DESCRIPTION AND ANALYSIS
OF SOCIO-ECC'NGMEC SYSTEHS
BY PHYSECAL SYSTEMS TECHNIQUES

THESIS FOR THE DEGREE 0F Ph. D.
MICHIGAN STATE UNIVERSITY

JACK BARRY ELLIS
1965

 

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ABSTRACT

THE DESCRIPTION AND ANALYSIS OF SOCIO—ECONOMIC SYSTEMS
BY PHYSICAL SYSTEMS TECHNIQUES

By Jack B. Ellis

Techniques of systems analysis which were originally evolved
for electrical networks have previously been generalized to enable
their use in analyzing physical systems of other types, such as
mechanical, hydraulic and mixed types. This thesis shows that they
may be further generalized to include the analysis of systems containing
social and economic components. The advantage of physical systems
analysis in this application is that it provides a consistent and
rigorous procedure for formulating mathematical models of a wide range
of social and economic phenomena, both of the static and dynamic types,
as shown by the two selected examples in the text. These models can
be formulated and solved on a digital computer, giving the analyst a

wide range of experience with his model in a short time.

The requirements for successful application of systems

analysis are:

l. The phenomenon must be identifiable as a collection of

components with discrete interfaces with one another.

2. TWO complementary variables must be found which satisfy
the two generalized Kirchoff postulates.
3. Each component of the system must be modeled quanti-

tatively in terms of these two complementary variables.

Jack B. Ellis

In the first example, the problem is to allocate the
attendances of campers at all state parks in Michigan in 1964. The
system is partitioned into three types of components; origin areas,
transportation links, and parks. Each component is modeled in terms
of the variable Y, the flow of campers, and the variable X, the
propensity to camp or the demand pressure for cauping. An algebraic
model was evolved which successfully described the operation of
the system for the year 1964, the only one for which complete data

was available.

The second model attempts to describe the dynamic growth of
a three-sector national economy by means of a discrete-time state-
space model. Each component is modeled in terms of the variable Y,
the flow of output, and the variable X, the stocks of accumulated
capital goods. Representative parameters are chosen from historical
United States data, and solutions are obtained for the growth of
investment in each productive sector over a five-year period. Appli—
cation of state-model theory in the field of economics enables a
complete analysis of an economy at one time, without having to cour
bine the many types of partial analysis usually employed. A new type
of production function results from cutset relationships, which
extends the Leontief inter-industry concepts to include the Marxian
linear labor-capital production function. This enables the model to

reflect all types of inputs to production processes.

THE DESCRIPTION AND ANALYSIS OF SOCIO-ECONOMIC SYSTEMS

BY PHYSICAL SYSTEMS TECHNIQUES

By

Jack 3? Ellis

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering

1965

ACKNOWLEDGMENTS

The author would like to acknowledge the guidance and
encouragement of his supervisor, Professor H.E. Roenig, during his
entire program. His enthusiasm for and encouragement of research
efforts in what appeared at the outset to be impenetrable and untrodden
areas will be a source of inspiration to the author for many years to
come. Thanks are also due to Professor D.N. Hilstein for his helpful
discussions and suggestions during the research leading to this thesis,
and for the foresight to instigate systems study effort in the recreational
field. The author would like to eXpress his appreciation of the
financial help received during his program from the Department of
Electrical Engineering, The Ford Foundation, and from the Michigan
Outdoor Recreation Demand Study in the Department of Resource Develop-

ment, Michigan State University.

Commendation is also due to the fortitude of my wife,
Barbara, without whose support and encouragement this research would
have been impossible, and to my mother and father for their constant

encouragement as well.

711-

TABLE OF CONTENTS

LIST OF TABLES 00....0.......0...OOOOOOOOOOOOCOOOOOOOOOO

LIST OF FIGIJRES OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

LISTmAPPENDICES 0.0.00.0000...OOOOOOOOOOOOOOOOOOOOOIO

Chapter

I. INRODUCTION OOOOOOOOOOOOOOOOOOOIOOOOO0.0.0...

II. AN ANALYSIS OF A STATEWIDE
PARK SYSTEM IOOOOOOOOOOOOOOO0.0.0....

III. AN ANALYSIS OF A NATIONAL
ECONOW I O O O O O O O O O O O O O O O O O O O O 0 O O O O 0
IV. CONCLUSIONS 0 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 0

REFERENCES OOOOOOOOOOOOOOOOOCOOOO0.000000'OOOOOOOOOOOOOC

- iii -

Page

iv

vi

60

91

LIST OF TABLES

Table

11-1. ERROR MEASURES FOR SELECTED
SYSTEMS MODEL RUNS aasaasaeseaoesaaeasaeaee

II-Z. PREDICTED ATTENDANCES AND
ERRORFm1964MTA 00.000.00.000...0.0.000...

III- 1. AGGREGATED INTER- INDUSTRY
TRANSACTIONS FOR 1947 eaaaaooeaoaseaaosaaeas

civ-

Page

25

27

51

Figure

1-1.

11-1.

III-l.

III-2.

LIST OF FIGURES

Page

CONCEPTUAL STAGES OF SYSTEMS ANALYSIS ........... 5

THE SYSTEM LINEAR GRAPH FOR THE MICHIGAN
STATE PARKS SYSTEM ................. 12

THE LINEAR GRAPH OF THE UNITED STATES
NATIONAL ECONOMIC SYSTEM................... 42

PREDICTED AND ACTUAL SECTOR INVESTMENTS FOR
THE UNITED STARS ECONOMY, 1947- 1952 s s a s a s a s s a 58

LIST OF APPENDICES

Appendix

A. VALUES OF THE ORIGIN AND LINK COMPONENTS
OF THE MICHIGAN STATE PARK SYSTEM..............

B. RESULTS OF THE FACTOR ANALYSIS AND
ATTRACTION INDEX CONSTRUCTION FOR THE
Pms OOOOOOOQOOOOOOOOOOO

Co PRmRAMPRKSYS 0.0.0.0000...OOOOOOOIOOOOOOOOOOOOOCO

-vi-

Page

64

78

8S

I. INTRODUCTION

For the last several decades, scientific workers in a wide
range of fields have addressed themselves to the problem of describing
and analyzing the phenomena of the natural universe, involving both
man and his environment. A comprehensive mathematical description
usually has been the most desired goal. Considerable progress has
been made in the mathematical analysis of man's physical environment,
and most physical phenomena are today amenable to same method or
other of physical systems analysis. However, when man enters a phe-
nomenon as an integral component, as is the case when social or econOmic
phenomena are considered, the analytical picture is still one of frag-
mentary quantitative results and a preponderance of qualitative theory,
supported in some cases by certain empirical observations. In the
most recent decade, several prominent workers in the social sciences
[1, 2, 3, 4, 5, 6, 7] have expressed their desire to overcome this
quantitative gap by employing the powerful conceptual tool of "system"
and the body of analytical procedures which flow forth from it. Their
goal, however, usually has been more than just a mathematical frame-
work for various phenomena; they desire also to establish a systems
analytical framework which is rigorous and consistent from discipline
to discipline and from phenomenon to phenomenon [1, 2, 3, 4, 6]. Need-
less to say, there has been as yet no unqualified success in achieving
this goal. This thesis does not purport to describe man's place in
the universe, or to have fully achieved the aforementioned aims and

desires. It does, however, present the foundations of a rigorous and

_1u

-2-

consistent formulation procedure for modeling certain classes of
socio-economic systems, along with the first large scale results

of the application of such models. The systems model formulation
techniques are those recently generated for electrical network analysis
[8], and generalized to include the analysis of other discrete physical
systems [9,10]. Their application to systems containing social and
economic components, and the prerequisite component modeling procedures,

are developed in the body of the thesis.

The aim of the research leading to this thesis was to describe
and analyze in mathematical terms the behavior of selected non-physical
complexes by application of the theory of physical systems. New
methods were evolved to use existing and new data to obtain mathematical
component models. ffiuachoice of examples was such that the application
of physical systems analysis in both a static and a dynamic case
could be shown, and also to illustrate the diversity of phenomena which
are amenable to such analysis. The model discussed in the first
example, a recreational travel model, is not paralleled in the literature
of the field to date. The results of the second model, of the United
State national economy, can be equalled by conventional methods
existing in economics [6, 34, 36], but the contribution of this thesis
lies in the explicit demonstration of the applicability of the rigorous

and general methods of physical systems analysis in this field.

The following postulates must be satisfied by any complex
before the methods of physical systems analysis are applicable [10].
Thus, any socio—economic system studied on this basis must also be

amenable to these requirements:

The system must be identifiable as a collection of component

parts or sub-phenomena.

The components must be discrete in nature. This implies that
interactions between components must be considered as taking
place only at points of interaction, called terminals. If

in fact an interface is a line or surface boundary, it must

be considered as collapsed to a single point.

TWO complementary variables, X and.Y, must be selected as a
basis of modeling all the individual components. This pair
of variables must satisfy postulates 5 and 6 below. An

equation relating X and Y for a component is referred to as

a component modelingiequation.

Measurements of painsof the complementary variables are
referable to the edges of a linear graph, which has an edge

to indicate the terminals of the component to which the
measurements refer. An N-terminal component is completely
specified as to performance by an arbitrarily chosen terminal
graph of N-l edges and a set of N-l component modeling equations,
relating the 2(N-1) complementary variables X and Y

i 1’
1 = 1,2)...,N..1.

The algebraic sum of the variables Y corresponding to the

i

directed edges of a cutset of a system linear graph must

vanish.

The algebraic sum of the variables X corresponding to the

i

directed edges of a circuit of a system linear graph must vanish.

- 4 -

When all of the above postulates are satisfied, a systems
analysis may be accomplished by the analytical and formulation
procedures presented in this thesis and in the literature [10].

The formulation procedure is consistent, irrespective of the social-
science field of study to which the components "belong". Only the
method of component modeling may vary from one field of study to
another, where phenomenological-specific component behavior theories
generally will be called upon. A schematic of the entire process

of a systems analysis for a non-physical system is shown in Figure 1-1,
which suggests that the systems model may finally be used as a basis

of stability, optimization and other analyses [11].

- 5 -

Figure 1-1.

 

Recognition
of System

 

 

Identification l
of Components

 

(Schematic) l

]

 

Conceptual Stages of Systems Analysis

 

 

* 'Modeling of
System Graph and Components
Interconnection Matrices (a) Form

 

(b) Parameter
Values

 

 

 

 

 

 

 

Formulation of
Appropriate

System Model

]

 

Propose Changes in System
Parameter Values Stimulation

 

 

+

 

 

 

\ Stability

‘Analysis

 

 

 

 

 

Performance
Optimization

 

[Recognition of Desired System I

Specify
Payoff Function

 

II. AN ANALYSIS OF A STATEWIDE PARK SYSTEM

II-l. Background

In recent years there has been a great awakening of interest
in America regarding problems of outdoor recreation for all members of
society. Over a quarter billion acres of public land are used in
recreation, and perhaps as much additional private land. Over 90 per-
cent of the population participates in some manner, patronizing a $20
billion a year industry which receives an additional $1 billion of
government investment per year [16]. For decades, however, such
entities as the National and State Park systems have been established
and their growth planned under quite crude intuitive assumptions about
the needs and desires of the population for features which these park
systems offer. The origin of such parks historically was to protect
a unique or especially attractive natural site from being despoiled by
noxious commerical or industrial development. Later, in the 1930's,
extensions to these systems often were made on the basis of make-work
projects to alleviate conditions of severe unemployment. In these
cases, most emphasis was placed upon the availability of publicly-
owned land resources and a pool of unemployed workers in the locality,
rather than upon specific development to provide demonstrably needed

facilities.

In the latter part of the 1950’s, mass availability of
leisure time created the need for a much more rational pattern of develop-
ment of outdoor recreation facilities. No longer is such development

<Hn1fined to preservation of "priceless" scenic sites which are part of

- 6 -

- 7 -

a "national natural heritage", although this is continuing, of course.
Rather, the creation of sites of acceptable scenic attractiveness which
are more accessible and therefore available to fill specific demands of
large population centers for leisure facilities is now receiving consider-

able emphasis [12,13,14].

While considerable progress has been made in the analysis of
demand and facility planning, there are still two broad questions which
remain to be answered before a completely satisfactory quantitative
analysis of outdoor recreation can be made. The first is; what is the
true measure of the demands of society for outdoor recreation activities .
of various kinds, and what economic value is placed upon them by society?
The second is; given a certain demand rate of the population, how do
people behave with respect to the particular sets of alternative facil-
ities open to them? In other words, given a demand rate and a set of

facilities, how can one determine the intensiveness of use of each?

Answers to the first broad question are still fairly far off.
Certain attempts have been made to measure demand [14,15,16], and an
analytic framework for the economic criteria to be evaluated has been
suggested [13]. However, there still has not been enough data gathered
to provide a reasonably accurate picture of the behavioral parameters
of society in this regard. For example, the main source of demand rates
and preferences is the report entitled "National Recreation Survey" by
the Outdoor Recreation Resources Review Commission [15], which is based
upon interviews with 3,647 persons out of approximately 180 million.
In a state the size of Muchigan, this consists of only about 120 persons.

While such a sample ordinarily is sufficient to overcome even the pro-

- g _

found regional, racial, ethnic and economic differences of groups of
society for a simple determination such as the preference for Candidate
X or Candidate Y for President of the United States, or television
programs A, B, or C, the immense variety of possible outdoor recreation
activities causes the results of such a survey to have disappointingly
high standard deviations. For example, the participation rate for
camping by females in the North-Central region of the United States
during the summer season is given as 6 percent [15]. However, the
table of standard error of estimated participation percentages([15],
Table IV, pg. 106) shows that based on a sample of 213 persons (the
number of females surveyed in the North-Central region), the 68 per-
cent confidence interval for this percentage is the interval from 3
percent to 9 percent. Actual participation rates by subgroups of

this group which vary by a factor of up to 3 may thus not be inconsis-
tent with the estimation. Very little planning or future estimation may
be meaningfully based upon such data. In the face of such uncertainty,

then, this thesis can add little to the knowledge of the first question.

The second question, that of the distribution of use of
facilities, given a demand level, will be the subject of the remainder
of this chapter. In the literature so far, various calls have been
made for an analytical model which would consider a given demand level,
road system and park facilities and allocate usage on some rational
basis [16,17,18], but no comprehensive solution has as yet been offered.
This thesis proposes an analytical framework for just such a model,
which is developed in subsequent sections. The activity of camping in
the state park system of Michigan was selected for a full-scale demon-

stration of such a model due to the availability of the largest amount

- 9 -

of data for any outdoor activity. For virtually all outdoor recreation
activities there is only fragmentary data, if it exists at all,
regarding the origin and destination of participants. For Michigan
State Park camping, a one year sample (for 1964 attendees) of a time
series of this data was available*, and the example choice was made on

this basis.

II-2. Definition and Modeling of Components

The system to be studied in this section consists of all
state parks in Michigan having campgrounds which issued more than 1000
camping permits in the 1964 calendar year, the road system of the State
of Michigan alongwith selected roads linking nearby states to Michigan,
and the population centers of Michigan counties plus selected population
centers in adjacent states. Three types of component are identified in

the system:

1. Origin areas of campers,
2. Highway links,

3. Destination areas, or state parks.

Each component is modeled by means of an oriented line segment

and an equation relating two variables, X and Y. The variable Y is taken

 

* A full determination of the origins and destinations of
Michigan State Park campers was made under supervision of the author
for the Michigan Outdoor Recreation Demand Study, Department of Resource
Development, Michigan State University.

- 10 -

to be the flow or attendance of campers in each component, measured on
a full-season basis in units of camper-days*. It would be quite feasible
to consider the measure of the Y variable to be camper-days per month
or per week rather than per season, if more data were available to the
analyst. The data required would have to include, among other things,
the average weather conditions for each period considered, the exact
timing of certain events such as the shut-down period of large manu-
facturing plantsand schools, and the timing of such holidays as
Independence Day and Labor Day. All of these considerations would
cause certain of the component parameters to be time-varying. Since
the methodology is new and unproven, and the quantitative effects of
such influences remains largely unknown, a seasonal measure is used so

that such effects may more reasonably be neglected.

The variable X is taken to be the propensity to camp, or the
demand pressure for camping. Propensity to camp is considered to be
annihilated by the process of making a trip to a campground and sub-

sequently camping there.

The origin areas are modeled as two-terminal flow drivers of

known magnitudes. Their component equations are taken as

YOi = known 1 = 1,2,...77 (II-l)

Their magnitude is taken to be the number of camper-days re-

corded for 1964 from that origin in the camper permit origin-destination

 

*One person camping over one night in a state park was considered
to equal one camper-day of use. The season length coincides with the
calendar year .

-11-

tabulations mentioned previously. Each origin considered is either
the geographical center of population of a Michigan county, or a
selected center of pOpulation in a nearby state. In order to reduce
the final number of components and equations with as little sacrifice
in accuracy as possible, in cases of counties with less than 10,000
population, the center of population of two or three counties is
considered as one origin component. Appendix A gives a list, description
and values of the 77 origin components used in the model. The line
segment representing an origin area is considered to be connected in
the system linear graph of Figure 11-1 from an arbitrary reference
point (not shown in Figure 11-1, for clarity) and the appropriate
center of population. The orientation is taken as away from the

reference.

It is postulated that the highway links connecting the

population centers and parks components could be modeled as

th : ijhj J = 1)2)000)208 (II-2.8)

The parameter Gj represents the reciprocal of "deterrence
to travel"; i.e., if l/Gj is large, the flow of campers for a given
propensity difference across the link is small. It is postulated
that for the purposes of making a trip to a campground two factors
of the route are influential; the time required to travel the route,
and the direct out-of-pocket expenses incurred in so doing. Note
that distance, as such, is considered to be a less representative
measure of trip deterrence than time. It has been discovered in many
studies that for many trip-making purposes people are willing to

choose: a longer route if they can save time by so doing. Cost of

- 13 -

gasoline and toll expenses only was considered, since most families seem
not to consider incremental depreciation and repairs of their auto-
mobiles caused by relatively short recreational trips. The time-cost
value of each link was raised to a power greater than unity in order to
weight longer trips with proportionally more deterrence than shorter

ones, in line with most traffic engineering findings [19,20,21].

A total of 208 links chosen from the Michigan State Highways
Map represent the state highway system. All centers of population and
state parks are connected by the most important road links on the map.
The importance of road links is determined from a traffic-count map for
1963, provided through the courtesy of the Michigan Department of High-
ways. Roads showing estimated daily volumes of fewer than 300 vehicles
are omitted, unless they are the only applicable links. Average driving
times were estimated for each link with the assistance of Mr. Clifford
Tiedemann of the Michigan Outdoor Recreation Demand Study staff. Average
direct cost is arbitrarily estimated on the basis of 14 miles per gallon
of 35¢ fuel, plus the toll charge for an automobile, plus one-half of
the toll surcharge for a single-axle trailer, since roughly 50 percent

of campers use trailers.

The parameter G. is evaluated as

 

k

Gj = 1 k3 j = l,2,...,208 (II-2b)

(T. + K C.
J 2])
where

Tj = estimated average driving time in hours

Cj = estimated direct cost of gasoline plus tolls in
dollars

k k k = constants

l’ 2’ 3

- 14 -

The presence of the constants k k2, and k above, and

1’ 3’
k4 in equation II-3a below is necessary to allow the model to be
scaled when in operation. With no firm data to go on, it cannot be
predicted in advance exactly how the propensity drop generated by

a flow of campers in a circuit from an origin back to the reference
point is distributed across the park and the highway link components.

The constants are inserted at this stage to allow them to be manipulated

for overall best fit of the model predictions at the solution stage.

A complete listing of the transportation links, with their
estimated time and cost parameters is given in Appendix A. Their
location is shown schematically in Figure 11-1, with arbitrary

orientations.

The parks are modeled as line segments connected from the
park location to the arbitrary reference point (omitted for clarity in
Figure 11-1), with their orientations taken as towards the reference.

It is postulated that the park components could be modeled as

ka = kaAkYk k = 1,2,...,55 (II-3a)

where k4 is a constant, and the parameter Ak is a measure of the
"attraction" or the "attractiveness" of park k. The measurement of this
parameter Ak in quantitative terms was the subject of considerable
deve10pment. Such concepts as "attractiveness" are usually considered
as being too subjective to be expressed in terms of hard numbers. There
is a certain amount of data available, however, with which the problem
of deriving a behaviorally-oriented attraction measure can be attacked.

0n the one hand, there is a certain amount of data on preferences of

people for the activities they engage in and thus the facilities they

- 15 -

tend to demand when they go camping [15]. On the other, there is an
inventory of the particular attributes and facilities possessed by

*
each of the 55 state parks included in the study .

From a study of the desires of people, reported in [15], it
is postulated that there are three main motivational factors which

lead to any park being considered "attractive" to campers. These are:

1. Its endowment of water-oriented features,
2. Its scenic aspects,
3. Its provision of comfort and convenience facilities.

Preference data from the same source [15], further suggests
that, if the water-oriented factor is given a relative importance
weight unity, since nearly every camper wishes to participate in water-
related activities, appropriate weights for the scenic aspects and
the convenience aspects would be 1/3 and 2/3 respectively. The problem
then is reduced to the following;find the relative quality ratings for
each park for each of these three features, and establish the total

park attraction rating as

Ak = R4 Ck(Ql + l/3Q2 + 2/303) k = 1,2,...55
where

k4 = a constant

Ck = the ratio of campsites at park k to the average

number of campsites in 55 parks

 

* This information was obtained by Mr. Carlton Van Doren
of the Michigan Outdoor Recreation Demand Study staff from the
Michigan Department of Conservation. Thirty-eight facilities and
attributes were considered.

- l6 _

Q1 = quality rating of water facilities
02 = quality rating of scenic attributes
Q3 2 quality rating of convenience facilities

For ready comparison of park attractiveness, it is desirable
to have an attraction value distributed such that an "average" park has

an attraction of unity. Thus we can let

61 = mean of Q1 for all parks
62 = mean of Q2 for all parks
O3 = mean of Q3 for all parks

and then write
Ak 2 k4 Ck(0.Sle/(%_ + 0.165xQz/Qg + 0.355xQ3/Q3)

k = l,2,...,55 (II-3b)

It still remains to be established that the 38 facilities and
attributes considered above actually do group into the three factors
postulated and also what quantitative contribution is made by each

particular attribute to the quality rating with which it is related.

At this stage, a factor analysis of the 38 facilities
considered for the 55 state parks is made using the FANOD 3 library
program on the CDC 3600 computer. The data is used in the analysis
in dichotomized form; i.e., a particular park is scored 1 if it
possessed facility i (i = l,2,...,38) and 0 if it does not,
yielding a 38 x 55 matrix, Z, of 0 or 1 scores. The program first

produces mean values and standard deviations for all of the 38

T
variables. It then produces a matrix R, 38 x 38, R = '%% , of

- 17 -

intercorrelations between these variables. It then produces a
principal-axis factor-loading matrix, A, 38 x 38 in size, such that

R = AAT is a close approximation to R [22]. The program then examines

the eigenvalues of R, and selecting them in descending order, rotates

the 2,3,...,k largest eigenVectors according to the varimax criterion [22].
This yields successive factor loading matrices, or k-factor solutions,

Ak’ 38 x k in size, from which the correlation matrix can be reconstructed
as 'Rk = Ak(Ak)T’ and compared with the observed matrix, R. The pro-
portion of variance "explained" by each of the k varimax-rotated factors
at each stage is determined, and can be considered a measure of the
goodness of fit to R.

The number of varimax factors considered necessary to "reproduce"

R withRk is usually measured by the behavior of the eigenvalues of
R [22]. Usually, k varimax factors are considered sufficiently des-
criptive of the variables if the first k eigenvalues are large, and a

relative drop in magnitude occurs with the (k+l)th eigenvalue.

The eigenvalues of R (given in full in Appendix B) are as

follows:
7.53, 5.05, 3.84, 2.43, 2.35, 2.00, 1.84, ...

This shows that, indeed, there is a sharp drop after the
3rd eigenvalue, followed by a much slower decline. The hypothesis
of only three significant factors is thus tentatively confirmed.
Fuller confirmation comes from comparing the detailed structure of the
three-factor solution with that of the four-factor solution. The first

three of the four factors are virtually identiCal to the original factors,

- 18 _
but the fourth is loaded by a subset of the variables loading the
second, or scenic, factor. Thus the three-factor solution is accepted,

and used to compute the three relevant quality indices.

The contribution of each facility to the makeup of the quality
score of the respective factors is taken as the factor loading coeff-
icient from the three-factor solution. For the first or water factor,
the highest value of loadings is obtained from the following ten facility
variables, after which there is .a noticeable gap before the remaining

loading values:

 

Variable Factor 1 loading
Great lakes shoreline ..8381
Inland lake shoreline -.7573
Great lakes swimming .8079
Inland lakes swimming -.9304
Sand beach on great lakes .6792
Sand beach on inland lake -.9102
Water skiing -.8448
Fishing -.5229
Boat rental -.5216
Lifeguard -.5569

The positive and negative loadings for great lakes and inland lake varie
ables’respectively, is considered to be the result of the special
situation of Michigan State Parks. The correlation between inland
lake and great lakes swimming, for example, is -.7600 since, for
geographical reasons a great lakes park does not usually have an in-

land lake shoreline as well. The exceptions, however, are some of the

- 19 -

most popular parks in the system, partly for this reason. Thus, the

quality score is taken on absolute values of the loadings:

Qli = E |alk|zik 1 = l,2,...,55 (II-4a)
where zik = 0, l; the score of the park i for variable k

a1k = the factor loading of variable k on factor 1

k = 13, 14, 20—24, 25, 26, 33, 36 from the numbering

of variables in Appendix B.

For the second or scenic factor, 8 variables are found to

have significant loadings. They are:

 

Variable Factor 2 loading
Virgin timber .5155
Wilderness areas .6502
Waterfalls .6895
Springs .5799
River frontage .5299
Historical site .6707
Interpretive program .6579
Hiking trails .6229
The quality score is taken as:
Q21 = E aZkZik i = l,2,...,55 (II-4b)
Where k = 7, 8, 10, 11, 12, 15, 16, 18, 19.

The third factor, representing comfort and convenience

- 20 -

facilities, is found to be highly loaded by the following 5 variables:

 

Variable Factor 3 loading
Store within 1 mile .5709
Shower facilities .8381
Flush toilets \ .8135
Laundry facilities .8301
Electricity service .7910

The quality score is taken as:

Q31 = Ea3kzik i = 1,2, 900,55 (II-4C)
Where k = 27, 28,..., 31.

The normalized scores are formed by dividing each Q

ji
(j = 1,2,3; 1 = 1,2,...,55) by
jS : Eajkzk J = 1.92:3 (11-46)
1 = l,2,...,55
where 2k = mean score of the 55 parks for variable k.

The Appendix B contains a full listing of the mean scores
of the variables, and the normalized factor scores of the parks, as

well as the final attraction indices calculated from relation II-3b.

Thus the entire system is now described by the system linear
graph of Figure 11-1 and the component equations II-l, II-2a, and II-3a,
with component values as given in the Appendices A and B. In the next

section, these are used to formulate a set of system equations which,

- 21 -

when solved, establish the desired flows; i.e. the predicted attendances

at the parks as measured in camper-days per season.

II-3. Formulation of the System Model

The system represented by the linear graph of Figure II-l
consists of a total of 340 elements and 146 vertices. The 340 edges

in the graph identify 77 origins, 208 highway links and 55 parks.

The 140 vertices are associated with 77 origins, 55 parks,

13 highway intersections not at a park or origin and 1 reference point.

The number of branch equations required to solve for the
system variables is [9], 146 - l = 145, and the number of chord equations
340 - 146 + l - 77 = 118. In spite of the slightly fewer equations
involved in chord formulation, a branch equation model is chosen, for the
reason that the fundamental cutsets and circuits were formulated manually
from the system graph. Since there are 145 fundamental cutsets and 195
fundamental circuits, there is a saving in manual error-prone work by
using the branch equations. Further, formulation of the 118 chord
equations requires the inverse of a matrix of order 118 with subsequent
post-multiplication by a 118x77 matrix. Because of core memory limitations,
multiplication of two such large arrays on the CDC 3600 computer can
only be performed in steps, with intermediate tape read-out read-in. To
solve the branch equations for the 55 park through variables, requires
the inverse of a matrix of order 145 but the first 55 rows only are pre-

multiplied by a diagonal matrix of order 55. This Operation can be performed in

- 22 -

the memory capability of the computer with only a few precautions.

For details of the compilation procedure, see Appendix C.

The nature of the graph is semi-Lagrangian, in the sense
that all of the flow drivers, and all of the edges corresponding to
parks share the reference point as a common vertex. Thus, all of the
flow drivers can be classified as chords of some tree, T, and the
edges corresponding to parks as branch elements. The tree thus
selected, is shown in heavy lines in Figure II-l and includes 90 high-

way links in addition to the parks.

The cutsets can be established by a self-checking procedure.
The first 55 cutsets must contain each chord element exactly twice, and
with opposite sign, since the first 55 branches correspond to parks

and each defines a separate sub-tree.

The details of the constraint matrices and system model
equations cannot be shown here because of their size, but in block

form the branch equations describing the system are:

X
G --P- : F Y : Y r (II’S)
xhb

where G is a 145 x 145 matrix with entries:

145
ij = Aj +121 ajici j = l,2,...,55
145 2
= Z 6.. J = 56,57,...,145
J1 1
1:1
and éji = 1 if element 1 is in cutset j and has

positive orientation,

-1 if element 1 is in cutset j and has

negative orientation,

- 23 -

= 0 otherwise.

145
G.. = 2 6 6 G 1;!3; j = 1,2,...,145

ji k=l 1k jk k
X is a 55x1 vector of park across
variables
th is a 90x1 vector of branch highway
link across variables

F is a 145x77 matrix with entries:

fij = 1 if origin area j is in cutset i
0 otherwise.

Y is a 77x1 vector of known origin through
variables

Y is a l45xl vector, the product of F and Y0.

It is a simple matter to perform the indicated matrix
multiplication on the right-hand side of II-S required to evaluate the

145xl vector of known quantities Y.

If G is partitioned in the form

.61 l 55xl45
""" 90x145

C)
H

the park across variables are given by

XP 2 [GI 1] Y (II-6)

- 24 -

The park through variables are then

X=AX =AGIlY II-7
p p [ ] ( )

where A = diag(A1,A ,...,A ), a 55x55 diagonal matrix

2 55

of park attraction indices

The solutions indicated in equations II-7 and II-6 were
carried out on the CDC 3600 computer, using Program PRKSYS shown
in Appendix C. The results are given and compared with the actual

attendances for 1964 in the next section.

II-4. Results of Solutions

The initial sets of solutions of the model are used to
determine what values of the constants k1,k2,k3 and k4 yield the
closest approximation to the actual 1964 attendances at the parks.

\,
Four measures are used to compare one set of solutions with another.

These are:

l. The average percent error of the estimates, defined as:
55
Average error = 53,2 I percent error of park i
i=1

2. The root-mean-square error of the estimates, definedaun
55

R.m.s. error = -EE 2 (percent error of park i)2
i=1

 

3. The number of parks with percentage error equal or less

than 20.

4. The number of parks with percentage error equal or greater

than 50.

- 25 -
Percent error was defined as follows:

predicted attendance at i - actual attendance
Percent error of = x100
park 1 attendance actual attendance

The latter two measures provide quick means of rejecting
obviously poor results. They are not nearly as discriminating when
the results are even moderately close. The r.m.s. error provides the
most acceptable test, since it is intolerant of large discrepancies.
Table II-l shows the error measures for a representative selection of

the solutions obtained with the indicated constants.

Table II-l

 

Error measures for selected systems model runs:

 

 

 

 

k1, k2, k3, k4 Avg. error R.m.s. error # errors 320% # errors 2 50%
‘7. ‘7.
1.0, 0.5,1.3, .01 28 33.7 26 10
1.0, 0.5,1 3, .02 29 36.5 24 11
1.0, 0.3,1.3, .01 32 34.5 23 10
1.0, 0.1,1.3, .01 33 35.3 22 10
1.0, 0.0,1.3, .01 34 36.1 22 11
1.0, 0.5,1.5, .01 32 34.3 22 11
1.0, 0.5,1.l, .01 30 34.1 24 10

The solution accuracy is found to be extremely dependent upon
the relative magnitudes of k1 and k4. These two constants determine

the relative magnitude of the propensity drop across the link and park

components, respectively, for a given flow. Subjectively, one would

- 26 -

postulate that the parks should "dominate" the links in this respect.
The solution obtained when k1 and k4 both equal unity yields
interesting, but rather inaccurate results. Parks in portions of the
state near large population centers are over-predicted by large per-
centages, while those in more remote northern areas are under-predicted
by a factor of 5 to 10. Good prediction is obtained when k4 is in

the order of 0.01 when k1 is unity. This result shows that, with
*
respect to camping, a highway link with unit conductance develops

only one percent of the prepensity drop across it for a given flow as

an "average" park with unity attraction index.

The constant k2 has the effect of including cost in addition
to time in determining the link "resistance". This is very rarely done
in the traffic engineering literature, and to the author's knowledge,
has never been attempted for such a large-scale extra-urban system

**
as this one . Table II-l indicates clearly that solution accuracy is

improved by this cost addition.

The full set of predicted attendances is given in Table II-2
below, using constants as indicated. The errors and percent errors are
expressed with respect to the 1964 attendance figures provided by the

Department of Conservation:

 

*
For example, a link having unit conductance could be one

involving 0.6 hours of time and 0.8 dollars of direct cost, yielding
Gj = 1.0 from relation II-2b.

Cost has been included as a route determining factor in
urban traffic flows in a study by Mr. Wallace McLaughlin in a forthr
coming thesis for the Ph.D. degree at Purdue University, Department
of Civil Engineering (from private communication).

10.

ll.

12.

13.

14.

15.

16.

17.

18.

19.

20.

- 27 -

Table II-2

 

Predicted Attendances and Error for 1964 Data:

(k = 1.0, k

1

Park

Baraga

Brimley

Fayette

Fort Wilkins
Gogebic Lake
Indian Lake
McLain
Muskallonge Lake
Porcupine Mts.
Straits
Tahquamenon Falls
Van Riper

Wells

Aloha

Bay City

Burt Lake

D.H. Day

East Tawas
Harrisville

Hartwick Pines

= 0.5, k

Predicted Attendance

3 _

 

(camper-days)

11,124
32,010

4,880
18,342
35,338
36,418

9,869
16,804
27,423
22,769
57,431
51,528
30,136
57,038
80,325
87,384
10,791
43,123
45,387

11,581

= 0.01)

Error

-3,002
-38,l77
-311
-6,479
5,797
-19,009
-12,604
5,557
-3,700
-28,214
5,372
11,093
-243
2,189
33,394

3,649

-18,144
-30,760
-13,906

-10,085

Percent Error

-21

-54

-26

20

-34

-56

49

-12

-55

10

27

71

-63

-42

-23

-47

21.
22.
23.
24.
25.
26.
27.
28.
29.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.

46.

Higgins Lake
Hoefft
Interlochen
Ludington
Mears
Mitchell
Muskegon
Onaway
Orchard Beach
Silver Lake
Traverse City
Wilderness
Wilson
Young
Benzie
Gladwin
White Cloud
Algonac
Brighton
Grand Haven
Hayes
Highland
Holland
Holly

Island Lake

- 28 -

174,924

30,586

143,111

136,159

21,718
50,806
86,942
24,840
30,090
60,543
79,365
59,793
33,619
38,577
10,287

7,834

6,393
32,122
67,850
68,545
75,076

9,724

110,096

51,172

29,758

-6,l8l
9,393
9,629

10,122
-3,119
2,959
7,787
4,574
-8,741
-25,666
-4,562

-14,236

-21,307

-14,497

-15,250

528
1,061
-6,378
30,347
-9,118
-4,748
-2,246
-22,192
10,266

"6,860

-13

10

23

-13

-30

-19

-39

-27

-60

20

-17

81

-12

-19

-17

25

30

47.
48.
49.
50.
51.
52.
53.
54.

55.

Lakeport
Metamora
Pinckney

Port Crescent
Proud Lake
Sleeper
Warren Dunes
Waterloo

Yankee Springs

- 29-

65,327
39,321
110,354
42,144
36,552
100,891
52,905
90,839

86,228

-15,598
-45,811
36,990
2,933
-34,918
34,320
-12,576
-33,693
-25,821

average error

r .m.s . error

The final results do not show any particular bias as

geographical area, such as the vicinity of metr0politan areas.

-19

-54.

50

-49

52

-19

-27

-23

 

28%
33.7%

to

The

positive and negative errors are reasonably uniformly distributed

over the geographical area of the state.

We find that there is a much

greater propensity drop across the park components than the link com-

ponents which verifies subjective considerations.

A measure of this

propensity drop, and some of its implications, can be found by solving

for

X for typical components:

for link 692 (see Figure II-l), with a flow of 100,000 on

this 20 mile two-lane highway,

X

692 = Y692 G692

= 100,000/1.4535 = 68,799

- 30..

for link 542, the Mackinac Bridge, with a flow of 100,000,

X

/ = 100,000/.2756 = 362,840

542 = Y542 G542

for park 21, Higgins Lake - the most popular park in Michigan,

6
x21 = Y21/k1A21 = 174,924/.0391 = 4.47x10

for park 31, Silver Lake - a "typical" Lake Michigan family
park,

60,543/.0137 = 4.42x106

II

x = YBl/klA

31 31

for park 37, Gladwin - a small out-of-the-way park,

6

Y = Y37/klA

37 7,834/.0017 = 4.6lxlO

37

The two examples of links indicate that the 20 mile link of
two-lane State highway is less "deterrent" to camping trips by a factor
of 53 than the Straits of Mackinac Bridge, with its present toll structure.
Alternatively, one could say that the deterrence provided by a 106 mile
stretch of two-lane State highway and the Mackinac Bridge are equal.
Furthermore, it is at first astonishing, but upon reflection becomes
intuitively obvious, that the prOpensity drop across such diverse types
of parks is very nearly equal. Given that the camper has arrived at
their gate, the wide differences in attendances (flows) is due solely

to their attraction indices.

As a check on the sensitivity of the solution to the attraction
indices (which will be the subject of further comment below), the model

was solved with all of the attraction indices set equal to C ' i.e. all

k!

quality factors were set effectively to unity for all parks. The result

- 31 _

showed an average error of 38 percent, an r.m.s. error of 56 percent,
with 17 parks having an error of s 20 percent, and 13 parks having
errors of 2 50 percent. These results are vastly inferior to those

obtained with the indices set at their computed values.

In comparison with other types of models which might be
constructed for such systems, the systems model as developed here shows
superior results. Gravity models have found almost universal use by
highway and traffic engineers for the study of many different types
of traffic flows [19,44]. A gravity model of the same park system
was made by Mr. Carlton Van Doren, a colleague of the author's,using the
same travel time values and the same links. Certain additional out-
of-state origins were included, since computer memory capacity was not
critical. The results of the best model run to date*, supplied by
Mr. Van Doren, showed an average error for the same 55 parks of 31 per-
cent, an r.m.s. error of 42.3 percent, with 24 parks having 3 20 percent

error and 11 parks 2 50 percent.

Those accustomed to the precision of modeling possible in
physical systems may find an r.m.s. error of 33.7 percent rather high
in an absolute sense. To provide an evaluation of the absolute per-
formance of the systems model, it is compared with a measure of the
effects of purely random or "happenstance" fluctuations in such a
phenomenon as camping. The r.m.s. value of the percentage attendance
fluctuations from 1963 to 1964 was considered, for the same 55 parks.

The total system camper-day increase was approximately 8 percent from

 

9:
April 28, 1965

- 32-

1963 to 1964. However, when one examines the figures for individual
parks, a vastly different picture appears. Some parks show an increase
of 158 percent (Aloha), 79.5 percent (McLain), and 74.5 percent (Van
Riper) others show a decrease of 47 percent (Island Lake) and 26.3
percent (Baraga). The average value of the percent changes from 1963
to 1964 was 18.2 percent, but the r.m.s. value is 33.4 percent. Thus,
it can be concluded that an overall r.m.s. error of 33.7 percent,as
obtained in this investigation, is not excessive, and may, in fact,
be extremely good.

Certain defects in the prediction of the systems model are
fairly easily traceable. Let us examine in detail some of the 10
"worst" predictions of the model to ascertain what, if any, effects not
accounted for in the component models used here could explain the large

discrepancies.

Benzie and D.H. Day parks were considerably under-predicted.
D.H. Day has a rather low attraction index, which undoubtedly does not
reflect the amenities immediately adjacent to the park, namely Glen
Lake and the Sleeping Bear Dunes area. Furthermore, both of these parks
are relatively exceptional in the Michigan state parks system in that
they do not require a vehicle permit for entry, almost certainly

adding to the "attraction".

Straits parks was also considerably under-predicted. This
park lies immediately at the northern end of the Mackinac Bridge, thus
providing perhaps a unique viewpoint from which to view the bridge
itself and the considerable shipping traffic in the Straits,and is

near the historic Mackinac Island. These features are perhaps inadequately

-33-

reflected in its relatively low attraction index of 0.740.

Two parks, Metamora and Sleeper, may have been influenced
by the choice of links in the system. Metamora is not served by a
road joining county centers of p0pu1ation, and since few other links
were added to conserve computer space, this may introduce an accessibility
barrier to this park which is not present in the real world. Sleeper
park is very close to Port Crescent, and arbitrarily was chosen as the
terminus of a road link. The over-prediction of Sleeper (52%) and
Port Crescent(6%) represents a 26 percent over-prediction for the parks

taken together as one unit, which they are functionally.

The over-prediction of Brighton (81%) and Pinckney (52%), may
well be due to the effects of the dichotomized facility data. The
fact that both parks have as high an overall quality index as some
very "attractive" northern Michigan parks may be due to the fact that
they possess many of the facilities loading the three quality factors -
but that these facilities individually are not really equivalent in
quality to those in some other parks. The dichotomized data cannot
discriminate between an "inland lake" and a"beautiful inland lake",

or between "fishing" and "good fishing".

While we cannot include exceptions such as those cited above
in the index structure, it appears that further refinement of the park
attraction indices would repay the effort - in particular, the gathering
of data in other than 0,1 form and a more careful inclusion of nearby
facilities commonly utilized by park users, even though technically they

may be outside the park.

III. AN ANALYSIS OF A NATIONAL ECONOMY

III-l. Background

Economics as a separate discipline has been studied by
various scholars over the last two centuries. The first studies
were mainly of a deductive or social-philosophic nature, where postu-
lates derived from common sense are set up as laws and the behavior
of economic systems is inferred from them. Such studies were carried
on by David Ricardo, Adam Smith, Thomas Malthus, Karl Marx, and many
others. Their impact upon later thinking and even upon real events
up until the present day attests to the widespread acceptance of their
studies, at least among certain segments of mankind 96], However
deeply convincing these philosophical structures may have been, they do
not embody the normal scientific precepts of hypothesis, test, veri-

fication or rejection, and re-hypothesis.

The truly scientific approach to economic analysis has
probably come about only in the last half-century. In 1899, Karl
Pearson published his Grammar of Science which introduced the discipline
of statistics and statistical inference to a wide group of scientific
fields. A few years earlier, Hollerith had introduced punched cards
into the record keeping system of the U.S. census, and thus for the
first time there existed both a large body of factual data regarding the
actual functioning of an economy, plus a rigorous method for drawing
inferences from it. Thus the stage was set for the economic philosophies

to be tested against the cold facts.
Various workers proceeded to do just this. For instance,

-34-

-. 35-

Edward Chamberlin [23] observed the behavior of supply, demand and
prices in actual marketplaces. He found that for a wide range of
goods, there was not a "perfectly competitive" supply or demand side

as had been postulated by the earlier social philosophers, and under
such realistic conditions, prices do_ggt always fall when demand drops.
Thus the Ricardo-Smith "Law of Supply and Demand" was shown to hold
only for a very limited range of economic goods. In the 1920‘s and
1930's the work of John Maynard Keynes [24,25] showed that the spending
of income by consumers does not take place automatically in such a
fashion as to tend constantly towards providing full employment, as had
been postulated by the "laissez-faire" school of economists. He

showed that a market economy often may balance itself by fluctuations
in employment, rather than in prices, wages and interest rates. He
introduced the hypothesis of an "investment multiplier", which states
that an investment of X dollars in productive capacity causes a flow,
mX dollars, of extra income, and that this effect obtains even when

the investment is made by a government operating with a budgetary deficit.

The scientific analysis of the growth of an economy was
attempted first by workers such as Colin Clark [26], Sir Roy Harrod [27]
and Wassily Leontief [28,29]. Professor Clark made two important dis-
coveries; firstly, that as of 1935 only seven countries in the world
enjoyed a truly high real standard of living, and secondly; that a
high percapita real income is almost always associated with an economy
having a high proportion of workers in tertiary employment, or service
industries, and a lower prOportion in primary employment such as agri-

culture, forestry and fishing. Clark offered no system-theoretic

- 36.-

explanation for these observations, but they are arrived at in a

scientifically sound manner.

Shortly thereafter, Harrod suggested a dynamic growth model
s
using empirically observed parameters in an equation which yields the
growth-path or development of an economy over time. Harrod used the

following variables:

Y1(t) = output or production at time t

new investment at time t

Y2(t)

X(t)

stock of capital goodsat time t

The relationships he postulated, and tested for several nations,

are:
Y1(t) 2121— X(t) (III-1)
Y2(t) = o Y1(t) (III-2)
dX .
dtt = 12(t) (III-3)

In the above, k is the "capital-output" ratio; i.e., the
amount of total capital investment required to produce one unit of
output, and o is the "savings ratio"; i.e., the preportion of income
which is saved. Equation III-2 states that investment equals savings.
Equation III-3 states that the growth rate of total investment equals
the amount of new investment.

The above equations are combined into a growth model as shown:

 

*
Usually termed the "Harrod-Domar growth model" [36,39] due
UDttle concurrent work of Domar.

fl“) _ -
dt _ Y2(t) (III 4a)
de (t)

1 = 0 Y (t) -
'7fir' 1 (III 4b)
dY (t) 'g
dtl = k Y1(t) (III-4c)

The latter equation, III-4c, when solved for Y1(t), gives

the time-path of output of the economy:

0'
Y1(t) = 9.1521(0) (III-4d)

Harrod estimated values of k betweei 2.5 and 3.5 for developed
economies, and o values of between 0.1 and 0.2, both on a per-year
basis. Values of ks3 and 0:0.15 yield a -% value of 0.05, implying
that such an economy would enjoy a growth rate of 5 percent per year,

which agrees quite well with the results for many industrialized nations.

However, such a model is useful only up to a certain point.
It aggregates the entire production of an economy into a single "sector".
It further assumes that only capital is required for production, and
labor does not appear in his relationship. Many industrialized nations
have shortages of labor, or of certain kinds of labor, thus making this
model unrealistic. Also, no depreciation of capital goods is allowed
for. This can be a serious omission in the case of a long-developed
economy which has considerable investment in obsolete production

facilities, or one in which technological obsolescence occurs rapidly.

The much more difficult problem of a structural analysis of

economic growth was first attacked by Wassily Leontief before World

- 38 -

War II [28]. He considered a large number of industries as separate
production sectors, and undertook the formidable job of estimating the
amount of each industry's deliveries to consumers(as final end-products)
.ggg to other industries as intermediate inputs to their own productive
processes. For instance, the amount and prOportion of the output of

the coal mining industry delivered to householdsfor final consumption
was shown separately from the amount delivered to the steel industry,
electric power industry, railroads, etc., for use in producing their
respective products.

Model construction of this type has been termed input-output
or inter-industry analysis, and in recent years has led to a considerable
sub-discipline of its own within economics. It has assumed such use-
fulness and importance for business and government purposes that the
U.S. Department of Commerce is now charged with the responsibility
of producing and up-dating just such an input-output table for the
entire U.S. economy. An early effort, for the year 1947, based on
the work of Leontief and colleagues was published in 1952, by the Bureau
of Labor Statistics [30]. Unfortunately, this version was not based
upon the same data as were the regular National Income Accounts for
the year in question, and some direct comparisons with other government
figures are not valid. The most recent work has been recently reported
on and published [31]. It consists of a full input-output table plus
supplementary data, evaluated for the year 1958 from the same data from
which the industry total production figures, to be found in the National
Income Accounts published by the U.S. Department of Commerce [32] are

computed.

-39-

A brief look at the results of Leontief's analytical endeavors
should now serve to clarify both what can and cannot be done with the
input-output matrix alone, and also what some of the underlying
assumptions implicit in its use are. Leontief allowed n different
industries to each produce a unique economic good, or output. He
traced the inputs of other goods needed to produce each output, as
well as the amount of each output which is consumed in its final
state. This latter amount was termed the final demand for a good.

He postulated that a fixed amount, yij’ of each good j would be

needed to produce one unit of output, yi. Thus, he wrote

n
where y1.- = output of good i

yij = input of good j required to produce y1 of i

yfi 2 final demand for i

As a further assumption, he considered that the quantity of
input yij demanded would be a function only of the volume of output
yj of good j; i.e. no internal scale economies are considered. Thus,

he wrote:

.. = a.. . III-6
le 13y] ( )

where

a a "technical" or'iechnological" coefficient

13

I]

which gives us the relationship

HMS

yi' =

- 40 -

or, in matrix form,

Y = AY + Yf (III-7b)

The above relationship can now be used to solve what economists
term an open static model. The term "open" implies that the model is
driven by some externally-given quantity, usually called "exogenous
variables". In this case, the variables required to be given exogenously
are the entries of the vector of final demands. The relation III-7b
can be used to solve for the production vector Y, the level of all
industries' production, which is required to satisfy some given final

demand vector, or "bill of goods", Y The solution is

f.

Y s (I - A)’1 Y (III-8)

f

There are two chief criticisms one can make of this simple
model [36,39]. Firstly the assumption that the aij entries are
constants is not valid if rapid technological change takes place,
enabling one input to be substituted for another. For example, an
extensive changeover from the use of steel to the use of aluminum for
automobile engines would cause a noticeable distortion in the present
input-output relations of the nationfis economy. Leontief studied
precisely this effect, the change of the matrix A over time, in one of
his earliest works [33]. His conclusion was that the A matrix is largely
stable over time periods shorter than a decade, during which time

significant changes are likely to occur only in a few technologically-

advancing industries.

Secondly, the model requires a forecast or estimate of final

- 41 -

demand levels, which may be difficult to make. It would be preferable
to "close" the model by making this exogenous variable a part of the
model itself, i.e. to have demand endogenously derived from some other
exogenous variable which is easier to predict, such as population.
Various efforts have been made to close the basic input-output model,
and all of them require theory in addition to the bare interéinduatry
matrix concept [34,35]. Generally speaking, concepts of a production
function, a capital accumulation function, and an investment generating
function are required. These concepts are clarified in the next section,
which will use a rigorous systems analysis formulation to construct a
closed model of economic growth. The latter two concepts will be seen
to be equivalent to the behavioral relations governing component
behavior; i.e., the component equations. The concepts of a production
function in the classic labor-capital value-added sense is a component
equation as well. knunlcutset conditions are applied in the formulation
procedure, inter-industry deliveries of intermediate inputs become a

part of the production function as well.

III-2. A National Economic System

Consider a national economy wherein production of three kinds
of goods takes place in three productive sectors. These sectors,
respectively, represent agriculture, mining and manufacturing, and ser-
vices, construction and government activities. Each sector component

is represented as a line segment in the system linear graph of Figure III-l.

-42—

Figure III-l. The Linear Graph of the United States
National Economic System

13

ll

 

- 43..

These sectors are considered to be joined together by linking components,

which are also shown as line segments in the system linear graph.

Each component is modeled in terms of an X and a Y variable,
satisfying the postulates given in chapter I. The Y variable is
considered to be a dollar-valued flow of income (or output, as will be
seen in certain cases) on a per-year basis. The X variable is taken

as the total dollar value of fixed capital investment.

The modeling equations for the production sector components

are:

Xk(n+l) = (l-dk) Xk(n) + YH§(n) (III-9a)
Yik(n) = okYk(n) (III-9b)
Yk(n) = kak(n) + AkLk(n) (111-96)
' k = 1,2,3

where xk = accumulated dollar value of fixed investment

in capital goods in production sector k; k = 1,2,3
n = time period (year)
d = depreciation rate per year of capital goods in
sector k; k = 1,2,3
Yi = dollar amount of new investment per year in
sector k; k = 1,2,3

= coefficient of investment in sector k; k = 1,2,3

Y = dollar value of annual net income of sector k;

k = 1,2,3*

 

*At this point, Y represents the net income to sector k, taken
as equal to the value added by sector k to its total output. When the
cutset relations of the system graph are considered (see below), Y be-
comes the gross income of sector k, telmm as equal to the total va~ue of

- 44-

k : capital-output ratio for investment in

sector k; k = 1,2,3

L = number of workers employed in sector k in

a year; k = 1,2,3

1 = labor-output ratio for labor employed in

sector k; k = 1,2,3

The significance of the relationship III-9a is in its
description of the growth of investment in a productive sector. The
relation states that the accumulated investment at the end of some
time period n+1 is equal to the amount of investment accumulated at
the end of the previous period n, less the amount of depreciation
suffered during the period n, plus the amount of new investment received
during the time period n (considering depreciation and new investment
to occur in lumps at the end of period n). Such a function is quite
Widely used in the literature on economic growth [35,36], and is known
as a capital accumulation function. Thus, capital accumulation is
considered to be part of the behavioral process modeled by the comr

ponent equations for a production sector.

Relation III-9b states that new investment in a sector will
be a linear function of the income of that sector. This assumption
is found in the Harrod-Domar type of rudimentary economic growth
models [27,35], in which is ok the "savings ratio" of sector k.

In the balance sheet of a company, this figure would be termed the
proportion of retained earnings. Thus, the assumption in III-9b is

twofold: firstly, that all earnings retained by industries in a

- 45 -

sector are invested entirely within that same sector; and secondly,
that a find proportion of income is retained for investment by each
sector, no other investment occurring. The work of Clark and

Harrod [26,27] suggests that a constant 0 value may be quite

k
apprOpriate for a model which is highly aggregated, as a three-sector
model is. When detailed investment patterns are considered, such as

by each of 100 industries, or say by each of 500,000 firms and private
investors, the picture is much more complex. In such a case, invest-
ment may be generated by expected rate of return, which is an extremely
difficult quantity to predict with any great degree of reliability [37].

Thus, the Harrod-Domar investment function is used here to model the

component.

The relationship III-9c is known as the production function
of a sector. The equation states that income to sector k is generated
because capital is invested and labor employed in the sector. Further,
it states that amounts are attributable to labor and captial separately.
The amount attributable to labor in a sector is taken as equal to the
amount of wages and salaries paid to workers in that sector. The
coefficient "k thus can be interpreted as the amount of output value
added per worker employed in sector k. The amount attributable to
capital invested in sector k is taken as the net sector income less
wages and salaries paid. In the accounting world, this would appear

83 initerest, dividends, and retained earnings. The coefficient kk
thuEB can be considered to represent the amount of net income per
dolléir invested in sector k - roughly, a gross rate of return to

capital invested in k.

- 46 -

Other forms of production function also occur in the economics
literature [24,26,27,29,30]. A common one is the so-called Cobb-Douglas

production function [24], which states

1: =1“)?

where A,B are constants

This function is non-linear and, if used, would introduce
serious difficulties into the solution of state equations. The production
function III-9c is referred to as the linear labor-capital function, and
was first proposed by Marx [39]. It is still used in economics today,
although the Cobb-Douglas function is more prevalent for the following

reason.

When differentiated with respect to X, the Cobb-Douglas function
yields B, which can be interpreted in economic theory as a measure of the
substitutability of capital for labor. Similarly, A is a measure of
the substitutability of labor for capital. Since any production function
can be regarded as a "recipe" for combining labor and capital, and a
wide range of allowable choices of ingredient combinations will yield
the same output, depending on technology, the degree of substitutability
of capital for labor is of great interest to social scientists studying
automation, for example. However, this feature of substitution of in-
puts is not allowed here, since it is not allowed in the Leontief input-

output framework of inuzrindustry deliveries.

The linkage components in the system graph are modeled using
Leontief's assumption of linear dependence of linkage flows upon out-

put levels of the sectors. Thus, the component models are written as

ij(n) (l + a j = 1, 2,3 (III-10a)

11’5""

j,k = 1,2,3; j¢k (III-10b)

ij(n) a kYk(n)

3

with the restriction that Y

Jk 2 o; j,k = 1’2’30

In electrical systems terminology, the linkage components are
dependent through drivers, whose value is dependent upon the value of
some other through variable. By making use of Leontief's linearity
assumption, we effectively cause subsequent models to be valid only
For the case of an

about some actual given value of the Y vector.

economy undergoing profound development, the coefficients a. will be

jk
subject to change.

The state vector for the system is selected as the vector
Applying the outset postulate at vertices a, m, and

of Xk; k = 1,2,3.

3 respectively, we see that after the interconnection links are joined

 

 

 

0r, in matrix form,

Y(n)

 

 

 

 

 

 

 

= KX(n) +-flLL(n) + AY(n)

 

 

 

 

to the sectors, the flows Yk become

r - - - - 3 ’ l r l

Y1(n) R1 0 X1(n) "1 0 L1(n)

Y2(n) = 0 k2 X2(n) 0 2 0 L2(n)

-Y3(n) O O LX3(n)‘ 0 A3 .L3(nl
all a12 a13 ‘ -Yl(""
a21 a22 a23 Y2(n) (III-11a)
a31 a32 a33.] Y3(“)

(III-11b)

- 48 -

The relation III-ll can be considered to be the set of
"complete" production functions of the sectors. They are complete in
the sense that III-ll specifies all inputs to each sector, i.e. labor,
capital, and intermediate goods, required for the total make-up of the
gross value of its output. This notion of a complete production function
is still very underdeveloped in the economics literature [36,39]. This
may be due to the fact that most economists view the value-added portion
(labor and capital) as more interesting and worthy of study, or to
the magnitude of thetask of estimating the inter-industry portion in
great detail. A private discussion by the author* with Dr. Jan Kmenta
of the Institute for Social Science Research, University of Wisconsin,
revealed that some economists regard such functions as an ultimate

ideal, not yet realized in a practical sense.
When equation III-11b is solved for Y, we obtain

Y(n) = (I-A)-1[KX(n) +/\L(n)] (III-11c)

which is needed for the derivation of the state model, shown below.

In deriving the state equation model of the system, we

start with the equations

 

 

 

 

 

 

 

 

 

 

[x1(n+l)] P(l-d1) 0 0 ' 'xl(n)] ‘61 0 0 ' 'Y1(n)‘

X2(n+l) = 0 (l-dz) 0 X2(n) + O 02 0 Y2(n)

X3(n+l)‘ L 0 0 (l-dl)‘ L.X3(n)‘ LO 0 03.. LY3(n)-
(III-12a)

 

*
At Michigan State University, Economic WorkshOp, February 19,
1965.

..49-

or, in matrix form,

X(n+l) = DX(n) + SY(n) (III-12b)

and eliminate the Y vector by substituting in equation III-11c. The

system state model is then:

X(n+l) _ DX(n) + S(I-A)-1[KX(n) +_/\_L(n)]

[D + S(I-A)-1K] X(n) + S(I—A)-1./\.L(n) (III-13)

The solution of this equation set is

X(n) = [n + S(I—A)-1K]nX(0)

“'1 -l n-1-' -l
+ 2 [D + S(I-A) K] JS(I-A) AL(j)
i=0
(III-14)
The relations III-l4 are thus a set of time paths of fixed
capital investment in the productive sectors, driven by a vector of
labor-force numbers, L(n), which must be known over time. It is
acceptable in such an aggregated model as this to assume that the
proportion of the total labor force employed in each sector is constant.
This assumption is valid over perhaps even a slightly longer time span

than the Leontief assumptions about constant interindustry delivery

proportions [32]. Thus we may write

 

DL1(n)" Pwlfi
L2(n) = w2 L(n) = WL(n) (III-15)
3‘3“) "3.

 

 

 

where w pr0portion of the total labor force employed in

sector k, k = 1,2,3

total labor force.

I."
ll

..50-

Further, we can take the total labor force to be a fixed

proportion of the total population of the country [32]; i.e., that
L(n) = 2 P(n) (III-l6)

where L

proportion of the total population in labor force

"U
[I

total population

When these relations are used in III-14, we have an expression
for the growth paths of sector investment which depends upon our knowh
ledge in addition to the parameters of the system, only of the initial
investment levels, and a time series of population. The deduction of
suitable parameters for such a model of the United States economy, as
obtained from literature which is widely available, and its solution

for the period 1947-1952 will be shown in the next section.

IIIr3. An Example: The U.S. Economy from 1947-1952

To provide a numerical example of the modeling concepts
discussed in the preceeding section, data from the United States
economy is used to provide the required parameters and to afford a
check on the results produced by the model. It was decided to choose
the period 1947-1952 for the analysis, since there was a 50-industry
input-output table showing all transactions in dollar amounts for

a
1947 [30].

 

*A similar study [31] gives an 82-industry input-output table
for the year 1958. However, the data is given only in A-matrix coefficient
form, making it impossible to determine the dollar value of the trans-
actions, which is necessary for re-aggregation of the matrix.

-51-
It 'is assumed that a five-year period would be short enough to val-

idate the previously-stated linearity of parameters assumptions,

yet would be long enough to provide an adequate test of the model.

An aggregation of the transactions between industries in

1947 is made from table 4 of [30], on the following basis:

Sector 1: Industries 1 and 2 (Agriculture)
Sector 2: Industries 3 through 29 (Mining and Manufacturing)
Sector 3: Industries 30 through 45, and 48

(Services, Construction and Government)

The rows and columns giving inventory changes, imports, and
exports are neglected in the aggregation. The following table shows the

results of this aggregation:

Table III - l

Aggregated Inter-Industry Transactions for 1947
(Amounts in millions of 1947 dollars [30])

 

 

 

 

 

Aggiculture Manufacturing, Services, etc. Final Demand
Agriculture 33,192 5,635 5,408 33,149
Manufacturing 5,127 63,381 36,662 49,689
Services, etc. 12,399 37,944 73,422 150,187
Value Added 25,428 56,327 105,470
Totals Y1: 76,146 Y2=162,287 Y3: 220,992

 

The above table yields the following Leontief-type input-

output matrix:

0.436 0.035 0.024
0.067 0.391 0.166

0.163 0.234 0.332

 

 

The entries in each column represent the ratio of the
corresponding entry in the columns of Table III-l divided by the total

at the foot of the same column in the table.

The system state model of equation III-13 requires a matrix

A which is the transpose of the above matrix. Thus, we can write

0.436 0.067 0.1631

A = 0.035 0.391 0.234

 

 

0.024 0.166 0.332
L 4

To determine the value of the 0 parameters, data on investment
in new capital equipment and structures purchased is taken from the
data in [32] for manufacturing and services, and from [42] for agri-

cultural investment. The figures are:

Investment in sector 1 3,203 (millionscf 1947 dollars)

Investment in sector 2 = 9,394

Investment in sector 3 11,218

These yield 0 values of

3,203/76,146 0.042

G
H
l

9,394/162,287 0.058

11,218/220,992 0.051

- 53 -

Values of the labor-output coefficients and the labor-force
vector are obtained by the use of the figures on the number of equivalent
full-time employees by industry, from [32]. The data for the agricultural
sector is supplemented by data from [42] on the number of family members
"employed" on farms as proprietors or workers, but who are not reported
in [32]. Their annual wage rate is estimated to be 1.33 times the
average for the employees reported, to allow for managerial and entre-
preneurial premiums. The values of k are obtained by dividing the
total sector wage bill (supplemented as described for agriculture) by
the total number of employees in the sector. The L vector is taken as

the number of reported employees and farm proprietors in the sectors.

 

 

 

 

 

Thus
'Ll‘ "10,382‘
L2 = 16,153 (thousand workers)
LLB. L28,9O3J
6 3 3 3
1 _ .13097x10 /2392x10 )2392x10 + 1.33 x 1290 x 7.285 x 10 _ 1 725
.. "' )
1 10,382 x 103
6 3
12 = 47,607x10 /l6,153x10 = 2,947
6/2 3
13 = 78,036x10 8,903x10 = 2,700

The total labor force, L, is seen to be

3

L = L + L + L = 47,448x10

l 2 3
The using the population figure
p(1947) = 143,136x103

we see that

- 54-

and we can write

 

 

 

 

PL 1 F ' 70 187]

1 w1 '
L2 = W2 2? = 0.291 0.388x143,136x103
L34 w3 £0.521‘

 

 

The nexttask is to determine the initial amounts of capital
invested by sector, i.e. the X(l947) vector. There are estimates of
the value of real net capital (structures, equipment, and inventory)

invested in manufacturing [32], which show
X2(l947) = 88.3 (billion 1947 dollars)

However, the other sectors are not explicitly covered in
standard data sources. An estimate of the total investment in agriculture
is made from data in [42], taking X1 to be the sum of value of land and

buildings on farms, value of machinery and equipment on farms, and

value of livestock owned by farmers. This yields a valuation giving
X1(l947) = 87.4 (billion.l947 dollars)

The total investment in the service sector is the most
difficult to determine. No estimate of it exists in the U.S. Government
literature cited, and none could be found elsewhere. There is also
no compatible time-series of new investment and depreciation charges
which would allow an explicit determination to be made. Thus a very

rough assumption is made that the total investment in service industries

- 55 -

is approximately the same proportion of the total investment in manu-
facturing that new construction plus new equipment expenditures in
service industries is of new construction plus new equipment expenditures
in manufacturing in 1946. This ratio is obtained from Table V-3 and
V-7, pg. 190 and 193 of [32], which uses classifications not entirely
compatible with the desired sector breakdowns, another caution to apply

to the following estimate of X2:

X2(1947) = 88.3 x 0.6 = 53.0 (billion 1947 dollars)

We can now determine the depreciation coefficients by

making use of the depreciation charged to the sectors in the National

Income Accounts [32]. These are as follows:

Depreciation charged to agriculture = 40 (million 1947 dollars)
Depreciation charged to manufacturing = 2,573

Depreciation charged to services, etc. = 2,667

whence
d1 = 40/87,400 = 0.005
d2 = 2,573/88,300 = 0.029
d3 = 2,667/50,000 = 0.053

The only remaining parameters to be determined are the capital-
output coefficients, k. These are determined by dividing the value
added attributable to capital by the total invested capital, for each
sector. The value added attributable to capital is taken to be the
total value added per sector less the amounts attributable to labor
in that sector, i.e. wages and salaries. For each sector, the wages

paid to employees are determined from [32]. To the agricultural wage

-56-

bill an amount equivalent to wages of proprietors and their families is
added, as explained previously. To the wages of the employees of
the service sector is added the remuneration received by self-employed

persons, such as professionals, which does not appear in the wage bill.
The amounts of value added attributable to capital, per

sector, are:

attributable to agriculture = 6,800 (million 1947 dollars)
attributable to manufacturing = 8,720

attributable to services, etc. = 7,500

which yields k coefficients of

k1 _ 6,800/87,400 = 0.078
k2 _ 8,720/88,300 = 0.099
k3 _ 7,500/50,000 = 0.150

The parameters determined above are all substituted into
the state equation model, III-l3, and the solution obtained for five
cycles; i.e., until 1952 figures are obtained. At each stage, the
appropriate yearly population figure is used for P. Also, in order that
all of the results can be expressed in "constant 1947 dollars", an
adjustment for dollar inflation is made to convert all values to 1947
dollars. The adjustment ratios used are the Gross National Product

Deflators series from [32].

The results are as follows:

-57-

 

 

- l - 1
X1(l952) 97.48
X2(l952) = 107.98 (billion 1947 dollars)
X3(1952)J 73.52

 

 

The complete series of results are plotted in Figure III-2,

which shows the changes in X over time.

In evaluating the results, the most soundly based comparison
of the model with reality is in the X2 variable, since [32] contains
exactly this variable, plotted as a time-series. The actual value

of X2 obtained from [32], is

X2(l952) = 104.1 (billion 1947 dollars)

The discrepancy here is 107.98 - 104.1 = 3.9 billion dollars,
or about 3.75 percent. The fact that the model prediction is on the
high side may be the reflection of the fact that 1947 tended to be a
somewhat singular year, with the immediate postwar industrial pressures
yielding higher than normal returns to capital, for example. However,
the fact that the error is not higher than it is, given that this was
the case, may well be a result of the further impetus to the economy

of the early 1950's given by the Korean war. The other values of X

l
and X2 cannot be compared to any convenient data, as previously stated,
but again a good estimate of X1 can be made from the Agricultural
Statistics [42]. The value of X1 for 1952 from 20 is

X1(1952) = 99.6 (billion 1947 dollars)

The X1 value was deflated to 1947 dollars by applying the

price-per-acre indices given in [42] to the farm land and buildings

- 5g -

 

X1, X2

(billions of | .
1947 dollars) 1
I

 

 

 

 

 

 

 

105 ’"_” """"
Manufacturing Investment, X2; //
predicted ———1 -
actual ’///
100

 

 

 

Agricultural Investment, X

l 5
if ’ _ ‘_

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

predicte ’
95 " “F44
. actual /// 7’/
90 a r
c+fihl
IF—‘=#=*—‘—4 1
+11“—- Predicted Services
85
- Investment, X3
80 ’"‘
1947 1948 1949 1950 1951 1952

YEAR

Figure III-2. Predicted and Actual Sector Investments for the
United States EconomyL 1947 - 1952

75

70

65

60

55

- 59 -

portion and the G.N.P. deflators of [30] to the remainder. The

error of the model is small in this case, being 99.6 - 97.5 = 2.1
billion low, or approximately 2 percent. This may well be due to

the underestimation of the a21 or the 02 coefficients from 1947 data.
At that early prewar stage, it is likely that farm machinery and
equipment investment was at an artificially low level, due to its short
supply so soon after the war. The 1952 total investment in farm
machinery and equipment was more than 2.5 times the 1947 total invest-

ment [42], a very rapid rise which suggests that the 02 coefficient,

in particular, may not have been constant over the period studied.

In any case, the solution results of the model confirm its
reasonable accuracy in describing the workings of the economy. Un-
doubtedly, more work in this area is needed, in particular with more
detailed models and for more recent time periods. The fact that a
situation as complex as the national economy of the United States
'can be conceived of in systems terms, the components modeled with
existing data, and that the dynamic model derived from rigorous
application of physical systems modeling procedures yields good
results when compared to actual figures, is considered to be one

of the major contributions of this thesis.

IV. CONCLUSIONS

IV-l. Regarding the Park Systems Model

There now exists a logical, consistent quantitative
framework for the analysis of spatial allocation of traffic flows,
given a facilities configuration and specified origin flows. The
requirements for component modeling make clear the sort of data which
must be gathered in order to model the attendance patterns for other
activities. Foremost, the origins of participants must be known.
Facilities of the destination areas must be carefully considered as
they pertain to the particular activity. The characteristics of the
links may be considered fixed, but the model scaling constants, will,
in general, assume new values. In all cases, the r.m.s. error criterion
provides a suitable measure for evaluating the sensitivity of the

solution to any parameter change which an analyst might wish to consider.

The existence of such a model for the Michigan state parks
system will allow a considerable amount of experimenting with possible
future configurations and parameters of the system. For instance, the
Department of Conservation might wish to estimate the demand for and
likely intensity of use of its state park camping facilities at some
future date. They could use demographically determined p0pu1ation
increases, with appropriate assumptions regarding leisure behavior,
to scale up the present day origin flow drivers, and make estimates
of the future highway link parameters based upon projected highway

improvement programs.
The solutions found under these conditions could be examined

- 60-

- 61-

for "critical" areas; i.e., those in which the projected attendances
were considerably above present capacity. They could thus select some
criterion [45] for expansion of their facilities based upon indicated
over-capacities. Presently used methods for determining expansion of
facilities usually rely upon such things as population trends alone,
and do not reflect the behavioral functioning of the system as a
collection of three quite different components, of which population

affects only one.

Future research might well apply the methods to some other
geographical region to see if the model which works so well for the
case of Michigan will also predict well in other circumstances. It
must be remembered that the bi-peninsula landform of Michigan is unique
in the United States, and this may have had some effect upon the model
accuracy and consistency. However, the singular nature of the
extensive water boundaries of Michigan can be considered to provide
a more stringent test of a model of recreational travel [44], as
opposed to a relatively homogeneous area with largely land boundaries,

such as Iowa or Indiana.

Another area for future investigation is undoubtedly the
extension of this modeling concept into other tpes of travel. The
analysis of urban journey-to—work patterns is an obvious example. The
modeling methods presented in this thesis can easily be used to solve
for the flows on the highway links, in the same way that the flows in
destination components are solved for. This would provide a one-

step solution of two of the most important classical problems in traffic

- 62 -

analysis; that of distribution, the construction by a model of the
destination flow values, given the origin flows, usually done by
means of a gravity model; and that of assignment, the allocation of
specific origin-destination flows to actual road links, usually done
by a linear programing technique. In view of the many billions of
dollars which our society expends on transportation facilities and
the increasing pressures upon urban land space, this problem area

is one of the most promising for new and advanced analytical techniques.

IV-2. Regarding the National Economic Model

It is fair to say that the model presented in this thesis is
only the first very small step of many which must be taken before a
comprehensive analytical model of the United States national economy
is perfected. Certain progress is offered to the application of systems-
type models in this area by the success of the relatively crude model

discussed.

A logical extension of the immediate model framework would
be to construct a detailed model which does not aggregate the data
(in particular, the input-output matrix) so grossly. If validated on
such a scale as a 50x50 economic model, this would surely provide a great
impetus to further behavioral studies of components and to the more

frequent gathering of basic input-output data.

A further, and most necessary step, is to include the effect

of prices in the model. On the scale of aggregation of a three-sector

- 63 -

model, prices are probably meaningless in view of the great discrep-
ancies in unit output values. However, on a truly inter-industry
basis, price changes could be considered to cause substitution of
one input for another, such as motor transport for rail transport,
aluminum for steel, capital investment for labor input, etc. Since
the possibility of substitution of inputs is not allowed by the
fixed-coefficient concept of the Leontief input-output analysis,

a new behaviorally-oriented framework of analysis may have to be

developed.

APPENDIX A

VALUES OF THE ORIGIN AND LINK COMPONENTS
OF THE MICHIGAN STATE PARK SYSTEM

A-l. The Origin Components

In connection with the Michigan Outdoor Recreation Demand
Study, a set of approximately 296,000 I.B.M. data cards were prepared,
one for each camper permit issued by Michigan State parks in 1964.
These were used for many other analyses besides the origin and destination
tabulations, such as length of stay studies, boat use by campers, type
of camping equipment used, size of camping party, etc. For input to the
park systems model, a listing was prepared showing the number of camper-
days recorded for each origin considered in the study. These origin areas
are described in the listing which follows. Counties in Michigan with
less than 10,000 p0pu1ation were combined with neighboring counties to
provide a reduction in the number of components in the system (to stay
within the bounds of computer memory space) with a small sacrifice in
accuracy. Nearby out-of-state origins were grouped as indicated in

the listing.

There is a certain error introduced by the exclusion of out-
of~state origins other than those given in the listing. The measure of
this error can be judged from the fact that total camper-days recorded
for the 55 parks considered was 3,050,000, whereas the total camper-
days for the 77 origin areas considered was 2,840,000 - a difference
of less than 6.9%. The areas not represented are:

-54-

- 65 -
North-eastern U.S.A.

Southern U.S.A.

Far West U.S.A.

Plains and Mountains U.S.A.

Iowa, Missouri, Kentucky, West Virginia
Canada, other than Ontario

Minnesota

Ontario

All of these origins with the exception of the latter two can
be considered as causing a constant error of under-prediction, since
their effect was nearly constant over the state. The latter two origins
tend to become appreciable proportions of attendance (over 3%) at only
a small number of parks in the Upper Peninsula, and are relatively con-
stant and small elsewhere. The effect of these attendances is not

included in the model results in Chapter II.

There was a consistent proportion of "unknown origin" cards
in the whole number of approximately 17%, caused by omission or in-
adequacy of information on the original camper permit. To allow for this
omission, it was postulated that the "unknown" campers were distributed
with respect to origin in the same way as "known" campers, and the figures

for each known origin were raised by 17% in each case.

The following is a description of the origin through driver com-
ponents, with their values (the element and node numbers correspond to the

numbering of Figure II-l):

Origin Area i

(country in Michigan,
if not otherwise.
specified)

1. Alcona and
Oscoda

2. Alger
3. Allegan

4. Alpena and
Montmorency

5. Antrim and
Otsego

6. Arenac
7. Baraga
8. Barry
9. Bay

10. Benzie and
Leelanau

ll. Berrien

12. Branch

13. Calhoun

14. Cass

15. Cheboygan

l6. Chippewa,
Luce and
Mackinac

l7. Clare

18. Clinton

19. Crawford and
Roscommon

20. Delta

- 66 -

Node #

(geographical
center of pop-
ulation)

001

002
003

004

005

006
007
008
009

010

011
012
013
014
016

017

018
019

020

021

Element #

(reference
point to node

i)

.401

402
403

404

405

406
407
408
409

410

411
412
413
414
416

417

418
419

420

421

Yoi

(camper-days
in 1964 cal-
endar year)

4,121

420
17,383

4,072

2,369

911
475
8,703
33,998

2,066

21,264
5,223
39,758
5,996
1,719

3,331

3,153
11,699

2,364

4,443

21.
22.

23.

24.
25.
26.

27.

28.
29.

30.

31.
32.
33.

34.

35.
36.
37.
38.
39.
40.
41.
42.
43.
44.

45.

Dickinson
Eaton

Emmet and
Charlevoix

Genesee
Gladwin
Gogebic

Grand Traverse
and Kalkaska

Gratiot
Hillsdale

Houghton and
Keweenaw

Huron
Ingham
Ionia

Iosco and
Ogemaw

Iron
Isabella
Jackson
Kalamazoo
Kent
Lapeer
Lenawee
Livingston
Macomb
Manistee

lMarquette

- 67 -
022
023

024

025
026
027

028

029
030
031

032
033
034
035

036
037
038
039
041
044
046
047
050
051

052

422
423

424

425
426
427

428

429
430

431

432
433
434
435

436
437
438
439
441
444
446
447
450
451

452

1,932
18,060

2,517

157,352
4,008
795

8,108

10,966
4,947
1,901

3,382
80,934
11,513

4,599

811
9,116
39,502
53,605
241,626
12,802
14,192
10,737
150,233
2,283

6,964

46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.

57.

58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.

70.

71.

Mason
Mecosta
Menominee
Midland
Monroe
Montcalm
Muskegon
Newaygo
Oakland
Oceana
Ontonagon

Osceola and
Lake

Ottawa
Presque Isle
Saginaw

St. Clair
St. Joseph
Sanilac
Schoolcraft
Shiawassee
Tuscola
Van Buren
Washtenaw
Wayne

Wexford and
Missaukee

Chicago Illinois and
Hammond-Gary Indiana

-68-

053

054

055

056

058

059

061

062

063

064

066

067

070

071

073

074

075

076

077

078

079

080

081

082

083

301

453
454
455
456
458
459
461
462
463
464
466

467

470
471
473
474
475
476
477
478
479
480
481
482

483

301

3,337
2,514
2,010
36,140
26,159
8,149
41,729
6,206
249,868
2,762
768

2,971

67,123
1,669
65,363
30,485
5,996
4,896
739
21,545
9,862
5,794
44,195
512,035

3,955

41,609

- 69 -

72. Rest of 301 306 149,526
Illinois

73. South Bend, 302 302 13,565
Indiana

74. Rest of Indiana 303 303 146,663

75. Toledo, Ohio 304 304 42,661

76. Rest of Ohio 305 305 228,770

77. Wisconsin 307 307 57,810

A-2. The Highway Links

The following list shows the values of estimated driving time
and estimated direct cost of gasoline plus tolls for the highway links
selected for the system study. The link numbers refer to the element

numbers on the graph of Figure II-l.

Link Number Time Direct Cost

(from Figure II-l) (estimated, in hours) (estimated, in dollars)

501 0.533 0.600
502 0.711 0.800
503 0.925 0.925
504 0.844 0.950
505 1.075 1.075
506 0.225 0.225
507 1.263 1.200
508 0.825 0.825
509 0.120 0.075
510 0.711 0.800

511

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

531

532

533

534

535

536

537

_ 70 -

1.524
0.267
1.260
0.938
0.575
0.578
0.750
0.666
0.625
0.657
0.500
1.375
0.600
0.553
1.400
1.119
1.100
0.225
0.750
0.820
1.160
0.300
0.750
1.000
0.800
0.480

0.875

1.600
0.300
1.575
1.125
0.575
1.650
0.900
0.750
0.625
0.625
0.625
1.375
0.750
0.525
1.575
1.175
1.100
0.225
0.750
1.025
1.450
0.375
0.750
0.750
0.800
0.600

1.000

538
539
540
541

542

543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562

563

- 71 -

0.889
0.200
0.700
0.067

0.120

0.400
0.417
0.300
0.356
0.250
0.167
0.600
0.571
0.450
0.533
0.889
0.100
0.560
0.428
0.778
1.750
0.500
0.800
0.143
0,700

0.450

0.750
0.250
1.050
1.200

5.150

0.300
0.625
0.375
0.400
0.250
0.250
0.900
0.500
0.450
0.600
1.000
0.075
0.700
0.375
0.875
1.750
0.500
0.900
0.125
0.700
0.450

(Mackinac
Bridge)

564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589

-72-

0.300
0.450
0.250
0.933
0.200
0.920
0.167
0.933
0.200
0.200
0.277
0.375
0.369
0.143
0.300
0.060
0.800
0.800
0.250
0.711
0.440
0.900
0.500
0.200
0.367
0.533

0.300
0.450
0.250
1.050
0.175
1.150
0.125
1.050
0.300
0.300
0.450
0.375
0.600
0.125
0.375
0.050
0.900
0.900
0.250
0.800
0.550
1.125
0.625
0.200
0.550
0.800

590

591

595

593

594

595

596

597

598

599

600

601

602

603

604

605

606

607

608

609

610

611

612

613

614

615

73 -

0.183
0.940
0.600
0.325
0.550
0.800
0.400
0.250
0.600
0.809
0.250
0.620
0.250
0.300
0.320
0.175
0.600
0.333
0.333
0.333
0.720
0.333
0.850
0.327
0.450

0.383

0.275
1.175
0.675
0.325
0.550
0.900
0.450
0.250
0.600
0.850
0.250
0.775
0.250
0.375
0.400
0.175
0.750
0.375
0.375
0.375
0.900
0.375
1.275
0.450
0.675

0.575

616

617

618

619

620

621

622

623

624

625

626

627

628

629

630

631

632

633

634

635

636

637

638

639

640

641

642

-74..

0.750

0.500

0.460

0.250

0.222

0.488

0.333

0.416

0.667

0.644

0.538

0.578

0.300

0.650

0.500

0.400

0.633

0.400

0.581

0.880

0.750
0.500
0.575
0.375
0.250
0.550
0.500
0.625
0.750
0.725
0.875
0.650
0.300
0.650
0.625
0.550
0.525
0.825
0.550
0.800
0.875
0.750
0.500
0.950
0.500
0.800

1.100

643
644
645
646
647
648
649
650
651
652
653
654
V 655
656
657
658
659
660
661
662
663
664
665
666
667
668
669

670

- 75 -

0.333
0.217
0.755
0.267
0.400
1.125
0.200
0.444
0.778
0.952
0.250
1.000
1.444
0.500
0.636
0.440
0.489
0.450
0.517
0.625
0.300
0.375
0.330
0.625
0.857
0.375
0.675

0.350

0.500
0.325
0.850
0.300
0.500
1.125
0.200
0.500
0.875
1.000
0.250
1.125
1.625
0.500
0.875
0.550
0.550
0.675
0.775
0.625
0.300
0.375
0.500
0.625
0.900
0.375
0.675

0.525

671

672

673

674

675

676

677

678

679

680

681

682

683

684

685

686

687

688

689

690

691

692

693

694

695

696

697

-76-

0.375
0.200
0.375
0.400
0.109
0.100
0.240
0.133
0.433
0.333
0.291
0.440
0.222
0.625
0.727
0.750
0.675
0.580
0.625
0.667
0.133
0.500
0.375
0.260
0.667
0.720

0.760

0.375
0.200
0.375
0.600
0.150
0.150
0.300
0.200
0.625
0.500
0.400
0.550
0.250
0.625
1.000
0.750
0.675
0.500
0.625
0.750
0.200
0.500
0.375
0.325
0.750
0.900

0.950

698

699

801

802

803

804

805

806

807

808

809

- 77 _

0.622
0.428
3.000
0.400
0.818
3.777
1.000
1.778
1.454
1.833

4.889

0.700
0.375
3.750
0.500
1.125
4.250
1.125
2.000
3.350
2.750

5.550

APPENDIX B

RESULTS OF THE FACTOR ANALYSIS AND
ATTRACTION INDEX CONSTRUCTION FOR THE PARKS

The descriptions of the 38 variables representing facilities
and attributes of each of the 55 parks, along with their mean values

(with 55 observations of each) were as follows:

Variable

Mean Score

1. Rolling terrain .5636
2. Mountainous or hilly terrain .4545
3. Evergreen vegetation .0909
4. Deciduous vegetation .4364
5. Mixed vegetation .4182
6. Barren terrain .0727
7. Virgin timber stand .0364
8. Wilderness areas .0727
9. Shade for over 50% of campsites .7636
10. Cliffs and overlooks .2545
11 Waterfalls .0909
12. Springs .1636
13. Great lakes shoreline .5455
14. Inland lake shoreline .6182
15. River frontage .4365
16. Historical site .1636

-78-

- 79

17. Contemporary interest site .0909
18. Interpretive program .4727
19. Hiking trails .4545
20. Swimming, great lakes .4545
21. Swimming, inland lakes .4545
22. Sand beach, great lakes .3818
23. Sand beach, inland lakes .4364
24. Boat launching facility .7636
25. Water skiing area .4545
26. Fishing .8000
27. Store within 1 mile .4545
28. Showers .7273
29. Flush toilets .8364
30. Laundry facilities .7091
31. Electricity service .8545
32. Pier .0727
33. Boat rental within 1 mile .2727
34. Horse rental within 1 mile .0364
35. Bathhouse .5273
36. Lifeguard .5091
37. Sports playground area .6909
38. Sand dunes .0909

The eigenvalues of the matrix of intercorrelations observed
between these 38 variables in the 55 cases were as follows (listed in

order of size):

- go _

7.5276, 5.0503, 3.8350, 2.4281,

2.3485, 1.9994, 1.8394, 1.3594,

1.2372, 1.1232, 1.0836, 0.9453,

0.8689, 0.8479, 0.6842, 0.6676,

0.5462, 0.4774, 0.4216, 0.3484,

0.3213, 0.3179, 0.2957, 0.2253,

0.2136, 0.1932, 0.1509, 0.1459,

0.1127, 0.0979, 0.0755, 0.0666,

0.0483, 0.0320 0.0279, 0.0205,

0.0085, 0.0069

The factor analysis solution containing three varimax
rotated factors was accepted for computation of the park attraction
indices, Ak' The rotated factor loadings are given below for the 38
variables. The proportions of variance of the observed correlation
matrix, R, were for factor 1, .1768; for factor 2, .1319; for factor 3,
.1232; making a total of .4319, which is considered to be reasonably
high for studies of this type with such diversified data observations*.

The underlined values were the loadings used in the computation of the

 

 

 

indices:
Variable Factor 1 Loading Factor 2 Loading Factor 3 Loading

1. Rolling terrain -.2662 .3540 -.1566
2. Mountainous terrain -.1819 .3728 -.l497
3. Evergreens .1738 .1156 .2147
4. Deciduous -.2960 -.l688 -.3831
5. Mixed vegetation .0767 .2841 .4108
6. Barren .2942 -.3591 .1582

 

*Private discussion of the author with Mr. Thomas Danbury,
Department of Communications, Michigan State University, Dec. 4, 1964

10.

ll.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

Virgin timber
Wilderness

Shaded camps
Cliffs, overlooks
Waterfalls

Springs

Great lakes shoreline
Inland lake shoreline
River frontage
Historical site
Contemporary interest
Interpretive program
Hiking trails

Swimming, great lake

. Swimming, inland lake

Beach, great lake
BeaCh, inland lake

Boat launching

Water skiing
Fishing
Store

Showers
Flush toilets
Laundry
Electricity

Pier

.1228

.2538

.1354

.0688

.1031

.2947

.0253

.1258

.1852

.1203

.1522

.0669

.0454

.1186

.0843

.2416

-.0345

-.1362
-.0340

.4728

.1380
.1987
.0150
-.2354
-.0976
-.2272
-.0901

-2741

.0744

.1602

.1818

-.0184

.3183

.1601

.1517
.0711
.1644
.0378
.0592
.1077
.2146

.2572

.1762

.3259
.1934

.1293

33. Boat rental -.§21§ .0310 .2510
34. Horse rental -.l933 .0870 -.3371
35. Bath house -.3938 -.3351 .3021
36. Lifeguard -2§§62 -.3148 .2712
37. Sports grounds -.l365 -.2709 .2295
38. Sand dunes .0237 .2145 .2091

The four-factor solution was not as satisfactory. The
fourth factor was loaded highly only by the variables for mixed
vegetation, sand dunes, and rolling terrain; a set which amounts to
perhaps a "mild scenery" factor. However, the increase in total
variance of R represented by the four factors was only .0639 over
the three-factor solution. Thus, the fourth factor contributes
barely half as much to the "explanation" of the variances as did the

third.

The following are the normalized factor scores of the parks,
computed from relations II-4a, II-4b, II-4c, and II-4d. The attraction
indices computed using relation II-3b are also given. In each column
a value of unity represents an "average" park, in the sense that unity

is the score computed using the mean values of the variables for the

 

 

55 parks.

Park Ql/Ql Q2/Q2 Q3,63 f:_
l. Baraga .796 .483 1.206 0.48
2. Brimley .796 .643 1.206 1.03
3. Fayette .796 1.535 0.0 0.14

4. Fort Wilkins .828 2.537 1.272 0.56

10.
ll.
12.
l3.
14.
15.

l6.

17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.

30.

Gogebic Lake
Indian Lake
McLain
Muskallonge Lake
Porcupine Mountains
Straits
Tahquamenon Falls
Van Riper

Wells

Aloha

Bay City

Burt Lake

D.H. Day

East Tawas
Harrisville
Hartwick Pines
Higgins Lake
Hoefft
Interlochen
Ludington
Mears
Mitchell
Muskegon
Onaway
Orchard Beach

Otsego Lake

-83-

1.250

1.106

.668

1.104

.817

.608

..940

1.398

.796

1.254

1.005

1.398

.668

1.088

.647

.359

1.398

.796

1.398

2.045

.944

1.254

1.348

.853

.647

1.348

.834

1.658

0.0

.662

3.362

1.041

2.569

1.642

1.658

.285

1.007

.965

.197

0.0

.475

2.460

.285

1.304

.475

2.380

0.0

.285

2.441

1.012

..190

.285

1.206

1.396

.719

0.0

.530

1.272

1.272

.843

1.396

1.083

1.396

1.396

0.0

1.206

1.083

1.272

1.396

1.206

1.396

1.272

1.396

1.206

1.396

1.206

1.083

1.396

1.16
1.18
0.30
0.54
0.82
0.74
1.65
1.54
0.95
1.75
1.90

1.70

0.25
0.97
0.96
0.25
3.91
0.67
3.35
3.04
0.47
1.08
1.82
0.50
0.72

1.26

31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.

55.

Silver Lake
Traverse City
Wilderness
Wilson
Young
Benzie
Gladwin
White Cloud
Algonac
Brighton
Grand Haven
Hayes
Highland
Holland
Holly

Island Lake
Lakeport
Metamora
Pinckney
Port Crescent
Proud Lake
Sleeper
Warren Dunes
Waterloo

Yankee Springs

1.631
1.032
.796
1.018
1.162
.382
.149
.149
.382
1.348
.944
1.348
1.106
1.015
1.254
1.348
.795
.869
1.348
.647
1.106
1.149
1.005
1.254

1.348

84-

.285
.285
2.392
0.0
.475
.936
.293
.490
.936
1.495
.190
1.041
1.399
0.0
1.495
1.519
.285
.475
1.692
.578
1.692
.727
.858
1.692

1.012

1.083
1.083
1.083
1.396
1.396
0.0
1.083
1.083
.530
.313
1.396
1.396
.109
1.396
.124
.124
1.272
1.083
.763
1.083
0.0
1.396
1.396
.530

.932

1.37
1.67
1.35
0.81
0.79
0.24
0.17
0.13
0.63
1.24
1.27
1.47
0.18
2.03
0.99
0.55
1.21
0.70
2.01
0.78
0.57
1.87
0.85
1.72

1.60

APPENDIX C

PROGRAM PRKSYS

The computer program given at the end of this appendix was
used to construct the branch equation matrix for the system of parks
and then to solve for the park through variables. In addition to

the program, the following data cards were used:

k

1 card, constants k k R3, 4

1’ 2’

55 cards, park attraction indices
208 cards, time and cost values for highway links

145 cards, a list of passive elements in each fundamental cutset,
with signs; first entry number of elements in cutset

for control of DO loops.
145 cards, a list of origin camper-day entries in Y vector

55 cards, a list of 1964 park attendances in camper-days

The routine to construct the G matrix was prepared with the
assistance of Mr. C. Hart. At first, a simple Gauss-Seidel method [43]
was used to solve for G"1 Y. This was found to introduce considerable
numerical error in the last 35 parks, and a more accurate matrix
inversion routine SUBROUTINE MATINV, was adopted from the Computer
Center Library. This routine was more accurate since it searched in

each case for the maximum diagonal entry as pivot element, and trans-

posed rows and columns accordingly.

Two measures of numerical accuracy were used, both rule-

- 35 -

-86..

of-thumb, since it was not feasible to compute GG-1 and compare it
to the unit matrix of order 145 due to the size of the matrices
involved. The first check was the size of the determinant of G.

This was typically in the order of 1060 for the constants chosen, a
large value which precludes the likelihood of calling for a division
by a small number. The second check was the application of the cut-
set postulate at the reference point of the graph. Here, all origins
"leave" and all parks "enter", and thus the attendances predicted
should equal the total of the origins entered. The total origins
entered was 2,863,227. The total attendance predicted varied between
2,806,000 and 2,820,000 approximately, giving a possible error of
from 1.5 to 2.0 percent. While it is not possible to state cons
clusively that the numerical error of any particular entry of the
inverse or in any particular attendance computation is this low, it

seems reasonable to expect any such error to be within, say, 3 to

4 percent.

The arrays and routines called in the program could not be
accommodated in the 32,768 word memory of the CDC 3600 computer in the
presence of either the SCOPE control unit programs or the normalfull
FORTRAN compiler. Thus, at the suggestion of Mr. P. Bintner of the Com-
puter Center Staff, the FORTRAN compilation was called by program cards
(*0 job card) to be done to tape and then from tape to core storage.
This enabled the use of more of the memory for the PRKSYS arrays and
routines than is normally available with an * job card, which compiles
FORTBIN to core. Average running time on the CDC 3600 computer was

five minutes and forty-eight seconds.

- 87-

*C550098 ELLIS
FORTBIN,C,L,4,P,P,P.
PROGRAM PRKSYS
DIMENSION G (145,146),KUTSET(145,l6),E(263),CONS(4),ATT(55),DIF(55)
1,PCT(55)
COMMON G,KUTSET
1000 FORMAT (11X,F4.l,2X,F3.2)
1001 FORMAT(15x,2F4.3)
1003 FORMAT(11x,1614)
1004 F0RMAT(15x,F8.0)
2000 FORMAT(6H PARK,12,14H ATTRACTION = ,F7.4,14H ATTENDANCE = ,F10.O,
110H ERROR IS ,F10.0,11H PCT OUT = ,F5.O/)
1005 F0RMAT(15x,4F5.O)
READ 1005, (CONS(I) ,I=1,4)
PRINT 3000,(CONS(I),I=1,4)
3000 F0RMAT(15H CONSTANTS ARE ,4F7.3///)
9001 FORMAT(15H1ERROR 0N CARD ,I3,27H OF CUTSET. ELEMENT INDEX =,12,
1 9H IS ZERO.///)
9002 FORMAT(16H1ERROR ON CARDS ,I3,5H AND ,I3,31H OF CUTSET. SIGNS DO N
lOT AGREE.///)
C READ MATRIX E FROM CARDS
Do 10 I = 1,55
READ 1000,ATTR,CSR
10 E(I)=CONS(1)*ATTR*CSR
PRINT 3001,(I,E(I),I=1,55)
3001 FORMAT (20H ATTRACTION OF PARK ,I2,4H IS ,F7.4/)
Do 11 I = 56,263
READ 1001, T, C
11 E(I) = CONS(2)/((T+CONS(3)*C)**CONS(4))
PRINT 4000,(I,E(I),I=56,263)
4000 FORMAT(6H LINK ,I3,10H VALUE IS ,F7.4/)

c READ IN CUTSETS
READ 1003,((KUTSET(I,J),J=1,16),I=1,145)

c BUILD DIAGONAL ELEMENT OF MATRIX G FROM ARRAY E
Do 40 I = 1,145
Do 20 J = 1,145

20 G(I,J) = O.
NI: KUTSET(I,1)+1
D0 23 J = 2, N1
M = KUTSET(I,J)
IF(M)21,991,22

21 G(I,I) = G(I,I)+E(-M)
GO TO 23

22 G(I,I) = G(I,I) + E(M)

23 CONTINUE

C BUILT OFF-DIAGDNAL ELEMENTS OF MATRIX - UPPER TRIANGLE ONLY

JJ = I+1
D0 40 J = JJ, 145
N2 = KUTSET(J,1)+1

sz = 0
DO 40 K = 2, N1
D0 40 L = 2, N2

IF(KUTSET(I,K)-KUTSET(J,L))30,32,30

30
32

33
34

35
36
37
38

39
40

41

2001

70

6000

60

5000

7000

2002

991

992
999

-88-

IF(KUTSET(I,K)+KUTSET(J,L))40,32,40
KPRODl = KUTSET(I,K)*KUTSET(J,L)
IF(KSW)33,34,33
IF(KPROD1*KPROD)992,992,35
sz = 1
KPRODzKPRODl
M = KUTSET(I,K)
IF(M)36,37,37
M = -M
IF(KPROD)38,39,39
G(I,J) = G(I,J) - E(M)
GO TO 40
G(I,J) = G(I,J)+E(M)
CONTINUE

FILL IN LOWER TRIANGLE OF MATRIX G
DO 41 I = 1, 145
N1: I+1
DO 41 J = N1,145
G(J,I) = G(I,J)
PRINT 2001,((G(I,J),J=1,5),I=1,5)
FORMAT (5(5E17.10/))
READ 1004, (G(I,146),I=l,145)
CALL MATINV (G,145,G(1,146),1,DETERM)
PRINT 2001,((G(I,J),J=1,5),I=1,5)

PREMULTIPLY BY MATRIX D

SUM = 0.
DISC = 0.
PCTO = 0.
D0 70 I=1,55
DIF(I)=0.
D0 60 I = 1,55
READ 6000, ATT(I)
FORMAT(15X,F8.0)
G(I,146)=G(I,146)*E(I)
SUM = SUM+G(I,146)
DIF(I)=G(I,146)-ATT(I)
PCT(I) = 100.*DIF(I)/ATT(I)
DISC = DISC + DIF(I)
PCTO = PCTO + PCT(I)
PRINT 2000, I , E(I), G(I,146), DIF(I) ,PCT(I)
PRINT 5000, SUM , DISC
FORMAT(7H SUM = ,2(E20.10)///)
PRINT 7000, PCTO
FORMAT(8H PCTO = ,F8.0//)
PRINT 2002,DETERM
FORMAT (12HODETERMINANT,,E20.10)
GO TO 999

ERROR ROUTINES
PRINT 9001, I, J
T0 T0 999
PRINT 9002, I, J
CONTINUE
END

-89...

SUBROUTINE MATINV(A,N,B,M,DETERM)
DIMENSION A(145,145),B(145,1),IPIVOT(145),INDEX(145,2),PIV0T(145)
INITIALIZATION
10 DETERM=1.0
15 DO 20 J=1,N
20 IPIVOT(J)=0
30 D0 550 I=1,N
SEARCH FOR PIVOT ELEMENT
4O AMAX=0.0
45 DO 105 J=1,N
50 IF (IPIVOT(J)-l) 60, 105, 60
60 D0 100 K=1,N
70 IF (IPIVOT(K)-l) 80, 100, 740
80 IF (ABSF(AMAX)-ABSF(A(J,K))) 85, 100, 100
85 IROW=J
90 ICOLUM=K
95 AMAX=A(J,K)
100 CONTINUE
105 CONTINUE
110 IPIVOT(ICOLUM)=IPIVOT(ICOLUM)+1
INTERCHANGE ROWS TO PUT PIVOT ELEMENT ON DIAGONAL
130 IF (IROW-ICOLUM) 140, 260, 140
140 DETERM=-DETERM
150 Do 200 L=1,N
160 SWAP=A(IROW,L)
170 A(IR0w,L)=A(IGOLUM,L)
200 A(ICOLUM,L)=SWAP
205 IF(M) 260, 260, 210
210 D0 250 L=1, M
220 SWAP=B(IROW,L)
230 B(IROW,L)=B(ICOLUM,L)
250 B(ICOLUM,L)=SWAP
260 INDEX(I,1)=IROW
270 INDEX(I,2)=ICOLUM
310 PIVOT(I)=A(ICOLUM,ICOLUM)
320 DETERM=DETERM*PIVOT(I)
DIVIDE PIVOT ROW BY PIVOT ELEMENT
330 A(ICOLUM,ICOLUM)=1.0
340 D0 350 L=1,N
350 A(ICOLUM,L)=A(ICOLUM,L)/PIVOT(I)
355 IF(M) 380, 380, 360
360 D0 370 L=1,M
370 B(IGOLUM,L)=8(ICOLUM,L)/PIVOT(I)
REDUCE NON-PIVOT ROWS
380 D0 550 L1=1,N
390 IF(LI-ICOLUM) 400, 550, 400
400 T=A(L1,ICOLUM)
420 A(L1,ICOLUM)=0.0
430 Do 450 L=1,N
450 A(Ll,L)=A(L1,L)-A(ICOLUM,L)*T
455 IF(M) 550, 550, 460
460‘DO 500 L=1,M
500iB(Ll,L)=B(L1,L)-B(ICOLUM,L)*T

550

600
610
620
630
640
650
660
670
700
705
710
740
750

CONTINUE

INTERCHANGE COLUMNS
D0 710 I=1,N

L=N+l-I

IF (INDEX(L,l)-INDEX(L,2)) 630, 710, 630
JROW=INDEX(L,1)
JCOLUM=INDEX(L,2)

DO 705 K=1,N
SWAP=A(K,JROW)
A(K,JROW)=A(K,JCOLUM)
A(K,JCOLUM)=SWAP
CONTINUE

CONTINUE

RETURN

END

END

(BLANK)

CALL,4.

RUN.

10.

11.

12.

13.

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