Memory, market stability and attractors coexistence in a nonlinear cobweb model

Abstract

We investigate the dynamics of a cobweb type model with nonlinear demand and supply curves in which producers make forecasts on future prices with a backward looking expectation formation mechanism: the expected price for the next period is obtained by a weighted average of the prices observed in the last two periods. The study herewith presents aims at confirming the existence of a locally stabilising effect due to the presence of memory, but an increase of memory in price expectations can be globally qualitatively destabilising, in the sense that it leads to coexistence of different attractors with their respective basins of attraction.

Keywords:

• Cobweb model
• memory
• stability
• bifurcation analysis
• attractors coexistence
• Arnold tongues

Acknowledgements

We thank the Editor in Chief and an anonymous Referee for useful comments and suggestions. The usual caveats apply.

Notes

No potential conflict of interest was reported by the authors.

1 In general a strictly convex cost curve leads to a nonlinear, increasing, supply curve. Our nonlinear supply curve can, for example, be derived from the convex cost function $c(q)=\frac{qlogq-q}{b}+q+e^{-b}$, where b is a positive constant. See also [14].

2 An exponential supply function like the one here proposed has been estimated in the context of electricity markets and energy prices. See [17] and [22].

3 See [18] for a detailed mathematical analysis about the dynamics around resonance points. See also [1] for a description of the dynamics around a 1 : 2 resonance point in a duopoly model.

4 For example, between two periodicity regions related to cycles with rotation numbers 1 / 2 and 1 / 3 there is a region corresponding to cycles with rotation number 2 / 5.