utwci
MIT. LIBRARIES -
{on
ALFRED
Mode
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Locking and Entrainment of Endogenous
Econonnic Cycles
Christian Haxholdt, Christian
Erik Mosekilde
WP#
Kampmann
and John D. Sterman
3646-94-MSA
January, 1994
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Mode
Locking and Entrainment of Endogenous
Economic Cycles
Christian Haxholdt, Christian
Erik Mosekilde
WP#
Kampmann
and John D. Sterman
3646-94-MSA
January, 1994
Ll.T.
FEB
LIBRARIES
1 1 1994
RECEIVED
D-43
I
Mode
Lockins:
o and Entrainment of
Endogenous Economic Cycles
Christian Haxholdt'
".
Christian
Kampmann"
Erik Mosekildc"'. and John D. Sterman'^
January 1994
Abstract
We
explore
t
he robustness of aggregation
wave. The original model aggregates
and generates a
50 years.
We
all
in
Sterman's model of the economic long
capital producing firms into a single sector
large amplitude self-sustained oscillation with a period of roughly
disaggregate the model into two coupled industries, one representing
production of plant and long-lived infrastructure and the other short-lived equipment
and machinery. While holding the aggregate equilibrium characteristics of the model
constant, we investigate
how mode-locking occurs
as a function of the difference in
capital lifetimes and the strength of the coupling between the sectors.
tion allows
new modes
of behavior to arise: In addition to mode-locking
Disaggrega-
we observe
cascades of period-doubling bifurcations, chaos, intermittency. and quasi-periodic behavior. Despite the introduction of these additional modes, the basic behavior of the
model
is
robust to the aggregation assumption.
We
consider the likely effects ot finer
disaggregation, the introduction of additional coupling mechanisms such as prices.
and other avenues
for the
exploration of aggregation in system dynamics models.
Copenhagen Business School. 1925 Frederiksberg C, stch^p2 cbs dk
-Physics Department, Technical University of Denmark. 2800 Lyngby, cekAhaos fldth dk
"Sloan School of Management, Massachusetts Institute of Technology, Cambridge, .MA 02142. jsterman
'Institute of Theoretical Statistics,
mit.edu
^
0-4375
Introduction: Aggregation in nonlinear systems
1
Macroeconomic models normally
a2;gre2;ate the
mdividual Hrms
ot
the
economy
with similar products, parameter values and decision functions. Sometimes.
sector
is
considered
all
I
it
is
impractical to portray separately
he products on the market, and arguing that the
are captured in sufficient detail by the aggregate formulation
.\evertheless, there are instances
(1988) shows
onl\' a
Samuelson 1939. Goodwin 1951). This simplification
(e.g.
on pragmatic grounds by noting that
an industry or
into sectors
how suppressing
where aggregation
is
(
all
not justified.
justihod
the lirms
phenomena
Forrester 1961.
is
Musle
in
of interest
Simon
19t)9l.
For example. Allen
individual variation by using aggregate population
means
can prevent a model from properly reproducing the dynamics of the system. Similarly, wdtk
by
e.g.
Bruckner,
et al. (1989)
shows how the representation of individuals
is
necessarv to
describe evolutionary processes.
However, the modelling literature
tion of
dynamic systems,
individual entities, as
is
is
weak
particularly
in
providing guidelines
when there
for
appropriate aggreg.i
are significant interactions between
tlic
usually the case in economic systems.
Consider the aggregation of different firms into a single sector. Such aggregate represen
tations are the mainstay of models of business fluctuations. Models of capital investment.
for instance, represent
each firm.
the average lead time and lifetime of the plant equipment used
In reality there are
many
\>\
types of plant and equipment acquired from maris
vendors operating under diverse conditions and possessing a wide range of lead times.
In
response to changes in external conditions, each firm will generate cyclic behaviors whos<'
0-4:57.-5
frequency, damping, and otlier properties are determined by the parameters charactenzins
the particular mix
of"
lead time.s and lifetimes
it
Because these individual tirms are
faces.
coupled to one another \ia the input-output structure of the economy, each acts as a source
of perturbations
How do
on the others.
the different lifetimes and lead times
ot plant
phase, amplitude, and coherence of economic cycles?
and equipment
How
ual firms into single sectors for the purpose of studying
issue of coherence or synchronization
is
valid
is
affect the trequencv.
aggregation of individ-
macroeconomic
particularly important.
fluctuations.'
The economy
as a
The
whole
experiences aggregate business fluctuations of various frequencies from the short-term business cycle to the long-term Kondratieff cycle (Sterman and Mosekilde. forthcoming j.
why should
cycle?
the cycles of the individual firms
Given the distribution
of
move
in
\ i^
phase so as to produce an aggregate
parameters among individual firms, why do we observe
only a few distinct cycles rather than cycles at
all
frequencies - cycles which might cancel
out at the aggregate level?
.\
common approach
to the question of synchronization
economic aggregates such
as
GDP
unemployment
or
sudden changes
in resource
Zarnowitz 1985
for a review). Forrester
arise
arise
supply conditions or variations
from the endogenous interaction
(
assume that fluctuations
in
from external shocks, such
as
to
is
in fiscal or
monetary policy
i
=cp
1977) suggested instead that synchronization could
of multiple nonlinear oscillators,
i.e..
that the cycles
generated by individual firms become reinforced and entrained to one another. Forrester
also proposed that such entrainment could account for the uniqueness of the
cycles. Oscillatory tendencies of similar periodicity in different parts of the
economic
economy would
0-4375
be drawn together to form a subset of distinct modes, such
and economic growth, and each
of these
as
busmess
eyries. Ions waves,
modes would be separated from the next
wide enough margin to avoid synchronization.
I'ntil recently,
1)\-
a
however, these suggestions
have not been subjected to rigorous analvsis.
Roughly speaking, synchronization occurs because the nonlinear structure
parts of a system creates forces that
"nudge" the parts of the system into phase with
For instance, two mechanical clocks, hanging on the
one another.
of the interacting
observed to synchronize their pendulum movements
same
wall, are often
was Huygens who hrst reported
(it
the entrainment of clocks). Each clock has an escapement mechanism, a highly nonlinear
mechanical device, that transfers power from the weights to the rod of the pendulum.
a
pendulum
faint click
is
close to the position
where the escapement
W
releases, a small shock, such as
from the release of the adjacent clocks escapement, might be enough
hen
i
he
to trigg'^r
the release. Hence, the weak coupling of the clocks through vibrations in the wall can bring
pendulum
individual oscillations into phase, as long as the two uncoupled
frequencies arc
not too different. In a similar manner, nonlinear interactions and dissipation working ovrr
millions of years have brought the rotational motion of the
its
orbital
motion with the
Synchronization
is
result the
moon
moon
into synchronization with
has a dark side never visible from earth.
only one manifestation of a more general
phenomenon known
as
locking or entrainment. In nonlinear systems, a periodic behavior usually contains a
of
harmonic frequencies
two nonlinear
of
one mode
is
at p times the
oscillators interact,
close to a
fundamental frequency, where p
mode-locking
harmonic frequency
will
is
an integer.
mode
serie-~
U
hen
occur whenever a harmonic frequen(\
of the other.
\s a
result, nonlinear oscillator^
D-4;375
tend to lock to one another >o thai one subsysteni ornpletes
each time
f)re(iseU- p rvrles
<
the other subsystem completes q cycles, with p and q being integers.
In a series ot papers,
phenomena
1992.
al.
we have analyzed how mode-locking and other nonlmear d\namic
arise in a simple, nonlinear
Sterman and
.Mosekilde.
model
of the
economic long wave
The model iSterman
forthcoming).
the long wave as a self-sustained oscillation arising from instabilities
An
production of capital.
increase
in
the
demand
for capital
m
i
.Mosekilde
ci
1985) explains
the ordering and
leads to further increases
through the investment accelerator or "capital self-ordering," because the aggregate capital
producing sector flepends on
Once
own output
its
to build
up
its
stock of productive capital.
a capital expansion gets underwav. self-reinforcing processes sustain
long-term equilibrium, until production catches up with orders.
the
economy has considerable
e.xcess capital, forcing capital
.\t
it
beyond
this point,
however,
production to remain below
the level required for replacement until the e.xcess has been fully depreciated, creatmg
a
its
:or
new expansion.
The concern
type.
The
different
tive
of the present paper
real
economy
consists of
amounts. Parameters, such
amounts
of different capital
isolation, the buildings-
significantly different
An
is
early study by
the simple model's aggregation of capital into a sinsic
many
sectors employing different kinds of capital
as the average productive
life
of capital
components employed, may vary from sector
and the
in
rela-
to sector.
In
and infrastructure-capital industry may show a temporal variation
from that
Kampmann
of. for
instance, the machinery industry.
(1984) took a
first
step in this direction by disaggregating
the simple long-wave model into a system of two or more capital-producing sectors with
D-4375
different characteristics.
Kampmann
siiowed that the multi-sector system could produce
a range of different behaviors, at times quite different from the original one-sector model.
The present paper
further investigates these results, using a two-sector model.
One
sector
can be construed as producing buildings and infrastructure with very long lifetimes, while
the other could represent the production of machines, transportation equipment, comput-
ers, etc..
with
much
shorter lifetimes.
oscillation with a period
ever,
when
In isolation,
each sector produces a self-sustained
and amplitude determined by the sectors parameter
values.
How-
the two sectors are coupled together through their mutual dependence on each
other's output for their
own
production, they tend to synchronize or lock together with
a rational ratio between the two periods of oscillation that depends on the differences
in
parameters.
Mode- Locking
2
in
Nonlinear Systems
The behavior
For linear systems the principle of superposition applies.
the system
is
a
sum
of distinct
to
exist
and develop independently
an external disturbance
will
any variable
in
modes, or elementary excitations, where the frequencies
and attenuation rates are determined by the eigenvalues
modes can
of
of
of the .Jacobian matrix. Different
one another, and the sensitivity of any mode
be the same irrespective of the phase of the others. Thus,
mode-locking cannot occur linear systems, because the individual oscillatory modes do not
affect
one another.
In nonlinear systems,
characteristics of one
on the other hand, different modes
mode may depend on
will interact,
and the behavioral
the phase of another. In particular, components
D-4:57-)
may
of difFcreni periodicities
adjust themselves until the frequencies coincide.
Synchronizations and mode-locking are uni\'ersal phenomena
et al.
phenomena
1983. 1984). .Such
and engineering systems
(e.g..
documented
are well
runilinear svstems
iti
between individual companies and
in
likely to
.Jensen,
in a variety of physical, biolosical.
Colding-.Jorgensen 1983. Glass,
Mosekilde 1988), and the same processes are
i
et
1986.
al.
be involved both
in
Toeebv and
the interaction
the coupling between the various sectors of the overall
economy.
To understand the phenomenon
where two self-sustained
illustrated in Figure
1.
of frequency-locking in
more
detail, consider hrst the case
oscillators with different periods operate without interaction.
the total trajectory
may
.\s
then be represented as a curve on a torus
For particular parameter values, namely those where the two periods are commensurate,
the trajectory forms a closed orbit, and the total motion
general, however, the periods will be
trajectory will
periodic.
Lb).
fail
to close
on
and the trajectory
itself.
will
incommensurate
In this case,
periodic, as in Figure la.
(their ratio
is
the total motion
irrational
is
I.
itself:
each cycle
however, a quasi-periodic motion
is
is
and
said to be
gradually cover the entire surface of the torus
Thus, quasi-periodic motion never repeats
to deterministic chaos,
is
unique.
i
In
iln-
(|ua.-i-
Figure
In contra.-i
not sensitive to the
initiai
conditions, since a small change in these conditions only shifts the entire trajectory h\
constant amount.
For the
same
reason, the largest
the divergence between nearby trajectories,
be positive (Wolf 1986).
is
zero.
<\
Lyaponov exponent, which measures
In chaotic
motion, this exponent
will
D-4375
Figure
A
I
cross-section ot the torus (see Figure
does not cross
itself).
L
i
defines a topological circle (a closed curve which
By noting the angular
passage of the trajectory, one can define a
position u^ along this curve of each subsequent
map
of the circle onto itself
0</^<l.
^n + l=/(^n).
111
^
where
=
u/'2:t
mod
1
measures the phase of one of the oscillatorv subsystems
stroboscopically with the period of the other.
period later
it
will
be &„+[
.
With
this
If.
at a certain
time
this
as
phase
obtained
is
0„.
one
procedure the steady-state behavior of a complicated,
multidimensional dynamic system can be represented compactly by a one-dimensional map.
as illustrated in Figure 2.
Figure 2
With no
interaction between the two oscillators, the phase shift of one oscillator during a
period of the other will always be the same, and the
map
is
linear,
where the winding number Q measures the
ratio
cases where
be periodic (Figure
is
irrational,
fi is
rational, the
motion
will
i.e..
between the two periods.
2. a).
In the particular
Generally, however.
and the map produces a never-repeating quasi-periodic motion (Figure
H
2.bl
It
one now allows the two oscillators
may depend on
period of the other
becomes nonlinear.
is
map
the sine circle
e^^i
K
where
=B„
.\
^n
phase
the initial phase of the
simple (wample to illustrate
<hift
one mode ilurins the
circle tnap
of introducing rionlinearit\-
tlie effects
(.VrnoTd 1065)
~ :^.^in{2-9j mod
shift of
of
mode, and the
first
\.
i:])
measures the strength of the nonlinear coupling while,
average phase
one oscillator per period of the other,
map.) The coupling strength
linear
to interact, the
!\
is
i
as before.
A'
=
H measure?
the
corresponds to the
analogous to the coupling parameter q governing
the interaction between the capital producing sectors in the model developed below while
the phase-shift parameter
oscillation of the
If
n
is
n
is
analogous to the difference
two sectors, determined by the difference
not too far from
0.
the
map may
produce a stable fixed point toward which
initial
conditions (see Figure
where the two
Q
The
the uncoupled periods
in capital lifetimes.
ot
Ar.
cut the diagonal of the {9n+\.9n)-p^^^c and
trajectories will converge, independent
all
fi.xed
oscillators have precisely the
changes slightly
move along the
2.c).
in
ui
point represents the synchronized solution
same
period.
in either direction, the fixed point will
.More importantly, however,
continue to exist (though
it
if
wiil
diagonal), and the stable I:I-solution therefore exists over a finite range
ut
parameters.
Figure 2.d illustrates a l;3-mode-locking, where one oscillator performs precisely 3 cycles
each time the other performs one cycle. The
map,
/'''(^). has 3 stable fixed points.
a finite range of parameters.
1
:
3
mode
arises
when the
thrice iterated
Note that these fixed points continue to
exist o\t"r
D-4375
Thus
in
coupled nonlinear systems periodic entrained) motion becomes more
i
quasiperiodic motion becomes
at all rational ratios of tfie
the phase-shift parameter
less
H
.\t least for
system can lock frequencies
In principle, the
two periods. Plotting the
In nonlinear
ratio of frequencies as a function ut
system, the curve
In a linear
illustrates the locking.
a straight line through the origin.
finite intervals for
common.
common and
each rational ratio of frequencies, resulting
small coupling strengths, the staircase
number.
.\n
example
wave model can be found
.\s
the nonlinearity
I\
is
in
sirnp[\'
systems, however, mode-locking pro(luc(^s
is
in
a so-called Devil's Staircase.
a fractal structure which repeats
under magnification, since between two given rational numbers, one can always
rational
is
of a DeviTs Staircase arising
another
from the forced one-sector long-
Mosekilde. Larsen. Sterman. and
increased, so does the range of
find
it'<eil
Thomsen
(1992).
over which a particular mode-
Vt
locked solution exists, producing a so-called .ArnoTd tongue diagram. Each tongue define^
the region in which a particular mode-locked solution exists.
collapses to a single point,
quencies.
is
.-\s
A'
is
i.e..
smooth and
invertible.
.As
long as A'
at
9ri
=
0.
=
I
for
the sine
map) where the map
and the slope becomes equal
to zero.
is
0.
the toneuc
to the ratio ol frc
small enough, the ma[j
and periodic and quasi-periodic motion-
are the only types of behavior that can occur. However, as
a critical value (A'
A =
namely the rational number corresponding
increased, the tongue flares out.
a diffeomorphism,
For
.\t this
K
(3)
continues to grow,
it
reaches
develops an inflection point
value,
one finds that the
.Arnol
<i
tongues start to overlap, indicating the simultaneous existence of two or more periodu
solutions.
Beyond
At the same time, quasi-periodic behavior ceases
K =
1.
the
map
is
to exist.
no longer invertible. folding occurs, and the motion ma\
10
0-4375
become
Within each Arnold tongue, a variety
chaotic.
above the
One
critical line.
doubling bifurcations.
classical
phenomenon
that can occur
For instance, a [:3-mode locking
and ultimately into chaotic motion.
represented on a torus.
.\
It
is
may
is
mav
tal<e
[jUuc
a cascade of period-
bifurcate into
'2:6.
1:12
(Period-doubling and rhaotic behavior carmot
cross-section of the "torus'"
folded or diffuse structure.!
of bifurcations
would reveal that
it
now
be-
lias
worth emphasizing, however, that period doubling
a
is
only one of several "routes to chaos". Other scenarios include intermittencv and a direct
transition from quasi-periodicity to chaos.
While the basic theory
behavior beyond the
in various
of frequency locking
critical line.
models, and
it
is
is
well established, less
is
known about
the
Several different types of behavior have been obser\ed
likely that the
behavior to some extent
will
be specific to the
individual system. Certain general results have been obtained, however, (see e.g. Knudsen.
et al.
3
We
1991) and
it
is
clearly of interest to pursue this type of investigation further.
The Model
have replicated Sterman's original (1985) model to portray two interacting sectors. The
original structure
is
maintained, except
for
minimal modifications necessary
to
I
)
extend
the one-sector structure to multiple sectors, and 2) replace the original model's piecewise
linear functions with analytic, infinitely differentiable functions.
The model
describes the flows of capital plant and equiment in two capital-producing
sectors.
Each sector uses capital from
itself
and from the other sector
11
as the only factors of produc-
D-43 )
I
liach sector receives orders tor capital, both from itself,
tion.
from the consumer H;oods sector. Product
and orders reside
The model
in
a backlog until capital
no inventories are maintamed
produced and delivered.
is
capital stock of each tvpe
"supply line" of capital) of each type
each sector
(2 x
Each sector
is
to order,
i
~
1).
The
in
m
each sector
each sector (2 x
(2 x 2). the capital
2).
and the
on order
t
he
total order backlog
m
i
state variables are indicated by capital letters.
1.2 maintains a stock
increased by deliveries of
new
/\,_,
capital
of
each capital type
7
=
1.2.
between the two sectors. Ar.
The
capital stock
and reduced by physical depreciation. The stock
of capital type j depreciates exponentially with an average lifetime of
in lifetime
rtnd
consists often ordinary differential equations, corresponding to ten stat(^ vari-
namely the
ables,
made
is
from the other sector,
is
Tj.
The
differrncr
used as a bifurcation parameter to e.xplore the
robustness of the aggregated model.
Output
is
distributed
sector
I
is
the share of total output from sector
sector
I
""fairly"
between customers,
has on order with sector
A'
=
I
j. x,.
j, S,j. relative to
the delivery of capital t\pe
i.e.
distributed according to
'd
/
how mu(
ti
sector j's total order backlog B,. Henrc.
^-^
'
I
and
where
o,_,
is
sector fs
new orders
for capital
Each sector receives orders from both
goods sector,
y,.
It
from sector
capital sectors,
accumulates these orders
in
12
j.
0,,
and
O;,,
a backlog 5, which
and from the consumer
is
then depleted b>
t
h<'
D-43
10
sector's deliveries of capital
=
B.
io„
-
-
Oj,
The consumer demands
The
latter
assumption
tors affects the
-
,7,)
Hence.
r..
.r,
7
:
recei\-ed b\-
is
dynamics
*
?.
,
each sector.
//,.
are exogenous, constant, and
(i
i
c(|iial.
not without consequence, since the relative size of the two sec-
i
Kampmann
Cqual demands
1984).
is
the most parsimonious
assumption.
Production capacity
each sector
in
is
determined by a constant-returns-to-scale Cubb-
Douglas function of the individual stocks of the two capital types, with a factor share
a G
of the other sectors capital t\'pe
[0. ll
type.
I
— q
of the sector's
own
capital
i.e..
;#^
= Kr'/v,r A-:
c.
where
and a share
/<,
is
The parameter q
a constant.
two sectors.
In the
between the
sectors,
simulation studies q
and
1,
I
i
thus determines the degree of coupling between
is
varied between
0.
i
!
lie
indicating no interdependenc\-
indicating the strongest possible coupling where each sector
i^
completely dependent on capital from the other sector.
The output from
sector
desired output x'.
ultimately to zero
If
if
i.
x,,
depends on the sector's capacity
desired output
no output
is
is
much
desired.
is
compared
to the sector
lower than capacity, production
Conversely,
if
formulated as
x,=/(^).c,.
13
is
'^
cut back,
desired output exceeds capacitv.
output can be increased beyond capacity, up to a certain
output
c,.
limit.
Specially, the sector
s
D-43
The
/
capacity-utilization function /(
=
fir)
where
'
-
'^^H-
'
a parameter set here to
is
=
/(O)
':([
=
0:/(l)
lim/(r)
1;
r
Thus, the parameter
degree line
for r
t
-
— CO
has the form
I
>
1.1.
=
ii))
1.
Note that
-:/(r)
>
r. r
t
[0.1].
determines the ma.ximum production possible, /(r)
[0. 1].
is
above the !>
implying that hrms are reluctant to cut back their output when
capacity exceeds demand.
Instead, they deplete their backlogs and lower their deli\er\-
delays.
Sector
I's
desired output x'
to deliver the capital
is
assumed
to
be the value that would allow firms
in that -ct tor
on order B, with the (constant) normal average delivery delay
<^.
Ii;r
that sector. Hence.
Sector
all
;s
desired orders for
new
capital of type j. o'^ consists of three components, first,
other things equal, firms will order to replace depreciation of their existing capital stork.
K,j/tj.
will
Second,
if
their current capital stock
is
below (above)
its
desired level
A,-',
tirrri'^
order more (less) capital in order to correct the discrepancy over time. Third, firms
consider the current supply line
supply line
is
5,^ of capital
and compare
below (above) desired, firms order more
the supply line over time. In total,
14
it
(less) in
to its desired level
s'^\
if
th('
order to increase (decrease:
D-4:57o
where the parameters
and the supply
r/"'
and
are the desired adjustment limes tor the (apital ^totk
r-
This decision rule
line, respectivelv.
(Senge 1980) and
e.\'periment;al
is
supported by extensi\-e empirical
iSterman 1989a. 1989b) work.
.Actual orders are constrained to be non-zero (cancellation of orders
the fractional rate of expansion of the capital stock
of
is
also
assumed
is
not permit
terli
and
to be limited beraust^
bottlenecks related to labor, market development, and other tactors not represented
in
the production function. These constraints are accounted for through the expression
where orders are expressed
as a multiple g\-) of depreciation.
The function
gi-) has the
form
g(rl
=
1
ii:5i
-
/iiexp-'^'t"-" -|i2e.xp-'^'f"-'>
where the parameters have the following values
J
=
6;
=
pii
27
—
^2
;
=
7
The parameters
^(1)
=
1;
8
-;
7
=
i^i
2
-;
3" =
i^T
ill'
3.
are specified so that
=
g\i)
\:
5"(1)
=
0.
Furthermore
lim g{r]
r—-OO
=
lim g{r)
3\
r
— — oo
Note that g{r) has a neutral
=
0.
interval
around the steady state point,
orders equal desired orders.
15
r
=
I.
where actual
D-43
The
(
desired capital ^tock
k'j
is
proportional to the desired production rate x' with a
constant capital-output ratio. Thus,
is
it
implicitly
two types of capital are constant, so there
assumed
no variation
is
that the relative prices of the
in
desired factor proportions.
Hence.
k'^
where
=
k,j
K,j
is
x'
I
the capital-output ratio of capital type
In calculating their desired
supply
line.
delivery delay for each type of capital.
j
firms are
.^',.
The
in
sector
assumed
target supply line
lo)
;.
to account for the current
is
taken to be the level
at
which the deliveries of capital, given the current delivery delay, would equal the current
depreciation of the capital stock.
sector's backlog divided by
''^
~
r,
for
The
is
the
output. Thus.
from consumers to each sector
both sectors. The
relative size of the
of the
delivery delay of capital from a sector
X
Finally, the orders
and equal
its
The current
latter
consumer demands
model considerably
(see
y,
assumption
for the
Kampmann
are
is
assumed
to be exogenous, constant,
not without consequence, since the
two types of capital can change the dynamics
1984).
capital-output ratios and average capital lifetimes are formulated in such a way that
the aggregate equilibrium values of these parameters for the model
remain constant and equal to the values
average capital lifetimes
in
in
economy
Sterman's original model.
the two sectors are
16
as a
whole
Specifically, the
D-4375
The
capital output ratios arc
.\",,
=
\
—
i
aK—
The average
(WK-
.
t
^
J
I
lifetime of capital. 7.
is
20 years and the average capital-output ratio,
k.
IS)
i-
:i
vears.
The formulation
assures that capacity equals desired output
their desired levels
and that the equilibrium aggregate
Hence
capital stocks i^qual
lifetime of capital
aggregate capital-output ratio equal the original parameters
K. respectively.
when both
in
and equilibrium
the one-sector model. - <ind
equilibrium we have
in
l')i
K.
Furthermore, parameters
produced by that
is
sector.
in the decision rules
When
there
is
were scaled to the average lifetime
no coupling between the sectors (a
thus simply a time-scaled version of the other.
way
to investigate the coupling of
two
We
felt
=
0).
ot capital
one
that this approach was the
sec tor
clean(>'«i
oscillators with different inherent frequencies.
1
liu'-.
the parameters, in years, are
r
T
T
where
T^'
=
1.5:
7^^
=
1.5:
6
=
'"-'
V.o.
17
0-4375
Figure J
Figure 3 shows a simulation of the hmit cycle of the one-sector model (a
Even with the modifications we have introduced, the behavior
mdistinguishable from that of the oneinal model
ters,
the equilibrium point
is
Sterman
1985).
The capital sector
is
capital, which, by further swelling order books, fuels the
The
self-ordering process
capital sector lead to falling
virtuaiU'
With the above parame-
thereby induced to order
upturn
in a self-reinforcing
Eventually, capacity catches up with demand, but at this point
the equilibrium level.
is
Each new cycle begins with a period of rapid 2:rowth.
where desired output exceeds capacity.
process.
model
of our
l)i.
unstable, and the system quickly settles into a limit c\Tle with
a period of approximately 47 years.
more
i
= O.Ar =
is
now
it
far
exceeds
reversed, as falling orders from the
demand, which further depresses the
capital sector's orders.
Consequently, output quickly collapses to the point where only the exogenous goods sector
places
new
orders.
.\
long period of depression follows, during which the excess capital
gradually depleted, until capacity finally reaches demand.
tor raises orders
enough
.-\t
to offset discards, increasing orders
this point,
is
however, the sec-
above capacity, and initiatins
the next cycle.
4
Simulation results
To explore the robustness
of the single-sector
model
to differences in the
paramters gov-
erning the individual sectors, we now simulate the model where some parameters differ
between the two
sectors. In spite of its simplicity, the
18
model contains a considerable num-
n-43
I
ber of parameters that might differ trom sector to sector.
the difference in capital lifetimes
described
rameters
in
in
Section
].
Ar
such a way that, when
all
0=0
we
\-arv
values of the coupling parameter n.
As
other parameters with the capital-lifetime
[)d-
for different
we have scaled
In the present stncl\'.
each sector
is
simply a time-scaled version of the
original one-sector model.
simulations that follow, sector
In the
1
is
always the sector with the longest lifetime of
capital output, corresponding to such industries as housing
2 has the shortest lifetime
its
and infrastructure, while sector
parameter, corresponding to the equipment sector. Introducins
a coupling between the sectors
apart from linking the behavior together, also change
will,
the stability properties of the individual sectors, taking the other sector as exogenous.
Thus, a high value of the coupling parameter a implies that the strength of the capital
self-ordering loop in any sector
not order any capital from
is
itself.
In the e.xtreme case
small.
If
a =
1.
each sector
will
the delivery delay of capital from the other sector
i-
taken as exogenous and constant, the behavior of the individual sector changes to a hishh'
damped
oscillation. Indeed, a linear stability analysis
of the individual sector
values of a.
As
will
around the steady-state equilibrium
shows that the equilibrium becomes stable
become evident below,
for sufficiently high
this stability effect of the coupling
parameter
has significant effects on the mode-locking behavior of the coupled system.
.As
(or
long as the parameters of the two sectors are close enough,
1
;
I
frequency locking) to occur,
i.e..
we expect that the
we expect synchronization
different cycles generated b\
the individual sectors will adjust to one another and exhibit a single aggregate economic
long wave with the
same period
for
both sectors. The stronger the coupling, a, the strong'T
19
D-4375
the torces of synchronization are expected to be.
Figure
4
As an example
of such synchronization. Figure
formed with a difference
coupling parameter q
periods of oscillation.
2 (the
=
in capital lifetimes
The two
0.25.
The
Ar =
sectors
now
2
:
1
2 solutioa
;
is
I
6 years
also the sector that leads in phase.
is
of 23 years.
increased to
mode through
is
Ar =
.-\gain.
lifetime dif-
we observe a doubling
9 years,
:
2
maxima
mode.
It
for their
ot th(^
product
has developed out of
(Feigenbaum
1978).
iwii
thi'
I
in'
both the temporal variations of the two
production capacities and the corresponding phase plot are shown. In the phase
now performs two
a
with the same coupling parameters,
a period-doubling bifurcation
5.
The
machinery capital of 17 years and
If.
referred to as a 2
illustrated in Figure
stationary solution
and a
sectors, although not quite in phase, have identical
alternate between high and low
capacities. This type of behavior
synchronous
is
and infrastructure
the difference in capital lifetimes
The two
Ar =
between the two sectors of
6 years corresponds to a lifetime for
lifetime of buildings
period.
shows the outconne of a simulation per-
larger excursions in production capacity are found for sector
"machinery" sector) which
ference.
f
loops before closing precisely on
plot, the
itself.
Figure 5
,A.s
the difference in lifetimes
is
further mcreased. the model passes through a Feigen
20
D-4375
l)aum cascade of period-doubling bifurcalions
at
approximatcl\-
Ar =
A- =
10.4 \'ears.
is
chaotic.
macroeconomic model which
I
:
1.
8
:
S.
etc.)
and becomes chaolic
Figure 6 shows the chaotic solution generated when
10.7 years. Calculation of the largest
the solution in Figure 6
i
We
Lyaponov e.\ponent
i
Wolf 19S6i conHrms that
conclude that deterministic chaos can arise
in a
aggregated form supports self-sustained oscillations,
in its
the various sectors, because of differences in parameter values,
fail
if
to synchronize.
Figure 6
.\
more detailed
Figure
7.
illustration of the route to chaos
Here, we have plotted the
is
provided by the bifurcation diagram
ma.ximum production capacity attained
each cycle as a function of the lifetime difference Ar.
spans the interval
capital stock
is
< Ar <
.'30
years.
When Ar =
The
in sector
is
30. the lifetime of the short-lived
first
in
is
35 years.
kept constant and equal to 0.2. Inspection of the figure shows
that the 1;1 frequency locking, in which the production capacity of sector
same maximum
over
difference in capital lifetimes
just 5 years, while the lifetime of the long-lived capital stock
The coupling parameter
1
in
each long-wave upswing,
period-doubling bifurcation occurs.
is
maintained up to
Ar
In the interval 6.4 years
1
reaches the
=« 6.4 years,
< Ar <
where the
8.0 years, the
long-wave upswings alternate between a high and a low maximum. Hereafter follows an
interval
etc.
up
to approximately
Within the
Ar =
8.1 years
interval approximately 8.2
are visible between regions of chaos, a
with 4:4 locking, an interval with 8:8 locking,
< Ar <
12.4 small
commonly observed
21
windows
of periodic
motion
behavior. In the region around
D-43
I
•
J
< Ar <
12.4
13.0 chaos gives
period doubling cascade
At %
about
and
1
:
1
1.5,2
:
12. etc.
:
:
S
.Another region of chaotic behavior follow^ until
\'isible as
around
Ar
mode-locked solution and the associated
3
1
2
:
Ar
continues to increase. Note that
27.6 years but then returns to
5;
in
these intervals, other stationary solutions
be reached from different sets of
Ar
The zones
and the coupling parameter
diagram are referred
a.
:
1
he
:
1
.
motion
3
1
,ii
1
:
of
n tongues,
the buildings industry completes precise
1
exist as well,
and these solution-
The phase diagram
in
Figure
S gi\(>
combinations of the lifetime difference
mode-locked
to as .ArnoTd tongues
tongue, the figure shows a series of
may
initial conditions.
for different
this
1
i
1
than cascading through further doublings to chaos.
an overview of the dominant modes
in
motion. Similarly the regions of
7
However,
may
6. S
entrainment are clearly
28. 3 rather
Figure
:
to the 2
where the system locks into
region bifurcates into 2
Ar %
4
way
(.Arnol'd
(i.e..
1965).
periodic) solution:-
Besides the
1
1
regions in parameter space wlierc
i.e..
long-wave oscillation each time the machmer\
industry completes n oscillations. Between these tongues, regions with other commensurate
wave periods may be observed. An example
a =
0.15 and
Ar =
is
12 years.
Figure 8
22
the 2
:
3
tongue found
in the
area around
D-4:57-)
Similar to ihe 2
is
a 2
It
is
_'
period-doubled solution on the right;-hand side of the
1
:
ron?u<?. there
1
period-doubled solution along part of the right-hand edge of the
1
:
;
likely thai similar
phenomena may be found along
the edges of the
'2
1
1
:
and
A
:
tongue.
1
tongues, etc.. producing a fractal, self-similar structure. However, at present we have
:
1
riot
yet investigated this structure in detail.
The phase diagram
in
Figure S also
re\'eals that
to the full range ot the lifetime differences
parameter
stable,
a.
when
When a
is
Ar
the synchronous
for sufficiently
I
:
1
solution e.xtends
high \'alues of the coupling
large enough, the equilibrium of the individual sectors
the delivery delay and
demand from
For reference, two curves have been drawn
in
the other sector
Figure
8.
is
becomes
taken as exogenous.
defining the regions in which one or
both of these individual equilibria are stable: For a given value of the lifetime difference,
values of
q above the curve
the overall behavior
is
more and more derived from the coupling between the
from the autonomous self-ordering mechanism
less
and
for
high values of q. there
less
oscillations,
is
less
in
in
amplitude, and.
altogether, resulting in a synchronous
The
1
:
1
in
Figure
8.
Thus.
1).
2)
However,
for sufficiently
completes several
a
as
high a's.
is
it
increased,
disappears
solution.
locally stabilizing effect of high values of
.\rnord tongues
and
For large differences in capital lifetimes and
cycles for each oscillation of the long-lived sector (sector
reduced
sectors,
each individual sector.
low values of the coupling parameter q. the short-lived sector (sector
is
increases,
competition between the two individual, autonomous
and stronger synchronization.
the short-term cycle
a
result in a stable individual sector equilibrium. .\s
a creates a complicated distortion
For instance, the figure reveals that both the
23
I
:
1
ot
the
region and
D-4375
the 2
:
1
down above
region are folded
Moreover,
it
appears rhat the phase diagram contains routes to chaos other
doubling bifurcations.
In particular, the region
tongue should be (>xplored.
5
is
between the 2
In particular, the areas of high
contain overlapping solutions: where
solution
the other regions for high values of q.
initial
conditions or
:
2
tlian period-
tongue and the
coupling strength
1
:
2
will likeh'
random shocks determine which
chosen.
Conclusions
By employing
only a single capital-producing sector, the simple long wave model represents
a simplification of the structure of capital and production. In
of diverse
components with
different characteristics.
the average lifetime of capital, and
it
is
clear
We
reality, "capitar"
is
composed
have focused on the difference
in
from our analysis that a disaggregate system
with diverse capital lifetimes exhibits a much wider variety of fluctuations. For moderate
differences in parameters
of
between the
sectors, the coupling
between sectors has the
merging distinct individual cycles into a more uniform aggregate
the cycle remains in the 50-year range, although the amplitude
cycle to the next.
of the simple
The behavior
model and
is
of the two-sector
may
cycle.
The period
ut
vary greatly from one
model thus retains the
robust to relaxation of the aggregation of
elft^n
all
essential features
firms into a single
sector.
Entrainment
in the
disaggregated model arises only via the coupling introduced by the
input-output structure of capital production. Other sources of coupling were ignored. The
most obvious
links are created by the price system.
24
If,
for instance,
one type of capital
i-
D-4375
in short supply,
one would expect the relative price of that lactor
to rise.
To the extent
that sectors can substitute one type of capital for another, one would expect
the relatively cheaper capital components to
Tims, the price-system
rise.
will
demand
for
cause local
imbalances between orders and capacity across the sectors to ecjualize thus brinamg the
individual sectors into phase.
(We have performed
a few preliminary simulations of a
version of the model that mcludes a price system, and these simulations show an increased
tendency
for
synchronization.)
production function
may
The degree
of substitution
well be an important factor:
of substitution to yield stronger synchronization.
between capital tvpes
One would expect
The next
m
the
high elasticities
step in our work therefore
involves introducing relative prices and differing degrees of substitution.
In light of the coupling effect of the price
(e.g.
system and of other macroeconomic linkages,
the Keynesian consumption multiplier.) we expect disaggregate models to show a
coherent long-wave motion
for a
wide range of parameter values, and the basic validity
the simple one-sector model seems intact. Thus, the fact that the simple model lumps
capital types into a single aggregate factor produced by a single aggregate sector
cause
for
is
not
ui
all
,i
doubts about the theory.
.Another,
more immediate extension
sectors.
On
of our study
would involve looking
at
more than two
the one hand, a wider variety of capital producers would introduce more
variability in the behavior and. hence, less uniformity.
On
the other hand, as the system
is
disaggregated further, the strength of the individual self-ordering loops within sectors
is
reduced to near zero, and overall cycles
between
sectors.
will
more and more
arise
from the interact kjii
Stronger intersectoral coupling leads to stronger entrainment and more
25
D-43
(
)
uniform behavior.
The
The
demonstrate the importance
results
intricacies ot such
phenomena
of studying non-linear
We
research in the area of economic cycles.
larger role in shaping
that there
suggest
in
the economv.
a vast unexplored
domam
of
suggest that non-linear entrainment [)lays a
economic cycles than the
mainstream business cycle theory
is
entrainment
random shocks on which much
e.xternal
of
relies.
References
.Allen. P.
(1988)
Dynamic Models
Systems Dynamics Review
of Evolving Systems.
\.
109-
130.
.Arnol'd,
V
.
(1965) Small Denommators.
I.
1.
American Mathematical Society Translations
Bruckner.
in
E.. VV. Ebeling,
and
.A.
Mappings
46:
.A.
Onto
Itself.
184-213.
Scharnhorst (1989) Stochastic Dynamics of Instabilities
Evolutionary Systems. Systems Dynamics Review
Colding-Jorgensen. M. (1983)
of the Circumference
Model
5.
176-191.
of the Firing Pattern of a
Paced Nerve
Cell.
Journal
of Theoretical Biology 101: 541-568.
Feigenbaum. M. (1978) Quantitative Universality
for a Class of
Nonlinear Transformations.
Journal of Statistical Physics\.9: 669-706.
Forrester, J. VV. (1961) Industrial
Forrester, J.
W.
(1977)
Growth
Dynamics Cambridge, MA: MIT
Cycles.
De Economist
26
Press.
125: 525-543.
0-4375
Glass.
(ed.)
A. Schrier and
L.;
Chaos .Manchester.
J.
Belair iI9S6) Chaotic Cardiac
.Jensen.
and Universality
50:
Holden.
.\.
\'.
and the Persistence
.\ccelerator
of Business Cvrlcs.
19:1-17.
IL: P.
.\I.
In
r.I\.:.\Ianchester Cniversity Press.
Goodwin. R. (1951) The .\onlinear
Econometnca
Rhythms.
Bak and
of
T.
Bohr (1983) Complete Devil's Staircase. Fractal Dimension,
Mode-Locking Structure
Physical Review Letters
the Circle .Map.
in
1637-1639.
Jensen. M. H.:
Bak and T. Bohr 1984 Transition
(
Systems.
in Dissipative
Kampmann.
P.
no.
Knudsen. C;
Sturis
in
J.
to
Chaos by Interaction
Circle .Maps. Physical Review
C. E. (1984) Disaggregating a Simple
Working Paper
Tongues
I.
)
.\
30:
Model
of
Resonances
1960-1969.
of the
Economic Long Wave
D-3641, Sloan School of .Management. M.I.T.. Cambridge. .Mass.
and
J. S.
Thomsen
(1991) Generic Bifurcation Structures of .Arnol d
Forced Oscillators Physical Review A
Mosekilde, E.; E. R. Larsen;
J.
44: 3503-3510.
D. Sterman and
J.
S.
Thomsen
(1992) Nonlinear .Mode-
Interaction in the Macroeconomy. Annals of Operations Research 37: 185-215.
Samuelson.
P.
.Acceleration.
Senge,
P.
A. (1939) Interactions Between the Multiplier Analysis and the Principle
The Review of Economic
ol
Statistics 21: 75-78.
M. (1980) A System D_ynamics .Approach
and Testing Socio- Economic Planning Science
27
14:
to
Investment-Function Formulation
269-80.
D-4:375
Simon. H. (1969) The Sciences of the
Sterman.
J.
D. (1985)
.\
J.
J.
6:
J.
Press.
Human
Dynamic Decision Making Organi-
Decision Processes 43: 301-335.
D. (1989b) Modeling .Managerial Behavior: Misperceptions of Feedback in a
Dynamic Decisionmaking Experiment Management Science
Sterman.
MIT
17-53.
D. (1989a) .Misperceptions of Feedback in
zational Behavior and
Sterman.
Cambridge. MA:
Behavioral Model of the Economic Long Wave. Journal of Eco-
nomic Behavior and Organization
Sterman.
Artificial.
35:
321-339.
D. and E. Mosekilde (forthcoming) Business Cycles and Long Waves:
havioral Disequilibrium Perspective.
In
Semmler. W.
(ed.)
.A
Be-
Business Cycles: Theory and
Empirical Methods. Kluwer Academic Publishers.
Togeby. M.. Mosekilde. E. and Sturis.
J.
(1988) "Frequency-Locking in a
Coupled Thermostatically Controlled Radiators'" Proceedings of
ing of the
the
Model
of
Two
Winter Annual Meet-
American Society of Mechanical Engineers. 88-WA/DSC-14.
Wolf, A. (1986) Quantifying Chaos with Lyaponov E.xponents. In Holden. A. V., ed. Chaos
Manchester, U.K.: Manchester University Press.
Zarnowitz. V. (1985) Recent
of theories
Work on
and evidence. The
J.
Econ.
Business Cycles in Historical Perspective:
Lit. 23:
28
523-580.
A
review
D-437.5
Appendix:
Equilibrium Conditions
The equations
in
for the
Two-Sector Model
the text completely describe the structure of the model.
ing equations give the equilibrium conditions for the model.
above the equilibrium
consumer demands,
is
y,.
=
6r
1.0
A, J
=
'^i;Xj
y,
=
yj
=
-
1.0.
toUow-
In the simulations reporrrd
unstable. To perturb the system a small change in the exogenous
are introduced, and the simulations are run long
transients to die out.
B,
I'he
k/t)
I
^
J
29
enough
for
any
D-43
(
1:
Torus representation of periodic and quasi-periodic behavior. Periodic motion
whenever there is a rational ratio between the two periodicities. Quasiperiodic
motion arises when the periodicities are incommensurate.
Figure
arises
Figure
2:
Mode-locking
the sine circle map.
in
nonlinear variation in the
map
parameter
exist over finite ranges of the
Figure
3:
9^ -^ fidn)-
Mode-to-mode interaction
Because of
gives rise to a
this nonlineanty. fixed points will
ft.
Simulation of the one-sector model. The steady state behavior
is
a limit cvclc
with a period of approximately 47 years. The plot shows production capacity, production.
and desired production
same
4:
Synchronization
(1:1
mode-locking)
shows the capacity of the two sectors
Ar
difference in capital lifetimes
variables are
shown on the
three figures.
Due
5:
6 years
as a function of
(i.e..
to the nonlinear coupling, the
the mode-locking ratio
Figure
is
is
Period doubling (2
1
:
coupled two-sector model.
in the
:
time
in
the steady state.
the lifetime of capital types
The coupling parameter a
23 and 17 years, respectively).
i.e.,
.\11
scale.
Figure
figure
of capital equipment, respectively.
is
1
and
The
The
1
is
0.25 in this and the following
two sectors are locked into a single cycle
1.
2 mode-locking) resulting
from increased lifetime difference
mode
1
As the difference in capital lifetimes Ar is increased to 9 years, the 1
replaced by an alternating pattern of smaller and larger cycles so that the total period idoubled. As in the previous figure, the coupling parameter a is 0.25. The two sectors still
complete an equal number of cycles in any interval of time, i.e., the mode-locking ratio i2:2.
l\t.
:
30
i'-
D-437.5
Figure
6:
Chaotic behavior. As the difference
in capital lifetimes
the model exhibits a progression of period doublings, which at
Ar
is
increased further.
some pomt become
infinitely-
beyond this point, as in this Hgure where Ar is 10.7 years, the behavior is
(The coupling parameter q is still 0.25.) The model shows no regular periodic
behavior, and initial conditions close to each other quickly diverge so that, in practice, the
behavior is unpredictable. Nonetheless, the two sectors remain locked together with a ratio
of unity between their periods.
dense.
.Just
chaotic.
Figure
Q.
The
7:
Bifurcation diagram for increasing lifetime difference
figure
shows the
local
maxima
Ar and
attained for the capacity of sector
constant coupling
1
(the longer-lived
capital producer) in the steady-state behavior for varying values of the lifetime difference
Ar. The coupling parameter a is held constant at 0.2. For a given Ar. a single value in
the diagram indicates a uniform limit cycle; two- values indicate a period doubling with a
smaller and larger cycle, etc. In chaotic regions, the number of local maxima is infinite
since no individual cvcles are identical.
Parameter phcise diagram. The figure summarizes the steady-state behavior
of the two-sector model for different combinations of the coupling parameter a and the
q" indicates the area in parameter space
lifetime difference Ar. A region labeled "p
where the model shows periodic mode-locked behavior of p cycles for sector 1 and q cycles
Figure
8:
;
for sector 2.
(However, other solutions may coexist at the same point
in
the diagram,
depending on the initial conditions of the model.) The question mark indicates that the
details of the diagram are still under exploration. In particular, regions of chaotic behavior
have not yet been outlined in detail. The dashed curves across the diagram indicate the
value of above which each sector in isolation (with the other sector treated as exogenous!
becomes stable. Above these lines the cycles are created solely by the interaction of the
two sectors, implying that, for large a, synchronous behavior becomes more and more
prevalent.
31
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Date Due
Lib-26-67