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

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## Abstract

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.

## Keywords

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

D84 G12 G41## References

- 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
- 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
- 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 - Barberis, N., Shleifer, A., Vishny, R.: A model of investor sentiment. J. Financ. Econ.
**49**, 307–343 (1998)CrossRefGoogle Scholar - 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 - Black, F.: Noise. J. Finance
**41**, 529–543 (1986)CrossRefGoogle Scholar - 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 - Chiarella, C.: The dynamics of speculative behavior. Ann. Oper. Res.
**37**, 101–123 (1992)CrossRefGoogle Scholar - Chiarella, C., Dieci, R., Gardini, L.: The dynamic interaction of speculation and diversification. Appl. Math. Finance
**12**, 17–52 (2005)CrossRefGoogle Scholar - 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 - Day, R., Huang, W.: Bulls, bears and market sheep. J. Econ. Behavior Org.
**14**, 299–329 (1990)CrossRefGoogle Scholar - 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 - De Long, B., Shleifer, A., Summers, L., Waldmann, R.: Noise trader risk in financial markets. J. Polit. Econ.
**98**, 703–738 (1990b)CrossRefGoogle Scholar - Dieci, R., Schmitt, N., Westerhoff, F.: Interactions between stock, bond and housing markets. J. Econ. Dyn. Control
**91**, 43–70 (2018)CrossRefGoogle Scholar - Dindo, P., Tuinstra, J.: A class of evolutionary models for participation games with negative feedback. Comput. Econ.
**37**, 267–300 (2011)CrossRefGoogle Scholar - Hofbauer, J., Sigmund, K.: The Theory of Evolution and Dynamical Systems. Cambridge University Press, Cambridge (1988)Google Scholar
- Lux, T.: Herd behaviour, bubbles and crashes. Econ. J.
**105**, 881–896 (1995)CrossRefGoogle Scholar - Medio, A., Lines, M.: Nonlinear Dynamics: A Primer. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
- Schmitt, N., Westerhoff, F.: Speculative behavior and the dynamics of interacting stock markets. J. Econ. Dyn. Control
**45**, 262–288 (2014)CrossRefGoogle Scholar - Schmitt, N., Westerhoff, F.: Managing rational routes to randomness. J. Econ. Behavior Org.
**116**, 157–173 (2015)CrossRefGoogle Scholar - Schmitt, N., Westerhoff, F.: Stock market participation and endogenous boom-bust dynamics. Econ. Lett.
**148**, 72–75 (2016)CrossRefGoogle Scholar - 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 - Shiller, R.: Speculative prices and popular models. J. Econ. Perspect.
**4**, 55–65 (1990)CrossRefGoogle Scholar - Shiller, R.: Irrational Exuberance. Princeton University Press, Princeton (2015)CrossRefGoogle Scholar
- Shleifer, A., Summers, L.: The noise trader approach to finance. J. Econ. Perspect.
**4**, 19–33 (1990)CrossRefGoogle Scholar - Shleifer, A., Vishny, R.W.: The limits of arbitrage. J. Finance
**52**, 35–55 (1997)CrossRefGoogle Scholar - Simon, H.: A behavioral model of rational choice. Q. J. Econ.
**9**, 99–118 (1955)CrossRefGoogle Scholar - Tversky, A., Kahneman, D.: Judgment under uncertainty: heuristics and biases. Science
**185**, 1124–1131 (1974)CrossRefGoogle Scholar - 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

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