Discrete Dynamics in Nature and Society
Volume 2008 (2008), Article ID 219653, 18 pages
A Stochastic Cobweb Dynamical Model
1Dipartimento di Istituzioni Economiche e Finanziarie, Università Degli Studi di Macerata, 62100 Mecerata, Italy
2Dipartimento di Matematica G. Castelnuovo, Università di Roma “La Sapienza”, 00185 Roma, Italy
Received 6 December 2007; Revised 31 March 2008; Accepted 29 May 2008
Academic Editor: Xue-Zhong He
Copyright © 2008 Serena Brianzoni et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the dynamics of a stochastic cobweb model with linear demand and a backward-bending supply curve. In our model, forward-looking expectations and backward-looking ones are assumed, in fact we assume that the representative agent chooses the backward predictor with probability , and the forward predictor with probability , so that the expected price at time is a random variable and consequently the dynamics describing the price evolution in time is governed by a stochastic dynamical system. The dynamical system becomes a Markov process when the memory rate vanishes. In particular, we study the Markov chain in the cases of discrete and continuous time. Using a mixture of analytical tools and numerical methods, we show that, when prices take discrete values, the corresponding Markov chain is asymptotically stable. In the case with continuous prices and nonnecessarily zero memory rate, numerical evidence of bounded price oscillations is shown. The role of the memory rate is studied through numerical experiments, this study confirms the stabilizing effects of the presence of resistant memory.