

  34. Since \(\pi ^{ij}_{0} = p^{ij}_{0} q^{ij}_{0}\) and \(q^{ij}_{1}\) has an effect such that \(p^{ij}_{0}\) is increased, when 𝜃 ∈ (1, 2), the ranking order of the profit of firm 0 is not definitely explained by the ranking orders of both \(q^{ij}_{0}\) and \(p^{ij}_{0}\), (i, j = p, q).

  35. We obtain that \(\pi ^{pp}_{0} > \pi ^{qp}_{0} \iff \left (1 - \theta \right )^{3} - 4 \alpha \theta \left (5 + 10 \theta + \theta ^{2}\right ) - 2 \alpha ^{2} \left (-1 + \theta \right )^{2} \left (1 + 3 \theta \right ) - 4 \alpha ^{3} \theta \left (-3 + 2 \theta + \theta ^{2}\right ) - \alpha ^{4} \left (-1 + \theta \right )^{3} > 0\). Here, we define

    $$f_{0} \left(\theta, \alpha \right) = \left(1 - \theta\right)^{3} - 4 \alpha \theta \left(5 + 10 \theta + \theta^{2}\right) - 2 \alpha^{2} \left(1 - \theta\right)^{2} \left(1 + 3 \theta\right) - 4 \alpha^{3} \theta \left(-3 + 2 \theta + \theta^{2}\right) - \alpha^{4} \left(-1 + \theta\right)^{3}. $$

    Moreover, we obtain that \(\pi ^{pp}_{1} > \pi ^{pq}_{1} > 0 \iff \left (1 - \theta \right )^{3} - 4 \alpha \theta \left (5 + 10 \theta + \theta ^{2}\right ) - 2 \alpha ^{2} \left (1 - \theta \right )^{2} \left (1 + 3 \theta \right ) + 4 \alpha ^{3} \theta \left (3 - 2 \theta - \theta ^{2}\right ) + \alpha ^{4} \left (1 - \theta \right )^{3} > 0\). Here, we define

    $$f_{1} \left(\theta, \alpha \right) = \left(1 - \theta\right)^{3} - 4 \alpha \theta \left(5 + 10 \theta + \theta^{2}\right) - 2 \alpha^{2} \left(1 - \theta\right)^{2} \left(1 + 3 \theta\right) + 4 \alpha^{3} \theta \left(3 - 2 \theta - \theta^{2}\right) + \alpha^{4} \left(1 - \theta\right)^{3}. $$

    In addition, we obtain the following result: f0 (𝜃, α) > f 1(𝜃, α) for any 𝜃 ∈ (0, 1) and α ∈ (0, 1).

  36. In the p-q game, we find that \(\pi ^{pq}_{0} > \pi ^{qq}_{0} \iff \left (-1 + \theta \right )^{3} - 4 \alpha \theta \left (5 + 10 \theta + \theta ^{2}\right ) + 2 \alpha ^{2} \left (1 - \theta \right )^{2} \left (1 + 3 \theta \right ) + 4 \alpha ^{3} \theta \left (3 - 2 \theta - \theta ^{2}\right ) - \alpha ^{4} \left (1 - \theta \right )^{3} > 0\) , whereas \(\pi ^{pq}_{1} > \pi ^{qq}_{1}\) , for any 𝜃 ∈ (1, 2) and α ∈ (0, 1). On the other hand, in the q-p game, we find that \(\pi ^{qp}_{0} > \pi ^{pp}_{0} \iff \left (-1 + \theta \right )^{3} + 4 \alpha \theta \left (5 + 10 \theta + \theta ^{2}\right ) + 2 \alpha ^{2} \left (1 - \theta \right )^{2} \left (1 + 3 \theta \right ) - 4 \alpha ^{3} \theta \left (3 - 2 \theta - \theta ^{2}\right ) - \alpha ^{4} \left (1 - \theta \right )^{3} > 0\) , while \(\pi ^{qp}_{1} > \pi ^{qq}_{1} \iff \left (- 1 + \theta \right )^{3} - 4 \alpha \theta \left (5 + 10 \theta + \theta ^{2}\right ) + 2 \alpha ^{2} \left (1 - \theta \right )^{2} \left (1 + 3 \theta \right ) + 4 \alpha ^{3} \theta \left (3 - 2 \theta - \theta ^{2}\right ) - \alpha ^{4} \left (1 - \theta \right )^{3}> 0\) . Here we define

    $$\begin{array}{@{}rcl@{}} g_{0} \left(\theta, \alpha\right) &\equiv& \left(-1 + \theta\right)^{3} + 4 \alpha \theta \left(5 + 10 \theta + \theta^{2}\right) + 2 \alpha^{2} \left(1 - \theta\right)^{2} \left(1 + 3 \theta\right) - 4 \alpha^{3} \theta \left(3 - 2 \theta - \theta^{2}\right) - \alpha^{4} \left(1 - \theta\right)^{3}, \\ g_{1} \left(\theta, \alpha \right) &\equiv& \left(- 1 + \theta\right)^{3} - 4 \alpha \theta \left(5 + 10 \theta + \theta^{2}\right) + 2 \alpha^{2} \left(1 - \theta\right)^{2} \left(1 + 3 \theta\right) + 4 \alpha^{3} \theta \left(3 - 2 \theta - \theta^{2}\right) - \alpha^{4} \left(1 - \theta\right)^{3} \end{array} $$

    In addition, we have g0 (𝜃,α)>g 1(𝜃, α) for any 𝜃 ∈ (1, 2) and α ∈ (0, 1).

  37. Throughout the simulation analysis of α and 𝜃, when α=0.000001, we have

    $$\pi^{pq}_{0} > \pi^{qq}_{0} \iff 1 < \theta < 1.04095. $$

    Thus, when the bargaining power of the manager relative to that of the owner, α, is sufficiently low, we find that it is optimal for firm 0 to choose its price contract, provided that firm 1 chooses a quantity contract if 𝜃 is sufficiently near 1.

  38. Note that when α is sufficiently low, the p-p game can become the equilibrium market structure.

  39. On the basis of a similar reasoning, and a comparison between \(\pi ^{qp}_{1}\) and \(\pi ^{qq}_{1}\) when 𝜃 ∈ (1, 2), it is optimal for firm 1 to choose a price contract, implying that the q-p game tends to be the equilibrium market structure.

  40. The situation considered in Section 4 is called the “asymmetric bargaining power case,” while the situation considered in Section 3 is referred to as the “symmetric bargaining power case.”

  41. We wish to thank an anonymous referee for indicating this topic.

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Correspondence to Yasuhiko Nakamura.

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I am grateful for the financial support of Inamori Grants and KAKENHI (25870113). All remaining errors are my own.

Appendix

Appendix

Comparison Between \(\pi ^{pp}_{0}\) and \(\pi ^{qp}_{0}\) with α ∈ (0, 1)

$$\begin{array}{@{}rcl@{}} \pi^{pp}_{0} \,-\, \pi^{qp}_{0} \,=\, \frac{ \left\{ \begin{array}{ll}2 a^{2} (1 - \alpha ) (1 + \alpha ) (1 - \theta )^{2} (1 + \theta ) [1 - \alpha (1 - \theta ) + 3 \theta ]^{2} [- (1 - \theta )^{3} - 4 \alpha \theta (5 + 10 \theta + \theta^{2} ) \\ + 2 \alpha^{2} (1 - \theta )^{2} (1 + 3 \theta ) + 4 \alpha^{3} \theta (3 - 2 \theta - \theta^{2}) - \alpha^{4} (1 - \theta)^{3} ] \end{array} \right\}} {[5 \,+\, 10 \theta \!+ \theta^{2} \!+ 2 \alpha (1 \,-\, \theta )^{2} \,-\, \alpha^{2} (3 \,-\, 2 \theta \,-\, \theta^{2} ) ]^{2} [1 \,+\, 10 \theta \!+ 5 \theta^{2} \!- \alpha^{2} (1 \!+ 2 \theta \!- \!3 \theta^{2} ) ]^{2}}. \end{array} $$

Thus, when 𝜃(0, 1) ∪ (1, 2), the ranking order between \(\pi ^{pp}_{0}\) and \(\pi ^{qp}_{0}\) is determined by the sign of −(1 − 𝜃)3−4α 𝜃(5 + 10𝜃 + 𝜃 2)+2α 2(1−𝜃)2(1 + 3𝜃) + 4α 3 𝜃(3−2𝜃𝜃 2) − α 4(1 − 𝜃)3.

Comparison Between \(\pi ^{pq}_{0}\) and \(\pi ^{qq}_{0}\) with α ∈ (0, 1)

$$\begin{array}{@{}rcl@{}} \pi^{pq}_{0} \,-\, \pi^{qq}_{0} \,=\, \frac{\left\{ \begin{array}{ll} 2 a^{2} (1 - \alpha ) (1 + \alpha ) (1 - \theta )^{2} (1 + \theta ) [1 + 3 \theta - \alpha (1 - \theta ) ]^{2} [(- 1 + \theta )^{3} - 4 \alpha \theta (5 + 10 \theta + \theta^{2} ) \\ + 2 \alpha^{2} (1 - \theta )^{2} (1 + 3 \theta ) + 4 \alpha^{3} \theta (3 - 2 \theta - \theta^{2} ) - \alpha^{4} (1 - \theta )^{3} ] \end{array}\right\} } {[5 \,+\, 10 \theta \!+ \theta^{2} \,-\, 2 \alpha (1\! -\! \theta )^{2} \!+ \alpha^{2} (3 \,-\, 2 \theta \,-\, \theta^{2}) ]^{2} [1 \,+\, 10 \theta \,+\, 5 \theta^{2} \,-\, \alpha^{2} (1 \,+\, 2 \theta \,-\, 3 \theta^{2} ) ]^{2}}. \end{array} $$

Thus, when 𝜃(0, 1) ∪ (1, 2), the ranking order between \(\pi ^{pp}_{0}\) and \(\pi ^{qq}_{0}\) is determined by the sign of (−1 + 𝜃)3 − 4α 𝜃(5 + 10𝜃 + 𝜃 2) + 2α 2(1−𝜃)2(1 + 3𝜃) + 4α 3 𝜃(3−2𝜃𝜃 2) − α 4(1 − 𝜃)3.

Comparison Between \(\pi ^{pp}_{1}\) and \(\pi ^{pq}_{1}\) with α ∈ (0, 1)

$$\begin{array}{@{}rcl@{}} \pi^{pp}_{1} \,-\, \pi^{pq}_{1} \,=\, \frac{]\left\{ \begin{array}{ll}8 a^{2} (1 - \alpha)^{3} (1 + \alpha) (1 - \theta )^{2} \theta (1 + \theta) [(1 - \theta)^{3} + 4 \alpha \theta (5 + 10 \theta + \theta^{2}) \\ + 2 \alpha^{2} (1 - \theta)^{2} (1 + 3 \theta) + 4 \alpha^{3} \theta (3 - 2 \theta - \theta^{2}) - \alpha^{4} (1 - \theta)^{3} ] \end{array} \right\}} {[5 \,+\, 10 \theta \!+ \theta^{2} \!+ 2 \alpha (1 \,-\, \theta)^{2} \,-\, \alpha^{2} (3 \,-\, 2 \theta \!- \theta^{2}) ]^{2} [1 \,+\, 10 \theta \!+ 5 \theta^{2} \,-\, \alpha^{2} (1 \,+\, 2 \theta \,-\, 3 \theta^{2}) ]^{2}}. \end{array} $$

Thus, when 𝜃(0, 1) ∪ (1, 2), the ranking order between \(\pi ^{pp}_{0}\) and \(\pi ^{qq}_{0}\) is determined by the sign of (1 − 𝜃)3 + 4α 𝜃(5 + 10𝜃 + 𝜃 2) + 2α 2(1 − 𝜃)2(1 + 3𝜃) + 4α 3 𝜃(3 − 2𝜃𝜃 2) − α 4(1 − 𝜃)3.

Comparison Between \(\pi ^{qp}_{1}\) and \(\pi ^{qq}_{1}\) with α ∈ (0, 1)

$$\begin{array}{@{}rcl@{}} \pi^{qp}_{1} \,-\, \pi^{qq}_{1} \,=\, \frac{\left\{ \begin{array}{ll} 2 a^{2} (1 - \alpha)^{3} (1 + \alpha) (1 - \theta )^{2} (1 + \theta)^{3} [(- 1 + \theta)^{3} - 4 \alpha \theta (5 + 10 \theta + \theta^{2}) \\ + 2 \alpha^{2} (1 - \theta)^{2} (1 + 3 \theta) + 4 \alpha^{3} \theta (3 - 2 \theta - \theta^{2}) - \alpha^{4} (1 - \theta)^{3} ] \end{array}\right\}} {[5 \,+\, 10 \theta \!+ \theta^{2}\,-\, 2 \alpha (1 \!- \theta)^{2} \,-\, \alpha^{2} (3 \,-\, 2 \theta \!- \theta^{2}) ]^{2} [1 \,+\, 10 \theta \!+ 5 \theta^{2} \,-\, \alpha^{2} (1 \!+ 2 \theta \!- \!3 \theta^{2}) ]^{2}}. \end{array} $$

Thus, when 𝜃(0, 1) ∪ (1, 2), the ranking order between \(\pi ^{pp}_{0}\) and \(\pi ^{qq}_{0}\) is determined by the sign of (−1 + 𝜃)3−4α 𝜃(5+10𝜃 + 𝜃 2)+2α 2(1−𝜃)2(1+3𝜃) + 4α 3 𝜃(3−2𝜃𝜃 2)−α 4(1−𝜃)3.

The Comparison Between \(\pi ^{pp}_{0}\) and \(\pi ^{qp}_{0}\) and Between \(\pi ^{pq}_{0}\) and \(\pi ^{qq}_{0}\) when 𝜃 ∈ (0, 1)

More precisely, when 𝜃 ∈ (0, 1), we have

$$\begin{array}{@{}rcl@{}} \pi^{pp}_{0} &\gtreqless& \pi^{qp}_{0} \iff \left(-1 + \theta\right)^{3} + 4 \alpha \theta \left(5 + 10 \theta + \theta^{2}\right) + 2 \alpha^{2} \left(1 - \theta\right)^{2} \left(1 + 3 \theta\right)\\ &&+ 4 \alpha^{3} \theta \left(-3 + 2 \theta + \theta^{2}\right) + \alpha^{4} \left(-1 + \theta\right)^{3} \gtreqless 0, \\ \pi^{pq}_{0} &\gtreqless& \pi^{qq}_{0} \iff \left(-1 + \theta\right)^{3} - 4 \alpha \theta \left(5 + 10 \theta + \theta^{2}\right) + 2 \alpha^{2} \left(1 - \theta\right)^{2} \left(1 + 3 \theta\right) \\ &&- 4 \alpha^{3} \theta \left(-3 + 2 \theta + \theta^{2}\right) + \alpha^{4} \left(-1 + \theta\right)^{3} \gtreqless 0. \end{array} $$

The Comparison Between \(\pi ^{pp}_{1}\) and \(\pi ^{pq}_{1}\) when 𝜃 ∈ (0, 1)

More precisely, when 𝜃 ∈ (0, 1), we have

$$\begin{array}{@{}rcl@{}} \pi^{pp}_{1} &\gtreqless& \pi^{pq}_{1} \iff \left(1 - \theta \right)^{3} - 4 \alpha \theta \left(5 + 10 \theta + \theta^{2} \right) - 2 \alpha^{2} \left(1 - \theta \right)^{2} \left(1 + 3 \theta \right)\\ &&+ 4 \alpha^{3} \theta \left(3 - 2 \theta - \theta^{2} \right) + \alpha^{4} \left(1 - \theta \right)^{3} \gtreqless 0. \end{array} $$

Thus, when the above conditions (1−𝜃)3+4α 𝜃(5+10𝜃 + 𝜃 2)+2α 2(1−𝜃)2(1+3𝜃) + 4α 3 𝜃(3−2𝜃𝜃 2)−α 4(1−𝜃)3>0 are simultaneously satisfied, the p-p becomes the equilibrium market structure.

The Comparison Between \(\pi ^{pp}_{0}\) and \(\pi ^{qp}_{0}\) and Between \(\pi ^{pq}_{0}\) and \(\pi ^{qp}_{0}\) when 𝜃 ∈ (1, 2)

More precisely, when 𝜃 ∈ (1, 2), we have

$$\begin{array}{@{}rcl@{}} \pi^{pp}_{0} &\gtreqless& \pi^{qp}_{0} \iff - \left(1 - \theta \right) \left(5 + 10 \theta + \theta^{2} \right)^{2} + 2 \alpha \left(15 - 45 \theta - 242 \theta^{2} - 210 \theta^{3} - 29 \theta^{4} - \theta^{5} \right) \\ &&+ \alpha^{2} \left(41 + 235 \theta + 74 \theta^{2} - 250 \theta^{3} - 99 \theta^{4} - \theta^{5} \right) - 4 \alpha^{3} \left(1 - \theta \right)^{2} \left(15 + 33 \theta + 17 \theta^{2} - \theta^{3} \right) \\ &&- \alpha^{4} \left(1\! -\! \theta \right)^{2} \left(7 + 67 \theta + 53 \theta^{2} \!+ \theta^{3} \right) \,+\, 2 \alpha^{5} \left(1 \!- \theta \right)^{2} \!\left(\!15 \,+\, 17 \theta \!+ \theta^{2} \,-\, \theta^{3} \right) \,-\, \alpha^{6} \left(1 \!- \theta \right)^{3} \left(3 \!+ \theta \right)^{2} \gtreqless 0, \\ \pi^{pq}_{0} &\gtreqless& \pi^{qq}_{0} \iff \left(- 1 + \theta \right)^{3} - 4 \alpha \theta \left(5 + 10 \theta + \theta^{2} \right) + 2 \alpha^{2} \left(1 - \theta \right)^{2} \left(1 + 3 \theta \right) + 4 \alpha^{3} \theta \left(3 - 2 \theta - \theta^{2} \right) \\ &&+ \alpha^{4} \left(- 1 + \theta \right)^{3} \gtreqless 0. \end{array} $$

The Comparison Between \(\pi ^{qp}_{1}\) and \(\pi ^{qq}_{1}\) when 𝜃 ∈ (1, 2)

More precisely, when 𝜃 ∈ (1, 2), we have

$$\begin{array}{@{}rcl@{}} \pi^{qp}_{1} &\gtreqless& \pi^{qq}_{1} \iff \left(- 1 + \theta \right)^{3} - 4 \alpha \theta \left(5 + 10 \theta + \theta^{2} \right) + 2 \alpha^{2} \left(1 - \theta \right)^{2} \left(1 + 3 \theta \right) \\ &&+ 4 \alpha^{3} \theta \left(3 - 2 \theta - \theta^{2} \right) + \alpha^{4} \left(-1 + \theta \right)^{3} \gtreqless 0. \end{array} $$

The Relative Bargaining Power is Fixed at 1/2 in Firm 1 and is Equal to α in Firm 0

  • pp game:

    $$\left\{\begin{array}{l} \pi^{pp}_{0} = \left[6 a^{2} \theta \left(1 + \theta \right) \left(3 + \alpha + \theta - \alpha \theta \right)^{2} \right] / \left[9 + 22 \theta + \theta^{2} - \alpha \left(5 - 6 \theta + \theta^{2} \right) \right]^{2}, \\ \pi^{pp}_{1} = 72 a^{2} \left(1 - \alpha \right) \left(1 + \alpha \right) \theta \left(1 + \theta \right)/ \left[9 + 22 \theta + \theta^{2} - \alpha \left(5 - 6 \theta + \theta^{2} \right) \right]^{2}. \end{array}\right. $$
  • pq game:

    $$\left\{\begin{array}{l} \pi_{0}^{pq} = 6 a^{2} \theta \left(1 + \theta\right) \left[1 - \alpha \left(1 - \theta \right) + 3 \theta \right]^{2} / \left[3 + 18 \theta + 11 \theta^{2} - \alpha \left(3 - 2 \theta - \theta^{2}\right) \right]^{2}, \\ \pi_{1}^{pq} = 8 a^{2} \left(1 - \alpha \right) \left(1 + \alpha\right) \theta^{2} \left(1 + \theta\right) / \left[3 + 18 \theta + 11 \theta^{2} - \alpha \left(3 - 2 \theta - \theta^{2}\right) \right]^{2}. \end{array}\right. $$
  • qp game:

    $$\left\{\begin{array}{l} \pi_{0}^{qp} = 6 a^{2} \theta^{2} \left(1 + \theta\right) \left[3 - \alpha \left(1 - \theta \right) + \theta \right]^{2} / \left(1 + \alpha + 22 \theta - 6 \alpha \theta + 9 \theta^{2} + 5 \alpha \theta^{2} \right)^{2}, \\ \pi_{1}^{qp} = \left[2 a^{2} \left(1 - \alpha\right) \left(1 + \alpha\right) \theta \left(1 + \theta\right)^{3} \right] / \left(1 + \alpha + 22 \theta - 6 \alpha \theta + 9 \theta^{2} + 5 \alpha \theta^{2}\right)^{2}. \end{array}\right. $$
  • qq game:

    $$\left\{\begin{array}{l} \pi_{0}^{qq} = 2 a^{2} \left(1 - \alpha\right) \left(1 + \alpha\right) \left(1 + \theta\right) \left(1+ 7 \theta\right)^{2} / \left[9 + 22 \theta + \theta^{2} - \alpha \left(5 - 6 \theta + \theta^{2}\right) \right]^{2}, \\ \pi_{1}^{qq} = 6 a^{2} \left(1 - \alpha\right)^{2} \left(1 + \theta\right)^{3} / \left[9 + 22 \theta + \theta^{2} - \alpha \left(5 - 6 \theta + \theta^{2}\right) \right]^{2}. \end{array}\right. $$

The Relative Bargaining Power in Firm 0 is Equal to 1/2

α ∈ (0, 1) and 𝜃 ∈ (0, 1) ∪ (1, 2)

  • pp game:

    $$\left\{\begin{array}{l} \pi^{pp}_{0} = 6 a^{2} \theta \left(1 + \theta\right) \left(3 + \alpha + \theta - \alpha \theta \right)^{2} / \left[9 + 22 \theta + \theta^{2} - \alpha \left(5 - 6 \theta + \theta^{2}\right) \right]^{2}, \\ \pi^{pp}_{1} = 72 a^{2} \left(1 - \alpha\right) \left(1 + \alpha\right) \theta \left(1 + \theta \right) / \left(9 + 22 \theta + \theta^{2} - \alpha \left(5 - 6 \theta + \theta^{2}\right)\right)^{2}. \end{array}\right. $$
  • pq game:

    $$\left\{\begin{array}{l} \pi^{pq}_{0} = 6 a^{2} \theta \left(1 + \theta\right) \left[1 - \alpha \left(1 \!- \theta\right) + 3 \theta \right]^{2} / \left[3 + 18 \theta + 11 \theta^{2} \,-\, \alpha \left(3 - 2 \theta - \theta^{2} \right) \right]^{2}, \\ \pi^{pq}_{1} = 8 a^{2} \left(1 - \alpha\right) \left(1 + \alpha\right) \theta^{2} \left(1 + \theta \right) / \left[3 + 18 \theta + 11 \theta^{2} - \alpha \left(3 - 2 \theta - \theta^{2} \right) \right]^{2}. \end{array}\right. $$
  • qp game:

    $$\left\{\begin{array}{l} \pi^{qp}_{0} = 6 a^{2} \theta^{2} \left(1 + \theta\right) \left[3 - \alpha \left(1 - \theta\right) + \theta \right]^{2} / \left(1 + \alpha + 22 \theta - 6 \alpha \theta + 9 \theta^{2} + 5 \alpha \theta^{2}\right)^{2}, \\ \pi^{qp}_{1} = 2 a^{2} \left(1 - \alpha\right) \left(1 + \alpha\right) \theta \left(1 + \theta\right)^{3} / \left(1 + \alpha + 22 \theta - 6 \alpha \theta + 9 \theta^{2} + 5 \alpha \theta^{2} \right)^{2}. \end{array}\right. $$
  • qq game:

    $$\left\{\begin{array}{l} \pi^{qq}_{0} = 2 a \left(1 + \alpha\right) \left(1 + \alpha\right) \left(1 + \theta\right) \left(1 + 7 \theta\right)^{2} / \left[9 + 22 \theta + \theta^{2} - \alpha \left(5 - 6 \theta + \theta^{2}\right) \right]^{2}, \\ \pi^{qq}_{1} = 6 a^{2} \left(1 - \alpha\right)^{2} \left(1 + \theta\right)^{3} / \left[9 + 22 \theta + \theta^{2} - \alpha \left(5 - 6 \theta + \theta^{2}\right) \right]^{2}. \end{array}\right. $$

Lemma 1–Comparison with 𝜃 ∈ (0, 1)

Comparison of γ 0 Among the Four Games with 𝜃 = 1/2

$$\left\{\begin{array}{l} \gamma^{pq}_{0} - \gamma^{qq}_{0} = 12 a \left(5 - \alpha\right) \left(1 - \alpha\right) \left(1 + \alpha\right/ \left(29 - 5 \alpha^{2}\right) \left(41 - 2 \alpha - 7 \alpha^{2}\right) > 0, \\ \gamma^{qq}_{0} - \gamma^{pp}_{0} = a \left(1 + \alpha\right) \left(697 + 47 \alpha - 217 \alpha^{2} + 49 \alpha^{3}\right) / 2 \left(41 + 2 \alpha - 7 \alpha^{2}\right) \left(41 \,-\, 2 \alpha \,-\, 7 \alpha^{2}\right) > 0, \\ \gamma^{pp}_{0} - \gamma^{qp}_{0} = 6 a \left(7 - \alpha\right) \left(1 - \alpha\right) \left(1 + \alpha\right) / \left(29 - 5 \alpha^{2}\right) \left(41 + 2 \alpha - 7 \alpha^{2}\right) > 0. \end{array}\right. $$

Thus, we have the ranking order of γ 0 such that \(\gamma ^{pq}_{0} > \gamma ^{qq}_{0} > \gamma ^{pp}_{0} > \gamma ^{qp}_{0} \) when 𝜃 = 1/2.

Comparison of γ 1 Among the Four Games with 𝜃 = 1/2

$$\left\{\begin{array}{l} \gamma^{qp}_{1} - \gamma^{qq}_{1} = 36 a \left(1 - \alpha\right)^{2} \left(1 + \alpha\right) / \left(29 - 5 \alpha^{2}\right) \left(41 - 2 \alpha - 7 \alpha^{2}\right) > 0, \\ \gamma^{qq}_{1} - \gamma^{pp}_{1} = a \left(1 - \alpha\right) \left(1 + \alpha\right) \left(205 + 84 \alpha - 49 \alpha^{2}\right) / \left(41 + 2 \alpha - 7 \alpha^{2}\right) \left(41 - 2 \alpha - 7 \alpha^{2}\right) > 0, \\ \gamma^{pp}_{1} - \gamma^{pq}_{1} = 24 a \left(1 - \alpha\right)^{2} \left(1 + \alpha\right) / \left(29 - 5 \alpha^{2}\right) \left(41 + 2 \alpha - 7 \alpha^{2}\right) > 0. \end{array}\right. $$

Thus, we have the ranking order of γ 1 such that \(\gamma ^{qp}_{1} > \gamma ^{qq}_{1} > \gamma ^{pp}_{1} > \gamma ^{pq}_{1} \) when 𝜃 = 1/2.

Comparison of q 0 Among the Four Games with 𝜃 = 1/2

$$\left\{\begin{array}{l} q^{pq}_{0} - q^{pp}_{0} = 6 a \left(1 - \alpha\right)^{2} \left(1 + \alpha\right)^{2} / \left(29 - 5 \alpha^{2}\right) \left(41 + 2 \alpha - 7 \alpha^{2}\right) > 0, \\ q^{pp}_{0} - q^{qq}_{0} = a \left(1 - \alpha\right) \left(1 + \alpha\right) \left(41 + 7 \alpha^{2}\right) / \left(41 + 2 \alpha - 7 \alpha^{2}\right) \left(41 - 2 \alpha - 7 \alpha^{2}\right) > 0, \\ q^{qq}_{0} - q^{qp}_{0} = 6 a \left(1 - \alpha\right)^{2} \left(1 + \alpha\right)^{2} / \left(29 - 5 \alpha^{2}\right) \left(41 - 2 \alpha - 7 \alpha^{2}\right) > 0. \end{array}\right. $$

Thus, we have the ranking order of q 0 such that \(q^{pq}_{0} > q^{pp}_{0} > q^{qq}_{0} > q^{qp}_{0} \) when 𝜃 = 1/2.

Comparison of q 1 Among the Four Games with 𝜃 = 1/2

$$\left\{\begin{array}{l} q^{qp}_{1} - q^{qq}_{1} = 3 a \left(1 - \alpha\right)^{2} \left(1 + \alpha\right) \left(7 + \alpha \right) / \left(29 - 5 \alpha^{2}\right) \left(41 +- 2 \alpha - 7 \alpha^{2} \right) > 0, \\ q^{qq}_{1} - q^{pp}_{1} = 48 a \left(1 - \alpha\right) \alpha \left(1 + \alpha \right) / \left(41 + 2 \alpha - 7 \alpha^{2}\right) \left(41 - 2 \alpha - 7 \alpha^{2} \right) \geq 0, \\ q^{pp}_{1} - q^{pq}_{1} = 4 a \left(1 - \alpha \right)^{2} \left(1 + \alpha \right) \left(5 + \alpha \right) / \left(29 - 5 \alpha^{2} \right) \left(41 + 2 \alpha - 7 \alpha^{2} \right) > 0. \end{array}\right. $$

Thus, we have the ranking order of q 1 such that \(q^{qp}_{1} > q^{qq}_{1} > q^{pp}_{1} > q^{pq}_{1} \) when 𝜃 = 1/2. Note that \(q^{qq}_{1} = q^{pp}_{1} \iff \alpha = 0\).

Comparison of p 0 Among the Four Games with 𝜃 = 1/2

$$\left\{\begin{array}{l} p^{qq}_{0} - p^{qp}_{0} = 9 a \left(1 - \alpha\right)^{3} \left(1 + \alpha\right) / 2 \left(29 - 5 \alpha^{2}\right) \left(41 - 2 \alpha - 7 \alpha^{2}\right) > 0, \\ p^{qp}_{0} - p^{pq}_{0} = a \left(1 - \alpha\right) \left(1 + \alpha\right) / 2 \left(29 - 5 \alpha^{2}\right) > 0, \\ p^{pq}_{0} - p^{pp}_{0} = 4 a \left(1 \alpha\right)^{3} \left(1 + \alpha\right) / \left(29 - 5 \alpha^{2}\right) \left(41 + 2 \alpha - 7 \alpha^{2}\right) >0. \end{array}\right. $$

Thus, we have the ranking order of p 0 such that \(p^{qq}_{0} > p^{qp}_{0} > p^{pq}_{0} > p^{pp}_{0} \) when 𝜃 = 1/2.

Comparison of p 1 Among the Four Games with 𝜃 = 1/2

$$\left\{\begin{array}{l} p^{qq}_{1} - p^{qp}_{1} = 3 a \left(5 - \alpha\right) \left(1 - \alpha\right)^{2} \left(1 + \alpha\right/ \left(29 - 5 \alpha^{2}\right) \left(41 - 2 \alpha - 7 \alpha^{2}\right) > 0, \\ p^{qp}_{1} = p^{pq}_{1} = 0, \\ p^{pq}_{1} - p^{pp}_{1} = 2 a \left(7 - \alpha\right) \left(1 - \alpha\right)^{2} \left(1 + \alpha\right) / \left(29 - 5 \alpha^{2}\right) \left(41 + 2 \alpha - 7 \alpha^{2}\right) > 0. \end{array}\right. $$

Thus, we have the ranking order of p 1 such that \(p^{qq}_{1} > p^{qp}_{1} = p^{pq}_{0} > p^{pp}_{1} \) when 𝜃 = 1/2. Note that \(p^{qp}_{1} = p^{pq}_{1}\) is always satisfied for any 𝜃 ∈ (0, 1) and α ∈ (0, 1).

Lemma 2–Comparison with 𝜃 ∈ (1, 2)

Comparison of γ 0 Among the Four Games with 𝜃 = 3/2

$$\left\{\begin{array}{l} \gamma^{pp}_{0} - \gamma^{qp}_{0} = a \left(1 + \alpha\right) \left(703 + 1269 \alpha - 727 \alpha^{2} + 45 \alpha^{3}\right) / \left(29 - 5 \alpha^{2}\right) \left(89 + 2 \alpha + 9 \alpha^{2}\right) > 0, \\ \gamma^{qp}_{0} - \gamma^{pq}_{0} = a \left(1 + \alpha\right) \left(125 + 2911 \alpha - 1541 \alpha^{2} - 55 \alpha^{3}\right) / 2 \left(29 - 5 \alpha^{2}\right) \left(109 + 11 \alpha^{2}\right) > 0, \\ \gamma^{pq}_{0} - \gamma^{qq}_{0} = 20 a \left(1 - \alpha\right) \left(1 + \alpha\right) \left(11 + \alpha\right) / \left(89 - 2 \alpha + 9 \alpha^{2}\right) \left(109 + 11 \alpha^{2}\right) > 0. \end{array}\right. $$

Thus, we have the ranking order of γ 0 such that \(\gamma ^{pp}_{0} > \gamma ^{qp}_{0} > \gamma ^{pq}_{0} > \gamma ^{qq}_{0} \) when 𝜃 = 3/2.

Comparison of γ 1 Among the Four Games with 𝜃 = 3/2

$$\left\{\begin{array}{l} \gamma^{pp}_{1} - \gamma^{pq}_{1} = a \left(1 + \alpha\right) \left(4901 + 597 \alpha + 403 \alpha^{2} + 99 \alpha^{3}\right) / 2 \left(89 + 2 \alpha + 9 \alpha^{2}\right) \left(109 + 11 \alpha^{2}\right) > 0, \\ \gamma^{pq}_{1} - \gamma^{qq}_{1} = 20 a \left(1 - \alpha\right) \left(1 + \alpha\right) \left(11 + \alpha\right) / \left(89 - 2 \alpha + 9 \alpha^{2}\right) \left(109 + 11 \alpha^{2}\right) > 0, \\ \gamma^{qq}_{1} - \gamma^{qp}_{1} = 3 a \left(1 + \alpha\right) \left(5 - 827 \alpha + 423 \alpha^{2} + 15 \alpha^{3}\right) / 2 \left(29 - 5 \alpha^{2}\right) \left(89 - 2 \alpha + 9 \alpha^{2}\right) > 0. \end{array}\right. $$

Thus, we have the ranking order of γ 1 such that \(\gamma ^{pp}_{1} > \gamma ^{pq}_{1} > \gamma ^{qq}_{1} > \gamma ^{qp}_{1} \) when 𝜃 = 3/2.

Comparison of q 0 Among the Four Games with 𝜃 = 3/2

$$\left\{\begin{array}{l} q^{pp}_{0} - q^{pq}_{0} = 10 a \left(1 - \alpha\right)^{2} \left(1 + \alpha\right)^{2} / \left(89 + 2 \alpha + 9 \alpha^{2}\right) \left(109 + 11 \alpha^{2}\right) > 0, \\ q^{pq}_{0} - q^{qq}_{0} \left(\theta = 3/2\right) = a \left(9 - \alpha\right) \left(1 - \alpha\right) \left(1 + \alpha\right) \left(11 + \alpha\right) / \left(89 - 2 \alpha + 9 \alpha^{2}\right) \left(109 + 11 \alpha^{2}\right) > 0, \\ q^{qq}_{0} - q^{qp}_{0} \left(\theta = 3/2\right) = 2 a \left(1 - \alpha\right) \left(1 + \alpha\right) \left(15 + 176 \alpha + \alpha^{2}\right) / \left(29 - 5 \alpha^{2}\right) \left(89 - 2 \alpha + 9 \alpha^{2}\right) > 0. \end{array}\right. $$

Thus, we have the ranking order of q 0 such that \( q^{pp}_{0} > q^{pq}_{0} > q^{qq}_{0} > q^{qp}_{0} \) when 𝜃 = 3/2.

Comparison of q 1 Among the Four Games with 𝜃 = 3/2

$$\left\{\begin{array}{l} q^{qp}_{1} − q^{qq}_{1} = a \left(1 - \alpha\right) \left(1 + \alpha\right) \left(221 - 18 \alpha + 181 \alpha^{2}\right) / \left(29 - 5 \alpha^{2}\right) \left(89 - 2 \alpha + 9 \alpha^{2}\right) > 0, \\ q^{qq}_{1} − q^{pp}_{1} = 80 a \left(1 - \alpha\right) \alpha \left(1 + \alpha\right) / \left(89 - 2 \alpha + 9 \alpha^{2}\right) \left(89 + 2 \alpha + 9 \alpha^{2}\right) ≥ 0, \\ q^{pp}_{1} − q^{pq}_{1} = 4 a \left(11 - \alpha\right) \left(1 - \alpha\right)^{2} \left(1 + \alpha\right) / \left(89 + 2 \alpha + 9 \alpha^{2}\right) \left(109 + 11 \alpha^{2}\right) > 0. \end{array}\right. $$

Thus, we have the ranking order of q 1 such that \(q^{qp}_{1} > q^{qq}_{1} > q^{pp}_{1} > q^{pq}_{1} \) when 𝜃 = 3/2. Note that the equality between \(q^{qq}_{1} = q^{pp}_{1} \iff \alpha = 0\).

Comparison of p 0 Among the Four Games with 𝜃 = 3/2

$$\left\{\begin{array}{l} p^{qq}_{0} - p^{pp}_{0} = a \left(1 - \alpha\right) \left(1 + \alpha\right) + \left(89 + 40 \alpha - 9 \alpha^{2}\right) / \left(89 - 2 \alpha + 9 \alpha^{2}\right) \left(89 + 2 \alpha + 9 \alpha^{2}\right) > 0, \\ p^{pp}_{0} - p^{pq}_{0} = 12 a \left(1 - \alpha\right)^{3} \left(1 + \alpha\right) / \left(89 + 2 \alpha + 9 \alpha^{2}\right) \left(109 + 11 \alpha^{2}\right) > 0, \\ p^{pq}_{0} - p^{qp}_{0} = 27 a \left(1 - \alpha\right) \left(1 + \alpha\right) \left(57 - 32 \alpha - \alpha^{2}\right) / 2 \left(29 - 5 \alpha^{2}\right) \left(109 + 11 \alpha^{2}\right) > 0. \end{array}\right. $$

Thus, we have the ranking order of p 0 such that \(p^{qq}_{0} > p^{pp}_{0} >p^{pq}_{0} > p^{qp}_{0} \) when 𝜃 = 3/2.

Comparison of p 1 Among the Four Games with 𝜃 = 3/2

$$\left\{\begin{array}{l} p^{qq}_{1} - p^{pp}_{1} = a \left(1 - \alpha\right)^{2} \left(1 + \alpha\right) \left(89 + 9 \alpha\right) / \left(89 - 2 \alpha + 9 \alpha^{2}\right) \left(89 + 2 \alpha + 9 \alpha^{2}\right) > 0, \\ p^{pp}_{1} - p^{pq}_{1} = 6 a \left(1 - \alpha\right)^{2} \left(1 + \alpha\right) \left(9 + \alpha\right) / \left(89 + 2 \alpha + 9 \alpha^{2}\right) \left(109 + 11 \alpha^{2}\right) > 0, \\ p^{pq}_{1} - p^{qp}_{1} = 27 a \left(1 - \alpha\right) \left(1 + \alpha\right) \left(57 - 32 \alpha - \alpha^{2}\right) / 2 \left(29 - 5 \alpha^{2}\right) \left(109 + 11 \alpha^{2}\right) > 0. \end{array}\right. $$

Thus, we have the ranking order of p 1 such that \(p^{qq}_{1} > p^{pp}_{1} > p^{pq}_{1} > p^{qp}_{1}\) when 𝜃 = 3/2.

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Nakamura, Y. Price Versus Quantity in a Duopoly with a Unilateral Effect and with Bargaining over Managerial Contracts. J Ind Compet Trade 17, 83–119 (2017). https://doi.org/10.1007/s10842-016-0232-1

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Keywords

  • Cournot
  • Bertrand
  • Managerial delegation
  • Delegation contract bargaining
  • Asymmetric demand functions

JEL Classification

  • L13
  • D43