‘THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986. 1.065
FOCUSING ON THE GROWTH RATE OF TECHNOLOGICAL ADOPTION
Frederick P, Wheeler
University of Bradford Management Centre
Bradford, UK
ABSTRACT
Models of substitution and adoption of consumer durable
technologies typically focus on the level of adoption or the
level of cumulative sales of the product. Although these
variables may be of interest, decisions on market entry and the
judgement of future return on investment are linked to the rate
of change in adoption level. The percentage change in the current
level of adoption, the growth rate, is more relevant, more
meaningful and more sensitive a measure of past and future trends
than is the level itself. This is an appeal for system modellers
and forecasters to focus their attention on growth in studies of
technological diffusion.
1. INTRODUCTION
Intuitively it is appealing to make analogies between the growth in
, popularity of a new technology and the spread of an epidemic. It is
common to talk of something new "catching on". The growth in adoption of
a technology can be thought of as a diffusion process, which has probably
been stimulated by some favourable change in the business environment.
If this change stimulates sufficient innovative purchasers to adopt the
new technology they will influence others to copy them and to set growth
in progress, Bass(1969) introduced a plausible model of this type. The
historical analysis of many successful technologies! patterns of growth
tends to confirm these notions. Of course, data on unsuccessful technologies
is not available over a long time span.
Models of technological adoption usually focus attention on prediction of
the level of adoption, or the surrogate measure of the cumulative sales.
Looking back in time over various innovationg characterisically a sigmoid
curve appears to describe the behaviour of the adoption level graph. The
graphs are so convincing that otherwise experienced forecasters are
apt to cast an uncritical eye over extrapolations of theSé curves into
the future. As a result, decision makers may be convinced of the existence
of a known and predictable saturation level toward which the curve of
adoption level often seems to be heading. It is not difficult to show
that, these extrapolations of sigmoid growth curves can be seriously in
error, But if mistakes are being made it is also necessary to show the
cause of the error, This is the problem tackled in this paper.
1,066 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986.
2. INFLUENCES ON GROWTH
Once the growth of the market for a new durable product is set in motion
the future changes in adoption level can be attributed to a few’ — strong
influences. Firstly, those people who have adopted the product influence
others, who have not, to imitate them. This is a positive influence to
buy and as a result of this influence the market for the product grows.
Secondly, the proportion of possible adopters who have not yet bought the
product will tend to decline as possible adopters become actual adopters in
the process of imitation of others. The rate of adoption at any time will
be a function of these two factors: it will be an inereasing function of
the number of adopters and a decreasing function of the proportion of actual
versus possible adopters.
But the number of possible adopters is not fixed in a changing business
climate. As the business climate changes so too the number of people who
are in a position to purchase the product will alter. Were conditions to
be fixed for all times it is clear that the level of possible adopters would
fall and the level of actual adopters would rise by an equal amount to
an equilibrium level. This equilibrium level would be the long term demand
for the product. The adoption-diffusion-imitation process represents the
delayed adjustment of demand to this equilibrium level. In practice the
equilibrium demand will be price-sensitive and price itself will reflect
business conditions as they fluctuate through the cycle and the seasons.
But real price will also show the steady influence of the accumulated
experience of manufacturers as more units are produced. This cumulative
production experience will push down costs (a phenomenon noted by Arrow, 1962)
and prices will be driven lower in a competitive environment. As the
real price falls so the number of possible adopters will rise. The
"equilibrium! demand will steadily increase in such a situation.
There are, of course, other influencés in operation. The link between
current adoption level and adoption rate is not.a simple one. Implied here
is the ability of adopters to diffuse within the possible adopter
population so as to induce imitation, In the early stages of growth the
chance of contact between each new adopter and those who may imitate him
is relatively high. But should adopters tend to cluster then new adopters
will be less effective in their influence. Adoption rate will be a non-
linear function of the number of adopters in this case. The discard of
old units is an example of past behaviour influencing the future. The
past adoption level will influence the current rate of discard of units
and adopters who are considering scrappage of their units form part of the
possible adopter population.
The current business environment will affect both adoption and discard rates.
4n influence on the environment is the industry's own forecast of growth.
When this does not marry with current behaviour there will be under or
over supply and then price adjustment. Finally the emergence of a substitute
technology will drain the level of possible adopters by removing those who
are attracted to the substitute by reason of cost, convenience or fashion.
INE 100 INTEMINATIUNAL GUNFEKENUE UF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986. 1.067
3. MODELLING THE RELEVANT VARIABLE
The foregoing discussion has indicated that the growth pattern of a
technology is a complicated process involving non-linear and lagged
dependencies together with fluctating disturbances from the business
environment. One might wonder how it is that rathér simple models of
adoption level, such as the logistic and the Gompertz curves, have been
used to describe this process, The reason is that these sigmoid models
have been used to describe the level rather than the rate of growth of the
level, Further, such models have fitted historical data but the relevant
question for policy makers is the behaviour of the process in the future.
It is not difficult to show that- such models are not reliably extrapolated,
and this will be demonstrated below.
To begin the discussion it will be helpful to review sigmoid models. In
the course of this review, waich will concentrate on a particular class of
sigmoids, a heuristic argument will be developed to show that a sigmoid
can describe the long term expected behaviour arising from the influences
of imitation by purchasers and experience of manufacturers. The present
approach differs from that of Bass(1980) who incorporated the experience
effect in a modification of his earlier work (Bass, 1969). It is not the
purpose of the present work to provide a genereal review of growth curves
and the interested reader is referred to a recent paper by Meade(1984)
for an appraisal of these models.
The growth inadoption of a technology, by analogy with the growth of a
Diological system, can be modelled by the equation:
as
at ) (1)
zers(-
elo
In this equation S is the level of adoption, r is the initial rate of
growth of the level, and A is an equilibrium level which is approached
asymptotically. Sometimes S is taken to be the level of cumulative sales,
which is a surrogate for the adoption level. When the level S is well below
the asymptotic value A equation (1) describes linear feedback as the
current adopters influence others with an effectiveness (a probability of
imitation) proportional to their number. The fraction of unsatiated
demand is (1 - S/A). As the level S increases so this fraction decreases
and the rate of growth slows. If A is treated as a constant equation (1)
integrates to give a logistic curve for S. This is a sigmoid which is
symmetric about its point of inflection, which is the point at which dS/dt
is a maximum,
The symmetric logistic curve is not sufficiently flexible to represent the
range of growth patterns observed in practice, Realising this, Easingwood
et al. (1981) suggested a modification which allowed for a parameter to
model asymmmetric growth. But it will be shown below that asymmetric
growth can be introduced naturally by a generalisation of the logistic
to include the effects of experience. The argument differs in important
respects from that of Sharp(1985).
1.068 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986.
Equation (1) describes the approach of adoption level to an asymptotic
equilibrium, A, which can be thought of as an equilibrium demand. This
description applies to a static equilibrium demand, but business conditions
are Imown to change and it can be presumed that the value of A will also
vary. An important cause for change is the feedback mechanism which links
cumulative experience in production of a technology to cost reduction and,
in a competitive market, to price reduction. As prices fall so more people
will be able to buy the product. They will not do so immediately because
awareness of the technology must first spread; the gradual adjustment to
equilibrium is described by an equation such as (1). The price in this
discussion is the real price of the product (deflated by an appropriate
index) and the-downward price trend is a long term average behaviour.
This phenomenon has been documented by the Boston Consulting Group (1968)
and Dino(1985) has made empirical analyses of electronic products recently.
The relationship between long term demand andprice is likely to be of the
form
4 @ peo (2)
where n> 0 measures the price elasticity of the equilibrium demand.
Price itself is expected to decline as manufacturers improve production
methods and costs fall. The empirical evidence suggests a relationship of
the form .
p « s7> (3)
where S has been taken to be proportional to cumulative experience. The
existence of such an experience curve closes the feedback loop between
the equilibrium level A and the current level $ to give
S . gh-™m (3)
A
The concept of an equilibrium demand is now best replaced by the idea
of a target level, a, so that
1- An Yi
s .(8 = (8
$ -(8} (8) a)
and now y = 1 - nA is a parameter which allows the symmetry of the growth
to be modelled. The inclusion of these effects in the simple model (1)
results in the modified equation
S-2 1 -(8)%s (5)
In (5) the initial rate of growth is equal to b/y. The equation can be
integrated (for y# 0) and the result is the generalised logistic:
oL
s(t) = af1+ cem(-bt)}"/Y 5 = (a/s(o)}¥- 1. (6)
Different sigmoids are obtained for different values of y . Recently
McGowan (1986) noted that this generalisation does indeed improve the
fit over the simple logistic. The simple logistic corresponds to y =
in other words a negligible value for the product of parameters, \n «
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986. 1.069
Ghemawat (1985) quoted typical experience curve slopes of around 40.85
in a large number of academic studies. If it is assumed that some price
reduction will result from experience then the simple logistic will apply only
when n = 0, that is when the long term equilibrium demand is perfectly
inelastic.
Equation (5}\has been deliberately written in the form
(7)
y # 90, (8)
in order to show that the right hand side of the equation contains a
power transformation of the fraction of target y = S/a. If, when y = 0,
the transform is
ty) = dog) 5 (Ba)
then the transform given by (8) and (8a) is the same family of transforms
introduced by Box and Cox (1964) to stabilise the variance of nonlinear
data. This suggests that equation (7) could be written in terms of the time
derivative of log(S) :
S rog(s)= -ve2%S) (9)
Equation (9) recommends itself on both heuristic grounds and statistical
“grounds as being a suitable starting point for the analysis‘of the non-
linear data observed in the adoption of new technologies. The important
difference between equation (9) as a starting point for analysis and
equation (6), which describes a level, is that in (9) the dependent
variable is the rate of change of the logarithm of level, usually referred
to as a growth rate. Not only is growth rate a more stable quantity in
statistical terms, it is also more relevant to a decision maker because it
is the same type of variable as a cost of capital, a wage inflation rate
and so on, In other words the growth rate can be compared directly with
the rate of return on investment and the rate of cost inflation in order
to assess the future direction of policy.
It should not be thought that equation (9) represents a complete description
of the growth in adoption. Quite clearly most of the influences discussed
in the previous section have been ignored in deriving the equation. Also
there has been no attempt to include chance effects. What is described by
equation (9) is the most probable path of growth rate for a process of
technological adoption in which imitation and experience effects dominate
and in which other influences are equally likely to push growth rate up or
down. The usefulness of such a model lies in its ability to illuminate the
past, to clarify the present and to make explicit the assumptios behind
predictions of the future. It is argued here that an analysis of growth
rate is able to do this whereas an analysis which emphasises the level
of adoption is not.
1.070 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986.
O-2E
e
e
e
. e
OL .°
e 2 °
0-0 1 1 f
1960 1970 1980
FIGURE 1. Cars per capita in Britain to 1970.
‘The reader is invited to judge upper and lower limits
for the future course of the adoption curve.
It 7986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986. 1.071
4. A PRACTICAL EXAMPLE,
It will be helpful’ to look at an example of technological adoption in
practice. The case considered is the adoption of private motor cars in
Great Britain. The data consist of the number of cars per head of population.
This quantity is computed from the U.K. government's official statistics on
population and on cars currently licensed, to be found in the publication
"Economie Trends!, for example.
Forecasts of the number of cars per capita influence policy decisions for
national transportation planning, road construction and so on. In Britain
the number of cars per capita has been increasing but the rate of increase
has shown signs of declining. The data up to 1970 are shown in Figure 1.
Before proceeding with the discussion, the reader is invited to examine
Figure 1 in detail, without reference to the other figures in the paper.
The adoption curve of Figure 1 certainly has the signoid form so
characteristic of a saturating market. The reader should attempt now to
sketch on the figure his judged extrapolation of the data, in particular,
to mark upper and lower limits to the pattern of adoption for the next
decade.
The actual course of events for the next decade is shown in Figure 2.
Most readers who have not seen this data previously will be surprised by
the pattern of adoption which actually ocurred. It is worth while checking
back on the simple extrapolations suggested above to see the extent of
any discrepancy. More sophisticated extrapolations of the curve, based on
some weighted least-squares criterion for example, are unlikely to give more
correct results. Meade(1985) has suggested an adaptive sigmoid fitting
procedure based on the Kalman filter-and Harvey(1984) has suggested a
local sigmoid trend-fitting procedure, These adaptive approaches will give
adjusted forecasts as new data becomes available but at any point in time
they represent an extrapolation of the adoption <curve which is similar to
that produced by eye.
It is necessary to examine the rate of growth of the adoption curve in
order to see why the growth pattern appears clear up to 1970. The percentage
growth over each year is shown in Figure 3. It is seen from this figure
that there was a steady dowmward trend in the years before 1970 and this
is the reason for the smooth appearance of the graph of adoption level
up to that time, But whereas one might be confident in extrapolating Figure 1,
ineorrectly, the growth rate data shows that there is noise affecting
the long term trend and so the extrapolation would be made with more care.
It would also be suspected from examination of Figure 3 that the changes in
growth are linked to the business cycle. Judgements of future business trends
are likely to be expressed in terms of rates of.interest, or rates of price
or cost inflation, and so on. These judgements are more naturally
incorporated within a forecast of the rate of growth in adoption level.
From these comments it would appear that extrapolations ofthe curve of ear
adoption level made during the 1970s would be doomed to failure. In fact
Brooks et al.(1978) showed that such forecasts were self-contradictory.
1.072 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986.
FIGURE 2.
0-3
Cars per capita in Britain to 1980.
°
N
7
0.1L
0-0
o-
FIGURE 3.
N _t f
1960: 1970 1980
Percentage Growth over year for data of Fig. 2.
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986, 1.073
5. CONCLUSIONS
Models of technological adoption should focus on the growth rate of the
adoption process. If growth rate is correctly modelled the adoption level,
which is often taken as the main object of the study, will follow. Models
which concentrate on the adoption level can miss the importance of the
patterns of consumer behaviour which changes in growth rate reveal.
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Review of Economic Studies, Vol. 29, pp. 155-173.
Bass, F.M. (1969). A new product growth model for consumer durables.
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Bass, F.M. (1980). The relationship between diffusion rates, experience
curves and demand elasticities for consumer durable technological
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Boston Consulting Group (1968). Perspectives on Experience. Boston
Consulting Group, Boston,
Box, G.E.P. and Cox, D.R. (1964). An analysis of transformations.
Journal of the Royal Statistical Society (B), Vol. 26, pp. 211-243.
Brooks, R.J., Dawid, A.P., Galbraith, J.I., Galbraith, R.F., Stone, M.
and Smith, A.F.M. (1978). Forecasting car ownership. Journal of the
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responding logistic model for forecasting technological substitution.
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Ghemawat, P. (1985). Building strategy on the experience curve.
Harvard Business Review, March-April 1985, pp. 143-149.
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