Abstract
This study examines endogenous determinations of both the degree of partial privatization and the timing of quantity competition under a homogeneous product duopoly with a public and either a domestic or a foreign private firm. We develop a three-stage game. In the first stage, a domestic government determines its partial privatization level. In the second stage, each firm decides the timing of its quantity setting. In the third stage, each firm sets its output level based on this timing. To select a single equilibrium for the timing game, we use the risk-dominance criterion. The main conclusion is as follows. When the private firm is domestic, partial privatization and private leadership emerge in equilibrium, whereas when the private firm is foreign, full nationalization and public leadership emerge in equilibrium.
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Notes
Among others, see Corneo and Jeanne (1994), Pal and White (1998), Bárcena-Ruiz and Garzón (2005), Matsumura et al. (2009), Mukherjee and Suetrong (2009), and Ghosh et al. (2013). Furthermore, some studies have analyzed the partial privatization problem by introducing trade policy (e.g., tariffs). See Chang (2005), Chao and Yu (2006), Han (2012), and Dijkstra et al. (2015).
The difference between the findings of Matsumura and Ogawa (2010) and those of Naya (2015) is related to the equilibrium in relation to a medium degree of privatization. The former argued that a private leadership game emerges in equilibrium, while the latter concluded that a public leadership game emerges in equilibrium.
Some studies have examined how various factors affect the privatization policy. For example, Kim et al. (2019) examined the privatization policy by considering corporate social responsibility (CSR). Liu et al. (2021) introduced corporate taxation and investigated how it affected the privatization policy.
That is, full nationalization is the equilibrium for the public leadership game, and partial privatization is the equilibrium for the private leadership game.
Matsumura and Ogawa (2009) found that the pay-off dominant equilibrium was either the unique equilibrium or the risk-dominant equilibrium. Therefore, it is sufficient to consider the risk-dominant equilibrium.
Even when we introduce product differentiation, the main results obtained in this study hold when the degree of product differentiation is small.
Differentiating \(q_0^t\) (\(t=pub,\,prl,\,sim\)) with respect to \(\theta\), we obtain \(\frac{\partial q_0^{pul}}{\partial \theta } =-\frac{3}{7 (\theta +2)^2}\), \(\frac{\partial q_0^{prl}}{\partial \theta } =-\frac{2 \left( 3 \theta ^2+9 \theta +7\right) }{(\theta +2)^2 (3 \theta +4)^2}\), and \(\frac{\partial q_0^{sim}}{\partial \theta } =-\frac{6}{(3 \theta +5)^2}\).
Note that this stems from the linear demand function.
Differentiating \(q_0^t\) (\(t=pub,\,prl,\,sim\)) with respect to \(\theta\), we obtain \(\frac{\partial q_0^{pul}}{\partial \theta } =-\frac{66}{(4 \theta +17)^2}\), \(\frac{\partial q_0^{prl}}{\partial \theta } =-\frac{4 \left( 2 \theta ^2+9 \theta +12\right) }{(\theta +2)^2 (\theta +6)^2}\), and \(\frac{\partial q_0^{sim}}{\partial \theta } =-\frac{3}{(\theta +3)^2}\).
The price order stems from the aggregate output order.
In the private leadership game, the first-mover advantage produces a large profit. In the public leadership game, firm 0’s encouraging firm 1 effect produces a large profit for firm 1.
Deriving the approximate value, we obtain \(SW^{prl} \approx 0.326834\) and \(SW^{pul} \approx 0.320542\).
Although Hamada (2020) also examined the optimal degree of privatization in a mixed duopoly model with endogeneity of timing, he focused on the case where the private firm is domestic, and considered whether the competition mode affects the optimal degree of privatization. In addition, Hamada (2020) did not select the equilibrium using the risk-dominance criterion, and either partial privatization or full nationalization emerged as the equilibrium. Thus, Proposition 3 provides a clearer result than Hamada (2020).
Note that \(Q^{prl}>Q^{pul}\).
Deriving the approximate value, we obtain \(SW^{pul} \approx 0.261615\) and \(SW^{prl} \approx 0.263615\).
Deriving the approximate value, we obtain \(SW^{prl} \approx 0.261594\) and \(SW^{sim} \approx 0.259689\).
Selecting the equilibrium to maximize social welfare, following Ou et al. (2016), we obtained the same results as Ou et al. (2016). Ou et al. (2016) assumed constant marginal costs, whereas in this study, we assumed increasing marginal costs. However, this difference did not affect the main results.
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Acknowledgements
The authors would like to thank the Editor and two anonymous referees for their constructive and helpful comments. This paper was presented at the 2020 autumn meeting of the Japanese Economic Association. We greatly thank Junichi Haraguchi, Toshihiro Matsumura, and Noriaki Matsushima. The authors gratefully acknowledge financial support in the form of Grants-in-Aid from the Ministry of Education, Culture, Sports, and Technology (Nos. 25245042, 26380340, 17K03734, and 18K01613). We also thank Geoff Whyte, MBA, from Edanz (https://jp.edanz.com/ac) for editing a draft of this manuscript.
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Kawasaki, A., Ohkawa, T. & Okamura, M. Optimal partial privatization in an endogenous timing game: a mixed oligopoly approach. J Econ 136, 227–250 (2022). https://doi.org/10.1007/s00712-022-00777-9
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DOI: https://doi.org/10.1007/s00712-022-00777-9