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Transcript
INFORMAnON Ta USERS
This manuscript has been reproduced fram the microfilm master. UMI films
the text directly from the original or copy submitted. Thus. some thesis and
dissertation copies are in typewriter face. while others may be tram any type of
computer printer.
The quallty of this reproduction is dependent upon the quality of the
copy submittecl. Broken or indistinct prin!. colored or poor quality illustrations
and photographs. print bleedthrough, substandard margins, and improper
alignment can adversely affed reproduction.
ln the unlikely event that the author did not send UMI a complete manuscript
and there are missing pages. these will be noted. Also. if unauthorized
copyright material had to be removed, a note will indicate the deletion.
Oversize materials (e.g.. maps, drawings, charts) are reproduced by
sectioning the original. beginning at the upper left-hand corner and continuing
frDm left to right in equal sections with small overtaps.
Photographs included in the original manuscript have been reproduced
xerographically in this copy. Higher qualïty 6" x 9- black and white
photographie prints are available for any photographs or illustrations appearing
in this copy for an additional charge. Contad UMI directly ta arder.
PIoQuest Information and leaming300 North Zeeb Raad, Ann Arbor, MI 48106-1346 USA
800-521-0600
UMI~
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Technological Progress and Technology Acquisition:Models With and Without Rivalry
Atiqur Rahman
Faculty of ManagementMcGill University
1001 Sherbrooke Street WestMontreal, Canada H3A IG5
November, 1999
A thesis submitted to the Faculty of Graduate Studies and Re5e8l'ch in partialfulfillment of the requirement for the degree of Ph. D. in Business
The author bas granted a non·exclusive licence allowing theNational Library ofCanada toreproduce, loan, distribute or sellcopies of this thesis in microfonn,paper or electronic formats.
The author retains ownership of thecop}Tight in this thesis. Neither thethesis nor substantial extracts from itMay be printed or othenvisereproduced without the author'spermission.
L'auteur a accordé une licence nonexclusive permettant à laBibliothèque nationale du Canada dereproduire, prêter, disttibuer ouvendre des copies de cette thèse sousla fonne de microfiche/film, dereproduction sur papier ou sur formatélectronique.
L'auteur conserve la propriété dudroit d'auteur qui protège cette thèse.Ni la thèse ni des extraits substantielsde celle~i ne doivent être imprimésouautrememreprodwtss~son
4.2 Pre-commitment Equilibrium with Uncertain Expectations
4.3 Subgame Perfect Equilibrium
4.4 Concluding Remarks .....
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95
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110
5 Technology Acquisition with Teclmological Progress: A Stochastic
Programming Approach 112
5.1 A Multi-Stage Stochastic Programming l\tlodel
5.2 Solution Procedure .
5.3 Experimental Results and Some Remarks ..
5.4 Conduding Remarks . . . . . . . . . . . .
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121
133
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6 Conclusion and Future Research Directions 144
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Abstract
In a technology driven world, technology acquisition decisions as to whenand which new technologies to acquire are becoming inereasingly eritical forfirms to survive and grow. The issue of technology acquisition is addressedwith three different focuses in the current dissertation.
In the first essay, we extend the results of some existing literature. Existing literature suggests that, in an oligopoly, identieal firms acquire the sametechnology at two different dates onder Nash or pre-commitment equilibrium,which assumes infinite information lag between two firms. The set of equilibrium dates tum out to be diflerent under subgame perfect or pre-emptionequilibrium that assumes zero information lag. We show that allowanee forasymmetry between firms leads to the same equilibrium dates under Nashand subgame perfect equilibrium.
In the second essay, a two-period technology game is considered to studythe effect of expectations regarding technological progress on a firm 's technology adoption decision in a duopoly. It is shawn that expectations ofbetter future technology retard adoption of the currently available technology. Uncertain future progreiS is shown to have either no effect or negativeeffect on the adoption of the currently available tecbnology when a Nashor open-loop equilibrium holds. However, under subgame perfection, uneertainty may actually encourage adoption of the current technology, contraryto what literature suggests.
In the third essay, a stochastic mathematieal programming framework isuse<! ta build a decision mode! ta solve for technology decisions facing rapidand uncertain technological progresse In our scenari~based approach, weallow uncertainties in both technological developments as well as in outputproduct market demands. Furthermore, the acquisition costs of the technol~
gies are assumed to be concave ta reftect economies of scale in acquisition. Anefficient procedure to solve the problem is propœed and implemented. Ournumerical results show that the expecatioD of future technologies impacts theacquisition of the current technology in a negative way, and highlights theimportance of incorporting expectations in a technology acquisition mode!.
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Sommaire
Dans un monde mené par le progrès technologique, les décisions d'aquisitionde technologies, notamment le choix du moment d'acquisition et du type denouvelles technologies à acquérir, sont d'une importance cruciale pour lacroissance, voir même la survie des entreprises. La présente dissertation considère la question d'acquisition de nouvelles technologies en mettant l'accentsur trois perspectives différentes.
Dans un premier essai, nous avons reconsidéré quelques résultats relcensés dans la littérature. Certains auteurs ont suggéré qu'en situationd'oligopole, deux firmes identiques acquièrent la même technologie à deuxdates différentes sous les équilibres Nash ou de pré-engagement; ce qui suI>"pœe un écart d'information infini entre les deux firmes. L'ensemble des datesd'équilibre est différent sous un équilibre de sous-jeux parfait ou préemptifqui suppose un écart nul d'information. Nous montrons que l'introductiond'assymétrie entre les deux firmes conduit aux mêmes dates d'équilibre sousles équilibres Nash et de sous-jeux parfait.
Dans le deuxième essai, nous considèrons un jeu technologique à 2 périodesen vue d'étudier l'effet des attentes relatives au progrès technologique sur ladécision d'acquisition de technologies d'une firme en situation de duopole.li est démontré que les attentes d'une meilleure technologie future retardentl'adoption de technologies disponibles immédiatement. L'incertitude relativeau progrès futur semble avoir un effet nul ou négatif sur la décision d'adoptiond'une technologie actuellement disponible sous les équilibres Nash ou enboucle-ouverte. Cependant, lorsque nous passons à un environnement desous-jeux parfait, le facteur incertitude semble plutôt encourager l'adoptionde la technologie existante, contrairement à ce qui est suggéré dans la littérature.
Le troisième essai utilise un cadre de programmation mathématique stochastique et propose un modèle pour la résolution des décisions technologiques ensituation de progrès technologique rapide et uncertain. Nous utilisons une al>"proche basée sur des scénarios et qui suppose de l'incertitude aussi bien dansles développements technologiques que dans la demande du marché pour leproduit final. De plus, nous supposons que la courbe des coûts d'acquisitiondes technologies est concave afin de refléter les économies d'échelle reliéesà l'acquisition. Nous proposons et mettons en application une procédureefficace pour résoudre ce problème de décision. Les résultats numériquesmontrent que les attentes de technologies futures ont un effet négatif sur
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l'acquisition de technologies présentes, d'où l'importance d'incorporer les attentes dans un modèle d'acquisition de technologies.
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Aclmowledgements
First and foremost, 1 would like to thank my thesis supervisor ProfessorRichard Loulou for bis contribution. Starting with the selection of the topieto the final ciraft, bis continuous and careful guidance has vastly improvedthe thesis. The thesis would not have been what it is without bis generoushelp.
1 would aIso like to express my sincere thanks to Professor Robert Cairnsof the department of Economies and a member of my thesis committee. Hetook enormous care in reading my write-ups and always suggested ways taimprove. 1 am specially thankful for bis help on the second essay (Chapter4 of the thesis).
Professor Shanling Li, another member of my thesis committee, has always been inspirational and continuously pushed me towards bringing thethesis to an end. Her help, in particular regarding the third essay (Chapter5), deserves special mention.
1 would also like to thank Professar Georges Zaccour of HEC, a memberof my thesis committe, for his careful reading and suggestions.
Three anonymous referees from the European Journal of Operations Research suggested ways ta improve the readability of the second essay. 1 takethis opportunity to thank them.
This thesis would not have seen the light of the day without the inspiration and support provided by my wife and friend Nayeema. 1 am grateful toher. Finally, this thesis is dedicated ta my parents, who set me up on a longbut delightful journey that started in Joradah, a tiny village in Bangladesh.
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CHAPTER 1
Introduction
Technology has playOO a great role in the development of human society. New
innovations have 100 to accelerated economic growth and greater heights of the living
standard. Recent trends in the globalization of the market place and the high rate
of growth in the United States and other economies are due, in significant part, ta
technological developments. Even in the early years of the 20th century, technical
progress accounted for about seven-eighths of the growth in the United States GDP,
while only one--eighth was due ta capital input (Solow [1957]). It is therefore not
surprising that scholars in many different disciplines have demonstrated their interest
in understanding what leads to innovation and the uses thereof. Economists and
management scientists in particular have shown a keen interest in the analysis of
innovation and adoption of new technologies.
Although a new stream of research by industrial organization economists views
technological development as endogenous l , traditionally, technological progress bas
been treated as exogenous by growth theory economists in neo-classical economics.
A significant part of technology research in economics deals with the diffusion of
innovations. It attempts, through use of empirical as weil as theoretical models,
l This view, which has 100 ta a substantial amount of research in research and development,holds that technological development is a resu1t of profit-driven investment in R&D (Grossman and
Helpman [19901) .
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to identify the factors that lead to a graduai process of adoption of a (exogenously
developed) technology by its potentiaI adopters. While a few papers deaI with the
characteristics of the supplying industry, in most part, this stream treats technology
as given and studies diffusion in terms of the buyer industry characteristics, such as,
firm size, market structure etc.
Management scientists, on the other hand, have focused primarily on manageriaI
decision making with respect to the acquisition of new technologies. Traditionally,
major investment decisions have been treated in management science as capacity
expansion and/or equipment replacement problem without explicit consideration of
technology choice. Increasing use of Flexible Manufacturing Systems (FMS) to replace
dedicated equipment 100 management scientists to incorporate technology choice in
capacity expansion and equipment replacement decisioDS. The recent literature in
management science on technology has studied issues sucb as: which technology to
buy, when to buy it, how much of it to buy etc.
As far as methodology is concerned, management scientists, until recently, have
almost exclusively used traditionaI optimization models and Markov process models
to study technology choice problems. One of the shortcomings of these single-firm
decision models is that the strategic interplay between rivais is not taken into con
sideration. These models fail to study wbat kind of effect rivaIry may have on the
decisions of a firm to acquire a new technology. Empirical findings suggest that in
industries where competition is intense, firms switch to new technologies faster than
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other industries (Tombak [1988}, Romeo [1977]). Despite the shortcomings, opti
mization models are popular among management scientists because of their ability to
capture numerous details of the environment that other modeling methods can not
capture.
Game theory has been gaining currency as a modeling tool among both economists
and management scientists in the last couple of decades. It has provided economists
with a much needed tool to study oligopolistic markets. For management scientists,
it provides a framework to study a firm's decision from a managerial decision making
perspective while incorporating the effect of rivalry between firms on the decisioDS.
However, in contrast to the optimization technique, the current state of game theo
retic modeling does not permit the inclusion of important operational details sucb as
capacity, economies of scale, etc. in the model.
In this dissertation we address some of the issues related to the technology adop
tion decision of a firm, which have not yet been adequate1y addressed and are of
interest to management scientists as weil as economists. More specifical1y, we study
the effect of rapid technological progress on a firm 's technology adoption decision and
some of its economic consequences. To this end, we use both optimization technique
and game theory. We employ game theory to analyze the technology decisions of
firms in an oligopoly and the economic impacts of such decisioDS. The optimization
technique is used to build a decision mode! for a firm in an uncertain and rapidly
changing technological environment.
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1.1 Motivation, Research Agenda and Organization of the Thesis
There are a hast of factors that play a raIe in a technology acquisition decision.
Sorne sucb factors are: the nature of the technology (cast reducing vs FMS), capital
requirement, uncertainty (in the output market and/or in technological development),
experience or learning e1fect, market structure, firm size etc. Some of the factors have
been addressed in some detail in the literature, while others need more examination.
The significance of the factors ta be studied is often dictated by the events taking place
in the real world. For example, the advent of FMS led ta a new stream of research
directed at examïning the advantages of FMS (sucb as the ability to produce more
than one product and to switch between them) over the more traditional dedicated
technologies. The choice between different technologies thus became one major issue
for management scientists.
Flexibility aIso enabled the producers to offer new products to the consumers
more often leading to a change in the consumers' taste, who now expect new and
improved varieties to be introduced in rapid successions. Life-cycles of products are
becoming increasingly short. Today, the speed to market a new product is considered
the most important strategic weapon for a manufacturing firme Therefore, although
a particular flexible technology may have the ability to produce a group of products
and may have enabled a manufacturer to add newer varieties in the first place, intense
competition in the market requiring the producer to introduce even newer varieties
(that the technology in question may not be able to produce) may force the man-
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ufacturer ta buy new and improved technologies. As a consequence, producers now
have ta replace their equipment more often than ever before ta keep up with the stiff
competition in the product market. This demand for new technologies~ coupled with
competition in the equipment supplying industry, has 100 to a situation wbere the
life-cyc1es of technologies (used to make the end products) are becoming short.
It bas therefore become a common phenomenon for new technologies to be in
troduced in rapid succession. While a decade or two ago, a manager could be quite
confident about using an equipment for as long as it would last, now he has ta take
into account the timeframe beyond which the equipment could become obsolete. This
is specially true for high-tech industries like computers and telecommunications wbere
new technologies become available in matters of months. The evolution of Intel mi
croprocessor chips for personal computers illustrates this point. While the lifetime for
the first chip, 80286, was about 10 years~ it bas been less and less for subsequent gen
erations; about 9 years for 80386, 7 years for 80486 and 5 years for the first Pentium
processor. Since the first Pentium processor, Intel has introduced improved versions of
Pentium, namely, Pentium ~IMX, Pentium Pro, Pentium II, and Pentium III, almost
8t the rate of one every year. A firm operating in such an industry must keep in mind
the issue of obsolescence while making 8 major investment decision in technology. A
plant manager of an IBl'tI plant, for instance, commented in 8 recent conversation
with us that they consider 8 timeframe of at mast two years while making investment
decisions in technologies.
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We believe that investments in technologies now must be viewed in this light.
That is, managers must look at possible future developments while making an invest
ment decision and have to weigh the benefits of acquiring a technology available now
vis-a-vis waiting for a while until the next and improved technology appears. Mode1s
dealing with technology choice 50 far have largely ignored successive improvements, or
the expectations thereof (with a few exceptions, such 88, Balcer and Lippman [1984j,
Gaimon [1989], Nair and Hopp [1992] and Rajgopalan et. al. [1997]). Traditional
methods of evaluating investment alternatives, sucb as Net Present VaIue (NPV) or
Internai Rate of Return (IRR), have been subject ta criticism as they can not capture
sorne of the subtle benefits that a new technology (e.g., FMS) may provide. However,
like NPV and IRR, most other models aIso fail to capture the henefits of waiting.
The decision to wait or not wait becomes even more interesting in the context
of an oligopoly. It may appear to be beneficial for a firm ta wait for some more
time before committing ta a new technology when considered in isolation (that is,
assuming that the firm is a price taker with a given demand). However, when the
strategie interplay between the rivais are taken into account, the same firm may no
longer be able to hold back its investment while the rivais go ahead and invest (as
the priee and the demand is now determined by the market).
There is a scarcity of research dea1ing with tecbnology adoption in an oligopoly.
With the exception of Gaimon [1989], most of the papers that have addressed the
issue, study the adoption of a new technology available in the market, asswning that
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the firms are identical and that no new technology will appear in the near future
(Reinganum [1981], Fudenberg and Tirole [1985], Kim et. al. [1994]). These models
identify the equilibrium adoption timings for a new technology. According to the
findings of these papers, the equilibrium timing is a function of the nature of the
technology, and the nature of the competition (pre-commitment versus pre-emption).
We address the issue of technological development in this dissertation. FUst, we
introduce asymmetry into the above mentioned models of timing, and investigate the
effect of asymmetry on the equilibrium dates. We show that asymmetry resolves some
of the problems associated with the equilibrium adoption dates proposed by these
papers, such as, no adoption of flexible technologies when pre-emption is allowed
(Kim et. al. [1994]). We then present a game-theoretic model to study the impact of
technologïcal progress on the investment decisions of two firms operating in a duopoly.
Finally, we build an optimization based decision model for a firm to decide when and
how much of successive generations of technologies ta buy given uncertainties in bath
technological developments and output product market demands. However, because
of the limitations of available techniques, the decision mode! ignores the presence of
rivalry.
The organization of the dissertation is as follows: we start with a comprehensive
review of the existing literature pertaining to the adoption of technologies in Chapter
2. We present a brief discussion on research that focuses primarily on the diffusion
process, and attempts to explain why diffusion, as opposed to simultaneous adoption,
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OCCUlS, and what factors determine the speed of diffusion. We also discuss papers that
caver single-finn decision making analyzed from a firm's perspective; the decisions
include when a firm should switch to a new technology, what kind of technology a
finn shouId acquire, etc. Finally, we review the papers that deal with technology
decisions in an oligopoly. Since some industry (as opposed to firm) characteristics
are taken into account, these models present a different perspective on diffusion and
other related issues, even though the primary focus is on managerial decision making.
ln Chapter 3, we present an extension to the game-theoretic models discussed in
Chapter 2. We introduce (cost and discount rate) asymmetries into the game-theoretic
models and show that the two diflerent equilibrium concepts, pre-commitment and
pre-emption (aIso referred to as closed- and open-Ioop equilibrium), produce the same
equilibrium adoption dates. We belleve this resuIt is important in so far as it resolves
the problem of confiicting dates arrived at in earlier works by the two different con
cepts. Our resuIt resembles the findings of other more static models of asymmetric
rivaIry.
We take up the issue of technological progress in chapters 4 and 5. In Chap
ter 4, using a tw~period tw~stage duopoly game, where firms have the option to
acquire or not a period-specific new (and impraved) technology, we investigate the
circwnstances under which different eql.lilibria will hold. We also examine the effect
of expectations on the adoption of the currently available technologies, and the effect
on social welfare. Furthermore, we add uncertainty (in expectations) to assess its
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impact on the equilibrium adoption decisions. Finally, we the study subgame perfect
equilibrium for such games in order ta investigate the effect of information structure
in terms of when the information regarding a firm 's decision is revealed to the rival.
We address the same issue of technological progress from a different perspective
in Chapter 5. We build a technology acquisition decision model for a finn facing
rapid technological changes. A stochastic programming &amework is used to deal
with uncertainties associated with the technology and with product demands and
prices. The acquisition cast is considered to he concave in the amount of technology
bought to reflect economies of seale. This makes our model a large-scale non-lïnear
non-convex mathematical programming mode!. We develop a solution procedure for
the mode! and a computer program to implement the procedure. Numerical examples
are used ta generate additional insights.
Chapter 6 concludes the dissertation.
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CHAPTER 2
Literature Review
The üterature on the adoption of new technologies can he classified into three
broad categories, as we have already pointed out in the introduction. They are, (i)
research on diffusion of new technologies, (ii) sing1e-firm managerial decision making
models (without considering rivalry between firms), and (iii) game-theoretic models.
We review these streams separately in the following.
2.1 Diffusion of New Technologies
The study of technology diffusion started in the sixties with Mansfield [1968]
and until recently, mast of the works done in the area were empirical in nature. The
empirical works have demonstrated that not all (potential) users adopt the technology
at the same time; rather a diffusion takes place. Furthermore, the diffusion path has
heen shown repeatedly to be sigmoid (S-shaped) (see Nasbeth and Ray [19741 and
Mansfield [1968] for example) , taking any of a number of mathematical fonns (logistic,
for example). Mansfield [1968] concludes that (i) the diffusion of a major technology
is fairly slow, (ii) during the process of diffusion, a "bandwagon effect" takes place
when, with growing number of users, the uncertainty associated with the technology
goes down and at the same time competitive pressure mounts, and (iii) the rate of
diffusion is directly related to the expected profitability of the innovation and inversely
related to the level of investment required for the adoption.
• The basic hypothesis of Mansfield's mode! is that the number of 'hold-outs' (firms
that have not yet adopted the technology) at time t, that adopt the technology by
t + 1, is a function of (i) the proportion of firms that have already adopted by time t,
(ii) the profitability of adoption, (iii) the level of investment required, and (iv) other
unspecified variables.
Let n he the number of firms in the industry, m (t) be the number of firms who
have adopted by t, 1r be profitability of the adoption and S be the investment required
as a percentage of total assets of the firm. Theo, the proportion of 'hold-outs' at t,
À (t) is given by
À (t) = m(t+l)-m(t)n-m(t)
Thus the hypothesis can be expressed as
À (t) = f (m~t) ,?r, S, ....)
(2.1)
(2.2)
Assuming m (t) to be continuous and using Taylor's series expansion (and drOI'
ping the third-and-higher-order terms) ta approximate À (t), Mansfield shows that
(2.3)
where l is a constant of integration and 4> is giveo by
(2.4)
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This is the logistic curve introduced by Mansfield to explain the S-shaped growth
curve. Mansfield further assumes that the unspecified terms in (2.4) are UDcorrelated
with 1t' and S and can be treated as a random error term. This way the rate of
diffusion is govemed by ooly one parameter 4>, which is a linear function of 7T and S.
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• According to bis hypothEiSis, a2 should tum out to be positive to reflect a positive
impact of profitability while a3 should turn out ta be negative reftecting a negative
impact of the level of invEiStment on the diffusion rate.
Mansfield tests bis model in two stages: in the first stage ifJij (i for industry and
j for innovation) is estimated by
(2.5)
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ThEiSe EiStimatEiS are then iDSerted into equation (2.4) and an ordinary-least-
squares regression is run to see how the equation fits the data. Mansfield finets that
equation (2.3) represents the data quite weIl, although he points out that a logistic
function is not the only one that might represent the data fairly adequately. He also
points out that it is Dot surprising that it fits reasonably well, as the plot of the data
is S-shaped anyway.
Romeo [1977} applies the same procedure ta study the diffusion of numerically
controlled machine-tools; in addition, he also studies inter-industry differencEiS. He
finds that two industry characteristics (the number of firms in the industry and the
variance of the logarithm of 6.rm size) play significant raies in explaining differences
in the rates of diffusion in different industries. These two variables, in effect, measure
concentration (a measure of competitiveness) in an industry. He finds that the number
of firms has a positive coefficient while the variance has a negative coefficient. The
first finding suggests that competitive pressure does facilitate diffusion.
Although Mansfield'8 study and approach opened the era of diffusion research,
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it has also been subjected ta serious criticisms. Some of the criticisms are based on
purely technical econometric grounds, such as, whether he should have used OLS in
the second stage. However, in bis development of the theory, he also assumes that 1r
and S were constant over time; this does not hold in reality. There is little argument
that the level of investment required goes down over time whereas the profitability
may go either go up or down, but seldom does it remain constant.
Somewhat along the saIne line AA Mansfield, David [1969] and Davies [19791 use
probit models to explain diffusion. Unlike Mansfield, who simply treats the number
adopters in a period as a function of the cumulative number of adopters, David and
Davies explicitly incorporate asymmetries among firms in order to explain diffusion.
According to their view, ''whenever or wherever some stimulus variate takes aD a value
exceeding a criticalleve~ the subject of stimulation responds by instantly determining
to adopt" (David [1969]). That Î5, the difference in adoption dates are due to the levels
of a stimulus variate and its criticallevel at different times. David uses firm size as the
stimulus variate (which remains constant for a firm over time in his madel), while the
critical level for it (at which point it becomes profitable for a firm to adopt) changes
aver time. For exampIe, it can be argued that as the acquisition cast goes down aver
time, it becomes profitable for smaller firms ta adopt. Davies uses a similar, but more
complex (and more realistic as well) framework using the payback periad (defined as
a function of firm size and other firm characteristics) as the stimulus vanate. Unlike
David, he lets bath the variate and the critical value of it change aver time.
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Davies concludes that the rate of diffusion is negatively related to the payback
period and positively related to the labor intensity and the rate of growth in the
industry. He aiso concludes that it is negatively related to both the number of firms
in the industry and the variance of the logarithm of firm size. Thus concentration
could have effect in either direction. Davies' results contradict Romeo's [1977) results
mentioned above, that the number of firms has positive effects on the rate of diffusion.
Other research l , empirical and theoretical, however, provide evidence for a positive
correlation between the number of firms and the adoption rate.
The probit models explicitly incorporate firm size, and therefore are more attrac-
tive than an epidemic-like model such as Mansfield's. However, according to these
probit modela, firms adopt a new technology whenever it becomes profitable to do 50.
Thus, they espouse a myopie view on the firm's part. But it cao be (and bas been2)
argued that firms adopt when it is most profitable to do sa, not when it becomes
profitable ta do 80. This requires the firms ta look ahead and base their decisions on
expectations about future developments. In fact, many of the above-mentioned em
pirical works provide strong evidence for an influence of expectations of future price
changes on the adoption dates. Expectations of favorable future changes, therefore,
may cause slowdown in diffusion. As a result, diffusion in reality will mest likely
be slower than what is proposed by the probit models, which do not account for
l see Hannan and McDowll [1984], Karshenas and Stoneman [1993] for the empirical works and
Stoneman [1990b], Waterson and Stoneman [1985], ReiDganum [1981b] for the theoretica1 works.
2 see for example, Rosenberg [1976} and Ireland and Stoneman [1986] .
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expectations.
Jensen [1982] shows that the diffusion curve will be sigmoid when the firms have
different prior beliefs about the success of a new technology; the shape is explained
solely by the differences in the beliefs of the firms. For competing new technologies
(the profitability of which was only probabilisticaIly known) , Jensen [1983] again
shows that the diffusion curve will be sigmoid (for the superior technology) due to
different prior beliefs among the firms.
Motivated by Rosenberg's [1976] assertion that future expectations about the
technology play a significant role in the technology decision of a firm, Ireland and
Stoneman [1986] propose a mode! ta study the effect of expectations about the price
of a technology on the diffusion rate. The mode! assumes that the suppliers play
a Cournot game in the quantity of the technology to he sold. However, it does not
incorporate the effect of rivalry in the buying industry. Ireland and Stoneman's mode!
is based on the following explicit assumptions:
(i) the potential users have expectations about both price and technology
(ii) the profitability is different for different users.
( iii) the users receive a constant ftow of services until obsolescence, and,
( iv) the number of potential users is large and fixed.
Technology expectations in assumption (i) concern both improvements in, and
obsolescence of, the technology. Ireland and Stoneman do not address the issue of
15
• future improvements separately. In their model, it is incorporated in the quality
adjusted price expectations. As for obsolescence, each user bas the same subjective
probability distribution. The price for the next period is forecast based on the current
price. The profitability in assumption (ii) defines firm characteristics in this model.
Fina1ly, the constant ftow in assumption (iii) is the (undiscounted) revenue that the
user receives. The cost of the technology however goes down over time making it
profitable for different firms to adopt at different times.
Note that assumption (i) is the major difference between the premise of this
model and that of the probit models.
Based on the above assumptions, the potential users are indexed in decreasing
arder of revenue flow (user with index x obtains a benefit of 9 (x)) and the expression
for expected benefit from adopting at time t for a firm is derived as
E (1r (t, x)) = -p (t) + ~,
where r = r + h, r being the certainty discount rate and h the discount rate
for hazard of obsolescence (h is the probability that obsolescence will oœur in the
interval {t + dt} given that it has not occurred by t); and p (t) is the purchase cast at
t. Ireland and Stoneman then contend that for a firm ta adopt at t, two conditions
have ta he satisfied: that the adoption is profitable (profitability condition) and that
waiting will not result in higher profits (arbitmge condition). Formally,
• where Dp (t) is the continuous-time representation of the users' expectation of
the change in price. Ireland and Stoneman's analysis of expectations is based on two
scenarios, myopia (when the users do not expect any change in price) and perfeet
foresight (when the buyers know exactly what was going to happen). Uoder myopia
the term D'fi (t) is zero and thus a firm adopts whenever it becomes profitable for it to
do so. Under perfectforesight however, if Dp(t) < 0 (reftecting a declining acquisition
cost, which has been observed to hold for mœt technologies), equality in condition (ii)
automatically satisfies condition (i) and determines the dynamic demand function.
Ireland and Stoneman then go on to model the supplier side. There are n quantity-
setting, profit-maximizing suppliers in the market. An individual supplier's expected
profit is given by
:x:E (1rs) = f (p (t) - c(t)) q (t) e-rtdt
o
where q is the quantity sold by the supplier, r is same as above, e is the unit cœt,
which falls until sorne time t and increases afterwards, 50 that De (t) ~O as t;t.
Substituting the priee in E (1rs), using the inverse demand function, and after
sorne rearranging, Ireland and Stoneman express g(x) for myopia and perfect foresight
as follows:
9 (x) = Te - De + n~t .Q.gz/r
9 (x) = rc - De - gz/n
(myopia)
(perfect foresight)
•where Q = Dx, the eurrent rate of industry sales.
Base<! on the above expressions for 9 (x), Ireland and Stoneman eonclude that:
17
•
•
(i) with a monopoly supplier (n = 1), "ownership of the new [technology] will be
less at all times under perfect foresight [than under myopia, as the third term
becomes zero]". This conclusion is quite intuitive as the users under myopia did
not wait once the technology became profitable. That is, expectation of lower
prices slowed down the diffusion process.
(ii) higher expectations of obsolescence (higher r) reduce usage along the diffusion
path under both scenario, and
(iii) when the buyers have perfect foresight, the higher the number of suppliers the
lower the value of the third term and, the higher the sales. As the number
of suppliers approaches infinity, the perfect foresight path approaches that of
myopic buyers with monopoly suppliers. Ireland and Stoneman rationalize this
by arguing that under perfect foresight (of the users) and perfect competition
(in the supplier market), all the rents go to the users who cboose a path to
maximize the total rent, while under myopia (of the users) with a monopoly
supplier, all the rents go to the supplier who again chooses the saIne path to
maximize the total rent (but now it all belongs to the supplier).
In addition to studYing the effect of expectations, this model provides ÏDSight
into the effects of competition in the supplying industry: the more competitive it is,
the higher the rate of diffusion in the user industry. More competition drives the
price down~ thus making it profitable for firms in the user industry to adopt it. The
18
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•
shortcoming of this model (and other such models) is that it assumes that a 6rm cao
gain a constant flow of revenue over time irrespective of rivaIs: actions.
Stoneman [1990a) extends the above mode! to incorporate an interesting aspect,
that of product differentiation by the suppliers. Quite olten it is found that the same
supplier sells different brands of the same technology (e.g., IBM) or different suppliers
(e.g., IBM and Apple) sell similar but not the same technology. Stoneman adds
one more assumption to the above model, that there is no obsolescence. The basic
framework is the SaIne, but here he defines the revenue function 9 (x) more explicitly
as a function of the distance between a buyer's ideal choice and what is supplied by
the brands available in the market. He argues that, for a monopolist supplier with a
single brand, as the priee falls over time, the users having more distant ideal choice
from the brand find it attractive to adopt (the distance is offset by lower cast) and
that is how the diffusion path is generated. ln other words, Stoneman reasons that the
relevant firm charaeteristics is the ideal choice from which a firm derives maximum
benefit, and which is different for different firms. For a monopolist supplier with
multiple brands, Stoneman shows that the diffusion rate increases with the number
of brands, and that the number of brands is endogenously determined. Relative ta
the situation where ooly a single brand is available, the eonsumers willingly pay a
higher priee in presence of multiple brands (sinee they get something doser to their
ideal choice). But at the same time the total launching oost for the supplier goes up
and, siuce more units are sold in the early years, total (discounted) production cost is
19
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•
higher. These two factors together determine the number of brands. Note that this
conclusion is hased on the assumption that the users are myopie; what would happen
under expectations, Stoneman conclude, is very much an open question.
The literature on diffusion helps us to identify the factors that lead to the adoption
decision of a firme Sorne of the factors, as discussed above, are firm size (reftected in
their investment or operational cast structure), expectations, market structure etc. In
the next two subsections, we review the literature on single-firm and game-theoretic
modeling of technology decisions involving examination of the above factors.
2.2 Single Finn Decision Models
Technology decisions from a firm's perspective, without consideration of rivalry,
have heen mostly studied by management scientists, and even that did not start
until recently. Traditionally, management scientists have studied major investment
decisions as either capacity-expansion problems, or equipment-replacement problems.
Sinee the 19608, however, with frequent introduction of new technologies, manage
ment scientists started building models incorporating the choice of a technology in
both capacity and replacement decisioDS. Not surprisingly, research on technology
choice addressed the issues related to technologies that could be used ta produce a
single product, since that was the only kind available in the market. The introduction
of Flexible Manufacturing Systems (FMS) triggered a new stream of research on tech-
20
•
•
nologies that could he used to produce more than one product3 • The rate at which
new technologies are introduced has risen sharply in recent times. This has led to
another very recent, and quite limited sa far, area of study that focuses on streams of
new technologies. However, because of the complexities in modeling the dynamics of
technology evolution, these models consider ooly single-product technologies, but not
8exible ones. We review the papers that address the issues of tecbnology acquisition
with and without future improvements below.
2.2.1 Technology Choice with No Expectations Research that ignores the appear-
ance of improved technologies in the future, and instead focuses on other related
issues, can be categorized into two groups. One addresses single-product technol-
ogy choice; the other deals with multi-product technology choice. The issues are
somewhat different in the two cases. A single product technology does Dot offer the
henefits of a multi-product technology, such as a wider product line, cushion agaînst
uncertain product demand, etc. Therefore, we present the two streams separately.
Single Product Technology Choice Manne [1961] proposes a model to examine the
capacity expansion path over an infinite horizon with linearly increasing demande
Studying the trade-off between economy of scale in investment (reflected by concave
cast function) and benefits of delayed investment, Manne shows that the optimal
3 Throughout this dissertation, dedicated and single product technology will be used synonymously to refel to technologies capable of producing a single product. FMS, flexible technology and
multiproduct technology will aIso be used synonymously.
21
• policy was to add capacity at regular intervals.
A number of papers since have analyzed capacity expansion decision; we brieBy
present the model proposed by Neebe and Rao [1983} as it captures the choice of
technology in a simple way. Neebe and Rao consider a discrete-time, 6nite-horizon,
capacity-expansion problem where a firm (producing ooly one product) has the option
ta add capacity (from a pool of alternative projects available) in different time periods.
The product demand is assumed to be deterministic and üùn-decreasing.
Let Tt = dt - dt. -I, be the iDcrease in demand from period t - 1 to period t, for a
planning horizon of T periods. Also let Cij be the fixed (and ooly) cast of activating
project i in period t, and Zi be the capacity of project i respectively. Now let us define
the decision variables Xlt and Yt as follows. Let
Xit = { 1 if project i is selected in Period tootherwise
and Yt be the excess capacity in period t before any expansion. The decision problem
for the adopting firm is then
n Tmin E E CitXit
t=l t=l
subject to
•
Xd E {O,l}
TE Xit ~ 1t=l
nE Xit ~ 1t=l
n
E %lX,t + Yt - Yt+l = Tti=l
Yt ~ 0
for all i and t
for i E 1
for t = 1.....T
for t = 1. .....T
for all t
22
•
•
The first constraint requires that a project he activated in full. The second and the
third constraints make sure that only one project is activated, and in ooly one period.
The fourth constraint ensures that demand increases are met by the expansion.
To solve the above, Neehe and Rao provide a Lagrangian-relaxation based alg~
rithm which works weil for a moderate-sized problem. Note that different projects in
the model cao be treated as alternative technologies available in the market. However,
the model as presented assumes that all the alternatives are available now; in reality
what is more likely to happen is that sorne of the alternatives would he available only
in future periods as the technology improves. Also, although the problem allows for
different prices in different periods for the same technology (which reflects one fonn
of improvement), it does not incorporate improvements in technology in terms of its
ability to perform its operations (e.g., low production cost or high capacity). Finally,
the model does not provide for choosing the amount of a particular technology.
Rajgopalan and Soteriou [1994] prERIlt a non-linear integer program and solution
procedure for linear relaxations (and beuristics for improving them) for modeling
capacity acquisition with discrete facility sizes. This is a discrete-time, finite-horizon
mode! where a number of equipment types are availahle that can be used to produce
the saIne product. The model also allows for declining cast of equipment in the future
periods. Capacity can he bought ooly in chunks. DisposaI (due to declining demand)
is allowed, at a salvage value minus cast of disposing, which may he either positive
or negative.
23
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•
Cohen and Halperin's [1986] paper is one of the few tbat deal with stochastic de
mand while studying the single-product technology decision. This is a discrete-time,
finite-horizon model, where the demand is specified by a probability distribution in
each period. Associated with each technology are the annual fixed cast, the unit
variable cast, an age-dependent salvage value, and a dynamic purchase cast. The
assumption that only one kind of technology may be used in one period, and that
capacity is defined by the choice of technology (which means that any capacity ad
justment means selling off existing technology and investing in a new one) make the
model too restrictive and unreal. The model bas a three-stage solution procedure.
In the mst stage, for each technology, the optimal quantity given the technology is
calculated using the newsboy solution procedure (this is possible as inventory and
sbortages are not permitted). ln the second stage, this is extended to multiple (but
finite) periods by finding the optimal production quantities in different periods, again
given a technology. In the third stage, a dynamic programming recursion is used to
find the optimal technologies in diff'erent periods.
Multi Product Technology Choice The introduction, and tben increasing use, of com
puterized process technologies and flexible manufacturing systems made management
scientists tum their attention to incorporate the choice of sncb technologies in major
investment decisioDS.
It bas been pointed out time and again that traditional methods to evaluate
alternative investment choices did not work very well for evaluating investments in
24
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•
advanced manufacturing technologies. Kaplan [1986] refers to the prevailing practice
among managers in North America as "justification by faith". Jaikumar [1986] and
Mansfield [1993] find that, relative to the Japanese firms, the US firms lag behind in
installing FMS. The failure of traditional discounted cash flow techniques, sucb as Net
Present Value (NPV) or Internal Rate of Return (IRR), in justifying advanced tech
nologies cao probably be attributed to two factors. First is the need for very high level
of investment. Secondly, many of the henefits of sucb technologies (improvements in
quality, leadtime, and f1exibility, and the way it serves as a cushion against uncertain
product demands) are not easily quantifiable in cash flow terms. Also, traditional
Discounted Cash Flow (DCF) methods place stronger emphasis on the short-tenn
henefits (by heavily discounting future cash fiows) , without recognizing the long-term
competitive advantage that flexibility cao provide. Ramasesh and Jayakumar [1993)
propose a multi-stage evaluation procedure, whereby, simple DCF is applied in the
first stage. If the NPV is found to he negative then, in later stages, other benefits
of sucb technologies, quantitative and qualitative, are given proper consideration to
narrow, and perhaps eliminate, the "justification gap" .
The recognition of the importance of sucb decision making 100 management scien
tists to model tecbnology acquisition incorporating the henefits of flexibility explicitly
in the model. Kalotay [1973] is among the first to examine the cboice between tech
nologies in a multiproduct case. In bis model, which is an extension of Manne's [1961]
model referred above, the firm cao either buy a dedicated tecbnology that can pro-
25
•
•
duce one particular product, or a flexible technology that can produce two different
products, or sorne amount of both. That is, the choice for the firm is to operate ei
ther in one market or in two markets. Kalotay examines the conditions under which
dedicated capacity should be used at ail when purchasing cast is concave (to reflect
economy of scale in investment), and the fixed cost of installing flexible capacity is
higher than that of dedicated capacity (while the variable cast is the same). He
concludes that for linearly growing demand, sorne amount of dedicated technology
should always be used, whereas, for exponentially growing demand, ''this may not
be the case". Kalotay does not explain why, under exponentially growing demand, it
might be optimal not to buy the dedicated technology at aIl. One plausible explana
tien can be as fellews: when demands for the two products grow at an increasing rate,
capacity has to be added more and more frequently. Therefore, it may be pœsible in
sorne situations ta incur only one installation cast (in the flexible capacity) everytime
capacity is added.
Luss [1979] presents a discrete-time, finite-horizon model with dynamic but de
terministic demand to examine the choice between two technologies, bath of which
are dedicated (can produce one type of product), but cao be converted from one type
to another at a conversion cost (which is non-decreasing and concave). The other
costs in bis model are the acquisition cast of new capacity and the holding cost of
idle capacity, which are also non decreasing and concave. The problem for the firm
is to decide, in each period~ how much of each capacity to add, and how much of one
26
•
•
type of capacity to convert into another type in order ta meet the demand at the
minimum cast. Luss shows that the problem cao be formulated as a network flow
problem. However, given the concavity of costs, even with this structure it is not
easy to solve the problem. Luss goes on to propose a dynamic programming alg~
ritbm for non decreasing demand situation. Lee and Luss [1987} extend Luss [1979}
to more than two products and propose solution procedures for two variations of the
model, one that allows shortage and the other that does not. They also show that
the computational complexity increases exponentially with the number of products.
Note that these models, while dealing with multiple products, do not really consider
"flexible" technologies.
Li and Tirupati [1994} consider the case of a truly flexible technology for the mul
tiproduct case: the technology can be switched from the production of one product
to the production of any other without any conversion time or cast. Dedicated tech
nologies are also available which can each produce exactly one product. The number
of technologies from which the firm can choose are thus N +1, where N is the number
of products under consideration. Bath the investment and the operational costs are
concave, reflecting economies of scale in investment as well as in production (thereby
encouraging more investment in one type). Li and Tirupati consider general demand
patterns that allow for increasing, decreasing or constant demand. The objective of
the model is to determine, in each period, how much of the difl'erent technologies ta
buy ta meet the demand with the minimum cast. The fonnulation is presented below:
27
• i
T
Xit
X iO
Yit
index for technology (and product) type; i = 0 means flexible technology
number of periods in the planning horizon
capacity of type i added in period t
initial capacity of type i
amount of 8exible capacity allocated to product type i in t.
~t demand for product type i in t
fIt (.) investment cast function for technology i in t
cd;t( .) operating oost function for product i with dedicated technology i in t
cflt (.) operating cast function for product i with 8exible technology in t
With these notations the formulation is
subject ta
t
EXiT + Yit ~ ~tï=1
N t
E Yit ~ E XOTr=1 r=O
i = 1,2, ....N; t = 1,2, ....T
t = 1,2, ....T
i=O,I, ....N;t= 1,2, ....T
i = 1,2, ....N; t = 1,2, ....T
•
Note that the consideration of a general demand pattern and no conversion cast
breaks down the network structure identified by Luss. Li and Tirupati, therefore,
present a two-stage heuristic-based procedure, where an initial solution is found in
the first step, and a good sub-optimal solution is obtained through improvements
made in the second. Computational results show that the procedure worles weIl.
28
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•
Based OD the computational rE~ults Li and Tirupati observe the following.
(i) Even when the flexible technology is significantly more expensive, investment in
it can be economically justified.
(ii) A more erratic demand pattern means higher iDvestment in the flexible technol
ogy,and
(iii) Investment in the flexible technology occurs early in the horizon (serving ~ a
"cushion" to absorb the fluctuations in demand).
Rajgopalan [1993) proposes a model that is somewhat similar to Li and Tirupati's,
and can be solved for optimality (recalI that Li and Tirupati can only be solved for
suboptimal solutions). The important differences are that (i) Rajgopalan considers a
fixed charge investment-cost function (unlike Li and Tirupati's general cost function),
(ii) Rajgopalan's model alIows only non-decreasing demand (unlike Li and Tirupati's
general demand patterns), and (iii) in terms of modeling, Rajgopalan is much more
specifie in the use of aequired technology. The capacity addition variables (X1t and
l'it in Li and Tirupati's model) have a third subscript in Rajgopalan's model; X itk
indicates the amount of the dedicated capacity of type i acquired in t to satisfy de
mand increment ~k of period k, and similarly Yitk indicates the amount of the flexible
capacity acquired in t allocated to satisfy ~k • Rajgopalan shows that a fonnulation
with such variables can he easily transformed into an uncapacitated plant location
problem which can be solved for optimality. However what Rajgopalan sacrifices for
29
•
•
optimality is the "fiexibility" of the flexible technology: once bought for a specifie
product, that part of the flexible technology remains committOO to that partieular
product for the rest of the planning horizon. Note that the "permanent eommitment"
is not a problem if demands for all the products of the firm are increasing. Therefore,
the problem is the assumption of non-decreasing demande In fset, if the demands
are non-decreasing for ail products, the flexible technology loses much of its appeal,
and sa, Li and Tirupati's model stands out in valuing flexibility among the models
discussed so far as most of them consider ooly non-decreasing demand.
As notOO earlier, one of the major benefits of the multiproduct or flexible tech
nology is its ability to serve as a eushion against uncert8Ïn product demande It is
therefore important to study how uneertainty may impact technology choice decisioDS.
Unfortunately there are not many research works that deal with this issue. This is
perhaps due to the diffieulty in stochastic modeling. Fine and Freund [1990] examine
the use of FMS to hedge against uncertainty in product demands. Their model is a
twO-stage stochastic program. The firm in question makes a technology decision in
the first stage facing uncertain produet demand. Theo in the second stage when the
uncertainty is resolved, the firm makes the production decisioDS. The model, however,
covers only one period (ail the future periods are roUed back into one). Like in Li
and Tirupati's model, the firm can either buy n different technologies to produce its
n different products, or it cao buy ooly the flexible technology that cao produce all n
different products, or any combination thereof. Here the simpler version of the mode!
30
• (that cao be easily solved) is briefly discussed. Let us use the following notation:
Ki amount of dedicated technology j, j = 1,2, ....n
K / amount of flexible technology
Tj per unit acquisition cast for technology j, j = 1,2, ....n
TF per unit acquisition cost for flexible technology
k number of possible states
Pi probability that state i OCCUIS, i = 1,2, ....k
Y;j production of product j on dedicated technology when i occurs
Zij production of product j on flexible technology when i occurs
~] (.) Revenue function
Cl production cost of j (same on both technologies)
The problem can now he fonnulated as
k n nmax EPi E [Rt j (Y;j + Zij) - Cj . (Y;j + Zij)] - TFKF - E TiKj
1=1 )=1 j=1
subject to
n"Z--KF<OL.... 1) _
j=1
Y:- > 01] _
z- > 0Il -
i = 1, ...k; j = 1, ...n
i = 1, ...k
i = 1, ....k; j = 1, ...n
i = 1, ...k; j = 1, ...n
j = 1, ...n
• The mode! provides very good insights with respect ta the role uncertainty plays
31
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•
in technology decisioDS. From a sensitivity analysis done on a tw~product case, Fine
and Freund observe that
(i) when the product demands are negatively correlated, the optimal quantity of the
flexible technology increases with the riskiness of the distribution
(ii) when the demands are positively correlated, increased riskiness lead to larger
purchases of the dedicated technologies, and
(iii) when the product demands are uncorrelated, the amoWlt of the flexible capacity
increases with the riskiness for some range, and then decreases.
Observations (i) and (ii) are quite intuitive; for observation (iii) the authors
suggest that "the need for flexible capacity is a complex function of the level of
demand in each of the future states and of the probability distribution governing those
future states". Fine and Freund serve an important purpose in showing what kind of
uncertainty favors the flexible technology. Note that this model is also different from
other models in using a revenue function that determines the price and the quantity
sold. Therefore the model applies to monopoly firms, and to firms in an oligopoly
where the quantities produced by the rivaIs are given and fixed.
Gupta and Buzacott [1993] model a similar situation to Fine and Freund's. How
ever, in this model there are sorne disadvantages associated with the flexible tech
nology. While the papers discussed above have assumed instant switching, Gupta
and Buzacott contend that the flexible technology takes sorne time to switch between
32
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•
products, which means a lcss of production time. The time that flexible technology
requires for switching also determines the production cycle; and hence the level of
inventory and related holding oost. In addition to the dedicated technologies, in this
model, more than one flexible technology is available and they differ in terms of their
switching times. Firms choose the cycle time, which in turn determines the particular
flexible technology with which this cycle time is optimum. Therefore, the decision in
c1udes the amount of each capacity to buy, and the degree of flexibility for the flexible
capacity. However, urùike Fine and Freund's model (and like other models above),
this model considers a price-taking firm. The observations made from experimental
resuIts reinforce the observations of Fine and Freund: while it is generally true that
negatively (positively) correlated demand favor the flexible (dedicated) technology,
there are many situations where intuition can he misleading and evaluations should
he made only after thorough investigation of the situation. Li and Tirupati [1995]
reconfirm these conclusions.
2.2.2 Technology Acquisition with Expectations Although research in this area can
he traced back to Hinomoto (1965], the interest bas grown ooly in recent years. Short
life-cycles of technologies bave contributed to the growing interest. As we have noted
before, expectations about technologies concem the timing of the appearance of the
next technology, the magnitude of improvement in terms of the technology's ability
to perform intended operations, and the acquisition cast. Note that some of the
models discussed in the previous section cao tackle declining acquisition costs over
33
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•
the planning horizon (which reftects one form of improvement). However, in this
section we refer to future expectations to mean enhanced ability of the technology,
which cao be reftected by lower marginal production cast, lower setup cost and time,
wider product tine etc.
Hinomoto [1965] makes the first attempt to explicitly incorporate technological
improvement in an investment mode!. He considers improvement both in terms of
investment cast and in terms of the ability of the technology to perform the same
operation more efficiently (reftected by lower operational cast). Hinomoto's mode!
however has two shortcomings. One, future improvements are known deterministi
cally. Second, both cast and performance improvements are continuous. In reality,
improvements usually appear in discrete jumps.
Other papers that have focused on expectations use semi-Markov process to
mode! the acquisition decision where the time till the arrivaI of the next technol
ogy is stochastic. In Balcer and Lippman [1984], the choice is between an immediate
adoption of the technology available DOW, and to wait until either a new technology is
available or it becomes profitable ta adopt the existing one given the expected delay
in arrivaI of the nOO one. The evolution process of the technology is reflected by (i)
the current state of knowledge (represented by marginal cast, for example) , (ii) the
current discovery potential and (iii) the number of periods since the last innovation.
The discovery potential follows a semÎ-Markov process represented by a one-step tran
sition matrix. The probability distribution for the time till the next discovery given
34
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•
some potential is known. Note that for exponential distribution of time till next dis
covery, an optimal stationary policy can he obtained. When innovation takes place,
potential jumps, and the state of knowledge changes, the magnitude of the change
having sorne known distribution. Balcer and Lippman assume that the per period
profit from adoption is linear in the technology lag of the firm (difference between the
level of the available technology and the level of the firm's own technology), 50 that
the process characterization does not require both levels. Using a recursive expression
for the optimal policy, Balcer and Lippman go on to establish some properties of the
optimal policy. Some of their conclusions reinforce our intuition, while others do not.
Among the conclusions are
(i) The firm adopts immediately if its technology lag is at least as large as a critica!
value which is a function of the current discovery potential and of the number
of periods since last innovation. The critical value is increasing in the potential
(expected rapid change deters adoption). The effect of the number of periods
since last innovation depends on the characterization of UDcertainty. An inno
vation may, on one hand, become more likely with time, but on the other, more
time mayalso mean that the research is heading in the wrong direction and
success less likely.
(ii) The critical value is increasing in the fixed cast, and,
(iii) depending on the characterization of UDcertainty, it is possible that a technology
rejected on arrivai will be adopted by a firm later on.
35
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•
The last conclusion may seem somewhat counter-intuitive at first, but it is possible
if, for example, the specification of uncertainty is such that after sorne period of time,
the success of current R&D becomes less and less likely. The 6rm could then face a
situation where acquiring the available technology is better than not acquiring at ail.
Baker and Lippman's model is more realistic than Hinomoto's as it deals with
discrete and, more importantly, uncertain improvements in technology. However, the
assumption of linear benefit (which allows the characterization to he based on the
lag, but not on bath levels) is simplistic.
Nair and Hopp [1992] and Naïf [1995] also use semi-Markov proœss to model ac
quisition decisions with uncert8ÏD expectations, and propose a dynamic programming
recursion to solve the problem. While these papers do propose algorithms ta solve the
problem (which Balcer and Lippman do not), they ignore one kind of uncertainty that
Baker and Lippman consider, that is the uncertainty in the degree of improvement.
In these models, only the timing of arrivai of the next technology is uncertain, while
the degree of improvements and acquisition oosts are aIl deterministic. Nair and Ropp
consider only one future improvement, and Nair extends it ta a n-improvement case.
Both of these models consider ooly finite horizon; however, the planning horizon is
based on identification of forecast horizon, for which the finite horizon results coïncide
with the infinite horizon results. Naïr [1995] is briefly presented below.
Let the technology in use and the technology now availahle in the market he in
dexed by 0 and 1 respectively. Another n future generatioDS of technologies (2, 3, ..... , n + 1)
36
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•
may appear in the future. Astate here is represented as (i, k), where i is the technol-
ogy in use, and k(k ~ i) is the latest technology available in the market. The actions
available in each state are Ki (keep technology i) or Ri (replace i with j, j ~ k ).
Let P~:11 be the probability that the next mode! (k + 1) appears in the next period
(t + 1). Let the acquisition cœt of technology i in t he Cit (Î. e., oost may vary from
period to period), the revenue from j in t he Tjt , and the discount factor in t be f3t .
It is assumed that Tlt is at least as large as Ttt for i < j (successive models represent
at least as good technologies). The current state is (0, 1) and the decision is to choose
between Ka and ~. The optimal value function is denoted by rr (0,1) where T is
the planning horizon~ and
fT (0 1) = {m :-CH + TH + f3t [(1- P~+1) f[;l (1.1) + pr+1f'[,.1 (1. 2)]t ~ max . [ .) T -2 TKa . TOt + f3t. (1 - Pi+d ft+1 (0, 1) + JJt.1ft+1 (0,2)]
Nair shows that by choosing appropriate boundary conditions, (for j > i, ff (j, k) ~
ff (i, k) ), the solution space can be reduced significantly, 50 that (i) it is oever opti-
mal to replace the technology in use with an oider technology, and (ii) if the optimal
decision is to replace j with l (l > j) in any t, then it is also optimal to replace i with
l (i < j) in same t. Such choice of boundary condition leads to the general recursion
T . {Jt1 :-Cjt + Tjt + l3t [(1 - p~:f) f~l (j, k) + p~:f ff+l (j, k + 1)]f (1 k) = maxt' t-«k<n K. ~ [( k+l) fi (. k) k+l fi (' k )]J_ - t • Tit + lJt 1 - Pt+l Jt+l 1, + Pt+l Jt+1 1, + 1
The above can be solved as long as T is not very large. However, an arbitrary
cboice of T may lead to non-optimal solution. This consideration leads to the identi-
37
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fication of the forecast horizon T, for which the finite horizon decision will he same as
the infinite horizon decision. Nair and Bopp use difference functions to identify the
horizon, but their approach does not work when n > 2. Nair's dynamic programming
recursion uses a non-unique set of boundary conditions ta find the forecast horizon
and solve the decision problem.
Rajgopalan [1998] also uses semi-Markov process to solve a finite horizon tech
nology replacement problem when the demand is growing over time. Recall that
Balcer and Lippman, Nair and Bopp, and Nair all address technology acquisition as
replacement problems; the issue of capacity is not considered in these papers. Ra
jgopalan, however, models capacity and replacement together and, therefore, unlike
the above profit maximizing models, this model minimizes the cast subject to meeting
the demand. Furthennore, Rajgopalan incorporates the uncertainty in the degree of
improvement as weIl. A regeneration-point-based dynamic program recursion salves
the problem. The state in the model is defined as (mt, kt), where mt is the latest
technology and kt is the period of its introduction. Technological evolution is there
fore represented as (mt+b kt+d = <P (mt, 1er), and the transition depends on one-step
transition matrix P for mt , and the time ta discovery distribution Qm (.) . At the
beginning of each period, the state of the technology becomes known, and based on
the state a finn decides whether and how much of the technologies currently in its
possession (of different vintages) to dispose off, and whether and how much of the
technologies available (of different vintages) to acquire. Let X t and It he the vectors
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representing used and unused levels of capacity respectively in possession in period
t. Also let ~, Zt, and Z: respectively be the vectors of capacity acquisition, total
disposai, and disposaI of unused capacities in t. Note that while technology evolution
is stochastic, the evolution of capacity is deterministic. The state can thus be parti-
tioned iota two sets, (mt, kt) and (Xe, lt). Let \{Ii (obtained from P and Qm (.)) be the
probability that the next period will be in state (mt+ l, kt+1)' Finally, let fpmt (.) he
the concave acquisition cast of vintage p in t when the latest tecbnology is m, hpt (. )
the concave carrying oost for unused technology, Cpt the per unit operating cast, and
gpmt (.) and g'"mt (.) the salvage cast of respectively unused and used technology p
in t when latest technology is m. Now assuming the terminal cost to be zero, and
denoting the expected total cast associated with the decision (~, Zt, Z:) (assuming
that al1 future decisions will he taken optimally) as Ct ((mt, kt) 1 (Xe, le) ,(l't, Zt, Z;)),
the stochastic dynamic program is
+ L 'IIi (me, kt) C;+l ((mt+l' ke+d ,(Xt+ll lt+d)i
and L t (.) is the sum of all casts incurred in period t,
+gpmct (Z;t) + g~ct (Zpt - Z~)]
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In arder ta reduce the state-space significantly, Rajgopalan uses a number of
properties of the optimal solution (properties derived from another fonnulation which
is identical ta the deterministic version of the above formulation). His recursive
algorithm is base<! on regeneration points, which are defined as the points when the
amount of unused capacity becomes zero. The algorithm takes advantage of two
properties of the optimal solution, that disposal of unused capacity is considered only
when a new technology appears, and that acquisition and replacement are considered
only when the firm has no unused capacity.
Rajgopalan's sensitivity analysis shows that higher uncertainty (variance) in the
interarrival time and in the number of technologies to appear in the future deters
adoption. This is consistent with Ba1cer and Lippman'S conclusion, as well as empir
ical findings of Antonelli [1989] and Karlson [1986].
Rajgopalan's model is richer than Nair's in addressing different types of uncertain
ties. But it uses an arbitrary finite horizon whereas Nair's model finds the appropriate
horizon for which the finite and the infinite horizon solutions coincide. However, using
numerical examples, Rajgopalan shows that the solution is not very sensitive beyond
sorne ''moderate'' finite horizon.
Before conc1uding this suœection, we would like to point out that there are other
papers that deal with the same or similar topic. Of particular interest is a stream of
papers spearheaded by Pindyck [1988] that addresses the issue of investments (not just
technology decisions) when the investments are irreversible and the future uncertain" .
-& see Pindyck [1988}, Pindyck [1991!, Pindyck [19931, Dixit and Pindyck [19941, Hubbard [19941,
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This stream is based on the contention that "one problem with the existing mode1s is
that they ignore two important characteristics of mast investment expenditures. First,
the expenditures .... are mostly sunk costs that can not be recovered. Second, the
investments cao be delayed, giving the firm an opportunity to wait for new information
to arrive about ... market conditions before it commits resources" (Pindyck [1991}).
Investment opportunities, in these papers, are therefore viewed 88 finandal options.
As with financial options, a finn with an investment opportunity has the choice to
spend money DOW (irreversibly), or in retum for an asset in the future (the value
of which is uncertain). These mode1s too, like Rajagopalao [1998] an Halcer and
Lippman {1984], predict that uncertainty about the future deters adoption.
Among other papers addressing the issue of technology choice for a finn, Tiru
pati and Vaitsos [1994] examine optimal timing for a firm ta switch from flexible ta
dedicated technology. The premise of the research is that many firms introduce new
product produced with a flexible technology. In the beginning the demand is low, and
as such using the flexible technology (that is shared with other products) makes sense,
but as demand grows, using a dedicated technology may be more profitable because
of economy of scale. McCardle [1985] mode1s a firm's decision problem when faced
with an adoption decision of a technology, the success of wbich is still uncertain. In
this model, the firm starts with some estimates of profitability and other information.
The firm's decision is either to adopt it, or to reject it altogether, or to continue to
and He and Pindyck [19921 .
41
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gather information (at a eost) in order to reduce uneertainty and ta update profitabil
ity estÏJDate and then to decide in the next period. McCardle proposes a stopping rule
for the mode!. Lippman and McCardle [1991] study a similar problem where a pool
of technologies (the values of which are uncertain) is available to a firm. The firm
sequentially purchases information, about a technology, to update its estimate. The
mode! develops ruIes for choosing from the following strategies: (i) pick a technology,
adopt it and quit searching, (ii) pick a technology, reject it and quit searching, and
(iii) pick a technology, reject it and gather more information. Oliva [1991] applies
catastrophe theory to build a mode! to compare McCardle's [1985] normative results
with what firm's actually do.
The models discussed sa far clearly indicate that some of the factors have been
studied in more detail than others. For example, economy of scale and cast functions
have been incorporated in many of the models. But studies on uncertainty and
expectations, two major influences on acquisition decisions, are very limited. More
studies in these areas are required in order for researchers to gain better understanding
of a firm's technology decision in an inereasingly complex environment.
2.3 Decision Models with Rivalry
Investment in a Dew technology often eonfers a significant competitive advantage.
Therefore it is important that the strategie aspects of rivalry he coosidered while
analyzing the adoption decision of a firm. Starting with Reinganum [1981a], there is
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a growing stream of research that focuses on the strategie decision making by firms in
analyzing adoption decisioDS. These decisions, in turn, determine the diffusion path.
Reinganum [1981a] shows that diffusion does not neœssarily require asymmetry
among firms. Using agame theoretie framework of timing, she shows that even for
identieal firms in a duopoly, it is not an optimal strategy to adopt a technology (the
acquisition cost of which declines over time) jointly. Rather, one firm should adopt
early and the other later. She argues that since the firms are identical, the different
adoption dates are a result of strategic behavior by the firms. Reinganum [1981b] ex
tends this model ta a symmetric oligopoly in arder to study the interaction between
market structure and the diffusion process. Here, too, she shows that in the equili~
rium, firms adopt the new technology at different dates. Sinee there are n identical
firms in the oligopoly, there will be n! Nash equilibria where the firms do not deviate
from any proposed arder, but maximize their payoffs by detennining the exact time
within the arder. Quirmbach [1986} uses the same modeling framework ta argue that
strategie behavior is inessential for diffusion to take place; rather ineremental bene
fits have to falI over time (in addition to the declining oost); and therefore comparing
diffusion in different markets was equivalent to comparing incremental benefits. He
uses this argument to demonstrate the effect of market power on the diffusion pr~
cess. He concludes that cooperation between potential adopters slows down diffusion
while under monopoly power in the supplier industry, the adoption dates are more
dispersed than under other market structures.
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Fudenberg and Tirole [1985] challenge the equilibrium dates proposed by Rein
ganum [1981a} by suggesting that it is not SUBtainable because whoever adopts early
obtains a higher payoff, and yet the model does Dot resolve which firm becomes the
leader. They suggest that an allowanœ for pre.emption resolves the problem and
show that in the equilibrium (they call it perfect equilibrium), the adoption dates
are either more disperse than in Reinganum's equilibrium, or in some cases, for tech
nologies with certain features, the equilibrium is delayed joint adoption. The rents
obtained by the 6.rms are equal in perfect equilibrium.
Kim, Roller and Tombak [1994} explicitly model the demand functioDS and the
nature of interaction between 6.rms ta analyze adoption decisions. They show that for
markets where the 6.rms are engaged in a Cournot quantity setting game, Reinganum's
(and Fudenberg and Tirole's) assumptions regarding the payoffs hold if the nature of
the technology is sucb that it still produces the saIne product (as the 6.rms currently
produce) but at a lower marginal oost. They calI it a single.product cost-reducing
technology, and foeus on a different kind of technology that enables a firm to produce
a wider product line (not necessarily at a lower cost) and to invade competitors'
markets. They conclude that when pre-emption is allowed, the 6.rms do not adopt
this kind of technology at al1 in the equilibrium.
Reinganum [1983] uses a static game model to study the effect of uncertain prof
itability of the new technology on the equilibrium decision. She establishes (mixed)
Nash equilibria for different configurations of parameters, sucb 88, current marginal
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cost, acquisition cast, discount rate etc.
Gaimon [19891 uses a dynamic differential game framework to obtain the Nash
equilibrium timing decisions for firms (in a duopoly with identical firms) to add new
technology over time. Unlike other papers discussed in this section, use of a differential
game framework allows Gaimon to mode! technological progress and addition of finite
quantity (capacity) in different time periods.
Mamer and McCardle [1987) model a duopoly where the firms can either adopt a
new technology with high uncertainty, reject the technology altogether, or continue ta
gather information till the next period in arder ta reduce uncertainty. They fonnulate
a dynamic program recursion to find the Nash equilibrium strategy.
Hendricks (1992) introduces uncertainty in Fudenberg and Tirole's [1985J model
where the firms are uncertain about the innovative ability (ability to lead) of the rival.
He demonstrates that in sucb uncertain situations, rents are oot necessarily equalized
as found in the deterministic equilibrium.
Stenbecka and Tombale [1994) introduce unœrtainty into Reinganum's [1981a] and
Fudenberg and Tirole's [1985) models where the time it took for a firm to successfully
implement a oew technology is exponentially distributed. They find that due to the
introduction of sucb uncertainty, the adoption dates of the leader and the follower
are not independent of each other 88 found in the basic models. They conclude that
uncertainty causes more dispersion in the adoption dates than the deterministic case.
In the following, we focus on the papers by Reinganum [1981a], Fudenberg and
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Tirole [1985], and Kim, RoUer and Tombale [1994] that use a somewhat similar frame
work to analyze the optimal timing of adoption under different scenario. We present
the analysis in detail and compare the findings of these papers. However, since the
modeling techniques used and the assumptions made in these papers are somewhat
different, we shall take the liberty of altering some of the assumptions and techniques
(without altering the results) in order to present a coherent discussion.
2.3.1 Acquisition of a Single Product (Cast Reducing) Technology Let us suppose
that two firms play a Cournot quantity game in a duopoly. For simplicity, let us
assume that the firms are identical in the sense that cunently they use the same
technology (resulting in the same marginal cast of c), and they have the saIne discount
factor, r. Let the market demand function be given by
p = 0. - f3Q,
where 0. and (3 are parameters and Q is the total quantity produced and sold
by the firms. Since the firms are identical, their (per-period) profits would are also
equal, and are given by (see Appendix A for derivation)
9~ (0. - c)2
A new technology is made available that, if adopted, wOlÙd lead to a marginal
cost of é < c. Clearly, then, there is incentive for both firms to acquire it. However,
there is aIso incentive to delay acquisition, as the (discounted) acquisition cast p(t)
declines aver time. The decision is further complicated by the fact that whoever
acquires it first will have a competitive edge aver the rival (thus higher profit) until
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the rival follows suit. We adopt the following notation
?roo profit per period ta each wben none bas adopted
?r1O profit per periad ta the first adopter, when the rival bas not yet adopted.
1ral profit per period ta the second adopter, wben it bas not yet adopted.
?rIt profit per period ta each when bath have adopted.
Given that they play Cournot quantity game, we have
?roo = 9~(O - C)2
1r1O = 9~(O + c - 2c')2
?rOI = 9~ (0 + c' - 2c)2
1rll = 9~ (0 - c')2
From the above expressions, we obtain the following payot! ordering:
1r1O > 1rIl > 1roo > 1rOI
ln addition,
Sînce c' < c,
( 1rlO - 1roo) > (?ru - 1r0l),
which means that tbe incentive for the leader (1rlO -?roo) is always greater than the
incentive for the follower (1ru - ?rad. Also, the higher the reduction in the marginal
cost, the higher the incentive differential.
Let TL and TF be the adoption dates of the first adopter (the leader) and the
second adopter (the follower) respectively. Theo the leader's payot! is
47
•and the follower's payoff is
The derivatives of the payoffs with respect to adoption dates (when the rival's
adoption date is fixed) are given by
As noted before, the discounted acquisition cost declines over time. That Î8,
Assumption 2.1. P'(t) < 0
In order to have strictly concave payoffs, let us assume
AssumptioD 2.2. P"(t) > r(7rlO - 7roo)e-rt
Finally, let us make two more assumptions to avoid a corner solution and an indefinite
postponement.
Assumption 2.3. -P'(O) > (7r1O - 7roo) (Immediate adoption is too costly.)
Assumption 2.4. lim p(t)e-rt <~t-oc r
(For very large f, adoption is the dominant strategy.)
•
We are now ready to analyze the equilibrium timing decisions. We shall discuss
two equilibrium concepts. One is pre-commitment (or open-Ioop) equilibrium, where
firms make irreversible commitments at time zero and can Dot pre-empt one another.
The preclusion of pre-emption is equivalent to an infinite information lag. It is as
if the firms can not observe the rivais' actioDS. Pre-emption, on the other hand,
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would require the firms to instantaneously react to the rival's actions. Therefore, pre
commitment equilibrium is likely to hold for technologies that take long to acquire and
install. The second is pre-emption (or closed-Ioop, or feedback) equilibrium, where
adoption is instantaneous and observable, and the firms can pre-empt one another.
Allowance for pre-emption is equivalent to a zero information lag. This applies to
technologies that do not take long ta acquire and install, so that the rival can observe
and react instantaneously. In general though, it can be argued that the pre-emption
equilibrium has characteristics of subgame perfection sinee it is void of empty threats
as opposed to pre-commitment equilibrium.
Pre-commitment Equilibrium Suppose the firms make irreversible decisions at time
zero. Now sinee VL(t, TF) and VF(TL, t) are strictly concave, there must be unique
Ti and TF , 0 < Ti, TF < 00, that maxiIDÎze VL(t, TF) and VF(TL, t) respectively.
Furthermore, "fit, V{(t,TF) < V;(TL,t) and sinee V;(TL,T;) = 0, VL(t,TF) must
reach it's peak before T;. Therefore TF > Ti.
That is, the leader's payoff is maximized by adopting at Ti, irrespective of the
follower's adoption date, and similarly, the follower's payoff is maximized by adopting
at T;, irrespective of the leader's adoption date; and the leader's optimal adoption
takes place before the follower's. Therefore the pair (Ti,T;' ) represents the Nash
equilibrium adoption dates. In fact, there are two symmetric equilibria where the
firms interchange their adoption dates. The terms Ti and TF can be found from the
first order conditions
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(7T'lO - 7T'oo)e-rTi - P'(TiJ = 0
(7T'1l - 7T'Ol )e-rT; - ri (T;) = 0
Note that the derivatives with respect ta rival's adoption dates
aVL:';L,t) = (7T1O - 7Too)e-rt > 0
aVF~,TF) = (7T'oo - 7rode-rt > 0
That is, the firms' payoffs are monotonically increasing in the rivals' adoption dates.
As long as the firms make irreversible commitments at time zero so that the raIes
of leadership and followership are determined heforehand (in which case we do not
have symmetrical or identieal firms any more), this equilibrium will hold. However,
if there is no such raIe predetermination and the firms are truly identieal, note that
VL(TL,T;') ~ VL(T;',T;) [Ti is the best response ta TF]
= VF(T;',T;) [due to symmetry]
> VF(Ti,T;') [payoff increasing in rival's adoption date]
That is, the leader's payoff is higher than that of the follower, although they are
otherwise identical! The question then is how the leader is determined. Fudenberg and
Tirole [1985] note that this equilibrium is suspect hecause "the firm which is able to
pre-commit itself to adopt first does best, yet any firm can adopt first in equilibrium" .
They contend that l'the strategie interactions suppressed by pre-commitment would
resurface in a competition to he the first to commit". They propose to resolve the
problem by allowing the firms to pre-empt.
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Pre-emption Eguilibrium While pre-commitment is equivalent ta an infinite informa
tion lag, allowance for pre-emption is equivalent to an information lag of zero, where
adoption is complete1y observable and înstantaneous.
H the films are able ta pre-empt, and if one firm knows that the rival is planning
to adopt at Ti, its best response may no longer be TF. Rather, it should pre-empt
the rival at Ti - E, 88 by doing sa it may be able to earn a higher payoff than it would
have earned by adopting at TF. It becomes the leader and leaves the rival with no
other option but to adopt at T;. However, if the rival knows that its rival is planning
to pre-empt, the rival can pre-empt by adopting at Ti - 2E. Where does this chain
of pre-emption lead to? To answer this, let us consider the following.
Once one of the firms has adopted, the other firm's decision is an optimization
problem yielding an adoption date of T;. Let VJ(t) be the payoff to each when they
jointly adopt at t. Then, with TF fixed,
VL(T;,T;) = VF(T;,T;) = VJ(T;)
VL(t,T;) > VJ(t) for t < 1'; [payoff increasing in rival's adoption date]
VF(t,T;) > VJ(t) for t < T; [TF is the hest response ta t]
This assumption ensures that, for different combinations of marginal costs, the
quantities produced by the rivais are positive. The Cournot quantity of a firm, when
its own cast is c and rival's d is given by (ct + d - 2c)/3/3 (see Appendix A for
the derivation of Cournot quantities and payofl5). The lowest quantity in our model,
(O+C2 - 2c)/3/3, will he produced by the firm that has a cost of c while the rival's cost
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is C2. Therefore, for quantities ta he always positive in our model, a > 2c - C2. We
strengthen this requirement by assuming a > 2c. The assumption cao be interpreted
as a restriction on the market size. However, as Lemma 4.1 indicates, assumption 4.1
almost always bolds.
Lemma 4.1. Assumption 1 i.s always verified if the absolute value of priee elasticity
of the produet, lei :5 0.5; othenoise, it is verified if the maryin ratio ~ > 21el x n,where q is the total quantity produœd and sold, and M is the total market when
P=o.
Praof: see Appendix A.
Note that q/M is always less than 1, and in most cases, it is much less than 1.
AIso, the margin ratio is always greater than 1 for a profitable firm. Therefore, the
assumption will hold for m08t cases, as the condition ~ > 21ej x iï is likely to be
satisfied.
4.1.1 Conditions for Cournot-Nash Equilihria The payoffs for different outcomes
when the firms play a Cournot quantity game in the second stage can now be com-
puted (see Appendix A for derivation of payoffs). For example, the two-period payoff
to a firm which acquires Tl, hut not T2, while the rival does the opposite (does not
acquire Tl, and acquires T2 in the second period) is
82
• For any of the outcomes in the payoff matrix ta be a Nash Equilibrium (whereby
a firm 's decision is conditioned on a given action by the rival), there should not be an
incentive for either player to deviate. This means that six different conditions have
to he satisfied for a given outcome to be a Nash equilibrium. For example, the payoff
table clearly indicates that for (BD, DB) ta be an equilibrium, the set of conditions
2(2+r)(Q-c)2 _ 4(2+r)(Q-Cl )~C-Cl) +9.8(1+r)2 9.8(l+r) f
= (2+r)(c-cd2 + f which is alw8lm positive.
6.8(1+r)2 'J ~
Similarly for (li), (ili) and (iv). Proved.
Proposition 4. 7.
It can he proved using the same procedure as proposition 6.
Proposition 4.12.
Refer ta Figure 4.1. In order ta prave that for 0 $ Pl ::; al and ~ ::; P2 ::; b;,uncertainty increases welfare, we need ta show that for ~ $ P2 $ b;, (expected)
welfare at equilibrium is higher when C2 is uncertain than when C2 is deterministic.
Suppose 0 $ Pl ::; al. Also suppose P2 = ~ + f, 0 < f < :~~~1~~.
For deterministic C2, clearly the unique equilibrium is (BD, BD). At this point,
total surplus, producers' plus consumers', (ignoring the acquisition cast of the cur-
rently available technology, as they are acquired in both situations under comparison)
uili'b . . . b TB 4(2+r)(Q-cd2at eq num lB glven;y BD,BD = 9.8(1+rr1 •
Now assume that C2 is distributed with 8 mean of C2 and a variance of Var(C2).
The unique equilibrium now is (BB, BD). Relative ta (BD, BD), one of the firms now
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acquires the future technology as weIl. Expected total surplus minus the acquisition
cost of one unit of the second technology is given by
Now,
TSBB.BD - TSBD.BD =
(a+Cl-rC2f1+4Var(C2) + (o+C:I-2câ1+Var(C2) + (2a-cl-ë-.l)=z+Var(C2)9~( l +r)2 9/3(1+r}2 18/3( l.r)2
4(O-~}(Cl-ë-.l) _ _ 4(2+r)(o-ctl2
g8(1.r)2 E 9/3(1+r)2
E h limi· f h 4Var(COl)ven at t e upper t 0 E W en E = 98(1+r)5,
TB TB (Ct-é2)2+Var(COll hich· al ·f· l' TBBB.BD- BD.BD = 6t3(1+r)'l , W 15 ways pOSl Ive, unp ymg BB.BD >
TSBD.BD .
This also must hold for any E < :~~1~~. Proved.
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Appendix B:
Chapter 5 • Proofs
Proof of Lemma 5.1
Hy Assumption 5.1, we know that there is no overlap in technology acquisitions;
that is, during the life span of one technology, only that technology is acquired, and no
other technology. AIso, the optimal acquisition amount for any technology in a period
depends on (i) the amount of the same technology acquired since it was introduced,
(ü) the amount of oider technologies on band, and (Hi) the expected product demands
during the life span of the technology (but not on demands beyond its life span, as is
implied by Assumption 2). Consider period 1. Since there is no previous technology,
the optimal amount of VI will depend on the demands only dwing its lire span. Once
the optimal amount of Vl is determined, when V2 arrives, since the capacity of Vl is
fixed, the optimal amount of V2 cao be optimally detennined by the demands during
its life span without considering Iater technologies. Proved.
Proof of Proposition 5.1
We prave the prapœition by contradiction.
The demands for products i = 1, ... ,1 producable by technology \Ii in periods t 1
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(with the elements of D sorted in ascending order). Finally, let Yn be the nth element
of D for n = 1, ... , N, where N is the number of elements in D.
New suppose the optimal amount at t l is x ~ D, and that the two closest elements
to x in D are Yn and Yn+l sucb that Yn < x < Yn+l. Then a fixed amount of demand
(x - Yn) is met at certain prices, say Pt, in periods t = t l, ... , t2'
Successive units of Vj upto Yn+ l will earn revenue at the same rate Pt in periods
t = t l, .. , t 2 , and will cast less and less because of the concavity of the cost function.
Therefore, if it is profitable to buy x units of Vj, then it is even more profitable to
buy Yn+ l units of Vj. Proved.
Proof of Proposition 5.2
Let ~T, i = 1, .. ,T, represent the amount of Vi bought in perlod i, when the
planning horizon is T. Also let the profit earned from amounts ~T be P(yt +Yi +
.... +Yi). Finally, let xt he the optimal amounts of Y/
The decision problem can be stated as . max P(~T + ~T + .... + Yi)yt.y,{.··,YI
