A STOCHASTIC COBWEB DYNAMICAL MODEL

Serena Brianzoni, Cristiana Mammana, Elisabetta Michetti, and Francesco Zirilli

Discrete Dynamics in Nature and Society

  We consider the dynamics of a stochastic cobweb model with linear demand and a backwardbending 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 q0, q1 and the forward predictor with probability (1 - q), so that the expected price at time t 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.

Introduction and Model

  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 who proposed a linear model with deterministic static expectation. The least convincing elements of this initial formulation are the linearity of the functions describing the market and their simple expectations. For these reasons, several attempts have been made over time to improve the original model. In a number of papers, nonlinearity has been introduced in the cobwebmodel (see Holmes and Manning while other authors have considered different kinds of price expectations see, among others, Nerlove, Chiarella, Hommes, Gallas and Nusse). In Balasko and Royer, Bischi and Naimzada and Mammana and Michetti, an infinite memory learning mechanism has been introduced in the nonlinear cobweb model. A more sophisticated cobweb model is the one proposed by Brock and Hommes where heterogeneity is introduced via the assumption that agents have different beliefs, that is, rational and naive expectations. The authors assume that different types of agents have different beliefs about future variables and provide an important contribution to the literature evaluating heterogeneity.

  Research into the cobweb model has a long history, but all the previous papers have studied deterministic cobweb models. The dynamics of the cobweb model with a stochastic mechanism has not yet been studied. In this paper, we consider a stochastic nonlinear cobweb model that generalizes the model of Jensen and Urban assuming that the representative entrepreneur chooses between two different predictors in order to formulate their expectations: backward predictor:

1. the expectation of future price is the weighted mean of past observations with decreasing weights given by a (normalized) geometrical progression of parameter p(0 =< p =< 1) called memory rate; (see Balasko and Royer);
2. forward predictor: the formation mechanism of this expectation takes into account the market equilibrium price and assumes that, in the long run, the current price will converge to it.

  Our study tries to answer the criticism of the economists regarding the total lack of rationality in the expectations introduced in dynamical price-quantity models. In fact, we assume that agents are aware of the market equilibrium price and therefore we associate forward-looking expectations to backward-looking ones.

  At each time, the representative entrepreneur chooses the backward predictor with probability q (0 =< q =< 1) and the forward predictor with probability (1 - q). This corresponds to introducing heterogeneity in beliefs, in fact we are assuming that, on average, a fraction q of agents uses the first predictor, while a fraction (1 - q) of agents chooses the second one.

  In recent years, several models in whichmarkets are populated by heterogeneous agents have been proposed as an alternative to the traditional approach in economics and finance, based on a representative (and rational) agent. Kirman argues that heterogeneity plays an important role in the economic model and summarizes some of the reasons why the assumption of heterogeneous agents should be considered. Nevertheless, it is obvious that heterogeneity implies a shift from simple analytically tractable models with a representative, rational agent to a more complicated framework so that a computational approach becomes necessary.

  The present work represents a contribution to this line of research: as in Brock and Hommes we assume different groups of agents even if no switching between groups is possible. Besides, our case can be related to the deterministic limit case studied in Brock and Hommes, when the intensity of choice of agents goes to zero and agents are equally shared between two groups. The new element with respect to such a limit case is that we admit random changes to the fractions of agents around the mean. Moreover, even though our assumption is the same as considering (on average) fixed time proportions of agents, the fraction of agents employing trade rules based on past prices increases as q increases. In fact the parameter q can be understood as a sort of external signal of the market price fluctuations. More specifically, increasing values of q correspond to greater irregularity of the market. In our framework, this means that for high values of q, a greater fraction of agents expect that the price will follow the trend implied by previous prices instead of going toward its fundamental value, and they will prefer to use trading rules based on past observed events.

  In the model herewith proposed, the time evolution of the expected price is described by a stochastic dynamical system. (Recent works in this direction are those by Evans and Honkapohja and Branch and McGough.) More precisely, since for simplicity we start considering a discrete time dynamical system, the expected price is a discrete time stochastic process. In particular the expected price is a random variable at any time.

  We note that the successful development of ad hoc stochastic cobweb models to describe the time evolution of the prices of commodities, will make possible to use these models to describe fluctuations in price derivatives having the commodities, has underlying assets. The stochastic cobweb model presented here can be considered as a first step in the study of a more general class of models.

  The paper is organized as follows. In Section 2, we formulate the model in its general form. In Section 3, we consider the case where the memory rate is equal to zero, that is, the case with naive versus forward-looking expectations, so that the agent remembers only the last observed price. In this case we proceed as follows. First, we approximate the initial model with a new one having discrete states. Consequently we obtain a finite states stochastic process without memory, that is, a Markov chain. We determine the probability distribution of the random variable of the process solution of the Markov chain and, using a mixture of analytical tools and numerical methods, we show that its asymptotic behavior depends on the parameter of the logistic equation describing the price evolution that we call. Second, we extend the analysis to the corresponding continuous time Markov process and we obtain the Chapman-Kolmogorov forward equation. In Section 4, we propose an empirical study of the initial model (i.e., the model with continuous states), that is, we do the appropriate statistics of a sample of trajectories of the model generated numerically. In particular, we obtain the probability density function of the random variable describing the expected price as a function of time and we study these densities in the limit when time goes to infinity. Numerical evidence of bounded price oscillations is shown and the role of the memory rate, that is, the role of backward-looking expectations, is considered. The results obtained on the stochastic cobweb model confirm that the system becomes less and less complex as the memory rate increases, this behaviour is similar to the one observed in the deterministic cobweb model see Mammana and Michetti. Note that the model considered in Section 4 is not a Markov chain.

  We consider a cobweb-type model with linear demand and a backward-bending supply curve (i.e., a concave parabola). (This formulation for the supply function was proposed in Jensen and Urban.) A supply curve of this type is economically justified, for instance, by the presence of external economies, that is by the advantage that businesses do not gain from their individual expansion, but rather from the expansion of the industry as a whole (see Sraffa).