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Multiple equilibria in a monopoly market with heterogeneous agents and externalities


  • Jean-Pierre Nadal
  • Denis Phan
  • Mirta Gordon
  • Jean Vannimenus


We explore the effects of social influence in a simple market model in which a large number of agents face a binary choice: to buy/not to buy a single unit of a product at a price posted by a single seller (monopoly market). We consider the case of positive externalities: an agent is more willing to buy if other agents make the same decision. We consider two special cases of heterogeneity in the individuals' decision rules, corresponding in the literature to the Random Utility Models of Thurstone, and of McFadden and Manski. In the first one the heterogeneity fluctuates with time, leading to a standard model in Physics: the Ising model at finite temperature (known as annealed disorder) in a uniform external field. In the second approach the heterogeneity among agents is fixed; in Physics this is a particular case of the quenched disorder model known as a random field Ising model, at zero temperature. We study analytically the equilibrium properties of the market in the limiting case where each agent is influenced by all the others (the mean field limit), and we illustrate some dynamic properties of these models making use of numerical simulations in an Agent based Computational Economics approach. Considering the optimization of the profit by the seller within the case of fixed heterogeneity with global externality, we exhibit a new regime where, if the mean willingness to pay increases and/or the production costs decrease, the seller's optimal strategy jumps from a solution with a high price and a small number of buyers, to another one with a low price and a large number of buyers. This regime, usually modelled with ad hoc bimodal distributions of the idiosyncratic heterogeneity, arises here for general monomodal distributions if the social influence is strong enough.

Suggested Citation

  • Jean-Pierre Nadal & Denis Phan & Mirta Gordon & Jean Vannimenus, 2005. "Multiple equilibria in a monopoly market with heterogeneous agents and externalities," Quantitative Finance, Taylor & Francis Journals, vol. 5(6), pages 557-568.
  • Handle: RePEc:taf:quantf:v:5:y:2005:i:6:p:557-568 DOI: 10.1080/14697680500362346

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    References listed on IDEAS

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    5. Gordon, Mirta B. & Nadal, Jean-Pierre & Phan, Denis & Vannimenus, Jean, 2005. "Seller's dilemma due to social interactions between customers," Physica A: Statistical Mechanics and its Applications, Elsevier, pages 628-640.
    6. Denis Phan & Stephane Pajot & Jean-Pierre Nadal, 2003. "The Monopolist's Market with Discrete Choices and Network Externality Revisited: Small-Worlds, Phase Transition and Avalanches in an ACE Framework," Computing in Economics and Finance 2003 150, Society for Computational Economics.
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    Cited by:

    1. Schütz, Gunter M. & de Almeida Prado, Fernando Pigeard & Harris, Rosemary J. & Belitsky, Vladimir, 2009. "Short-time behaviour of demand and price viewed through an exactly solvable model for heterogeneous interacting market agents," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(19), pages 4126-4144.
    2. Sebastian Goncalves & M. F. Laguna & J. R. Iglesias, 2012. "Why, when, and how fast innovations are adopted," Papers 1208.2589,
    3. Paolo Dai Pra & Fulvio Fontini & Elena Sartori & Marco Tolotti, 2011. "Endogenous equilibria in liquid markets with frictions and boundedly rational agents," Working Papers 7, Department of Management, Università Ca' Foscari Venezia.
    4. repec:eee:phsmap:v:486:y:2017:i:c:p:192-205 is not listed on IDEAS
    5. Vladimir Belitsky & Antonio L. Pereira & Fernando P. de Almeida Prado, 2009. "Stability analysis with applications of a two-dimensional dynamical system arising from a stochastic model of an asset market," Papers 0909.4815,
    6. Paolo Pellizzari & Elena Sartori & Marco Tolotti, 2014. "Trade-in programs in the context of technological innovation with herding," Working Papers 04, Department of Management, Università Ca' Foscari Venezia.
    7. Mirta B. Gordon & Jean-Pierre Nadal & Denis Phan & Viktoriya Semeshenko, 2012. "Entanglement between Demand and Supply in Markets with Bandwagon Goods," Papers 1209.1321,, revised Dec 2012.
    8. Szybisz, Martín A. & Szybisz, Leszek, 2017. "Extended nonlinear feedback model for describing episodes of high inflation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 465(C), pages 91-108.
    9. Viktoriya Semeshenko & Alexis Garapin & Bernard Ruffieux & Mirta Gordon, 2010. "Information-driven coordination: experimental results with heterogeneous individuals," Theory and Decision, Springer, pages 119-142.
    10. Vincenzo Atella & Jay Bhattacharya & Lorenzo Carbonari, 2008. "Pharmaceutical Industry, Drug Quality and Regulation: Evidence from US and Italy," NBER Working Papers 14567, National Bureau of Economic Research, Inc.
    11. Y. P. Ma & S. Gonçalves & S. Mignot & J.-P. Nadal & M. B. Gordon, 2009. "Cycles of cooperation and free-riding in social systems," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 71(4), pages 597-610, October.
    12. D. Sornette, 2014. "Physics and Financial Economics (1776-2014): Puzzles, Ising and Agent-Based models," Papers 1404.0243,
    13. Gunter M. Schutz & Fernando Pigeard de Almeida Prado & Rosemary J. Harris & Vladimir Belitsky, 2007. "Short-time behaviour of demand and price viewed through an exactly solvable model for heterogeneous interacting market agents," Papers 0801.0003,, revised Jun 2009.
    14. Denis Becker & Alexei Gaivoronski, 2014. "Stochastic optimization on social networks with application to service pricing," Computational Management Science, Springer, vol. 11(4), pages 531-562, October.
    15. Fontini, Fulvio & Sartori, Elena & Tolotti, Marco, 2016. "Are transaction taxes a cause of financial instability?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 450(C), pages 57-70.
    16. Paolo Pellizzari & Elena Sartori & Marco Tolotti, 2015. "Optimal Policies In Two-Step Binary Games Under Social Pressure And Limited Resources," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 18(05n06), pages 1-16, August.
    17. Christian Borghesi & Jean-Philippe Bouchaud, 2007. "Of songs and men: a model for multiple choice with herding," Quality & Quantity: International Journal of Methodology, Springer, pages 557-568.
    18. Jean-Philippe Bouchaud, 2012. "Crises and collective socio-economic phenomena: simple models and challenges," Papers 1209.0453,, revised Dec 2012.
    19. Kindler, A. & Solomon, S. & Stauffer, D., 2013. "Peer-to-peer and mass communication effect on opinion shifts," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(4), pages 785-796.
    20. Pigeard de Almeida Prado, Fernando & Belitsky, Vladimir & Ferreira, Alex Luiz, 2011. "Social interactions, product differentiation and discontinuity of demand," Journal of Mathematical Economics, Elsevier, vol. 47(4-5), pages 642-653.
    21. G'erard Weisbuch & Vincent Buskens & Luat Vuong, 2007. "Heterogeneity and Increasing Returns May Drive Socio-Economic Transitions," Papers 0706.1454,
    22. Semeshenko, Viktoriya & Gordon, Mirta B. & Nadal, Jean-Pierre, 2008. "Collective states in social systems with interacting learning agents," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(19), pages 4903-4916.

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