Natural Disasters in a Two-Sector Model of
Endogenous Growth∗
Masako Ikefuji† Ryo Horii‡
January 3, 2008
Abstract
This paper studies sustainability of economic growth considering the riskof natural disasters caused by pollution in an endogenous growth model withphysical and human capital accumulation. We consider an environmental taxpolicy, and show that economic growth is sustainable only if the tax rate on thepolluting input is increased over time and that the long-term rate of economicgrowth follows an inverted V-shaped curve relative to the growth rate of theenvironmental tax. The social welfare is maximized under a positive steady-state growth in which faster accumulation of human capital compensates theproductivity loss due to declining use of the polluting input.Keywords: natural disasters, human capital, endogenous depreciation, eco-nomic growth.JEL Classification Numbers: O41, O13, E22.
∗We are grateful to Kazumi Asako, Koichi Futagami, Tatsuro Iwaisako, Kazuo Mino, Takumi
Naito, Tetsuo Ono, Yoshiyasu Ono, Makoto Saito, Yasuhiro Takarada, and seminar participants at
Fukushima University, Nagoya University, Osaka University, and Toyama University for their helpful
comments and suggestions. This study is partly supported by JSPS Grant-in-Aid for Scientific
Research (No. 16730097). All remaining errors are naturally our own.
†Fondazione Eni Enrico Mattei, Corso Magenta, 63, 20123 Milano, Italy. E-mail:
‡Graduate School of Economics, Tohoku University, Kawauchi 27-1, Aoba-ku, Sendai 980-8576,
Japan. E-mail: [email protected]
1
1 Introduction
Natural disasters have a large impact on economic activity primarily through de-
struction of capital stock. For example, National Oceanic and Atmospheric Admin-
istration (NOAA, 2005) estimates that Hurricane Katrina, occurred on August 2005
in the Gulf of Mexico, caused over 100 billion dollars’ worth of damage mainly on
physical capital. This magnitude of destruction might seem unusual, but the losses
caused by landfalling hurricanes in the United States in the previous year, 2004, were
also considerably large—approximately 45 billion dollars, as reported by NOAA. This
magnitude of damage is not negligible even in comparison to the whole size of the
U.S. physical capital stock.
Despite various forms of preventive efforts, the occurrence of natural disasters is
not declining but in a growing trend.1 The long-term consequence of the increased
possibility of such disasters critically hinge on whether their occurrences are purely
exogenous phenomena to the economic system, or they are caused for some part by
economic activity. That is, if the latter is true, economic growth itself creates a threat
to economic activity, making the sustainability of economic growth questionable.
Recent meteorological research shows that, unfortunately, the latter argument is
likely to be true. For example, it is clearly stated in the Intergovernmental Panel on
Climate Change third assessment report (IPCC, 2001) that “Emissions of greenhouse
gas and aerosols due to human activities have a great impact on global climate
change.” Global warming, or more specifically, increasing sea surface temperature,2
1In Hoyois et al. (2005), the global number of reported disasters is 1.55 times higher in 2000-
2004 period than in 1995-1999 period, and the number of reported extreme temperature disasters,
floods, and people affected by disasters are almost 2 times, 1.74 times, and 1.33 times higher,
respectively. A disaster in the database must be fulfilled at least one of the following criteria: 10
or more people reported killed; 100 people reported affected; declaration of a state of emergency;
call for international assistance.
2The tropical sea surface temperature in the North Atlantic shows a large upswing in the last
decade. Emamuel (2005) argues such a upswing is related to the El Nino and reflects the effect of
global warming.
2
is in turn suspected to increase hurricane frequency and intensity (Emamuel, 2005;
Webster et al., 2005).
Those observations suggest that there is a two-way causality between economic
activities and the occurrence of natural disasters. This paper investigates the sustain-
ability of economic growth in the presence of this two-way causality, by introducing
the endogenous risk of natural disasters into a Uzawa-Lucas type of endogenous
growth model. Following the literature (e.g., Copeland and Taylor, 1994; Bovenberg
and Smulders, 1995; Stokey, 1998), we assume that polluting inputs such as fossil
fuels are necessary for economic activities and those inputs are subject to an envi-
ronmental tax.3 Differently from earlier studies, however, this paper examines the
case in which the use of polluting inputs raises the probability that capital stocks
are destroyed by natural disasters. Agents make saving decisions taking into account
the possibilities of loss of asset due to natural disasters.
Using the model, we show that the economic growth is, in fact, not sustainable
if the (per-unit) tax rate on polluting inputs is kept constant. Intuitively, under the
constant tax rate, firms are willing to use increasing amounts of polluting inputs as
the economy grows. However, as increased use of polluting inputs raises the risk of
natural disasters, it reduce incentive agents to invest in capital stock since they face
a higher possibilities of asset loss. Thus, even in the long run, capital stock cannot
exceed a constant threshold under a constant tax rate.
To overcome this limitation, we next consider a time-varying tax on polluting in-
puts. If the per-unit tax rate is raised over time, private firms owing a Cobb-Douglas
production technology will increase the use of other inputs, including human cap-
ital, relative to polluting inputs. As a result, the use of polluting inputs can be
bounded above, making the economic growth sustainable. It is also shown that the
growth rate of environmental tax has both positive and negative effects on economic
growth. The faster the rate at which the environmental tax is increased, the lower is
3Copeland and Taylor (1994) and Stokey (1994) assumed that there are a continuum of technol-
ogy generating a different level of pollution. However, the technology can be written with capital
and the total level of pollution as inputs.
3
the asymptotic amount of pollution and therefore the lower is probability of disasters.
This gives households more incentive to save, which promotes growth. However, the
increased cost of using polluting input faced by private firms reduces their produc-
tivity at each date, which has a negative effect on growth. Due to those opposite
effects, the rate of economic growth rate is shown to follow an inverted V-shaped
curve relative to the growth rate of environmental tax.
Having shown that the sustained growth is feasible, this paper then examine
whether it is desirable or not. This question may seem trivial, but in a AK-growth
model with pollution Stokey (1998) shows that, even when production technology
allows sustained growth, it is theoretically possible that agents prefer a no-growth
state with a good environment.4 Contrary to Stokey’s analysis, we show that the
social welfare is maximized on a steady-state growth path, where the environmental
tax is raised at a positive rate, although this does not coincide with the growth
maximizing path.
The difference of our result from Stokey’s stems not from our assumption that the
use of polluting input only affects the risk of natural disasters without directly affect-
ing consumer’s utility.5 Rather, it comes from the our two-sector specification that
the growth is driven both by physical and human capital accumulation. This paper’s
analysis shows that sustained growth with a Cobb-Douglas technology is feasible
under a limited use of polluting input because human capital stock is accumulated
much faster than the rate at which output is increased. In fact, provided that human
capital stock has lower degree of vulnerability to disasters, as suggested by Skidmore
4Stokey (1998) assumed additive separable preference for consumption and pollution in which
the marginal utility from consumption is declining whereas the marginal disutility from pollution is
increasing. Then, as consumption increases due to capital accumulation, reducing pollution becomes
more important than increasing consumption. She shows that further growth is not optimal at a
high level of capital stock.
5In section 5, we examine the case in which pollution directly causes disutility to consumers, in
addition to raising the risk of disasters. It is shown that a positive steady-state growth is compatible
to welfare maximization even in this case.
4
and Toya (2002), the risk raises incentives for more investment in human capital
stock relative to physical capital stock.
To our best knowledge, this study is a first attempt to examine the consequences
of natural disasters in an endogenous growth model. This does not mean, of course,
that our study is independent from the previous literature. In fact, great attention
has been paid to the sustainability of economic growth in the literature of growth
theory, by noticing that the finite nature of natural environment may potentially
restrict sustainability of economic growth. With regard to the finiteness of natu-
ral resources, Aghion and Howitt (1998), Scholz and Ziemes (1999), Schou (2000),
Grimaud and Rouge (2003), and Agnai, Gutierrez and Iza (2005) examined sus-
tainability of economic growth in endogenous growth models with non-renewable
resources.6
Complementary to those studies, Stokey (1998) and Uzawa (2003) examined sus-
tainability focusing on emission of pollutants. Since the global atmosphere is finite,
the negative effects from pollutants will become unacceptably serious when the us-
age of polluting input increases without bound. Therefore, both Stokey (1998) and
Uzawa (2003) concludes that, without exogenous technological change, it is opti-
mal for the economy to converge to a no-growth steady state. Their conclusions
are different from ours because they consider only accumulation of physical capital,
whereas we consider both human and physical capital. In this respect, our study is
more related to a two sector model developed by Bovenberg and Smulders (1995),
where accumulation of knowledge is explicitly incorporated. Specifically, Bovenberg
and Smulders (1995) assumed that the amount of pollution cannot be increased in
the long-run and showed that, if output can grow at the same rate as that of physical
and human capital for a constant level of pollution,7 sustained growth is both feasible
6For example, Grimaud and Rouge (2003) analyzed sustainability of economic growth intro-
ducing a non-renewable natural resource into a Schumpeterian endogenous growth model. They
showed that whether both optimal and equilibrium growth is positive at the steady-state depends
on the value of the subjective discount rate relative to the productivity of R&D.
7They focused on the case in which the production function exhibits constant rate to scale
5
and optimal.
While their conclusions are similar to ours, the settings in the model differs in two
respects. First, we does not assume that pollution cannot increase in the long run,
but show that a path in which pollution increases without bound is not chosen in
equilibrium since it reduces the incentive to save of consumers, who face increasingly
high possibility of asset losses due to disasters. Second, we consider a more natural
setting in which the production function exhibits constant return to scale with respect
to all production factors—including polluting inputs. This means that, if the amount
of pollution is fixed, the production function faces decreasing returns to scale (DRS).
We show that, even under this more severe condition, sustained growth is both
feasible and desirable since the DRS can be overcome by accumulating human capital
more rapidly than the rate of economic growth.
The rest of the paper is organized as follows. Section 2 presents the model
and proves that growth cannot be sustained under a constant tax rate on polluting
inputs. The steady-state growth with an increasing environmental tax is analyzed in
Section 3. The social planners’s problem is examined in Section 4 so as to investigate
the desirability of sustained growth. Section 5 considers an extension of the model
in which pollution harms the utility of consumers as well as increases the risk of
disasters. Section 6 concludes.
2 The Model
This section presents a model of natural disasters and economic growth. In the first
subsection, the risk of natural disasters is introduced into a two-sector growth model
based on Uzawa (1965) and Lucas (1998). In the second subsection, the behavior of
households and firms is examined. The third subsection summarizes the equilibrium
(CRS) with respect to Kt and (HtPt), where Pt represents the level of pollution. This means that,
if all production factors Kt, Ht and Pt, are doubled, the output is more than doubled; i.e., the
production function exhibits increasing returns to scale with respect to Kt, Ht and Pt. In contrast,
we maintain a standard setting in which output is CRS with respect to all production factors.
6
conditions. The final subsection proves that sustained growth is not possible when
the environmental tax rate is kept constant.
2.1 The risk of natural disasters
In the model, output is produced by a constant-returns-to-scale production technol-
ogy using physical capital Kt, human capital Ht, and polluting input Pt such as fossil
fuels that emits pollutants or greenhouse gases. The production function is given by:
Yt = F (Kt, utHt, Pt) = AKαt (utHt)
1−α−βP βt , (1)
where ut ≥ 0 is the time share devoted to production of goods, α ∈ (0, 1/2) is a
constant share of physical capital,8 and β ∈ (0, 1− α) is that of the polluting input.
Note that production function (1) exhibits constant returns to scale with respect to
all inputs including Pt. Output is either consumed or added to physical capital stock.
Since we focus on the risk of natural disasters, we ignore the extraction cost and/or
production cost of polluting inputs and finite nature of natural resources. The risk
of natural disasters on capital stock is assumed to be inevitable and to depend on
the amount of pollution.
Specifically, suppose that economy consists of continuum of local areas. Let qt be
the arrival rate of natural disasters per unit of time at each area:
qt = q + qPt, (2)
where q and q are positive constants. Equation (2) says that the arrival rate raises
as the amount of aggregate polluting inputs increases, as in the case of hurricanes
and fossil fuels. When a natural disaster occurs at an area, it causes damage to
physical capital. For example, if natural disasters occur at an area where the existing
aggregate physical capital stock is Kt, the expected loss of physical capital is ϕKt,
where ϕ > 0 is the average damage to physical capital stock. Note that, for various
8We assume α < 1/2 for simplicity of the stability analysis. In a model where human capital is
considered explicitly, this assumption seems quite reasonable.
7
reasons, a natural disaster harms human capital stock as well.9 Similarly to ϕ, define
ψ > 0 as the average damage to human capital stock. The damage on human capital
measured in relative to the size of the existing stock is typically smaller than the
damage on physical capital, which implies that ϕ would be smaller than ψ.
For simplicity, each area is assumed to be small enough and the occurrence of
natural disasters in one area is assumed not to be correlated to others.10 By the
law of large numbers with (2), the aggregate damage to physical capital stock and
human capital stock are respectively:
qt · ϕKt = (ϕq + ϕqPt)Kt, (3)
qt · ψHt = (ψq + ψqPt)Ht. (4)
Let δK and δH be the constant rates of depreciation of physical capital stock and
human capital stock, respectively. Then, similarly to Lucas (1988), the resource
constraints for physical and human capital stocks are written as:
Kt = F (Kt, utHt, Pt) − Ct − (δK + ϕPt)Kt, (5)
Ht = B(1 − ut)Ht − (δH + ψPt)Ht, (6)
where δK ≡ δK + ϕq, ϕ ≡ ϕq, δH ≡ δH + ψq, ψ ≡ ψq, and Ct, B, and 1 − ut are
the aggregate consumption, the constant productivity of human capital accumula-
tion, and the fraction of time devoted to production of human capital, respectively.
Equations (5) and (6) shows that the risk of natural disasters effectively augments
the depreciation rates of physical and human capital stocks, in proportion to the use
of polluting input.
9The death toll in Katrina rose to over 1000 (NOAA, 2005) and the number of the injured was
much more. In addition, many education institutions are forced to remain closed for extended
periods of time and a large number of data and documents storing valuable knowledge are lost after
the disasters pass.
10This assumption is not realistic when considering large scale disasters. Without it, the law of
large numbers does not apply, and disasters cause short-term fluctuations. However, since we are
focusing on long-term behavior of the economy, analysis of such fluctuations are out of the scope
of this paper.
8
Observe that, unlike standard endogenous growth models, the right hand sides of
equations (5) and (6) are not homogenous of degree one in terms of quantities. This
implies that balanced growth that exhibits a homothetic expansion is not feasible,
reflecting the finiteness of natural environment.
2.2 The market economy
Since firms do not take into consideration the externality that the use of polluting
inputs increases the risk of natural disasters, the market equilibrium does not cor-
respond with the solution of the social planner’s problem. The followings consider
explicitly the market economy where per-unit tax τt, in terms of final goods, is levied
on the use of polluting inputs. The government balances its budget at each moment
and equally distribute the tax revenue Tt = τtPt among households in a lump-sum
fashion.
Households
The economy is populated by a unit mass of infinitely lived homogeneous house-
holds. Each household owns physical capital stock, kt, and human capital stock, ht.
However, due to natural disasters, they are faced with the risk of damages to both
types of capital stock. The insurance market is assumed to be complete. Under
this assumption, it is optimal for households to take out insurance that cover the all
losses associated with natural disasters. Since their expected damages to physical
capital stock and human capital stock are qtϕkt and qtψht, respectively, the budget
9
constraint of households can be written as:11
kt = rtkt + wtutht − (δK + ϕPt)kt − ct + Tt, (7)
ht = B(1 − ut)ht − (δH + ψPt)ht, (8)
where rt, wt, and ct denote the real interest rate, the real wage rate, and the amount
of consumption, respectively. Note that the cost associated with depreciation and
insurance is paid by the owner of the capital.
The utility function of the representative household is given by:∫ ∞
0
c1−θt − 1
1 − θe−ρtdt, (9)
where θ > 1 is the inverse of the elasticity of intertemporal substitution and ρ is the
rate of time preference. We assume B − δH > ρ so that households have enough
incentive to investment in human capital. Given the time paths of rt, wt, Pt, τt and
Tt, each household maximizes (9) subject to the constraints (7) and (8). From the
first-order condition for maximization problem, we obtain the Keynes-Ramsey Rule:
−θct
ct
= ρ + ϕPt + δK − rt. (10)
This condition is similar to that obtained in the original Uzawa-Lucas model, except
that the depreciation rate is augmented by the risk of natural disasters, ϕPt.
The shadow price of human capital relative to that of physical capital is wt/B,
which equals to the market price of human capital measured by physical capital
stock. Hence, the arbitrage condition between human capital investment and physical
capital investment is given by:
wt
wt
= rt − (ϕ − ψ)Pt − (δK − δH) − B. (11)
11Equations (7)-(8) implicitly assume that damaged human capital is compensated in the form of
human capital. Obviously, a more realistic setting is that this compensation is done in the form of
goods. Nonetheless, as long as the amount of compensation in terms of goods is calculated using the
appropriate price of human capital, wt/B, the equilibrium outcomes do not change in the aggregate
level.
10
In (11), the left hand side implies the rate of change in the relative shadow price of
human and physical capital while the right hand side implies the difference between
the marginal return to investment in physical capital and human capital. Note that
this condition must be satisfied in the long run. If it is not, the solution would be
either ut = 0 or ut = 1 for all agents, and therefore one of the two kinds of aggregate
capital stock approaches zero due to depreciation. However, it raises the shadow
price of the unaccumulated type of capital stock, which is at odds with the decision
of agents that they do not invest in that type of capital stock. The transversality
conditions (TVCs) for physical capital stock and human capital stock, respectively,
are
limt→∞
ktc−θt e−ρt = 0, and (12)
limt→∞
ht(wt/B)c−θt e−ρt = 0, (13)
where c−θt and (wt/B)c−θ
t represent shadow prices of physical and human capital.
Firms
There are a continuum of firms producing final goods in competitive market. We
consider a representative firm maximizing its profit. The firm pays the wages for
labor input, the rental rate for physical capital input, and the environmental tax as
well. Given factor prices rt, wt and τt, the profit maximization problem is:
maxKt,Nt,Pt
F (Kt, Nt, Pt) − rtKt − wtNt − τtPt,
where Nt ≡ utHt is the amount of human capital employed by the firm. The first-
order conditions for this problem are rt = αYt/Kt, wt = (1 − α − β)Yt/Nt, and
τt = βYt/Pt. Substituting the profit maximizing polluting input, Pt = βYt/τt, into
production function (1), the output can be written as:
Yt =(Aτ− β
1−β
)K bα
t N1−bαt , (14)
where A ≡ ββ/(1−β)A1/(1−β) and α ≡ α/(1 − β). When written in the form of (14),
it becomes clear that the environmental tax lowers the effective total factor pro-
ductivity, Aτ−β/(1−β). Using this notation, the first order conditions are expressed
11
as
rt = αYt
Kt
= αAτ− β
1−β
t
(Kt
Nt
)bα−1
, (15)
wt = (1 − α − β)Yt
Nt
= (1 − α − β)Aτ− β
1−β
t
(Kt
Nt
)bα
. (16)
2.3 Equilibrium conditions
In this economy, the equilibrium path is characterized by the motions of aggregate
physical capital stock Kt, aggregate human capital stock Ht, aggregate labor supply
ut, aggregate consumption Ct, and the use of polluting input Pt. This subsection
summarizes the equilibrium conditions for those variables.
Note that, since the population is homogenous and normalized to unity, Kt = kt,
Ht = ht, Ct = ct, and ut = Nt/Ht hold in equilibrium. Substituting factor prices
(15) and (16) as well as the lump-sum transfer Tt = τtPt into the budget constraint
of households (7) yields the evolution of physical capital stock:
Kt
Kt
=Yt
Kt
− Ct
Kt
− (δK + ϕPt). (17)
From the production function of human capital (8), the evolution of aggregate human
capital stock is given as:
Ht
Ht
= B(1 − ut) − (δH + ψPt). (18)
Substituting factor prices (15) and (16) into the arbitrage condition (11), we obtain
the evolution of labor supply ut:
− β
1 − β· τt
τt
+ α(Kt
Kt
− Ht
Ht
− ut
ut
)= α
Yt
Kt
+ (ψ − ϕ)Pt − (δK − δH) − B. (19)
The dynamics of consumption is given by the Keynes-Ramsey Rule (10) where rt is
replaced by (15):
−θCt
Ct
= ρ − αYt
Kt
+ δK + ϕPt. (20)
Finally, from the firm’s f.o.c. (see the previous subsection), the amount of polluting
input is determined by:
Pt = βYt/τt. (21)
12
The equilibrium dynamics is determined by equations (17)-(21), exogenously given
time path of τt, initial levels of K0 and H0, and the TVCs (12) and (13).
Note that the TVCs can be simply stated using equilibrium conditions. From
(17) and (20), the growth rate of ktc−θt e−ρt is (1 − α)(Yt/Kt) − (Ct/Kt). Similarly,
from (10), (11) and (18), the growth rate of ht(wt/B)c−θt e−ρt is −But. Therefore, a
sufficient condition for the TVC is that those growth rates are negative in the long
run:
(12), (13) ⇐ limt→∞
(1 − α)(Yt/Kt) − (Ct/Kt) < 0, limt→∞
ut > 0. (22)
Condition (22) implies that the TVCs are satisfied when more than fraction 1 − α
of output is consumed and the fraction of time used for production is positive. For
later use, we also present necessary conditions for the TVC which are slightly weaker
than (22):
(12) ⇒ limt→∞
(1 − α)(Yt/Kt) − (Ct/Kt) ≤ 0, (23)
(13) ⇒ limt→∞
ut/ut ≥ 0. (24)
Condition (24) implies that, if the fraction of time used for production converges
toward zero, it must do so very slowly. That is, if limt→∞ ut/ut < 0, the attrition
rete of ht(wt/B)c−θt e−ρt, which is But, decreases toward zero very rapidly (i.e., ex-
ponentially). In that case ht(wt/B)c−θt e−ρt cannot reach zero in the long run, and
therefore the TVC (13) is violated.12
2.4 Sustainability under a constant tax rate
Observe, from (21), that pollution increases in proportion to output Yt if the gov-
ernment do not change the environmental tax rate. Since the increasing usage of
polluting inputs makes natural disasters more and more frequent, it seems that eco-
12Let V h(t) ≡ log(ht(wt/B)c−θt e−ρt). Then (13) is equivalent to limt→∞ V h(t) = −∞. Note
that, for arbitrary T > 0, limt→∞ V h(t) = V h(T )−B∫ ∞
Tu dt. The first term is finite. In addition,
when limt→∞ ut/ut < 0, the integral of the second term is also finite. Therefore the TVC is violated
if limt→∞ ut/ut < 0.
13
nomic growth is not sustainable under such a static environmental policy. This
subsection formally proves that this insight is correct.
Proposition 1 If the per-unit tax on polluting input is constant, then economic
growth is not sustainable in the sense that aggregate consumption cannot grow in the
long run.
Proof: The proof goes via reductio ad absurdum. When the government sets a con-
stant environmental tax rate (i.e., τt = τ0 for all t), the Keynes-Ramsey Rule (20)
can be rewritten, from (21), as:
−θCt
Ct
= ρ + δK −(
α − ϕβ
τ0
Kt
)Yt
Kt
.
This equation states that, if consumption grow in the long-run (i.e., Ct → ∞ as
t → ∞), the sign of the value in the parentheses must be positive. Hence, Kt
must be bounded above by a constant value τ0α/ϕβ (i.e., limt→∞ Kt < τ0α/ϕβ).
To interpret this result, observe that that, from (14) and (15), the rental price of
physical capital is rt = αYt/Kt and therefore the last term of represents marginal
rate of return of holding capital net of insurance cost ϕPt = ϕβYt/τ0. As physical
capital accumulates, the insurance cost increases in relative to interest rate due to
increased risk of natural disasters. Since this lowers the incentive to save, the stock of
physical capital should not become too large in order to maintain sustained growth.
This raises another question, however, of maintaining output growth under a lim-
ited size of physical capital. From (14), the positive growth rate of output requires
the positive growth rate of human capital stock devoted to production due to the
supremum of Kt. That is, limt→∞ Nt ≥ 0 must hold in order to support increas-
ing consumption. Under a constant environmental tax rate equation, (19) can be
rewritten as:
α(Kt
Kt
− Nt
Nt
)= α
Yt
Kt
+ (ψ − ϕ)Pt − (δK − δH) − B.
Consider the behavior of both hands of the above equation in the long-run. The left
hand side implies the growth rate of wage, which eventually becomes negative value,
14
−αNt/Nt. Conversely, the right hand is given by:(α − ϕβ
τ0
Kt
)Yt
Kt
− δK + ψPt − B + δH ,
which is the difference between the marginal rate of return on both types of capital
stock. From the condition for the positive growth rate of consumption derived above,
the sign of the value in the parentheses must be positive, and thus, the value of the
right hand side goes to infinity as Yt → ∞. These results imply that the equality in
(19) fails to hold, and therefore Ct and Yt cannot grow in the long run. ¥The proof of proposition clarifies that, under a constant environmental tax rate,
economic growth is not sustainable since agents lose the incentive to save when output
and the risk of natural disasters increases to a certain level. The risk of disasters
rises proportionally with output, and this follows from the fact that firms face a
constant tax rate on the polluting input Pt (see equation 21). This result provides
an anticipation that, in order to sustain economic growth, it might be necessary to
increase the rate of environmental tax increased over time so as to prevent the risk
of disasters to rise excessively when output grows. In the remaining of the paper, we
consider such a time-varying tax policy.
3 Sustainable Growth
This section examines the possibility of long-run growth under a time-varying tax
policy. In the literature of endogenous growth, long-term analysis is usually done by
focusing on balanced growth paths, which is sometimes called steady-state growth
paths, where the growth rates of all variables are constant. However, in the present
model, the economy do not typically have a steady-state growth path primarily
because the introduction of the endogenous risk of natural disasters (and therefore
the endogenous effective depreciation rate of capital) makes the structure of the
model intrinsically non-homothetic. Nonetheless, it does not rule out the possibility
that, under an appropriate tax policy, the economy converges, or asymptotes, to a
steady-state growth path as will be examined in this section.
15
Specifically, we seek to find a tax policy realizes an asymptotically steady-state
growth path, which is defined as follows.
Definition 1 An equilibrium path is said to be an asymptotic steady-state growth
path if the limiting values of the growth rates of output, inputs, and consumption all
exist and finite. That is, g∗ ≡ limt→∞ Yt/Yt, gK ≡ limt→∞ Kt/Kt, gH ≡ limt→∞ Ht/Ht,
gu ≡ limt→∞ ut/ut, gP ≡ limt→∞ Pt/Pt, and gC ≡ limt→∞ Ct/Ct are well-defined and
finite. Furthermore, an asymptotically steady-state growth path is said to be sustain-
able if gC is strictly positive.13
In the remaining of the paper, we focus on the sustainable, asymptotically steady-
state growth paths and refer them simply as steady-state growth paths unless it
causes any ambiguity. Note that the requirements for a sustainable, asymptotically
steady-state growth also restricts the asymptotic behavior of tax rate τt because
Pt = βYt/τt (equation 21) must be satisfied in the long run. In particular, for g∗ and
gP to be well-defined and finite, the asymptotic growth rate of tax rate
gτ ≡ limt→∞
τt/τt
must also be well-defined and finite. This means, in the long run, the per-unit tax
rate on the polluting input must be changed at a constant rate. The main task of
this section is to examine the dependence of long-term rate of economic growth g∗
on the (long-term) growth rate of environmental tax gτ . In the first subsection, we
13The notion of growth path presented here is essentially the same as the notion of a nonde-
generate, asymptotically balanced growth path proposed by Palivos et al. (1997, Definition 2). We
call it steady-state growth rather than balanced because an important property of the equilibrium
path which will be derived below is that it exhibits different growth rates among production inputs.
Correspondingly, the notion of sustainability in our definition is weaker than the notion of nonde-
generate growth by Palivos et al. (1997) in that we do not require every production input to grow
at positive rate. In fact, we show an important case of the analysis is that in which the growth rate
of one production input (namely, polluting input Pt) is negative and converges to zero. Even in
this case, the growth rates of output and consumption can be positive if the growth rates of other
inputs are positive and more than offset the declining use of a certain type of input.
16
present the conditions that must be satisfied on the steady-state growth path. In the
second and third subsections, we examine two different possibilities of steady-state
growth. The final subsection summarizes.
3.1 Conditions for Sustainable, Asymptotically Steady-State
Growth Path
We first show that, in the long run, the economy cannot grow faster than the growth
rate of environmental tax.
Lemma 1 On any asymptotically steady-state growth path, g∗ ≤ gτ .
Proof: in Appendix.
Intuitively, if production grows so fast that the usage of polluting input Pt = βYt/τt
become infinite in the long run, natural disasters occurs increasingly frequently. In
such a situation, however, both physical and human capital deteriorate at an accel-
erating rate, contradicting with the initial assumption that output can grow. One
implication of Lemma 1 is that sustainable steady-state paths (with g∗ > 0) are
obtained only when gτ > 0; i.e., only when the per-unit tax rate is increased expo-
nentially. This confirms the anticipation provided in the end of Section 2.4.
Another implication is that g∗ ≤ gτ leads to gP ≡ limt→∞ Pt/Pt ≤ 0 from (21).
Since the amount of polluting input Pt is nonnegative, this means that Pt converges to
a constant value in the long-run. We denote this asymptotic value by P ∗ ≡ limt→∞ Pt.
Note that P ∗ = 0 if Pt/Pt < 0. Even though we limit our attention to sustainable
growth paths, we should not rule out this possibility. It is true that output Yt is
zero if Pt = 0 given the Cobb-Douglas function (1) in which polluting inputs such
as fossil fuels are necessary; that is, a steady-state growth path in a conventional
sense with Pt = P ∗ = 0 is obviously inconsistent with sustainable growth. However,
we are considering asymptotically steady-state growth path in which Pt asymptotes
to P ∗, and therefore Pt does not necessarily coincide with P ∗ = 0 at any date.
Furthermore, limt→∞ Pt = 0 does not necessarily mean limt→∞ Yt = 0 since other
production factors in (1), namely Kt and Ht, can grow unboundedly.
17
We next show that these requirements and transversality conditions determine
the growth rates of ut, Kt, and Ct.
Lemma 2 On any sustainable, asymptotically steady-state growth path,14
(i) ut, zt ≡ Yt/Kt, and χt ≡ Ct/Kt are asymptotically constant.
(ii) gu = 0 and gK = gC = g∗.
Proof: in Appendix.
The result that physical capital and consumption grow in parallel with output
is a common property in models of endogenous growth (e.g., Lucas 1988, Palivos et
al. 1997). However, in our model, the growth rate of human capital is not the same
as output. Differentiating production function (14) logarithmically with respect to
time gives g∗ = − β1−β
gτ + αgK + (1 − α)(gu + gH), where we used Nt = utHt. This
equation implies that conditions for the steady-growth path (i.e, gK = g∗ and gu = 0)
are satisfied only when
gH = g∗ +β
1 − α − βgτ . (25)
Equation (25) says that, in a steady state growth path, human capital must accu-
mulate faster than physical capital and output, and the difference is larger when the
growth rate of environmental tax is higher.
To see why agents are willing to accumulate human capital faster in equilibrium,
observe that, as the tax rate on the polluting input is increased over time, the effective
productivity of private firms Aτ−β/(1−β) gradually falls (see equation 14). This means
that, if human capital is accumulated at the same speed as physical capital, output
can only grow slower than the speed of physical capital accumulation. As a result,
interest rate rt = αYt/Kt falls, which discourages agents from investing in physical
14Observe that property (ii) of Lemma 2 is a stronger statement than (i); i.e., property (i) holds
whenever gu ≤ 0, gK ≥ g∗, and gC ≤ gK . In the proof of the lemma on Appendix, we show that
all of gu ≤ 0, gK ≥ g∗, and gC ≤ gK must hold with equality since otherwise either transversality
conditions (their necessary condition are given by equations 23 and and 24) or sustainability (gC >
0) is eventually violated.
18
capital.15 This induces agents accumulate human capital relatively more rapidly than
physical capital. When human capital becomes increasingly abundant in relative to
physical capital, it can compensate the declining productivity so that the marginal
productivity of physical capital rt = αYt/Kt is kept constant (which is a necessary
condition for a steady-state growth).
Now we are ready to summarize the conditions that must be satisfied by any
sustainable, asymptotically steady-state growth path. For convenience, let us denote
asymptotic values by u∗ ≡ limt→∞ ut ∈ [0, 1], z∗ ≡ limt→∞ Yt/Kt ≥ 0, and χ∗ ≡
limt→∞ Ct/Kt ≥ 0. Substituting gu = 0, gK = gC = g∗ and (25) for (17)-(21), the
equilibrium conditions that must hold in the long run can be represented as follows.16
Evolution of Kt: g∗ = z∗ − χ∗ − (δK + ϕP ∗), (26)
Evolution of Ht: g∗ +β
1 − α − βgτ = B(1 − u∗) − (δH + ψP ∗), (27)
Arbitrage condition: − β
1 − α − βgτ = αz∗ − B + (ψ − ϕ)P ∗ − (δK − δH), (28)
Keynes-Ramsey rule: − θg∗ = ρ − αz∗ + (δK + ϕP ∗), (29)
Asymptotic pollution: P ∗
≥ 0 if g∗ = gτ (Case 1),
= 0 if g∗ < gτ (Case 2).
(30)
Given gτ > 0 set by the government, the five conditions, (26)-(30), determine five
unknowns, g∗, z∗, χ∗, u∗, and P ∗, in the asymptotically steady-state growth path.
In the following, we explicitly calculate the values for unknowns as a function of
gτ . Note that, however, there is a complementary slackness condition (30), and we
cannot know whether P ∗ = 0 or g∗ = gτ holds in advance. Thus we need to examine
two possible cases in turn, and then to determine which case actually occurs in
equilibrium under a particular tax policy.
15Recall that the rate of return from physical capital investment is rt − δK − ϕPt, whereas that
from human capital investment is B − δH − ψPt.
16Since ut = u∗ is constant, we use Nt/Nt = ut/ut + Ht/Ht = Ht/Ht in deriving the LHS of
(28).
19
3.2 Case 1: P ∗ ≥ 0 and g∗ = gτ
Let us first examine the possibility that the equilibrium long term rate of growth
coincides with the growth rate of environmental tax on the steady-state equilibrium
path. Substituting g∗ = gτ into (28) and (29), we obtain the asymptotic value of
polluting input:
P ∗ =1
ψ
[B − δH − ρ −
(θ +
β
1 − α − β
)gτ
], (31)
which is decreasing in gτ . Recall that, as shown by (30), the asymptotic value must
be nonnegative: P ∗ ≥ 0. From (31), we can see that the condition is satisfied if gτ
is within the following region:
gτ ≤ B − δH − ρ
θ + β1−α−β
≡ gmax, (32)
where gmax is positive from the assumption that B − δH > ρ. Hence, Case 1 (i.e.,
P ∗ ≥ 0 and g∗ = gτ ) is possible only if gτ ∈ (0, gmax].
Using g∗ = gτ , we obtain the asymptotic values of other variable from (26)-(29):
z∗ =1
α(θgτ + δK + ϕP ∗ + ρ) , (33)
χ∗ =1
α
((θ − α)gτ + (1 − α)(δK + ϕP ∗) + ρ
), (34)
u∗ =1
B
(B − (δH + ψP ∗) − 1 − α
1 − α − βgτ
). (35)
Substituting (31) into (33)-(35) and using gτ ∈ (0, gmax], it can be confirmed that
z∗ > 0, χ∗ > 0, u∗ ∈ (0, 1), and (1 − α)z∗ − χ∗ < 0. The last two inequalities
imply that the sufficient condition for the transversality condition, given by (22), is
satisfied. In addition, we show in Appendix that the steady-state growth path is
saddle stable given that the impacts of disasters on human capital (ψ) is relatively
small compared to that on physical capital (ϕ). The following lemma states the
obtained results.
Lemma 3 A sustainable, asymptotically steady-state growth path with P ∗ ≥ 0 and
g∗ = gτ exists if and only if gτ ∈ (0, gmax]. It is characterized by g∗ = gτ and (31)-
(35), and satisfy equilibrium conditions (26)-(30) and the transversality conditions.
20
In addition, if ψ/ϕ < (1 − 2α)/(1 − α − β), this equilibrium path is locally saddle
stable.
Proof: Stability is examined in Appendix.
We assume parameters satisfy ψ/ϕ < (1 − 2α)/(1 − α − β) in the remaining of the
paper.
3.3 Case 2: P ∗ = 0 and g∗ < gτ
Next, we examine the possibility that the amount of polluting input asymptotically
converges toward zero on the steady-state growth path. Substituting P ∗ = 0 for (28)
and (29) yields the steady-state growth rate:
g∗ =1
θ
(B − δH − ρ − β
1 − α − βgτ
). (36)
Contrary to Case 1 in which g∗ = gτ , equation (36) shows that the long-term rate of
growth is decreasing in gτ . Recall that, for the amount of polluting input Pt = βYt/τt
to converge toward zero as assumed, growth rate g∗ must be lower than gτ . Equation
(36) shows that, for condition g∗ < gτ to be satisfied, the rate of environmental tax
must be raised faster than gmax, where gmax is defined in (32). However, equation (36)
also implies that economic growth cannot be sustained when gτ is too high: g∗ ≤ 0
if gτ ≥ glim ≡ (1 − α − β)β−1(B − δH − ρ). Therefore, a sustainable, asymptotically
steady-state with P ∗ = 0 obtains only if gτ ∈ (gmax, glim), for which range of policies
g∗ ∈ (0, gτ ) holds.
Substituting P ∗ = 0 and (36) into (26)-(29), we obtain the asymptotically steady-
state values for other variables:
z∗ =1
α
(B + δK − δH − β
1 − α − βgτ
), (37)
χ∗ =( 1
α− 1
θ
)(B − δH − β
1 − α − βgτ
)+
1 − α
αδK +
ρ
θ, (38)
u∗ =1
Bθ
[(θ − 1)
(B − δH − β
1 − α − βgτ
)+ ρ
]. (39)
From gτ ∈ (gmax, glim), it can be confirmed that z∗ > 0, χ∗ > 0, (1−α)z∗−χ∗ < 0, and
u∗ ∈ (0, 1), implying that transversality condition (22) is satisfied. In addition, we
21
show in Appendix that this steady-state growth path is saddle stable. The following
lemma states the results.
Lemma 4 A sustainable, asymptotically steady-state growth path with P ∗ = 0 and
g∗ < gτ exist if and only if gτ ∈ (gmax, glim). It is characterized by P ∗ = 0 and (36)-
(39), and satisfy equilibrium conditions (26)-(30) and the transversality conditions.
In addition, this equilibrium path is locally saddle stable.
Proof: Stability is examined in Appendix.
3.4 Summary
Lemma 3 and Lemma 4 shows that there are two possible patterns of sustained
growth. Observe that those two possibilities are mutually exclusive—that is, under
a given time path of environmental tax at most one lemma is applicable. Therefore,
the sustainable, asymptotically steady-state growth path is always unique; it is char-
acterized by Lemma 3 if gτ ∈ (0, gmax], and by Lemma 4 if gτ ∈ (gmax, glim). The
following proposition states the main result.
Proposition 2 A sustainable, asymptotically steady-state growth path exists if and
only if the per-unit tax on polluting input is raised over time so that its asymptotic
growth rate, gτ , is strictly positive and less than glim ≡ (1− α− β)β−1(B − δH − ρ).
When it exists, it is unique and saddle stable. The long-term rate of economic growth
follows an inverted-V shape against gτ ∈ (0, glim), and it is maximized at gτ = gmax ≡
(B − δH − ρ)(θ + β1−α−β
)−1.
Figure 1 illustrates how asymptotic growth rate of environmental tax, which is
an exogenous policy variable, affects growth rates of other variables as well as the
level of pollution in the long run.17 This result can be interpreted as follows.
17Figure 1 is derived as follows. gC and gK are the same as g∗, which is given by g∗ = gτ for
gτ ∈ (0, gmax) and (36) for gτ ∈ (gmax, glim). Given g∗, gH and gP are determined by (25) and
gP ≡ Pt/Pt = g∗−gτ (recall equation 21). Finally P ∗ is given by (31) for gτ ∈ (0, gmax) and P ∗ = 0
for gτ ∈ (gmax, glim).
22
g¿
gmax
glim
gH
gK = gC = g¤
gP
P ¤P
gmax
g¿
0
Figure 1: Growth rate of environmental tax and the sustainable, asymptotically
steady-state growth. The upper panel shows the relationship between the growth rate of envi-
ronmental tax (gτ ) and that of human capita (gH), physical capital (gK), output (g∗), and pollution
(gP ). The lower panel shows the level to which the level of pollution converges to in the long run
(Pt → P ∗). Parameters: α = .3, β = .2, θ = 2, ρ = .05 B = 1, ψ = .005, ϕ = .01, δH = .09, and
δK = .1.
When the environmental tax rate is asymptotically constant (i.e., gτ = 0), the
asymptotic growth rates of all endogenous variables are zero, which means that the
economy settles to a no-growth steady state. In this steady state, the amount of
pollution converges to P ∗ = (B − δH − ρ)/ψ ≡ P , which causes the probability of
natural disasters (i.e., the risk of losing their physical and human capital) so high
that agents loses incentive to save. Interestingly, observe that the asymptotic level
of Pt does not depend on the level of the environmental tax rate, τt, as long as it
is constant. Nonetheless, since Yt = τtPt/β from (21), a higher tax rate induces the
economy to converges to a higher output level by influencing the equilibrium path
in transition.
When the government raises the per-unit tax rate on polluting inputs at an
exponential rate (gτ > 0), the asymptotic level of Pt can be kept below P so that
economic growth may be maintained without bound. When gτ in increased within
23
the range of (0, gmax], the long-run amount of pollution P ∗ decreases, and so does the
risk of natural disasters. The reduced risk of natural disasters encourages agents to
accumulate capital at a faster speed. As a result, the growth rate of physical capital
gK increase in parallel with gτ (i.e., gK = gτ ). The growth rate of human capital, gH ,
also increases with gτ , and more rapidly than physical capital. (Recall the discussion
in Section 3.1 for why agents are willing to do so.) This makes possible sustained
growth under asymptotically constant use of polluting input.
However, accelerating the rise of tax rate does not necessarily enhance economic
growth because it also has a negative effect on growth through limiting the use of
Pt, which is a necessary input for the Cobb-Douglas production process. Specifically,
when the environmental tax rate is raised so rapidly that gτ exceeds gmax, the use
of polluting inputs Pt is continually reduced, converging asymptotically to zero level
(Pt → P ∗ = 0). In this case, a further acceleration of gτ no longer can reduce the
asymptotic probability of natural disasters (because is is already at the lowest level),
but only accelerates the fall of the effective productivity of private firms Aτ−β/(1−β).
As a result, g∗ is no longer increasing in parallel with gτ , but decreasing in gτ . An
extreme case is that of gτ ≥ glim, for which case, even though the risk of natural
disasters is a the lowest level, the fall of the effective productivity Aτ−β/(1−β) is
so fast that it cannot be compensated by a faster accumulation of human capital,
resulting in zero or negative growth.
4 Welfare
The previous section established that sustained growth is feasible by raising the
rate of environmental tax rate over time. It is yet to shown, however, such an
environmental policy is desirable in terms of welfare. This section investigates the
social planner’s problem so as to derive the welfare-maximizing environmental policy,
and compares it with the growth maximizing policy derived in the previous section.
24
The social planner maximizes (9) subject to the following constraints:
Kt = F (Kt, utHt, Pt) − Ct − (δK + ϕPtKt), (40)
Ht = B(1 − ut)Ht − (δH + ψPtHt). (41)
From the first-order conditions for optimality, it can be confirmed that the dynamics
of Kt, Ht, and Ct, and the arbitrage condition between two types of capital stocks are
exactly the same as (17)-(20), which are parts of market equilibrium conditions. Since
the social planner takes into consideration the externality of polluting emissions, that
is, the risk of natural disasters, it chooses the amount of polluting input according
to the following rule:
Pt = βYt
(ϕKt + ψHt ·
(1 − α − β)Yt
ButHt
)−1
. (42)
Equation (42) says that the use of polluting input should be determined so that its
marginal product (βYt/Pt) is equal to the marginal disadvantage of using polluting
inputs. The marginal disadvantage is given by the expression in the large parenthesis,
which is the sum of increases in damage to physical capital stock and in damage to
human capital stock, both measured in terms of final goods (in particular, (1 − α −
β)Yt/(ButHt) is the shadow price of human capital in terms of final goods).
Recall that, in the market economy, firms choose Pt according to (21); i.e., Pt =
βYt/τt. This means that if the tax rate at each date is determined by
τ optt = ϕKt + ψ
(1 − α − β)Yt
But
, (43)
then the optimality condition for Pt, equation (42), is satisfied. Recall again that
other conditions for social optimum is the same as that for the market equilibrium.
Therefore, if the government set the environmental tax rate by following rule (43),
then the market equilibrium path completely traces the social optimal path.18
18Generally speaking, even when they look similar, a time-varying policy (a function only of time
such as considered in the previous section) and a policy rule (a function of state variable such
as equation equation 43) may result in different outcomes if agents behaves strategically. Such a
difference is studied in the literature of differential games: the difference between the open-loop
equilibrium and the Markovian equilibrium. However, in the present model, all agents are price
takers and therefore both policies results in the same outcome.
25
Although it may be convenient in practice to follow a policy rule than calculating
the time path of tax rate at the outset, equation (43) is not so informative about
the equilibrium path that result from such a policy. Equation (43) implies that τt
is function of Kt, Yt, and ut. However, Kt, Yt, and ut are endogenously determined
depending the future path of τt that is expected by consumers. That is, the welfare
maximizing path is given as a solution to a dynamic fixed point problem.
Since it is excessively difficult to solve this problem directly, we limit our attention
to sustainable, asymptotically steady-state growth paths and examine whether there
is a path that satisfies optimality condition (42) within this family of paths. Note
that on any sustainable, asymptotically steady-state growth path, both the LHS and
the RHS of condition (42) asymptotes to constants. Specifically,
P ∗ = β
(ϕ
1
z∗ +ψ(1 − α − β)
Bu∗
)−1
(44)
must hold in the long run, where P ∗, z∗ and u∗ represents the asymptotic values of Pt,
zt ≡ Yt/Kt and ut. We know from Proposition 2 that, whenever the equilibrium path
asymptotes to a sustainable steady-state growth path, the asymptotic growth rate of
the optimal tax rate, goptτ ≡ limt→∞ τt/τt, is well defined and that gτ ∈ (0, glim). In
addition, P ∗, z∗ and u∗ are determined as a function of gτ . Therefore, we examine
whether there exist a value of gτ within (0, glim) such that (44) holds.
Figure 3 plots the RHS and the LHS of equation (44) against gτ . The actual
amount of asymptotic pollution (the LHS) is positive but decreasing in gτ for gτ ∈
(0, gmax), and it is zero for gτ ≥ gmax. On the other hand, the optimal amount of
asymptotic pollution (the RHS) is positive for all gτ > 0. The both curves are are
continuous in gτ . In addition, it can be shown that the intercept of the RHS is lower
than that of the LHS whenever ρ or β is sufficiently small.19 Therefore, unless both β
and ρ are large, the two curves have an intersecting point within gτ ∈ (0, gmax), which
19Observe from (31), (33) and (35) that P ∗ = (B − δH − ρ)/ψ ≡ P , z∗ = (δK + ϕP + ρ)/α and
u∗ = ρ/B when gτ = 0. Substituting these into both sides of (44) and rearranging terms shows
that the prerequisite is satisfied if and only if β < (αϕ/(δK + ϕP + ρ) + ψ(1 − α − β)/ρ)P , which
is satisfied whenever ρ or β is sufficiently small.
26
g¿glim
P ¤
P
gmax0
gopt¿
Actual (LHS)P ¤
Optimal (RHS)P ¤
Figure 2: Determination of the optimal growth rate of environmental tax. The figure
plots the RHS and the LHS of condition (44) against gτ . The asymptotic growth rate of optimal
environmental tax is goptτ , is given by the intersection, which is lower than the growth maximizing
rate, gmax. The parameters are the same as in Figure 1.
represents the growth rate of optimal tax rate, goptτ . Note that, since gopt
τ ∈ (0, gmax),
the long-term rate of economic growth g∗ coincides with goptτ , and therefore it is
positive. The following proposition states the obtained result.
Proposition 3 Suppose that the welfare of consumers is given by (9) and that either
discount rate or or the share of polluting input is sufficiently low. Then among
sustainable, asymptotically steady-state growth paths, there exists a unique optimal
growth path with strictly positive rate of economic growth. The asymptotic growth
rate of optimal per-unit tax, goptτ , is positive but slower than the growth maximizing
rate, gmax.
Thus the above analysis shows that the sustained growth implemented by raising
the environmental tax rate is not only feasible but also desirable. It is also notable,
however, that the optimal policy does not coincide with the growth maximizing policy
since goptτ < gmax. That is, if the government care about welfare it should employ
milder policy for protecting environment than when growth is their only concern.
This result may seem at odds with the usual growth vs. environment arguments, but
its reasoning is similar to the modified golden rule argument familiar to economists.
27
Although an aggressive environmental policy that aims to eliminate the emission of
pollutants in the long run (i.e., P ∗ = 0) may maximize the economic growth rate in
the very long run, the cost in the form of reduced effective productivity that must
be incurred in the transition can overwhelm the benefit that cannot be reaped until
far future.
5 Disutility of Pollution
Our result that the sustained growth is both feasible and desirable is in contrast to
the previous literature. Notably, Stokey (1998) has shown that even when sustained
growth is feasible, it is not desirable when production of goods emits pollutants that
harm the utility of consumers. The difference of the results of course comes from
the setting of models. More specifically, our model significantly differs from Stokey
(1998) in two aspects; (i) we are considering human capital accumulation differently
from Stokey’s AK model; (ii) so far pollutants are assumed to cause disasters but do
not directly give disutility.
In this subsection, we clarify that the critical reason behind the difference in the
results is not (ii) but (i). To this end, we present an extended model in which agents
suffer from not only damages to capital stocks caused by natural disasters but also
disutility of pollution. Suppose that consumers has an utility function of∫ ∞
0
(c1−θt − 1
1 − θ− P 1+γ
t
1 + γ
)e−ρtdt, γ > 0. (45)
The only difference of (45) from (9) is that it includes a disutility term P 1+γt /(1+γ),
which is isoelastic and convex with respect to the use of polluting input Pt. Since
function (45) is separable with respect to ct and Pt, behavior of all agents, who takes
Pt as given, does not change. That is, the equilibrium outcome is exactly the same
as in analyzed in sections 2 and 3.
Let us examine how the planner’s problem is affected. Under resource constraints
(40) and (41), the planner maximizes function (45). Then, the optimality condition
28
with respect to polluting input becomes
Pt = βYt
(ϕKt + ψHt ·
(1 − α − β)Yt
ButHt
+P γ
t
C−θt
)−1
. (46)
When compared to (42), there is an additional marginal cost of polluting input that
comes from disutility.
Similarly to the previous section, we again limit our attention to sustainable,
asymptotically steady-state growth paths. In the long run, condition (46) implies
that
P ∗ = β
(ϕ
z∗+
ψ(1 − α − β)
Bu∗ +(χ∗)θ
(z∗)θlimt→∞
P γt Y θ−1
t
)−1
. (47)
The value of the RHS critically depends on the behavior of P γt Y θ−1
t . Note that, from
Pt = βYt/τt, its asymptotic growth rate is γgP +(θ−1)g∗ = (γ+θ−1)g∗−γgτ . Using
the fact that g∗ is a function of gτ , we can show that there is gτ ∈ (gmax, glim) such
that the asymptotic growth rate of P γt Y θ−1
t is strictly positive when gτ ∈ (0, gτ ) and
it is strictly negative when gτ ∈ (gτ , glim). This means that the RHS of (47) is zero
for gτ ∈ (0, gτ ) and strictly positive when gτ ∈ (gτ , glim). Therefore, as illustrated
in Figure , condition (47) holds for all gτ ∈ (gmax, gτ ). Whenever the asymptotic
growth rate of per-unit tax is between gmax and gτ , it is optimal to reduce the use of
polluting input toward zero in the long run, and Pt actually converges toward zero.
However, this does not necessarily mean that every tax policy with gτ ∈ (gmax, gτ )
is optimal, because (47) is merely a necessarily condition for (46). A more strong
condition is that both the LHS and the RHS of (46) falls toward zero at the same
speed. When gτ ∈ (gmax, gτ ), the asymptotic growth rate of the RHS of (46) is
−(γ + θ − 1)g∗ + γgτ (the negative of the growth rate of P γt Y θ−1
t ). As shown in
Figure , it coincides with the growth rate of Pt, which is g∗ − gτ , if and only if
gτ = goptτ ≡ (γ + θ)(1 − α − β)(B − δH − ρ)
θ(1 + γ)(1 − α − β) + β(γ + θ)∈ (gmax, gτ ).
Thus the following proposition obtains.
Proposition 4 Suppose that the welfare of consumers is given by (45). Then among
sustainable, asymptotically steady-state growth paths, there exists a unique optimal
29
g¿glimgmax0
¡gP
eg¿
bgopt¿
g¿
P ¤
Actual
Optimal
Actual
Optimal
Figure 3: Optimal growth rate of environmental tax when disutility of pollution is
accounted for. The upper panel plots the asymptotic levels of RHS and the LHS of condition
(46), whereas the lower panel depicts thier asymptotic growth rates. The asymptotic growth rate of
optimal environmental tax is goptτ , is higher than the growth maximizing rate, gmax. The parameters
are the same as in Figure 1 and γ = .2.
growth path with strictly positive rate of economic growth. The asymptotic growth
rate of optimal per-unit tax, goptτ , is higher than the growth maximizing rate, gmax.
The main result is that the desirability of sustained growth does not change even
when disutility of pollution is introduced into the model. However, the desirable
speed at which the environmental tax is increased is now higher than the growth
maximizing speed, gmax. This implies that, if pollution affects the utility of agents
directly, the emission of pollutant should be eliminated in the long run even at the
cost of accepting a slower (although positive) rate of economic growth.
6 Concluding Remarks
The sustainability of economic growth has been analyzed in a two-sector model of
endogenous growth, taking into account the risk of natural disasters. Polluting inputs
30
are necessary for production, but they intensify the risk of natural disasters. In this
setting, we obtained following results.
First, the long-run economic growth can not be sustained if the private cost of
using the polluting input is kept constant.20 Since, for simplicity, we do not con-
sider the cost associated with extracting resources or the finiteness of those inputs,
this result implies that the environmental tax rate should be increased over time.
However, it should be noted that if the private cost changes for some ignored rea-
sons, the environmental tax rate must be adjusted to absorb those changes. More
substantially, a next step in the research agenda would be to integrate the analysis
of natural disasters with the studies of finiteness of natural resources, although it is
beyond the scope of this first endeavor.
Second, the rate of the economic growth rate follows the inverted V-shaped curve
relative to the growth rate of the environmental tax. When the rate of environ-
mental tax is initially slow growing, its acceleration will reduce the long-run level of
emission and the risk of natural disasters, which enhances the incentive to save and
hence promotes economic growth. When the rate of environmental tax is already
fast growing, the amount of polluting input at the steady state is fairly small so that
further acceleration of environmental tax excessively impair the productivity of pri-
vate firms, which works against economic growth. Therefore, the economic growth
can be maximized with choice of the most gradual increase in environmental tax rate
that minimizes the amount of pollution in the long-run.
Third, the sustained growth, realized by ever increasing tax rate on polluting
inputs, is not only feasible but also desirable. Although economic growth ceteris
paribus induces private firms to use more polluting input, an appropriate environ-
mental policy can lead firms to use more of human capital (e.g., by investing in
alternative technology), which decreases their reliance on polluting inputs. The op-
20When the damage to physical capital stock is much larger than that to human capital stock
(i.e., ϕ >> ψ), the steady-state value of P ∗ is rather large (see Equation 31). In this case, the speed
of convergence to the steady state is slow, the economic growth declines gradually, and amount of
pollution increases during the transition to the steady state.
31
timal speed at which the environmental tax rate is increased is lower than the growth
maximizing speed if pollution only causes disasters, while it is higher when direct
disutility from pollution is accounted for.
Appendix
A Proof of Lemma 1
Suppose that g∗ > gτ (i.e., limt→∞ Yt/Yt > limt→∞ τt/τt). Then, Pt = βYt/τt → ∞.
From (18) and ut ≤ 1, this means Ht/Ht ≤ B − δH − ψPt → −∞. It con-
tradicts with the definition of the asymptotically steady-state growth, in which
gH ≡ limt→∞ Ht/Ht is finite.
B Proof of Lemma 2
On the asymptotically steady-state growth path, in which Ht/Ht and Pt are asymp-
totically constant, equation (18) implies that ut must also be asymptotically constant.
It means that the growth rate of ut is zero or negative (i.e., in the case of ut → 0),
but from (24) we know that the TVC for human capital accumulation is satisfied
only when the growth rate of ut is nonnegative. Therefore gu = 0. Next, since Ct/Ct
and Pt are asymptotically constant, equation (20) implies that the value of Yt/Kt
must also be constant in the long run. This means that the growth rate of Yt/Kt
is zero or negative. However, if Yt/Kt → 0, equation (17) says Kt/Kt < 0, which
means that Yt = (Yt/Kt) · Kt → 0. With output converging to zero, the sustainable
steady-state growth path, where gC ≥ 0, is clearly in contradiction. Therefore, the
growth rate of Yt/Kt must be zero; i.e., gK = g∗. Finally, given that Kt/Kt and
Yt/Kt are asymptotically constant, equation (17) in turn imply Ct/Kt must also be
asymptotically constant. Recall that that the TVC for physical capital (23) requires
Ct/Kt must not be smaller than (1−α)(Yt/Kt), which converges to a strictly positive
constant as shown above. Therefore, the growth rate of Ct/Kt must not be negative
but zero; i.e., gC = gK = g∗.
32
C Stability of the equilibrium path around the steady-state
growth path
In this subsection, we establish the stability of the equilibrium path stated in Lemma
3 and Lemma 4. The equilibrium path is characterized by a four-dimensional dy-
namics system of Kt, Ht, ut, Ct, where the laws of motion for those variables are
given by (17)-(20).21 In this dynamic system, Kt and Ht are predetermined state
variables, whereas ut and Ct are jumpable. Therefore, the system is both stable and
determinate when it has a stable manifold of dimension two.
For convenience we transform this system into another four-dimensional system
of ut, χt, zt, Pt. where χt ≡ Ct/Kt, z ≡ Yt/Kt and Pt ≡ βYt/τt. This transformed
system is equivalent to the original system since Kt, Ht, ut, Ct can be represented
in terms of ut, χt, zt, Pt.22 Therefore, saddle-stability (and determinacy) can be
established by confirming that this transformed system has a two dimensional stable
manifold. Using (14) and (17)-(21), we can write the dynamics of the system as
ut = ut
(But − χt + βzt + ΛPt +
1 − α − β
α(B + δK − δH) − β
αgτ
), (48)
χt = χt
(χt −
θ − α
θzt +
θ − 1
θϕPt −
ρ
θ+
θ − 1
θδK
), (49)
zt = zt
(−(1 − α − β)zt + ΛPt +
1 − α − β
α(B + δK − δH) − β
αgτ
), (50)
Pt = Pt
(−χt +
α + (1 − α − β)β
1 − βzt + ΩPt +
1 − α − β
αB − α + β
αgτ
+(1 − 2α − β)δK − (1 − α − β)δH
α
)(51)
where Λ and Ω are constants defined by Λ ≡ (1 − α − β)(ϕ − ψ)/α and Ω ≡
((1 − 2α − β)ϕ − (1 − α − β)ψ) /α.
21Note that, by making use of (14), (21) and Nt = utHt, Yt and Pt appearing in the LHS of (17)-
(20) can be expressed in terms of Kt,Ht, ut and τt, where the motion of τt is given exogenously by
the government.
22Equivalence is confirmed by that the inverse transformation is well defined. Specifically, Kt =
τtPt/(βzt), Ct = τtPtχt/(βzt), and Ht =(τ1/(1−β)−bα/A
)1/(1−bα)
zbα/(1−bα)t Pt/(βut).
33
We first examine the stability of the steady state equilibrium path for the case
of gτ ∈ (0, gmax]. As examined in Section 3.2, the (asymptotic) steady state of
the transformed system, denoted by u∗, χ∗, z∗, P ∗, is given by (31) and (35)–(33).
Applying a first-order Taylor expansion of equations (48)–(51) around this steady-
state yields ut
χt
zt
Pt
≃
u∗B −u∗ βu∗ Λu∗
0
0 J1
0
ut − u∗
χt − χ∗
zt − z∗
Pt − P ∗
, (52)
where
J1 ≡
χ∗ − θ−α
θχ∗ (θ−1)ϕ
θχ∗
0 −(1 − α − β)z∗ Λz∗
−P ∗ α+β(1−α−β)1−β
P ∗ ΩP ∗
.
We want to show that the Jacobian matrix of (52) has two positive and two negative
eigenvalues. From the block-triangular structure of the matrix, one eigenvalue is
u∗B > 0, and the other three are given by the eigenvalues of the submatrix J1. The
characteristic equation for J1 is
−λ3 + TrJ1λ2 − BJ1λ + DetJ1 = 0, (53)
where
TrJ1 =
θ + β − α
α− (1 − 2α)ϕ
αψ
(θ +
β
1 − α − β
)gτ +
β
αδK +
α + β
αρ
+
(1 − 2α)ϕ
αψ− 1 − α − β
α
(B − ρ − δH),
MJ1 =χ∗ − θ−α
θχ∗
0 −(1 − α − β)z∗+
−(1 − α − β)z∗ Λz∗
α+β(1−α−β)1−β
P ∗ ΩP ∗+
χ∗ (θ−1)ϕθ
χ∗
−P ∗ ΩP ∗,
= −1 − α − β
α
(ϕ − ψ)(z∗ − χ∗) +
αϕχ∗
θ(1 − α − β)
−1 − α − β
α· z∗
P ∗
(θ − α)gτ + (1 − α)δK + (1 − 2α)ϕP ∗ + ρ
,
DetJ1 =ψ(1 − α − β)
θz∗χ∗P ∗,
34
where MJ1 denotes the sum of the principal minors of J1. We determine the sign of
the real parts of the roots of (53) based on Theorem 1 of Benhabib and Perli (1994).
Theorem 1 (Benhabib-Perli) The number of roots of the polynomial in (53) with
positive real parts is equal to the number of variations of sign in the scheme
−1 TrJ1 − MJ1 +DetJ1
TrJ1
DetJ1.
Under the assumption that ψ/ϕ < (1−2α)/(1−α−β), we have TrJ1 > 0,23 MJ1 < 0,
and DetJ1 > 0. Thus the above theorem implies that there is only one eigenvalue
with positive real parts in the matrix J1. Combined with Bu∗ > 0 obtained before,
we have two positive eigenvalues in total. This completes the stability analysis for
that case of gτ ∈ (0, gmax] (and therefore the proof of Lemma 3).
Turning to the case of gτ ∈ (gmax, glim], the (asymptotic) steady state of the
transformed system for this case is given by P ∗ = 0 and (37)–(39) in Section 3.3. The
Taylor expansion of equations (48)–(51) around this steady-state yields essentially
the same expression as (52), with the only difference that submatrix J1 is replaced
by
J2 =
χ∗ · · · · · ·
0 −z∗(1 − α − β) · · ·
0 0 g∗ − gτ
,
where g∗ is the asymptotic growth rate of output, which is defined by (36). Since
J2 is a triangular matrix, its eigenvalues are simply given by its diagonal elements.
Observe that g∗ − gτ represent the asymptotic growth rate of Pt = βYt/τt. As
discussed in Section 3.3, it is negative in this case (i.e., when gτ ∈ (gmax, glim)).
Therefore, J2 has one positive eigenvalue (χ∗) and two negative ones (−z∗(1−α−β)
and g∗ − gτ ). This completes the stability analysis for that case of gτ ∈ (gmax, glim)
and the proof of Lemma 4.
¥
23This can be confirmed by noting that TrJ1 is linear in gτ and confirming TrJ1 > 0 at both
gτ = gmax and gτ = 0.
35
Reference
Aghion, P., Howitt, P.: Endogenous Growth Theory, MIT Press, Cambridge, MA.
(1998)
Agnani, B., Gutierrez, M.-J., Iza, A.: Growth in overlapping generation economies
with non-renewable resources, 50, 387-407 (2005)
Benhabib, J., Perli, R.: Uniqueness and indeterminacy: on the dynamics of endoge-
nous growth. Journal of Economic Theory, 63, 113-142 (1994)
Bovenberg, A.L., Sumlders, S.: Environmental quality and pollution-augmenting
technological change in a two-sector endogenous growth model. Journal of Public
Economics, 57, 369-391 (1995)
Copeland, B. R., Taylor, M. S.: North-South trade and the environment. Quarterly
Journal of Economics 109, 755-787 (1994)
Emanuel, K.: Increasing destructiveness of tropical cyclones over the past 30 years.
Nature 436, 686-688 (2005)
Grimaud, A., Rouge, L.: Non-renewable resources and growth with vertical inno-
vations: optimum, equilibrium and economic policies. Journal of Environmental
Economics and Management 45, 433-453 (2003)
Hoyois, P., Below, R., Guha-Sapir G.: World Disasters Report 2005. Annex 1,
Disaster Data. Geneva. International Federation of Red Cross and Red Crescent
Societies.
IPCC (2001): Summary for Policymakers, A Report of Working Group I of the
Intergovernmental Panel on Climate Change.
Lucas Jr., R. E.: On the mechanics of economic development. Journal of Monetary
Economics 22, 3-42 (1988)
36
National Climatic Data Center: Technical Report 2005-01. U.S. Department of
commerce, National Oceanic and Atomospheric Administration
Parivos, T., Wang, P., Zhang, J.: On the existence of balanced growth equilibrium.
International Economic Review, 38, 205-224 (1997)
Scholz, C. M., Ziemes, G.: Exhaustible resources, monopolistic competition, and
endogenous growth. Environmental and Resource Economics, 13, 169-185 (1999)
Schou, P.: Polluting Non-renewable resources and growth. Environmental and Re-
source Economics, 16 211-227 (2000)
Skidmore, M., Toya, H.: Do natural disasters promote long-run growth? Economic
Inquiry 40, 664-687 (2002)
Stokey, N. L.: Are there limits to growth? International Economic Review 39, 1-31
(1998)
Uzawa, H.: Optimal technical change in an aggregative model of economic growth.
International Economic Review 6, 18-31 (1965)
Uzawa, H.: Economic theory and global warming, Cambridge, UK: Cambridge
University Press (2003)
Webster, P. J., Holland, G. J., Curry, J. A., Chang, H.-R.: Changes in Tropical
Cyclone Number, Duration, and Intensity in a Warming Environment. Science 309,
1844-1846 (2005)
37
7 Referee Appendix
Optimization of the Household (Section 2.2)
The current value Hamiltonian for the maximization problem is:
H =c1−θt − 1
1 − θ+νt (rtkt − (δK + ϕPt)kt + wtutht − ct + Tt)+µt (B(1 − ut)ht − (δH + ψPt)ht) ,
where νt and µt are the shadow prices associated with the accumulation of physical
capital and human capital, respectively. The optimality conditions are
νt = c−θt , (54)
µt =wt
Bνt, (55)
νt
νt
= ρ + ϕPt + δK − rt, (56)
µt
µt
= ρ − νt
µt
wtut − B(1 − ut) + δH + ψPt. (57)
The transversality conditions for physical capital stock and human capital stock,
respectively, are
limt→∞
ktνte−ρt = 0, and
limt→∞
htµte−ρt = 0,
Substituting (55) into (57) yields
µ
µ= ρ − B + δH + ϕPt. (58)
From (54) and (56), the the Keynes-Ramsey Rule is
−θct
ct
= ρ + ϕPt + δK − rt.
Differentiating logarithmically with respect to time in (55) and using (56) and (58),
we obtain the arbitrage condition between human capital investment and physical
capital investment:
wt
wt
= rt − (ϕ − ψ)Pt − (δK − δH) − B.
38
Optimization of the Social Planner (Section 4)
The current value Hamiltonian is
H =C1−θ
t − 1
1 − θ+νo
t [AKαt (utHt)
1−α−βP βt −Ct−(δK+ϕPt)Kt]+µo
t [B(1−ut)Ht−(δH+ψPt)Ht],
where νot and µo
t are the planner’s shadow prices associated with the accumulation
of physical capital and human capital, respectively. The necessary conditions for
optimality
Hc = c−θt − νo = 0, (59)
Hu = νot (1 − α − β)
Yt
ut
− µotBHt = 0, (60)
HP = νot β
Yt
Pt
− νot ϕKt − µo
tψtHt = 0, (61)
νot
νot
= ρ −(
αYt
Kt
− (δK + ϕPt)
)= 0, (62)
µot
µot
= ρ − νot
µot
(1 − α − β)Yt
Ht
−(B(1 − ut) − (δH + ψtPt)
)= 0, (63)
with the transversality conditions for physical capital stock and human capital stock
as follows:
limt→∞
Ktνot e
−ρt = 0, and
limt→∞
Htµote
−ρt = 0.
Substituting µot = νo
t (1 − α − β)Yt/ButHt from (60) into (61) yields
νot β
Yt
Pt
− νot ϕKt − νo
t ψHt(1 − α − β)Yt
ButHt
= 0.
Hence, we obtain the rule for polluting input (42) in the text.
Optimization of the Social Planner with Disutility of Pollu-
tion (Section 5)
The current value Hamiltonian
H =C1−θ
t − 1
1 − θ− P 1+γ
t
1 + γ
+νot [AKα
t (utHt)1−α−βP β
t − Ct − (δK + ϕPt)Kt] + µot [B(1 − ut)Ht − (δH + ψPt)Ht].
39
The necessary conditions for optimality are (59), (60), (62), (63), and
HP = −P γt + νo
t βYt
Pt
− νtϕKt − µotψHt = 0. (64)
Substituting µot = νo
t (1 − α − β)Yt/ButHt from (60) into (64) yields the optimality
condition with respect to polluting input (46).
40