Modelling Dynamic Economic Problems on an Analog Computer
MODELLING DYNAMIC ECONOMIC
PROBLEMS ON THE ANALOG COMPUTER
by
Alfred Engel
and
John J. Kennedy
ABSTRACT
The basic application of analog computer met hods to the solution (modelling) of dynamic economic problems
is discussed. Consideration is given, especially, to both the necessity and feasibility of utilizing for this
purpose the built-in expansion equipment available in modern analog computers. In addition, the solution
to several fundamental problems of economic modelling are presented and discussed in their relation to
dynamic computation in the broad area of economic analys is.
INTRODUCTION
Recent developments in analog computer technology -- improved dynamic accuracy in operational amplifiers, higher resolution in attenuators,
high-speed repetitive operation capability ... all
the way to the built-in expansion equipment usually
provided -- now make possible a serious attempt
to meet this goal of true economic modelling. This
goal , set up by Leontief in his "Studies in the
Structure of the American Economy" (4), indeed,
seems to set the guide line for future development
by its contrasting of static vs dynamic theory.
During the past several decades, considerable effort
has been devoted to the mathematical analysis and
description of economic concepts, and to the construction of analytical economic models. Fairly
good results have been obtained, for example, in
determining parameters from experimentallyfound statistics, and in predicting behavior in
finite difference models. To the latter, it might
be noted, the analytic work done by Frisch -phenomenon of collinearity, (1) -- and Haavelmo-" th e s tatis tic al me th od use d must be de ri ved fro m a model
STATIC AND DYNAMIC THEORY
th at speci fies the rela tio n s among the jointly de pendent
variables," (2) -- contributed significantly in the
The major difference between static (sequence)
models (implemented on purely digital computers)
and dynamic models (implemented on analog or,
if more involved, on hybrid computers) is stated
c learly in the following definition by Leontieff:
methodology of economic modelling.
Along this same line, the pinpointingofthe ' 'prankster phenomenon" by Koopmans (3) eliminated
much ambiguity from data cor relation: "the hy pothe tical
"A static th eory de rives the c hanges in the variables o f
a give n system fr om th e observed changes in the unde rlying s truc tura l re lations hip; dynamic th eo ry goes furth e r
and s hows how certain c hanges in the variab les c an be
ex plain e d on the ba s is o f fixe d ii.e. invariant) struc tural
charac te risti cs o f the sys tem" .
mode l specifying the re lations among th e var iables s houl d
be such as to all ow fo r identifica tion o f param e te rs ."
Quite naturally, however, despite the development
of analytic al techniques such as these, there ha s
been little work done in dynamic modelling, pe r se .
As with any 'young' SCience, with new vistas of
re searCh , it i s essential that thorough study
proceed from idealized 'sample' systems toward
the more realist ic -- complex and cont inuous -syst em s of de sc ription.
From the 1953 edition (4, above), he continues :
"Dynamic theory thu s e nab les u s to de rive th e e mpirical
law o f c hange o f a parti cular eco nom y from information
ob tained th ro ugh th e o bservation o f its structural charac te risti c s at one single point o f tim e . This possibility,
me th odo lo gic ally ra ther obvious and prac ti c all y very
1
<9 El ect ronic Assoc i ates, Inc. 1965
All Righ ts Rese rve d
Printed in U.S.A. 055
Bu l le t in No. A L Ae 65087
causes for inaccurate economic predictions from
sequence models.
important, has unfortunately been obscured by the fact
that most of the recent attempts to determine the structural
characteristics of actual economic systems have been
based on some kind of statistical time-series analysis,
thus giving rise to the erroneous impression that ihe
empirical laws of change necessarily must be derived
from direct observations of past development."
MODEL BUILDING AND INTERPRETATION
The feasibility of dynamic economic modelling is
enhanced by the following features of presently
available computational accessories, over and
above the familiar summing and integrating amplifiers which provide the firm basis on which all
analog model building is grounded:
The implication here is that finite difference equations of sequence models are to be replaced by
sets of differential and difference-differential equations. Differential equations occur, for example,
when the "lag" of an input variable can be considered as an improper one, such as with extended
payments for sold products or 'gambling', say,
with a tax refund sure to come within the next year.
Difference-differential equations, on the other hand,
are a necessity in the event of a genuine lag which
is almost always the result of a behavioral phenomenon due to people in business. In more involved systems, indeed, this lag feature acts as a
moderator and stabilizer and usually influences
system characteristics considerably.
a.
LogiC branching of dynamic quantities can be
accomplished at computed points of time by
comparators (COMP) ... which compare desired and attained values and make 'decisions'
by error signal outputs ... or arbitrarily in
time by function switches (FS) ... which permit
random inputs to be made manually.
b.
Genuine lags can be mechanized bytrack-andstore amplifiers (TS) -- These amplifiers can,
if desired, be used also to smooth out unrealistic discontinuities from their sampled
inputs.
c.
Exogenous variables whose analytic form is
unknown or only approximated can be simulated continuously and with high accuracy as
experimental functions on variable diode function generators (VDFG).
d.
Sensitivity and vulnerability can be tested
with scaled and normalized step inputs--such
occurrences as the injection or rejection of
money from a system, for example -- at
computed times by comparators or arbitrarily
by function switches.
e.
.The feature of repetitive operation (Rep Op)
permits display of solutions on an oscilloscope
while parameters are being changed manually,
thus providing a truiy dynamic system model.
f.
In larger systems, master controlforthelogic
It is worthwhile, here, to provide some general
clarification to Leontieff's definition. Given a
difference-differential equation equivalent to a
genuine lag relation, and known initial conditions,
it is possible to achieve a quasi-continuous solution, x(t), for the system being modelled. However,
the same equation written in terms of a finitedifferenced sequence model-- with one step equivalent to a whole unit of time -- gives rise only to
an 'approximate guess' for one point of time .•• it
drops all samples in between and lacks a criterion
of convergence depending on how fine the time grid
is with regard to the input variations.
Granted, the dynamic model sometimes requires
an entire function stored among its initial conditions -- but only once in the entire simul!i.tion -but it provides another function for any time, t,
in the solution, thus displaying the dynamic characteristics of the simulated system. The sequence
model yields only a datum, or set of data, whose
mutual connection and continuation are unknown
and, actually, undecidable as long as the grid
violates Shannon's Sampling Theorem*(5). The result is a framework of static points in problem
space which deviate more from the accurate solution as the number of successive time steps increases. This source of error is one of the major
states and modes of a program can be executed
either by a memory logic group (MLG) .••
which directly governs track-and-store modes,
the state of AND and OR gates, etc .•.. or by
a digital operations system (DOS) ... which
provides for digital-analog communication and
bidirectional converSion, data buffering, and
logic control. MLG is a feature of the EAI
PACE® 231R-V GeneralPurposeAnalogComputing System. A digital operations system
(DOS 350) provides programmable logic and
communication as an integral part of the EAI
'This Theorem states that: Sampling must occur at such a rate or
density that !1E "information of relevance (i.e. information r~quired
because of being characteristic) is lost. In other words, the sampled
function must. allow for reconstruction of the original function with a
prescribed measure of accuracy.
2
HYDAC ®
System.
2400 Combined Hybrid Computing
Obviously, a function switch also can be used to
test stability, sensitivity and vulnerability of a system or whether the system can or cannot stand a
sudden, percentagewise change without oscillation
or breakdown. Operating a function switch eventually can cause a comparator to change system
policy. In this regard, a suffiCiently fast comparator to pick up accumulated exogenous demands
used in combination with the Repetitive Operation
feature provides one of the most powerful tools
for the dynamic study of simulated economic systems. Selected portions of a run can be displayed
on the oscilloscope, showing several output variables simultaneously, while manual modification of
parameter values provide a quick check of the
dynamic response of the system.
The class of problems embraced by the equipment listed under Item f is beyond the scope of
this paper, and will not be discussed further
here.
It can be seen from this brief review of computational features that the analog computer can be
an ideal tool for dynamic economic model simulation. In view of the national -- and international -implications of economic competition and rivalry,
such activity in dynamic modelling and vulnerability
testing is certainly justified.
2.
Open Systems: The usage of this term is that
a mathematically inhomogeneous system of ordinary differential equations (ODE) is equivalent to
an economic system with an exogenous function.
That is, the function cannot be derived within the
given system; it comes from 'outside' and the
system is called 'open'. In mechanizing open systems, the number of 'openings' equals the number
of variable diode function generators required to
set up these experimental functions. The VDFG
can be adjusted with an amazing degree of accuracy in such systems even though the function
might appear rather wild in spikes or slopes.
There are obstacles, however, in contemporary
economic model building, and several examples will
be given later showing how these obstacles can
be overcome by means of modern analog computer
equipment. It will be helpful, though, to discuss
these problems briefly at this time especially
since they are closely related to model building
and interpretation.
1.
Policy Decisions: Leontief's definition: "Questions of policy can have an operational meaning only if one
assumes that the structure of certain sectors of the economy
can be changed" , regarding the effect of policy
In open systems, the exogenous function remains
the same for each run. If it were partially subject
to changes due to variables computed in the rest
of the system if would be partially derivable
within the system and thus not entirely exogenous
no matter whether its analytical form were known
or not. Of course, it would have to be known how to
derive the function in general terms whenever a
statement relative to change were made. Otherwise, the system would be unrealistic and, thus,
unjustified.
changes on the economic system is alreadyformulated in such a way as to fit into analog computer
philosophy. Thus, a "question" regarding the effect
of a policy change can be asked of a comparator
whose decision, based on predetermined levels of
achievement, will continue with existing business
and/or economic principles (as simulated) or will
switch to an alternative, pre-patched simulation
embracing those changes in principle which are
assumed necessary in theory to accommodate the
policy change.
3.
Forced Goal Systems: This type of system applies if the final state of an economy is prescribed,
such as a national emergency in case of war,
survival conditions of vital industries, etc. Mathematically speaking, it represents a boundary value
problem but it should be noted that in the field of
economics this does not imply by itself that the
goal must be reached along an optimized path.
The mechanization of policy decisions on comparators is always applicable whenever the need or
demand for a decision is endogenous within the
economic structure. In addition, there might be
randomly distributed (and rare) exogenous demands
for changes in policy requiring similar kinds of
connecting and disconnecting also mechanizable
on a comparator. These can include such events
as unexpected injection or rejection of money,
stocks, or materials, etc., orboomsordepressions.
Of course, these random inputs also can be scaled
so as to represent defined quantities and can be
put onto the system manually by function switches
at the discretion of the investigator.
Essentially, there are two techniques available for
dealing with Forced Goal Systems. One is simple
trial and error with repeated runs, preferably in
high speed repetitive operation; it can be mechanized to be fully automatic also and applies to the
original system.
3
The other technique is an approach with adjoint
variables as treated in the theory of differential
equations and in the simulation of engineering
systems; it implies optimized paths. The increase
in required equipment usually is negligible if the
adjoint variables are selected carefully, based on a
good understanding of the original system.
Accordingly, the most time-consuming part of
economic model-building is to satisfy Haavelmo's
prinCiple concerning the establishing of a truly
individual model. Above all, the only proper beginning is from precise definitions of qualitative
and quantitative variables. A single gap in the chain
of definitions can, and usually will, void all further
analysis.
This so-called "adjoint technique", although not
too complicated itself, is beyond the scope of this
present paper in which only the basic applications
are discussed and will not be treated further.
Next, by following, say, an incoming order through
every detail between its receipt and completion
(Le., receiving payment), a number of individual
structures and parameters will be revealed about
(It is these details which count in
a company.
competition, not the general framework which is,
after all, the common mold.) The numerical value
of these parameters can be obtained by assigning
a departmental 'work factor' for production and
clerical operations based on average handling time,
average cost per job, efficiency under various
workload conditions. etc.
4.
Irreversi bility:
Strict or partial irreversibility
of goods in stock, invested capital or real estate
bought can give rise to situations calling for policy
decisions such as, for example, when the market
volume drops. Usually, such a policy change demand pinpoints an error in past decisions and it
can create contradictions in small systems, calling
for a simulation on a broader basis in order to find
out how the fault could be avoided. Leontief's discussion of this problem under the general heading
"defects of multi-phase theory" gives a general
theorem which proves that surplus stock in these
cases is to be tolerated just as entropy is in, say,
thermodynamic systems. He argues that' 'in terms
of these principles, the deviation into the alternative
path would be ... as one actually has reason to believe
... explicitly rejected".
Finally, superimposed on this structure of
internal workload processing are factors reflecting the general policy of the management, the
internal and external economics of plant management, the company's accomodation to the environment (the market itself, suppliers, competitors,
taxes, etc.) and the individual emergency decision lines (personnel, price relations, weighting of
advertising, etc.).
There is one solution, however, which can be simulated very nicely on an analog computer. Provided
that market volume is governed by a variable which
is independent of the endangered (irreversible)
quantity -- governed, say, by advertising or by
changes in another (competitive) product -- then,
instead of changing the phase of Simulation, it is
possible to decrease an otherwise constantparameter in the precarious equation (mechanized on a
servopotentibmeter) while increaSing market support until market volume increases again. (Cure the
cause instead of submitting to the results.) In larger
systems, of course, the very loose relationship
existing between different products will avoid bottlenecks of this kind.
Such analysis by itself, if carefully done, can tell
an amazingly accurate story of what is really
going on in a company businesswise.
If these
data and structures are investigated in their
dynamics ... run on the computer, tested in various
forms, changed to account for hypothetical situations, etc .... there can hardly be a loss to a company in any business competition.
THE MECHANIZATION OF SYSTEM EQUATIONS
This section will deal with the actual mechanization of specific economic models. Here, interpretation is of the essence. For example, in
sequence models, functions are always encountered in terms of differenced arguments, such as
X(t-l) or X_l' Sometimes the function itself is
differenced; for example, X(t) - X(t-l) or VX o ,
X(t-l) - X(t-2) or VX_ l . BeSide the fact that
differences are derivable from differentials and
from differential quotients, there still remain
here alternative interpretations of these symbolic
notations depending upon the intention of their
authors.
5.
Individual vs. General Models: It is ofimportance
here that the greater relative value of individual
models as opposed to general ones be stressed.
The purpose of economic modeliJ:?g in the first place
is to analyze the economic structure of a single
entity, not to examine that entity's profile against
the overall or general economic structure of the
market place, the financial community, or even the
nation as a whole.
4
That is, it seems natural in economics to approximate differential quotients only by backward
differences (\7) instead of by central (/) or forward
(\7) differences*. Thus ...
On this basis, it is justified to transform Equation
(8) into an ordinary differential equation and, since
the most-shifted argument is (t-2) , a second-order
ODE must be expected.
A general and simple
scheme for this transformation is:
:t X(t) = )I:(t) corresponds to \7X
d = Y (t) - a (I-b) Y (t-l) + ab Y (t-2)
(1)
Def X(t) - X(t-I) or X
o
- X
+ 2ab Y (t-l) - 2ab
-1
-abY (t)
2
d 2 X(t) = X(t) corresponds to \7 (\7X)= t::,.2 X =
dt
+ ab
Y (t-2)
Y (t)
+ a (1-3b) Y (t)
(2)
I
I
to acc.?unt
for Y
to acc?unt
for Y
-a(I-3b) Y (t)
--------------------------------~--
X(t) - 2X(t-I) + X(t-2) or Xo - 2X -1 + X -2'"
d = (l-a+2 ab) Y (t) + a (1-3b)[Y(t) - Y (t-l)J +
However, it is with these approximations where the
alternative in interpretation arises. The validity
of equations (1) and (2) depends, quite naturally,
upon whether the relationship
'YX
X
«
1
Y
ab[Y (t) - 2Y (t-l) + Y (t-2)J
,
/
Y"
Y
(3)
The final equation is
. (1b - ). (1b 1)
is satisfied or not. The following sections will show
the implications in economics of this interpretive
choice.
Y(t) = -
denoting Income
(4)
C(t) = aY (t-I)
denoting Consumption
(5)
+IOV
I(t) = b [C(t) - C (t-I) J + d
Y(t) -
2-
+ ab Y(t) + d (9)
This equation is the ODE-equivalent of Equation
(8); written in this manner, with the highest derivative on the L.H.S., it is possible to construct the
unsealed circuit diagram of Figure 1 using the
"bootstrap method" of analog computation. (Figure
2 describes the symbols used in the diagram.)
1. Non Genuine Lags: Samuelson's Second Interaction
Model is described by Beach (2) with the following
set of equations:
Y(t) = C (t) + I(t)
3
-IOV
denoting Investment
(6)
denoting Initial
Conditions
(7)
By substitution
Y(t) =aY(t-I) + ab [Y(t-I) - Y(t-2)J + d
(8)
Figure 1. Mechanization of equation (9)
Applying equations of type (1) and (2) to Equation
(8) is equivalent to denying genuine lags in Samuelson's Model. The unit of time (t=I) , say, would be
equivalent to one month and all variables (Y, C,
and I) would have to satisfy Equation (3). Then and
only then would his solutions be correct [refer to
Equations (HI), (112) and (113) in Reference (5) J .
-[>- SIGN INVERTING INTEGRATOR
---{>-- SIGN INVERTING AMPLIFIER
--0-- POTENTIOMETER
*Contrary to mathematical usage/! most authors use the forward difa
Ference symbol, !J., when actualfy backward d;fference~ ~, ;s meant.
Figure 2.
5
Symbols used in Analog Computer Diagram
Potentiometers 1, 2 and 3 should be adjusted, according to Equation (9), so as to represent the values
of d, (2 - lib + II ab), and (lib - 3) respectively,
assuming all of these quantities to be positive.
(It should be noted that positive coefficients are
almost always assured if the feedback loops contain
an odd number of amplifiers (negative feedback)
and the system equations have a realistic meaning.
ill 'i,is case, there are two loops, onewith integrator 01, and one with integrators 01 and 02 and
amplifier 03, thus assuring positive potentiometer
settings. Generally, due to sign inversion, negative
feedback stabilizes the solution while positive feedback offers the explosive models and noisy solutions worthy of dynamic simulation.)
supplies Y(O) = Yo and potentiometer 06 supplies
Y(O) =Yo .
It can be seen now that Equation (7) of the model
actually states a boundary value problem to be met
at time t = 1. This, however, is an incremental
value; otherwise, the ODE approach would not be
valid. This apparent contradiction would be revealed, on detailed examination, to be inherent
in the static conception of sequence models and,
in fact, to be the basis on which such models are
built.
The dynamic concept, on the other hand, escapes
such difficulties by employing the I.C.' s as required by the theory of ODE's. Thus, if there
were a real boundary value problem, then t = T
at the boundary could not, of course, be allowed
to be the unit step in time.
Scaling: Depending on problem complexity or on
the objectives of a particular simulation, it is necessary to 'scale' the equations to fit properly on the
computer. Thus, even though the three potentiometers described above represent all of the problem
coefficients, an additional potentiometer, 04, was
used to enter integrator 02. The reason for this is
as follows:
Continuing, then, on the basis of this dynamic
concept, it is necessary to mechanize the remaining system equations, viz, (5) and (6), to
complete the model.
It can be derived easily
that the following equivalent equations for them
are valid:
ill computer time, one second corresponds to one
year of problem time (based on the original set of
equations). ill the simulation, however, it is desired to have one second on the computer correspond to one month of problem time. Accordingly,
it is necessary to reduce the inputs to the integrators to 1/12 of their previous values by adjusting
potentiometers 01, 02 and 03 appropriately, and
by adding potentiometer 04 to the circuit set at
1/12 = 0.0833.
C(t)
I(t)
1
1
- b C (t) + b Y(t)
d
b
= Y(t) - C (t)
(10)
(11)
Expanding the circuitry of Figure 1 to include
these equations, as shown in Figure 3, the mechanization of Samuelson's Second illteraction Model
is complete through the simultaneous solution of
Equations (9), (10), and (11). Here, potentiometers
07 and 09 refer to lib, and 08 to d/b. A third
initial condition, C (0) = Co, is needed which, due to
the finite difference assumptions, cannot be deri ved from Equation (5). Potentiometers 11 and 12
ill addition to time scaling, there is amplitude
scaling by which it is assured that the maximum
problem value is represented by the maximum
voltage available on the computer.
Thus, to
achieve maximum resolution, the variables of
interest here, viz Y, Y and Y, would be represented on the EAI TR-20 General Purpose Analog
Computer by the full :1:10 volts available on the
machine.
It is beyond the scope of this general presenta-
tion to go into the details of time and amplitude
scaling.
The reader is referred to the Primer
on Analog Computation (Application Study: 1.1.2a,
Bulletin No. ALAC 64002-1) from the EAI Applications Reference Library for a complete description of these subjects.
illitial Conditions: The remaining potentiometers,
05 and 06, supply initial condition (I.C.) values
for the two integrators. Thus, potentiometer 05
Fi gure 3. Samuel son' 5 Second Interaction Model
6
are required only if I, C and Y have different
amplitude scaling factors. Of course, the time
scaling factor, say {3 == 12, enters potentiometers
07, 08, and 09 if it is contained in the setting of
the other potentiometers since time scaling must
be consistent throughout the whole circuitry.
in Equation (13b) where consumption is concerned.
Because of. the positivity of EKI, in fact, it will
increase + C and thus + C as opposed to Equation
(12d). Hence, this is no cure for the problem of
decreased investments but only makes it worse by
increaSing consumption. Consequently, there is no
return from Phase II into Phase I without an
exogenous correction not yet in the system. (This
is one point criticized by Leontief.)
Indeed, if there were 10 or more differential
equations describing the system, they would lead
to circuits similar to that shown in Figure 3 wh~re,
for instance, one function (- Y) enters into -C of
another function to establish all the mutual connections required by the system equations. The
only limitation, in fact, is the availability of integrators, amplifiers and potentiometers in the
equipment complement.
Example:
Of course, lowering the consumption parameter,
E, is a cure but this obviously is exogenous,
implying, as it does, control of the price index.
The system, therefore, is not self-regulating as
suggested (ideal business cycle); the coefficients
of the problem must be subjected to control. From
the computational point of view it is the servo-set
potentiometer that quarantees dynamic equilibrium
under the law of policy decisions.
The effect of PoliCy Decisions on the
Economy:
In order to give a simple example of the effect
of policy decisions on the economy, it should be
recalled that Samuelson's Second Interaction Model
contains savings as a part of the investment, Le.,
there is only a partial irreversibility. Furthermore, investment as an input must be a positive
quantity.
Moreover, people will stop investing
money whenever consumption exceeds a specified
fraction, F, of their income, Le., when C ~ FY.
(At this point, in a modelled situation, a comparator in the computer would throw in order to
change the circuit connections.) They will even
withdraw savings from investments gradually when
this occurs (partial reversibility). As a result, two
different phases are created in the system:
By mechanizing the sets of equations, (12) and (13),
it can be shown how an unregulated economy breaks
down (provided E is large enough) even though all
the money saved previously (reversible money)
should be spent. That the extraction of Kt from I
could be terminated if the simulation proceeded
toward a minimum 10 (controlled by a second
comparator) is immaterial with this phenomenon.
Because of Equation (13c), it is necessary to hold
the value of I beginning from the instant Phase II
takes. ove,r. ~herefore, it is desireable to mechanize I == Y - C and have the integral I remain constant after its inputs are removed.
Figure 4 is seen to be the proper modification to
Figure 3 to .effect this mechanization. Here, the
derivative - Y is av.ailable as it was in Figure 3;
however, to make -C available it was necessary to
add summer 06. In order to hold I(T) == IT, amplifier 05 had to be patched as an integrator whose
I.C. on potentiometer 16 has already been defined.
Amplifier 07 is required to provide the proper sign
of C for -6 as prescribed by Equations (12b) and
(13b) .
PHASE I
Y == -AY - BY + D [as in (9)J (12a)}
6 == -EC + EY - G [as in (10)J(12b)
I == Y - C
if C>FY (12d)
[as in (l1)J (12c)
PHASE II
Y == -AY - BY + D
(13a)
C == -EC + EY - G + EKI
(13b)
In addition, potentiometer 13 provides the value
of F from Equations (12d) and (13d), and the comparator throws at t == T to its positive output when
C - FY ~ O. Consequently, the comparator relay
disconnects or connects the eight contacts as
indicated. (Usually, two comparators are required
to govern .four separate pairs of contacts.) In fact,
set (12) now is changed into set (13). Potentiometer
14 with K* is the sole input into amplifier 05 whose
output starts from IT; amplifier 06 has an additional term on potentiometer 15.
if C~Y (13d)
I == (1- F) YT - K* (t-T) ==
IT - K* (t-T)
(13c)
The withdrawal of KI and adding it to income has
no effect, of course, on the income cycle; compare
(12a) and (13a). However, it does add to income
7
z
z
~
I
~
I
I
I
I
I
I
I
I
I
I
I
~
I
I
~
(B)
(Al 10 YEARS
z
RANGE
OF
8'5
1/2 HOUR
RANGE OF 8'5
~
I
I
I
(C)
c:
I
I
~
I MONTH
+
0---
Figure 5. (a) no genuine lag, (b), (c) genuine lags indicated
by ranges
8~
e
-1
item. The sampled lag, (b), occurs if the product
is large enough for one unit to be a sample of
item ... a six-jet airliner, for example.
Fi gure 4. Samuel son's Second Interaction Model wi th Leontief's
Two-phase Policy (all switches down in Phase I)
z
_
",-
It should be noted that during Phase II the comparator never can receive a voltage C - FY < 0
to throw the system back to minus, Le. to Phase I.
2. Genuine Lags: A genuine lag with regard to
variable time, t, occurs if at time, t, a dependent
variable, z, is required with its value equal to Z
(t-T) of time (t-T) and simultaneously there exists
a variable, (), such that
o S () S T
with
Z (t) - Z (t-()
Z (t - () /2)
o
LAGGED
- ..... -t"UNCTION
,,
,
"
" ......
T
(A)
z
1
Figure 5 illustrates the implications of this definition. It can be seen in (a), for example, that a
system with a genuine lag can be unaffected for
long periods of time, whereas in other cases ...
the stock market, say ... one-half hour of the day
(b) or one day of an otherwise constant month (c)
can create definite disturbances or even outright
market shock.
T
( B)
Figure 6. (a) continuous lag, (b) sampled lag
One of the simplest systems with genuine lags is
given by W. Leontief:
Two classes of lagged inputs can be distinguished,
as shown in Figure 6. The continuous lag, (a),
occurs with the delay of a production line manufacturing very large quantities of a product ...
thumbtacks, say, or other small mass-produced
(14)
(15)
8
Differentiation of Equation (15) and substitution of
it into Equation (14) establishes a differencedifferential equation for Xl; a similar procedure
yields the equation for X 2 :
..
Thus, when the output of integrator 03 exceeds
+ 10 volts, maximum time, T1 has elapsed and the
comparator throws to plus, resetting integrator 03
to its initial conditions.
.
This should be interpreted as equivalent to Yi*'s
being periodic in T -- although Xi and Xi must
not be so -- and unaltered by the function Xi'
Consequently, repeated intervals of length T always start with the first value of function Yi*
while the initial conditions of Xi and Xi in the nth
run are different by their respective final values
of run (n-1).
Xl (t) =AX 1 (t-T) - AY1 (t-T) - B2 Y2 (t-T2)
(16)
= AX 1 (t- T) +
Y1 * (t)
..
.
X2 (t) =AX 2 (t-T) - AY2 (t-T) - B1Y1 (t-T 2)
(17)
=AX 2 (t-T) + Y2* (t)
where A = l/blb2, B1 = 11b 1 , T = Tl + T 2 , and
Y1 *, Y2* as indicated.
There is another situation possible with this model.
Assuming that Yl, Y2 ... etc. are not periodic in
T, they would have to be generated in some dynamic
fashion, a definition of which would have to be
provided for in additional system equations. This,
of course, would transform their model character
from exogenous to endogenous. Then, satisfying
Haavelmo's Principle, there would have to be four
mutually independent relations in Xl, X 2, Y1, Y2
and their derivatives instead of only the two relations shown in Equations (14) and (15).
Functions Y1 and Y2 can be considered as exogenous and, after transformation into Y1 * and Y2*,
are set up on two separate VDFG's for t = + 0
back to t = -T. A symbolic circuit diagram would
appear as in Figure 7, and applies for each one of
the two functions, Xi (i = 1,2), separately.
The righthand half of Figure 7 merely simulates
a second-order ODE. General methods of mechanizing continuous, stepped-sampled, and smoothedsampled lags will be discussed later. These are
labelled here, for convenience, only as two-part
networks "Delay T" .
3. The Mechanization of Lags:
Among the three
kinds of genuine lags in economic model simulation,
the easiest to mechanize is the stepped-sampled
lag. This is accomplished with so-called' 'trackstore amplifiers" (TS) ... integrators alternating
between their I.C. and HOLD modes. Chains of n
TS amplifiers can be patched in order to delay
samples by n/2 TS intervals in time.
TS amplifier modes are controlled by a master
clock. Figure 8 shows a chain oftwo TS amplifiers;
the first input and last output correspond to the
terminals of the "Delay T" box of Figure 7. The
performance of TS I and TS II versus time is best
explained by the diagram shown in Figure 9.
+t
During the tracking period (TR), the amplifier
attains an output voltage equal to the inverted
input very quickly; during the store period (ST),
RESET
a.
TO T. C.
::E
-10
8
0---
~t2:T
,-- ----1
+
I
I
Figure 7. A possible interpretation of equations (16), (17).
I
I
I
I
I
I
I
I
I
I
The lefthand half of the figure symbolizes the
VDFG for Yi*, and a time base integrator, (03),
supplies an input voltage whose slope -- speed of
problem run -- is determined by potentiometer 06.
L_
I
__-l
Figure 8. Stepped-sampled lag with two TS amplifiers.
9
INPUT TO I
o
MODE T
+
OUTPUT OF I
o
MODE II
+
it to TS II which tracks while TS I stores. On the
next tracking period of TS I, amplifier TS II will
store this difference, thus causing integrator B to
integrate at precisely this rated voltage. If the
integrator potentiometer is adjusted properly, the
output of B attains fn always at TS seconds later
than if occurred at the input of TS I. Through this
repetitive process, the delayed function between fn
and f n -1 is approximated by smoothed samples in
a straight line segment.
OUTPUT OF II 0
It can be seen from Figure 10 that equipment
requirements soon would become excessive should
the delay problem occur in several places in a
system. With such systems, where"the delay could
be judged as a continuous lag, the use of a hybrid
system is justified. This would mean that the data
to be delayed would be converted from analog to
digital (AD), stored digitally for the required period
of time -- which can even be a varying and computed T -- and then converted back digital to
analog (DA) and fed into the analog circuit.
Figure 9. Time characteristics of a TS chain.
the amplifier preserves its latest value from the
TR period no matter how its I.e. changes. Thus,
after the specified "lag" time, the properly
delayed input value is returned to the simulation
as a 'staircase function' .
In some cases, however, because system requirements are opposed to such an essential staircase function of abruptly varying height, it is
necessary to smooth out this final delayed output
-- smoothed-sampled lag. Here, the final stage is
terminated by an ordinary integrator with a summing feedback, as shown in Figure 10, thus producing a continuous lagged output as close to the
original function as possible within the limited
density of sampling (sampling rate). The objective
here, of course, is to obey Shannon's Sampling
Theorem as well as possible by loosing only a
minimum of vital "information". Accordingly, the
intimate details of any delays are not shown; only
their cumulative effect on the function itself is
produced.
The hybrid approach permits the simulation of
tremendous systems, handles multiple channels of
information easily, and can be programmed to obey
Shannon's Theorem at less expense than a purely
analog simUlation of a large economic system
would be. Thus, delayed functions as shown in
Figure 5a can be achieved very easily and in great
number.
4. r=urther Improvements Toward Reality:
In the interest
of keeping simulated models within the bounds of
reality at all times, there are several important
features of the analog computer -- even in smalland medium-sized installations -- which are very
helpful in this regard.
A comprehensive, non-mathematical explanationof
the performance of this circuit is as follows:
assume that TS I has just tracked for sample fn of
the original function, f(t) , and has gone into store
while the integrator slowly builds up to the value
of the preceeding sample, fn -1. Then, summing
amplifier, A, forms their difference and supplies
The function switch is one of these features. As a
manual switch, operable at random, it provides
not only for alternating between different phases
of the simulation but also (and of most importance)
for the sudden injection and rejection of quantities
such as can occur in even the most sophisticated
of economies. It is this Simple instrument which
tests such important dynamiC characteristics as
sensitivity and vulnerability of systems. Indeed,
the beginning of fluctuations due to injection can
be determined as easily as can be the limited
amount of capital needed to be withdrawn from a
system in order to force it into phase change.
This type of analysis can be especially gratifying
when system stability cannot be determined with
paper and pencil, i.e. when due to nonlinear
phenomena, experimental tests become a necessity.
SMOOTHED
OUTPUT
Figure 10. Smoothed sampled lag with two TS amplifiers.
10
The much discussed question ofthe ceiling problem
is another dynamic characteristic which requires
realistic analysis. Warranted and artificial disturbances in business (fluctuations, depressions,
booms, etc.) can cause this dynamic problem to
occur whenever .. , as a result ... a "quantity
of merit" (e.g. effectiveness, rentability, gross
national product, etc.) approaches its absolute
maximum in a non-asymptotic way. Assuming that
some relationship defining this maximum value
either statistically or dynamically is known, the
question is: How can asymptotic merging be
enforced?
part of the new and sophisticated hybrid computers. These computers can proceed with digital
subroutines and data processing simultaneously
with analog Simulations, and all are under the
master control of a digital operations system.
The simulation of the dynamics of a national
economy, say, incorporating 25 ODE's with 10
genuine lags, each one requiring 100 samples per
unit of delay, and 6 analog decisions to be instrumented at digital speed, plus provision for manual
or digital computed switching among three separate
phases is well within the capabilities of modern
hybrid computers.
A comparator (the automatic analagon of an FS)
together with a servo-controlled limiter can be
used to mechanize the necessary control function.
(It should be noted that in a free economic system
a "limiter" is nothing more or less than the free
and conscientious decision of a responsible executive -- a human being voluntarily serving the
common good if vital objects are at stake.)
REFERENCES
1.
Frisch, R., Propagation Problems and Impulse Problems in Dynamic Economics, London 1933 (Allen)
2.
Beach, E. F., Economic Models, New York
1957 (J. Wiley's Sons)
3.
Koopmans, T. C., Achieving Analyses of Production and Allocation, New York 1951 (Wiley).
and Essays on the State of