A Nonlinear Macrodynamic Model with Fixed Exchange Rates : Its Dynamics and Noise Effects

In this paper, we formulate a discrete time version ofthe Kaldorianmacrodynamic model in a small open economy with fixed exchange rates. The model is described by a system of the three-dimensional nonlinear difference equations with and without stochastic disturbances (noise effects). We study the local stability/instability properties analytically by using the linear approximation method, and chaotic dynamics with and without noise effects are investigated by means ofnumerical simulations. In general, it is believed that the effect ofthe noise is to obscure the basic structure of the system. But, this is not necessarily the case. We show by means of numerical analysis that the noise can reveal the hidden structure of the model contrary to the usual intuition in some situations.


INTRODUCTION
The purpose of this paper is to formulate a macrodynamic model which is described by a sys- tem of the three-dimensional nonlinear difference equations with and without stochastic disturbances (noise effects), and to investigate its behavior by means of analytical method and numerical simula- tions.The model presented in this paper is a discrete time version of the Kaldorian business cycle model in an open economy which was formulated by Asada (1995) as a continuous time model, Con- trary to Asada (1995)'s original model, we intro- duce the noise effects.
Generally speaking, the economy is not isolated system, but it is subject to the interactions with other subsystems of the society.One of the effective methods to model such influences is to introduce the 'noise' (stochastic disturbance).In our model, it is supposed that another subsystem named 'foreign country' exists outside the system, and the dynamics of the economy are affected by the Corresponding author.E-mail: asada@tamacc.chuo-u.ac.jp.
The original version of the Kaldorian business cycle model in a closed economy was presented by Kaldor (1940)'s classical paper, and it was later refined by several authors.See, for example, Chang and Smyth (1971), Gabisch and Lorenz (1989), and  Lorenz (1993).
transactions with 'foreign country'.We assume that the parameter/3 which reflects the 'degree of capital mobility' is subject to the stochastic disturbances, and study the effects of the noise on the dynamics of the system by means of numerical simulations.
A seminal paper which introduced noise into the Kaldorian business cycle model in a closed econo- my is Kosobad and O'Nell (1972)'s model.Dohtani et al. (1996)'s work is a more recent contribution.In particular, Dohtani et al. (1996) introduced the noise effects into the Kaldorian business cycle model which is described by the two-dimensional nonlinear difference equations, and showed by means of numerical experimentation that the noise can reveal rather than obscure the hidden structure of the system in some situations contrary to the usual intuition.In this paper, we show that such a conclusion also applies in an extended version of the three-dimensional Kaldorian system in an open economy with fixed exchange rates.In particular, it is shown by means of numerical approach that the noise can reveal the hidden chaotic attractors at the vicinity of the 'window', and two separate chaotic attractors can be combined under the influence of the noise.
In this paper, we shall consider a stochastic version of Asada (1995)'s model of fixed exchange rates.We can describe the basic system of equations as follows: Ks+ Ks Is; (2) c, c( + Co; 0<c< ,c0>0; Is I(Ys, Ks, rs); Iy-OIs/OYs > O, IK-OIs/OKs < O, Ir OIt/Ors < 0; (4)   Tt -"rY,-To; 0 < -< 1, To > 0; (5) M,/p-L(Ys, r,); Lr-OLs/OYs > O, Lt OLs/Ors < 0; (6) Jt J(Yt, Et); Jy-OJt/OYt < O, JE OJ, /OEs > 0; (7) Qt ( + crTt){rt rf (E Es)/Et); >0, or>0; 2. THE MODEL Asada (1995) tried to extend the Kaldorian type of the nonlinear business cycle model to the small open economy by using the deterministic continuous time model.In Asada (1995), both the system of fixed exchange rates and that of flexible exchange rates were formulated and investigated by analytical method and numerical simulations.In parti- cular, the effect of the change of the parameter/3 which represents the 'degree of capital mobility' were analyzed, and it was shown by means of the Hopf bifurcation theorem that the cyclical fluctua- tion can occur at some parameter values in the system of fixed exchange rates. At-Js+Qs; (9) Es E; (1 O) where the meanings of the symbols are as follows: Y net real national income, C real consumption expenditure, I-net real private investment expen- diture, G=real government expenditure (fixed), K=real physical capital stock, T=real income tax, M-nominal money supply, p=price level The model of flexible exchange rates will be considered separately in another paper.
(fixed), r-nominal domestic rate of interest, rf-nominal foreign rate of interest (fixed), E value of a unit of foreign currency in terms of domestic currency (exchange rate), Ee-expected exchange rate of near future, J--balance of current account (net export) in real terms, Q--balance of capital account in real terms, Atotal balance of payments in real terms, c adjustment speed in the goods market,/3-parameter which represents the "degree of capital mobility" (/3 > 0),normal pseudo-random number N(0, 1), cr standard devi- ation parameter (or> 0).The subscript denotes time period.
Equation (1) formulates the quantity adjustment process in the goods market, i.e., Yt fluctuates according as the excess demand in the goods market is positive or negative.Equation ( 2) is the capital accumulation equation.Equations ( 3)-( 5) are con- sumption function, investment function, and income tax function respectively.Equation ( 6) is the equilibrium condition in the money market.Equation (7) is the current account function.Equation (8) says that the balance of capital account depends on the difference between the rates of return of domestic and foreign bonds.It is assumed that the parameter 3 (degree of capital mobility) is fluctu- ated by noise.Equation ( 9) is the definition of the total balance of payments.Equations ( 10) and (11) express the institutional arrangement of the system of fixed exchange rates.Equation (12) says that money supply endogenously fluctuates according as the total balance of payments is positive or negative under the system of fixed exchange rates.These equations can be reduced to the following set of three-dimensional nonlinear difference equations: We shall call the system ($1) 'model 1'.By the way, Chang and Smyth (1971)'s version of the Kaldorian business cycle model adopts the follow- ing type of the saving function: St S(Yt, Kt); > Sy-OSt/OY > O, SK-OSt/OK < O. (3) Since the saving St is the difference between the disposable income and the consumption Ct, Eq. ( 13) implies the following type of the consumption function: This consumption function represents a sort of the 'wealth effect', i.e., the increase of the real capital stock stimulates the consumption expendi- ture.If we adopt this type of consumption function, we must replace Eq. ( S1)-(i) with the following equation: In this case, we have In particular, in the special case of Cx-IIKI, we obtain =0. (17) The expression r(Yt, Mr) is the 'LM equation' which is derived from Eq. ( 6).It is easy to see that r.=_Or,/OY, >0 and ra4 Ort/OMt < O.
In other words, the negative effect of the change of Kt on Yt+l through the negative effect on the investment expenditure tends to be canceled out by the positive effect on the consumption expenditure when the 'wealth effect' exists.If Eq. ( 17) is satisfied, the system (S1) must be modified as follows: (i) Yt+ F( Y, Mr; c); (ii) Kt+l F2( rt, Kt, Mt); ($2) (iii) Mt+l F3( Yt, Mt; , or).
In this system, Yt+ is independent of Kt so that the system becomes 'decomposable'.In other words, the path of Kt depends on the paths of Yt and M,, but the movements of Y and Mt are independent of the path of Kt.We shall refer to the system ($2) as 'model 2'.

(--)
In these expressions, m -Jr > 0 is the 'marginal propensity to import'.The characteristic equation of this system is expressed as /1 (/) I-J11 k 2 k + a + a2A + a3 0, (19)   where al -trace J1 -Fll (ct) F22 F33 (/), ( 20) First, let us consider the local stability-instability analysis of 'model 1' by assuming or=0 (no stochastic disturbance).Asada (1995) proved that the system (S) has the unique equilibrium point (Y*,K*,M*)>(O, 0, 0) under some reasonable conditions.In this paper, we shall assume that such an equilibrium point in fact exists.The Jacobian matrix of this system which is evaluated at the equilibrium point can be written as follows: + F13(ct)F22F31 (fl) + F12(oz)F21F33(fl). ( 22 The Cohn-Schur conditions for local stability can be expressed as follows: where F,,(c) See Gandolfo (1996), p. 90.In fact, the condition ( 25) is redundant because this condition can be derived from other two conditions.However, for our purpose, this expression is convenient.
From these local stability conditions we can derive a very simple sufficient condition for local instability, i.e., a2 > 3.By using this local instability condition, we can derive the following proposition.

PROPOSITION
Suppose that If< c(1 r) + m.Then, the equilibrium point of the system (S) is locally unstable if c > 0 and/3 > 0 are sufficiently large.
Proof Differentiating Eq. ( 21), we have (26) From Eq. ( 26) we have lim_+ Oa2/Oc +oc so that Oa2/Oc becomes positive for sufficiently large/3 > 0. In this case, a2 > 3 for sufficiently large c and/3.Proposition implies that under certain conditions, the increase of the adjustment speed in the goods market (c0 and the degree of capital mobility (/3) tends to destabilize the system under the system of fixed exchange rates.This conclusion is in line with the result which was derived by Asada (1995)'s continuous time version of the model of fixed exchange rates.

LOCAL STABILITY-INSTABILITY ANALYSIS OF 'MODEL 2'
Next, we shall consider the local stability-instabil- ity analysis of 'model 2'.We also assume in this section that -0 (absence of noise effect).The Jacobian matrix of the system ($2) becomes Fll (o) 0 F13 (a) ] J2 F21 F22 F23 F31 (fl) 0 F33 (fl) (27) where the meanings of the symbols are the same as those of the previous chapter.The characteristic equation of this system is /2(/) --I ;-J l- F22)(/2 -bl, + b2) 0, We can express the Cohn-Schur conditions for local stability as follows --1-b2 > bl I, Equation ( 31) is equivalent to the condition IIKI < 1.We assume that in fact this condition is satisfied.By the way, we can easily see that b 2 > is a sufficient condition for local instability.
From Eq. (34), we have lim/+ Ob2/Oct so that we have b2 > for sufficiently large c > 0 and /3>0.This proves that Proposition also applies to the system ($2).

NUMERICAL EXPERIMENTATION OF 'MODEL 1'
Analytical approach by means of linear approx- imation of the system without stochastic distur- bance which was developed in the previous sections gives us relatively little information on the behavior of the original nonlinear system with stochastic disturbance.Numerical approach will provide us some useful insight, which cannot be obtained if we stick to the analytical approach.In this section, we shall summarize the results of our numerical experimentation of'model 1' We specify the functional forms of the relevant functions and the parameter values as follows: I(Yt, K,, rt) --f(Y) 0.3K rt + 147; (35)    (i) Yt+ Yt c{-0.66Yt +f Yt) 0.3Kt + 147-lOv/Yt + mt + 165, (ii) Kt+I Kt f Yt) 0.3Kt 10V/Yt + Mt + 147, f(Y;) (80/70 Arc tan{ (2.25/20) x (Y-165/0.66)}+ 35; (36) (iii) M,+I M, -0.3 Yt + 50 + (/+ cr'Tt x (IOv/Y, Mt-6).
If we assume that/3 and cr 0, the equilibrium solution of the system (41) becomes Y*, K*, M *) _ (250, 503, 127).The equilibrium national income Y* is independent of the values of c and .On the other hand, K* and M* depend on the values of c and /3.Our numerical simulation shows that the behavior of this model can be very complex even if the noise does not exist (or-0), and the hidden structure of the system may be sometimes revealed rather than obscured when the system is fluctuated by noise.

Dynamics of National Income
In this subsection, we shall consider the dynamics of national income (Y).First, let us consider the case without noise (or=0).Figures 2 and 3 are the bifurcation diagrams of national income with respect to the parameters c and/3 respectively.
Figure 2 shows that the period of income fluctua- tion increases rapidly as the adjustment speed in the goods market (c0 increases, and eventually the chaotic behavior emerges.However, as the 400 parameter a increases furthermore, the 'window' which represents the periodical behavior emerges, and then the chaotic region reappears.We can confirm this statement by observing the largest Lyapunov exponent (see Fig. 4).Figures 5 and 6 are the bifurcation diagram and the largest Lyapunov exponent with small stochastic distur- bance (a 0.01).We can see from these figures that the window of periodical solution disappears and It is assumed that/3 when c is selected as a bifurcation parameter, and c--is assumed when/3 is selected as a bifurcation parameter.
-0.5 0.5 0.5 the behavior of the system becomes more chaotic because of the noise effects.
Now, let us compare the system with noise effects and that without noise effects.Figures 2 and 4 show that the behavior of the system without noise is chaotic when c= 2.0, while the 'window' of the periodical solution appears when c 2.1.However, the behavior of this system becomes chaotic again when c 2.2.Figures 7-9 give the attractors of the system in K-Y plane in these three cases.Figure 10 is the attractor of the system with small noise effects (or 0.01) when c 2.1.The shape of the attractor in Fig. 10 is similar to that in Fig. 7 or 9.If there is no stochastic disturbance, the periodical trajectory is stable and chaotic trajectory is unstable at the 'window'.This implies that the chaotic structure is invisible and hidden at the 'window' if there is no stochastic disturbance.However, our numerical experimentation shows that the stochastic noise can make visible this hidden chaotic structure in some situations.Hence, it is not correct to say that the noise only obscures the basic structure of the system.Figure 11 is the bifurcation diagram of Y with respect to the parameter/3 when/3 is subject to the small stochastic disturbance (or 0.01).Also in this case, some 'windows' of the periodical solution  '..

FIGURE
Bifurcation diagram of Y with noise (or 0.01).

NUMERICAL EXPERIMENTATION OF 'MODEL 2'
We can construct a numerical example of'model 2' by slightly modifying Eq. ( 41).In fact, we can obtain such a model by replacing 0.3 K in Eq. ( 41) (i)   with zero and keeping other two equations of (41)(ii) and (iii) intact.However, this slight modification changes the behavior of the system considerably.Figure 16 is the bifurcation diagram of Y with respect to the changes of the parameter c without noise effects, and Fig. 17 shows the largest Lyapunov exponent in this case.The equilibrium point is stable when c is small, but it becomes unstable and two period cycle becomes stable when c exceeds 2.25.Then, the period-doubling bifurca- tions occur rapidly, and the behavior of the system becomes chaotic.
It is worth to note that this system has two equilibrium points.In fact, we obtained Fig. 16  adopting the initial condition (Y0, K0, M0), which is near from the equilibrium point of 'model 1', i.e.
(Y*, K*, M*) _ (250, 503, 127).If we adopt another initial condition, we can obtain another attractor and another bifurcation diagram.However, Fig. 16 shows that the fusion of two attractors occurs so that the fluctuating area of Y expands suddenly when exceeds 2.5. Figure 17 shows that there are several 'windows' of periodic solutions in the area ofc > 2.5.
Figure 18 is the bifurcation diagram which is fluctuated by noise (or=0.08).This figure shows that the 'windows' of the periodic solutions dis- appear because of the noise effect.Furthermore, in this figure the fusion of the attractors occur even if c < 2.5.We can interpret this phenomenon that the hidden structure of the system is revealed because of the noise effects.For convenience, let us say that the economy is in 'boom' when Yt > 250 and it is in 'slump' when Yt <  we can conclude that even if the economy is in boom, some stochastic disturbance can bring abut slump if the depressing structure is hidden in the system.
Figures 19 and 20 compare the trajectories of Yt without noise and that with noise in the case of c 2.5.Figures 21 and 22 show the result of the similar experimentation in the case of c=2.55.These examples show that the noise may transform the lasting booms into the violent alternations of booms and slumps, but the opposite case is also possible.This is one of the important lessons of our numerical simulations. (iii) dM/dt pJ( Y, E) +/p{r( Y, M) rf} =f3(Y,M;).
Asada (1995) derived the following results analytically under some assumptions: (1) The equilibrium point of the system S is locally stable if/ > 0 is sufficiently small, and it becomes a saddle point if/ is sufficiently large.(2) There exists the parameter value /0>0 at which the Hopf bifurcation occurs.In other words, there exist some nonconstant periodic solutions at some values of/ which is suffi- ciently close to fl0.
Asada (1995) also presented some numerical simulations which support the above analytical results.However, Asada (1995)'s original version could not produce chaotic motion, but it produced rather 'regular' movement.Compared to Asada (1995)'s version, the discrete time version with and without noise which is presented in this paper can produce much complex and richer behavior, and it provides us a foundation to further research.corresponds to Eq. ($1) in this paper, is given as follows: (i) d Y/dt c[c(1 -) Y + cTo + Co + G + I(Y,K,r(Y,M)) + J(Y,) ] =f(Y,K,M), (ii) dK/dt-I(Y,K,r(Y,M)) =--f2(Y,K,M), In this paper, we investigated the discrete time version of the Kaldorian business cycle model in an open economy with and without noise effects by means of analytical method and numerical simula- tions.As a result, we could find some interesting behaviors of the system including chaotic move- ment.However, the model which was presented in this paper is restricted to the system of fixed ex- change rates.While in the system of fixed exchange rates the money supply becomes an endogenous variable, in the system of flexible exchange rates we can consider the money supply as the exogenous variable which is controlled by the central bank.## Obviously, the next step must be the analysis of the system of flexible exchange rates.This is the theme which we shall study in another paper.
FIGURE 8 Attractor in Y-K plane without noise when c=2.1.

FIGURE 9
FIGURE 9 Attractor in Y-K plane without noise when c 2.2.
FIGURE 10 Attractor in Y-K plane with noise when oe=2.1.

FIGURE 12
FIGURE 12 Bifurcation diagram of K without noise (param- eter:
FIGURE 21 Trajectory of Y without noise when c= 2.55.