Abstract
This paper explores a nonlinear Cournot oligopoly with n firms displaying general isoelastic demand. The marginal profits-based gradient rule and the expectation rule Local Monopolistic Approximation were employed in two Cournot oligopoly games. Nash equilibrium stability analysis is carried out on each of the two games to throw light on the effects of demand elasticity and other parameters on the dynamics of the game. Our results show that the influence of demand elasticity on stability depends on firms’ expectation rules.
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Notes
The boundary equilibria of system (2) are unstable, consisting either of repelling nodes or saddle points depending on the parameter values.
Since we are considering the symmetric case of model (2), we have \(\frac{q_i^{*} }{Q^{{*}}}=\frac{1}{n}\) and then \(\frac{\partial ^{2}\prod _i }{\partial q_i \partial q_j }\left( {E^{{*}}} \right) =-\frac{1}{\eta }Q^{{*-(1+\eta )}/\eta }\left( {1-\frac{1+\eta }{\eta n}} \right) >0,\forall \eta \in \left( {\frac{1}{n},\frac{1}{n-1}} \right) \).
The fulfillment of the sufficient condition is guaranteed by \(\frac{\partial ^{2}\prod _{i,t+1}^e }{\partial q_{i,t+1}^2 }=\frac{-2Q_t^{{-(1+\eta )}/\eta } }{\eta }<0\).
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Acknowledgements
The authors wish to thank the Spanish Ministry of Economics and Competitiveness (ECO2016-74940-P) and the Government of Aragon and FEDER (consolidated group S40_17R) for their financial support. The authors would like to express their thanks to the anonymous referees for their comments on earlier versions of this work.
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Andaluz, J., Elsadany, A.A. & Jarne, G. Dynamic Cournot oligopoly game based on general isoelastic demand. Nonlinear Dyn 99, 1053–1063 (2020). https://doi.org/10.1007/s11071-019-05333-7
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DOI: https://doi.org/10.1007/s11071-019-05333-7