Mathematical Models in Marketing: A Collection of Abstracts

534 Pages·1976·10 MB·English

by  Prof. Dr. Ursula H. Funke (auth.)

Lecture Notes in Economics and
Mathematical Systems 132
Ursula H. Funke
Mathematical
Models in
Marketing
A Collection of Abstracts
Lecture Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and H. P. Künzi
Operations Research
132
Ursula H. Funke
Mathematical Models
in Marketing
A Collection of Abstracts
With aPreface by M. Beckmann
Springer-Verlag Berlin Heidelberg GmbH 1976
Editorial Board
H. Albach · A. V. Balakrishnan · M. Beckmann (Managing Editor)
P. Dhrymes · J. Green · W. Hildenbrand · W. Krelle
H. P. Kilnzi (Managing Editor) · K. Ritter · R. Sato · H. Schelbert
P. Schtinfeld
Managing Editors
Praf. Dr. M. Beckmann Prof. Dr. H. P. Kilnzi
Brown University Universităt Zilrich
Providence, Rl 02912/USA 8090 Zilrich/Schweiz
Author
Prof. Dr. Ursula H. Funke
Department of Economics
Brown University
Providence R.l. 02912/USA
Library of Coagress Ca\alogiag In P11blic:atlon Data
Funke, Urs ula, 1939-
Mathematical models in ma.rketing.
(Lecture notes in economica and mathematical systems :
Operations resea.rch ; 13~)
Bibliography: p.
Includea indexes.
1. Marketing--Ma.thematical ODdels, I. Title,
II. Serles: Lecture notes in economica a.nd ma.thematica.l.
system.s ; 132.
HF5415.l25.F86 658.8'001'51 76-44509
AMS Subject Classifications (1970): 90A10, 90A15, 90899
ISBN 978-3-540-07869-2 ISBN 978-3-642-51565-1 (eBook)
DOI 10.1007/978-3-642-51565-1
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© by Springer-Verlag Berlin Heidelberg 1976
Originally published by Springer-Verlag Berlin Heidelberg New York in 1976
Among ~e most active fields in operations Rese~rcn in
re cent years Marketing nas been outstanding. One index of
operations Researcn activity in marketing is the number of books
and articles on marketing subjects containing mathematical models.
This collection of abstracts includes from the
1940's 1 model,
1950's 12 models,
1960's 80 models,
1970's (so far) 58 models.
Here and in the following we have excluded mere statistical
moqels of the regression or analysis of variance types. A diffi
cUlty facing the reader is not just the abundance of articles
and the reappearance of basically the same models in various
forms, but their dispersion over a great number of journals both
in the marketing and in the Operations Research fields. (22 have
been covered here) •
As one remedy to this situation there have appeared
several books of readings in which key articles have been
reprinted, sometimes with editorial comment. Same of
these are:
F.M. Bass, et. al., (eds.), Mathematical Models and Methods
in Marketing, Homewood, Illinois: R.D. Irwin, Inc., 1961.
R.E. Frank, A.A. Kuehn, and W.F. Massy, Quantitative
Techniques in Marketing Analysis, Text and Readings, Homewood,
Illinois: R.D. Irwin, Inc., 1962.
B. Montgomery and G.L. Urban, Applications of Management
Science in Marketing, Englewood-Cliffs: Prentice-Rall, !nc.,
1970.
R.L. Day and T.E. Ness, Marketing Models: Behavioral Science
Applications, Scranton-Toronto-London: International Textbook
Company, 1971.
L. Day and J. ~arsons, Marketing Models: Quantitative
Applications, Scranton-Toronto-London: International Textbook
Company, 1971.
The aims of the present monograph are more ambitlous. It
is an attempt to collect all significant articles and books
that contain mathematical models on marketing subjects. It
seems to us that the greatest usefulness is achieved not by
reprinting but by abstracting from these articles the types of
mathematical model used. This has been done, wherever possible,
in the authors' own notation and words (but without their
assistance). The format used and the cla5sification that has
emerged are explained in the introduction.
It is impossible to summarize the state of the art as it
emerges from these abstracts in a few paragraphs, and no such
attempt will be made here. The field is in astate of flux,
and no common framework is apparent in terms of which all
theoretical ideas can be readily organized. This would signal
the need for efforts to develop such a general theoretical
framework on the basis of a unified theory of behavior. As
far as sellers behavior goes, the well-known economic model
of profit maximizing behavior and its variants of the sales
maximization and satisficing types would seem to offer promising
starting points.
Things are more tricky at the consumer's or buyer's end.
Here it would seem that the economic theory of choice as
formulated in utility theory would have to be brought in. But
at present, utility theory i5 not formulated in a way that allows
for brand switching, or responses to advertising, packaging or
other forms of marketing effort.
Ursula Funke and I have made an attempt at reformulat~ng
utility models for purposes of marketing theory in the paper,
"~roduct Attraction, Marketing Effort and Sales: Towards a
v
Utili ty Model of Market Eehav;lQr," U41J to which we refer
the intere$ted reader.
It remains for me to reccrd the pleasure I had ~n
collaborating with Ursula Funke on this project. Both the
article, which summarizes our theoretical results, and this
collection cf abstracts, which was preliminary to our
theoretical efforts, have been sponsored by Grant No. Be 272124
from the "Deutsche Forschungsgemeinschaft. " We should like
to express our sincere gratitude for this support.
Providence, April 1976
Martin J. Beckmann
INTRODUCTION
Mathematical models can be classified in a number of ways,
e.g., static and dynamic; deterministic and stochastic; linear
and nonlinear; individual and aggregate; descriptive, predictive,
and normative; according to the mathematical technique applied
or according to the problem area in which they are used.
In marketing, the level of sophistication of the mathe
matical models varies considerably, so that a nurnber of models
will be meaningful to a marketing specialist without an extensive
mathematical background. To make it easier for the nontechnical
user we have chosen to classify the models included in this
collection according to the major marketing problem areas in
which they are applied.
Since the emphasis lies on mathematical models, we shall
not as a rule present statistical models, flow chart models,
computer models, or the empirical testing aspects of these
theories. We have also excluded competitive bidding, inventory
and transportation models since these areas do not form the
core of ·the marketing field.
The mathematical formulation of human behavior and particularly
consumer behavior is only in its infancy. In marketing the
chief development of mathematical models beg an after World War 11
with a major thrust in the early 1960's. So far, it seems to
have been mainly an Anglo-Saxon affair and we have, therefore,
concentrated on the Arnerican and British literature. (See, however,
entries #41, 88, 94, 113, 118.)
The models included in this collection have been selected
on the basis of being either meaningful in terms of actual applica
ti on or historically important or worthwhile for their own sake.
Within the nine subject categories they are arranged chronologically
except in cases where one article is an extension of another.
References at the end of the abstract refer to closely related
models.
VIII
Since the model categories frequently overlap and some
models treat several subjects simultaneously (example:
"Advertising and Promotion Effects on Consumer Response to New
Products"), the classification of some models is somewhat
arbitrary. The reader is referred to the subject index.
Part 1 contains models of consumer purchasing behavior,
the center of marketing. The first subclass consists of brand
choice models in general; new product models are listed in
Part 4.1.
Brand choice models are as a rule stochastic models.
The development of stochastic models of buying behavior began in
the late 1950's. There have been four major model-building
approaches: zero-order models, Markov models, linear learning
models, and probabilistic diffusion models. Each presumes a
different consumer behavior process. Whereas the zero-order
model assumes that past brand choices do not affect future bran~
choices, the two-state Markov model postulates that only the
most re cent purchase affects the current brand choice decision.
The learning model presumes that brand choice is dependent upon
the complete history of past purchases. The probability diffu
sion model, finally, is a zero-order model in which the probability
of choosing a particular brand may change between purchases;
it allows nonstationarity in the probability of brand choice, but,
in contrast to the linear learning and Markov models, assumes
that this nonstationarity is not due to purchase event feedback.
In the late 1960's marketing theorists began to look to
the behavioral sciences and formulated buyer attitude models.
These deal with the relationship between consumers' attitude
toward a product and brand choice behavior.
The category Other Consumer Behavior Models contains product
attribute models, models where the demand is affected by dealer
or store location, consumer utility models, a model of demand
with variable consumer preferences, and a model of consumer behavior
with and without knowledge of the price distribution.
Part 2 considers Advertising and Sales Promotion models
where the term sales promotion covers a wide variety of sales
stimulating devices, such as temporary price reductions,
premiums, coupons, and sampling. The technique of programming
dominates the models in Media Selection. The abundance of
models in Advertising Expenditure reflects the importance of
advertising in marketing research and in most firms' budgets.
Some theoretical results justify the common empirical use of a
percentage-of-sales-rule in advertising budget decisions (see,
for instance, model #56). A large number of models in the
category Sales Response to Advertising treats a central question
in advertising: the measurement problem, as well as the dura
tion and stability of advertising's effects. The category Other
Models of Advertising and Sales Promotion comprises models of
promotional competition and the diffusion of knowledge.
Part 3 considers the subject of pricing whose importance to
the firm is not mirrored by the number of models represented
here. There are two reasons for this. First, the mathematical
models of general price theory are widely published and assumed
known. Second, many advertising, new product, and marketing mix
models contain the pricing variable.
Four out of five new products fail. With market research
focussing on this problem it is not surprising that the category
Product Models consists mostly of new product models. They range
from the simple Fourt-Woodlock model to Massy's STEAM model.
Part 5 is directed at the problem area of sales forecasting.
Sales forecasting or rnarket share expressions are found in models
of all categories. Besides the five models presented here, other
fore casting models are to be found especially in the New Product
category (e.g., #96, 108, and 109).
x
The Faeility Loeation Models of Part 6 treat the loeation
of a warehouse or retail outlet as adecision variable of the
firm.
In most economies personal selling represents a larger
marketing expenditure than advertising. Model building started
early in this field, and the Sales Force Model category contains
the earliest paper in this collection (1943). Most articles
deal with salesmen scheduling and routing, i.e., with the problem
of allocating sales effort among customers, geographical areas,
and time, whereas the Montgomery-Silk-Zaragoza model investigates
the problem of allocating selling effort across a firm's
product Une.
Part 8, Marketing Mix Models, comprises models of market
response to more than one marketing variable. Models which
consider several variables but with, the emphasis on one variable,
for example advertising, are listed under that variable.
Part 9, finally, contains two models of market simulation of
which MATE represents business games, and one model of trade area
boundaries.
This collection does not claim to be exhaustive, rather it
is intended to be representative. Its main purpose is to
provide a survey of the state of the art which may be useful to
the marketing researcher, to the general Operations Research
practitioner, and to the teacher of management scisnce courses.
I am grateful to the Economics Department of Brown University
for providing a stimulating environment and the facilities to
undertake this research. My gratitude extends to Nancy Kimelman
for helping with the indices and to the department's capable
secretary, Marion Anthony.
I am deeply indebted to Martin J. Beckmann, colleague, "Meister,"
and friend. I thank Wolfgang for his understanding and patience
and dedicate this volume to my parents, Hertha and Erwin Rehbinder.
Providence, April 1976
Ursula H. Funke