The heterogeneity of expectations among traders introduces an important non-linearity into the financial markets. In a series of papers, Brock and Hommes, propose to model economic and financial markets as adaptive belief systems. Asset price fluctuations in adaptive belief systems are characterized by phases of close-to-the-fundamental-price fluctuations, phases of optimism where most agents follow an upward price trend, and phases of pessimism with small or large market crashes. In this paper will be discussed the EMH benchmark and forecasting rules of fundamentals and trend extrapolators. Some illustrative examples are supplied.heterogeneity of expectations, adaptive belief systems, forecasting rules, fundamentals, trend extrapolators equations, limit cycles, asymptotical stability

Author Miloslav

English

Research Papers in Economics

  1 
Bifurcation Routes in Financial Markets
∗∗∗∗ / 
 
Miloslav Vosvrda 
Department of Econometrics 
Institute of Information Theory and Automation 
Academy of Sciences of the Czech Republic 
 
 
Abstract
 
The heterogeneity of expectations among traders introduces an important non-linearity into the financial 
markets. In a series of papers, Brock and Hommes, propose to model economic and financial markets as adaptive 
belief systems. Asset price fluctuations in adaptive belief systems are characterized by phases of close-to-the-
fundamental-price fluctuations, phases of optimism where most agents follow an upward price trend, and phases 
of pessimism with small or large market crashes. In this paper will be discussed the EMH benchmark and 
forecasting rules of fundamentals and trend extrapolators. An illustrative example is supplied. 
 
Keywords 
Heterogeneity of expectations, adaptive belief systems, forecasting rules, fundamentals, trend 
extrapolators equations, limit cycles, asymptotical stability 
 
JEL: G10, G12, G14  
 
1. Introduction 
As a macroeconomic model describing conditions for equilibrium between the commodity market and 
the security market it is possible to use, among others, an IS-LM model. In a traditional Keynesian 
model, the commodity market is described by the so-called IS model (Investment-Saving Model), i.e. 
by means of the functions of savings S and investment I. The savings S(Y,R) are assumed to be an 
increasing function of the product Y and the nominal interest rate R. Similarly, the investment I(Y,R) is 
again an increasing function of the product Y, but a decreasing function of the nominal interest rate R. 
Considering a static IS model, we are looking for an equilibrium point, say (Y
*,R
*), given by the 
following equation 
S(Y,R)=I(Y,R),  (1) 
i.e. for the equilibrium point we have S(Y
*,R
*)=I(Y
*,R
*). 
Similarly, the security market is described by the nominal supply of money M
s that must be equal 
to the nominal demand for money M
d. In a Keynesian model is assumed that the nominal money 
demand depends only on the product Y and the nominal interest rate R. The equilibrium in the money 
market is then given by 
M
s = M
d (Y,R).  (2) 
In dynamic settings, any dynamic economic system is possible to be considered as an expectation 
feedback system. A theory of expectation formation is therefore a crucial part of any economic model. 
The rational expectation hypothesis is the paradigm in the economic theory. The rational expectation 
hypothesis is very close related to the efficient market hypothesis. By the efficient market hypothesis 
there is no forecastable structure in asset returns because rational traders process information very 
quickly and this is reflected immediately in asset prices. Thus the value of a risky asset is completely 
determined by its fundamental price, equal to the present value of the expected future dividends. By 
this hypothesis all traders are rational and changes in asset prices are random. The hypothesis that all 
traders are rational is a little bit limiting. Hence a new alternative to the efficient market hypothesis, 
the heterogeneous market hypothesis (Hommes 2000), was introduced. Traders choose their trading 
strategy according to a measure expressing a coupling between market equilibrium dynamics and 
evolutionary updating of traders beliefs. For using formation of expectation in a security market we 
introduce the non-linear framework used by Chiarella (Chiarella(2000)) based on a portfolio 
adjustment imposed by non-linear asset demand functions. The security market model involving 
heterogeneous agents is used. We assume that on the security market exist two groups of agents: 
                                                 
∗ / The support by the Grant Agency of the Czech Republic under grants 402/00/0136, 402/01/0034, and 
402/01/0539 is gratefully acknowledged.   2 
tt t xn Fn C h =−
() ( )
22
,,, , t a n h ( () ()
2
t tt F tt tt C htt xEE c F c C h
δ
πππ π −∆ −∆ −∆ −∆  =⋅− − − + −  
() () () ()
22
,, , , exp ( ) exp ( )
,
tt t t F t t t t C h t t
tt
tt
tt tt
NF NCh E cF E cCh
NF NCh
nF nCh
NF NCh NF NCh
δπ π δπ π −∆ −∆ −∆ −∆    + = ⋅− + + ⋅− +      
==
++
11
,.
22
tt
tt
xx
nF nCh
−+
==
() ( )
*
,, 1.
FF Ft t t t t Ep p ππ πβ β π +∆ +∆ =− ∆ ⋅ ⋅ − + ∆ ⋅ ⋅ !
()
2 *
exp
F
t pp
s
π β
 −  ∆⋅ = −


, tt π +∆ ! ()
* . t pp −
() ,, 1
Ch Ch
tt
Ch t t Ch t t
pp
EE ππ πβ π β
−∆
+∆ −∆
−  = −∆⋅ ⋅ +∆⋅ ⋅  ∆ 
() 0,1
Ch π β ∆∈
a) fundamentalists  (F) who possess varying degrees of information about their economic background 
and using this in forming expectations, 
b)  chartists (Ch) who only use past price to form their expectations. 
We firstly verify whether the adaptively evolving expectations of both fundamentalists and 
chartists in the heterogeneous market can lead to global stability. Using the approach of Brock and 
Hommes (Brock&Hommes (1997)) and we introduce 
  (3) 
with 
   
(4) 
where 
   
 
(5) 
the quantity δ  is the intensity of choice variable measuring how quickly agents switch between 
fundamentalism and chartism, cF, cCh are costs incurred by each group in forming expectations. By 
some simple algebra we get  
   
(6) 
The formation of expectations for fundamentalists is rational in the sense of knowledge of equilibrium. 
It means that fundamentalists form their estimation  , tt π +∆ ! based on knowledge of the model. Thus 
  (7) 
Fundamentalists anticipate that the further prices deviates from their fundamental value, the more 
likely they are to revert toward their fundamental value and the more likely to revert towards 
equilibrium. Thus  
   
(8) 
where s is the rate at which the fundamentalists switch between  
 and     
The classical adaptive scheme for chartists of the following form  
   
(9) 
is assumed. The value of the 
Ch π β ∆  belongs to the interval (0,1), i.e., 
   
(10) 
We assume that δ  > 0, cF > cCh, i.e., the fundamentalists incur a greater relative costs in forming their 
expectations. 
In the next sections we present the complete macroeconomic model and the dynamics of this model 
in the section 2. A numerical example can be found in the section 3. 
 
2. Dynamics in the complete macroeconomic model 
Introducing a dynamic IS-LM model, we assume that the variables of the IS-LM model depend also 
on the current time t. Recall that in a neo-Keynesian approach it is assumed that the nominal interest 
R(t) depends also on the expected inflation π (t), hence R(t)=r(t)+π (t) where r(t) is the real interest 
rate. The dynamic IS model is then given by the following differential equation   3 
exp( ( ))
d
tt t mp f π =⋅
()/ ( ).
sd dp t dt m m β =⋅ −
01 2 (,) ( )
d
tt t t t t ml y rl l y l r π == + ⋅ − ⋅ +
exp( )
ss
tt t mm σξ −∆ =⋅ ⋅ ∆ ⋅
 
 
(11) 
or by introducing the so-called propensity to invest i(• ,• )=I(• ,• )/Y(• ) and the propensity to save 
s(• ,• )=S(• ,• )/Y(• ), also expressed by 
 
 
(12) 
where y(t)=ln(Y(t)) is called the output of the economy, and a positive constant α  signifies the speed 
of adjustment. 
Observe that for an equilibrium point Y(t) = Y
*, y(t) = y
*, r(t) = r
* we have I(Y
*,r
*) = S(Y
*,r
*) or i(y
*,r
*) 
= s(y
*,r
*) (see (1)). We assume that 
 
(13) 
where the parameters si, (i=0,1,2, with s1, s2>0) are known. It is convenient to assume that the 
propensity to invest i(y,r) is a product of 1/(r+1) and so-called logistic function (Sladký&all(1999)). 
Hence the propensity to invest is assumed to be given analytically as  
 
 
(14) 
where the parameters k, a>0 and b is an arbitrary real number. 
As for the security market, we let m
d denote the demand for money and m
s be the supply of money. 
The market dynamics is then described here by the money supply and money demand of Cagan's 
model of monetary dynamics. The money supply is supposed to vary randomly according to 
  (15) 
where ∆  is the length of the  time interval, σ  is the volatility of the money supply and ξ ~N( 0,1) is a 
random variable with a normal distribution, mt = ln( Mt), Mt is a nominal money at time t. The demand 
for money can be described either by the traditional Keynesian demand-for-money function in the 
following form 
  (16a) 
or by Cagan's model as the money demand of time t being in the following form 
  (16b) 
where  pt = ln Pt is an aggregate price level of time t,  π t =E( dpt / dt) and the function f is taken to be 
linear (Chiarella&Khamin(2000)) 
 
(17) 
The price dynamics of the security market is governed by price adjustment to excess money demand 
as follows 
  (18) 
The expected inflation π (t) is assumed to be adaptive as shown by the following differential equation 
 
(19) 
where γ   is a positive constant. Moreover, in a detailed analysis, as shown in Kodera (1996) or Sladký 
and all (1999), there exists a constant µ  such that 
 
(20) 
From (19) we can conclude that 
{}
()
( () , () ) ( () , () )
dY t
IYt rt SYt rt
dt
α =−
{}
()
( () , () ) ( () , () )
dy t
iyt rt syt rt
dt
α =−
01 2 (,) syr s sy sr =+ +
1
(,)
1 1 exp( )
k
iyr
rb a y
=⋅
++ ⋅ −
()/ ( ()/ () ) d t dt dp t dt t πγ π =⋅ −
()/ ()/ . dp t dt dy t dt µ =⋅
() . tt fc d ππ =−⋅  4 
 
(21) 
Inserting from (13), (14) into (12) and from (16a) into (22b), we can conclude that the dynamics of the 
complete model is given by the following system of the four differential equations: 
 
 
(22a) 
 
 
(22b) 
 
 
(22c) 
 
 
(22d) 
Now let us suppose that the system given by (22 a-d) is in equilibrium, i.e. there exist equilibrium 
points y
*, r
*, π
*, and p
* such that 
 
(23) 
To calculate the values y
*, r
*, π
*, and p
* the system of equations (22) take on the following form: 
 
 
(24a) 
 
(24b) 
 
(24c) 
 
 
 
(24d) 
A solution of this system is a base for the fundamentalist forming expectation. Adaptively on this 
part is associated the part of the chartists. On the following example is shown that such system is 
global stable. We have to compute the Jacobian of the dynamical system (22a-d) at its equilibrium 
point and eigenvalues of this Jacobian. The Jacobian of the system (22) in equilibrium point has the 
following form (25). 
 
 
 
 
(25) 
where  
()/ ( ()/ () ). d t dt dy t dt t πδ µ π =⋅ ⋅ −
[]
[]
[]
01 2
01 2 0
0
01 2
0
() 1
() ()
1 1 exp( ( ))
()
() () () e x p ( )
() 1
() () ()
1 1 exp( ( ))
()
t
s
tl
l
s
dy t k
ss y ts r t
dt r b ay t
dr t
ll y tlr t t m
dt
dt k
ss y ts r t t
dt r b ay t
dp t
m
dt
α
βπ σ ξ
π
δµ α π
β
−∆
=

=⋅ ⋅ − + + 
++ ⋅ − 

=⋅ + ⋅ −⋅ + − ⋅ ⋅ ∆ ⋅ 

 
=⋅ ⋅⋅ ⋅ − + + −   ++ ⋅ −  
=⋅ ⋅
∑
0
exp( ) exp( ( ) )
t
tl t t t
l
pf σξ π σ ξ −∆
=

⋅∆ ⋅ − ⋅ −⋅∆ 
 ∑
** * * lim ( ) , lim ( ) , lim ( ) , lim ( )
tt tt
yt y rt r t pt p ππ
→∞ →∞ →∞ →∞
=== =
**
01 2 **
** * * *
01 2
** *
01 2 **
0
* 0
*
1
1 1 exp( ))
exp( )
1
1 1 exp( )
exp( )
exp( ( ) )
t
t
s
l
l
k
ss ys r
rb a y
ll yl r p c d
k
ss ys r
rb a y
m
p
f
ππ σ ξ
µα π
σξ
πσ ξ
=
 ⋅= + +  ++ ⋅ −
 ++ += ⋅ − ⋅ − ⋅ ∆ 

 ⋅⋅ ⋅ − + + =   ++ ⋅ − 
⋅⋅ ∆ ⋅
=
−⋅∆
∑
()
12 2
12
1 exp( ( )) 1
00
1 1 1 exp( ( )) 1 exp( ( ))
10
00 1
00 0
kb ayt k
ss
rr b a y t ba y t
ll
X
h konst σ
⋅⋅ −⋅
⋅− ⋅ +
++ + ⋅ − ⋅ +⋅ −⋅
−− =
−
⋅⋅
A  5 
432
123 4 0 ccc c λλλλ −⋅ +⋅ −⋅+ =
4
1
1
ii
i
ca
=
= ∑
3 c
()
()
2
22
22
2
2
22
22
2
11
tanh 9.8 9.8
44 2
19.6
11
tanh 9.8 9.8
44 2
19.6
ss
ss
mm
XF C c F c C
ss
CF u C
s
mm
FC c F c C
ss
FC u F
s
δ
ππ
δ
ππ
δ
ππ
δ
ππ
      =− + − − − + − ⋅            
⋅
⋅− ⋅ +
      +− − − − +− ⋅            
⋅
⋅+ ⋅
 
 
 
 
 
 
(26) 
 
A stable or unstable behavior over time depends exclusively on roots of the characteristic equation. In 
our case the characteristic equation has the following form: 
  (27) 
where  
 
elements of the matrix A 
sum of all second order principal minors of the 
matrix A 
 
sum of all principal minors of the third order of the 
matrix A 
Routh-Hurwitz method (Gandolfo (1997), p.221) gives necessary and sufficient stability conditions for 
this system, i.e., eigenvalues of this characteristic equation have to possess negative real parts. The 
characteristic equation possesses negative real parts if and only if the following expressions are held. 
 
 
 
 
Numerical results are introduced in the following section. 
 
3. Numerical example 
Let in the expressions (22a), (22b), (22c), (22d), α  = 0.20, ββ  = γ  = 0.05, µ  = 1-2 and, k=0.4 in (22a) 
and constant m in (22b), all the remaining parameters are given as follows: 
a = 0.5, b = 1, c = 0.4, cC = 0, cF  = 0.5, d = 1, h = 0.008, l0 = 0.92, l1 = 0.65, l2 = 7.99, δ   = 0.5, β  = 
2.15, β C = 30, σ  = 0.01, s = 0.99, s0  =-0.16, s1 = 0.1, s2 = 0.016, π C = 0.9, π F = 1.1.  
Then the Jacobian matrix has the following form at equilibrium point (y
*=2.04, r
*=0.093, π
*=1.55, 
and p
*=2.48): 
 
 
 
 
(28) 
A determinant of the matrix A is equal 0.00018. By Routh-Hurwitz condition, the values of the ∆ 1, ∆ 2, 
∆ 3, ∆ 4 are positive as is demonstrated on the following row 
()
2
1 0.4 exp( 0.5 2.04) 1 0.4
0.1 0.016 0 0
1.093 1.093 1 exp( 0.5 2.04) 1 exp( 0.5 2.04)
0.65 7.99 1 0
0 0 1 0.0019
00 0 0 . 0 1 0 . 0 0 8 konst
⋅− ⋅
⋅− ⋅ +
+− ⋅ +− ⋅
−− =
−
⋅⋅
A
2 c
11 0 c ∆= − > 13
2
2
0
1
cc
c
−
∆= >
13
32 4
13
0
10
0
cc
cc
cc
−−
∆= >
−−
13
24
44 3
13
4
00
10
0
00
000
cc
cc
c
cc
c
−−
∆= = ⋅ ∆>
−−
() 4 det c = A  6 
    
therefore the eigenvalues are negative real parts. The eigenvalues are (-0.035, -8.007, -1, -0.00065). 
Thus the system is a globally stable. It is possible to show that the stability condition is fulfilled for a 
wide region of the parameter space. In the figures 1 and 2 is shown the dynamics of real product and 
inflation. Here is demonstrated that at the beginning of the period was a percentage of the chartists 
greater than a percentage of the fundamentalists and the market system has had in this period the high 
amplitudes of cycles. An adjusting mechanism has acted towards increasing of the percentage of 
fundamentalists and this situation causes a stabilization activity in the economic system. The stable 
economic system with low amplitudes of cycles enables to make price – adaptive – pursuit strategy 
and thus a percentage of chartists in security market is increasing. Again a period of the high 
amplitudes of cycles is started. 
 
012345
0
10
20
Dynamics of the Real Product
yp
t
   
 
0246
10
0
10
20
Dynamics of the Inflation
inf
t
          
 
Fig. 1  Fig. 2 
In the figures 3, 4, and 5 this basic mechanism is observed when the fundamentalists put increasing 
weight on a reversion to the fundamental as the real product is very low, as inflation accelerates, and at 
the same time an increasing fraction of chartists is switching to fundamentalism. This mechanism 
performs stabilizing effects for this economic system. In the fig. 5 it is possible to observe the price-
trend pursuit strategy of chartists whilst fundamentalists hold the fundamental-model position. There 
are higher amplitudes of cycle behavior of real product when chartists have a more weight on the 
heterogenous market. 
 
  
Fig. 3  Fig. 4 
 
1 4.041 ∆= 2 12.322 ∆= 3 0.136 ∆= 4 0.00018 ∆=  7 
012345
0
1
2
3
4
Chartists
Fundamentalists
prices
moneysupply
Heterogeneous Market
nC
nF
p
m
t
                
x
 
Fig. 5 
 
References 
 
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65, 5, pp 1059-1095, 1997 
[2] Hommes, H., C.: Financial Markets as Nonlinear Adaptive Evolutionary Systems,  
CeNEF, University of Amsterdam, 2000 
[3] Chiarella, G. and Khamin, A. : The Dynamic Interaction of Rational Fundamentalists  
and Trend Chasing Chartists in a Monetary Economy, Springer, pp. 151-65, 2000 
[4] Gandolfo, G.: Economic Dynamics, Springer, 1997 
[5] Guckenheimer, J. and Holmes, P. :  Non-Linear Oscillation, Dynamical Systems and  
Bifurcations of Vector Fields. Springer Verlag, New York, 1986 
[6] Hahn, F: Stability. In: Handbook of Mathematical Economics, vol. II (K. J.Arrow  
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Economics, Cheb, 59-64, 1998 
[9] Sladký, K., Kodera, J. and Vosvrda, M.: Sensitivity and Stability in Dynamical  
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[10] Sladký, K., Kodera, J. and Vosvrda, M.: Macroeconomic Dynamical Systems:  
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[12] Tobin, J.: Keynesian Models of Recession and Repression. American Economic  
Reviews 195-202, 1975 
[13] Vosvrda M.: Bifurcation Routes and Economic Stability, 7th International Conference of 
the Society for Computational Economics, Yale University, June 28-30, New Haven, 
2001 
 
 

A Dynamic Model of Inflation.

Analytical Methods in Economics. Harvester Wheatsheaf, Hertfordshire,

and Hommes, C.H.: A Rational Route to Randomness,

Bifurcation Routes and Economic Stability,

C.: Financial Markets as Nonlinear Adaptive Evolutionary Systems, CeNEF,

Economic Dynamics,

Keynesian Models of Recession and Repression.

Macroeconomic Dynamical Systems: Analytical Treatment and Computer Modelling,

Non-Linear Oscillation, Dynamical Systems and Bifurcations of Vector Fields.

Nonlinear Dynamics in IS-LM Model. In:

Sensitivity and Stability in Dynamical Economic Systems.

The Dynamic Interaction of Rational Fundamentalists and Trend Chasing Chartists in a Monetary Economy,

Bifurcation Routes in Financial Markets

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Bifurcation Routes in Financial Markets

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