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8
Dynamics and Replacement of
U.S. Transport Infrastructures
NEBOJSA NAKICENOVIC
This chapter uses several indicative examples to assess the time needed
to build new and replace old transport infrastructures and energy transport
systems. As the examples will show, both the growth and senescence of
transport infrastructures evolve as regular processes, which are describable
by Sshaped logistic curves. Not all growth and senescence phenomena
can be described by simple logistic functions, however. Sometimes more
complex patterns are observed, which are often described by envelopes
that can be decomposed into a number of Sshaped growth or senescence
phases. Two typical cases are (1) successive growth pulses with inter
vening saturation and a period of change, and (2) simultaneous substitution
of more than two competing technologies. In the first case, successive S
shaped pulses usually represent successive improvements in perfor
mance for example, aircraft speed records in which the first pulse is
associated with the old technology, the piston engine, and the second with
the new technology, the jet engine. In the second case, simultaneous
substitution of competing technologies is usually described by increasing
market shares of new technologies and decreasing market shares of old
technologies. This chapter will demonstrate that the evolution of transport
infrastructure can be described both by the performance or productivity
of competing technologies and by their market share (see also chapter 71.
For both of the cases just mentioned, it will be shown that the devel
opment of transport networks is subject to regular patterns of change,
which can in principle be used for forecasting future trends. Most of the
examples are taken from the United States because it is one of the few
175
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176
NEBOJSA NAKICENOVIC
countries to have welldocumented experience of most of the technological
changes that have occurred over the last 200 years. Because most of these
changes subsequently diffused throughout the world, the examples also
indicate the dynamics of these processes elsewhere.
In this chapter, the use of the term infrastructure is rather narrow,
referring only to transportation and energy grids and networks, and other
components of these two systems. These systems are interesting because
they have played a crucial role in the economic and technological devel
opment process, are very capital intensive, and, in general, have long
lifetimes. Analysis of the historical development of these two systems will
include a quantitative description of performance improvement, the gen
eral evolution of a particular infrastructure, and the replacement of old
technologies and infrastructures by new ones in terms of their relative
market shares. The term performance is used as a multidimensional con
cept (i.e. as a vector rather than a scalar indicator), and, where appropriate,
the size of an infrastructure is measured as a function of time.
The first section of this chapter describes the evolution of transport
systems and their related infrastructures. The analysis starts, somewhat
unconventionally, with the youngest technologies, aircraft and airways,
and ends with the oldest transport networks, canals and waterways. The
second section describes the evolution of energy consumption and pipe
lines as an example of dedicated transport infrastructures; The appendix
briefly describes the methods and statistical tests used for this analysis.
TRANSPORTATION
Aircraft
Aircraft are the most successful of today's advanced modes of trans
portation. Other concepts of rapid transport such as highspeed trains have
shown limited success, but they are not used as universally as aircraft. In
fact, the rapid expansion of air travel during recent decades has its roots
in the developments achieved in aerodynamics and other sciences many
decades ago and especially in the engineering achievements made between
the two world wars. The DC3 airliner is often given as the example of
the first "modern" passenger transport because in many ways its use
denotes the beginning of the aircraft age.
The use of aircraft for transportation has increased ever since, and their
performance has improved by about two orders of magnitude (about a
hundred times). Figure 81 shows the increase in air transport worldwide
measured in billions of passengerkilometers per year (passkm/year). It
includes all carrier operations, including those of the planned economies.
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U.S. TRANSPORT INFRASTRUCTURES
1 800
1 600
1400
1200  _

y
a.' 1 000 
cn
Q)
u,
(A
cow
~800
o
600
400
l
i/
At= 29yr
1940 1950 1960 1970 1980 1990 2000
Year
FIGURE 81 Air transport worldwide, all operations.
177
The logistic function has been fitted to the actual data, and it indicates
that the inflection point in the growth of air carrier operations occurred
about 10 years ago (around 1977~. Thus, after a period of rapid exponential
growth, less than one doubling is left until the estimated saturation level
is achieved after the year 2000. The growth rate has been declining for
about 10 years. If the projection here is correct, it will continue to do so
until the total volume of all operations levels off after the year 2000 at
about a 40 percent higher level than that experienced at present.
Figure 82 shows the same data and fitted logistic curve transformed
as X/(K X), where x denotes the actual volume of all operations in a
given year (from Figure 81, given in millions of passengerkilometers
per hour [passkm/in]) and K iS the estimated saturation level. The data
and the estimated logistic trend line are plotted in Figure 82 as fractional
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178
1n2 ~
Year
FIGURE 82 World air transport, logistic plot.
NEBOJSA NAKICENOVIC
~ 
/
/ (K=200) ~
iB747
/\B707 ~ t = 29yr
~B377
1 1 '1 1 1 1 1 1 1
1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020
 0.99



_ _ 0.90
_ _
_ _ 0.70 0
IL
0.30
0.10
0.01
shares of the saturation level,f = X/K, which simplifies the transformation
to/1  A. Transformed in this way, the data appear to be on a straight
line, which is the estimated logistic function.]
Perhaps the most interesting result is that it took about 30 years for
world air transport to reach the inflection point (about half of the estimated
saturation levelly and that within two decades the saturation level will be
reached. This raises a crucial question: What will happen after such a
saturation? Can we expect another growth pulse, a decline, or the insta
bility of changing periods of growth and decline? Most likely a new period
of growth associated with new technologies will follow the projected
saturation.
To understand the implications of a possible saturation in world air
transport operations and future developments, one must look at the trans
port system in general, comparing aviation with other modes of transport,
and analyze the various components of the air transport system itself.
Aircraft, airports, and ground services are the most important components
of the air transport infrastructure. The commercial aviation infrastructure
differs, however, from the infrastructures of other competing transport
systems such as roads or railways in that airports are not continuously
connected physically as are pipelines, roads, and railroad stations.
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U.S. TRANSPORTINFRASTRUCTURES
179
The global fleet of commercial aircraft and how it developed can be
described in a number of ways. An obvious descriptor of the fleet is the
number of aircraft in operation worldwide. This number increased from
about 3,000 in the l950s to almost 10,000 in the 1980s. During the same
time, however, the performance or carrying capacity and speed of aircraft
increased by about two orders of magnitude. Thus, the size of the fleet
is not the most important descriptor, because much of the traffic is allocated
to the most productive aircraft operating among the large hub airports
while other aircraft constitute the feeder and distribution system for des
tinations with a lower traffic volume. The analogy between air transport
systems and electrical grids or road systems is very close: large aircraft
correspond to highvoltage transmission lines or primary roads.
Figure 83 shows the improvement over time of one important perfor
mance indicator for commercial passenger aircraft: carrying capacity and
speed (often called productivity) measured in passengerkilometers per
hour. Each point on the graph indicates the performance of a given aircraft
when used in commercial operations for the first time. For example, the
:,

1o1
// ( K= 1200}
B747/, /
/ TU144
B707 / . DCA130O 1011
DC8' ·. Concorde
TU114~ ~ B527
DC 7 ·CV880
DC7C .;Caravelle
L649,4 /
DC4/ · ·. CV340
DC3 ~ /
~/ ·:/
1o3 / i/ 1 1
1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020
At = 30.8 yr
Lag = 9 yr
Year
/ /
_ 0.99
_ 0.90
_
0.70
.
 0.10
0.01
. _
0.001
0.50 O
0.30 ce
11
FIGURE 83 Passenger aircraft performance. Source of data: Angelucci and Ma
tricardi (1977), Grey (1969~.
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180
NEBOJSA NAKICENOVIC
DC3 was introduced in 1935 with a performance of about 7,400 pass
km/h (21 passengers at 350 km/h); the B747 was introduced in 1969 with
a performance of about 50O,000 passkm/in (500 passengers at 1,000 km/
hi. The largest planned B747s can carry almost 700 passengers. They
are therefore about a hundred times as productive as the DC3 50 years
ago.
The upper curve in Figure 83 represents a kind of performance fea
sibility frontier for passenger aircraft because the performance of all other
commercial air transports at the time they were introduced was either on
or below the curve. Longrange aircraft on the performance feasibility
curve were commercially successful; all other planes, whether they were
successful or not, are below the curve. Thus, at any given time there
appears to be only one appropriate (best) productivity specification for
longrange passenger planes. Because the more recent jet transports all
fly at about the same speed, the appropriate design for a new longrange
transport should allow for a capacity of more than 700 passengers. Ac
cording to the estimated curve, the asymptotic capacity for the largest
aircraft is about 1,200 passengers at subsonic speed (1.2 million passkm/
h or 10 billion passkrn/yr). This implies that the next pulse in aircraft
performance, if it occurs, would take place after the saturation phase. It
would start at about 1.2 million passkm/in and grow to much higher levels
by at least one if not two orders of magnitude. Thus, after the year 2000,
longrange aircraft might be larger and faster. Another interesting feature
of Figure 83 is that the productivity of all passenger aircraft is confined
to a rather narrow band between the performance feasibility curve and a
"parallel" logistic curve with a lag of about nine years. This logistic
curve represents the growth of world air transport from Figure 82.
It took about 30 years for the performance of the most productive aircraft
to increase from about 1 percent of the estimated asymptotic performance
to about half that performance (for example, the DC3 represents the 1
percent achievement level and the B747 roughly the 50 percent mark).
In many ways the achievement of the 50 percent level represents a struc
tural change in the development of the entire passenger aircraft industry
and airlines. With Sshaped growth (that is, a logistic curve) the growth
process is exponential until the inflection point or the 50 percent level is
achieved. Thus, at the beginning, the productivity of aircraft is doubled
many times within periods of only a few years. Once the inflection point
is reached, however, only one doubling is left until the saturation level
is reached. In Figure 83, for example, this development phase occurred
in 1969 with the introduction of the B747. Because the B747 can, in
principle, be stretched by about a factor of two, it could remain the largest
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U.S. TRANSPORTINFRASTRUCTURES
181
longrange aircraft over the next two decades. Thereafter, a new growth
phase with either larger or faster longrange aircraft is conceivable; the
more distant future may possibly see both.
Before the inflection point, stretching does not help for more than a
few years, given the rather frequent doublings in productivity. Conse
quently, new solutions are necessary. The necessity for new solutions also
means that having a model that performs poorly on the market can be a
crucial but nevertheless reversible mistake, provided the manufacturer is
able to launch a new, improved performance model after a few years.
After the inflection point, however, this strategy is no longer possible
because those aircraft that are successful can be stretched to meet market
demand through the saturation phase.
Thus, in the late 1960s the introduction of new, longrange aircraft
suddenly became riskier because it was no longer possible to launch a
new model after a few years. The B747, for example, had the appropriate
productivity, whereas two other competitors, the DC 10 and L 101 1, were
too small. The introduction of a 700 to 800passenger aircraft in the near
future would be risky because a stretched B747 can do the same job; the
introduction of a smaller longrange aircraft would probably fail because
it would fall short of the primary market requirements. The only routes
open to smaller, longer range transport such as the MD11 or Airbus 340
are those for which the B747 is too large.
In the long run, most of the longrange traffic will be between large
hubs and will be less in the form of direct traffic between smaller airports.
The market niche for longrange aircraft with fewer than 500 passengers
will therefore be very limited. A feasible alternative might be to redesign
the MD11 and the Airbus 340 as widebody, shorter range transports
suitable for frequent cycles. The design of a cruise supersonic aircraft able
to carry 300 to 400 passengers, or a hypersonic transport able to handle
200 to 300 passengers, may therefore be a better strategy for the first
years of the next century. To some extent, these principles also explain
why the Concorde cannot be a commercial success: with a capacity of
100 passengers, it is 150 passengers short of being a serious competitor
for the B747.
How probable is the development of a large cruise supersonic or hy
personic transport? Spulses do not occur alone but usually in pairs. Thus,
at saturation, structural change will occur, leading to a new growth pulse
(probably Sshaped) and in turn to new productivity requirements and
therefore to supersonic or hypersonic transport. The alternative is that air
transport will saturate pe~anently over the next few decades, and the
widebodied families of subsonic transports will actually constitute the
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182
1ol
10°
10 1_
10  2
NEBOJSA NAKICENOVIC
1o2
·/ :,

Jets
Pistons (hp.) Max Thrust (kg)~
( ~ = 3800) ( K = 29,000) 
~
1936 /
~k.
. ~ ~9 =
· ~
.
.
1966 ·/
I/
30 yr/
i
hit= 30yr
. , , , , , , ,
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
Year
.  099
0.90

y
0.70 ~

0.50 0
0.30
0.10
 0.01
FIGURE 84 Performance of aircraft engines. Source of data: Angelucci and
Ma~icardi (1977), Grey (19691.
asymptotic technology achievement. This alternative is unlikely, however,
because air transport may become the dominant mode of passenger (long
distance) travel in the next century.
Figure 84 shows the improvement in the performance of civil aircraft
engines since the beginning of aviation. The first piston engine visible on
the plot is the French Antoinette engine, rated at 50 horsepower (hp.) in
1906, and the last is the American Wright Turbo Compound, rated at
3,400 hp in 1950. These engines represent an improvement in power of
almost two orders of magnitude over 44 years and about 90 percent of
the estimated saturation level for piston engines (about 3,800 hp.). A
parallel development in the maximal thrust of jet engines follows with a
lag of about 30 years, starting in 1944 with the German Junkers Juno 004
(rated at 900 kg) and ending with the American Pratt and Whitney JT9D
in the early 1980s with 90 percent of the estimated thrust saturation level
at about 29,000 kg. Both pulses in the improvement of aircraft engines
are characterized by a time constant (fit) equal to about 30 years. The
midpoints (inflection points) of the two pulses, which occurred in 1936
and 1967, respectively, coincided with the introduction of the DC3 and
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U.S. TRANSPORTINFRASTRUCTURES
183
the B747. In fact, the Pratt and Whitney Twin Wasp, introduced in 1930,
became the power plant for the DC3. This engine also served as the basis
for subsequent and more powerful derivatives such as the Wasp Major
introduced in 1945 toward the end of the aircraft piston engine era.
Thus, certain parallels are evident in the dynamics of passenger aircraft
development beyond the obvious similarity in the time constant (/\t =
30 years). Those engines introduced slightly before the inflection point,
such as the Twin Wasp, are "stretched" by doubling the cylinder rows
to increase the power. In the Wright Turbo Compound, representing the
last refinement in aircraft piston engines, turbocharging was used to derive
shaft power. Otherwise, the engine was a direct derivative of the Wright
Cyclone series introduced in the early 1930s with Cyclone 9 (which orig
inally powered the DC24.
The time constant for the development of passenger aircraft is therefore
about 30 years. Thirty years after the standard industry design emerged
during the 1930s, the B747, the first widebody jet to enter service,
became one of the most significant improvements in commercial transport,
and its productivity represents half of the estimated industry saturation
level that may be approached toward the end of this century. Thus, within
a period of 60 years the life cycle will be completed: from standardization
and subsequent rapid growth characterized by numerous improvements
through the inflection point, when the emphasis changes to competition
characterized by cost reductions and rationalization. These characteristics
of the industry, including steady improvement in performance, are well
illustrated by the development of the B747 family.
The emergence of a new aircraft engine at the beginning of a new phase
of growth is very likely toward the end of the century after the projected
saturation has been reached. Furthermore, the characteristics of the long
range aircraft in this hypothetical growth pulse are implicit in this analysis.
The saturation in the thrust of turbojet and turbofan engines indicates that
a new generation of engines will emerge over the next few decades, a
generation having much higher thrust ratings and growth potential. The
only realistic candidates are the ramjet or scram engines with efficient
offdesign flight engine characteristics, or perhaps the HOTOLtype air
breathing rocket engines. Thus, a new pulse in the improvements of aircraft
engines will most likely be associated with some variant of the variable
cycle airbreathing engine with a high Mach number and perhaps even
ballistic flight potential. This trend would single out methane as the fuel
of choice between Mach 3.5 and Mach 6, with a small overlap of hydrogen
starting at Mach 5. Thus, cryogenic methanepowered cruise supersonic
aircraft may become the mode of transport with the highest productivity
after the year 2000.
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184
NEBOJSA NAKICENOVIC
The Automobile
At the beginning of this century, few proponents of the automobile
envisaged that its use would spread as rapidly as it did throughout the
world. As a commercial and recreational vehicle the motor car offered
many advantages over other modes of transportation? especially animal
drawn vehicles. Perhaps the most important advantage was the possibility
of increasing the radius of business and leisure transport.
At first the railroads were not challenged by the automobile. Rather,
railroads helped the spread of the automobile by offering efficient long
distance transport that combined well with the use of motor vehicles for
local urban and rural road transport. Within a few decades, however, the
automobile had become an important form of transport for both local and
longdistance passenger travel in the United States. Since the 1930s the
total mileage traveled by motor vehicles in general has been divided almost
equally between rural and urban travel.
The automobile had a relatively late start in the United States in relation
to European countries (for example, France, Germany, and the United
Kingdom). According to the records, four motor vehicles were in use in
the United States in 1894. This was followed, however, with an impressive
expansion of the automobile fleet: 16 vehicles in 1896; 90 in 1897; 8,000
in 1900; almost 500,000 10 years later; and more than 1 million after
another 2 years. Thus, the United States quickly surpassed the European
countries both in production and in the number of vehicles in use.
Figure 85 shows the rapid increase in the number of cars used in the
United States. The automobile fleet is characterized by two distinct secular
trends, with an inflection in the 1930s followed by less rapid growth rates.
Because the two secular trends of the curve appear to be roughly linear
on the logarithmic scale in Figure 85, the automobile fleet evolved through
two exponential pulses. Thus, in this example the growth of the automobile
fleet did not follow a simple, single Sshaped growth pulse.
The working hypothesis in this case is that the two trends indicate two
different phases of the dissemination of motor vehicles in the United States.
The first characterizes the substitution of motor vehicles for horsedrawn
road vehicles, and the second the actual growth of road transport after
animaldrawn vehicles essentially disappeared from American roads (see
Nakicenovic, 19861. Only after the completion of this substitution process
did the automobile emerge as an important competitor of the railroads for
the longdistance movement of people and goods and perhaps as a com
petitor of urban transportation modes such as the tram, subway, or local
train. Thus, the first expansion phase was more rapid because it represents
a "market takeover" or expansion in a special niche; the second represents
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U.S. TRANSPORTINFRASTRUCTURES
105
_%
cn
° 4
_'
in
.~
o 3
a) 10
Q
1o2
1o1
1 850
~,
/' Cars

Horses _~:


185

/
1 900
1950
2000
Year
FIGURE 85 Number of road horses and mules and automobiles in the United
States.
the actual growth of the road vehicle fleets and their associated infra
structures.
The lack of historical records of the exact number of horsedrawn
vehicles in the United States soon after the introduction of the automobile
in 1895 makes it difficult to describe accurately the assumed substitution
of the motor car for the horse during the first, more rapid expansion phase
of the motor vehicle fleets. The number of draft animals (road horses and
mules) and automobiles given in Figure 85 therefore are a rough ap
proximation of this substitution process. Horse and saddle were sometimes
used as a "road vehicle," but often more than one horse was used to pull
buggies and wagons. City omnibuses used about 15 horses in a single
day, and a stagecoach probably used even more because the horses were
replaced at each station. Figure 85 therefore may exaggerate the number
of horsedrawn vehicles if the number of draft animals is used as a proxy
for the number of vehicles actually in use. On the other hand, farm work
animals are not included in Figure 85, although certainly they were also
used for transport, especially in rural areas.
Although estimates of nonfarm horses and mules are not very accurate
and are unevenly spaced in time, Figure 86 indicates that the automobile
replaced horse and muledrawn road vehicles during a relatively short
period and that the substitution process proceeded along a logistic path.3
OCR for page 175
U.S. TRANSPORT INFRASTRUCTURES
211
() 1 +exp(~t~)
(1)
where t is the independent variable usually representing some unit of time,
and a, is, and K are the unknown parameters. Alternatively, the logistic
growth curve can be expressed as a linear function of time by moving the
(usually) unknown parameter K to the left side of the equation and taking
logarithms of both sides:
in ~ (r) ~ ~ = cat +
(2,
This is a convenient form for showing the logistic growth process on
semilogarithmic paper because the historical data indicate a linear secular
trend (assuming that K can be estimated from the data or that it is a priori
known).
The three parameters are interpreted as follows in terms of the under
lying growth process: ~ denotes the rate of growth; ~ is the location
parameter (it shifts the function in time, but it does not affect the function's
shape); and ~ is the asymptote that bounds the function and therefore
specifies the level at which the growth process saturates (as t tends to
infinity, x(t) approaches K). Thus, all three parameters have clear physical
interpretations, although the values of ~ and ~ are not necessarily clear
intuitively.
The logistic function is a symmetrical Sshaped growth function, with
an inflection point, to, at which the growth rate reaches a maximum, x~to)
CX K/4. Symmetry implies that the value of the function is half the
asymptote at the point of inflection, ditto) = K/2. Thus, the location
parameter of the function can be defined as the point of inflection, to =
Q/c~, or, alternatively, as the time when 50 percent of the saturation
level K is reached. The growth rate of the function can be defined alter
natively as the length of the time interval needed to grow from 10 to 90
percent of the saturation level a. The length of this interval is At =
(ln 81~/~. This second set of parameters K, /\t, and to also specifies
the logistic growth curve (in the same way that K, CX, and ~ do), but
these alternative parameters have, in addition, clear intuitive interpreta
tions.
A nonlinear leastsquares regression method was used to estimate the
three unknown parameters of the logistic function from the empirical
observations. Alternative estimation algorithms were then used to test the
sensitivity of the estimated parameter values to different assumptions about
the errors and weighing of observations.6 The estimation methods used
in the analysis are reported in Grubler et al. (19871.
The values of the estimated parameters for the logistic growth processes,
=
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312
NEBOJSA IVAKICENOVIC
as well as the correlation coefficient R2 and uncertainty ranges for the
parameters, are given in Table 81. Estimation of the uncertainty ranges
is based on a Monte Carlo simulation approach by Debecker and Modis
(1986) for the threeparameter logistic function. A total of 33,693 different
Scurves were generated and subsequently fitted by the logistic function
providing values for K, At, and to, with known and varying distributions
of statistical fluctuations. Debecker and Modis concluded that the value
of the estimated asymptote K is the most sensitive (it varies the most),
depending on the amount and accuracy of the data available. Their results
indicated that as a rule of thumb the uncertainty of parameter K will be
less than 20 percent within a 95 percent confidence level, provided that
at least half of the data are available (at least up to the point of inflection)
and that they are at least 10 percent accurate. A more comprehensive
treatment of uncertainty ranges and estimation methods is given in Grubler
et al. (19871.
In all cases, except Figure 818, the values of the estimated parameters
were within the specified uncertainty ranges. Even alternative algorithms
used to estimate the parameters provided values within the specified ranges.
The Technological Substitution Models
and Parameter Estimates
In general, the examples of technological substitution are inherently
more complex than the determination of single logistic growth pulses.
From a statistical point of view, however, the estimation of substitution
processes is much simpler. By setting K = 1, a threeparameter logistic
function x~t) can be normalized by setting fats = X(t)/K. For given values
of K, it then reduces to a function with two unknown parameters. In the
examples of technological substitution, this known asymptote specifies
the size of the "market" in which old technologies are replaced by new
ones. In the simplest case, there are only two technologies. The market
shares are fate for the new technology and 1  fats for the old technology
(Fisher and Pry, 19711. Thus, only the values of to and At must be
determined from the observations.
Fisher and Pry used the twoparameter logistic function to describe a
large number of technological substitution processes. They assumed that
once substitution of the new for the old had progressed as far as a few
percent, it would proceed to completion along a logistic substitution curve
fats
At ~ = at + ~
(3)
Ordinary least squares were used to estimate parameters At and to in
OCR for page 175
213
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OCR for page 175
214
NEBOJSA lVAKICElVOVIC
all cases in which one old technology is replaced by a new. Table 82
gives the estimated parameter values, the estimation period, and the cor
relation coefficients for the three examples given in this chapter.
In dealing with more than two competing technologies, a generalized
version of the FisherPry model was used because logistic substitution
cannot be preserved in all phases of the substitution process. Every given
technology undergoes three distinct substitution phases: growth, satura
tion, and decline. The growth phase is similar to the FisherPry substi
tution, but it usually ends before complete market takeover is reached. It
is followed by the saturation phase, which is not logistic but which en
compasses the slowing of growth and the beginning of decline. After the
saturation phase of a technology, its market share declines logistically (for
example, see the path of railway and road substitution in Figure 814 and
coal substitution in Figure 8171.
It is assumed that only one technology saturates the market at any
given time, that declining technologies fade away steadily at logistic
rates "uninfluenced" by competition from new technologies, and that
new technologies enter the market and grow at logistic rates. The current
saturating technology is then left with the residual market share and is
forced to follow a nonlogistic path that curves from growth to decline
and connects its period of logistic growth to its subsequent period of
logistic decline. After the current saturating technology has reached a
logistic rate of decline, the next oldest technology enters its saturation
phase, and the process is repeated until all technologies but the most
recent are in decline.
For example, n competing technologies are ordered chronologically
according to their appearance in the market, technology 1 being the oldest
and technology n the youngest (i.e., i = 1, 2, . . ., n). Thus, all
technologies with indices k, where k < j, will saturate before the tech
nology with index j, and technologies 1, where I > j, will saturate after
technology j.
The historical time series is denoted by aft), where the indices i = 1,
2, . . ., n represent the competing technologies and t the time points
(year, month, etc.) of the historical period for which data are available.
The fractional market shares of competing technologies, fi (t), are obtained
by normalizing the sum of the absolute shares to one:
x;(t)
jets
(4)
By applying the linear transform of the logistic function to the fractional
market shares,
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215
l
o
._
~_
C~
._
U:
._
C~
o
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216 NEBOJSA l!lAKICENOVIC
(I) l [ f (') ] (s)
a transformed time series with piecewise linear secular trends can be
obtained. In fact, only three distinct possibilities exist: a decreasing or an
increasing linear trend or a phase of linear increase that is connected by
a nonlinear saturation phase to a phase of linear decline. The oldest tech
nology (i = 1) always displays a declining linear trend, and the youngest
technology (i = n) an increasing linear trend (see Figure 817). These
linear trends can be estimated, including the increasing linear trends of
technologies that enter the saturation phase during the historical period.
Ordinary least squares were used to estimate the linear trends for each
competing technology. Table 82 gave the estimated parameter values for
the three multiple substitution processes used here, as well as the esti
mation period (historical time interval for which the parameters were
estimated) and the correlation coefficient. Parameter values were not given
for cars in Figure 815 because they have been saturating during the entire
historical period. Consequently, their substitution path is specified by the
model.
Each estimated linear equation with estimated parameters At and to can
be transported into a logistic function with coefficients ~ and ¢:
1 + exp(ait Pi)
(6)
where f(t) is now the estimated fractional market shares of technology i.
Because such a logistic function does not capture the saturation phases
and represents only growing or declining logistic trends,
n
Ifi(t)
i=1
may exceed 1 for some value of t, although it must be equal to 1 for all
t. Thus, the n 1 estimated logistic equations were left in their original
form (6) that is, as specified by coefficients Hi and Pi and one of the
n equations was defined as a residual
1
Aft) = 1 ~j 1 + exp (cxitQi)
(7)
that is, as the difference between 1 and the sum of the n 1 estimated
market shares f(t). The latter equation represents the oldest still growing
technology, j, such that of ' 0 where off_ ~ 1. The selected
technology cannot, however, be the oldest technology (i.e., j ~ 1), be
cause the oldest technology is replaced by the newer technologies and,
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U.S. TRANSPORT INFRASTRUCTURES
217
consequently, its market shares decline logistically from the start (i.e.,
ox < 01.
Thus, initially, there are n 1 technologies denoted by indices i ~ j
that follow logistic substitution paths, and one technology, j, that reflects
the residual of the market that is, the complement of the sum of other
technologies and 1. Based on the point in time, tj, at which technology j
is defined as a residual, application of the linear transform of the logistic
function to the market shares of technology j, defined above, produces a
nonlinear function that can be written in the form of equation (59.
This function has a negative curvature. It increases, then passes through
a maximum at which technology j has its greatest market penetration, and
finally decreases. After the slope becomes negative the curvature dimin
ishes for a time, indicating thatfj~t) is approaching the logistic form. But
then, unless technology j is shifted into its period of logistic decline, the
curvature will begin to increase as newer technologies acquire larger mar
ket shares. Phenomenological evidence from a number of substitutions
suggests that the end of the saturation phase should be identified with the
point at which the curvature of yet), relative to its slope, reaches its
minimum value. This criterion is taken as the final constraint in this
generalization of the substitution model, and from it the coefficients for
the jth technology in its logistic decline are determined.
Thus, the point in time at which the rate of decrease of yet) approximates
a constant is determined. From this point on, the rate of change is set
equal to this constant, thereby defining a new logistic function. This point
of constant slope is approximated by requiring that the relative change of
slope is minimal,
y (t)
.~( ~ = minimum
(8)
for tj ' t < te' yet) < 0, and yet) < 0.
If this condition is satisfied (the point in time at which this occurs is
tj~ ~ > tj), the new coefficients for technology j can be determined as
~j=yj~tj+l,
and
(9)
A = yj~tj+~)yj~tj+~)tj+~. (10)
After time point tj, technology j + 1 enters its residual phase. The
process is then repeated until either the last technology n enters the sat
uration phase or the end of the time interval (te) is encountered.
These expressions, which have been developed in algorithmic form,
determine the temporal relationships between competing technologies.
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218
NEBOJSA A7AKICENOVIC
 PENE (fj(t), c;, Bj, iYo, no) 
t, 1901 +iyo+no 
t:  1900 + iyo J
I t:t+1 1
i
,
NO ~ ~ YES
tr ~)
Y
 f`(t)1/(1+exp( ~;'t§j)) 
NO /: YES
fj(t) $~:
~t)] 2 _ y "(t) ~ y (t)
aj ~ y'(t)
,Bj ~ y(t)  y'(t) · t _
\/ ~
(t)~1  :: f`(t)
L: l°g [fj(t)/(1  fj(t))]
FIGURE 820 Flowchart of the logistic substitution algorithm.
Figure 820 is a flowchart of the algorithm that describes the logistic
substitution process. A more detailed description of this procedure and
the software package for the generalized logistic substitution model is
given in Nakicenovic (1979, 1984~.
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U.S. TRANSPORTINFRASTRUCTURES
219
NOTES
1. One general finding of a large number of studies is that many growth processes follow
characteristic Sshaped curves. A logistic function is one of the most widely applied
Sshaped growth curves and is denoted by X/(K  X) = exp(cxt + Q), where t is the
independent variable usually representing some unit of time; a, if, and K are constants;
x is the actual level of growth achieved; and K  X iS the amount of growth still to be
achieved before the (usually unknown) saturation level K is reached. Taking logarithms
of both sides results in the lefthand side of the equation being expressed as a linear
function of time so that the secular trend of a logistic growth process appears as a
straight line when plotted in this way. Substituting f = X/K in the equation expresses
the growth process in terms of fractional share f of the asymptotic level K reached
that is, the equation becomes[(1  f) = exp(at + B).
2. This period of elapsed time we call At, and we define it as the time elapsed between
the achievement of 1 and 50 percent of the saturation level K (in this example At =
30.0 yr). Given the symmetry of the logistic function the same time is required for the
increase from 50 to 99 percent of the saturation level. An alternative definition of At
is the time elapsed between the achievement of the 10 and 90 percent level. This
definition of At differs slightly from its first definition, but for all practical applications
both definitions can be used interchangeably.
3. A large number of studies have found that the substitution of a new technology for an
old one, expressed in fractional terms, follows characteristic Sshaped curves. Fisher
and Pry (1971) formulated a simple but powerful model of technological substitution
by postulating that the replacement of an old technology by a new one proceeds along
the logistic growth curvef/(1 f) = exp(cxt + p) where t is the independent variable
usually representing some unit of time, ~ and ~ are constants, f is the fractional market
share of the new competitor, and 1  f is that of the old one.
4. The fractional shares (f) are not plotted directly but as the linear transformation of the
logistic curve that is, p( 1  f ) (in this more general cases if is the fractional market
share taken by a given energy and (1  f) is the sum of the market shares of all other
competing transport infrastructures). This form of presentation reveals the logistic sub
stitution path to be an almost linear secular trend with small annual perturbations. Thus,
the presence of some linear trends in Figure 814 indicates where the fractional substi
tution of transport infrastructures follows a logistic curve.
In dealing with more than two competing technologies, the FisherPry model must
be generalized because in such cases logistic substitution cannot be preserved in all
phases of the substitution process. Every competitor undergoes three distinct substitution
phases: growth, saturation, and decline. This process is illustrated by the substitution
path of railway tracks, which curves through a maximum from increasing to declining
market shares (see Figure 814). In the model of the substitution process, it is assumed
that only one competitor is in the saturation phase at any given time, that declining
technologies fade away steadily at logistic rates, and that new competitors enter the
market and grow at logistic rates. As a result the saturating technology is left with the
residual market shares (i.e., the difference between one and the sum of fractional market
shares of all other competitors) and is forced to follow a nonlogistic path that joins its
period of growth to its subsequent period of decline. After the current saturating com
petitor has reached a logistic rate of decline, the next oldest competitor enters its
saturation phase, and the process is repeated until all but the most recent competitor
are in decline. A more comprehensive description of the model and assumptions is given
in Nakicenovic (1979).
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220
NEBOJSA NAKICENOVIC
5. As in Figure 814, the fractional shares f are not plotted directly but as the linear
transformation of the logistic curve that is, f/(1  f ) (as the ratio of the market share
taken by a given energy source over the sum of the market shares of all other competing
energy sources). The form of presentation in Figure 817 reveals the logistic substitution
path as an almost linear secular trend with small annual perturbations. Thus, the presence
of some linear trends in Figure 817 indicates where the fractional substitution of energy
sources follows a logistic curve.
6. In this particular application the difference was in the assumptions about weighing of
the observations in the estimation procedure for example, whether unit or data
dependent weights are used.
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