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Bisaggregating a Simple Model of the Economic Long Wave
Christian E. Kampmann
System Dynamics Group
Massachusetts Institute of Technology
ABSTRACT
A multi-sector, input-output version of Sterman’s simple Long Wave
Model is developed to investigate the validity of the capital self-
ordering theory for a more realistic system with diverse capital
types. Simulation experiments with varying capital lifetimes and
input-output coefficients tend to reproduce the characteristic
fluctuations in capital production, caused by self-ordering, with a
period in the 30 to 70 year range. However, complex patterns of
oscillation with wide variance in period can emerge, explained by
varying dominance of self-ordering loops. The analysis thus
confirms the destabilizing effect of self-ordering and its signifi-
cance for long term fluctuations while raising issues and generating
new insights about the-long wave.
INTRODUCTION
The System Dynamics National Model project at M.I.T. represents a unique
approach to macro-economic theory, in that it aims to explain the aggregate
behavior of the economy from the bounded rationality of internal decision
rules of firms and households (Forrester, 1979). But while such decision rules
are represented with great richness and detail in this approach, whole
industries and sectors in the economy are aggregated into single "firms", to
allow for internal detail while keeping the model at a manageable size.
This aggregation, which occurs in both the National Model and the simple long
wave model as well as in a host of other system dynamics economic models, is
based on the assumption that interactions occurring within each individual
firm in a particular sector are more or less “in phase" with that of other
firms in the sector, and that these interactions are more important than the
interactions among firms for the sector's overall behavior. While this
assumption may be adequate, there have been few attempts to test it
explicitly.
One of the major outcomes of the National Model is the comprehensive theory
of the economic long wave, the large cycles in economic activity with a period
of about half a century (see e.g. Sterman, 1984a). The mechanism of “capital
self-ordering" seems to play an important role in’the long wave. A simple
model of a one-sector capital production system built by John Sterman shows
how the self-ordering of capital is sufficient to cause long waves.
Sterman'’s model provides a convenient focus for a first attempt to tackle tha
aggregation issue; apart from being simple, it also raises some obvious
questions about capital aggregation: In the simple model, the physical and
technical features of the system, namely the capital/output ratio and the
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average lifetime of capital, are very important parameters: they significantly
affect the period and amplitude, and even the existence of fluctuation. But
at the same time, these aggregate parameters relate to an aggregate capital
stock that is supposed to represent the real economy, where production takes
place in a complex input-output structure of different industries, producing
and using many different kinds of capital with widely different physical
characteristics. The average lifetime of capital, for instance, may vary from
a few years, in the case of automobiles and tools, to fifty years or more, in
the case of buildings, roads, and other infrastructure.
To address the importance of the physical diversity of capital in the real
economy, it is necessary to replace the one-sector model with a multi-sector
system, where each sector may use several different kinds of capital in
production and may ship its own output to several other sectors, in addition
to shipments to the “household”, or final demand sector. The economy becomes
an input-output system of capital producing firms, where the technical
coefficients, average lifetimes of the capital product, and substitution
possibilities may be different in each sector.
It is assumed in the following that the reader is already somewhat familiar
with Sterman’s model (for a more extensive description, see Sterman, 1984b).
The next section only briefly recapitulates the model. The following section
discusses some issues in the approach of this study and sketches the disaggre-
gate model. Some simulation results are then presented, followed by conclu-
sions and suggestions for future work.
THE SELF ORDERING THEORY
The notion of self-ordering, i.e. the idea that it takes capital to produce
capital, is well known in economics, mostly under the name of the “accelera-
tor" mechanism (see e.g. Samuelson, 1939). Mathematically, it can be expres-
sed as follows: if production is proportional to capital stock by a factor of
1/COR (capital-output ratio), and capital depreciates at a rate of 1/ALC
(average lifetime of capital) then, to produce a net flow of capital goods HO,
the capital sector must produce a gross flow PR oft
R = HO/(1-(COR/ALC)).
However, this observation per se does not explain the existence of cycles. The
long wave is inherently a disequilibrium phenomenon arising in the transient
adjustment, where the effect of self-ordering is greater than in equilibrium,
due to several factors, the most important being the need to restore backlogs
after an unanticipated change in demand.
““Sterman’s model consists of a single capital=producing firm which receives an
exogenous stream of orders for capital. In addition to the orders from out-
side, the aggregate capital sector orders capital from itself. All orders
are accumulated in a backlog, which is then depleted as the capital is deli-
vered. The firm's production capacity is proportional to its capital stock,
but production is cut back below capacity when there is insufficient demand,
i.e. when the order backlog falls below what is compatible with the existing
capacity. The firm's orders for capital are based on the need to replace
depreciating capital and the desire to increase or decrease capacity. In its
Planning, the firm takes into account the orders already placed (i.e. the
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capital under construction) to avoid double ordering. The desired production
capacity is based both on (adaptive) expectations of future orders and the
need to adjust order backlogs to normal levels.
When, from an initial equilibrium, this system is perturbed by a small change
in exogenous demand, it exhibits large oscillations in capital production and
capacity with a period of about 50 years-~much longer than a typical business
cycle fluctuation-~that persist without continuous outside triggering.
Self-ordering is directly responsible for the persistence and long period of
fluctuation, through several channels. First, if there is a perceived shor-
tage of capacity, more orders are placed, swelling the backlog, reinforcing
the perception of shortage of capacity. Second, as delivery delays rise,
capacity arrives more slowly than expected, widening the initial discrepancy
between desired and actual capacity. Third, to compensate for higher delivery
delays, orders are increased, resulting in still higher backlogs and delays.
These self-inforcing mechanisms cause an initial small shortage of capacity to
result in a foot-race between desired and actual capacity, leading to a large
overshoot of capacity over its equilibrium value. Eventually, capacity
catches up with demand, and the process is reversed, causing a rapid fall in
demand and production, followed by a prolonged depression with capacity far
exceeding demand. The depression lasts for about two decades, as excess
capacity is depreciating, until orders once again catch up with capacity to
start a new cycle.
DISAGGREGATING THE MODEL
Because of the prevalence of non-linear, highly complex models in system
dynamics, it is virtually impossible to derive general mathematical results
that could indicate the validity of aggregation. Moreover, system dynamicists
stress the fact that validity is a relative concept. Philosophers of science
have long emphasized that there is no objective way of judging the performance
of 4 model. Ultimately, then, the validation of a model is a question of
using common sense. Moreover, good judgement can only be exercised when one
has a clear idea of the purpose of the model (see e.g. Forrester, 1961).
The purpose of Sterman’s model was to investigate how self-ordering by itself
is sufficient to cause a long wave, and what factors may control the period
and amplitude of the cycle. In comparing results from a disaggregate model,
three questions should thus be asked:
o Will the instability of the simple model persist, and for the same
causes?
o Will the cycles in production be similar, i.e. be of about 35 to 70
years in period with a large amplitude?
o And will fluctuations in different sectors be in phase so that one can
meaningfully speak of a single "wave"?
The simplest approach to the aggregation issue is to construct a disaggregate
input-output version of the simple model and compare the results. However, in
making the transition from one to several sectors, the question arises of how
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the sectors should interact. In the one-sector model, the “firm" and the
"household" have no choice in what capital to order and where to order it
from, since there is only one supplier and only one type of capital. This is
not the case in a multi-firm setting. How, for instance, will a firm or a
consumer react to changing availability (and/or price) of factors, given
factor substitution possibilities? Will the decision rules vary depending on
what type of capital is in question and what kind of industry the firm is in?
For example, the decision to build a new plant may be influenced by other
factors than the decision to buy more typewriters for the office.
To keep the focus on the physical aspects of capital, the simple assumption
has been adopted that the decision making in all sectors is identical in
structure to that of the simple model, i.e. the decision rules, adjustment
times, shapes of non-linear functions, and the structure of ordering and
production equations are the same, with only minor modifications to allow for
a multi-input multi-output situation. Each sector is assumed to order from
the other sectors based on a desired production capital mix that remains
fixed, regardless of the relative availability of factors and the possiblities
for substitution. The only difference between sectors is the production
techonology, lifetime of capital product, and the share of this product in
“household” or final demand.
While such simplification is dome partly to limit the scope of the project,
there are also limits to how far one can depart from the decision functions in
the simple model, since they are an inseperable part of the theory it
represents; at what point does a disaggregate model no longer represent the
same theory?
The question of when the two models are “equivalent” relates closely to the
old problem of capital aggregation in economics. In order to compare a simple
and a disaggregate model in a meaningful way, one must require that the
agoregate parameters of the disaggregate model are the same as those of the
simple model. But the question is what aggregation rule is appropriate. The
economics literature has shown that unfortunately, it is impossible to give
exact rules and that the aggregation procedure must be taylored to the parti-
cular aspect of capital one wishes to investigate (see e.g. Fisher, 1969).
It is common usage to define the aggregate capital/output ratio as the
aggregate dollar value of capaital divided by the aggregate production
capacity measured in dollars per year. Likewise the aggregate lifetime of
capital can be defined as the aggregate dollar value of capital stock divided
-by the aggregate depreciation rate in dollars per year. However, prices are
not included in the model, and even if they were, they would change over time,
making it difficult to interpret the results. The only way around this is to
aasume that relative prices of different capital types are constant. Moreover,
units of capital can be defined in such a way as to allow direct addition of
the physical capital stocks to obtain aggregates. (see Kampmann, 1984.)
A series of simulation experiments were made with a Two-sector and a Five-
sector model, respectively, in which all sectors were assumed to have a
Cobb-Douglas production function, thus relegating the issue of capital
substitution to future studies. Given the Cobb-Douglas production functions
and the number of sectors, n, one can completely describe the system by the
following parameters:
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o The coefficient of capital type i in sector j’s production function,
RVS(i J j)s i=1,...,m3 jutl,...,n. CWhen production cost is minimized, this
parameter is equal to the flow value share of capital i in sector j's
production cost, hence the acronym RVS for “Reference Value Share")
o The capital/output ratio for sector j, FOR(j); j=l,...,n (In the model,
the more general name “Factor/Output Ratio" was used, hence the acronym
FOR(j))
o The flow share of sector j in final demand, IVSED(j); j=!,...,n (The
acronym stands for “Initial Value Share of End Demand")
o The average lifetime of capital type i, AL(i); i=1,...,n
Appendix 1 contains a computer listing of the two-sector model. For a more
detailed explanation and description of the models, see Kampmann (1984).
SOME SIMULATION RESULTS
A couple of the many experiments performed with the model are described and
reproduced below to give an impression of the range of possible behavior. The
experiments centered around two major themes:
o How will variations in input-output structure (including the capital
intensity of individual sectors) affect behavior, assuming the lifetime
of each capital type is the same? This corrensponds to variations in the
paramters RVS(i,j), IVSED(j) and FOR(j)
o How will diversity in capital lifetimes affect behavior in systems with
identical (flow) input-output structures? This corresponds to varying the
parameters AL(i)
While the experiments show a wide range of behavior, there are certain
features which persist throughout. The examples below illustrate some of
these features, which are summarized under the conclusions below.
Variations in the Input-Output Structure.
If all capital has the same lifetime and all sectors have the same overall
capital intensity, how will a disaggregate input-output system behave
differently from a simple one-sector system? Figure 1! shows an example of
what can happen when the input-output structure of the system is varied.
It turns out that.the symmetry of the system is of crucial importance for the
result. A symmetrical input-output system is defined as one where an increase
in orders for any capital product will require an increase in each sector's
gross output of the same proportion for all sectors. This condition is
satisfied if any given sector’s cost share is the same everywhere, i.e. if:
RVS(i,j) = IVSED(i) for all i,j.
A symmetrical system would behave exactly as the simple model if household
orders were perturbed from equilibrium in the same proportion. Even when
incoming orders are not exactly proportional at all times, the interdependence
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implied by the input-output structure results in a very strong entrainment,
bringing all sectors to fluctuate uniformly. This entrainment is illustrated
in Figure f(a), which shows a simulation of a symmetrical two-sector system
where each sector has an equal share of end demand and uses the two capital
types in equal proportions in the production function, i.e.
RVS(i,j) = «6 «5 IVSED(j) = .5 .8
5 5
At time 1@, the household demand for sector 1's product only is increased 5
percent. If both household demands were stepped uniformly, the response would
be identical to a one-sector system, but even still, the two sectors move
quickly into phase to give a fluctuation pattern virtually identical to that
of the simple model.
In figure 1(b) and (c), the symmetry is broken by changing the shares of the
two sectors in final demand, IVSED(j). The split is changed from fifty-fifty
in Figure {(a) to 2:1 in (b) and the extreme in (c), where Sector 1, hence-
forth called sector “F" for “Final" thus delivers all final demand, while
Sector 2, henceforth called sector "I" for "Intermediate", produces only
intermediate goods.
The result is a more complex pattern of fluctuation. In Figure f(b) the long
wave breaks into a combination of a 7@ year and a smaller 35 year cycle, and
Figure 1(c) shows a double cycle with the longer period of about 5@ years.
Moreover, the relative amplitude of fluctuation will be different for each
sector and for each cycle. The change in relative fluctuation comes from the
change in the composition of the incoming orders in each sector. Thus, when
sector "F" holds a larger share of household demand, the “self-ordering”
component of its incoming orders is relatively less than in sector "I". It is
not surprising that the sector whose orders thus fluctuate the most also shows
the most fluctuation in production and capacity.
The difference in relative amplitudes of the two sectors also explains the
double cycle: After the initial disturbance, the "I" sector will expand
relatively more above the equilibrium capacity than the "F" sector. In the
subsequent downturn, there will therefore be ample availability of sector
"I"'s product, and when demand meets capacity in sector "F", sector "I" will
still have unused capacity. As sector "“F" expands again, the excess of the "I"
product will reduce the strength of the self-ordering mechanism that fuels the
expansion, and production will catch up faster with demand. In Figure 1(b),
demand eventually catches up with capacity in sector “I", but at such a late
stage that the new peak is significantly lower than the previous one. Both
sectors will therefore have less excess capacity in the next downturn, setting
the stage for a larger expansion in the following upturn. In i(c) the
disparity is so pronounced that sector "I“ has unused capacity even at the
peak of the intermediate cycle. Through the duration of the long cycle
downturn, sector "F" therefore effectively behaves like a one-sector model
with a capital/output ratio half of normal, since only half of sector "F"'s
orders for capital fall upon itself. Such a system would show a very lightly
damped cycle of about 20 years (see Kampmann, 1984, App.1). The intermediate
cycle observed in !(c) is thus an internal dynamic of sector 1.
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Figure 1
Variations of Input-Output Structure in a Two-Sector System
The sectors are named “F"inal, “I“ntermediate and "H“ousehold, respectively.
In all three runs, the initial equilibrium is perturbed by stepping up house-
hold orders for product “F" 5% at time 1@. In the symmetrical system in (a),
where sector “F" and "I" hold equal shares of household demand, strong
entrainment results in behavior almost identical to the simple model. But as
the split of end demand between product "F" and "I" is changed in (6b) and (c),
the long wave breaks into a double cycle, due to a change in the strength of
the self-ordering effect in the two sectors.
Figure J(a)
Cost in Sector
Share FoI OH
af F WS OB OS
Product I .5 .5 .5
Figure i(b)
Cost in Sector
Share FoI oH
of F .& .5 .67
Product I .5 .S .33
ear ray +
Figure lic)
Cost in Sector
Share FT oH
of F .S 45 1
Product I 5 .5 @
J |
|e
U =
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In the above variations in input-output structure, the capital/output ratio
was kept the same for each sector. In the simple model, this parameter is
very important because it determines the strength, or the gain, of the self-
ordering loop. An increase in this parameter will increase the self-ordering
effect, causing fluctuations of longer period and higher amplitude. Ina
disaggregate system, however, there is not one but many self-ordering loops,
and they will in general not be of equal gain if the capital intensity of
production varies significantly among sectors, or if capital lifetimes differ.
In the particular case where the system is symmetric, as defined above, a
difference in capital intensity has no effect on behavior. Only when non-
symmetry is allowed will there be any effect. It is possible, for instance,
to increase the capital intensity in all sectors while keeping aggregate
capital intensity constant by increasing the less capital intensive sectors’
share of end demand. The effect is similar to increasing the capital/output
ratio in the simple model: the period and amplitude of fluctuation both in-
crease.
On the other hand, even quite extreme variations in capital intensity have not
nearly the effect a change in the capital/output ratio has in the simple
model. Variations by a factor of 1@ of the sectoral capital/output ratios
produce fluctuations with a period between 28 and 6@ years and only mode-
rately different amplitudes. Moreover, the qualitative behavior is the same
as in the simple model.
Variations in Capital Lifetimes
Table | summarizes the result of varying the lifetimes of the two kinds of
capital in a model with the same completely symmetric input-output flow struc-
ture as in Figure 1(a) above. The variation is done so as to keep the aggre-
gate equilibrium average lifetime at 2@ years as in the simple model. In this
particular input-output configuration, the equilibrium aggregate lifetime is
simply the arithmetic mean of each capital lifetime.
In all cases, the outcome is still a very regular limit cycle with the same
basic characteristics as that of the simple model. However, because of the
faster dynamics inherent in the short-lived capital stocks, there is now the
possibility of an additional shorter cycle. The double cycle occurring in the
previous example (Figure 1(b) and (c)) had its root in the change in the
relative strength of self-ordering for the two sectors (a "gain" component).
The intermediate cycle now results from the difference in the time it takes
for excess capacity to depreciate (a “delay" component).
As Table | testifies, there is no simple relationship between the diversity of
lifetimes and the period and amplitude of fluctuation. The period stays ina
range from about 40 to about 6&5 years, and the amplitude generally declines as
the diversity of lifetimes increases, but there are several exceptions to that
rule. These irregularities are due to the non-linear frequency entrainment of
the two sectors (see Kampmann, 1984, App.4).
The last run of Table 1 is reproduced in Figure 2. Here, sector "S” for Short
and sector "L" for Long produce a product with a lifetime of 5 and 35 years,
respectively.. The system shows a combination of two modes, a long cycle of a
85 year period, and a shorter, damped cycle of about 15 years. The
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Figure 2
Introducing a Split in Capital Lifetimes in a Two-Sector Mode
Utilization "S$"
{@ - 100%
Capacity "S"
(.55 -,95 E12]
Utilization "L"
[@ - 100% 3
‘
Capacity "L"
(@- 4 E12)
'
Year
@ 30 60 30 120 150 180
Cost Share Average
Able 4 in Sector Lifetime
Results of Varying Average Lifetines of S§ L oH (years)
Capital in Tvo-sector Model of cS .S £ 5
Run # = AL(1)—aL(2)—s Period® = Fea Value® Product Lo «5 «5S 35
. Po(t) Pc(2)
years
years
% of Base Case
19 i . ig ° 98 Regardless of whether the flow
20° 16 24 45/31 1/32 93/39 input-output structure of a system
ae 4 26 63 72/66° 95 is- symmetrical , introducing '
220 12 28 61 57 92 substantial differences in capita
23erd 10 30 60/49 49/55 92/57 lifetimes can result in composite
28 5 6 66 31 3 cycles. The figure shows the last
case of Table 1, where sector "L"
and sector "S" produce capital that
has a 35 and a 5 year average
lifetime, respectively. Sector
fluctuates the most in the long
cycle, while only sector “S"
exhibits the shorter cycle.
a) Limit cycle steady state values
b) The base run is identicel to run 5 “pe
c) The cycle is not uniform; every other peak is larger.
Both peak velues are shown, and where the time
distance betweer peaks alternates in a similar manner,
both of these sre shown in the “period” column.
4) In addition to the long cycle, there is a short
intermediate Puctuation in PC(1).
©) Every other peak in PC(1) ie lerger, and both peek
values are shown. PC(2) shows @ uniform cycle.
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Figure 3
Variations in Capital Lifetimes in a Five-Sector System
Large differences in capital lifetimes in a five-sector model typically leads
to rather irregular fluctuations, where long-lived sectors are hit the worst.
'
‘
'
‘
'
eet wetedn ye HERE OE
- i
Capacity 1
(@> 1 £12) :
' Capacity 5
«22-.30 E12)_§
'
‘
'
1
Capacity 3
[.2-.68 E12
Utilization
_ Utilization
Utilization
Utilization
Utilization
[@ - 100% 1
antennae ieee: [ASR OREN CE a RNEEEREEaY HanioeNnlONmeSete:
Cost Share Average ff
in Sector Lifetimele
oF 12 3 4 65 H (years)
Product
oBiecmeitiicnnm: temsmansemenesee 12 B8 BB 2 BY) cpt
' ‘ ZiosZ Bia Buk «2 25
} 4 BioeBeiv2 Bic? Qo 08
‘ i Ro Bg? Pa @ 7
: : GS aQete2ali2 2 2
. Year
Uy) 30 60 90 120 150 188
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fluctuations in overall production are a great deal less than in the simple
model, but vary significantly between the two sectors; sector "S" shows
relatively less fluctuation than sector "L". Moreover, only sector "S""
capacity exhibits the short cycle.
The result is thus quite parallel to the difference between sector "I" and
sector "F" in figure 1, though for different reasons. The excess of factor
"L" takes too long to depreciate for sector "L" ta show a short cycle. In
contrast, the excess of factor “S" falls quickly in the long cycle downturn
and sector "S" then operates essentially like a one-sector system with a
factor/output ratio of 1.5 and an average lifetime of 5 years. The result is
the damped 15 year cycle. When the excess of factor 2 has finally been
eliminated, self-ordering regains its full strength, causing a new major
upsurge in demand and production.
To get an impression of the possible range of behavior in a more complicated
input-output system, runs were performed with a five-sector model, letting the
average lifetimes vary widely while keeping the aggergate lifetime equal to 20
years. As an typical example, Figure 3 shows the response of a system where
all five sectors have equal shares in both end demand and in each others
Production functions, but where the lifetime varies between products from 50
years in sector | to 5@ years in sector 5.
The behavior is much less regular than in simpler systems. One sees a number
of occasional major surges in production and capacity, and a shorter, more
regular fluctuation. The sectors with the longest lived products fluctuate
uniformly more in large surges than the sectors with shorter-lived products.
The long-lived sectors remain depressed for longer periods because the the
excess of the factor they produce takes longer to decay. During this period,
the other sectors, like in the previous examples, behave like a system of
production with shorter lived products and with a lower overall capital/output
ratio than the full system, giving a short, damped fluctuation. However, as
the excess of the long-lived products is eliminated, the strength of self-
ordering increases, and the system becomes more prone to self-created major
surges in demand. Thus, the short cycle fluctuations start to get larger, and
sooner or later all sectors simulataneously expand in a traditional long wave
peak.
SUMMARY AND CONCLUSIONS
The simulation experiments have produced a wide range of behavior, but also
some recurrent features!
1) Like the simple system, all the disaggregate systems tested show major
fluctuations in production and capacity that persist without continuous
exogenous disturbances, and that are much too large and long to be
explained by simple adjustment delays.
A disaggregate system tends to exhibit a combination of major expansions
and contractions, occurring over a long time period, and a shorter, more
regular oscillation. While the small cycles, if they occur, may only
affect some sectors, the major surges occur in all sectors simulaneously.
z
3) The major fluctuations in production are persistently more violent in
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sectors whose incoming orders come in the main from the production system
rather than from consumers. And the fluctuations in a sector are larger
the longer the lifetime of its product. Thus, the more a sector has the
characteristics of a capital producer, the more vulnerable that sector is
to long wave fluctuations.
4) Variations in the input-output structure tend to change the behavior from
a uniform to a compound cycle. The period of the long cycle tends to fall
in the 3@ to 70 year range. The more the household/non-household eplit
of incoming orders is the same for each sector, the closer the behavior
will be to a one-sector system. Moreover, the sectors will quickly
entrain their fluctuations. Thus, a group of sectors who have largely
the same mix of production capital and whose incoming orders vary roughly
in proportion can be well approximated by a single sector.
5) Variations in sector capital intensity can change the period of fluctua-
tion, but not dramatically. The higher the symmetry of the system, the
less discrepancies in sector capital intensity will alter the behavior.
6) With very large variation in capital lifetimes, the major fluctuations
can be quite irregular. Indeed, it becomes difficult to speak of a single
“oycle"; rather, self-ordering creates a potential in the economy for
large self-created surges in demand and production.
The most fundamental feature of the simple model is its inherent tendency to
create very large long term fluctuations in capacity and production that
persist without continuous outside triggering. This feature has reappeared in
all the simulations of the disagaregate model, and it is evident that the
surges and collapses of demand are created in the disaggregate model by the
same basic self-ordering mechanism. Based on the evidence so far, one must
therefore conclude that the self-ordering hypothesis retains its validity for
a disaggregate system.
A conspicuous difference between the simple and the disaggregate model occurs
in the shape of fluctuations. While the simple model shows a completely
regular cycle with a well defined period, the disaggregate version generally
generates more complex and sometimes quite irregular patterns. Such results
suggest that the long wave should be sought of as an inherent (endogenous)
ability in the economy to create large fluctuations, rather than as a reguler
cycle.
On the other hand, there are several factors which may shape the irregular
fluctuations of an input-output system into a more coherent pattern. Foremost
among these is probably the effect of substitution in production capital. For
instance, if a factor is in excess supply, its delivery time and/or price will
be low, causing firms to shift to a more intensive use of that factor. The
resulting increase in orders will draw down the excess supply faster--an
important way to bring long-lived sectors more into phase with the rest of the
economy. The most important next step in further research will therefore bé
to introduce a more sophisticated ordering rule in the model that reflects
relative availability of factors and the elasticity of substitution.
Work on the National Model suggests that there are several structures other
than capital self-ordering involved in the long wave (Sterman, 1984a). It is
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conceivable that these structures, involving prices, interest rates, wages and
other factors, would mold the fluctuations of the capital sector into a more
regular pattern. It would be interesting to test this hypothesis by
introducing in the National Model a disaggregate capital structure that in
isolation would give highly irregular fluctuations.
Another next step could be to examine the role of non-linearities in the
disaggregate model, and compare it to the simple model. In some simulations
(Kampmann, 1984), a change in non-linearities which is sufficient to stabilize
the one-sector system does not stabilize a disaggregate system. The self-
ordering theory could thus be further refined, and possibly strengthened.
REFERENCES
Fisher, F.M. (1969),"The Existence of Aggregate Production Functions",
Econometrica, 37, 553-577.
Forrester, J. (1961), Industrial Dynamics, Cambridge: MIT Press.
Forrester, J. (1979), “An Alternative Approach to Economic Policy: Macrobeha~
vior from Microstructure", in Economic Issues of the Eighties, Kamrany
and Day, eds., Baltimore: John Hopkins Univ. Press.
Kampmann, C. (1984) "Disaggregating a Simple Model of the Economic Long Wave",
Working Paper D-3641, System Dynamics Group, MIT, Cambridge, Mass.
Samuelson, P. (1939), "Interactions Between the Multiplier Analysis and the
Principle of Acceleration", Review of Economic Statistics, 21, 75-9.
Sterman, J. (1984a), “An Integrated Theory of the Economic Long Wave", Working
Paper WP-1563-84, Sloan School of Management, M.I.T., Cambridge, Mass.
Sterman, J. (1984b), "A Behavioral Model of the Economic Long Wave", Journal
ef Economic Behavior and Organization, vol. 5.
-451-
LISTING OF TWO-SECTOR MODEL
APPENDIX 1
Phuation Listing of the fvo-aector Model, KOR?
wore
NOTE,
® DISAGGREGATED TVO-SECTOR KORDRATIEFY MODEL,
NOTE
OTE DEFINITION OF SECTORS
NOTE
To are
1 ee,
Por Pet, TP/Pet, IP
OTE
NOTE PDUCTION SECTORS
ROTE
NOTE PRODUCTION RATE
OTE
ct
A PRIK(P)=PC-R(P)*CU.K(F)
A CU.X(P) TABHL( CUT, 17. (P) /PC.X(P) ,0,2,-2)
7 CUTO/.5/-55/. ISSN ANNAN
A IPK(P)*B-RCP) /mDD(P)
WOTE
\OTE PRODUCTION CAPACITY
OTE
x
x
A PO-R(P)#519PC1.K(P)4S2"PC2.K(P)+S3°PC3.K(P)
A FOLK(P)#RP-K (MIR (SK (1, P)/RS-K(1,P) S.K (2,P)/RS-K (2,P))
A PO2.K(P )aRP. K(P)*EXP(SPC.K(P))
A PO3.x(P)@RP-R(P)®( (RVS(1,P)°S.K(1,P)/RS-K(1,P))*
X — (RVS(2, P)*S.K(2, P)/RS.K(2,P))).
A SPCAR(P)oRVS(1,P)*LON(S.K(1, PURER PD)
X___-RVS(2,P)*LOON(S.R(2,P)/RS.K(2,P))
NOTE
NOTE STOCK AND SUPPLY LINE OF PACTOR
NOTE
Lente phos alts Phe net a2 P)-B.3(7,P))°PERE- JIT.)
A DAKCE,P)*SDL(,P #8. (FP )/ALCT)
ARG pbssnbtr pbegbeRte pirareCr)
L SLK(F,P)+SL.J(P,P) e098 (0.9(F,P)-A-3(P,P) )*PEVSLAI(?,P)
x
‘OTE
OTE ORDERS POR FACTOR
NOTE
r
a
t
s
nN
a
a
x
r
a
O.K(F, P)eSDL(F-P)*S-K(P,P)*PO-RCF,P)
PO.K(F,P)*TARKT( POT, IFO.K(F,P) =< 1455-05)
POT*0/0/0/.05/.1/.15/.2/.25/.5/.35/.38/-4/.8
Troe (Tobe C kU Pe eSU RGR Po sceck OP PNN/SHKCT.P)
OSL.K(F,P)*(DSL. UF, P)-SL-K(F,P))/TASL(F)
DEL.K(F,P)*PAT-R(F)*D.KCP SF)
PAS. K(P)oRAT(P)SEATSL. RCP)
EATSL- KP) oTARM:(TEATSL_AT-KCF) /MAT(),0, 34-5)
TEATSL*0/-5/1 1.5/2 /2.5,
€S.K(P,P)* (DS.K(F.P)-S.K(P,P))/TaS(F)
NOTE
NOTE DESIRED STOCK OF FACTOR
NOTE
A DS.K(F,P)*RS.R(P,P)*RDRS-K(Y,P)
A -RDRS.K(F, P)oPABRT(PRDRS, 15.X(7,P)/RS.R(7,P),=+5,7+5)05)
7 —-TRDRS+0/0/.5/1/1.5/2/2.5/3/3.5/4 14. 5/5/5-4/5.7/5.9/6/6
A IS.KCT,P)=19C.K(P)RS.K(P,P)/RP-X (P)
A TPC.K(P)=ETO.R(P)+CB.K(P)
A CB.K(P)=(B.R(P)-TB.K(P))/20B(P)
‘A IB.K(P)«RDD(P)#ETO-K(P)
L__ ELO.K(P)=BI0. 3(P)+ (D4/A10(P))(10,.3(P)-E10. 9(P))=PEVEO- 0(P)
OTE
OTE COUPLING EOUATIONS
NOTE
A 10.4(P)oSurY(0.K(P,*),1, TF)+HO.K(P)
ABS (9) S01 (70), NVTP)ONSL.K(P)
A DD.R(P OB. K(P)/PR. I
A aT KCF)*DD.RF)
ROTE
NOTE NOUSENOLD SECTORS
OTE
Lo HSL.K(P )eHSL.9(P)-D7*(HO.9(P )-HA. J(P))*PEVHSL. J(P)
A -HO.K(P)=ssH0.K°RO(P)# (1 +STEP(FSHO, TSHO)-:
A SSHO.ReCLIP(1,O,TIME.X,0)
A RALK(P)SHSLR(P)/AT.K(P)
rEP( PSHOP(P) ,TSHO))
HOTE PARAMETERS
AT(F) oRDD(?)
SDM P)CLIP(s.0,RS(P.P) 18-2)
1.5
Y,
ns
TAB()O1.5/1.5
T¥SED(*)*.5/.5
TATO(*) «2/2
RO(P)oTHO*IVSED(P)
THO 412
PSHO*.05
TSHO*1E9
SYMAX(O, 1-HAZ(SALT-1, 1-SALT))
S2-MAK (0, 1-RAT(SALT~2,2-SKT))
SD=AK(O, 1-MAX(SALT-3, 3-SALT))
SALT#2
FSHOF(#)=0/0
POR(*) 95/3
PROCEDURE T0 IRITIALIZE 1” ROUILTRIM
SUP P)=1B-2
ExO(P)o1B-2
BRS: X(r.P)-rS. 3,7 SDL(Y.P)*CDING. I, P)-1R5.3(7,7))
InS(Y,P
DINS.K(P, P)=RVS(?,P)*DIP. Sposa artemis
DIP.K(P)>SUMV(INO.K(P,*) 1 37)
2w0-RU7 P)ea9s-xC7_ 79/047)
PEYS.K(F,P)*PULS.X*DINS. (P,P)
PEVSL.K(F,P)*POLS, reagan. (P,P)
PEVEO.(P) »PULS-K®DIP.K(P)
PEWISL-R(P)=POLS eHOUP HDD)
PULS. KeSTEP(1, -DP)-STEP(1 01
RP.K(P Jot E-geSTEP (DIP. K(P},0)
BS-K(P,P)*1E-9¢STEP(I8S.K(F,P) ,0)
AGGREGATES
ESAL(P)=SUmV(CESAL(®.P),1,7F)
CESAL(F,P)@RYS(Y,P)*AL(P)
EFOR-KeSTEP(APOR.K,0)
EAL.KeSTEP(AAL-,0)
A70H, SAS R/APE
ARLAK AS.)
AS-KeSUMY(SAS-K(9),1, TF)
AD. oStOW (SAD-K(#), 1,77)
SAS.K(P)=SURV(S.K(9,P),1,7F)
SAD.K (P)oSUm (D.4(*,P) TF)
APC. KeSUMV(PC.x(*), 1, TF)
SAL.K(P) #SA5.K(P)/SAD-n(P)
APR.KeSURV(PR-K(®), 4,.7P)
ADK
STAULATION CONTROL PARAMETERS
PLIPER.K*CLIP(PLPER, IPLPER, TIME. K, TCP)
PLPEP st
IPLPEP=O
Ter
FRIPER .KeCLIP(PRPER, IPRPER, TIME. K, 0)
PRPEP+O
IPRPER =D
TGs -RITRDT
MITP=10
LERCTHO/DT*. 125,
Paranaters:
Variable List Symbols for unite: X = Dinenstoniess
AL AVERAGE LIFETIME OF FACTOR PRODUCT (YR)
U = Unite of product FOR FACTOR OUTPUT RATIO (YR)
4 YR ~ Year FSHO FRACTION STEP IN ALL HOUSEHOLD ORDERS (x)
ia ARRIVAL OF FACTOR (U/YR) FSHOP FRACTIONAL STEP IN HOUSEHOLD ORDERS FOR SPECIFIC PRODUCT (x)
aL AGGREGATE AVG. LIFETIME (YR) IPLPER INITIAL PLOT PERIOD (YR)
AD AGGREGATE DEPRECIATION (U/YR)
IPRPER INITIAL PRINT PERIOD (YR)
IVSED INITIAL VALUE SHARE OF END DEMAND (x)
NAT NORMAL ARRIVAL TIME (YR)
NOD NORMAL DELIVERY DELAY (YR)
AFOR AGGREGATE FACTOR/OUTPUT RATIO (YR)
APC AGGREGATE PRODUCTION CAPACITY (U/YR)
APR AGEREGATE PRODUCTION RATE (U/YR)
AS —ABGREGATE STOCK OF FACTOR (UN) PLPER PLOT PERIOD (YR)
AT ARRIVAL TIME (YR) PRPER PRINT PERIOD (YR)
8 BACKLOG (U) RO REFERENCE HOUSEHOLD ORDERS (U/YR)
CB © CORRECTION FOR BACKLOG ¢U/YR) RUS REFERENCE VALUE SHARE (1)
CS CORRECTION FoR STOCK (u/yR)
TAB TIME TO ADJUST BACKLOG (YR)
TAIO TIME TO AVERAGE INCOMING ORDERS (YR)
TAS TIME TO ADJUST STOCK (YR)
TASL TIME TO ADJUST SUPPLY LINE (YR)
TCP TINE TO CHANGE PERIOD (YR)
THO TOTAL HOUSEHOLD ORDERS (U/YR)
YSHO TIME TO STEP HOUSEHOLD ORDERS (YR)
CSL CORRECTION FOR SUPPLY LINE (U/YR)
CU CAPACITY UTILIZATION (x)
cUuT CU TABLE
D DEPRECIATION OF FACTOR (U/YR)
oo DELIVERY DELAY (YR)
os DESIRED STOCK (U)
OSL DESIRED SUPPLY LINE (U)
EATSL EFFECT OF ARRIVAL TIME ON DESIRED SUPPLY LINE (x)
Variables for equilibrium initialization and special parameters:
E10 EXPECTED INCOMING ORDERS (U/YR)
-452-
LIST OF VARIABLES
APPENDIX
F FACTOR INDEX (1) CESAL COMPONENT IN ESAL (YR)
FO FRACTIONAL ORDERS (2/YR) DINS DESIRED INITIAL STOCK (U)
FOT FO TABLE DIP DESTRED INITIAL PRODUCTION (U/YR)
WA HOUSEHOLD ARRIVALS (U/YR) EAL EQUILIBRIUM AGGRAGATE FACTOR LIFETIME (YR)
HO HOUSEHOLD ORDERS (U/YR) FOR EQUILIBRIUM FACTOR/OUTPUT RATIO (YR)
HSL HOUSEHOLD SUPPLY LINE (U) ESAL~ EQUILIBRIUM SECTOR AGGREGATE LIFETIME (YR)
1B INDICATED BACKLOG (U) INO INITIAL ORDERS (U/YR)
IFO INDICATED FRACTIONAL ORDERS (1/YR) INS INITIAL STOCK (U)
10 INCOMING ORDERS (U/YR) NITR NUMBER OF ITERATIONS IN INITIALIZATION (X)
1p INDICATED PRODUCTION (U/YR) PEVEO PULSE OF EQUILIBRIUM VALUE OF ETO (U/YR)
IPC INDICATED PRODUCTION CAPACITY (U/YR) PEUMSL PULSE OF EQUILIBRIUM VALUE OF HSL (U)
IS INDICATED STOCK (U) PEVS PULSE OF EQUILIBRIUM VALUE OF § (U)
° ORDERS FOR FACTOR (U/YR) PEUSL PULSE OF EQUILIBRIUM VALUE OF SL (U)
P PRODUCT INDEX (2) PULS PULS FUNCTION (1)
PAT PERCEIVED ARRIVAL TIME (YR) RP REFERENCE PRODUCTION (U/YR)
Pe PRODUCTION CAPACITY (U/YR) (1 :LEONTIEFF 2:COBB-DOUBLAS 3: INF. SUBST.) RS __REFERENCE STOCK (U)
PR PRODUCTION RATE (U/YR) S1,S2,S3 SWITCHES FOR CHOICE OF PRODUCTION CAPACITY
RORS RATIO OF DESIRED TO REFERENCE Stock (x) SALT " SUITCH FOR ALTERNATIVES IN PRODUCTION CAPAICITY
8 STOCK OF FACTOR (U) SDL SWITCH FOR DELTVERY LINK
SAD SECTOR AGGREGATE DEPRECIATION (U/YR)
SAL SECTOR AVERAGE LIFETIME (YR)
SAS SECTOR AGGREGATE STOCK (U)
SFC SUM OF FACTOR CONTRIBUTIONS (x)
SL SUPPLY LINE OF FACTOR (U)
SSHO_ SWITCH TO START HOUSEHOLD ORDERS (%)
TEATSL TABLE FOR EATSL
TF TOTAL FACTORS (x)
TP TOTAL PRODUCTS <x)
TRORS TABLE FOR RORS
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