THE APPLICATION OF
LINEAR ECONOMIC MODELS TO MARKETING
JIM JOSEPH TOZZI
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
To the members of my committee, Professors Ralph B. Thompson,
Chairman; Willard 0. Ash, Ralph H. Blodgett, C. Arnold .Matthewp,and
Lowell C. Yoder, I wish to express my sincere appreciation for their
interested efforts in my behalf. I wish to express my special indebt-
edness to the Chairman of my committee, Professor Ralph B. Thompson,
who has been unsparing of his time and energy and who has been hard
upon the weaknesses of this dissertation.
Appreciation is also expressed to the following: Professor
Carter C. Osterbind, Director of the Bureau of Economic and Business
Research, for initiating a study which formed the basis for this
dissertation and for providing the resources needed for its completion;
Professor John N. Webb for valuable suggestions; Professor Henry Mine
for his comments on certain mathematical topics; Professor R. G. Self-
ridge and the University of Florida Computing Center for their help in
inverting the interindustry matrices; Professor John E. Maxfield, Head,
Department of Mathematics, for providing teaching assignments which
served as an invaluable background for the preparation of the manuscript;
and to Senator George Smathers for making the necessary arrangements
with the Second United States Army so that I could complete this disser-
tation prior to reporting to active duty.
A special note of thanks is due to Miss Dorothy G. Rae for
proofreading the manuscript and to Mrs. Philamena Pearl for typing it.
TABLE OF CONTENTS
ACKNOWLEDGMENTS . . . . . . . . ... . . . ii
LIST OF TABLES . . . . . . . . ... . . . viii
LIST OF FIGURES . . . . . . . . ... . . . x
INTRODUCTION. . . . . . . . . ... . . . 1
1. THEORETICAL CONSIDERATIONS IN THE DEVELOPMENT OF
MARKETING MODELS . . . . . . . . . 5
1.1 The Need for Mathematical Models in Marketing .. 5
The Increase in Marketing Data
Marketing, A Maturing Discipline
The Advantages in Using Models in Marketing
1.2 The Concept of Mathematical Marketing Models. . 9
The Role of Model Building in Marketing Analysis
Mathematical Models in the Natural Sciences
A Natural Science Model
Assumptions of the Ideal Gas Model
Analogies Between the Natural Science Models
and Marketing Models
Basic Methodological Differences Between the
Construction of Natural Science and Marketing
Mathematical Models in Marketing
The Classification of Marketing Models
Abstract or Physical
Conceptual, Arithmetical, or Axiomatic
Deterministic or Stochastic
Static or Dynamic
Linear or Nonlinear
Stable or Unstable
The Components of Axiomatic Marketing Models
Rules of Inference
Theorems and Lemmas
Necessary Steps in the Construction of Marketing
Technological, Definitional, and Behavioral
Systematic and Random Variables
Endogenous and Exogenous Variables
Outline for Subsequent Chapters
2. THE THEORY OF INPUT-OUTPUT MODELS . . . . .. 34
2.1 Historical Note. . . . . . . . .. 34
2.2 Leontief's Model: A Special Case of the Walrasian
Model .................. 38
Walras's System of Equations
Deficiencies in the Walrasian System
Linear Homogeneous Production Functions
A Static Equilibrium Model
The Existence of a Unique Solution for
the Walrasian System
Stability of the Walrasian Model
A Purely Competitive System
The Significance of Walras's Contribuitions
Marginal UtiliLy functions
The Derivation of Leontief's Model from the
2.3 The Basic Model. . . . . . . . ... 53
A Mathematical Statement of the Model
3. THE PROCEDURE FOR THE CONSTRUCTION OF AN INPUT-OUTPUT
TABLE . . . . . . . . . . .. 65
3.1 Introduction . . . . . . . . . 65
Exclusive Use of Secondary Data Sources
Limitations on the Use of Secondary Data Sources
The Advantages in Using Secondary Data Sources
The Purpose of a Numerical Example
3.2 The Construction of the Input-Output Table . 71
Selection of Industries
Estimation of Gross Outputs
A Comparison of Three Methods
Estimating Endogenous Flows
Estimation of Final Demand
3.3 The Conversion of Total Output to Employment and
Income Data. . . . . . . . ... 80
4. EMPIRICAL RESULTS . . . . . . . . ... .83
4.1 Basic Data for the Construction of an Input-Output
Table. . . . . . . . .... 83
Major Industries in Duval County, Florida
Gross Outputs of the Major Industries
Estimation of Endogenous Flows
Example of the Calculations of Endogenous
Estimation of Final Demand
A Numerical Example for the Duval County
4.2 The Input-Output Table . . .. . . . 88
The Row Entries
The Column Entries
4.3 Interdependence Coefficients . . . .. 92
4.4 Supplementary Transformation Functions . . 95
5. MARKETING APPLICATIONS OF INPUT-OUTPUT ANALYSIS . .. 98
5.1 Introduction . . . . . . . .... .. 98
5.2 Modifications of the Basic Model . . ... 99
Forecasting Levels of Output, Employment, and
The Effect of Governmental Spending on Output,
Employment, and Income
Projected Output Levels for a Given Schedule of
The Multiplier and its Effects on Output,
Employment, and Wages
An Aggregate Model Constructed from the Basic
The Import Problem
The Extension of the Basic Model to the Wholesale,
Retail, and Service Sectors
6. MARKETING INTERPRETATIONS OF THE BASIC MODEL AND ITS
MODIFICATIONS . . . . . . . .... . 138
6.1 Marketing Uses of Interindustry Data . . . 140
7. THE LIMITATIONS OF INTERINDUSTRY ANALYSIS ...... 145
7.1 A General Statement of the Assumptions .... 146
7.2 Model Error. . . . . . . . . ... 147
The Constancy of Activity Coefficients
Calculation of the Activity Coefficients
Direct Input Estimates
Successive Linear Approximations
Two-Point Estimates: Nonhomogeneous
The Problem of Aggregation
Some General Principles of Aggregation
Perfect Proportion in the Product Mix
Similarity of Production Functions
Minimal Distance Criterion
The Acceptability of Aggregation
7.3 Statistical Error. . . . . . . ... 164
Lack of Experimental Design
Faulty Design of Questionnaires
The Non-Reproducibility of Economic Data
7.4 Computational Error. . . . . . . .. 173
7.5 The Effect of Model, Statistical, and Computational
Errors on Interindustry Projections. ... . 180
Standard Errors of Prediction for Various
Methods of Output Projection
The Variation of Activity Coefficients
Between 1919 and 1939
The BLS Test, 1929 to 1937
Barnett's Test 1950 Full Employment Projection
Ghosh's Use of Nonhomogeneous Production Functions
8. CONCLUSION: THE USE OF INTERINDUSTRY MODELS AS A STATEMENT
OF FUNCTIONALISM. ...... . . . . . ... 189
8.1 Limitations on Functionalism Resulting from the
Use of Interindustry Models. . . . ... 190
Constancy of Interregional Distribution Systems
The Absence of Non-Economic Behavioral Equations
Modification of Economic Models
The Construction of Isomorphic Models
The Limited Use of Natural Science Models
APPENDIX. . . . ... .. . ....... . . . 195
BIBLIOGRAPHY. . . . .. . . ........ . ... 217
LIST OF TABLES
4.1-1 Major Industries in Duval County, Florida . . .. 83
4.1-2 Gross Outputs of the Major Industries . . .. 84
4.1-3 Activity Coefficients . . . . . . ... 86
4.2-1 Interindustry Flows (Dollars) . . . . ... 90
4.3-1 Interdependence Coefficients. . . . . ... .. 93
4.4-1 Output per Worker by Industry for Duval County. . 96
4.4-2 Average Income by Industry for Duval County ..... 97
5.2-1 Current Levels of Final Demand, Output, and Employment
by Industry Type. . . . . . . . ... 100
5.2-2 Hypothetical Final Demand for the Year 19xx by
Industry Type . . . . . . . . . 101
5.2-3 Output, Employment, and Income Based on 19xx
Final Demand. . . . . . . . . ... 102
5.2-4 Hypothetical Governmental Expenditures by
Industry Type . . . . . . . . . .104
5.2-5 The Government's Contribution to Output, Employment,
and Income. . . . . . . . .... . 105
5.2-6 Estimated Levels of Consumer Expenditures and
Their Corresponding Output by Industry Type in
Duval County. . . . . . . . . ... 108
5.2-7 Output Multipliers for the Basic Model. ..... . 111
5.2-8 Change in Output Induced by a Ten Percent Change
in Final Demand . . . . . . . ... 112
5.2-9 Change in Employment Induced by a Ten Percent Change
in Final Demand . . . . . . . ... 117
5.2-10 Change in Income Induced by a Ten Percent Change in
Final Demand. . . . . . . . . ... 121
5.2-11 Interindustry Flows (Dollars) . . . . ... 124
5.2-12 Activity Coefficients. . . . . . . ... 126
5.2-13 Aggregated Interindustry Flows (Dollars) . . .. 130
5.2-14 Distribution Coefficients. . . . . . .. 131
5.2-15 Activity Coefficients. . . . . . . ... 132
5.2-16 Interdependence Coefficients . . . . . . 132
5.2-17 Imports Generated by the Output of the
Endogenous Industries. . . . . . . ... 134
7.5-1 A Comparison of the Actual and Indirect Demands
for 1929 and of Estimates Based on the Structure
of 1939 (Millions of Dollars, 1939 Prices) . . 183
7.5-2 Standard Errors of Prediction of Thirteen Industry
Outputs in 1919 and 1929 from 1939 Data (Millions
of Dollars). . . . . . . . . ... 184
7.5-3 Statistical Characteristics of Changes in Activity
Coefficients . . . . . . . . ... 185
7.5-4 Average Errors of Alternative Projection 1929-1937
(Billions of Dollars) . . . . . . . . 186
7.5-5 Average Errors of Alternative Projections, 1950
(Billions of Dollars) . . . . . .... .187
7.5-6 Output Estimates for 1939 and 1947 (Millions of
Dollars, 1929) . . . . . . . ... .188
LIST OF FIGURES
1.2-1 A Schematic Classification of Marketing Models. . 22
1.2-2 Model Building. .................. 28
2.3-1 A Schematic Arrangement of a Transaction. Table . 57
The conception of the term "marketing" is usually attributed
to Ralph Starr Butler, who, in 1910, published six printed pamphlets
titled Marketing Methods which were later published as a textbook
titled Marketing. Butler states, "In brief, the subject matter that
I intended to treat was to include a study of everything that the
promoter of a product has to do prior to his actual use of salesmen
and of advertising." In order to examine the above subject matter,
marketing scholars have developed the institutional, functional, and
commodity approaches for the formulation of marketing principles.
The application of these approaches to marketing problems has
generated a wealth of information. However, the results of these
approaches have not been integrated into any logical frame of reference
so as to form the foundations for the statement of marketing principles.
In the last decade, considerable research has been performed in an
attempt to accomplish such an integration. The domain of marketing
thought is no longer confined to Butler's interest in the promotion
of products, but has expanded to a multi-dimensional discipline.
To date, the major part of the research done in the field of
marketing belongs to the structural dimension, i.e., an examination
of the basic components of the marketing system. The commodity,
R. Bartels, The Development of Marketing Thought (Homewood,
Ill.: R. D. Irwin, 1962), p. 225.
institutional, and functional approaches to marketing provide an
analysis of the structural dimension of marketing. However, such an
analysis does not consider the influence of the temporal dimension on
marketing phenomena. In addition to neglecting the temporal dimension,
traditional market analysis also fails to emphasize the effects of
geographical location, the spatial dimension, in its explanation of
the market mechanism. Until recent years the advances made in the
other social sciences were not used to any large extent in the field
of marketing. A combination of this interdisciplinary dimension of
marketing with the previous dimensions will provide a basis for the
development of marketing principles, the intellectual dimension of
In recent years, many marketing scholars have attempted to
synthesize this multi-dimensional approach to marketing so as to
provide a conceptual frame of reference for the solution of marketing
problems. A leading exponent of this approach is Professor W.
Alderson whose functionalism is an integration of the commodity,
functional, and institutional approaches.2 In addition to integrating
these traditional approaches, Alderson also increases their scope.
He emphasizes the structural relationships between marketing units and
insists that such relationships are dynamic in contrast to the tradi-
tional static approaches to marketing.
Functionalism is a recent innovation in marketing theory,
consequently, it is not stated in a rigorous fashion. Although
2W. Alderson, Marketing Behavior and Executive Action (Homewood,
Ill.: R. D. Irwin, 1957).
Alderson's functionalism has not been defined by a mathematical model,
conceptually, it has emphasized the structural interdependence of
marketing activities. However, in order that functionalism may be
used in the solution of marketing problems, it must be stated in
quantitative terms. As one critic states, "In sum, the challenge
facing the student of marketing is that of operationalizing the concep-
tual schemata suggested by functionalisi:i--that is, of translating them
into empirical research instruments."3
The object of this thesis is to use a mathematical model,
which exemplifies the concept of functionalism,for an examination of
current marketing problems. The model to be used is an economic mrdel
developed by Professor W. Leontief. The model, which is usually
called an input-output model, was chosen for a variety of reasons.
First, the traditional approaches to marketing may be used for collect-
ing and organizing in a logical and consistent manner the enormous
amount of data required for the use of an input-output model.
Second, Alderson's functionalism emphasizes the structural
relations between marketing units. The interdependency of firms is
inherent in an input-output model. Another property of functionalism
is that it offers a dynamic interpretation of marketing activity as
opposed to the static theory represented by the institutional, com-
modity, and functional approaches to marketing. Given specific
conditions, the Leontief model can be used for a dynamic analysis of
F. M. Nicosia,"Marketing and Alderson's Functionalism,"
Journal of Business, XXV (1962), 412.
marketing activities. The remainder of this thesis involves the
application of this linear economic model to the functionalist
approach in marketing.
THEORETICAL CONSIDERATIONS IN THE DEVELOPMENT
OF MARKETING MODELS
1.1 The Need for Mathematical Models in Marketing
The need for mathematical models in marketing is an outgrowth
of two recent marketing phenomena. First, there is a need for a
theoretical framework which will provide a means for integrating the
enormous amount of existing information concerning various facets of
the market. Second, the scope of marketing analysis has increased
as a result of its maturing as an academic discipline.
The Increase in Marketing Data
Consider the increase in marketing data. Since the development
and use of the electronic computer, marketing data have increased expo-
nentially. Not only is there an increase in the availability of
national data but there is also a corresponding increase in the data for
the individual firm in the market. With the introduction of electronic
computers, firms now collect and store more data concerning their inter-
nal operations. The problems confronting the market analyst are to
determine the best way to interpret and analyze these data. When the
data are meager, the market analyst has only to consider a few vari-
ables before he renders a decision. However, when the data become
quite detailed, the interrelationships between each set of data are not
obvious; therefore the analyst needs some conceptual frame of reference
into which he can place the data.
A model provides the market analyst with such a conceptual
frame of reference. For the purpose of this discussion, a model is
defined as a set of symbolic relationships which abstracts some phase
of the marketing process.
The connection between the use of models and the growth in
empirical information which describes the marketing process is obvious.
The use of a model will enable the market analyst to synthesize all of
the relevant data in a concise way so that it will form a conceptual
frame of reference. Without the use of a model, it would be impossible
to analyze all the interdependencies between the different sets of data
in many complex marketing systems.
For example, consider the problem of distributing a sales pro-
motion budget over different regions. Prior to the recent advances in
the collection of marketing data, such decisions could be made without
the use of models since the available data upon which the decision
depended were meager. However, the present day market executive has
reams of information upon which he draws before rendering a decision.
As the data become more detailed, the executive cannot make optimum use
of all the information without the use of a model. For example, it is
one problem to allocate advertising expenditures over the given sales
regions by knowing only each region's total sales and profit; however,
the problem becomes more difficult when additional data are known such
as: the sales promotion activities of the competitors, income levels
of the average customer, employment trends, consumer preferences, the
educational level of the consumers, the nature of the product mix,
geographic sales distribution within the region, and customer sales by
nature of employment. By using models, the market analyst is able to
determine the quantitative effects of the change of one variable on the
other variables in the system.
Marketing, A Maturing Discipline
The other need for models in marketing results from the new
domain of marketing inquiry. :To the surprise of many academicians,
marketing is no longer limited to study of salesmanship, retailing,
wholesaling, and sales promotion in a given firm. The market analyst
is now concerned with consumer behavior beyond the level of an indi-
vidual firm. He is a specialist whose primary interest is in the nature
and functioning of the market, the most basic component of a capital-
In order to study the market, the market analyst must consider
many interrelationships between place, product, promotion, price, gov-
ernment control, income, employment, and consumer demand. This list is
by no means exhaustive; it merely illustrates the changing interests
of those engaged in marketing.
One of the greatest advances in marketing methodology is the
tendency towards a macro approach to marketing problems. Until recent
years, the primary interests of those engaged in marketing involved the
marketing activities of the individual firm, i.e., a micro approach was
used in lieu of a macro approach. Studies on the micro level furnish
basic data for the construction of general concepts, which, when stated
in mathematical form, become models. Therefore, the development of
marketing concepts and marketing models is closely related; i.e., the
latter is a symbolic statement of the former.
The use of models permits a holistic approach to the study of
marketing in that they can handle the interrelationships between the
many economic and social systems in the market. Such a synthesis of
socio-economic systems was not needed on the micro level thereby re-
stricting the use of mathematical models.
The following chapters contain an application of a linear eco-
nomic model to a macro study of the market mechanism. However, it
should not be inferred that the use of mathematical models in marketing
is restricted to the macro level. The need for marketing models arising
from an increase in the number of variables to be considered before
rendering any market decision was discussed in the aforementioned adver-
tising example. The number of variables to be considered can increase
on the micro level as well as on the macro level as evidenced by the
need for models in the previously discussed sales promotion problem.
The Advantages in Using Models in Marketing
In the majority of instances, the need for mathematical models
in marketing is actually a necessity since certain problems could not
be solved without the utilization of a model. This section contains a
discussion of the advantages in using models in lieu of the traditional
qualitative approaches to marketing. The implication of this section is
that the market analyst should use models, when applicable in his
analysis, even though they are not an absolute necessity. Some of the
advantages in using mathematical models in marketing are:
1. Synthesis-- A mathematical model synthesizes the
relevant variables so that they may be subjected to a
variety of statistical inferences.
2. Comprehension-- The use of models permits a more
rigorous and clearer statement of marketing principles.
3. Explicitness-- In order to construct a model, the
assumptions used in its formulation must be stated in
symbolic form thereby making them explicit.
4. Description-- Mathematical models use symbolic
notation which generally permits a more accurate de-
scription of marketing phenomena than those descriptions
given by qualitative marketing concepts.
5. Computation-- A model is easily adapted to statis-
tical techniques which may be programmed on an electronic
computer thereby reducing the time involved for a numer-
6. Data Collection-- The model states explicitly the
data needed for the analysis therefore eliminating the
problem of collecting extraneous data.
7. Depth of Analysis-- The use of a model permits the
use of advanced mathematical techniques which increase
both the scope and depth of the analysis.
8. Extension-- Additions to or modifications of the
basic model can be made quite easily in order that it
may be made more comprehensive.
1.2 The Concept of Mathematical Marketing Models
The Role of Model Building in Marketing Analysis
The first step in the quantitative solution of marketing problems
by the use of mathematical models is model building or specification.
The other steps are: estimation, verification,and prediction. This
chapter concentrates on model building, i.e, specification. Chapters 3,
4, 5, and 6 discuss and illustrate the techniques involved in the steps
of estimation and prediction. Chapter 7 concentrates on the problem
In the previous section, a marketing model is defined as a
set of symbolic relationships which abstracts some phase of the market-
ing process. The process of expressing market phenomena in terms of
mathematics is called specification or model building. Inaccurate
specification procedures lead to model error which is the most diffi-
cult to correct of the three major sources of error in model building.
Chapter 7 contains a detailed analysis of the major sources of error
in model building.
The problem of specification is quite complex in the construc-
tion of models in the social sciences, in particular, marketing models.
Each marketing transaction represents a countless number of consider-
ations on the part of the consumer. The successful model builder must
choose the most pertinent variables out of this great number. This
process is one of the most difficult problems in model building; it is
the central problem in the specification of marketing models. "Success-
ful model building requires an artist's touch, a sense of what to leave
out if the set is to be kept manageable, elegant, and useful with the
collected data that are available."' The problem of estimation,
verification,and prediction are defined and illustrated later in
IS. Valavanis, Econometrics (New York; McGraw-Hill, 1959),
Mathematical Models in the Natural Sciences
Since the physical scientist has been building mathematical
models for many years, it is natural for the social scientist to in-
vestigate the methodology of the physical scientist and to use it,
when applicable, for the construction of models in the social sciences.
The dispute concerning the validity in paraphrasing natural
science models by the use of economic analogies is a lengthy one.
Some economists state dogmatically that such paraphrasing is without a
doubt an excellent approach to the development of economic models.
"Some writers (economists) attempt to distinguish between statics and
dynamics by analogy with what they understand to be the relationship in
theoretical physics. That this is a fruitful and suggestive line of
approach cannot be doubted."2
Other economists state that the use of natural science models
in economics is useless.
Examples of situations where mechanical behavior models
are applicable are: chemical systems, biophysical systems,
and types of structure analyzed in classical physics.
Unfortunately, however, the conditions under which the
application of a mechanical type behavior model is most
useful do not exist in these aspects of the world which
are studied by economics. The attempt to describe its
behavior by one of these mechanical models may be con-
venient and desirable for many purposes, but it does
not make for successful prediction, or any kind of
Logically the position of the author should be one of reconcil-
iation. However, the author is in support of the latter argument
P. A. Samuelson, Foundations of Economic Analysis (Cambridge,
Mass.: Harvard University Press, 1947), p. 311.
3S. Schoeffler, The Failures of Economics: A Diagnostic
Study (Cambridge, Mass.: Harvard University Press, 1955), p. 19.
subject to a few minor modifications, i.e., in the opinion of the
author, paraphrasing natural science models leads to the development
of erroneous marketing models. Since the current dispute is one of
methodology, no attempt will be made to prove the above statement.
The author is well aware of the fact that it may be possible
to paraphrase a natural science model and arrive with a meaningful
marketing model. Although this has never been accomplished, it is ab-
surd to state that it may be impossible to do so. However, in economics,
the Leontief model has had more empirical uses than any other model and
at no part of its derivation have analogies been drawn between natural
science models and market phenomena. It was constructed from an exam-
ination of pertinent market forces; not a quasi-scientific approach
based on forced analogies between economic and natural science pheno-
mena. Similarly, those who state that without doubt, the models of the
natural sciences will eventually lead to significant advances in mar-
keting models are also astray. Although it is impossible to construct
marketing analogies of all models in the natural sciences, the con-
struction of a particular model in the natural sciences is discussed
in order to give an example of the limitations of the methodology used
in the construction of social science models. The model to be discussed
is the well known ideal gas law. The selection of this model is not
the result of a haphazard choice. The primary reason for the choice
of this model is that its assumptions are common to many natural
science models. Therefore many of the criticisms which apply to the
ideal gas model also apply to natural science models in general.
Although the author has a prejudice before the analysis is undertaken,
he attempts to describe those analogies between natural science concepts
and marketing phenomena which are useful in addition to those which
lead to the development of invalid marketing models.
A Natural Science Model.-- The ideal gas law is a natural science
model which relates three important properties of any gas, namely its
pressure, volume, and temperature. The problem of specification is
quite simple in the model, i.e., the number of known physical concepts
which describe a gas is considerably smaller than the number of variables
which may influence the attitude of a consumer towards the purchase of
a product in the market.
Therefore, the problem of constructing a model for an ideal gas
is reduced to obtaining a functional relationship between the specific
variables, pressure, temperature,and volume.
Consider a rectangular box of volume v containing n molecules.
The gas exerts a pressure by the bombardment of its molecules on the
walls of the box. Since the molecules move in all directions, and
since there is no preferred direction, the pressure on any one face of
the box is the same as the pressure on the other faces. By definition,
the pressure P exerted by the molecules is equal to the rate of change
of momentum per square centimeter of wall.
If i is the mean velocity component in the x direction, the
change of momentum resulting from the impact of a single molecule under
consideration is 2m3, where m is the mass of the molecule. All molecules
within a distance x should reach each square centimeter of the wall in
unit time. Since there are n molecules in the volume v, it follows
that n7 molecules strike the wall in unit time. Therefore
S(22mn 'kt nkT
e = ( \2m. )
If N is the number of molecules in 1 mole, and V is the corresponding
volume, it follows that
R = Nk
(1.2-1) PV = RT
Equation 1.2-1 is the equation of state for an ideal gas and is
frequently called the ideal gas law. Actually equation 1.2-1 is a nat-
ural science model, i.e., it is a symbolic statement expressing a
unique relationship between specified variables. It can be compared
with the Leontief model, X = (I-A) Y, which is described in Chapter 2.
Assumptions of the Ideal Gas Model.-- The construction of the
ideal gas model involves several assumptions which are analogous to the
assumption of constant production coefficients and homogeneous output in
the Leontief model. The basic assumptions of the ideal gas model are:
1) Intermolecular forces are negligible.
2) The volume of each molecule is a lot smaller than the
volume of the space it inhabits.
3) Different forms of energy are separable.
4) The energy of the system is a quadratic function of
its corresponding momentum.
Analogies Between the Natural Science Models and Marketing Models
The aim of the previous section was to present a description of
a typical model in the natural sciences in a non-technical manner. The
assumptions could have been stated in a more rigorous fashion by the
use of mathematics. However, the entire discussion is presented merely
as a foundation for the topic of this section which is an examination
of the marketing interpretations of the assumptions of the ideal gas
Assumption I. The existence of liquids is a real example of
the fact that attractive intermolecular forces do exist between molecules
even though they do not interact chemically. These forces are usually
called Van der Waals forces. However, there are a few real gases in
which the intermolecular forces are so small that the Van der Waals
forces can be neglected. Therefore, although an ideal gas is nonexist-
ent, the rodal can be used to explain the physical characteristics
of real gases by relating them to deviations from the ideal gas.
.'n analogous assumption in the construction of marketing
models is an absence of interaction between the component parts. In
very few situations are the marketing activities of one firm independ-
ent of the actions of others. Therefore, a marketing model which
utilizes this assumption of non-interaction might be internally con-
sistent, as is the ideal gas model, but it would have only a limited
degree of applicability in the solution of marketing problems, whereas
the ideal gas model can be used in the physical sciences.
Assumption 2. The assumption of the small size of each indi-
vidual atom as compared to the space which it inhabits is analogous to
the marketing assumption of pure competition, i.e., each firm in the
model is so small that it cannot influence price. In the majority of
the industrial applications of the ideal gas model the assumption of
the relative smallness of the atom is not overly restrictive. Similarly,
the assumption of pure competition is not a serious constraint on the
general applicability of Leontief models.
Assumption 3. This assumption states that the total energy of
a system may be separated into its component parts, translational,
vibrational, and rotational. The actual separation of the total energy
of any system into these unique categories is an empirical impossibility.
Similarly, the division of all industries in the economy into sectors
which produce a homogeneous product, as done in the Leontief model, is
also an empirical impossibility.
Assumption 4. This assumption states that the energy of a
system is a quadratic function of its corresponding momentum. The
importance of this assumption should not be underestimated. Glasstone,
an authority in physical chemistry, states that "the equipartition
principle (which leads to the calculation of 3) depends on the partic-
ular form of the energy being an exact quadratic function of the
corresponding coordinate or momentum equation. If this is not the case,
the principle must inevitably fail, even at high temperatures.4
A similar relationship in Leontief models is the assumption of
a linear relationship between input and output. Fortunately in the
case of the Leontief model, the model will not fail in the absence of
this linear condition. However, due to the lack of data on the nature
of production coefficients by industry type, the assumption of constant
production coefficients is justified.
Basic Methodological Differences Between the Construction
of Natural Science and Marketing Models
The previous section contains an analysis of the similarities
between the ideal gas model, one of the more popular models in the
natural sciences, and the Leontief model, probably the most widely used
econometric model. However, even with this similarity in the nature
of the assumptions used in both models, there still exist some large
1. The application of linear systems, e.g., network
analysis, mass transfer systems, and linear equations
of state, has more validity in physical science models
than they have in marketing models.
2. The number of variables in the physical science
models is smaller than the number used in marketing
models, i.e., the specification problem is more complex
in the construction of social science models.
3. There is a higher amount of noise (uncertainty) in
marketing models than there is in the natural science
S. Glasstone, Theoretical Chemistry (New York: D. Van
Nostrand, 1961), p. 302.
4. The market analyst is often concerned with transient
analysis. While engineering models also deal with this
subject, generally the classical models of the natural
sciences do not.
5. The most rewarding models that may be used by the
market analyst are closed-loop systems, i.e., they allow
for information-feedback in the system. Classical
models in the natural sciences are usually open loop
6. Many of the natural science models are micro models;
for example, the ideal gas model was concerned with the
bombardment of the sides of a cell wall by individual
molecules. However, many marketing models are concerned
with aggregate levels of economic activity.
7. The physicist usually deals with large numbers of
elements in his models so that the introduction of one
additional element may be neglected. Consider the
Maxwell-Boltzman distribution law which gives the most
probable distribution of molecules among the various
possible individual energy values, at statistical equi-
librium, for a system of constant total energy. In the
derivation of this law the Sterling approximation is
used. Since n is very large, an assumption is made that
n + 1/2 is nearly equal to n, i.e., (n + 1/2) In n =
n In n.
Unfortunately, the market analyst is not always in
such an ideal position. Usually, he is constructing
models in an oligopolistic market in which the number
of firms are small and hence have a direct effect on
the marketing activities of the other firms.
8. Some of the earlier models in classical mechanics
assume that the position and the value of a moving par-
ticle can be uniquely determined at a given time.
Similar conditions never exist in the development of a
marketing model. The dynamics of particle movement in
classical mechanic models can be compared with the
changing preferences of consumers. However, to determine
the unique position of a consumer on his preference scale
is impossible without making a probability statement,
i.e., the market analyst can designate the "neighborhood"
in which the consumers' particular preference level
exists, but he cannot give it a unique position.
The theoretical physicist argues that this criticism
of the lack of probability statements in classical me-
chanics is unwarranted with the development of wave
mechanics, commonly referred to as quantum mechanics.
To a degree the argument of the physicists is correct.
Frequently, to determine the position and the
velocity of a moving particle, a beam of light is
directed upon it. The earlier models of classical
mechanics assume that the particle is of macroscopic
size so that the momentum of the particle is not changed.
However, with the introduction of electrons and atomic
nuclei into theoretical physics, the particle is no
longer macroscopic but microscopic. With the presence
of microscopic particles in the system, a beam of light
would alter its momentum.
With the uncertainty introduced into the analysis,
due to the deflection of the particle by the light beam,
the physicist turned to probability statements regarding
the position and the velocity of a moving particle. The
results of their investigations have been generalized by
Heisenberg in the form of the Uncertainty Principle which
states that if p and q represent two conjugate variables,
such as momentum and position of any particle, the prod-
uct. of the uncertainties Z p and A q in the determi-
nation of their respective values is approximately equal
to the Planck constant, i.e.,
(6 P)(nq) h.
Even with the introduction of stochastic elements
into the classical model, it still cannot be applied
directly to marketing problems. The Uncertainty Prin-
ciple will offer estimates of the relative uncertainty
in conjugate observations given the Planck Constant.
Unfortunately, there are no such parameters which de.-
scribe marketing phenomena. At best there are certain
structural parameters which describe different types of
marketing activities; however, they are not invariant
with time and would not have the same position in mar-
keting models as does the Planck constant in quantum
Mathematical Models in Marketing
The author knows of no textbook which describes the basic prin-
ciples underlying the construction of marketing models. Therefore the
purpose of the remainder of this thesis is twofold; namely, to offer
a rigorous statement of the basic principles which a marketing model
should meet and then offer an illustrative example of the method in
which these principles are used in the construction of a mathematical
In the past there has been considerable skepticism regarding
the use of models in marketing. Many members of the "old school"
believe that mathematical models similar to those used in the natural
sciences have no role in marketing theory since marketing phenomena
are a result of humans who are not amenable to mathematical law.
Another argument against the use of models in marketing is that
intuition and past experience are the most important determinants of
marketing decisions. Therefore, since the intuition of an individual
is not likely to be transformed into a mathematical equation, the use
of marketing models in actual business situations is of little value.
To the mathematically trained, the second argument has little
foundation. If the intuition of an individual is reliable enough to
generate a series of successful judgements, i.e., if there is a unique
correspondence between a given event and a given outcome, then this
relationship can be expressed by a mathematical relation. The only
problem is that the relationship between the event and the action may
be so complicated that the model used to represent it might have to be
oversimplified, thereby, generating crude estimates of the actual
The first criticism regarding the incompatibility of human
behavior and mathematical models is out of context. First, no one
mathematical model is intended to describe all marketing phenomena.
What is needed is a plurality of models, i.e., a series of models each
of which describes one particular facet of human behavior in the market.
"Mathematics," said the American Physicist Gibbs, "is a language." "If
this is true any meaningful proposition can be expressed in a suitable
mathematical form, and any generalizations about social behavior can
be formulated mathematically."5
If one accepts the idea that mathematics is a language, then
the usefulness of mathematics in marketing should not be questioned.
It is the belief of the author that if marketing is to generate any
true principles which are not tautologies and which are to have direct
empirical use, then the use of mathematical models is inevitable.
The Classification of Marketing Models
Models may be classified in a variety of ways depending on the
discipline in which they are used. The following table represents a
possible classification of marketing models. It was constructed from
a synthesis of the major models used in economics with some of the more
important conceptual models used in the physical sciences.
Abstract or Physical.-- Physical models are replicas of some
actual object or group of objects. An example of a physical model in
marketing is a scale model of stockrooms available for inventory. The
manipulation of prototypes representing certain types of merchandise
aids in an effective use of existing facilities.
D. Lerner and H. D. Lasswell (ed.), The Policy Sciences
(Stanford, Calif.: Stanford University Press, 1951), p. 129.
Conceptual Arithmetical Axiomatic
Linear Nonlinear Linear Nonlinear
Figure 1.2-1. A Schematic Classification of Marketing Models
An abstact model is one which uses symbols or concepts to
describe some segment of the marketing process. Reilley's Law of
Retail Gravitation, which states that two cities attract retail trade
from any intermediate city in the vicinity of the breaking point approx-
imately in direct proposition to the populations of the two cities and
in inverse proportion to the square of the distances from these two
cities to the intermediate town, is an example of an abstract model.6
Conceptual, Arithmetical, or Axiomatic.-- Conceptual models are
those stated in the terms of marketing principles. For example, "Adver-
tising by itself serves not so much to increase demand for a product as
6E. J..Kelley, and W. Lazer, Managerial Marketing (Homewood,
Ill.: R. D. Irwin, 1958), p. 422.
to speed up the expansion of a demand that would come from favoring
conditions, or to retard adverse demand trends due to unfavorable
conditions," is an example of a conceptual model.7
Arithmetical models are constructed by counting results. For
example, many retail establishments use an automatic re-order plan.
Such a plan involves the automatic reordering of certain types of mer-
chandise when the inventory falls below a previously designated level.
Axiomatic models are mathematical statements which are deduced
from a set of axioms or unproved statements. They are the form of "if
X exists then Y follows." An example of an axiomatic model used in
market analysis is the Leontief model which is described in detail in
the subsequent chapters. It is the belief of the author that axiomatic
models will lead to the greatest contributions to marketing analysis.
An acceptance of Figure 1.2-1 substantiates this conclusion. The table
shows that the more important mathematical models are an extension of
the axiomatic model.
Deterministic or Stochastic.-- Deterministic models may be
traced back to the early models of Newtonian mechanics. In essence
these models show an absence of probability statements. In marketing,
deterministic models are those taking place under absolute certainty.
Deterministic models are compatible with those marketing models which
assume perfect competition in the market since under this market struc-
ture, each entrepreneur has complete knowledge of his production costs
and the demand for his product at a given instant of time.
7R. Bartels, The Development of Marketing Thought (Homewood,
Ill.: R. D. Irwin, 1962), p. 60.
With the development of quantum mechanics, probability theory
played a dominant role in the development of models in the natural
sciences. Similarly, with the consideration of oligopolistic market
structures, the market analyst was forced to consider decision making
under conditions of uncertainty, i.e., the entrepreneur no longer had
a complete knowledge of the market conditions at a given instant of
time. For example, the demand for consumer goods may respond to
expected (uncertain) prices instead of known prices. In order to treat
the various uncertainties in the market, the market analyst is forced
to develop models with random variables, i.e., stochastic models.
Static or Dynamic.-- Static models are concerned with variables
which are not expressed as a function of time. This does not mean
that the variables do not change with time, it merely means that
the observations are taken at a given instant of time. Static models
show their greatest validity in short-run models in which the importance
of the time element may be minimized.
Dynamic models are concerned with an explicit consideration of
the effect of time on the component variables. Dynamic models are of
particular interest in studying the changes in industrial capacity
since these are usually long-run phenomena in which time plays an
Linear or Nonlinear.-- Linear models possess two unique charac-
teristics; namely, additivity and homogeneity. If a variable xl pro-
duces an effect al when it is used alone, and variable x2 produces an
effect a2 when it is used alone, and if both variables xl and x2 used
together produce an effect equal to al, a2, then the variables are
said to be additive. Similarly, if the variable xl produces an effect
a1 and if klx, produces an effect klal, then x, is homogeneous.8
Absence of either additivity or homogeneity denotes nonline-
arity. Usually nonlinear models offer a more accurate description of
reality. Unfortunately, the mathematical analysis involved in nonlinear
systems is very complex since general solutions to nonlinear systems
are nearly impossible to obtain.
Stable or Unstable.-- The problem of stability is an auxiliary
problem of dynamic models. A system or model is stable if after being
subjected to some divergence around its equilibrium position, it returns
to its initial or static condition. Similarly, a system is unstable
if after a dynamic movement of its components it does not return to its
The stability in marketing models can be related to supply and
demand schedules for consumer goods. Equilibrium between these quanti-
ties occurs when the supply price equals the demand price. If a change
in price is accompanied by a corresponding change in output, then the
given equilibrium position is stable.
The Components of Axiomatic Marketing Models
Every marketing model regardless of the degree of its complex-
ity has several common components, namely, undefined terms, definitions,
G. Hadley, Linear Algebra (Reading, Mass.: Addison-Wesley,
1961), p. 2.
axioms, postulates, rules of inference, theorems, and lemmas. This sec-
tion describes each of these components. The methodology used in the
development of marketing models is deductive; a deductive system is
one which begins with a group of assumptions from which theorems are
Undefined Terms.-- Basic to any marketing model are the intro-
duction and use of some undefined terms. The ambitious but mathemat-
ically naive market analyst may attempt to define all the terms in the
model. Such an attempt will give rise to two different situations.
First, the definition of one term will lead to the introduction
of additional terms which must then be defined. This process may
continue until there is an infinite number of terms in the model.
Second, an attempt may be made to define each term by relating it to
the existing variables in the system. The result of this approach will
lead to a circularity of definitions which offer no information. "In
general no attempt is made to analyze undefined terms; they are
accepted as given."9
Definitions.-- As the model increases in complexity new con-
cepts are introduced. In order to offer a rigorous statement of these
concepts, they are defined in terms of the undefinables. For example,
Euclid's definition of a line is that which has no breadth. In this
definition of a line, breadth is taken as an undefinable.
9J. R. Feibleman, "Mathematics and Its Applications in the
Sciences," Philosophy of Science, XXIII (1956), 204.
Axioms.-- Both the undefined and the defined terms are com-
bined into statements known as axioms. "Axioms are unproved statements
which are never submitted to any test of truth. The axioms taken
together are known as the axiom-set. It is the purpose of the axiom-
set to yield a great many theorems."10 In essence the axioms of a
model represent the underlying assumptions upon which it is constructed.
Postulates.-- Both postulates and axioms comprise all the
assumed statements of a model. The test determining
whether a certain assumption is an axiom or a postulate
will be defined as follows: "If the statement contains
no undefinables or defined terms of the science itself,
but only the terms of presupposed sciences, then it is
an axiom; but if it contains a term which is an unde-
finable of the system or is defined by the undefinables
of the system, then it is a postulate."ll
Rules of Inference.-- After the set of undefined terms and
axiom-sets are formulated, they must be manipulated in such a way as to
lead to the derivation of theorems and lemmas. There are two generally
accepted rules of inference, namely those of substitution and detach-
ment. The principle of substitution states that equals may be substi-
tuted for equals. Detachment implies that whatever follows from a
true proposition is true.
Theorems and Lemmas.-- Theorems are statements which are deduced
from the axioms, postulates, definitions and undefined terms, of the
system. In mathematical systems, these statements are usually written
10Ibid., p. 205.
11C. W. Churchman, Elements of Logic and Formal Science (New
York: J. B. Lippincott Co., 1940), p.10.
in symbolic form. In marketing, the mathematical statements of these
deduced theorems are called models. A lemma is an auxiliary or support-
ing theorem which is proved in order that the principal theorem may be
Necessary Seeps in the Construction of Marketing Models
Figure 1.2-2 is a schematic representation of the steps invol-
ved in the construction of marketing models. The primary steps are
those of specification, estimation, verification, and prediction. These
steps are perfectly general and generate the basic format for the
construction of many types of models. These steps are used in both the
construction of natural science models and econometric models.12 The
methodology illustrated in Figure 1.2-2 is not exhaustive; however,
it does represent the important steps involved in the construction of
(1) (2) (3) (4)
Specification ---- Estimation- -Verification-- Prediction
I Least Maximum
Structural Equations Squares Likelihood
Technological Definitional Behavioral
Figure 1.2-2. Model Building
Specification is the most basic step in the model building
process. In economics, the specification problem is usually solved by
the mathematical economist whereas the econometrician is usually inter-
ested in the last three steps: estimation, verification, and prediction.
In the construction of a marketing model, specification is the process
by which a marketing theory is developed and then expressed in symbolic
form. Accuracy in specification is of utmost importance since it
involves an explicit consideration of those variables to be included
in the model. In this step of the model building process, the market
analyst must make a choice as to which variables, out of the countless
number that describe marketing activity, he believes to be the most
important for the solution of the given problem.
The pinpointing of the variables to be considered in the model
is only the first step toward the completion of the specification
problem. The next step involves the formation of a quantitative
relationship between the selected variables. The mathematical rela-
tionships between the pertinent variables are called structural equations
because they describe the basic framework of the model.
Technological, Definitional, and Behavioral Equations.-- Each
structural equation may be classified into one of three classes;
namely, technological, definitional, or behavioral. Technological
equations are classified according to the nature of their constraints;
if the constraints on the equation are determined by the technological
conditions in the market, then the structural equation is a technolog-
ical equation. Technological equations are transformation functions,
i.e., given a value for the independent variable, the equation makes a
mechanical transformation on it with the result that the value of the
dependent variable is uniquely determined. The production function
discussed in neo-classical economics is an example of a technological
Definitional equations owe their existence to the identities
used in the model. They represent truisms which are invariant with
time. Unlike the technological or the behavioral equations, the defi-
nitional equations are not influenced by the conditions of the market.
An example of a definitional equation used in marketing is that the
total flow of consumer goods is equal to the sum of the individual
flows of specialty, consumer and convenience goods.
Behavioral equations describe the actions of the fundamental
units, households, and business firms, in the market. Examples of
behavioral equations are the demand curves for the consumer and the
profit maximization equation for the firm. Generally, the accuracy of
the behavioral equations exerts the greatest influence on the accuracy
of the entire model since the error in the technological or defini-
tional equations is not likely to be as large as the error in the
Systematic and Random Variables.-- A set of variables is system-
atic if the variables are defined in terms of each other. For example,
if Y (the dependent variable) is always equal to a times X, the value
of Y is related to the value of X in a unique manner. A random vari-
able is one which has a probability distribution and is the independent
variable in the system.
Usually, the proper use of random variables increases the
realism in marketing models. For example, assume that the demand
function for some consumer goods is of the following form: D = f(P).
Although demand is a function of price, iL is also a function of other
variables such as income, taste, and prices of substitutable quantities.
Therefore, the equation may be written as:
D = f(P) + e, where e designates the variation due to these
Endogenous and Exogenous Variables.-- All systematic variables
must be either endogenous or exogenous. Endogenous variables are
determined by the model whereas exogenous variables are determined by
forces outside the model. Therefore, the values of the exogenous
variables are assumed to be given. Exogenous variables are independent
variables whereas endogenous variables are dependent.
Closed models are self-contained, i.e., all the variables are
endogenous. Therefore values for all the variables are determined
within the model. Open models employ both exogenous and endogenous
variables. The exogenous variables relate behavioral patterns which
are external to the system to the endogenous variables.
In the first step of the model building process, specification,
pertinent variables (D and P in the previous example) are selected and
then related in a unique manner. Estimation is the second step in the
model building process. Every model contains certain structural param-
eters whose values must be estimated. A demand equation maybe of the
D = aP + b
contains the structural parameters a and b and the specified variables
D and P. It is beyond the scope of this thesis to go into an elaborate
description of the statistical techniques of estimation since several
excellent books have been written on this subject.1 Two of the most
fruitful approaches are the use of least squares and maximum likelihood
Verification, the third step in the model building process, in-
volves a study of the accuracy of the preceding steps. The results
generated by the model are compared with observed values and their
difference is recorded. In order to determine if this difference is
significant certain statistical tests must be employed and a criterion
for acceptability must be designated.
Verification is an important step since it gives the market
analyst some idea of the accuracy of the results generated by the model.
It gives him additional information so that he can express the results
of the model in terms of probability statements.
13Valavanis; T. C. Koopmans (ed.). Statistical Inference
in Dynamic Economic Models (New York: John Wiley, 1950).
Prediction, the fourth step in the model building process, is
primarily an organizational step. It involves a rearranging of the
model so that it may be suited for empirical studies. A typical action
in this step involves a statement of the model so that it may be
programmed on a large scale electronic computer.
Outline for Subsequent Chapters
The following chapters contain an application of the Leontief
interindustry model to marketing. The general outline follows the
previously discussed steps in the model building process. The next
chapter contains the basic theory underlying the construction of the
model. This is the step of specification. Following this step the
next chapter discusses the statistical estimation of pertinent param-
eters. The final chapters contain studies concerning the validity of
the model which is the important part of verification. The model is
then used to predict industry outputs in Duval County, Florida.
THE THEORY OF INPUT-OUTPUT MODELS
2.1 Historical Note
Economic historians usually state that Quesnay developed the
first and most basic input-output model when he published the Tableau
Economique in 1758. Schumpeter states that Cantillon was the origi-
nator of the table. Although Cantillon did not present his analysis
in tabular form, his Tableau is essentially the same as Quesnay's.l
Quesnay's use of the table was quite different from its present
use. His purpose in formulating the table was to show that only the
agricultural industry produced a net profit. Presently, the primary
interest in the Tableau is due to its contribution to economic method-
ology; therefore, the conclusions drawn by Quesnay are not examined.
Probably the greatest contribution of the Tableau to economic-analysis
was that it presented for the first time, a general theory of economic
equilibrium. It showed the relationship between money flows and the
flows of goods and services, thus illustrating the interdependence
between the various sectors of the economy. This concept of general
economic equilibrium was to have far reaching effects on the writing
of future economists.
J. A. Schumpeter, History of Economic Analysis (New York:
Oxford University Press, 1954), p. 222.
After the publishing of the Tableau, there were no significant
contributions to general equilibrium theory until 1874 at which time
Walras published his monumental Elements d'Economie Politique Pure.
The details of Walras's Model are discussed in the following section.
In essence the model determines the prices and the consumer demand for
economic goods, given their supply and demand schedules, production
functions, and the utility functions of the consumers. Walras is often
criticized for not presenting a model which could be substantiated by
empirical analysis. The difficulty in using the model for the solu-
tion of actual economic problems is apparent if an attempt is made to
obtain all the relevant information for the construction of the utility
functions for each consumer and the supply and demand schedules for all
of the commodities in the economy. However, Walras was an economic
theorist and did not intend that the model be used for the solution of
empirical problems. His objective was to show the structural relation-
ships between pertinent economic variables and to use these general
relationships for an illustration of the interdependency between the
various sectors within an economy.
It was not until the publication of The Structure of the Ameri-
can Economy, 1919-1929 by Professor Leontief that the theoretical
system of Walras was modified in order to provide solutions to some of
the most difficult problems ever encountered by economists. The tech-
nical differences between the Leontief and Walrasian models are the
topic of the next section. The input-output model of Leontief is one
of three basic models used in interindustry economics. The other two
are linear programming and process analysis.
"Input-output analysis or the quantitative analysis of inter-
industry relations has in recent years absorbed more funds and more
professional resources than any other single field of applied eco-
nomics."2 However, the model has not been accepted by many members of
the profession. Ironically, the General Theory by Keynes and Leontief's
paper Quantitative Input and Output Relations in the Economic System
of the United States were both published in 1936. Both the works of
Keynes and Leontief have made notable contributions to economic anal-
ysis. The spectacular success of the Keynesian model is now compared
with the limited success of the Leontief model.
First, the statistical methods involved in verifying the Keynes-
ian model are trivial compared to those of Leontief's Model. The
simplicity in the mathematical statement of the Keynesian model permits
the use of basic statistical tools in determining its validity. However,
in the case of the Leontief model, empirical verification is quite
difficult. The data required for the execution of the Leontief model
are of enormous magnitudes resulting in large expenditures of human and
financial resources. With the development of the electronic computer
the statistical advantage enjoyed by the Keynesian model diminishes.
Another factor leading to the increased popularity of the
Keynesian model over that of Leontief's is that the Keynesian model
deals with the problem of unemployment, which was of primary interest
2National Bureau of Economic Research, -Input-Output Analysis:
An Appraisal, Conference on Research and Wealth (Princeton, N.J.:
Princeton University Press, 1955), p. 3.
in the thirties. The Keynesian model presented a possible solution to
a current problem whereas Leontief's model was concerned with the
balance between industrial output and consumer demand.
The third factor contributing to the popularity of the Keynesian
model over that of Leontief's is that Keynes, as did many of his
readers, believed that his model was revolutionary since it appeared
to be in direct contradiction to many of the prevailing neo-classical
theories. Hence his model provided the anti-neo-classicists with a
nucleus around which they could expound their theories. Leontief's
model, which was formulated on the theories of the classical and neo-
classical economists, was not of this nature. Leontief's model did
not refute the Walrasian model but it did simplify the model so that
it could be used as an empirical tool for economic analysis. Some
economists doubted that the Leontief model could be used in an empir-
ical analysis and therefore classified it as a model which was not as
general as that of Walras in addition to having the same empirical
Although the two models were competing for popularity, the
previous discussion should not imply that they were totally unrelated.
Both models are involved with an efficient allocation of resources,
hence their ends are identical. The differences between the two models
are in their means to these ends. The Keynesian model is concerned
with the effective allocation of resources by the use of monetary and
fiscal policies. The Leontief model is also interested in such an
allocation and attempts to do so by co-ordinating industrial output
with consumer demand. A proper use of the Keynesian model provides a
solution to the cyclical unemployment problem. The Leontief model
provides a means for maintaining this level of employment.
2.2 Leontief's Model: A Special Case of the Walrasian Model
Schumpeter refers to the Walrasian model as the "Magna Carta of
Economics" and this is exactly what it is. In this model Walras gave
a precise theoretical statement of the manner in which the prices and
the quantities of goods and services demanded in the market are deter-
mined. His approach is to be distinguished from the partial equilib-
rium approach of Marshall.
"Economics, like every other science, started with the investi-
gation of local relations between two or more economic quantities, such
as the relation between the price of a commodity and the quantity that
is available in the market; in other words, it starts with a partial
analysis, A general equilibrium theory explains the interdependence
between all economic phenomena. Marshall admits that his theory is
one of partial equilibrium; unfortunately Keynes's is not as explicit.
The General Theory of Keynes does not offer an explanation of equilib-
rium throughout the entire economy; it concentrates on one sector of
the economy and is primarily interested in the equilibrium of national
aggregates. Marshall's approach is usually classified as a partial
equilibrium analysis since it is concerned with the determination of
Schumpeter, p. 242.
prices for the output of a firm which is too small to affect national
aggregates such as national income and total employment.
It is interesting to examine Marshall's contribution to the
theory of general equilibrium. Quite frequently Marshall is accused of
not contributing any new tools to economic analysis. Supposedly, he
merely synthesized the theories of the classical and marginal utility
schools. An example of incorrectness of this statement is given in the
Mathematical Appendix of the Principles. In this discussion he is using
a system of simultaneous equations to show that the wage received by a
carpenter tends to equal the marginal productivity of his output. He
It would be possible to extend the scope of such systems
of equations as we have been considering, and to increase
their detail, until they embraced within themselves the
whole of the demand side of the problem of distribution.
But while a mathematical illustration of the mode of
action of a definite set of courses may be complete in
itself, and strictly accurate within its clearly defined
limits, it is otherwise with any attempt to grasp the
whole of a complex problem of real life, or even any
considerable part of it in a series of equations.4
It can be concluded that Marshall was aware of the relation of his par-
tial analysis to a general theory of equilibrium. However, he was
doubtful of the results of such an approach; this is obvious when he
referred to Walras only three times in his book, all of which were
brief unimportant statements.
Walras's System of Equations
The system of equations used by Walras represents his model.
Alfred Marshall, Principles of Economics, 8th ed. (New York:
Macmillan Co., 1948), p. 850.
It is important to note that these equations, which constitute his
general equilibrium model, are concerned with the general equilibrium
of production and exchange. Basic to this model is the assumption that
all firms are so small that they cannot influence prices; i.e., the
model is developed under the conditions of pure competition. Walras's
system of equations describes an economy in which each consumer maxi-
mizes his material welfare with the land, labor, and capital which are
available to him. Walras's model does not have any assumptions re-
garding the distribution of wealth, it merely describes the equilib-
rium determination of prices.
The Walrasian model can be represented by a set of simultaneous
equations. Let there be n factors of production (productive services)
used in the production of m products. These n factors and m products
will be subject to the following conditions:
The quantity of each productive service demanded is equal
to the quantity supplied.
The quantity of each productive service supplied is a
function of its price.
The price of each product is equal to its cost of production.
The quantity of each product demanded is a function of its
In Conditions 1 and 2 there are n equations since there are n
factors of production. Conditions 3 and 4 both have m equations since
there are m products. Therefore the total number of equations is
n + n + m + m = 2 n + 2 m. However, Walras expresses the prices of all
goods in terms of a unit account called a numeraire whose price is
equated to unity. Since the price of this commodity is given by defi-
nition, there are only 2 n + 2 m 1 unknowns (n + n + m + m 1).
Hence the four conditions generate a system of simultaneous equations
in which there are 2 n 2 m equations in 2 n +2 m 1 unknowns. This
problem can be solved by recognizing that the m equations in Condition
4 are not independent. Therefore one of these m equations can be
written for the numeraire. Since the price of this product is given,
the equation for the numeraire in Condition 4 can be derived from the
equations for it in Conditions 1 and 3. Therefore, the system reduces
to 2 m 2 n 1 equations in 2 m 2 n 1 unknowns.
Although the number of equations is equal to the number of
unknowns, the solution may not be unique, i.e., there may be no solution
or there may be an infinite number. Mistakes of this nature are fre-
quently found in the literature. Marshall is guilty of this mistake.
He states, "However complex the problem may become, we can see that it
is theoretically determinate, because the number of unknowns is always
exactly equal to the number of equations we obtain."5 The necessary
mathematical properties of a system of simultaneous equations, which
insure a unique solution to the system are discussed later on in this
5Ibid., p. 856.
The eminent mathematician, Abraham Wald, modified the Walrasian
model and stated the conditions under which the system would have a
unique solution. Wald concluded that the quantities and sources of
both the products and the factors of production can be determined
given the form of the demand and supply functions for the market, the
number of factors and goods in the system, the coefficients of produc-
tion and the amounts of productive services held by the individuals at
the beginning of the period.6 The complexity of the system increases
as soon as non-linear equations enter the model. In this case, the
equations may support the thesis of consistent equilibrium but there
may be no real solutions. Even if the equilibrium position does exist,
there is no guarantee of its stability.
The problem of stability is discussed in Appendix A but a few
remarks are in order. Usually the problem of stability is studied by
a partial analysis, i.e., all of the variables in the static model are
fixed except one. By allowing one variable to change within a known
range, its effects on the other variables in the system can be deter-
mined. This method of partial analysis in general equilibrium models
is one of Leontief's great contributions.
Deficiencies in the Walrasian System
This section contains a discussion of the shortcomings of the
Walrasian model. Walras was aware of these deficiencies, however, he
had to make certain assumptions in order to construct the model. The
0. Morgenstern, Economic Activity Analysis (New York:
John Wiley, 1954) p. 13.
restrictions in the Walrasian model are of interest for several reasons.
First, Walras's model is one of the first models of general equilibrium,
therefore his model serves as a basis for many economic models. Unfor-
tunately many of the deficiencies inherent in his model are passed on
to other economic models. Second, with the recent advances in mathe-
matical theory and electronic data collection, some of the restrictions
may be eliminated and lead to a more general theory of equilibrium.
Third, many of the restrictions may be solved by a partial equilibrium
analysis; traditionally, economics has used this approach in lieu of a
general equilibrium analysis. Therefore, an examination of the restric-
tions and limitations of the Walrasian model may lead to a closing of
the widening gap between the mathematical economist and his normative
Linear Homogeneous Production Functions.-- The Walrasian system
is determinate, if the production functions are linear homogeneous
entities. The assumption of linear homogeneous production functions is
an excellent example of the deficiencies in the Walrasian model which
are passed on to other models, the most striking example being the
Leontief model. Consider the following production function
x = f(xl, x2, .. . Xn)
where x is equal to the output produced from the input (xl, x2, . .. xn).
"A function f of n variables xl, x2, . . xn is called homogeneous
of degree m, if upon replacement of each of the variables by an arbitrary
parameter k times the variable, the function is multiplied by km."7
71. S. Sokolnikoff, Advanced Calculus (New York: McGraw-Hill,
1939), p. 75.
Therefore, the above production function is homogeneous of degree m if
f(kx1, kx2, . kx) = km f(xl, x2, . )
This implies that
x + f mkf(xl, x2 . xn)
If the production function is homogeneous
x = km f(xl, x2, . n)
-cx is a constant.
The economic interpretation of this result is that a linear homogeneous
production function leads to a constant return to scale. The above
analysis assumes that the partial derivatives are continuous, i.e.,
the inputs and outputs are variable so as to lead to a continuous
production function. This implies that there is a continuous substi-
tution of the factors of production. The Walrasian model does not stop
with the assumption of constant returns to scale, it also assumes that
the production coefficients are constant. Generally with the assumption
of constant coefficients of production, there is no substitution between
inputs in the production function. Consider the general function
xij = aijX
The term aij is the technical coefficient which is assumed to
be fixed. The greatest disadvantage in assuming constant activity
coefficients is that it prohibits substitutability among inputs and
changes in productivity. The validity of this assumption is discussed
when a similar assumption made by Leontief is subjected to an empirical
In both the Leontief and the Walrasian systems the firm has al-
ready made its choice regarding the nature of the production process
to be used. Therefore, the assumption of fixed production coefficients
is not as stringent as it appears to be. The development of linear
programming has eased the rigidity of the constant productivity assump-
tion. The application of linear programming techniques to all the
possible processes available for the production of the output, leads to
an optimum combination of the inputs, given some objective function to
be maximized or minimized.
A linear homogeneous production function coupled with the
assumption of constant activity coefficients will guarantee a unique
solution to the Walrasian system; however, this is not the only condi-
tion for which the system is determinate. Rogin states that
with complete variability in the proportion of inputs
(premised on complete divisibility and mobility of
economic resources) Walras's mechanism of competition
still assures a determinate solution for price and
output in the markets for all commodities premised
upon the determinate equilibrium output of each firm.
The essential condition is that in the context of each
firm the dosing of each factor service be accompanied
by decreased incremental returns.8
Incremental returns should be distinguished from proportionate returns.
The former refer to the rate of variation of output associated with
the variation of input of a factor, the latter to the ratio of the
proportional (percentage or average) change of output to the propor-
tional (percentage or average) change of input.
8Leo Rogin, The Meaning and Validity of Economic Theory (New
York: Harper Brothers, 1956), p. 439.
A Static Equilibrium Model.-- The model of Walras is a static
equilibrium model since none of the equations is a function of time.
This is a serious restraint since many sectors of the model are depend-
ent on the time variable. Consider the investment sector. Future
investments are a function of future expectations concerning the
general price level and the interest rate. These future expectations
are a function of time which is not present in the Walrasian model.
Therefore, the elimination of the time element from the Walrasian
model seriously restricts its use as an empirical tool of analysis.
Previously, the conditions were stated upon which the Walrasian
model was constructed. Two of the conditions were an explicit state-
ment of the assumption that supply and demand were a function of only
price; thereby implying that these schedules were determined only by
current prices and were invariant with future prices.
The Existence of a Unique Solution for the Walrasian System.--
Walras assumed that since his system had a number of independent and
consistent equations equal to the number of unknowns the solution would
be unique. However, the solution may be unique but there is no guaran-
tee that the solution will be non-negative which is essential for a
proper economic interpretation of the results. Wald modified the
Walrasian system by ignoring the marginal utility functions and assum-
ing that the factors of production were given. He then proved that if
the following conditions are fulfilled the Walrasian system has a unique
non-negative solution set.9
R. E. Kuenne, "Walras, Leontief, and the Interdependence of
Economic Activities," The Quarterly Journal of Economics, LXVIII (1954),
1. The production coefficients are non-negative.
2. The supplies of the factors of production are non-negative.
3. At a minimum, one factor of production is used in the pro-
duction of each commodity.
4. The rank of the activity matrix is equal to the number of
5. The demand functions are continuous and positive for any
Stability of the Walrasian Model.-- Another criticism of the
Walrasian model is that it fails to relate the changes in the supply
and demand functions to the original state of equilibrium. Walras was
confused in his attempt to relate stability to the existence of a
unique solution set. A static approach to both problems is overly
restrictive, i.e., a unique solution to the system is not a sufficient
criterion for stability. The problem of stability can be properly
analyzed by the consideration of dynamic extensions of the model.
Walras did consider the effects of disturbances such as price
changes in his system, but he failed to state explicitly the path of
the transition. He did not describe the relation of the new equilib-
rium position to the original one. There is no guarantee that a
divergence from the original equilibrium position in the Walrasian
model will move back to the original static position.
The problem of divergences around a given equilibrium position
stresses the need for a dynamic interpretation of the Walrasian model.
A stable equilibrium position when subjected to disturbances is one in
which the end results of the divergences lead back to the static equilib-
rium position. However, these divergences are dynamic and must be
subjected to one analysis; not by the static approach used by Walras
in which he attempted to relate the existence of a unique solution to
A Purely Competitive System.-- Basic to the construction of
the Walrasian model is the idea of pure competition. The previous
analysis contains a discussion of the determinancy of the system given
this basic assumption. However, with the introduction of oligopolistic
elements into the model, the solution is no longer determinate.
In the oligopolistic version of the Walrasian model the
assumptions of fixed production coefficients and constant returns to
scale, along with the other conditions previously described, no longer
lead to a determinate solution for several reasons. First, there is
an increase in the number of variables in the system due to the variety
of products in an oligopolistic market. Therefore, depending on the
nature of the model, many additional variables will be introduced.
Second the structural relationships in the economy become complex
due to the coalitions in the oligopolistic market. Many of the mathe-
matical relationships in the model would have to be expressed as non-
A partial answer to the oligopoly problem is to use game theory
for the construction of the model. Conceptually game theory could deal
with the problems introduced by labor unions, consumer cooperatives,
and other coalitions in the market. However, game theory as it exists
today, is in a primitive stage and not directly applicable to the
Walrasian model since the number of participants in the game are
limited to a small number. However, it is simpler to construct a
new model from the theory of games than it would be to modify the
existing Walrasian model, so that it would be applicable to an oligop-
The Significance of Walras's Contribution
The previous discussion of the shortcomings in the Walrasian
system should not imply that the model is not a valuable tool of
economic analysis; it merely describes those assumptions in the model
which limit its generality. One of the greatest contributions of the
model is that it synthesized many of the partial equilibrium theories
into one unified body of thought. For example, the theory of marginal
productivity may be obtained from the general model of Walras by
eliminating the assumption of constant production coefficients.
Partial equilibrium theorists are aware of the limitations of
their analysis. Walras provided a model which would overcome the
major objections of partial equilibrium analysis since his model con-
siders the interdependence of economic phenomena. Although the various
activities in the economy are related by naive mathematical functions,
the general relationships do exist.
The model also specifies some of the more influential vari-
ables on the level of economic activity. The scope of the model can
be expanded to cover a larger number of sectors in the economy; however,
extensions of this type would involve the addition of many variables to
the model. Since the equilibrium model of Walras is general, these
additional sectors may be added without any conceptual changes in the
construction of the model.
The model of Leontief which is discussed in the next section,
is derived from the Walrasian model. Essentially, Leontief converted
the theoretical model of Walras into an empirical tool for economic
analysis. A practical application of the Walrasian model is of extreme
importance since it is useful for prediction, the goal of any science.
In order to construct his model, Leontief had to make some general
modifications of the Walrasian model which included:10
Aggregation.-- Walras did not consider the problem of aggrega-
tion, the details of which are discussed in a subsequent chapter. With-
out aggregating the input data to the model, each product and each
factor of production within the model would need an equation. For
example, if there were 1,000 products and 100 factors, which is very
small for any realistic economy, there would be 2,199 equations and
unknowns in the system. Even with high speed electronic computers,
the solution to this system of equations would be tedious, if not
Marginal Utility Functions.-- Walras paid little attention to
the accessibility of information on marginal utility functions which
10. Morgenstern, p. 22.
are difficult to measure in practice. Market demand and supply
functions may be directly related to utility functions by assuming
each consumer maximizes his utility.
Free Competition.-- Since the assumption of free competition is
made in the Walrasian model, this divergence from reality should be
corrected when the model is placed into use. One method is to use
empirical demand, supply, and production functions. The advantage in
this approach is that these functions are descriptions of reality;
they can take into account the existence of partial and complete
The Derivation of Leontief's Model from the Walrasian System
The object of this section is to show that a proper modifi-
cation of the Walrasian model will lead to the Leontief model.
Leontief's model is the first modification of Walras's general equili-
brium model which can be subjected to empirical verification. However,
this modification is achieved by a loss of generality. Walras's model
is a closed static model since it does not contain an exogenous sector.
An exogenous sector is one in which the flows are obtained from an
independent analysis before they are used in the model. Leontief's
model is open since it has an exogenous sector. One of the primary
differences between the closed model, which has only an endogenous
sector, and the open model, is that all of the production functions
are not considered to be constant in the open model. Presently,
research is being performed with the objective of closing the Leontief
model in order to approach Walras' general theory of equilibrium.
In an early part of this section four conditions were stated as
the basic foundations of the Walrasian system. Although not explicitly
stated, but inherent in the analysis, was the assumption that each
individual acted in such a way as to maximize his position (for example,
profits, or utility). Walras also assumed that the institutional
setting, quantity and quality of productive resources, consumer tastes,
and other pertinent data were given.
Leontief made the following modifications to the Walrasian
1. Each industry produces only one product.
2. There is no explicit use of utility functions.
3. The quantities of factor services supplied are not
expressed as a mathematical function of price.
4. The quantities of products demanded are not expressed
as a mathematical function of price. They are usually
placed in the final demand category and are taken as
5. The technical coefficients of each industry are fixed.
6. The interest rate is known.
7. Labor is the only primary (non-produced) input.
8. The number of inputs is equal to the number of ouputs.
9. For the period under consideration, the total output of
any good is equal to the consumption of that period.
10. The input of capital is a fixed proportion of other inputs.
Therefore, inventories may be counted as one of the other
industries in the model or it may be included as an
element of the final demand figure.
By applying these assumptions to the system of equations representing
the Walrasian system, Leontief derived the model which is described
in the next section.
2.3 The Basic Model
As an economic unit increases its productive capacity there is
a tendency for the division of labor to show a comparable increase.
An increase in the division of labor for any society produces an inter-
dependence between the producing and consuming sectors. Interdependence
is the system of relationships which describes the complex activities
in an economy.
An infinite number of statements can be written to describe a
particular segment of the economy. The combination of all statements
leads to a general model of social behavior. The Leontief model does
not pretend to have this universality, since it concentrates on the
economic sector of society. The relationships in the model are not
explicitly related to the ideals, motivations, or the noneconomic inter-
est of the individuals within a society.
The primary objective of the model is to relate by quantitative
relationships, the demand for finished products with the available
agents of production, land, labor, and capital. In a complex indus-
trial society, there is no established connection between the demand
for products and the aggregate amount of resources required for the
fulfillment of this demand. Usually the production of any product a1
requires raw materials in the form of other products bl, b2, . b
However, the production of each of the original input products, bl,
b2, . bn requires input products of the form cl, c2 . c.
This process may continue ad infinitum. There are several advantages
in using a mathematical model for expressing the complex chain reaction
between inputs and outputs. The model is quite useful in forecasting
output, employment, and income levels for a given level of demand.
Information of this type indicates which sectors of the economy have
an excess of deficiency in manpower and financial resources. For
example, in the early thirties there was a controversy between the Works
Project Administration and the Public Works Administration concerning
the effect of different types of expenditures on employment levels.
The Works Project Administration wished to spend the majority of its
funds on wages and only a limited amount on materials. Conversely, the
Public Works Administration was in favor of large expenditures for
construction materials and limited expenditures on direct wages. To
resolve the problem the Construction Division of the Bureau of Labor
Statistics was asked to compare the effect of direct work relief expen-
ditures and expenditures for public construction projects on employment
levels. The use of basic interindustry techniques solved the problem.
In the last 20 years, considerable research has been undertaken
to extend the use of Leontief's model. The first major extension was
the application of linear programming techniques to input-output anal-
ysis. The use of this technique permits a consideration of the sub-
stitutability of inputs in the model in order to maximize some economic
objective function. Given a consumer demand schedule, there are a
variety of input schedules which will meet the given demand require-
ments. The linear programming approach will generate a solution which
determines the optimum input schedule given specific criteria to meet,
such as maximum employment.
The second and most recent innovation to input-output analysis
is the method of process analysis, which is concerned with a detailed
study of the technical processes of production and relates such activ-
ities to output, employment,and income levels. Process analysis is not
limited to the use of linear models. However, the majority of work to
date involves such an assumption.
Basic to any input-output model is a transactions table. The
table merely represents a double entry bookkeeping system applied to
industry purchases and outputs. The transactions table is often called
a transaction matrix since it leads to the development of equations
which are usually expressed in matrix form. Table 2.3-1 is a schematic
arrangement of a transaction table. For purposes of discussion, the
table is divided into four arbitrary sections called quadrants which
have no effect on the analysis.
Quadrant II.-- The total value of the output of industry 1 is
represented by X1. The corresponding x12 represents the output of
industry 1 that goes to industry 2; and x13 represents the output of
industry 1 that goes into industry 3. The general term, xij, is the
output of industry i going to industry j. The same analysis applies to
the industries in the final demand sector. Therefore an examination of
all the x i's describes the nature in which the output of each industry
is distributed. This analysis may be applied to each row in the trans-
The preceding analysis is based on a consideration of the row
entries in the table. However, a similar analysis may be applied to
the column entries. In column 3, X3 represents the total purchases of
industry 3. In the analysis of row entries, it was concluded that x13
represented the value of goods produced by industry 1 and purchased by
industry 3. In the analysis of column entries it represents the pur-
chases of industry 3 from industry 1. Likewise x23, which is the next
entry in column 3, represents the purchases of industry 3 from industry 2.
An analysis of column entries shows the distribution of an industry's
purchaseswhereas the corresponding row entries show the distribution
of the industry's output.
An examination of the table shows that the total output of an
industry is equal to its total purchases. This occurs if the inter-
industry flows are measured in money terms, not physical terms. If
such a convention were not adopted, column totals would have no meaning
since the output of one industry may be in tons and the other in
thousands of units. Therefore, since every sale is a purchase, row and
column totals are equal. In a later discussion, the above condition is
found to generate a matrix in which the columns are stochastic.
-4 C * -4 *
4-1 e. 4-
C -4 >4
X >< < > >
Another definitional problem to be examined is that of interin-
dustry flows. Each xij represents an interindustry flow, i.e., the
flow of goods from industry i to industry j. However, for i=j, i.e.,
xii, the term becomes an intraindustry flow. The term x22 represents
the output of industry 2 used by industry 2. For example, the chemical
industry produces polyhydric alcohols such as ethylene glycol and
pentaerythiritol, both of which are used in the manufacturing of many
other chemicals. In some interindustry models, the term xii is consid-
ered zero and the corresponding changes are made to the input and output
totals; this convention is not used in the present model.
Another point to remember is that although there is an equal
number of rows and columns, the matrix is not symmetrical, i.e., xij
j xji. This is reasonable, since there is no reason that the output
of industry i to industry j should equal the output of industry j to
In the preceding discussion, the interindustry flows were called
x i. They represent the input from industry i to industry j. In the
equation i = aij, aij is called the activity coefficient. It repre-
sents the input from industry i per unit of output from industry j.
Therefore, the activity coefficient may be obtained by dividing each
element in the transactions table by its column total. The preceding
equation can be written as xij = aijXj. The economist will recognize
this as a linear homogeneous production function. Although activity
coefficients may be calculated for all industries, only those in the
endogenous sector (Quadrant II) are considered to be constant. This
is a very important assumption and is discussed in detail in subsequent
chapters. The only reason for including it in the present discussion
is to offer some meaningful criteria for distinguishing the endogenous
sector from the exogenous sector, Quadrant I.
Quadrant III.-- As an aggregate, the elements in this quadrant
approximate the value added by each industry. Since Xj represents the
value of the total output and xij (j=l . . n) represents interin-
dustry purchases, their difference is equal to the value added by
manufacturing, i.e., the difference between the value of shipments and
cost of materials, including other related costs such as containers and
supplies. The entries in Quadrant III, with the exception of Govern-
ment and Imports, are often called primary inputs since they usually
give rise to profits, wages, and salaries.
Quadrant IV.-- The data in Quadrant IV are not used in the
basic model or any of its applications. The only reason for including
it is to illustrate the complete breakdown of interindustry transactions;
in many transaction tables it is not listed. The entries in this quad-
rant represent the primary inputs to the final demand sectors. Although
this information has little use in interindustry analysis, it is quite
important in analyzing national income data.
Quadrant I.-- In the following chapters, the column entries in
this quadrant comprise the exogenous sector of the model since they are
external to the industries which form the transactions matrix. The
industries in the transactions matrix, the first n columns and the
first n rows, are usually denoted as the endogenous sectors of the
model since they are the only manufacturing industries in the trans-
actions matrix. The exogenous sector is occasionally called the final
demand or bill of goods. The endogenous sectors are often called the
processing sectors. As stated in the discussion of Quadrant II, the
activity coefficients for the final demand sector are not considered
to be constant, i.e., the input to the industry is not a linear func-
tion of its output.
The elements in the exogenous sector of the model are often
called "autonomous" elements to emphasize their independency from the
endogenous sector. They are independent since there is no predeter-
mined output from a given input. An excellent example of a final demand
sector is the government entry. The column titled Government indicates
that portion of an industry output which is purchased by the govern-
ment. Since the actions of the government do not depend on any tech-
nical process, its activity coefficient is not considered to be
constant. The government's purchase of the output of any industry is
influenced by political, social, and financial pressures. However, in
the endogenous sectors these conditions do not exist. For example, the
input of steel to the automobile industry is more likely to be a constant
percent of its output than will be governmental expenditures for the
products of the steel industry.
A Mathematical Statement of the Model
Consider a transactions table with four industries in the
endogenous sector. The final demand is considered as one sector and is
expressed as an aggregate value. The following equations are obtained
by adding all of the elements in each row:
xll + x12 + x13 + x14 + YI = X1
x21 + x22 + x23 + x24 + Y2 = '2
x31 + 32 + x33 + x34 + Y3 = X3
x41 + x42 + x43 + X44 + Y4 = X4
xij = the output from industry i to industry j
Yi = the final demand for the products of industry i
Xi = gross output of industry i
One of the primary objectives of input-output analysis is to
determine the output required from each industry given the final demand
for its products, i.e., given Y1, Y2, Y3, and Y4, determine X1, X2, X3,
and X4. An examination of the equation system will reveal the presence
of twenty unknowns and four equations.
Assuming that the equations are independent and consistent,
sixteen additional equations are needed in order to obtain a unique
solution. These additional equations may be obtained from the produc-
tion function for each industry.
Since a. = Xi
xj = aXj
X11 = all 1 ; x12 = a12X2; x13 = a13X3;
x21 = a21Xl; x22 = a22X2; x23 = a23X3;
x31 = a31X; x32 = a32X2; x33 = a33X3;
x41 = a41Xl; x42 = a42X2; x43 = a43X3;
Substituting equation Set II into equation Set
allXI + al2X2 + a13X3 + a14X4 + Y1 =
a21X1 + a22X2 + a23X3 + a24X4 + Y2 =
a31X1 + a32X2 + a33X3 + a34X4 + Y3 =
a41X1 + a42X2 + a43X3 + a4X4 + Y =
Equation Set III may be written as:
Define, a, as
-al3X3 -a14X4 = Y1
-a23X3 -a24X4 = Y2
+ (l-a33)X3 -a34X4 = Y3
-a43X3 + (1-a44)X4 = Y4
formed by the activity coefficients
I equal the identity matrix
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
Y equal the column vector of final demands
X equal the column vector of outputs
Therefore in matrix form equation Set IV may be written as:
(I-a) X = Y
(2.3-1) X = (I-a)-Y
Equation (2.3-1) is the input-output model. Given a vector Y
representing the final demand for each industry and the inverse of (I-a),
the output of each industry is obtained from its product. Equation
(2.3-1) applies to a n by n matrix and is not restricted to the four-
This section has discussed the elementary properties of the
model. No attempt has been made to discuss the static, dynamic, open,
closed, or other important properties of the model. These topics, in
addition to other applications of the basic model, are discussed in
THE PROCEDURE FOR THE CONSTRUCTION OF AN INPUT-OUTPUT TABLE
Basic to the application of any interindustry model is the con-
struction of an input-output table. This chapter discusses the method-
ology involved in the construction of such a table; and the next
chapter contains a numerical example of the procedures outlined in
this chapter. The data in an input-output table are classified into
one of three general sectors: endogenous (interindustry flows),
exogenous (final demand), and gross output. Therefore, the procedures
described in this chapter begin with an estimation of gross output and
then proceed to a discussion of the methods involved in estimating the
endogenous and exogenous flows. One of the primary objectives of this
thesis is to discuss the applications of the interindustry table to
market analysis; therefore this chapter contains a description of the
methodology involved in converting gross outputs to employment and
income data for a given region.
Exclusive Use of Secondary Data Sources
The input-output table is constructed from secondary data. Pri-
mary data need no transformation and may serve as input to the model in
its existing form; secondary data must be modified before they can
serve as input to the model. Secondary data are modified by applying
specific transformations to them with the hope that the transformed
data will be the same as primary data. Unfortunately the majority of
the transformations are not 100 percent efficient, i.e., the modified
secondary data are not identical to the primary data.
Obviously, primary data are used when they are available or
when they may be obtained at a reasonable cost. When this condition
does not exist, secondary data are substituted for primary data. In
the present study, the principle reason for using secondary data in
lieu of primary data is the element of cost. The required primary data
would have to be acquired by interviewing each firm in the sample and
then aggregating the data by industry type. This operation is very
expensive since the interviewer may have to spend several days with
each firm in order to receive accurate responses in the questionnaire.
For the numerical example in the next chapter the scope of such a
study has been compared with that of similar studies in order to ob-
tain a rough estimate of the cost involved in such an undertaking. As
a result of this investigation, it is concluded that the anticipated
study would cost $25,000. The basic model was constructed from secon-
dary sources since these funds are not available at the present time.
The substitution of secondary data for primary data is not a
serious restraint on the analysis of interindustry models since the
major objective of this study is to examine the methodology involved
in analyzing the results obtained from an interindustry model. The
methodology involved in the analysis of the results of the model are
perfectly general and independent of the data sources. Therefore, the
methodology employed in this thesis is unique in that the entire input-
output table was constructed from secondary data, which eliminated the
need for a costly questionnaire survey.
Limitations on the Use of Secondary Data Sources
From the previous discussion it is evident that in general the
use of primary data obtained from a survey is preferable to data ob-
tained from secondary data sources. However, the use of primary data
does not insure a greater accuracy in the results generated by the
model than does the use of secondary data. The use of data obtained
by the use of a poorly designed questionnaire or an unrepresentative
sample may introduce greater error into the results than does the use
of secondary data. There are several reasons for this phenomenon.
First, the major sources of secondary data are governmental agencies
who have the resources and experience needed for an accurate collection
of data. Usually those engaged in local regional studies are limited
to the amount of resources available for data collection and therefore,
cannot be as exhaustive in their analysis as comparable studies con-
ducted by governmental agencies. Second, local regional studies are
usually conducted by small interindustry groups who have to supervise
many steps of the construction of the model. Therefore, the data
collection procedures may be subjected to various inaccuracies resul-
ting from a lack of experience and the improper supervision of field
personnel by the interindustry group.
However, if the sample is chosen with a reasonable degree of
accuracy, an input-output table constructed from primary data is far
superior to that constructed from secondary data. Primary data are
obtained by a survey of the firms within the region; secondary data
usually apply to some area larger than the region under consideration
and must be scaled down so as to apply to the particular region for
which the model is being constructed. The disadvantage in construc-
ting an input-output table based on primary data is the time and
expense involved in the collection and aggregation of the information
obtained from the field survey.
The most serious limitation in the substitution of secondary
data for primary data is that the scope of the inferences based on the
output of the model may be limited. This presents no restrictions on
the methodology but it does limit the applicability of the results
for the solution of specific problems.
The Advantages in Using Secondary Data Sources
One of the greatest advantages in the use of secondary data as
a substitute for primary data is that it permits an individual or a
very small interindustry group to construct a preliminary model which
does not require the vast quantity of resources demanded by a field
Another advantage is that a pilot study based on the construc-
tion of a preliminary model using secondary sources is invaluable. A
pilot study requires a thorough investigation of the existing sources
of data; the outcome of such an investigation permits a more intelli-
gent design of the questionnaire to be used in the survey. A pilot
study which is made previous to performing the survey will also pin-
point some of the major problems that the interindustry group are going
to encounter. It is almost impossible to foresee all the problems
which will occur by merely examining other interindustry studies. De-
pending on the objectives of the study, a pilot study may offer a
solution to the given problem and thereby eliminate the survey in its
The Purpose of a Numerical Example
It is possible to describe the operation of the model without
using a numerical example; however, there are several advantages in
this approach. First, the use of a numerical example permits a de-
tailed description of the methodology to be used in analyzing the
results of the model. Second, the use of a numerical example, in ac-
dition to providing a basis for the description of the statistical
methodology, also gives insight into the nature of empirical problems
to be encountered. Without a numerical example it would be difficult
to give an economic interpretation of the results generated by the
Therefore, there are two alternatives for obtaining the data for
the model, namely the use of hypothetical data or secondary data. As
previously stated, the methodology is general and would apply to both
types of data. The advantage in using hypothetical data is that the
properties of the model can be examined without an extensive data
search. The use of secondary data also has definite advantages. First,
the results based on secondary input data will approximate the results
that would be obtained from the use of primary data. Second, this
information will permit the development of ideas concerning the inter-
dependence of business activities in the given region before performing
the survey. After making the appropriate transformations on the secon-
dary data, the problems encountered in constructing the model and
making extensions to it will be similar for both data sources. The
data used in the construction of the model were obtained from an anal-
ysis of Duval County in the State of Florida. The purpose in obtaining
these data was to overcome the criticism that the use of hypothetical
data in a theoretical model restricts the scope of the analysis.
Although the transactions table is derived from national coeffi-
cients, it is quite valuable. In fact, many of the major input-output
studies, such as Maryland, Utah, California, and Texas, have also used
the national coefficients in the construction of regional models. An
analysis which is based on the data obtained from the national coeffi-
cients will, in addition to illustrating the methodology to be used in
the construction of an interindustry model, give some insight into the
functioning of Duval County economy.
Fortunately Duval County is one of the Standard Metropolitan Sta-
istical Areas utilized by the Bureau of the Census in the classification
of statistical information. Therefore, the 1958 Census of ManufacturersI
will be a major source of the data utilized in the construction of
IU. S. Department of Commerce, Bureau of the Census. 1958 Census
of Manufacturers: 1961. Area Statistics, Vol. III.
the input-output table. Volume III, Area Statistics, of the
1958 Census of Manufacturers2 furnishes data for selected counties in
Florida, in which Duval County statistics are included. Therefore, by
combining the above sources of information, it is possible to obtain a
relatively accurate account of the aggregate economic activity in
The term "aggregate" is used to refer to the type of data fur-
nished by the Census since it merely gives the value added by a given
industry, not the distribution of a given industry's output to other
industries within the complex. A knowledge of the distribution of a
given industry's output to other industries in the complex would form
the foundation for the construction of an input-output table from pri-
mary data sources. Nonetheless, the use of secondary data from Duval
County Area will increase the realism in the model even though they are
scaled down by the use of coefficients which are calculated on a national
basis and are not entirely applicable to a specific region such as
3.2. The Construction of the Input-Output Table
The input-output (transactions) table is constructed in the
following manner. First, the gross outputs of each industry in the
model are estimated. Second, the sum of the endogenous flows are esti-
mated by the use of national coefficients. Third, the exogenous flows
are calculated by subtracting the sums of the endogenous flows from the
estimated output of each industry.
Using the notation employed in Section 2.3, the above outline
may be summarized in the following manners
1. Estimate Xj
2. Estimate xij by computing aijXj, where aij is obtained
from the 1947 table4
3. Calculate Yi, Yi = Xi aijxj
Selection of Industries
Before making any estimates of industry output, it is mandatory
to determine the dominant industries in Duval County. Dominant indus-
tries in a given area are those industries which contribute the most
to employment and income in the given region.
The industries in this model are limited to the manufacturing
sector since the government has published the activity coefficients for
this sector. The 1958 Census of Manufacturers5 has a list of the major
industries in Duval County. This source provided a means for selecting
the industries to be considered in the model.
Estimation of Gross Outputs
In order to obtain a realistic estimate of the elements in the
final demand sector, the corresponding value of shipments or gross out-
put of each producing industry within the manufacturing sector must be
W. D. Evans, and M. Hoffenberg. "The Interindustry Relations
Study for 1947." The Review of Economics and Statistics, XXXIV (1952),
1958 Census of Manufacturers, p. 9-11.
1958 Census of Manufacturers, p. 9-11.
calculated. The gross outputs of an industry refer to the total value
of shipments at producer's prices.
Unfortunately, the Bureau of the Census in its publications does
not furnish the value of shipments for each industry in the manufactur-
ing sector. Mr. Maxwell R. Conklin, Chief of the Industry Division,
Bureau of the Census, states in a private letter,
the value of shipments does not appear in the Census data
because the figures for 3-digit6 and 2-digit industry
groups would include a considerable amount of duplication
since, in many industry groups, the shipments from estab-
lishments in a constituent 4-digit industry are often used
as materials by another 4-digit industry within the same
group. As a matter of policy the Bureau of the Census does
not publish such summary values of shipment figures.
Value added, which is the difference between cost of
materials and value of shipment, is considered a more
accurate measure of the economic contribution of establish-
ments when figures are summarized to industry group totals.7
Therefore, the immediate problem is to estimate the value of
shipments from the value added by manufacturing. The following discus-
sion suggests three methods for determining the gross output of each
industry in the model. Generally, there is no one method which always
offers the most accurate estimate of gross output. The relative merits
of each method are compared after all three of the methods have been
6The Standard Industrial Classification is an industrial
classification of all activities into broad industrial divisions
(manufacturing, mining, retailing, etc.). It further subdivides each
division into major industry groups(two-digit industries) then into
industry groups(three-digit industries) and finally into detailed
7Letter from Maxwell R. Conklin, Chief, Industry Division,
Bureau of the Census, Washington, D.C., April 2, 1963.
Method I.-- In this method, the value of shipments and its
accompanying value added by manufacturing are combined in order to ob-
tain an average ratio of the value of shipments to the value added by
manufacturing for selected industries within a given industry aggregate.
Although the Bureau of the Census does not publish the total value of
shipments for each two-digit industry, it does so for the majority of
the three-digit industries. The difference between a two-digit and
three-digit industry is that the former is the aggregate of all three-
The above ratios must be calculated for the State of Florida,
and not Duval County since the Area Statistics list only the value
added by manufacturing for the three-digit industries. Therefore,
there is the additional problem of converting the estimate of the state
value of shipments to a regional value of shipments. The entire conver-
sion of state value added by manufacturing to a regional value of ship-
ments may be accomplished in the following manner:
1. For each three-digit industry within a given two-digit
industry, compute the ratio of the state value of
shipments to the state value added by manufacturing,
and then calculate the average of the ratios for
each two-digit industry.
2. The second ratio is calculated by dividing the
number of employees in each industry within the
given complex by the number of employees within the
state who are in the industry classification.
3. Multiply the results of operation one by the results
of operation two, and then multiply this product by
the value added for each two-digit industry. The
value added for the two-digit industry is based on
that industry's total for the state, not the value
added by that industry in Duval County. This final
product will give an estimation of the value of
shipments for each industry in Duval County.
V = Vms E
V. = Gross output of jth industry in Duval County.
-- -- = The average of the ratio of the value of shipments
to the value added by manufacturing for all three-
digit industries within the state.
Vms = Value added by manufacturing for each two-digit
industry within the state.
E, = Number of employees in the jth industry in Duval Count:
Es = Number of employees in the jth industry in the State o
Method II.-- Method II is merely an extension of Method I in
that the ratio of the value of shipments to the value added by manufac-
turing based on state data, is applied directly to the value added by
manufacturing for each industry within Duval County. The difference
between Method I and Method II is that the former worked with the value
added by manufacturing in the state, and scaled it down to Duval County
by utilizing a constant ratio equal to the number of employees in the
state divided by the number of employees working in Duval County. In
Method II, the value added by manufacturing was not scaled down since
it is the value added by the given industry in Duval County.
V ( Vs (Vc)
V. = Gross output of the jth industry in Duval County.
(V = The average of the ratio of the value of shipments
to the value added by manufacturing for all three-
digit industries within the state.
Vmc = Value added by manufacturing for each two-digit
industry within the county.
Method III.-- Previously, the values of shipments for the sec-
tors under consideration have been calculated by developing a general
relationship between value added in a given industry and the corre-
sponding value of shipments. However, there is another method of
accomplishing the same task, namely, the use of industry productivity
data. It is possible to obtain accurate data regarding the number of
workers in Duval County in 1958. The average productivity of the wor-
kers in a given industry based on the national data of the years 1947
and 1950 has been calculated. Therefore, in order to estimate the
value of shipments by Method III one merely multiplies the aver-
age productivity by the number of employees.
General Formula: -
Vj = (P)(N)R
Vj = Gross output of the jth industry in Duval County.
S= Average productivity of the employees in a given industry.
N = Number of employees.
R = Ratio of price index for the current year to the base
Method III assumes that the average productivity of the workers
in each industry is constant before multiplying by the ratio of the
price indexes. This assumption should not be alarming since a similar
assumption was made in assuming that the activity coefficients in 1947
are applicable in 1962. Therefore, the assumption of constant pro-
ductivity gives more consistency to the analysis even though it may
widen the gap from the 1962 data.
A Comparison of Three Methods
As previously stated, no one of the three methods is always
more accurate than the remaining two. However, of the three methods
described, Method III is probably the most accurate in calculating the
total value of shipments for the major industries in Duval County. The
reason for this assertion is that one of the two components in the
determination of the value of shipments is known quite accurately, i.e.,
the number of employees by industry in the manufacturing sector in
Duval County. The inaccuracies in this approach are found in the pro-
ductivity term since the average productivity of each worker in each
manufacturing sector in Duval County can be calculated only from secon-
dary sources. The problems involved in obtaining the average produc-
tivity for each industry are described in detail when output data are
converted to employment data.
In the construction of the Duval County Input-Output Table,
Methods I and II were not used for several reasons. First, there was
a large discrepancy between the ratio of the value of shipments to the
value added by manufacturing for the three-digit industries within
each two-digit industry classification. Second, there were not
sufficient data to provide an adequate weighting of three-digit indus-
try output in order to reconcile the aforementioned discrepancies.
Third, in many cases, the number of the three-digit industries for
which output data were furnished, represented a small portion of the
total number of three-digit industries.
Estimating Endogenous Flows
The empirical problems involved in the estimation of the endog-
enous flows are greatly reduced by the use of national activity coeffi-
cients. Given the outputs of each industry, i.e., the values obtained
by the estimation procedures described in the previous section, the
endogenous flows are calculated quite easily. The endogenous flows
for each cell of the transactions table are obtained by multiplying its
coefficient times the output of the industry, i.e., xij=ajX..
Estimation of Final Demand
After estimating the gross outputs and the endogenous flows of
each sector in the model, the next step is to allocate this aggregate
figure to the various input industries. The current problem is to
distribute that amount which is equal to the difference between the
total value of shipments and the inputs into the endogenous sector
among the competing sectors in the final demand sector, that is, the
The mere fact that a given industry is included in the exoge-
nous sector implies that its activity coefficients are not to be con-
sidered constant. However, the allocation of this total amount among
the competing industries must be made with some accuracy if any eco-
nomic inferences are to be made from the results. As previously stated,
the methodology used in interpreting the results of the model is quite
general and independent of the magnitude of the data utilized in its
The allocation of the gross output over the endogenous sectors
is solved by assuming that the national activity coefficients apply to
Duval County. The current problem is to allocate the aggregate final
demand figure over its component parts; three different approaches
were used. The first approach is one in which the aggregate final
demand figure was not broken down into its component parts. In this
method the national coefficients are applied to the estimates of gross
output for each sector in the model. By using these coefficients, the
sum of the flows in the endogenous sector are determined. The value
of the exogenous flows is calculated by subtracting the sum of the
endogenous flows from the gross output.
The second approach assumes a hypothetical breakdown of the
final demand sector. The purpose in doing this is to illustrate the
methodology involved in analyzing the contribution of each component to
the aggregate levels of output, employment, and income.
In the third approach, the aggregate final demand figure was
not distributed over its component parts. However, independent esti-
mates were made of one segment of the final demand sector, namely,
consumer expenditures by industry type. The results of this analysis
relate consumer expenditures to industrial output, employment, and
3.3 The Conversion of Total Output to Employment
and Income Data
The purpose of this section is to describe the mechanics involved
in converting output levels to employment data. The conversion of
output to employment is one of the important extensions of the model.
Although a knowledge of the output required to produce the final bill
of goods is quite important to those involved in marketing, finance,
and regional planning, a knowledge of the employment required to pro-
duce this output is equally important.
In order to make the conversion from output to employment
figures, the level of output corresponding to a given bill of goods is
divided by the output for each worker. Hence,
Output per worker = Total output for industry
Total employment (including the clerical
workers within the industry)
Total employment Total output
Output per worker
Output per worker may be calculated in several ways. Prefer-
ably, the total output of each industry in Duval County would be
divided by its number of employees. Although the denominator of this
fraction is known quite accurately, the numerator must be estimated.
Since the numerator is estimated by making transformations on the value
added by manufacturing, there might be considerable error in obtaining
any output per worker from these data. For this reason the productivity
calculations are based on national statistics which give accurate esti-
mates of both output and employment for a given year.
Total output by industry, on a national basis, is given in the
1947 table. Therefore, the productivity for the workers in a given
industry is calculated by dividing these industry outputs by the number
of employees given in the 1950 Census. In all subsequent operations
concerning the conversion of output to employment figures, the output
level of each industry is divided by its average productivity in order
to obtain the number of employees corresponding to that level of output.
Thus far the analysis has been discussed in the following
sequence: given a bill of goods, the required output was calculated;
knowing the output per worker the level of employment induced by the
change in output was determined. The induced employment due to the
increase in output can be related to income figures by having a know-
ledge of the average income of the workers in each industry.
8Evans and Hoffenberg, passim.
9U.S. Department of Commerce, Bureau of the Census. U. S.
Summary, U. S. Census of Population: 1950.
In subsequent discussions in which employment data are converted
to income figures, the following transformatiorsare used:
Industry Income = (Number of workers) (Average income for
This chapter contains the empirical results obtained by
applying the statistical procedures described in Chapter 3 to the
secondary data of Duval County, Florida. Generally, no additional
comments or restatements of the information contained in Chapter 3
are made in this chapter.
4.1 Basic Data for the Construction of an Input-Output Table
Major Industries in Duval County, Florida
TABLE 4.1-1--MAJOR INDUSTRIES IN DUVAL COUNTY, FLORIDA
Industry Number of Number of Value added by
Establish- Employees manufacturing
Food 88 4,195 49,677
Lumber 63 1,107 5,691
Furniture 38 682 4,835
Paper 19 1,909 29,142
Printing 71 1,560 11,630
Chemicals 30 1,093 12,947
Stone, Clay, and Glass 27 1,227 12,407
Fabricated Metal 59 1,144 10,500
Transportation 23 2,746 15,773
Eating and Drinking 583 1,341a 31,988b
Source: U S. Department of Commerce, Uureau of the Census.
1958 Census of Manufacturers: 1961; Area4Statistics, Vol. III.
Gross Outputs of the Major Industries
TABLE 4.1-2--GROSS OUTPUTS OF THE MAJOR INDUSTRIES
Industry Averagea Number ofb Output 1947 Output 1962C
Productivity Employees (Dollars) (Dollars)
Food 26,859 4,195. 112,673,505 149,855,762
Lumber 6,975 1,107 7,721,325 10,269,362
Furniture 8,757 682 5,972,274 7,943,124
Paper 16,937 1,909 32,332,733 43,002,534
Printing 7,506 1,560 11,709,360 15,573,448
Chemicals 21,317 1,093 23,299,481 30,988,309
Stone, Clay, and
Glass 10,487 1,227 12,867,549 17,113,840
Fabricated Metal 2,762 1,144 3,159,728 4,202,438
Transportation 16,397 2,746 45,026,162 59,884,795
Drinkingd 7,870 1,341 10,553,670 14,036,380
Source: U. S. Department of Commerce, Bureau of the Census.
Statistical Abstract of the United States: 1962. Table 465.
CRatio of price index for 1962 to price index for 1947 is 1.33.
dThe output of this sector applies only to male help.
Estimation of Endogenous Flows
Activity Coefficients.-- Activity coefficients represent the
value of input used per dollar of output. The national activity
coefficients were applied to Duval County. The activity coefficients
are the basic elements in the estimation of endogenous flows since the
product of an activity coefficient and its corresponding industry
output, gives an estimation of interindustry flows, i.e., xi = ai X..
The activity coefficients for the major industries in Duval County are
given in Table 4.1-3. From Table 4.1-3 one can determine the amount
of goods required from each industry in the model in order to produce
$1.00 of output. For example, it is observed that the Chemical indus-
try must purchase the following amounts of goods to produce $1.00 of
Chemical products: 4.9 cents from the Food industry, .3 cents from the
Lumber industry, nothing or a negligible amount from the Furniture indus-
try, 2.4 cents from the Paper industry, .1 cent from the Printing indus-
try, 19.1 cents internally, 1.9 cents from the Stone, Clay and Glass
industry, and nothing or a negligible amount from the Fabricated Metal,
Transportation, and Eating and Drinking industries. The inputs from
the exogenous sectors of the model per dollar of chemical output are
not calculated since the activity coefficients in the exogenous sectors
are not considered to be constant.
Example of the Calculation of Endogenous Flows.-- From the
xij = aijXj
it follows that:
(4.1-1) xll = allX
(4.1-2) x74 = a74X4
Equation (4.1-1) states that the output of the Food industry
going to the Food industry is equal to the appropriate activity coeffi-
cients times the output of the Food industry. (.131870)($149,855,262) =
* *. i
40 L00 0
C'M 0 0 04-
, 0 00
$.4*J *l. U
.0 a 0
E 4 0a4 0)
c u o
o0 o O
4-J a '
0 D .-4
0 C' C'M
00 en o
4t CM4 0 00
CM m o
-n -00 o0
S(cO c- O
a% 00 ( I
LrD 0 nOo
-4 U 4J 1-
U) U 0.
-4 0) M-4 C)
0) 0.0 C
Similarly, equation (4.1-2) states that the output of the Stone,
Clay, and Glass industry which goes to the Paper industry is equal to
the appropriate activity coefficient times the output of the Paper
industry, (.003641)($43,002,534) = $156,672. A repetition of this
procedure for all industries in the model will lead to the estimation
of all the exogenous flows in the model.
Estimation of Final Demand
Final demand estimates are obtained from an algebraic solution
of the basic model.
Sxij + Yi = Xi
Yi = Xi xij
For example, consider the four industry examples in Chapter 2;
the final demand for industry 1 is given by
(4.1-3) Y1 = X1 xl x12 x13 x14
Equation (4.1-3) states that the final demand for industry 1 is
equal to the total output of industry 1 less all interindustry (endog-
A Numerical Example for the Duval County Model.-- For the Food
X1 = $149,855,762 (Table 4.1-2)
The interindustry endogenouss) flows for the Food industry
were calculated as described in the previous section. The following
are a result of these calculations:
Y = 149,855,262 (19,762,006+585+1,998+163,581+825+1,525,168
+6,982+366,906) = 128,028,516
A similar calculation for each industry in the table will give
the final demand for each industry in the model.
4.2 The Input-Output Table
The empirical results of Section 4.1 may be summarized in a
table commonly called an input-output or transactions table. The con-
struction of an input-output table, without any additional modifications
or extensions, gives an insight into the functioning of a given complex.
In essence, it is a method for summarizing interindustry flows in a
concise logical manner. The use of these interindustry flows in market
analysis is described in a following chapter. Table 4.2-1 is the input-
output table for Duval County.
The table is constructed by filling either n rows or n columns.
Each row total represents the sales of that given industry; each
column total represents the purchases of the industry, i.e., every
sale is a purchase. This symmetry in the flow table gives rise to the
name input-output table. By examining the fifth row in Table 4.2-1,
the Printing industry has a total output of sales of $15,573,448 of
which $35,078 was purchased by the Chemical industry. Likewise by
examining the sixth column it is observed that the Chemical industry
purchased $35,078 of materials from the Printing industry.
The Row Entries
Each row total represents the total output or sales of the
industry, valued at producer's prices. For example, reading across the
fourth row, the Paper industry sold $1,823,044 to the Food industry,
$7,786 to the Lumber industry, and $5,980 to Eating and Drinking places.
Out of a total output of $43,002,534, $22,690,764 is sold in the exog-
enous or final demand sector of the economy. The total sold in the
endogenous sector for any given industry is equal to the difference
between total output and the total value of exogenous flows (final
demand) i.e., $20,311,770.
The final demand figure is made of many elements, In this model,
the following sectors: Government, Inventory Change, Gross Private
Capital Formation, Construction, Households, and Imports comprise the
aggregate value of exogenous flows. This breakdown is quite arbitrary
and may change from model to model depending on the available data and
the objective of the analysis. The present discussion is concerned