Rational versus non-rational agents in a heterogeneous agent model : two type agent model

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Rational versus non-rational agents

in a heterogeneous agent model

Two type agent model

Bette Donker

Econometrics and Operational Research

University of Amsterdam

Studentnumber: 10791396

Supervised by dhr. prof. dr. Cars Hommes

June 2018

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Statement of Originality

This document is written by Student Bette Donker who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been

used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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1 Introduction

The main paradigm in the agent market is the representative agent and the efficient market. More recently heterogeneous models have been introduced. In early research there was a main focus on rational agents, whereas later on this shifted to bounded rational agents. An important statement was made by Milton Friedman (1953), he said rational agents will drive non rational agents out off the market. This way eventually prices will go back to their fundamental values. In the heterogeneous market on the other hand, several types of investors can be separated in two groups. The first unit contains rationalists (e.g. fundamentalists), who believe the price of an asset is determined by the efficient market hypothesis (EMH) fundamental. This theory states that the price of an asset is the discounted value of all future dividends. The second group contains non rational traders, which are also being called ‘noise’ traders. Within this cluster many different types of trading behaviour arise. Most of these traders do not believe that the price of an asset can solely be determined by fundamentals. This heterogeneity in the agent market leads to dynamics and instability. Therefore, it is important to study the fractions of different types of traders and subsequently how these fractions may cause disturbances of markets. The follow up question would be if despite these disturbances there eventually will arise a market equilibrium again. The named instability in markets is caused by stabilizing and destabilizing traders. Most often rationalists and fundamentalists are seen as stabilizing factors to the market equilibrium. Alternatively, trend chasers, contrarians and feedback traders which are also called non rationalists are seen as destabilizing trader behaviour types.

The caucus on heterogeneous agent models regarding behavioural economics is evolving at high speed. Computational and analytical tools are constantly improving which sets the belief that there is a more convenient way to learn about the dynamics that are shown in heterogeneous markets. As a result of the complexity of these markets due to the infinite number of types of behaviour that traders can show, a simple yet advanced model is needed. This model has to provide a good explanation on a micro economic level and reflect well on the macro economic level. The key questions that will be answered by the dynamics following from that model are whether rational agents perform better compared to ‘noise’ traders and consequently whether these non rational agents survive in the market.

To uncover this new approach, a model with two types of trader behaviour is developed in this paper. Every type of trader behaviour will be displayed as a deviation from a formula as used by Brock and Hommes (1998). After adapting the model for several forms of traders, the fitness measure is calculated taking possible costs into account. Brock and Hommes (1998) compared a fundamentalist with a certain type of ‘noise’ trader. Their model with a rational trader has a forward looking element, which keeps us from analysing global dynamics. Therefore, in this paper an agent with perfect foresight instead of the fundamentalist is chosen as the first type. Using the new model,

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primary and secondary bifurcation diagrams are being used to display how to reach several asset price dynamics.

After this introduction the paper is structured as follows. In the second section the theoretical background of the heterogeneous agent model, the asset pricing model as developed by Brock and Hommes (1998) and the design of the research in this thesis are discussed. The results of the study and analysis of these are outlined in section three. Lastly, in section four there is a conclusion and recommendations regarding the main question are stated.

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2 Theory and Model description

In this section the theory required to fully comprehend this research is described. At first several ideas regarding the scientific view on agent markets are evaluated. Hereby the efficient market hypothesis and heterogeneous markets are described. Secondly, different types of traders are explained. Thirdly, the asset pricing model of Brock and Hommes (1998) is elucidated. Lastly, the two type model used in this thesis is specified.

2.1 From Efficient Market Hypothesis to Heterogeneous markets

As mentioned earlier, the paradigm around agent markets has been the efficient market hypothesis for a long time. Friedman (1953) was named as one of the early researches. However, several years before Friedman (1953), Keynes (1936) published an article that discussed behavioural agent models. He argued that there exists a possibility of having perfect expectations. Thereby he stipulated the importance of psychology in the decision making of agents.

Friedman (1953) nonetheless, did not agree with Keynes (Keynes, 1936) and preferred a different approach of the agent market. He set up the Friedman Hypothesis. It says that rational agents will push the non rational agents out of the market. Eventually this process will lead to asset prices going back to their fundamental values (Brock & Hommes, 1998).

The efficient market hypothesis, that thrived until the 1980s, is discussed by Fama (1965). He stated that there is no possible way to differentiate from an efficient market. When an investment opportunity arises, rational agents will participate immediately which causes the agent to sell an asset at a lower price. In this way the price evolves back to the fundamental value. This is also called arbitrage, which in the case of agent markets is performed by rational agents.

The paradigm shifted from the efficient market hypothesis and representative agents to hetero-geneous markets as of the early nineties. There are several reasons why this happened, of which three will be discussed.

To start with the high frequency trading seen in markets. If everybody in a market is rational, there would be no trading. That is; if a trader possesses all information available and therefore wants to buy an asset, nobody would sell the asset to him. This can be explained by the trait of rational agents that they know that another agent has good information. This theory is discussed by Milgrom and Stokey (1982) among others.

Furthermore, several survey data show different types of behaviour in agent markets. One of those studies was done by Frankel and Froot (1986). They described a difference between long run and short run trading. In the long run, prices will go back to fundamentals and therefore the equilibrium. On the short run however, agents will look at the past results and follow the ongoing trend. This coincides with other researches, where short run traders make their decisions based on extrapolative trading rules (Hommes, 2006).

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Lastly, as said in the introduction, technology evolves at high speed and thereby the markets evolve at high speed. Additional computational tools to make investment decisions became acces-sible in time. This development improves the manner in which scientists are able to research agent markets.

2.2 Different types of traders

Several types of trading behaviour are discussed. First rationalists and noise traders are being dis-cussed. Next fundamentalists, chartists and pure trend chasers are described.

To begin, rationalists are defined. Sargent (1993) stipulates two ways to recognize a rationalist. The first property is that the beliefs of a rationalist are exactly the same as realizations and there are no forecasting errors involved. This concept is also called rational expectations. The second property is that every decision a rational agent makes is made by optimization and is based on micro economics. Rationalists are also called smart money traders and have perfect foresight (Brock & Hommes, 1998).

Now, the noise traders are portrayed. Noise traders, also called liquidity traders, think they have special information about an asset even though they do not. Thus they are not making decisions based on events, but on misplaced assumptions. One of the ways they misinterpret the market is by looking at signs from other agents in the market in a wrong way (Hommes, 2006).

Here, fundamentalists are explained. Fundamentalists are comparable to rationalists. Brock and Hommes (1998) discuss in their paper that fundamentalists have the same information about past prices and transactions as traders with perfect foresight. However, they do not possess any knowledge about the behaviour of other agents in the market. Therefore, they do not take that specific information into account. Fundamentalists trade around the fundamental value, which signifies that they buy an asset with a price below the fundamental value and sell the assets that have a price above that value.

At this point, chartists are outlined. These traders do not look at the fundamental value and only make trades based on the history of trades and predictions. To do so, they extrapolate data concerning trades in the past. Zeeman (1974) write about a stock market where fundamentalists and chartists are compared. The study explains how markets go from bull markets to bear markets and the other way around. According to him the amount of chartists enlarges whilst bull markets arise. Bull markets are a positive indication to chartists, hence the stock index increases. The latter leads to the selling of assets by fundamentalists, resulting in a bear market.

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Lastly, pure trend chasers are clarified. Pure trend chasers will follow trends in the trading envi-ronment, as Brock and Hommes (1998) write in their paper. Therefore, their beliefs and investment choices depend on a trend.

2.3 The asset pricing model of Brock and Hommes

In this subsection the present discounted value asset pricing model of Brock and Hommes (1998) combined with the more recent model of Hommes (2006) is discussed. This model, derived from the maximization of mean-variance, is an Adaptive Believe System (ABS) when stretched to Heteroge-neous beliefs cases. The model, describing an asset market with heterogeHeteroge-neous agents, is set up at first. Later on the fitness measure is reviewed. It shows the way agents switch strategies based on their relative performance. Lastly, the intensity of choice demonstrates at what speed an agent will deviate from its strategy decision.

Both a risky asset and a risk free asset are stated in the model. The price of a risky asset at time t is denoted by pt and yt is stated as the stochastic dividend process of the asset at time t.

The risk free asset on the other hand is perfectly elastically supplied with gross return R > 1, with R = r + 1.

As of now, every letter displayed in a bold type of letter is a random variable. At first the dynamics of wealth are given by

Wt+1= RWt+ (pt+1+ yt+1− Rpt)zt. (1)

The amount of shares the agent purchased at time t is stated by ztand pt+1+ yt+1− Rptis denoted

as the excess return per share.

EtVtis the conditional expectation and conditional variance operator. Specified to a specific type

of investor h, this translates to EhtVht. Brock and Hommes (1998) believed that the conditional

variance is the same for every agent type in the market. Therefore, each type of investor maximizes their mean variance:

M axz{EhtWt+1] − (

a

2)VhtWt+1]}. (2)

This leads to the next demand of assets for any type of investor:

zht= Eht(pt+1+ yt+1− Rpt)/aσ2. (3)

All traders are supposed to have the same risk aversion a and Vht = σ2 needs to be equal and

constant.

Now that the demand for shares is clarified, zs _{is stated as the supply of outside shares so that}

the equilibrium of demand and supply can be formulated:

H X h=1 nht {Eht(pt+1+ yt+1− Rpt)} aσ2 = z s_. ₍₄₎

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Here the fraction of type h traders at time t, nht, is required and H states the amount of different

traders.

In case there is a homogeneous market, the pricing equation yields Rpt= Eht(pt+1+ yt+1) − aσ

2_z

st. (5)

Now (5) is specialized to the case when the supply of outside shares is zero, zs_{= 0, additionally p} t

is changed to the fundamental price ,p∗_t =P∞

k=1 Et y_t+k] Rk , which gives Rp∗t = Et(p∗t+1∗ +yt+1). (6)

The deviation between the fundamental (6) and (5) is equal to xt, as seen by

xt= pt− p∗t. (7)

The objective of this paper however is not to look at homogeneous agent models, but at a hetero-geneous one. Thus, the pricing equation without changing the asset price to the fundamental price is

Rpt=

X

nhtEhtpt+1+ yt+1]. (8)

Hommes (2006) assumes that all beliefs concerning the risky future price of an asset that contains risk are of the form

Eht(pt+1) = Eht(pt+1∗ ) + fh(xt−1, ..., xt−L). (9)

The way in which a trader thinks that an asset price is going to deviate from the fundamental price is indicated by fh. As of this moment fh(xt−1, ..., xt−L) is written as fht.

As a consequence, the pricing equation involving deviations is formulated. Hereby the assumption that Brock and Hommes (1998) made, ρhtis EhtRt+1, (6) andP nht= 1 are exploited. In addition,

several researches described the process of the adaption of beliefs. The adaption can be described as the dynamics of the fractions of trader types and is discussed more later on. Including this by changing the time period in the deviated pricing equation, leads to

Rxt= H X h=1 nh,tEhtxt+1] = H X h=1 nh,tfht. (10)

The earlier stipulated expression t−1 indicates the early start of period t, just before the equilibrium price is known. At this point it is achievable to compute the realized excess return in period t to t + 1,

Rt+1= pt+1+ yt+1− Rpt= xt+1= p∗t+1+ yt+1− Rxt− Rp∗t

= xt+1− Rxt+ p∗t+1+ yt+1− Et(pt+1∗ + yt+1) + Et(p∗t+1+ yt+1) − Rp∗t

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Using the formula of the realized excess return, the profits of strategy h are defined as follows

πh,t= (xt+1− Rxt+ δt+1)z(ρht). (12)

The earlier mentioned part of the model where the adaption of beliefs over time is described, is explained more sufficiently below.

First, the fraction of different types of traders is stated nht=

exp[βUh,t−1]

Zt

. (13)

Hereby the multinomial logit probabilities of a discrete choice are used, Zt−1=

X

exp[βUh,t−1]. (14)

The fitness measure of strategy h just before period t is defined as Uh,t−1. Consequently, according

to Hommes (2006) the measure of (accumulated) realized profits is the best function for the measure of fitness at time t,

Uht= (pt+ yt− Rpt−1)

Eh,t−1pt+ yt− Rpt−1]

aσ2 + wUh,t−1− Ch. (15)

The memory strength is stated by 0 ≤ w ≤ 1 and Ch is the notation of costs. As of now w is set

at 0, so the choice of strategy is only depending on recently realized profit. Therefore, the fitness measure is modified using xt:

Uht= (xt− Rxt−1)

fh,t−1− Rxt−1

aσ2 − Ch, (16)

When β, as indicated in the above functions, is high it tells us that most traders use the strategy with the highest fitness. Then again, a low β tells us that most traders stick with there chosen strategy.

In summary, this section explains the asset pricing model by Brock and Hommes (1998) and describes the dynamics of trader types fractions using the fitness measure.

In the next section the model of Brock and Hommes (1998) and Hommes (2006) is adjusted to the model that is used in this study. The fundamentalists need to be replaced by agents with perfect foresight, who are compared with trend chasers.

2.4 The two belief types model

Every type of belief that a trader can have is shown in a linear form by Brock and Hommes (1998),

fht= ghxt−1+ bh. (17)

The trend is stated by gh and bh is defined as the bias parameter

If the deviation of both the fundamental price and the bias parameter are zero, the forecast of fundamentalists seems to be fht = 0. In this paper, a model is discussed where perfect foresight

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agents are being compared with pure trend chases. Function (17) with g > 0 is stated as the belief of a pure trend chaser. When the bias is said to be zero, the belief is given by

f2t= gxt−1. (18)

Rationalists however, have a different linear form of belief:

f1t= xt−1. (19)

This belief contains a perfect foresight. The expectations of rationalists are always correct because they have all the information that is existing in and about the market.

The fitness measure for both sorts of agents is found by combining (17) and (18) with (3). The profits, (12), that are realized by pure trend chasers then are

πj,t=

1

aσ2(xt− Rxt−1)(ghxt−2+ bh− Rxt−1). (20)

In deviations from the fundamentalist, the Adaptive Believe System is set by the following formulas:

(1 + r)xt= H X h=1 nht(ghxt−1+ bh) + t (21) nh,t= exp(βUh,t−1) PH h=1exp(βUh,t−1) (22) Uh,t−1= (xt−1− Rxt−2)( ghxt−3+ bh− Rxt−2 aσ2 ) − Ch (23)

On the other hand, the profits that are realized by rational agents are πR,t=

1

aσ2(xt− Rxt−1) 2

− Ch. (24)

The costs are stated by C and being a rational agent is costly, therefore C > 0. The profits in the past are accumulated like as

Uj,t−1= πj,t−1+ Uj,t−2. (25)

Brock and Hommes (1997) stipulated that an infinite memory, η, would drive overreacting investors out of the market if there would be a limitless amount of rationalists that do not hold any cost. This model sticks to perfect rationality.

Agents with perfect foresight and trend chasers linked with (10) gives

Rxt= n1,txt+1+ n2,tgxt−1. (26)

The new fractions of trader types, (nht), are set by a combination of (19), (20) and (21):

n1,t= exp[β(

1

aσ2(xt−1− Rxt−2) 2_{+ ηU}

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n2,t= exp[β(

1

aσ2(xt−1− Rxt−2)(gxt−3− Rxt−2) + ηUP,t−2)]/Zt (28)

The agent with perfect foresight is indicated as type 1 and the pure trend chaser as type 2. Hommes (2006) described the following steady state fractions:

n∗₁= exp

−βC

1 + exp−βC (29)

n∗₂= 1

1 + exp−βC. (30)

If β = 0 above fractions are equal and if β increases, the fraction of the trend chasing traders rises. Eventually, the trend chasers fraction will even equal 1 when β approaches 1.

This section described the adaption of the asset pricing model in section 2.3. The fundamentalists were replaced by agents with perfect foresight. Therefore the pricing equation as well as the agent type fractions were adjusted.

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3 Results and analysis

In this chapter the results of the research are displayed and analyzed.

To obtain an answer to the main question of this thesis, do rationalists perform better then trend chasers, the computer program EF Chaos is used. This is a software package for simulation of nonlinear dynamic models to investigate stability of steady states and the presence of periodic orbits and chaos by standard numerical simulation techniques. The techniques that are displayed in the remainder of this research are time series, phase plots and bifurcation diagrams. The package is developed by C. Diks, C. Hommes, V. Panchenko and R. van der Weide.

The asset pricing model, as described in chapter 2, is reviewed using different values for several parameters by the script noted in appendix A.

At first, the values of Brock and Hommes (1998) set are used to create dynamics. These values represen the values they used when they compared fundamentalists with trend chasers. The result is an overflow, which is most likely due to the fact that fundamentalists have been replaced by traders with perfect foresight.

The height of the parameter β is one of the reasons of the overflow, as a high β will cause the fraction of trend chasers to be too high. Therefore, the β needs to be set low. When the trial and error method is applied with a low value β however, there still seems to be an overflow.

For that reason the interest is lowered, the argument therefore is the bifurcation diagram seen in figure 12b. By using the method of trial and error again, the maximal interest rate was set at r < 1.082 and 0.1 < β < 0.5.

The two belief types asset pricing model is tested in this research on three types of interest rates: low, medium and high. The high interest rate will be 8.2%, the medium interest rate 4.0% and the low interest rate is determined to be 0.5%. For each of those interest rates, several parameter values will be adjusted. The values alternated are the coefficient of the pure trend chaser, b2 and the memory strength w.

Next, a specific section will pay attention to the costs involved in being a trader with a perfect foresight. Both zero and negative costs are further explored for a low, medium and high interest rate.

Every section contains a time series diagram where the movements of the price of an asset, as well as the fitness measure of both the rationalist and the pure trend chaser are displayed against time. The first is presented by a blue line, the second by a pink line and the third by a purple line.

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3.1 High interest rate

Figure 1: Time series, r = 8.2%

In this section the dynamics resulting from the two believe types asset pricing model with a high interest rate 8.2%, are discussed. The coefficient of the trend chasers and the memory strength are adapted, but the other parameters set at the standard values as seen in figure 12b. The costs at present time and one, two and three periods ago are among these parameters. Others are the fractions of both traders with perfect foresight as trend chasers and trends.

Figure 1 shows the model with no adaptions of parameters. When the amount of perfect foresight traders grows, the price of an asset elevates a little. When the fitness measure of traders with perfect foresight decreases, the price of the asset decreases as well. It seems that the growth of the fitness measure of perfect foresight traders stimulates either the positive and negative growth of the asset price. The fitness measure of the pure trend chasers always acts in the exact opposite way of the perfect foresight traders’ fitness measure.

3.1.1 Coefficient trend chasers

The value of the coefficient of the trend chasers is set at zero, but when adapted it is changed to a negative value, −0.5, and a positive value, 0.5. A negative coefficient stipulates when a change in trend chasers is noticed, that it has an opposite influence on the price. A positive coefficient however, points out when that change is noticed, the asset price changes in the same directions. So when the coefficient of trend chasers increases, the asset price will do so as well.

With the interest rate r = 8.2%, changes in the coefficient of trend chasers result in an overflow. It might be interesting for next studies to discover why this overflow arises.

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3.1.2 Memory strength

Figure 2: Time series, r = 8.2%, w = 0.5

The memory strength is set at zero, but when adapted it is changed to the positive value 0.5. As mentioned earlier in this thesis, when the memory strength is going to infinity it will push the trend chasers out of the market. That is, if the traders with perfect foresight would have no costs. In the standard settings however, they do have costs. Nevertheless, the expectation for a higher memory strength still is to be a good feature for the traders with perfect foresight.

In figure 2 it is described that when the fitness measure of the rationalists decreases, the price of an asset rises. However, this is not the case in every movement downwards. Therefore, it is difficult to find a pattern in this figure. Further studies might be able to explore this more.

3.2 Medium interest rate

Now, the interest rate is adapted to 4.0%, the other parameters are set at the standard values as shown in figure 12a. Figure 3 shows when the fitness measure of the rationalist increases, the price of an asset will either go down or up more further. In other words, the fitness measure strengthens the movement of the price whatever way it is going. However, the movement of the asset price is delayed in comparison to the fitness measure of the perfect foresight trader.

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(a) b = −0.5 (b) b = 0.5

Figure 4: Time series, r = 4.0%, adapted coefficient of trend chasers

In figure 4a it is shown that when the fitness measure of the perfect foresight traders rises, the asset price goes down. This coincides with the expectations that when the coefficient of trend chasers is negative the costs will behave in the opposite way. As a result of the fitness measure of trend chasers acting the exact opposite way of the fitness measure of perfect foresight traders.

In figure 4b the opposite of figure 4b is displayed. This again corresponds with the expectations of a positive coefficient of pure trend chasers. The asset price moves along with the movement of the fitness measure of rationalists.

Figure 5: Time series, r = 4.0%, w = 0.5

The movements in figure 5 again indicate that when the fitness measure of the perfect foresight trader rises, the asset price movement will be enlarged. Most likely due to the presence of memory strength, the timing of the empowering done by the movement of the rationalists’ fitness measure is more accurate compared to zero memory strength. Namely, the difference in timing of the fitness

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measure movement and the asset price movement is less. Further, decreasing the asset price is less stimulated in comparison to increasing of the asset price.

3.3 Low interest rate

In this section the interest rate is set low, at a rate of 0.5%.

Figure 6 demonstrates the same type of movements to occur as earlier seen with a medium interest rate. This is, when all the other parameters are set at standard values. However, the difference between the low and medium interest is shown by the maximum and minimum values of the asset price. The values of the asset price at a small interest rate, 2.7 and −2.7, seem to be closer to zero then when a medium interest rate is set, 3.8 and −3.8. It implies that the boost given to the asset price movements by a positive growth of the fitness measure of a rationalist is smaller when the interest rate is set at a lower rate.

(a) b = −0.5 (b) b = 0.5

Figure 7: Time series, r = 0.5%, adapted coefficient of trend chasers

In figure 7a the same dynamics as earlier mentioned exist. A rise in the fitness measure of the perfect foresight trader causes an enlarged growth of the asset price. However, only the positive growth of

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the asset price is strengthened by an increase in the rationalists’ fitness measure. Thus, the negative coefficient of the pure trend chaser causes the enlargement of negative growth of the asset price to cease.

Differently, figure 7b lacks the presence of the asset price movement. The absence of the latter might be appealing to research in further studies.

As seen in several earlier figures, figure 8 appears that a high fitness measure of traders with a perfect foresight enforces the positive and negative growth of the asset price. In contradiction to the earlier figures however, the movement of the asset price follows the movement of the fitness measure intensely. Again, the decreasing of the asset price is less triggered then the increasing of the asset price.

3.4 Costs

In this section the costs that go hand in hand with being a trader with perfect foresight are further discussed. At first sight, the main question seems to be straightforward. When one trader has all the information available and another trader does not, it seems obvious that the first trader defeats the second when it involves the agent market.

However, to obtain all the information sacrifices need to be made. One of these are the costs a trader with perfect foresight needs to pay to gain the vacant information. For that reason the parameter costs in the two type asset pricing model is adjusted. It is adjusted for zero costs, C = 0 as well as high costs, C = 5.

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3.4.1 High interest rate

(a) C = 0 (b) C = 5

Figure 9: Time series, t = 8.2%, adapted costs

Figure 9a presents an image that seems almost identical to figure 1. This stipulates that with a high interest rate, there is no difference in the impact that fitness measures and the asset price have on each other between zero costs and low costs.

Figure 9b however, demonstrates dynamics that make it challenging to find a pattern. This might be an interesting case to explore in further studies.

3.4.2 Medium interest rate

(a) C = 0 (b) C = 5

Figure 10a shows the same system as mentioned earlier several times. The high fitness measure of traders with perfect foresight empowers the positive and negative growth of the asset price. Yet, it appears that a small rise of that fitness measure causes a relatively big movement of the asset price. The strength of the impact caused by the fitness measure seems tremendous. This can be explained by the fact that no costs are involved.

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Figure 10b on the other hand, shows that a big change in the fitness measure only leads to a moderate change in the asset price. Thus, the impact of the fitness measure appears to be less powerful. When compared to zero costs, this implies the strength of the impact of the fitness measure on the asset price movement to be larger when costs are lower.

3.4.3 Low interest rate

(a) C = 0 (b) C = 5

Here, the interest is set at the low rate 0.5%. Figure 11a displays the dynamics appearing when the costs are zero for a trader with perfect foresight. As seen twice beforehand, the asset price line does not show. Therefore, this also might be something to investigate in further studies.

Figure 11b demonstrates that to achieve alike values of the asset price as with medium interest rates, an even higher fitness measure of the trader with perfect foresight is needed. This implies the lower the interest rate, the bigger the impact of high costs on the relation between fitness measures and the asset price.

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4 Conclusion

The paradigm in the agent market shifted from the homogeneous market to the heterogeneous market. As of that moment, more questions arised concerning the trader types in the agent market. Among others, this was caused by the high speed evolvements of technology and markets, as this improved the way research could be done.

This paper focuses on the asset pricing model developed by Brock and Hommes (1998) and the more recent version of the model as described in studies by Hommes (2006). They compared funda-mentalists with pure trend chasers. In this thesis the fundafunda-mentalists were replaced by traders with perfect foresight. Therefore, the main question stated; do traders with perfect foresight eventually drive pure trend chasers out of the agent market?

This question was looked at from different angles. The interest rates in the asset pricing model, as well as the coefficient of the pure trend chaser, memory strength and costs have been researched. The question on the answer remains partly unanswered. As the fitness measure of the perfect foresight agent rises, the growth of the asset price enlarges. This growth however enlarges both positive as negative growth in asset price.

High costs seem to reduce the impact of the high fitness measure of perfect rationalist agents on the negative and positive growth of the asset price. This conforms with the expectation of high costs being a disadvantage of an agent with perfect foresight.

On the other hand, there are several situations put on display where a high fitness measure of perfect foresight agents mostly has effect on the positive growth, which will eventually drive pure trend chasers out of the market. This occurs when the interest rate is set medium or low and the coefficient of pure trend chasers is negative or memory strength is present.

Several dynamics seem interesting to investigate in further studies. This includes the two belief types asset pricing model with a relatively high interest rate. The thesis does not deliver concrete conclusions about the dynamics that follow from the interest rate 8.2%. Furthermore, it might be appealing to do research on the case where perfect foresight traders have zero costs.

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A

Script

Initial values:

xt=0.1, xtmin1=0.1, xtmin2=0.2, xtmin3=0.1 n1=.2 u1=0, u2=0 Parameters: beta=0.1-0.5, R changes b1=0, b2=0 g1=0, g2=1.2 C=1, w=0

Auxil. plot vars: n2 Start script u1=w*u1+(xtmin1-R*xtmin2)*(xtmin1-R*xtmin2)-C u2=w*u2+(xtmin1-R*xtmin2)*(g2*xtmin3+b2-R*xtmin2) ztmin1=math.exp(beta*u1) ztmin1=ztmin1+math.exp(beta*u2) n1=math.exp(beta*u1)/ztmin1 n2=1-n1 xtplus1=(R*xt-n2*(g2*xtmin1+b2))/n1 xtmin3=xtmin2 xtmin2=xtmin1 xtmin1=xt xt=xtplus1 End script

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B

Trials

B.1 Brock and Hommes values

(a) Parameter values

(b) Bifurcation Diagram

Figure 12: Standard settings

B.2 High interest - R=1.082

Figure 13: Phase plots, r = 8.2%

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Figure 14: Time series, r = 8.2%, w=0.5

B.3 Medium interest - R=1.04

(a) Phase plot (b) Bifurcation diagram

Figure 15: r = 4.0%

Change in b b=-0.5

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(a) Phase Plot (b) Bifurcation Diagram

Figure 16: r = 4.0 and b = −0.5

b=0.5

Figure 17: r = 4.0 and b = 0.5

Change in w w = 0.5

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B.4 Small interest - R=1.005

Figure 19: r = 0.5

Change in b b=-0.5

Figure 20: r = 0.5 and b = −0.5

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Figure 21: r = 0.5 and b = 0.5

Change in w w = 0.5

Figure 22: r = 0.5 and w = 0.5

B.5 Costs

Negative costs R = 1.082

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Figure 23: Phase Plot, r = 8.2 and c = −2

R = 1.04

Figure 24: r = 4.0 and c = −2

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Figure 25: r = 0.5 and c = −2

Zero costs

R = 1.082

Figure 26: r = 8.2 and c = 0

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Figure 27: r = 4.0 and c = 0

R = 1.005

Figure 28: r = 0.5 and c = 0

High costs

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Figure 29: Phase Plot, r = 8.2 and c = 5

R = 1.04

Figure 30: Phase Plot, r = 4.0 and c = 5

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Rational versus non-rational agents in a heterogeneous agent model : two type agent model