Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue5-7.Jul2000:
Journal of Economic Dynamics & Control
24 (2000) 935}963
High order disequilibrium growth dynamics:
Theoretical aspects and numerical features
Carl Chiarella!, Peter Flaschel",*
!School of Finance and Economics, University of Technology, Sydney, P.O. Box 123, Broadway, NSW
2007, Australia
"Faculty of Economics, University of Bielefeld, Bielefeld, P.O. Box 10 01 31, 33501 Bielefeld, Germany
Accepted 30 April 1999
Abstract
We investigate an open monetary growth model with sluggish prices and quantities.
The model combines the dynamics of Rose's employment cycle and Metzler's inventory
cycle with internal nominal dynamics of Tobin and external nominal dynamics of
Dornbusch type, implying eight laws of motion, four for the real sector and four for the
nominal part. These intrinsically nonlinear 8D-dynamics are asymptotically stable for
low adjustment speeds of prices and expectations, give rise to Hopf-bifurcations as
adjustment parameters are increased and explosive behavior thereafter. Extrinsic nonlinearities are therefore added, one in capital #ows and one in wage behavior. These
nonlinearities modify the dynamics radically, limiting them to domains with economically plausible outcomes, also for extreme parameter choices, where the dynamics may
become chaotic. ( 2000 Elsevier Science B.V. All rights reserved.
JEL classixcation: E12; E32
Keywords: Monetary growth; Open economies; Employment cycles; Relaxation
oscillations; Complex dynamics
* Corresponding author. Tel.: #49-521-106-5114/6926; fax: #49-521-106-6416.
E-mail addresses: [email protected] (C. Chiarella), p#[email protected]
(P. Flaschel)
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 3 1 - 7
936 C. Chiarella, P. Flaschel / Journal of Economic Dynamics & Control 24 (2000) 935}963
1. Introduction
In this paper we motivate and analyze a disequilibrium monetary growth
model of a small open economy. The model consists of the dynamic interaction
of a real sector and a nominal one. The dynamics of the real sector are
determined by employment and labor intensity dynamics and an inventory
dynamics. In the nominal part of the model price and in#ationary expectations
dynamics interact with dynamics of the foreign exchange rate and expectations
of exchange rate depreciation. The resulting model is expressed as an 8D
dynamical system which incorporates sluggish price and quantity adjustments,
allows for #uctuations in both capital and labor utilization and allows for
international trade in goods as well as "nancial assets.
Our aim is to understand the main stabilizing and destabilizing economic
forces driving the dynamics of the model and to analyze their potential to
generate complex dynamic behavior.
In Section 2 we lay out and motivate the eight di!erential equations governing the dynamics of our model. In Section 3 we discuss the "ve main economic
feedback chains, the Rose e!ect, the Mundell e!ect, the Metzler e!ect, the
Dornbusch e!ect and the Keynes e!ect and show that their con#icting stabilizing and destabilizing in#uences drive the dynamic behavior of the model.
We show that eigenvalue analysis indicates that local stability is lost via
Hopf bifurcations in a way that is dependent in particular on the speeds of
adjustment of prices and expectations. In this section we also discuss the
intrinsic (or &natural') nonlinear features of the model. Simulations reveal,
however, that the aforementioned intrinsic nonlinearities are generally
not su$cient to bound the dynamics when the equilibrium is locally
unstable. Therefore, in Section 4 we introduce (and motivate) an extrinsic
nonlinearity into the function modeling net capital #ows by taking account of
the fact that these are bounded by international wealth. This extrinsic nonlinearity in conjunction with rapid speeds of adjustment of exchange rates and of
expectations of exchange rate depreciation give rise (close to the limiting case of
myopic perfect foresight) to a relaxation oscillation between the exchange rate
and its expected rate of depreciation. Simulations reveal that movements away
from the locally unstable equilibrium remain bounded on some sort of complex
attractor. However, high frequency of movements in the foreign exchange sector
here lead to unrealistic high-frequency movements in the real sector of the
model.
In Section 5 the frequency of the movements in the real sector are made more
realistic by introducing a ninth di!erential equation which allows for sluggish
adjustment of the trade balance based on a sluggish adjustment of the terms of
trade that governs imports and exports. Simulations here reveal motion to
high-order limit cycles, but now with #uctuations in the real sector which exhibit
more realistic frequencies.
C. Chiarella, P. Flaschel / Journal of Economic Dynamics & Control 24 (2000) 935}963 937
In Section 6 we introduce a further nonlinearity, this time both into the real
and the nominal part of the economy, namely that nominal wage de#ation (and
thus part of real wage determination) is subject to some kind of #oor. This
nonlinearity also gives rise to high-order limit cycles and also complex attractors, in particular when coupled with the other modi"cations of the model
discussed above. Period-doubling routes to such complex attractors are considered in Section 7. Section 8 draws some conclusions and makes suggestions
for further research.
2. Disequilibrium growth in small open economies
We consider a disequilibrium model of monetary growth of an open economy
with sluggish adjustment of all prices and of output (coupled with imbalances in
the utilization rates of both labor and capital) and where real and "nancial
markets interact, here primarily by way of international capital mobility and the
trade balance. The state variables of the model are u"w/p, the real wage,
l"¸/K, the labor}capital ratio, p, the price level, n, the expected rate of
in#ation, ye">e/K, sales expectations per unit of capital, l"N/K, inventories
per unit of capital, e, the nominal exchange rate, and e, the expected rate of
change of the exchange rate. These state variables are fundamental for any
disequilibrium approach to monetary growth with sluggish price as well as
quantity adjustments on the market for labor, for goods and also to some extent
on the market for foreign exchange. We show in Chiarella and Flaschel (1999b)
that the evolution of these state variables is governed by the dynamical system
(1)}(8) below.
These equations are based on growth laws in four cases (x( the growth rate of
a variable x) and on simple time derivatives in the four remaining laws of
motion. They have to be inserted into each other in three cases. One has to make
use of the static relationships shown below in addition in order to obtain an
explicit representation of this system as an autonomous eight-dimensional
system of di!erential equations.
u( "i[(1!i )b (/K, for aggregate demand per
unit of capital yd">d/K and for the trade balance per unit of capital nx the
expressions:
y"ye#nb dye#b (b dye!l),
n
n n
(9)
yd"(1!q )uy/x#c (g)(1!s )(oe!tn)#cH(g)
1
w
c
c
# i (oe!r#n)#i (;!;M )#n#d#g,
1
2
nx"cH(g)!(1!c (g))(1!s )(oe!tn),
1
c
c
(10)
(11)
C. Chiarella, P. Flaschel / Journal of Economic Dynamics & Control 24 (2000) 935}963 939
and have employed as abbreviations in the presentation of the above dynamics:
24 (2000) 935}963
High order disequilibrium growth dynamics:
Theoretical aspects and numerical features
Carl Chiarella!, Peter Flaschel",*
!School of Finance and Economics, University of Technology, Sydney, P.O. Box 123, Broadway, NSW
2007, Australia
"Faculty of Economics, University of Bielefeld, Bielefeld, P.O. Box 10 01 31, 33501 Bielefeld, Germany
Accepted 30 April 1999
Abstract
We investigate an open monetary growth model with sluggish prices and quantities.
The model combines the dynamics of Rose's employment cycle and Metzler's inventory
cycle with internal nominal dynamics of Tobin and external nominal dynamics of
Dornbusch type, implying eight laws of motion, four for the real sector and four for the
nominal part. These intrinsically nonlinear 8D-dynamics are asymptotically stable for
low adjustment speeds of prices and expectations, give rise to Hopf-bifurcations as
adjustment parameters are increased and explosive behavior thereafter. Extrinsic nonlinearities are therefore added, one in capital #ows and one in wage behavior. These
nonlinearities modify the dynamics radically, limiting them to domains with economically plausible outcomes, also for extreme parameter choices, where the dynamics may
become chaotic. ( 2000 Elsevier Science B.V. All rights reserved.
JEL classixcation: E12; E32
Keywords: Monetary growth; Open economies; Employment cycles; Relaxation
oscillations; Complex dynamics
* Corresponding author. Tel.: #49-521-106-5114/6926; fax: #49-521-106-6416.
E-mail addresses: [email protected] (C. Chiarella), p#[email protected]
(P. Flaschel)
0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 3 1 - 7
936 C. Chiarella, P. Flaschel / Journal of Economic Dynamics & Control 24 (2000) 935}963
1. Introduction
In this paper we motivate and analyze a disequilibrium monetary growth
model of a small open economy. The model consists of the dynamic interaction
of a real sector and a nominal one. The dynamics of the real sector are
determined by employment and labor intensity dynamics and an inventory
dynamics. In the nominal part of the model price and in#ationary expectations
dynamics interact with dynamics of the foreign exchange rate and expectations
of exchange rate depreciation. The resulting model is expressed as an 8D
dynamical system which incorporates sluggish price and quantity adjustments,
allows for #uctuations in both capital and labor utilization and allows for
international trade in goods as well as "nancial assets.
Our aim is to understand the main stabilizing and destabilizing economic
forces driving the dynamics of the model and to analyze their potential to
generate complex dynamic behavior.
In Section 2 we lay out and motivate the eight di!erential equations governing the dynamics of our model. In Section 3 we discuss the "ve main economic
feedback chains, the Rose e!ect, the Mundell e!ect, the Metzler e!ect, the
Dornbusch e!ect and the Keynes e!ect and show that their con#icting stabilizing and destabilizing in#uences drive the dynamic behavior of the model.
We show that eigenvalue analysis indicates that local stability is lost via
Hopf bifurcations in a way that is dependent in particular on the speeds of
adjustment of prices and expectations. In this section we also discuss the
intrinsic (or &natural') nonlinear features of the model. Simulations reveal,
however, that the aforementioned intrinsic nonlinearities are generally
not su$cient to bound the dynamics when the equilibrium is locally
unstable. Therefore, in Section 4 we introduce (and motivate) an extrinsic
nonlinearity into the function modeling net capital #ows by taking account of
the fact that these are bounded by international wealth. This extrinsic nonlinearity in conjunction with rapid speeds of adjustment of exchange rates and of
expectations of exchange rate depreciation give rise (close to the limiting case of
myopic perfect foresight) to a relaxation oscillation between the exchange rate
and its expected rate of depreciation. Simulations reveal that movements away
from the locally unstable equilibrium remain bounded on some sort of complex
attractor. However, high frequency of movements in the foreign exchange sector
here lead to unrealistic high-frequency movements in the real sector of the
model.
In Section 5 the frequency of the movements in the real sector are made more
realistic by introducing a ninth di!erential equation which allows for sluggish
adjustment of the trade balance based on a sluggish adjustment of the terms of
trade that governs imports and exports. Simulations here reveal motion to
high-order limit cycles, but now with #uctuations in the real sector which exhibit
more realistic frequencies.
C. Chiarella, P. Flaschel / Journal of Economic Dynamics & Control 24 (2000) 935}963 937
In Section 6 we introduce a further nonlinearity, this time both into the real
and the nominal part of the economy, namely that nominal wage de#ation (and
thus part of real wage determination) is subject to some kind of #oor. This
nonlinearity also gives rise to high-order limit cycles and also complex attractors, in particular when coupled with the other modi"cations of the model
discussed above. Period-doubling routes to such complex attractors are considered in Section 7. Section 8 draws some conclusions and makes suggestions
for further research.
2. Disequilibrium growth in small open economies
We consider a disequilibrium model of monetary growth of an open economy
with sluggish adjustment of all prices and of output (coupled with imbalances in
the utilization rates of both labor and capital) and where real and "nancial
markets interact, here primarily by way of international capital mobility and the
trade balance. The state variables of the model are u"w/p, the real wage,
l"¸/K, the labor}capital ratio, p, the price level, n, the expected rate of
in#ation, ye">e/K, sales expectations per unit of capital, l"N/K, inventories
per unit of capital, e, the nominal exchange rate, and e, the expected rate of
change of the exchange rate. These state variables are fundamental for any
disequilibrium approach to monetary growth with sluggish price as well as
quantity adjustments on the market for labor, for goods and also to some extent
on the market for foreign exchange. We show in Chiarella and Flaschel (1999b)
that the evolution of these state variables is governed by the dynamical system
(1)}(8) below.
These equations are based on growth laws in four cases (x( the growth rate of
a variable x) and on simple time derivatives in the four remaining laws of
motion. They have to be inserted into each other in three cases. One has to make
use of the static relationships shown below in addition in order to obtain an
explicit representation of this system as an autonomous eight-dimensional
system of di!erential equations.
u( "i[(1!i )b (/K, for aggregate demand per
unit of capital yd">d/K and for the trade balance per unit of capital nx the
expressions:
y"ye#nb dye#b (b dye!l),
n
n n
(9)
yd"(1!q )uy/x#c (g)(1!s )(oe!tn)#cH(g)
1
w
c
c
# i (oe!r#n)#i (;!;M )#n#d#g,
1
2
nx"cH(g)!(1!c (g))(1!s )(oe!tn),
1
c
c
(10)
(11)
C. Chiarella, P. Flaschel / Journal of Economic Dynamics & Control 24 (2000) 935}963 939
and have employed as abbreviations in the presentation of the above dynamics: