Studies of-a Model of Entrainment between
Economic Cycles
Juan Herndndez-Guerra. *
April 1996
Summary
This work carries out an analytic study of a two-sector model previously
developed by Kampmann et al. (1993). This model explains why two capital
producing sectors with initially different cyclical motions can entrain to only
one cyclical mode. Some new insights into the motion of the model will be
shown. All the introduced assumptions and results were tested by simulations.
1 Introduction
The model consists of ten ordinary differential equations including the following
variables: K’p,, that is the capital stock produced by sector q located in sector p,
By, that is the sector p’s total orders backlog, and 5,,, that indicates the supply
line of capital produced by sector q to sector p (p,q € {1,2}). The two sectors are
connected by two specific parameters. They are the difference between each sector’s
capital lifetime, Av, and the dependence between sectors a, 0< a <1.
The objective of our work is to study analytically the extreme cases of the model
(a=0, a=1). Case a=0 is important because the system changes from a totally
independent sectors to dependent ones. Then, possibly this point will define a
bifurcation. Case a=1 is also relevant. In this situation, each sector’s production
depends on the capital produced by the other sector solely. It is observed by
simulations (Kampmann et al. 1993, Haxholdt et al. 1994) that synchronization
fills all the (a, Ar) phase portrait for high values of a. By means of the analytic
“Universidad de Las Palmas de Gran Canaria. Phone: (34) 28451803.
E.mail:juanh@empresariales.ulpgc.es
148
treatment, we will show the cause of this phenomenon.
2 Analytic Study of the Model
Using similarities with Sterman’s model (1985), we can consider the following
equation:
By =Sep+Spp+Sa, 3 Wp, € {1,2}, p44 (2.1)
where Sg, is the supply line from sector p to goods producing sector. The equality
has been verified by simulations run in the original model.
We will also take one sector as exogenous. So, we will reduce the number of the
equations and emphasize essential properties in the model. This will be observed in
each particular analysis.
After incorporating the new assumptions, the ten equations are transformed into:
(Note the sector treated as exogenous using subindex i, and the sector treated as
non exogenous using subindex j)
Kj; ES -ZpGf a
K
Kjj = ay a5 (2.2)
Be = Oj — (ti -2f
By = 03; — (tj - 25
where x (h=i or j) is the proportional part of sector h’s production corresponding
to the exogenous sectors. We will consider it as exogenous.
e Independence between Sectors, a = 0:
Incorporating a=0 in 2.2, we obtain two groups of decoupled equations. One
group includes the variables (B;, Kj;) and the other group the variables (B;, K;:).
It is easy to observe that (B;, Kj;) system achieves a period variable limit cycle
depending on the parameters governing sector j. On the other hand, (B;, K;:)
system includes only one globally asymptotically stable equilibrium point, that is
(Bj, Kj) = (Se,,0)-
However, it can be proven that the equilibrium point is not hyperbolic, so the
structural stability of the model with a=0 is not assured. We will go back on this
1944
item later.
¢ Total Dependence between Sectors, (a = 1):
Forcing a=1, we obtain four no decoupled equations. The only equilibrium point
in the system is (Bj, Kj;, Bi, Kj:) = (Se,,0, 6:(2? + 224), #5.;)- It can be proven
that this equilibrium point is locally asymptotically stable if the following conditions
over the parameters are verified:
5 E> ts (2.3)
Realistic values of the parameters verify this condition (Sterman 1989).
In order to study the global motion of the system in this point we are going to
analyze the groups of variables (B;,Kj;) and (B;,Aj;) separately. So, it is easy to
prove that (B,.K,,) = (S.,,0) is the only equilibrium point in the first group and it
is globally asymptotically stable. The other group admits only one equilibrium point,
note (B,,A,,) (if Bj = By, then (B;, Kj) = (By, Ky:)). So, using some Lemmas in
Theory of Dynamical System (compare Hirsch and Smale 1974), we can assert that
the equilibrium point is globally asymptotically stable. Then, (B;,Kj;, Bi, Aji) is
globally asymptotically stable.
It can also be proven that the motion of the model is qualitatively persistent
under small perturbations in the parameters defining the system. So, when a small
change of parameter a occurs (a < 1, a = 1), only one equilibrium point of the
new model can still find out. This point is close to (B;, Kj;, Bi, Ky) and globally
asymptotically stable too.
In accordance with above, we can assert that when one sector is in equilibrium
(exogenous), the other one will be necessarily in equilibrium. Thus it is shown that
the motion of each sector pulls the motion of the other sector. That is the cause of
why synchronization fills the (a, \r) phase portrait for high values of a.
3 Discussion
The analytical study is a quite good tool to find out some properties that simulation
can not do. Here the model was not easy to attack, however we can draw some new
08
results from this paper.
First, we have delimited a region in the parametric space where the equilibrium
point in case a=] is asymptotically stable (equations 2.3). Moreover, starting from
the equilibrium point when a = 0 is not hyperbolic, a necessary condition of a
bifurcation is obtained. Hence, we can support the idea that entrainment arise since
any value of a > 0. On the other hand, structural stability of the system in case
a=1 lead us to infer that synchronization fills the (a, Ar) phase portrait for a no
much less than one and any Ar.
The model can be extended including more sectors and some new economic
linkages. There are some new results on this line (Kampmann 1996). To incorporate
another macroeconomic linkages is our next goal. Then the use of simulated and,
if it was feasible. analytical methods would help us to study the motion of the new
model.
4 References
Haxholdt. C.. Nampmann, C., Mosekilde, E. and Sterman, J.D. 1994. Mode locking
and entrainment of endogenous economic cycles. WP-3646-94-MSA, Sloan
School of Management. MIT.
Hirsch. M.W. and Smale, S. 1974. Differential equations, dynamical systems, and
linear algebra. Academic Press, Inc.
Kampmann. C., Haxholdt, C., Mosekilde. E. and Sterman, J.D. 1993. Entrainment
in a disaggregated economic long-wave model. Loet Leydesdorff and Peter Van
den Besselaar. ed. “Evolutionary economics and chaos theory” London: Pinter
Publishers.
Kampmann. C. 1996. Preprint.
Sterman, J.D. 1985. A behavioral model of the economic long wave. Journal of
Economic Behavior and Organization 6: 17-53.
—_—. 1989. Misperceptions of feedback in dynamic decision making.
Organizational Behavior and Human Decision Processes 43: 301-335.
o\
CC BY-NC-SA 4.0