A cobweb model with gradient adjustment mechanism: nonlinear dynamics and multistability

ABSTRACT

A cobweb model, characterized by boundedly rational producers with a production adjustment mechanism based on the gradient rule, is described by a nonlinear discrete time dynamical system of the plane. Firms do not have a complete knowledge of the demand function and try to infer how the market will respond to their production changes by an empirical estimates of the marginal profits. Analytical conditions for local stability of the market equilibrium are provided, showing that the stability loss of the market equilibrium may give rise to chaotic dynamic as well. When memory is introduced in the production adjustment mechanism, a locally stabilizing effect is revealed as well as a globally qualitatively destabilizing role for memory. This is related to the occurrence of period doubling and Neimark–Sacker bifurcations, the latter being of supercritical nature as analytically proved. Endogenous fluctuations and multistability, with consequent loss of predictability in the long run dynamics, are observed.

KEYWORDS:

• Cobweb model
• nonlinear dynamics
• gradient rule
• bifurcation analysis
• multistability

Acknowledgments

The authors wish to thank the Editor and two anonymous Referees for their valuable comments and suggestions. The usual caveats apply.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. As before, we remark that we are still considering the system only in .