IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v143y2021ics0960077920309954.html
   My bibliography  Save this article

Global Analysis, Multi-stability and Synchronization in a Competition Model of Public Enterprises with Consumer Surplus

Author

Listed:
  • Li, Wen-na
  • Elsadany, A.A.
  • Zhou, Wei
  • Zhu, Yan-lan

Abstract

We research global dynamic behaviors, multi-stability and synchronization in a dynamic duopoly Cournotian model with consumer surplus on the basis of nonlinear demand function and bounded rationality. This game generalizes the traditional dynamic duopoly Cournot game. The aim of enterprises is to optimize their own profits and the social welfare. This game is characterized by discrete difference equations incorporated in the competition model's optimization problem. The stability of equilibrium points and complicated dynamical phenomena are studied, such as flip bifurcation for unique Nash equilibrium point. The global bifurcation of non-invertible two-dimensional map is analyzed through critical curves, which is an important method to further study global properties. Multi-stability is observed due to the presence of multiple complex attractors. An inspection of the basins of attraction is provided for the situation of several attractors coexisting. Since this game has symmetry, it can be stated that a one-dimensional invariant sub-manifold is the diagonal of the system by analyzing the nonlinear map x(t+1)=μx(t)(1−x(t)). The properties of this nonlinear map is completely different from the logistic map although both of them have similar form. Along the invariant diagonal, the synchronization phenomenon of the game is discussed in the end.

Suggested Citation

  • Li, Wen-na & Elsadany, A.A. & Zhou, Wei & Zhu, Yan-lan, 2021. "Global Analysis, Multi-stability and Synchronization in a Competition Model of Public Enterprises with Consumer Surplus," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
    • Handle: RePEc:eee:chsofr:v:143:y:2021:i:c:s0960077920309954
    DOI: 10.1016/j.chaos.2020.110604
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077920309954
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2020.110604?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Bischi, Gian Italo & Kopel, Michael, 2001. "Equilibrium selection in a nonlinear duopoly game with adaptive expectations," Journal of Economic Behavior & Organization, Elsevier, vol. 46(1), pages 73-100, September.
    2. A. A. Elsadany & A. M. Awad, 2019. "Dynamics and chaos control of a duopolistic Bertrand competitions under environmental taxes," Annals of Operations Research, Springer, vol. 274(1), pages 211-240, March.
    3. Naimzada, Ahmad K. & Sbragia, Lucia, 2006. "Oligopoly games with nonlinear demand and cost functions: Two boundedly rational adjustment processes," Chaos, Solitons & Fractals, Elsevier, vol. 29(3), pages 707-722.
    4. Fanti, Luciano & Gori, Luca, 2012. "The dynamics of a differentiated duopoly with quantity competition," Economic Modelling, Elsevier, vol. 29(2), pages 421-427.
    5. Luciano Fanti & Domenico Buccella, 2017. "Corporate social responsibility, profits and welfare with managerial firms," International Review of Economics, Springer;Happiness Economics and Interpersonal Relations (HEIRS), vol. 64(4), pages 341-356, December.
    6. Cavalli, Fausto & Naimzada, Ahmad, 2016. "Complex dynamics and multistability with increasing rationality in market games," Chaos, Solitons & Fractals, Elsevier, vol. 93(C), pages 151-161.
    7. Agiza, H.N. & Elsadany, A.A., 2003. "Nonlinear dynamics in the Cournot duopoly game with heterogeneous players," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 320(C), pages 512-524.
    8. Andaluz, Joaquín & Jarne, Gloria, 2015. "On the dynamics of economic games based on product differentiation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 113(C), pages 16-27.
    9. William Novshek, 1985. "On the Existence of Cournot Equilibrium," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 52(1), pages 85-98.
    10. Agliari, A. & Naimzada, A.K. & Pecora, N., 2016. "Nonlinear dynamics of a Cournot duopoly game with differentiated products," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 1-15.
    11. J. Laugesen & E. Mosekilde & Yu. L. Maistrenko & V. L. Maistrenko, 1998. "On-Off Intermittency and Riddled Basins of Attraction in a Coupled Map System," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 1(02n03), pages 161-180.
    12. Nirvikar Singh & Xavier Vives, 1984. "Price and Quantity Competition in a Differentiated Duopoly," RAND Journal of Economics, The RAND Corporation, vol. 15(4), pages 546-554, Winter.
    13. Seung-Leul Kim & Sang-Ho Lee & Toshihiro Matsumura, 2019. "Corporate social responsibility and privatization policy in a mixed oligopoly," Journal of Economics, Springer, vol. 128(1), pages 67-89, September.
    14. Dana, Rose-Anne & Montrucchio, Luigi, 1986. "Dynamic complexity in duopoly games," Journal of Economic Theory, Elsevier, vol. 40(1), pages 40-56, October.
    15. Rand, David, 1978. "Exotic phenomena in games and duopoly models," Journal of Mathematical Economics, Elsevier, vol. 5(2), pages 173-184, September.
    16. Zhou, Jie & Zhou, Wei & Chu, Tong & Chang, Ying-xiang & Huang, Meng-jia, 2019. "Bifurcation, intermittent chaos and multi-stability in a two-stage Cournot game with R&D spillover and product differentiation," Applied Mathematics and Computation, Elsevier, vol. 341(C), pages 358-378.
    17. Hommes,Cars, 2015. "Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems," Cambridge Books, Cambridge University Press, number 9781107564978, October.
    18. Jean Magnan de Bornier, 1992. "The “Cournot-Bertrand Debate”: A Historical Perspective," History of Political Economy, Duke University Press, vol. 24(3), pages 623-656, Fall.
    19. Matsumura, Toshihiro & Matsushima, Noriaki & Cato, Susumu, 2013. "Competitiveness and R&D competition revisited," Economic Modelling, Elsevier, vol. 31(C), pages 541-547.
    20. Dixit, Avinash K, 1986. "Comparative Statics for Oligopoly," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 27(1), pages 107-122, February.
    21. Agiza, H.N. & Hegazi, A.S. & Elsadany, A.A., 2002. "Complex dynamics and synchronization of a duopoly game with bounded rationality," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 58(2), pages 133-146.
    22. Arghya Ghosh & Manipushpak Mitra, 2014. "Reversal of Bertrand-Cournot Rankings in the Presence of Welfare Concerns," Journal of Institutional and Theoretical Economics (JITE), Mohr Siebeck, Tübingen, vol. 170(3), pages 496-519, September.
    23. Xavier Vives, 2001. "Oligopoly Pricing: Old Ideas and New Tools," MIT Press Books, The MIT Press, edition 1, volume 1, number 026272040x, April.
    24. Duan, Xi-Chao & Yin, Jun-Feng & Li, Xue-Zhi, 2017. "Global Hopf bifurcation of an SIRS epidemic model with age-dependent recovery," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 613-624.
    25. Jun, Byoung & Vives, Xavier, 2004. "Strategic incentives in dynamic duopoly," Journal of Economic Theory, Elsevier, vol. 116(2), pages 249-281, June.
    26. Ahmed, E. & Elsadany, A.A. & Puu, Tonu, 2015. "On Bertrand duopoly game with differentiated goods," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 169-179.
    27. Rahman, Aminur & Blackmore, Denis, 2016. "Neimark-Sacker bifurcations and evidence of chaos in a discrete dynamical model of walkers," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 339-349.
    28. Myatt, David P. & Wallace, Chris, 2015. "Cournot competition and the social value of information," Journal of Economic Theory, Elsevier, vol. 158(PB), pages 466-506.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Singh, Piyush Pratap & Roy, Binoy Krishna, 2022. "Chaos and multistability behaviors in 4D dissipative cancer growth/decay model with unstable line of equilibria," Chaos, Solitons & Fractals, Elsevier, vol. 161(C).
    2. Fang Wu & Junhai Ma, 2023. "Research Trend, Logical Structure and Outlook on Complex Economic Game," Mathematics, MDPI, vol. 11(5), pages 1-16, February.
    3. Ye, Yi & Guo, Qiang & Nie, Jiajia, 2023. "Retailer information strategies with a dual-purpose manufacturer," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 180(C).
    4. Li, Hui & He, Ruichun & Zhou, Wei, 2022. "Dynamic behaviors in a Cournot duopoly model with knowledge spillover effect based on constant conjectural variation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 201(C), pages 305-323.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Sameh Askar & Abdulaziz Foul & Tarek Mahrous & Saleh Djemele & Emad Ibrahim, 2021. "Global and Local Analysis for a Cournot Duopoly Game with Two Different Objective Functions," Mathematics, MDPI, vol. 9(23), pages 1-14, December.
    2. Askar, S.S., 2022. "Nonlinear dynamic investigations and global analysis of a Cournot duopoly game with two different objectives," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    3. Andaluz, J. & Elsadany, A.A. & Jarne, G., 2017. "Nonlinear Cournot and Bertrand-type dynamic triopoly with differentiated products and heterogeneous expectations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 132(C), pages 86-99.
    4. Sameh S Askar & Abdulrahman Al-Khedhairi, 2020. "Local and Global Dynamics of a Constraint Profit Maximization for Bischi–Naimzada Competition Duopoly Game," Mathematics, MDPI, vol. 8(9), pages 1-16, August.
    5. Georges SARAFOPOULOS & Kosmas PAPADOPOULOS, 2017. "On A Cournot Duopoly Game With Differentiated Goods, Heterogeneous Expectations And A Cost Function Including Emission Costs," Scientific Bulletin - Economic Sciences, University of Pitesti, vol. 16(1), pages 11-22.
    6. Fanti, Luciano & Gori, Luca, 2012. "The dynamics of a differentiated duopoly with quantity competition," Economic Modelling, Elsevier, vol. 29(2), pages 421-427.
    7. Lorenzo Cerboni Baiardi & Ahmad K. Naimzada, 2018. "An evolutionary model with best response and imitative rules," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 41(2), pages 313-333, November.
    8. Troy Tassier, 2013. "Handbook of Research on Complexity, by J. Barkley Rosser, Jr. and Edward Elgar," Eastern Economic Journal, Palgrave Macmillan;Eastern Economic Association, vol. 39(1), pages 132-133.
    9. Li, Hui & He, Ruichun & Zhou, Wei, 2022. "Dynamic behaviors in a Cournot duopoly model with knowledge spillover effect based on constant conjectural variation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 201(C), pages 305-323.
    10. Mikhail Anufriev & Davide Radi & Fabio Tramontana, 2018. "Some reflections on past and future of nonlinear dynamics in economics and finance," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 41(2), pages 91-118, November.
    11. Antonio Doria, Francisco, 2011. "J.B. Rosser Jr. , Handbook of Research on Complexity, Edward Elgar, Cheltenham, UK--Northampton, MA, USA (2009) 436 + viii pp., index, ISBN 978 1 84542 089 5 (cased)," Journal of Economic Behavior & Organization, Elsevier, vol. 78(1-2), pages 196-204, April.
    12. Villena, Marcelo J. & Araneda, Axel A., 2017. "Dynamics and stability in retail competition," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 134(C), pages 37-53.
    13. Luciano Fanti & Luca Gori, 2013. "Stability Analysis in a Bertrand Duopoly with Different Product Quality and Heterogeneous Expectations," Journal of Industry, Competition and Trade, Springer, vol. 13(4), pages 481-501, December.
    14. Tomasz Dubiel-Teleszyński, 2010. "Complex Dynamics in a Bertrand Duopoly Game with Heterogeneous Players," Central European Journal of Economic Modelling and Econometrics, Central European Journal of Economic Modelling and Econometrics, vol. 2(2), pages 95-116, March.
    15. Lorenzo Cerboni Baiardi & Ahmad K. Naimzada, 2019. "An evolutionary Cournot oligopoly model with imitators and perfect foresight best responders," Metroeconomica, Wiley Blackwell, vol. 70(3), pages 458-475, July.
    16. PAPADOPOULOS Kosmas & SARAFOPOULOS Georges, 2019. "Dynamics of a Cournot Game with Differentiated Goods and Asymmetric Cost Functions based on Relative Profit Maximization," European Journal of Interdisciplinary Studies, Bucharest Economic Academy, issue 02, June.
    17. Fanti, Luciano & Gori, Luca, 2011. "Cross-ownership and stability in a Cournot duopoly," MPRA Paper 34574, University Library of Munich, Germany.
    18. Tramontana, Fabio, 2010. "Heterogeneous duopoly with isoelastic demand function," Economic Modelling, Elsevier, vol. 27(1), pages 350-357, January.
    19. Zhu, Yan-lan & Zhou, Wei & Chu, Tong, 2022. "Analysis of complex dynamical behavior in a mixed duopoly model with heterogeneous goods," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    20. Ueda, Masahiko, 2019. "Effect of information asymmetry in Cournot duopoly game with bounded rationality," Applied Mathematics and Computation, Elsevier, vol. 362(C), pages 1-1.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:143:y:2021:i:c:s0960077920309954. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.