# Dynamic Cournot oligopoly game based on general isoelastic demand

## 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.

This is a preview of subscription content, access via your institution.

1. 1.

The boundary equilibria of system (2) are unstable, consisting either of repelling nodes or saddle points depending on the parameter values.

2. 2.

Since we are considering the symmetric case of model (2), we have $$\frac{q_i^{*} }{Q^{{*}}}=\frac{1}{n}$$ and then $$\frac{\partial ^{2}\prod _i }{\partial q_i \partial q_j }\left( {E^{{*}}} \right) =-\frac{1}{\eta }Q^{{*-(1+\eta )}/\eta }\left( {1-\frac{1+\eta }{\eta n}} \right) >0,\forall \eta \in \left( {\frac{1}{n},\frac{1}{n-1}} \right)$$.

3. 3.

The fulfillment of the sufficient condition is guaranteed by $$\frac{\partial ^{2}\prod _{i,t+1}^e }{\partial q_{i,t+1}^2 }=\frac{-2Q_t^{{-(1+\eta )}/\eta } }{\eta }<0$$.

## References

1. 1.

Andaluz, J., Elsadany, A.A., Jarne, G.: Nonlinear Cournot and Bertrand-type dynamic triopoly with differentiated products and heterogeneous expectations. Math. Comput. Simul. 132, 86–99 (2017)

2. 2.

Bandyopadhyay, S.: Demand elasticities, asymmetry and strategic trade policy. J. Int. Econ. 42(1–2), 167–177 (1997)

3. 3.

Beard, R.: N-firm oligopoly with general iso-elastic demand. Bull. Econ. Res. 67(4), 336–345 (2015)

4. 4.

Bergstrom, T., Varian, H.: Two remarks on Cournot equilibria. Econ. Lett. 19, 5–8 (1985)

5. 5.

Bischi, G.I., Naimzada, A.: Global analysis of a dynamic duopoly game with bounded rationality. In: Filar, J.A., Gaitsgory, V., Mizukami, K. (eds.) Advances in Dynamics Games and Application, vol. 5, pp. 361–385. Birkhäuser, Boston (2000)

6. 6.

Bischi, G.I., Naimzada, A., Sbragia, L.: Oligopoly games with Local Monopolistic Approximation. J. Econ. Behav. Organ. 62, 371–388 (2007)

7. 7.

Bischi, G.I., Chiarella, C., Kopel, M., Szidarovszky, F.: Nonlinear Oligopolies. Stability and Bifurcations. Ed. Springer (2010)

8. 8.

Cerboni Baiardi, L., Naimzada, A.K.: Imitative and best response behaviors in a nonlinear Cournotian setting. Chaos: an interdisciplinary. J. Nonlinear Sci. 28, 5 (2018). https://doi.org/10.1063/1.5024381

9. 9.

Chirco, A., Scrimitore, M., Colombo, C.: Competition and the strategic choice of managerial incentives: the relative performance case. Metroeconomica 62(4), 533–547 (2011)

10. 10.

Collie, D.: Collusion and the elasticity of demand. Econ. Bull. 12, 1–6 (2004)

11. 11.

Cournot, A.: Recherches sur les principes mathématiques de la théorie des richesses. Hachette, Paris (1838)

12. 12.

Fanti, L., Gori, L., Sodini, M.: Nonlinear dynamics in a Cournot duopoly with iso-elastic demand. Math. Comput. Simul. 108, 129–143 (2015)

13. 13.

Goodwin, R.M.: Chaotic Economic Dynamics. Oxford University Press, Oxford (1990)

14. 14.

Hommes, C.: Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems. Cambridge University Press, Cambridge (2013)

15. 15.

Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, vol. 112. Springer, New York (2004)

16. 16.

Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)

17. 17.

Lorenz, H.W.: Nonlinear Dynamical Economics and Chaotic Motion. Ed. Springer. Berlin (Vol. 334) (1993)

18. 18.

Matouk, A.E., Elsadany, A.A., Xin, B.: Neimark-Sacker bifurcation analysis and complex nonlinear dynamics in a heterogeneous quadropoly game with an isoelastic demand function. Nonlinear Dyn. 89, 2533–2552 (2017)

19. 19.

Moon, F.C.: Chaotic and Fractal Dynamics: Introduction for Applied Scientists and Engineers. Ed. Wiley. New Jersey (2008)

20. 20.

Neary, J.P.: International trade in general oligopolistic equilibrium. In: CESinfo Area Conference on Global Economy, CESinfo, Munich (2009)

21. 21.

Novshek, W.: On the existence of Cournot equilibrium. Rev. Econ. Stud. 52(1), 85–98 (1985)

22. 22.

Puu, T.: Chaos in duopoly pricing. Chaos Solitons Fractals 1, 573–581 (1991)

23. 23.

Puu, T.: Complex dynamics with three oligopolists. Chaos Solitons Fractals 7, 2075–2081 (1996)

24. 24.

Seydel, R.: Practical Bifurcation and Stability Analysis, Interdisciplinary Applied Mathematics, vol. 5. Springer, New York (2010)

25. 25.

Shone, R.: Economic Dynamics: Phase Diagrams and Their Economic Application. Cambridge University Press, Cambridge (2002)

26. 26.

Tramontana, F., Elsadany, A.A.: Heterogeneous triopoly game with iso-elastic demand function. Nonlinear Dyn. 68, 187–193 (2012)

27. 27.

Tversky, A., Kahneman, D.: Rational choice and the framing of decisions. In: Multiple Criteria Decision Making and Risk Analysis Using Microcomputers, pp. 81-126. Springer, Berlin, Heidelberg (1989)

## 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.

## Author information

Authors

### Corresponding author

Correspondence to J. Andaluz.

## Ethics declarations

### Conflict of interest

The authors declare that they have no conflict of interest.