Cobweb

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A COBWEB MODEL WITH RANDOM EXPECTATIONS
Serena Brianzoni, Università degli studi di Macerata
Cristiana Mammana, Università degli studi di Macerata
Elisabetta Michetti, Università degli studi di Macerata
Francesco Zirilli, Università di Roma ‘La Sapienza’
EXTENDED ABSTRACT
1. Introduction
The cobweb model is a dynamical system that describes price fluctuations as a result of the interaction between demand function depending on current price and supply function depending on expected price.
A classic definition of the cobweb model is the one given by Ezekiel (1938) who proposed a linear model with deterministic static expectation. The least convincing elements of the initial formulation is the linearity of the functions describing the market and its simple expectations. For these reasons several efforts have been made over time to improve the original model. In a number of works nonlinearities have been introduced in the cobweb model (see Holmes and Manning (1988)) while other authors considered different kinds of price expectations (see, among others, Nerlove (1958), Chiarella (1988), Hommes (1994),
Gallas and Nusse (1996)). More recently in Mammana and Michetti (2003, 2004) an infinite memory learning mechanism has been introduced in the nonlinear cobweb model.
In this work we consider a stochastic nonlinear cobweb model that generalizes the model of Jensen and Urban (1984) assuming that the representative entrepreneur chooses between two different predictors in order to formulate his expectations:

backward predictor: the expectation of future price is the arithmetical mean of past observations with decreasing weights, according to a geometrical progression of ρ region;1

forward predictor: the formation mechanism of this expectation takes into account the market equilibrium price while considering that the current price will converge to it only in the long run.
The representative entrepreneur chooses the backward predictor with probability q
( 0 < q < 1 ) and



References: [1] Ahmad, N., Bischi, G. I., Gardini, L., 1996. Infinite distributed memory in discrete dynamical systems [2] Aicardi, F., Invernizzi, S., 1992. Memory effects in discrete dynamical systems. [3] Branch, W., McGough, B., 2005. Misspecification and consistent expectations in stochastic non-linear economies [4] Brock, W. A., Hommes, C. H., 1997. A rational route to randomness. Econometrica 65, 1059-1160. [5] Chiarella, C., 1988. The cobweb model:its instability and the onset of chaos. Economic Modelling 5, 377-384. [6] Evans, G. V., Honkapohja, S., 1998. Stochastic gradient learning in the cobweb model. [7] Ezekiel, M., 1938. The cobweb theorem. Quarterly Journal of Economics 52, 255-280. [8] Gallas, J. A. C., Nusse, H. E., 1996. Periodicity versus chaos in the dynamics of the cobweb models [9] Holmes, J. M., Manning, R., 1988. Memory and market stability: the case of the cobweb. [10] Hommes, C. H., 1994. Dynamics of the cobweb model with adaptive expectations and nonlinear supply and demand [11] Jensen, R. V., Urban, R., 1984. Chaotic price behaviour in a nonlinear cobweb model. [12] Mammana, C., Michetti, E., 2003. Infinite memory expectations in a dynamic model with hyperbolic demand [13] Mammana, C., Michetti, E., 2004. Backward and forward-looking expectations in a chaotic cobweb model [14] Mann, W. R., 1953. Mean value methods in iterations. Proceedings of the American Mathematical Society 4, 506-510. [15] Michetti, E., 2000. Chaos and learning effects in cobweb models. Rivista di politica economica 12, 167-206. [16] Sraffa, P., 1986. Le leggi della produttività in regime di concorrenza. In P. Sraffa (ed.), Saggi (pp

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