The Dynamics of the Cobweb When Producers are Risk Averse Learners

  • Carl Chiarella
  • Xue-Zhong He

Abstract

In this paper we investigate the dynamics of the traditional cobweb model where producers are risk averse and seek to learn the distribution of asset prices. We consider the subjective estimates of the statistical distribution of the market prices based on L.-step backward time series of market clearing prices. With constant absolute risk aversion, the cobweb model becomes nonlinear. Sufficient conditions on the local stability of the unique positive equilibrium of the nonlinear model are derived and, consequently, we show that the local stability region is proportional to the lag length L. When the equilibrium loses its local stability, we show that, for L = 2, the model has a strong 1: 3 resonance bifurcation and a family of fixed points of order 3 becomes unstable on both sides of criticality. For general lag lengths, numerical simulations suggest that the model displays a variety of complex dynamics.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. Arrowsmith, J. Cartwright, A. Lansbury, and C. Place. The Bogdanov map: Bifurcations, model locking, and chaos in a dissipative system. Int. J. Bifurcation and Chaos, 3:803–842, 1993.CrossRefGoogle Scholar
  2. [2]
    Z. Artstein. Irregular cobweb dynamics. Economics Letters, 11:15–17, 1983.CrossRefGoogle Scholar
  3. [3]
    J.-M. Boussard. When risk generates chaos. Journal of Economic Behavior and Organization, 29:433–446, 1996.CrossRefGoogle Scholar
  4. [4]
    C. Chiarella. The cobweb model, its instability and the onset of chaos. Economic Modelling, 5:377–384, 1988.CrossRefGoogle Scholar
  5. [5]
    C. Chiarella and X.-Z. He. Learning about the Cobweb, pages 244–257. 1998. Complex Systems’98, eds, Standish, R., Complexity Online Network, 1998, ISBN 0 7334 0537 1.Google Scholar
  6. [6]
    S. Elaydi. An Introduction to Difference Equations. Springer, New York, 1996.CrossRefGoogle Scholar
  7. [7]
    J Hale and H. Kocak. Dynamics and bifurcations, volume 3 of Texts in Applied Mathematics. Springer-Verlag, New York, 1991.CrossRefGoogle Scholar
  8. [8]
    J. Holmes and R. Manning. Memory and market stability, the case of the cobweb. Economics Letters, 28:1–7, 1988.CrossRefGoogle Scholar
  9. [9]
    C. Hommes, Adaptive learning and roads to chaos: The case of the cobweb. Economics Letters, 1991:127–132, 1991.CrossRefGoogle Scholar
  10. [10]
    G. Iooss. Bifurcations of maps and applications, volume 36 of Mathematics studies. North-Holland, Amsterdam, 1979.Google Scholar
  11. [11]
    R. Jensen and R. Urban. Chaotic price behaviour in a nonlinear cobweb model. Economics Letters, 15:235–240, 1984.CrossRefGoogle Scholar
  12. [12]
    Y.A. Kuznetsov. Elements and applied bifurcation theory, volume 112 of Applied mathematical sciences. SV, New York, 1995.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Carl Chiarella
    • 1
  • Xue-Zhong He
    • 1
  1. 1.School of Finance and EconomicsUniversity of TechnologySydneyAustralia

Personalised recommendations