HD28
.M414
no.
c|-5
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Business Cycles and Long Waves: A Behavioral
Disequilibriun^ Perspective
John D. Sterman
and
Erik Mosekilde
WP#
3528-93-MSA
January, 1993
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Business Cycles and Long Waves: A Behavioral
Disequilibrium Perspective
John D. Sterman
and
Erik Mosekilde
WP# 3528-93-MSA
January, 1993
^
J
1993
D-4308
Business Cycles and Long Waves:
A Behavioral Disequilibrium
Perspective*
John D. Sterman
Sloan School of Management
Massachusetts Institute of Technology
Cambridge,
MA 02139 USA
and
Erik Mosekilde
Physics Laboratory
Technical University of Denmaric
m
2800 Lyngby
Denmark
December 1992
* Forthcoming in Semmler,
W.
Business Cycles: Theory and Empirical Methods.
Kluwer Academic Pubhshers.
(ed.)
EX)rdrecht:
Please address ccMrespondence to John Sterman at the address above orjstennan@mit.edu.
This work was supported in part by the Sponsors of the MTT System Dynamics National Model
Project Jesper Skovhus Thomsen assisted in the preparation of the computer simulations.
0^308
Introduction
1.
The evolution of
mode
the
of behavior
macrocconomy
we mean
caused by a particular
nential
set
reflects the interaction of multiple
a particular pattern of
modes
dynamic behavior, such
of feedback processes. The most important
of behavior.
By
a
as growth or fluctuation,
mode
is
the long-term expo-
growth of the world economy. This exponential growth, both cause and consequence of
industrialization, population growth, capital accumulation, technological advance,
and
historical
accident, has accelerated dramatically since the beginning of the industrial revolution, transforming
virtually every aspect
of our world, including economic,
pxjlitical, cultural,
and even biogeochemi-
caJ systems.'
Yet economic development around the growth trend
tions are a persistent feature of
economic
life.
is
far
Economic
from steady. Indeed, cyclical
fluctua-
historians have identified several distina
cycles, including the shon-term business cycle (3-7 years), the construction or Kuznets cycle (15-
25 years), and the long wave or Kondratieff cycle (40-60
years).
The
existence of these cycles
is
not without controversy, however. Debate continues today about the causes of the short-term
business cycle, the most extensively studied mode.
longer cycles are
still
more
The causes and even
controversial. In part, the uncertainty
is
the existence of the
empirical:
we
data for fewer long cycles than short ones. Yet in large measure the controversy
is
necessarily have
due
to a lack of
appropriate theory to account for disequilibrium dynamics that can persist for years or even
decades.
Of course,
theory and data are entwined in a feedback loop: without theory to guide
empirical
tests, little
evidence for disequilibrium dynamics such as long cycles was collected; with-
out compelling evidence of long cycles, there was litde motivation to develop
new
theory. Recent
years have wimessed a dramatic change in both the theories available to model nonlinear, disequi-
librium dynamics such as long waves and the data supporting their existence.
'
The existence and
Ultimaiely, of course, growth of population and material production will cease as the world makes a transiuon to
economy consistent with various social, environmental and ecological limits. Debate continues as
a post-industriaJ
to the
proximity of the limits
to
growth, the likely dynamics of the transition from growth, and the susiainability of
(Meadows, Meadows, and Randers 1992).
different cconoenic and social systems
IM308
causes of the long
how
wave
are
the different cyclical
One of the
now
modes
reasonably well established, and theory
in the
economy
interact with
why
principal mysteries theorists have faced is
periodicities rather than cycles at all frequencies.
fiindamentaUy,
emerging
to understand
one another.
there
seem
And how might the
to be only a
few
different cyclical
distinct
modes
Schumpeter (1939) argued, the coincident downturn of the business
interact? Could, as
constructicMi cycle,
is
cycle,
and long wave account for the severity of the Great Depression? More
why
should the frequencies of these cycles have (roughly) commensurate periods,
so that their downturns might coincide? Indeed, even
if
one admits the
possibility that individual
firms might generate cycUcal movements, the different parameters characterizing the structure and
decision
making processes of different firms would cause them
frequencies and phases.
Why then
to oscillate with different
should there be aggregate cychcal movements at all?
UnfOTtunately, macroeconomic theory has been largely silent on the issues of multiple modes, synchrcHiization,
ity
and entrainment. The problem resides both
in the prevailing
assumptions of rational-
and equilibrium, neither of which are good approximations to actual economic systems, and
the tools used to analyze
economic dynamics. Over
the past
evidence has accumulated documenting the bounds on
in
few decades an impressive body of
human
rationality
(Simon 1982).
Experimental and field studies in psychology, economics and other social sciences have docu-
mented a wide range of heuristics people use
many
to
make
decisions in complex environments, and the
systematic errors and biases that result (Kahneman, Slovic, and Tversky 1982, Hogarth
1987). Appropriate theories of economic dynamics, and economic behavior in general, should
embody models of decision making consistent
and
field study as well as
actually use
The
ing.
with empirical knowledge (including qualitative data
econometric analysis) of the processes of judgment and choice managers
(Simon 1979, Sterman 1987, Morecroft 1985).
analytical tools traditionally used to study
economic dy-namics have also slowed understand-
Though many macroeconomic models of the business
cycle exist, few address the issue of
D^308
muluplc cyclical modes.
analysis (see
pan, this
Samuelson 1947,
identify the unit of time
structures, parameters,
that the cycles
More
In
p.
is
because difference equations have dominated dynamic
380), and
between periods'
many
difference equation models
and behavior of such models cannot be validated.
of these models are the short-term business cycle (sec
most nxxlels of econonuc cycles were
Semmler 1989
1989,
for
modem
Goodwin
feedback loops. Well
Low
nonlinear approaches). But linear theory
kiwwn examples
in the natural
simply presumed
1980
linear or nearly linear (see e.g.
themselves from most systems considered
fKjsitive
It is
that the
for a critique).
1951, Kaldor 1940), until
foundation for the study of economic dynamics (Forrester 1987).
gtiish
not explicidy
Samuelson 1939, GoodvMn 1951) so
(e.g.
important, despite notable early exceptions (e.g.
recently
do
First,
is
Day
1982, Lorenz
not an appropriate
economic systems
distin-
sciences by the prevalence of
include the accelerator and multiplier loops of
Keynesian theory. Other positive loops operate through extrapolative expectations, agglomeration
effects, increasing returns, the effect of inflation expectations
on
real interest rates
and thus
aggregate demand, speculation and financial crises, and synergies and standards formation
among
and within technologies for production, communication, and organization (Sterman 1986a,
Graham and Senge
1980, Arthur 1988,
Semmler
1989). Such positive feedbacks create the
possibility of strongly nonlinear behavior, the positive loops
may
destabilize otherwise convergent
processes of adjustment which then grow in amplitude until constrained by various nonlinearities.
Such phenomena cannot be understood by means of
Furthermore,
tries,
if
the
economic system were
linear or nearly-linear models.
linear, the cycles
produced by different firms, indus-
and nations would evolve independentiy of one another and the
linear superposition of the independent
the aggregate of
many
total
behavior would be the
modes. While individual firms might exhibit fluctuations,
independently oscillating firms might be quite constant - there would be no
business cycle as a macroeconomic phenomenon. While diffusion of business cycles has received
considerable empirical attention, theoretical understanding of synchronization has lagged. Thus
many
theories find the cause of synchronization in
common
sources of external variation, either
D^308
government monetary and
fiscal poUcies,
(Bums
shocks and expectations
changes
in
aggregate demand, or highly correlated
1969, Mitchell 1927;
Zamowitz 1985 provides a
Modem dynamical theory offers another explanation:
nonlinear
mode
survey).
locking. In nonUnear sys-
tems, superposition does not hold. Instead, the periodicities of coupled oscillators
one another
to achieve a rational ratio, or winding number.
Mode
considerable interest in the natural sciences, especially since
it
may
adjust to
locking has recendy attracted
has been established that
mode
locking possesses a number of universal features independent of the particular system under study
(Jensen, Bak, and
Bohr 1983, 1984). The same processes of entrainment have been observed,
for
instance, in paced nerve cells (Colding-Jorgensen 1983), externally stimulated heart cells (Glass,
Shrier,
and Belair 1986),
radiators (Togeby, et
fluid
dynamics (Glazier et
1986), coupled thermostatically controlled
al.
1988), and forced microwave diodes (Mosekilde et
al.
1990).
al.
locking provides an explanation for the entrainment of economic fluctuations that
than prior explanations, and creates the possibility of nonlinear
phenomena such
is
Mode
more robust
as period-
doubling bifurcations, simultaneous multiple periodic solutions, and deterministic chaos.
locking also gives rise to the
'devil's staircase',
an unusual fractal structure
we
Mode
describe below.
We begin by reviewing the stylized facts of the different cycles, then discuss the behavioral foundations for each
mode at
the micro level.
We focus on the longer cycles as these are the most
controversial and least understood, particularly the economic long wave.
theories of the
economic long wave
thcOTy devel(^)ed by the
theories.
To illustrate
that
have emerged
the type of behavioral, nonlinear disequilibrium theory
wave and show how
anoong locally rational decision rules embedded
we use the model
trainment and
mode
in the past decade, specifically the integrated
MIT System Dynamics Group and the neo-Schumpeterian innovation
present a simple nxxlel of the long
Next,
We survey the principal
to consider interactions
locking sheds light on
why
in
the
wave
we
advocate,
we
arises through interactions
a nonlinear feedback system.
among
the modes.
there are a small
The theory of nonlinear en-
number of modes
rather than cy-
0^308
clcs of
all
frequencies, and
level cycles that
wash
why
there are aggregate
macroeconomic
out at the
movements
level.
than firm or industry
at all rather
We conclude
with implications for the
development of empirically grounded, behavioral, disequilibrium theories of economic dynamics.
2.
Economic Dynamics:
The most thoroughly analyzed
cycle (Mitchell 1927.
industrial production
phase
shifts
Modes
Multiple
cyclical
mode
in the
of Behavior
economy
is
the short-term (3-7 year) business
Gordon 1951, Moore 1961. Zamowitz 1985).
illustrated in figure
by
US
and civilian unemployment for the period 1947 to 1992. With characteristic
and ampliuides, the short-term business cycle manifests clearly
and industry level data including capacity
interest rates, etc.
1
Among
the
utilization, inventory
well-known
phenomenon of amplification,
in
in a host
of aggregates
coverage, help wanted advertising,
characteristics of the short-term cycle
is
the
which tbe amplitude of the cycle increases as one moves from
production of consumer goods to intermediates to raw materials (figure
2).
the
Theories of the shon-
term cycle should explain these details as well as generate a fluctuation in output with the appropriate period, amplitude, phase relations,
Many
and
variability.
time series also provide evidence for the existence of a 15-25 year construction (or Kuznets)
cycle (Riggleman 1933, Hoyt 1933.
showing the vacancy
rate
Long
1940. Kuznets 1973).
of commercial office space
in
Boston from 1952
in
capacity utilization of the world
oil
given
to 1990.
in figure 3
Similar cycles
tanker fleet (Bakken 1992). At the industry or regional level,
the amplitude of the construction cycle
is
often so high that nonlinearities are clearly involved, For
example, during bust periods the rate of new construction
at
is
production capacity of the paper industry (Randers 1984) or in
can also be found, for instance,
and excess capacity declines
An example
falls
nearly to zero for extended periods,
a rate constrained by the lifetime of capital stocks.
fib. 3
t^^^
Let us consider the processes that produce these two distinct modes.
dynamics of individual
firms, and later consider
how
such firms
We
focus, initially, on the
may become
entrained with one
another and with the government, consumer, and financial sectors to produce a coherent aggregate
D^308
cycle. Consider a manufacturing firm in equilibrium,
relative to the labor, capital
stant.
and other input markets, so
will eventually
expand output
sumption, woik force and capital stock
to
meet
that the firm is small
can be considered con-
incoming orders. The
orders. In the long run, production, material con-
all rise in
prqK>Ttion to incoming orders.
The question
is
the transient will unfold.
If all inputs
However,
could be adjusted immediately, the transient would be
factor inputs cannot
from short term variations
line
that factor prices
Now consider the firm's response to an unanticipated step increase in
company
how
assuming for simplicity
change
in
instantly.
demand
becomes
to provide time for efficient adjustment of inputs. In
may
not change production at
all, until it
clear that incoming orders will jjemain at the new, higher level. Inventories necessarily
Optimal inventory
fall.
and nonoscillatory.
Backlogs and inventories buffer the production
immediately following the demand shock the firm
fact,
fast
may also increase
as the expected throughput rises.
To restore
inventory to
desired levels, the firm must increase production above the rate of incoming orders for at least
some period of time. As a consequence, orders
for materials
and intermediate goods must also
in-
crease above the rate of incoming orders, passing a larger disturbance on to the supplying industries.
This process, the familiar inventory accelerator, provides an explanation for the amplification
of the business cycle from the consumer goods sector through the intermediate goods and finally to
the
raw materials sector
The
(T. Mitchell 1923,
Metzler 1941, Forrester 1%1, Mass 1975).
amplification of demand shocks at each stage of production
three fundamental features of production:
in adjusting production to
demand shocks
( 1)
is
an inevitable consequence of
the existence of decision-making
(e.g.
and physical delays
forecasting and administrative lags, lags in factor
acquisition); (2) the existence of stocks such as inventories,
work
in process,
and backlogs which
buffer the difference between orders and output; and (3) the need to adjust these stocks towards
target values
when shocks occur (to restore
increase, output
must
rise
inventory to
above shipments for a time).
initial levels after
an unanticipated demand
D-1308
The
inevitability of amplification,
(by oscillation
is
meant
a
system
however, does not mean
that is less than critically
frequency of oscillatory response to
that oscillation is similarly inevitable
damped). The existence,
demand shocks depends on
stability,
the nature of the feedback
processes by which a firm adjusts output to demand, as well as the myriad couplings
firm,
its
suppliers, customers,
Provided
that
and other actors
in the
in output trjay
From
through more intensive use of existing employees (overtime).
workweek
loop which
is
to regulate inventory creates
non-oscillatory and adds
workweek response
ductivity after long
is
nonlinear:
it
is
work weeks, and
damping
to the
the
be accomplished quickly
a control-theoretic px)int of
an effectively first-order negative feedback
system (Sterman 1988). However, the
limited by the cost of overtime, by decreasing worker pro-
ultimately by the length of the day. Thus, while small ampli-
demand can be ascommodated through
tude changes in
among
economy.
needed materials are available, small changes
view, the use of
and
overtime, larger and more persistent
changes saturate the workweek feedback, requiring expansion of the work force.
Expanding
the
work
new employees
force,
however, involves significant delays. Vacancies must be authorized,
hired and trained, and rime must pass before productivity rises to that of experi-
enced workers (comparable delays exist
in the
case of an unexpected decrease in demand). The
use of employment to control inventory levels and respond to
demand shocks
creates a negative
feedback loop, but unlike the work week loop, the employment adjustment loop involves delays on
the order of several
oscillatory.
ments with
The
months or more. Negative feedback loops with such phase
characteristic behavior of
realistic decision
(Forrester 1961,
Mass
parameters
is
models
damped
that portray
force adjust-
oscillations with a period of 3 to 7 years
employment, inventories, delivery delay, vacancies,
labor accession and separation flows, and other variables.
is
workweek and work
1975). These models also generate the phase (lead and lag) and amplitude
relationships observed in the data for output,
erate
lag elements are
The business cycle
these models gen-
robust as the boundary of the model expands to include consumer demand, interest rates.
D^308
monetary and
fiscal policies,
and other elements of the
demand model (Mass 1975, N.
traditional aggregate supply/aggregate
Forrester 1982).
Regulation of output by workforce adjustment
is
also limited
due
to diminishing returns as labor
expands relative to existing plant and equipment In the long run, capital stocks must also be
increased.
planning
However,
for,
capital investment involves
ordering and constructing
even longer delays arising from the process of
new plant and equipment Adjustment of capital
thus involves a negative feedback loop with substantially longer delays.
tal
Models
that integrate capi-
investment with inventory and work fence management tend to produce oscillations with periods
of 15-25 years in addition to the short-term cycle (Mass 1975, N. Foirester 1982,
The theory described so
far
assumes agents have bounded
1982). Agents seek to take appropriate decisions, but
sources necessary to approach optimality, even in the
The theory of bounded
do not possess
weak
the plant level, while pricing
may
availability,
nated.
and
be decided
may
Simon (1979,
the cognitive and other re-
rational expectations sense,
in
due
at
The theory of bounded
is
to the
which they operate.
typically influenced
problem
by decisions
at
be the responsibility of senior divisional management and capital
corporate headquarters.
attentional resources,
rationality does not
the couplings anoong subsystems were
making within
Due
to limitations
of time, information
management of the subsystems may be imperfectly
irrational but rather locally or intendedly rational
factoring and decision
1980).
raticmality, as applied here, recognizes that firms partition the total
of optimizing the enterprise into subproblems. Production
investment
Low
rationality in the sense of
complexity of the high order, nonlinear, randomly-excited dynamic system
if
stocks
-
weak and
the firm
assume
that
is,
that the individual
coordi-
managers are
they use heuristics that would
work well
the separability assumption implicit in task
were valid (Sterman 1985, 1987; Morecroft 1985).
Extensive experimental evidence shows that the bounds on rational decision making in dynamic
systems are severe. In simple experimental economies such as the classical multiplier-accelerator
model (Sterman 1989a), inventory management (EHehl 1992), or distribution of a commodity
0^308
(Sierman 1989b), subjects perform well below optimal and generate systematic, persistent and
costly oscillations.
These systematic decision errors become nx)re severe as
the feedback
com-
plexity of the environment increases, particularly as delays lengthen (Diehl 1992, Paich and
Sterman 1992, Brehmer 1990). Experience, incentives, and market
not eliminate these errors (Paich and Sterman 1992,
Kampmann
institutions
moderate but do
and Sierman 1992, Smith,
Suchanek and WUliams 1988).
3.
The economic long wave
The
third
cycle.
is
main
cyclical
The long wave
mode of economic
is
the
most important
also the
most controversial and
TTie long
such great duration that the stresses
political
it
wave
is
the
least
far larger in
economic long wave or Kondratieff
understood of the three cyclical modes.
It
amplitude than the business cycle, and of
generates cannot be contained within the market system, but
in,
the institutional strucmre of the
system (Sterman 1992, 1986a). The Russian economist N.D.
Kondratieff (1928/1984, 1935) was one of the
industrial
is
and sometimes the revolutions
rather influence the evolution of,
world economic and
behavior
first
to
draw
attention to the wave-like character of
development, with alternating periods of relative affluence and economic hardship.
Using data on commodity prices,
interest rates, industrial production,
raw materials consumption,
and foreign trade, Kondratieff argued for the existence of a roughly 60 year cyclic motion, and
speculated that
it
was
related to investment in long-lived capital.
The economic
stagnation and crises of the last
nomic pobcies
to restore former balances have
many new
theories of
its
origin
two decades and
the inability of conventional eco-
prompted renewed
interest in the long
wave and
(Freeman 1982, van Duijn 1983, Vasko 1987). However,
long wave remains controversial
among
the
economists. Most have taken a rather agnostic stance
concerning the existence of long waves, maintaining that historical evidence for long fluctuations
of sufficient regularity to be considered cyclic
is
Rosenberg and Frischtak 1983). While few deny
unconvincing (Garvy 1943, Mansfield 1983,
that the
performance of industrialized economies
D-4308
10
experiences significant long term variations,
particular historical events such as
many economists
see these
wars or gold discoveries than as a
more
result of
as the
endogenous
processes. In contrast, recent studies by Bieshaar and Kleinknecht (1984) and by
al.
(1989) designed to
results,
test the
To
et
and Sterman (1986a) repcms a wide range of data consistent with the long wave hypothe-
and 1840s, the 1870s through
through
Rasmussen
Kondratieff hypothesis in real series arrive at generally positive
Today, most students of long cycles agree
sis.
outcome of
(at least) the eariy
illustrate.
late
were the
that the historic depression periods
1
830s
1890s, the 1920s and 1930s, and the period firom about 1974
1990s (van Duijn 1983, Vasko 1987, Goldstein 1988).
Figure 4 shows detrended real
GNP in the United States from
1947 to 1992. After
removal of the long-term exponential growth trend what remains are the cyclical modes, particularly the short-term business cycle
with
GNP growing faster than
and the long wave. The post-war long wave
trend
from the end of World War
slower than trend since. The business cycle, with
smaU
appears as
wave conditions
ripples
on
much
clearly visible,
through about 1970, and
smaller amplitude than the long wave,
the great swell of the long wave.
Note also how the phase of the long
the apparent severity of the business cycle. During the expansion of the long
wave, periods of business cycle expansion seem
to
be long and vigorous, while recessions are
thought to be short and mild, as the rising tide of the long wave
turn phase of the long wave, recessions
business cycle appears to be weaker.
the character of the business cycle
Fifc
II
is
seem
An
to
lifts all
boats.
During the down-
be longer and deeper, and the growth phase of the
analyst unaware of the long
wave would conclude
had changed as the long wave peaked and began
that
to decline.
4-
Kondratieff viewed the long
and argued
particular,
that a
wave
as a manifestation of essential forces in the capitalist economy,
broad spectrum of social and economic phenomena were shaped by the wave. In
each burst of capital expansion would allow a new
set
of technologies to be exploited.
While accepting the general ideaof endogenously generated long waves, Schumpeter (1939)
articulated the opposite causality
between economic growth and technological innovation. For
Schumpeter, innovations create the products and markets that drive economic growth.
1
D-1308
Both
1
lines
One
of thought continue today.
els of the long
1979, 1981,
wave has been developed
Graham and Sengc
at
tary policy, debt,
wave
the long
ing
A
MITs System Dynamics Group
(Forrester 1976, 1977,
1980, Sicrman 1985. 1986a, 1986b, 1987, 1988, 1989a, 1990,
1992). TTic theory integrates a variety of
capital investment,
of the earliest and nx)St thoroughly tested formal mod-
economic processes, both
employment, work force
real
and nominal, including
participation, wages, inflation, interest rates,
MIT model
and consumer demand, among others. The
endogenously generates
as well as the short-term business cycle, construction cycle, and other
economic growth and
the expansion of the
simple version of this model
In parallel with this line
is
government sector
mone-
modes
relative to the private
includ-
economy.
analyzed below.
of economic modeling, neo-Schumpeterian theories stressing the role of
technological innovation as causes of the long
fundamental scientific discoveries and
new
wave have been developed. Mensch (1979) argues
inventions occur
more or
less
randomly. But for an
in-
vention to acquire economic significance, innovation, or the commercialization of the invention,
must occur. The
which plant the seeds of new
rate of basic innovations, those
ditioned by the state of the economy. During long
existing infrastrucmre
the
same
ficult to
arise
is
wave
upturns,
highly productive: incentives to invest in
time, fx>sitive network externalities and
commitment
industries,
economic growth
new
con-
rapid and the
technologies are small. At
to existing infrastructure
introduce alternative transport, communication or energy systems. Lx)ng
when
is
is
make
it
dif-
wave downturns
the potential of existing technologies saturates. Switching costs then decline, produc-
ing a burst of basic innovation as
find practical application.
The
many
of the inventions accumulated during the upswing
resulting
swarm of
innovations launches
new
industries
now
and pro-
vides the impetus for the next upswing. Formal mathematical models of these neo-Schumpeterian
theories include Montario and Ebeling (1980), Mosekilde and
Rasmussen (1986), Silverberg
(1988) and Dosi (1988); Kleinknecht (1984) provides some empirical
novation theories of the long
wave
is
and markets should reach saturation
explaining
in
why
synchrony
tests.
One
difficulty in in-
disparate technologies in disparate contexts
after
40-60 years, cycle
after cycle.
Addressing
D^308
this
12
gue innovaticm rates are entrained by the endogenous economic processes
wave. Other authors have related the long wave to changes
al.
MIT model
problem, Graharn and Senge (1980) integrated innovation theories with the
in
and
ar-
that generate the long
employment and wages (Freeman
1982), resource scarcity (Rostow 1978), class struggle (Mandel 1980), and
et
war (Goldstein
1988).
4.
A
simple behavioral model of the long wave
A control-theoretic explanation for the long wave emerging firom the MIT theory can be divided
into
two
parts: first, as described above, acquisition
of capacity in individual firms involves inher-
damped
endy
oscillatory processes. In isolation, these processes are stable, producing
when
excited by exogenous changes in demand. However, a wide range of self-reinforcing pro-
cesses exist in the linkages between firms and
among
oscillations
the production, financial, household and
government sectors of the economy, destabilizing the cycle and lengthening
its
Demand
period.
for capital increases the capacity needs of the capital producing industries, further boosting orders
for capital.
For example, expansion by
to substitution
of capital for labor and
demand and wages,
capital producers raises labor
still
greater
demand
for capital. Rising aggregate
leading
demand
boosts prices, reducing real interest rates and further stimulating investment. Rising output boosts
income and aggregate demand, further boosting output. Expansion leads
growth, leading to further investment and output growth. Rising credit
boom causes monetary accommodation,
so on. These positive loops include
multiplier, the
model
Mundell
integrates these
effect,
additional inflation,
many
and
still
to expectations of future
demand
lower
to finance the
real interest rates.
familiar processes including the Keynesian
and Fisher's (1933) debt/deflation
spiral.
The
full
income
MTT national
and other feedback processes (Sterman 1986a and 1988 provide
Model analyses (Rasmussen, Mosekilde and Sterman 1985, Br0ns and
Sturis 1991)
unstable.
Any
details).
show
these positive feedbacks cause a Hopf-bifurcation through which the equilibrium of the
becomes
And
that
economy
perturbations cause divergent oscillations that are eventually bounded by
nonlinearities such as the nonnegativity of gross investment and limits
on capacity
utilization
of the
DM308
13
capital stock,
that,
producing a limit cycle. The long wave appears
to
be a self-sustaining oscillation
although influenced by shocks and perturbations, does not require external excitation to
persist.
In contrast, the short-term business cycle appears to be a stable,
damped mode
that
requires external excitation, as in Frisch (1933).
One
of the most fundamental self-reinforcing feedbacks
'capital self-ordering', the fact that in the
new
factories,
the capital investment multiplier, or
aggregate the capital producing sector of the
orders and acquires plant and equipment fiDm
services increases, the
is
If the
itself.
demand
consumer goods industry must expand
machinery, vehicles,
producing sector must also expand
machines, rolling stock, trucks,
etc.
its
etc.,
To
its
for
consumer goods and
capacity and so places orders for
supply the high volume of orders, the capital
capital stock
causing the
and hence places orders for more buildings,
total
demand
for capital to rise
self-reinforcing spiral of increasing orders, a greater need for expansion, and
In equilibrium, the multiplier effect of capital self -ordering is
the long
wave
is
economy
still
still
further in a
more
orders.
modest (Sterman 1985). However,
an inherently disequilibrium phenomenon, and during transient adjustments the
strength of self-ordering
becomes much
greater than in equilibrium. This
is
partly a
consequence
of the classical investment accelerator. During disequilibrium a variety of additional positive feed-
back loops further augment the demand for
(i)
capital.
These include:
Amplification caused by inventory and backlog adjustments: Rising orders deplete the
tories
and swell the backlogs of capital-sector firms, leading
more
orders.
expand and
still
During the downturn, low backlogs and involuntary inventory accumulation further
depress demand, leading to
(ii)
to further pressure to
inven-
still
more excess
inventory.
Amplification caused by rising lead time for capital: During the long wave expansion, the
demand
acquire
for capital outstrips capacity. Capital producers find
new
it
takes longer than anticipated to
capacity, causing capacity to lag further behind desired levels, creating
sure to order, and further swelling the
demand
for capital.
still
more
pres-
EM308
(iii)
14
Amplification caused by growth expectations: Growing demand, rising backlogs, and long
lead times during the long
mand
wave expansion
all
encourage expectations of additional growth
for capital. Expectations of growth lead to additional investments, further swelling
in a self-fulfilling prophecy.
in de-
demand
During the downturn, pessimism further undercuts investment.
Sterman (1985) developed a behavioral model capturing the destabilizing positive feedback caused
by capital self-wdering. The model
erate the long
wave with
included in the
realistic
the model.
It is
(Mosekilde
et al.
The focus
parameter values.
It
minimum
structure sufficient to gen-
does not include the
produced by the simple model
also possible to find
is
full
range of feedbacks
robust to structural elaboration of
more complicated modes of behavior
1992) and disaggregated
creates a two-sector
is
designed to isolate the
MIT model. However simulations with more comprehensive versions have shown
that the characteristic behavicw
The model
is
(Kampmann
economy with
the capital investment accelerator.
as the
model
is
extended
1984).
a capital producing and goods producing sector.
Goodwin
(1951, 4) notes that the traditional
acceleration principle assumes
...that actual, realized capital stock is maintained at the desired relation with output We know in reality that it is
seldom so, th^e being now too much and now too little capital stock. For this there are two good reasons. The rate
of investment is limited by the capacity of the investment goods industry ....At the other extreme there is an even
more inescapable and effective limit Machines, once made, cannot be unmade, so that negative investment is
limited to attrition from wear.... Therefore capital stock cannot be increased fast enough in the upswing, nor decreased
fast enough in the downswing, so that at one time we have shortages and rationing of orden and at the other excess
capacity with idle plants and machines.
A single factw of production (capital plant and equipment) is considered.
The model
however, an explicit representation of the capital acquisition delay (construction
ity
of the investment goods seaor. As a
sarily equal,
and
at
any moment there
result, orders for
will typically
and the capac-
and acquisition of capital are not neces-
be a supply line of capital under construction.
For simplicity, the demand for capital of the goods-producing sector
representation of the consumption multiplier.
lag)
includes,
is
exogenous, and there
is
no
CM 308
Wc
15
first
describe the equations for the capital producer, then the couplings between sectors.
model allows
of production capacity C. Utilization
Desired output P*
to capacity.
is
mal delivery delay A*. Capacity
P = u{P*/C)C
The
Thus production P depends on
for variable utilization of the capital stock.
,
u(0) =
utilization
a nonlinear function of the ratio of desired production
is
determined by the
is
The
total
backlog of unfilled orders
B and
P*
the nor-
proportional to the capital stock K, with capital/output ratio tc
0, u{
1
=
)
1.
u
^
0, u"
<
0,
u{«} =
uniax
q)
P* = B/A*
(2)
C
(3)
= K/K.
capital stock of the capital sector
is
augmented by acquisitions
D. Discards arc exponential with average lifetime
(d/dt)K
=
A D
-
A and diminished by discards
x:
(4)
,
D = K/T.
The
(5)
acquisition of capital depends
on the
firm's supply line of unfilled orders for capital S
and the
capital acquisition lag A:
A
= S/A
The supply
(6)
line of capital
under construction represents the orders for capital plant and equipment,
C\, the firm has placed but not yet received:
(d/dt)S
Thus
far the
=
Ok-A
(7)
model describes the stock and flow
capacity utilization.
structure of the firm
The key behavioral formulation
Ok = Kf{Ok*/K),
is
5f{.) <P"'";f'>0.
and the physical
limits
on
the decision rule for capital orders Ok:
(8)
D-4308
16
Ok* =
D
+ ak(K*
-
K) + as(S*
-
S)
(9)
here the actual order rate depends nonlinearly on die indicated order rate
Ok*
of existing capital stock K, ensuring that orders remain nonnegative even
of capital
Due
to limits
on
if
as a fraction per year
there is a large surplus
e.g. financing, absorption capacity, etc., orders are limited to a
mum fraction of existing capacity P"^, as in Goodwin (1951).
maxi-
Three motivations for investment
are assumed: (1) to replace discards; (2) to correct any discrepancy between the desired capital
stock
K* and
the actual stock K; and (3) to oxrect any discrepancy between the desired supply line
of capital under construction S* and the actual supply
cxs
line S.
The adjustment parameters
determine the aggressiveness of the response to discrepancies.
acquisition rate of
new
capital, firms
Thus
in acquiring capital.
the desired sujl^ly line
the current capital discard rate
evidence supporting
must maintain a supply
D (see Sterman
is
To
ttk
and
ensure an appropriate
line proportional to the
delay they face
proportional to the capital acquisition lag
1989a and 1989b for details and experimental
this formulation):
S* = A-D
The
A and
(10)
desired capital stock
K*
is
a nonlinear function of desired output P*:
K* = Kog{KP*/Ko),
Desired capital stock
is
g{0)=0,g{l) = l,g'>0,g"<0
assumed
(11)
to rise proportionately with desired output for small deviations
from the equilibrium value Ko, but diminishing returns
to capital are
assumed
to limit capital
expansion when kP*/Ko becomes large.
Finally, the backlog of the firm
(d/dt)B
is
augmented by customer orders O and reduced by output
= 0-P
and the actual delivery delay of the firm. A,
P:
(12)
is
determined by the average residence time of orders
D-1308
17
backlog,
in the
A
= B/P.
Equations (1)
-
(13)
(13) describe a simple
model of a
firm.
The model includes an
explicit delay in
acquiring capital stock and realistic nonlincarities representing basic physical processes such as
nonnegativity of gross investment and limits to utilization of existing capacity. Sterman (1985)
shows
the individual decision rules of the
are intendedly rational, and investigates
Ok =
3,
and Os =
3)
and exogenous orders O, the transient response of the model
a highly damjjcd oscillation with a period of about
from
by which output
the negative feedback loop
20
is
years.
As described above,
regiilated through
capacity, wath a lag caused by the capital acquisition delay.
The model does
term business cycle because labo»
is
not explicitly treated; production
desired rate P* as long as the firm
is
not capacity constrained.
To
see
how
the long
wave might
changes
arise through capital self-ordering,
P
in
to
sensi-
its
A=A=
parameters. With realistic parameters for a capital producing firm (k=3,
tivity to
20,
model
1
t
.5,
shocks
=
is
the cycle arises
production
not produce the short-
instantly adjusts to the
we now modify
the
model
to
represent the entire capital-producing sector of an economy. In the aggregate the capital sector
orders capital from
sum
itself,
so the
total rate at
which new orders for
capital are received
O is now the
of the capital sector's oiden for capital, (\, and orders for capital placed by the goods sector,
Og, which represents
all
other purchasers of capital plant and equipment:
O = Ok -fOg
The backlog of unfilled orders
goods
(14)
for capital
is
now
the
sum of the supply
lines
of the capital and
sectors:
B = S + Sg
where Sg
is
the supply line of unfilled orders for capital placed
(12)
by the firms
in the
goods
sector:
D^308
18
(d/dt)Sg
The
rale at
= (Og - Ag)
(15)
which the goods sector acquires
and the delivery delay of the capital sector
capital
depends on the goods
sector's
supply line Sg
A
Ag=Sg/A
(16)
Likewise, since the capital sector acquires capital from
own
is its
itself,
the capital acquisition lag. A,
it
faces
delivery delay. A:
A=A
Finally, the
The
full
and
Sg).
demand
model
It
(17)
as
from the goods sector of the economy Og
captures
some of the
its
positive feedbacks created
own output As shown
in
exogenous.
above and constant orders from the goods
by the dependence of the
capital sector
Sterman (1985) and Br0ns and Stuns (1991),
to these positive feedbacks the equilibrium of the
oscillations
is
a third order nonlinear differential equation system (the state variables are K, S,
is
of any economy on
due
for capital derived
sector,
model
is
unstable.
With
the
same parameters
Og, a small perturbation produces expanding
which are ultimately bounded by the nonlinear constraints associated with
ment function g {
}
•
and capacity
utilization function
u{
•
)
.
The steady
a limit cycle with a period of approximately 50 years (Figure 5).
model has many of the
features of the long
wave generated by
state
the invest-
behavior of the model
The long wave generated by
the full
MIT model,
is
die
including phase
relationships and relative amplitudes for output, capital stocks, capital orders, acquisitions and
discards, delivery delay,
tions,
pit.
^
and
and capacity
sensitivity tests arc
found
utilization.
in
A full equation listing, explanation of formula-
Sterman (1985).
^
The
cycle arises via the lagged negative feedback loop described in the discussion of the construc-
tion cycle.
To
understand
how
the oscillation sustains itself, consider the processes that produce
the upper and lower turning points in the cycle. Imagine that the equilibrium
is
disturbed by a
EM308
19
small increase in the
demand derived from
the
goods
has insufficient capacity and therefore increases
demand
total
for capital thus increases
still
its
The
sector.
own
capital
producing sector finds
orders above the replacement
further above capacity, stimulating orders
The
rate.
still
it
more.
Total orders rise faster than capacity due to the construction delay, so the backlog of unfilled orders
rises,
and capital producers find
gap berween desired and acnial
These feedbacks generate
and
still
more
expand are slowed by
their attempts to
capital
widens
further,
causing
still
more orders
a self-reinforcing spiral of increasing orders, a greater
orders. Evenniaily, the various nonlinearities limit the increase in
Production capacity gradually overtakes orders. The backlog then stans to
positive loops that
powered
the expansion drive the
backlogs, desired production capacity starts to
fall,
economy
fall.
into depression.
to
be placed.
need for capital
demand.
Now
capital
for capital and leading to
still
same
leading to a reduction in orders. Falling deliv-
the capital sector finds itself with^xcess capacity and cuts
demand
the
With decreasing
ery delays reduce orders by accelerating acquisitions and reducing the required supply
the
The
rising delivery delays.
its
Thus
orders for capital, funher decreasing
At
nrore cutbacks in orders.
producing sector has severe excess capacity and cuts
line.
its
own
the
end of the upswing, the
orders to zero. Capital pro-
duction must remain below the level required for replacements until the excess capacity depreciates
-
a process
The lower
rationality.
which may take a decade or more due to the long
turning point and initiation of the next cycle are direct consequences of
The model assumes
capital producers build capacity to
and do not uiKJerstand or invest to
satisfy the general
tem's equilibrium because the capital sector itself
demand
bounded
the order rate they forecast
full
economy.
for capital
is
less than the sys-
ordering less than discards. Evenniaily capac-
approaches the level required to meet the demand of the goods sector. Capital producers then
increase their orders in order to offset discards.
demand
fill
is
meet
equibbrium of the
Specifically, during the depression phase of the long cycle
ity
lifetime of the capital stock.
for capital
However,
above capacity, and backlogs begin
incoming orders from the goods sector and
its
own
the increase in orders boosts the total
to rise.
Faced now with capacity too low
orders, the capital sector
must increase
its
to
D^308
own
20
OTders further above replacement needs, and the next expansion begins.
Thus
the long
sectOT,
and
wave
is
generated endogenously by the investment behavior of the capital producing
persists without
exogenous
excitation.
for instance, the capital/output ratio or the
Changing the parameters of the model such
maxima of the nonlinear functions may change
amplitude and period of the wave. However, the characteristic
period on the order of 50 years
is
range various bifurcations
changes
(Rasmussen
The model,
et al. 1985,
(i.e.
self- sustained oscillation
robust over most of the realistic parameter range.
in the steady-state
as,
the
with a
Beyond
this
behavior of the model) occur
Szymkat and Mosekilde 1989, Brons and Stuns 1991).
particularly the critical decision rule for capital investment, has been tested both
econometrically and experimentally. Senge (1980) showed that a disequilibrium investment function similar to the rule here provides a better account of post-war
US
data for a variety of industries
than the neoclassical investment function. Sterman (1987, 1989a) converted the model into an
experiment
in
which
subjects, including
some experienced managers, made
decision for the capital producing sector. Despite
generated long
wave
full
information, the vast majority of the subjects
cycles ccHresponding closely to those of the model. Econometric estimation
of the subjects' decisions showed they conformed well
orders. Simulation
showed
to the
assumed decision
that the estimated decision rules for about
produced the limit cycle behavicw, and about
25%
Subsequent experiments have shown these effects
Interacting cycles:
to
cyclical
omy
modes
were
and the
in the
total
Kampmann and Sterman
mode
1992).
locking
disequilibrium, behavioral foundation for each of the three
economy. Thus
linear, the cycles
of the subjects
be robust to financial incentives, training,
Nonlinear entrainment and
The discussion above provides a
40%
rule for capital
yielded deterministic chaos (Sterman 1989c).
experience, and the presence of maricet institutions (Ehehl 1992,
5.
the capital investment
far,
each
mode
has been discussed separately.
If the
main
ecwi-
generated by each firm would evolve independently of one another,
behavior would be a simple superposition. Each firm might generate fluctuations with
rM308
21
a characteristic
power spectrum
in
response to various disturbances
in the
environment. But to the
extent such variations were imperfectly correlated across firms, the cyclical
pendent firms would tend to average out
at the
industry and
movements of
macroeconomic
world the only way a coherent aggregate business cycle could come about
is
levels.
such a
In
through
inde-
common
sources of exogenous variation, such as government monetary and fiscal policies or highly correlated shocks or expectations,
1969, Mitchell 1927;
However,
and indeed there are many such theories of business cycles (Bums
Zamowiu
1985 provides a survey).
there are strong theoretical arguments to suggest that nonlincarity plays a crucial role in
bringing about interaction between the
modes and thereby shaping
the level of the individual firm, the nonlinear limits
processes tend to couple the shon and loag term
the overall behavior.
Even
on the workweek and work force adjustment
modes
to
one another. Other nonlinearities
from nonnegativity constraints on gross investment, shipments of goods from inventory,
from upper
limits to capacity utilization, hiring
at
and investment
rates;
arise
etc.;
and because these decisions
depend nonlinearly on multiple cues.
The empirical evidence
for nonlinear interactions
example figure 6a shows the variation
between the various modes
in oil-tanker spot rates
is
As an
also strong.
from 1950 through 1991. Spot
rates
are characterized by series of sharp peaks and deep valleys occurring at 3 to 5 year intervals, sepa-
rated by periods of 10-15 years in
which
often last for only a few nwnths, rates of
periods rates are as low as 40.
rates
and
their variance are low.
more than 400
The altemating
During the
f)eaks,
which
are attained while during die depression
pattern of
calm punctuated by wild swings
reflects
the nonlinear interaction of the tanker construction cycle with business cycle variations in the
demand
for oil transportation.
ordering and building of
Pl(5.
^^^
new
The construction cycle
in this case arises
from
the long delays in the
tankers.
t
Econometric, experimental and
are primarily based
on
field studies
show
that ship-owner's decisions to order
the recent tanker rate (Zannetos 1966,
new
Randers 1984, Bakken 1992).
tankers
D-4308
22
Suppose demand for oil shipment
will be high.
entry of
new
The
high relative to the capacity of the world fleet Tanker rates
resulting high profits induce existing operators to
players into the market Orders for
struction delay (2-4 years),
tional
is
new orders
demand
will
new
expand
their fleets
However, due
ships swell.
and cause
to the long
con-
remain high for several years, during which rime addi-
are placed by existing players and
sioned excess capacity develops and tanker rates
nearly reaching zero), but since the service
life
new entrants. When
fall.
these ships are
New orders drop below
of typical tankers
is
commis-
scrap rates (often
construction remain depressed for years, until capacity once again drops below demand, rates
and the next cycle begins. Consistent with the theory of bounded
new
15-25 years, spot rates and
rise,
rationality, this description
assumes shipowners do not have complete information about the global shipbuilding market or
understanding of long-term market dynamics, but rely primarily on current profit potential (spot
rates relative to costs of
new
ships) in placing orders (Zannetos 1966,
The nonlinear interaction of the
to figure 6b. Spot rates are
capacity, since
elasticity
of supply
the wcffld fleet
variations in
in spwt rates.
demand
Suez
demand
is
is
business and construction cycles
low and
fluctuations are easily
High
utilization
demand caused by
shown by comparing
figure 6a
insensitive to the business cycle in periods of surplus tanker
accommodated by higher
high). Conversely, rates arc high
high.
is
Bakken 1992).
means supply
and
is
volatile
utilization (the short
when
mn
capacity utilization for
quite inelastic in the short run; small
the business cycle or by geopolitical shocks yield dramatic changes
The parameters governing
the response of the maricet to short term variations in
including the business cycle depend on the phase of the long construction cycle. Thus the
crisis,
coming
at
a time of high fleet utilization, caused surges in rates, while the Iran-Iraq
war, coming during a time of excess capacity,
is
barely visible in the data.
Nonlinear dynamical theory also suggests that the different cyclic modes
may
through the process of mode-locking. Specifically, oscillatory modes
nonlinear systems with
similar frequencies tend to adjust to
The
classical
example
is
one another such
that their periods
in
entrain
become
the synchronization of the rotational motion of the
one another
precisely the same.
moon
to its orbital
mo-
D-4308
23
tion, so that the
same hemisphere of
the
moon
perpetually faces the earth. Other well-known ex-
amples arc the synchronization of the circadian rhythm of many organisms
night and day, the synchronization of (mechanical) clocks
chronization of menstrual cycles between
different oscillators can thus explain
parameters and
cies
initial states
why
at the
phase relations (Forrester 1977).
to relative price
with different parameters and
there are aggregate business cycles
and two modes
As
and the syn-
wall,
when
the differing
Homer
(1980) shows
Couplings between firms cause the
into a coherent aggregate cycle with stable
how
basic market processes such as con-
and availability provide sufficient coupling
initial
to
synchronize firms
phases.
only one manifestation of the more general phenomenon of frequency-locking
is
1982, Mosekildc et
the other.
level.
drawn together
or nonlinear entrainment (Amol'd 1965, Glass et
ics,
same
living in close contaa. Nonlinear coupling of
macrocconomic
cycles generated by different firms to be
Synchronization
the
24 hour cycle of
of different firms might cause them to oscillate with different frequen-
and phases, averaging out
sumer response
women
hangmg on
to the
al.
al.
1984, Jensen et
1990). In nonlinear systems, an oscillatory
may
synchronize whenever a harmonic of one
a result, nonlinear oscillators tend to lock to
al.
mode
mode
1983, 1984,
Rand
et al.
contains various harmonis
one another such
close to a harmonic of
that
one
oscillator
completes precisely p cycles each time the other oscillator completes q cycles, where p and q are
integers.
Such mode locking might explain Schumf)eter's (1939) observation
construction cycle
was approximately
that the period of the
three times the period of the business cycles, and the period
of the long wave was approximately three times the period of the construction cycle.
To
illustrate
nonlinear entrainment and explore
nxxlify the long
wave model so
soidally with period
Og = Og*(l
The
the different cyclical
that orders for capital derived
T and fractional
-t-
how
amplitude
A
modes might
interact,
from the goods sector fluctuate
we
sinu-
around a constant level Og*:
(18)
Asin(27Ct/D).
sinusoidal forcing models in a simple fashion the other cyclical
modes generated by
the econ-
D^308
24
omy. Faced with
this
fwcing, the frequency of the long wave will adjust
in a
manner that depends
both on the amplitude and frequency of the external forcing. The adjustment will tend to lock the
two cycles
into an overall periodic motion in
which the long wave completes precisely p cycles
each time the forcing signal completes q cycles, where p and q are integers.
As an example
figure 7a
shows
the results obtained
when
the
model
(A = 0.20) sinusoidal modulation with a forcing period T = 22.2
quency
is
is
wave has increased
its
40%
period by close to
fre-
so as to accommodate precisely
3 periods of the faster cycle. Moreover, within the interval 19.9 years
in the period
pit.
Here the forcing
years.
representative of the construction cycle. Relative to the unforced limit cycle behavior
(figure 5), the long
that the
perturbed by a 20 per cent
1 :3
of the forcing signal will cause a precisely proportional
entrainment
is
maintained.
<
T<
24.8 years, a change
shift in the
long wave such
^
^A,^t
H€^^
A clear illustration of the periodio nature of the mode-locked solution is shown in phase
jections of the steady-state behavior of the system. Figure
ing to the time-domain behavior in figure 7a. Here,
capital sector capital
7b shows
we have
space pro-
the phase f)ortrait correspond-
plotted simultaneous values of the
K and the goods goods sector capital orders Og over many cycles.
The
hori-
zontal axis represents the external forcing, and the vertical axis the response of the model.
Production capacity of the coital sector builds up and decays precisely once for each three swings
of the external signal.
Figure 8 shows the results obtained with the same amplitude of the forcing signal (A = 0.20), but
with the fwcing period
eccMiomic long
T = 4.6 years.
wave and
wave completes
the short-term business cycle, produces 1:10 entrainment.
precisely one oscillation for each 10 business cycles.
tion exists in the internal 4.47
ment
H'^^
ratios
To illustrate
of
This case, which could represent the interaction between the
< T < 4.70
years.
Near
this interval
we
The long
The 1:10 mode-locked
solu-
find intervals with entrain-
1:9, 1:11, 2:19, 2:21, etc.
the variety of different behaviors that can result
from
relatively
weak
perturbations of
D-4308
25
wave model,
the long
the first
200
figure 9
shows
years, the nxxlel runs with a constant
the undisturbed long
wave
oscillation.
In
demand
1
is
for capital to the
waves
a result of a period-doubling of the above
we may
16.4 years, and the half-period (which
= 0.20 and
T=
19.4 years. For
goods
sector,
showing
year 200, the external forcing begins. After a short
transient the nxxicl locks into a 2:6 solution, with 2 long
forcing. This
A
the results obtained with
still
1
for each
6 cycles of the external
The
:3 solution.
true period
wave
identify as the long
period)
now
is
is
58.2
years.
v":^^
A more complete
picture of the entrainmcnt process
is
obtained by plotting the observed mode-
locking ratio as a function of the forcing period. Figure 10 shows an example of such a construc-
(Mandelbrot 1977). The period of the external forcing has here
tion, a so-called devil's staircase
been varied from 5
to
series of intervals with l:n
numbers
surate winding
mode-locked
By
observe a
Between
these, intervals with other
commen-
from 27 <
T<
at
A
= 0.025.
37 years, for example,
ad infmitum on a smaller and smaller
between the
of figure 10. Here,
1
and the
:3
random shocks
1
we
:2 steps.
that continuously
scale.
The
fractal nature
that causes
of the devil's staircase
it
as, for instance,
1
:3
and
1
to repeat
is illustrated
have plotted some of the principal mode-locked solutions
In practice, the finer details will be
bombard
the
economy
washed out by noise -
:4 are stable
over
the
will not allow the trajectory to settle in the
neighborhood of one of the more complicated solutions. However, the more fundamental
such
find
more and more resonances covering narrower and narrower
For small values of A the phenomenon has a self-similar structure
in the insert
we
and 5:6 entrainment.
refining the calculations one finds
intervals.
solutions.
are observed. In the region
intervals with 3:5, 2:3, 3:4, 4:5
\'(^~^^
We
54 years while keeping the amplitude constant
much broader
intervals.
Mode
ratios,
locking can thus
be robust to perturbations and noise that cause individual cycle shape and timing to vary.
If the
1 1
amplitude of the forcing signal
shows
the principal
is
changed, the intervals of entrainment also change. Figure
mode-locked zones as a function of forcing period and amplitude. For
A
=
0^308
26
there can, of course, be
intervals of
no entrainment
As
at all.
A is increased, however, wider and wider
mode-locked behavior develop, and the regions of mode locking, known as Arnold
The
tongues, broaden. For small amplitudes quasiperiodic behavior exists between the tongues.
As
tcmgues cannot continue to grow, however.
tongues begin to overlap, and quasiperiodic behavior then vanishes. In our model
=>
pife-
Above
0.025.
grows
the amplitude of the forcing signal
the critical value the trajectory
is
this
the
occurs at
A
either periodic or chaotic.
»1
Figure 12 shows an example of chaotic behavior in the model. The period and amplitude of each
long wave are
years and
now
different.
The period and
A = 0.20, respectively.
the amplitude of the perturbing signal are
Chaotic behavior
conditions such that two simulations with
initial
is
characterized by
its
T=
16.1
sensitivity to initial
conditions differing only slightly will diverge
exponentially until the position of one bears no relation to that of the other.
H€»^
A variety of complex nonlinear phenomena arise where the Amol'd tongues overlap, including
period doubling, intermittency, and frustration. Figure 13 shows a bifurcation diagram in which
the 1:2 mode-locked solution
is
transformed into 2:4, 4:8,
tude increases from 0.0475 to 0.0625 while maintaining
along the vertical axis in this diagram
long wave.
When
is
the
T=
19.6 years.
maximal production
the forcing amplitude is less than 0.048
forcing ampli-
8:16,... solutions as the
all
The
variable plotted
capital reached at the
maxima
are equal.
For slighdy
higher amplitudes, however, the model bifurcates into a behavior where low and high
alternate.
At about
peak of each
maxima
A = 0.0552 a new bifurcation occurs so that the model now shifts between 4
different
maxima. The period-doubling cascade continues
becomes
chaotic.
As A
is
increased further
behaviOT (Feigenbaum 1978) until
we
finally, at
until at
A = 0.0570 the behavior
observe the characteristic windows of periodic
about
A = 0.0597, a sudden expansion of the chaotic
attractor occurs. This represents a so-called crisis (Grebogi et
al.
1982), where the
model now
generates a complicated behavior in which intermittency chaos due to the interaction of the
solution with period-doubling chaos arising
from the
1
:2 solution.
1
:3
CM308
In
27
other regions of the phase diagram, two or nx)rc periodic solutions coexist, and
tions (or subsequent pcnurbations) determine
stance, the case in the region around
cross.
T=
29.4 years and
K
and the capital sector supply
tions that lead to the 2:3 [jcriodic solution,
the 3:5 solution.
The boundary between
ing periodic solutions
qualitative
6.
changes
is
clearly fractal.
in the
A
= 0.05, where
line S.
initial
the 2:3
conditions for the capital
and white points indicate those conditions
Minor changes
for in-
is,
and 3:5 tongues
Black points indicate those
the basins of attraction for the
condi-
that lead to
two simultaneously
in initial conditions
condi-
initial
exist-
cause unpredictable,
steady state behavior.
Conclusion
Recent developments
nomics have joined
to
in
nonlinear dynamics, behavioral decision theory, and experimental eco-
form
the basis for empirically testable, nonlinear, disequilibrium theories of
economic dynamics grounded
The
which solution the system chooses. This
Figure 14 shows a 200 x 200 point scan over the plane of
sector capital stock
initial
in
experimental
test
and
field study
integration of these disciplines sheds significant light
movements
cyclical
decision
at different
the structure of
on the origin of aggregate cyclical
frequencies, as well as the interaction of these nxxles. In particular,
movements of different
making with
of economic decision making.
periodicities can arise through the interaction of
the time delays, stock
economic
and flow
structure,
boundedly rational
and nonlinearities fundamental to
activity.
Behavioral nxxicls of disequilibrium dynamics show
how
firms can generate cycles that closely
resemble the short term business cycle and the 15-25 year construction cycle. Incorporating positive
feedback processes arising from macroeconomic couplings between firms and among the pro-
duction, consumption, financial, and government sectors explains
how
the long
wave can
arise.
Unlike the short term business cycle, the long wave appears to be a self-organized cycle that does
not require continuous exogenous excitation to persist.
The approach described here
also sheds light
on
the interaction of the different cyclical
modes. The
D^308
28
systematic coincidence of different cyclical
modes
in
economic dynamics was suggested long ago
by Schumpeter (1939), and Forrester (1977) proposed nonlinear entrainment as the explanation for
the apparent
mode-locking among macroeconomic cycles. However, formal investigation of such
macroeconomic entrainment processes with
modem
nonlinear theory does not appear to have been
attempted before. Though the model investigated here
entrainment
may
arise in a
system
that captures basic
is
highly simplified,
we have shown how
macroeconomic feedback processes and
fundamental nonlinearities such as nonnegativity and ci^acity constraints.
More
generally, entrainment can cause different oscillatory processes with approximately similar
periods to
move
phase
in
at a single
frequency, producing aggregate business fluctuations.
Nonlinear entrainment also accounts for the existence of a small number of relatively well-defined
periodicities: oscillatory tendencies of sigiilar periodicity in different parts of the
drawn together in
1
1
:
synchrony to form a single mode, and each of these modes
from the next by a wide enough margin
to avoid entrainment at the
omy exhibits clearly distingiiishable modes economic
the Kuznets cycle,
economy
historians
same
period.
is
are
separated
Hence
the econ-
have dubbed the business cycle,
and the economic long wave, rather than fluctuations equally distributed
at all
frequencies and phases, fluctuations that would wash out in the aggregate.
Even with
enough
may
its
relatively
to lock
wide separation
them together such
in periodicity, the interaction
that they
between modes may be strong
have commensurate periods. Nonlinear interactions
thus pull the Kuznets cycle and business cycle into phase with the long
peaks ot downturns. AdditionaUy, since mode-locking
stable over a finite range of individual cycle periods,
at
wave and accentuate
a given rational winding
mode-locking
is
number is
robust with respect to varia-
ticms in the parameters governing the individual cycles, allowing entrainment to persist over long
time periods despite technological and institutional change, perturbations, and other sources of
variation in
economic
life.
D-4308
29
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D-4308
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Figure captions
Figure
US
1.
and
Industrial Pixxluction
civilian
the system reflects the interaction of multiple
cycles,
unemployment
1947-1992. The evolution of
rate,
modes of behavior: long-term growth, business
and the long wave (production and unemployment grow on average and fluctuate with the
short-term business cycle.
and the slowdown
in
The long wave causes
the rise in the average level of
economic growth since about 1970;
this is
more
unemployment
easily seen in the detrended
data (figures 2 and 4).
Figure
Detrended
2.
US
industrial production at three levels
characteristic amplification of business cycle fluctuations
of the distribution chain, showing the
from production of consumer goods
through intermediate goods to raw materials. Note also the phase lag as the cycle propagates from
consumer goods production
removed
Figure
to
raw
materials.
The long-term trend of =.29%/month has been
to highlight the business cycle.
3.
The
real-estate or construction cycle, illustrated here
office space in
Figure 4.
US
since 1800.
downtown
real
GNP,
Bostoil,
-
1990 (see also figure
6).
rate for
commercial
Source: Bakken 1992.
detrended by removing the long-term average exponential growth rate
The deviations
fixjm the long-term trend reflect the interactions of the cyclical
particularly the business cycle
faster than trend
1952
by the vacancy
and long wave (and noise). The long wave causes growth
from 1947 through about 1970, and slower than trend
since.
modes,
to
be
Note the large
amplitude of the long wave compared to the business cycle. During the long-wave expansion
business cycle recessions seem to be short and mild; during the long-wave decline recessions
appear to be longer and deeper.
Figure
5.
Limit cycle behavior of the simple long wave model. The steady
state
behavior
is
shown. There are no exogenous sources of variation.
Figure
6.
(a)
utilization
in
demand
Oil tanker spot rates (worldscale units; 100
of the world tanker
fleet,
for oil transport with the
Source: Bakken (1992).
showing nonlinear
= normal
profitability); (b)
Capacity
interaction of the short-term business cycle
endogenous 20 year construction cycle
in tanker supply.
D-4308
37
Figure
7.
(a)
Time domain and and
exogenous forcing
demand
in
(b)
phase space behavior of long wave model with 22.2 year
for capital
from
(a)
8.
demand
ing in
Time domain and
for capital
from
(b)
the
The period of the long wave
sector.
exogenous fluctuation
adjusts to maintain 1:3 entrainment with the
Figure
goods
the
phase space behavior of long wave model with 4.6 year forc-
goods
sector.
Now
the f)eriod of the long
wave
adjusts to
maintain 1:10 entrainment with the forcing fluctuation.
Figure
In year
9.
200
the
demand
tude .20 and period 19.4 years.
for capital of the
goods sector begins
The system quickly
to fluctuation with ampli-
entrains with a 2:6 solution.
Figure 10. Devil's Staircase, showing dependence of mode-locking ratio on the frequency of the
forcing function.
Mode
locking exists
interval of the forcing frequency.
Figure
1
1.
The
at
every rational winding number and
staircase is a self-similar fractal as
Amol'd diagram showing def)endence of
tude of the forcing function.
The
so-called
the intervals of
Figure 12. Chaotic behavior of the model with
Figure 13. Bifurcation diagram showing
of the forcing amplitude A.
the
As
A
until they
= .20 and
maximum capital
stable over a finite
shown
solutions increase in
begin to overlap
T=
in the inset.
mode-locking on the ampli-
Amol'd tongues of mode-locked
width as the amplitude of the forcing signal increases,
is
at the critical line.
16.1 years.
stock in each long
wave
as a function
the amplitude of the forcing signal, and thus the coupling
endogenous and exogenous cycles, grows,
the
model experiences
between
a period doubling cascade to
chaos.
Figure 14. 200 x 200 point scan over the plane of
stock
K
and the capital sector supply
line S,
initial
conditions for the capital sector capital
normalized so base case values =
1.0.
Black points
indicate steady-stale entrainment with a 2:3 ratio; white points indicate 3:5 entrainment.
boundary
is fractal.
Here
A
= .05 and
T=
29.4 years.
The basin
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Commercial vacancy rate,
Downtown Boston (%)
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Date Due
APR.
KIAR.
2S
3
i;
195
96
8
Lib-26-67
MIT
3
"1060
LIBRARIES
OOfiMbiaM 7