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SOME NOTE ABOUT THE EXISTENCE OF CYCLES AND
CHAOTIC SOLUTIONS IN ECONOMIC MODELS
by
Beatrice Venturi
Department of Economics
University of Cagliari
Italy
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EXISTENCE OF CYCLE AND CAOTIC SOLUTIONS IN
ECONOMIC-FINANCIAL MODELS
We analyze the global structure of a tree-dimensional abstract continuous time stationary
economic model that includes some
determinates parameters.
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We shall consider a generic
non-linear first order system with some structural parameters:
nii
xxxi
fx ...2,1,,,,,,,3211
3212
,,,, xxxgxi
3213
,,,, xxxhxi
THE MODEL
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Where f, g and h are complicate non-linear functions of class C2 (twice continuously
differentiable) in all their arguments.
The parameters:
0, i nii ...2,1,
are real and positive.
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A stationary (equilibrium) point of our system is any solution of :
0321 xxx
Assuming the existence of such
a solution at some point
*3
,*2
,*1
* xxxP
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THE JACOBIAN MATRIX
The local dynamical properties of system from (2.1) to (2.3), at a hyperbolic
equilibrium point P*, can be described in terms of the Jacobian matrix , for brevity.
In fact, the nature of the eigenvalues of J*, plays a key role.
J P J* *
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We consider the system as a one-parameter
family of differential equations dependent
of the parameter .
We fixe the other parameters .
We assume that in our modelexists a parameters set where the Jacobian
has two eigenvalues complex conjugate :
),(
J P J* *
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EMERGENCE OF STABLE CYCLES
First we use rigorous arguments to show as anequilibrium of the model could be destabilizedinto a stable cycle in the dynamic of R3 under the
following two alternative assumptions:
a) A steady state has three stable roots.
b) A closed orbit has a two dimensional manifolds in which it is asymptotically stable .
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APPLICATIONS
Next we apply these results to a general non-linear fixed-price disequilibrium
IS-LM model as formulated by
Neri U. and Venturi B. (2007).
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APPLICATIONS
Neri U. and Venturi B. (2007) discuss the effect of a change of the adjustment parameter in the money market, via the Hopf bifurcation’s approach in a three dimensional fixed-price disequilibrium IS-LM model.
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THE ECONOMIC MODEL
mwyrLr ,,.
S = savings, T= tax collections, G=government expenditure, B = interests payment on
perpetuities, L = liquidity preference
function m = is the real money
supply. I = investment, r =interest rate, y = output (income) w =wealth α >0 and µ>0are the adjustment parameters
in their respective markets
ByTGwySyrIy D ,,.
)(.
yTBGm
Specifically, our “generalized model” is a non-linear systemin the independent state variables r, y and m.
(1)
(2)
(3)
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(1) describes the traditional disequilibrium of dynamic adjustment in the product market; (2) describes the corresponding disequilibrium in the money market;
(3) represents the governmental budget constraint.
THE ECONOMIC MODEL
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THE ECONOMIC MODEL
Next, we define the disposable income y and the wealth w as follows:
ByTByyD
r
Bmw
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THE ECONOMIC MODEL
We assume that the functions: I, S, T are of class C2
Recall that a stationary (equilibrium) point of our system is any solution of.
Assuming the existence of such a solution at some point P*(y*,r*,m*), we want to analyze its local properties (e.g. stability, etc.) around P*.
0 mry
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myrfmwyrLr ,,,,
myrgByTGwyySyrIy D ,,,)(,
yhyTBGm )(
We shall rewrite our system in the equivalent form:
with h: R R
THE ECONOMIC MODEL
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Since G and B are fixed a choice of the policies, implies:
because all deficit must be financed either by creation
of money or by creation of new debt.
y
Tyh '
Remark1: Unlike Schinasi G.J. ,1982 , we allow the functions L and S (liquidity and savings) to depend on
wealth w.
THE ECONOMIC MODEL
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THE ECONOMIC MODEL
The monotonicity, of the restrictions:
r I r ,.,. r L r ,.,. )(.,.,wLw ,.., ySy wSw .,.,
is assumed and these assumptions imply the (economic) conditions:
0rI
,
0rL
,
0wL
,
0yS 0wS
.
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Dynamical analysis
Theorem 1. A hyperbolic stationary point of the system (1), (2), (3) is locally asymptotically stable if the following assumptions hold at:
Ngrf y 1
with N given by (*) at P* :
N = the first integer such that N a2> a3 , at P*,
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Dynamical analysis
yT
yTyS
yI
1 0yg
rww gLrfS )1( 10 wL gr 0
yT
wS
yL
rg
yg
rf 1
i) the marginal propensity to invest out of income is greater than unity
ii)
iii)
10 yT
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Dynamical analysis
Corollary 1. Let the hypotheses of Theorem 1 hold at a hyperbolic stationary point of the our system.
Then the steady state for each α >0 and all
0**1
PNP
yg
rf
is locally asymptotically stable.
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Dynamical analysis
Corollary 2. Assume Corollary 1 and let J*=J(P*) as before.
Then, there exists a value µ>0 for which
J* has a pair of purely imaginary eigenvalues.
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Theorem 2. Assume the hypotheses of Theorem1 except that Ngrf y
1
is now replaced by :
*
*
20
Pr
f
Py
g
Then, there exists a continuous function µ(δ) with µ(0)=μ’ and for all δ small enough , there exists a continuous family of non-constant positive periodic solutions
[r*(t ,), y*(t,), m*(t,)]
for the dynamical system (1), (2), (3) which collapse to the stationary point P * .
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REMARK For a three-dimensional non linear dynamical model the general version
of the Hopf bifurcation theorem, ensures the existence of a
small amplitude periodic solutions bifurcating
from the steady state only in the center manifold
(a two-dimensional subspace of 3).
THE ECONOMIC MODEL
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Dynamical analysis
From an economic point of view, sub-critical or super-critical orbits are both reasonable.
Since the third real root of the Jacobian matrix is negative the existence of a super-critical Hopf bifurcations becomes very interesting in the analysis of macroeconomic fluctuations for a IS-LM model
A stable economy, by the increase or decrease of its control parameters, could be destabilized into a stable cycle in the dynamic of R3
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CONCLUSIONS
The sub-critical Hopf bifurcations may correspond to the Keynesian corridor (Leijonhufvud, 1973):
the economy has stability inside the corridor while it will loose the stability outside the corridor.
In such a case the dynamics are either converging to an equilibrium point or the trajectories go somewhere else, and it is also possible that another attracting set exists, but often the alternative is diverging trajectories.
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CONCLUSIONS
We have seen that fluctuations derive from the mechanisms through which money markets reflect and respond to the developments in the real economy.
Our analysis provides an example of the classical thesis concerning endogenous explanations to the existence of fluctuations in some real world economic variables
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For a three-dimensional non linear dynamical model the general version of the Hopf
bifurcation theorem, ensures the existence of a small amplitude periodic solutions
bifurcating from the steady state only in the center manifold
(a two-dimensional subspace of 3).
The general version of the
Hopf bifurcation theorem
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Shil’nikov showed that if the real eigenvalues has large magnitude than the real part of the
complex eigenvalues of the Jacobian of system,
then there are horseshoes present in return maps near the homoclinic
orbit of the model.
Shil’nikov Theorem
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The orbits, in the dynamics of the center manifold, can generally be either
attracting or repelling..In the case of an attracting orbit
(the so- called sub-critical case) trajectory trajectory on the center manifoldcenter manifold are locallylocally
attracted by this this orbit, which becomes a limit set.
.
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In this situation the stationary pointstationary point is an unstable solutionunstable solution
meanness from an economic point of viewmeanness from an economic point of view(unless the initial conditionsinitial conditions happen to coincidecoincide with the stationary value).
Conversely, if the cyclecycle is unstableunstable((the so- called super-critical case)
the steady state is attracting.
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The study of stability of emerging orbits on the centre manifold can be performed
by calculating the sign up of-up-third order derivative of the nonlinear part of the system,
when written in normal form.
This case is particularly relevant from an economic point of view
only if the initial conditions imply that
the economy fluctuates right from the beginning.
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In fact, the Hopf bifurcation theorem proves the existence of closed orbits but it gives no information on their number and
their stability.
Using the non linear parts of an equation system,
a stability coefficient
(as formulate for example by Guckenheimer J.- Holmes P.,1983)
may be calculated in order to determine
the stability properties of the closed orbits
(see Foley , 1989, Feichtinger ,1992, Mattana - Venturi,1999, Anedda C. -Venturi B. ,2003).
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Mattana P.- Venturi B. , 1999, analyzed a simplified three-dimensional version of Lucas’s model,
a two sector endogenous growth model with externality.
Considering the externality as a bifurcation parameter, they proved, the existence of small
amplitude periodic solutions, Hopf bifurcating from the steady state in the center manifold.
Venturi B., 2002, have elaborated a numerical
simulation of this model.
.
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In Fig.1 is plotted the dynamics of one orbit Hopf bifurcating from the steady state
of the reduced version of
the Lucas model
(see Venturi B., 2002)
The orbits is super-critical
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Figure 1
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THE ECONOMIC MODELS
We review a generalized two sector models of endogenous growth, with externalities, as formulated by Mulligan B.- Sala-I-Martin X.,1993.
We show that in this class of economic models, considering the externalityexternality as bifurcation parameterbifurcation parameter, the conditionsconditions of existenceexistence of periodic orbitsperiodic orbits and chaotic chaotic solutionssolutions come true.
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The Mulligan B. - Sala-I-Martin X. model deal with the maximization of a standard utility function:
where
c = is per-capita consumption
= is a positive discount factor
= is the inverse of the intertemporal elasticity of substitution.
dtec t
0 1
1max
1
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The constraints to the growth process are represented by the following equations
00
)0(,00
)0(
)(
)(ˆ)(ˆ))())(1)(()())(1((
)()(
)(ˆ)(ˆ))()()()()((
ˆˆ
ˆˆ
hhkk
thh
tkthtktvthtuBh
tctkk
tkthtktvthtuAk
khkuhu
khkvhu
k =is the physical capitalphysical capital,
h =is the human capitalhuman capital
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vkuh ,
k h k h 1 1
k, , h are the private share of physicalphysical and the
humanhuman capital in the output sectoroutput sector
vkuh ,
k , h are the private shareprivate share of humanhuman
capitalcapital in the educationeducation sector sector
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u and v =are the fraction of aggregate human human and physical capitalphysical capital
used in the final output sectorfinal output sector at instant t
(1- u) and (1- v)
are the fractionsfractions used in the education sector, education sector,
A and B
are the level of the technologylevel of the technology in each sectorsector,
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is a positive externalityexternality, parameterparameter in the production of production of physical capital
k
h
is a positive externality parameterexternality parameter in the production of production of human capital
All the parameters:
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live inside the following set:
= (0, 1)(0, 1) (0, 1) (0, 1) (0, 1) (0,1) (01) (0 1) (0 1) 4.
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The representative agent’s problem is solved by defining the current value Hamiltonian:
)()(ˆ
)(ˆˆ
)(ˆ))()()()()((1
1
11
tctkk
ktkhthktkvtvhthutuA
cH
)(
ˆ)(ˆ
ˆ)(ˆ))())(1)(()())(1((
2th
hktkhthktkutvhthutuB
i, i = 1, 2 =Lagrange multipliers
(co-state variables). = is a depreciation factor
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Plus the usual two transversality conditionstransversality conditions::
lim ( ) ( )
lim ( ) ( )
te t t k t
te t t h t
1 0
2 0
and obtaining the first-order necessary conditions for an interior solutionsinterior solutions.
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The general model just presented collapses to Uzawa-Lucas’ when depreciation is neglected and the following
restrictions are imposed: v k
00ˆˆ kvhk
u h k 1
u h 1
A B 1
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According to the strategy used by Mulligan B.C. - Sala-I-Martin X.,1993,we express the two multipliers 1 and 2
in terms of their corresponding control variables c and u
and obtain an autonomous system of four differential equations in the four variables k, h, c, u.
A solution of this autonomous system is called a Balanced Growth Path (BGP) if it entails a set of functions of time
solving the optimal control problem presented such that
k, h and c grow at a constant rate and u is a constant.
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We choose a standard combination of the original variables that is stationary on BGP.
khx
1/1
1
ux 2
kcx
3
We get a first order autonomous system of
three differential equations
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B.G.P that does not include positive externalities admits only the saddle-path stable solutions
(see Mulligan B.C. - Sala-I-Martin X.,1993).
We get a first order autonomous system of three differential equations
311)21(
12
111 xxxxxxx
322
1222 xxxxx
3312
11
233 xxxxxx
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Where:
;
.1
1
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We put our model into normal form(see Guckenheim - Holmes 1983, pp. 573).
The system becomes:
,13
,1
,111
3232
32322
32
,,*
,,*
,,*
3
2
1
wwwFwwBJw
wwwFwBJww
wwwFwTrJw
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In the equilibrium point is translated the origin P*(0,0,0) of the model and the eigenvalues are
linearized in P*.
The real eigenvalue is
*TrJR
and the complex conjugate eigenvalues are
2,1 jBij
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We consider the dynamic of the system when the parameter values are evaluated in a parameters
set where emerges a super-critical Hopf bifurcation.
We choose the value (the exponent of the externality factor )very close to the bifurcation value *.
Setting: = 0.75; = 0.055; = 0.054; = 0.1 and = 0.05,
We found that the system has a homoclinic orbit.
(See Fig. 3.)
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Figure 2
00.05
0.10.15
0.20.25
-40
-20
0
20-35
-30
-25
-20
-15
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The loss of stability and the presence of periodic solutions Hopf bifurcating from
the steady state can lead to the occurrence of an homoclinic orbit, Fig. 2,
and, under Shil’nikov assumption,
of chaotic solutions
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The aim of this work is to point out some basic ideas that may be useful to prove the transition to
bounded and complex behavior, and to explain how
the presence of Hopf bifurcations in a general class of economic-financial models can be
interesting from an economic and dynamic point of view.
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