# Updating Wealth in an Asset Pricing Model with Heterogeneous Agents

Authors: Serena Brianzoni; Cristiana Mammana; Elisabetta Michetti
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

We consider an asset-pricing model with wealth dynamics in a market populated by heterogeneous agents. By assuming that all agents
belonging to the same group agree to share their wealth whenever an agent joins the group (or leaves it), we develop an adaptive model which characterizes the evolution of wealth distribution when agents switch between different trading strategies. Two groups with heterogeneous beliefs are considered: fundamentalists and chartists. The model results in a nonlinear three-dimensional dynamical system, which we have studied in order to investigate complicated dynamics and to explain wealth distribution among agents in the long run.


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## Body

## 1. Introduction

The traditional approach in economics and finance is based on a representative rational agent who knows the market equilibrium equations and is able to solve the model. Simon [1] documents that knowledge of the economic environment is an extreme assumption. Moreover, it would be difficult to compute the rational expectations equilibrium in nonlinear market equilibrium models, even if the agent knew all the equilibrium equations.As a consequence, many recent studies model agents as boundedly rational (see, e.g., Conlisk [2] for a survey on bounded rationality) and assume that they are heterogeneous (see [3, 4] for an extensive survey of heterogeneous agent models).Many authors have introduced heterogeneous agent models in order to investigate some important facts in financial markets, see, for example, Brock and Hommes [5], Hommes [6]), Chiarella and He [7–9], Chiarella et al. [10], Anufriev et al. [11], Anufriev [12], and Anufriev and Dindo [13]. Examples of the impetus behind these kinds of models are (i) the question of whether the behavior of agents can be described as if they are rational (claimed by Friedman [14]) and (ii) some stylized empirical findings such as the excess volatility exhibited by markets (difficult to explain by means of a representative agent model).The key aspect of these models is that they consider expectations feedback and the adaptiveness of agents (see, e.g., Brock and Hommes [5, 15]). To be more specific, at each time agents choose from a set of different types of trading strategies by looking at their past performance (measured by the profits they have made). Brock and Hommes [5] introduce adaptive beliefs to the present discounted value asset-pricing model and observe endogenous price fluctuations similar to those observed in financial markets. Chiarella and He [7] extend the model of Brock and Hommes [5] by assuming that agents have different risk attitudes and different expectations for both the first and second moments of price distribution.A common feature of this kind of heterogeneous agent model in asset-pricing theory is the independence of optimal demand for the risky asset from agents' wealth, as a result of the assumption of the constant absolute risk aversion (CARA) utility function (such as the exponential one). Nevertheless, some authors document that a framework in which investors' optimal decisions depend on their wealth is more realistic, see, for example, Levy et al. [16–18], and Campbell and Viceira [19]. This framework is in line with the assumption of a constant relative risk aversion (CRRA) utility function. The only utility function with CRRA property is the power utility function, of which the logarithmic utility function is a special case. The use of CRRA utility functions in financial markets is important in capturing the interdependence of price and wealth dynamics.For this reason, in recent years, several models have focused on the study of the market equilibrium price and wealth distribution when the economy is populated by boundedly rational heterogeneous agents with CRRA preferences. Chiarella and He [8] study an asset-pricing model with heterogeneous agents having logarithmic utility functions. In the case with two groups of agents, that is, fundamentalists and chartists, the authors prove the existence of multiple equilibria and the convergence of the return and wealth proportions to the steady state with the higher return under the same variance. The model shows volatility clustering as well as other anomalies observed in financial market data. Nevertheless, the authors focus on the case with fixed population fractions. In order to obtain a more appealing framework, Chiarella and He [9] allow agents to switch between different trading strategies and show the profitability of momentum trading strategies over short-time intervals and of contrarian trading strategies over long-time intervals. Chiarella and He [20] develop a model able to characterise asset price dynamics, the evolution of population proportions and wealth dynamics. In order to describe the evolution of wealth, the authors investigate the framework of heterogeneous agents using a selection of numerical simulations.Chiarella et al. [10] consider a market-maker model of asset price and wealth dynamics and introduce a growing dividend process and a trend in the fundamental price of the risky asset. The authors consider explicitly the interdependence between price dynamics and the evolution of wealth distribution among agents and show that fundamentalists do not accumulate more wealth than chartists.Other examples of the analytical exploration of the CRRA framework with heterogeneous agents are Anufriev et al. [11] and Anufriev [12]. More recently, Anufriev and Dindo [13] provides an analytical derivation of the results of the Levy et al. model. This work incorporates the feedback of past prices with investment strategies.Our paper follows on this wide stream of research by introducing a switching mechanism and studying its effects on wealth distribution. We observe that many contributions to the development and analysis of financial models with heterogeneous agents and CRRA utility consider fixed proportions of agents. Moreover, models which allow agents to switch between different trading strategies (such as Chiarella and He [9]) make the following assumption: when agents switch from an old strategy to a new strategy, they agree to accept the average wealth level of agents using the new strategy. More precisely, the switching agent leaves his wealth to the group of origin.Motivated by such considerations, we develop a model based upon a new switching mechanism. We assume that all agents belonging to the same group agree to share their wealth whenever an agent joins the group (or leaves it). When agents switch between different prediction strategies, the wealth of the new group takes into account the wealth coming from the group of origin. In other words, agents who change group bring their wealth to the new group. As a consequence, the wealth of each group is updated from periodt to (t+1) not only as a consequence of portfolio growth of agents adopting the relative strategy, but also due to the flow of agents coming from the other group.In line with theevolutionary finance literature (see, e.g., [21–25]) we analyze the survival of agents in a financial market. In contrast with the evolutionary finance approach, we incorporate the feedback on past prices with the investment strategies, as in the recent contribution of Anufriev and Dindo [13].As in many interacting agent models, we focus on the case where there are two groups of agents: fundamentalists and chartists. Among others, Chiarella et al. [10], Chiarella and He [8], Brock and Hommes [5] distinguish between fundamentalists and chartists in order to explain excess of volatility and to analyze the agent survival. Fundamentalists believe that the price of an asset is determined by its fundamental value. The fundamental price is completely determined by economic fundamentals. Fundamental traders sell (buy) assets when their prices are above (below) the market fundamental value. In contrast, chartists, or technical analysts, do not take the fundamental value into account, rather they look for trends in past prices and prediction is based upon simple trading rules. For a long-time, chartists have been viewed as irrational and, according to the Friedman hypothesis, they would be driven out of the market by rational traders. We will see that both types of agents can survive in the market in the long-run.The new switching mechanism we have introduced leads the final system to a particular form in which the wealth of agents is defined by a continuous piecewise function and the phase space is divided into two regions. Nevertheless, our final dynamical system is three dimensional and all the equilibria are present. We will prove that it admits two kinds of steady state, fundamental steady states (with the price being at the fundamental value) and nonfundamental steady states. In performing the stability analysis, we are limited by the atypical form of our system, but we prove the existence of a trapping set which allows us to study the stability of the fundamental steady state. Several numerical simulations supplement the analysis and show complexity, which is mainly due to wealth dynamics.In Section2, we present our general framework describing an asset-pricing model where agents use different beliefs about future price. We obtain a three-dimensional dynamical system where wealth distribution can be explicitly considered. In Section 3, we focus on the case in which the market is populated by fundamentalists and chartists. The resulting map has a particular structure, being piecewise smooth, although we analytically find the steady states and we prove the existence of a trapping set in which all the wealth is owned by fundamentalists. In order to consider the possibility of complex dynamics to be exhibited. In Section 4 we perform a series of numerical simulations showing the great variety of qualitative behaviors which our model can present and their relation to a number of parameter values. As in the related literaure, our heterogeneous agent model leads the market to periodic or even chaotic fluctuations in prices. In addition, the new switching mechanism involves complexity in the long-run wealth distribution. Section 5 concludes.


## 2. The Model

Consider an economy composed of one risky asset paying a random dividendyt at time t and one risk free asset with constant risk free rate r=R-1>0. We denote by pt the price (exdividend) per share of the risky asset at time t.  In order to describe the wealth dynamics, we assume that all agents belonging to the same group agree to share their wealth whenever an agent joins the group (or leaves it). (This assumption was introduced by Chiarella and He [9], the main motivation is that the model would not otherwise be tractable.) According to such an assumption, the wealth of agent type h at time t, denoted by w¯h,t, is given by the total wealth of group h in the fraction of agents belonging to this group. Generally speaking, we are assuming homogeneity between agents within the same group, while heterogeneity is introduced between agents belonging to different groups. As a consequence, the wealth dynamics of investor h is described by(2.1)Wh,t+1=(1-zh,t)w¯h,tR+zh,tw¯h,t(1+ρt+1)=w¯h,t[R+zh,t(ρt+1-r)],
where zh,t is the fraction of wealth that agent-type h invests in the risky asset and ρt=(pt+yt-pt-1)/pt-1 is the return on the risky asset at period t. Observe that Wh,t+1 represents the wealth earned by agent h at time t+1 later on the investment made at time t.The individual demand functionzh,t is derived from the maximization problem of the expected utility of Wh,t+1, that is, zh,t=maxzh,tEh,t[uh(Wh,t+1)], where Eh,t is the belief of investor-type h about the conditional expectation, based on the available information set of past prices and dividends. Since each agent is assumed to have a CRRA utility function, investors' optimal decisions depend on their wealth. In line with Chiarella and He [8], the optimal (approximated) solution is given by(2.2)zh,t=Eh,t[ρt+1-r]λhσh2,
where λh is the relative risk aversion coefficient and σh2=Varh,t[ρt+1-r] is the belief of investor h about the conditional variance of excess returns.In our model, different types of agents have different beliefs about future variables and prediction selection is based upon a performance measureϕh,t. Let nh,t be the fraction of agents using strategy h at time t. Hence, as in Brock and Hommes [5], the adaptation of beliefs, that is, the dynamics of the fractions nh,t of different trader types, is given by(2.3)nh,t+1=exp⁡[β(ϕh,t-Ch)]Zt+1,Zt+1=∑hexp⁡[β(ϕh,t-Ch)],
where the parameter β is the intensity of choice measuring how fast agents choose between different predictors and Ch≥0 are the costs for strategy h. When β increases, more and more agents use the predictor with the highest fitness. In the extreme case β=+∞, all agents choose the strategy with the highest fitness, while in the other extreme case β=0, no switching at all takes place and both fractions are equal to 1/2.Let us define the performance measureϕh,t. To this end we observe that at time t+1 agent h measures the performance he has achieved and then chooses whether to stay in group h or to switch to another one. With this consideration, we measure past performance as the personal wealth coming from the investment in the risky asset with respect to w¯h,t: In this framework, at any time the wealth w¯h,t represents the initial endowment of agent h(2.4)ϕh,t=zh,t(ρt+1-r).
Agents revise their beliefs in a boundedly rational way in the sense that, at any time, most agents choose the predictor which generates the best past performance. In other words, the fraction nh,t+1 of traders using strategy h at time t+1 will be updated according to ϕh,t.In this work, we focus on the case of a market populated by two groups of agents, that is,h=1,2. In order to ensure that the model remains tractable, we assume that agents can move from group i to group j at any time, with i,j=1,2 and i≠j, while both movements are not simultaneously possible. The simplified assumption that switching is unilateral is in line with our framework, in which agents can only switch to the group which generates the best past performance.We defineΔnh,t+1=nh,t+1-nh,t as the difference in the fraction of agents of type h from time t to time t+1. Note that, in a market with two groups of agents, it follows that Δn1,t+1=-Δn2,t+1. As a consequence, we can have two different cases:  (1)
Δn1,t+1≥0, if Δn1,t+1 fraction of agents moves from group 2 to group 1 at time t+1,(2)
Δn1,t+1<0, if Δn1,t+1 fraction of agents moves from group 1 to group 2 at time t+1.Following Brock and Hommes [5], we define the difference in fractions at time t, that is, mt=n1,t-n2,t, so that n1,t=(1+mt)/2 and n2,t=(1-mt)/2. As a consequence(2.5)mt+1=tanh[β2(ϕ1,t-ϕ2,t-C1+C2)].
Conditions Δn1,t+1≥0 and Δn1,t+1<0 can be replaced by mt+1≥mt and mt+1<mt, respectively.In order to describe the wealth dynamics of each group, we defineW̃h,t as the share of the wealth produced by group h to the total wealth:(2.6)W̃h,t=nh,tw¯h,t,h=1,2,
which represents the wealth of grouph.Hence, we have to distinguish two different cases to define the wealth of group 1 at timet+1:  (1)
ifΔn1,t+1 fraction of agents moves from group 2 to group 1, the wealth W̃1,t+1 is given by the wealth coming from group 2 and the wealth generated by traders of type 1, otherwise,(2)
ifΔn1,t+1 fraction of agents moves from group 1 to group 2, the wealth W̃1,t+1 is simply given by the wealth of agents which do not leave the group.Summarizing, the wealth of group 1 is defined as(2.7)W̃1,t+1={Δn1,t+1W2,t+1+n1,tW1,t+1=n1,t(W1,t+1-W2,t+1)+n1,t+1W2,t+1ifmt+1≥mtn1,t+1W1,t+1ifmt+1<mt.In a similar way we can derive the wealth of group 2:(2.8)W̃2,t+1={n2,t+1W2,t+1ifmt+1≥mt,n2,t(W2,t+1-W1,t+1)+n2,t+1W1,t+1ifmt+1<mt.
Note that, in this way, we ensure that the wealth of both groups is updated at all times. More precisely, when Δn1,t+1≥0 (i.e., strategy 1 performs better) then some agents switch to the first group. In such a case, at time t+1 new agents joining the group bring the wealth made in group 2 to group 1. Differently, the wealth of the second class is simply given by the wealth generated by type-2 agents who do not leave the group. A similar reasoning applies when Δn1,t+1<0.In order to summarize how the wealth distribution changes as a consequence of the switching mechanism, let us focus on thetiming of the model.  (i)
At timet: the market is made up of n1,t (n2,t) fraction of traders belonging to the first (second) group. Agents have different expectations about the returns on the risky asset and, hence, different demand functions.(ii)
At timet+1: type-h agent generates his personal wealth Wh,t+1. At the same time, the new fractions of agents n1,t+1 and n2,t+1 are determined according to the performance measures generated by the investment in the risky asset. Hence, from time t to t+1 switching occurs and some traders move from one group to the other. Agents leaving group i bring the wealth they have generated (Wi,t+1) to class j and the wealth W̃j,t+1 of group j is determined. Finally, as all agents agree to share their wealth whenever an agent joins the group, w¯h,t+1=W̃h,t+1/nh,t+1 is the wealth of agent h at time t+1. Then, the story repeates.Finally, we definewh,t as the wealth of group h in the total wealth, that is, wh,t=W̃h,t/∑hW̃h,t (where W̃h,t=nh,tw¯h,t and h=1,2), then wh,t represents the relative wealth of group h. In the following, we will consider the dynamics of the state variable wt:=w1,t-w2,t, that is, the difference in the relative wealths. To this end, we recall (2.7) and (2.8) and analyze both the cases mt+1≥mt and mt+1<mt.Case 1   (mt+1≥mt).
From (2.7) and (2.8) and after some algebra we obtain
(2.9)wt+1=w1,t+1-w2,t+1=W̃1,t+1-W̃2,t+1W̃1,t+1+W̃2,t+1=-2n2,t+1W2,t+1+n2,tW2,t+1+n1,tW1,t+1n2,tW2,t+1+n1,tW1,t+1,
where we have made use of relation n1,t+n2,t=1.
Considering (2.1), it follows(2.10)wt+1=-2n2,t+1w¯2,t[R+z2,t(ρt+1-r)]+n2,tw¯2,t[R+z2,t(ρt+1-r)]n2,tw¯2,t[R+z2,t(ρt+1-r)]+n1,tw¯1,t[R+z1,t(ρt+1-r)]+n1,tw¯1,t[R+z1,t(ρt+1-r)]n2,tw¯2,t[R+z2,t(ρt+1-r)]+n1,tw¯1,t[R+z1,t(ρt+1-r)].
Remembering that w¯h,t=W̃h,t/nh,t and wh,t=W̃h,t/(W̃1,t+W̃2,t), hence w¯h,t=(W̃1,t+W̃2,t)(wh,t/nh,t), we divide both numerator and denominator for W̃1t+W̃2t to obtain
(2.11)wt+1=-2n2,t+1(w2,t/n2,t)[R+z2,t(ρt+1-r)]+n2,t(w2,t/n2,t)[R+z2,t(ρt+1-r)]n2,t(w2,t/n2,t)[R+z2,t(ρt+1-r)]+n1,t(w1,t/n1,t)[R+z1,t(ρt+1-r)]+n1,t(w1,t/n1,t)[R+z1,t(ρt+1-r)]n2,t(w2,t/n2,t)[R+z2,t(ρt+1-r)]+n1,t(w1,t/n1,t)[R+z1,t(ρt+1-r)].
Finally, recalling that w1,t=(1+wt)/2, w2,t=(1-wt)/2 and n1,t=(1+mt)/2, n2,t=(1-mt)/2, we have
(2.12)wt+1=-((1-mt+1)/(1-mt))(1-wt)[R+z2,t(ρt+1-r)]+((1-wt)/2)[R+z2,t(ρt+1-r)]((1-wt)/2)[R+z2,t(ρt+1-r)]+((1+wt)/2)[R+z1,t(ρt+1-r)]+(1+wt)/2[R+z1,t(ρt+1-r)]((1-wt)/2)[R+z2,t(ρt+1-r)]+((1+wt)/2)[R+z1,t(ρt+1-r)]=-2((1-mt+1)/(1-mt))(1-wt)[R+z2,t(ρt+1-r)](1-wt)[R+z2,t(ρt+1-r)]+(1+wt)[R+z1,t(ρt+1-r)]+1.Case 2   (mt+1<mt).
Following the same steps as in the previous case, we arrive at(2.13)wt+1=w1,t+1-w2,t+1=W̃1,t+1-W̃2,t+1W̃1,t+1+W̃2,t+1=2n1,t+1W1,t+1-(n2,tW2,t+1+n1,tW1,t+1)n2,tW2,t+1+n1,tW1,t+1,
and by using (2.1),
(2.14)wt+1=2n1,t+1w¯1,t[R+z1,t(ρt+1-r)]n2,tw¯2,t[R+z2,t(ρt+1-r)]+n1,tw¯1,t[R+z1,t(ρt+1-r)]-(n2,tw¯2,t[R+z2,t(ρt+1-r)]+n1,tw¯1,t[R+z1,t(ρt+1-r)])n2,tw¯2,t[R+z2,t(ρt+1-r)]+n1,tw¯1,t[R+z1,t(ρt+1-r)].
Dividing both numerator and denominator for W̃1t+W̃2t and recalling that w¯h,t=(W̃1,t+W̃2,t)(wh,t/nh,t), we obtain
(2.15)wt+1=2n1,t+1(w1,t/n1,t)[R+z1,t(ρt+1-r)]n2,t(w2,t/n2,t)[R+z2,t(ρt+1-r)]+n1,t(w1,t/n1,t)[R+z1,t(ρt+1-r)]-(n2,t(w2,t/n2,t)[R+z2,t(ρt+1-r)]+n1,t(w1,t/n1,t)[R+z1,t(ρt+1-r)])n2,t(w2,t/n2,t)[R+z2,t(ρt+1-r)]+n1,t(w1,t/n1,t)[R+z1,t(ρt+1-r)].
Finally, we have
(2.16)wt+1=((1+mt+1)/(1+mt))(1+wt)[R+z1,t(ρt+1-r)](1-wt)/2[R+z2,t(ρt+1-r)]+((1+wt)/2)[R+z1,t(ρt+1-r)]-(((1-wt)/2)[R+z2,t(ρt+1-r)]+((1+wt)/2)[R+z1,t(ρt+1-r)])(1-wt)/2[R+z2,t(ρt+1-r)]+((1+wt)/2)[R+z1,t(ρt+1-r)]=2((1+mt+1)/(1+mt))(1+wt)[R+z1,t(ρt+1-r)](1-wt)[R+z2,t(ρt+1-r)]+(1+wt)[R+z1,t(ρt+1-r)]-1.As a consequence, the dynamics of the state variablewt can be described by (2.17)wt+1={F1G+1ifmt+1≥mt,F2G-1ifmt+1<mt,
where(2.18)F1=-21-mt+11-mt(1-wt)[R+z2,t(ρt+1-r)],F2=21+mt+11+mt(1+wt)[R+z1,t(ρt+1-r)],G=(1-wt)[R+z2,t(ρt+1-r)]+(1+wt)[R+z1,t(ρt+1-r)].
Notice that the function defining wt+1 is continuous.


### 2.1. Price-Setting Rule

In this work, we assume that price adjustments are operated by a market-maker who knows the fundamental price. The price-setting rule of the market-maker is given by (see [10])
(2.19)pt+1-pt=Et,f(pt+1⋆-pt⋆)+ptHt(NtD-NtS).
Under the assumption of an i.i.d. dividend process the fundamental value is constant and given by E(yt+1)/r=y¯/r so that we obtain (pt+1-pt)/pt=Ht(NtD-NtS), where NtD is the total number of shares demanded at time t and NtS denotes the supply of shares at time t.LetNh,tD be the number of shares of the asset that investor h purchases at price pt, that is, Nh,tD=zh,tw¯h,t/pt so that the total demand is given by (2.20)NtD=n1,tz1,tw¯1,t+n2,tz2,tw¯2,tpt.
Moreover we focus on the case with zero supply: NtS=0.Notice thatHt(NtD-NtS) is a strictly increasing function such that Ht(0)=0. Following Chiarella et al. [10], we consider that the agents' total demand can be rewritten as(2.21)NtD=W̃1,t+W̃2,tptn1,tz1,tw¯1,t+n2,tz2,tw¯2,tW̃1,t+W̃2,t
and that the market-maker rule is not affected by the level of (W̃1,t+W̃2,t)/pt. Consequently, we obtain Ht(NtD)=Ht((n1,tz1,tw¯1,t+n2,tz2,tw¯2,t)/(W̃1,t+W̃2,t)).After introducing the formHt(·)=α(·) with α>0, we can rewrite (2.19) as (2.22)pt+1-ptpt=αn1,tz1,t(W̃1,t/n1,t)+n2,tz2,t(W̃2,t/n2,t)W̃1,t+W̃2,t,
hence: (2.23)pt+1-ptpt=αz1,tW̃1,t+z2,tW̃2,tW̃1,t+W̃2,t.Recalling thatwh,t=W̃h,t/(W̃1,t+W̃2,t) and wt=w1,t-w2,t (consequently w1,t=(1+wt)/2 and w2,t=(1-wt)/2) the price-setting rule of the market-maker becomes:(2.24)pt+1-ptpt=α(z1,t1+wt2+z2,t1-wt2).Observe that prices today influence prices tomorrow through agent demand.The final dynamical system is obtained by using (2.5), (2.17), and (2.24) as stated in the following proposition.Proposition 2.1.
Under the assumption of an i.i.d. dividend process{yt} such that Et(yt+k)=y¯  for all  k=1,2,…, the dynamics of the deterministic skeleton of the model is described by the following three-dimensional system:
(2.25)pt+1=[α2(z1,t+z2,t+(z1,t-z2,t)wt)+1]pt,(2.26)mt+1=tanh{β2[(z1,t-z2,t)[α/2(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r]-C]},(2.27)wt+1={F1G+1ifmt+1≥mt,F2G-1ifmt+1<mt,
where
(2.28)F1=-4(1-wt){R+z2,t[(α/2)(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r]}(1-mt)[exp⁡{β[(z1,t-z2,t)[(α/2)(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r]-C]}+1],F2=4(1+wt){R+z1,t[(α/2)(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r]}(1+mt)[exp⁡{-β[(z1,t-z2,t)[(α/2)(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r]-C]}+1],G=2R+[z1,t+z2,t+(z1,t-z2,t)wt]⋅[(α/2)(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r].Proof.
From (2.24), we immediately obtain (2.25) and
(2.29)ρt+1-r=α2[z1,t(1+wt)+z2,t(1-wt)]+y¯pt-r,
where we have assumed that dividends evolve in a deterministic way according to their expected value y¯.
Putting (2.29) in equations of F1 and F2 and G, we rewrite such functions as in (2.30)F1=-21-mt+11-mt(1-wt)[R+z2,t(ρt+1-r)]=-21-mt+11-mt(1-wt){R+z2,t[α2(z1,t+z2,t+(z1,t-z2,t)wt)+y¯pt-r]},F2=21+mt+11+mt(1+wt)[R+z1,t(ρt+1-r)]=21+mt+11+mt(1+wt){R+z1,t[α2(z1,t+z2,t+(z1,t-z2,t)wt)+y¯pt-r]},G=(1-wt)[R+z2,t(ρt+1-r)]+(1+wt)[R+z1,t(ρt+1-r)]=2R+(ρt+1-r)[z1,t(1+wt)+z2,t(1-wt)]=2R+{α2[z1,t(1+wt)+z2,t(1-wt)]+y¯pt-r}[z1,t(1+wt)+z2,t(1-wt)]=2R+[z1,t+z2,t+(z1,t-z2,t)wt]⋅[α2(z1,t+z2,t+(z1,t-z2,t)wt)+y¯pt-r].
Consider now the dynamics of the difference in fractions of agents, that is, (2.5). Equation (2.26) is trivially derived putting C=C1-C2 and recalling (2.4) and (2.29). Moreover:(2.31)1-mt+1=2exp⁡{β[(z1,t-z2,t)[(α/2)(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r]-C]}+1,1+mt+1=2exp⁡{-β[(z1,t-z2,t)[(α/2)(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r]-C]}+1,
where we have made use of relations 1-tanhx=2e-x/(ex+e-x)=2/(e2x+1) and 1+tanhx=2ex/(ex+e-x)=2/(e-2x+1). Finally, introducing (2.31) into the expressions of F1, F2 we arrive at (2.27).Notice that in the previous proposition we have introduced the difference between costs, that is,C=C1-C2.In order to study the system defined by Proposition2.1, we have to specify the individual demand functions (2.32)zh,t=Eh,t[ρt+1-r]λσ2=1λσ2{1pt[Eh,t(pt+1)+rp⋆-pt]-r},∀h=1,2,
where we have assumed that beliefs about variance and risk aversion coefficients are constant and equal for all traders, that is, Varh,t[ρt+1-r]=σ2 and λh=λ, for all h=1,2. In making this assumption we follow Brock and Hommes [5]. Notice that p⋆ is the fundamental solution, that is, the long-run market clearing price path when homogeneous beliefs about expected excess return are considered. Under the assumption of an i.i.d. dividend process {yt} with Et(yt+1)=y¯, the fundamental solution is constant and given by pt⋆=p⋆=y¯/r. Brock and Hommes [5] derive endogenously the fundamental solution satisfying the no-bubbles condition, in the particular case of zero net supply of shares.As in many interacting agent models (see, e.g., [5, 8, 10]), in order to explain why prices deviate from their fundamental values for a long-time and to analyze agent survival, in the following section, we assume that agents of type 1 are fundamentalists while agents of type 2 are chartists.


## 2.1. Price-Setting Rule

In this work, we assume that price adjustments are operated by a market-maker who knows the fundamental price. The price-setting rule of the market-maker is given by (see [10])
(2.19)pt+1-pt=Et,f(pt+1⋆-pt⋆)+ptHt(NtD-NtS).
Under the assumption of an i.i.d. dividend process the fundamental value is constant and given by E(yt+1)/r=y¯/r so that we obtain (pt+1-pt)/pt=Ht(NtD-NtS), where NtD is the total number of shares demanded at time t and NtS denotes the supply of shares at time t.LetNh,tD be the number of shares of the asset that investor h purchases at price pt, that is, Nh,tD=zh,tw¯h,t/pt so that the total demand is given by (2.20)NtD=n1,tz1,tw¯1,t+n2,tz2,tw¯2,tpt.
Moreover we focus on the case with zero supply: NtS=0.Notice thatHt(NtD-NtS) is a strictly increasing function such that Ht(0)=0. Following Chiarella et al. [10], we consider that the agents' total demand can be rewritten as(2.21)NtD=W̃1,t+W̃2,tptn1,tz1,tw¯1,t+n2,tz2,tw¯2,tW̃1,t+W̃2,t
and that the market-maker rule is not affected by the level of (W̃1,t+W̃2,t)/pt. Consequently, we obtain Ht(NtD)=Ht((n1,tz1,tw¯1,t+n2,tz2,tw¯2,t)/(W̃1,t+W̃2,t)).After introducing the formHt(·)=α(·) with α>0, we can rewrite (2.19) as (2.22)pt+1-ptpt=αn1,tz1,t(W̃1,t/n1,t)+n2,tz2,t(W̃2,t/n2,t)W̃1,t+W̃2,t,
hence: (2.23)pt+1-ptpt=αz1,tW̃1,t+z2,tW̃2,tW̃1,t+W̃2,t.Recalling thatwh,t=W̃h,t/(W̃1,t+W̃2,t) and wt=w1,t-w2,t (consequently w1,t=(1+wt)/2 and w2,t=(1-wt)/2) the price-setting rule of the market-maker becomes:(2.24)pt+1-ptpt=α(z1,t1+wt2+z2,t1-wt2).Observe that prices today influence prices tomorrow through agent demand.The final dynamical system is obtained by using (2.5), (2.17), and (2.24) as stated in the following proposition.Proposition 2.1.
Under the assumption of an i.i.d. dividend process{yt} such that Et(yt+k)=y¯  for all  k=1,2,…, the dynamics of the deterministic skeleton of the model is described by the following three-dimensional system:
(2.25)pt+1=[α2(z1,t+z2,t+(z1,t-z2,t)wt)+1]pt,(2.26)mt+1=tanh{β2[(z1,t-z2,t)[α/2(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r]-C]},(2.27)wt+1={F1G+1ifmt+1≥mt,F2G-1ifmt+1<mt,
where
(2.28)F1=-4(1-wt){R+z2,t[(α/2)(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r]}(1-mt)[exp⁡{β[(z1,t-z2,t)[(α/2)(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r]-C]}+1],F2=4(1+wt){R+z1,t[(α/2)(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r]}(1+mt)[exp⁡{-β[(z1,t-z2,t)[(α/2)(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r]-C]}+1],G=2R+[z1,t+z2,t+(z1,t-z2,t)wt]⋅[(α/2)(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r].Proof.
From (2.24), we immediately obtain (2.25) and
(2.29)ρt+1-r=α2[z1,t(1+wt)+z2,t(1-wt)]+y¯pt-r,
where we have assumed that dividends evolve in a deterministic way according to their expected value y¯.
Putting (2.29) in equations of F1 and F2 and G, we rewrite such functions as in (2.30)F1=-21-mt+11-mt(1-wt)[R+z2,t(ρt+1-r)]=-21-mt+11-mt(1-wt){R+z2,t[α2(z1,t+z2,t+(z1,t-z2,t)wt)+y¯pt-r]},F2=21+mt+11+mt(1+wt)[R+z1,t(ρt+1-r)]=21+mt+11+mt(1+wt){R+z1,t[α2(z1,t+z2,t+(z1,t-z2,t)wt)+y¯pt-r]},G=(1-wt)[R+z2,t(ρt+1-r)]+(1+wt)[R+z1,t(ρt+1-r)]=2R+(ρt+1-r)[z1,t(1+wt)+z2,t(1-wt)]=2R+{α2[z1,t(1+wt)+z2,t(1-wt)]+y¯pt-r}[z1,t(1+wt)+z2,t(1-wt)]=2R+[z1,t+z2,t+(z1,t-z2,t)wt]⋅[α2(z1,t+z2,t+(z1,t-z2,t)wt)+y¯pt-r].
Consider now the dynamics of the difference in fractions of agents, that is, (2.5). Equation (2.26) is trivially derived putting C=C1-C2 and recalling (2.4) and (2.29). Moreover:(2.31)1-mt+1=2exp⁡{β[(z1,t-z2,t)[(α/2)(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r]-C]}+1,1+mt+1=2exp⁡{-β[(z1,t-z2,t)[(α/2)(z1,t+z2,t+(z1,t-z2,t)wt)+y¯/pt-r]-C]}+1,
where we have made use of relations 1-tanhx=2e-x/(ex+e-x)=2/(e2x+1) and 1+tanhx=2ex/(ex+e-x)=2/(e-2x+1). Finally, introducing (2.31) into the expressions of F1, F2 we arrive at (2.27).Notice that in the previous proposition we have introduced the difference between costs, that is,C=C1-C2.In order to study the system defined by Proposition2.1, we have to specify the individual demand functions (2.32)zh,t=Eh,t[ρt+1-r]λσ2=1λσ2{1pt[Eh,t(pt+1)+rp⋆-pt]-r},∀h=1,2,
where we have assumed that beliefs about variance and risk aversion coefficients are constant and equal for all traders, that is, Varh,t[ρt+1-r]=σ2 and λh=λ, for all h=1,2. In making this assumption we follow Brock and Hommes [5]. Notice that p⋆ is the fundamental solution, that is, the long-run market clearing price path when homogeneous beliefs about expected excess return are considered. Under the assumption of an i.i.d. dividend process {yt} with Et(yt+1)=y¯, the fundamental solution is constant and given by pt⋆=p⋆=y¯/r. Brock and Hommes [5] derive endogenously the fundamental solution satisfying the no-bubbles condition, in the particular case of zero net supply of shares.As in many interacting agent models (see, e.g., [5, 8, 10]), in order to explain why prices deviate from their fundamental values for a long-time and to analyze agent survival, in the following section, we assume that agents of type 1 are fundamentalists while agents of type 2 are chartists.


## 3. Fundamentalists versus Chartists

### 3.1. The Map

Let us move on to analyze the case in which agents of type 1 are fundamentalists, believing that prices return to their fundamental value, while traders of type 2 are chartists, who do not take into account the fundamental value but base their prediction selection upon a simple linear trading rule. In other words, we assume thatE1,t(pt+1)=p⋆ and E2,t(pt+1)=apt with a>0. Trivially, for a>1 (a<1) agents of group 2 believe that the price will increase (decrease) in the next period, while they expect the same price in the next period when a=1 (in this last case naive expectations are considered).Therefore, the demand functions are given by:(3.1)z1,t=1λσ2(1+r)(xt-1),z2,t=1λσ2[a-1+r(xt-1)].Following the framework of Chiarella et al. [10], we introduce a new state variable, given by the fundamental price ratio: xt=p⋆/pt. Consequently: xt+1=p⋆/pt+1=p⋆/pt·pt/pt+1=xt(pt/pt+1).The final nonlinear dynamical systemT is written in terms of the state variables xt, mt and wt: (3.2)xt+1=f1(xt,wt)=xt(α/2λσ2)(xt-2+a+2r(xt-1)+(xt-a)wt)+1,(3.3)mt+1=f2(xt,wt)=tanh⁡{β2[(1/λσ2)(xt-a)[(α/2λσ2)(xt-2+a+2r(xt-1)+(xt-a)wt)+r(xt-1)]-C]},(3.4)wt+1=f3(xt,mt,wt)={F1G+1ifmt+1≥mt,F2G-1ifmt+1<mt,
with(3.5)F1=-4(1-wt){R+(1/λσ2)[a-1+r(xt-1)][Y]}(1-mt)[exp⁡{β[(1/λσ2)(xt-a)[Y]-C]}+1],F2=4(1+wt){R+(1/λσ2)(1+r)(xt-1)[Y]}(1+mt)[exp⁡{-β[(1/λσ2)(xt-a)[Y]-C]}+1],G=2R+1λσ2[xt-2+a+2r(xt-1)+(xt-a)wt]⋅[α2λσ2(xt-2+a+2r(xt-1)+(xt-a)wt)+r(xt-1)],
where 𝒴 denotes (α/2λσ2)(xt-2+a+2r(xt-1)+(xt-a)wt)+r(xt-1)and we have made use of
(3.6)pt+1={α2λσ2[a-b-2+2r(xt-1)+(a+b)wt]+1}pt.Notice that the function defined by (3.4) is continuous and piecewise smooth. In particular, mt+1 being defined by (3.3), the phase space is divided into two regions by the surface of equation f2(x,w)-m=0. Observe that all the equlibria must belong to the border surface for any range of the parameter values.Finally, we wish to underline that our model is characterized by two differentsuccess indicators: the difference in the fractions of agents, m, and the difference in the relative wealths, w. More precisely, a strategy h can be successful both in terms of the number of agents using it or in terms of the wealth of group h.


### 3.2. Steady States

In order to find the steady states owned by the system, we put(xt,mt,wt)=(x,m,w) for all  t. Recalling that xt=p⋆/pt and under the assumption of i.i.d. dividend process, we already know that any equilibrium fundamental price ratio x is different from zero. Afterwards, we have: x-2+a+2r(x-1)+(x-a)w=0 (see (3.2)) and it trivially follows that G=2R.Consequently, (3.4) shows that w must solve:
(3.7)w=12R{-4(1-w){R+(1/λσ2)[a-1+r(x-1)]r(x-1)}(2/(eM+1))(eM+1)}+1,
where 1-m has been rewritten as 1-m=2/(eM+1) with(3.8)M=β[1λσ2(x-a)[α2λσ2(x-2+a+2r(x-1)+(x-a)w)+r(x-1)]-C],
that is, M=β[(1/λσ2)(x-a)r(x-1)-C]. Hence, we obtain (3.9)w=1R{(w-1){R+1λσ2[a-1+r(x-1)]r(x-1)}}+1,
that is,(3.10)R(w-1)=(w-1){R+1λσ2[a-1+r(x-1)]r(x-1)}.It follows that the steady state valuesw and x must satisfy (3.11)x-2+a+2r(x-1)+(x-a)w=0,R(w-1)=(w-1){R+1λσ2[a-1+r(x-1)]r(x-1)},
and we can identify two types of steady states:(i)
fundamental steady states characterized by x=1, that is, by the price being at the fundamental value,(ii)
nonfundamental steady states for which x≠1.More precisely, fora≠1 the fundamental steady state Ef of the system is such that wf=1 and there exists a nonfundamental steady state Enf such that wnf=-1, xnf=((1-a)/r)+1. Notice that the equilibrium Enf exists for a<1+r (i.e., xnf>0). In fact, though such a steady state has been derived analytically for any a, for a≥1+r it is outside the economic meaning of x and numerical evidence confirms that it is nonattracting. Observe that the equilibria Ef and Enf are characterized by w=1 and w=-1 respectively. In other words, at the fundamental (nonfundamental) equilibrium the total wealth is owned by fundamentalists (chartists).Otherwise, whena=1 the fixed point Enf becomes a fundamental steady state. Moreover, every point E=(1,tanh{-Cβ/2},w¯) is a fundamental equilibrium, that is, the long-run wealth distribution at a fundamental steady state is given by any constant w¯∈[-1,1]. In other words, a continuum of steady states exists: they are located in a one-dimensional subset (a straight line) of the phase space. Notice that this is a natural result, as for a=1, the expectations schemes are equivalent at the fundamental price.Summarizing, the following lemma deals with the existence of the steady states.Lemma 3.1.
The number of the steady states of the systemT depends on the parameter a.  (1)
Leta≠1, then
(a)
fora<1+r there exist two steady states: the fundamental equilibrium:
(3.12)Ef=(xf=1,mf=tanh{-Cβ2},wf=1)
and the nonfundamental equilibrium:
(3.13)Enf=(1-ar+1,tanh{β2[1λσ21+rr(1-a)2-C]},-1),(b)
fora≥1+r the fundamental steady state Ef is unique.(2)
Leta=1, then:
(a)
Every pointE=(1,tanh{-Cβ/2},w¯) is a fundamental equilibrium.The basin of attraction in Figure1 shows that for a=1 the steady state wealth distribution which is reached in the long-run by the system depends on the initial condition. More precisely, for different initial conditions (m0,w0) in the gray region, the system converges to different equilibria E with w¯∈(-1,1), providing that relative wealths converge to some mixture.Figure 1
Basin of attraction in the plane(m,w) for the initial condition x0=1 and parameter values a=1, α=4, λ=1, σ2=1, r=0.02, C=0.5 and β=1. The white point corresponds to Ef while the black point corresponds to Enf.


### 3.3. Trapping Set and Stability Analysis

Given the atypical form of our three-dimensional system, in which the function definingwt+1 is piecewise smooth, so that the phase space is divided into two regions, we look for appropriate restrictions of our map. We recall that a set X is trapping for a map T if T(X)⊆X. The following proposition proves the existence of a trapping set characterized by wt=1, for all t.Proposition 3.2.
For allα,r,λ,σ2 with α(1+r)/λσ2≤1, there exists a¯=2λσ2/α(r+1)-1 such that for all  a≥a¯ the set X={(xt,mt,wt):xt≥1,wt=1} is trapping for any initial condition (x0,m0,w0) with 1≤x0≤(a+1)/2 and m0=-1+ϵ (ϵ≥0 small enough).Proof.
Looking at (3.4) for mt+1≥mt, we find that wt=1 implies wt+1=1 for all xt,mt. Therefore we require mt+1≥mt for all t. From (3.2) and (3.3) it is easy to obtain xt+1=f1(xt) and mt+1=f2(xt) for wt=1, so that condition mt+1≥mt becomes f2(xt)≥f2(xt-1) and it must be verified if f2 is a decreasing function and xt≤xt-1. Function f2 is decreasing if and only if zt=(β/2)[(1/λσ2)(xt-a)(xt-1)(α(1+r)/λσ2+r)-C] is decreasing, that is, if and only if zt′=2xt-(a+1)≤0 (xt≤(a+1)/2). Notice that xt=f1(xt-1) is increasing if α(1+r)/λσ2≤1 and upper bounded for all xt-1≥1 with limx→+∞f1(x)=λσ2/α(r+1), then it must exists a¯=2λσ2/α(r+1)-1 such that 2xt-(a+1)≤0  for all a≥a¯.
Our second requirement, that is,xt≤xt-1, can be rewritten as f1(xt-1)≤xt-1 or equivalently:(3.14)f1(xt)-xt=(α/λσ2)(1+r)xt(1-xt)(α/λσ2)(1+r)(xt-1)+1≤0
which must hold if xt≥1. Finally, looking at (3.2) for wt=1 and α(1+r)/λσ2≤1, it follows that xt≥1 implies xt+1≥1 for all t.
Notice that functionsf1 and f2 do not depend on mt, thus both conditions “f2  decreasing” and “xt≤xt-1” satisfy mt+1≥mt for all t≥1, as a consequence it is necessary to consider an i.c. m0 small enough to obtain mt+1≥mt for all t≥0. Similarly, we require x0 such that 1≤x0≤(a+1)/2.Observe that the previous proposition defines parameter values and initial conditions such thatmt+1≥mt for all t, that is, at any time the system uses the first equation defining f3(x,m,t) (see (3.4)) which leads to: wt=1⇒wt+1=1. Following the same steps of Proposition 3.2, it is possible to see that there are no parameter values such that mt+1<mt for all t. In other words, for any parameter values and initial conditions the system sooner or later will use the first equation defining f3(x,m,t). This means that a movement from class 2 (chartists) to class 1 (fundamentalists) always occurs.The trapping setX defined by Proposition 3.2 allows us to study the local asymptotic stability of the fundamental steady state in the case in which the dynamical system is restricted to the subspace X. Then, the map TX:(xt,mt)→(xt+1,mt+1) is defined by:(3.15)xt+1=f1(xt)=xt(α/λσ2)(xt-1)(1+r)+1,mt+1=f2(xt)=tanh{β2[1λσ2(xt-a)(xt-1)(α(1+r)λσ2+r)-C]}.The Jacobian matrix evaluated at the fundamental steady stateEf is:(3.16)J(Ef)=(∂f1∂xt(Ef)0∂f2∂xt(Ef)0)
which implies that one eigenvalue is 0 (and thus smaller than one in modulus), while the other eigenvalue is (∂f1/∂xt)(1,tanh{-Cβ/2},1)=1-(α/λσ2)(1+r). Under the hypothesis of Proposition 3.2, this eigenvalue is smaller than one in modulus as well. In other words, if α(1+r)/λσ2≤1, the fundamental equilibrium wf=1, xf=1, mf=tanh{-Cβ/2} is locally asymptotically stable for high values of a and for any initial condition (x0,m0,w0) such that 1≤x0≤(a+1)/2, m0=-1+ϵ (ϵ≥0 small enough) and w0=1. Summarizing, in the case in which, at the initial time, the price is below the fundamental value and the market is dominated by chartists while fundamentalists own the total wealth, the system converges to the fundamental steady state Ef.


## 3.1. The Map

Let us move on to analyze the case in which agents of type 1 are fundamentalists, believing that prices return to their fundamental value, while traders of type 2 are chartists, who do not take into account the fundamental value but base their prediction selection upon a simple linear trading rule. In other words, we assume thatE1,t(pt+1)=p⋆ and E2,t(pt+1)=apt with a>0. Trivially, for a>1 (a<1) agents of group 2 believe that the price will increase (decrease) in the next period, while they expect the same price in the next period when a=1 (in this last case naive expectations are considered).Therefore, the demand functions are given by:(3.1)z1,t=1λσ2(1+r)(xt-1),z2,t=1λσ2[a-1+r(xt-1)].Following the framework of Chiarella et al. [10], we introduce a new state variable, given by the fundamental price ratio: xt=p⋆/pt. Consequently: xt+1=p⋆/pt+1=p⋆/pt·pt/pt+1=xt(pt/pt+1).The final nonlinear dynamical systemT is written in terms of the state variables xt, mt and wt: (3.2)xt+1=f1(xt,wt)=xt(α/2λσ2)(xt-2+a+2r(xt-1)+(xt-a)wt)+1,(3.3)mt+1=f2(xt,wt)=tanh⁡{β2[(1/λσ2)(xt-a)[(α/2λσ2)(xt-2+a+2r(xt-1)+(xt-a)wt)+r(xt-1)]-C]},(3.4)wt+1=f3(xt,mt,wt)={F1G+1ifmt+1≥mt,F2G-1ifmt+1<mt,
with(3.5)F1=-4(1-wt){R+(1/λσ2)[a-1+r(xt-1)][Y]}(1-mt)[exp⁡{β[(1/λσ2)(xt-a)[Y]-C]}+1],F2=4(1+wt){R+(1/λσ2)(1+r)(xt-1)[Y]}(1+mt)[exp⁡{-β[(1/λσ2)(xt-a)[Y]-C]}+1],G=2R+1λσ2[xt-2+a+2r(xt-1)+(xt-a)wt]⋅[α2λσ2(xt-2+a+2r(xt-1)+(xt-a)wt)+r(xt-1)],
where 𝒴 denotes (α/2λσ2)(xt-2+a+2r(xt-1)+(xt-a)wt)+r(xt-1)and we have made use of
(3.6)pt+1={α2λσ2[a-b-2+2r(xt-1)+(a+b)wt]+1}pt.Notice that the function defined by (3.4) is continuous and piecewise smooth. In particular, mt+1 being defined by (3.3), the phase space is divided into two regions by the surface of equation f2(x,w)-m=0. Observe that all the equlibria must belong to the border surface for any range of the parameter values.Finally, we wish to underline that our model is characterized by two differentsuccess indicators: the difference in the fractions of agents, m, and the difference in the relative wealths, w. More precisely, a strategy h can be successful both in terms of the number of agents using it or in terms of the wealth of group h.


## 3.2. Steady States

In order to find the steady states owned by the system, we put(xt,mt,wt)=(x,m,w) for all  t. Recalling that xt=p⋆/pt and under the assumption of i.i.d. dividend process, we already know that any equilibrium fundamental price ratio x is different from zero. Afterwards, we have: x-2+a+2r(x-1)+(x-a)w=0 (see (3.2)) and it trivially follows that G=2R.Consequently, (3.4) shows that w must solve:
(3.7)w=12R{-4(1-w){R+(1/λσ2)[a-1+r(x-1)]r(x-1)}(2/(eM+1))(eM+1)}+1,
where 1-m has been rewritten as 1-m=2/(eM+1) with(3.8)M=β[1λσ2(x-a)[α2λσ2(x-2+a+2r(x-1)+(x-a)w)+r(x-1)]-C],
that is, M=β[(1/λσ2)(x-a)r(x-1)-C]. Hence, we obtain (3.9)w=1R{(w-1){R+1λσ2[a-1+r(x-1)]r(x-1)}}+1,
that is,(3.10)R(w-1)=(w-1){R+1λσ2[a-1+r(x-1)]r(x-1)}.It follows that the steady state valuesw and x must satisfy (3.11)x-2+a+2r(x-1)+(x-a)w=0,R(w-1)=(w-1){R+1λσ2[a-1+r(x-1)]r(x-1)},
and we can identify two types of steady states:(i)
fundamental steady states characterized by x=1, that is, by the price being at the fundamental value,(ii)
nonfundamental steady states for which x≠1.More precisely, fora≠1 the fundamental steady state Ef of the system is such that wf=1 and there exists a nonfundamental steady state Enf such that wnf=-1, xnf=((1-a)/r)+1. Notice that the equilibrium Enf exists for a<1+r (i.e., xnf>0). In fact, though such a steady state has been derived analytically for any a, for a≥1+r it is outside the economic meaning of x and numerical evidence confirms that it is nonattracting. Observe that the equilibria Ef and Enf are characterized by w=1 and w=-1 respectively. In other words, at the fundamental (nonfundamental) equilibrium the total wealth is owned by fundamentalists (chartists).Otherwise, whena=1 the fixed point Enf becomes a fundamental steady state. Moreover, every point E=(1,tanh{-Cβ/2},w¯) is a fundamental equilibrium, that is, the long-run wealth distribution at a fundamental steady state is given by any constant w¯∈[-1,1]. In other words, a continuum of steady states exists: they are located in a one-dimensional subset (a straight line) of the phase space. Notice that this is a natural result, as for a=1, the expectations schemes are equivalent at the fundamental price.Summarizing, the following lemma deals with the existence of the steady states.Lemma 3.1.
The number of the steady states of the systemT depends on the parameter a.  (1)
Leta≠1, then
(a)
fora<1+r there exist two steady states: the fundamental equilibrium:
(3.12)Ef=(xf=1,mf=tanh{-Cβ2},wf=1)
and the nonfundamental equilibrium:
(3.13)Enf=(1-ar+1,tanh{β2[1λσ21+rr(1-a)2-C]},-1),(b)
fora≥1+r the fundamental steady state Ef is unique.(2)
Leta=1, then:
(a)
Every pointE=(1,tanh{-Cβ/2},w¯) is a fundamental equilibrium.The basin of attraction in Figure1 shows that for a=1 the steady state wealth distribution which is reached in the long-run by the system depends on the initial condition. More precisely, for different initial conditions (m0,w0) in the gray region, the system converges to different equilibria E with w¯∈(-1,1), providing that relative wealths converge to some mixture.Figure 1
Basin of attraction in the plane(m,w) for the initial condition x0=1 and parameter values a=1, α=4, λ=1, σ2=1, r=0.02, C=0.5 and β=1. The white point corresponds to Ef while the black point corresponds to Enf.


## 3.3. Trapping Set and Stability Analysis

Given the atypical form of our three-dimensional system, in which the function definingwt+1 is piecewise smooth, so that the phase space is divided into two regions, we look for appropriate restrictions of our map. We recall that a set X is trapping for a map T if T(X)⊆X. The following proposition proves the existence of a trapping set characterized by wt=1, for all t.Proposition 3.2.
For allα,r,λ,σ2 with α(1+r)/λσ2≤1, there exists a¯=2λσ2/α(r+1)-1 such that for all  a≥a¯ the set X={(xt,mt,wt):xt≥1,wt=1} is trapping for any initial condition (x0,m0,w0) with 1≤x0≤(a+1)/2 and m0=-1+ϵ (ϵ≥0 small enough).Proof.
Looking at (3.4) for mt+1≥mt, we find that wt=1 implies wt+1=1 for all xt,mt. Therefore we require mt+1≥mt for all t. From (3.2) and (3.3) it is easy to obtain xt+1=f1(xt) and mt+1=f2(xt) for wt=1, so that condition mt+1≥mt becomes f2(xt)≥f2(xt-1) and it must be verified if f2 is a decreasing function and xt≤xt-1. Function f2 is decreasing if and only if zt=(β/2)[(1/λσ2)(xt-a)(xt-1)(α(1+r)/λσ2+r)-C] is decreasing, that is, if and only if zt′=2xt-(a+1)≤0 (xt≤(a+1)/2). Notice that xt=f1(xt-1) is increasing if α(1+r)/λσ2≤1 and upper bounded for all xt-1≥1 with limx→+∞f1(x)=λσ2/α(r+1), then it must exists a¯=2λσ2/α(r+1)-1 such that 2xt-(a+1)≤0  for all a≥a¯.
Our second requirement, that is,xt≤xt-1, can be rewritten as f1(xt-1)≤xt-1 or equivalently:(3.14)f1(xt)-xt=(α/λσ2)(1+r)xt(1-xt)(α/λσ2)(1+r)(xt-1)+1≤0
which must hold if xt≥1. Finally, looking at (3.2) for wt=1 and α(1+r)/λσ2≤1, it follows that xt≥1 implies xt+1≥1 for all t.
Notice that functionsf1 and f2 do not depend on mt, thus both conditions “f2  decreasing” and “xt≤xt-1” satisfy mt+1≥mt for all t≥1, as a consequence it is necessary to consider an i.c. m0 small enough to obtain mt+1≥mt for all t≥0. Similarly, we require x0 such that 1≤x0≤(a+1)/2.Observe that the previous proposition defines parameter values and initial conditions such thatmt+1≥mt for all t, that is, at any time the system uses the first equation defining f3(x,m,t) (see (3.4)) which leads to: wt=1⇒wt+1=1. Following the same steps of Proposition 3.2, it is possible to see that there are no parameter values such that mt+1<mt for all t. In other words, for any parameter values and initial conditions the system sooner or later will use the first equation defining f3(x,m,t). This means that a movement from class 2 (chartists) to class 1 (fundamentalists) always occurs.The trapping setX defined by Proposition 3.2 allows us to study the local asymptotic stability of the fundamental steady state in the case in which the dynamical system is restricted to the subspace X. Then, the map TX:(xt,mt)→(xt+1,mt+1) is defined by:(3.15)xt+1=f1(xt)=xt(α/λσ2)(xt-1)(1+r)+1,mt+1=f2(xt)=tanh{β2[1λσ2(xt-a)(xt-1)(α(1+r)λσ2+r)-C]}.The Jacobian matrix evaluated at the fundamental steady stateEf is:(3.16)J(Ef)=(∂f1∂xt(Ef)0∂f2∂xt(Ef)0)
which implies that one eigenvalue is 0 (and thus smaller than one in modulus), while the other eigenvalue is (∂f1/∂xt)(1,tanh{-Cβ/2},1)=1-(α/λσ2)(1+r). Under the hypothesis of Proposition 3.2, this eigenvalue is smaller than one in modulus as well. In other words, if α(1+r)/λσ2≤1, the fundamental equilibrium wf=1, xf=1, mf=tanh{-Cβ/2} is locally asymptotically stable for high values of a and for any initial condition (x0,m0,w0) such that 1≤x0≤(a+1)/2, m0=-1+ϵ (ϵ≥0 small enough) and w0=1. Summarizing, in the case in which, at the initial time, the price is below the fundamental value and the market is dominated by chartists while fundamentalists own the total wealth, the system converges to the fundamental steady state Ef.


## 4. Numerical Simulations

In this section we move to the study of the asymptotic dynamics by using numerical simulations.Firstly, we consider the case in which Proposition3.2 holds. In Figure 2(a) we present a diagram of the state variable wt with respect to a. We choose parameter values such that the condition α(1+r)/λσ2≤1 of Proposition 3.2 holds, hence, if a is great enough, our system admits the trapping set X. Furthermore, we consider an initial condition belonging to X, that is, at the initial time, the market is dominated by chartists while all the wealth is owned by fundamentalists and also, being x0≥1, the price is below the fundamental value. The diagram confirms the convergence to the fundamental value wf=1. Notice that if a is low enough, the figure is dominated by the black region in which the trajectory does not converge to a cycle of period k (k>1) nor to a complex attractor, but we observe a long transient (LT where with LT we indicate the case in which the trajectory converges to a fixed point after a very high number of iterations). In fact, if a is low enough, the state variable wt tends to the fundamental steady state very slowly and, after a large number of iterations, the attractor has not yet been reached.(a) Diagram of the state variablewt with respect to a for the initial condition x0≥1, m0=-1 and w0=1 and parameter values α=0.5, λ=1, σ2=1, r=0.02, C=0.5. We ignore the first 1000 iterations and we plot the following 5000 values of wt for each value of a. (b) Trajectory of the state variable wt versus time. We consider a=0.3 and we plot the first 20000 iterations.
(a)(b)In Figure2(b)
we observe the trajectory with a=0.3, starting from the same initial condition. After 20000 iterations, the fundamental steady state has not yet been reached (wt is still far from wf), however numerical computations show that it will be approached very, very slowly (as t→+∞).Let us go on to consider parameter values such that the setX is nontrapping and, at the initial time, the market is dominated by one class of agents owning the greater fraction of the total wealth. We first consider the case in which, at the initial time, the market is dominated by fundamentalists, while the initial price is above its fundamental value. In Figure 3, we present some bifurcation diagrams of the state variable wt with respect to β (the intensity of choice) for different values of a. We found an a-typical bifurcation sequence due to the particular form of our system (piecewise smooth) which admits high period cycles or complex dynamics. Notice that, as a increases, the final dynamics becomes simpler. In fact, in panel (d) the state variable converges to the value w=-1, such that the total wealth is owned by chartists.One-dimensional bifurcation diagram of the state variablewt with respect to β for the initial condition x0=0.9, m0=0.8 and w0=0.8 and parameter values α=1, λ=1, σ2=1, r=0.02, C=0.5. In panel (a) a=0.5, in panel (b) a=0.9, in panel (c) a=1.2 and in panel (d) a=2.
(a)(b)(c)(d)A similar feature is also observed by considering the state variablex. The graphs presented in Figure 4 show the evolution of xt=p⋆/pt versus time. Being p⋆ constant, if xt fluctuates then also pt fluctuates (i.e., the qualitative dynamics of xt completely describes that of prices). As for the relative wealth, prices show fluctuations for low values of a. In panel (a), a 10-period cycle is observed while in panel (b) the period of the attracting cycle is 23. However, if a is great enough, the final dynamics becomes simpler (in panel (c) prices convrge to the fundamental value). Numerical simulations show a similar feature also in the opposite case, that is, m0 and w0 are close to −1.State variablext versus time for the initial condition x0=1.1, m0=0.8 and w0=0.8 and parameter values α=1, λ=1, σ2=1, r=0.02, C=0.5, β=1. In panel (a) a=0.35, in panel (b) a=0.4, in panel (c) a=0.45.
(a)(b)(c)Such evidence confirms the main results of the relevant literature. In fact, models with heterogeneous agents and a switching mechanism are able to reproduce the stylized facts observable in real markets, that is, periodic or even chaotic fluctuations in prices, excess of volatility, bubbles and crashes.In order to better investigate the previous feature, we move towards the extreme case in which, at the initial time, all agents are fundamentalists and they own the total wealth. Moreover, in the following, we focus on the more interestinga range of values, that is, a<1+r.In Figure5(a), we present the trajectory of the state variable wt while assuming that, at the initial time, m0=w0=1 and the price is above its fundamental value. We choose two different values of a and observe that if a is low enough the system fluctuates (a 12-period cycle is observed) while, as a increases, the long-run dynamics becomes simpler (wt→1 after a LT).State variablewt versus time for an i.c. x0=0.8, m0=1 and w0=1 and parameter values α=0.5, λ=1, σ2=1, r=0.02, C=0.5, β=1. (a) a=0.1. Periodical fluctuations. (b) a=0.5. Convergence.
(a)(b)Numerical simulations show the existence of a valueã such that for all a∈(ã,1+r) the state variables converge to Ef, while more complex dynamics are exhibited if a<ã, that is, for a low enough (see Figure 6). Notice that the system preserves similar features, that is, the stabilizing effect of a in the long-run, also in the extreme case w0=-1=m0, that is, at the initial time, the market is dominated by chartists who own all the wealth.Bifurcation diagrams of the state variables w.r.t.a. Initial condition and parameter values as in Figure 5.
(a)(b)(c)In Figure6 we observe a succession of periodic windows which occur at border collision bifurcations. These noncanonical bifurcations have mainly been studied in the context of piecewise linear maps. Hommes and Nusse [26] showed, for instance, that a “period three to period two” bifurcation occurs for a class of piecewise linear maps. More recent interesting contributions on this topic are from Jain and Banerjee [27], Avrutin et al. [28–30]. These periodic windows have the following properties:  (i)
the width of the periodic window reduces monotonically as the parametera increases(ii)
the windows are characterized by the same periodicity(iii)
fora>ã, the periodic windows terminate in convergence.Notice how complexity is mainly due to border collision bifurcations which are involved by wealth dynamics. This means that the new mechanism which explains the evolution of wealth plays an important role in the qualitative dynamics observed in the long-run.In order to consider also the role of the parameterβ, in Figure 7 we present a two-dimensional bifurcation diagram in the parameter plane (a,β) in the extreme case m0=w0=1 (at the initial time, all agents are fundamentalists and they own all the wealth) and the price x0 is above the fundamental value. Rich dynamics is exhibited and the final behavior increases in complexity as β increases (according to what happens in asset-pricing models with heterogeneous agents and adaptiveness).Figure 7
Two-dimensional bifurcation diagram in parameters' plane(a,β) for i.c. x0=0.8, m0=w0=1. The other parameter values as in Figure 5.A similar behavior is observed in Figures8 and 9. In such cases, we consider two different i.c., that is, m0=-1 and w0=1 (all agents are chartists but all the wealth is owned by fundamentalists) and m0=1 and w0=-1 (all agents are fundamentalists but all the wealth is owned by chartists). Observe that, also in these cases our system may exhibit fluctuations or aperiodic patterns and the final dynamics strictly depends on the parameter value. This study confirms the evidence that the system is more complicated for high values of β and low values of a.Figure 8
Two-dimensional bifurcation diagram in the parameter plane(a,β) for and i.c. x0=1.3, m0=-1w0=1. The other parameter values as in Figure 5.Figure 9
Two-dimensional bifurcation diagram in the parameter plane(a,β) for and i.c. x0=1.3, m0=1w0=-1. The other parameter values as in Figure 5.By comparing Figures7, 8 and 9, we also observe that the final dynamics is affected by the choice of the initial conditions. In order to focus on the role of the initial condition, we fix a low enough. In Figure 10, we present the basins of attraction in the plane (m0,w0), where the initial price is below the fundamental. Observe that, if we move parameter β, the structure of the basin also undergoes a change. More precisely, it seems to become more complex as β increases, confirming the evidence previously obtained. In fact, the structure of the basin becomes fractal providing the strong dependence of the final dynamics w.r.t. the initial conditions (in panel (b) outside the stability region the system may converge to a 10-period cycle or to a more complex attractor).Basins of attraction in the plane(m0,w0) for x0=1.3, a=0.1 and the other parameter values as in Figure 5. (a) β=1.8, (b) β=2.
(a)(b)Basins of attraction in the plane(m0,x0) for w0=0, a=0.1 and the other parameter values are as in Figure 5. (a) β=1.25, (b) β=2.
(a)(b)Finally, the basins of attraction in the plane(m0,x0) are presented for an initial condition w0=0, that is, at the initial time, the wealth is equally shared between the two groups. Again, the structure seems to increase in complexity as β increases. Observe that the stability region (the blue one) reduces as β increases and that outside this region different attractors coexist (i.e., a 10-cycle or a 11-cycle and a more complex attractor).In contrast with the existing models in the CRRA framework, which suggest that wealth dynamics will lead to amonomorphic behavior in the long-run, our simulations show complexity in long-run wealth evolution. A possible explanation is based on the key new concept of this paper, that is, our assumption about wealth redistribution. In fact, a great number of simulations with constant proportions of agents (i.e., the model without switching) shows that the relative wealths converge to 0 and 1, in other terms some classes survive, some classes do not. We obtain the same result by simulating the model for high values of the intensity of choice β: the relative wealths still converge to 0 and 1. In other terms, wealth-driven selection works for the simulation without switching or for high β, differently from the previous simulations of the general model (with switching). This suggests that the outcome wealth dynamics, able to exhibit complexity, strictly depends on the new switching mechanism introduced, hence it is responsible for the discrepancy with the literature concerning heterogeneous agent models.


## 5. Conclusions and Further Developments

Motivated by recent developments in the class of heterogeneous agent models of asset price and wealth dynamics, we developed an analytical adaptive model with two types of heterogeneous agents.The model is developed in the discrete time setting of standard portfolio theory, in that agents are allowed to revise their portfolios over any time interval.Both expectation feedback and adaptiveness are common features of recent heterogeneous asset-pricing models. In addition, our model is able to characterize the evolution of the distribution of wealth when agents switch from an old strategy to a new strategy, according to their past performances.The new contribution of our model comes from the assumption that all agents belonging to the same group agree to share their wealth whenever an agent joins the group (or leaves it). This assumption allows us to characterize equilibrium price and wealth evolution among heterogeneous agents. Moreover, it leads to differentsuccess indicators of each strategy, the difference in the fraction of agents and the difference in the relative wealths of the groups. In other words, a certain strategy can be successful in terms of the number of agents using it or in terms of the wealth of the respective group.In performing the analysis, we focus on the case with fundamentalist and chartist agents. The final system is defined by a three-dimensional piecewise map which is not easily tractable. Nevertheless, we are able to derive all the steady states analytically, proving the existence of two kinds of equilibria: fundamental steady states and nonfundamental steady states. Moreover, we prove the existence of a continuum of fundamental steady states for some parameter values. We also show that the system admits a trapping region. The map restricted to this subset of the phase space is tractable and allows us to perform the stability analysis of the fundamental equilibrium.The asymptotic dynamics is studied by using numerical simulations which show the great variety of qualitative behavior presented by our model and their relation to a number of parameter values. Several border collision phenomena are observed, which are due to the new mechanism introduced into wealth dynamics. To examine this in greater detail, numerical simulations suggest that wealth-driven selection (some classes survive, some classes do not) works for the model without switching or for highβ, in contrast with our general framework (with switching). In this last case, the wealth dynamics shows complexity in the long-run, mainly due to the key new concept of this paper, namely the assumption about wealth redistribution.In line with existing heterogeneous agent models, our framework is able to explain some economic issues such as: the survival ofirrational agents, the role of market forces (wealth-driven selection) and some stylized facts (fluctuations, excess of volatility, bubbles and crashes).An interesting further contribution would be to consider the possibility of a mutual switch between strategies.



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*Source: 676317-2010-11-28.xml*