Decisions in Economics and Finance

, Volume 41, Issue 2, pp 357–378 | Cite as

Steady states, stability and bifurcations in multi-asset market models

  • Roberto Dieci
  • Noemi Schmitt
  • Frank WesterhoffEmail author


We provide a full analytical treatment of a multi-asset market model in which speculators have the choice between two risky and one safe asset. As it turns out, the dynamics of our model is driven by a four-dimensional nonlinear map and may undergo a transcritical, flip or Neimark–Sacker bifurcation. While the first bifurcation is associated with an undervaluation of the risky assets, the latter two may trigger (complex) endogenous dynamics. To facilitate our analysis, we first study a simpler two-dimensional setup of our model in which speculators can only switch between one risky and one safe asset.


Multi-asset markets Replicator dynamics Nonlinear maps Stability and bifurcation analysis 

JEL Classification

D84 G12 G41 


  1. Agliari, A., Dieci, R.: Coexistence of attractors and homoclinic loops in a Kaldor-like business cycle model. In: Puu, T., Sushko, I. (eds.) Business cycle dynamics: models and tools, pp. 223–254. Springer, Berlin (2006)CrossRefGoogle Scholar
  2. Agliari, A., Bischi, G.-I., Gardini, L.: Some methods for the global analysis of closed invariant curves in two-dimensional maps. In: Puu, T., Sushko, I. (eds.) Business Cycle Dynamics: Models and Tools, pp. 7–49. Springer, Berlin (2006)CrossRefGoogle Scholar
  3. Agliari, A., Naimzada, A., Pecora, N.: Boom-bust dynamics in a stock market participation model with heterogeneous traders. J. Econ. Dyn. Control 91, 458–468 (2018)CrossRefGoogle Scholar
  4. Barberis, N., Shleifer, A., Vishny, R.: A model of investor sentiment. J. Financ. Econ. 49, 307–343 (1998)CrossRefGoogle Scholar
  5. Bischi, G.I., Lamantia, F., Radi, D.: An evolutionary Cournot model with limited market knowledge. J. Econ. Behavior Org. 116, 219–238 (2015)CrossRefGoogle Scholar
  6. Black, F.: Noise. J. Finance 41, 529–543 (1986)CrossRefGoogle Scholar
  7. Brock, W., Hommes, C.: Heterogeneous beliefs and routes to chaos in a simple asset-pricing model. J. Econ. Dyn. Control 22, 1235–1274 (1998)CrossRefGoogle Scholar
  8. Chiarella, C.: The dynamics of speculative behavior. Ann. Oper. Res. 37, 101–123 (1992)CrossRefGoogle Scholar
  9. Chiarella, C., Dieci, R., Gardini, L.: The dynamic interaction of speculation and diversification. Appl. Math. Finance 12, 17–52 (2005)CrossRefGoogle Scholar
  10. Chiarella, C., Dieci, R., He, X.-Z.: Heterogeneous expectations and speculative behavior in a dynamic multi-asset framework. J. Econ. Behavior Org. 62, 408–427 (2007)CrossRefGoogle Scholar
  11. Day, R., Huang, W.: Bulls, bears and market sheep. J. Econ. Behavior Org. 14, 299–329 (1990)CrossRefGoogle Scholar
  12. De Long, B., Shleifer, A., Summers, L., Waldmann, R.: Positive feedback investment strategies and destabilizing rational speculation. J. Finance 45, 379–395 (1990a)CrossRefGoogle Scholar
  13. De Long, B., Shleifer, A., Summers, L., Waldmann, R.: Noise trader risk in financial markets. J. Polit. Econ. 98, 703–738 (1990b)CrossRefGoogle Scholar
  14. Dieci, R., Schmitt, N., Westerhoff, F.: Interactions between stock, bond and housing markets. J. Econ. Dyn. Control 91, 43–70 (2018)CrossRefGoogle Scholar
  15. Dindo, P., Tuinstra, J.: A class of evolutionary models for participation games with negative feedback. Comput. Econ. 37, 267–300 (2011)CrossRefGoogle Scholar
  16. Hofbauer, J., Sigmund, K.: The Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge (1988)Google Scholar
  17. Lux, T.: Herd behaviour, bubbles and crashes. Econ. J. 105, 881–896 (1995)CrossRefGoogle Scholar
  18. Medio, A., Lines, M.: Nonlinear Dynamics: A Primer. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  19. Schmitt, N., Westerhoff, F.: Speculative behavior and the dynamics of interacting stock markets. J. Econ. Dyn. Control 45, 262–288 (2014)CrossRefGoogle Scholar
  20. Schmitt, N., Westerhoff, F.: Managing rational routes to randomness. J. Econ. Behavior Org. 116, 157–173 (2015)CrossRefGoogle Scholar
  21. Schmitt, N., Westerhoff, F.: Stock market participation and endogenous boom-bust dynamics. Econ. Lett. 148, 72–75 (2016)CrossRefGoogle Scholar
  22. Schmitt, N., Tuinstra, J., Westerhoff, F.: Side effects of nonlinear profit taxes in a behavioral market entry model: abrupt changes, coexisting attractors and hysteresis problems. J. Econ. Behavior Org. 135, 15–38 (2017)CrossRefGoogle Scholar
  23. Shiller, R.: Speculative prices and popular models. J. Econ. Perspect. 4, 55–65 (1990)CrossRefGoogle Scholar
  24. Shiller, R.: Irrational Exuberance. Princeton University Press, Princeton (2015)CrossRefGoogle Scholar
  25. Shleifer, A., Summers, L.: The noise trader approach to finance. J. Econ. Perspect. 4, 19–33 (1990)CrossRefGoogle Scholar
  26. Shleifer, A., Vishny, R.W.: The limits of arbitrage. J. Finance 52, 35–55 (1997)CrossRefGoogle Scholar
  27. Simon, H.: A behavioral model of rational choice. Q. J. Econ. 9, 99–118 (1955)CrossRefGoogle Scholar
  28. Tversky, A., Kahneman, D.: Judgment under uncertainty: heuristics and biases. Science 185, 1124–1131 (1974)CrossRefGoogle Scholar
  29. Westerhoff, F., Dieci, R.: The effectiveness of Keynes-Tobin transaction taxes when heterogeneous agents can trade in different markets: a behavioral finance approach. J. Econ. Dyn. Control 30, 293–322 (2006)CrossRefGoogle Scholar

Copyright information

© Associazione per la Matematica Applicata alle Scienze Economiche e Sociali (AMASES) 2018

Authors and Affiliations

  • Roberto Dieci
    • 1
  • Noemi Schmitt
    • 2
  • Frank Westerhoff
    • 2
    Email author
  1. 1.Department of Mathematics and School of Economics, Management and StatisticsUniversity of BolognaBolognaItaly
  2. 2.Department of EconomicsUniversity of BambergBambergGermany

Personalised recommendations