This dissertation has been — microfilmed exactly as received 67>-16f329
QUINN, Michael Thomas, 1938- SOME EFFECTS OF GEOGRAPHIC PRICE POLICIES ON SELECTED VARIABLES IN THE STEEL AND BELT INDUSTRIES.
The Ohio State University, Ph.D., 1967 Business Administration
University Microfilms, Inc., Ann Arbor, Michigan SOME EFFECTS OF GEOGRAPHIC PRICE POLICIES
ON SELECTED VARIABLES IN THE
STEEL AND BELT INDUSTRIES
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By Michael Thomas Quinn, B.A., M.B.A.
******
The Ohio State University 196?
Approved by 1A)'(Lit- Adviser Department of Business Organization ACKNOWLEDGMENTS
I take this opportunity to thank the members of my com mittee, who have provided counsel and guidance throughout the writing of this dissertation: Dr. James A.. Black,
Dr. Emilio Casetti, Dr. Alvin E. Coons, Dr. Fred E. Kindig, and Dr. Leslie J. King.
In addition, I would like to thank my original adviser,
Dr. J. L. Heskett, for his many valuable suggestions and the impetus he provided in the early stages of this dissertation.
Especial thanks go to Dr. W. Arthur Cullman, whose con
stant encouragement and Interest have been an Indispensable
aid.
I wish to acknowledge that I have had free use of the
computing facilities of the Numerical Computation Laboratory
of The Ohio State University and the Western Data Processing
Center at the University of California, Los Angeles.
Lastly, I would like to thank Katharine G. Hoch and
Kristin H. Quinn for editing and typing assistance so help
fully provided.
ii VITA
July 1, 1938 Born - Philadelphia, Pennsylvannia
1959..*..... B.A., Cornell University, Ithaca, New York
1961...... M.B.A., Cornell University, Ithaca, New York
1964-1965.... Teaching Assistant, Department of Business
Organization, The Ohio State University, Col
umbus, Ohio
1965-1967*••• Acting Assistant Professor, Graduate School
of Business Administration, University of
California at Los Angeles, Los Angeles,
California
FIELDS OF STUDY
Major Field: Business Logistics, Professor J, L, Heskett
Minor Field: Economics. Professor C. L. James
Quantitative Methods. Professor J. A. Black
Mathematics. Professor Jesse Shapiro
ill TABLE OF CONTENTS
Page
ACKNOWLEDGMENT li
VITA ill
LIST OF TABLES vl
LIST OF CHARTS vii
LIST OF ILLUSTRATIONS viii
Chapter I. INTRODUCTION 1 Purpose Demand for Transportation Services Geographic Price Policies Hypothesis Research Methodology Model Construction Limitations of the Study
II. LITERATURE REVIEW...... 16
Introduction Argument Related Models Theory of the Firm Location Models Optimal Models Trading Models Summary of Related Models
iv Chapter III. THE MODEL
Introduction Simulation Model Characteristics Important Relationships Sellers' Behavior Buyers' Behavior Experiments Model Equations Solution of Equations Model Inputs Output Sequence of Operations
IV. RESULTS 92
Graphical Representation of the Data Effects of the Model Variables Effects of Parameter Changes Transportation Costs and Cross-Hauling Production Costs Baslng-Point Profits Prices Summary of Analysis Summary
V. CONCLUSTIONS...... 137
Introduction Geographic Demand Stimulation Geographic Demadn Stimulation Factors of the Price Policies Conclusion about Hypothesis Variables Hypothesis Validation Summary of Hypothesis Validation Real World Conclusions Conclusions about Key Model Variables Conclusions about the Model Conclusions about the Use of the Iterative Method Suggestions for Further Work Implications for Business and Government Policy
APPENDIX A. 161
APPENDIX B 175
BIBLIOGRAPHY 191
v LIST OF TABLES
Table Page 1. Experiment Parameter Settings...... 82
2. Average Distance Between Sellers and Buyers...... 95
3. General Rankings of Transportation Costs and Cross-Hauling...... 112
4. General Rankings of Production Costs...... 120
5. Production Costs...... 123
6. General Rankings of Prices...... 134
7. Experiment 1 Data ...... 162-163
8. Experiment 2 Data...... 164-165
9. Experiment 3 Data...... 166-16?
10. Experiment 7 Data ...... I68-I69
11. Experiment 8 Data...... 170-171
12. Experiment 9 Data ...... 172-173
13. Basing-Point Profits...... 174
vl LIST OF CHARTS
Charts Pages 1-3 Belt Industry Average Transportation Costs...... 99-101
4-6 Steel Industry Average Transportation Costs...... 102-10^
7-9 Belt Industry Cross-Hauling...... 105-107
10-12 Steel Industry Cross-Hauling...... 108-110
13-15 Belt Industry Average Production Costs...... 113-115
16-18 Steel Industry Average Production Costs...... 116-118
19-20 Belt and Steel Industry Average Profits...... 125-126
21-23 Belt Industry Average Prices...... 128-130
2^-26 Steel Industry Average Prices...... 131-133
vli LIST OF ILLUSTRATIONS
Figure Page 1. A Diagram Showing F.o.b. Mill and Basing-. Point Arguments...... 21
2. The Model of Hotelling and Chamberlin Showing Positions of Sellers A. and B. on the Linear Market L of Length a+x+y+b...... 25
3. A Graphic Representation of von Thunen's Theory...... 29
4. Weber's Location Theory in Terms of Transport and Non-Transport Factors...... 31
5. A. Construction of Weber's Isotims and Isoda- panes Showing the Interrelated Influence of Transportation and Production Costs on the Selection of a Production Site...... 33
6. The Launhardt-Palander Construction of Transport-Orientation...... 35
7. Machlup's Grid Construction Showing the Placement of Four Mills...... k2
8. A Diagram Showing Production Cost Functions of the Belt and Steel Industries...... 59
9. A Matrix Diagram Showing the Locations of the Buyers and Sellers...... 6l
10. A. Diagram Showing the Potential Behavior of a Function Given an Initial Price...... 79
11. Main Model Flowchart...... 86
12. Cost Subroutine...... 87
13. Iteration Subroutine...... 89
14. F.o.b. Subroutine...... 90
15. A Hypothetical Seller at X and Buyers at A and B for the Demand Stimulation Example 139
viii CHAFTEB I
INTRODUCTION
Purpose
Geographic price policies vary greatly, yet the rela tive impact of these policies on sellers' costs, prices and profits has not yet been determined, largely because of in sufficient data. The main purpose of this dissertation is, by generating reasonable data on the subject", to study the effects of these geographic price policies on transportation and production costs, cross-hauling, prices and profits.
A further purpose is to provide a possible framework
for the study of problems having geographic dimensions, such as location of facilities, routing of transportation equip ment, and regulation of transportation modes. In so far that distance through product availability affects product dif-.
ferentiation, business promotion problems are included.
As an independent variable, transportation cost has a
direct effect on the development of geographic areas and on
the performance of geographic operations. As a dependent
variable, actual transportation charges for a buyer vary
according to a seller's geographic price policies. There
fore, transportation cost is the key dependent variable in
this dissertation. 2
Demand for Transportation Services
There are many factors in an economy which determine the demand for transportation services. This demand is largely a derived demand based on the utility of services to com modities. The following factors affect the amount and flow of commodities transported:
1. spatial characteristics of the economy,
2. size and type of economy,
3. governmental policy,'*'
These factors, in turn, affect the quantity and quality of transportation services.
Spatial Characteristics
Spatial characteristics of the economy have a great effect on the. amount and flow of commodities because the spatial array of raw materials, factories, and markets largely determines shipping patterns. For Instance, fac tories may locate close to raw materials, perhaps at long distances from dispersed market areas, and may thus require
The sources for this section are as follows: C. J. Zwick, Demand for Transportation Services in a Growing Ec onomy, P-26&2, Rand Corporat1on~Paper 1, J. L. Heskett, Robert M. Ivie and Nicholas Glaskowsky, Jr., Business Logistics (New York: The Ronald Press Company"] 196*0", Frank H. Mossman and Newton Morton, The Logistics of Dis tribution Systems (Boston: Allyn and Bacon, inc., 19&5). and Walter I sard, Location and Space Economy (New York and Cambridge: John Wiley and Sons, Inc., and the Technology Press of the Massachusetts Institute of Technology, res pectively, 1956). long shipments from the factories to the markets. If mar kets are generally concentrated, factories need not be located near their raw material sources, thus manufacturers may locate close to their markets and transport raw materials instead. In the United States, refineries have been located either close to the sources of crude oil or close to markets
for refined products, requiring long shipment of refined pro ducts in the first case and crude oil in the second. As ad vancing technology changes industrial processes and materials
incorporated in finished products, the types, amounts, and
sources of raw material shipments change accordingly. For
example, the use of fuel oil for heating purposes has dras
tically reduced the demand for anthracite coal, eliminating
a large amount of traffic on railroads built to carry this
coal.
Regionalization, or the development of relatively self-
suffic.lent regions, may reduce long shipments by lessening
the need for inter-regional hauling. The Soviet Union has
attempted to regionalize its economy for defense reasons and
has reduced the amount of cross-hauling raw materials and 2 finished products. Urbanization may reduce self-sufficiency
and generate more traffic, as food and raw materials are
shipped into and finished products out of urban areas. For
2 Harry Schwartz, Russia’s Soviet Economy (Englewood Cliffs, N. H. : Prentlce-Kail,“i nc77~1958) 7 p “ . 400. example, in countries which have had their transportation networks essentially destroyed in a war, cities have great difficulty in acquiring minimum food and coal requirements.
The topography of the area has an.effect on the move ment of commodities. For instance, mountain chains hinder
commodity movement to some extent, while, on the other hand,
numerous navigable rivers greatly facilitate it.
Size and Type of Economy
Another factor is the size of the economy and its
growth rate. The bigger the economy, the more commodities
and people there will be to move around. Business fluctua
tions affect the amount of traffic, causing an increase dur
ing expansion and a decrease during contraction. Forecasters
use freight carloadings as a good indicator of present bus
iness activity.3
The development of the economy affects demand for
specific commodities. If the economy is in the primary or
agricultural stage, as the United States was in the early
l800»s, a large movement of bulk commodities to production
centers or shipping ports is required, with a corresponding
backhaul movement of lighter, more valuable manufactured
goods. When the economy enters the secondary or manufactur
ing stage, as the United States was in the late 1800’s,
3 •^Robert Aaron Gordon, Business Fluctuations (New York: Harper and Brothers, 1952), p. 139. manufactured goods make up a growing share of commodities hauled. When the economy reaches the tertiary or service stage, as in the post-World War II United States, services or Intangibles make up a larger share of the economy’s out put. These items do not create a large demand for transpor tation services. These stages of the economy affect the types of commodities hauled and the types of equipment util ized. Disposable income in the economy also affects the amount of movement. As disposable income grows, markets for such localized commodities as caviar can be expanded be cause consumers can afford the higher costs of long-distance transportation. On the other hand, perhaps a significant portion of an increase in disposable income may be spent on services or luxuries which, except for personal travel, do not involve transportation services.
Governmental Policies
Governmental policies concerning the regulation of in ternal and external trade Influence the demand for transpor tation services.
A policy of free external trade may result in a heavier
traffic through ports, stimulating both ocean and surface
transportation. A. restrictive policy generally has the
opposite effect.
Furthermore, regulation of transportation agencies in
fluences the movement of commodities. The rate-making policy determines how much it costs to move each commodity class. For example, the Soviet Union has stimulated region alization of its economy by penalizing long hauls with very k high rates. Conversely, in the United States, the Inter state Commerce Commission has granted relatively low rates to carriers soliciting long haul business.^ The rate policy can affect the length and direction of commodity movements; it can, in fact, partially determine whether a commodity will move at all.
The amount of government control of the economy has an effect on the amount and direction of commodity movement. A planned economy may have a smaller commodity movement than an unplanned economy, in part because of reduced market inter penetration. Market Interpenetration may occur in an un planned economy as competitive firms try to increase or pro tect their market positions. This interpenetration results from the desire of firms to reduce their dependency on a certain geographic market, to increase sales by Invading new market areas, and to offset Invasion by competitors.
Regardless of the reason for market Interpenetration, sellers must adopt a geographic price policy which will determine who selects modes of transportation, and the
4 . Schwartz, op. clt., p. 402.
•^John R. Meyer, et al, The Economics of Competition in the Transportation Industries (Cambridge: Harvard University Press, 19597, P* 177. routing of shipments, and the amount and payment of trans portation charges. This dissertation is concerned with how
the amount of transportation charges billed to the buyer will
affect the costs, prices, and profits in two types of in
dustries.
Geographic Price Policies
The geographic price policies considered in this disser
tation are the f.o.b. mill, basing-point, zone and uniform
policies.^ The policies were defined to be:
F.o.b. Mill.— Normally, sellers in an Industry quote
buyers a price f.o.b. (free on board) at their respective
producing plants. The buyer pays any transportation charges
incurred in the transaction.
Baslng-Folnt.— Normally, sellers in an industry quote
buyers a price which is composed of the basing-point stan
dard production cost plus published transportation charges 7 from the basing-point to the buyer..1' The location of the
seller is immaterial in determining the price because the
^Adapted from Joel Dean, Managerial Economics (Engle wood Cliffs, N.,.J.: Prentice-Hall, Inc., 195T) , pp. 5^1-5^8. 7 For the basing-point, zone and uniform policies, it is assumed that Industry use of each respective policy has be come customary over time. It is also assumed that the in dustry trade association publishes facts concerning industry costs which each seller Independently decides to use as a basis for setting geographic prices. price is not based on actual transportation charges between the seller's plant and the buyer.
If these actual transportation charges are higher than those included in the basing-point price, then the seller absorbs freight. If the actual charges are smaller, the seller receives phantom freight.
Zone.--Normally, the total market is divided into agreed-upon geographic zones and all buyers in a given zone pay identical delivered prices.
Uni form.— Normally, sellers in an Industry quote buyers a single delivered price regardless of the buyer’s location.
It should be noted that the last three policies do not incorporate the actual transportation charges in the prices.
Hypothesis
The hypothesis of this dissertation is that an f.o.b. mill price policy when compared with three other geographic price policies will result in:
A. the lowest average transportation costs,
B. the lowest average cross-hauling,
C. the lowest average production costs,
D. the lowest average profits, and
E. the lowest average prices.
This hypothesis was chosen because of the debate, which will
be discussed in Chapter II, over the relative efficiency of
basing-point and f.o.b. mill pricing. Part of the debate focused on the incidence of cross-hauling under the basing- point price policy. Cross-hauling has various definitions.
U.S. Steel’s definition of cross-hauling is that products going past each other in different directions involve cross- hauling only if: (1) the products are identical, (2) ship- g ments occur at substantially the same time. Cross-hauling in this dissertation is defined as two or more sellers ship ping similar products into each others’ home markets, or as
Stratton puts it, market interpenetration.^
Profits are defined here to be earnings of the sellers in excess of interest on their owners’ invested capital.
This interest function is assumed to be incorporated in the sellers’ production functions. Scholarly opinion supports the contention that the f.o.b. mill price policy leads to lower prices and costs than the basing-point policy. In the model discussed in Chapter III, parameters are specifically selected to enable analysis of key Issues developed during this controversy.
Research Methodology
The research methodology for this study was experimen tation through the manipulation of a mathematical model.
^U. 8. Steel Corporation, T.N.E.C. Papers (Pittsburgh: United States Steel Corporation,“l'940) , p. 1^. 9 ■ Carrol R. Daugherty, Melvin G. de Chazeau and Samuel S. Stratton, The Economics of the Iron and Steel Industry (New York: McGraw-Hill Book Co., Vol. II, 1937). P* 6?5. 10
This model describes or simulates the competition of two sellers in markets located on a plane.
Simulation is.commonly defined as the construction and manipulation of a model which, in turn, embodies and illus trates a process. As Chorafas states, a simulation model of this kind might be called a working analogy of a physical reality.^
The purpose of the model was to develop this disserta tion, since it would have been impossible to conduct the ex periments in real business situations. Therefore, in place of an actual business, this model was constructed in which real business operations were reduced to mathematical equa tions calculated to create an environment sufficiently close to reported business practices to permit valid conclusions.
It was thereby possible to observe and control the behavior of the model during several different iterations and by
this means answer the purpose of the model.
To meet its purpose, the model in Chapter III is a
compromise between the generality of an abstract model and
the immediacy of a case study. It is first of all mostly
abstract in order for its results to have the widest pos
sible applicability. At the same time, the decision rules
and cost functions used within such a model are taken from
real business situations without sacrificing the applicability
■^Dimitris N. Chorafas, Systems and Simulation (New York: Academic Press, 19^5)t p. 1?. and generality of the whole. This combination, then, of an abstract scheme into which more or less actual data are put makes this model all the more useful as a guide in studying related business practices outside the immediate concern of this dissertation and, at the same time, more easily relat- able to the business situation it was specifically con structed to illustrate.
Model Construction
The process followed in constructing the mathematical model for the simulation process was as follows:
1. Review and analysis of the existing mathematical models dealing with transportation costs to determine the model form best suited to the purposes of this dissertation.
2. Exploration of business practices reported in the literature to determine which functional relationships should be included in the model. Special attention was focused on sellers' and buyers' practices dealing with:
a. production processes,
b. cost accounting and price determination,
c. geographic price policies,
d. market participation,
e. price elasticity of demand.
3. The reduction to-verbal statements of specific de cision processes concerning prices, production, demand, and market participation. 12
4-. The translation of these verbal statements of be havior into mathematical statements.
5. Through a survey of the literature on the steel and belt industries and a regression analysis, the choice and estimation of model parameters for transportation and produc tion functions and the price elasticities of demand.
6. The examination of the mathematical statements to ensure that once they were initialized, the first iteration would generate data for the next.
7. The translation of the mathematical statements into the FORTRAN IV compiler language and the trial testing of the feasible program on an IBM 709^' to examine model incon sistencies.
8. The running of several experiments, each with dif ferent sets of parameters, for each of the four price pol icies to generate data for testing the hypothesis. The value of the key dependent variables (when the model reached positions of equilibirum) were the data of importance to this study. These variables were:
a. transportation cost per unit,
b. unit miles per unit (a measure of cross-
hauling) ,
c. production cost per unit,
d. average price, and
e. profits per unit (1 - (2 + 3)). 9. Data collection, graphing, and analysis to under
stand the effect of the different parameters on the models and to analyze the output in light of the hypothesis using
the process of comparative dynamics. The f.o.b. mill price policy results were accepted as the norm, to which the re
sults of the basing-point, zone and uniform price policies
were compared.
10. The hypothesis was tested and accepted or rejected.
Limitations of the Study
The first limitation of the dissertation is that it is
not concerned with the promotional aspects of competition.
This becomes Important under f.o.b. mill price policy when
one seller is suffering drastic market share slippage in
several markets because of his non-competitive prices. In
this situation, the only solution to this slippage would be
product differentiation through heavy promotional expendi
tures.
The second limitation of this study is that it is not
concerned with the question of interdependency. The duopoly
potential resulting from there being only two sellers was
not developed. Thus the kinked demand curve, collusion and
product differentiation were not included in the model.
The third limitation is that the structure of the mar
keting channels in the model is very simple; there is only 14 the manufacturer-seller and the ultimate customers. Thus the whole question of the effects of wholesalers and re tailers on the market is not considered.
A related limitation is that inventory was not Included in the model. Ordering, manufacturing, and delivery times are assumed to "be zero, which avoids all inventory control problems.
The fifth limitation is that the legality of the dif ferent price policies was not of interest here, despite the fact that much of the literature on geographic price pol icies was concerned with influencing court and regulatory agency decisions concerning the basing-point policy.
The spatial framework is simple and Idealized in that market locations are symmetrical, and discrete, and there are not barriers to efficient transportation. This might be an additional limitation. It was not a purpose of this dis sertation to explore how concentrated or dispersed markets, physical barriers, or specific transportation modes affect the price policies vrhich in turn modify costs and prices.
The last limitation is due to the iterative process of
searching for a solution. This process can neither give a \ general solution to the set of equations in the model, nor
can it give the exact values of the particular solution.
Thus, this dissertation does not produce fmictions which
would determine which geographic price policy would contribute 15 most to the goal of efficient allocation of resources. It does try to show, however, given certain conditions, the possible allocations of the price policies.
The broad purpose of this dissertation is to gain know ledge about some of the effects of geographic price policies.
The specific purpose is to test the hypothesis about the effects of these policies on certain production and transpor tation operations of two sellers competing on a spatial plane. To test the hypothesis, a set of experiments was devised to be run on a deterministic, mathematical model.
The model is relatively abstract, but was parameterized specifically to add insight to the controversy about the relative efficiency of geographic price policies. This argument was carried on without the benefit of much empirical data. Since the collection of empirical data on the effects of geographic price policies is difficult and expensive, it is hoped that this dissertation can help to provide a theoretical basis to determine these effects. CHAPTER II
LITERATURE REVIEW
Introduction
In this chapter the literature used as a guide for the selection of the hypothesis and the construction of the model is examined. In.the search for the selection of the hy pothesis, one fact was readily apparent: There is a great deal of literature concerning the relative efficiency of the f.o.b. mill price policy as opposed to the basing-point price policy because the controversy was a semi-political
Issue and the subject of numerous hearings of the Federal
Trade Commission. Most of this voluminous literature is not
supported by empirical data. Few scholars attempted to col lect empirical data to support their arguments on the ef
ficiency of the two price policies and their data have been
criticized as being inaccurate. The problem is compounded
by disagreement on definitions of cross-hauling. Below,
there is a brief exploration of the essentially intuitive s argument concerning f.o.b. mill versus basing-point price
policies and some discussion of the very few attempts to
collect empirical data on the subject.
The second part of the chapter is concerned with the
description of mathematical models incorporating distance 16 17 and transportation costs as both independent and dependent variables. Attention will be focused upon models most closely meeting the purpose of this dissertation, specifi cally, the models of Enke, and Machlup, and the Cobweb model.^
Argument
The argument for the superiority of the f.o.b. mill price policy is based on the belief that market interpene- 2 tration will be a minimum under such a policy-
The basing-point price policy was devised to facilitate market interpenetration, which may cause a large amount of cross-hauling. Cross-hauling, as defined above, is two or more sellers shipping comparable products into each other's market territories. George W. Stocking states that "...(the
basing-point price policy) increased the average cost of
shipping steel by increasing the average distance which steel
was shipped."-^ This Increase in transportation costs will
result in higher prices. As Charles M. Schwab stated,
■^Stephen Enke, "Equilibrium Among Spatially Separated Markets: Solution by Electric Analogue," Econometrlca, Vol. IXX (January, 1951). PP* 4-0-47, Fritz Machlup, The Baslng-Polnt System (Philadelphia: The Blakiston Company, X949"), "pp. 190-202, and R. G. D. Allen, Mathematical Economics (London: Macmillan and Co., Ltd., 1959), PP. 2-12. 2 Machlup, loo, clt.
^George U. Stocking, Baslng-Polnt Pricing and Regional Development (Chapel Hill: The University of North Carolina Press, 1954') » P» 128. 18
"...the net result of the cross-hauling of materials has not been to increase the output of the individual producers by any appreciable amount. It has merely served to dissipate a h part of their profits in unnecessary transportation." Ac cording to Stigler, however, not much cross-hauling should transpire under the basing-point price policy.
Unfortunately, the little empirical data bearing on these arguments is not acceptable to many writers. The work of the National Recovery Administration, however, tends to show that there was a large amount of market interpene tration in the steel industry. This study, however, was based on the examination of actual railroad bills of lading and has been Judged inconclusive because of the lack of exact product specifications.^
In another study based on empirical evidence, McGee examined steel shipment records to determine, if possible, the extent of cross-hauling of hot and cold rolled sheet
A Charles M. Schwab, Report of the Federal Trade Com mission to the President with Respect to the Baslng-Polnt .. System In the Iron and Steei Industry, ("Washington: 193*0 . p T T X T
■^George J. Stigler, "A Theory of Delivered Price Systems," American Economic Review. Vol. XXXIX (194-9), pp. 114-3-59.
^See Samuel S. Stratton's interpretation of the results published in the Supplement No. 1 to the Report of the National Recovery Administration on the Operation of the Baslng-Polnt System in the Iron and Steel Industry, Nov. 30, 193^.in"Daugherty, de Chazeau, and Stratton, op. clt., p. 667. 19 7 between Detroit and Chicago, His data show a substantial
amount of cross-hauling during the period which he studied.
He felt his data were not complete enough for a flat state ment on the existence of cross-hauling between these two markets.
The argument for the superiority of the basing-point price policy concedes the cross-hauling issue. For, as the
U, S. Steel source book states, "the only,. .saving., .which would result froEi an f.o.b, mill price system would be in O its elimination of certain amounts of transportation costs."
However, the key argument in favor of a basing-point price policy lies in the nature and effect of the production func
tion of the plant which, in the case of the steel Industry, has relatively high fixed costs and low variable costs.^
Supposedly, if the plant has high fixed costs and low vari
able costs, the extra business brought in by market inter
penetration and attendant cross-hauling can be produced at
relatively low cost. Simon states that
sellers generally expand the territory in which they sell their goods because they seek a greater volume of production. In creased volume spreads the overhead and
7 J. S, McGee, "Cross-Hauling: A Symptom of Incomplete Collusion under Basing-point Systems," The Southern Economic Journal. Vol. XX, No.(April, 195*0, 8 United States Steel Corporation, United States Steel Corporation T.N.E.C. Papers (3 vols.; New York: United States Steel Corporation, 19*J-0) , p.. 98.
^Stocking, op. clt., p. 23. 20
usually results in manufacturing savings which reduce production costs...The econ omies In production resulting from in creased volume may be expected to more than equal the cost of.absorbing freight to obtain that volume.
Proponents of the basing-point price policy argue that increased unit transportation costs will be more than offset by decreased unit production costs, resulting in lower prices.
(See Figure 1).
This argument has been attacked by Machlup.^ In the case of the steel Industry, where the assumption is made that the price elasticity of demand for steel is less than unity, what one seller gains by market penetration is at the expense of other sellers. Thus, it would appear that, on an Industry basis, sellers could not gain extra business through market interpenetration.
There is no suitable empirical data on production costs which would bear on the critical issue of whether the in
creased transporation costs resulting from the basing-point price policy can be offset by production cost savings, which
is a critical issue in the hypothesis.
Related Models
Several types of models have been developed which deal
with transportation and production costs. Most models treat
■^William Simon, Geographic Pricing Policies (Chicago: Callaghan and Company, T 950I , pp. SIT-29.
^Machlup, op. c-lt., p. 225. Costs F.o.b. Argument Baslng-Polnt Argument (lower Trans. Cost) (lower Production costs) Basing- Point F.o.b. Costs Costs
Basing- F.o.b. Point Costs Costs $
$ \
& Production Costs
Transportation Costs
FIGURE 1
A Diagram Showing F.o.b. Mill and Basing-Point Arguments 22 transportation costs as one of the independent variables, as , for example:
1. In the theory of the firm: Production Costs =
f(Market Structure),
2. In various location models: Least Cost Loca
tion = f(Transportation Costs, Other Variables),
and
3. In optimal models: Lowest Transportation and
Production Costs = f(Locations, Constraints).
These model types will be discussed below, with stress on their relevance to the purpose of this dissertation.
Theory of the Firm
Production Costs = f(Market Structure) 1 * 1 1 ' ■ T r'' r 1-1 - ''r m~~" ■" ■ ■ r i Perhaps one of the most basic models in economics is the 12 theory of the firm. This model is based on a rational firm's maximizing a set of outputs with a set of inputs, sub
ject to constraints. Under the goal of profit maximization, a firm will force production up to the point where marginal costs equal marginal revenue. However, all activity in this market takes place at a point. This eliminates the aspect of distance and therefore transportation costs.
Without consideration of distance, trade theory econ-
12 Paul Anthony Samuelson, Foundations of Economic Anal ysis (Cambridge: Harvard University press, 1948”) , p.' 88". 23 omlsts had trouble explaining international trade phenomena, restricting their reasons for national' advantage to the loca tion of raw materials, technological factors, and labor costs. They could not discuss transportation cost differen tials due to locational advantages. Thus they discussed:
a one point world, which somehow or other is conceived as divided into n points, representing n nations between which trade and trade barriers exist. ^
The theory of the firm is inadequate for the purposes of this project because it does not include transportation costs.
Locational Models
Least Cost Location = f(Transportation Costs, Other Variables)
Linear Models
Economists such as Hotelling and Chamberlin conceived a linear model along which two or more competitors could 14 operate. The objectives of these linear models were to determine the most advantageous location for maximizing pro fits. Market areas, profits, and prices can be determined analytically with this model. The model's purpose is norm ative, that is, to show how to compete on a linear market with transportation costs as the key independent variable.
13 "\Jacob L. Mosak, General Equilibrium Theory in Inter national Trade (Cowles Commission Monograph No. 7; Blooming ton, Indiana: 1§44), pp. 64— 65. 14 Edward Chamberlin, The Theory of Monopolistic Com petition (Cambridge:- Harvard Unlveriity Press, 1936), Appendix A. 24
In general, the functional relationships of these models are as follows:
Location = f(Transportation Costs)
These models have one dimension, a straight line. The
purpose of these linear models is to find market areas and
profits and to predict the sellers' equilibrium locations
and other competitive decisions which would be possible and
necessary because of the effect of the sellers' different
relative positions on the line. In other words, they would
show how to compete in a linear market, with transportation
costs the determinant of market boundaries.
In Figure 2, buyers are uniformly distributed along
the line L. A and B are two sellers. Transportation costs
are c per unit distance; costs of production are zero. One
unit of commodity is consumed at each unit of length of line
for each time period, and demand is extremely inelastic. The
lowest transportation cost, using f.o.b. mill price policy,
gets all the sales:
Let: P^ be A's price
T?2 be B's price
be A's quantity sold
Qg be B's quantity sold
Let a + x + y + b be the total market.
To find the length of x and y, that is, A's and B's
sales region, set the delivered prices equal
P^ + cx = P 2 + cy 25
A B .1.., _L_ a
FIGURE 2
The Linear Model of Hotelling and Chamberlin
Showing Positions of Sellers A and B on the
Linear Market L of Length a+x+y+b. 26
solving: x=|(L-a-b-P1 -P2^ c /
y = -| (L - a - b + Px - P2 j profits and prices can be similarly computed.
This model can be expanded to determine the best loca tion on the line for each seller relative to the other seller(s). Hotelling concludes that two sellers will bunch in the middle of the line instead of locating at the quartlle points which would maximize buyer convenience. If more sellers are added, then dispersion will take place because some sellers can gain an advantage over the others by moving.
Although it treats transportation cost as an independent variable, the linear model can be used to study problems similar to the one in this dissertation. The advantage of this approach is that sellers and buyers are located on a straight line simplifying the problem of locational inter dependency, Given this simplification and a relatively non complex set of demand and market share relationships, the model equations can be reduced to a single difference equa tion and solved analytically.
Location Theory School
Many economic geographers are primarily interested in
spatial location and development with emphasis on transporta tion costs (distance) as a key independent variable. In general, their models are normative, that is, they try to show what would be the rational locational configurations of economic units given a certain set of circumstances.
Generalized Model
The generalized model of the economic geographers is interesting in that it deals with the important variables of the dissertation model (except for the price policies).
A3.though the economic geographers are primarily concerned with location as a function of transportation costs, it is the purpose of this dissertation, in part, to determine if location will modify the effects of geographic price pol icies on transportation costs. The important variables of the economic geography model are as follows:
Location = f(Transportation Costs, Production Costs
and Demand Location)
Economic geographers since von Thunen have recognized the importance of the effect of transportation costs (dis tance) on the location of facilities. V/eber added to the model by adding the factors of production costs and the location of demand, in terms of marketing advantages. Re
cently, Walter I sard has consolidated the previous work of
the economic geographers, and has proposed a means of bal
ancing the above factors. 28
Background
Von Thunen.— -Von Thunen was the founder of the location theory school concerned with the rational location of agricul tural commodities around a central market place.^ The farm ing area and market place are located on a homogeneous plane.
Demand, production costs per unit, and the transportation rate are constant, and an f.o.b. mill price policy is used.
The model shows that the products least capable of ab
sorbing transportation costs to the central market place would be cultivated closest to the market; products more capable of absorbing transportation costs would be culti vated around the first products, resulting in a series of concentric cultivation rings about the market place. Rent absorbs any surplus remaining after transportation charges are paid. Extensive work has been done by E. Dunn and others on the relationship of rent to transportation costs, dealing with the fact that the greater the distance of a piece of land from the market, the lower the rent this land -
can command. In brief, von Thunen*s model is two-dimen
sional, placing heavy emphasis on transportation costs.
Specifically, in Figure 3t where 0 is the central market
15 ^Johann Heinrich von Thunen, Per Isollerte Staat in auf Landwirtschaf t und Nati onalokono'mi'e l~3rd ed. ; Berl 1 n•" Schumacher-Zarchlin, 1875IV
Edgar S. Dunn, The Location of Agricultural Produc tion (Gainesville: University of Florida Press, 195*0 . X K H 0 L J X' . A Graphic Representation of von Tinmen's Theory. Source: Melvin L. Greenhut, Plant Location in Theory end in Practice (Chapel Hill, N.C.: University of North Carolina Press, 1956), p. 254.
FIGURE 3 place, OA Is the cost of growing potatoes and A'S is the
cost of transporting these potatoes from J to 0, OB is the
cost of growing wheat and B'M is the cost of transporting
wheat from X’ to 0. Thus, because of the higher transporta
tion rate on potatoes A1S B'M. Potatoes would be grown AA.1 > BB'
between H and L and wheat from L to X* and H to X, with L
and'H being the( points where it is cheaper to produce wheat
than potatoes. Note that von Thunen assumes the location is
given; the question is what to produce at this location. 17 Weber.— Weber developed a more complex model. He
felt that there were two factors of location, that is,
transport and non-transport costs, and the problem was to
find the least cost location for a particular Industry by
trading off the two cost factors. His approach differed
from von Thunen* s in that he assumed that the industry was
given and the problem was where to place installations on
a heterogeneous plane. He also included demand as a factor
in location.
Figure ^ illustrates Weber's trade-off approach in
finding the least cost location for a given Industry. Note
that transport costs Include both transportation and agglom
eration factors (including marketing advantages) while non
transport costs include production factors.
. j. Freldrich, Alfred Weber* s Theory of the Loca- tlon of Industries (Chicago: University of Chicago Press, !92Fr: 31
Y
0 Non-tr3ns?ort Costs X Weber’s Locution Theory in Terms of Transport and Non- Transporl Factors. Source: Melvin L. Grecnluit, Plant Location in Theory and in Practice (Chapel Hill, N.C.: University of North Carolina Tress; 1956), p. 13.
FIGURE 4 32
On the isosale curve SS*, point D is the least cost site
because movement away from D on SS' increases total costs,
that is, savings in the one cost will be out-weighed by in
creases in the other cost. Note here that demand is very much a factor in this model. Given equal demand this model
shows where the least cost location is.
Weber's approach in a spatial form was this: Find the
equal transportation cost lines around the raw materials and market locations. Using these cost lines, or isotims, one
can find a tentative production point at which the transpor
tation costs incurred in bringing in raw materials and ship
ping out finished products is at a minimum. Around this
point, rings called isodapanes can be drawn showing succes
sively higher total transportation costs. The values for
the isodapanes can be combined with the production costs of
different feasible locations to determine the optimum plant
location. (See Figure 5)* Weber provided a more enriched
location model. Much of the modern vrork in location theory
has depended on this formulation and approach.
Isard.— Recently, Walter Isard has consolidated much of
the work of the earlier economic geographers, including 1 8 Weber* ' His treatment of the effect of transportation on
T ft Walter Isard, Location and Space-Economy (New York and Cambridge: John Wiley and Son's, Inc.' ’and'^the Technology Press of Massachusetts Institute of Technology, respectively, 1956). 33
— Isofims — Isodapanes
4 — ,3L-
< 0
\/ /^-— j 37y / 38
4 4 __ L
FIGURE 5
A construction of Weber's Isotims and Isodapanes Showing the
Interrelated Influence of Transportation and Production Costs
on the Selection of a Production Site.
Source: Heskett et al, op..clt., p. 130. location is based on work done by Launhardt and Palander
(see Figure 6). Basically, he assumes that there are two sources of raw materials, and Mg, and a consumer at C,
The problem is where to locate production so as to minimize total transportation costs. Constructing a location triangle
^1CK2’ ^en weight triangle M^OMg and using 0 as a
Launhardt pole, one circumscribes a circle passing through
0, .. M^, and Mg. The best production location to serve any given customer located between the angle M^OMg will lie on the segment M-jJ^ on straight line from 0 to the market.
Hence, Pg is the correct location to serve Cg. Note that with this method, the best location to serve any consumer location can be ascertained.
Isard has also shown how economies of scale can per haps compensate for the transportation cost disadvantages IQ of a given location. y Isard's comments relate to the hypothesis of this dissertation in that the savings in pro duction cost may offset higher transportation costs under the basing-point price policy.
The location models of the economic geographers are concerned with the likely placement of facilities based on transportation costs as an important independent variable.
Since, in this dissertation, this relationship is reversed, these models are not suitable for its purpose.
l9Ibld., Chapter 8, pp. 173-199. 35
■To
The LaunluircU-Palandcr construction.
FIGURE 6
The Launhardt-Palander Construction of Transport-Orlentation
Source: Walter Isard, Location and Space Economy (New York: John Wiley and Sons, Inc'.', 1956), p. Optimal Models
Lowest Transportation and Production Costs = f(Locations,
Production and Transportation Costs, Constraints)
In earlier theory, maximization of profits was assumed to be forced by competition. The newer tools, notably linear programming, are intended to enable a firm to maxi mize its net revenue, given certain inputs and subject to certain constraints. Linear programming might be said to make the theory of the firm operational.
The typical linear programming formulation is as fol- 20 lows:.
Maximize:
f = C..X, + c0x0 +...+ C X 1 1 2 2 n n Subject to:
+ allxl a12X2 + *‘*+ alnxn - bl + a21Xl a22X2 +,,,+ a2nxn - b2 + amlxl am 2x2 + ***+ amnxn S bm ii • H' 0, for j • • • * o' • • * 1b are the set of inputs
I—1 m the method of combining these inputs to maximize net revenue
according to the revenue contributions c^,...,c of the out
puts x^,... , xR .
20 From Kenneth E. Boulding and VI. Allen Spivey, Linear Programming and the Theory of the Firm (New York: The Mac millan Company, i960), p. 71. 3? One of the Inputs in a linear program can be space, and in the transportation algorithm it is added in the form of transportation costs to production costs to form various route costs. The objective function usually minimizes these two costs.
Let: x1j = amount shipped from factory i to warehouse
j.
Clj ~ Produc^i°n an^ shipping cost from i to j
(f.o.b. mill price policy assumed),
a^ ~ total capacity of I.
b^ = total requirement of j.
Minimize:
m n E E c, ,x, , i=l 5=1 13 10
Subject to:
n E x, , = a. , for i=l,2,...,n. 3=1 13 1
E x, , = b,, for i=l,2,...,m. 1=1 13 3
m n E a. ^ E b . * 1=1 1 “ j=l 3
Here space has been added to the theory of the firm, but in an optimal, not descriptive model.
Naturally, this method can be used to explore optimal location configurations. Koopmans and Beclsmann have 38
utilized linear programming to explore optimal locational 21 and pricing configurations.
Another group of scholars have combined mathematical
programming and sequencing algorithms to search for an opti mal locational array. These models generally employ heur
istic techniques which enable a good approximation of the
results of all practical combinations of facilities to ap
proach an optimal array. Transportation costs here are ex
plicitly considered and for the most part the problem
focuses.on finding the optimal array of facilities given
certain constraints, such as economies of scale or customer
service.
Alan S. Manne has developed a model that would locate 22 plants subject to an economies of scale constraint.
Minimize:
Total System Costs ~ L a. y. + L b. .x. , 1 1 1 i.j
Subject to:
£ xiJ = Rj (j = 1,2 J)
If Y, = 0 then x, . = 0 (i = 1,2..... 1) 1 = 1 1J > 0
21 Tjailing C. Koopmans and Martin Beckmann, “Assign ment Problems and the Location of Economic Activities,“ Eoonometrlca, Vol. XXV, No. l(January, 1957). PP« 53-76. 22 Alan S. Manne, “Plant Location Under Economles-of- Scale— Decentralization and Computation," Management Science, Vol. XI, Mo. 2 (November, 1964), pp. 213-235* 39 Where:
xi;J > 0. = 0 or 1.
x^j = annual rate of manufacturing at source i for
shipment to location j.,
y^ = fraction (actually an integer) for fixed
charges for a plant at i, = 0 or 1).
tfj = proportional transport cost per unit from i
to
c^ = proportional manufacturing cost per unit at
source i .
b ^ » c^ + t^j = total proportional cost per unit
produced at i for shipment to j.
a^ = fixed annual charge for a plant at 1, if
there is one.
R^ = annual rate of market requirements for lo
cation j.
Manne attempts to find an array which is near optimal, through the use of a heuristic technique, the steepest-ascent one point move algorithm, called SAOPMA, in place of an eval uation of a complete enumeration of all possible location arrays. He considers the difference in cost incurred by this array and the optimal array is small enough to be considered acceptable given the use of a special bidding technique.
Given the array of transportation costs, sources and destinations, and restrictions, the optimal model can be a 40 useful tool for optimizing transportation costs when routing and scheduling shipments. Generally, this model can be thought of as choosing the best routing and scheduling plan from among several feasible alternatives. This dissertation is not concerned with the optimization of transportation . costs, but with generating costs likely to result from the use of different, price policies. The choice of the optimal geographic price policy given the array of transportation costs, sources and destinations, and restrictions would per haps be a logical linear programming application.
Trading Models
Some scholars have developed trading models which are concerned with sellers producing goods and shipping these goods to other areas, Machlup and Enke have developed trad ing models, but for different purposes. The purpose of
Machlup*s model was to help illustrate and visualize the reasoning behind his attack on the basing-point price policy.
The purpose of Enke's was to give a theoretical basis for finding equilibrium prices in inter-regional trading. Both models are related to this dissertation, which is to find equilibrium prices (and costs) resulting from sellers com peting in a geographical framework. For this reason they have served as the basis for the dissertation model, and are described below. • ^ Machlup^ Model: Prloe, Transportation Costs = f (Geographic price policies, Location)
Machlup suggests a two-dimensional framework for study ing how geographic price policies affect transportation costs. (See Figure 7).2^ This framework consists of a sim ple Cartesian grid with markets and factories located at var ious nodes. Demand is fixed and the railroad can move only north and south or east and vrest. The basic equation is:
Price^. = f(Production Cost^ + Transportation Cost^)
Machlup, with a few examples, illustrates how this grid is used. However, he does not base his argument of the rela tive efficiency of the basing-point price policy and f.o.b. mill price policy on the data which could be generated using this dimensional grid to study.
Enke«s Model; Inter-regional Trading - f(Transportation
Costs)
Enke developed a model to study the effect of transpor tation costs on trading between regions geographically
separated. With his model, one can determine
equilibrium prices and quantities that will result in a static model when • a number of interdependent trading units stand ready to buy or sell a homogeneous good, according to known trading functions, and when there are significant freight costs per unit between each trading unit
^Machlup, op. clt., p. 9 FIGURE 7
Machlup»s Grid Construction Showing the Placement of Four Kills
Source: Fritz Machlup, The Basing-Polnt System (Philadelphia The Blakston Company, 19^9), p. 9« 43 24 and every other.
Furthermore,
The regions of each possible pair of regions are separated— but not isolated-- by a transportation cost per physical unit which is independent of volume. There are no legal restrictions to limit the actions of the profit-seeking traders in each region. For each region the functions which relate local production and local use to local price are known, and consequently the magnitude of the difference which will be exported or im ported at each local price is also known. Given these trade functions and transpor tation costs, we wish to ascertain: (1 ) the net price in each region, (2) the quantity of exports or im ports for each region, (3) which regions export, import, or do neither, (4) the aggregate trade in the commodi ty, (5) the volume and direction of trade between each possible pair of regions.
The equations are
E1 = bl^Pl “ ’ E2 = b2^P2 “A 2^’ etc* Where:
Rjl = region 1 / P1 = price in
= exports of R^ at P^
b^ = trading function of R^
= price in Rj^ at which local production minus
local use is zero, or = 0
24 1 En^:e» op. clt., p. 40.
^-*Ibld., p. 41. 44
Tig = transportation cost from R^ to Rg.
The delivered price in Rg from R^ is equal to P^ + T^g.
If this is equal to or lower than Pg, then R^ will export to
Rg according to the value of the trading function bg of Rg.
Enke feels that his model could be solved by the use of an electric circuit because
if there are more than three trading regions the problem is mathematically soluble, in the sense that there are enough equations to match the unknowns, but except by Iteration the proper method of solution has not been apparent to those mathematicians who were consulted.
Enke’s model incorporates transportation costs as a factor in determining prices and has an elasticity of demand factor to enrich'the model environment. This rich environ- ment(however, adds complexity to the model equations. A method of solving complex equations by an Iterative process is illustrated by the Cobweb Model.
Cobweb Model: Demand, Supply - f(Price)
The cobweb model is a dynamic model which relates demand, supply and price by difference equations which control the 27 change in these variables over time. Demand and supply are both a function of price.
In this model:
The demand schedule D = D(F)
26Ibld.
^Allen, loc. clt. The supply schedule S = S(p)
x = equilibrium sales
P = equilibrium price
The equilibrium price is set to clear the market, or
D(P) = S(P)
x = D(P) = S(P)
If there is a delay of one period, for supply to adjust to price changes then*.
Dt = D(P) and S t = S(Pt_1 )
The general equation is therefore:
S t ■ D(Pt ) = S(Pt_1)
Given P , the equations give P and S, governed by the dif ference equation:
D (pt> = s(pt-i> Given certain values for the demand and supply func
tions, one can find the explicit equilibrium price using analytical methods. One can also use an iterative method
to reach equilibrium points in the same model, given certain values for the functions. This iterative method would start
in this manner:
1. Initialize
2. Pt determines
3‘ St+1 = Dt+1 4. Dt+1 determines Pt+^ 46
If no, continue process.
If yes, stop.
The cobweb model illustrates how the iterative method
mentioned by Enke might be handled by a computer. However,
the cobweb model does not deal with transportation costs and
could not be used as a vehicle for the dissertation experi
ments.
Summary of Related Models
The models discussed In this chapter primarily are con
cerned with transportation costs as an independent variable.
The purpose of several of the models is to show how trans
portation costs should be considered in making operating de
cisions. Most of these models attempt to minimize trans
portation costs. The Theory of the Firm ignores transporta
tion costs altogether. The locational models are concerned
with how transportation costs should affect the choice of
location. The purpose of the economic geography models is
to demonstrate how transportation costs affect geographic
development. The optimal models show the best amount of
_ -transportation input to maximize profits. The linear models
show how sellers would locate in relation to each other when
all buyers and sellers are located on a straight line. Since
geographic relationships can be expressed in a linear form
in this type of model, the equations, if not too complex, k 7 can be solved explicitly. This type of model could be used as a basis for the dissertation model.
The purposes of the trading models were, in part, to
examine the effects of distance on business and economic
operations. Machlup’s model is descriptive and it treats
transportation costs as a dependent variable and location
and geographic price policies as independent variables.
Enke's model considers transportation costs primarily as an
independent variable which partially determines equilibrium
positions between several geographically separated trading
areas. *Machlup and Enke’s models were developed for purposes
similar to that of this dissertation, which is to illustrate
the effects of geographic price policies on prices and costs
and, therefore, were chosen to be the basis for the disserta
tion model.
i CHAPTER III
THE MODEL
Introduction
The data for this dissertation were generated by the manipulation of a mathematical model which simulates certain market behavior. The background, of the model, the equations, input and output, and the experiments conducted with it are discussed in this chapter.
Simulation
The model was drawn from the trading models discussed in Chapter II. It is a simulation model and hopefully a useful representation of certain market behavior reported in the literature. As Chorafas states,
...a simulation model might be called a working analogy of a physical reality. Models of mathematical simulation have been used to date (i) for purposes of experimentation or evaluation, that is, in try ing to predict the consequences of changes in policy, conditions, or methods without having to spend the money or take the risk or actually make the change in real life;
(ii) as a means of learning about new systems in order to redesign or refine them;
48 (iii) as a tool In familiarizing per sonnel with a system or a situa tion which may, as yet not exist in real life;
(iv) for the verification or demon stration of a new idea, system, or approach; and
(v) as a means for projection into the future ah? thus providing quantitative basis for plan ning and forecasting.
This specific simulation model was developed both for use in experiments and for use in expanding present know ledge about the system. Central to creating a good work ing analogy of reality, i.e., a simulation model, was the creation of an environment close enough to reality to be useful. Usefulness would be measured in terms of how well it provides experimental data which can serve as a basis for valid conclusions and also how well it contributes to the
understanding of certain market behavior.
Model Characteristics
From the search in the literature dealing with various models, it was decided that to answer the questions posed in
the dissertation, a model with the following characteristics
must be constructed: The model should be dynamic, that is,
the values of its variables may change over time; its loca
tional relationships should be two-dimensional; and its
^Chorafas, loc. clt. 50 environment should be rich enough so that the conclusions will be relevant. Therefore, it was decided to construct a model combining elements of Hachlup1 s and Enke's models, and
solved by the iterative method suggested by Enke and explained 2 in the Cobweb Model. That is, to develop a system of deter ministic difference equations which could be used to generate
the necessary data.
The adoption of a Cartesian grid similar to Machlup’s,
as opposed to the line of the linear model, was based on two
reasons. The first is that most of the literature on geo
graphic price policies uses a two-dimensional framework to
demonstrate locational Interdependencies. Machlup portrays
the different locations of buyers and sellers on his grid.
The T.N.E.C. papers use maps of the northeastern United
States to illustrate geographic monopolies which might re
sult if the steel industry had to quote all prices on an 3 f.o.b. mill basis. Others, discussing the economic ef-
ficlency of the basing-point price policy use similar maps.
Therefore, to be consistent with the geographical framework
in the literature, it was felt that a Cartesian grid system
should be used in the model. Thus, the results of this dis
sertation would have more bearing on the controversy sur-
2 Allen, loc-. clt. 3 U. S. Steel Corporation, op. clt., pp. 88 and 91. 4 Stocking, loc. clt. Daugherty etal, loc. clt. 51 rounding the geographic price policies because the geographic
framework of the model would be in harmony with those de veloped during this controversy.
The second reason is that any applications this model may have in solving transportation and logistics problems
would dictate the use of a Cartesian grid system. This grid
system would be most helpful in dealing with problems such
as.the location of facilities and inventories, scheduling
and utilization of equipment, and geographic price policies.
Unfortunately the selection of a Cartesian grid means that
the linear model advantages are sacrificed. First, the
potential of finding a particular solution to the model
equations is lessened. Secondly, any insight' in the lit
erature on linear models bearing on locational interdepen
dency is abandoned by the choice of a Cartesian grid.
Machlup's grid, used for demonstrating the effects of
geographic price policies on transportation costs, pro
vided a suitable basis for constructing a locational frame
work. Unfortunately, his model was relatively simple in
that it did not include a price elasticity of demand factor
or a market share factor.
Enke's model has a much richer environment with price
elasticity of demand and market share factors incorporated
in his inter-regional trading function. These factors were
necessary for the purposes of the dissertation. Their 52 exclusion from the dissertation model would seriously de tract from the validity of the conclusions.
Enke*s suggestion of solution by iteration is explained clearly in the Cobweb Model. This model shows that for cer tain values of the functional relationships in a set of dif ference equations, equilibrium points may be searched for by the iterative method. This is crucial if the analytical sol ution to the equation would be difficult to discover and evaluate.
Important Relationships
The basic functional relationships of the dissertation model are as follows:
1. D (t) “ e(D(t-l)’ KP(t)’ MP(t-l)^ 2.
3. M (t) - h(H(t-1)i sp(t))
4. SC(t) = k(D(t), M(t), C)
5 ‘ SP(t+l) = f(i) SG(t) (i=l'-4)
6 - P (t-KL) = q(SP(t+l)) Where:
SP = sellers* price
D = market demand
M = market share
C = equation parameters
SC = sellers* cost S = supply
MP = average price
f(i) = price functional relationship
g = price elasticity of demand functional relation-,
ship
h = market share functional relationship
k = cost functional relationship
q = price functional relationship
These equations form a simple system. First,the market de mand is a function of the previous market demand and the change in market prices. Supply equals demand. Next, each seller’s market share is a function of its previous share and both sellers* prices. Each seller’s cost is a function of its market share, the market demand, and cost parameters.
Then, each seller’s price for the next period is a function of its last period costs, depending on the geographic price policy. Finally, the possible equilibrium price is a func
tion of the seller’s prices. If the equilibrium test does not work, then the system starts over again.
Assumptions
There are important assumptions which served as a basis
for buyer’s and seller’s behavior.
1. There are two sellers selling a homogeneous product
directly to buyers in discrete markets. 5^ 2. Each seller’s goal is to maximize sales subject to a maximum cost constraint.
3. The sellers use geographic price policies.
4. Each seller has infinite productive capacity.
5. Production and transportation variable costs are a linear function of output. A normal profit is incorporated
in the production function.
6. Direct transportation between all locations is avail
able.
7. There is no collusion.
8. Demand in each discrete market is a non-linear func
tion of price.
9. The market shares for the seller in each market are
a non-linear function of their relative prices.
Seller’s Behavior
Sellers
One half of the experiments dealt with sellers in the
leather belt industry and the other half with sellers in the
steel industry. These two industries were chosen because of
their contrasting production functions, and because, as dis
cussed in Chapter II, the controversy over the baslng-point
price policy focused on the steel industry. 55 Product
The sellers make a single, homogeneous product consistent
with the theory of the firm and with the models of the linear
economists and with those of the economic geographers. 5 it
is Important that the product be homogeneous because there
is to be no product competition. The sellers compete only
through geographic price policies.
Seller’s Goals
The goal of the sellers is to maximize sales and market
shares, subject to a minimum profit level constraint. This
follows Baumol’s suggestion that sales growth is really the
primary goal of any organization.^ His suggestion implies
that a seller’s sales will either grow or shrink, and that
a firm that wishes to survive must constantly seek to grow
either through merger or increased sales. It follows, there
fore, that a seller will be hesitant to drop its selling op
erations in a given market even if it is supposedly losing
money (according to full cost accounting techniques) in that
market. The full cost technique might not be the best ap
proach for determining the profitability of individual sales.
5see Boulding, loc. cit. and Chamberlin, op. cit., p. 185.
^William J. Baumol, Business Behavior, Value, and Growth (New York: The Macmillan Company, 19 5 9 ) . p. 56
However, it is widely used.? If a seller drops out of a mar ket, he will have lost his sales contacts, advertising impact, and physical distribution set up. Leaving can involve large shut-down and.set-up costs.
Maximizing sales until production capacity is approached will helt> to pclve a seller any existing economies of scale.
Actual economies will vary according to the industry because of the relative size of fixed costs required for efficient plant size.
Of course, the seller must not let any market area create a drain on profits Indefinitely. Therefore, at some point, the seller would decide to get out of a market when the costs of supplying that market become unreasonably high. In the model, this market participation rule was set arbitrarily and was in the form of a branching statement which elimin ated a seller from a market if its standard costs in this market were more than $20.00 per unit.
Price Policies
In the uniform price policy, it is assumed that all sellers have the same price and equal shares of the market.
The resulting costs are used as Initial standard costs for the remaining price policies. After the first price is
7 R. L. Hal.l and C. J. Hitch, "Price Theory and Bus iness Behavior," Oxford Economic Papers, No. 2 (May, 1959). pp. 18-19. arbitrarily set, or initialized, all following prices under
this policy are based on industry average total costs of the
previous Iteration; or the market price of the present period
is equal to the average total cost of the previous periods.
The iterations are stopped when there is no significant change
in price.
The initial costs of the f.o.b.:inlll price policy are
supplied by the first model run under the uniform price pol
icy. Each seller1s market price is based on these total
average costs to supply each market of the previous model
run. This cost is called the market cost. Again, the iter
ation is stopped when there is no significant change in the
industry average prices.
In the basing-polnt price policy, initial standard costs
are, once more, supplied by the first iteration of uniform
price policy. All market prices are equal to the total aver
age cost of the basing-polnt supplier to each market, and all
sellers have equal market shares. When the basing-polnt price
does not change significantly, equilibrium is reached.
Finally, the first uniform price iteration provides the
initial standard costs for the zone price policy. In this
case, all market prices depend on the zone in which the mar
ket is placed by the seller. Zone prices depend on the aver
age total costs of the zone's previous iteration. Zone
costs are an average of all costs of the markets which it 58 contains. Here, as In basing-polnt price policy, market shares are equal. When the average industry price does not change significantly, equilibrium is reached.
Production Functions
In the model, sellers have linear production functions drawn from the literature on two real industries. Point competition assumes zero production costs, and the economic O geographers assume linear production functions. Most of the empirical evidence points to linear f u n c t i o n s . ^ This model uses steel and leather belt production functions, because a steel firm has relatively high fixed costs and a leather belt firm lower fixed costs. (See Figure 8). Thus the element of fixed costs can be studied.
Each seller has the capacity to supply the whole mar ket economically, This is opposed to point competition, but in keeping with linear competition.^ This condition is im portant because, if one seller is driven from some of the markets, then the other seller must be able to supply them.
Sellers do not sell through middlemen, eliminating arbitrage, or reselling in other markets to take advantage of price differentials.
0 Boulding, op. clt., p. 98 and Chamberlin, loc. cit. 9 The Oil, Steel, and Rail industries assume constant marginal costs.
^Chamberlin, op. clt., p. 184. 59 Production Function
Belt Production Function ($2,97^ + $.77/ lb.)/
/Steel Production / Function (§25,000 + 027/lb.)
Pounds Produced
FIGURE 8
A DIAGRAM SHOWING PRODUCTION COST FUNCTIONS OF. THE BELT
AND STEEL INDUDTRIES USED IN THE EXPERIMENTS
Source: Joel Dean, The Relation of Cost to Output for a Leather Belt Shop (New York: National Bureau of Economic Research, 19^1), p. 25 and United States Steel Corporation, op. cit., II, p. 52. The steel fixed cost element was re duced for ease of manipulation. 60
Transportation Functions
The railroad is assumed to be the only mode of trans portation available, because the mode is to be a control var iable, All markets are connected by direct routes to all other markets. The rail rate functions were based on actual rate tariffs.
Time and Inventory
The time period between iterations, t and t+1, is not defined; it could be a day or week or a month. The time
for ordering, manufacturing and delivering the product is assumed to be zero. There is no inventory in the system,
either at the location of sellers or buyers, or in transit.
Sellers Locations
The sellers are located on the nodes of a four-by-
four grid, vrith sixteen potential locations for each seller.
(See Figure 9). These sixteen locations also contain buyers,
which are discussed later in this section.
This locational framework is simple and idealized. The
buyers and sellers are located discretely at the nodes of a
Cartesian grid, with no barriers to transportation between
these locations. Raw materials are considered to be of the
same quality and are ubiquitous at all locations. There are
no labor cost or market potential differentials at any of
the locations and demand is not clustered in any group of FIGURE 9
A MATRIX DIAGRAM SHOWING THE LOCATIONS
OF BUYERS AND SELLERS 62 markets. These conditions result in a simple and idealized locational framework. The purpose of introducing locations as an independent variable in the model was to examine how different locational configurations would affect the results of the geographic price policies. If the locational frame work becomes too complex, then these added complexities would become independent variables and threaten the generaliza tions obtained by the use of the model.
. The rectangular distance between the locations on the grid was set at one thousand miles,creating a total market of three thousand miles, somewhat bigger than the United
States. This choice of distance was based on the desire to get compara.ble output for the belt and steel industries to
facilitate the analysis of the effects of the industry pro duction functions.
To explain how the location pairs were examined, the
locational framework must be explained. As mentioned above,
there are sixteen locations in a four-by-four gird. These
locations are designated by matrix notation (See Figure 9).
The number of pair combinations for a four-by-four grid is
156, including the cases In which both sellers take the same
location. Because of the symmetry of the grid, however,
there are only twenty-four unique location pairs since most
of the combinations are equivalent to each other. For in
stance, the location pair of 1,1-1,4 is the equivalent of the location pair 1,4-1,1 and also the location pairs of 4,1-4,4,
4,4-4,1, 4,4-1,4 and 1,4-4,4. Thus the examination of the pair 1 ,1-1,4 removes the need to examine the remaining equi valent pairs. There are 132 equivalent pairs which could be removed from consideration. For each experiment, the unique location pairs were examined in this order;
1,1 - 1,1 1.2 1.2 2,2 - 2,2 1,1 - 1,2 1.2 - 1.3 2,2 - 3,2 1,1 - 1,3 1.2 o. 2,1 2,2 - 3,3 1,1 - 1,4 1,2 - 2,2 1,1 - 2,2 1,2 - 2,3 1,1 - 2,3 1.2 — 2,4 1.1 - 2,4 1.2 — 3,2 1,1 - 3.3 1,2 - 3,3 1,1 - 3,4 1.2 — 3.4 1,1 - 4,4 1,2 - .4,2 1.2 — 4,3
Buyer’s Behavior
The buyers ’ demand is related to price through the standard price elasticity of demand equation and under the f.o.b. mill price policy, the sellers1 share of the market is related to price through a market share factor equation.
Price Elasticity of Demand
Market Demand has various price elasticities of demand.
The linear models assumed only an infinte price elasticity of demand.^ However, the theory of the firm recognized a
Hotelling says that demand would be a continuous func tion of price in this artical "Stability in Competition," The Economic Journal, Vol. XXXIX, No. 113 (March, 1929), pp. 133* Whereas in the typical linear model, the lowest price in each market captures the one unit of demand. 64 possible range in price elasticity of demand and the economic geographers, particularly I sard, have shown that an impor tant factor in location theory should be price elasticity
i p of demand.
Arc price elasticity of demand has been defined as follows:
F = „ (D(j)(t) ~ D (j)(t~l))(MP (j)(t) * KP(,1)(t-l)j ®U)(t) - MP( j)(tH7^D( ^(tp^UHt-i)) Where:
~ first price in market ( j) for
period (t-1).
MP(j)(t) ” seconii price in market ( j) for
period (t).
D ( j)(t-l) = first quantity in market ( j) for period (t-1 ).
D (^)(t) = second quantity in market ( j) for period (t).1^
This shows the change in revenue due to a change in price.
Note that both sides of the conventional definition have been multiplied by -1 to give a positive E to facilitate
12 Isard, op. clt., p. 126. 13 ^William J. Baumol, Economic Theory and Operations (2nd ed.; Englewood Cliffs, N. J.: Prentlce-Hall, 1965). P« 175. ( Qi „ Qn) (P i *t P o ^ As Baumol* s definition is actually - E = 7-=,---- BTT77;— T ~ 7 T T ' ' 1 “ 0 '' Q1 lQ ' noted in the text, both sides i^ere multiplied by -1. manipulation. An E less than one indicates that demand is inelastic, and an E greater than one that it is elastic, and an E equal to one that demand is unitary.
Market Share Factor
The market shares of the sellers in each market change according to a market share factor S. This factor governs how fast market, shares will change. Given relative prices of the sellers, a large S will cause their market shares to change more abruptly; a small 3 will cause their shares to change more slowly. This market share factor is defined as
follows:
s = 3) W '5 " SP(lH.D(t) Sum P U ) ( t ) Where:
M (i)(j)(t) = market share of seller (i) for period
(t).
SP(i)(j)(t) = seller (i) price for period (t).
Sum = sum ^oth seH ers prices in market
(j) for period (t).
Since I'1(i)(j)(t) is to be found
M (l)(J)(t) = H (i)(J)(t-l) + s /•5 " sp(l)( j)(t)’ V Sum P( J) (t)
This factor controls how abruptly a firm's share changes with 66 a difference In Its market price In relation to the industry market price. S is a critical factor in that it regulates how fast market shares will change.
It is evident that before equilibrium is reached, one of the companies involved might experience some drastic slip page in market shares. Since this project is concerned with geographical price competition only, the companies involved are not able to differentiate their products through non- price means in order to insulate them from price competition.
Location of Buyers
Traditional micro-economic theory has heretofore been
unconcerned with competition over space. The location of
this competition was and is assumed to be a point. All
buyers and sellers were located at this point, and all busi
ness was conducted there. This point location made perfect
knowledge more feasible, and also removed any product differ- I/4, entlatlon due to distance, as Sraffa has noted. Some lo
cational theorists maintain that this frictional force of
competition automatically created a situation of imperfect 1*5 competition,tending, at least, to monopolistic competition. J
It has been pointed out that Hotelling and Chamberlin
14 Piero Sraffa, "Laws of Returns under Competitive Conditions," The Economic Journal, XXXVI (December, 1955). P. 5^3.
^M. L. Greenhut, Microeconomics and the Space Economy (Chicago: Scott Foresman and Company, I963T. tried to introduce distance as a differentiating force by- building a model of linear competition. Buyers are located continuously along a line, and all sellers would locate some where on this line. The economic geographers developed spatial models, but these models were concerned with the geo graphical development of industry and cities, and not so much with competition at the micro-economic level.
. Some of these models, such as Lbsch’s, assume that de mand is located at one point or continuously either on a line or on a plane. Other economic geography models, such as Isard*s and the optimal models, assume that demand is 17 located at discrete points. ' In linear programming models 1 R it is assumed that demand is located at discrete points.
It is one of the main purposes of this dissertation to gen erate data bearing on the geographic price policy contro versy, Therefore, a two-dimensional array of buyers and sellers is necessary to achieve consistency with the geo graphic framework-developed in this controversy.
^ A u g u s t Lbsch, The Economics of Location (New Haven: Yale University Fress,' 195^. Translated by Willi am H. wog- lam with the assistance of Wolfgang F. Stopler from the 2nd rev. ed, ; Jena: Gustaf Fischer Verlag, 19^), pp. 11^- 115. 17 11 sard, op. clt., Chapter 11 and Boulding and Spivey, loc. cit. 1 R Walter V/. Garvin, Introduction to Linear Programming (New York: McGraw Hill Book Company, Inc., 19^0) , pp. ffj~. 68
Experiments
The model was used for twelve experiments. The exper iments were so constructed that all contingent factors would be included , and thus protect conclusions that would be weak ened by the omission of such factors.
In discussing the hypothesis, one could say that the relative efficiency of the price policies may depend on at least four factors:
1. the shape of the production functions,
2. the price elasticity of demand for the product,
3. the location of the sellers,
4. the market share change parameter,
These factors all have been used in the literature jus tifying either baslng-point or f.o.b. mill price policies, and would have to be examined in a study of these and other price policies.
Therefore, in each experiment the variables in the model equations were given a certain set of values or parameters.
To Isolate the effect of the above factors, these equations were run using a process in which the equilibrium points are found for the twenty-four pairs of locations for all four policies. Thus each experiment consists of one set of para meters processed by the computer for all location pairs for all policies. Each industry experiment had its production and transportation functions and required parameter settings 69 for market share and price elasticity of demand variables.
Model Equations
Below are the model equations which together make up a set of difference equations. There, are equations for demand, market share factor, cost, and prices.
Definitions
SC(i)(J)(t) = seller (1) cost per unit in market ( j) for
period (t).
SP(i)(J)(t) = sel^er (*•) P^ice Pe*1 unit in market ( J) for period (t).
MP(j)(t) = the average price in market ( j) for period
(t).
M(i)( j)(t) = sel^-er market share in market ( j) for period (t).
D( j) (t) s demand in Blanket ( J) for period (t).
^(t) ~ averaSe Prlce of all of the markets for per
iod (t).
Sum ?(j)(t) = ^ e sum sellers1 prices in mar
ket ( j) for period ( t).
E = price elasticity of demand.
S = market share factor 70
Demand Equation
Since the price elasticity of demand was
can be solved for algebraically. Thus,
D ( 3) (t) = ° U)(t) ~ (~E(AP) *
Where:
(a P) = MPuMP ) (t) - MP( J)( t-1)
Demand in a market is a function of the demand of the previous period and modified by the change in the market price through the price elasticity of demand parameter.
Market Share \ Sum P
Under the uniform, basing-point, and zone price policies, market shares for policies are fixed at 50/£. Under f.o.b. mill price policy, a seller's market share can range from
0% to 100;£. The market share for the present period is the
sum of the last period's market share and the amount the
seller's price in the previous period deviates from a price
which would give it a 50fo share. 71 Cost
scd ) ( j )11) - (d)(t) (i) ( j) (t)
Where:
VP = variable production costs
VT = variable transportation costs
L = distance
K == constants
The seller's market cost for the present period is the sun of the fully allocated transportation and production costs in volved in supplying this market. Scholarly evidence points to the prevalence of the "full cost" method of pricing in industry. The important independent variable in this equa tion is distance.
Market Price
The market price for the present period is the average of both of the sellers' market prices of the same period. Only under the f.o.b. mill price policy can the sellers' prices
be different.
Sellers' Market Prices
2 16 L Z Uniform Policy: SF J=l 32 72
The uniform market parice for the present period is the weighted average cost of supplying all markets in the pre vious period. All prices in all locations are equal.
F.o.b. Mill Policy: SP(1)u)(t) = SC(1)( j)(t_i,.
The f.o.b. mill market.price of the present period is simply the average cost of supplying that market in the previous period.
Baslng-Point Policy: = Baslng-Point Company
SC(i)(3)(t-1)' The basing-point market price for the present period is the basing-point company’s average cost of supplying that market in the previous period.
A A SC(D (j)(t-D Zone Policy: SP(1)(j)(t) = i-T— J-i-g------•
Under the zone price policy, there are four zones having four markets each. The market price for the present period is both sellers' average cost of supplying all four markets in the zone.
Price
A (D(J)(t))(MP( Prlcer!(t) . . = r6— — *
The present price is the weighted average price of all six
teen markets during the present period. • Sum of Prices 2 Sum P (t) = if1 SP(i)(J)(t)
The sum of the prices for the present period is the summation
of both sellers' prices in the market.
Solution of the Equations
One can choose between two methods for solving the set
of model equations as described above. The basis for these
methods and their machinations are described below.
An .Analytical Solution
One method for finding the equilibrium price resulting
from the interaction of the sellers and buyers over time iq would be the analytical method. 7 For this method, the
equations describing this behavior can be combined so that
the price^t+1j (p(t-KL)^ is a functlon of price^ ^P(t)^
and the other variables x(2)* x( 3) ^ of thls differ ence equation. Thus one would have one difference equation
P(t+D = f(p
If the price at time t is desired, one can then dir
ectly solve for given and a, b, and c. However,
19 'This section is based on Samuel Goldberg, Introduc tion to Difference Equations (New York: John Wiley and Sons, 1 9 5 8 pp. 5 0 - 8 77 95*03• in this dissertation, the equilibrium price (p) is desired.
This equilibrium price is the limit of the sequence assuming the sequence converges. Therefore, the analytical method of solution would entail finding the limit P of the price sequence as t goes to infinity or
Thus the sequences resulting from different initial prices and different equation parameters can readily be analyzed by merely finding their limits. One can determine which initial prices and equation parameters will cause the sequence to 20 converge. To test the hypothesis of this dissertation, the limits of the experiment sequences would be found and could easily be compared.
However, the search for an analytical solution of the model equations was not attempted because of the complexity of the model. This complexity is itself the result of many factors. A situation in which two sellers sell to sixteen markets using one of the four geographic price policies is merely the basic difficulty in calculation. The difficulty and the complexity are compounded when one calcualtes the demand in each market. Demand is taken to be a function of
both a market share factor and price elasticity of demand.
The set of equations that result from such a number of
20 See Goldberg, op. clt., p. 85 for examples of the behavior of a solution sequence. 75 variables is extremely complex and highly interdependent.
The complexity of these interrelationships would give the analytical solution exponential functions which, according 21 to Tompkins, would be very difficult to evaluate.
Apart from the problems of evaluation, the discovery of the analytical (or explicit) solution can be made more ar duous than necessary because of the non-linear relationships of the equations. In the price elasticity of demand equa tions, demand is not a linear function of price, and in the market share factor equation, market share is not a linear function of price. Goldberg maintains that "such problems as these non-linear relationships are difficult at 22 best and in general incapable of explicit solution."
Iterative Solution
The iterative process uses the basic equations or com bined equation and enumerates sequentially each value of
instead of finding the limit P of the sequence. However, usually the limit of the sequence will not be reached by the iterative method and one has to be satisfied with an
21 This is the opinion of Charles E. Tompkins, Professor of Mathematics at the University of California at Los Angeles, after he inspected the model equations on Decem ber 2, 1966. 22 Samuel Goldberg, op. clt., p. 182. However, there are examples -of solution methods to second degree equations in the literature. See T. W. Chaundy and E. Phillips, "The Convergence of Sequences by Quadratic Recurrence-Formulae," Quarter Journal of Mathematics (Oxford Series), Vol. LX, No. 4 (ApriT, 1953). p p 7 T 5 5 = 2 59. approximation If the sequence converges P(T) can be as close to P as one wishes if T is big enough. The itera tion is stopped when the absolute value of | P ^ “ p(t 1 is smaller than an arbitrary constant.
The best reason for using the iterative method is its
feasibility in terms of finding equilibrium positions and in evaluation of results. We are assured of results, then, with this method (again assuming the sequence converges).
Despite many short-comings, it will, under certain condi
tions, generate data which can be useful in terms of the pur pose of the dissertation.
Another important reason for using an iterative method
is to enable one to observe the behavior of the model over
time, even on an iteration-*by-iteratlon basis if necessary.
This observation of the variables would contribute greatly
to the understanding of the model and, hopefully, the pro
cesses it is based on. Because of the lack of empirical data
on the subject of the geographic price policies, not much is
known about their effect on prices and important cost var
iables. Less is known about the reasons for these effects.
It is the purpose of the iterative method not only to dis
cover how these policies affected the dependent variables, t but why they did so. It is therefore Important to be able
to look at the model behavior when significant changes in
output occur. In being able to so Isolate and examine in
dividual changes, one can also determine, it is hoped, 77 the significant changes in model output.
There are disadvantages resulting from the use of the iterative method. The main disadvantage is that it will not produce an explicit solution to the model equations, instead of being able to find the limit of the resulting sequence and to analyze the effects of various Initial prices and equation parameters, one must find by trial and error which prices and which parameters will cause the sequence to converge. Thus instead of finding the limit P of a sequence analytically, one must enumerate every value of the sequence to find an equilibrium price
Finally, the initialization process in the iterative method supplies definite values for the functions in prepar ation for the attempt to discover an equilibrium point. This, naturally enough, can have a great effect on the success of the search for an equilibrium point. Conversely, a success ful analytical solution can give the entire feasible ranges for the Important functional relationships of the equations.
The iterative solution was chosen because it is an ex cellent vehicle for understanding the model, and more feas ible than an analytical solution to the model. If the model were not as complex and the non-linear functions were removed,
then an analytical solution would, perhaps, also be feasible. 78
Equilibrium
Each run of the model generates cost figures which be come standard costs used to determine the next period’s prices. The relationship between iterations is
Prices^« Standard Costs^^j
This is consistent with the actual practice of using standard costs to compute guidelines for pricing. In this model, costs will be the only basis for prices.
The iterations continue until the change in average price is smaller than an arbitrary constant of one penny.
Thus,
if | ? (-t-l) “ ^(tjj > continue.
j p (t~l) “ P (t)J - st°P and select the next pricing policy.
The iterations were run three times after a price change
smaller than $.01 to see if the position reached was not
simply a local equilibrium and not the true equilibrium, if
one exists. Each iteration generates all the required data
for the testing of this dissertation's hypothesis.
One of the problems of finding an equilibrium value may be that an equilibrium value, in the context of the ex
periments, may not exist. That is, the output may oscillate.
The functions may have several humps as shown in figure 10.
For other initilization values, the function may diverge
instead of converge to an equilibrium position. During Price
Diverging Function
Initial Price
Converging '~-~^Functi on
Analytic, Solution
Non-Converging Oscillating Function
Number of Iterations
FIGURE 10
A DIAGRAM SHOWING THE POTENTIAL BEHAVIOR
OF A. FUNCTION GIVEN AN
INITIAL PRICE 80 preliminary model runs, workable initialization values had to be found.
Model Inputs
For each experiment, values for the model equations had to be supplied in the form of computer input. Variables which had to be parameterized at the start of each experiment are as follows:
1. market share factor (S)
2. price elasticity of demand (E)
3. old price
old market price
5 . market participation rule
6 . old market demand
7 . production cost function
8. transportation cost function
The parameters supplied for three of these variables,
D (J)(t-l)’ KP(j)(t-1 )* and p (t-l) are values, that is, parameters for these variables for the following iterations will be generated by the model. The model inputs were adapted from functions reported in the literature and are explained below.
Market Demand
The initial values were arbitrarily chosen; however, an attempt was made to select demand figures which would 8 1 keep the output from both Industries in the same range, that is, between one and ten dollars. This was done so that-com parisons between relative output figures would be easier
(see Table 1).
Market Price Elasticities of Demand
This is a key variable, but unfortunately there is no
usuable data on the price elasticities of demand for either
product. This is particularly surprising in the case of the
steel industry, an industry which has been extensively
studied. A s the National Bureau of Economic Research points
out,
although (steel) producers may make rough estimates of the probable effects upon sales of slight changes in price, demand elasticity is probably one of the most vacuous of our "empty economic boxes" in the industry.
Because of this lack of empirical data, a simple range was
used for these parameters.
Market Share Factors
Because of the lack of available specific data in the
literature on the two Industries, it was necessary to be
arbitrary and develop two parameters, one which would cause
the market share to change slowly, while the other would
cause it to change rapidly.
2 3 ^National Bureau of Economic Research, Price Research in the Steel and Petroleum Industries (New York: National Bureau of Economic Research, 1939). p. 63. 82
TABLE 1
EXPERIMENT PARAMETER SETTINGS
Equation EXPERIMENTS Variable
LEATHER BELT S;TEEL 1 2 3 4 5 6 7 8 9 10 11 12 Market Share .5 .5 1.5 1.5 '1.5 1.5 .5 .5 .5 1.5 1.5 1.5 Factor
Price Elas ticity of .5 1.0 1.5 .5 1.0 1.5 • 5 1.0 1.5 .5 1.0 1.5 Demand.
Old $10.00/lb. $10.00/lb. Price
Market $10.00/lb. $ . /lb. m o e (t - D 10 00
Market Par ticipation Cost > $20.00/unit Cost > $20.00/unit Rule
Market Demand 250 lbs. 500 lbs.
Production Cost $2,973.00 + $.77/lb. $25,000.00 + $.027/lb. Function
Transporta tion Cost $.843 + $.00295/lb.-mile $.143 -5- $.00108/lb.-mile Function 83 Production Functions
The belt Industry production function x^as extracted from oh Dean's work. The steel industry function was the one.de veloped by U. S. Steel.^ Although the computational approach of U. S. Steel has been questioned, no alternative figures are available. It was necessary to decrease the size of the fixed cost element "in the U. S. Steel figure to make it easier to compare the output of this industry with the output of the belt industry.
It must be remembered that the two industries were chosen for the model because of the contrast in the size of both their fixed and variable production costs. Exact, up- to-date figures were not necessary. What was important was to insure that the slope and intercept of the production functions did Indeed contrast.
Transportation Functions
The data for both the Industries were taken from rail- road rate tariffs. Specifically, the rates were for leather goods and cold rolled sheet. Upon inspection, the functions
2 k Joel Dean, The Relation of Cost to Output for a Leather Belt Shop TNew~YorF! National Bureau of Economlc Re search",' I9V1 ) , p . 25. 2*5 ^United States Steel Corporation, op. clt., II, p. 52. 26 The source for the Steel (cold rolled sheet) rates were Trunk Line Central Territory Railroads, Freight Tariff ^3*4— E, 1965. The source for the leather belt rates wrere Trunk Line Central Territory Railroads, Freight Tariff E-1009-A, 1965. 8^ appeared to be linear. Regression lines were fitted to the rates as a function of distance.
The leather goods transportation function is:
rate/lb. = $.843 + $.0029 5/mile.
The cold rolled sheet transportation function is:
rate/lb. = $.143 + $ .00108/mile.
Surprisingly good fits were obtained. The coefficients of correlation were, respectively, .99 and .9 7 .
Output
Much of the model output generated by the experiments was specifically designed for use in testing the hypothesis.
This output was as follows:
1 . average transportation costs,
2 . average cross-hauling, actually pound-miles
divided by pounds,
3 . average production costs,
4. average profits, and
5 . average prices.
The average profit is over and above interest on owners* capital assumed to be contained in the production function.
Cross-hauling is defined as pound-miles divided by pounds, giving a measure in terms of miles which is an indicator of the relative amount of market Interpenetration. A high pound-mile/pound figure would indicate a larger amount of market interpenetration than a low figure. 85 Sequence of Operations
Each experiment generated price and cost data for twenty-four location pairs for all four price policies. The policies were run in this order: uniform, f.o.b. mill, bas- ing-point, and zone. The price policy routines and the twenty-four location pairs were set for all of the experi ments. For each experiment, the price elasticity of demand, market share factor, and production and transportation functions were fixed. The following is an explanation of the computer program sequence of operations.
Starting with the first iteration under the uniform policy, the program chooses the first seller location pair
(see Figure 11). Then it initializes the old price (P(t_i))* old market price (Kp(j)(t - l ) ^ d-emand ^ D( j) (t-1) ^ * Next it enters the distance loop where the shipping distances from each seller location to all of the buyers' locations are calculated using the Pythagorean Theorem. When this is com pleted the program enters the demand loop and both sellers'
Market Price (sP(i) ( j) (t) ^ resulting Market Demands
(D(j)(t)) are calculated. Using these figures, the sellers' revenue in these markets are calculated.
The program then transfers to the Cost Subroutine (see
Figure 12) and enters the cost loop where the average trans portation cost and cross-hauling for each seller in each mar ket are calculated. Then the average production cost is Z' T O PICK Y IN FIRST I i t e r a t i o n POLICY XSUBROUT lMS^
FIGURE 11 EQUILIBRIA MAIN MODEL .ITERATION FLOWCHART PICK Ye s WE'jCT POLICE NO p i c k , f i r s t SELLER LOCATION PAF? I INITIALIZE
O (j)(t-0 Pc-t-o = £ = PICK CALCULATE DISTANCES
TO D IN F. 0. B. SUBROUTINE
CALCULATE KE'Y PICK S P e.lii)(t'l O U M t i FROM COST NOD12- SUBROUTINE
FINISH ALL NODES FROM i t e r a t i o n / TO B ©s u b r o u t in e SUBROUTINE SUBROUTINE I e. i F.O.B. V ^ J s FIGURE 12 PICK FIRST COST SEL-L&R SUBROUTINE
PICK FIRST MARKET
CALCULATE TRANSPOTOSH COSTS AND CROSS* HAUUWG
IN l£?Hv NO ALL
CALCULATE PRODUCTION COSTS
NO
Ye s
CALCULATE r T O F N TOTALS IN AND MAIN M O D E L P K i w T o y i v FLOWCHA^IV 88 calculated for each seller. The transportation cost and al located production cost for each seller in each market are s m m e d to give the Seller's Market Cost (3C( i) ( j) (t) ^ * Fin“ nally, the new Price ( P ^ ) is calculated and the values for
SP(i)(3)(t)’ D(^)(t)’ averaSe transportation and production cost, S3(i)(j)(t)' p(t) are P I’tn te,i out. After this is finished the program returns to P in the Main Model Flow chart and the equilibrium test is applied. If equilibrium has not been reached, the program transfers to the Itera tion Subroutine (see Figure 13)•
Here the Sellers' Market Costs ^ ) are tested to see if they will force either seller out of a market. If a seller is forced out, its market share is 0^ and the other
sellers' is 100$. When this is completed, the program re
turns to C in the Main Model Flowchart and the model is ready
for the next iteration.
However, if the f.o.b. mill policy is being run, then
the program must transfer to the F.o.b. Subroutine (see
Figure 1*0 and enter the market share loop. The new market
shares ( j) (t) ^ are calculated,based on the sellers'
relative prices. If a seller has been driven out of a market because of its price, its share of this market goes
to 0% and the other seller's share goes to 100$. When this has been completed the program returns to E in the Main
Model Flowchart and the next iteration may now begin. FIGURE 13
ITERATION
SUBROUTINE
^ K s c ^ . , ;fc)'
NO
NO PICK NErKT
Y e s
NO
/ T O C ^ m MAIN MOOSU V FL O W c h a r -jv FIGURE 14
F.O.B
SOUBROUTINE
PICK FIRST MARKET
CALCULATE
i \
N o
NO
Yes ^ T O E > IN MPiIN m o d e l yFLG\f\f GHP&Ys If equilibrium has been reached, then the model chooses the next location pair. After equilibria have been found for the remaining location pairs, the next price policy is chosen using the uniform policy first iteration output as initial izing values. After all four price policies are finished, the experiment has been completed. CHAPTER IV
RESULTS
In this chapter the results obtained from the computer
iterations are explained and analyzed. It is important to
remember that these results were the output of a model which hopefully is close enough to reality to provide meaningful
insights into the actual workings of a segment of business
operations. These results were also directly affected by
the parameters placed in the equations and the initializ
ations used to start the model iterations.
In each section of this chapter the explanation and
analysis will focus on how the results relate to each part
of the hypothesis. The hypothesis is that an f.o.b. mill
price policy when compared with three other geographic price
policies will result in the lowest
A. average transportation costs,
B. average cross-hauling (pound-mlles/pound),
C. average production costs,
D. average profits, and
E. average prices.
Concurrently, the effects of changes in these variables
will be studied:
1. location of the sellers,
92 2. price elasticity of demand,
3. industry cost function, and
4. market share factor.
In keeping with the hypothesis, the f.o.b. mill price results are considered the norm. The analysis is based on the comparison of the results of the other price policies with those of f.o.b. mill, that is, of their corresponding equilibrium points.
Recall that half of the experiments were run with a market share factor of .5 and the other half with a market share factor of 1.5» This parameter change only affected the results of the f.o.b. mill policy. However, since the effect of the change in the market share factor was neglig- able, the presentation of the results is concerned only with the results of half of the experiment, since both halves turned out to be essentially the same. These experiments are 1-3 and 7-9 •
Graphical Representation of the Data
In the discussion of the results, reference is made to several bar charts. These were constructed so as to show as much information as possible.
The dependent variable for all four price policies is shown on the y-axis. The different location pairs of the sellers are on the x-axls. These location pairs are grouped 9^ in decending order according to the average distance from both seller locations to the sixteen buyer locations. Within
these groups, the pairs are ranked in descending order accord ing to the distance between the two sellers (see Table 2).
The output for all four policies for each location is shown
in bar form to facilitate the visualization of the price pol
icy rankings.
For three location pairs, 1,1-1,1, 1,2-1,2, and 2,2-2,2
the results for the f.o.b. mill and basing-polnt policies
are the same. For these location pairs, both sellers are
located in the same market. When both sellers are in the
same market, the f.o.b, mill and basing-point policies are
effectively identical. That is, f.o.b. mill prices in all
of the markets are identcal to the basing-point prices in
these markets. Thus not only the prices but also the other
output for the two policies are Identical. Since the output
for the two policies ’was the same for the three location
pairs, the lengths of their bars in the bar charts were the
same.
Effects of the Model Variables
The variables modified to some extent the effect of the
price policies on costs and prices. Certain of these vari
ables were important, others not so important. Following
are the variables. 95
TABLE 2
AVERAGE DISTANCES BETWEEN SELLERS AND BUYERS
AND BETWEEN SELLERS
AVSRA AVERAGE DISTANCE DISTANCE LOCATION PAIR GROUP BUX&Ro Ift BETWEEN SELLERS IN MILES
1,1 - 4-,4 2,400 4.240 HHHNNNN 1,1 - 1,4 V 3,000 1,1 - 1,1 It 0 1,1 - 3.^ 2,240 3,610 1,1 - 2,4 11 3,160 1,1 - 1,3 it 2,000 1,1 - 1,2 it vnVH'AWO'O VO1,000 1,2 - 4,3 2,080 3.160 1,2 - 4,2 It 3,000 1,2 - 3,4 It 2,830 1,2 - 2,4 ft 2 , 240 1,2 - 2,1 It 1.4-10 1,2 - 1,3 11 1,000 1,2 - 1,2 tt 0 1,1 - 3.3 1,990 2,830 1,1 - 2,3 ft 2.240 1,1 - 2,2 It 1,410 1.2 - 3,3 1,830 2,230 1.2 - 3,2 It 2,000 1,2 - 2,3 ft 1.4-10 1,2 - £.2 tf 1,000 2,2 - 3,3 1,590 1.4-10 2,2 - 2,3 ft 1,000 2,2 - 2,2 It 0 96
Location of Sellers
The location of the sellers affects the value, but not the ranking of the output. There are two main effects of the location of the sellers on the output. The first is that, as the distance between the sellers and buyers de creases, the lower the output becomes. The level of the out put drops as the locations progress from groups one through six. Thus the closer the sellers are to the center of gravity of the grid, the lower the output.
Secondly, the larger the distance between sellers given the same closeness to the center of gravity, the lower the output for the basing-point and zone policies. However, the larger the distance between sellers under the f.o.b. mill policy, the higher the output. The distance between sellers has no effect on the uniform policy output.
Price Elasticity of Demand
For all policies, the higher the price elasticity of demand, the lower the model output because the higher the price elasticity of demand, the more demand would react to changes in price. Since the model initialization caused an increase in demand for the second iteration of the sol utions, the direction for the change in prices would be downward.
I 97 Industry Cost Functions
The industry cost functions, especially the production functions, affected the range of the output. With high fixed and low .variable costs, the steel industry would have high production costs when demand-was low and low costs when demand was high. The belt industry would have lower costs when demand was'low and higher costs when demand was high.
The transportation functions, because they were diver gent, with the belt function higher, contributed to the dif ferences in the ranges.
Market Share Factor
The market share factor, used only in the f.o.b. mill price policy equations, had a negligible effect on the model output. The small differences it did cause were due to
sellers' being forced out of markets on different iterations,
resulting in oscillations in average prices.
Effects of Parameter Changes
These model equation parameters, if not kept constant
in comparing the results of the price policies, can alter
the rankings of the price policy output. For example, two
sellers using a uniform price policy, but located close to
the center of the grid, perhaps might have lower prices than
if they used a zone price policy, but were located at one of
the corners of the grid. In the following analysis, it will 98 be assumed that the equation parameters are the same for all the price policies when comparisons are made.
Transportation Costs and Cross-Hauling
In all of the experiments, transportation costs and
cross-hauling were lowest under f.o.b. mill, second lowest
for basing-point, third lowest for z o n e , and highest for uni
form price policy (see Charts 1-12). The absolute size of
these costs and cross-hauling for each policy was modified by
the effect of the price elasticity of demand and the location
of sellers.
As noted before, when the price elasticity of demand in
creases, the transportation costs decrease (see Charts 1-3)
except for the uniform price policy. This effect was
strongest for the f.o.b. mill policy.
The location of the sellers strongly affected the trans
portation costs and cross-hauling for all policies. The
closer the sellers were to the center of gravity of the grid,
the lower the costs and cross-hauling. The. larger the dis
tance between sellers, the lower the costs and cross-hauling
for the basing-point and zone policies. The distance be
tween sellers had no effect on the uniform costs and cross-
hauling. B’or the f.o.b. mill policy, however, the larger
the distance between sellers, the larger the costs and cross-
hauling. However, since the price elasticity of demand and Uniform CHART. 1 Zone BELT INDUSTRY AVERAGE TEAKSFORTATIOR COSTS Baslng-Foint DEMAND ELASTICITY: .5 F.o.b. Kill
W i-i & cj iH I m Q 5 C i i 43 A ra o o o> 3- fcfc ..ai ; Is > 0 „ ^ *
fV LOCATIOMJ1,1 n.n/iji fLiniji n,z' 11,21/1,21(2,2 F A I R S [ 4 ( l3<4Hz,4|\l,3jil,2)14,3 4 ZJ 13,41 1Z,4J 12,1 Z,3j l2,2J [3,3] 2,3112,2
*FOR LOCATION P A lR s jj'jj,1 y {22} ’ &NP E5AS1NG-P01NT ARE EQUAL \D so SOURCE-’ APPENP1X A ? TABLE 7 , P. I<°2. Uni foma CHART 2 Zone BELT INDUSTRY AVERAGE TRANSPORTATION COSTS 3asinc-polnt DEMAND ELASTICITY: 1.0 F.o.b. Mill
LOCATION]!,!! 11,11 fl/Il J 1.11/1,11 Jl.11 J 1,1 l.Zl/l,2l(2,Z) J2.2I/2V PAIRS [4,4] II,4J ]1,1J (3,4] (Z,4j 11,3J |1,Z 4,ZJ p,4j Z,3j{2,2Jl3,3} {2,3)12,1
FO R LOCATION PAIRS , F.O.B. ANP BASING-POINT ARE EQUAL O o SOURCE: APPENP1X A , TABLE © , P. CHART 3 Uni form BELT INDUSTRY AVERAGE TRANSPORTATION COSTS Zone Easing-point DEMAND ELASTICITY: 1.5 F.o.b. Mill
11 W 6 M . as r-l t-i - O Q C «c4 4 •p « o o <1> 3 $ : f-i © 2 > <5
i ;
ni L O C A T I O N '1,ZH1.ZH1,Z)(2,Z) f2,2)/2, FAIRS 3,2/12,3112,2)13,3/ l2,3jU
/ 2,2 FO R LOCATION PAIRS , F.O.B. ANP BASING-POINT APE EQUAL u m z ’ 12,2 101
SOURCE: APPENPIX A,TABLE 9 , P.lfcb CHART 4 Uni form STEEL INDUSTRY. AVERAGE TRANSPORTATION COST: Zone ■ *rT«?r 4btMcg| Easintr-point ■DEMAND ELASTICITY: .5 F.o.b. Mill
g 2.201 ! L eg £ 2 Si 'ill iilii 9 200} I Q
-P 100 01 o £ g $ o IIS
to !
1,20
i n s F N ' r s QE E 3 n n a j m i Q l o c a t i o n /I, p a i r s | 4 f‘5 . {! {!;!} fti} fei} {I;!} 11,2} ft: {z',4] M l ft!} {!;!} flii {2.3} fez} fell {3; fl,':
fT 0 R LOCATION PAIK5 j} '] { f 2 , F.O.B. ANP BASING-POINT ARE EQUAL o SOURCE: APPENPIX A, TABLE 10, P. »US to FT:rrr. CHAR? 5 Uni Yomn Zone . ■ f> 2,6 STEEL INDUSTRY -AVERAGE TRANSPORTATION COSTS Basing-point j' S DEMAND- ELASTICITY: 1.0 F.o.b. Hill
tc 2^0
o &Q
©LAO
l,2in,Zl/2,Z) 2,21/2,2 3/112) 14,3/14,2/13,4/ {Z,4j R l /{l,3 /R zJ {3,3
*FOR LOCATION PAIRS/ j, j , F.O.B. ANP BASING-POINT APE EQUAL o SOURCE-'APPENPIX A, TABLE U , P 170 VjJ Uni form CHART 6 . .. Zone STEEL INDUS THY AVERAGE TRANSPORTATION COSTS Basing-point DEMAND. ELASTICITY: 1.5 F.o.b. Mill
$ 2160.
“ 2.«:
^ 160 r 120
L O C A T I O N FAIRS
* F 0 R LO CA TIO N p a i k s |J 'J ? {z'l} 7 R aB* ANP PASING-POINT APE EQUAL H* o SOURCE: APPENPIX A, TABLE it, F lit -P- Uniform CHART 7 Zone BELT INDUSTRY AVERAGE CEOSS-HAULING Basing-point jj DEMAND ELASTICITY: .5 F.o.b, Hill , « 24I i i
L O C A T I O N'Jm U m uui/ui/i.1! fi,ii jz. 21/2,21 PAIRS 14,411,4/ 11,1 J XiM\z,M (1,3/1UJ 14,3114,2/(3,4] (2,4] (2,1 J (l,3 /\ \ z ] (3,3)l2,3J (2,2}(3,3}\3,2j jz,3Jl2,2j(3,3/ (2,3}\2,2]
*FO R LOCATION PAIRS F.O.B. ANP BASING-POINT ARE EQUAL o SOURCE: APPENPIX A , TABLE ~t ? P. I to 2 La Unif o i m .. . CHART 8 : ; . Zone BELT INDUSTRY AVERAGE CROSS-HAULIRC- Basing-point DEKAND ELASTICITY: 1.0 I F.o.b. Mill
LOCATION FAIRS
*FO R LOCATION p a i r s , F.O.B. ANP BASING-POINT ARE EQUAL M O SOURCE: APPENPIX A, TABLE &, P. IC»4 ON Uni form ;chart 9 . .. BELT INDUSTRY AVERAGE CROSS-HAULING Zone - Basing-point • ■ dsialto' El a s t i c i t y : i.5. F.o.b. Kill
i
LOCATION PAIRS ft?} {!;!} ft?} fti} {1,3} {i,z} ft 1} ft 2} ft?] ft?] ft?] ft 1} {!;!} ftft] ft?} ftftjft?} ft lift;!} ft; 2} ft?} (Ilift?}
*FO R LOCATION PAIRS { ] 'j} ? { | 2ft {| 2} ’ F.O.B. ANP BASING-POINT ARE EQUAL M o >3 SOURCE: APPENPIX A, TABL.E P. Uni for® "S'"! CHART 10 Zone STEEL' INDUSTRY AVERAGE CROSS-HAULING Basing-point M DEM AND' ELAS TI Cl TY : .5 MlAJJU F.o.b. Mill'
« 20
v * oCSX^JfiXSJ LOCATIONl1,l| J1,1I Jl,l m m m
{121 {2,2} sFOR. LOCATION PAIRS ri,n , F.O.B. ANP BASING-POINT ARE EQUAL V .hf’lu. ’ [2,2. M O 00 SOURCE: APPENPIX A, TABLE P. x ~ CHART 11 Uniform Zone STEEL INDUSTRY.AVERAGE CR03S-HAULING Basing-point DEM .AND ELASTICITY: 1.0 F.o.b. Mill
REEIkQHlnS
L O C A T I O N 1.Z1/L21 jl.Zl Jl.21 n.Zl fl.Z’ 1 ,2 1 /1 ,2 ) (2.Z} (2,21 PAIRS 4, Z J [3,4j (z,4j 12,1 j {1,3J (1 ,Z Z.3J (2,2J (3,3j (2,3)
FOR LOCATION PAIRS F.O.B. ANP BASING-POINT ARE EQUAL o vo SOURCE: APPENPIX A, TABLE 11 ; P. 170 CHART 12 Uni form STEEL INDUSTRY AVERAGE CROSS-HAULING Zone DEMAND ELASTICITY: 1.5 Basing-point p: F.o.b. Mill n
I
LOCATION] I, II j ] , 1 l,4J 11,1 J Z, 11,3/11,2. 4 ,3 /l4 -,2 / \Z,4J 11,3.
*FO R LOCATION PAIRS j j ' [22 , F.O.B. ANP BASING-POINT APE EQUAL M H1 SOURCE: APPENPIX A, TABLE II, P. H 2 O Ill the location of the sellers did not affect the rankings of the price policy output, there appears to be a strong func tional relationship between the goegraphic price policies and transportation costs and cross-hauling (see Table 3).
In general (for the same model variables):
1. Trans. Costf>0>b> < Trans. Costbaslng_polnt <
Trans. C o s t ^ < Trans. C o s t ^ ^
2. Cross-hauling^0. b. < 0r°3s-hauilnSbaslng_polnt-
Cross-haulingzone < Cross-hauling^ fQ M
3. Trans. Cost and Cross-hauling = f^ (Geographic
Price Policy)
Production Costs
In general, production costs were lowest under f.o.b. mill, second lowest for basing-point, third lowest for zone, and highest for uniform (see Charts 13-18). This ranking is the same as for transportation costs and reflects the in direct effect of transportation costs on production costs, because transportation costs affect prices, which in turn affect demand and therefore production costs.
There are two cases in which the above rankings did not hold. For some location pairs (see Charts 13, 16, 17) basing-point production costs we re higher than those of the uniform and zone price policies because the basing-point 112
TABLE 3
GENERAL RANKINGS OF TRANSPORTATION COSTS AND CRCS S-HA.ULING
PRICE BASING- ELASTICITY INDUSTRY F.O.B. ZONE UNIFORM OF DEMAND POINT
Belt 1st 2nd 3rd 4th .5 Steel 1st 2nd 3rd 4th 3rd 1.0 Belt 1st 2nd 4t’n Steel 1st 2nd 3rd 4th
Belt ! 1st 2nd 3rd 4th 1.5 Steel 1st 2nd 3rd I 4th CHART 13 Uniform \Y-;S BELT INDUSTRY AVERAGE PRODUCTION COSTS Zone M DEMAND ELASTICITY: .5 Basing-point F.o.b. Mill
ED n j £ l LOCATION PAIRS
FOR LOCATION PAIRS- 12}’ {22} ’ ANP PAS1NG-P0JNT a r e e q u a l M VjJ SOURCE: APPENPIX A, TABLE t 7 P. I<©3 . CHART 1** Zone BELT INDUSTRY AVERAGE FRCDUCTION COSTS Basing-point DEMAND ELASTICITY: 1.0 F.o.b. 4! 220 m lc; ;i I-..'
m 14C
be 1
LOCATION PAIRS
1,2 2 , Z *F 0R LOCATION PAIRS [Ul F.O.B. ANP BASING-POINT APE EQUAL .UJ’ 1.2 512.2
SOURCE: APPENPIX A , TABLE S ; P. I<*.5 Uniform 'wU/i;U.-»■ * r^fr» J--?. *! £ .... Zone BELT 'I ND US T?. 'I PJRQQUCTION COS TS -1- ^ 0 .
DEMAND El A3 TI CITY: 1.5 Basi nr-point F.o.b. Mill
m m 0 I
PAiRTs 0N{i;i} {I;!} {!;]} fei} {z$ {[3} {1,2} fell {m ) {z 51 ft’ !}{!;i} R il {2,3} fe ll {y } { iiifc il fei) (3,3} {IIH II}
*F0R LOCATION F.O.B. ANP 13ASING-P0I NT ARE EQUAL 115
SOURCE: APPENPIX A , TABLE <3, P. >(*7 CHART 16
steel industry average production cost ; 'DEMAND ELASTICITY: .5 Uni fore Zone Basing-poirt ■F.o.b. Mill I
r i n LOCATION! F A IR S |4f4j [l(4J M 1J 3,4 z,4j 11,3J 1,2J [4,3 Z.3J [2,2) [3,3j [2,3} 12,2}
2,2 *F O R LOCATION PAIRS-j }'] j ? } ] ' ! ?12,2 , F.O.B. ANP BASING-POINT APE EQUAL M M ON SOURCE: APPENPIX A , TABLE lO } P. 10-3 CHART 17 — STEEL- EffBHSTRJr A?SJASE~PHeBGCTI«H-. COSTS DEMAND ELASTICITY: 1.0 U n i f o r m Z o n e Basing-point | j j:. ■ F.o.fc. Kill
u n n : n r i r— i r m i— 11—inmmr—inn. LOCATION PAIRS
TO R LOCATION p a i k s |J'J|,|]^|?||'||, F.O.B. ANP PASlNG-POl NT ARE EQUAL M SOURCE-* APPEN PIX A , TABLE u , P. H i ■>3 Uni fora CHART 18 STEEL INDUSTRY AVERAGE PRODUCTION COSTS Zone
DEMAND ELASTICITY: 1.5 ' Basing-poi“ t F.o.t.
V 2,00
LOCATION FAIRS f e W ; ! }
*FOR LOCATION PAlKsjj'j],{]'!}, {II}, F.O.&. ANP £ASING-P01NT ARE EQUAL
SOURCE-'APPENP1X A, TABLE t* ; P. 113 119 price does not reflect the true transportation charges of an advantageously located non-basing-point seller. This results in phantom freight for this seller, and its low costs do not stimulate demand. This leaves production unit costs higher than those of the uniform and zone price policies
(see Table . These do, in this situations, allow the transportation costs of an advantageously located seller to be reflected somewhat in prices. This increased demand, thereby decreasing production costs enough to be lower than
those of the basing-point policy.
In the other case, for some locational combinations, baslng-point costs were even lower than f.o.b. mill costs
(see Chart 15). These situations are the direct result of
the market price calculation for the low price market under
the two price policies. Recall that the market price is
the average of both sellers' prices in a particular market.
In some situations, this low price market can dominate the
other markets and influence the industry average price.
Under the baslng-point price policy, the basing-polnt
market has the lowest price because the basing-point seller
does not incurr any transportation costs in his home market.
The non-basing-point seller's transportation costs to the
baslng-point market are ignored in determining the market
price. Therefore the market price of the basing-point mar
ket is equal to the basing-polnt seller's price which is es
sentially the basing-point seller's production cost. 120
TABLE 4
GENERAL RANKINGS OP PRODUCTION COSTS
PRICE ELASTICITY INDUSTRY F.O.B. BASING-POINTZONE UNIFORM OP DEMAND
Belt 1st 2nd,3rd 2nd,3rd 4th .5 Steel 1st 2nd, 3rd, 4th 2nd,3rd 3rd,4th
Belt 1st 2nd 3rd 4th 1.0 Steel 1st 2nd, 3rd 2nd, 3rd 3rd,4th
Belt 1st,2nd 1st,2nd 3rd 4th 1.5 Steel 1st 2nd 3rd 4th 121
Under the f.o.b. mill price policy, both sellers’ home markets are the low price markets. At least at the start, both sellers can participate economically in the other's home market. Since the price each seller quotes in the other's home market includes transportation costs to this market, these costs will make market prices in these two mar kets higher than they would be if each were a home market under the basing-point price policy. Thus
MP^of basing-point seller market = basing-point
seller's production cost.
MP(t) °P seller market = seller's pro
duction cost + the other seller's production
cost and its transportation cost to this
market.
Given a high price elasticity of demand, this lower market price in the basing-point home market stimulates de mand in this market more than the higher market prices in the f.o.b. home markets. This stronger stimulation of de mand forces the basing-point production costs to be lower than the f.o.b. mill costs. This differential in production costs is large enough to compensate for the basing-point price policy's higher transportation costs, and will result in lower prices. These lower prices will stimulate demand
even more and sustain the difference in production costs.
In these situations the basing-point market has the 122 lowest price, because under this policy, transportation costs are not included in any shipments to the basing-point regardless of the shipping point. In general, demand is stimulated in this market because it has the lowest price for both sellers. Under an elasticity of 1.5» demand in creases rapidly, especially at the baslng-point market, and production costs drop markedly. Under the f.o.b. mill price policy, what would be the baslng-point company location is not stimulated as, much as under the baslng-point price policy.
Therefore, in the belt industry, with the price elas ticity of demand of 1.5 and for most locational combinations, this effect dropped production costs lower than f.o.b. pro duction costs (see Table 5)*
Summary
Production costs, in general, had the same ranking as transportation costs. However, In some situations, this result was Influenced strongly by the price elasticity of demand which partly caused basing-point production costs to be lower than f.o.b. production costs.
In general (for the same model variables):
4. Production Cost- . < Production Cost, . < .f.o.b. bas:.-pt.
Production Cost ^ < Production Cost,. , _ zone uniform
5. Production Cost = fg(Geographic Price Policy) 123
TABLE 5
PRODUCTION COSTS
ELASTICITY OF DEMAND =1.5 LOCATION PAIRS BELT INDUSTRY STEEL INDUSTRY F.O.B BASING-POINT BASING-FOINT
1.1
1.1 01
1.1
1.1 97. . 124-
Baslng-Polnt Profits
In this model, the basing-point price policy is the only one that generates profits and losses (see Charts 19,
20). Profits in the remaining policies were reduced to, or close to, zero in the equilibrium process. Under the basing- point price policy, all market prices are determined by the basing-polnt seller's transportation and production costs to these markets. If the non-basing-point seller has a more ad vantageous location than the baslng-point seller, then its transportation costs are probably lower than the basing-point seller's, thus giving it a profit. This type of profit is called phantom freight. If, however, the relative locations are reversed, the non-basing-point seller’s transportation costs will be higher than those of the basing-point seller’s.
This results in freight absorption by the non-basing-point seller. If, in the case of high price elasticity of demand, the non-basing-point seller is located far away from the bas ing-point, it must ship a tremendous quantity into the bas ing-point area, resulting in drastic losses for the non-bas ing point seller.
In this model, profits are the result of phantom freight, which means the buyer pays for fictitious transportation services. Losses are the result of freight absorption and presumably could not be sustained indefinitely, forcing the non-basing-point seller to move to the basing-point or go out of business, CHART 19 Demand Elas. .5 BELT INDUSTRY AVERAGE PROFITS
si.oo
3 -2.0GT^
a i - 3 . 0 ' d
4 . 0 0 U
S.T5.00S
LOCATION! 1 ,1 1 11,1 4,3j 14,2] 13,4J1Z,4J 12,1] 11,3
2,2 fFOK LOCATION PAIRS j} ' j j ,j 2 F.O.B. ANP BASING-POINT APE EQUAL M IN> SOURCE: APPENPIX A , TABLE 13; P. 174 vn CHART 20 Demand Zlas.
i- $ .5 0 STE2L INDUSTRY AVERAGE PROFITS
■ m
f-fljr "WO'
*$1,401 LOCATION PAIRS
*FOR LOCATION PAiKsjj'J? { f 'l } ’ R a B - ANP BAS1NG-P0I NT ARE EQUAL I-* ro S O U R C E : APPENP1X A, T A B L E P. ITA ON 127
Generally, there is a relationship between profits and a geographic price policy such that
6. Profits = f3(Prtoepolloy x)
Prices
In general, prices will be lowest under f.o.b. mill, second lowest under basing-point, third lowest for zone, and highest for uniform (see Charts 21-26 and Table 6). This is consistent with the rankings of transportation and production costs, since prices, according to the price policy, were de termined by the transportation and production costs and pro fits, or losses, if any. The rankings would not be changed, except in situations in which basing-polnt production costs were lower than f.o.b. production costs. The baslng-point prices were also lower because the lower f.o.b. transporta tion costs did not offset lower basing-point production costs. In the other situations, basing-polnt prices are highest because they include large amounts of phantom freight.
It must be noted that these two‘types of situations are the result of freight absorption and phantom freight.
Without phantom freight, basing-point prices would drop to second lowest, which is consistent with the other basing- polnt output rankings. Where basing-polnt prices are lowest, there is a large amount of freight absorption. Without this freight absorption, these prices would be second lowest. C E i H T 21...... Zone BELT INDUSTRY AVERAGE PRICES ! 10 Basing-polnt 'DEMAND ELASTICITY: .5 F. o.b.
<& H r-i G O
CQ O O •H(4 Ph O 60 <8 G : m n m LOCATION PAIRS f t 4} {]',!} {!;!} fti} ftl} { ift ftz} f t 1} ftzH y } {z,^} ftf} fl,'!} {!;!} f tii f lf t ftz} {^1} { ilife llfe i} {Is} ( til ft!} cF0R LOCATION PAIRS , F.O.&. ANP 0ASING-POINT ARE EQUAL 128 SOURCE-'APPENPIX f\, TABLE “I , P. i<#3 Uni form CHART 22 Zone 10 BELT INDUSTRY AVERAGE PRICES Basing-polnt [jiff] DEMAND ELASTICITY: 1.0 F.o.b. Mill 01 ua <—i i—i -o Q tQ a> o u p* © tc "Uj © $ L rcr-T r ' i . r s r s r s n r a LOCATIONirijinji ri.iui.nn.ii fijnijui^iazULZMui n,zi n.ziaqp.ii n,ii PAIRS M ii ,4J u j j k 3 ! hzj (3.4) \im (2,1 i m M f o j U M FOR LOCATION PAIRS F.O.B. ANP BASING-POINT ARE EQUAL l\> SOURCE: APPENPIX A, TABLE P. »<•* VO Uniform .• L.:__; CEAHT :23.... -. Zone Basing-point DEMAND ELASTICITY; 1.5 F.o.b. Kill t LOCATION PAIRS FOR LOCATION PAIRS |J'J{f 2 }» F-O-5 - ANP gASlNG-POVNT ARE EQUAL H u> o SOURCE: APPENPIX A , TABLE 9, P. ffe? Uni form .CHABf 2 4 Zone Basing-point BE3EAND ELASTICITY: F.o.b. Hill niiiauin n i r t ■ wjrxx* liman'll ill fa2b 1(M3 LOCATION FAIRS f y l {2,3} {zi} {3^3} {s Iljz .l} {2,2} { | 3} cFORLOCATlON PAlRsjj'j|,|j^J,||'|}, F.O.B. ANP BASING-POINT A R E EQUAL M U) H* SOURCE-* APPENPIX A , TABLE |0 ; P. Ife* rr U niform ' ' : ' ' CTTAT?^ 25 ; Zone i i :s -feet. in d u st r y average ph t c e s : Basing-polnt : DEMAND ELASTiGITY: 1.6 F.o.b. Mill i I I H l l i ! f I LOCATION PAIRS ftJftiHMiaittSHBmHfellWeSfaiHMWHHHH *FOR LOCATION P A IP sjj' !H!M :ib F 0-B- ANP EASING-POINT ARE EQUAL M Vo) ro SOURCE: APPENPIX A ? T A B L E |( , P. 17/ Uastng-point « LOCATION PAIRS {U} {!;!} feilfzii} {1,3} {i ,2} {4,3} ft } {3,4) {z',4] f!;I}{!:i} f y l (2,3) {z^} {ilifeilfk l} (I,'!} {I;l}fl'i} \kk 1 \ fifi \k?} £FOR location p a irs| J'j {2'!} ’ F.O.B. ANP BAS»NG-PO»NT A R E E Q U A L M VuJ VjJ SOURCE: APPENPIX A, TABLE ? P. IT3 TABLE 6 GENERAL RANKINGS OF PRICES PRICE ELASTICITY INDUSTRY F.O.B. BASING-POINT ZONE UNIFROM OF DEMAND Belt 1st 2nd,3rd,4th 2nd,3rd 3rd,4th .5 Steel 1st 2nd,4th 2nd,3rd 3rd,4th Belt 1st 2nd 3rd 4th l.o Steel 1st 2nd,3rd,4th 2nd,3rd 34d,4th Belt 1st,2nd 1st,2nd 3rd 4th 1.5 Steel 1st 2nd 3rd 4th 135 In general (for the same model variables): 7. From equations 1 and 4: Price_ ^ . < f.o.b. •Pr* cebasing-point < Pricezone ■price uni form 8. ■Pr^cepoiicy x = Transportation and Produc- tlon Costspolloy x) Summary of Analysis In reviewing the functional relationships discovered in the analysis of the data, we see that there is a strong rela tionship among the geographic price policies and transporta tion costs, cross-hauling, production costs, and prices. These relationships were stated as follows (model var iables kept constant): 1. Trans. Costf<0_b_ < Trans. Costbaslng_poln(. < Trans. Costzone < Trans- Costuniform 2. Cross-haullnsf_0_b_ < Cross-haullngbaslng_polnt < Cross-hauling^, < Cross-hauling,^,^, 3. Trans. Cost and Cross-hauling = f^( Geographic Price Policy) k. prod. Costfi0_b_ < Prod. Costbaslng_polnt < Prod. Costzone •= Prod- Costunifor,n 5. Prod, Cost - fgCGeographlc price Policy) 6. Profits = f>3(Pricep0iiCy y ) * basing-polnt only 136 7. Prioef>0-b- < Prloebaslng. point < Prfcezon9 < prlceuniform 8- Prloepolloy x = f^(Trans. and Prod. Costspolloy p . Summary The dissertation model was manipulated to find equili brium positions for a prescribed set of experiments. The results of the experiments have been presented in this chap ter with emphasis on portraying them in the context of the hypothesis. The factors which may affect the impact of the geographic price policies are arranged so that their effects are highlighted. The result is a portrayal of the results which will facilitate the testing of the hypothesis and the understanding of the model, and possibly,' the business prac tices' on which it is based. CHAPTER V CONCLUSIONS Introduction In this chapter, the results of the experiments will be analyzed, the hypothesis tested, and the conclusions made possible by the data discussed* In drawing conclusions from - the results discussed in Chapter IV, it is to be remembered that these xesults were determined by the equation parameters and initialization values used in the experiments. Therefore, conclusions are limited in scope to situations similar to those found in the model and the experiments. Geographic Demand Stimulation Factor In explaining the results, the geographic demand stim ulation factor of geographic price policies is most helpful. The geographic demand stimulation factor of a geographic price policy stimulates or depresses demand in a market through the amount of transportation costs charged to buyers in these markets. The strength of this factor for a geo graphic price policy depends on the extent to which actual transportation costs are incorporated in the buyer’s bill. Since actual transportation costs are a function of distance, this factor can affect demand in different markets according 137 138 to their distance from the seller. If the geographic price policy one uses closely incorporates actual transportation charges in the buyer's bill, then the geographic demand stim ulation factor will be strong. If, on the other hand, actual transportation charges are not incorporated in the buyer's bill, the geographic demand stimulation factor vdll be very weak. Since transportation costs in this disserta.tion model are a function of distance, it will be possible to illustrate how this geographic demand stimulation factor would work. In figure 15, a seller at X sells to both buyers at A and B. Since XB is longer than XA, actual transportation costs will be higher for the sale to B than the sale to A. If the seller's geographic price policy Incorporates these actual charges in its prices to A and B, the buyers at A will have to pay a lower price than those at B. Demand at A will be stimulated more than at B and, therefore, the average distance shipped will decrease because of relatively greater demand at A than B. This will cause the average price to A and B to decrease because the average transportation cost per shipment will decrease. If the price elasticity of de mand is high, then: it is possible in the above example not only to have prices decrease at A, but to have prices in crease at B, which will increase the change in relative de mand at A and B, giving more impetus to the geographic de mand stimulation factor. 139 FIGURE 15 A HYPOTHETICAL SELLER AT X AND BUYERS A.T A AND B FOR THE DEMAND STIMULATION FACTOR EXAMPLE 140 Geographic Demand Stimulation Factors of the Prlc-e Policies F.o.b. Mill The f.o.b. mill price policy incorporates the exact transportation charges In the buyers’ bill, and allows a more advantageously located seller to capture a large share of the market. The f.o.b. mill price policy has the strongest geographic demand stimulation factor, hereafter called demand stimulation factor. Baslng-Point The basing-point price policy incorporates the exact transportation charges from the basing-point to the seller, regardless of the actual distance shipped. If the non-bas- ing-point seller is located more advantageously than the bas ing-point seller, these reduced transportation charges will not show up in the buyer’s bill. Thus, the baslng-point price policy has the second strongest demand stimulation factor. Zone The zone price policy incorporates a zone average trans portation charge in the buyer's bill, regardless of the dis tance shipped. The zone price policy market cost equation (just focusing on distance) is as follows: K Z £ Z Dlst ( J) 1=1 - ______KZ Where: K = number of sellers Z - number of markets in a zone. The market cost within a zone is partially a function of the average distance shipped by both sellers to all markets with in that zone. The zone price policy has the third strongest demand stimulation factor. Uni form This price policy incorporates the average transporta tion charge in the buyer's bill regardless of the distance shipped. Again, just focusing on distance, the price policy mar ket cost equation is K N L L Dist ( •)) KN Where: K = number of sellers N = number of markets. Thus the market cost is partially a function of the average distance shipped by both sellers to all markets. The uniform price policy has the weakest demand stimu lation factor. In general, the f.o.b. mill policy has the strongest demand stimulation factor, baslng-point the next strongest, zone the next, and uniform the weakest. ]>2 Conclusions about Hypothesis Variables The demand stimulation factor is valuable in understand ing the effects of geographic price policies on transporta tion costs, cross-hauling, production costs, profits, and prices. From the analysis of the results and the demand stimulation factor, it is apparent that transportation cost is the most sensitive dependent variable in the hypothesis, owing to the effect of the demand stimulation factor on the average distance shipped. This points up that the most sen sitive variable in the model is distance and that the var ious price policies through demand stimulation will have a great effect on the average distance shipped. Except for a few situations in the belt industry with a high price elasticity of demand, f.o.b. mill pricing leads to the lowest transportation ->osts, cross-hauling, produc tion costs, and prices. The primary reason for f.o.b. mill pricing resulting in lower costs is that it has the highest demand stimulation factor. Under conditions of the model, basing-point pricing leads to the second lowest transportation costs, cross-haul ing, production costs, profits, and prices. In the model, profits only occurred under basing-point pricing and varied according to location and price elasticity of demand. Bas- ing-point costs and prices were second lowest because it had the second strongest demand stimulation factor. 1^3 The significant exceptions, in which basing-point prices were either highest or lowest, were discussed, in Chapter IV, It is interesting to note that basing-point pricing is the only price policy in the model that results in profits and losses. These profits or losses are directly attribu table to the existence of phantom freight and freight absorp tion. When the non-basing-point seller is located more fav orably than the basing-point seller with respect to the six teen markets, it will receive phantom freight. If, on the other hand, the non-basing-point seller is disadvantageously located, it will have to endure freight absorption. As can be seen from the charts, the non-basing-point company faces severe losses and presumably will be forced out of business if it does not move to the basing-point location. Under conditions of the model, zone pricing in the model resulted in the third lowest transportation and produc tion costs, cross-hauling, profits, and prices because zone pricing has the third strongest demand stimulation factor. The only exception to this occurs when basing-point pricing results in higher prices making zone prices second lowest. Under conditions of the model, uniform price policy will result in the highest transportation and production costs, cross-hauling, profits, and prices because uniform price policy has the weakest demand stimulation factor. 144 Hypothesis Validation The validity of the hypothesis will be tested by apply ing the results of the analysis to the corresponding parts of the hypothesis. Of interest will be the discussion of how the results of the analysis bear on the key areas of scholar ly conflict which were discussed in the second chapter. Recall that the hypothesis was that, when compared with three other geographic price policies, an f.o.b. mill price policy will result in the lowest A. average transportation costs, B. average cross-hauling, C. average production costs, D. average profits, and E. average prices. Fart A. and B of Hypothesis A. An f.o.b. mill price policy will result in the lowest transportation costs. B. An f.o.b. mill price policy will result In the lowest cross-hauling. Under conditions of the model, f.o.b. pricing wTill lead to the lowest transportation costs and lowest cross-hauling because it has the highest demand stimulation factor. Bas ing-point pricing will lead to the second lowest transporta tion costs and cross-hauling, primarily because it has the 1^5 second highest demand stimulation factor. Zone pricing will lead to the third lowest transportation costs and cross-haul ing, because it has the third highest demand stimulation fac tor. Uniform pricing, because of the lowest demand stimula tion factor, will lead to the highest transportation costs and cross-hauling. Part A of the hypothesis is accepted: An f.o.b. mill price policy will result in the lowest average transportation costs. Part B of the hypothesis is accepted: An f.o.b. mill price policy will result in the lowest cross-hauling. Part C of the Hypothesis C. An f.o.b. mill price policy will result in the low est average production costs. Under conditions of the model, f.o.b. mill pricing will lead to the lowest production costs. This is due to the in direct effect of the demand stimulation factor which de creases transportation costs, leading to lower prices, higher demand; and therefore lower production costs. However, there was one case in which f.o.b. mill produc tion costs were higher than basing-point production costs. It occurred, not because of the broader market penetration resulting from the basing-point price policy, as put forth by Simon and U.S. Steel, but because of the effect of the price elasticity equation. This equation, which was parameterized 3*6 at 1.5i allowed demand to expand tremendously In the basing- point sellers home market. This tremendous and unrealistic expansion of demand is due to the definition of price elas ticity of demand used in the model. Despite the fact that in most of the experiments f.o.b. mill production costs were lowest, one experiment resulted in basing-point production costs being lowest. Since part C of the hypothesis cannot be accepted, it must be rejected: An f.o.b. mill price policy will result in the lowest average production costs. Part D of the Hypothesis D. An f.o.b. mill price policy will result in the low est average profits. In this model the basing-point price policy was the only policy producing profits or losses, because this policy did not induce actual transportation charges in prices to buyers. On an industry basis, the net result was either phantom freight in which more was paid for transportation than was actually used or freight absorption in which less was paid for transportation than was used. This result depends on the location of the non-basing-point seller. In the model, profits are considered to be over an above a normal profit or interest function which is assumed to be Included in the production function. The profits under the basing-point price policy are due to phantom freight, which 1^7 simply inflates the prices buyers must pay. Therefore, any profits due to the basing-point price policy result in a loss to buyers because of inflated prices. Losses due to freight absorption do not benefit the buyers because this freight is absorbed by a seller to enable the seller to participate in a market. The remaining price policies did not result in profits as defined by the model. Thus, f.o.b. profits in this model are no higher than the zone policy or milform policy profits. As the results are Inconclusive, part D of the hypothesis is rejected: An f.o.b. mill price policy will result in the lowest average profits. Part E of the Hypothesis E. A n f.o.b. mill price policy will result in the low est average prices. Under conditions of the model, f.o.b. mill pricing will lead to the lowest average prices, due both to the dir ect and Indirect effect of the demand stimulation factor on transportation and production costs. However, basing-point production costs were lowest. Machlup and Stocking both felt that f.o.b. mill pricing would lead to lower prices because of lower transportation costs. The U. S. Steel source book and Simon agree that the higher transportation costs due to the basing-point price policy in the steel industry would be more than offset by 148 decreased production costs and result In lower prices. In the model, basing-point prices were lower than f.o.b. prices in some situations; however, this occurred under the belt in dustry conditions. Again, zone pricing and uniform pricing resulted in third and fourth lowest prices, because of the relative strength of their demand stimulation factors. Part E of the hypothesis is rejected: An f.o.b. mill price policy will result in the lowest average prices. Summary of Hypothesis Validation The key section of the hypothesis is part C, which con cerns production costs. Not only is it a critical issue in the literature on the subject, but it also has an important effect on part E of the hypothesis, which deals with prices. The data in this dissertation has shown that the effects of the geographic price policies can be modified by other variables, such as the location of sellers and the price elasticity of demand. Real World Conclusions The writer concludes that for real world situations, similar to those of the model, an f.o.b. price policy will lead to lower 1. average transportation costs, 2. average cross-hauling, 3. average production costs, and ty. • average prices. and would lead to highest buyer profits. Even though part C of the hypothesis was rejected, this rejection was caused primarily by a combination of the effect of the model elasticity equation, which would allow demand to rise to Infinity under a price elasticity of demand of 1.5, and the effect of the market participation rule which allowed the non-basing-point seller to continue to compete in the basing-point market at a large loss. These two factors caused the basing-point production costs to be lower than f.o.b. production costs in certain situations. It is interesting to note that the case in which this happened was in the belt industry which has low fixed costs and high variable costs. The basing-point argument rests on the low variable production costs of the steel Industry, For the same reasons given above, the situations in which the basing-point prices were lowest would not be likely to occur in the real world. If it is true that the f.o.b. mill price policy would result in the lowest prices, buyers would be better off under this policy, or in other words, buyer profits would be high est, other things remaining equal. 150 Conclusions about Key Model Variables It follows that the two most important model variables are the location of the companies and the price elasticity of demand. The location of the companies had an important ef fect on average distance shipped and therefore on costs and prices. The price elasticity of demand has an effect on how much a given price policy’s demand stimulation factor will be allowed to operate. As can be seen, the combination of f.o.b. mill pricing and high price elasticity of demand can result in drastically lower prices and costs. It is interesting to note that the market share factor was relatively insignificant. However, it must be noted that this might be due to the model construction and the fact that the model searched for a close approximation to the final equilibrium position and not the final equilibrium position Itself. Conclusions about the Model The question as to whether the model is an adequate portrayal of reality is a difficult one to answer. For the purposes of this dissertation, the model is useful; it has generated data that appears to be acceptable within the re strictions of the model. However, the model has some im portant limitations, which are discussed below. These 151 limitations are: A. model environment, B. model initialization, C, model equilibrium, and D , model d e bugging. Model Environment It is obvious upon detailed study of the general model equilibria that some model rules and equations create some, perhaps, unrealistic results. These rules and equations were as follows: 1. market participation rule, 2. demand equation, and 3. market share equation. Market Participation Rule.— Under the f.o.b. mill situa tion, the poorly located competitor is driven from the mar ket in some Instances because costs are too high or market share drops to zero. Given the market share slippage and possible market elimination, it would seem probable that the poorly located competitor would try to rescue lost market share by a number of methods, such as sales promotion, pro duction differentiation or even an industry geographic price policy switch. More drastic would be a possible location shift. All of these alternatives have been eliminated from the model. Under the basing-point price policy, the badly located 152 competitor (and this is usually the non-basing-point company) incurs large transportation costs which are not recovered. This unfortunately located company has no opportunity in the model to avoid- freight absorption. This situation seems to justify the claim made by many writers that under a basing- point price policy there is a tendency to locate new plants at or near the basing-point location. Demand equatlon,— -The equation for calculating the de mand change due to price changes does not contain a factor which would impede drastic growth or shrinkage over a series of iterations. Thus, in some situations, demand grew to be many times its initial value, which perhaps is not realistic for the purposes of the model. Market share equation.— Because of the form of the mar ket share equation, it is possible for unequal market shares to become stabilized if the prices of the two sellers become equal and remain so over time. Under the experiment condi tions, this did not create any difficulty, however, because when one seller's higher price started to decrease its mar ket share this price would usually remain higher and would never stabilize at equality with the other sellers price. Initial!zatlon During preliminary runs, it was apparent that initiali zation can drastically affect the model output. The use of certain initializing values would cause prices to drop to a 153 very low level causing demand to increase monatonically. Other initializing values would increase prices enough to cause a negative demand. In one case, an old Price of $15.00 caused the sequence to oscillate under a steel industry pro duction function, a market share parameter of 1.5* and a price elasticity of demand of 1,5# Care was taken during prelim inary model runs to insure that the model was properly in itialized, that is, the sequence would converge to a stable equilibrium.. In general, the final initializlngvalu.es created a decrease in prices with a corresponding increase in demand. Figure 10 on page seventy-nine shows the general behavior with the prices decreasing at a decreasing rate, that is, converging to an equilibrium position. Equilibrium As was stated in Chapter III, the equilibrium point was defined as that point at which the difference between the average price of the present iteration and that of the pre vious iteration was smaller than $.01 for the third straight time. . 1 From analysis of the output, it appears that the number of Iterations was more than sufficient for the basing-point, zone and uniform price policies, but perhaps not sufficient in a few instances for the f.o.b. price policy where there appeared to be some fluctuation around the true equilibrium point, if, Indeed, this true equilibrium exists. In one 154 trial run, this iterative process could not find an equili brium point. However, because of the general behavior of the output, it is doubtful that the results of this dissertation would be modified by a change in the number of iterations in the definition of the equilibrium point. In most of the runs, the prices of the uniform, basing- point, and zone price policies seemed-to approach stable equilibrium points. Analytical solutions might be possible for these policies within certain initialization ranges. The f.o.b. mill prices for most runs oscillated somewhat, pro bably because the market participation routine would change the values of the Iteration when ever a seller was driven from a market. (See FlgurelO). Debugging In debugging the model, It was discovered that some I sellers were being driven out of a market even though their prices were lower than the market prices. In rechecking the iterations it became evident that once a seller had lost a substantial share of the market, it would not stay in the mar ket even- if its relative price became lower than the other seller's. It was necessary to dampen the change in the mar ket share component of the equation by subtracting it from .5. In some f.o.b. mill price policy Iterations very large price changes generated negative demand figures. Because of these price changes the numerator of the elasticity 155 component of the demand equation became negative. As these values were very small, a statement was added multiplying all resulting negative demand figures by -1. In other f.o.b. mill price policy iterations, it was discovered that both sellers would be driven out of some mar kets. The first seller, it happens, was driven out because its market share became zero. Several iterations later, the second seller left the market because its costs were too high. The market itself was closer to the second seller than to the first. Thus, the first seller’s transportation cost and. accordingly its prices became too high to compete with the second seller in that market and therefore lost its en tire share. However, the first seller is more advantageously positioned in reference to all the markets. Therefore, even though the first seller has lost one market to the second, its prices overall remain lower than the second seller. This overall price advantage increased its market share at the ex pense of the second seller's costs in all of the markets. Eventually the second seller is driven out of some of the same markets it drove the first seller from, leaving no sellers in these markets. To compensate for the loss of both sellers to this mar ket, the whole market was given to the first seller at the run initialization values for price, demand, and costs. 1 5 6 Conclttslons about the Use of Iterative Method Aside from the problems of determining whether it was possible to arrive at an analytical solution to the model equation, the infinite loop mentioned above adds a note of complexity to the task of finding this solution. If such a solution exists, it is no doubt valid only under certain conditions, that is, the sequence will not converge for all values of the initial prices. Despite the problems involved in the initialization process, the iterative method has been an adequate tool for the purposes of this dissertation, Al though it would be of interest to know the exact analytical expressions which govern the functional relationships set forth in the analysis of the data, it was not the purpose of this dissertation to quantify the relationships. The ana lytical solution to the equations would, perhaps, be a log ical topic of further research. Even so, in that it was necessary on several occasions to consult the whole set of iterations to explain the behavior of a price policy, the ability to observe the model over several time periods was helpful in analyzing results. Suggestions for Further Work Suggestions for further work are brokln down into two categories. The first category is concerned with the elab oration of the present model and the second category is 157 concerned with the exploration of spatial competition. Elaboration of the Present Model Routes and obstructions may be added for the sake of realism and to highlight possible locational advantages. Channels of dlstributuion, such as wholesalers, retailers, and corresponding inventory needed to carry out their func tions, may be added. This would introduce time lags and the problem of inventory control to the model. Demand may be varied according to market location and possible time. Pro motional aspects, such as selling and advertising, might be added to introduce the dimension of non-price competition to the model. As noted above, the exact functional relation ships between the price policies and the output may be derived for the sake of conciseness. The inclusion of these factors would make the model more complex. Such a model could be used for studying the effects of geographic price policies on specific industries. The data developed with these models could be of use in govern mental regulation of business, or in the development of com pany strategies. Spatial Conflict The most interesting area of future work appears to lie in the area of spatial competition. With the Inclusion of of the above factors and possibly more, the present model may 158 be modified for use as a means of finding and exploiting ad vantageous location situations. There are several ways of approaching conflict situations. The first would be game theory which is too restrictive in the area of the assump tions of perfect knowledge and hedging philosophy. The sec ond approach would be business gaming in which two or more companies are in direct competition for market share. Pre sently most of these games are concerned with sales promo tion and production planning. The addition of locational competition could make these games more effective learning devices. The largely unexplored field of spatial competition may be exploited by introducing the factor of distance into the typical business game. After enough experience has been built up, generalizations concerning locational competition may be developed for use in formulating military strategies. Implications for Business and Government Policy Business Policy Given the conditions of the research model, if bus inesses "used an f.o.b. price policy, they would be able to give their customers lower average prices. In a truly com petitive economy, an f.o.b. price policy is in the best in terest of the businessman. However, in economies character ized by monopolistic and oligopolistic market structures, it x59 is not clear that lower average prices are good for business men. For, if they face a kinked demand curve, price competi tion might lead to lower revenues. Secondly, under an f.o.b. mill price policy, it would be extremely difficult to deter mine what competitors1 prices should be to any given customer., Because of this, trade associations would have a difficult " time discovering and disciplining price cutters. An f.o.b. mill price policy would reintroduce price competition in many markets with all of the uncertainty this entails. Moreover, under an f.o.b. mill price policy, poorly located facilities would be priced out of the market and would have to be moved or abandoned. Some businessmen, upon adoption of an f.o.b. mill price policy, would be forced to write off certain plant and equipment and therefore retained earnings. The introduction of effective price competition would to some extent diminish Investments in advertising, sales forces, and good will. Government Policy It would appear to be in the public interest to have all businesses, operating under conditions similar to the research model, adopt an f.o.b. mill price policy. Care would have to be taken by the relevant governmental regula tory agencies to see that the effects of price competition were not evaded through any means of collusion open to the businessman--legal, semi-legal, or illegal. For instance, to individual producers in order to negate their transporta tion cost advantages. If the regulatory agencies are suc cessful in establishing an f.o.b. mill price policy, there would be other benefits aside from the general lowering of the costs of living to the consumers. One would be a more rational placement of new manufacturing facilities. These facilities would have to be placed so as to minimize overall movement costs. If freight were to reflect the true costs of movement, there would be a more rational allocation of freight among the transportation modes because producers would have to seek the best means of moving raw materials and finished products. Again, care would have to be taken to see that the carriers did not circumvent the f.o.b. mill price policy by giving Important producers unjustified rate concessions so that they may Increase their market areas. To sum up the implications, it is possible that an f.o.b. mill price policy for business would not be in bus iness's interest because price competition would introduce more uncertainty and would destroy some investment in adver tising, sales forces, and plant and equipment. Because a mandatory f.o.b. mill price policy would be in the best in terests of the public, but against the best interests of some major companies, agenecles would have to continually be on guard against effective evasion of this policy. APPENDIX A EXPERIMENT DATA TABLE 7 EXPERIMENT 1 DATA AVERAGE TRANSPORTATION COST AVERAGE CROSS-HAULING XT.OCj V ri.xATTONT J. w * PAIRS BASING- UNIFORM F.O.B. ZONE UNIFORM F.O.B. BASING- POINT POINT ZONE 1,1 - 4,4 7.10 4.04 7.04 7.10 2406 1370 2382 2404 1,1 — 1,4 7.10 4.33 6.76 6.84 240 6 1466 2291 2317 1,1 - 1,1 7.10 6 .3 4 ’ 6.34 6.84 2406 2146 2146 2317 1,1 - 3.4 6.53 3.66 6.4o 6.49 2212 1239 2166 2163 1,1 - 2,4 6.53 3.78 6.30 6.39 2212 2180 2132 2163 1,1 - 1,3 6.53 ■ 5.04 5.89 6.22 2212 14 66 2052 2105 1,1 - 1,2 6.53 5.04 5.89 6.22 2212 1708 1996 2105 1,2 - 4,3 5-96 3.30 5.87 5.95 2019 1116 1974 2017 1,2 - 4,2 5.96 3.29 5.83 5.95 2019 1116 1974 2017 1,2 - 3,4 5.96 3.51 3-79 5.90 2019 1187 1962 1999 1,2 - 2,4 5.96 3.69 5*66 5.79 2019 1250 1916 1958 1,2 - 2,1 5.96 4.21 5.49 5.78 2019 1424 1859 1958 1,2 - 1,3 5.96 4.33 5.44 5.58 2019 1468 1841 I890 1,2 - 1,2 5.96 5.34 5.34 5.58 2019 1809 ' 1809 I890 1,1 - 3,3 5.90 3.33 5.67 5.85 1998 1128 1920 1982 1,1 - 2,3 5.90 3.78 5.67 5.73 1998 1214 1920 1940 1,2 - 2,2 5.90 4.06 5.38 5.73 1995 1376 1833 1940 1,2 - 3,3 5.33 3.15 5.11 5.12 1804 1068 1731 1784 1,2 - 3,2 5.33 3.31 5.06 5.12 1804 1120 1712 1784 1,2 - 2,3 5.33 3.63 4.94 5.87 1804 1230 1673 1733 1,2 - 2,2 • 5.33 4.02 4.87 5.27 1804 1363 1648 1733 2,2 - 3,3 4,69 3.02 4.50 4.62 1589 1058 1525 1566 2,2 - 2,3 4.69 3.28 4.42 4.56 1589 1112 1498 1543 \-j 2,2 - 2,2 4.69 4.32 4.32 4.56 1589 1463 1463 1543 £ TABLE 7 CON'T AVERAGE PRODUCTION COST AVERAGE PRICE LOCATION PAIRS BASING- BASING- UNIFORM F.O.B. ZONE UNI FORM F.O.B. ZONE POINT POINT 1,1 - 4,4 2.20 1.89 2.11 2.19 9.31 5^94 8.45 9.30 1,1 - 1,4 2.20 1.92 2.11 2.17 9.31 6.24 8.45 9.01 1,1 - 1,1 2.20 2.11 2.11 2.17 9.31 8.45 8.45 9.01 1.1 - 3,4 2.16 1.85 2.11 2.14 8.69 5.51 8.45 8.60 1,1 - 2,4 2.16 1.87 2.11 2.14 8.69 5.65 8.45 8.52 1,1 - 1,3 2.16 1.92 2.11 2.12 8.69 6.24 8.45 8.33 1,1 - 1,2 2.16 2.00 2.11 2.12 8.69 6.95 8.45 5.33 1,2 - 4,3 2.11 1.82 2.02 2.11 8.07 5.12 7.37 8 . Oo 1,2 - 4,2 2.11 1.82 2.02 2.11 8.07 5.11 7.37 8.06 1,2 - 3,4 2.11 1.84 2.02 2.10 8.07 5.35 7.37 8.00 1,2 - 2,4 2.11 1.85 2.02 2.08 8.0 7 5.54 7.37 7.86 1,2 - 2,1 2.11 1.90 2.02 2.08 8.07 6.11 7.37 7.86 1,2 - 1,3 2.11 1.91 2.02 2.0 6 8.07 6.25 • 7.37 7.64 1,2 - 1,2 2.11 2.02 2.02 2.06 8.0 7 7.37 7.37 7.64 1,1 - 3,3 2.10 1.83 2.11 2.09 8.00 5.51 8.45 7.95 1,1 - 2,3 2.10 1.87 2.11 2.08 8.00 5.42 8.4 5 7.81 1,1 - 2,2 2.01 1.89 2.11 2.08 8.00 5.95 8.45 7.81 1,2 - 3,3 2.05 1.80 2.02 2.04 7.38 4.96 7.37 7.31 1,2 - 3,2 2.C5 1.82 2.02 2.04 7.38 5.12 7.37 7.31 1,2 - 2,3 2.05 1.86 2.02 2.02 7.38 5.47 7.37 7.14 1,2 - 2,2 2.05 1.90 2.02 2.02 7.38 5.92 7.37 7.14 2,2 - 3.3 1.99 1.80 1.93 1.98 6.6 8 4.92 6.25 6.61 2,2 - 2,3 1.99 1.82 1.93 1.97 6.6 8 5.11 6.25 6.52 2,2 - 2,2 1.99 1.93 1.93 1.97 6.68 6.25 6.25 6.52 H1 o\ V*) TABLE 8 EXPERIMENT 2 DATA AVERAGE TRANSPORTATION COST AVERAGE CROSS-HA.ULING T AfPTnM C\H J-U1n PAIRS UNIFORM F.O.B. BASING- BASING- ZONE UNIFORM F.O.B. ZONE POINT POINT 1,1 - 4,4 7.10 2.94 6.89 7.09 2406 996 2335 2403 J,1 - 1,4 7.10 3.05 f, 6.24 6 .58 2406 1034 2112 2227 1,1 - 1,1 7.10 5.04 I 5.04 6.48 240 6 1806 1806 222 7 1,1 - 3,4 6.53 2.73 6.17 6.45 2212 924 2040 2184 1,1 - 2,4 6.53 2.77 5.93 6.24 2212 939 2008 2115 1,1 - 1,3 6.53 3.07 5.34 5.89 2212 1041 1807 1995 1,1 - 1,2 6.53 3.42 4.89 5.89 2212 1159 1655 1995 1,2 - 4,3 5.96 - 2.51 5.6 9 5.95 2018 852 1926 2015 1,2 - 4,2 5.96 2.51 5.19 5.95 2018 851 2015 1,2 - 3,4 5-96 2.60 5.50 5.85 2018 882 1981 1,2 - 2,4 5.96 2.72 5.18 5.60 2018 920 1756 1897 1,2 - 2,1 5.96 3.01 3.76 5.60 2018 1021 1612 1897 1,2 - 1,3 5.96 3.05 4.61 5-18 2018 1034 1561 1754 1,2 - 1,2 5.96 4.31 4.31 5.18 2018 1459 1459 1754 1,1 - 3,3 5.90 2.54 5.32 5.81 1998 862 1803 1912 1.1 - 2,3 5.90 2.72 5.00 5.56 1998 923 1695 1883 1,1 - 2,2 5.90 3.36 4.59 5.56 1998 1139 1556 1883 1.2 - 3,3 5.33 2.4 5 3.76 5.21 1804 829 1613 1763 1.2 - 3,2 5.33 2.54 4.63 6.21 1804 859 1568 1763 1,2 - 2,3 5.33 2.70 4.34 4.90 1804 914 1471 1661 1,2 - 2,2 5.33 2.89 4.15 4.90 1804 978 1404 1661 2,2 - 3,3 4.69 2.41 4.13 4.54 1489 919 1399 1539 2,2 - 2,3 4,69 2.51 3.93 4.40 1589 849 1332 1491 3.62 3.62 4.40 1589 1226 1226 1491 2,2 - 2,2 4.69 w TABLE 8 CON'T AVERAGE PRODUCTION COST AVERAGE PRICE LOCATION PAIRS BASING- BASING- UNIFORM F.O.B. ZONE UNIFORM F.O.B. POINT POINT ZONE 1,1 - 4,4 2.15 1.42 1.78 2.14 9.25 4.35 6.82 9.24 1,1 - 1,4 2.15 1.45 1.78 2.05 9.25 4.50 6.82 8.63 1,1 - 1,1 2.15 1.78 1.78 2.0 5 9.52 6.82 6.82 8.63 1.1 - 3,4 2.05 1.38 1.78 2.03 8.58 4.12 6.82 8.48 1.1 - 2,4 2.0 5 1.30 1.78 2.00 8.58 4.17 6.82 8.24 1,1 - 1.3 2.0 5 1.4 5 1.78 1.93 8.58 4.53 6.82 7.82 1,1 - 1,2 2.05 1.53 1.78 1.93 8.58 4.94 6.82 7.82 1,2 - 4,3 1.95 1.35 1.66 • 1.94 7.91 3.87 5.96 7.89 1,2 - 4,2 1.95 1.35 1.66 1.94 I 7.91 3.86 5.96 7.89 1,2 - 3,4 1.95 1.37 1.66 1.93 7.91 3.97 5.96 7.77 1,2 - 2,4 1.95 1.38 1.66 1.88 1 7.91 4.11 5:96 7.49 1,2 - 2,1 1.95 1.43 1.66 1.88 7.91 4.45 5.96 7.49 1.2 - 1,3 1.95 1.45 1.66 1.81 7.91 4.50 5.96 6.99 1,2 - 1,2 1.95 1.66 1.66 1.81 7.91 5.96 5.96 6.99 1.1 - 3,3 1.93 1.96 1.78 1.92 7.83 3.90 6.82 7.77 1,1 - 2,-3 1.93 1.38 1.78 1.88 7.83. 4.11 6.82 7.43 1,1 - 2,2 1.93 1.13 1.78 1.88 7.83 4.49 6.82 7.43 1,2 - 3,3 1.83 1.34 1.66 1.81 7.16 3.78 5.96 7.02 1,2 - 3,2 1.83 1.35 1.66 1.81 7.16 3.89 5.96 7.02 1,2 - 2,3 1.83 1.40 1.66 1.76 7.16 4.09 5.96 6.67 1,2 - 2,2 1.83 1.43 1.66 1.76 7.16 4.32 5.9§ 6.67 2,2 - 3.3 1.72 1.35 1.54 1.70 6.42 3.78 5.16 6.24 2,2 - 2,3 1.72 1.35 1.54 1.67 6.42 3-86 5.16 6.07 2,2 - 2,2 1.72 1.54 1.54 1.67 6.42 5.16 5.16 6.07 OS U\ TABLE 9 EXPERIMENT 3 DATA. AVERAGE TRANSPORTATION COST AVERAGE CROSS-HA.ULING LOCATION PAIRS BASING- UNIFORM F.O.B. ZONE UNIFORM F.O.B. BASING- POINT POINT ZONE 1,1 - 4,4 7.10 .67 6.37 7.09 2406 226 2159 2401 1,1 - 1,4 7.10 1.02 4.72 6.31 240 6 346 1600 2137 1,1 - 1,1 7.10 .85 .85 6.31 240 6 290 290 2137 1,1 - 3,4 6.53 .75 6.41 2212 2 55 1851 2170 1,1 — 2,4 6.53 .91 4.87 6.11 2212 309 1660 2068 1,1 — 1,3 6.53 1.15 3.33 - 5.56 2212 402 1130 1882 1,1 — 1,2 6.53 1.24 2.02 5.56 2212 421 686 1882 1,2 - 4,3 5.96 .84 4.88 5.94 2018 284 1656 2012 1,2 - 4,2 5.96 .92 4.67 5.94 2018 310 1583 2012 1,2 - 3,4 5.96 .96 4.45 5.69 2018 326 1508 1962 1,2 - 2,4 5.96 1.09 3.67 5.42 2018 368 12-4-5 1837✓ 1 1,2 - 2,1 5.96 1.23 2.60 5.42 2018 415 882 I837 1»2 - 1,3 5.96 1.20 2.09 4.75 2018 407 ' 707 1610 1,2 - 1,2 5.96 .87 .87 4.75 2018 293 293 1608 1.1 - 3,3 5.90 .96 4.36 5.76 1998 14 77 1951 1,1 - 2,3 5.90 1.15 3.57 5.39 1998 2 1 841 1826 1,1 - 2,2 5.90 1.31 2.48 5.39 1998 442 841 1826 1,2 - 3,3 5.33 1.08. 3.50 5.14 180 4 367 1519 1743 1,2 - 3.2 5.33 1.13 3.28 5.04 1804 382 1113 1743 1,2 - 2,3 5.33 1.23 2.53 4.69 1804 418 857 1588 1,2 - 2,2 5.33 1.23 1.99 4.69 1804 417 675 1588 2,2 - 3,3 4.69 1.17 2.59 4.45 1589 397 876 1508 2,2 - 2,3 4.69 1.16 2.06 4.22 1589 392 '599 1431 2,2 - 2,2 4.69 .88 .88 4.22 1589 296 296 1431 5 Os TABLE 9 CON'T AVERAGE PRODUCTION COST AVERAGE PRICE LOCATION I PAIRS BASING- BASING- UNIFORM F.O.B.- ZONE UNIFORM F.O.B. ZONE. POINT POINT 1,1 - 4,4 2.08 .87 .91 2.07 9.18 1.54 1.76 9.17 i ’i - 1,4 2.08 .94 .91 1.90 9.18 1.95 1.76 8.21 i,i - 1,1 2.08 .91 .91 1.90 9.18 1.67 1.76 8 .21'‘ i,i - 3,4 1.93 .90 .91 1.90 8.46 1.64 1.76 8.30 i.i - 2,4 1.93 .91 • 91 1.83 8.46 1.82 1.76 7.94 1,1 - 1,3 1.93 .97 .91 1.71 8.46 2.15 1.75 7.29 1,1 - 1,2 1.93 .99 • 91 1.71 8.46 2.23 1.76 7.27 1,2 - 4,3 1.78 .90 .90 • 1.78 7.74 1.74 1.76 7.72 1,2 - 4,2 1.78 .90 .90 1.78 7.74 1.82 1.76 7.7 2 1.2 - 3.4 1.78 .91 .90 1.75 7.74 1.88 1.76 7.54 1,2 - 2,4 1.78 • 95 .90 1.67 7.74 2.03 1.76 7.09 1,2 - 2,1 1.78 .98 .90 1.67 7.7 4 2.20 1.76 7.09 1,2 - 1:, 3 1.78 .98 .90 1.53 7.74 2.18 1.76 6.18 1,2 - 1,2 1.78 .90 • 90 1.53 7.74 1.76 1.76 6.28 1,1 - 3,3 1.80 .91 .91 1.74 7.55 1.8 7 1.76 7.56 1,1 - 2,3 1.77 .94 .91 1.66 7.66 2.06 1.76 7.0 5 1,1 - 2,2 1.77 .97 .91 1.66 7.66 2.29 1.76 7.05 1,2 - 3.3 1.63 .93 .90 1.59 6.95 2.01 1.76 6.74 1,2 - 3,2 1.63 .94 .90 1.59 6.95 2.07 1.75 6.74 1,2 - 2,3 I .63 .96 .90 1.50 6.95 2.19 1.76 6.19 1,2 - 2,2. I .63 .96 .90 1.50 6.95 2.19 1.76 6.19 2,2 - 3,3 1.49 .95 .89 1.44 6.17 2.12 1.77 5.89 2,2 - 2,3 1.49 .96 .89 1.40 6.18 2.11 1.77 5.62 2,2 - 2,2 1.49 .89 .89 1.40 6.18 1.77 1.77 5.6 2 On -O TABLE 10 EXPERIMENT 7 DATA. AVERAGE TRANSPORTATION COST AVERAGE CROSS-HAULING BASING- BASING- UNIFORM F.O.B. ZONE UNIFORM F.O.B. ZONE POINT POINT 2.60 1 . 6k 2.59 2.60 2k0 6 1518 2399 2 *40 5 2.60 1.83 2.56 2.56 2k0 6 1697 3/466 3369 2.60 2.51 2.51 2.56 2k0 6 2321 2321 2369 2.39 2.03 2.37 2.38 2212 1882 2197 2206 2.93 2.03 2.36 2.37 2212 1772 217/4 2192 2.39 2.04 2.33 2.34 2212 1885 2157 2169 2.39 2 . 0k 2.31 2.34 2212 1885 2139 2169 2.18 1.33 2.17 2.18 2018 1233 2010 2018 2.18 1.36 2.17 2.18 2018 1262 2005 2018 2.18 1.4-3 2.16 2.17 2018 1325 2000 2011 2.18 2.03 2 . 1k 2.15 2018 1875 198/4 199^' 2.18 1.77 2.12 2.15 2018 1636 1965 199^ 2.18 2.01 2.12 2.13 2018 1865 1959 1969 2.18 2.11 2.11 2.13 2018 1951 •1951 1969 2.16 1.63 2.13 2.15 1998 1510 1971 1991 2.16 1.63 2.11 2.13 1998 1509 1955 197^ 2.16 1.63 2.09 2.13 199.8 1510 1937 197^ 1.95 1.63 1.92 1 .9 k 180/4 1509 1780 1796 1.95 1.63 1.92 1 .9 k 180k 1509 1773 1796 1.95 1.63 1.90 1.92 180 k 1510 1760 1776 1.95 1.63 1.89 1.92 180/4 1510 1752 ■ 1776 1.72 1.27 1.70 1.71 1589 1179 1570 158I 1.72 1.37 1.69 1.70 1589 1271 1562 1572 1.72 1.68 1.69 1.70 1589 1271 1562 . 1572 TABLE 10 CON'T AVERAGE PRODUCTION COST AVERAGE PRICE LOCATION PAIRS BASING- UNIFORM F.O.B ZONE UNIFORM F.O.B. BASING- POINT POINT ZONE 1,1 - 4.4 5-73 5.26 5.66 5.73 8.33 6.90 8 .1? 8.33 1.1 - 1,4 5-73 5.40 5.66 5.70 8.33 7.23 8.17 8.26 1.1 - 1,1 5.73 5.66 5.66 5.70 8.33 8 .1? 8.17 8.26 1,1 - 3.4 5.62 2.01 5.66 5.62 8.01 4.04 8.17 8.00 1.1 - 2,4 5.62 2.01 5.66 5.60 8.01 4.04 8.17 7.97 1,1 - 1.3 5.62 2.02 5.66 5.59 8.01 4.06 8.17 7.93 1.1 - 1,2 5.6 2 2.01 5 • 66 5.59 8.01 4.05 8.17 7.93 1,2 - 4,3 5.51 5.01 5.45 5*51 7.69 6.34 7.56 7.69 1,2 - 4,2 5.51 5.11 5.45 . 5.51 7.69 6.48 7.56 7.69 1,2 - 3,4 5.51 5.07 5.45 5.50 7.69 6.50 7.56 7.67 1,2 - 2,4 5.51 2.02 5.45 5.49 7.69 4.04 7:56 7.64 1,2 - 2,1 5.51 5.29 5.45 5.49 7.69 7.06 7.56 7.64 1,2 - 1.3 5.51 5.46 5.45 5.47 7.69 7.4 7 7.56 7.59 1,2 - 1,2 5.51 5.46 5.45 5.47 7.6 9 7.47 7.56 ■ 7.59 1.1 - 3,3 5.50 1.89 5.66 5.49 7.65 3.52 8.17 7.64 1,1 - 2,3 5.50 1.89 5.66 5.48 7.65 3.52 ■ 8.17 7.61 1,1 - 2,2 5.50 1.89 5.66 5.48 7.65 3.52 8.17 7.61 1.2 - 3,3 5.28 1.88 5.45 5.37 7.33 3.51 7.56 7.31 1.2 - 3.2 5.38 1.89 5.45 5.37 7.33 3.82 7.56 7.31 1.2 - 2,3 5.38 1.89 5.45 5.35 7.33 3.52 7.56 7.2 7 1.2 - 2,2 5.38 1.90 5.45 5.35 7.33 3-53 7.56 7.27 2,2 - 3,3 5.24 4.99 5.21 5.24 6.96 6.26 6.88 6.94 2,2 - 2,3 5.24 5.12 5.21 5.53 6.96 6.49 6.88 6.93 2,2 - 2,2 5.24 6.21 5.21 5.23 6.96 6.88 , 6.88 6.93 CF\ TABLE 11 EXPERIMENT 8 DATA. AVERAGE TRANSPORTATION COST AVERAGE CROSS-HA.ULING LOCATION BASING- PAIRS UNIFORM F.O.B. ZONE UNIFORM F.O.B. BASING- POINT POINT ZONE 1,1 - 4,4 2.60 1.48 2.58 2.60 2406 1366 2387 2404 1,1 - 1,4 2.60 1.59 2.49 2.50 2406 1471 2301 2317 1,1 - 1,1 2.60 2.35 2.35 2.50 2306 2178 2178 2318 1,1 - 3.4 2.39 1.66 2.35 2.37 2212 1533 2172 2198 1,1 - 2,4 2.30 1.66 2.31 2.34 2212 1533 2139 2162 1,1 - 1,3 2.39 1.66 2.23 2.2 7 2212 1533 2066 2014 1,1 - 1,2 2.39 1.66 2.18 2.27 2212 1533 2019 2014 1,2 - ^,3 2.18 1.20 2.15 2.18 2018 1107 1992 2016 1,2 - 4,2 2.18 1.19 2.14 2.18 2018 1104 1977 2017 1,2 - 2.18 1.26 2.12 2.16 2018 1170 1965 1999 1,2 - 2,4 2.18 1.36 2.07 2.11 2018 1261 1919 1955 1,2 - 2,1 5.18 1.56 2.01 2.11 2018 1445 1864 1955 1,2 - 1,3 2.18 1.62 2.00 2.04 2018 1498 1847 1886 1,2 - 1,2 2.18 1.96 1.96 2.04 2018 1819 1819 1886 1,1 - 3,3 2.16 1.36 2.08 2.14 1998 1258 1927 1981 1,1 ~ 2,3 2.16 I .36 2.04 2.19 1998 1258 1886 1937 1,1 - 2,2 2.16 1.36 1.99 2.09 1998 1258 I838 1937 1,2 - 3,3 1.95 1.36 1.87 1.93 1804 1258 173^ 1782 1,2 - 3,2 1.95 I .36 1.85 1.93 1804 1258 1715 1782 1,2 - 2,3 1.95 1.36 1.81 1.87 1804 1258 1676 1728 1,2 - 2,2 1.95 I .36 1.79 1.87 1804 1258 1653 1728 2,2 - 3,3 1.72 1.14 1.65 I .69 1589 1053 v 1624 1538 2,2 - 2,3 1.72 1.21 1.62 1.66 1589 1121 1495 1538 2,2 - 2,2 1.72 1.58 1.58 1.66 1689 1459 1459 1538 TABLE 11 CON'T AVERAGE PRODUCTION COST AVERAGE PRICE LOCATION BASING- BASING- PAIRS ZONEUNIFORM F.O.B. ZONE UNIFORM F.O.B. POINT POINT 1,1 - 4,4 4.41 2.53 4.00 4.40 7.01 4.01 6.35 7.00 1,1 - 1,4 4.41 2.79 4.00 4.25 7.01 4.38 6.35 6.75 1,1 - i,i 4.41 4.00 4.00 4.25 7.01 5.35 6.35 6.75 l.i - 3 A 4.06 .79 4.00 4.03 6.45 2.45 6.35 6.4l 1,1 - 2 A 4.06 .79 4.00 3.97 6.45 2.45 6.35 6.31 1,1 - 1.3 4.06 .79 4.00 3.86 6.45 2.45 6.35 6.14 1,1 - 1,2 4.06 .79 4.00 3.86 6.45 2.45 6.35 6.14 1.2 - 4,3 3.71 2.12 3.35 3.71 5.89 3.32 5.32 5.89 1,2 - 4,2 3-71 2.11 3-35 3-71 5.89 3.30 5.32 5.89 1.2 - 3,4 3.71 2.34 3.35 3.67 5.89 3.51 5.32 5.84 1,2 - 2,4 3.71 2.34 3.35 3.59 5.89 3.67 5.32 5.71 1,2 - 2,1 3.71 2.78 3.35 3-59 5.89 4.36 5.32 5.71 1,2 - 1,3 3.71 2.89 3-35 3-47 5.89 4.81 5.32 5.51 1,2 - 1,2 3.71 3.35 3-35 3.47 5.89 5.32 5.32 5.51 1.1 - 3,3 3.67 . 66 4.00 3.64 5.83 2.02 6.35 5.79 1.1 - 2,3 3.67 .66 4.00 3.56 5.83 2.02 6.35 5.60 1,1 - 2,2 3.67 . 66 4.00 3.56 5.83 2.02 6.35 5.66 1,2 - 3,3 3.32 .66 3-35 3-28; 5-27 2.02 5.32 5.21 1.2 - 3,2 3.32 . 66 3.35 3.28 5.27 2.02 5.32 5.21 1,2 - 2,3 3.32 .66 3.35 3.19 5.27 2.02 5.32 5.0 5 1,1 - 2,2 3.32 . 66 3.35 3.19 5.27 2.02 5.32 5.05 2,2 - 3,3 2.94 2.07 2.70 2.89 4.65 3.21 4.28 4.58 2,2 - 2,3 2.94 2.23 2.70 2.81 4.65 3.44 4.28 4.51 2,2 - 2,2 2,94 2.70 2.70 2.84 4.65 4.28 " 4.28 4.51 171 TABLE 12 EXPERIMENT 9 DATA. AVERAGE TRANSPORTATION COST AVERAGE CROSS-HAULING BASING- BASING- ZONE UNIFORM F.O.B. ZONE UNIFORM F.O.B. POINT POINT 2.60 .03 2.4-8 2.59 2406 28 2294 24-03 2.60 .03 2.13 2.37 2406 28 1972 2196 2.60 1.4-5 1.4-5 2.57 2406 1346 1346 2196 2.39 .04 2.19 2.36 2212 42 2028 2178 2.39 .05 2.06 2.26 2212 44 1910 2095 2.39 .04 1.75 2.10 2212 42 1616 194-2 2.39 .06 1.50 2.10 2212 52 1385 19^2 2.18 .03 1.96 2.18 2018 29 1811 2014 2.18 .03 1.90 2.18 2018 28 1755 2014 2.18 .03 1.84 2.13 2018 28 1704 1971 2.18 .03 1.64 2.01 2018 29 1515 1864 2.18 .03 1.36 2.01 2018 29 1255 1867 2.18 .03 1.24 1.80 2018 29 1149 1668 2.18 .99 .99 1.80 2018 914 914 1668 2.16 .05 1.85 2.12 1998 43 1712 1959 2.16 ,06 1.68 2.00 1998 1555 1851 2.16 .05 1.46 2.00 1998 % 1349 1851 1.95 .04 1.55 1.89 1804 36 1431 1750 1.95 .04 1.46 1.89 1804 40 1352 1750 1.95 .05 1.27 1.74 1804 45 1175 1612 1.95 .06 1.13 1.74 1804 51 1050 1612 1.72 .03 1.08 1.64 1589 29 1003 1517 1.72 .03 .92 1.56 1589 30 848 14-4-0 1.72 .56 .56 1.56 1589 518 518 14-4-8 TABLE 12 CONf T AVERAGE PRODUCTION COST AVERAGE PRICE LOCATION PAIRS BASING- BASING- UNIFORM F.O.B. POINT ZONEUNIFORMF.O.B.. POINT ZONE 1,1 - 4,4 1.93 .04 .77 1.93 4.53 .07 2.23 4.52 1,1 - 1,4 1.93 .04 .77 1.63 4.53 .07 2.23 4.00 1,1 - 1,1 1.93 .77 .77 1.63 4.53 2.23 2.23 4.00 1,1 - 3.4 1.60 .04 .77 1.55 3.99 .09 2.23 3.91 1,1 - 2,4 1.60 .04 .77 1.45 3.99 .09 2.23 3.71 1,1 - 1,3 1.60 .04 .77 I .27 3.99 .08 2.23 3.37 1,1 - 1,2 1.60 .04 .77 1.2 7 3.99 .09 2.23 3.37 1,2 - 4,3 1.31 .04 .43 1.31 3.49 .07 1.42 3.49 1,2 - 4,2 1.31 .04 .43 1.31 3.49 .07 1.42 3.49 1,2 - 3.4 1.31 .04 .43 1.41 3.49 .0? 1.42 3.49 1,2 - 2,4 1.31 .04 .43 1.15 3.49 .07 1.42 3.16 1,2 - 2,1 1.31 .04 .43 1.15 3.49 .07 1.42 3.16 1,2 - 1,3 1.31 .04 .43 .95 3.49 .07 1.42 2.76 1,2 - 1,2 1.31 .43 .43 .95 3.49 1.42 1.42 2.76 1,1 - 3.3 1.29 .03 .77 1.24 3.44 .08 2.23 3.36 1,1 - 2,3 1.29 .03 .77 1.13 3.44 .09 2.23 3.13 1,1 - 2,2 1.29 .03 °77 1.13 3.44 .09 2.23 3.13 1.2 - 3,3 1.04 .04 .43 .99 2.99 .08 1.42 2.89 1,2 - 3.2 1.04 .04 .43 .99 2.99 .09 1.42 2.61 1,2 - 2,3 1.04 .04 .43 .87 2.99 .09 1.42 . 2.61 1,2 - 2,2 1.04 .04 .43 .87 2.99 .09 1.42 2.6l 2,2 - 3,3 .82 .04 .21 .76 2.53 .07 .77 2.40 2,2 - 2,3 .82 .04 .21 .71 2.53 .07 .77 2.2 7 2,2 - 2,2 .82 .21 .21 .71 2.53 .77 .77 2.27 TABLE 13 BASING-POINT PROFITS LOCATION EXPERIMENT EXPERIMENT PAIRS ■ 1 ■■ ...... - 1 2 3 7 8 9 1,1 - 4,4 — . 70 -1.86 -5.51 -.08 -.22 -1.02 1,1 - 1,4 -.45 -1.20 -3.87 -.05 -.13 - .67 1,1 - 1,1 .00 .00 .00 .00 .00 .00 1,1 - 3,4 -.06 -1.13 -4.61 .13 .01 - .73 1,1 - 2,4 .04 - .89 -4.02 .15 .04 - .61 1,1 - 1.3 .28 - .30 -2.48 .18 .12 - .29 1,1 - 1,2 .44 .15 -1.17 .20 .17 - .91 1,2 - 4,3 -1.38 -4.02 -.06 -.18 - .97 1,2 - 4,2 -.49 -1.28 -3.80 -.06 -.17 - .97 1,2 - 3,4 -.45 -1.20 -2.08 -.05 -.15 - .85 1,2 - 2,4 -.32 - .88 -2.81 -.14 -.11 - .65 1,2 - 2,1 -.15 - .43 -1 .74 -.02 -.05 - .37 1.2 - 1,3 -.09 - .30 -1.22 -.01 -.03 - .25 1,2 - 1,2 .00 .00 .00 .00 .00 .00 1.1 - 3,3 .67 - .29 -3.50 .38 .27 - .39 1.1 - 2,3 .80 .03 -2.71 .40 .32 - .22 1,1 - 2,2 .96 .44 -I.63 .42 .37 .00 1.2 - 3.3 .23 - .46 -2.73 .19 .09 - .56 1,2 - 3,2 .29 - .32 -2.42 .19 .11 - .47 1.2 - 2,3 .40 .03 -1.66 .21 .16 - .28 1,2 - 2,2 .48 .16 -1.13 .22 .18 - .14 2,2 - 3.3 -.18 - .51 -1.71 -.02 -.07 - .52 2,2 - 2,3 -.10 - .31 -1.19 -.02 -.07 - .35 2,2 - 2,2 .00 .00 .00 .00 .00 .00 I—1 -vJ ■p- APPENDIX B SOURCE STATEMENTS IN FORTRAN IV MAI N - EFIM SOURCE STATEMENT - I FN(S 3 - INTEGER X,Y,Z,P DIMENSI0NX(80),Y(80) ,DEMAND 180) ,SHMARK(80) ,REV(8C) ,8NGDE(80 ), 2TRANSC(80),01 ST(80),AVEPR(80) ,OUT(80) ,TOT(80),PR ICE(80),ZOSUM(80), 3ANODE(80),FUNCP(80),CONSP(80),HULD(80),SAVCST(80),PR00C0(8G), 4TUTCST(80),LXI80),LYI 80) , TPPU(80) ,CTPO(80) ,TREV(80),TANDIS(80), 5) UlDEM(80), ION( 80) , AVEZUim ( 80 ) ,SAVTOT(80) ,OEMAN!(80) R EA D ( 5, 1 ) IM 1 1 FORMAT(14) R EAD(8,2)K 3 2 FORMAT(12) RE AD ( 8,3) i\N 5 3 FORMAT(12) REAu(5»4)(X(I),Y(I),I=l,N) 7 4 FORMAT 1212) wklTElo,14)(X(I),Y(I),I=1,N) 15 14 Format{1hu,2 i2) READ(5,5)(LXIM),LY(M),M=1,KN) 23 8 FORMATU12) WRITE to,6)(LXIM) ,LY(M),M=1,RN) 31 6 FORMAT I1HO,212) READ( 8,7 .) (UEMANT (I ) ,1 = 1 ,N) 39 7 FuRMA T(F6.2) Wk ITE(6,Id)(DEMANTII),1=1,N) 46 16 FORMA T(iMU,F6.2) READ(8,8){FUNCP(I),CGNSP(I),I=1,K) 5 3 8 FURMATt2F10.4) wRITE(6, 15 ) ( FUiMCP ( I ) ,CUNSP( I ) ,1=1 ,K) 61 15 FURMAT(1H0»2P10.4) READ(5,9 ) PR1C 69 9 FORMAT(FS.2) _ READ I 5, 10 )S ... 70 10 FORMAT(F5.2J R EAD (5, 11 ) E 71 MAIN EFN SUURCfc STATEMENT I FN{ S } il FORMATtFIG.2) KEAO<3 ,1 2 i ^ ETPK 72 12 i - o k ! (V 10. 2) WRITE(0, 13) 7 3 13 FORMAT( 1H0 »1 1 MC CiUKO i N ATE S ) iiiK i T t! o i 17)S»E»StlPK 74 17 FORMAT!zFb.2,F10.2) NONLRU=2_____ ’______PKUL-PK 1 C STukt = StI PR ...... LUUR T = • 0 L = 0 lei 1 = 0 ...... 1 v 1 = i +i______.______' l"="l + 1 OUT (L 1=0.0 ^______GlYUUE I L ) =SETPK hOLO ( L ) =UEMANT ( I ) IF(i.LT.N) GO TO IS ...... ------...... 1 HL .II .32 ) GOTO 1 6______PKt- SPK=G .0 91 R U N = G . O ______* ’TOTAL=0.0 .TLiYDtM = U.O TREV=0 .0 " ; TO I ST =0.0______TKANS=u.0 FONMI=0. 0______T O T M1L = 0.0 SUMKE V = u .0.______■______SUMUEM=u .0 ’ ” ’ " ’ "'fj _ SUHCST = 0 .0______-o SPRGFI = G .0 SUM PRO=0 .0 MAIN - EFN SUURCE STATEMENT - IFNIS) - SUMTRA=0.0 ’ l, Z0SUM = 0. 0 SO T = 0•0 T ICK = 0 .0 P R iO P R O C M = 1______I UEP = 0 L = 0 20 1=0 21 L=L+1 1=1+1 HUB I=({L X(M J-X( I ) )**2+< LY(M) — Y BNODE(L)=ANGOE{L ) U=K OEM AN 01 L )=HGLD ( L ) / U 27 TGTCEM J)=TuTDEM{J J+DEMAND(L) TU I Ai_ = TU TAL +OEMANO ( L ) KEV(L)=PKICfc(L)*DEMANDlL) TREV(J)=IREVlJ)+REV(L) SUMARK(L)=1,0/U lF(I.LT.N) GUTU2 6 IF(L.L T.3 2)GUT025 L = 1 M = I J = i 1-1 GUIUb2 28 L = 0 29 L=L + 1 SAVCSTlL )-TOTCST(L) 1F(L.LT.(K*N)) GUTG29 TICK-1.0 GGTU30 30 1 = 1+1 L=L +1 GUT 062 ’ 31 L = 0 32 1=0 33 1 = 1+1 L = L +1 UliT(L)=TAKUlS{L) --- PRILE(L)= T U TC S T( I) ANGOE(L)=PRICE{L) H IF(I.LT.Im) GOTU33 \0' IF tL.LT.U*N)) GUTQ32 GUTU24 MAIN - EEN SOURCE STATEMENT - IFN(S) ''34 L = 0 35 1=0 36 1=1+1 L=L +1 UlSTtlj=TAKOISiL) Pk 1 C£ (1_ )=TUTCST(1+16) ANUDE(L)=PRICE(L) iF(I.LT.N) GOTO36 IE( L.LT. {'k *N) ) G0T035 G0TUZ4 37 i_ =0 ZTAP = 4. 0______Y T A P = Z T A P 38 P = 0 1 = 0 39 P=P+1 1 = 1+1 L = L_+_l______Z03UM=T0T{ 1 ) + ZO S U M AP = P I M AP «LT . Y TAP ) GCJTU3 9 AVEZON(P)=ZGSUM/(ZTAP*U) J = { AP+i .0 )-Z IAP 40 PR IC E1J )=AVEZON(P )______ANUDE(J)=PRlCt(J) a NUUE(J+161=PklCE(J) Pk 1 C E ( J + 16 ) = A VE ZON I P ) J=J + 1 iF(J.LE.P) GOTO4 0 ’ YTAP = YTAP + ZTAP______Z0SUM=0.0 IP(P.LI.N) GOTO39 G0T0Z4 41 L = 0 MAI N - EFIM SOURCE STATEMENT - 1FN{S) M= 1 J=0 42 J = J+1 1=0 43 1=1+1 L=L +1 T0TC S T { L ) =C TPU(lT+TPPU( J ] IF ( UUTIL ) .t'J. 1.0 ) GUI045 I H rorCSKL) .LE.25.0) G 0 T 0 4 6 44 1F(UUT(L+16J.LT.1.0) GO TG43 TUTCST(L+16)=SAVCST(L+16) DiSr{L+lb)=TAk OIS(L+16) UUT(L + lo)=0.0 GNUDEIL + lo} = S E TPR 43 TUTCST(L)=U.O OUTIL) = 1 .0 46 IF ( I • L T . N} GUT043 I F { L . L T .(K*N) ) GUT042 L=0 J = 0 47 U = J + 1 1=0 46 1=1+1 L=L + 1 49 PR1CL{L)=T0TCST(L) T U T U )=TUT1 I)+TO TCST(L ) IFU.LT.N) GUT04E I F { L . L T . (K*N)) GUT047 iPl riCK.tCj.O.O) GUT028 30 1F{RUN.EG.3 .0)G0T091 IF(RUN.to*4.0)GUT037 I F(RUN.to.3.0) G0T034 00 h-> 1FIKUN.E0.2.0) G0TU31 1F(RUN.EO.0.0J Gu TU22 MAIN - EFN SOURCE STATEMENT - IFN(S) - TOTAL=0.0 J =0 L = 0 51 J=J + 1 . I=U 52 1 = 1 + 1 L = L +1 IFITOTCSTCLJ.LT.TOTII)) G0T053 SUMARK(L )=1.0 ANOUETLJ=PRICETL) G0T05 7 53 IF{TU TCS T(L J•GT•0•0) C 0 T 0 5 4 SUMARK ( L ) ■= 0 .0 GQT059 54 SrlMAkK ( L ) •= iHMARK (L J +S* (.5-1TUTC ST(L)/TOT (I ) ) ) 1FISHMARKIL) .GT.O.GT GOTU55 6HMAR K< I. ) -0 . 0 OUT(L j =1.0______ANODE(L i =0. 0 GUT05b ~55~IFT S HM AKK (J_ )~LT * 1 • 0 ) G0T05o ' : SHM ARK { L ) = 1 .0 _ .______:______ANODfclL)=PKICEIL) GOT 057______56 ANODE(L)= T0T{I)/U 5 7 HOLD ( L )=H0LU(L ) » (~E»( ANODE ( L ) - B NODE ( L ) )+ANODE ( L ) +3NOPE ( L ) )/ ( E*T ANU______2DETL)-6N00E(L))+ANODE(L)+6N0DE{L)) . 58 bNGDET L ) =AN00t I L.)______’______T 59 DEMANDiL ) = 5HMARK(L)*H0LL>(LJ 60 TOTUEM(J J=TOTOEM < J)+0EMa ND(L} TUTAL=TUTAL+DfcMAND(L) ' — - - ^ PRlLfctL)=T01CSt(L)______£ 61 KEV (L)=RRiCET L)*UEMANU(L) ■ TkbVTJ}=IKEV1J)+REV(L) MAIN - EFN SOURCE STATEMENT - IFN(S) IFll.LT.N) GGTU5 2...... ' ...... , " ... IF(J.l T.K) GOTO51 * L = i ...... ".. * ...... i_=i ______J=i 62 TKANSL{I )={DEMANU(L)*DIST(L)*0.00108)+.143...... IF{LiUT(L).LT.i.0)GUTU63 i; i SI ( L ) = u. 0 ...... ; 1 k A N 5 C ! I ) = 0 • 0 6 3 T km IK Aivj SJiT.KA N_S C.LU______I'D 1ST- T U ISl+DiSTtL) icr ii j—u .o .______.______uTPU(L ) = Ti TGTPU=TuTC/TUTDEM(J) PKUF1T=TREV(J) — TUTC ‘ PRPU=PRGFIT/TUTUfcM(J) " .. ... If I SOT.L1.2.0) GUTU68 ” WR I TE i 6, 93 ) AVEPR ( J ),lOTDEMJJ) , TUTSH ,TRE V {J),TUIST f TRANS , TTPU, PRODC ^ 2,IPPU{ J) rTOTC, ruiPUtPKOFi I,PRPU,TQNMI______458 93 FURMATI1HG,F5.2,2X,F7.1,2X,F4.2 r2X,F8.1,2X,F8.1,2X,F8.1r 2X,F4.1,2X 2 » F d . 1»2X»F4. l»2X*Fd.l» 2 X »F 4. 1» 2 X f F 7 . 1f2X»F4.1fl4X?F12.1) 68 TUNM1=0.0 TU1ST=0.0 TREV{J ) = Q.O TUIDEM(J )=0 .0 SU.MCST = TkAf\IS + PRUDC + SUMCST PkIC=SUMCST/SUMDEM s p r u f i =s p k u f i +pkof IT J = J + 1 M=NUMERO SUMPRO=SUMPRO+PRODC S UM TKA=SUMTRA + 1 KANi> TRANS=0•0 1=0 IF(J.Lt.K) GO T030 TRa NU=SUMTKA/SUMDEM PRUDU=SUMPRU/SUMDEM COUNT=CU UNT + 1.0 UN 1X = SPRUE I/SUMDEM Ci WRITE(6,94)PRESPR,TRANUrPRODU,SUMDEM,SUMCST,TOTMIL,RUN,NUMERO,SUMR...... 2EV,SPRUF I,UNIX,CROSS _ _. . . .. 472____ 94 FURMAT ( 1 HO , F 7 .2 , 2X ,F 5 . 2» 2X* F 5 . 2,2 X, FI 0 . I , 2X, F10 . 2 ,2 X, F11 . 1, 2X , F 5|. 2, 2 X, 2,2X,12,2X,F7.1,2X,Fa.2,2X,F6.2,2X,F8.2)______SUMCST-0.0 SPROF 1=0 .0 „ _...... SUMD£M=0.0 T U T MIL=0 » 0______SUMREV=U.G ZEP=PRESFR—ULOPR______ZEPX=AbS(ZEP) IF t zEHX. Gt ,.(..01).)..GO TO 7.1 __ SU T = Su 1 + 1.0 IF ( SUT. .L T . 3.0 )_.GOTO 71______SU T = G .0 SETPR=S1 ORE______'CCUNT=0.0 B IN = 0 • 0 L = 0 69 1=0 70 1=1+1 L = L +1______UU T(L1=U .0 OiSTIL)=TARDIS(L) hOL D(L )=DEMANT(I ) bNUGE(L )=SETPR TUTGST(L)=SAVCSTTlT IF( I.Lf.N) G 0 T 0 7 G ______1 H L.LT. (K*N) J G0TU69 j--.RUN=RUN+1.0 L= ■■ ■ 1 - • ■ ------,-Joo 1 = 1 Xn IF ( RUN .L T .5.0) G0T049 ______MAIN - FEN SOURCE STATEMENT - IF N f S ) 71 IF{RUN.fcQ.5.0J GUTG72 G0TU41______.72 NUMtKU=NUMtRQ+l i F(NUMERU.L T.KN+i) G U T 0 4 9 STOP END M co O n 187 BIBLIOGRAPHY Ackley, Gardner. '’Spatial Competition in a Discontinuous Market," The Quarterly Journal of Economics. Vol. XXV, No. 1 (January, 1957), PP. 53-75’T Alexander, Ralph S., Cross, James S., and Cunningham, Ross M. Industrial Marketing. Homewood, 111.: Richard D. Irwin, Inc., 195^ ' Allen, R. G. D. Mathematical Economics. London: Macmillan and Company, Ltd., 1959. PP. 2-12 American Management Association, Inc. Pricing: The Critical Decision. AMA Management Report 6F7~”Mewt’York: American Management Association, Inc., 1961. Balderston, Frederick E. and Hoggatt, A.utln C. Simulation of Market Processes. Berkeley: University of Califor- hia, instTEute oT”Business and Economic Research, 1962. Bass, Frank N. et el. Mathematical Models and Methods in Marketing. Homewood.”," Ill'7: Richard D.' Irwln, Inc.’’ 196l7~“ ^ Baurnol, William H. Business Behavior, Value and Growth. New York: The Macmillan Company, 1957. . Economic Theory and Operations Analysis. Engle wood Cliffs, N. J . : Prentice-Hall, 19^5. Baumol, William J. and Wolfe, Philip. "A. Warehouse Loca tion Problem," Operations Research, Vol. VI (March- April, 1958), pp. 252-63. Beckmann, Martin. Duopoly in Two Markets— A. Game Model. A. paper given at Seattle m¥eting\''’’^g u s t ’"23, 1956 of the West Coast Region of Econometric Society. Beveridge, V/. I. B. The Art of Scientific Investigation. New York: Random HouseT-X957» Bingham, H. H. "The Uniform Delivered Pricing Method in the Grocery Manufacturing Industry." Journal of Marketing, Vol. XIV, No. 1 (January, 1950), p p T ^ ^ ^ O O T 188 Bonini, Charles P. Simulation of Information and Decision System In the Firm, Englewood. Cliffs, “N.~~J.7 PrervET'ce- Hall, Inc.", 19637“ Boulding, Kenneth E. and Spivey, V/. Mien. Linear Program ming and the Theory of the Firm. New York: The Mac millan Company, I960T ~ Chaundy, T. W. and Phillips, E. "The Convergence of Se quences by Quadratic Recurrence-Formulae," Quarter Journal of Mathematics (Oxford Series), Vol. Vll, No. 2 5™ Cm a rch7~193^) , PP. 7^-80. Chinltz, Benjamin and Vernon, Raymond. “Changing Forces in Industrial Location," Harvard Business Review, Vol. XXXVIII, No. 1 (January^ebHar5T~1950). Chorafas, Dimitris, N. Systems and Simulation. New York: Academic Press, 1965. ~ Chow, Gregory C. Demand for Automobiles in the United States. Amsterdam: North-Holland Publishing Company, 1957, Clark, Colin. The Conditions of Economic Progress. London: Macmillan and Company, Ltd., 1957* Clark, John Maurice. "Imperfect Competition Theory and Bas- ing-Polnt System," The Quarterly Journal of Economics, Vol. XXXVIII, No. 2 TT u ^ T ^ ^ I ) , PP. )~ . “Law and Economics of Basing-Points," The Amer- lean Economic Review, Vol. XXXIX, No. 1 (March,1 19*657. pp. 430- ^ ' Cohen, Dalman J. Computer Models of. the Shoe, Leather, Hide Sequence. Englewood Cliffs, N. J.: Prentice-Hall, Inc., i960. Coyle, John J. "Simulation: discussion and Application," Transportation Journal, Vol. IV, No. 3 (Spring, 1965), p p n a r a o : ~ Daggett, Stuart. Principles of Inland Transportation, ^th ed. New York*i Harper and Ro~w,"~1933~, Daugherty, Carroll R., de Chazeau, Melvin G., and Stratton, Sammuel S. The Economics of the Iron and Steel Industry. New York: McGraw-Hill Book Co., Vol. II, 1937. Dean, Joel, Managerial Economics. Englewood Cliffs, J. J,: Prentice-Hall, Inc., 1951. . The Relation of Cost to Output for a Leather Belt Shop. New York: National Bureau of Economic Research, L9$L. . "Statistical Cost Functions of a Hosiery Mill,*' The' Journal of Business, Vol. XLVIII, No. 6 (December, 1940), pp. 777-821. Dean, Joel and Warren, James R, The Long Run Behavior of Costs In a Chain of Shoe Stores. Chicago:. The Univer sity of Chicago Press, 1942. DeHayes, Daniel Wesley, Jr. The Validity of a Linear Dls- ance-Transportation Rate Relationship Assumption In the Location of a Single FlxedH^acilPo.y. Unpublished Master’ Dissertation, The Ohio State University, 1964. Dewey, D. "A Reappraisal of F.o.b. Pricing and Freight Ab sorption," The Southern Economic Journal, Vol. XXII, No. 1 (July, 1955) , pp71T8-54r Dunn, Edgar S. The Location of Agricultural Production. Gainesville: University of Florida Press, 1 9 5 4 7 ~ ~ Edwards, C. D. "The Effect of Recent Basing-Point Decisions Upon Business Practices," The American Economic Review, Vol. XXXVIII, No. 4 (December, 1958), pp. 8TCT2. Bnke, Stephen. "Equilibrium Among Spatially Separated Mar kets: Solution by Electric Analogue," Econometrlca, Vol. IXX (January, 1951), pp. 40-47. ______. "Space and Value," The Quarterly Journal of Eco nomics, Vol. LVI, No. 3 (August, 1942) 7 p p • " 6 2 7 -3 7 • Ferber, Robert and Verdoorn, P. J. Research Methods in Eco nomics and Business. New York: The Macmillan Company, Fetter, Frank A. "The Economic Law of Market Areas, " The Quarterly Journal of Economics, Vol. XXXVIII, No. 2 Flagle, Charles D., Huggins, William H., and Roy, Robert H. Operations Research and System Engineering. Baltimore: The Johns Hopkins Press, I^TTo. 190 Friedrich, J. J. (ed and trans.). Alfred Weber’s Theory of Location of Industries, ChicagoT'njnfvefsTTy*ol^THTcago Press, 19587 Translated from Alfred Weber. Uber den Standort der Industrien. Part 1, "Reine Theorie des Standort." Tubingen: 1909. Garvin, Walter. W. Introduction to Linear Programming. New York: KcGraw-Itlll Book C677 Inc., pp. 1, 86-87. Goldverg, Sammuel. Introduction to Difference Equations. New York: John Wiley and Sons, Inc77' 19313• Gordon, Robert Aaron. Business Fluctuations. New York: Harper and Brothers, 1952, Gort, Michael. Stability and Change in Market Shares, Chi- cago: Uni v e r 7 1 7 7 7 7 * Ch i c a go Press, 1963 ♦' Greenhut, H, L. "A. General Theory of Plant Location," Metro- economics, Vol. VII (August, 1955). PP. 59-72. . "Integrating the Leading Theories of Plant Loca- tion," The Southern Economic Journal, Vol. XVIII, No. 4- (April, ' 195277 PP . 32 6 - 3 8 7 . Mlcroeconoralcs and the Space Economy. Chicago: Scott Foresman and Company, 1963. " ______» Plant Location in Theory and Practice. Chapel Hill: University of North Carolina Press, 1956. Greenhut, K L. and pfouts, R. W. "Space and Economic Theory," Regional Science Assoclatlon Papers and Proceedings, VoT. V (1959), PP. 237-80. Guthrie, J. A. "Impact of Geographical Price Discrimination on the Buyers," Journal of Marketing, Vol. XIV, No. 1 (January, 1950), pp. 538^3. Hall, R. L. and Hitch, C. J. "Price Theory and Business Be havior," Oxford Economic Papers. No. 2 (May, 1959). pp. 18-19. Hawkins, E. R. "Price Policies and Theory," Journal of Marketing, Vol. XVIII, No. 1 (January, 19f&)'» PP. ?33~4-0. Helburn, A. "Location of Industry," Land Economlcs, Vol. XIX, No. 3 (August, 194-3)# PP. 253-637 19i Heskett, J. L., Ivie, Robert M., and Glaskowsky, Jr., Nich olas A. Business Logistics. New York: The Roland Press Co., 1957}“ Hildreth, Clifford and Jarrett, F. G. A Statistical Study of Livestock Production and Marketing. New York: John Wiley and Sons, 1953* Hoggatt, Austin Curwood and Balderston, Frederick D. (eds.). Symposium on Simulation Models: Methodology and Appll- ca£ions~l;o tiTe~~6 ehavloraI~l?clences. Cincinnati: South western Publishing C., Inc., 196~37 Hoover, Edgar H. The Location of Economic Activity. New York: McGraw-Hill Book Company, Inc., ’'Spatial Price Discrimlne,tion," Review of Eco nomic Studies, Vol. 4 (June, 1937), pp. 122-91. Hoover, Edgar M. and McLaughlin, E. E. "Strategic Factors In Plant Location," Harvard Business Review, Vol. XX, No. 2 (March-April, 1942), pp. 133. Hotelling, H. "Stability in Competition," The Economic Journal. Vol. XXXIX, No. 113 (March, 1929), pp. 51^57. Isard, Walter. "A General Location Principle of an Optimum Space-Economy," Econometrica, Vol. XX, No. 3 (July. 1952). pp. 406-30: . "The General Theory of Location and Space-Economy," The Quarterly Journal of Economics, Vol. LXIII, No. 4 (November, 1941), pp7V?6~50ZT~ . Location and Space-Economy. New York and Cam bridge: John Wiley and” Sons, Inc., and The Technology Press of Massachusetts Institute of Technology, respec tively, 1956. Isard, Walter et^al. Methods of Regional Analysis: An In troduction €o~~Reglonai Science. New York”: ^JoHn^Wiley and Sons, Inc., 1 9^07 Kiaman, M. S. "Geometric Convergence," American Mathemat ical Monthly. Vol. LX, No. 4 (Aprilr W S T T T v V - 256-259. Koopmans, Tjailing C. and Beckmann, Martin. "Assignment Pro blems and the Location of Economic Activities," Econo metrica, Vol. XXV, No. 1 (January, 1957), pp. 53-7^7 192 Lanzillottl, R. F. ''Pricing Objectives in Large Companies,” The American Economic Review, Vol. XLVIII, No. 4 (dec- ember, 195'B77“pp. 921^ 0 . Learned, Edmund P. and Ellsworth, Catherine C. Gasoline Pricing In Ohio. Eoston*. Harvard University Press, 1959* Lefeber, L. .Allocation In Space. Amsterdam: North-Holi and Publishing Company, 195&. Lfcsch, August. The Economics of Location. New Haven: Yale University Press, Translated by William H. Woglam with the assistance of Wolfgang F. Stopler from the 2nd. rev. ed. Jena: Gustaf Fischer Verlag, 1944. Machlup, Fritz. The Basing-Polnt System. Philadelphia: The Blaklston Comp any", i’9 4’$'. . The Economics of Sellers1 Competition. Baltimore: The Johns Hopkins Press7~T9'55» Manne, Alan S. "Plant Location Under Economles~of-Scale-~ Decentralization and Computation," Management Science, Vol. XI, No. 2 (November, 1964), pp. 213-235. Marshall, Alfred. Principles of Economics. London: Mac millan and Co., 8th ed., Bk*V, Ch. XV, Sec. 1, 1936. McGee, J. S. "Cross Hauling: A. Symptom of Incomplete Col lusion Under Baslng-Polnt Systems," The Southern Eco nomic Journal. Vol. XX, No. 4 (April,T^TT"PP.'"396-79. McLean, John G. and Halgh, Robert V/. The Growth of Inte- rated Oil Companies. Harvard: Graduate School of fusiness AHSiii^raTion, 1954. Meyer, John R. et al. The Economics of Competition in the Transportation- Industrl "e'sT Cambridge: Harvard'lJniver- sity Press, 1959. Mills, Edwin S. and Law, Michael R. "A. Model of Market Areas with Free Entry." Journal of Political Economy. Vol. LXXII, No. 3 (June, 19 # 0 7 " Moore, J. F. and Levy, L. S. "Price Flexibility and Indus trial Concentration," The Southern Economic Journal, Vol. XXI, No. 4 (April,"1955), ppT^J^ffT ' Mosak, Jocob, L. General Equilibrium Theory in International Trade. Cowles'Commission MohograpTT No. 7. Bloomington, Ind., 19l<4. Moses, L. N. ”Locs.tion and the Theory of Production,” The Quarterly Journal of Economics, Vol. LXXII, No. 2 (May, 195§T7"PP. 259-7^7 ' Mossman, Prank H., and Morton, Newton. Logistics of Distri bution Systems. Boston: Allyn and Bacon, Inc., 19&57 National Bureau of Economic Research. Price Research in the Steel and Petroleum Industries. New York: National Bureau ‘of Economic^Research, 'T939• National Recovery Administration. Report of the National Re covery Adralni strati on on the Operation of the "Basing" Point System in the Iron and Steel Industry." Supplement No. 1 (November JO, 1934). Oxenfeldt, Alfred R. Industrial Pricing and Market Prac tices. New York: Prentice-Hall, Inc., 1951. Pierce, John F. Jr.. Some Large-Scale Production Scheduling Problems in the Paper industry. Englewood "Cliffs. N. J. T 9 oTT' Poole, Ernest C. Costs~~A Tool for Ral1road Management. New York: Simmons-Boardman PuBITsElng Corp. Riley, Vera and Gass, Saul I. Linear Programming and Associ ated Techniques. Baltimore: fEe^oEnsTIoplilns Press, 1956. Rodwin, L. "Two Approaches to Industrial Location Analysis,” The Economic Journal, Vol. XI, No. 162 (June-September, 194TJ, p p . 270-75. Samuelson, Paul Anthony. Foundations of Economic Analysis. Cambridge: Harvard University Press, 19^-8. _____ . "Spatial Price Equilibrium and Linear Programming, The American Economic Review, Vol. XLII. No. 3 (June. T75ZT7rvpr^=yST. ----- Scarborough, J. G. Numerlcal Matheinatlcal Analysts. Balt imore: The Johns Hopkins’Press, 1^58. Schwartz, Harry. Russ 1 a!_s Sov 1 et Economy. Englewood Cliffs, N. J . : Prent'i ci^Hall, lncT7’T95B". 19^ Schultz, Henry. The Theory and Measurement of Demand. Chi- ago; The University of Chicago Press, 1938. Schwab, Charles M. Report of the Federal Trade Commission to the President with Respect to the Basing-Point Sys tem TrPEhe Iron and Steel Industry. WashTngtohl T 934. Seidler, C. ”Geographical Price Relations and Competition,” Journal of Marketing, Vol. I, No. 1 (January, 1937), ppTT9'8^20T; Shapley, L. S. "A Duopoly Model with Price Competition,” A paper given at Seattle meeting of the West Coast Region of the Econometric Society, (August 23, 1958). Schycon, Harvey N. and Kaffei, Richard B. "Simulation-- Tool for Better Distribtuion,” Harvard Business Re view, Vol. XXXVIII (November-December, I^5UT7~PP. 65-75* Simon, William. Geographic Pricing Practices. Chicago: Callahan and Company, l 9 5 T « Singer, II. VI. ”A. Note on Spatial Price Discrimination," Revlew of Economic Studies, Vol. V (October, 1937). PP. 7 £ T 7 . Smithies, A. "Optimum Location in Spatial Competition,” The Journal of Political Economy, Vol. XLIX, No. 3 1 June, 19~4l), pp. 4 2 3-39. Smykay, Edward W., Optimum Location in Spatial Competition,” Physical Distribution Management. New York; The Mac millan Company, I 9 T H Sraiffa, Piero. "The Laws of Returns under Competitive Con ditions,” The Economic Journal, Vol. XXXVI (December, 1926), pp. T f T Stevens, Benjamin H. "An Application of Game Theory to a Problem in Location Strategy,” The Regional Science Association— Papers and Proceedings, Vol. VII (1961), p p . 1WPT57: “ Stigler, George J. "A Theory of Delivered Price Systems,” American Economic Review, Vol. XXXIX (19^9). PP. 1143-59. ______. The Theory of Price. New York: The Macmillan Company, 1952” " 195 Stocking, G. W, Basing-Polnt Pricing and Regional Develop ment. ChapeT~Hill: The University of North Carolina Press, 1954. Tocher, K. D. The Art of Simulation. London: The English Universities Press, Ltd., 19S3™ U. S. Steel Corporation, T.N.E.C. Papers. Pittsburgh: United States Steel Corporation, 1940. Valavanis. "Lbsch on Location," The American Economic Re view, Vol. XLV, No. 3 (September, 1955). pp . ^3 7 “ 44." von Thunen, Johann Heinrich. Per Isollerte Staat in Bezle- hung auf Landwirtschaft und Natlonalokonoxnles. 3rd ed. Berlin: Schumacher-Zarchlin, lS~75. Wallace, Donald H. Market Control in the Aluminum Industry. Cambridge: Harvard University Press,~1937. Warntz, William.. Toward a Geography of Price. Philadel phia: University of Pennsylvania Press, 1959. Welntraub, Sidney. Price Theory. New York and London: Pitman Publishing Corp., 1949. Whitman, Roswell A. "The Statistical Law of Demand for a Producers Goods as Illustrated by the Demand for Steel," Bconometrlca. Vol. IV (1936). Wilson, T. and Andrews, P. W. S. Oxford Studies in the Price Mechanism. London: OxforH~University Press, 1951* Wylie, Kathryn N. and Ezekiel, Mordecai. "The Cost Curve for Steel Production," The Journal of Political Econ omy, Vol. XLVIII, No. 6 (December, 1946)", pp. 777-821. Zeuthen, F. "Theoretical Remarks on Price Policy: Hotel ling’s Case with Variations," The Quarterly Journal of Economics, Vol. XLVII, No. 1 (February, 1 9 j 3 T T p p r 2 3 f ^ 53. Zwick, C. J. Demand for Transportation Services in a Grow ing Economy, P^2tTSiT, Rand Corporation Paper 1.