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This is to certify that the
dissertation entitled

AN INVESTIGATION OF THE IMPACT OF CONTINUOUS
QUALITY IMPROVEMENTS ON OPTIMAL PRICING

presented by

DAVID MENDEZ EMILIEN

has been accepted towards fulfillment
of the requirements for

EhID..____degree in _anine.ss_Adminis trat ion

)
(flak 7‘ K' it)! \/3\filv. .

Major professor

Date /////6/ 05‘
/

MSU i: an Affirmative Action/Equal Opportunity Institution 0—12771

 

 

LIBRARY l
Michigan State!
University ‘

 

 

PLACE N RETURN BOXtoromovothb chockomfrom yomrocord.
TO AVOID FINES mum on or More data duo.

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU Is An Affirmative Action/Equal Opportunity Irntltwon

 

AN INVESTIGATION OF THE IMPACT OF CONTINUOUS QUALITY
IMPROVEMENTS ON OPTIMAL PRICING

By

David Mendez Emilien

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Management

1995

ABSTRACT

AN INVESTIGATION OF THE IMPACT OF CONTINUOUS QUALITY
IMPROVEMENTS ON OPTIMAL PRICING

By

David Mendez Emilien

This research develops a dynamic model that relates price and product
quality to the sales rate of a durable good. The model treats quality and
price as control variables or inputs, and sales rate and profit (measures of
corporate performance) as outputs. The model incorporates elements that
have not been considered by existing research such as the notion that the
consumers’ perceptions of quality improvements of a product change more
slowly than the product’s actual quality. The model also considers how the
products’ durability affects the relationship between price-quality policies and
performance. The model is validated with data from the automobile
industry. Incorporating the relationship between cost and quality according
to the prevailing ideas in the literature, the model is used to compute price
trajectories that maximize profits at a specified target time while quality is
continuously improved. The relationship among product durability, quality

perceptions and optimal prices is discussed.

Copyright by
DAVID MENDEZ EMILIEN
1995

To my mother

Dulce Maria Emilien

iv

ACKNOWLEDGMENTS

I would like to express my gratitude to the many individuals who made
possible the completion of this dissertation.

I specially thank my dissertation chair, Dr. Ram Narasimhan, for his
tutoring, encouragement, and continuing support over the years. His
unselfish sharing of his time and knowledge, and genuine concern for my
development as a scholar are much appreciated.

I would also like to thank the members of my dissertation committee,
Drs. Paul Rubin, Peter Schmidt and Soumen Ghosh for their valuable
contributions to this research. I specially thank Dr. Rubin for all the help he
offered throughout my doctoral program.

I also wish to thank my fellow doctoral students and the secretarial
staff of the Management Department for their support and friendship.

Finally, I would like to acknowledge the unfailing support I received
from my wife Hildegarde and my son Jose David, and express to them my
loving gratitude for their patience, understanding and encouragement

throughout the time that this work was in progresss.

TABLE OF CONTENTS

LIST OF TABLES ............................................................... viii
LIST OF FIGURES ................................................................ ix
CHAPTER 1: INTRODUCTION ................................................. 1
CHAPTER 2: LITERATURE REVIEW ......................................... 4
CHAPTER 3: FOCUS OF RESEARCH ........................................ 11
CHAPTER 4: MODEL OUTLINE .............................................. 14
CHAPTER 5: MODEL DEVELOPMENT ..................................... 18
5.1 : Market Potential as a Function of Price ....................... 21
5.2 : Recognition of Product’s Finite Life—Span .................... 23
5.3 : Effect of Quality on Sales Rate ................................. 32
5.4 : Proposed Model ................................................... 39
CHAPTER 6: MODEL VALIDATION ......................................... 43
6.1 : Validation of System Dynamics Models ....................... 43
6.2 : Consistency Evaluation: Face Validity ......................... 47
6.3 : Consistency Evaluation: Stability Analysis .................... 48
6.4 : Fit to Historical Data .............................................. 63

vi

6.5 : Conclusions ......................................................... 68

CHAPTER 7: OPTIMAL CONTROL PROBLEM ............................. 72
7.1 : Problem Definition ................................................. 72
7.2 : Solution Method .................................................... 75
7.3 : Scenario Analyses .................................................. 79
CHAPTER 8: OPTIMIZATION RESULTS ..................................... 83
8.1 : Price elastic demand; average life of product, D1 =10 ...... 83
8.2 : Price elastic demand; average life of product, D1 =5 ........ 85
8.3 : Price elastic demand; average life of product, D1 =3 ........ 87

8.4 : Price inelastic demand; average life of product, D1 = 10. . . .90

8.5 : Price inelastic demand; average life of product, D1 =5 ...... 92

8.6 : Price inelastic demand; average life of product, Dl =3 ...... 93
CHAPTER 9: DISCUSSION OF RESULTS AND CONCLUSIONS ........ 94
LIST OF REFERENCES ........................................................... 100

vii

Table 1:

Table 2:

Table 3:

Table 4:

LIST OF TABLES

Taxonomy of Current Literature ................................... 4
Stability Conditions of the Model’s Equilibrium Points ....... 56
Ford Mustang Data Used in Model Validation .................. 64

Ford Mustang. Actual vs. Estimated Sales ...................... 70

viii

1',

 

 

 

 

LIST OF FIGURES

 

 

 

Figure 1: Conceptual Overview of the Model .............................. 18
Figure 2: Graphical Representation of a Product’s Life-span ............ 23
Figure 3: Graphical Representation of a Product’s
Quality Life-span .................................................... 32
Figure 4: Signal Flow Diagram of Proposed Model ....................... 41
Figure 5a: Phase Portrait Diagram of Proposed Model --
Case q > 1 .................................................... 57
(11102
Figure 5b: Phase Portrait Diagram of Proposed Model --
Case q < 1 .................................................... 58
aAflDZ
Figure 5c: Phase Portrait Diagram of Proposed Model ~-
1
Case q = ..................................................... 58
(11102
Figure 6: Sales Rate over Time for Different Levels of Quality .......... 62
Figure 7: Procedure Used in Fitting the Model to Historical Data ....... 69
Figure 8: Ford Mustang. Actual vs. Estimated Sales ...................... 71
Figure 9: Cost as a Function of Quality ....................................... 74
Figure 10: Interaction Between the Model and GRG2 Algorithm for
Optimal Control ....................................................... 78

ix

 

 

 

 

 

 

 

Figure

Figure

 

 

Figure 11: Optimal Controls Derived Under Non-Linear Quality
Costs and Elastic Demand (e = 1.3)
A: D2 = 0.75D1 B: D2 = 0.50D1 C: D2 = 0.25D1 ....... 88

Figure 12: Optimal Controls Derived Under Non-Linear Quality
Costs and Inelastic Demand (e = 0.7)
A: D2 = 0.75D1 B: D2 = 0.50D1 C: D2 = 0.25D1 ....... 89

 

ISIS

compet
manage
prioriti.
quality.
dealing
quantita
perform:
at al., 1

interact

nature

has yiel
this relr
between

MCClUr

 

:Ahkl- ‘ [0 0F .1!!_.IMP; TOF O O _ C ._ITY

I VEME N PTIMAL RIIN

l. INTR D TI

The past decade has seen quality emerge as a critical element of
competitive success. Surveys indicate that both American and European
managers now rank improving product and process quality among their top
priorities (Fortuna, 1990; Kim, 1994). Despite all the recent interest in
quality, much of the research in this area is anecdotal, especially when
dealing with the influence of quality on product demand. Even though
quantitative research addressing the strategic effects of quality on market
performance has appeared in the literature (Banker and Kosla, 1992; Tapiero
et al., 1987), this body of research has not explicitly modeled the dynamic
interaction among price, quality and sales response.

The literature is particularly controversial when dealing with the
nature of the relationship between quality and prices. Correlational analysis
has yielded inconclusive results with respect to the existence and direction of
this relationship. However, existing research has focused on the relationship
between quality and actual prices, not optimal prices (Gerstner, 1985;

McClure and Spector, 1991). A consistent relationship between quality

 

 

 

levels
chargi
particr
ration.
opdnr
betwe
relatio
not in
proces
I992;

effect 1
its [335
import:
quality
that a p
in the j

market.

 

Concerr

dllrable

 

2
levels and prices is more likely to become evident when the manufacturer is

charging ‘optimal prices’ which are independent of the decision-maker’s
particular shortcomings, such as limited information and/ or bounded
rationality. Actual prices can deviate without a discernible pattern from
optimal prices and this deviation can obscure any existing relationship
between quality and prices. Another issue that may contribute to cloud the
relationship between price and quality is the fact that existing research does
not incorporate dynamics that are important to the determination of the sales
process. For example, current research on quality (Banker and Khosla,
1992; Tapiero et al., 1987; Lee and Tapiero, 1986) has not considered the
effect of market saturation on sales rate of a product and the extent to which
its past quality continues to influence current sales. These effects are
important to consider because they can modify the impact of price and
quality policies on the product’s sales rate. For example, it can be argued
that a policy that calls for an agressive increase in product quality will have,
in the short run, little effect on the sales rate of a product in a saturated
market.

This dissertation seeks to fill the existing gap in the literature
concerning quality by exploring the nature of optimal price trajectories for

durable goods, when product quality is continuously improved. By

 

lg. -§|

 

modelir
strategi
profit,
quality

resear C

 

3
modeling the dynamic interaction among price, quality (i.e. elements of

strategic control) and measures of market performance, such as sales rate and
profit, this study examines the relationship between price and increasing

quality, incorporating factors that have not been considered in previous

research.

 

‘

 

this di
existin

quality

interac
of the
depend

charact

Table l

W
t\

Norton
llromps
Teng gm
Horsky
Bass(l9

 

 

 

 

 

 

LW

In this chapter, current research relevant to the issues addressed by

this dissertation is reviewed. The purpose of the chapter is to show the

existing gap in the literature concerning the dynamic interaction among

quality, price and sales rate that this dissertation seeks to fill.

Table 1 presents a taxonomy of the relevant literature concerning the

interaction among price, quality and the sales rate. To facilitate the analysis

of the literature, the studies reviewed have been classified into four groups

depending on their focus.

The remainder of the chapter will discuss the

characteristics of these groups and their relevance to this dissertation.

Table l: Taxonomy of Current Literature

 

Research Stream

Focus

Characteristics

 

 

ro 1

Norton and Bass (1987)

. Thompson and Teng (1984)
Teng and Thompson (1983)
Horsky and Simon (1983)
Bass(1969)

 

Dynamic Nature
of the Sales
Process

 

Studies that capture the
dynamic nature of the
sales process have
concentrated on the
marketing mix decision
of product pricing and
advertising levels. This
body of research is
remiss in incorporating
quality as a determinant
of sales.

 

 

 

 

Table 1

Narasir
Banker
Tapierc
Lee ant

Besankc
MCCIUI:
Gerstnei
Monroe

 

 

Table l: Taxonomy of Current Literature (cont.)

5

 

Gretna

Narasimhan and Ghosh (1994)
Banker and Khosla (1992)
Tapiero et al. (1987)

Lee and Tapiero (1986)

Strategic
Dimension of

Quality

These studies can be
classified into two
categories: Those which
postulate a static model
to identify an optimal
equilibrium level of
quality (and price) and;
those which formulate a
dynamic model to study
the transient behavior
of the sales rate due to
(cont.)

changes in quality (and
price). None of the
studies in the latter
group has considered

the effect ofm
l l' 1 l °li

on the sales rate.

 

 

Simian!

Besanko and Winston (1991)
McClure and Spector (1991)
Gerstner (1985)

Monroe and Dodds (1988)

 

Price-Quality
Relationship

 

Studies that address the
price-quality
relationship are based
on correlational
analysis. These studies
have not modeled the
dynamic nature of the
sales process. Also, this

stream of research has
focused on 2913131, not

gptimal prices.

 

 

 

 

Table l:

/

Narasiml
Cook (19*.
KOhil ant
Mesak (1 :
Dockner !
Khalish “I
Dolan an
RobinsorI

 

G
necessar
dynamic
literatur
advenis
Quantitz
of new
mOdel,
Potentia

pUrChaS

 

 

6
Table l: Taxonomy of Current Literature (cont.)

 

 

Group 4

Narasimhan and Ghosh (1994) Optimal Pricing This research is

Cook (1992) reviewed to assess the

Kohli and Mahajan (1991) existing methodological

Mesak (1990) framework concerning

Dockner and Jorgensen (1988) optimal pricing. Most

Khalish (1983) (1985) of these studies do not

Dolan and Juland (1981) consider the effect of

Robinson and Lakhani (1975) product quality in their
formulation, and none
of them considers the
effect of produgtgnd
W on the
sales rate.

 

 

 

Group 1- For the kind of study proposed in this research, it is
necessary to investigate the literature for existence of models that capture the
dynamic nature of the sales process. There is an abundance of models in the
literature addressing the marketing mix decisions of product pricing and
advertising levels (particularly in the area of diffusion of new products).
Quantitative modeling of sales response functions dealing with the diffusion
of new product innovations was first examined by Bass (1969). Bass, in his
model, treated the diffusion of innovations as a function of the market
potential for that product, and the buying characteristics of two types of

purchasers: “innovators” and “imitators”. His model captured the sales

 

 

of new (11
proportio
the prod
number t
of the re
of the B:
Such ex‘
Teng an
(1987),

Sethi (1

 

decision
concernt
lllCOl‘pm
models.
quality :

in

quality

 

the 5a] C

does 110

 

Perceit

 

' 7

of new durable goods as the sum of purchases made by innovators, which is
proportional to the number of potential customers who do not already own
the product, and imitators whose purchases are proportional to both the
number of people who have the product as well as those who do not. Much
of the recent effort in this area has involved the extension and modification
of the Bass (1969) model to incorporate pricing and/or advertising decisions.
Such extensions can be found in Ozga (1960), Horsky and Simon (1983),
Teng and Thompson (1983), Thompson and Teng (1984), Norton and Bass
(1987), Nasimento and Vanhonacker (1988) and others. Little (1979) and
Sethi (1977) provide good reviews of models dealing with advertising
decisions, while Rao (1984) presents a comprehensive survey of works
concerned with pricing. While substantial progress has been made in
incorporating price and advertising decisions in dynamic sales response
models, this body of literature is surprisingly remiss in incorporating product
quality as a determinant of product sales.

Group 2- Quantitative research dealing with the strategic dimension of
quality is scarce. Lee and Tapiero (1986) attempt to link quality control and
the sales process; however, their research postulates a static model which
does not capture the delay associated with the time it takes for the market to

perceive changes in a product's quality. Tapiero et al. (1987) develop a

 

 

model tt
model i
be a fur
last per
potentia
model I
study e'
affected
therefor
and sale

G
price-qu
0f the p
price as‘
excellen
t0pic. '
incompii

quality

 

W’estig

reiiition

 

8
model to link price, quality management and the concept of reliability. The

model is dynamic in the sense that it considers the demand for a product to
be a function of the current price and the number of defective products from

last period. The model, however, does not capture the process by which

potential sales become actual sales. Banker and Khosla (1992) develop a ,

model to investigate the impact of quality in a competitive market. This
study evaluates how the optimal equilibrium values of price and quality are
affected by increasing levels of competition. The model is static and
therefore the study does not consider the dynamic interaction between quality
and sales over time.

Group 3- Another stream of research in the literature investigates the
price-quality relationship. The main focus of these papers is the investigation
of the presumed positive relationship between price and quality, and higher
price as a signal of higher quality. Monroe and Dodds (1988) provide an
excellent review and assessment of marketing literature dealing with this
topic. The authors conclude that investigation of price-quality relationship is
incomplete and suggest that the dynamic relationship between price and
quality merits a richer conceptualization. Gerstner (1985) empirically
investigated the relationship between price and quality and concluded that the

relationship was weak. The author did not explore the effect of price-quality

9
relationship on the sales response of products. Additional examples of this

implicit and indeed, implied relationship between price (as a signal of
quality) and quality can be found in McClure and Spector (1991) and
Besanko and Winston (1991)

Group 4- The literature on optimal pricing for a profit maximizing
firm is extensive. Mesak (1990) has discussed the optimal strategic pricing
of technological innovations. Kohli and Mahajan (1991) discuss conjoint
analysis approach to examining pricing decisions for new products. Cook
(1992) presents a framework for pricing to maximize profit taking into
consideration price elasticity of demand in the telecommunications industry.
Investigations into optimal pricing decisions include Robinson and Lakhani
(1975) Dolan and Juland (1981), Khalish (1983), (1985), and Dockner and
Jorgensen (1988). Narasimhan and Ghosh (1994) develop an optimal
control model to study the effect of quality on optimal pricing and advertising
decisions. The authors present qualitative characterizations of the nature of
optimal price and advertising policies. None of these studies has considered
the effect of product and quality durability on the sales rate.

This chapter has reviewed and discussed the current literature on the
interaction of price, quality and the sales rate. It has been shown that there

is a need for a model that considers the dynamic relationship among price,

7;

 

 

 

 

10
quality and sales rate, and the moderating effects of product and quality

durability.

3. FOCUS OF RESEARCH

While there is little dispute as to the importance of product quality to
competitive performance, the relative effect of different dimensions or
attributes of quality on such performance is still unclear. Even though the
evaluation of the costs associated with quality management has drawn a lot of
attention from researchers, the strategic impact of quality on market
performance indicators such as sales rate and profits, has not received the
same attention in the literature. Current research, scarce as it is, dealing
with this strategic dimension of quality (Banker and Khosla, 1992; Tapiero et
al., 1987; Lee and Tapiero, 1986) does not consider the time lag between
quality changes (i.e. , improvements) and the effects of those changes on
consumers' purchases. Consideration of the dynamic interactions among
price, quality and market response is necessary to understand the full impact
of quality improvements on performance and to optimize performance
through appropriate pricing strategies. The relationship among price, quality
and sales rate is essentially dynamic. The impact of price-quality policies on
sales rate is not instantaneus. There is a time lag between managerial actions
and market effects; therefore, to develop a better understanding of the effect

of quality on market performance, both the timing and levels of attainment

11

12
for the performance measures of interest arising from quality improvements

must be considered.

Existing research also fails to consider other market related influences
that can modify the effect of quality on sales rate. Elements such as the
products’ life span, the level of market saturation and the effect of the
difference between actual and perceived quality on sales have not been
incorporated in a formal model.

The foregoing discussion suggests that there is a need for explicitly
incorporating these considerations into a formal, quantitative model that can
be used to study the dynamic process by which quality and price together
affect sales of a product. This research develops a dynamic model that
treats quality and price as control variables or inputs, and sales rate and
profit (measures of corporate performance) as outputs. The model
incorporates elements that have not been considered by existing research such
as the notion that the consumers’ perceptions of quality improvements of a
product change more slowly than the product’s actual quality. The model
also considers how the products’ durability affects the relationship between
price-quality policies and performance.

Incorporating the relationship between cost and quality according to

the prevailing ideas in the literature, the model is used to compute price

13
trajectories that maximize profits at a specified target time. Quality, which

is also a control variable in the model, is set to a pre-specified trajectory of
continuous growth throughout the planning horizon. The model is not used
to compute optimal quality trajectories. Overall quality depends on many
factors within the firm; some of them difficult to measure and/or control.
Also, the exact nature of the relationship between such factors and a single
number signifying quality is hard to establish. The task of coordinating all
the variables that affect quality at any given point in time to achieve a pre-
established quality level is an extremely difficult if not impossible task; it can
be argued that the computation of optimal quality policies has limited
applicability. Instead, we have opted to postulate a general steady quality
growth pattern, in agreement with current business tendencies, and examine

the behavior of optimal price policies while quality is increasing.

13
trajectories that maximize profits at a specified target time. Quality, which

is also a control variable in the model, is set to a pre-specified trajectory of
continuous growth throughout the planning horizon. The model is not used
to compute optimal quality trajectories. Overall quality depends on many
factors within the firm; some of them difficult to measure and/ or control.
Also, the exact nature of the relationship between such factors and a single
number signifying quality is hard to establish. The task of coordinating all
the variables that affect quality at any given point in time to achieve a pre-
established quality level is an extremely difficult if not impossible task; it can
be argued that the computation of optimal quality policies has limited
applicability. Instead, we have opted to postulate a general steady quality
growth pattern, in agreement with current business tendencies, and examine

the behavior of optimal price policies while quality is increasing.

15
In this research, the term quality is represented by a composite

measure that captures the characteristics of the product that provide
consumer's satisfaction (other than durability, which is treated explicitly in
the model). Consumer’s Report publishes such a " quality index" for the
automotive industry based on the response of hundreds of thousands of
questionnaires sent to its readers (Automotive News, June 1, 1992). Data on
this index is used to validate the model.

Profit is used as a performance measure. Profit at any time is
computed as a function of the sales rate, price, and average cost. Sales rate
is computed by the model, and the specification of average cost as a fraction
of quality is consistent with current literature.

A modification of the Bass’ (1969) diffusion model for new products is
used to describe the dynamic interactions that drive sales rate. The sales
response of a product is treated as a function of its market potential, price,
durability, quality, and quality weighted volume of the product currently in
the market to capture the effect of perceived quality. Price is assumed to
influence the market potential for the product, that is, the number of units of
the product that consumers are willing to buy at any given time (Lilien et al. ,
1992). The functional form chosen to describe the relationship between

market potential and price is similar to the traditional demand function

4. MODEL OUTLINE

In order to study the nature of optimal price strategies, first, a
continuous time dynamic model is developed that provides the basis for the
optimization efforts. The dynamics of the model are driven by price and
quality signals that comprise specific managerial policies. The system's
response is evaluated using sales rate and profit as business performance
measures. Most of the terms used in the definition of the model are well
defined concepts in the literature and do not deserve any further explanation.
However, this is not the case for the term "quality", which may take
different meanings. For some, quality implies the number of features or
attributes of a product. For others, quality means product reliability.
Broadly speaking, the quality of any product can be divided into two primary
dimensions: "fitness for use" and "conformance to specifications" (Juran and
Gryna 1980). Fitness for use consists of those product attributes which meet
the needs of the customer and thereby provide product satisfaction. Product
attributes may include such quality characteristics as performance, reliability,
features, durability, serviceability, and aesthetics. Conformance to
specifications, on the other hand, refers to the degree to which a product

meets its design specifications.

14

16
encountered in economics. Market potential and quality-weighted cumulative

sales are assumed to influence the sales rate for the product. Similar to the
Bass model, we assume that sales rate is proportional to the total quality
weighted units in the market as well as to the difference between market
potential (at a given price) and the quantity of products in the market at time
"t" . Durability (average life) of the product is a parameter that enters the
model in the computation of the number of units still in service at any time
point in time. The inclusion of durability in the model makes it possible to
recognize the difference between cumulative sales and the actual number of
units in the market. This distinction is important. Cumulative sales are used
in the Bass model to compute market saturation. This approach carries with
it the implicit assumption that no unit of the product leaves the market within
the time horizon represented in the model. Under such conditions, for a
sufficiently long time horizon, the market will saturate driving the sales rate
to zero, a situation that is not in agreement with ‘real’ market behavior. As
will be recognized later in the analysis, the inclusion of actual number of
units in the market, instead of cumulative sales, guarantees that the sales rate
for a durable good will remain positive even in the " steady state." This will
permit the analysis of the product’s behavior in the market through an

indefinite time horizon.

17
The model also introduces the concept of “quality durability” or

“quality life”, which represents the length of time the quality of a product
exerts influence in the generation of new sales through a diffusion process.
Even though throughout the dissertation the model focuses on durable
goods, this choice was made to stress the influence of product durability on
the price-quality-sales rate relationship. The constructs of the model also
apply to consumer goods, as long as they face a downward sloped market
potential curve. However, due to the short life-spans of the products, the
influence of product durability on the sales rate is much less relevant for
consumer goods than for durable goods. We also chose not to model
competition explicitly, to keep the model tractable and focused on the issues
addressed by the dissertation. This is not a serious limitation, because the
effect of competition on the ability of a firm to set prices is indirectly
captured through the price elasticity of demand (Pindyck and Rubinfeld,
1995) in the market potential formulation. As we show in chapter 6, with the
help of historical data for a good, the constructs of the model describe

appropriately the interrelationship among price, quality and the sales rate.

5. MODEL DEVELOPMENT

This chapter develops the dynamic model outlined in the previous
chapter. The model establishes a dynamic relationship among price, quality
and the sales rate of a durable product. In the ensuing discussion, it is
assumed that the firm has certain control over the quality level of its
products. It is also assumed that the firm is able to influence the market
potential for its products through its pricing decisions, and so, it faces a
downward sloped market potential curve (Varian, 1992). Figure 1 shows a

conceptual overview of the model.

 

 

 

 

 

 

    

Units of Product Product
Price Market \ in the Market Durability
Potential -

 

 

 

 

 

 

 

Units of Quality Quality
in the Market Durability

Figure 1: Conceptual Overview of the Model.

 

 

 

 

The sales rate at any time “t” is modeled by modifying Bass' diffusion

model for the computation of sales rate of a durable good. This

18

19
modifications allow the evaluation of the impact of price, quality and

durability of the product on its sales rate. Consider the Bass diffusion

equation for sales rate:

sto=ggm= aQ(t)[M -Q(t)]
where S(t) and Q(t) represent the sales rate and the cumulative sales,
respectively. (Notice that, because this model does not consider the finite
life-span of the products, Q(t) is also equivalent to the number of units in the
market at time “t”.) The parameter M denotes the product's market
potential, or the number of units of the product the market can absorb and at
is a proportionality constant. In modifying the Bass model's diffusion
equation, the contribution to sales from the "innovation" component has been
deleted, since this research deals with "established," not new products.
Also, it is known that the effect of the innovation component on sales rate
disappears over time (Norton and Bass, 1987). Bass’ equation is based on

9’

the premise that sales rate at time “t is proportional to the quantity of the
product in the market, Q(t), because the larger the value of Q(t) the greater
the chance that potential customers will come into contact with the product

(i.e., diffusion effect). Also, in Bass’ model, sales rate is proportional to the

gap between the market potential M and the quantity of the product in the

20
market Q(t) representing the so called saturation effect. This implies that,

given a fixed market potential, each unit of the product sold represents a

reduction in the chance to sell an additional unit.

The proposed model extends Bass' equation by:

0 modeling the market potential as a function of price;

- recognizing the limited life-span of products (which after
some time free up potential buyers); and

0 recognizing the effect of the market’s perception of product
quality on sales rate.

A more detailed explanation of these modifications and extensions

follows.

21
5.1 Marketletentialmfiunrtionnu’me

The model presented here assumes that market potential depends on
the price of the product. The term M in Bass' equation is replaced by M(P),
the number of units that customers are willing to buy at price P. To establish
the relationship between price and market potential, we assume that the firm
possesses the power to set price for its product in its market. This
assumption is reasonable for such goods as automobiles and other consumer
durables, over which the producer enjoys certain level of monopoly power
(Lilien et al., 1992).

The definition of market potential used in this research is similar to the
concept of a demand function in economics. As in the standard demand
function, we assume an inverse relationship between price and market
potential (i.e., the lower the price, the more the customers who will be
willing to buy the product).

Given the analogy between market potential and demand functions, we
can conveniently express the (M(P), P) relationship in terms of the price
elasticity of market potential. The price elasticity of market potential can be
defined as the ratio of the proportional change in market potential to the
proportional change in price leading to the relationship in (5-1). Equation (5-

1) characterizes the functional relationship between M and P as one of

22
constant price elasticity for all levels of M and P. The price elasticity, “e”,

characterizes how sensitive the market potential for the specific product

being modeled is to changes in price:

M(P,) =[%)'M, e > o (5-1)

where (Mo, P0) is a m demand-price pair for the product. It can be
noted that the relationship in (5-1) gives the right behavior between market
potential and price as price increases.

Introducing the market potential modification into the Bass’ equation

yields:

S(t)=aQ(t)[M(Ii)-Q(t)]

23
5.2R 'i n fPr ct' Finit Lif an

The second modification to Bass' sales rate model is the inclusion of
durability of the product as a determinant of sales rate. In the proposed
model, units "enter" the market when they are sold to customers and "leave"
the market when they cannot perform their functions any longer because of
either failure or obsolescence. A unit leaving the market is assumed to free
up a potential customer. The life-spans of the units of the product
(durability) are assumed to be independent and identically distributed random
variables.

In order to establish the connection between the durability of the
individual product units and the aggregate sales and obsolescence rates, we
follow the procedure described in Manetsch (1966).

Consider the following system, that represents the life of a unit in the

 

 

 

 

market:
Input ftnction for (>um function for
prodmti’ S.|=O(t-ti) product i,Yi=5(t-ti'Tl)

 

 

 

Figure 2: Graphical Representation of a Product’s Life-span.

24
The ‘birth’ of a product unit, that is, the introduction of a unit into the

market at time “t,” , is represented by a Dirac Delta (impulse) function of the
form S'i(t) = 8(t-ti). The impulse function is ‘defined’ in the following way:

6(t) = 0 Vt at 0
6(0) = undefined

[160) dt =1
The exit from the market of the unit that was sold at time t, and
remained in service for I, years, is represented with another impulse function
as Y:(t) = 5(t-ti-ti). Adding the “N” (“N” being the total number of units
ever sold) impulse functions that correspond to the ‘birth’ and ‘death’ of the
individual units, we obtain the sales and obsolescence rate functions
respectively.
t=N .-=~
s‘tr)=§s:(t)= get-r.)

t'=N

Y‘(t) = 251/11) = get - — .,.)

To show that S'(t) and Y'(t) are indeed rate functions, consider for

instance the following expression:
1 i=N t i=N
T(t) = L S'(£)d£ = ZL6(£— 4.) d8 = ZUQ — t,)

where U(t) is the unit step function, defined as:

25
U(t)=l for 120
=0 for t<O

Then, 7(1), the integral of 5"(1) represents the total number of units
that have been sold up to time t, and so, S'(t), can be considered as a sales
rate function. A similar argument can be used to show that Y (t) can be
considered as the product’s exit rate.

We seek to determine the relationship that exists between S. (t) and
Y. (t). In order to facilitate the task, we will first study this relationship in
the frequency domain, by applying the Laplace transform to the relevant
variables, and then we will translate the results back to the time domain.

The Laplace transform of a function fit) defined for t _>_ 0 can be

expressed as:

L{ f (t)}(s) = I: exp(—st) f (t) dt = f(s)

Observing that the individual output functions Y*i(t) are identical to the
input functions S‘,(t) shifted 1, years, applying the Laplace transform to both
variables, and translating the time shift relationship that exists between them
to the frequency domain, we obtain the following relationship:

1T0) = exrho-«$95.15)
Then, the relationship between the aggregate input and output flow

rates can be expressed as:

26

Yozr‘ozptm)

In this research, the product’s “birth” epochs “ts ’ ” are considered to
be random variables; the life-spans of the units in the market “1’s” are
considered as independent and identically distributed (iid) random variables
and; the product’s life-spans are assumed to be independent of the product’s
“birth” epochs. Under these assumptions, the output flow Y'’ is also a
random variable. To obtain a deterministic measure of Y’, we compute its
expected value. Note that because the I, are iid we can discard the subscript

(Si 9’

on taking expectations.

E{17’(s)}= E{ZY. (3)}: :5 (s))E E={exp(—rs)} S(s)E{exp(—rs)}

i=N
where S(s) = E(Z§,'(s)] is the expected value of the sales rate and, in this

research it will be used as a deterministic approximation to the sales rate
function.

Let f 0 be the probability density function (pdf) of the random variable
‘L’. Applying both the definition of the expectation of a non-negative random

variable and the definition of the Laplace transform, we arrive at the

following expression for E {exp(—rs)}:

27

E{exp(—rs )} = J: exp(—rs)f(t)dr = L{f(r)}(s) = f(s)

and the expression for the expected value of the obsolescence flow rate in the

frequency domain becomes:
E{r*(s)} = 17(s)= sun/"(n

In order to develop this expression further, we need to specify the
probability distribution of the product’s life-span. In this research, the life-
spans of units of the product in the market is represented as independent,
identically exponentially distributed random variables. The model developed
here is intended to be applicable to a wide range of products, with potentially
different life-spans distributions. The exponential distribution enables us to
consider the aspects of durability that are relevant to the determination of the
sales rate while offering mathematical tractability.

Under the assumption of exponentially distributed life-spans, the pdf of

where 11., is the expectation of T. The Laplace transform of this expression

yields:

28
f (S) Lif (7)}(3) ’

 

ps+1

and the expression for the expectation of the output rate Y becomes:

7(s)-_-_S_(f)_

s+1

From this expression we obtain, after some manipulation:
s Y (s)— - (S (s) Y _(s))

and this expression translates into the time domain (for a sufficiently smooth

S) as:
%1’(t)= —(S(t)- Y(t))

In our model, we denote the average life-span of the units with the

parameter D1, and the above differential equation becomes:

d 1

—Yt=—St—Yt 5-2

,0 mm 0) ( )
To fully describe the model, we need to develop an expression for the

number of units of the product present in the market at a given time. This
expression can be obtained from (5-2). Let Q(t) be the number of units in
the market at time “t”. Q(t) can be computed as the integral of the product

inflow rate minus the outflow rate:

29

Q(‘)= titre)- Yte» d.

If we integrate both sides of (5-2) we obtain:

Y(t)=fil$(S(E)-Y(e))de

Combining both equations, we arrive at the following expressions for
Q(t):

Q(t) = D1 Y (t) (5-3)

We then modify the sales rate equation

sto= ago) {mm-Q(t)}
by recognizing that Q(t) represents the number of units present in the market
at time “t”; this value is different from cumulative sales, given the finite
life-spans of the units. We provide, in (5-2) and (5-3), a mechanism to
compute Q(t).

Before we introduce the effect of quality into the study, it is useful to
examine the equilibrium behavior of the model as it has been developed so
far (i.e., equilibrium behavior of S, Q and Y) to assess its internal validity,
and also to gain insights into the relationships among sales response, price,
and product durability. The objective is to examine the structure of the

model for its internal consistency. Assuming fixed price (i.e. constant

30
market potential), combining the sales rate expression and equations (5 - 1)

and (5-2) we obtain the following differential equation:

%Q(t>=iaM—blri9‘ ‘32 Q(°)=Q°

This can be solved to yield the following closed form solution for Q(t),

(we-h)

—a+atw-siexpt-tw-ai+a

From this result, we find that the long run behavior (as t -) 00) of the

 

Q(t)=
i

system, when aM z i is:
D1

1
Q°° (1."; Q(t) aDl

Yoo = limQ_(t_2=M_.-__1__
t—>oo D1 D1 ale2

M 1
Sm: - St =Yw=——
tleoo () D1 6:012

 

It can be seen from these equations that the sales rate S(t) and quantity
in the market Q(t), approach equilibrium values asymptotically. However,

the equilibrium value for the quantity in the market Q(t) is not market

potential (M), but a lower value [M — fi] , due to the limited life of the
a

31
product. The shorter the durability of the product, the lower the maximum

attainable value for Q(t) , due to the fact that new potential customers are
being created faster. This result is consistent with what one would expect the
effect of durability to be on realized market potential (i.e. Q(t)).

It is important to note that, according to the previous derivations,

there is a positive value for D1 that maximizes S(t). This value can be
. . d . . e 2

determined by equating —S,,o to zero, resultmg 1n D] = —. The second
le aM

order condition shows that

d2 S _—a3M4 <0
arm2 °° 8

 

 

asserting that D1*is indeed a maximizer for so...
Substituting D1* in (8), we get
2
max _ a M
Sm _ _4—
as the maximum steady state value for sales rate.

If aM < g: , the long run equilibrium value for Q and S is zero. This

condition implies the existence of a ceiling price for the sales rate to remain
positive in the steady state. This fact is addressed in chapter 6, where we

discuss the equilibrium properties of the full model.

32
5-3 WM:

As discussed previously, market potential M(P) represents the total
number of units that can potentially be sold at a given price P. In the model
presented in this research, the rate at which those potential sales become
actual sales is assumed to depend not just on the number of units in the
market, but also on the quality level of those units.

We seek to compute the total number of quality weighted units present
in the market at a given time in order to introduce this value into the sales
rate equation. In the following derivations, a procedure similar to the one
used in the analysis of the products’ life-spans is followed.

Consider the product unit ‘i’, that enters the market with a certain

level of quality. The process is depicted in the following figure:

 

 

 

 

 

 

 

 

 

Quality input function for Quality (11th function for
producti,S'q.=qx8(t-t.) WOdUCti-Xi.=qxa(t'tl'1qi)
.' MARKET F
t‘ ti 4' Tqi

 

 

 

Figure 3: Graphical Representation of a Product’s Quality Life-span.

33
The quality level of the ‘1’ unit, denoted by q,, is assumed to be a

deterministic function of the time at which the unit entered the market and

so, henceforth, this quantity will be denoted q(t,).

In the model presented here, the units in the market are assumed to
influence the generation of new sales according to their individual quality
levels. The higher the quality level, the more likely it is that a specific unit
in the market stimulates new sales. The length of the interval during which a
product’s quality influences the generation of new sales, tqi, is modeled here
as a random variable, independent of the life-span of the product. On the
one hand, products’ owners, rather than the products themselves, are the
agents who report to other individuals about the quality of the units they
own. They can report on units they currently own, or on the ones they used
to own. This implies that the life-span of the product does not necessarily
constitute an upper bound for its quality to be reported to other individuals in
the market. On the other hand, a product’s potential to generate new sales
may dissipate before the product is taken out of the market. This situation
could occur when newer models of the product are introduced into the market

and the public recognizes that the quality of an old product is not a reliable

34
indicator of the quality of the new ones. These arguments support the reason

for modeling the quality life independently of the product’s durability.
Going back to figure 3, the entry of the quality weighted unit ‘i’ into
the market at time ti, is modeled as an impulse function at time ti multiplied

times the quality level of the unit ‘i’ (S; (t) = q(t,.)6(t — t,)). The end of the

influence of product ‘i’ in the generation of new sales is modeled as the

product of the quality of the unit and another impulse function at time 1“,

(atom->604,- 4.9)).

Following the same procedure we employed in the study of the
products’ life-spans, we add the quality input and output functions over all

products to obtain:

Sm):-2’s;(t)=:qo>a(r-o
m:impgqewo—t -r:’)

We will investigate the nature of S’Jt) by taking the same approach

that was used with S’(t). Consider the expression:

i=N t=N
.=1

73(1) = [0' 5:18) d6 = ;(q(t,)j;a(e- ti) d3) = z (1,.) up _ t)

i=l

35
Then Tq(t) represents the total number of quality units that have

entered the market up to time “t”, and S; (t) , which may be expressed as
q(t)§(t), can be considered as the rate of quality units entering the market at
time “t. ” Following a similar derivation, it can be shown that X*(t) can be
regarded as the rate of quality units ceasing to influence new sales at time
“tn

As in the case of the sales rate, we will study the relationship between

51:0) and 1(0) in the frequency domain, with the aid of the Laplace
transform. Noting the time shift between S") (t) and X :(t), the following

expression relating these two variables in the frequency domain can be

written as:
JETS) = exp(- 73S) 5&0)
Y‘(s)= ”iron Talent-m) 3;; (s)
i=l i=1 1'
Being a function of random variables, X" is itself a random variable.

By taking as a deterministic measure of X. its expected value, from the

previous equation we obtain:

ego) E{exp(-rqs) }= i(s>f'.(s)

E{}r“(s)} = E{Z: Z.‘(s)} = E(

i=N
i=l

36
where Sq(s) is the expectation of if; (s), and fq( ) stands for the pdf of
I=l t

1".

Again, for mathematical tractability, the random variable ‘tq is

assumed to be exponentially distributed, and its pdf can be written as:

M“) = fllqexr)[- :1)
1'

T

 

 

where pr, is the expected value of 1“. The Laplace transform of this pdf

can be expressed as:

l
jurqs+1

 

fq<s>=elfqtrqh =

Substituting this term in the expression for the expectation of X‘, and

denoting this expectation by X, after some manipulation we obtain:

 

s ids) = #1 q (s,(s) — )7(s))

T

For a sufficiently smooth 5., we can apply the inverse Laplace
transform to both sides of this expression, to obtain the following differential
equation:

gxokflem-xtrn

T

37
Given that S q (t) = E (S q (1)) = E (q(t)S (t)) = q(t)E (S (t)) = q(t)S(t) the

above differential equation becomes:

57m) =;.'—(s(r>q<r>-X<t>>

1.4

In our model, we denote the average ‘quality life-span’ with the

parameter D2, and so, the equation becomes:

§X(t)=-,§§(S(t)q(t)-X(t)) <s-4>

In (5-4), S(t)q(t) represents the quality weighted units that enter the
market; X(t) represents the flow of quality weighted units that are no longer
influential in the generation of new sales, i.e. these units are now too old to
influence the customer's perception of quality of that product; and D2 is the
average duration of the products in the market that still influence customer's
perception of the quality level of the units in the market in the generation of
new sales.

Let EQ(t) be the number of quality weighted units present in the
market at time “t”. EQ(t) can be computed as the integral of the quality

weighted units inflow minus outflow rates:

EQ(t) = J: (S(e')q(£) — X (8)) d8

Integrating both sides of (5-4) we obtain:

38
X(t)= grandad— X(s))de
Finally, combining both equations, we arrive at the following
expression for EQ:

EQ(t) = D2 X (t) (5-5)
Recognizing that it is the quality weighted units, rather than just the
number of units in the market, that influence potential buyers of the product,

results in the final modification to the Bass’ diffusion equation:
S(t) = a EQ(t)[M(Pt) -- Q(t)l (5-6)
‘ where S represents the sales rate of the product, M the market potential, Q
the quantity of products in the market and EQ the quality weighted units of

the product in the market.

39
5.4 MM

The complete mathematical specification of the sales rate model

proposed in this research is:
P e
M (Pt)=(;°) Mo (54)
t
d 1
—Y —— S -Y -2
dt ‘ D1(‘ ‘) (5 )
Qt =Dl><Y. (5-3)
d 1
EX! = 550]: X St — Xt) (5‘4)
EQ, =D2>< X, (5-5)
S = E M — 1 5-
t “X Qt"( P Qt)’< [MP>Q,] ( 6)
Where:
Pt = Price [$]

q, = Quality index [0 to 1]

Mt = MarketPotential [units]

Q. = Quantity of units in the market [units]
8. = Sales Rate [units/time]

Yt = Rate at which units leave the market [units/time]

4o
Xt = Rate at which the quality weighted quantity of goods in the

market ceases to influence consumers' behavior [units/time]
EQt= Quality weighted quantity of goods in the market. [units]
D1 = Average life of the units [time]
D2 = Average time of the effect of quality of goods on consumer's
buying behavior (quality persistence delay).
a = Proportionality constant used to calculate sales rate [time-unitsl'1
e = Price elasticity of demand.
(The indicator function is added to (5-6) to show that sales rate will vanish
rather than become negative, if the quantity in the market exceeds market
potential. This situation can occur due to a sudden raise in prices.)
The system can also be represented as a signal flow diagram, as shown

in Figure 4:

41

 

 

 

(l

 

 

 

 

 

 

Figure 4: Signal Flow Diagram of Proposed Model.

The model shown in Figure 4 captures the dynamic interactions among
price, quality and sales implied by the mathematical model. Price (P) and
quality (q) are inputs to the system. The price level (P) determines the
market potential (M) through a demand type function with constant price
elasticity. The number of units currently in the market (Q) is subtracted
from the market potential (M) in order to compute the number of units that
can possibly be sold. An indicator function is applied to the (M-Q) value, to I

prevent potential negative sales. The product of (M-Q) with the quality

42
weighted units in the market (EQ) generates the sales rate (S). The sales rate

(S) is used to compute two quantities in the model: The sales rate, multiplied
times the quality level of the units (q) is used as the input of the “life of
quality” process described by equation (5-4). This process determines the
products’ “quality exit rate” (X). By integrating the difference between the
input and output rates of the “life of quality” process, we obtain the quality-
weighted number of units in the market (EQ). Also, the sales rate by itself is
used as input to the process described by equation (5-2) that represents the
life-span of the products. This process determines the products’ exit rate
from the market (Y). By integrating the difference between the input and
output rated of this “life of product” process, we obtain the number of units
currently in the market (Q).

Before closing this chapter, we introduce the state space representation
of the model, which will be useful in subsequent analyses concerning the
stability properties of the equilibrium points of the model. Taking Q and EQ

as state variables, the state space representation of the model is:

d P ‘ ,
7? = “(Wrifij M0 ‘ “(EQJ Qt ' %_1 Q(O) = Q0

dE P e E 1
gang) armada—Lei We

6. DEL I

In this chapter we discuss the tests that were conducted on the
proposed model to assess its validity. The chapter is organized as follows:
First, the issue of model validation is discussed. Then, the specific
validation procedures that were adopted in this research are discussed; and

finally, the results of the validation tests are discussed.

6.1 Validation of System Dmamig Models

Gilmour (1971) and Forrester (1980) describe several validation tests
for system dynamics models such as the one proposed in this research. For
the purposes of this investigation, these tests are classified into three major
groups: Consistency tests, fit to historical data tests, and policy implication
tests.

Consistency tests refer to how well the structural specifications and
the behavior of the model agree with the existing theory and/or experience
gathered about the real system. This testing includes subjective as well as
analytical procedures. The subjective methods, known as " face validity
tests", require that people with experience in the area of investigation

examine both the output and the structural specification of the model. These

43

 

44
experts must decide whether the structural design of the model is in

conformance with the existing knowledge about the actual system.

The analytical tests employed in consistency validation examine the
structure of the model to infer its behavior under circumstances that can be
easily predicted in the actual system.

Fit to historical data tests deal with how well the model can
reproduce historical behavior of the actual system, under an appropriate
choice of the structural parameters. In order to decide whether a model has
passed or failed the test of conformance to past behavior of the system,
criteria for the evaluation of the fitness must be specified a priori. A vast
number of criteria can be used to evaluate goodness of fit. The simplest
criterion is a judgmental visual comparison between the actual and
reproduced time series. Analytical criteria can be divided in two categories:
the ones that measure the overall " closeness" of the model response to the
actual data, and those that assess the correctness of the general shape of the
model's response. Within the first group, common measures of goodness of
fit are the Mean Square Error (MSE) and the Coefficient of Determination
(R2). In the second category, spectral analysis is used to confirm whether the

model is able to generate a time series with approximately the same spectral

45
densities at different frequencies. This analysis is useful to detect time shifts

between the actual data and the model’s response.

Policy implication tests examine the ability of the model to predict the
behavior of the real system under sets of exogenous inputs and/or structural
parameters different from the ones used to fit the model to historical data.

The model proposed in this research was subjected to the three types
of validation mentioned above. Face validity evaluation was carried out with
the help of an expert from the Ford Motor Company. The model was also
submitted to analytical consistency tests to further investigate the correctness
of the structural specification. Fitness to past behavior was verified with
price, quality and sales data collected on Ford Mustangs. The procedure
used to fit the model to the historical data also produced estimates for the
structural parameters of the model. These parameters values were also
evaluated for consistency. The policy implication tests conducted on the
model will not be described in this chapter. They were performed in the
latter part of the research, during the derivation of optimal price policies.
The parameter settings used in the derivation of those policies are identifiable
‘real world’ situations, and care was taken to compare, when possible, the

results of the optimization efforts with actual business practices. These

46
comparisons will be discussed along with the presentation of the optimal

price policies, in chapter 8.

47
6.2 Oomsg’ tangy Evaluation; Fag: Validity

For a face validity evaluation of the proposed model, we interviewed
Mr. Joseph Verga, Thunderbird project leader at the Ford Motor Company.
Mr. Verga offered his opinion regarding the appropriateness of the structural
specification of the model and sample outputs he reviewed. Mr. Verga also
commented on the pricing policy of the Ford vehicles and the relationship
between cost and quality. His comments on the last two issues, while not
directly related to the validation of the structure of the model, are relevant to
the development of the latter part of the research, and, for conciseness, will
be included in this chapter.

Mr. Verga agreed with the general specification of the model. In
particular, he expressed that the inclusions of product durability and quality
durability were relevant and done appropriately. He also reviewed sample
outputs from the model, including a fitting of the model to Ford Mustang
data. He thought that the general behavior of the model was correct, and
was particularly impressed with the fit of the model’s response to the actual
data. Mr. Verga suggested that the model be expanded to include the effect
of advertising on sales response. He thought that, given its dynamic
specification, the model could be used to determine appropriate timing for

advertising efforts.

48
When asked about the quality management policies in Ford, Mr. Verga

expressed that, although he could not reveal detailed quality data, Ford is
committed to a policy of continuous quality improvement. The company
measures quality with a composite measure of “things going right”, “things
going wrong”, and warranty records. He felt that increases in quality tend to
reduce rather than increase cost; however, he admitted that neither he nor
anyone else in the company has compiled data to verify this assumption.
With regard to the pricing policy of the firm, Mr. Verga said that Ford
follows different pricing policies depending on the car line. In the particular
cases of the Mustang and Thunderbird, Ford tries to keep constant the price
adjusted for changes in the consumer price index (CPI). Changes in prices
can also occur because of changes in manufacturing cost (when, for example,
an additional feature is added to the, car). The Company has not tried a

proactive price management strategy.

6.3 Ogmistgncy Evalaation; Stability Analygis

In this section the long run behavior of the model is studied and its
performance is compared with the expected behavior of the real system. To
facilitate the task, we analyze the model under conditions of constant price

and constant quality. In particular, we want to assess:

49
0 whether the response of the model is bounded;

0 whether the model approaches an equilibrium point(s)
asymptotically and;

0 the dependence of the model’s long run behavior on the initial
conditions and parameter values.

In order to investigate the equilibrium points of the system under
conditions of constant price and quality, we set to zero the time derivatives
of the state variables. The following is a restatement of the state space
representation of the model. Given that market potential (a function of price)
and quality are both constant, we have omitted in the representation of these

variables any notation that indicates time dependency.

12 = a(EQ,)M _ a(EQ,) Q, — 7%] = o

d:
dE E ,
—dtQ = aq(EQ,)M - aq(EQ,) Q, — —( DQZ) = 0

Let it} = [Q EQ]'. Then, the above system of equations can be
expressed as ti} = f(&')) = O , with solutions:

Q“=0

a) EQ”=O

50
1
an2

EQb‘M D_2__l_— D_2(M_ 1 )
qDl aDl qDl an2

 

Q‘=M-
b)

 

Notice that Q" , EQ" and (M —:q%) are of the same sign. From

the last expression, we can derive the following relationship between the

sign of Q" and EQ", and the quality level q:

 

1

aMD2
l

aMD2

1 b b
= :> ,1? =0
q aMD2 Q Q

 

q> :>Q",EQ">0

:> Q",EQ" < 0

 

q<

 

Loosely speaking, an equilibrium point is said to be stable if all
solutions starting at nearby points stay nearby; otherwise it is unstable. It is
asymptotically stable if all solutions starting at nearby points not only stay
nearby but also converge to the equilibrium point as time goes to infinity
(Khalil, 1992). In order to investigate the stability properties of the

equilibrium points it)" and as" , we examine the signs of the real parts of the
eigenvalues of the Jacobian matrices A = 2L(&) .. and B = 21(5)) _. to
— 6&3 a)“ “ 521) a)‘ ’

apply Theorem 3.7 in Khalil (1992). In essence, the theorem states that an

equilibrium point of a nonlinear system is asymptotically stable if all the

 

51

eigenvalues of the Jacobian matrix defined above have negative real parts. If

at least one of the eigenvalues has a positive real part, the equilibrium point

is unstable.

If all of the eigenvalues have zero real parts, or some of the

eigenvalues have zero real parts, and the rest have negative real parts, the

procedure fails to determine the nature of the equilibrium point.
the theorem to study the stability properties of (13" and if)" , we start by

writing matrices A and B explicitly:

 

_é’f .. =
4'35)“, a)“
_6f .. I _
E'aa3(w) 55"“

Applying

wfiuxm' - m -:

1 and

aqM-E

l

0

l—

: anD2
D1
_ -aqEQ"

1
qDZ
O

 

 

First we examine the stability of the equilibrium point Ei‘by looking at

the eigenvalues of matrix A. A is a triangular superior matrix with (real)

eigenvalues 11‘; = _Dl—l’ and A", = aqM -—g§.

can be characterized as:

The signs of these eigenvalues

 

 

,i‘;<0
1
A“ 0 f
2< l q<aMD2
>0 if q 1
aMD2
=0 if q: 1

 

 

52
To analyze the eigenvalues of B, we apply the property that the sum of

the eigenvalues of a matrix is equal to its trace, and the product of the

eigenvalues equals its determinant. Then,

 

 

 

’1‘} +2." = —anD2 and
D1
2‘; x ,t” = “EQb
D2
If EQ” <0 (q < l ), the eigenvalues of B are real (because the
aM D2

product of complex conjugates cannot be negative), and of different sign.

 

If EQ" > 0 (q> ), the eigenvalues of _B can be either real or

1
aM D2
complex. If they are real, they are of the same sign (because their product is
positive), and their sum is negative so they have to be both negative. If the

eigenvalues are complex conjugates, their real parts have to be negative

because the sum of the eigenvalues is negative.

1
aMDZ

 

If EQ" = 0 (q = ), the eigenvalues of matrix B are both real;

one zero and the other negative. In this case, matrices A and B are identical
because they both refer to an equilibrium point at (0,0). Under these
conditions (having one negative and one zero eigenvalue) theorem 3 .7

(Khalil, 1992) cannot determine the stability properties of the equilibrium

point.

53
For completeness, we now analyze the stability conditions of the origin

For this analysis, we apply the ‘Center Manifold

 

inthecaseof = .
q aMD2

Theorem’, as described in Khalil (1992), section 4.7.

l
, matrix A becomes [-31 aM] , with eigenvalues —fi ,
0 0

 

1
If =
q aMD2

and 0. We seek a change of coordinates under which matrix A becomes
diagonal. To accomplish this, we construct matrix I, whose columns are the

eigenvectors of A.

Then

and the change of variables we seek is

lil=r‘la%l=l? -.:.alla%l=la-.£%nal

This change of variables puts the system into the form

= -aqy(z+aMDly)= 02+gr(y,2)

SIR-SIS-

: aMy — ay(z + aMDly) - Bl-l-(z + aMDly) + aqumly(z + amny) =

1
= -‘D—IZ+82(%Z)

54
Notice that g1 and g2 have the following properties:

_. fl =. 28a :
g,(0,0)-0, 5y (0,0) 0, 02 (0,0) 0

To apply the ‘center manifold theorem’, we need to find the center

manifold z = h(y) , where h is a solution of the partial differential equation
52 1 . fl _

and study the stability of the equilibrium point (0) in the reduced system

:1 = g,(y,h(y)). The ‘center manifold theorem’ (Khalil, 1992) asserts that the
origin of the reduced system has inherited the stability conditions of the

origin of the original system.

We start the analysis of the reduced system, by choosing a simple
solution for the center manifold equation: h(y) = 0( |y|‘) , and investigate the
stability of the reduced system’s origin. If the stability. of the origin cannot
be investigated with this choice of h(y), other solutions will be tried.

(In the preceding paragraph, we have used the notation h(y) = 0( lylz).

f0) = 0( ltvll’) if troll s kllyll" for sufficiently small Hill 0

Under our choice of h(y), the reduced system becomes

:12:

dt ~aquDly2 + 0( M3)

55
-a2qMDly2 is the dominant term in the reduced system, implying that

a sufficiently small negative deviation from the origin will push the state of
the system further into negative coordinates; therefore the origin of the
reduced system is unstable and, by the ‘center manifold theorem’ (Khalil,
1992), the origin of the original system is also unstable. Notice, however,
that following a small positive deviation from the origin, the reduced system
will return to zero. Given that negative values for y (EQ) are infeasible in
the ‘real’ system we are modeling, we conclude that following any ‘feasible
small’ deviation from the origin, the reduced system will return to the
equilibrium point. Then, for our purposes, we can treat the origin as a stable

equilibrium point.

The following table summarizes our findings regarding the stability of
the equilibrium points of the proposed model, when price and quality are

held constant:

56
Table 2: Stability Conditions of the Model’s Equilibrium Points.

 

 

 

 

 

E‘l‘llllbrlum Pomt q > (ZMlD2 q < aMlD2 q = aMlD2
Q“ = 0
EQa = 0 unstable asymptotically ‘asymptotically
stable stable’
(in feasible
system)
Q" = M - _1__
“402 asymptotically unstable ‘asymptotically
b D2 1 stable (infeasible) stable’
59 = ‘1 ”137i” " 5.1—DE) (in feasible
system)

 

 

 

 

The stability analysis has shown the existence of a stable equilibrium
point in the system. To investigate how ‘close’ to this point the state of the
system has to be in order to converge to it, the model is parameterized and a
phase portrait diagram showing the trajectories followed by the state of the
system for various initial conditions is constructed. Alternatively, this can be
done through analytical derivations. It is hoped that this visual analysis will
make it easier to discern the region of attraction of the stable equilibrium
point.

The parameters values chosen for this investigation were:

D1 =7 years, D2= 2 years, M=2,000,000 units, and 0t= 0.001424
[units-years]". The value chosen for or was estimated by fitting the model to

a real product’s historical data. This analysis will be presented in the next

 

57
section. Quality (q) was set to three different levels: .8, 0.1, and 0.1756,

which correspond to the cases q >,<, and =

 

1 .
aMDZ respectively. The
following figures show, for each of the three values of q, the state

trajectories following different initial conditions.

 

mmu

Stable
Equiibn'un

Mbb
Eqdlbriun

 

0 500 1.0m 1.500 2“!) 2500 1ND

 

 

n
[Produd Unite 11 Low]

 

Figure 5a: Phase Portrait Diagram of Proposed Model -- Case q > Zia

58

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.x
i... \\\ J
/
000 Stabb / /
m H equbtttm
o 0< t W M < 500 t 2.000 < 500 < 3G”

 

 

[Product Unit-x 1.0001

 

Figure 5b: Phase Portrait Diagram of Proposed Model -- Case q < —1——

 

2.000

 

/

 

 

 

 

.i‘... - It
.. h t] /

200 /; $1
“/3 - - - a

V ‘ ‘ ‘ V ‘7 V

o 500 1,000 1,500 2.000 2.500 3,000
a

[Product Unlll I 1.000]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5c: Phase Portrait Diagram of Proposed Model -- Case q = BUZZ—

59
The figures suggest that, with the exception of the Q axis, all the

feasible states are included in the region of attraction of the stable

equilibrium point. When q > , the state of the system will be

aM—D2

attracted to an unstable equilibrium at the origin if EQ ever vanishes.

However, any small perturbation that drives the state into positive EQ
coordinates, will make the system converge to the stable equilibrium point.

The analysis of the equilibrium points, under conditions of constant

price and quality, reveal a number of interesting properties of the system.

Some of them coincide with our intuition about the ‘real world’, and lend

validity to the model; others are nonintuitive, and provide important insights

into the relationships embedded in the model. The most relevant

observations that can be derived from the study of the equilibrium points are:

0 The response of the model is bounded and, in fact, the model

predicts that the state of the system will asymptotically converge to

a stable equilibrium point. This implies that, regardless of the

initial conditions, sales rate (a function of the state of the system)

will eventually stabilize at a constant level. Explicitly, the long run

sales rate implied by the equilibrium point

w’is S"=—L[M- 1 J, for (M- 1 )20.
D1 an2 an2

 

 

60
0 The equilibrium point depends on the quality level. The higher the

quality level, the higher the equilibrium values of Q, EQ, and S.
The model implies the existence of a minimum level of quality for

the equilibrium sales rate to remain positive. A quality level lower

than 351% will drive the sales rate to zero. A quality level higher

than ——1——implies that the long run sales rate level is positive. It
aMD2

can be observed that this is a major distinction from the Bass’
diffusion model, in which product units never die, and eventually
saturate the market, driving the sales rate to zero. Figure 6 shows
the behavior of the sales rate over time for different levels of
quality. In all the runs presented in Figure 6, the same parameter
values were used to derive the phase portrait diagrams shown in
Figures 5a—50. Initial values for Q and EQ were set to 1.5 million
and .2 million respectively.

0 The model also implies a maximum price level for the sales rate to

remain positive. Given that q,,,,.,, = —l—2 , and assuming that there
is a maximum attainable level for quality (In... (in fact, in this

research quality is specified to assume a value between 0 and 1),

61
then, the minimum level of market potential for which asymptotic

 

sales rate is positive is M. = 1 1 Assuming an

m aqu2 = a02'

inverse monotonic relationship between price (P) and market

potential (M), the last relationship implies P < M'l(fi) (where

M 1 is the inverse of the market potential function).

The model indicates that the equilibrium state of the system
depends on the product’s durability (D1), and the persistence of
quality perceptions (D2). However, increases in these variables
have opposite effects on the equilibrium value of the sales rate. An
increase in D2 implies an increase in S, because of a larger number
of quality units able to stimulate sales by diffusion. On the other
hand, an increase in D1 suggests a decrease in sales rate, due to a
higher saturation level. It is important to note, however, that the
model does not account for a possible positive relation between D1
and market potential. This positive relation might be able to offset
the depressive effect on the sales rate exerted by a higher saturation

level

62

 

 

300

WK

 

 

 

 

 

 

 

 

 

 

 

T //q-I.l \
i 200 "i"
g 150 h /
E
n: 100
g N:
so \\

 

 

 

 

 

 

 

0 5 10 15

 

 

20

 

Figure 6: Sales Rate over Time for Different Levels of Quality

 

63
6.4 W

The model defined in equations 5-1 to 5-6 was set up for numerical
solution in an Excel spreadsheet. Euler integration with a time step of 0.1
years was used to solve the differential equations numerically. To evaluate
the fitness of the model to a real product’s historical data, a time series of the
production history of Ford Mustang between 1968 and 1985 was collected.
Data indicating production volume and price were found in Automotive News
while Consumer Reports (Consumer Reports, 1991) served as the source for
the quality index of the cars between 1968 and 1985. Consumer Reports ’
quality index is an approximation to the overall measure of quality used in
the development of the model in that it does capture various attributes of
quality such as features, fuel economy, serviceability and ease of
maintenance. The index is a five points ordinal scale. For the purposes of
this research, the index was rescaled to a value between 0 and 1, 1 being the
highest attainable quality level.

Prices for the cars were deflated to 1967 prices to account for changes
in the consumer price index (CPI). The CPI values for private
transportation were obtained from the "Monthly Labor Review and
Handbook of Labor Statistics", published by the US Bureau of Labor

Statistics (Monthly Labor Review, 1993). The data are presented in Table 3.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3: Ford Mustang Data Used in Model Validation.
YEAR PRICE CONSUMER DEFLATED SALES QUALITY N ORMALIZED
PRICE PRICE INDEX QUALITY
INDEX INDEX
$2,712 1.03 $2,633 345,194 3 0.75
$2,635 1.06 $2,486 275,391 1 0.25
1970 $2,771 1.11 $2,496 165,414 1 0.25
$2,91 l 1.17 $2,488 130,488 2 0.50
$2,729 1.18 $2,313 118,972 2 0.50
$2,297 1.22 $1,883 193,129 2 0.50
$2,895 1.37 $2,] 13 338,136 2 0.50
1975 $3,529 1.50 $2,353 187,454 2 0.50
$3,525 1 .65 $2,136 183,369 2 0.50
$3,678 1.77 $2,078 170,315 2 0.50
$3,614 1.85 $1,954 240,162 2 0.50
$4,187 2. 12 $1,975 365,357 3 0.75
1980 $5,262 2.49 $2413 232,507 3 0.75
$6,230 2.77 $2,249 153,719 3 0.75
$6,345 2.88 $2,203 127,371 3 0.75
$6,727 2.94 $2,288 124,225 3 0.75
$7,089 3.07 $2,309 140,338 3 0.75
1985 $6,989 3.14 $2,226 187,776 2 0.50

 

 

 

 

 

 

 

 

65
In order to fit the model to the historical data, we used the Solver

routine, embedded in the Microsoft Excel program. The Solver has
integrated a nonlinear optimization algorithm, GRGZ, that was used to
minimize the squared difference between the observed and the estimated Ford
Mustang’s production history. The minimization was performed with respect
to the following parameters of the model: M, (the market potential
corresponding to a base price P0 of $2,000), D1, D2, and a. The price
elasticity of market potential, e, was set to the value of 1.3, consistent with
the value reported in the "Monthly Labor Review and Handbook of Labor
Statistics," (Monthly Labor Review, 1993) as the price elasticity for
automobiles. Initial conditions for the model were set as follows: From the
1992 Ward’s Automotive Yearbook the number of automobiles in operation
in the US by the end of 1967 was obtained. From the same source, the
market share of the Ford Mustang in 1967 was also obtained. Combining
these two values, the number of Ford Mustangs in operation by the end of
1967 was estimated to be 1.9 million; Q, was set equal to this value.
Looking at the Mustang’s quality index in Consumer’s Report, average
quality was estimated at 0.25 for the period 1964-1967. This figure,
multiplied by Q0, gave us an estimate of 0.475 million for EQO, the initial

number of quality weighted units in the market.

66
The mathematical specification of the minimization problem is as

follows:

Let Pm, 21‘0“, and R0,, be the vectors of the observed price, quality,
and year sales of the Ford Mustangs, between 1968 and 1985 respectively.
We use the index “j” (j = 1 18) to denote each element of these vectors.

Let also P0 = 2,000, Q, = 1.9 million and EQ, = 0.475 million. We define
a! ={Mo’a’D1’DZ}’ and f.22 : {fimsqobstpo’Qo’EQo} -
Let 5(5,,O,,t) be the sales rate at time ‘t’, which corresponds to the

solution of the model presented in 5-125-6, under the parameter values,
initial conditions, and control values specified in f), and (2,. Then, the fit of

the model to the historical data was accomplished by solving the following

problem:

. F“ . I ~ - 2
Ml” 2{ Retail) _ L_15(91,Qz,t) df}
Q, 2 0 H
The solution method is illustrated in figure 7.

The values of the elements of Q, , obtained by solving the

minimization problem were: Mo= 2.4 million, D1 = 12.9, D2 = 2.26 and
or = 0.001424. The goodness of fit between the model response and the

actual data was measured by calculating the Corrected R2. This value was

67
found to be 0.74. The estimated value of 12.9 years for D1 agrees closely

with the average of 11 years for cars mentioned by automotive manufacturers
(Consumer Reports, 1991). The estimated value of 2.26 for D2 appears to
be a reasonable value for this parameter. As discussed below, this estimated
parameter value of D2 is consistent with generally held views on how long
the quality of a car continues to influence future sales. In fact, quoting from
the 1991-92 edition of Consumer Reports New Car Buying Guide (Consumer
Reports, 1991): “By the time a car is six years old, the model’s overall
record starts to be less informative than an individual car’s condition, which
depends mainly on how far and how hard it has been cared for.” This
statement implies that the quality of a car older than 6 years is not a reliable
indicator of the quality of its model line. If consumers act rationally, the
quality of cars older than 6 or 7 years should not influence their decision to
buy or not a car of similar model. Interestingly, the value of D2 that we
estimated for the Ford Mustangs, is consistent with this assertion. An
average ‘life of quality’ of 2.26 years, estimated under the assumption that
the ‘life of quality’ is exponentially distributed, implies that just 7% of the
cars older than 6 years (and 4.5% of the cars older than 7 years) influence

new car sales.

‘T Y

 

68
The goodness of fit (Corrected R2) measure between actual sales and

predicted sales shows that the hypothesized relationships in the model capture
the dynamics of sales response fairly accurately. Table 4 shows the actual
and the estimated sales for the years included in the fitting of the model.
Figure 8 shows graphically, the correspondence between actual and estimated

sales.

6.5 Qonclmigns

In this chapter, we discussed the validation tests performed on the
model proposed in this dissertation. Tests of face validity and stability
showed that the constructs embedded in the model behave appropriately.
Historical data on Ford Mustangs were used to appraise the ability of the
system to replicate a real market situation. The results of parameter
estimation yielded estimates for the parameters that are consistent with
reported values form automobiles. The fit of the model based sales to actual
sales data was shown to be good. The model can be deemed to be a valid

representation of the dynamic relationships among price, quality and sales.

wfkm‘m" _-

 

69

 

 

Observed A Observed
Price Function Quality Function

   

 

Start at time o

> Advance time step dt

Compute values for state
variables (X and Y)
at time t using Euler
Integration

Compute values for remaining
variables at time t

._.. Repeat until terminal time T ls

 

 

 

 

 

reached

1

I Estimated sues Rate
Observed
Sales Rate .

 

 

  

MODEL PARAMETERS GRGZ

 

Figure 7: Procedure Used in Fitting the Model to Historical Data.

 

70

Table 4: Ford Mustang. Actual vs. Estimated Sales.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Year Observed Sales Estimated Sales % Difference
1968 345,194 312,189 -9.56
1969 275,391 217,190 -21.13
1970 165,414 175,686 6.21
1971 130,488 125,854 -3.55
1972 1 18,972 130,264 9.49
1973 193,129 231,883 21.62
1974 338,136 315,293 -6.76
1975 187,454 225,777 20.44
1976 183,369 175,831 -4.11
1977 170,315 215,524 26.54
1978 240,162 265,551 10.57
1979 365,357 314,312 -13.97
1980 232,507 292,980 26.01
1981 153,719 175,391 14.10
1982 127,371 110,428 -13.30
1983 124,225 102,919 -17. 15
1984 140,338 90,161 -35.75
1985 187,776 108,723 -42. 10

 

 

71

 

Units [11 1000]

OPN Vino 190
§§sss§sss§ss§
I-I-v-v-I-v-i-Pv-v-v-v-I-

Yu-

 

1981

1982

3;

 

1985

 

Figure 8: Ford Mustang. Actual vs. Estimated Sales.

 

7. R BLEM
7.1 Emblemmfinitian

In this chapter, we use the model developed in the previous chapters to
investigate the nature of optimal prices in the presence of continuous quality
improvements. It is assumed that the durable goods manufacturer wants to
maximize discounted profits over a specified planning horizon.

Our goal is to compute P1031, that is, the optimal price trajectory, that
maximizes total discounted profits at the specified target time [T] given the

formulation in 5-1:5-6.

rpm T) = j :[pm — C(t)] S(t) exp(-rt) dt
where C(t) represents the average cost function and r the rate of discount.

To evaluate total discounted profits in the above objective function, we
need to specify a functional form for cost as quality changes over time. There
is some controversy in the literature regarding the functional relationship
between quality and cost. Plunkett and Dale (1988) present empirical
analysis of cost data which leaves the specification of cost as a function of
quality inconclusive. The results of their study suggest that cost could
increase as a function of quality. Other studies in the literature argue that
cost decreases with increasing levels of quality (see, for example, Shank and

Govindarajan, 1993). In this paper, quality is viewed broadly to include its
72

73
various aspects (Garvin, 1988). In the ensuing optimization scenario

analyses, the composite measure of quality is varied from an initial value of
.25 to a maximum attainable value of 1, following a prespecified linear
trajectory. Accordingly, the cost function used decreases as quality increases
over a range and then reverses direction as quality approaches a value of 1.
The cost function was constructed by assuming a 40% reduction in cost as
quality increases from .25 to .9, and a 40% increase as quality approaches
the maximum value of l. The percentage change in cost of this magnitude is
alluded to in Shank and Govindarajan (1988) and was confirmed as a
‘reasonable value’ by Mr. Joseph Verga, the expert from the Ford Motor Co.
who reviewed the specification of the model. This nonlinear, asymmetric
specification is a plausible representation of the cost function which captures
the different perspectives on the behavior of cost in current literature. In this
research, the percentage changes shown in the quality-cost function are

applied to a base-cost of $2,000.

74

 

 

30

Percentage cha es

 

 

20 are applied to a Taco

‘\ cost of 82.000
1 0

 

 

 

 

/

 

-10 \

 

 

 

 

 

 

.20 \ ,

 

 

-30
awry LEVEL pm 041

 

 

 

Figure 9: Cost as a Function of Quality

75
7.2 W

The complexity of the model makes it difficult to solve the
optimization problem by the classical calculus of variations approach. Even
though the necessary conditions for optimality can be derived, there is no
guarantee that a closed form solution for the optimal price trajectory or even
a meaningful interpretation of the necessary conditions for optimality can be
obtained.

An alternative to the variational approach consists of specifying a finite
number of points in time at which the ordinates of the control functions are
allowed to change. The problem becomes one of obtaining a finite number
of values of the control functions at specific epochs, to maximize an
objective function. This is, in general, a non—linear optimization problem
(Kirk, 1970).

Even though we do not have a closed form expression for the objective
function, for each set of proposed controls we can numerically solve for the
behavior of the system through time, solving the differential equations that
constitute the model. Then the performance measure can be readily
computed.

To obtain the vector of optimal prices that maximizes the objective

function, a numerical optimization routine can be linked to the dynamic

76
model to search the space generated by all possible combinations of control

values at the specified times. Many numerical optimization procedures are
available. We chose the Generalized Reduced Gradient (GRG) method to
solve the nonlinear optimization problem. The reason for this choice is that
the generalized reduced gradient method has been used successfully to solve
optimal control problems (Abadie, 1972). Besides, a current version of the
algorithm, GRG2 (Lasdon et al., 1978) (Smith and Lasdon, 1991) is readily
available as part of the Microsoft EXCEL spreadsheet software.

The differential equations were numerically solved using Euler
integration with a time step equal to 0.1 years. This numerical solution was
carried out in an EXCEL worksheet. The values for price, were specified
for epochs six months apart; linear interpolation was used to join the
semiannual prices into a continuous control function. It is important to note
that the interpolation procedure allows prices to change continuously
throughout the planning horizon. The number of degrees of freedom of the
optimization problem (semiannual prices) was chosen according to the
limitations of the EXCEL Solver routine. The performance measure was the
discounted profit at various terminal times, using a discount rate of 7%.

The nonlinear optimization algorithm GRGZ iteratively varied the

price vector to maximize total discounted profit. In essence, the Euler

77
solution of the differential equations behaved as an implicit nonlinear

function connecting a vector of prices (indexed by time) to total discounted
profit at the terminal target time (a scalar). The nonlinear routine was used to
maximize this implicit function.

The interaction between GRGZ and the dynamic model whose
performance we seek to optimize is illustrated in Figure 10.

Linear interpolation was performed on the price vectors from GRGZ
which contained values for price at specified epochs to create a price
trajectory. Using the price trajectory thus derived, specified values for the
parameters and the quality function, the dynamic model was numerically
solved using Euler integration and total discounted profit at terminal time
TPFT(T) was computed. Total profit was fed back to GRG2, which
generated another vector of prices. The process was repeated until the GRGZ
convergence criteria were met. For details of the GRG2 algorithm, the

reader is referred to Lasdon et al. (1978).

78

  
  

 

 

  
 

Perform Linear _
Interpolation on Price Input Quality
Vector to Create
Continuous Function I Set Parameter

  

Function

    

 

   

 

 

 

   

 

   

Values

 

  
 
 

 

 

  

Start at time 0
Advance time step dt
Compute values for state
variables (X and Y)
at timet using Euler
Integration
Compute values for remaining
variables at time t

__ Repeat until terminal time T is
reached

 
       
   
   
   
 

 

   

 

 

  
   
 
 

 

PROFIT“)

PRICE“), i = 0, 0.5,],1.5,...,T

 
 

Figure 10: Interaction Between the Model and GRG2 Algorithm for
Optimal Control

79
7.3 W

Given a profit maximization objective, the research questions of
interest were:

0 Is the optimal price trajectory influenced by the average life of a
product (which is related to durability, an aspect of quality)? If yes,
what is the nature of this relationship?

0 How does the optimal price trajectory behave in the presence of
continuous quality improvements?

0 Does the optimal price trajectory depend on the persistence of
quality perceptions? If yes, what is the nature of this relationship?

0 Does the optimal price trajectory depend on the elasticity of
demand? and,

0 Does the qualitative nature of these relationships change when a
short planning horizon versus a long planning horizon is
considered?

The motivation for these research questions stems from the importance
of the pricing decisions to the competitiveness of the firm. We examine the
case of a durable goods producer who is increasing product quality over time
(quality disimprovements are not considered in our model) and must set

prices to maximize profits. However, it is to be noted that quality is a

80
control variable. The specification of a linear trajectory for quality over time

captures the spirit of continuous quality improvement without being overly
restrictive.

As a firm increases quality, it is not obvious what pricing strategy is
optimal (Narasimhan and Ghosh, 1994). For example, price increases in
view of quality improvements may be appropriate under certain conditions
(such as pursuing niche markets with premium products). Prices can also be
lowered, if the objective is to maximize value to the customer and to achieve
rapid market penetration. Similarly, it is not obvious whether prices should
be increased as the durability of a product is increased. It is also important
to understand the relationship between the ‘quality persistence’ effect and
optimal prices, because it is reasonable to believe that advertising and
promotion can be used to modify the durability of quality perceptions. The
research questions were intended to shed some light on these issues and to
generate insights. about the behavior of optimal prices.

Several optimization runs were carried out to address the research
questions of interest by varying different parameters in the model and
assessing their effect on the optimal price strategies. The parameters that
were varied in the optimization runs were D1, the average life of the product

D2, corresponding to the persistence effect of quality improvements (i.e. , the

81
average length of time that the quality of a unit continues to affect sales by

influencing customer perception of quality), and the demand elasticity of the
product. The parameter for demand elasticity is set at two levels, .7 and 1.3,
to represent price inelastic demand and price elastic demand. We
acknowledge that price elasticity of demand could vary over the life of a
product. However, we have chosen to use these parameter values in the
sense of averages corresponding to price inelastic and price elastic demand
cases. It is to be noted that the constant demand elasticity value of 1.3 was
used to validate the model. Also, to study the impact of a short-term versus
a long-term orientation on optimal price trajectories, the length of the
planning horizon was varied in the experimental runs. All possible
combinations of the following parameter levels were investigated:

Average life of product (D1): 3, 5, and 10 years

Planning horizon (T): .5*Dl, 2*Dl years

Quality Persistence (D2): .25*Dl, .5*Dl, .75*Dl years

Demand elasticity (e): .7 (inelastic), 1.3 (elastic)

In total, 36 optimal control problems were solved to derive the optimal
price trajectories. The parameter a was not altered in the experimental
runs. It was set at the value estimated in the fitting of the model to the Ford

Mustang data. a is a measure of the speed at which a product diffuses

82
through a population. There are no external sources of data that can provide

values for this parameter, so it has to be estimated within the model from
historical data. Varying a would not provide any managerial insight because
there are no measures that link this parameter with interpretable market
conditions. As indicated before, the cost function was modeled as a non-
linear function of the current quality level of the products. Other factors
that can affect the cost function such as economies or diseconomies of scales
or learning effects were not included in the formulation to keep the analysis
sharply focused on issues of interest in this research.

In our research, we treat the composite measure of quality as an index
between 0 and 1, 1 being the maximum attainable level of quality. In all the
runs, quality increases linearly from a "low" level (0.25) at time 0, to a
maximum level of 1 at terminal time T. Since T is varied in the numerical
computations, implicitly, the rapidity of quality improvements is also varied
in the various scenarios considered (that is, scenarios with lower values of T

correspond to more rapid quality improvements than those with higher values

ofT).

 

8. MIZ N T
In this section we present the result of the experiments described in the
previous chapter. The results (optimal price trajectories) are shown

graphically in Figures 11 and 12.

8.1 W

Comparing runs llF-A and llE-A where D2/D1= 0.75, it can be
seen that for the case T=20 (corresponding to a longer time horizon), price
increases initially and then decreases by the terminal time to 2,200 which
is lower than the initial price of 3,300. For the case T=5 (corresponding to a
shorter planning horizon), price declines from approximately 3,450 to 2,100
by the terminal time. A comparison of runs llF-B and llE-B, where
D2/D1=0.50, shows that qualitative behavior of the optimal price
trajectories for these two cases is similar to that of the previous two cases
for the planning horizons. However, even though initial and terminal
optimal prices are approximately the same as in runs llF-A and llE-A, the
transition prices are lower for these two cases. Comparison of runs llF-C
and llE—C, where D2/ D1 = 0.25 shows that prices decline throughout the

planning horizon, and the transition prices are lower than those in the

83

84
previous runs. The following observations can be made about the behavior of

optimal price trajectories:

0 regardless of the magnitude of the ratio D2/D1, optimal prices
generally decline over the planning horizon (i.e. , the optimal
terminal prices are consistently lower than the optimal initial
prices);

0 optimal pricing decisions depend on the planning horizon. A
longer term orientation leads to optimal price trajectories that call
for an increase in price followed by decreases. As time elapses and
quality improvements diffuse through the market, price is
decreased to accelerate sales and to achieve market penetration
through “value pricing.” This price behavior conforms to
observations made by researchers in marketing and in the area of
manufacturing strategy. Hayes and Wheelwright (1984) have
discussed the switch from "quality sensitivity" to "price sensitivity"
as a product moves through its life cycle. Hill (1989) refers to the
shift from quality as an order winner to an order qualifier over the
product life cycle;

0 optimal price trajectory depends on the D2/D1 ratio as well as the

length of planning horizon. Higher values for D2 which

85
correspond to longer duration over which quality diffusion effects

persist are associated with higher initial prices and higher transition
prices during the expansionary phase of the product lifecycle. This
optimal price behavior is observed consistently in runs 11F-
(A,B,C) and 11E-(A,B,C); and

0 when we examine shorter term horizon results in Figure 11E, rates
of price increases and decreases are higher compared to the results
in Figure 11F. Initial and terminal prices are about the same in

Figures 11E and 11F.

8.2 W

Comparison of runs llD-A, llC-A, llD-B, and llC-B,
corresponding to D2/D1 ratios of 0.75 and 0.50, and T values of 10 and 2.5
shows that optimal price trajectory declines throughout the planning horizon.
The initial optimal price is dependent on the D2/D1 ratio. The larger the
value of D2/D1 ratio, the higher the initial optimum prices (compare, for
example, runs llD-A and llD-B. Comparison of runs llD-C and 11C-C
corresponding to a D2/D1 ratio of 0.25 shows that the optimal price
trajectory declines from an optimum price initially followed by a slight

increase in the latter stages in run llC-C. The price increase observed in

86

run llC-C towards the end of the planning horizon is not seen in run llD-C

corresponding to the longer planning horizon (T= 10 years). The following

observations can be made in reviewing the behavior of the optimal price

trajectories:

optimal price behavior for D2/D1 = 0.75 and 0.50 is different
from the behavior corresponding to D1 =10 for the same D2/D1
ratios. Therefore it can be inferred that the average life of the
product (an aspect of quality that is influenced by product design,
manufacturing quality and materials used) does affect the trajectory
of optimal prices. The observed differences in optimal trajectories
also reflect the influence that average life of the product has on the
proportion of the market that is saturated at any given time;
although the optimal trajectories are similar for D2/D1 values of
0.75 and 0.50, the behavior is different for the case when D2/D1
= 0.25 . This further underscores the linkage between optimal
prices and product quality; and

optimal initial prices and price trajectories are related to the value
of D2. Higher values of D2 are consistently associated with higher

initial prices.

87
8.3 W

Runs 11A-(A,B,C) and llB-(A,B,C) correspond to this combination of
parameter settings. Runs llB-A, llA-A, llB-B, and llA-B, corresponding
to D2/D1 ratios 0.75 and 0.50 indicate that optimal prices decline from an
initial price throughout the time horizon. However, the optimal price
behavior is markedly different when D2/Dl = 0.25 (compare llA-C and
llB-C). For the shorter planning horizon, optimal price trajectory initially
declines, reverses course and increases for the remainder of the planning
horizon. For the longer planning horizon (1 lB-C), price declines initially and
exhibits slight fluctuating behavior as the terminal time is approached. The
dependence of optimal prices on the ratio D2/D1 and the value of D2 can be

seen in these runs also.

 

88

 

 

 

 

 

0 1 2 3 4 5

no 11‘ m "I
our-1.5 MI. 7‘
m one
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2500
m m
1m 1m
0 0.5 m 1 1.5 0 I 2 3 4 I
no "C no 110
01‘ m

 

 

 

 

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01-1. m
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8”
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0 01 1 1.5 2 21 3 8.5 4 6.5 I

 

 

m 11!
01-10 1120

0 2 4 C I ‘0

M

i.

1.

 

 

Figure 11: Optimal Controls Derived Under Non-Linear Quality

Costs and Elastic Demand (e = 1.3).
A: D2 = 0.75D1 B: D2 = 0.50D1 C: D2 = 0.25D1

 

89

 

no 12A no 12.
MI! 7.1.! 01-3 1"

‘

‘

   

 

PB 12: In 120
01-5 Til!» 01-! 7'10

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01-10 1’" 0101. rue

   

 

 

0 05 1 1.5 2 2.0 3 3.5 4 45 5 o 2 4 0 I 10 ‘2 M 10 1. 20

 

 

Figure 12: Optimal Controls Derived Under Non-Linear Quality
Costs and Inelastic Demand (e = 0.7).
A: D2 = 0.75D1 B: D2 = 0.50Dl C: D2 = 0.25D1

89

 

no 12A
0102 NJ

 

M12.
01-3 1“

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no 12c
01‘ THIS

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01-5 TI1I

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D1I1I M

 

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no 12'
01-10 TI2I

I 2 4 I I 10

 

 

Figure 12: Optimal Controls Derived Under Non-Linear Quality
Costs and Inelastic Demand (e = 0.7).
A: D2 = 0.75D1 B: D2 = 0.50D1 C: D2 = 0.25D1

 

90
8.4 ' in i mn° if f Dl=1

Runs 12F-(A,B,C) and 12E—(A,B,C) show the optimal price
trajectories for a relatively inelastic product for D2/D1 ratio values of 0.75 ,
0.50 and 0.25. The optimal price trajectories in runs 12F-A, 12E-A, l2F—B,
and l2E-B differ markedly from those corresponding to an elastic product.
Optimal price trajectory increases from an initial price, attains a maximum
and then decreases to an optimal terminal price that is less than the initial
optimum price. The magnitude of the price increases relative to the initial
prices are larger in these runs compared to similar runs for the higher
elasticity case. In Run 12E-C corresponding to D2/D1 = 0.25 for the
shorter planning horizon (T=5) , price decreases from the initial optimal
price over the planning horizon. However, in run 12F-C, that corresponds to
the same D2/D1 ratio for the longer time horizon (T=20) , optimal price
increases for more than half of the planning horizon, which is very different
from the corresponding elastic-demand case. The following observations can
be made about the relationship of optimal price trajectory to the parameters
in the model:

0 Optimal price policies are related to the demand elasticity of the

product, as shown by comparison of runs llF—A and 12F-A.

Relative inelasticity of demand, affords an opportunity to charge

91
higher prices immediately following quality improvements, thus

allowing faster recovery of investments made in quality
improvements. Although this conclusion is intuitive in a static
setting, the dynamic interrelationship among price, quality, sales
rate and durability makes this result less obvious. For instance,
without the model it could be argued that the elasticity of demand
has little effect on optimal pricing in a saturated market.

price increase is sharper when D2/D1 ratio is larger. For example,
when D2/D1 = 0.75, price increases from 4400 to 7500 in 9.5
years (see run 12F-A) as compared to an increase from 4400 to
6700 in 9 years when D2/D1 = 0.50 (see run l2F-B).

as observed previously for the case of high elasticity, there is
interaction effect between D1, D2 and the planning horizon.
Comparing runs 12F-A, 12E—A, 12F-B, and 12E-B, shows that
higher values for D2 (corresponding to scenarios where quality
diffusion effect persists longer) are accompanied by generally
higher prices regardless of the length of the planning horizon.
These results underscore, as in previous runs, the relationship

between quality and optimal price trajectories.

92
8.5 W

Runs 12D-(A,B,C) and 12C-(A,B,C) corresponding to these parameter
settings show a variety of optimal price behavior. When D2/ D1 = 0.75 and
the planning horizon is long (run 12D-A), price behavior is concave. Price
increases from 10800 to 11800 and decreases for the remainder of the
planning horizon. This differs from the optimal price trajectory under
similar conditions for D1 = 10, in that the initial price is considerably higher,
price increases relative to the initial price are lower and price increases are
not sustained as long as in D1 =10 case, suggesting a strong influence from
the average life of the product. Runs 12C-A and 12C-B for a shorter time
horizon, show that optimal price trajectory declines from an initial optimal
price. Runs 12D-C and 12C-C corresponding to D2/D1=0.25 show that
optimal prices should decrease at first and possibly increase during the
planning horizon. In 12D—C, price declines after year 7. The length of time
over which prices decline and the magnitude of the decline are related to the
length of the planning horizon. The following observations can be made
about the behavior of optimal price trajectories:

0 optimal prices depend on D1, D2 and the length of the planning

horizon. For higher values of D2/ D1 , higher prices are suggested;

and

93
0 the duration over which prices decline (or increase) and the

magnitude of the decrease (increase) is influenced by the planning

horizon as well as D2, the length of time over which quality

diffusion persists.
8.6 WWWLEQ

Runs 12B-(A,B,C) and 12A-(A,B,C) correspond to these parameter
settings. Runs corresponding to D2/D1 ratios of 0.75 and 0.50 indicate that
the optimal prices generally decline over the planning horizon. Starting
optimal prices are dependent on the value of D2 (16500 versus 13500 for
runs 12B-A and 12B-B corresponding to D2 values of 2.25 and 1.5
respectively). As discussed in previous sections, when D2/D1=0.25,
optimal price behavior is markedly different from that observed in runs 12B-
A, 12A—A, 12B-B, and 12A-B. When D2/D1 = 0.25, price initially

decreases and then increases for the remainder of the planning horizon.

9.1.1..-_0\|'-‘1L.i-;l' 0 -0
Chapter 8 has presented the results of solving the optimal control
model for a number of scenarios by varying the parameters in the model.
The results of analyzing the individual runs can be synthesized into the
following conclusions:

1) Optimal price trajectories are dependent on demand elasticity of the
product which coincided with our prior expectation. While it can be
reasoned that when demand elasticity is high, price increases will be
detrimental to sales and profits (and that there will be greater price flexibility
when demand elasticity is low), the exact nature of the optimal price
trajectories cannot be readily discerned. The numerical solutions not only
show the behavior of the different optimal trajectories for the two cases
(high versus low elasticity) but also the timing and magnitude of price
increases and decreases. For both cases of demand elasticity, prices decline
as well as increase depending on the values of the other parameters. Demand
elasticity interacts with the average life of the product and the persistence
effect of quality diffusion.

2) The ratio of average life of product (D1) to the persistence of
quality diffusion parameter (D2) seems to play an important role in the

behavior of optimal price trajectories. Regardless of the elasticity of

94

95
demand, there is an interaction effect (on optimal prices) between the length

of planning horizon and the ratio of D2 to D1.

To interpret this result managerially, D1 and D2 need to be viewed
more conceptually than as delay parameters in the model. D1 which
corresponds to the average life of the product can be construed to represent
the effect of product design, quality of materials used and the effectiveness of
manufacturing processes. Although these aspects were not explicitly
considered in the model, the level of aggregation represented by the
parameter D1 admits of such an interpretation. D2 which captures the
persistence effect of quality improvements is (can be) influenced by a firm's
marketing, sales and promotion efforts. That is, through its marketing
efforts (aimed at creating brand loyalty and image) a firm can influence the
"quality life" of a product. Intensive efforts in this regard extend the quality
life of a product (i.e. , increase D2 and prolong positive perception of the
product in the minds of customers).

When interpreted in this manner, this result suggests that when
engineered product quality is high (high D1) and the product is not heavily
promoted (low D2) so that D2/D1 is low, optimal price strategy is to price
the product aggressively and let the diffusion effect of quality take hold

before raising prices in the wake of quality improvements. This insight

96
regarding the interaction of D1, D2 and optimal pricing is interesting from a

theoretical as well as practical perspective;

3) The average life of a product affects the optimal price trajectories
in important ways. In general, higher values for D1 support higher initial
optimal prices, thus suggesting that a strong dependence between "quality"
and pricing decisions.

4) The persistence effect of quality has an influence on optimal
pricing trajectories. The higher the value of D2 in the runs, the higher the
prices (initial prices, price increases etc.). The value of this parameter can
be influenced 'by advertising and promotion. The model characterizes the
optimal pricing strategies to follow when such a tactic is pursued.

5) Conclusions (3) and (4) above indicate that there is a relationship
between pricing decision and product quality. In the marketing literature
(see, for example, Gerstner, 1985), correlational analysis of price-quality
data has been used to assert that the relationship between price and quality
may be weak. The optimal trajectories for price, derived in this paper, when
viewed in conjunction with the quality trajectory suggest that there is a
relationship between price and quality. One possible explanation for this
difference in results is that empirical data analyses (using cross-sectional

data) have attempted to correlate quality with actual prices, not necessarily

97
optimal prices. The results based on our model affirm the conclusions

reached by Rao and Monroe (1989) and Dodds, Monroe and Grewal (1991).
A dynamic model of the type analyzed in this paper is better at capturing the
intricacies of the dynamic, nonlinear relationships between quality and price,
and therefore is useful for generating insights.

6) The results also suggest that important interrelationships exist
between pricing, positioning and promotion decisions, and manufacturing
quality. Better understanding of this interrelationship will enable a, firm to
adopt appropriate pricing strategies to maximize its profit in the short or the
long term.

This research has investigated the optimal pricing decision in the
presence of continuous quality improvements for a profit maximizing firm.
A dynamic (optimal control) model incorporating demand elasticity, life of a
product and diffusion effect of quality improvements was developed and
validated. While the constructs embedded in the model were shown to have
validity, it is important to highlight certain assumptions in the formulation of
the model that may act as limitations in its applicability to certain products:

0 Markat Potential was ngt mmalfl aa a fimgtign Qf quality. For certain
products, quality may have an impact on the demand function and not just

on the speed of diffusion.

98

1‘ rug.“ 1...; 1... tron‘l'! a uni: o --. -LO-‘!0-

MW. For certain products, price elasticity may

change throughout their life-cycles.

WWW. Changes in market Potential may

arise from growing or declining market size, not just from changes in

price and quality.
hm. .. c. 9.11; i wr mlon'fn. -. foo'llwin {_L ‘ m ‘1...
mm. This choice was made because of the mathematical

tractability of the exponential distribution. Even though very few
products will have exponentially distributed life-spans, sensitivity
analysis has shown that the results of the model are robust to the choice of
the distribution.

Cgmmtitign was ngt mflslfi explisitly. The degree of competition was
modeled indirectly through the choice of price demand elasticity. For
example, in very competitive markets, market potential is very sensitive
to changes in prices; this situation can be modeled by assigning a high
value to the price elasticity of demand. This modeling, however, does not
capture the reaction of the competition to changes in the product’s price
and/ or quality, and the impact of this reaction on the product’s market

potential .

99
In spite of these limitations, we believe that the model proposed in this

dissertation is applicable to a wide range of products and market conditions ,
and can be easily fine-tuned to cope with specific situations.

The model was solved numerically using the generalized gradient
procedure to derive optimal price trajectories. The solutions show that
important linkages exist among optimal prices, average life of the product
and the persistence effect of quality perception. The results affirm some
ideas from existing literature and suggest that there is a relationship between
quality and price. The results also suggest that more careful consideration
of quality aspects in pricing and promotion decisions is warranted. Given the
large impact of the ‘quality persistence delay’ (D2) on optimal pricing, 3
natural extension of this research could be an investigation of the possibility
of controling D2 through managerial actions, possibly through advertisement.
Other possibilities for further research include the extension of the model to
incorporate quality in the market potential function, the explicit modeling of

competition and the investigation of other cost-quality relationships.

LIST OF REFERENCES

LIST OF REFERENCES

Abadie, J. , "Applications of the GRG Algorithm to Optimal Control
Problems" , in Nonlinear and Integer Programming, J. Abadie, Ed. ,
North-Holland Pub. Co., Amsterdam, (1972), 37-47

Abraham, B. and Ledolter. 1., W John
Wiley & Sons, Inc., 1983.

Banker, RD. and LS. Khosla, "Quality and Competition", Working Paper,
Dept. of Operations and Management Science, Carlson School of
Management, University of Minnesota, Minneapolis, Minnesota,
1992.

Bass, F.M., "A New Product Growth Model for Consumer Durables,"
Management Science, 15 (January 1969), 215-227.

Bergstrom, A.R. , Continuous Time Econometric Modeling. Oxford
University Press, 1990.

, editor, Statistical Inference in Continuous Time Economic
Models. North-Holland Publishing Company, 1976.

, and Wymer, C.R.,"A Model of Disequilibrium Neoclassical
Growth and its Application to the United Kingdom", in Statistical

Inference in Continuous Time Economic Models, ed. by AR.
Bergstrom. North-Holland Publishing Company, 1976.

,"Monetary Policy in a Model of the United Kingdom", in
Stability and Inflation, ed. by Bergstrom A.R., Catt A.J.L., Peston
M.H., and Silverstone B.D.J. John Wiley & Sons, 1978.

 

Besanko, D., and Winston, W.L., "Optimal Pricing of a Product with
Periodic Enhancements", European Journal of Operational Research,
55(2), (November 1991), 268-278.

100

101

Consumer Reports, “New Car Buying Guide 1991-1992 Edition” Consumers
Union, Yonkers, NY, 1991.

Cook, T., "Pricing for Maximum Profit", Telephone Engineer &
Management, 96(10), (May, 1992), 50-53.

Department of Labor, “Monthy Labor Review” , Bureau of Labor Statistics,
Washington, DC, 1993.

Dockner, E. and S. J orgensen, "Optimal Advertising Policies for Diffusion
Models of New Product Innovation in Monopolistic Situations,"
Management Science, 34 (January 1988), 119-130.

Dolan, R. J. and A. P. J euland, "Experience Curves and Dynamic Demand
Models: Implications for Optimal Pricing Strategies," Journal of
Marketing, 45 (1981), 52-73.

Durbin, J. , "Efficient Fitting of Linear Models for Continuous Stationary
Time Series from Discrete Data," Bulletin of the International
Statistical Institute, vol. 38, pp. 273-82, 1961.

Forrester, J.W., and Senge, P.M., "Tests for Building Confidence in System
Dynamics Models." System Dynamics. Edited by A.A. Legasto, Jr.,
J .W. Forrester and J.M. Lyneis. TIMS Studies in the Management
Sciences 14 (1980) 209-228.

Forrester, Jay W., Principles of Systems, Wright-Allen Press, Cambridge,
MA (1968).

Fortuna, R. , "The Quality Imperative", in Total Quality: an Executive Guide
For The 19908, Richard D. Irwin, Inc., Homewood, IL (1990).

Gerstner, E., "Do Higher Prices Signal Higher Quality", Journal of
Marketing Research, XXII (May 1985), 209-215.

Gilmour, P. , Development of a Dynamic Simulation Model for Planning
Physical Distribution Systems: Validation. Ph.D. dissertation.
Department of Management, Michigan State University, East Lansing,
MI, 1971.

102

Graham, A.K. , " Parameter Estimation in System Dynamics Modeling."
System Dynamics. Edited by A.A. Legasto, Jr., J.W. Forrester and
J .M. Lyneis. TIMS Studies in the Management Sciences 14 (1980)
125-142.

Hayes, R. , and Wheelwright, S. , Restoring Our Competitive Edge:
Competing Through Manufacturing, John Wiley & Sons, (1984).

Hill, T., Manufacturing Strategy: Text and Cases, Irwin, (1989).

Horsky, D. and LS Simon, "Advertising and the Diffusion of New
Products," Marketing Science, 2 (Winter 1983), 1-18.

Jonson, P.D., Moses E.R.,and Wymer C.R., "The RBA76 Model of the
Australian Economy", in Conference in Applied Economic Research,
Reserve Bank of Australia, 1977.

Judge, G.G. et al. , Introduction to the Theory and Practice of Econometrics.
Second Edition. John Wiley & Sons, Inc. 1988.

Kalish, S., “A New Product Adoption Model with Price, Advertising, and
Uncertainty," Management Science, 31 (December 1985), 1569-1585.

Kalish, S. , "Monopolist Pricing with Dynamic Demand and Production
Cost," Marketing Science, 2 (Spring 1983), 135-160.

Khalil, H., Nonlinear Systems, Macmillan Publishing Company, 1992.

Kirk, D.E., Optimal Control Theory, Prentice-Hall, Inc. 1970.

Knight, M.D., and Wymer C.R., "A Macroeconomic Model of the United
Kingdom," International Monetary Fund Staff Papers, Vol. 25,
pp.742-78, 1978.

Kohli, R, and Mahajan, V., "A Reservation-Price Model for Optimal Pricing
of Multiattribute Products in Conjoint Analysis", Journal of Marketing
Research, 28(3), (August 1991), 347-354.

Lasdon, L.S., Waren, A.D., Jain, A., and Ratner, M., "Design and Testing
of a Generalized Reduced Gradient Code for Nonlinear

103

Programming", ACM Transactions on Mathematical Software, 4(1),
(March 1978), 34-50.

Lee H.L. and CS. Tapiero, "Quality Control and the Sales Process," Naval
Research Logistics Quarterly, 33, 1986, 569-587.

Lilien, G.L., Kotler, P. and Moorthy K.S., Marketing Models, Prentice
Hall, Englewood Cliffs, New Jersey, (1992).

Little, J.D.C., "Aggregate Advertising Models: The State of the Art,"
Operation Research, 27 (July-August 1979), 629-667.

Mahajan, V., Muller, E., and Bass, F.M., "New Product Diffusion Models
in Marketing: A Review and Directions for Research", Journal of
Marketing, 54, (January 1990), 1-26.

Manetsch, T.J . , “Transfer Function Representation of the Aggregate
Behavior of a Class of Economic Processes.” IEEE Transactions on
Automatic Control., Vol. ag-ll, No. 4, (1966).

Manetsch, T]. and G.L. Park, Systems Analysis and Simulation with
Applications to Economic and Social Systems, Department of

Electrical Engineering and System Science, Michigan State University,
East Lansing, MI, 1982.

, Planning to Meet Economic and Nutritional Goals in Developing
Countries. An Application of Multi-Criterion Optimization. Working
Paper. Department of Electrical Engineering and System Science.
East Lansing, MI, 1985.

McClure, J.E., and Spector, L.C., "Joint Product Signals of Quality",
Atlantic Economic Journal, (19)4, (December 1991), 38—41.

Mesak, H.I. , "Impact of Anticipated Competitive Entry and Cost Experience
on Optimal Strategic Pricing of Technological Innovations",
Computers and Operations Research, 17(1), (1990), 27-37.

Monroe, KB. and W.B. Dodds, "A Research Program for Establishing the
Validity of the Price-Quality Relationship", Journal of the Academy of
Marketing Science, 16 (Spring 1988), 151-168.

104

Narasimhan, R. and Mendez, D. , "An Investigation of the Effect of Product
Durability on Monopolistic Pricing." 1992 Midwest Decision Sciences
Institute Conference Proceedings, Kansas City, MO, 1992.

Narasimhan, R. and S. Ghosh, "A Dynamic Model of Manufacturing
Quality's Effect on Optimal Advertising and Pricing Policies,"
European Journal of Operational Research, 72, 1994, 485-502.

Norton, J .A. and RM. Bass, "A Diffusion Theory Model of Adoption and
Substitution for Successive Generations of High-Technology
Products," Management Science, 33 (September 1987), 1069-1086.

Ozga, S.A. , "Imperfect Markets Through Lack of Knowledge," Quarterly J.
Economics, 74 (February 1960), 29-52.

Peterson, D.W. , "Statistical Tools for System Dynamics. ", Elements of the
System Dynamics Method. Edited by J. Randers. MIT Press,
Cambridge, Mass., 1976.

, " Parameter Estimation for System Dynamics Models."
Proceedings of the Summer Computer Simulation Conference.
Washington, DC. pp. 584-586, 1976.

Phillips, A.W. , "The Estimation of Parameters in Systems of Stochastic
Differential Equations", Biometrica, vol. 46, pp. 67-76, 1959.

Plunkett, J .J . and Dale, B.G., "Quality Costs: A Critique of Some Economic
Cost of Quality Models", International Journal of Production
Research, 26(11), (1988), 1713-1726.

Pindyck, R. and D. Rubinfeld, Microeconomics, Third edition, Prentice
Hall, Englewood Cliffs, NJ (1995).

Richard, D.M. , "A Dynamic Model of the World Copper Industry,"
International Monetary Fund Staff Papers, Vol. 25, pp. 779-833,
1978.

Robinson, B. and C. Lakhani, "Dynamic Price Models for New Product
Planning," Management Science, 21 (June 1975), 1113-1122.

105

Sethi, S.P., "Dynamic Optimal Control Models in Advertising: A Survey,"
SIAM Review, 19 (October 1977), 685-725.

Simon, H., Price Management, New York: North Holland (1989).

Smith, S., and Lasdon, L., "Solving Large Sparse Nonlinear Programs Using
GRG," ORSA Journal on Computing, 4(1), (1991), 1-15.

Tapiero, C.S., P.H. Ritchken, and A. Reisman, "Reliability, Pricing and
Quality Control," European Journal of Operational Research, 31,
1987, 37-45.

Teng, J .T. and G.L. Thompson, "Oligopoly Models for Optimal Advertising
When Production Costs Obey a Learning Curve," Management
Science, 29 (September 1983), 1087-1101.

Thompson, G.L. and J .T. Teng, "Optimal Pricing and Advertising Policies
for New Product Oligopoly Models," Marketing Science, 3 (Spring
1984), 148-168.

Van Horn, R., "Validation." The Design of Computer Simulation
Experiments. Edited by T. H. Naylor. Durham, NC: Duke
University Press, 1969.

Varian, H.L., Microeconomic Analysis. 3rd edition. W. W. Norton &
Company, Inc., 1992.

Wymer, C.R. , "A Continuous Disequilibrium Adjustment Model of the
United Kingdom Financial Market", in Econometric Studies of Macro
and Monetary Relations, ed. by A.A. Powell and RA. Williams.
North-Holland, Amsterdam, 1973.

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