Faculty Economics and Business, Amsterdam School of Economics
The interaction between different
forecasting rules
Studying the interactions between Fundamentalists, Contrarians and Trend followers
Koen Kempff - 10750088
Supervisor: dr. D´avid Kop´anyi University of Amsterdam
Econometrics and Operational Research June - 2018
Contents
1 Introduction 1
2 Theoretical framework 3
2.1 Set-up of the model . . . 3
2.2 Comparison to Brock and Hommes . . . 7
2.3 Dynamical system theories . . . 8
3 Results 9 3.1 Examples of different dynamics . . . 10
3.2 The effect of α . . . 14
3.3 The effect of β . . . 18
3.4 The effect of g . . . 20
4 Conclusion 21
Statement of Originality
This document is written by Student Koen Kempff 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.
1
Introduction
Whenever an economic crisis arises, an economic model is tested. In November 2010, the Governor of the European Central Bank at that time, Jean-Claude Trichet, made a statement about available economic models and tools in time of economic crisis. In this statement he called the models of little help and felt abandoned by conventional tools. As a response to this, Farmer, et al. (2012) proposed a program called FuturICT which looks at the economy in a different way than most eco-nomic models used to do at the time. The tools and approaches used for this program focused more on the crisis periods than the calm periods, therefore responding directly to Jean-Claude Trichet’s statement.
One of the assumptions used in most of the models is that agents on a market are perfectly rational, meaning that they have all the information available and make the best possible decisions to maximize their utility. Even though this assumption is used in a substantial amount of models, most of the time it does not concur with reality. ”The comparison of the I.Q. of a computer with that of a human being is very difficult. If one were to factor the scores made by each on a comprehensive intelligence test, one would undoubtedly find that in those factors, on which the one scored as a genius, the other would appear a moron – and conversely” (Simon H.A., 1955, p.114). Simon argues that, instead of using perfectly rational agents, one can use agents of limited knowledge and ability which could correspond better with reality. Using agents of limited knowledge and ability is also known as bounded rationality. Brock and Hommes (1998) use bounded rationality and give agents a finite set of choices to predict future prices. In the financial market described by Brock and Hommes, agents can have different decision rules in mind and they tend
to choose the rules that have performed better in the past. So instead of having perfect rationality, agents are bound to one of the following three choices. First, they can choose to be a Fundamentalist. Funda-mentalists believe that the price is purely determined by its fundamental value which is given by the present discounted value of the future divi-dends. Secondly, one can be a Trend follower. A Trend follower does as it is called and solely follows the trend of the price: when it has risen last time period they will believe it shall rise again and the other way around. The third and final choice is to be a Contrarian. A Contrarian will do the exact opposite of a Trend follower: whenever a price has risen last time period, a Contrarian believes the price will fall and vice versa. In their paper they compare these choices and their influences on the price, after which the results could be used for policy recommendations. Yet to be used for policy recommendations, a model has to be robust and its predictions should not change substantially when some assumptions are slightly modified.
In this paper, I modify the set-up of the original model and study the dynamical properties of the new model. Both models are compared to see whether they lead to the same kind of dynamics. The focus of this paper is to check to which extent the results of the original model are robust, in specific when comparing the interactions between Fundamentalists and Contrarians and Fundamentalists and Trend followers. In their paper, Brock and Hommes (1998) use a number of specifications, some of which have been modified in this paper. Instead of using either today’s or tomorrow’s price as done by Brock and Hommes (1998), both prices are used. Another slight modification lies in the way an agent can invest its wealth. Instead of choosing an amount of shares to buy, an agent can invest a certain amount of his wealth in my model. In this paper I
find that the paramaters α, β and g all have a significant effect on the stability of the system. For some interval of these parameters the system dynamics will be stable but as these parameters become more extreme they tend to destabilize the system. The results that are obtained in this paper support the robustness of the model by Brock and Hommes.
The set-up of this paper is as follows: Section 2 presents the theoret-ical framework and discusses the set-up of the model, the differences to the model by Brock and Hommes and looks at dynamical system theo-ries. Section 3 shows some examples of different system dynamics and analyzes the effect of α, β and g on the system dynamics. Finally, Sec-tion 4 concludes and gives a recommendaSec-tion on the use of the original model.
2
Theoretical framework
In this paper I have modified the asset pricing model used by Brock and Hommes (1998). Therefore, in this section, the new modified model will be discussed and compared to the model of Brock and Hommes (1998). Furthermore, theories of dynamical systems are discussed.
2.1
Set-up of the model
An asset pricing model with two assets is used: a risky asset and a risk free asset. When looking at the first step of the set-up of the model by Brock and Hommes (1998), it can be seen that the formula for wealth (1) depends on the amount of shares one buys of the risky asset. When speaking from a logical point of view, this might not be the best choice. It might be more natural to decide what amount of your own wealth to invest in this risky asset. The formula for wealth given by Brock and
Hommes is as follows:
Wt+1= (1 + r)Wt+ (pt+1+ yt+1− (1 − r)pt)zt. (1)
Whereas in the adjusted model used in this paper the formula for wealth will be given in the following way:
Wt+1= (1 + r)(Wt− Wa) +
Wa
pt
(pt+1+ yt+1), (2)
where Wa is the amount of wealth invested in the risky asset, with
Wa ∈ [0, W
t]. By dividing this by the price of the risky asset in time
t, pt, one can essentially get a different kind of look at how to invest
their money. The dividend the risky asset pays is denoted by yt and the
risk-free asset pays an interest of r%.
In this model it is also assumed that every investor type has a mean-variance preference. This means that all of them choose Wa such that it
maximizes Et(Wt+1) −a2Vt(Wt+1). Filling in the formula for Wt+1 in this
function, gives the following objective function:
(1 + r)(Wt− Wa) + Wa pe t (pet+1+ y) −a 2(W a)2V t pt+1+ yt+1 pt , (3) where pet and pet+1 are the expected prices and y the expected value of yt.
Now look at the first order condition while assuming that Vt(pt+1p+yt t+1) =
σ2: −(1 + r) + p e t+1+ y pe t − aWaσ2 = . (4)
Solving this first order condition for Wa gives the optimal investment of
Wta= p e t+1+ y − (1 + r)pet aσ2pe t . (5)
We now need a way to derive the price at time t, pt of the risky asset.
The price of the risky asset is determined by market clearing, meaning that the total demand equals the total supply. In this market there are
I investors, leading to the following expression: PI i=1 Wa i pt = z s. Here zs
is the constant supply of assets. Solving this for pt leads to:
pt = 1 zs I X i=1 Wia = I X i=1 pe i,t+1+ y − (1 + r)pei,t aσ2zspe i,t . (6)
In other words, the price pt equals the summation of the amount of
wealth each investor type wants to invest in the risky asset divided by the constant supply of assets.
In this paper I will look at the interactions between Fundamental-ists and Contrarians and the interactions between FundamentalFundamental-ists and Trend followers. That is why I can rewrite pt in a slightly different way
as follows: pt= nFt p∗+ y − (1 + r)p∗ aσ2zsp∗ +(1−nFt ) pe ct,t+1+ y − (1 + r)pect,t aσ2zspe ct,t (7) Here p∗ denotes the fundamental price which will be explained later and pect,tand pect,t+1 denote the expected price for Contrarians or Trend follow-ers in their respective time period. The variable nF
t denotes the fraction
of people that are Fundamentalists. Because the sum of all fractions is equal to 1, it means that 1 − nF
t denotes the fraction of people that are
Contrarians or Trend followers. The expected prices for Contrarians and Trend followers are obtained by following these rules:
pect,t = pt−1+ g · (pt−1− pt−2), (8)
pect,t+1 = pt−1+ α · g · (pt−1− pt−2). (9)
To determine whether I am dealing with Contrarians or Trend followers, all that has to be done is to look at the parameter g. Trend followers fol-low the direction of the last price change and therefore have g > 0, while Contrarians predict that the price will change in the opposite direction
which means that g < 0. Here α is a logically chosen parameter around 1. When α = 1 both Contrarians and Trend followers expect pt+1 to be
equal to pt. When α < 1 and the price change from pt−1 to pt is positive,
Trend followers will expect pt+1 to be lower than pt and when α > 1
they will expect pt+1 to be higher than pt and vice versa for a negative
price change. For Contrarians this is slightly different, when α < 1 they expect the change from pt to pt+1 to be in the opposite direction from
the change of pt−1 to pt and when α > 1 they expect the price to rise or
drop further depending on what it did before.
The next step will be to calculate the fundamental price p∗ that is used in the simulations. To do this I will look at the equation for the price (7) and fill this in as if everyone was a fundamentalist. Solving this for the fundamental price gives the following result:
p∗ = −r +pr
2+ 4 · θ · y
2 · θ , (10)
where θ = aσ2zs. Evidently the fundamental price only depends on mar-ket parameters but not on the different forecasting rules. Now that it is known how the prices for the two rules and fundamental price came to be, I can look at the next step which will be to determine the two respective amounts of wealth invested in the risky asset, WFa and WCT,ta .
WFa = p ∗+ y − (1 + r)p∗ aσ2p∗ (11) WCT,ta = p e ct,t+1 + y − (1 + r)pect,t aσ2pe ct,t (12) Now that Wa
F and WCT,ta are known, the price at time t can be calculated
by dividing both WFa and WCT,ta by zs and multiplying these by their respective fractions. Although this is the same as filling in equation (23), it will be useful to calculate WCTa as these will be saved and used in the
next step. With pt known, the next step will be to calculate the profits
Ut−1F and Ut−1CT as these are used to calculate the new fractions.
To calculate these profits at time t-1, the optimal choices for Wa at time
t-1 are used. This leads to the following expressions:
Ut−1F = WFa −(1 + r) + pt+ y pt−1 − C (13) Ut−1CT = WCT,t−1a −(1 + r) + pt+ y pt−1 , (14)
where C denotes the cost for rational expectations used by Fundamental-ists. The final step is calculating the fractions for the next period which is done in the same way as in Brock and Hommes. The formula for the updated fractions is given by:
nFt+1 = e
(βUF t−1)
e(βUt−1F )+ e(βUt−1CT)
(15) Calculating the fractions in this manner makes sure that they are always between 0 and 1.
With the newly acquired pt and nFt+1, the whole process can be repeated
for the next time period giving new prices and fractions.
2.2
Comparison to Brock and Hommes
The first difference lies with the formulas for wealth (1) and (2). As mentioned before, Brock and Hommes (1998) let the wealth depend on the amount of shares one buys of the risky asset, whereas the model I use looks at the amount of wealth one wishes to invest. Another difference is, instead of using either today’s or tomorrow’s price as done by Brock and Hommes, both prices are used. The amount of wealth to invest dif-fers per forecasting rule and eventually gives equations (11) and (12). With these two Wa the realized profits will be calculated and used as
a fitness measure. The fitness function used by Brock and Hommes is different and uses the difference in price from the fundamental price and the gross returns on the risk free asset. Therefore the wealth and profits are calculated differently. In their paper Brock and Hommes also use the parameter η to represent the memory strength of the model. In this paper the memory strength has not been taken into consideration. A fi-nal difference lies with the forecasting rules used by Brock and Hommes. They investigate market competition between some simple linear fore-casting rules with one lag. They do this by defining the deterministic function fht as follows:
fht= ghxt−1+ bh. (16)
Here gh represents the trend and bh the bias for each trader type ’h’.
Depending on the values of gh and bh, fht defines the following belief
types. When gh = 0 and bh = 0, equation (16) reduces to zero and a
trader is called a Fundamentalist who believes that the prices return to their fundamental value p∗. There are two different cases when gh 6= 0
. When g > 0, an invest of type ’h’ is a Trend Follower and if g < 0, an investor of type ’h’ is a Contrarian. The difference with this paper is that Brock and Hommes make use of the bias bh and do not set this at
0. They also investigate purely biased traders versus Fundamentalists.
2.3
Dynamical system theories
To analyze the model and its stability, methods from dynamical sys-tem theories will be used. One common method used for analyzing a dynamical system that is nonlinear starts off by looking at a stability analysis of the steady states. It is important to determine how a steady state changes and becomes unstable when a parameter of the model is
changed, as this can reveal possible bifurcations. A bifurcation is a qual-itative change in the structure or dynamics of a system with regards to, for example, the presence of a steady state or its stability. The type of bifurcation depends on the eigenvalues (λ) of the Jacobian matrix that is found through the steady states. In particular, in the way these eigen-values cross the unit circle when a parameter in the model is changed slightly. If we look at the eigenvalues, a couple of possibilities occur. First, when the value of λ is between -1 and 1 a stable steady state ex-ists. Secondly, when λ = 1, a saddle node bifurcation will occur. In this case one steady loses its stability and two new steady states are created: a saddle and a node. Thirdly, when λ = −1, a period doubling bifur-cation will occur and a 2-cycle is created. Fourthly and finally, when λ is complex a Hopf bifurcation can occur. In this case an invariant circle with periodic or quasi-periodic dynamics is created. Further details and theoretical background of dynamical systems can be found in Kuznetsov (2004) and Hommes (2013).
3
Results
One of the most important things that needs to be done for this model to work is to choose valid and logical values for all parameters. By choosing reasonable values for the parameters θ and ’r’, a reasonable fundamental price will be obtained after which two logical starting prices can be chosen. These two starting prices are necessary, as the rules for Contrarians and Trend followers use the prices of one and two periods lagged.
In the following results some of these parameters are kept constant, whereas others are changed. The interest rate is kept at 1% meaning
that r = 0.01. The dividend of the risky asset y and costs C are also not changed and kept at 0.05 and 0.3 respectively. The final parameters that are not changed are a, σ2 and zs, which are kept at 0.5, 0.003 and
20 respectively. Using these values gives us a fundamental price, p∗, of 1.1350. In all of the results the starting prices will be the same. p1 = 1.20
and p2 = 1.15 and the model will be simulated for 500 periods. Because
all the parameters used for calculating Wa
F are constant over time, WFa
will also be constant over time.
In the simulations I will investigate how α, β and g affect the dynamics of the model. For more convenient results the parameter mtis introduced
which denotes the difference in fractions such that:
mt= nFt − (1 − nFt) = 2nFt − 1. (17)
To get a clear view of the results obtained, the effects of α, β and g will be split into different sections.
3.1
Examples of different dynamics
When varying α, β and g, an infinite amount of results can be obtained as these three can be varied infinitely. To demonstrate which situations can occur some example figures are given and analyzed.
0 5 10 15 20 25 30 35 40 45 50 Time 1.1 1.12 1.14 1.16 1.18 1.2 Price 0 5 10 15 20 25 30 35 40 45 50 Time -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 m
Figure 1: Time series of price (upper panel) and difference in fractions (lower panel)
In figure 1 two different graphs can be seen. The first graph plots price against time and the second graph plots the difference in fractions m against time. In this example the values for the varying parameters are as follows: β = 2, g = −0.8 and α = 1. One can see that when these val-ues are used, the price converges to a steady state very quickly. Within the first 50 time periods the price is steady at the fundamental price which is still 1.1350. Something to be noted is that whenever the system reaches a steady state the corresponding price will be equal to the fun-damental price.
450 455 460 465 470 475 480 485 490 495 500 Time 1.05 1.1 1.15 1.2 1.25 Price 450 455 460 465 470 475 480 485 490 495 500 Time -0.1 0 0.1 0.2 0.3 0.4 m
Figure 2: Time series of price (upper panel) and difference in fractions (lower panel)
In figure 2 the first two graphs are the same kind of graphs that are used in figure 1. In this example the varying parameters are: β = 2, g = −0.8 and α = 1.07. Unlike the first example, the price and fractions do not converge to a steady state. Instead what we have is a 2-cycle. Only the last 50 time periods are shown in the graphs to give a better view of the 2-cycle. This way you can easily see that the price switches only between two values and so do the fractions. If a phase plot would be made to illustrate a 2-cycle, it would only show us two dots as for every single time period there are only two possible combinations of price and difference in fractions.
300 320 340 360 380 400 420 440 460 480 500 Time 1.05 1.1 1.15 1.2 1.25 Price 300 320 340 360 380 400 420 440 460 480 500 Time -0.4 -0.2 0 0.2 m 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 Price(t) -0.4 -0.2 0 0.2 m(t)
Figure 3: Time series of price (upper panel), difference in fractions (middle panel) and phase plot (lower panel)
In the example of figure 3 the varying parameters are: β = 2, g = −0.8 and α = 0.97. From these graphs it can be deducted that there is no steady state nor is there a 2-cycle. Instead the system has quasi-periodic dynamics. The prices and fractions will periodically follow a certain pattern which can be seen in the phase plot.
300 320 340 360 380 400 420 440 460 480 500 Time 0.5 1 1.5 Price 300 320 340 360 380 400 420 440 460 480 500 Time -1 0 1 m 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Price(t) -1 0 1 m(t)
Figure 4: Time series of price (upper panel), difference in fractions (middle panel) and phase plot (lower panel)
Figure 4 shows an example of chaotic dynamics. In this condition the system has no stable state nor does it cycle between a number of points such as a 2-cycle would. The phase plot shows that unlike the quasi-periodic dynamics, the system does not quasi-periodically follow a pattern and that the prices tend to differ from each other.
3.2
The effect of α
Figures 1 until 4 were all examples of what conditions the system could be in, although these are not the only options. The system could also be
0.9 0.95 1 1.05 1.1 1.15 1.2 0.6 0.8 1 1.2 1.4 1.6 1.8 Price
Figure 5: Bifurcation diagram with respect to α
in a 3- or 4-cycle or even more. To get an overview of when this occurs and how often it occurs, a bifurcation diagram can be computed. We do this by keeping two out of the three varying parameters constant, while only changing the third and saving only the last one hundred values of the price. This will give us three different bifurcation diagrams: one for α, one for β and one for g. In this subsection I will look at the effect of α on the dynamics of the model. A clear distinction will be made between Contrarians and Trend followers, meaning that a positive and negative value for g will be used.
Figure 5 shows the bifurcation diagram where α ∈ [0.9, 1.2] and β and g are kept constant at 2 and -0.8 respectively. This bifurcation di-agrams represents the dynamics for Contrarians as g < 0. It appears that varying α has a significant effect on the dynamics of the system.
0.9 0.95 1 1.05 1.1 1.15 1.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 Price
Figure 6: Bifurcation diagram with respect to α
When α ≈ 0.9 the system appears to be in a somewhat chaotic state, after which it turns into a 6-cycle at α ≈ 0.925 followed by a 4-cycle at α ≈ 0.93. Then, from around 0.95 until 0.98, the system is quasi-periodic, after which it converges to the steady state with a price equal to that of the Fundamentalists. The steady state remains until α ≈ 1.05, after which a period doubling bifurcation occurs and the system is turned into a 2-cycle. After the pitchfork bifurcation a Hopf bifurcation takes place, leading to quasi-periodic and chaotic dynamics.
Figure 6 shows the bifurcation diagram for Trend followers where: g = 0.8, β = 2 and α is varied between 0.9 and 1.2. It appears that α has a significant effect on the dynamics of the system for Trend followers as well. The system starts out in a chaotic state after which the system gets quasi-periodic. When α is slightly larger then 1 a Hopf bifurcation leads
the system to a 2-cycle. Then as α approaches 1.05, the system gets into a steady state in which it stays until α gets larger than 1.1, after which again a Hopf bifurcation occurs, making the system quasi-periodic again. Finally the system turns into a 4-cycle, after which it gets into a chaotic state again.
When looking at the bifurcations diagram for Contrarians (Figure 5) it seems that when α is close to 1, meaning that Contrarians expect pt+1
to be almost equal to pt, the dynamics are stable. But as α becomes more
extreme, the system is destabilized. As for the bifurcation diagram for Trend followers (Figure 6), it seems that when α ∈ [1.05, 1.1], meaning that Trend followers expect pt+1 to be almost equal to pt, the dynamics
are stable, but when α diverges from this interval the system is destabi-lized. An interesting thing to note is that the bifurcation diagrams for Contrarians and Trend followers look as if they were mirrored. The same kind of bifurcations take place but in reversed order.
5 6 7 8 9 10 11 12 13 14 15 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 Price
Figure 7: Bifurcation diagram with respect to β
3.3
The effect of β
In figure 7 the bifurcation diagram for Contrarians is shown where: α = 1, g = 0.8 and β is varied between 5 and 15. It can be seen that for values of β up until around 9, the system is in a steady state. When β > 9, a Hopf bifurcation occurs and the system shows periodic and quasi-periodic dynamics. When these results are compared to those obtained by Brock and Hommes in 1998, it can be seen that even though the dynamics shown are not identical, both show a similar kind of periodic and quasi-periodic dynamics.
1 1.5 2 2.5 3 3.5 4 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 Price
Figure 8: Bifurcation diagram with respect to β
Figure 8 shows the bifurcation diagram for Trend followers where: α = 1, g = 0.8 and β is varied between 1 and 4. When β = 1, the system is a 2-cycle after which it has periodic, quasi-periodic and even somewhat chaotic dynamics for β around 1.5 till around 2.6. When β is around 2.6, the system again has cyclic dynamics and some period doubling bifurcations occur. At β = 2.75, the system is even in an 8-cycle. When β becomes larger than 3 the system mostly shows chaotic dynamics but sometimes shows periodic dynamics again.
When looking at the bifurcations diagram for Contrarians (Figure 7) it shows that when β ≤ 9, meaning that Contrarians expect pt+1 to
be almost equal to pt, the dynamics of the system are stable. But as
β becomes bigger than 9, the dynamics are periodic and quasi-periodic. As for the bifurcation diagram for Trend followers (Figure 8), it seems that there is a 2-cycle when β is around 1, but when β becomes more
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 g 0.5 1 1.5 2 2.5 3 3.5 Price
Figure 9: Bifurcation diagram with respect to g
extreme the system is destabilized. When comparing Figure 8 with the bifurcation diagram with respect to β for Trend followers by Brock and Hommes (1998), it can be seen that the two are very different. Yet in both diagrams Hopf bifurcations occur and (quasi-)periodic dynamics are present. Figure 8 also does not show any bifurcations or dynamics that were not mentioned or expected by Brock and Hommes. Another thing to note is that β has a more destabilizing effect for Trend followers than it has for Contrarians.
3.4
The effect of g
Figure 9 shows the bifurcation diagram with respect to g where: α = 1 and β = 2. The right side of the graph with g > 0 represents Trend followers whereas when g < 0 Contrarians are represented. It can be seen
that whenever g gets close to zero the system is in a steady state. On the side for Contrarians, meaning g < 0, the system stays in a steady state for a much bigger interval of g than for Trend followers. For Contrarians the system loses its steady state whenever g approaches -1.2. As for Trend followers, the system almost immediately loses its steady state when a period doubling bifurcation occurs at g ≈ 0.1 and the system turns into a 2-cycle. Afterwards Hopf bifurcations occur and the system has quasi-periodic dynamics followed by chaotic dynamics. When g < −1, a Hopf bifurcation occurs and the system has quasi-periodic and cyclic dynamics. Something to note is that the bifurcation diagram with respect to g is very similar to the bifurcation diagram with respect to α for Contrarians. The same kind of bifurcations take place in the same order. This also means that the same kind of bifurcations take place as in the bifurcation diagram with respect to α for Trend followers but in reverse order. This is not a strange occurrence as both parameters α and g affect only the forecasting rules for Contrarians and Trend followers and therefore it makes sense that both affect the system dynamics in a similar manner.
It appears that when g is close to zero, meaning that both Contrarians and Trend followers expect pt+1to be almost the same as pt, the dynamics
are stable. But as g becomes more extreme, the system is destabilized. Judging by figure 9, the system becomes destabilized sooner for Trend followers than it does for Contrarians.
4
Conclusion
The main purpose of this thesis was to determine to which extent the results of the original model by Brock and Hommes (1998) are robust. In this paper I modified the set-up of the original model and studied the
dynamical properties of the new model using different forecasting rules. By varying the three parameters α, β and g, five bifurcation diagrams were obtained. The two bifurcation diagrams with respect to α have shown that α has a significant effect on the dynamics and stability of the system for both Contrarians and Trend followers. When α is around 1 for Contrarians and 1.07 for Trend followers the dynamics are stable, but for more extreme values of α the system destabilizes. The bifurcation di-agram with respect to g shows similar results to that of α. Both contain Hopf and period doubling bifurcations and similar dynamics. Just like with α, more extreme values of g tend to destabilize the system. When varying β, the results obtained in this paper can be compared to those obtained by Brock and Hommes. Even though the bifurcation diagrams for Contrarians and Trend followers obtained by Brock and Hommes do not look very similar to those obtained in this paper, they do contain the kind of dynamics one would expect in such a system. The bifurcations with respect to β that have occurred in this paper are familiar and have occurred in different cases in the paper by Brock and Hommes. After modifying the model, the results obtained are still valid and logical, sup-porting the robustness of the model by Brock and Hommes. This gives rise to the suggestion that the model could still be used when circum-stances are different than what they are supposed to be.
References
Brock, W. A. and Hommes, C. H. (1997). A Rational Route to Random-ness. Econometrica, 65(5), 1059
Brock, W.A. and Hommes C.H. (1998). Heterogeneous beliefs and routes to chaos in a simple asset pricing model. Journal of Economic Dynamics and Control, 22(8-9), 1235-1274
Farmer, J.D. et al. (2012). A complex systems approach to construct-ing better models for managconstruct-ing financial markets and the economy. The European Physical Journal Special Topics, 214(1), 295-324.
Hommes, C., Kiseleva, T., Kuznetsov, Y., & Verbic, M. (2012). Is more memory in evolutionary selection (de)stabilizing? Macroeconomic Dy-namics, 16(3), 335-357
Hommes, C. (2013). Behavioral Rationality and Heterogeneous Expecta-tions in Complex Economic Systems. Cambridge University Press. Kuznetsov, Y.A. (2004). Elements of Applied Bifurcation Theory, third edition(Vol.112). Springer-Verslag New York Inc.
Simon, H. A. (1955). A Behavioral Model of Rational Choice. The Quar-terly Journal of Economics, 69(1), 99-118
