# A Stochastic Cobweb Dynamical Model

Authors: Serena Brianzoni; Cristiana Mammana; Elisabetta Michetti; Francesco Zirilli
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

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 probabilityq_,__0_q_1, 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.


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## Body

## 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 [1] 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 cobweb model (see Holmes and Manning [2]) while
other authors have considered different kinds of price expectations (see, among
others, Nerlove [3], Chiarella [4], Hommes
[5],
Gallas and Nusse
[6]). In Balasko and Royer [7],
Bischi and Naimzada [8] and Mammana and Michetti [9, 10] 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 [11] 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 [12] assuming that the
representative entrepreneur chooses between two different predictors in order
to formulate their expectations:(1)
backward predictor: the expectation of future
price is the weighted mean of past observations with decreasing weights given
by a (normalized) geometrical progression of parameterρ (0≤ρ≤1) called memory rate; (see
Balasko and Royer [7]);(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 probabilityq(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 which markets 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 [13] 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 [11] 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 [11],
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 asq 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 [14] and Branch and
McGough [15].) 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 Section2, 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 μ.
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 [10]). Note that the model considered in
Section 4 is not a Markov chain.


## 2. The Model

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 [12].) 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
[16]).According to the previous considerations, demand and
supply are given by(2.1)Dt+1=D(pt+1)=a+bpt+1,St+1=S(Pte)=c+dPte+e(Pte)2,where Dt+1, St+1 are, respectively, the amount of goods
demanded by consumers and the amount of goods supplied by the entrepreneurs at
time t+1; and a,b,c,d,e are real constants such that b<0, d>0 and e<0.
In our model pt, Pte are, respectively, the price of the goods at
time t and the expectation at time t of the price at time t+1.The market clearing equationDt+1=St+1 gives a quadratic and convex relationship
between pt+1 and Pte,
that is:(2.2)pt+1=D−1(S(Pte))that
can be rearranged to obtain
the well-known logistic map(the logistic map can be
obtained from (2.2) by a linear transformation in the
variable Pte):(2.3)pt+1=fμ(Pte)=μPte(1−Pte),1<μ<4.This last formulation is the
same as the one reached by Jensen and Urban [12]. As a matter of fact, we
will consider pt∈[0,1],
for all t, where interesting dynamics occurs.The mapfμ is a function of expectations so that we are
considering two different formulations for the expectation formation mechanism.
Despite the fact that in an uncertain context, economic agents do not have a
perfect forecasting ability and therefore need to apply some sort of
extrapolative method to formulate hypotheses on the future level of prices, we
assume that they are aware of the market equilibrium level.


### 2.1. Backward-Looking Expectation

We consider the backward-looking component to extract
the future expected market price value from the prices observed in the past
through an infinite memory process (this iterative
scheme is known as Mann iteration, see Mann [17]). The use of this type of learning mechanism
is rather “natural” if the agent uses all the information available, that is,
all the historical price values. In our model Pte is the weighted mean of the previous values
taken by the real variable pt,
measured with decreasing weights given by the terms of a (normalized) geometric
progression of parameter ρ (0≤ρ≤1), called memory rate:(2.4)Pte=∑k=0tρt−kWtpk,ρ∈(0,1),Wt=∑k=0tρt−k=1−ρt+11−ρ.Note that when ρ→0,
naive or static expectations are considered; while when ρ→1, (2.4) becomes the arithmetic mean of past
prices. This kind of expectations have been used by several authors such as
Balasko and Royer [7], Bischi and Naimzada [8],
Invernizzi and Medio
[18], and Aicardi and Invernizzi [19]. They have also been studied
in Michetti [20] and in Mammana and Michetti
[10] where the model studied here
has been considered in the deterministic contest.
(Equation (2.3) with backward expectation as in (2.4)
has an equivalent autonomous limit form describing the expectation prices
dynamics, that is Pt+1e=−μ(1−ρ)(Pte)2+[μ(1−ρ)+ρ]Pte that can be used to study the asymptotic
behaviour of the sequence {Pte}t=0∞ and consequently of the sequence {pt}t=0∞ through fμ.
See Aicardi and Invernizzi [19] and
Bischi et al. [21].)


### 2.2. Forward-Looking Expectation

We consider a forward-looking expectation creation
mechanism. In order to introduce a more sophisticated predictor, we assume that
the representative supplier of the goods knows the market equilibrium price,
namelyP⋆,
but at the same time he knows that the process leading to the equilibrium is
not instantaneous. In other words, we do not assume that prices come back to
their equilibrium value P⋆ instantaneously, as was assumed in Mammana and
Michetti [10].Our assumption is consistent with the conclusion
reached in many dynamical cobweb models (see, e.g., Hommes [5] and Gallas
and Nusse [6]) where it is proved that prices converge to the steady state,
that is, to the equilibrium price, only in the long-run. According to such
considerations we use the following equation to describe the forward-looking
expectation:(2.5)Pte=P⋆+γ(pt−P⋆),γ∈(0,1),P⋆=μ−1μ,μ>1,where P⋆>0 is the long-run equilibrium price. We note that P⋆ is independent of the expectation mechanism
introduced. According to (2.5), the agent expects a weighted mean between the
last observed price and the equilibrium price. The economic intuition behind
the choice made in (2.5) is the following: if pt>P⋆ (pt<P⋆) the agent expects that the price decreases
(increases) toward its equilibrium value; in other words, the expected price
moves in the direction of the equilibrium value.


### 2.3. Choosing between Expectations

We assume that the representative agent chooses
between the two predictors (the backward and the forward predictors) as
follows:(2.6)Pte={∑k=0tρt−kWtpk,withprobabilityq,0≤q≤1,P⋆+γ(pt−P⋆),withprobability(1−q),that is, Pte is a random variable and consequently the
dynamical system (2.3) describing the price evolution in time is a stochastic
dynamical system, in fact the sequence of the prices pt+1,t=0,1,…,
is a sequence of random variables, that is, it is a discrete time stochastic
process obtained through the repeated application of fμ to Pte as shown in (2.3). Via (2.6), heterogeneity in
beliefs is introduced, more specifically (2.6) translates the assumption that on
average, a fraction q of the agents is “backward-looking” while a
fraction 1−q of the agents is “forward-looking.”In this paper, we want to study the stochastic processpt,t=0,1,…,
both from the theoretical and the numerical points of view (in this last case
we will use elementary numerical methods and statistical simulations). In
particular we want to describe the probability distribution of the discrete
time stochastic process pt,t=0,1,… as a function of the parameters defining the
model.


## 2.1. Backward-Looking Expectation

We consider the backward-looking component to extract
the future expected market price value from the prices observed in the past
through an infinite memory process (this iterative
scheme is known as Mann iteration, see Mann [17]). The use of this type of learning mechanism
is rather “natural” if the agent uses all the information available, that is,
all the historical price values. In our model Pte is the weighted mean of the previous values
taken by the real variable pt,
measured with decreasing weights given by the terms of a (normalized) geometric
progression of parameter ρ (0≤ρ≤1), called memory rate:(2.4)Pte=∑k=0tρt−kWtpk,ρ∈(0,1),Wt=∑k=0tρt−k=1−ρt+11−ρ.Note that when ρ→0,
naive or static expectations are considered; while when ρ→1, (2.4) becomes the arithmetic mean of past
prices. This kind of expectations have been used by several authors such as
Balasko and Royer [7], Bischi and Naimzada [8],
Invernizzi and Medio
[18], and Aicardi and Invernizzi [19]. They have also been studied
in Michetti [20] and in Mammana and Michetti
[10] where the model studied here
has been considered in the deterministic contest.
(Equation (2.3) with backward expectation as in (2.4)
has an equivalent autonomous limit form describing the expectation prices
dynamics, that is Pt+1e=−μ(1−ρ)(Pte)2+[μ(1−ρ)+ρ]Pte that can be used to study the asymptotic
behaviour of the sequence {Pte}t=0∞ and consequently of the sequence {pt}t=0∞ through fμ.
See Aicardi and Invernizzi [19] and
Bischi et al. [21].)


## 2.2. Forward-Looking Expectation

We consider a forward-looking expectation creation
mechanism. In order to introduce a more sophisticated predictor, we assume that
the representative supplier of the goods knows the market equilibrium price,
namelyP⋆,
but at the same time he knows that the process leading to the equilibrium is
not instantaneous. In other words, we do not assume that prices come back to
their equilibrium value P⋆ instantaneously, as was assumed in Mammana and
Michetti [10].Our assumption is consistent with the conclusion
reached in many dynamical cobweb models (see, e.g., Hommes [5] and Gallas
and Nusse [6]) where it is proved that prices converge to the steady state,
that is, to the equilibrium price, only in the long-run. According to such
considerations we use the following equation to describe the forward-looking
expectation:(2.5)Pte=P⋆+γ(pt−P⋆),γ∈(0,1),P⋆=μ−1μ,μ>1,where P⋆>0 is the long-run equilibrium price. We note that P⋆ is independent of the expectation mechanism
introduced. According to (2.5), the agent expects a weighted mean between the
last observed price and the equilibrium price. The economic intuition behind
the choice made in (2.5) is the following: if pt>P⋆ (pt<P⋆) the agent expects that the price decreases
(increases) toward its equilibrium value; in other words, the expected price
moves in the direction of the equilibrium value.


## 2.3. Choosing between Expectations

We assume that the representative agent chooses
between the two predictors (the backward and the forward predictors) as
follows:(2.6)Pte={∑k=0tρt−kWtpk,withprobabilityq,0≤q≤1,P⋆+γ(pt−P⋆),withprobability(1−q),that is, Pte is a random variable and consequently the
dynamical system (2.3) describing the price evolution in time is a stochastic
dynamical system, in fact the sequence of the prices pt+1,t=0,1,…,
is a sequence of random variables, that is, it is a discrete time stochastic
process obtained through the repeated application of fμ to Pte as shown in (2.3). Via (2.6), heterogeneity in
beliefs is introduced, more specifically (2.6) translates the assumption that on
average, a fraction q of the agents is “backward-looking” while a
fraction 1−q of the agents is “forward-looking.”In this paper, we want to study the stochastic processpt,t=0,1,…,
both from the theoretical and the numerical points of view (in this last case
we will use elementary numerical methods and statistical simulations). In
particular we want to describe the probability distribution of the discrete
time stochastic process pt,t=0,1,… as a function of the parameters defining the
model.


## 3. Naive versus Forward-Looking Expectations

### 3.1. The Discrete Time Stochastic Process

We consider the discrete time stochastic processpt,t=0,1,…,
with memory rate ρ equal to zero, that is, static and
forward-looking expectations are assumed. Lack of memory is the crucial
assumption to obtain the price dynamics described by a Markov process.Letxt be a possible value of the price, then our
stochastic process evolves according to the conditional probability of going
from state xt at time t to state xt+1 at time t+1.
Let this Markovian conditional probability be denoted by ψ(pt+1=xt+1∣pt=xt).According to the Markov property, sometime called
memoryless property, the state of the system at the future timet+1 depends only on the system state at time t and does not depend on the state at earlier
times. The Markov property can be stated as follows: ψ(pt+1=xt+1∣pt=xt,…,p0=x0)=ψ(pt+1=xt+1∣pt=xt), for all possible choices of the states x0,x1,…,xt,xt+1 and of the time t.
In order to obtain a Markov process given by a random variable Pt which takes only discrete values at time t=0,1,…,
we first discretize model (2.3) assuming(3.1)P¯t+1=f¯(Pte)={0,ifpt+1=fμ(Pte)=0,n,ifpt+1=fμ(Pte)∈(n−110,n10],n=1,2,…,10,so that the price assumes 11 values (0,1,…,10).
Consider the following new variables resulting from a discretization of the
interval [0,1] in the way such that(i)
p˜t=0 if pt=0 and p˜t=n/10 if pt∈((n−1)/10,n/10,],n=1,2,…,10,
while(ii)
P¯t=0 if pt=0 and P¯t=n if pt∈((n−1)/10,n/10,],n=1,2,…,10.
Then(3.2)Pte={p˜twithprobabilityq,γp˜t+(1−γ)P⋆,withprobability(1−q).Finally we associate each value of the price with a
state(3.3)Pt=siffP¯t=s−1,s=1,2,…,11so that we obtain the state set S={1,2,…,11}.Note that the corresponding process may be treated as
a discrete-time Markov chain, whose state space isS={1,2,…,11} (we use a state space S made of eleven states numbered from 1 to 11 for simplicity). The transition matrix is
given by A=[ψij],i,j∈S,
where ψij=ψ(Pt+1=j∣Pt=i),i,j∈S,
is the transition probability of going from state i to state j,i,j∈S.
We have ψij∈[0,1],i,j∈S and ∑j∈Sψij=1,i∈S.It is easy to see that the transition probabilitiesψij of models (2.3), (3.1),
(3.3) depend only on the
value of the current state i and the value of the following state j,
regardless of the time t when the transition occurs, consequently the
Markov chain considered is homogeneous. The homogeneity property implies that
the m-step state transition probability(3.4)ψij(m)=ψ{Pt+m=j∣Pt=i},i,j∈S,m=1,2,3,…,is also independent of t and it can be defined as(3.5)ψij(m)=∑k∈Sψik(m−1)ψkj(m−1),i,j∈S,m=2,3,…,so that the m-step transition matrix A(m) is given by(3.6)A(m)=Am.We are interested in the probability thatPt is in state j∈S at time t,
when at time 0P0 it is in state k with probability one.Let the probability vector be denoted byπ(m)=(π1(m),…,π11(m)),
where πj(m) is the probability that Pt is in state j after m steps, j∈S.
The probability distribution can be obtained through(3.7)π(m)=π(m−1)A=π(0)Am.Note that π(0)=(π1(0),π2(0),…,π11(0)) and that πi(0)=δik,i=1,2,…,11, where δik is the Kronecher delta.Recall that a subsetΛ⊂S is closed if ψij=0,
for all (i,j):i∈Λ,j∉Λ,
while Λ is irreducible if for all pairs (i,j)∈Λ there exists a positive integer k such that ψij(k)>0.
The subset Λ is reducible if it is not irreducible.
Finally we recall the notions of recurrent and transient states. A state i is called recurrent (transient)
if ∑m=1∞ψii(m)=∞(<∞).
(Regarding these definitions see, e.g., Feller [22].)We now assumeγ=0.5 and we prove a general result about the Markov
chain for some values of μ.Proposition 3.1.
For allμ∈(1,3.6), the transition matrix A is reducible.Proof.
ConsiderPt+1=j=11.
If no state Pt=i≠j∈S exists such that fμ(Pte)∈(0.9,1]⇒μPte(1−Pte)>0.9 then A is reducible. This last inequality has no
solution if μ∈(1,3.6), hence j=11 cannot be reached by another state. The
proposition is proved.We now fix the value of the parameterμ in (3.3), (3.1) in order to determine the
transition matrix A and to obtain the invariant distribution. In
fact, some numerical insight will be helpful to draw some general conclusions
on the qualitative properties of the process considered.In a first experiment, we assumeμ=2 and γ=0.5,
in this case we obtain a transition matrix A(11×11) whose nonzero entries are ψ1,1=ψ2,3=ψ3,5=ψ9,5=ψ10,3=ψ11,1=q, ψ1,5=ψ2,6=ψ3,6=ψ9,6=ψ10,6=ψ11,5=1−q and ψi,6=1,4≤i≤8.As previously provedA is reducible, moreover by simply rearranging
the order of the states in A,
it is possible to see that {6} is the unique closed set of the chain and that
it is an absorbing state.  This result holds for other
values of μ as stated in the following
proposition.Proposition 3.2.
Assumeγ=0.5.
For all μ∈(μk,μk+1], k=0,1,…,4,
where μk=10/(10−k) and μk+1=10/(10−(k+1)),
the state k+2 is an absorbing state.Proof.
Assumeμ∈(μk,μk+1], k=0,1,…,4.
Let(3.8)pt∈(k10,k+110]⇒p˜t=k+110,P¯t=k+1,hence Pt=k+2.
First consider that with
probabilityq also Pte=p˜t hence fμ(Pte)=fμ((k+1)/10).
Notice that fμ is increasing in μ then fμ(Pte)∈(fμk(Pte),fμk+1(Pte)].
Trivial computations show that(3.9)fμk(Pte)=k+1109−k10−k>k10,ask=0,1,…,4;fμk+1(Pte)=k+110.As a consequence fμ(Pte)∈(k/10,(k+1)/10] and consequently P¯t+1=k+1 implying that Pt+1=k+2.
Second, with probability(1−q),
we have(3.10)Pte=0.5p˜t+0.5P*=k+120+μ−12μ,hence fμ(Pte)=fμ(0.5p˜t+0.5P*);
again fμ is increasing in μ and, trivially, we have(3.11)fμk(Pte)=2k+12019−2k20−2k>k10,whilefμk+1(Pte)=k+110.Using the same arguments
employed before, we conclude that the state k+2 is mapped into the state k+2.
Henceψk+2,k+2=1.As proved in Proposition3.2, we can conclude that our
chain admits an absorbing state if μ∈(0,2],
confirming the result obtained with our simulations.Let us recall some mathematical results concerning the
stability of Markov chains (for further details
see, among others,
Feller [22] and Meyn and Tweedie [23]). A stationary distribution is a
probability distribution π¯ verifying the equation π¯A=π¯.
If there exists one, and only one, probability distribution π¯ such that ∥π(0)Am−π¯∥→0 as m→∞ for every initial probability distribution π(0),
we say that the Markov chain is asymptotically stable .In our case, forμ=2,
the Markov chain is asymptotically stable. In fact the limit distribution π¯=(π¯1,π¯2,…,π¯11),
where π¯j=limm→∞πj(m),j∈S,
is the distribution π¯j=δj,6,j∈S,
and this result holds for every initial probability distribution π(0).The asymptotic behaviour of the probability
distribution changes if we consider a different value ofμ,
for example μ=2.8 (notice that for this value of μ Proposition 3.2 does not hold). In this case
we obtain a transition matrix A whose nonzero entries are as
follows: ψ1,1=ψ2,4=ψ3,6=ψ4,7=ψ8,7=ψ9,6=ψ10,4=ψ11,1=q, ψ1,8=ψ2,8=ψ3,8=ψ4,8=ψ8,8=ψ9,7=ψ10,6=ψ11,6=1−q and ψi,8=1,5≤i≤7.Rearranging the order of the states inA, it is easy to deduce that A is reducible (as proved in Proposition 3.1):
the set Λ={7,8}⊂S is closed and irreducible, consequently the
states j=7 and j=8 are recurrent while all the other states are
transient. Moreover, considering the irreducible matrix(3.12)A¯=(01q1−q)we obtain the invariant
distribution given by (the nonzero elements of the
invariant distribution are obtained by looking at the left eigenvectors of
matrix A¯
)  (3.13)π¯=[0,0,0,0,0,0,q/(q+1),1/(q+1),0,0]. The following simulations support our analysis. We choose q=0.6 and we calculate the probability distribution
of the state variable after  m-steps for several initial conditions.First of all we consider the caseμ=2.
In Figure 1 we start from the initial condition P0=2 with probability one, that is from πi(0)=δi,2,i∈S.
After the first step two states can be reached with different probabilities.
After the third step we have that the state j=6 is reached with probability equal to one as
supported by the previous considerations. This situation does not change if the
number of steps taken is increased since j=6 is the unique asymptotic state. It should be
kept in mind that for a given initial distribution π(0) we define an asymptotic state as a
state j∈S such that π¯j≠0.Probability
distributions afterm-time steps obtained using function (3.3) with μ=2, γ=0.5, q=0.6 and initial condition P0=2 with probability one, that is πi(0)=δi,2,i=1,2,…,11.
(a)(b)(c)In Figure2 we calculate the probability distribution
for m=1 and m=150 when μ=2.8.
Our simulation proves that two equilibrium prices will be approached in the
long run with different probabilities.Probability distributions afterm-time steps obtained using (3.3) with μ=2.8, γ=0.5, q=0.6, and initial condition P0=2 with probability one, that is, πi(0)=δi,2,i=1,2,…,11.
(a)(b)Moreover, considering simulations for different values
ofμ it seems that as μ increases the process becomes more
complicated, this was already known in the case of the deterministic logistic
map (see, e.g., Devaney [24]). In fact several asymptotic states can be
reached as μ increases. More precisely, starting from the
same initial condition we have found different numbers of asymptotic states
corresponding to different values of μ.The diagram of Figure3 represents a sort of final-states
diagram: for μ chosen in the interval I=[1,4],
the probability distribution after  m-time steps, with m big enough (we have considered m=500) and an arbitrarily chosen initial state (we
have chosen P0=3 with probability one, that is πi(0)=δi,3,i∈S), has been
calculated and depicted versus the correspondent value of μ.
Moreover, after a large number of simulations, we have observed that the
final-states diagram does not change when different initial conditions are
considered so that Figure 3 seems to hold for every initial distribution π(0).
The empirical result described here is in agreement with the result proved in Proposition
3.2, furthermore we can conclude that the absorbing state existing for μ≤2 is also an asymptotic state.Figure 3
Asymptotic
states versusμ for γ=0.5, q=0.6, and an arbitrary initial condition.The situation is quite different for greater values ofμ.
For example, our calculation shows that there exists a value μ¯∈(3.33,3.34) such that the process converges to a unique
asymptotic state if μ∈[3.33,μ¯) while five asymptotic states are possible as
soon as the parameter value μ¯ is crossed. According to these considerations, μ¯ can be understood as a sort of bifurcation
value or critical value since the number of long-run states changes
going across μ=μ¯.
In Figure 4, we simulate the two cases μ=3.33 and μ=3.34 using a large number of time steps m,
in order to show how the asymptotic probability distribution obtained changes.Probability
distributions afterm-time steps obtained using (3.3) with γ=0.5, q=0.6, and initial condition P0=2 with probability one, that is, πi(0)=δi,2,i=1,2,…,11,
with different values of  μ:μ=3.33 (panel (a)) and μ=3.34 (panel (b)).
(a)(b)This study allows us to conclude that, in the case
with naive versus forward-looking expectations, there exists a unique
asymptotic distribution whose behaviour becomes more complicated asμ increases, according to the fact that
nonlinearity implies nontrivial dynamics.Obviously, the quantitative results are not
independent of the number of states considered, in any case it is possible to
verify that the qualitative results (i.e., the increase in complexity asμ increases) still hold. As a matter of fact,
note that a similar scenario occurs when we consider the model in a
deterministic contest with naive expectations. In fact, in this case, the price
sequence converges to a fixed point or to a 2-period cycle depending on the value of μ,
and the process as μ increases produces orbits tending towards
high-period cycles (see Devaney [24]).


### 3.2. The Continuous-Time Stochastic Process

In this section we move on to the continuous-time
Markov process, we calculate the probability distribution solving the
appropriate system of ordinary differential equations and we compare it with
the probability distribution obtained by statistical simulation of the
appropriate continuous-time limit of the stochastic
process defined by (2.3), (3.1),
(3.3). (About the
mathematical concepts related to continuous Markov processes see
Ethier and Kurtz [25].)In particular, we start looking at the transition
matrixB=B(Δt) over the time interval Δt>0.
Using (3.7) and taking time steps of length Δt, we obtain that the following relation holds π((m+1)Δt)=π(mΔt)B(Δt).AssumingmΔt=t,
we have π(t+Δt)=π(t)B and when Δt goes to zero and m goes to infinity in order to guarantee mΔt=t with t>0 fixed, it is easy to deduce the following
system of ordinary differential equations called Chapman-Kolmogorov's forward
equations:(3.14)dπdt(t)=π(t)Q,t>0,where the initial condition of
(3.14) is π(0)=π0, where π0 is a given probability
distribution and Q=limΔt→0((B(Δt)−I)/Δt) is called infinitesimal generator matrix,
whose entries are the transition rates.It is well known that the solution of (3.14) is given
by(3.15)π(t)=π0eQt,where eQt=∑n=0∞((Qt)n/n!).
In order to define the transition matrix over the time interval Δt, B(Δt),
in terms of the infinitesimal generator matrix Q,
we consider  that the probability vector π(t+Δt) can be expressed as follows:(3.16)π(t+Δt)=π(t)+dπdtΔt+O(Δt2)=π(t)[I+QΔt]+O(Δt2),Δt→0,where O(.) is the Landau symbol.Since the interval[t0,t] is divided into  m-time steps of length Δt,
we have(3.17)π(t)[I+QΔt]+O(Δt2)=π0[I+QΔt]t/Δt+1+O(Δt2),Δt→0,so that we have(3.18)π(t+Δt)=π0[I+QΔt]t/Δt+1+O(Δt2),Δt→0.Then, the transition matrix B is given by(3.19)B(Δt)=I+QΔt.From (3.15) we have that calculating the exponential of
the generator matrix Q times t;
and acting with this “exponential matrix” on π(0) we obtain the probability vector π(t) for an arbitrary value of t>0.
Moreover, we observe that the matrix Q=[rij]i,j∈S defined as Q=A−I,
where A is the one-step transition matrix defined in
the previous section and I is the identity matrix, is an infinitesimal
generator matrix. In fact it is easy to verify that the following properties
hold (see Inamura [26]):(1)
∑j=111rij=0,1≤i≤11;(2)
0≤−rii≤1,1≤i≤11;(3)
rij≥0,1≤i,j≤11,  with i≠j.We present some simulations to compare the numerical
solution obtained from computing (3.15) with the
statistical distribution obtained from simulations of the appropriate
continuous time limit of the stochastic process defined by (2.3),
(3.1), (3.3).In Figure5, we consider the case μ=2.8 and we observe that as Δt→0 and m→+∞ in such a way that the product Δt⋅m is equal to a constant value t>0,
the statistical distribution approaches the solution obtained solving
numerically (3.14) with the appropriate initial condition, that is computing
(3.15).Solution obtained using finite differences and
solution obtained using statistical simulation at timet=4 for μ=2.8, γ=0.5, q=0.6, and initial condition P0=3 with probability one, that is πi(0)=δi,3,i=1,2,…,11.
We compare the results obtained using different values of the time
discretization step Δt.
(a)(b)(c)Figure6 confirms that the same conclusion holds if we
consider the parameter value μ=3.34.Solution obtained using finite differences and
solution obtained using statistical simulation forμ=3.34, at time t=8 for μ=3.34, γ=0.5, q=0.6, and initial condition P0=3 with probability one, that is πi(0)=δi,3,i=1,2,…,11.
We compare the results obtained using different values of the time
discretization step Δt.
(a)(b)


## 3.1. The Discrete Time Stochastic Process

We consider the discrete time stochastic processpt,t=0,1,…,
with memory rate ρ equal to zero, that is, static and
forward-looking expectations are assumed. Lack of memory is the crucial
assumption to obtain the price dynamics described by a Markov process.Letxt be a possible value of the price, then our
stochastic process evolves according to the conditional probability of going
from state xt at time t to state xt+1 at time t+1.
Let this Markovian conditional probability be denoted by ψ(pt+1=xt+1∣pt=xt).According to the Markov property, sometime called
memoryless property, the state of the system at the future timet+1 depends only on the system state at time t and does not depend on the state at earlier
times. The Markov property can be stated as follows: ψ(pt+1=xt+1∣pt=xt,…,p0=x0)=ψ(pt+1=xt+1∣pt=xt), for all possible choices of the states x0,x1,…,xt,xt+1 and of the time t.
In order to obtain a Markov process given by a random variable Pt which takes only discrete values at time t=0,1,…,
we first discretize model (2.3) assuming(3.1)P¯t+1=f¯(Pte)={0,ifpt+1=fμ(Pte)=0,n,ifpt+1=fμ(Pte)∈(n−110,n10],n=1,2,…,10,so that the price assumes 11 values (0,1,…,10).
Consider the following new variables resulting from a discretization of the
interval [0,1] in the way such that(i)
p˜t=0 if pt=0 and p˜t=n/10 if pt∈((n−1)/10,n/10,],n=1,2,…,10,
while(ii)
P¯t=0 if pt=0 and P¯t=n if pt∈((n−1)/10,n/10,],n=1,2,…,10.
Then(3.2)Pte={p˜twithprobabilityq,γp˜t+(1−γ)P⋆,withprobability(1−q).Finally we associate each value of the price with a
state(3.3)Pt=siffP¯t=s−1,s=1,2,…,11so that we obtain the state set S={1,2,…,11}.Note that the corresponding process may be treated as
a discrete-time Markov chain, whose state space isS={1,2,…,11} (we use a state space S made of eleven states numbered from 1 to 11 for simplicity). The transition matrix is
given by A=[ψij],i,j∈S,
where ψij=ψ(Pt+1=j∣Pt=i),i,j∈S,
is the transition probability of going from state i to state j,i,j∈S.
We have ψij∈[0,1],i,j∈S and ∑j∈Sψij=1,i∈S.It is easy to see that the transition probabilitiesψij of models (2.3), (3.1),
(3.3) depend only on the
value of the current state i and the value of the following state j,
regardless of the time t when the transition occurs, consequently the
Markov chain considered is homogeneous. The homogeneity property implies that
the m-step state transition probability(3.4)ψij(m)=ψ{Pt+m=j∣Pt=i},i,j∈S,m=1,2,3,…,is also independent of t and it can be defined as(3.5)ψij(m)=∑k∈Sψik(m−1)ψkj(m−1),i,j∈S,m=2,3,…,so that the m-step transition matrix A(m) is given by(3.6)A(m)=Am.We are interested in the probability thatPt is in state j∈S at time t,
when at time 0P0 it is in state k with probability one.Let the probability vector be denoted byπ(m)=(π1(m),…,π11(m)),
where πj(m) is the probability that Pt is in state j after m steps, j∈S.
The probability distribution can be obtained through(3.7)π(m)=π(m−1)A=π(0)Am.Note that π(0)=(π1(0),π2(0),…,π11(0)) and that πi(0)=δik,i=1,2,…,11, where δik is the Kronecher delta.Recall that a subsetΛ⊂S is closed if ψij=0,
for all (i,j):i∈Λ,j∉Λ,
while Λ is irreducible if for all pairs (i,j)∈Λ there exists a positive integer k such that ψij(k)>0.
The subset Λ is reducible if it is not irreducible.
Finally we recall the notions of recurrent and transient states. A state i is called recurrent (transient)
if ∑m=1∞ψii(m)=∞(<∞).
(Regarding these definitions see, e.g., Feller [22].)We now assumeγ=0.5 and we prove a general result about the Markov
chain for some values of μ.Proposition 3.1.
For allμ∈(1,3.6), the transition matrix A is reducible.Proof.
ConsiderPt+1=j=11.
If no state Pt=i≠j∈S exists such that fμ(Pte)∈(0.9,1]⇒μPte(1−Pte)>0.9 then A is reducible. This last inequality has no
solution if μ∈(1,3.6), hence j=11 cannot be reached by another state. The
proposition is proved.We now fix the value of the parameterμ in (3.3), (3.1) in order to determine the
transition matrix A and to obtain the invariant distribution. In
fact, some numerical insight will be helpful to draw some general conclusions
on the qualitative properties of the process considered.In a first experiment, we assumeμ=2 and γ=0.5,
in this case we obtain a transition matrix A(11×11) whose nonzero entries are ψ1,1=ψ2,3=ψ3,5=ψ9,5=ψ10,3=ψ11,1=q, ψ1,5=ψ2,6=ψ3,6=ψ9,6=ψ10,6=ψ11,5=1−q and ψi,6=1,4≤i≤8.As previously provedA is reducible, moreover by simply rearranging
the order of the states in A,
it is possible to see that {6} is the unique closed set of the chain and that
it is an absorbing state.  This result holds for other
values of μ as stated in the following
proposition.Proposition 3.2.
Assumeγ=0.5.
For all μ∈(μk,μk+1], k=0,1,…,4,
where μk=10/(10−k) and μk+1=10/(10−(k+1)),
the state k+2 is an absorbing state.Proof.
Assumeμ∈(μk,μk+1], k=0,1,…,4.
Let(3.8)pt∈(k10,k+110]⇒p˜t=k+110,P¯t=k+1,hence Pt=k+2.
First consider that with
probabilityq also Pte=p˜t hence fμ(Pte)=fμ((k+1)/10).
Notice that fμ is increasing in μ then fμ(Pte)∈(fμk(Pte),fμk+1(Pte)].
Trivial computations show that(3.9)fμk(Pte)=k+1109−k10−k>k10,ask=0,1,…,4;fμk+1(Pte)=k+110.As a consequence fμ(Pte)∈(k/10,(k+1)/10] and consequently P¯t+1=k+1 implying that Pt+1=k+2.
Second, with probability(1−q),
we have(3.10)Pte=0.5p˜t+0.5P*=k+120+μ−12μ,hence fμ(Pte)=fμ(0.5p˜t+0.5P*);
again fμ is increasing in μ and, trivially, we have(3.11)fμk(Pte)=2k+12019−2k20−2k>k10,whilefμk+1(Pte)=k+110.Using the same arguments
employed before, we conclude that the state k+2 is mapped into the state k+2.
Henceψk+2,k+2=1.As proved in Proposition3.2, we can conclude that our
chain admits an absorbing state if μ∈(0,2],
confirming the result obtained with our simulations.Let us recall some mathematical results concerning the
stability of Markov chains (for further details
see, among others,
Feller [22] and Meyn and Tweedie [23]). A stationary distribution is a
probability distribution π¯ verifying the equation π¯A=π¯.
If there exists one, and only one, probability distribution π¯ such that ∥π(0)Am−π¯∥→0 as m→∞ for every initial probability distribution π(0),
we say that the Markov chain is asymptotically stable .In our case, forμ=2,
the Markov chain is asymptotically stable. In fact the limit distribution π¯=(π¯1,π¯2,…,π¯11),
where π¯j=limm→∞πj(m),j∈S,
is the distribution π¯j=δj,6,j∈S,
and this result holds for every initial probability distribution π(0).The asymptotic behaviour of the probability
distribution changes if we consider a different value ofμ,
for example μ=2.8 (notice that for this value of μ Proposition 3.2 does not hold). In this case
we obtain a transition matrix A whose nonzero entries are as
follows: ψ1,1=ψ2,4=ψ3,6=ψ4,7=ψ8,7=ψ9,6=ψ10,4=ψ11,1=q, ψ1,8=ψ2,8=ψ3,8=ψ4,8=ψ8,8=ψ9,7=ψ10,6=ψ11,6=1−q and ψi,8=1,5≤i≤7.Rearranging the order of the states inA, it is easy to deduce that A is reducible (as proved in Proposition 3.1):
the set Λ={7,8}⊂S is closed and irreducible, consequently the
states j=7 and j=8 are recurrent while all the other states are
transient. Moreover, considering the irreducible matrix(3.12)A¯=(01q1−q)we obtain the invariant
distribution given by (the nonzero elements of the
invariant distribution are obtained by looking at the left eigenvectors of
matrix A¯
)  (3.13)π¯=[0,0,0,0,0,0,q/(q+1),1/(q+1),0,0]. The following simulations support our analysis. We choose q=0.6 and we calculate the probability distribution
of the state variable after  m-steps for several initial conditions.First of all we consider the caseμ=2.
In Figure 1 we start from the initial condition P0=2 with probability one, that is from πi(0)=δi,2,i∈S.
After the first step two states can be reached with different probabilities.
After the third step we have that the state j=6 is reached with probability equal to one as
supported by the previous considerations. This situation does not change if the
number of steps taken is increased since j=6 is the unique asymptotic state. It should be
kept in mind that for a given initial distribution π(0) we define an asymptotic state as a
state j∈S such that π¯j≠0.Probability
distributions afterm-time steps obtained using function (3.3) with μ=2, γ=0.5, q=0.6 and initial condition P0=2 with probability one, that is πi(0)=δi,2,i=1,2,…,11.
(a)(b)(c)In Figure2 we calculate the probability distribution
for m=1 and m=150 when μ=2.8.
Our simulation proves that two equilibrium prices will be approached in the
long run with different probabilities.Probability distributions afterm-time steps obtained using (3.3) with μ=2.8, γ=0.5, q=0.6, and initial condition P0=2 with probability one, that is, πi(0)=δi,2,i=1,2,…,11.
(a)(b)Moreover, considering simulations for different values
ofμ it seems that as μ increases the process becomes more
complicated, this was already known in the case of the deterministic logistic
map (see, e.g., Devaney [24]). In fact several asymptotic states can be
reached as μ increases. More precisely, starting from the
same initial condition we have found different numbers of asymptotic states
corresponding to different values of μ.The diagram of Figure3 represents a sort of final-states
diagram: for μ chosen in the interval I=[1,4],
the probability distribution after  m-time steps, with m big enough (we have considered m=500) and an arbitrarily chosen initial state (we
have chosen P0=3 with probability one, that is πi(0)=δi,3,i∈S), has been
calculated and depicted versus the correspondent value of μ.
Moreover, after a large number of simulations, we have observed that the
final-states diagram does not change when different initial conditions are
considered so that Figure 3 seems to hold for every initial distribution π(0).
The empirical result described here is in agreement with the result proved in Proposition
3.2, furthermore we can conclude that the absorbing state existing for μ≤2 is also an asymptotic state.Figure 3
Asymptotic
states versusμ for γ=0.5, q=0.6, and an arbitrary initial condition.The situation is quite different for greater values ofμ.
For example, our calculation shows that there exists a value μ¯∈(3.33,3.34) such that the process converges to a unique
asymptotic state if μ∈[3.33,μ¯) while five asymptotic states are possible as
soon as the parameter value μ¯ is crossed. According to these considerations, μ¯ can be understood as a sort of bifurcation
value or critical value since the number of long-run states changes
going across μ=μ¯.
In Figure 4, we simulate the two cases μ=3.33 and μ=3.34 using a large number of time steps m,
in order to show how the asymptotic probability distribution obtained changes.Probability
distributions afterm-time steps obtained using (3.3) with γ=0.5, q=0.6, and initial condition P0=2 with probability one, that is, πi(0)=δi,2,i=1,2,…,11,
with different values of  μ:μ=3.33 (panel (a)) and μ=3.34 (panel (b)).
(a)(b)This study allows us to conclude that, in the case
with naive versus forward-looking expectations, there exists a unique
asymptotic distribution whose behaviour becomes more complicated asμ increases, according to the fact that
nonlinearity implies nontrivial dynamics.Obviously, the quantitative results are not
independent of the number of states considered, in any case it is possible to
verify that the qualitative results (i.e., the increase in complexity asμ increases) still hold. As a matter of fact,
note that a similar scenario occurs when we consider the model in a
deterministic contest with naive expectations. In fact, in this case, the price
sequence converges to a fixed point or to a 2-period cycle depending on the value of μ,
and the process as μ increases produces orbits tending towards
high-period cycles (see Devaney [24]).


## 3.2. The Continuous-Time Stochastic Process

In this section we move on to the continuous-time
Markov process, we calculate the probability distribution solving the
appropriate system of ordinary differential equations and we compare it with
the probability distribution obtained by statistical simulation of the
appropriate continuous-time limit of the stochastic
process defined by (2.3), (3.1),
(3.3). (About the
mathematical concepts related to continuous Markov processes see
Ethier and Kurtz [25].)In particular, we start looking at the transition
matrixB=B(Δt) over the time interval Δt>0.
Using (3.7) and taking time steps of length Δt, we obtain that the following relation holds π((m+1)Δt)=π(mΔt)B(Δt).AssumingmΔt=t,
we have π(t+Δt)=π(t)B and when Δt goes to zero and m goes to infinity in order to guarantee mΔt=t with t>0 fixed, it is easy to deduce the following
system of ordinary differential equations called Chapman-Kolmogorov's forward
equations:(3.14)dπdt(t)=π(t)Q,t>0,where the initial condition of
(3.14) is π(0)=π0, where π0 is a given probability
distribution and Q=limΔt→0((B(Δt)−I)/Δt) is called infinitesimal generator matrix,
whose entries are the transition rates.It is well known that the solution of (3.14) is given
by(3.15)π(t)=π0eQt,where eQt=∑n=0∞((Qt)n/n!).
In order to define the transition matrix over the time interval Δt, B(Δt),
in terms of the infinitesimal generator matrix Q,
we consider  that the probability vector π(t+Δt) can be expressed as follows:(3.16)π(t+Δt)=π(t)+dπdtΔt+O(Δt2)=π(t)[I+QΔt]+O(Δt2),Δt→0,where O(.) is the Landau symbol.Since the interval[t0,t] is divided into  m-time steps of length Δt,
we have(3.17)π(t)[I+QΔt]+O(Δt2)=π0[I+QΔt]t/Δt+1+O(Δt2),Δt→0,so that we have(3.18)π(t+Δt)=π0[I+QΔt]t/Δt+1+O(Δt2),Δt→0.Then, the transition matrix B is given by(3.19)B(Δt)=I+QΔt.From (3.15) we have that calculating the exponential of
the generator matrix Q times t;
and acting with this “exponential matrix” on π(0) we obtain the probability vector π(t) for an arbitrary value of t>0.
Moreover, we observe that the matrix Q=[rij]i,j∈S defined as Q=A−I,
where A is the one-step transition matrix defined in
the previous section and I is the identity matrix, is an infinitesimal
generator matrix. In fact it is easy to verify that the following properties
hold (see Inamura [26]):(1)
∑j=111rij=0,1≤i≤11;(2)
0≤−rii≤1,1≤i≤11;(3)
rij≥0,1≤i,j≤11,  with i≠j.We present some simulations to compare the numerical
solution obtained from computing (3.15) with the
statistical distribution obtained from simulations of the appropriate
continuous time limit of the stochastic process defined by (2.3),
(3.1), (3.3).In Figure5, we consider the case μ=2.8 and we observe that as Δt→0 and m→+∞ in such a way that the product Δt⋅m is equal to a constant value t>0,
the statistical distribution approaches the solution obtained solving
numerically (3.14) with the appropriate initial condition, that is computing
(3.15).Solution obtained using finite differences and
solution obtained using statistical simulation at timet=4 for μ=2.8, γ=0.5, q=0.6, and initial condition P0=3 with probability one, that is πi(0)=δi,3,i=1,2,…,11.
We compare the results obtained using different values of the time
discretization step Δt.
(a)(b)(c)Figure6 confirms that the same conclusion holds if we
consider the parameter value μ=3.34.Solution obtained using finite differences and
solution obtained using statistical simulation forμ=3.34, at time t=8 for μ=3.34, γ=0.5, q=0.6, and initial condition P0=3 with probability one, that is πi(0)=δi,3,i=1,2,…,11.
We compare the results obtained using different values of the time
discretization step Δt.
(a)(b)


## 4. The Role of the Memory Rate: Numerical Simulations

We now come back to the initial model with discrete
time, continuous states,ρ is not necessarily
zero and performs some numerical simulations for
several choices of the parameters. In this way it is possible to compare
different markets made up by agents applying naive and
infinite-memory expectations, each of them versus
forward-looking ones.Our simulations have been performed as follows:(i)
we want to depict a trajectory starting at
time zero from an initial conditionP0,
concentrated with probability one in one state, from time t=0 to time t=m.
We extract an m-dimensional random vector q¯′ whose i th component is denoted with qi′ made of m random numbers independent and uniformly
distributed in the interval [0,1].
At step i of the trajectory simulation we compare qi′ to the given value of q and we choose consequently the expectation-mechanism
formation and we apply the map (2.3). Repeating this procedure at each time step,
we depict the trajectory of the stochastic dynamical system in the plane (t,Pt).
Obviously, different extractions of the random vector q¯′ correspond, in general, to different
trajectories of the stochastic dynamical system;(ii)
we repeat the previous procedure several
times, that is, we extract a large number of vectorsq¯′ of random numbers so that we obtain a
sufficiently large number of trajectories of the stochastic dynamical system.
In this way for any given time t we have constructed a sample of the possible
outcomes for Pt.
It is then straightforward to draw an approximate probability distribution of
the random variable Pt.This procedure is in agreement with the consideration
of a market made of a large number of agents and with the heterogeneity of
beliefs, that is, with the assumption that, on average, agents distribute
between the two predictors in the fractionsq and (1−q), respectively.We first assumeρ=0 as done in the previous section. Our goal is
to find numerically the distribution of the random variable Pt at time t and to consider the behaviour of this
distribution as t→∞.We represent the evolution in time ofPt for γ=0.5 and q=0.6.
In Figure 7, in panel (a),
three different trajectories are depicted for μ=2.9;
in panel (b) the probability distributions are presented
for several values of the time variable t.
Note that when t is big enough, the equilibrium price P⋆≃0.65517 is approached by the stochastic cobweb model.(a) Three trajectories produced by the stochastic
process defined by (2.3) for P0=0.6 with probability one, μ=2.9, γ=0.5, q=0.6, and ρ=0; (b) Probability distributions at different time
values for the model considered in panel (a).
The probability distributions are approximated starting from a sample made of 1000 individuals.
(a)(b)Completely different behaviours are observed if we
change the value of the parameterμ.
For example, let us choose μ=3.4 while the other parameters are the same as
those used in the previous simulations. In Figure 8, we show for these new
parameter values the results of the same simulation shown in Figure 7.(a) Three trajectories produced by the stochastic
process defined by (2.3) for P0=0.5 with probability one, μ=3.4, γ=0.5, q=0.6, and ρ=0; (b)   (c) probability distributions at different time
values for the model considered in panel (a).
The probability distributions are approximated starting from a sample of 1000 individuals.
(a)(b)(c)Whenμ=3.4 the trajectories do not approach the
equilibrium price but they continue to move into a bounded interval even in the
long run.This behaviour is closely related to that exhibited by
the deterministic cobweb model with naive expectations and with the dynamics of
the logistic map. In fact asμ increases the deterministic cobweb model
produces more and more complex dynamics. Similarly, in the stochastic contest,
prices no longer converge to the equilibrium value, but fluctuate infinitely
many times.  Consider now ρ≠0.
We want to understand how the memory rate affects the asymptotic probability
distribution of our stochastic model. Assume μ=3.4 as in the previous simulation and let ρ increase from zero toward one. An interesting
observation is that as the memory rate ρ increases, the unique equilibrium price is
reached after a number of time steps which
decrease, thus confirming the stabilizing effects
of the presence of resistant memory. The same effect has been observed in the
deterministic contest. This consideration is confirmed by the simulation shown
in Figure 9 representing the trajectories depicted for several increasing
values of ρ.Trajectories produced by the stochastic process
defined by (2.3) for P0=0.5 with probability one, μ=3.4, γ=0.5, q=0.6, and different values of ρ: (a)  ρ=0.01, (b)  ρ=0.1, (c)  ρ=0.2.
(a)(b)(c)Similar results have been observed considering several
other parameter values. All the experiments confirm the well-known result
obtained for the deterministic cobweb model with infinite memory expectation,
that is the fact that the presence of resistant memory contributes to
stabilizing the price dynamics.


## 5. Conclusions

We studied a
nonlinear stochastic cobweb model with a parabolic demand function and two
price predictors, called backward-looking (based on a weighted mean of past
prices) and forward-looking (based on a convex combination of actual and
equilibrium price), respectively.  The representative agent chooses
between expectations, this fact may be interpreted as a population of economic
agents such that, on average, a fraction chooses a kind of expectation while a different kind of expectation is chosen by the remaining fraction of agents.  Since
a random
choice between the two price expectations is allowed (a possible motivation is
the assumption of heterogeneity in beliefs among agents), we have considered a
new element that is the stochastic term in the well-known cobweb model. In
fact, even if research into the cobweb model has a
long history, the existing literature is limited to the deterministic contest.
As far as we know, the dynamics of a cobweb model with expectations decided on
the basis of a stochastic mechanism have not been
studied in the literature, so our paper may be seen as a first step towards
this direction, and our results may trigger further studies in this field.In order to describe the features of the model, we
have concentrated on the case in which the backward predictor is simply a
static expectation, so that the stochastic dynamical system is a Markov
process, considered both in discrete and continuous time.By using an appropriate transformation, it has been
possible to study a new discrete time model with discrete states. In this way
we have been able to apply some well-known results regarding Markov chains with
finite states and to prove some general analytical results about the
reducibility of the chain and the existence of an absorbing state for some
values of the parameters. We have also presented numerical simulations
confirming our analytical results. From an empirical point of view, we have
observed that the absorbing state is also an asymptotic state if parameterμ is small enough. On the other hand, for
increasing values of μ,
the chain is still reducible although its closed set may be composed of more
than one state providing that cyclical behaviour is admitted in the stochastic
cobweb model, as in the deterministic one. Other evidence is related to the
sensitivity of the invariant distribution to little changes of the parameter μ.
The study done for the discrete time stochastic model with finite states
enables us to conclude that as μ increases the process becomes more complicated
confirming the well-known results of the deterministic logistic map.In a following step we moved to the continuous-time
Markov process in order to use analytical tools on differential equations that
made it possible to obtain the exact probability distribution at any timet.
By solving the appropriate system, the probability distribution has been
obtained and some numerical simulations have been presented by calculating the
time limit of the discrete-time Markov chain. We have shown that as the time
discretization step Δt goes to zero, the result of the statistical
simulation approaches the probability distribution obtained analytically.Finally, we come back to the case with backward
expectations with memory. Since the model become quite difficult to be treated
analytically, we presented some numerical simulations that enable us to
consider the role of the memory rate in the stochastic cobweb model. We have
found that the presence of resistant memory affects the asymptotic probability
distribution: it contributes to reducing fluctuations and to stabilizing the
price dynamics thus confirming the standard result in economic literature.



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*Source: 219653-2008-07-07.xml*