# Numerical Exploration of Kaldorian Macrodynamics: Enhanced Stability and Predominance of Period Doubling and Chaos with Flexible Exchange Rates

Authors: Toichiro Asada; Christos Douskos; Panagiotis Markellos
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

We explore numerically a discrete Kaldorian macrodynamic model of an open economy with flexible exchange rates, focusing on the effects of variation of the model parameters, the speed of adjustment of the goods marketα and the degree of capital mobility β, on the stability of equilibrium, and on the possible existence of business cycles. We determine by a numerical grid-search method the stability region in the parameter space and find that, by comparison to fixed exchange rates, flexible exchange rates cause increased stability of equilibrium with respect to variations of the model parameters. We identify analytically the Hopf-Neimark bifurcation curve along which business cycles may be generated, as well as the flip bifurcation curve along which period-doubling cascades leading to chaotic behavior are generated. We find that period-doubling cascades leading to chaos are the dominant behavior of the system of flexible exchange rates outside the stability region, persisting up to large values of the degree of capital mobility β. Cyclical behavior of noticeable presence is detected for some extreme values of a state parameter. Bifurcation and Lyapunov exponent diagrams are computed illustrating the complex dynamics involved. Examples of attractors and trajectories are presented. The effect of the speed of adaptation of the expected rate on the stability of equilibrium is also briefly discussed. Finally, we explore the special case (Model 2) incorporating the so-called wealth effect, which in the present case of flexible exchange rates is found to behave similarly to the basic model, contrary to the case of fixed exchange rates in which incorporation of the wealth effect causes an entirely different behavior of the system.


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## Body

## 1. Introduction

In this
paper, we consider a three-dimensional
discrete time model of an open economy with flexible exchange rates. The model
is in fact the Kaldorian small open economy model with flexible exchange rates,
expressed by Asada et al. [1] as a three-dimensional system of nonlinear
difference equations. The corresponding discrete model of the system of fixed
exchange rates was studied in Asada et al. [2, 3], while a similar Kaldorian-type
model in continuous time was studied in Asada [4]. The Kaldorian business
cycle theory based on the original idea of Kaldor [5] has been developed
recently by Lorenz [6], Gandolfo [7], Agliari and Dieci [8], among others.
For a general discussion of analytical and numerical methods in the study of
nonlinear dynamical systems in economics, we refer to Lorenz [9].The present work is an extension of
the work presented in Asada et al. [1]. We carry out an extensive
numerical exploration of the model and consider the effects of the changes of
parameter values on the stability of equilibrium of the system. The main model parameters considered
are the speed of adjustment of the goods market α and the degree of capital mobility β. Using as our main tool a numerical grid-search method,
we determine the
stability region in the parameter space. We also identify analytically the Hopf
bifurcation curve and the flip bifurcation curve, which form parts of the
boundary of this region. We further compute bifurcation and Lyapunov exponent
diagrams to obtain information on the asymptotic behavior of the system when
equilibrium is no longer stable, that is, outside the stability region in the
parameter space.The model exhibits complex dynamics.
Certain conclusions are drawn regarding the effects of parameter
changes on the stability of equilibrium and on the predominant asymptotic behavior
of the system of flexible exchange rates outside the stability region. We find
that, by comparison to fixed exchange rates, flexible exchange rates cause increased
stability of equilibrium with respect to variations of the model parameters. The predominant asymptotic behavior
of the system of flexible exchange rates outside the stability region is found
to be period doubling cascades leading to chaos. Examples of this behavior of
the system are illustrated by means of bifurcation and Lyapunov exponent
diagrams. For certain extreme values of a state parameter, cycles of noticeable
size and persistence are found to occur and are similarly illustrated by means
of bifurcation and Lyapunov exponent diagrams as well as diagrams of the cycles
themselves in two-dimensional projections. The effect of the speed of adaptation of the
expected rate on the stability of equilibrium is also briefly discussed. Finally, we explore briefly the special case (Model
2) incorporating the so-called wealth effect, which in the present case of
flexible exchange rates is found to behave similarly to the basic model,
contrary to the case of fixed exchange rates in which incorporation of the
wealth effect causes an entirely different behavior of the system.The
paper proceeds as follows. In Section2, we present the equations of the model
and the specifications adopted. In Section 3, we determine the position and stability
of equilibrium, and identify analytically the Hopf-Neimark bifurcation curve and
the flip bifurcation curve, which constitute parts of the boundary of the
stability region. The results of our exploration of the occurring asymptotic
dynamical behavior of the model outside the stability region are presented, in
the form of bifurcation and Lyapunov exponent diagrams, and are discussed in Section
4. In Section 5, we present some examples of chaotic attractors and some trajectories.
In Section 6, we explore the occurrence of cycles; and in Section 7, we consider
briefly the effect of the parameter γ, representing
the speed of adaptation of the expected rate, on the stability of equilibrium. Finally,
in Section 8, we explore briefly
the special case (Model 2) incorporating the so-called wealth effect, and Section
9 concludes.


## 2. Model Equations-Variables and Parameters

We consider the three-dimensional
model of open economy Kaldorian dynamics proposed by
Asada et al. [1]. This is obtained from the following system of equations: (2.1)Y(t+1)−Y(t)=α[C(t)+I(t)+G+J(t)−Y(t)],α>0,(2.2)K(t+1)−K(t)=I(t),(2.3)C(t)=c[Y(t)−T(t)]+C0,0<c<1,C0>0,(2.4)I(t)=I[Y(t),K(t),r(t)],∂I∂Y>0,∂I∂K<0,∂I∂r<0,(2.5)T(t)=τY(t)−T0,0<τ<1,T0>0,(2.6)M(t)=pL[Y(t),r(t)],∂L∂Y>0,∂L∂r<0,(2.7)J(t)=J[Y(t),E(t)],∂J∂Y<0,∂J∂E>0,(2.8)Q(t)=β[r(t)−rf−Ee(t)−E(t)E(t)],β>0,(2.9)A(t)=J(t)+Q(t),(2.10)A(t)=0,(2.11)Ee(t+1)−Ee(t)=γ[E(t)−Ee(t)],γ>0,(2.12)M(t)=M¯, where t denotes the time period and the meanings of the symbols are
as follows. Y= net real national income, C= real consumption expenditure, K= real physical capital stock, T= real income tax, M= nominal money supply, I= net real private investment expenditure on
physical capital, G= real government expenditure (fixed), p= price level (fixed), E= exchange rate, Ee= expected exchange rate in near future, J= balance of current account (net export) in
real terms, Q= balance of capital account in real terms, A=J+Q= total balance of payments in real terms, r= nominal rate of interest, rf= nominal foreign rate of interest (fixed), α= adjustment speed of the goods market, and β= degree of capital mobility. Finally, the parameter γ represents the “speed
of adaptation” of the expected rate.Equation (2.1)
expresses the Kaldorian adjustment process of the goods market, and (2.2)
is the equation of capital accumulation, stating that the physical capital
stock increases or decreases according to whether net investment is positive or
negative. Equation (2.3)
is the standard Keynesian consumption function, and (2.4)
is the Kaldorian investment function. Equations (2.5)
and (2.6)
are the income tax function and the condition of equilibrium for the money
market, respectively. Equation (2.7) is a standard type of the
current account equation, stating that the current account is determined by Y(t) and E(t). Equation (2.8)
expresses the idea that the capital account is positive or negative according
to whether the difference between the rates of return of domestic and foreign
exchange bonds is positive or negative. Equation (2.9)
is the definition of the total balance of payments. Equation (2.10)
characterizes the flexible exchange rate system, stating the assumption that the
exchange rate is adjusted instantaneously to keep the equilibrium of the total
balance of payments [A(t)=0]. Equation (2.11)
expresses the “adaptive expectation hypothesis” concerning the expected
exchange rate. Contrary to Asada et al.
[1, 2], in the present paper, we do not
assume that the speed of adaptation γ is fluctuated by noise. Finally, (2.12)
expresses the institutional arrangements of the system of flexible exchange
rates, stating that under flexible exchange rate the domestic monetary
authority can control money supply contrary to the case of fixed exchange rate
system, so that the money supply M can be considered as an exogenous variable (to
be assumed fixed: M=M¯). The dynamical system can be described by the
following equations: (2.13)Y(t+1)−Y(t)=α[c(1−τ)Y(t)+cT0+C0+G+I{Y(t),K(t),r(Y(t),M¯)}+J[Y(t),E(t)]−Y(t)],(2.14)K(t+1)−K(t)=I[Y(t),K(t),r(Y(t),M¯)],(2.15)Ee(t+1)−Ee(t)=γ[E(t)−Ee(t)],(2.16)A(t)=J[Y(t),E(t)]+β{r(Y(t),M¯)−rf−Ee(t)E(t)+1}=0. The first three of these equations are the
difference equations describing the evolution of our discrete dynamical system,
while the important quantity E(t) is found at every “step” from (2.16). Further details
on the present discrete model of flexible exchange rates and its similarities to
a continuous time version are given in Asada et al. [1].In the present paper, for our numerical
exploration we adopt
the same specifications of Asada et al. [1] as follows: (2.17)I[Y(t),K(t),r(t)]=f(Y(t))−0.3K(t)−r(t)+100,r(t)=r(Y(t),M)=10Y(t)−M,J[Y(t),E(t)]=−0.3Y(t)+100−100E(t),c=0.8,τ=0.2,p=1,rf=6,M=M¯=100,γ=1.2,     except that we will consider the quantity cT0+C0+G as a state parameter denoted by Z, and we will explore the following values for
it: (2.18)cT0+C0+G=Z=238,120,60,50. The first value is the value proposed in
Asada et al. [1]. Also, in Section 7, we will consider variation of the speed of adaptation γ of the expected rate. We, thus, obtain
the dynamical system described by the following nonlinear difference equations:  (2.19)Y(t+1)−Y(t)=F1=α[−0.66Y(t)+300+Z+f(Y(t))−0.3K(t)−10Y(t)−100E(t)],(2.20)K(t+1)−K(t)=F2=f(Y(t))−0.3K(t)−10Y(t)+200,(2.21)Ee(t+1)−Ee(t)=F3=1.2[E(t)−Ee(t)]. At every time step,
the exchange rate is obtained from (2.16): (2.22)E(t)=100+βEe(t)100−105β+10βY(t)−0.3Y(t). This system is called Model 1 to distinguish it from its variant, called Model
2, which we consider in Section 8. The function f is
a particular case of the Kaldorian S-shaped direct dependence of the investment
function on income (see also Dohtani et al., [10], Agliari and Dieci, [8],
Asada et al., [3]) given by  (2.23)f(x)=80πarctan[9π80(x−112)]+35.


## 3. Equilibrium Position and Stability

It is easily found that
the system is in equilibrium at(3.1)Y*=2581[9Z+2β{477+625β−259Z+β(954+625β)}],(3.2)K*=103[200−10Y*+f(Y*)],(3.3)Ee*=E*=10001000−1060β+100βY*−3Y*. Indeed, (3.2) is obtained by
solving F2=0 for K, while (3.3) is similarly obtained by
observing from (2.21) that the
equilibrium values of Ee and E must be equal, and solving (2.22) for Ee=E. To find the equilibrium value of Y we first solve F2=0 for f(Y), obtaining f(Y)=0.3K+10Y−200, and substitute this and (3.3) into F1=0. We, then, obtain (3.4)9Y*+250βY*−25(Z+106β)=0, giving (3.1). Thus, for Z=238,β=10, we find that Y*≅154.36,K*≅496.86, and E*≅0.424. All three equilibrium values depend on β, and they are shown graphically as functions of β in Figure 1. We note that Y* and K* are always positive for β>0. However, E* is positive for β>0 only when Z<120. For Z>120,
the value E* is positive only when β>βmin>0, where βmin is the root of the denominator in (3.3):
(3.5)1000−1060β+100βY*−3Y*=0. For example, when Z=238, the minimum value of β for which we have a positive equilibrium is βmin≅0.446. More generally, it is easily found, by
eliminating Y* from (3.4) and (3.5), that (3.6)βmin=15(Z−120)5066Z+100−3498. This is plotted in Figure 2.Equilibrium values as functions ofβ, for Z=238.
(a)(b)Figure 2
The minimum
value ofβ for positive equilibrium, as function of Z.Stability of the
equilibrium is determined by the roots of the characteristic polynomial of the
Jacobian of the mapping:(3.7)(1+∂F1∂Y∂F1∂K∂F1∂Ee∂F2∂Y1+∂F2∂K∂F2∂Ee∂F3∂Y∂F3∂K1+∂F3∂Ee), evaluated at the equilibrium. The
characteristic equation is of the form (3.8)P3(λ)=λ3+a2λ2+a1λ+a0=0, and the Cohn-Shur conditions
for stability are (3.9)1+a1−|a2+a0|>0,1−a1+a2a0−a02>0,a1<3. These conditions
represent relations between the parameters β and α of the model which are satisfied in the shaded regions of the (β,α) plane shown in Figures 3 and 4 for the first two values of the
parameter Z explored here.Region of stability of the equilibrium and its
bounding curves, the flip bifurcation curve
(dashed), and the Hopf-Neimark bifurcation curve (continuous), in the(β,α) parameter plane for Z=238.
(a)(b)Region of stability of the equilibrium and its bounding curves, in the(β,α) parameter plane for Z=120.
(a)(b)Our
basic tool for the numerical determination of the region of stability is a
grid-search method. This involves computing the characteristic polynomial (3.8) and its roots at
all node points of a dense grid covering a region in a two-dimensional space of
parameters, and storing the points at which the equilibrium is stable. The
technique is employed to determine the stability region diagrams of Figures 3 and 4. The part of
the stability region in which the roots of the characteristic equation are all
real is shown dark shaded, while the part in which two of the roots are complex
conjugate is shown light shaded.The flip bifurcation curve,(3.10)P3(−1)=1+a1−(a2+a0)=0, related to the first
Cohn-Shur condition
for stability (3.9), is drawn in as a dashed
curve, and the Hopf bifurcation curve is drawn in as a continuous curve. We
have found that the Hopf bifurcation curve can be determined in the present
case by requiring simply that two of the roots of the characteristic polynomial
have unit product: (3.11)λ3+a2λ2+a1λ+a0=(λ−ρ)(λ2+μλ+1). The coefficients of the left-hand side
are functions of the model parameters α and β. We, then, equate coefficients of same powers of λ, eliminate ρ and μ, and obtain the condition: (3.12)1−a1+a2a0−a02=0. This is the implicit
equation of the Hopf bifurcation curve in the present stability diagrams, and
we note that it is related to the second Cohn-Shur condition (3.9).The intersection
of the two curves, forming parts of the boundary of the stability region,
occurs at the point indicated, in the diagrams of Figures3 and 4, by its
coordinate values: (140.7,0.081) for Z=238, and (52.8,0.206) for Z=120. This shows that up to such large values of the degree of capital mobility β stability of equilibrium is lost through a
flip bifurcation. Therefore, up to such large values of the degree of capital mobility β period doubling can be expected to occur
outside the stability region.In Asada et al. [1], the conclusion was drawn by analytical
considerations regarding the stability of the system in the (β,α) plane, that under certain conditions the increase of the adjustment speed of
the goods market α and the degree of capital mobility β tends to destabilize the system of flexible
exchange rates, in the sense that the equilibrium point becomes unstable for
sufficiently large α and β. In the present paper, we see that in fact, for
the specifications adopted in the present formulation of the model, increase of
the parameter α destabilizes the equilibrium quickly for large
values of β, while high levels of the adjustment speed α are required for destabilization for
relatively low levels of capital movement β. On the other hand, increase of the parameter β, to very high values, does not destabilize the equilibrium for sufficiently small values
of α. We recall that a small value of α (<1) implies prudent reaction by firms,
while a large value (α>1) implies rash reaction and coordination failure
(see Agliari and Dieci, [8]). Thus, one might interpret the stability region
by saying that capital movement, even at very high levels, does not destabilize
the system of flexible exchange rates so long as firms are sufficiently prudent
in their reactions.These results are contrary to the corresponding
behavior of the fixed exchange rates system, in which the equilibrium is
destabilized at much lower levels of capital movement and/or adjustment speed
(see Asada et al., [3]). Thus, flexibility of the  exchange rates causes enhanced stability of
equilibrium with respect to variations of the basic parameters of the
model.


## 4. Bifurcation and Liapunov Exponent Diagrams

To illustrate the
complicated dynamics involved, we have employed numerical simulations of the trajectories to obtain characteristic bifurcation and
Lyapunov exponent diagrams representing the asymptotic state of the system. In Figures5 and 6, we present such diagrams for Z=238 and some sample values of the parameter β, with the parameter α as the bifurcation parameter. Similarly, in Figure 7, we present
such diagrams for Z=120 and sample values of the parameter β. Some details of the period-doubling process for Z=238 and β=20 are shown in
Figure 8.Bifurcation diagrams and Lyapunov exponent forZ=238: β=1 (left) and β=2 (right).
(a)(b)(c)(d)(e)(f)(g)(h)Bifurcation diagrams and Lyapunov exponent forZ=238: β=5 (left) and β=20 (right).
(a)(b)(c)(d)(e)(f)(g)(h)Bifurcation diagrams and Lyapunov exponent
forZ=120: β=2 (left) and β=5 (right).
(a)(b)(c)(d)(e)(f)(g)(h)Details of the bifurcation diagram (Figure6)
of E for Z=238,β=20 (two levels of magnification).
(a)(b)These
diagrams show that, as expected from the region of stability diagrams, the
predominant behavior of the system of flexible exchange rates for parameter
values outside the stability region is that of period-doubling cascades leading
to chaos. The existing windows of periodicity are also visible within the
chaotic regions. The Lyapunov exponent diagrams clearly indicate chaotic behavior
when it occurs. We also note in Figures5 and 6 that for Z=238 period-doubling behavior persists longer (for larger intervals of α) at
higher levels of capital mobility.


## 5. Examples of Attractors and Trajectories

In Figure9, we present two examples of chaotic attractors of the system of flexible exchange rates, in the case
of Z=120 and β=5. The first example corresponds to a two-piece
attractor, while the second example shows the evolution of that attractor when
its two pieces have merged into a single chaotic attractor.Chaotic attractors in their projections in the(Y,K) plane (left) and the (Y,E) plane (right), for Z=120,β=5, and α=0.83 (top), α=0.9 (bottom).
(a)(b)(c)(d)The two dominant types of dynamical behavior
exhibited by the system are shown in Figure10 in terms of the actual time
paths–-trajectories–-of the capital stock and the exchange rate. In
the top row, we see the case of a periodic trajectory (the trajectory of a 7-periodic orbit), while,
in the bottom row, we see the case of a chaotic trajectory.Trajectories ofK and E for Z=120,β=5.  Top: regular
(7-periodic), α=0.87. Bottom: chaotic, α=0.9.
(a)(b)(c)(d)


## 6. The Occurrence of Cycles

In Figures3 and 4, for β>140.7 and β>52.8, respectively, cycles would be expected to
occur when crossing the Hopf bifurcation curve, but such cycles were found to
be difficult to compute, being small in size and existing very briefly, that
is, for very small interval of values of the parameter α after crossing the bifurcation curve.
Nevertheless, we have found cycles of noticeable size, and existing for
intervals of α values of noticeable length, at smaller values
of the state parameter cT0+C0+G=Z. For this, we have used the stability region diagrams
for Z=50 and Z=60 (Figure 11), which we determined by the same
grid-search method as before. In these cases, the flip bifurcation curve forms
part of the stability region boundary only at relatively low values of β.Region of stability of the equilibrium and its bounding curves, the flip bifurcation curve
(dashed) and the Hopf-Neimark bifurcation curve (continuous), in the(β,α) parameter plane for Z=50 (left) and Z=60 (right).
(a)(b)Note that the
region of stability diagrams forZ=238,120, and 60 are qualitatively similar as determined
by the behavior of the bounding curves, but at Z=50 we meet with a qualitatively different behavior
of the bounding curves. Also,
it is at Z=50, and to a lesser extent at Z=60, among the four values considered for Z, that cycles of noticeable size and
existing for intervals of α values of noticeable length were found. This can
be seen, for example, in the bifurcation diagrams of Figures 12 and 13. Note
also the characteristic flatness at the zero value of the Liapunov exponent
diagrams in Figure 13, a landmark of cyclical behavior. The cycles themselves for Z=50 are shown for two different values of β in Figures 14 and 15. A chaotic attractor
showing the chaotic asymptotic behavior of the system after the development of
the cycles, for Z=50,β=5,α=0.8, is also shown in Figure 15.Bifurcation diagrams ofY for Z=60: β=15 (left) and β=20 (right).
(a)(b)Bifurcation diagrams and Lyapunov exponent
forZ=50: β=3 (left) and β=5 (right).
(a)(b)(c)(d)(e)(f)(g)(h)Figure 14
Development of cycles forZ=50,β=3,2.7≤α≤3.  Projection in the (K,E) plane.Left: development of cycles forZ=50,β=5,0.37≤α≤0.56.  Right: chaotic attractor
for Z=50,β=5,α=0.8. Projection in the (K,E) plane.
(a)(b)As a matter of
interest, we note that, in the caseZ=50 and β=5, the bifurcation appears to be of the
subcritical type. This type of bifurcation is of particular interest to
economists as well as to mathematicians, as it may be associated to a double
bifurcation process, creating coexistence of attractors, related hysteresis
effects and policy questions (see, e.g., Kind, [11], Agliari, [12]). The
occurrence of such a bifurcation in the present case is supported by the
numerical evidence of Figure 16.Graphical evidence of subcriticality of the Hopf bifurcation forZ=50,β=5. Left: three successive attractors (the first being a point) for α=0.363380,0.363385,0.363390. Right: the same for α=0.3645,0.3650,0.3655. Projections in the (K,E) plane.
(a)(b)A description of
this evidence is as follows. The right diagram shows three attractors found by
starting the iteration process very near the equilibrium point, for three
values of the bifurcation parameter near the valueα=0.3647 at which stability is lost for the present
values of the other parameters. The first value α=0.3645 corresponds to stable equilibrium and the
iterations end at a single point, the equilibrium point. The next two values, α=0.3650 and 0.3655, correspond to unstable equilibrium and the
iterations accumulate on cycles, but these cycles appear by their sizes to represent a
discontinuous jump from the equilibrium point, and this is made apparent by the
fact that they are also very close to each other in spite of the fact that the
change in parameter value between them is the same as the change of parameter
value between the first cycle and the equilibrium point. This apparent
discontinuity suggests that the bifurcation is subcritical. The evidence is
more conclusive in the left diagram of Figure 16, produced by starting the
iterations far from the equilibrium. In this diagram, we see that the same
situation occurs at three values of the bifurcation parameter all of which
correspond to stable equilibrium. The cycles have appeared inside the stability
region. Thus, the evidence of Figure 16 points to a double bifurcation process
in which a global bifurcation occurs inside the stability region and a local bifurcation
occurs when equilibrium loses stability. During this process, we have coexistence
of a point attractor and a cyclical attractor for an interval of values of the
bifurcation parameter. For further discussion on the subcritical Hopf
bifurcation, see Puu [13].


## 7. Effect of the Speed of Adaptation of the Expected Rate

We now
consider briefly the effect of the parameterγ, representing
the speed of adaptation of the expected rate, on the stability of equilibrium. To
do this, we use the same grid-search method, together with the analytic
determination of the flip bifurcation curve and the Hopf bifurcation curve, to
determine the region of stability in the (β,γ) plane for Z=238 and several values of the parameter α. Some of the stability regions obtained are
shown in Figure 17, while qualitatively similar results are obtained for other
values of the state parameter Z.Stability region in the(β,γ) plane for Z=238, and α=0.05,0.1,0.5,1,1.5,2.2, from left to right and top to
bottom (note the different scales).
(a)(b)(c)(d)(e)(f)We
recall here that in Asada et al. [1] the conclusion was drawn by analytical considerations regarding the
stability of the system in the (β,γ) plane, that under certain conditions the equilibrium is locally unstable when
the parameters β and γ are sufficiently large. In the present paper,
we see that in fact, for the specifications adopted in the present formulation
of the model, this conclusion is confirmed although one of the sufficient
conditions (∂I/∂K<−1) specified in Asada et al. [1] does not hold.
Further, we see that for most levels of the speed of adjustment α≥0.5 stability of equilibrium persists for values
of the speed of adaptation γ of the expected rate such that γ<2 approximately.We
also note that after a certain value of the adjustment speedα, the entire stability region boundary is formed
by the flip bifurcation curve (3.10), which now has the form (7.1)P3(−1)=ψ0(α,β)+ψ1(α,β)γ=0. This relation can be written as (7.2)γ=−ψ0(α,β)ψ1(α,β)=ζ0(β)+ζ1(β)αχ0(β)+χ1(β)α, and represents the two branches of a hyperbola-like
curve (dashed) in each frame of Figure 17.Due to the linearity of numerator and
denominator of the right-hand side with respect toα, (7.2) represents a hyperbola in
the (α,γ) plane. We see that the topology of the curves
in the (β,γ) parameter plane changes at a critical value of α, between α=1.5 and α=2.2. This happens when the two branches of the hyperbola
in the (α,γ) plane meet, and the condition for the branches
to meet is (7.3)|ζ0(β)ζ1(β)χ0(β)χ1(β)|=0. Solving this numerically we obtain the critical
value of β: (7.4)β0≅1.164, and the corresponding
values of α and γ are (7.5)α0=−ζ0(β0)ζ1(β0)≅2.079,γ0=ζ1(β0)χ1(β0)≅2.026.  In the present case  (Z=238) and for values of α sufficiently larger than α0, specifically for α>2.0851, the Hopf bifurcation curve no longer crosses the flip bifurcation curve and the entire boundary of the stability region is formed by the latter.


## 8. Model 2-the “Wealth Effect”

We now discuss, in the present case of flexible exchange rates, the special
two-dimensional variation of the model, called Model 2 (Asada et al.,
[2, 3]). The system of equations for model 2 is
identical to  (2.19)–(2.22), except that the
term −0.3K(t) is
missing from (2.19). This means that
the system decouples with (2.19) and (2.21) becoming independent of K.
For economic theoretical justification of Model 2 see Asada et al.
[2], where relevance of
this model is discussed in relation to a special case of a type of savings
function considered by Chang and Smyth [14], leading to a consumption
function representing the “wealth effect” (according to which increase of the
real capital stock stimulates the consumption expenditure).Although Model 2
is of lower dimension than Model 1, consisting essentially of (2.19) and (2.21), determination of
the equilibrium values as functions of the parameters is more difficult.
However, we can still compute the equilibrium value of E from (3.3) when the
equilibrium value of Y is known, and substituting (3.3) into F1(2)=0, where F1(2) is F1 for Model 2 (i.e., without the term −0.3K(t)); we obtain the following equation that can be
solved numerically for the equilibrium value Y* of Y:(8.1)200+106β+Z−10(1+β)Y*−0.36Y*+f(Y*)=0. Equations (3.3), (8.1), and (3.2) provide the
numerical values of all three variables at equilibrium for each value of β (and given Z). Thus, for Z=238,β=10, we find Y*≅187.11,K*≅457.51,
and E*≅0.284. The equilibrium values are shown graphically as
functions of β in Figure 18 for Z=238. As in Model 1, Y* (and K*) is always positive for β>0. However, E* is positive for β>0 only when Z<Z0≅27. For Z>Z0,
the value E* is positive only when β>βmin>0, where βmin is the root of  (3.5). For example, when Z=238,
the minimum value of β for which we have a positive equilibrium is βmin≅0.531, and for Z=50 this minimum value is βmin≅0.086. The relevant plot of βmin as a function of Z shown in Figure 19 is produced by solving numerically
the system of (3.5) and (8.1) for Y* and β for each value of Z.Equilibrium values of Model 2 as
functions ofβ, for Z=238.
(a)(b)Figure 19
The minimum
value ofβ for positive equilibrium of Model 2, as function of Z.With a valid
scheme for the computation of the equilibrium values of the variables, we now
proceed to examine the stability of the equilibrium globally by applying the grid-search
method for the determination of the stability region in a two-dimensional
parameter space.Stability in
this case is determined by the roots of the characteristic polynomial of the
Jacobian matrix of the two-dimensional mapping defined by the equations of
model 2 forY and Ee:(8.2)(1+∂F1(2)∂Y∂F1(2)∂Ee∂F3∂Y1+∂F3∂Ee), evaluated at the equilibrium. The
characteristic equation is now quadratic: (8.3)P2(λ)=λ2+a1λ+a0=0, and the conditions for stability are (8.4)1+a0−|a1|>0,a0<1. The region of stability of
equilibrium of Model 2, as obtained by the grid search method for Z=238, is shown in Figure 20 (for β>βmin).Figure 20
Region of stability of the equilibrium for Model
2 and its bounding curves in the(β,α) parameter plane for Z=238.The part of
the stability region in which the two roots of the characteristic equation are
real is shown dark shaded, while the part in which the roots are complex
conjugate is shown light shaded.
We note that in Figure20 the first condition (8.4) is satisfied below the flip bifurcation curve: (8.5)P2(−1)=1+a0−a1=0, which is shown as a dashed curve. The second
condition (8.4) is satisfied below
the Hopf-Neimark bifurcation curve, which is shown as a continuous curve and is
given in this case by (8.6)a0=1.In the case of fixed exchange rates (Asada et al., [3]), we found that Model 2 had an entirely different dynamical
behavior compared to Model 1, that is, introduction of the wealth effect in the
fixed exchange rates system
produced a much larger region of equilibrium stability in the parameter space
and caused business cycles to disappear. We now see, by comparison of Figures 3 and 20, that in the present case of flexible exchange rates Model 2 behaves in
the same way as Model 1, in so far as the behavior of the system can be
determined from the formation of the stability region and the nature of its bounding
curves (although Model 2 is for most values of β destabilized somewhat faster than Model 1 when
the speed of adjustment α is increased). In particular, as in Model 1,
up to large values of the degree of capital mobility β the bounding curve of the stability region is
the flip bifurcation curve; thus, up to such large values of β the predominant behavior of the system of Model
2 outside the stability region is period doubling and chaos, with
characteristic absence of business cycles.


## 9. Summary and Conclusions

We explored numerically a discrete
Kaldorian macrodynamic model of an open economy with flexible exchange rates,
focusing on the effects of variation of the model parameters, the speed of
adjustment of the goods marketα and the degree of capital mobility β, on the stability of equilibrium, and on the
possible existence of business cycles. Using a numerical
grid-search method as our main tool, we determined the stability region in the
parameter space. We also identified analytically the Hopf bifurcation curve and
the flip bifurcation curve, which form parts of the boundary of this region. We
further computed bifurcation and Lyapunov exponent diagrams to obtain information
on the asymptotic behavior of the system when equilibrium is no longer stable, that
is, outside the stability region in the parameter space.The model exhibits complex dynamics. Certain conclusions are drawn regarding the effects of parameter changes on the stability of
equilibrium and on the predominant asymptotic behavior of the system of
flexible exchange rates outside the stability region. We found that at high
levels of capital mobilityβ stability is lost quickly, that is, at low
levels of the adjustment
speed. In other words, capital
movement, even at very high levels, does not destabilize the system of flexible
exchange rates for small values of the speed of adjustment of the goods market α, that
is, so long as firms are sufficiently prudent in their reactions,
but high levels of the adjustment speed α are required for destabilization for
relatively low levels of capital mobility β. This contrasts with the behavior of the fixed
exchange rates system (see Asada et al., [3]), in which the equilibrium is
destabilized at much lower levels of capital movement and/or adjustment speed.
Thus, flexible exchange rates cause increased stability of equilibrium with
respect to variations of the basic parameters of the model.The predominant asymptotic behavior
of the system of flexible exchange rates outside the stability region is found
to be period doubling leading to chaos, and persists up to large values of the
degree of capital mobilityβ. Examples of this dominant behavior of the system are illustrated by
means of bifurcation and Lyapunov exponent diagrams. For certain extreme values
of a state parameter, cycles of noticeable size and persistence were found to
occur and were similarly illustrated by means of bifurcation and Lyapunov
exponent diagrams, as well as diagrams of the cycles themselves in two-dimensional projections. Numerical evidence for the occurrence of subcritical
bifurcations and coexistence of attractors was also provided, and examples of
attractors and trajectories were presented.We
also explored briefly the effect of the parameterγ, representing
the speed of adaptation of the expected rate, and found that in most cases
increase of γ destabilizes
the system when it exceeds the value γ=2 approximately. Finally, we explored the
special case (Model 2) incorporating the so-called wealth effect, which in the
present case of flexible exchange rates was found to behave similarly to Model
1, contrary to the case of fixed exchange rates in which incorporation of the
wealth effect causes an entirely different behavior of the system. In other
words, flexibility of the exchange rates drastically reduces the significance
of the wealth effect.



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*Source: 529164-2008-09-25.xml*