Or this.......... LOL
What is Sustainable Transport?
Chuan-Zhong Li
Department of Economics, University of Dalarna
S-781 88 Borlänge, Sweden
March 1, 2000
Abstract
In recent years there has been increasing interests in transport and
sustainability studies. However, there has been no formal definition on
the concept of sustainable transport. In this paper, we attempt to formalize
the concept in a framework of dynamic welfare analysis. Using
a control-theoretic model, we characterize a sustainable transport as the
development of the transport sector that can support a maximum sustainable
social welfare measured by an augmented net national product
with environmental values taken into account. Dynamic interactions between
transport, production, consumption and technological changes are
also discussed.
1 Introduction
Sustainable development has become a popular catchphrase nowadays. In the
well-known Bruntland Commission Report, sustainable development was defined
to be ”a development that meets the needs of the present generation without
compromising the ability of future generations to meet their needs”. However,
1
the concept of needs is a rather complex one in economics and to embed it in the
issue of sustainability seems to make the already complex definition even more
intractable. Accordingly, economists have attempted to narrow the definition
of sustainable development to refer to an economy in which future growth is not
compromised by that of the present (Goldin andWinters, 1994). This is evidenced
by two emerging branches of research on the subject. One is to extend the wellknown
Weitzman (1976) theory on welfare significance of net national product by
taking into account natural and environmental resources (Mäler, 1991; Aronsson
et. al., 1997; Weitzman and Löfgren, 1997). The other is to adapt optimal
growth models with hyperbolic discounting so as to achieve a maximum long-run
sustainable welfare (Chichilnisky, 1996, 1997; Heal, 1998; and Li and Löfgren,
2000).
In addition to the general concept of sustainable development, researchers and
policy makers have in recent years also concerned about its ramifications in various
production sectors and geographical areas. There have been concerns about,
for example, sustainable agriculture, sustainable forestry, sustainable cities, sustainable
coast, sustainable environment, and sustainable transport. While these
concepts seem to be widely accepted, their precise contents have remained rather
elusive. For instance, what is sustainable transport? The transport sector has
been one of the largest emitters of greenhouse gases which may lead to dramatic
climatic changes in the future. It is also known that other pollutants from transport
damage the natural environment which is our life-support system. With
increasing demand for consumption, transport services and environmental quality,
it is natural that sustainability related to transport has come to the top list of
policy agenda. However, to the best of our knowledge, there has been no formal
models for the concept of sustainable transport.
In this paper, we attempt to formalize the concept of sustainable transport
in a framework of dynamic welfare analysis. Using the control-theoretical model
developed by Weitzman (1976) and extended by Aronsson et. al. (1997), we
2
will highlight the transport sector by explicitly taking into account transport infrastructure,
transport means, energy supply, and environmental quality in our
model. It is shown that in a broad sense, sustainable transport cannot be separately
defined irrespective of the rest of the economy as transport is an integral
part of the whole system. At best, the concept of sustainable transport can be
defined as a development of the transport sector that is supported by production
and energy sectors on the one hand and that supports a maximum sustainable
social welfare on the other hand. In a narrow sense, one may conceptualize sustainable
transport as an adaptation of transport services that can satisfy some
other goals such as environmental objectives. However, a sustainable transport
defined as such may not support any overall sustainability or maximum social welfare.
We also extend our analysis with technological progresses of fuel efficiency
and non-stationary time preferences with a hyperbolic discounting criterion.
The remaining part of the paper is structured as the following. In section 2,
we develop the dynamic model with a highlighted transport sector and outline the
optimal solutions, and in section 3 we perform sustainability and welfare analysis
with special reference to sustainable transport. Section 4 extends the analysis
with technical progresses and the hyperbolic discounting criterion, and section 5
sums up the findings in this study.
2 The model
To make the problem analytically tractable, we consider that the economy consists
of only four sectors, a transport sector, an energy sector, an environmental
sector, and a production sector. It is assumed that transport services q can be
produced using transport means (vehicles) m, energy (fuel) h, and emissions of
pollutant e as inputs for a given infrastructure z, i.e. q = q(m, h, e, z). Further,
we differentiate transport services to households and firms by q1 = q(m1, h1, e1, z),
and q2 = q(m2, h2, e2, z), respectively, where m = m1 + m2, h = h1 + h2, and
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e = e1 + e2. The production sector is assumed to produce all other goods and
services y with a neoclassical production function1 y = f(k, q2) with respect to
capital stock k and transport service q2. Let c(t), i(t) and δk(t) denote, respectively,
consumption, investment in infrastructure and capital depreciation rate at
any time t. Then the dynamics of the capital stock can be described by
˙k
= f(k(t), q2(t)) − δkk(t) − c(t) − m(t) − i(t) (1)
with initial stock k(0) = k0. Let z(t) represent the transport infrastructure at
time t, which depreciates at a rate δz. Then, its dynamics can be described by
˙ z = g(i(t)) − δzz(t) (2)
where g(i) is a infrastructure production function with respect to investment
i. Environmental quality at time t, E(t), is negatively related to the stock of
pollution p(t), which evolves as
˙ p = e(t) − δpp(t) (3)
where e(t) is the flow of emission at time t and δp is the assimilative rate of the
natural environment. To simplify matters, we may think that E(t) = ¯p − p(t)
with ¯p as the ”largest” stock of pollutants so that dE/dp = −1. Energy flow at
time t, h(t), is extracted from an exhaustible resource with stock x(t) ≥ 0 so that
˙ x = −h(t) (4)
Note that we have for notational ease not introduced any extraction cost or oil
refinery processes as these were not of out essential concern in this paper.
Without loss of generality, we normalize the population to unity and thus simply
consider a representative household that derives utility u from consumption c,
1Note that we omitt the emission input from the production function here in order to
highlight the transport ptoblem. The essential results on welfare analysis will not be altered
by this simplification.
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transport service q1, and environmental quality E such that u = u(c, q1, E). It is
assumed that utility increases with all three arguments but at a decreasing rate.
As E is negatively related to p, with E(t) = ¯p − p(t), it is obvious that utility
decreases with pollution stock p. Then, we may rewrite the utility function as
u = u(c, p, q1) with ∂u/∂p ≤ 0. For a given rate of time preference r, we can now
formulate the society’s optimization problem (the time argument t is suppressed
for national convenience) as to maximize
U = Z ∞
0
u (c, p, q1) exp(−rt)dt (5)
with respect to the control variables c,i,m1,m2,h1,h2,e1,e2, and the state variables
k,z,p,x, subject to the dynamics equations (1) to (4). To facilitate the analysis, we
define three new variables 0 ≤ α, β, γ ≤ 1 such that m1 = αm, m2 = (1 − α)m,
h1 = βh, h2 = (1− β) h, e1 = γe, and e2 = (1− γ) e. By so doing, we can
maximize the objective functional (5) with respect to the control variables c, i,
m, α, h, β, e and γ. The current-value Hamiltonian then becomes
H = u (c, p, q1) + λk [f (k, q2) − δkk − c − m − i]
+λz (g(i) − δzz) + λp (e − δpp) − λxh (6)
where λk, λz, λp, λx are the co-state variables associated with capital stock,
transport infrastructure, pollution stock and energy resource, respectively. Following
the standard procedure, we derive the first-order optimality conditions
with respect to the control variables as
uc − λk = 0 (∂H/∂c = 0)
−λk + λzg0(i) = 0 (∂H/∂i = 0)
αuq∂q1/∂m1 + (1 − α) λkfq∂q2/∂m2 − λk = 0 (∂H/∂m = 0)
uq∂q1/∂m1 − λkfq∂q2/∂m2 = 0 (∂H/∂α = 0)
βuq∂q1/∂h1 + (1 − β) λkfq∂q2/∂h2 − λx = 0 (∂H/∂h = 0) (7)
uq∂q1/∂h1 − βλkfq∂q2/∂h2 = 0 (∂H/∂β = 0)
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γuq∂q1/∂e1 + (1 − γ) λkfq∂q2/∂e2 + λp = 0 (∂H/∂e = 0)
uq∂q1/∂e1 − βλkfq∂q2/∂e2 = 0 (∂H/∂γ = 0)
The first two equations in (7) have the usual interpretations that the shadow
price of capital equals both to the marginal utility of consumption and to the
marginal value of investment in transport infrastructure. For the third one, while
the first term on the left-hand-side measures marginal contribution to household
utility of a unit input in transport means, the second term reflects that to firm
productions, and the sum of them equals marginal cost of providing a unit infrastructure.
The forth equation determines the optimal share of transport means
used in household and production sectors at which their marginal values are
equalized. Similarly, the rest of the four equations in (7) determine the optimal
levels of energy input and emissions, as well as their shares in the house household
and production sectors. Euler equations for the co-state variables are
˙λ
k = λk (r + δk − fk)
˙λ
z = λz (r + δz) − (uq∂q1/∂z + λk∂q2/∂z)
˙λ
p = λp (r + δp) − up (8)
˙λ
x = rλx + ν
which together with the dynamic system equations (1) to (4), the equation system
in (7) and the conventional transversality conditions constitute a complete set
of neccesarry optimality conditions. Under certain regularity conditions on the
utility and production functional forms, it can be proved that there exists an
optimal solution path for all the control, state and co-state variables c∗(t), m∗(t),
i∗(t), i∗(t), i∗(t), α∗(t), β∗(t), γ∗(t), k∗(t), z∗(t), p∗(t), x∗(t), λ∗k(t),λ∗z
(t), λ∗p(t)
and λ∗x(t), for all t ≥ 0.
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3 Sustainability and welfare analysis
In his seminal paper on Quarterly Journal of Economics,Weitzman (1976) showed
that the current-value Hamiltonian at ant time t corresponds to a constancy
equivalent measure of themaximumsustainable future utilities in a dynamic economy.
On the one hand, the Hamiltonian measures the national income (though
in utility terms) being the value of all consumptions and investments, and on the
other hand, it reflects the interest on an aggregated ”social capital” being the
discounted present value of all future utilities. Thus, if this Hamiltonian can be
made constant over time, then the ”social capital” would be kept intact while the
interest on it would support a constant stream of welfare over time. The model
in this famous paper has been later extended in several directions for green accounting
with natural resources and environmental externalities (Aronsson et.
al., 1997; Mäler, 1992).
In this paper, we attempt to formalize the concept of sustainable transport
by taking advantage of the analytical tools developed by Weitzman (1976). The
upshot is to study the current-value Hamiltonian function, its components and
their trends over time. The exercise to totally differentiate the Hamiltonian function
(6) with respect to time is straightforward but rather tedious. Fortunately,
by help of the first-order conditions in (7) and (8), we can simplify the resulting
expression as
dH(t)
dt
= r hλk(t) ˙k
+ λz(t) ˙ z + λp(t) ˙ p + λx(t) ˙ xi (9)
that is the interest on the aggregated value of ”investment” in all types of
capitals, the production capital, transport infrastructure, pollution and energy
stocks. The shadow values of λz(t), λp(t), and λx(t), for instance, can be derived
from (8) as
λz(t) = Z ∞
t Ãuq
∂q1
∂z
+ λkfq
∂q2
∂z !exp (−(r + δz) (s − t)) ds
λp(t) = Z ∞
t
up exp (−(r + δp) (s − t)) ds (10)
7
λx(t) = Z ∞
t
ν exp (−r (s − t)) ds
It is seen that the shadow value of an extra unit of transport infrastructure
is simply the present value of its overall contribution to household utility and
firm production in the future, discounted with an effective rate being the sum of
the rate of time preference and that of the infrastructure depreciation. Similarly,
the shadow prices of pollution and energy stocks are their presents values of
marginal pollution damages and use costs of the energy resource, discounted at
their appropriate rates.
Using the Hamiltonian function in (6), we can rewrite (9) as
dH(t)
dt
= r [H(t) − u (c, p, q1)] (11)
that has the well-known Bernoulli equation form. The solution of this differential
equation is simply
H(t) = r Z ∞
t
u (c, p, q1) exp(−r (s − t)) dt (12)
that can be interpreted as the static equivalent of the maximum sustainable future
welfare. To verify this, we denote the hypothetical constant welfare over time by
¯H
and then equate the present value of this stream to that of the stream of
optimal solution u (c∗(t), p∗(t), q1∗(t)), i.e.
Z ∞
t
¯H
exp (−r (s − t)) dt = Z ∞
t
u (c, p, q1) exp(−r (s − t)) dt (13)
then, it is straightforward to shown that ¯H
= H(t) as defined in (12). Thus, we
have
Proposition 1 Given the model setup from (1) to (5), the current-value Hamiltonian
is a static equivalent of the maximum sustainable future welfare along the
optimal path, which equals the interest on the present value of the whole stream
of future utilities.
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If we define the present value of future utilities as a ”super capital”, then
the static equivalent of future maximum welfare would remain constant over
time if this super capital is kept constant. A development path as such may
be regarded as sustainable according to the weak sustainability criterion. From
(9), it can be seen that this would also imply a zero value of the aggregated
”investment” in all types of capitals - the productive capital k, infrastructure z,
pollution p and energy stock x over the development path. It is worth mentioning
that there is no requirement for each type of capital to remain intact as the
value decrease in one type of capital may be compensated by value decreases
of other capital types. The main thing is that the sum of investment values
in all capital types remain non-decreasing over time. For example, if the value
of a transport infrastructure investment can sufficiently compensate a loss in
environmental value caused by production activities, the development path may
also be claimed to be sustainable, provided that they are appropriately priced
according to (10). Similarly, if the decrease in the exhaustible energy stock can be
compensated by increases in the value of other capital types including renewable
energy, then we may also claim that the development follows a sustainable path
(Hartwick 1977, 1996; Dixit and Hoel, 1980; and Solow 1986, 1992). Loosely
speaking, in a word with continuous changes, it may not be neccesarry to sustain
the size of each type of capital, say, trees, minerals, fishes, and railroad, but to
sustain the aggregated value of them.
From (4) and (12), we obtain
H(t) = u (c, p, q1) + λk(t) ˙k
+ λz(t) ˙ z + λp(t) ˙ p + λx(t) ˙ x
= r Z ∞
t
u (c, p, q1) exp(−r (s − t)) dt (14)
i.e. the sum of instantaneous consumption and investment values along the optimal
growth path equal the interest on the present value future utilities. We
are now interested in how the concept of sustainable transport enters the whole
picture. The question is whether it is possible to break the Hamiltonian into
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components on sector levels or at least to separate a transport-sector specific
sub-Hamiltonian function ˜H
(t) from the rest of the economy in the sense that
˜H
(t) = r Z ∞
t
u(q(s)) exp (−r (s − t)) dt (15)
where u(q(s)) denotes the utility of transport services at time s. If this were
true, then we would define sustainable transport as a development that ensures
the sub-Hamiltonian in (15) non-decreasing over time, i.e. d ˜H
(s)/ds ≥ 0 for
all s ≥ t. However, this is in general not the case since transport is always an
integral part of the whole system. First, transport services directly enter the
utility function, which may not be separable from consumption of other goods
and services including environmental quality. Second, the value of net changes in
the different types of capital on the right-hand-side of the first equation in (14)
depends on their shadow prices which hinges upon future interactions between
transport, energy, production and the environment. Even if we impose separability
conditions on the utility and production functional forms, it would be hardly
possible to decompose the overall Hamiltonian into an exclusive and exhaustive
set of sub-Hamiltonians, one of those, say, is for the transport sector.
Thus, we are inclined to define sustainable transport as an integral part of the
overall sustainable development. Specifically, we consider sustainable transport
as a development of the transport sector that is on the one hand supported by the
rest of the economy and that on the other hand supports a maximum sustainable
welfare over time. More formally, we state:
Proposition 2 A sustainable transport from time t onwards is characterized by a
development of transport services for households and firms, q∗1 [m∗1
(s), h∗1
(s), e∗1
(s), z∗(s)]
and q∗2 [m∗2
(s), h∗2
(s), e∗2 (s), z∗(s)], respectively, such that the overall Hamiltonian
in (12) satisfies dH(s)/ds ≥ 0 for all time s ≥ t.
According to this definition, one has to solve the overall sustainability problem
in order to know how a sustainable transport would look like. Among other
10
things, one has to assess the shadow prices of the different types of capitals
including pollution and energy stocks as shown in (10). This is obviously an
extremely challenging work since such shadow prices depend on the marginal
social damage of pollution and scarcity value of energy etc. in the future.
It is worth mentioning that, in various case studies, sustainable transport was
considered to be some ad hoc adaptation of the transport sector that can satisfy
certain environmental goals within a given period of time (Rothengatter, 1997;
Whitelegg, 1992). Although it is easier to do such an exercise, the adaptation
here provides only a partial picture to overall sustainability. A sustainable transport
defined as such would not ensure any overall optimality or sustainability
conditions. If it would do, then the optimal development path discussed above
would not be optimal, which is a contradictions.
As argued by some cost-benefi analysts, it may be more useful to try to measure
the right things though it is difficult than to accurately measure the wrongs
things. With the definition in proposition 2, the task would be to gather information
to better understand the complex system such as the economic contribution
of transport infrastructure to transport services, the effect of transport services
to households and firms, the relationship between transport and the environment,
energy supply, and the assimilative ability of the environment etc. With better
knowledge on these relations, we will be one step closer to operationalize the
concept of sustainable transport and overall sustainable development discussed
above.
4 An extension with technical progress and hyperbolic
discounting
An important result from the previous analysis is that along the optimal solution
path for the maximization problem (5), welfare is properly measured by the
11
current-value Hamiltonian, and sustainable transport is thereby defined as an endogenous
solution for the transport sector that supports a maximum sustainable
welfare. However, there have been two major omissions concerning technological
progress and time preferences. With a constant technology and fixed energy
stock, it may not be possible to sustain welfare if the exhaustible energy resource
is essential for transport and transport services are essential for the economy. A
simple remedy is to reinterpret the variable m in the transport service function q
as a composite factor input including some renewable energy energy sources, i.e.
a substitute for the exhaustible energy source. In so doing we will still stick to the
sustainability results derived above. Now, we explicitly introduce technological
progress in the model and examine how this will modify the sustainability conditions.
Although, technological progresses can take place in various sectors, here
we simply focus on energy efficiency for illustrative purposes as the qualitative
properties would remain the same with more sectors modified. Let us define the
effective energy use as ˆh(t) = φ(t)h(t) with φ(t) ≥ 0 and φ0(t) ≥ 0. Then, the
production of transport services will be q1 = q(m1, φ(t)h1, e1, z) for households
and q2 = q(m2, φ(t)h2, e2, z) for firms, with which we can use the same procedure
as in the previous sections to derive
dH(t)
dt
= r ³λk(t) ˙k
+ λz(t) ˙ z + λp(t) ˙ p + λx(t) ˙ x´+ λkhfq∂q2/∂h2φ0(t) (16)
and
H(t) + Ω(t) = r Z ∞
t
u (c, p, q1) exp(−r (s − t)) dt (17)
where
Ω(t) = Z ∞
t
λkhfq∂q2/∂h2φ0(t) exp(−r (s − t)) dt (18)
is the present value of marginal product increases due to the technical progress
function φ(t).
With technical progress explicitly introduced in the model, it is now the sum
of the conventional Hamiltonian and the value of the technical progress that measures
welfare. With a prospect of more efficient energy uses, the static equivalent
12
of maximum future sustainable welfare exceeds the current-value Hamiltonian
that is the value of current consumption plus that of investment in all capital
types (Aronsson et. al. 1997; Nordhaus, 1993). Note that the transport sector is
still non-separable from the rest of the economy and thus we still need to define
sustainable transport as an integral part of overall sustainable development.
Proposition 3 With technical progress explicitly introduced in the model, a sustainable
transport from time t onwards is characterized by a development of transport
services for households and firms, q∗1 [m∗1
(s), h∗1
(s), e∗1
(s), z∗(s)] and q∗2[m∗2
(s),
h∗2
(s),e∗2
(s),z∗(s)], respectively, such that the sum of the current-value Hamiltonian
in (12) and the value of technical progresses in (18) does not decrease over
time, i.e. dH(s)/ds + Ω0(s) ≥ 0 for all time s ≥ t.
Finally, we briefly analyze the effect of non-stationary preferences on welfare
measurement and sustainable development. In optimal growth models, it is often
assumed that people have stationary time preferences in that they discount the
future with a constant exponential rate. However, strong empirical evidences
suggest that people discount the future hyperbolically, applying a larger annual
discount rate to near-future returns than returns in the distant future (Ainslie,
1992; Loewenstein and Prelec, 1992; Cropper, Portney and Aydede, 1994; Laibson,
1996). Here we are to examine the implications of such non-stationary
preferences on sustainable development and sustainable transport. Let Λ(t) denote
the discount function with Λ0(t) < 0 and limt→∞ Λ(t) = 0, then the society’s
problem will be to optimize
U = Z ∞
0
Λ(t)u (c, p, q1) dt (19)
subject to the dynamic equations (1) to (4). As the integral of the discount
function over an infinite time horizon may not be bounded, the problem may
need to be solved using a more general optimization criterion (Seierstad and
13
Sydsaeter, 1987). After the usual derivations, we obtain the time trend of the
current-value Hamiltonian as
dH(t)
dt
= r(t) hλk(t) ˙k
+ λz(t) ˙ z + λp(t) ˙ p + λx(t) ˙ xi (20)
where r(t) = −Λ0(t)/Λ(t) is the instantaneous rate of discount at time t, and the
expression for the Hamiltonian itself as
H(t) = R∞t Λ(s)u (c, p, q1) ds
R∞t Λ(s)ds
= ¯r(t) Z ∞
t
Λ(s)
Λ(t)
u (c, p, q1) ds (21)
where ¯r(t) = Λ(t)/ R∞t Λ(s)ds is the long-run rate of discount.
When the exponential discount function is used as a special case with Λ(t) =
exp(−rt), we have r(t) = ¯r(t) = r, i.e. both the short and the long-run rate of
discount equals the constant exponential rate. The Hamiltonian in (21) would
in this case reduce to the conventional expression in (12). Since Λ0(t) < 0 and
limt→∞ Λ(t) = 0, the asymptotic long-run discount rate converges to zero, i.e.
limt→∞ ¯r(t) = 0. Following Radner (1967), the very long-run welfare would be
lim
t→∞
H(t) = lim
t→∞,T→∞
1
T Z t+T
t
u (c, p, q1) ds (22)
i.e. the greatest welfare that can be sustained forever. Such an ideal state may be
termed as the golden-rule (Phelps, 1961) or the green golden rule (Chichilnisky,
1996, 1997).
Proposition 4 Given the hyperbolic discount function Λ(t) with Λ0(t) < 0 and
limt→∞ Λ(t) = 0, a sustainable transport from time t onwards is characterized by a
development of transport services for households and firms, q∗1 [m∗1
(s), h∗1
(s), e∗1
(s), z∗(s)]
and q∗2[m∗2
(s), h∗2
(s), e∗2
(s), z∗(s)], respectively, such that the Hamiltonian expressed
in (21) does not decrease over time, i.e. dH(s)/ds ≥ 0 for all time s ≥ t, and it
eventually converges to the largest possible sustainable welfare defined in (22) as
time goes to infinity.
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Note that in the long-run, it is the optimal configuration of consumption,
transport services and environmental quality that provides the maximum welfare.
Wether the transport sector at this ideal state is larger or smaller than at other
states would depend on the exact functional forms and parameters in the model.
Following the optimal development path, the economy would eventually converge
to this state at which the overall welfare derived from a combination of goods
and services is maximized.
5 Concluding Remarks
In this paper we have attempted to formalize the concept of sustainable transport
in a control-theoretic framework. This was done by an explicit introduction of
the transport sector in the neoclassical optimal growth model. We first formulate
the Hamiltonian function, which corresponds to an augmented national product
in utility terms, as the sum of instantaneous utility and value of investment in
all types of capital including transport infrastructure, environmental quality, and
exhaustible energy stock. Then, following Weitzman (1976), we show that this
augmented national product along the optimal path is a static equivalent of the
maximum sustainable future welfare. With general sustainability conceptualized
as a non-declining sustainable welfare over time, we have characterized sustainable
transport as a development of the transport sector that is supported by the
rest of the economy and that supports an overall maximum sustainable welfare
over time. Since transport is an integral part of the whole economy, it is not
possible to extract a sub-Hamiltonian for the transport sector from the overall
welfare measure. As a result, we cannot define sustainable transport as a development
which sustains the ”value” of transport corresponding to some kind of
sub-Hamiltonian functions.
We have also extended the optimal growth models with exogenous technological
progress and hyperbolic discounting and examined their implications on
15
sustainable transport. It is shown that with technical progress, it is the sum of
the Hamiltonian (national product) and the value of technical progress that corresponds
to the static equivalent of maximum sustainable future welfare. However,
since the transport sector remains non-separable from the rest of the economy,
the definition of sustainable transport in this case is simply adapted to be a
development of the transport section that sustains the sum of Hamiltonian and
the value of technical progress. With hyperbolic discounting, the dynamic system
from the growth model becomes non-autonomous and therefore the resulting
national-product-like Hamiltonian measure depends explicit on the time variable
t. As time goes to infinity, we show that the long-run steady state welfare
is maximized at what may be called the Phelp’s golden rule or Chichilnisky’s
green golden rule. Sustainable transport is according defined as a development of
the transport sector that sustains the national-product-like Hamiltonian welfare
measure.
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