RATIONAL EXPECTATIONS, OPTIMAL CONTROL
AND INFORMATION TECHNOLOGY ADOPTION
Yoris A. Au
Assistant Professor
Information Systems Department, College of Business
University of Texas at San Antonio, San Antonio, TX 78249
yau@utsa.edu
Robert J. Kauffman
Professor and Director, MIS Research Center
Chair, Information and Decision Sciences, Carlson School of Management
University of Minnesota, Minneapolis, MN 55455
rkauffman@csom.umn.edu
Last revised: April 23, 2004
Forthcoming in Information Systems and E-Business Management.
____________________________________________________________________________________
ABSTRACT
The existing economics and IS literature on technology adoption often considers network
externalities as one of the main factors that affect adoption decisions. It assumes that potential
adopters achieve a certain level of expectations about network externalities when they have to
decide whether to adopt a particular technology. However, there has been little discussion on
how the potential adopters reach their expectations. This article attempts to fill a gap in the
literature on adoption of information technology (IT), by offering an optimal control perspective
motivated by the rational expectations hypothesis (REH) and exploring the process dynamics
associated with the actions of decision makers who must adjust their expectations about the
benefits of a new technology over time due to bounded rationality. Our model primarily
addresses technologies that exhibit strong network externalities. It stresses adaptive learning to
show why different firms that initially have heterogeneous expectations about the potential value
of a technology eventually are able to arrive at contemporaneous decisions to adopt the same
technology, creating the desired network externalities. This further allows the firms to become
catalysts to facilitate processes that lead to market-wide adoption. We also discuss the
conditions under which adoption inertia will take over in the marketplace, and the related
managerial implications.
KEYWORDS: Adaptive learning, adoption, bounded rationality, economic theory, network
externalities, optimal control, rational expectations, technology adoption.
______________________________________________________________________________
ACKNOWLEDGEMENTS. The authors wish to thank the participants in the Information and Decision Sciences
Workshop at Carlson School of Management of University of Minnesota, and the anonymous reviewers of this
article and a related paper for helpful suggestions. We also appreciate input offered by Norm Chervany, Eric
Clemons, Rajiv Dewan, Fred Riggins, Paul Glewwe, Paul Johnson, Rajiv Sabherwal, Jan Stallaert and Chuck Wood.
An earlier version of this research appeared in the Proceedings of the 2002 INFORMS Conference on Information
Systems and Technology, and reflects the input of several helpful reviewers. A related paper is Au, Y. A., and
Kauffman, R. J., “What Do You Know? Rational Expectations and Information Technology Investment,” Journal of
Management Information Systems, 20, 2 (Fall 2003), 49-76.
1
INTRODUCTION
This article attempts to fill a gap in the literature on adoption of information technology (IT)
[Choi and Thum, 1998; Jensen, 1982; Katz and Shapiro, 1985 and 1986 ; Kauffman, McAndrews
and Wang, 2000] by offering a perspective motivated by the rational expectations hypothesis
(REH) [Muth, 1961; Sheffrin, 1996] and exploring the process dynamics associated with the
actions of decision makers who adjust their expectations about the benefits of a new technology
over time in line with bounded rationality. Evans and Honkapohja [2001] define rational
expectations as the mathematical conditional expectation of the relevant variables in a decision
making or forecasting setting. We will focus on the adoption of technologies that exhibit strong
network externalities, and hence, rational expectations will apply to the conditional expectations
of decision makers who are interested in technology adoption-related variables.
Muth [1961, p. 316] states the REH as follows: “I should like to suggest that expectations,
since they are informed predictions of future events, are essentially the same as the predictions
of the relevant economic theory.” Begg also comments [1982, p. 30]: “The hypothesis of
rational expectations asserts that the unobservable subjective expectations of individuals are
exactly the true mathematical conditional expectations implied by the model itself.” REH
suggests that economic agents form expectations based on the “true” structural model of the
economy in which their decisions are made. Their subjective expectations are informed
predictions of future events and are the same as the conditional mathematical expectations based
on the “true” probability model of outcomes in the economy.
Although REH has since been widely used in other areas of microeconomics and
macroeconomics, and notwithstanding the fact that rationality is commonly assumed in business
and economic analyses [Zimmerman, 1986], this article is one of the first attempts to explicitly
2
bring rational expectations economics thinking into the IS field and to use the theory for
understanding the complex issues of technology adoption. We will examine three research
questions in this article:
□
How do we represent a technology adoption situation where network externalities
considerations require potential adopters to adjust their expectations based on what they
observe in the market over time?
□
How can we characterize such a situation and model it with a control theory model based
on the REH? How can we conceptualize the optimal control in an adoption setting?
□
What are the possible actions that the potential adopters may take, as a result?
Although technology researchers have long analyzed the role of expectations about perceived
business value and network externalities in technology adoption, how potential technology
adopters reach their observed level of expectations has not been fully understood. So being able
to understand the process through which this occurs will help to shed new light on the conditions
that enable adoption inertia to either take over or be overcome in the marketplace.
We will first discuss a multi-period model that represents the kind of optimal control decision
a potential technology-adopting firm has to make. This involves determining at which point in
time the firm should make the decision to adopt in order to maximize the net benefits from the
new technology. We then look into a situation where potential adopting firms, prior to making
an adoption decision, can freely share information with each other and utilize the information to
obtain a better understanding of the value of the new technology and other related issues.
Since the focus is on technologies with strong network externalities, our analysis will be
based on an important assumption: that firms prefer to adopt a technology at about the same time
in order to be sure that they will obtain a more attractive level of network benefits. To this end,
3
we develop a model in which all potential adopters as well as the technology supplier—as
economic agents—perform adaptive learning, and adjust their forecast rules for the accrual of
network externalities as new data become available over time. Our analysis of the resulting
equilibrium is motivated by the works of Bray and Savin [1986], Fourgeaud, Gourieroux and
Pradel [1986], and Evans and Honkapohja [2001], and suggests the kinds of outcomes that may
be observed in the marketplace. The optimal control perspective [Sethi and Thompson, 2000]
that we develop is intended to deal with the question of how the evaluation of technology
benefits and costs are evaluated over the time horizon in which adoption may occur.
Our adaptation of the REH to technology adoption in a control theory model allows us to
treat the technology adoption issues using a bounded rationality perspective. Technology
adoption decisions are generally complex in nature due to the continuous development of the
technology, the competitive interactions of firms, and the path-dependent growth of externality
value. As a result, managers need some time before they can really decide what to do with any
new technology. Many adoption decision making processes, therefore, can be seen as ongoing
processes that involve observations in the marketplace over time, rather than quick decisions.
Consequently, we offer a new approach to improve our understanding of technology adoption.
LITERATURE
A brief summary of the role of expectations in the formation of economic theory and analysis
is offered by Evans and Honkapohja [2001]. They maintain that expectations began to play a
major role in economics as early as the 1800s, with Henry Thornton’s 1802 treatment of paper
credit, and Émile Cheysson’s 1887 formulation of a framework which had features of the
4
“cobweb model.” 1 Hicks [1939] is recognized to have offered the key systematic exposition of
the temporary equilibrium approach, in which expectations about future prices influence demand
and supply in a general equilibrium context. Muth [1961] formulated the REH.
The REH would not be as popular as it is today without extensive additional work
undertaken by macroeconomic theorists. The papers of Lucas [1972, 1973, 1975], Sargent and
Wallace [1976], Lucas and Sargent [1981] and others with new explanations of output and
inflation pushed Muth’s concept to the vanguard of contemporary economic thinking. This has
led to developments in macroeconomic theory and time-series econometric research that have
opened up new ways of thinking about the management of modern economies [Pesaran, 1987].
Lucas [1975] interprets the REH as a hypothesis that assumes that every economic agent
optimally utilizes available information in forming expectations. Grossman [1981] has argued
that the essence of rational expectations is that economic agents have some economic model of
the economy which relates the exogenous variables to the endogenous variables which must be
forecasted. The REH is based on two assumptions. Economic agents form beliefs based on a
given set of information. Their expectations will be conditioned on all available information.
Also, agents will know the stochastic process that generates the equilibrium condition.
The second assumption is what makes the REH unique. However, this assumption often is
considered to be too strong. Why? It requires economic agents to have full knowledge of the
structure of the relevant models and their parameter values. Considering that decision makers as
economic agents have bounded rationality, Sargent [1993] suggests an alternative notion,
1
The basic idea behind the cobweb model is that an agricultural producer who grows a small amount of
grain will benefit because high demand will permit him to obtain an attractive price for his crop. The
natural response, then, is for the producer to grow more the following year. But every producer will also
respond in a similar manner, driving supply up and reducing prices relative to demand, and thus,
negatively impacting producers’ profits. All producers will subsequently adjust production, but it
typically will take a number of periods before a stable equilibrium can be reached.
5
adaptive learning, in which agents are assumed to be willing and able to update their
expectations about relevant parameter values on the basis of newly-received information.
We adopt an assumption consistent with REH adaptive learning: the technology supplier and
potential adopters know about the economic structure (e.g., the supply of and demand for) of the
technology over time but must learn the parameter values that reflect each potential adopter’s
willingness-to-pay. Stoneman and Ireland [1983] have shown the importance of supply-side
factors in the adoption of technological innovations. The willingness-to-pay of other firms can
be estimated by a potential adopter through its communications with other firms in the market.
To support our readers’ understanding of the concepts that we are introducing from an area
of economic theory that is new in IS research, we include a table of definitions for the primary
terms and concepts that are drawn from rational expectations economics theory that we will
shortly use to develop our new perspective on IT adoption. (See Appendix 1.)
OPTIMAL CONTROL MODEL DEVELOPMENT
In the optimal control model [Sethi and Thompson, 2000] that we develop, we treat the
adoption of technologies with strong network externalities from the perspective of REH and
adaptive learning. We first discuss the role of first-mover advantage in technology adoption and
contrast it with the impacts of network externalities on what is observed to occur in the market.
First-Mover Advantage, Network Externalities and Technology Adoption
There are many different reasons why a firm may choose to adopt a new technology earlier
or later. Earlier adoption of emerging technologies gives entrepreneurial firms first-mover
advantage based on a strong market reputation, technology leadership, and brand loyalty.
Shapiro and Varian [1999] maintain that first-mover advantage can be powerful and long-lasting
for firms that can establish an installed base before the competition arrives to make it difficult for
6
later entrants to achieve the scale economies necessary to compete. This implies that first-mover
advantage can be short-lived or even fail to materialize if early entrants are unable to maintain
their dominant position. First-mover advantage corresponds closely with the risk-taking
behavior of decision makers. The common wisdom from the strategy literature suggests that
market pioneers will have higher returns if they are successful, but at the same time they bear a
higher risk of failure [Shepherd, 1999]. Early technology adopters are more willing to take risks,
whereas late adopters are likely to more risk-averse. Decision makers who are more averse to
risk, by the same token, will be more likely to make their decisions later, when enough
information has become available and they can process it properly. In fact, an important
advantage that later adopters have is the ability to learn more about a technology and to observe
market acceptance before irrevocably committing limited resources.
The model that we will propose will treat situations where most of the benefits of a new
technology come from network effects (e.g., electronic bill payment and presentment, high
definition television, etc.). We will investigate a case where some firms (that share similar
characteristics or serve similar markets, for example) choose to adopt the new technology at
about the same time to avoid the risks of being “stranded.” Stranding occurs when only a firm or
two adopt the technology and the other firms decide not to, consequently diminishing the
expected network benefits. Our model will incorporate an assumption that the network benefits
will significantly outweigh other kinds of potential benefits, such as the first-mover advantage.
When potential benefits of a technology mainly come from network externalities, adoption
may be delayed because it may take some time before decision makers reach a consensus
among themselves about the economic benefits that an emerging IT may bring. This creates
interesting dynamics in the adoption decision making process. Firms must now observe each
7
other’s actions and take cues from each other before making a decision. In fact, they may share
private information about their costs, or even signal to other firms their interest in order to
achieve earlier accrual of the network externalities.
An Optimal Control Model for Network Technology Adoption
We begin with an optimal control model in which a firm seeks to maximize the net benefits
from adopting a new technology. Since our focus is on an IT with network externalities, the total
benefits can be formulated based on a function of stand-alone and non-network value, and
network externalities benefits, as a + b(N). In this expression, the term a represents the standalone and non-network benefits that result from the features of the IT artifact that is involved;
b(N) denotes the benefit from a network of size N, with the function b representing the strength
of network externalities [Farrell and Saloner, 1986; Kauffman, McAndrews and Wang, 2000].
In our model, we use the term at to represent the non-network benefits (which include, among
others, the stand-alone benefit and the first-mover advantage) in period t after implementation
occurs in period t0. We denote the fixed implementation cost as ft0, and the recurring cost in each
period t following implementation as ct. We also consider the time interval [t0, T] over which an
adopting firm optimizes value, with T representing the lifespan of the technology. Finally, we
suppress the notation for the time value of money, and assume that there is a one-period lag
before the firm starts realizing the benefits. The resulting objective function for each firm is to
maximize the net benefit:
T
Maximize [ ∑ at + b( N t ) − ct ] − f t 0 for each t from t0 + 1 to T, and for each
t =t0 +1
t0 ∈ {0, K, T − 1} , subject to N t ≥ 0 , t0 ∈ {0, 1, 2, K, T − 1} and t ∈ {t0 + 1, K, T }
The variables are defined as follows:
(1)
8
Nt
=
number of adopting firms in period t
b( N t )
=
network benefits associated with Nt firms adopting by period t
t0
=
the period in which the technology is adopted and implemented
ft0
=
technology implementation cost incurred in adoption in period t0
ct
=
technology variable cost in period t ∈ {t 0 + 1, K, T ]
T
=
adoption valuation duration or lifespan of the technology
The model suggests that each firm will dynamically compute the sum of the net benefits for
each potential adoption period. Starting from period t = 0, each firm will perform the
computation for time t0 = 0, t0 = 1, …, t0 = T - 1 and determine which period will bring the
maximum sum of net benefits. Therefore, in period t0 = 0, a firm will compute the following:
T
t0 = 0 : π 0 = [∑ at + b( N t ) − ct ] − f 0
t =1
:
:
T
t0 = T − 1 : π T −1 = [∑ at + b( N t ) − ct ] − fT −1
(2)
t =T
It will also determine the maximum profit opportunity among π0, π1, …, πT-1. If the
maximum is π0, then the firm will adopt the technology at t0 = 0. Otherwise, it will wait until the
next period, t0 = 1, and compute the following:
T
t0 = 1 : π 1 = [∑ at + b( N t ) − ct ] − f1
t =2
:
:
T
t0 = T − 1 : π T −1 = [∑ at + b( N t ) − ct ] − fT −1
t =T
The firm will then determine the maximum of π1, π2, …, πT as before. Similarly, if the
(3)
9
maximum is π1, then the firm will adopt the technology at t0 = 1. Otherwise, it will wait until the
next period t0 = 2 and recalculate the sum of net benefits for each period starting with t0 = 2,
determine the maximum, and see if the maximum is π2. If not, the process will be repeated until
the maximum coincides with the current period, signifying the optimal adoption period, t0*.
The need for recalculation each period is due to the expected number of adopters, as well as
their individual expectations about the benefit and cost components, may be updated over time
based on new information they receive. New information may result from a firm’s effort in
monitoring the environment or it may come from another firm motivated to persuade others to
adopt the technology to realize the network benefits. This is known as signaling [Spence, 1973].
A firm should adopt the technology only if the total benefits minus the recurring costs during
T
the period [t0+1, T] exceed the implementation cost, ft0, i.e., [ ∑ at + b( N t ) − ct ] > f t 0 .
t =t0 +1
Furthermore, to maximize the net benefits in the optimal control problem, the firm should adopt
and implement the technology as soon as the total benefits at + b(Nt) exceed the recurring cost ct,
but not any earlier. Why? Because any additional positive net benefit (i.e., at + b(Nt) – ct > 0)
will increase the sum of the net benefits, but an adoption and implementation that is done too
early before the net benefit becomes positive will only reduce the sum of the net benefits.
New technologies show increasing benefits and decreasing costs over time. They are
enhanced and include more features as they become cheaper due to competition and efficiency.
When a technology exhibits network externalities, most of the benefit increases will come from
the growth of adopters over time. As soon as the expected number of adopters reaches a certain
level that generates a large enough network benefit that, together with the stand-alone benefit,
will offset the recurring cost, a firm should implement the technology to maximize net benefit.
10
Signaling and Information Processing
Signaling occurs when firms are still in the midst of their adoption decision making
processes. Due to the desire to realize the network benefits, a firm that would like to see a new
technology adopted will signal a favorable opinion of the technology to the other firms. A firm
may make a positive signal because it has experience dealing with similar technologies, has
better forecasting infrastructure and skills, or has a more upbeat managerial outlook.
Signaling may or may not incur costs depending on how the signals are sent by the firm.
When a firm can transmit information freely through a process known as cheap talk, regardless
of how other firms accept and process the information, the signaling costs will be insignificant.
When there are significant costs involved—an extreme case is signaling via a pilot
implementation of the technology under consideration—a firm will be more selective in
transmitting any information and may wait until it is certain that other firms are ready to process
the information it transmits. This readiness is measured by a firm’s information processing cost.
The higher the information processing cost, the less ready the firm to process the information it
receives.
Our general optimal control model can be rewritten to include various types of signaling,
from costless to costly signaling. A firm that is willing to signal will incur some cost if it can
reasonably expect that the effort will accelerate the adoption rate, allowing the firm and other
firms to enjoy a higher sum of net benefits during the life cycle of the technology. The
signaling cost creates real options since it allows the firm to make an investment and then
decide whether to invest fully in the technology based on the resulting signals.
A RATIONAL EXPECTATIONS PERSPECTIVE ON TECHNOLOGY ADOPTION
As we have seen in the optimal control model, a necessary condition for adoption and
11
implementation would be that the total net benefits over the period exceed the implementation
cost. We now place this model in the rational expectations context with costless signaling.
Modeling Preliminaries
Our rational expectations model consists of a group of N potential technology-adopting firms
and one technology supplier firm. The size of the group N is the minimum number of adopters
that will trigger the adoption and implementation. Although we assume there is only one
supplier, other suppliers will come into play with newer and better technologies. Considering
that the markets for new and emerging ITs are generally contestable and have low barriers to
entry [Baumol, Panzer and Willig, 1982], the supplier in our model cannot exercise monopoly
power since the potential adopting firms will have choices with the arrivals of future suppliers of
a similar technology. Moreover, as we have stated earlier, the technology that we consider
exhibits strong network externalities characteristics, yielding higher value to adopters as their
number increases.
Our main conjecture is that all N potential adopters will make the adoption decision at about
the same time: when they learn that all adopters in their group are ready to adopt. This group
may consist of firms that share similar characteristics or serve similar markets, signifying the
actual potential for realizing the network benefits. For example, in the new technology arena of
electronic bill presentment and payment (EBPP), a group may include financial services,
telecommunications, insurance and utilities firms that serve the same geographical market.
These firms can expect network externalities benefits to be created when they adopt the
technology together since consumers will be more likely to adopt the EBPP service as the
number of firms (i.e., billers) offering the service increases [Au and Kauffman, 2001 and 2003].
12
Willingness-to-Pay, Bounded Rationality, and Rational Expectations Adoption
A learning process is necessary since it will take some time before each of the firms reaches
a “comfort level” to make a decision due to the fact that they may initially have different value
expectations about the technology, reflecting different levels of willingness-to-pay. This might
be because they can convert IT investments into value more quickly than other firms, or because
they experience different levels of implementation costs or recurring maintenance and
enhancement costs for the technologies and systems they adopt. When signaling is free, any firm
can transmit information freely, regardless of other firms are going to process the information.
In our rational expectations model, the decision rule for each firm in every period (i.e., either
to adopt or not adopt the technology) is based on the expected value of the network of adopting
firms, adjusted for fixed implementation and variable recurring costs for maintenance and
enhancement, as described previously in the optimal control model. We operationalize this for
the rational expectations analysis in terms of the price expectation of each of the potential
adopting firms relative to the new technology. The price expectation captures information about
the business value of the new technology, in the context of market supply and demand. If all
firm’s price expectations (and hence, their willingness-to-pay) are at least equal to the vendors’
selling prices of the technology, then, according to our modeling interpretation, every firm will
decide to adopt the technology in the period; otherwise, none of the firms will adopt. Based on
the concepts of REH and adaptive learning—and due to bounded rationality—firms may start out
with different price expectations and different levels of willingness to pay, some of which may
be below the selling price set by the technology supplier.
For example, some (and possibly all) of the price expectations may be lower than the selling
price set by the technology supplier. In every period, each firm will adjust its willingness-to-pay
13
to reflect changes that may have occurred during the past period. In addition, the technology
supplier may also adjust its selling price in each period to reflect its own reading of market
demand. As the constraint in the optimal control model suggests, the adoption of the technology
will take place only when all potential adopters’ willingness-to-pay levels are at least equal to the
selling price of the technology. As we will shortly show, this will occur in a process that reaches
the rational expectations equilibrium (REE) point [Fryman, 1982]. There, all acts of learning
are complete, in the sense that there is no more incentive on the part of economic agents to
change the beliefs they hold about their economic environment.
The Rational Expectations Model Analysis Process
In our modeling analysis, we first will look at how the technology supplier reaches the REE.
We will then discuss how the heterogeneous willingness-to-pay levels—stated as price
expectations for the new technology—of N potential technology-adopting firms converge to the
REE. Once the relevant constraint in the optimal control model is satisfied, all firms should
adopt, though they have the option to wait longer to see if their price expectations are correct.
This is based on the idea that firms have unique levels of willingness-to-pay, the assessments of
which may be perceived in the market as being “accurate-within-an-error.” The error results
from the information processing limitations that most firms have. As a result, they typically will
know that they may not know enough to make the best estimate, which is common with decision
problems involving infinite regress [Kelly, 2000; Melberg, 1999].
All N potential adopting firms share the same interest: each would like to see the others adopt
to maximize the benefits of the technology. This guides the behaviors of the decision makers of
the firms to encourage the free flow of information. Although willingness-to-pay cannot be
observed directly, potential adopters will be able to utilize as a proxy some of the observable
14
economic indicators at the firm, industry, and macro level. In addition, we assume that each firm
is willing to share its assessment of the new technology with other firms, directly through interfirm exchanges and indirectly through market signaling. We further assume that the signaling is
costless. However, since each firm may have different information processing capabilities, it will
take some time before their expectations converge to reach an equilibrium.
THE TECHNOLOGY SUPPLIER’S RATIONAL EXPECTATIONS MODEL
We consider a multi-period scenario in which the new technology supplier predicts the price
at which all potential adopters are willing to pay for the technology. Similar to the role of price
expectations relative to willingness-to-pay on the adopter side, the price the supplier establishes
also can represent the technology supplier’s assessment of what adopters are willing to pay.
Modeling Assumptions
We will use a market-clearing condition for supply and demand to show that, under certain
conditions, the technology supplier will be able to eventually arrive at the correct prediction of
the price that represents the willingness-to-pay of all the potential adopters. We also will model
how potential adopters arrive at the same price expectations under learning, despite their initially
heterogeneous levels of willingness-to-pay.
The technology supplier must predict the willingness-to-pay of the potential adopters of its
technology in every period t −1 to plan for the resources (e.g., people, equipment) needed to
serve the market in period t. The supply of the technology will depend on the supplier’s
expectation of the market’s average willingness-to-pay. In the period when adoption occurs, this
price is the actual price that each of the adopters will pay for the technology.
We further assume that all potential adopters—as well as the technology supplier—will be
able to assess others’ willingness-to-pay. If each of the potential adopters (through whatever
15
means are available to them in the marketplace) were able to reveal its willingness-to-pay and
adjust its valuation, all will reach a consensus to purchase the technology at the average price.
Although we assume free flow of information, in the early periods adoption still may not occur.
Why? Because each potential adopter’s bounded rationality prevents it from immediately
adjusting its willingness-to-pay. Firms adjust their expectations based on new available
information, which includes price expectations of other firms and other observable variables.
These willingness-to-pay levels only become known to each of the agents at the end of
period, when more information will become available causing each potential adopter to adjust its
own willingness-to-pay. This process will repeat itself until an equilibrium is reached, and the
willingness-to-pay levels across firms in the market converge. Our assumption that each
potential adopter attempts to learn the others’ willingness to pay is supported by the extensive
contacts that occur among firms whose senior managers, strategic planners and IT professionals
sample the market’s attitude relative to the adoption of a technological innovation.
Pseudo-Prices, Willingness-to-Pay, Demand and Supply
We let ptp be the pseudo-price in period t. The word pseudo-price is intended to highlight
the contrast between demand based on the actual price that potential adopters will pay if each
eventually decides to purchase the technology, as opposed to the expected demand that is
revealed prior to actual adoption. Thus the pseudo-price for a period indicates the price that
would prevail if adoptions occurred in that period. Recall that one of our main assumptions is
that information flows freely due to the interests that potential adopting firms share. This
creates the willingness for firms to send signals to each other either directly or indirectly,
enabling each to know the others’ willingness-to-pay for the new technology and, thus, the
pseudo-price in every period prior to the adoption. However, this pseudo-price will change from
16
period to period, as firms adjust their willingness-to-pay based on newly-available information.
(Appendix 1 provides details for the modeling notation used in the remainder of the paper.)
We represent the technology demand and supply functions in period t with dt and st, and
analyze the implications of demand-supply equilibrium in a manner similar to Evans and
Honkapohja [2001]. Technology demand in period t is given by:
d t = β 0d − δ ptp + ε td , with δ > 0
(4)
β0d is the intercept of the demand function, and εtd represents unobservable drivers of new
technology demand. The parameter δ indicates demand is decreasing in willingness-to-pay.
Just like the potential adopters, the supplier will be able to observe the pseudo-price in period
t - 1. The supplier can use this to estimate the supply price, pts , which also is an estimate of the
adopters’ willingness-to-pay. The expected price pts that the supplier estimates in period t − 1 will
determine the pseudo-price in period t as ptp . The supply function in period t is:
s t = β 0s + β s X t −1 + σ p ts + ε ts , with σ > 0.
(5)
Here X t −1 is a vector of observable exogenous variables in period t – 1 that affect the level of
supply. β 0s is the intercept of the supply function, β s is a vector of parameters for the variables
that affect supply, and ε ts represents the unobservable drivers of the new technology supply. The
parameter σ indicates that the supply for the new technology is increasing in price.
Another interpretation of the pseudo-price discussed above is the market equilibrium price of
the new technology under consideration for adoption. In equilibrium, each potential adopter is
willing to pay this price, and the supplier is willing to sell the technology at the same price. The
equilibrium price itself is determined by the intersection of the supply and demand curves,
17
dt = st. Substituting Equations 4 and 5 into the demand-supply equality and rearranging terms
yield the following expressions:
β 0d − δ ptp + ε td = β 0s + β s X t −1 + σ pts + ε ts , p tp = 1 ( β 0d − β 0s ) − σ pts − 1 ( β s X t −1 ) + 1 (ε td − ε ts ) ,
δ
1
ptp = w + v pts + u X t −1 + ε t , with w = ( β 0d − β 0s ) ; v = −
δ
δ
δ
δ
1
σ
1
; u = − ( β s ) ; and ε t = (ε td − ε ts ) (6)
δ
δ
δ
In these expressions, the parameter w is the proportional difference between the intercept of
the demand and supply curves. The greater the proportional difference, w, the higher the market
clearing price and quantity. The parameter v is the ratio of the slope of the supply curve to that
of the demand curve. Any plausible economic demand and supply situation will have an
absolute value, |v| < 1. The parameter u represents the magnitude to which the exogenous
variables will negatively affect the price. The parameter, εt, incorporates ε td and ε ts , the
unobservable drivers of demand and supply that result in the market-clearing price.
The Technology Supplier’s Rational Expectations Model: Analysis
We next show that the technology supplier learns the potential adopters’ expectations and
adjust its own expectation to reach a condition that allows the adoption of its new technology.
Recall that pts is the supplier’s expected price determined in period t – 1. The REH maintains that
the expected value of pts is the optimal prediction made in period t – 1: the best linear predictor
which minimizes the expected prediction’s squared error. This assessment is based on a set of
information, It-1, a vector of observable variables, Xt-1, some of which may be random variables.
Following Fourgeaud, Gourieroux and Pradel [1986], we define the optimal prediction of pts as a
linear regression of ptp on I t −1 . The random variables reflect real world drivers (e.g., firm
18
announcements, a technology toward becoming standard, etc.) that may affect willingness-to-pay
for potential adopters, and hence, the observed pattern of technology adoption decision making.
Substituting E ( ptp | I t −1 ) for pts yields another expression for the pseudo-price:
ptp = w + v E ( ptp | I t −1 ) + u X t −1 + ε t
(7)
Taking the optimal linear prediction of both sides of the Equation 7 gives:
u X t −1
E ( ptp | I t −1 ) = w + v E ( ptp | I t −1 ) + u X t −1 or E ( ptp | I t −1 ) = w +
1− v
1− v
(8)
Plugging Equation 8 into Equation 7 leads to:
ptp =
w
u
+
X t −1 + ε t , v ≠ 1
1− v 1− v
ptp = y + z X t −1 + ε t , with y =
w
u
and z =
.
1− v
1− v
(9)
This gives a unique rational expectations equilibrium (REE) because ptp does not depend on
the expectation of future prices for the new technology. As pointed out previously, a REE refers
to a state in the marketplace where all acts of learning are complete; so there is no more incentive
on the part of agents to change the beliefs they hold about their economic environment. Pesaran
[1987, p. 33] maintains that a REE can be characterized by three main features: all markets clear
at equilibrium prices; every agent knows the relationship between equilibrium prices and private
information of all other agents; and, the information contained in equilibrium prices is fully
exploited by all agents in making inferences about the private information of others.
We now introduce adaptive learning into the process that occurs as buyers form their
willingness-to-pay levels for adoption of the new technology. When learning is possible, the
supplier’s prediction of price, pts, is not optimal; however, it is expected to improve over time.
The supplier believes that pts follows the process described in Equation 9 and corresponding to
19
the REE, but must estimate y and z. This signifies that in forecasting the price, the technology
supplier acts like an econometrician or a forecaster:
ptp = y + z X t −1 + ε t
(10)
Equation 10 depicts the perceived law of motion (PLM) of the firm [Evans and Honkapohja,
2001], suggesting that the firm attempts to estimate the parameters y and z. PLM is the
stochastic process generating the equilibrium condition that economic agents believe in. This is
the key bounded rationality assumption. In forecasting the price, the decision makers in the firm
act in aggregate like an econometrician who believes that the economy has reached equilibrium.
They use all available data to estimate the parameters of Equation 10. We further assume that the
technology supplier estimates y and z using an ordinary least squares (OLS) regression of pip on
X i , ( i = 1, ..., t − 1 ). This is called least squares learning. As a result, in period t − 1 , the forecast
is made using the following least squares regression equation:
ptp = yt −1 + zt' −1 X t −1
(11)
Convergence and Expectational Stability
Next we will see that the parameters y and z indeed converge to certain values over time and
that the convergence is stable, indicating that the supplier’s price estimate (or expected price)
will eventually match the potential adopters’ average willingness-to-pay. Recall that this is one
of the important assumptions that we made for technology adoption to occur in the first place: all
potential adopters must eventually arrive at the same willingness-to-pay and the technology
supplier must be willing to sell its technology at the price that all the potential adopters are
willing to pay.
Bray and Savin [1986] have shown that the convergence of least squares learning will be
obtained if and only if v < 1. To show that the convergence that characterizes the rational
20
expectations equilibrium is stable, it is necessary to use the expectational stability principle of
Evans and Honkapohja [2001].
DEFINITION (THE EXPECTATIONAL STABILITY PRINCIPAL). A rational expectations
equilibrium is said to be e-stable if, given a small deviation of expectations functions from
rationality, the system returns to the equilibrium under a natural updating rule.
Imposing e-stability can narrow the number of acceptable solutions, and in some but not all
cases, there will a unique locally stable equilibrium [Evans, 1985].
Thus, under least squares learning, the technology supplier in our model will be able to learn
the potential adopters’ expectations and adjust its expectations so convergence results between
its price for the new technology and the adopters’ willingness-to-pay so adoption occurs. The
managerial implication is clear. To be successful, a technology supplier that enters the market
with a new technology that shows the appropriate network externalities characteristics should try
to learn about willingness-to-pay in the market and keep adjusting its prices based on signals
from the potential adopters and other relevant indicators. The model also reveals that, through
information sharing, it is possible for the technology supplier to eventually cause technology
adoption to occur despite the initial differences in expectations among potential adopters.
THE HETEROGENEOUS ADOPTERS’ RATIONAL EXPECTATIONS MODEL
We will now show how heterogeneous adopters may eventually reach the same willingnessto-pay, despite their initially different expectations. Since there are N potential adopters, there
can initially be as many as N different levels of willingness-to-pay or price expectations. Muth
[1961] suggested that the price expectations of firms tend to be distributed even for the same
information set. As a result, we presume that a distribution of willingness-to-pay levels will be
observed. The distribution will change over time as each firm adjusts its expectations based on
newly-revealed information in each period. Such stochastic changes are captured by the
21
dynamic system in our model. These different expectations also are due to bounded rationality:
each firm may have a different ability to access and process information. However, all firms
learn asymptotically over time to reach the REE.
Analysis of the Heterogeneous Potential Adopters Model
At the beginning of the period t, each potential adopting firm i holds its own price
expectation, pi, which is formed based on the set of available information conditional on each
firm’s bounded rationality. The price reflects the potential adopter’s willingness-to-pay for the
technology, as before. Now since the information available to each potential adopter may vary,
the prices that they hold can be different from each other. We assume that each potential
technology adopter is able to forecast according to a linear rule:
pie, t = zi , t −1 X t −1
(12)
Each potential adopter arrives at different parameter estimates, however. As with the technology
supplier, all potential technology adopters also learn from past prices, as the past perceptions of
the market’s willingness-to-pay (represented by ptp−1 ), and the observable variables X t −1 .
We further assume that the average potential adopters’ expectations in Equation 13 affect the
market price which is given by the model in Equation 14:
p te =
1
N
N
∑p
i =1
e
i, t
ptp = v pte + u X t −1 + ε t
(13)
(14)
Substituting Equations 12 and 13 into Equation 14 results in:
ptp =
N
1
v (∑ zi , t −1 ) X t −1 + uX t −1 + ε t
N i =1
(15)
To compute the coefficient vector zi ,t of the dynamic system in Equation 12, we can use a
22
recursive least squares algorithm. The analysis of convergence is carried out by deriving the
associated ordinary differential equation as follows:
zi , t = zi , t −1 + λt Ri−, 1t X t −1 ( ptp − zi , t −1 X t −1 )
(16)
with the moment matrix, Ri ,t = Ri , t −1 + λt ( X t2−1 − Ri , t −1 ) , i = 1, K, N . The system is recursive and
the above set of equations is a globally stable system. Its stability is determined by the stability
of the change in zi with respect to a small period of time τ:
dzi
v
=u+
dτ
N
N
∑z
i
− zi
(17)
i =1
The coefficient matrix of this linear system has N – 1 eigenvalues of –1 and one eigenvalue
of v - 1. Therefore, it is globally asymptotically stable with each zi converging to z , provided
v < 1. The condition occurs when the demand curve crosses the supply curve from above. This
indicates that if the potential adopters and the technology supplier both act normally in the
economic sense (i.e., demand increases when price decreases and vice versa, and by the same
token, supply increases when price increases and vice versa), then the rational expectations will
reach an equilibrium that is stable under learning. When v > 1 the supply curve crosses the
demand curve from above, an economically implausible case occurs. This causes the rational
expectations equilibrium to be unstable under learning. When v = 1 the supply curve is parallel
to the demand curve, so no rational expectations equilibrium will exist.
MANAGERIAL DISCUSSION
Our optimal control model suggests that a firm should adopt and implement a new
technology that is subject to network externalities as soon as the discounted total benefits offset
the recurring cost. For a technology with strong network externalities, the majority of the
benefits will come from the network effects. This means that a firm should look closely at the
23
other potential adopting firms and observe their behaviors. Indeed, this proves to be the main
challenge for each firm in our model. The ability of all of the potential adopters in the model to
eventually reach a consensus in the estimation of the willingness-to-pay is the primary condition
that must be satisfied. As we have seen, the consensus willingness-to-pay is also what the
technology supplier continues to observe and use as a basis for adjusting its price for the new
technology. We have shown that it is possible for all the potential adopters to reach a
consensus, allowing technology adoption to occur. The main assumption of the REH is that
economic agents know the true equilibrium relations of the economy [Pesaran, 1987]. The
equilibrium is the future state of the economy in which the new technology under consideration
reaches its maturity, i.e., when the benefits of the new technology are confirmed and accepted by
the market. With adaptive learning, agents can learn over time the actual values of the
parameters in the equilibrium relations, and the assumption that the agents know the true
equilibrium also will hold.
The model we use indicates that as long as the potential adopters and the technology supplier
act rationally in the usual economic sense (e.g., with value maximum in mind, and logically,
based on bounded rationality), the equilibrium will be reached and it will be stable under
learning. There are cases, however, where a potential adopter may not act as others expect. For
example, a firm may ignore fixed implementation and variable costs, and adopt a new
technology based on strategic considerations, an opportunity to preempt other firms, or a desire
to be a market leader with technological innovation. Another example is when a large firm can
afford to adopt a new technology based on its size and market share, setting the standards for its
industry. Rational expectations may not apply in such cases in the way we have described.
For any new and emerging technology, there is a phase of over-enthusiasm and unrealistic
24
projections due to a flurry of well-publicized activity by technology vendors, resulting in some
successes, but more failures, as the technology is pushed to its limits. During this phase, which
Gartner Research has called the “peak of inflated expectations” [Linden, Fenn and Haley, 2001],
it is possible that self-fulfilling expectations will be developed when market participants fail to
understand the true economy relationship. This occurs when everyone acts as if “the model” is
correct, and knows that every other firm believes in the same model and acts the same way. It is
also possible that only a subset of the market participants maintains a belief in the “false model.”
In either case, our results suggest that the adoption of a new technology based on expectations
about network externalities benefits could occur prematurely, depending on how fast the
potential adopters reach a consensus in terms of their willingness-to-pay.
This is because our model allows the set of potential adopting firms that benchmark against
each other to be of any size. So it is possible to have a small set of firms with similar
characteristics (e.g., firms in the same industry, firms serving the same geographical location
and/or market segment, etc.) that share similar interests and beliefs in the potential of the
technology. Each of these firms in the set will observe the behaviors and actions of the other
firms within the same set and base their adoption decisions on such observations. Firms then
will tend to act rationally, and make economically plausible adoption decisions. However, if the
firms use the same evaluation model, then convergence to the rational expectations equilibrium
can occur in a relatively short amount of time because of the small set, resulting in adoption.
The model that we have proposed based on REH and adaptive learning is different from what
happens under the rational herding behavior theory of Bikchandani, Hirshleifer and Welch
[1992, 1998]. This theory suggests, as ours does, that everyone will do what everyone else is
observed to be doing—only with different treatment of privately-held information. With
25
herding, if one firm decides to adopt the new technology, then the second firm will follow suit
and adopt the same technology, ignoring its own information. Other firms will do the same,
joining the herd. Banerjee [1992] argues that the decision of the second firm to ignore its own
information and join the herd imposes a negative externality on the rest of the group. If the
second firm had used its own information, then its decision would have encouraged the rest of
the firms to use their own information as well, leading to a rational expectations equilibrium.
The key difference between the REH and herd behavior theories lies in the amount of
information that is used. Although herding theory involves individual rationality, there is no
meaningful information sharing among the decision makers prior to adoption [Li, forthcoming].
Herding behavior defies a very basic assumption of economic behavior: that decision makers as
economic agents do the best they can with whatever information they have [Au and Kauffman,
2003]. Rational expectations and adaptive learning, in contrast, assume that decision makers will
utilize all available information efficiently and will be able to learn the true value of a
prospective technology adoption investment over time.
The firms whose behavior we modeled are likely to be a fraction of the total number of
potential adopting firms. The REE convergence that leads to the adoption of a new technology
among a group of firms may trigger adoption decisions for the same technology by the rest of the
firms in the market. This is because the network externalities characteristics may tip the market
to favor the technology. Why? Because the market may see the adoption by a few firms—
especially if the firms are considered to be the most informed—as a positive signal that
reinforces the private signals that each of the firms sends and receives in the market. This is
called an informational cascade, and has been described by economists [Bikhchandani,
Hirshleifer and Welch, 1992 and 1998; Zhang, 1997] and IS researchers [Li, forthcoming].
26
This suggests that the ability of a technology supplier to quickly convince as many potential
adopters as possible of the potential of the technology is critical to the new technology’s
adoption success in the market. If the technology vendor is able to persuade potential adopters
to use an evaluation model which favors adoption, this may be all it takes to plant the seeds for
the growth of network externalities. It may not even be necessary to wait until the technology
has actually proven itself by demonstrating that it can deliver the promised real world benefits.
CONCLUSION
We offer a relatively new theoretical approach to analyze the adoption of ITs that exhibit
network externalities characteristics. We have used optimal control theory thinking to introduce
the general adoption decision-making model that involves multiple periods, and the rational
expectations hypothesis (REH) and related theory to explain why firms with heterogeneous
expectations about the potential value of a new technology are eventually able to arrive at
contemporaneous decisions to adopt the same technology, creating the desired network
externalities. We have described how different levels of willingness-to-pay of the potential
technology adopters may eventually converge. We have also shown that, if the technology
supplier in our model keeps observing the signals and perceptions that the market (i.e., the
potential adopters) provides and adjusts its expected price accordingly, it will eventually arrive at
the same price expectations (at the rational expectations equilibrium) as the technology adopters.
This ability to reach the same price expectation level, we have argued, is essential for obtaining
network externalities through simultaneous or time-clustered adoption. In addition, we have
argued that a technology supplier can capitalize on the network externalities characteristics of its
technology and draw in many adopters by making them believe in a certain model that favors the
new technology.
27
Our model reflects both bounded rationality and adaptive learning. This is necessary to treat
some of the real issues that arise in new technology adoption. For example, potential
technology adopters need to learn relevant information about the newly-introduced technology
and other developments related to it (e.g., the status of competing technologies, alternative
vendor commitment levels, etc.). Moreover, the technology supplier must continuously adjust
its pricing based on the feedback that it gathers from its marketplace of potential adopters. We
have assumed that, in the learning phase, the potential adopters and the technology supplier are
able to know the equilibrium adoption model, but may be ignorant of its parameter values. This
is due to bounded rationality, but that will be overcome as adaptive learning occurs.
Our model incorporates commonly used OLS regression expressions. Although they clearly
are a simplification of how the real world works, we nevertheless believe that they are
representative of the new technology adoption settings that involve the kind of learning that we
have discussed. The setup can be interpreted as a linear regression model with time-varying
parameters, for instance. Other possible functions—such as univariate/multivariate or
linear/non-linear, among others—can also be considered for other learning situations. However,
the estimates may or may not converge, or they may result in multiple rational expectations
equilibria. These have been characterized in the rational expectations macroeconomics literature
that we have read using various terms, such as coordination failures, sunspots, bubbles,
endogenous fluctuations, and indeterminacy of equilibria. The potential existence of multiple
equilibria adds to the complexity of a model that actually may suggest non-adoption of the new
technology. The arrival in the market of a new competing technology typically will make the
choices that firms must choose among clearer, and may result in a new adoption equilibrium [Au
and Kauffman, 2001]. In the particular case of OLS regression that we have used in our model,
28
we are able to show that all the potential adopters and the technology supplier will eventually
reach an equilibrium that is stable under learning as long as they act normally in the economic
sense: all their actions should be in synch with the economic laws of supply and demand.
Muth [1961] maintains that from a purely theoretical perspective, there are good reasons for
assuming rationality. One of them is because it is a principle that is applicable to all dynamic
problems. However, the only real test of this assertion is to examine the empirical implications
of the rational expectations hypothesis in practical situations. Thus, our efforts here should be
seen as the first step toward applying rational expectation economics thinking to further
understand new technology adoption issues when network externalities are present. Going
forward, it will be critical to understand the economic variables that potential technology
adopters observe during their learning process. This will enable us to predict the kinds of
outcomes when other new technologies with similar characteristics are introduced to the market.
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Appendix 1. Definitions of Key Terms from Rational Expectations Economics Theory
TERM
Rational
Expectations
Hypothesis
(REH)
Adaptive
Expectations
Bounded
Rationality
Adaptive
Learning
Rational
Expectations
Equilibrium
(REE)
DEFINITION
A theory formulated by Muth [1961] that suggests that economic agents form their
expectations on the basis of the “true” structural model of the economy in which their
decisions are made, and that on average, these expectations are essentially the same as
the predictions of the relevant economic theory.
A hypothesis that states that in any one period, expectations are revised (linearly) in
light of past errors in expectations [Pesaran, 1987].
Recognizes the limited cognitive capacity of humans in decision making when they
face problem complexity under the constraints of time and lacking information.
Framework based on REH. Assumes that economic agents know the true equilibrium
structural relations of economy but—due to bounded rationality—are not allowed to
learn the actual values of the parameters in the equilibrium relations [Pesaran, 1987].
Equilibrium condition characterized by three features: all markets clear at equilibrium
prices; every agent knows the relationship between equilibrium prices and private
information of all other agents; and the information in equilibrium prices is exploited
by all agents in making inferences about private information of others [Pesaran, 1987].
Appendix 2. Rational Expectations Model Notation Appendix
NOTATION
pt p , pt-1 p
pi,t e ; pt s
DEFINITION
Pseudo-price for new technology in period t and t – 1; indicates the prices that
would prevail if adoptions occurred in the respective period
Nt ; N
zi,t, zi,t-1
y, z; yt-1, zt-1
Potential adopter i’s period t price expectation estimated in t - 1; new technology
supplier’s period t price expectations estimated in period t – 1
Average period t price expectation for all potential adopters estimated in period t – 1
Intercepts for demand and supply functions for new technologies
Vector of parameters for exogenous variables that affect supply
Demand and supply of new technology in period t
Observable variables vector in period t – 1; the related information set in period t -1
Slope of new technology demand and supply functions; δ < 0 and σ > 0
Unobservable drivers of technology demand and supply; εt = 1/δ (εtd − εts)
u = 1/δ (β s), the magnitude to which the exogenous variables negatively impact price;
v = - σ /δ, the ratio of the slope of the supply curve to that of the demand curve;
w = 1/δ (β0 d − β0 s), the proportional difference between the intercept of the demand
and supply curves
Number of adopters of a technology in period t; total number of potential adopters
Parameter vectors in linear forecasting rule of potential adopter i for period t and t − 1
Parameters in the pseudo-price forecasting function in general; and for period t – 1
y, z
Parameters, y = w , z = u , in the pseudo-price forecasting function at the point of
pt e
β 0d, β0s
βs
dt , st
Xt-1, It-1
δ, σ
d
εt , εts; εt
u, v, w
Ri,t , Ri,t-1
1− v
1− v
rational expectations equilibrium.
Moment matrix in the least squares formulas for firm i in period t and t-1