SYSTEM DYNAMICS AND INPUT-OUTPUT ANALYSIS
Charles H. Braden
School of Physics
Georgia Institute of Technology
Atlanta, Georgia 30332
Abstract
Input-output analysis for an "open" system relates production rates
for various sectors of an economy to stipulated final demands. However, it
is well known that the conventional dynamic analysis usually does not yield
results which approach smoothly to those of the static analysis. In this
work, the dynamic analysis is cast into the form of a system dynamics model.
A modification of the rule which governs sector production rates is intro-
duced so that stable results are obtained which do approach those of the
usual static input-output analysis. The system equations are further modi-
fied to incorporate time-lagged stock indices and damping in the production
rate rule. Prices are handled throughout as in conventional input-output
analysis.
Introduction
Input-output analysis, as formulated by Wassily Leontief [1], is
a valuable tool in the analysis of economic systems. In an "open" system,
the production rates for the sectors of an economy may be determined in terms
of the stipulated exogenous final demands for the products. A set of first-
order, differential equations expresses the conservation of goods. Goods
produced’ by a particular sector go to: (1) inputs needed: by: the. productive.
sectors. for production purposes, (2): stecks of goods; held: by, the productive
sectors, and (3) the stipulated: final demands for the goods:
“BY + (I-A) Y= X (and ¥, = AL ¥) « @):
The economy is: resolved into N sectors, with sector N the "kabor"
or "final demand" sector; this sector furnishes: the tahor utilized by the
other sectors, and consumes the final products. The production rates. for the.
N-1 productive sectors are. given by the. column vector ¥,, with, ¥ denoting the.
time derivatives of the production rates. ¥y. ts. the rate. at which Labor is
supplied by sector N. The exogenous final demands are the N-1 elements. of
the column veetor X. The r-s element of the structural matrix A is. the
ratio of the input of goods furnished to sector s by; sector r to the output
of goods produced by sector s. Al is a row vector, with N-1 elements, that
is defined similarly except it is a measure of the labor supplied to.
Sector s by sector N, The r-s element of the stock coefficient matrix B
is the ratio of the stock of goods produced by sector r being held by sector
s to the output of sector s.
Another set of conservation equations applies to the values of the
goods. Money received by one sector for its output must balance payments
made by that sector to other sectors for inputs plus payments of wages
to labor,
P= Py (iA*ez BY) al® Q)
The prices for goods produced by the productive sectors are given by the
N-1 elements of the column vector P, Py is the exogenously stipulated wage
rate, and z is a diagonal matrix with elements Paes aa The notation E*
denotes the transpose of the matrix, or colum vector, E; Et denotes the
inverse of the square matrix E.
In this exposition, many simplifications are made in the interest
of improved clarity of the essential points. It is assumed that the labor
sector does not itself employ labor or accumulate stocks, and stock coeffi-
cients will be assumed time independent. Often, the simplifications can be
removed without damage to the analysis, albeit with some added complexity
in the. formulation.
The static solutions to these equations, corresponding to ¥=0, have
probably proved of most utility in the analysis of economic systems.
For the static case
x
(a-ay7+ x (and Y= Al ¥) (a)
P
Py aay? a* (3b)
It is well known that solutions to the dynamic. equations (1). are
likely to diverge [2]. Hence, the dynamic analysis is usually not helpful
either in the determination of the existence of an equilibrium state, to which
the static solutions would apply, or in tracing the time evolution of'a
system from arbitrarily stipulated initial conditions. The dynamic equations
do demonstrate the existence of an unstable equilibrium; viz. if the initial
state coincides with the static solutions, the system remains in that state.
Of course, a perturbation of the system would upset the equilibrium.
In view of the generality of the conceptual nature of input-output
analysis and its close relationship with conventional economics parameters,
it is an attractive goal to cast the analysis into system dynamics form,
as developed by Jay Forrester [3]. The intent is to do this with as little
disturbance of the conventional input-output analysis as possible. The
resultant, basic system dynamics formulation can then be used as the point
of departure for such refinements as are indicated for modelling a specific
system. It may prove advantageous to view the same system from the points
of view of both methodologies, with the expectation that some features of
the system may be more transparent in one formulation, while other features
may be clearer in the other formulation.
Modified Input-Output Analysis
In order to achieve a satisfactory system dynamics formulation of
the input-output analysis, two matters must be attended to. The dynamical
equations (1) are now phrased in terms of production rates, rather than
proper system levels, and the equations should have stable solutions which
approach the equilibrium state described by the static equations (3).
The former matter can be handled through a change in notation, which intro-
duces the stocks as the appropriate system levels, while the latter matter
will necessitate a change in the structure of the system.
In ‘the convent ional formulation, the stock of goods "r" (ise. goods
produced by sector r) held by sector s, Si3? is related to the production
vate of sector s according to Ss brs ’s* The further development of this
modified analysis is facilitated by introduction of dimensionless stock
andices for. the N-l productive sectors, denoted as the elements of the
column vector S. New stock coefficients, which play the role of the previous
B coefficients, but which are not equal to them, are introduced according to
Sis ns 85° (4)
‘The ‘stock indices will have a "desired" value, which may be :chosen
arbitrarily, and ‘these desired values ‘of ‘S are ‘denoted ‘by ‘the column vector
R. For example, one may choose all elements of R equal ‘to unity. The choice
of R affects ‘the values of ‘the b coefficients according to
8, , (desired) = bis RS = (5)
A connection between ‘the conventional 8 ‘and ‘the :new ‘b coefficients
may be useful. To do this, we can associate the desired stock levels with
the equilibrium state and express the B coefficients as $,(desired)=B, Y°,
where Y° is the solution of the static equations (3). Then, one has
a e
b Bp, /Ry> and also b.,/>.
a /By
t's Mrs Bytg*
Although the notation now departs from the usual input-output nomen-
clature, the substance of the analysis has undergone no change. In order to
clarify the problem of stability of the solutions, it can prove helpful to
examine a trivial, conventional dynamic input-output model in which all
sectors are uncoupled from one another; this can be accomplished by making
the A and B matrices diagonal. The influence diagram for a typical produc-
tive sector is shown in Fig. 1, with attention given to the upper quantities
on the diagram. The control rule gives positive feedback, and solutions of
the equations may be expected to diverge exponentially with time.
Dyn. IO
\ 7
~ K(I-A)(R-S)-" Mod. 10
Fig. 1. Influence diagrams for model with uncoupled production sectors.
Quantities on the upper part of the diagram refer to conventional input-
output analysis.’ Quantities on the lower part of the diagram refer to the
modified analysis.
A modified control rule will be introduced. The sense of the
modified rule is that the production rate for a sector has a norm, which is
attuned to what would be equilibrium conditions if that equilibrium currently
existed, and then adjusted to departures of the various stock levels from
their desired values:
y=yv'ab K (RS). (6)
The elements of the diagonal matrix K are the “recovery constants”
for the proportional control rule. The intent is te introduce a simple,
reasonable control rule and regard this rule not as an inviolable component
of the modified formulation but as a base point for whatever rule seems
appropriate to: the specific system under study. ‘The corresponding influence
diagram is depicted in Fig. 1, with attention to the lower quantities on the
diagram. The feedback is now negative, and solutions for the uncoupled case
are given by
s. = R. + constant EXP[-K,(1-A__)t] : 2)
where t denotes the time variable. A smooth approach to the equilibrium
state, where S=R, occurs.
The change in notation to the stock indices § and the introduction of
an essential modification in the production rate control rule yield the system
equations
s =p! (1a) b K (RS) (8a)
Yy = ALY (8b)
Pep (raz byt al. (8c)
Z is a diagonal matrix with elements 275 /Y + At equilibrium, R=S and
y=y®, where Y® has been stipulated as the solution to the static equations,
viz. ¥°=(1-a) tx,
Another, redundant price equation can be derived. Its validity
depends upon the validity-of the other system equations. Although the
equation yields no new information, it can be helpful as a check on internal
consistency of solutions during a computational procedure,
*
PoXK-PYYy =O; (9)
Lagged Stock Indices and Damping
It is straightforward to incorporate some added features into the
basic system dynamics formulation. As it stands, the analysis can accommo—
date certain parameters which depend on values of other system variables.
The structural matrices A and Al, the wage rate Pye the final demands X,
and the stock PacnueEy constants K may be related to other variables without
alteration of the system equations. Additional terms must be incorporated
in some of the equations in order to accommodate other changes, including
utilization of labor by the labor sector, the holding of stocks of goods
by the labor sector, and time variation of the stock coefficients b.
Often, it is deemed necessary to incorporate time-lagged variables
into a model in order to account for inevitable delays in the gathering of
data, the making of decisions, and the implementation of the decisions. The
inclusion of delays can have a profound effect on the qualitative behavior of
the model. A model which evidences good stability in the absence of delays
may demonstrate an oscillatory behavior when delays are included. Because of
the expected important role of delays in model behavior, some attention
will be given here to their inclusion in the basic system dynamics formulation.
The control rule for sector production rates can be altered to include
time-lagged values of the stock indices, denoted as the column vector S'.
These lagged values are related to the stock indices S according to a first-
order exponential delay. Furthermore, in order to provide a mechanism that
can be adjusted to cope with the expected oscillatory tendency, derivative
feedback, or damping, will also be incorporated into the conttol rule [4, 5,
6]. The new control rule, to replace Eq. (6), is then
yev°+bK (RS!) -bDS'. «10
The time-lagged stock indices obey
3" =L (s-8') . qi)
The elements of the diagonal matrix D are the damping coefficients, and the
elements of the diagonal matrix L are the delay factors, which govern the
lag of the S" indices relative to the prompt stock indices, S. The first
of the system equation (8a) is now replaced by ,
S=o? (1-a) [b K CS") -b DS}. 12)
In addition to variables and parameters common to both the convention-
al and modified onput-output analyses, the additional parameters K (stock
recovery constants), L (delay factors), and D (damping coefficients) have
been introduced. These additional parameters are rather removed from the
usual economics data. The effects these additional parameters are intended
to simulate must be ascribed to a complex series of decisions and actions.
Accordingly, it will probably be difficult to empirically determine values
in the same sense as one can do for the A and B coefficients. A combination
of a feeling for the working of an actual system coupled with test studies
of the simulation model will probably be necessary in order to arrive at
reasonable values.
Model Behavior
In order to provide examples of characteristic behavior of the basic
system dynamics formulation of the input-output analysis, a simple, test
system of three coupled sectors, two productive sectors and the labor or
final demand sector, has been studied. All parameters are held constant.
The system was coded as a conventional system-dynamics computer algorithm,
although in this most simple form, analytic solutions are possible. The
6000
O TIME 5 10
Fig. 2. Behavior of production rate of sector #1 for a test model.
Curve A is for a conventional dynamic input-output model, and curve
B is for the modified model.
behavior of a conventional dynamic input-output model and the modified model
are compared in Fig. 2, for a case where the initial stock levels are lower
than the desired values. As suggested by the previous consideration of
the model with uncoupled sectors, the conventional model diverges from
and the modified model converges smoothly to the equilibrium state.
In Fig. 3, the effects of the time-lagged stock indices and damping
are demonstrated. Curve A was obtained from a model which included time-
lagged stock indices but‘with the L parameters set to large values so that
10
the behavior is similar to the simpler model with neither delay nor damping.
— | ! nl
O 4 TIME 8 l2 I6 |
Fig. 3. Behavior of the stock index for sector #1 for the modified model
with inclusion of time-lagged stock indices and damping. In curve A, the
effects of delay and damping are negligible. Curve B includes delay ef-
fects but no damping. Curves C and P include both delay effects and damping}
In curve B, delay effects lead to oscillation, and the damping coefficients
are set to zero. The heavily damped nature of the oscillations is due to
the presence of "natural" damping in the system equations, i.e. terms which
persist when D=0. In curve C, enough damping has been included to largely
control the oscillatory tendency. More damping is added in curve D.
11
Concluding Remarks,
An input-output problem can be converted into an equivalent system
dynamics model. The model tends to be quite stable, provided a modified but
reasonable production control rule is substituted for the usual rule of
conventional dynamic input-output analysis. With the model in system dynamics
format, additional features that have proven useful in the modelling of.
specific systems can be incorporated as desired. The system equations
have been provided for inclusion of time~lagged stock variables and damping
in the production rate control rule.
12
References
{1]
(2]
[3]
{4]
[5]
[6]
Wassily Leontief, Input-Output Economics (Oxford University Press,
New York, 1966).
R. G. D. Allen, Mathematical Economics (Macmillan, London, 2nd edition
1959).
Jay Forrester, Industrial Dynamics (M.I.T. Press, Cambridge, Massachu-
setts, 1961).
Doug Tengdin, "P-I-D Control in System Dynamics Models," Plexus-System
Dynamics News, Vol. 1, pp. 8-11, November 1980.
Charles Braden, "Decision Procedure to Minimize Marginal Production
Cost in a System Dynamics Model," Dynamica, Vol. 5, pp. 24-34, Autumn
1978.
R. G. Coyle, Management System Dynamics (John Wiley & Sons, London,
1977).
attachment tot
SYSTEM DYNAMICS AND INPUT-OUTI'UT ANALYSIS
TOOPQGRLS Charles H. Braden
of Model: INPUT-OUTPUT TIT
. and current address of the senior technica) Charles Braden, School of Physies
person responsible for the model's construction: Ga.
. 30332
who funded the model development? institution (Ga, Tech)
In what language is the program written? BASIC
at computer system is the model currently
ented? __ CDC Cyber 74/6400
"oadext not included
What is the length of time obit 1A for one wares about 0.4 s wee laverecive * "step"
run of the model? for 10 levels (strong dependence on # of levels); 400 sec for 10 year run
Is there a detailed user's manual for the model? ___no
2 PURPOSE OF THE MODEL:
For what individual or institution was the model
designed? Charles Braden
What were the basic variables included in the model?
levels are stocks of goods held by N sectors of an economic system
sector #N is labor (or final demand)
Qver what time perjod is the model supposed to provide useful information on real
world behavior? 1 - 50 years °
was the model intended to serve as the basis of:
an academic exercise designed to test the implications of a set
of umptions or to see if a specific theory would exylain his-
! torical behavior
| communication with others about the nature and inplicaeticns cf an
! PHSCEY interacti x
i important set of interactions, exemplify metho hte for conversion of _ a
: i r inpu put moi to system dynamics model
projecting the general behavioral abate of the real system
nt{s) et future
predicting the value of some system elen
point in time
3. MODEL SPECIFICATION AND THEORETICAL JUSTIFICATION:
Provide two Giacrams illustrating the extreme behavior modes exhibited by the major
model elements:
~& (Braden)
If they are not included in the body of the paper indicate where the reader
may find:
a model boundary diagram that indicates the important
endogenous, exogenous and excluded variables noted in paper
a causal influence diagram, a flow diagram, the com- a *
} 4 wailable, i
puter program and definitions of the program elements yequoct @ 2B Pare, ‘upon
Is the model composed of: ‘,
simultaneous equations x
difference or differential equations x
procedural instructions
Is the model deterministic ~ x or stochastic
continuous x or discrete
DATA ACQUISITION
What were the primary sources for the data and theories incorporated in the model?
Data only exemplary, test data run to date
Theory Leontief dynamic input-output analysis for "open" ‘system,
conventional system dynamics methodology
What percent of the coefficients of the model were obtained from:
measurements of physical systems
inference from social survey data
I
j
{
econometric analyses |
{
expert judgment
the analyst's intuition i
What was the general quality of the data? exemplary, test data only.
PARAMETER .ESTIMATION
nea. (test data only)
If they are not given in the publication, where may the reader obtain detailed /infor-
mation on the data transformations, statistical techniques, data acquisition proce-
dures, and results of the tests of fit and significance used in building and analyzing
the model?
MODEL PERFORMANCE AND TESTING nea. (test data only)
Over what period was the model's behavior compared with historical data?
What other tests were employed to gauge the confidence deserved by the model?
(Braden).
“Where may the reader obtain a detailed discussion of the prediction errors and the
dynamic properties of the model?
7. APPLICATIONS
What other reports are based upon the model?
Name any analysts outside the parent group that have implemented the model on another
computer system.
List any reports or publications that may have resulted from an evaluation of the
model by an outside source.
Has any decision maker responded to the recommendations derived from the model?
Will there be any further modifications or documentation of the model?
Where may information on these be obtained?
XX modifications planned in pricing structure of model (currently, conventional
input-output analysis is used); a more detailed description, with additional
exemplary runs, is then planned
