Abstract
In this paper, we assume a market with an isoelastic price function and construct a three-country model with two active governments and two firms. The purpose of this study is to consider dynamic behavior of the sequential subsidy game in which the governments determine their optimal trade policies and, accordingly, the firms choose their optimal outputs. We first show the existence of an optimal trade policy under realistic conditions. Our main results are summarized as follows: (1) when the production costs are identical, then a trade policy and the corresponding optimal output are stable if the demand is elastic while multistability (i.e., coexistence of multiple attractors) and complex dynamics occur if the demand is inelastic; (2) when the production costs are different, then a stable trade policy induces chaotic output fluctuations regardless of demand elasticity; (3) policy dynamics can be chaotic if demand is elastic while multistability still occurs if the demand is inelastic.
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Notes
- 1.
It can be checked that the second-order conditions are satisfied for any x and y that solve the first-order conditions.
- 2.
A two-stage game with unit-elastic demand (i.e., \(\lambda \,=\,1)\) is considered in Matsumoto and Szidarovszky (2012).
- 3.
R is a measure of the concavity of demand curve defined by \(Q\frac{{P}^{{\prime\prime}}} {P} ,\alpha \) and α ∗  are the market shares of firm 1 and firm 2 at the Cournot point, respectively, \(\alpha \,=\, \frac{{x}^{C}} {{Q}^{C}}\) and \({\alpha }^{{_\ast}}\,=\, \frac{{y}^{C}} {{Q}^{C}}\) where \({Q}^{C}\,=\,{x}^{C} + {y}^{C}\).
- 4.
These are the same as Case 2a and Case 3a of Bandyopadhyay (1997).
- 5.
Suppose that \({D}_{1}({s}_{2}) > 0\). Multiplying both sides of (19) by \(2{(1 - \lambda )}^{2}\) and then moving the first two terms of the right hand side to the left, we obtain
$$2{(1 - \lambda )}^{2}{s}_{ 1,L} + \lambda ((1 - \lambda ){c}_{1} + (3 - 2\lambda ){c}_{y}) = (2 - \lambda )\sqrt{{D}_{1 } ({s}_{2 } )}.$$The left hand side is equal to \(-df({s}_{1,L},{s}_{2})/d{s}_{1}\) and the right hand side is positive. Thus \(df({s}_{1},{s}_{2})/d{s}_{1} < 0\) for \({s}_{1} = {s}_{1L}.\)
- 6.
\({s}_{1}^{m} = \lambda \left (2{\lambda }^{2} - \lambda - 2 - \lambda (3 - 2\lambda )\sqrt{\frac{\lambda +1} {\lambda -1}}\right ){c}_{1}.\) It also holds that \({r}_{1}({s}_{2}^{(-)}) = {s}_{1}^{m}.\)
References
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Acknowledgements
The authors are grateful to G. I. Bischi and an anonymous referee for constructive comments and suggestions on an earlier version of this paper, which was prepared when the first author visited the Department of Systems and Industrial Engineering of the University of Arizona. They are also grateful to Junichi Minagawa for his supporting computational study and want to acknowledge the encouragement and support by Kei Matsumoto for the research leading to this paper. They appreciate financial supports from the Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) 21530172), Chuo University (Joint Research Project 0981) and the Japan Economic Research Foundation. The usual disclaimer applies.
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Matsumoto, A., Szidarovszky, F. (2013). A Little Help from My Friend: International Subsidy Games with Isoelastic Demands. In: Bischi, G., Chiarella, C., Sushko, I. (eds) Global Analysis of Dynamic Models in Economics and Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29503-4_5
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DOI: https://doi.org/10.1007/978-3-642-29503-4_5
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