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International trade and the possible occurrence of chaos
Economics Letters North-Holland
23 (1987) 135-138
INTERNATIONAL Hans-Walter
135
TRADE AND THE POSSIBLE
OCCURRENCE
OF CHAOS
LORENZ
Georg-August-Uniuersitiit Giittingen, 34 Gijttingen, Federal Republic of Germany University of Southern California, Los Angeles, CA 90089-0035, USA Received
8 October
1986
The paper demonstrates that attractor and hence of chaos. considered as a perturbation Newhouse, Ruelle and Takens
international trade among fluctuating economies may involve the occurrence of a strange Once autonomous economies display cyclical behavior, international trade activities can be of the motion of the autonomous economies, possible implying chaos via a theorem by (1979).
1. Introduction Recent work on chaotic dynamical systems ’ in economics has concentrated mainly on the investigation of non-linear systems framed in one-dimensional difference equations [e.g., Day (1982, 1983), Day and Shafer (1985,1986), Grandmont (1985) and Stutzer (1980)]. While the establishment of chaotic trajectories in these models can be relatively easily performed by means of the Li-Yorke theorem or Sarkovskii’s theorem, the mathematical requirement of non-invertible maps usually implies model structures which do not seem to be generic in economics. * Alternatively, the literature on dynamical systems in physics and chemistry deals with some examples of higher-dimensional non-linear differential equation systems displaying chaotic trajectories in which non-linearities are involved that seem to be more compatible with standard assumptions in dynamical economics than those involved in one-dimensional models. While the occurrence of chaos in continuous-time systems is restricted to the case of at least three dimensions, the analytically most accessible case seems to be that of a six-dimensional system consisting of three coupled non-linear two-dimensional limit cycle oscillators. In the following, a simple example will be provided showing how such a system naturally emerges in a model of international trade among three different economies.
2. Oscillations in autonomous
economies
In order to generate chaotic dynamics in the international trade model presented below, the involved economies must display oscillations in the case of autonomy. The dynamic behavior of the following model of an autonomous economy was extensively studied by Torre (1977) who established the existence of closed orbits by means of bifurcation theory. Consider a very simple ’ Compare ’ However,
Day (1982) for a verbal description of discrete chaotic dynamics. compare Day and Shafer (1986) for an intrinsic non-linearity in the standard
0165-1765/87/$3.50
0 1987, Elsevier Science Publishers
B.V. (North-Holland)
IS-L&Z-model.
136
H.-W. Lorenz / International
trade and
thepossible occurrence of chaos
Keynesian macroeconomic model of a single economy with Y as income, r as the interest rate, M as the (constant) nominal money supply, and assume that the goods prices, p, are fixed during the relevant time interval. Suppose that gross investment, I, and savings, S, depend both on income and the interest rate in the familiar way, i.e.,
I=I(Y, r),
I,<0
I,>O,
and
s, z=-0,
S=S(Y, r), Income
adjusts
I;= cY(l- S),
according
s, > 0. to excess demand
in the goods market,
i.e.,
a > 0.
(1)
The set of points {(Y, r) 1Z( Y, r) = S( Y, r)} constitutes the IS-curve of the model. Let L( Y, r) denote the liquidity preference with L y > 0, L, < 0 and assume that the interest adjusts according to
r)-Wp),
+=@(y,
P>O,
rate
(2)
with the set of points {(Y, r) 1L(Y, r) = a/p} forming the LM-curve of the model. Let (Y*, r * ) be the unique equilibrium of the system, i.e., Y, ( ,,*, r * ) = i, cu*, r* ) = 0, and consider the Jacobian of the system evaluated at (Y *, Y* ):
J=
41,
- s,> PL,
4
Ir - s,.> PL,
1.
(3)
As is well known, the equilibrium (Y *, r *) is asymptotically (locally) stable if tr J < 0 and det J > 0. It can be demonstrated by means of the Poincare-Bendixson theorem 3 or the Hopf-bifuris able to generate oscillating behavior. By means of the cation theorem 4 that system (l)-(2) Poincare-Bendixson theorem the following result can be established: Proposition 1. Consider a compact subset D c R2. If the unique equilibrium (Y *, r *) is locally unstable and if the vector field (l)-(2) points inwards the set D on the boundary of D, then there exists at least one closed orbit of (I)-(2) in D which encloses the equilibrium. In most cases such a compact set D can easily be found if the trace of J, i.e., LY(I, - S,) + PL,, changes its sign twice in an appropriate set W c R2. Compare, e.g., Chang and Smyth (1971) or Gabisch and Lorenz (1987) for details on the proof of the existence of limit cycles. Alternatively, the Hopf-bifurcation theorem can be applied to (l)-(2) if, e.g., (Y is interpreted as a bifurcation parameter, and if the eigenvalues of (l)-(2) cross the imaginary axis, i.e., if tr J = 0 for a bifurcation value (Ye. Obviously the requirements are fulfilled if I( Y, . ) and S( Y,.) are Kaldor-type sigmoid functions and if the parameters cx and /I possess appropriate values. Besides the Kaldor-assumption about the ’ Compare Hirsch and Smale (1974, p. 248). 4 Compare Guckenheimer and Holmes (1983, p. 151 ff).
137
H.-W. Lorent / International trade and the possible occurrence of chaos
relation between ZY and S, other specifications of the functions I, S, and L exist for which the interactions of the partials I,, S,, and L, and the parameters lead to the same result.
3. International trade as a perturbation of internal oscillations Consider three economies, each of which is described different numerical specifications of the functions, i.e.,
c =q(z,(r,, r,>- Si(X> r,>>7
J,=&(L,(y,,
by equations
r,)-M/p,)
like (l)-(2)
with possibly
i=l,2,3.
Eq. (4) constitutes a six-dimensional differential equation system which can also be written as a system of three independent two-dimensional limit cycle oscillators. If all three economies are indeed oscillating, the overall motion of eq. (4) constitutes a motion on a three-dimensional torus T3, which is a geometric object in R6. By introducing international trade with linear functions Ex, = Ex,( Y,, Y,), i #j, k and Zm, = Zm, (y ), eq. (4) becomes
c=a,(z,(r,,
r,)-S,(y.,
r,)+Ex;(Y,,
with i, j, k = 1, 2, 3; j, k Z equilibria. Eq. (5) constitutes demonstrated by Newhouse, three-dimensional torus may following properties: 5 _ _ _
Y,)-h,(X)),
t,=P,(L,(Y,,~,)-@/zA)~
(5)
i, and d, as the money supplies reflecting balance of payments a system of three linearly coupled limit cycle oscillators. It has been Ruelle and Takens (1978) that a perturbation of a motion on a face a strange attractor. A strange attractor is characterized by the
It is an attractor. It does not consist of a single point or a closed orbit. It is indecomposable. There exists a sensitive dependence on initial values. The existence
of a strange
attractor
implies
chaotic
Proposition 2. Zf all three autonomous economies may imply the existence of a strange attractor.
trajectories.
are oscillating the introduction
of international
trade
Eq. (5) constitutes a system of coupled non-linear oscillators which can be understood as a perturbation of the motion of the autonomous economies on a three-dimensional torus. The Newhouse-Ruelle-Takens theorem therefore implies that the international trade system (5) may face a strange attractor. Whether or not there are indeed strange attractors and hence chaotic trajectories in a specified model can be established only by numerical techniques. Simulation results in Lorenz (1987) indicate that economically reasonable specifications in a model formally similar to the present one can be found which indeed imply broad band noise in the associated power spectra. However, the emergence of strange attractors is not exclusive in models like these: some variations in the parameters can lead to the occurrence of other phenomena like quasi-periodic motion or phase-locking. ’ Compare Ruelle (1979) for a detailed description. chaotic motion.
See also Eckmann (1981) for a survey of different scenarios leading to
138
H.-W. Lorenr / International
trade and the possible occurrence of chaos
4. Conclusion It has been demonstrated that it is easily possible to construct economic models whose dynamic behavior is described by a motion on an n-dimensional torus (n 2 3). Perturbations of this motion in the form of, e.g., introducing international trade among at least three oscillating economies can lead to the occurrence of strange attractors and hence of chaotic motion. The result is intuitively not implausible because the trade activities, i.e., especially the export quantities, can be understood as external forces disturbing the harmonic internal oscillation. However, the chaotic motion should not be confused with noise superimposed on the harmonic motion: the chaotic trajectories may display amplitudes which are drastically different from those of the harmonic oscillation. The semantically tieak expression possible occurrence has not been chosen in order to express a rather improbable emergence of strange dynamics in this model. On the contrary, the occurence of strange dynamics can be the typical phenomenon. However, it cannot be excluded that other kinds of dynamic behavior emerge in the same model as well. The result of the model presented above does not rely on the special structural form of the functions (l)-(2). Basically, every two-dimensional oscillator describing the dynamic behavior of an economic entity like an autonomous economy, and isolated market, or a self-sustaining sector of an economy, can serve as the starting point in an appropriate interactive model featuring strange attractors.
References Chang, W.W. and D.J. Smyth, 1971, The existence and persistence of cycles in a nonlinear model: Kaldor’s 1940 model re-examined, Review of Economic Studies 38, 37-44. Day, R.H., 1982, Irregular growth cycles, American Economic Review 72, 406-414. Day, R.H.. 1983, The emergence of chaos from classical economic growth, Quarterly Journal of Economics 98, 201-213. Day, R.H. and W. Shafer, 1985, Keynesian chaos, Journal of Macroeconomics 7, 277-295. Day, R.H. and W. Shafer, 1986, Ergodic fluctuations in deterministic economic models, MRG working paper no. 8631 (University of Southern California, Los Angeles, CA). Eckmann, J.P., 1981, Roads to turbulence in dissipative dynamical systems, Reviews of Modern Physics 53, 643-654. Gabisch, G. and H.-W. Lorenz, 1987, Business cycle theory (Springer, New York). Grandmont, J.-M., 1985, On endogenous competitive business cycles, Econometrica 53, 995-1045. Guckenheimer, J. and P. Holmes, 1983, Non-linear oscillations, dynamical systems, and bifurcations of vector fields (Springer, New York). Hirsch, M.W. and S. Smale, 1974, Differential equations, dynamical systems, and linear algebra (Academic Press, New York). Lorenz, H.-W., 1987, Strange attractors in a multisector business cycle model, Journal of Economic Behavior and Organization, forthcoming. Newhouse, S., D. Ruelle and F. Takens, 1978, Occurrence of strange axiom A attractors near quasi-periodic flows on Tm, M 2 3, Communications in Mathematical Physics 64, 35-40. Ruelle, D., 1979. Strange attractors, Mathematical Intelligence 2, 1266137. Stutzer, M., 1980, Chaotic dynamics and bifurcation in a macro-model, Journal of Economic Dynamics and Control 2, 253-276. Torre, V., 1977, Existence of limit cycles and control in complete Keynesian systems by theory of bifurcations, Econometrica 45. 1457-1466.
23 (1987) 135-138
INTERNATIONAL Hans-Walter
135
TRADE AND THE POSSIBLE
OCCURRENCE
OF CHAOS
LORENZ
Georg-August-Uniuersitiit Giittingen, 34 Gijttingen, Federal Republic of Germany University of Southern California, Los Angeles, CA 90089-0035, USA Received
8 October
1986
The paper demonstrates that attractor and hence of chaos. considered as a perturbation Newhouse, Ruelle and Takens
international trade among fluctuating economies may involve the occurrence of a strange Once autonomous economies display cyclical behavior, international trade activities can be of the motion of the autonomous economies, possible implying chaos via a theorem by (1979).
1. Introduction Recent work on chaotic dynamical systems ’ in economics has concentrated mainly on the investigation of non-linear systems framed in one-dimensional difference equations [e.g., Day (1982, 1983), Day and Shafer (1985,1986), Grandmont (1985) and Stutzer (1980)]. While the establishment of chaotic trajectories in these models can be relatively easily performed by means of the Li-Yorke theorem or Sarkovskii’s theorem, the mathematical requirement of non-invertible maps usually implies model structures which do not seem to be generic in economics. * Alternatively, the literature on dynamical systems in physics and chemistry deals with some examples of higher-dimensional non-linear differential equation systems displaying chaotic trajectories in which non-linearities are involved that seem to be more compatible with standard assumptions in dynamical economics than those involved in one-dimensional models. While the occurrence of chaos in continuous-time systems is restricted to the case of at least three dimensions, the analytically most accessible case seems to be that of a six-dimensional system consisting of three coupled non-linear two-dimensional limit cycle oscillators. In the following, a simple example will be provided showing how such a system naturally emerges in a model of international trade among three different economies.
2. Oscillations in autonomous
economies
In order to generate chaotic dynamics in the international trade model presented below, the involved economies must display oscillations in the case of autonomy. The dynamic behavior of the following model of an autonomous economy was extensively studied by Torre (1977) who established the existence of closed orbits by means of bifurcation theory. Consider a very simple ’ Compare ’ However,
Day (1982) for a verbal description of discrete chaotic dynamics. compare Day and Shafer (1986) for an intrinsic non-linearity in the standard
0165-1765/87/$3.50
0 1987, Elsevier Science Publishers
B.V. (North-Holland)
IS-L&Z-model.
136
H.-W. Lorenz / International
trade and
thepossible occurrence of chaos
Keynesian macroeconomic model of a single economy with Y as income, r as the interest rate, M as the (constant) nominal money supply, and assume that the goods prices, p, are fixed during the relevant time interval. Suppose that gross investment, I, and savings, S, depend both on income and the interest rate in the familiar way, i.e.,
I=I(Y, r),
I,<0
I,>O,
and
s, z=-0,
S=S(Y, r), Income
adjusts
I;= cY(l- S),
according
s, > 0. to excess demand
in the goods market,
i.e.,
a > 0.
(1)
The set of points {(Y, r) 1Z( Y, r) = S( Y, r)} constitutes the IS-curve of the model. Let L( Y, r) denote the liquidity preference with L y > 0, L, < 0 and assume that the interest adjusts according to
r)-Wp),
+=@(y,
P>O,
rate
(2)
with the set of points {(Y, r) 1L(Y, r) = a/p} forming the LM-curve of the model. Let (Y*, r * ) be the unique equilibrium of the system, i.e., Y, ( ,,*, r * ) = i, cu*, r* ) = 0, and consider the Jacobian of the system evaluated at (Y *, Y* ):
J=
41,
- s,> PL,
4
Ir - s,.> PL,
1.
(3)
As is well known, the equilibrium (Y *, r *) is asymptotically (locally) stable if tr J < 0 and det J > 0. It can be demonstrated by means of the Poincare-Bendixson theorem 3 or the Hopf-bifuris able to generate oscillating behavior. By means of the cation theorem 4 that system (l)-(2) Poincare-Bendixson theorem the following result can be established: Proposition 1. Consider a compact subset D c R2. If the unique equilibrium (Y *, r *) is locally unstable and if the vector field (l)-(2) points inwards the set D on the boundary of D, then there exists at least one closed orbit of (I)-(2) in D which encloses the equilibrium. In most cases such a compact set D can easily be found if the trace of J, i.e., LY(I, - S,) + PL,, changes its sign twice in an appropriate set W c R2. Compare, e.g., Chang and Smyth (1971) or Gabisch and Lorenz (1987) for details on the proof of the existence of limit cycles. Alternatively, the Hopf-bifurcation theorem can be applied to (l)-(2) if, e.g., (Y is interpreted as a bifurcation parameter, and if the eigenvalues of (l)-(2) cross the imaginary axis, i.e., if tr J = 0 for a bifurcation value (Ye. Obviously the requirements are fulfilled if I( Y, . ) and S( Y,.) are Kaldor-type sigmoid functions and if the parameters cx and /I possess appropriate values. Besides the Kaldor-assumption about the ’ Compare Hirsch and Smale (1974, p. 248). 4 Compare Guckenheimer and Holmes (1983, p. 151 ff).
137
H.-W. Lorent / International trade and the possible occurrence of chaos
relation between ZY and S, other specifications of the functions I, S, and L exist for which the interactions of the partials I,, S,, and L, and the parameters lead to the same result.
3. International trade as a perturbation of internal oscillations Consider three economies, each of which is described different numerical specifications of the functions, i.e.,
c =q(z,(r,, r,>- Si(X> r,>>7
J,=&(L,(y,,
by equations
r,)-M/p,)
like (l)-(2)
with possibly
i=l,2,3.
Eq. (4) constitutes a six-dimensional differential equation system which can also be written as a system of three independent two-dimensional limit cycle oscillators. If all three economies are indeed oscillating, the overall motion of eq. (4) constitutes a motion on a three-dimensional torus T3, which is a geometric object in R6. By introducing international trade with linear functions Ex, = Ex,( Y,, Y,), i #j, k and Zm, = Zm, (y ), eq. (4) becomes
c=a,(z,(r,,
r,)-S,(y.,
r,)+Ex;(Y,,
with i, j, k = 1, 2, 3; j, k Z equilibria. Eq. (5) constitutes demonstrated by Newhouse, three-dimensional torus may following properties: 5 _ _ _
Y,)-h,(X)),
t,=P,(L,(Y,,~,)-@/zA)~
(5)
i, and d, as the money supplies reflecting balance of payments a system of three linearly coupled limit cycle oscillators. It has been Ruelle and Takens (1978) that a perturbation of a motion on a face a strange attractor. A strange attractor is characterized by the
It is an attractor. It does not consist of a single point or a closed orbit. It is indecomposable. There exists a sensitive dependence on initial values. The existence
of a strange
attractor
implies
chaotic
Proposition 2. Zf all three autonomous economies may imply the existence of a strange attractor.
trajectories.
are oscillating the introduction
of international
trade
Eq. (5) constitutes a system of coupled non-linear oscillators which can be understood as a perturbation of the motion of the autonomous economies on a three-dimensional torus. The Newhouse-Ruelle-Takens theorem therefore implies that the international trade system (5) may face a strange attractor. Whether or not there are indeed strange attractors and hence chaotic trajectories in a specified model can be established only by numerical techniques. Simulation results in Lorenz (1987) indicate that economically reasonable specifications in a model formally similar to the present one can be found which indeed imply broad band noise in the associated power spectra. However, the emergence of strange attractors is not exclusive in models like these: some variations in the parameters can lead to the occurrence of other phenomena like quasi-periodic motion or phase-locking. ’ Compare Ruelle (1979) for a detailed description. chaotic motion.
See also Eckmann (1981) for a survey of different scenarios leading to
138
H.-W. Lorenr / International
trade and the possible occurrence of chaos
4. Conclusion It has been demonstrated that it is easily possible to construct economic models whose dynamic behavior is described by a motion on an n-dimensional torus (n 2 3). Perturbations of this motion in the form of, e.g., introducing international trade among at least three oscillating economies can lead to the occurrence of strange attractors and hence of chaotic motion. The result is intuitively not implausible because the trade activities, i.e., especially the export quantities, can be understood as external forces disturbing the harmonic internal oscillation. However, the chaotic motion should not be confused with noise superimposed on the harmonic motion: the chaotic trajectories may display amplitudes which are drastically different from those of the harmonic oscillation. The semantically tieak expression possible occurrence has not been chosen in order to express a rather improbable emergence of strange dynamics in this model. On the contrary, the occurence of strange dynamics can be the typical phenomenon. However, it cannot be excluded that other kinds of dynamic behavior emerge in the same model as well. The result of the model presented above does not rely on the special structural form of the functions (l)-(2). Basically, every two-dimensional oscillator describing the dynamic behavior of an economic entity like an autonomous economy, and isolated market, or a self-sustaining sector of an economy, can serve as the starting point in an appropriate interactive model featuring strange attractors.
References Chang, W.W. and D.J. Smyth, 1971, The existence and persistence of cycles in a nonlinear model: Kaldor’s 1940 model re-examined, Review of Economic Studies 38, 37-44. Day, R.H., 1982, Irregular growth cycles, American Economic Review 72, 406-414. Day, R.H.. 1983, The emergence of chaos from classical economic growth, Quarterly Journal of Economics 98, 201-213. Day, R.H. and W. Shafer, 1985, Keynesian chaos, Journal of Macroeconomics 7, 277-295. Day, R.H. and W. Shafer, 1986, Ergodic fluctuations in deterministic economic models, MRG working paper no. 8631 (University of Southern California, Los Angeles, CA). Eckmann, J.P., 1981, Roads to turbulence in dissipative dynamical systems, Reviews of Modern Physics 53, 643-654. Gabisch, G. and H.-W. Lorenz, 1987, Business cycle theory (Springer, New York). Grandmont, J.-M., 1985, On endogenous competitive business cycles, Econometrica 53, 995-1045. Guckenheimer, J. and P. Holmes, 1983, Non-linear oscillations, dynamical systems, and bifurcations of vector fields (Springer, New York). Hirsch, M.W. and S. Smale, 1974, Differential equations, dynamical systems, and linear algebra (Academic Press, New York). Lorenz, H.-W., 1987, Strange attractors in a multisector business cycle model, Journal of Economic Behavior and Organization, forthcoming. Newhouse, S., D. Ruelle and F. Takens, 1978, Occurrence of strange axiom A attractors near quasi-periodic flows on Tm, M 2 3, Communications in Mathematical Physics 64, 35-40. Ruelle, D., 1979. Strange attractors, Mathematical Intelligence 2, 1266137. Stutzer, M., 1980, Chaotic dynamics and bifurcation in a macro-model, Journal of Economic Dynamics and Control 2, 253-276. Torre, V., 1977, Existence of limit cycles and control in complete Keynesian systems by theory of bifurcations, Econometrica 45. 1457-1466.