Discussion Papers In Economics
And Business

Natural Disasters in a Two-Sector Model of
Endogenous Growth
Masako Ikefuji Ryo Horii

Discussion Paper 06-13

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANNatural Disasters in a Two-Sector Model of
Endogenous Growth
Masako Ikefuji Ryo Horii

Discussion Paper 06-13

May 2006
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANNatural Disasters in a Two-Sector Model of
Endogenous Growth∗
Masako Ikefuji†

Ryo Horii‡

Osaka University
May 11, 2006

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 tax
policy, 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 economic
growth follows an inverted V-shaped curve relative to the growth rate of theenvironmental tax. The social welfare is maximized under a positive steadystate growth in which faster accumulation of human capital compensates the
productivity loss due to declining use of the polluting input.
Keywords: natural disasters, human capital, endogenous depreciation, economic growth.
JEL Classification Numbers: O41, O13, E22.
∗

We are grateful to Koichi Futagami, Tatsuro Iwaisako, Kazuo Mino, and Tetsuo Ono, andseminar participants at Osaka University and Nagoya University for their helpful comments andsuggestions. All remaining errors are naturally our own.
†

Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka 560-0043,

Japan. E-mail: cg043im@srv.econ.osaka-u.ac.jp‡

Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka 560-0043,

Japan. E-mail: horii@econ.osaka-u.ac.jp1

1

Introduction

Natural disasters have a large impact on economic activity primarily through destruction of capital stock. For example, National Oceanic and Atmospheric Administration (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, werealso considerably large–approximately 45 billion dollars, as reported by NOAA. Thismagnitude 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 byeconomic 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 islikely to be true. For example, it is clearly stated in the Intergovernmental Panel on
Climate Change third assessment report (IPCC, 2001) that “Emissions of greenhousegas and aerosols due to human activities have a great impact on global climatechange.” Global warming, or more specifically, increasing sea surface temperature,2

1

In 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.
2

The 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 Niño 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 sustainability 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 environmental tax.3 Diﬀerently from earlier studies, however, this paper examines thecase in which the use of polluting inputs raises the probability that capital stocks
are destroyed by natural disasters. Agents make saving decisions taking into accountthe 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 theconstant 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 ofnatural disasters, it reduce incentive agents to invest in capital stock since they facea 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
inputs. If the per-unit tax rate is raised over time, private firms owing a CobbDouglas production technology will increase the use of other inputs, including humancapital, relative to polluting inputs. As a result, the use of polluting inputs canbe is bounded above, making the economic growth sustainable. It is also shownthat 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,3

Copeland 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 capitaland the total level of pollution as inputs.

3the lower is the asymptotic amount of pollution and therefore the lower is probabilityof 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 productivity 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-growthmodel with pollution Stokey (1988) shows that, even when production technology
allows sustained growth, it is theoretically possible that agents prefer a no-growthstate with a good environment.4 Contrary to Stokey’s analysis, we show that thesocial 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 theuse of polluting input only aﬀects the risk of natural disasters without directly aﬀecting consumer’s utility.5 Rather, it comes from the our two-sector specification thatthe 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 humancapital stock has lower degree of vulnerability to disasters, as suggested by Skidmore

4

Stokey (1998) assumed additive separable preference for consumption and pollution in which

the marginal utility from consumption is declining whereas the marginal disutility from pollution isincreasing. 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 ahigh level of capital stock.
5

In section 4.1, we examine the case in which pollution directly causes disutility to consumers, inaddition 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 capitalstock relative to physical capital stock.
To our best knowledge, this study is a first attempt to examine the consequencesof 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 growththeory, by noticing that the finite nature of natural environment may potentiallyrestrict sustainability of economic growth. With regard to the finiteness of natural resources, Aghion and Howitt (1998), Scholz and Ziemes (1999), Schou (2000),
Grimaud and Rougé (2003), and Agnai, Gutiérrez and Iza (2005) examined sustainability of economic growth in endogenous growth models with non-renewable
resources.6
Complementary to those studies, Stokey (1998) and Uzawa (2003) examined sustainability focusing on emission of pollutants. Since the global atmosphere is finite,
the negative eﬀects from pollutants will become unacceptably serious when the usage of polluting input increases without bound. Therefore, both Stokey (1998) andUzawa (2003) concludes that, without exogenous technological change, the economy
should converge to a no-growth steady state. Our study is related to their studies inthat it is focusing on pollutants, but is more closely related to Bovenberg and Smulders (1995) who explicitly considers accumulation of knowledge. When accumulationof knowledge improves the productivity of other inputs (including polluting inputs),
the economy can grow using constant or even declining amounts of polluting inputs.
Their prediction that sustained growth is possible under an appropriate policy also
holds in our model.
The rest of the paper is organized as follows. Section 2 presents the model andproves that growth cannot be sustained under a constant tax rate on polluting inputs.
6

For example, Grimaud and Rougé (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.

5

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 thedesirability of sustained growth. Section 5 concludes.

2

The ModelThis 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 ofhouseholds and firms is examined. After presenting equilibrium 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 technology 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 , ut Ht , Pt ) = AKtα (ut Ht )1−α−β Ptβ ,where ut is the time share devoted to production of goods, α is a constant share ofphysical capital, and β is that of the polluting input. 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 + qbPt ,(1)where q and qb are positive constants. Equation (1) says that the arrival rate raises

as the amount of aggregate polluting inputs increases, as in the case of hurricanes
6and fossil fuels. When a natural disaster occurs at an area, it causes damage tophysical capital. For example, if natural disasters occur at an area where the existinge t,
e t , the expected loss of physical capital is φK
aggregate physical capital stock is K

where φ > 0 is the average damage to physical capital stock. Note that, for various

reasons, a natural disaster harms human capital stock as well.7 Similarly to φ, define
ψ > 0 as the average damage to human capital stock. The damage on human capital
measured in relative to the existing stock is, however, smaller than the damage on
physical capital stock, and therefore it is reasonable to assume φ > ψ > 0.
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.8 By the lawof large numbers with (1), the aggregate damage to physical capital stock and humancapital stock are respectively:
q Pt )Kt ,qt · φKt = (φq + φb(2)

qt · ψHt = (ψq + ψb
q Pt )Ht .

(3)

Let δ K and δ H be the constant rates of depreciation of physical capital stock andhuman capital stock, respectively. Then, similarly to Lucas (1988), the resource
constraints for physical and human capital stocks are written as:
K̇t = F (Kt , ut Ht , Pt ) − Ct − (δK + φPt )Kt ,(4)

Ḣt = B(1 − ut )Ht − (δH + ψPt )Ht ,(5)q , δH ≡ δ H + ψq, ψ ≡ ψb
q , and Ct , B, and 1 − ut is thewhere δK ≡ δ K + φq, φ ≡ φbaggregate consumption, the constant productivity of human capital accumulation,
7

The death toll in Katrina rose to over 1000 (NOAA, 2005) and the number of the injured wasmuch 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.
8

This assumption is not realistic when considering large scale disasters. Without it, the law oflarge numbers does not apply, and disasters cause short-term fluctuations. However, since we arefocusing on long-term behavior of the economy, analysis of such fluctuations are out of the scopeof this paper.

7

and a fraction of time devoted to production of human capital, respectively. Equations (4) and (5) shows that the risk of natural disasters eﬀectively augments thedepreciation rates of physical and human capital stocks, in proportion to the use ofpolluting input. We assume that α + β > φ so that the sensitivity of the augmented
depreciation rate to the polluting input is not too high.
Observe that, unlike standard endogenous growth models, the right hand sides ofequations (4) and (5) are not homogenous of degree one in terms of quantities. Thisimplies that balanced growth that presents a homothetic expansion is not feasible,
which reflects the finiteness of natural environment.

2.2

The market economy

Since firms do not take into consideration the externality such that the use of polluting inputs increases the risk of natural disasters, the market equilibrium does notcorrespond 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, islevied on the use of polluting inputs. Since there is no uncertainty, consumers pre9
dicts {τt }∞t=0 at the initial time under perfect foresight. The government balances

its budget at each moment and equally distribute the tax revenue Tt = τt Pt amonghouseholds in a lump-sum fashion.
Households
The economy is populated by a unit mass of infinitely lived homogeneous households. 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 bothtypes 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 qt phikt and qt psiht , respectively, the9This paper does not consider the issue of dynamic inconsistencies.

8budget constraint of households can be written as:10k̇t = rt kt + wt ut ht − (δK + φPt )kt − ct + Tt ,(6)ḣt = B(1 − ut )ht − (δH + ψPt )ht ,(7)

where rt , wt , and ct denote the real interest rate, the real wage rate, and the amountof consumption, respectively. Note that the cost associated with depreciation andinsurance is paid by the owner of the capital.
The utility function of the representative household is given by:
Z ∞ 1−θ
ct − 1 −ρt
e dt,
1−θ
0

(8)

where θ > 1 is the inverse of the elasticity of intertemporal substitution and ρ isthe rate of time preference. We assume B > ρ so that households have enoughincentive to investment in human capital. Each household maximizes (8) subject tothe constraints (6) and (7). From the first-order condition for maximization problem,
we obtain the Keynes-Ramsey Rule:−θċt= ρ + φPt + δK − rt .
ct

(9)This condition is similar to that obtained in the original Uzawa-Lucas model, exceptthat it depends on 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 capitalstock. Hence, the arbitrage condition between human capital investment and physicalcapital investment is given by:ẇt= rt − (φ − ψ)Pt − (δK − δH ) − B.
wt
10

(10)

Equations (6)-(7) 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 ofgoods. Nonetheless, as long as the amount of compensation in terms of goods is calculated using theappropriate price of human capital, wt /B, the equilibrium outcomes do not change in the aggregate
level.

9In (10), the left hand side implies the rate of change in the relative shadow price ofhuman 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 beeither ut = 0 or ut = 1 for all agents, and therefore one of the two kinds of aggregate
capital stock approaches zero due to depreciation, clearly inconsistent with the CobbDouglas production function. The transversality conditions are limt→∞ kt νt e−ρt = 0
for physical capital stock and limt→∞ ht µt e−ρt = 0 for human capital stock, whereand µt = (wt /B)c−θνt = c−θ
t
t .
Firms
There is a continuum of firms producing final goods in competitive market. We
consider a representative firm maximizing its profit. The firm pays the wages forlabor input, the rental rate for physical capital input, and the environmental tax aswell. Given factor prices rt , wt and τt , the profit maximization problem is:
max F (Kt , Nt , Pt ) − rt Kt − wt Nt − τt Pt ,Kt ,Nt ,Pt

where Nt ≡ ut Ht is the amount of human capital employed by the firm. The firstorder conditions for this problem are rt = αYt /Kt , wt = (1 − α − β)Yt /Nt , andτt = βYt /Pt . Substituting the profit maximizing polluting input, Pt = βYt /τt , into
the production function, the output can be written as:³´βe − 1−β K αb N 1−bα ,Yt = Aτtt(11)e ≡ β β/(1−β) A1/(1−β) and α
where Ab ≡ α/(1 − β). When written in the form of (11),it becomes clear that the environmental tax lowers the eﬀective total factor pro-e −β/(1−β) . Using this notation, the first order conditions are expressed
ductivity, Aτ
as

´αb−1
β ³
e t− 1−β Kt,
rt = αAτNt−βe t 1−βwt = (1 − α − β)Aτ
10(12)
³ K ´αb
t

Nt

.

(13)

2.3

Equilibrium conditions

Since the population is homogenous and normalized to unity, Kt = kt , Ht = ht , Ct =ct , and ut = Ht /Nt hold in equilibrium. From the budget constraint of households,
the evolution of physical capital stock is given by:
Yt
Ct
K̇t
=
−
− (δK + φPt ),
Kt
Kt Kt

(14)

From the production function of human capital stock, its evolution is given by:Ḣt= B(1 − ut ) − (δH + ψPt ).
Ht

(15)

From (10) and (13), the evolution of labor supply ut must satisfy:
−

³ K̇
β
τ˙t
Yt
Ḣt u̇t ´tb−−+ (ψ − φ)Pt − (δK − δH ) − B.
· +α=α
1 − β τt
K t H t ut
Kt

(16)

The dynamics of consumption is given by the Keynes-Ramsey Rule:
−θ

Yt
Ċt
=ρ−α
+ δK + φPt .
Ct
Kt

(17)

Finally, from the firm’s f.o.c., the amount of polluting input is determined by
Pt = βYt /τt .

(18)

A necessary condition for the transversality condition associated with physicalcapital stock is K̇t /Kt + ν˙t /νt − ρ < 0 as t → ∞. We will show in Section 3that K̇t /Kt = Ċt /Ct in the long run. Thus, from ρ > 0 and νt = c−θt , we obtainK̇t /Kt + ν˙t /νt = (1 − θ)Ẏt /Yt so that the transversality condition associated withphysical capital stock is satisfied so long as Ẏt /Yt ≥ 0 (i.e., unless the economy
is shrinking toward zero). Likewise, the transversality condition associated withhuman capital stock is satisfied whenever Ḣt /Ht + µ̇t /µt − ρ < 0. Diﬀerentiating

logarithmically with respect to time in µt = (wt /B)c−θt and using (9), (10), and (15),
we obtain Ḣt /Ht + µ̇t /µt − ρ = −But . Because B > 0, the transversality condition
associated with human capital stock is satisfied if ut > 0.

11

2.4

Sustainability under a constant tax rate

Observe, from (18), that pollution increases in proportion to output Yt if the government do not change the environmental tax rate. Since the increasing usage ofpolluting inputs makes natural disasters more and more frequent, it seems that economic growth is not sustainable under such a static environmental policy. This
subsection formally proves that this insight is correct.
The proof goes via reductio ad absurdum. When the government sets a constantenvironmental tax rate (i.e., τt = τ0 for all t), the Keynes-Ramsey Rule (17) can be
rewritten, from (18), as:
µ
¶
φβ
Yt
Ċt
−θ
= ρ + δK − α −
Kt
.
Ct
τ0
Kt
This equation states that, if consumption grow in the long-run (i.e., Ct → ∞ ast → ∞), the sign of the value in the parentheses must be positive. Hence, Kt
must be bounded above by a constant value τ0 α/φβ (i.e., limt→∞ Kt < τ0 α/φβ).
To interpret this result, observe that that, from (11) and (12), the rental price ofphysical 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 physicalcapital accumulates, the insurance cost increases in relative to interest rate due toincreased risk of natural disasters. Since this lowers the incentive to save, the stock ofphysical capital should not become too large in order to maintain sustained growth.
This raises another question, however, of maintaining output growth under a limited size of physical capital. From (11), the positive growth rate of output requires
the positive growth rate of human capital stock devoted to production due to thesupremum of Kt . That is, limt→∞ Ṅt ≥ 0 must hold in order to support increasing consumption. Under a constant environmental tax rate equation, (16) can be
rewritten as:
³ K̇

Yt
Ṅt ´
α
b
−
+ (ψ − φ)Pt − (δK − δH ) − B.
=α
Kt Nt
Kt
t

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,12−bαṄt /Nt . Conversely, the right hand is given by:
µ
¶
φβ
Yt
Kt
− δK + ψPt − B + δH ,α−
τ0Ktwhich is the diﬀerence between the marginal rate of return on both types of capitalstock. 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 theright hand side goes to infinity as Yt → ∞. These results imply that the equality in(16) fails to hold, and therefore Ct and Yt cannot grow in the long run. To summarize,
Proposition 1 If the environmental tax rate is constant, economic growth is notsustainable.

3

Steady-state growth

Given Proposition 1, this section considers a time-varying tax policy. In order tofocus on the long-term behavior of the economy, the following considers a policythat has the constant growth rate of environmental tax:
τt = τ0 exp(gτ t),

τg : positive constant.

The main task of this section is to examine the dependence of long-term rate of
economic growth, denoted byẎt,t→∞ Ytg ∗ ≡ limon the growth rate of environmental tax gτ . Note that, for any gτ chosen by thegovernment, the corresponding equilibrium value of g ∗ cannot be larger than g τ
since it means the usage of polluting input, Pt = βYt /τt , become infinite in the longrun, and therefore is impossible for the same reason as explained in subsection 2.4.
Hence, g ∗ ≤ gτ , which leads to limt→∞ Ṗt /Pt ≤ 0 from (18). Since the amount ofpolluting input is nonnegative, this means that Pt converges to a constant value,denoted by P ∗ , in the long-run.
To examine the long-term property of the equilibrium dynamics, we consider thesteady-state growth path in which the various quantities grow at constant rates:
13

K̇t /Kt = gK , Ċt /Ct = gC , Ḣt /Ht = gH , and u̇t /ut = gu . In this steady state, (15)and (17) implies that the value of ut and Yt /Kt must be constant. In addition, from(14), these properties in turn imply Ct /Kt must also be constant. Thus we denote
those constant values by ut = u, Yt /Kt = z, Ct /Kt = χ.
Since gK = gC = g ∗ and gu = 0 are already known, the following focuses on thedetermination of g ∗ and gH . Diﬀerentiating logarithmically with respect to time in(11) and using gK = g ∗ yield the growth rate of human capital stock:gH = g ∗ +

β
gτ .
1−α−β

(19)

Equation (19) shows that, in the long run, the use of human capital grows fasterthan physical capital (and output). It compensates for the decreasing use of polluting
input due to the increasing environmental tax. Constancy of u leads to Ṅ /N = Ḣ/H.
Hence, from (13), the rate of change in wage is given by:
ẇ
β
=−
gτ .
w
1−α−β

(20)

Recall that the price of a unit of human capital stock measured in goods is wt /B.
Equation (20) shows that it decreases as human capital stock increases.
Applying the above discussion, the equilibrium conditions (14)-(18) can be simplified along the steady-state growth path, as follows.
evolution of Kt : g ∗ = z − χ − (δK + φP ∗ ),

(21)

β
(22)
gτ = B(1 − u) − (δH + ψP ∗ ),1−α−ββarbitrage condition: −gτ = αz − B + (ψ − φ)P ∗ − (δK − δH ), (23)1−α−βevolution of Ht : g ∗ +Keynes-Ramsey rule: − θg ∗ = ρ − αz + (δK + φP ∗ ),≥ 0 if g ∗ = gτ (case 1),∗
asymptotic value of Pt : P

= 0 if g ∗ < gτ (case 2).

(24)
(25)

Those five conditions, (21)-(25), determine five unknowns, g ∗ , z ≡ Yt /Kt , χ ≡ Kt /Ct ,u, and P ∗ , in the steady state growth path. Note that, however, they include acomplementary slackness condition (25), and we cannot know whether P ∗ = 0 or14

g ∗ = g τ holds in advance. Thus we need to examine two possible cases in turn, andthen to determine which case actually occurs in equilibrium under a particular setof exogenous parameters.
Let us first derive the steady-state growth path assuming that Case 1 actuallyoccurs (whether this assumption is appropriate will be checked immediately). Substituting g ∗ = gτ into (24) and (23), we obtain the steady-state value of polluting
input:
P∗ =

³
´
i
1h
β
B−ρ− θ+gτ − δH .
ψ
1−α−β

(26)

In (26), we can see that the condition, P ∗ ≥ 0 is satisfied if gτ is within the following
region:gτ ≤B − ρ − δH
≡ g max .
βθ + 1−α−βHence, Case 1 is possible only if gτ ≤ g max . From (21) to (24), we obtain the valueof other variable as follows:1(θgτ + δK + φP ∗ + ρ) ,α´1³∗(θ − α)gτ + (1 − α)(δK + φP ) + ρ ,
χ=
α
´
³
1
u=
(θ − 1)gτ + ρ .
B
z=

(27)
(28)
(29)

Next, we examine the possibility of steady-state growth in Case 2. Suppose thatthe amount of polluting input is zero at the equilibrium. The same procedure as theabove yields the steady-state growth rate:
g∗ =

´
1³
β
B − δH −gτ − ρ ,
θ
1−α−β

(30)

we can see that the condition g ∗ < gτ is satisfied if gτ > g max . The steady-state valueof other variables are:´β1³
B + δK − δH −
gτ ,
α
1−α−β
³ 1 1 ´³
´ 1−α
β
ρ
χ=
−
B − δH −gτ +δK + ,
α θ
1−α−β
α
θ
³
´
i
1 h
β
u=
(θ − 1) B − δH −gτ + ρ .
Bθ
1−α−β
z=

15

(31)
(32)
(33)

It can be confirmed that all z, χ and u in (31)-(33) are positive if gτ is lower thang lim ≡ (1 − α − β)β −1 (B + δK − δH ).
Note that the possibilities of Case 1 and Case 2 are mutually exclusive–that is,under a given set of parameters, only one case is possible. Therefore, the steadystate path is always unique, and it is characterized by g ∗ = gτ and (26)-(29) ifgτ ∈ [0, g max ], and by P ∗ = 0 and (30)-(33) if gτ ∈ (g max , g lim ). In Appendix, weshow that the steady state is saddle stable in both cases, given that either δK is not
too large or gτ is below a certain level:gτ ≡e(γ + θ)(1 − α − β)(B − δH ) + ρ(1 − α − β).
(θ − 1)βSince g lim and geτ are considerably high under reasonable parameter values, we restrict
our attention to gτ < min{g lim , egτ }.

Figure 1 illustrates the relationship between the growth rate of environmental

tax and the long-run growth rate of human capital stock in (22), physical capitalstock and output in (21), and the growth rate of polluting inputs or pollution which
is given by Ṗ /P = g ∗ − gτ . The asymptotic amount of pollution, P ∗ , is largest whenthe environmental tax rate is constant (i.e., gτ = 0), which results in the highest
probability of natural disasters and zero growth. When gτ in increased within therange of [0, g max ], the long-run amount of pollution, P ∗ decreases while the growth
rates of Y and K increase in parallel with gτ . The growth rate of human capital gH
also increases, at a faster pace. This enables sustained growth under (asymptotically)
constant use of polluting input. When the environmental tax rate is raised so rapidlythat gτ exceeds g max , the use of polluting inputs Pt is continually reduced, converging
asymptotically to zero level. In this case, the growth rates of Y and K can nolonger increase in parallel with gτ , but decreasing in gτ . Nonetheless, provided thatgτ < g lim , the economy can grow at a positive rate growth by accumulating gH much
faster than output grows. The following proposition summarizes the obtained results.
gτ ),Proposition 2 If the environmental tax rate is raised at constant rate gτ ∈ (0, ethere exists a unique and saddle-stable steady-state growth path with positive rate of16

Figure 1: Growth rate of environmental tax and the steady state growth. The upper
panel shows the relationship between the growth rate of environmental tax (gτ ) and that of humancapita (gH ), phisical 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 = .4, β = .1,
θ = .2, ρ = .98, B = 2, ψ = .005, φ = .01.

growth. The long-term rate of growth follows an inverted-V shape against the growthrate of the environmental tax rate, and it is maximized when gτ = g max .
It is possible to interpret Proposition 2 in terms of the Keynes-Ramsey rule. Note
that, from the arbitrage condition between human capital investment and physicalcapital investment, the rates of return to both of capital net of insurance payments
are equal. Therefore, using the arbitrage condition (10) and the Keynes-Ramsey
Rule (9) and the fact that gC = g ∗ , the growth rate at the steady state can be
rewritten as:
1
g =
θ
∗

µ

β
B − ψPt −gτ − ρ − δH
1−α−β

¶

.

(34)

The second term in the parenthesis of (34) expresses the expected loss of humancapital due to natural disasters. If the growth rate of environmental tax is accelere falls, the value of human capital
ated, the eﬀective productivity of private firms, A
17

is reduced, and individuals invest less in human capital. Given P ∗ , thus, the growthrate of economy decreases. As a result, the growth rate of pollution, Ṗt /Pt = g ∗ − gτ
decreases. However, when P ∗ is positive (Case 1), the risk of damage to humancapital stock decreases due to a reduction in P ∗ , which allows the growth rate of theeconomy to increase, as shown in (34). This boost continues until g ∗ catches up withgτ . However, when Pt is already approaching P ∗ = 0 (Case 2), the reduction in Ṗ /P
accelerates the speed at which Pt converges to P ∗ = 0, but it does not change theasymptotic value P ∗ . Thus, the steady-state growth rate remains low.

4

WelfareThe previous section established that sustained growth is feasible by raising therate of environmental tax rate over time. It is yet to shown, however, such anenvironmental policy is desirable in terms of welfare. This section investigates thesocial planner’s problem so as to derive the welfare-maximizing environmental policy.
The planner maximizes (8) subject to the following constraints:
K̇t = F (Kt , ut Ht , Pt ) − Ct − (δK + φPt Kt ),(35)

Ḣt = B(1 − ut )Ht − (δH + ψPt Ht ).

(36)

From the first-order conditions for optimality, the dynamics of Kt , Ht , and Ct , andthe arbitrage condition between two types of capital stocks are the exactly same as(14)-(17), which are parts of market equilibrium conditions. Since the social plannertakes into consideration the externality of polluting emissions, that is, the risk ofnatural disasters, it chooses the amount of polluting input according to the following
rule:(1 − α − β)Yt
βYt
= φKt + ψHt ·.
Pt
But Ht(37)

Equation (37) says that Pt is determined such that the marginal benefit is equal tothe marginal cost of using polluting inputs. The left hand side of (37) is the marginal
productivity of polluting input whereas the right hand side is the sum of increases18

in damage to physical capital stock and in damage to human capital stock, bothmeasured in terms of final goods.
Recall that Pt is determined according to (18) in the market equilibrium. Substituting (18) into (37) reveals that the optimal growth path can be realized inequilibrium by setting the optimal environmental tax rate:τtopt = φKt + ψ(1 − α − β)Yt
.
But

(38)

In order to implement such a policy, it is necessary for the government to knowequilibrium paths of Kt , Yt , and ut , which appear in the RHS of (38). However,Kt , Yt , and ut are endogenously determined depending the future path of τt that is
expected by consumers.
Although it seems excessively diﬃcult to solve this dynamic fixed point problem,the long-term property of the optimal policy can be conveniently analyzed by focusing
on the usage of polluting inputs, Pt , which approaches a constant steady state value,rather than in terms of the time-varying tax rate imposed on it. Note that optimality
condition (37) can be stated as
Ptopt = βµ¶−1YtKt ψ(1 − α − β)
=β φ
+
.
Yt
But
τtopt

(39)

In the previous section, we have shown that Yt /Kt and ut appearing in the RHS of(39) become constant in the steady-state growth path, and their values are derived
explicitly in terms of g τ (see equations 26-28 and 31-32). Therefore, the welfaremaximizing amount of polluting input P opt in the long run can be calculated as afunction of gτ . As depicted in Figure 2, P opt is positive and continuous in gτ for allgτ ∈ [0, g lim ).
The actual amount of polluting inputs used in equilibrium is, however, determined
not by (39) but by (26) for gτ < g max and P ∗ = 0 for g ∗ ≥ g max , which is also depicted
in Figure 2. Unless consumers heavily discount the utility in the future, it can beshown that the intercept of P opt curve is smaller than that of P ∗ . Therefore, there
exists a level of gτ ∈ (0, g max ), denoted as gτopt , such that P ∗ = Ptopt in the long run.
This means that the optimal growth path can be (asymptotically) implemented byraising the environmental tax rate at the rate of gτopt .
19

Figure 2: Determination of the optimal growth rate of environmental tax. The optimal
growth rate of environmental tax, gτopt , is given by the intersection of P ∗ and P opt , and is lower
than the growth maximizing rate, g max . The parameters are the same as in Figure 1.

Note that, since gτopt ∈ (0, g max ), the long-term rate of economic growth g ∗ coincides with gτopt , and therefore it is positive. Thus the above analysis shows thatthe 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 gτopt < g max . That is, if the government care about welfare it should tighten the environmental policy more slowly
than when growth is its first priority. 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. Although an aggressive environmental policythat 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 in far future.

4.1

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 that20

harm the utility of consumers. The diﬀerence of the results of course comes fromthe setting of models. More specifically, our model significantly diﬀers from Stokey(1998) in two aspects; (1) we are considering human capital accumulation diﬀerently
from Stokey’s AK model; (2) 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¶
Z ∞ µ 1−θct − 1 Pt1+γ−e−ρt dt, γ > 0.
1
−
θ
1
+
γ
0

(40)

Since function (40) is separable with respect to ct and Pt , behavior of all agents, whotakes Pt as given, does not change. That is, the equilibrium outcome is exactly thesame as in analyzed in sections 2 and 3.
Let us examine how the planner’s problem is aﬀected. Under resource constraints(35) and (36), the planner maximizes function (40). Then, the optimality condition
with respect to polluting input becomes(1 − α − β)Yt
Ptγ
βYt
= φKt + ψHt ·
+ −θ
.
Pt
But HtC(41)

When compared to (37), there is an additional marginal cost of polluting input interms of utility. Multiplying both sides of (41) by Pt /Yt yields the conditionµ ¶θµ¶(1 − α − β)Ct
Kt+ψYtθ−1 Pt1+γ ,
β= φ
Pt +
Yt
But
Yt

(42)

which is convenient since the LHS is constant.
Our goal is to find the environmental tax policy such that (42) holds in the longrun. Recall that on any steady-state growth path, Kt /Yt , ut , and Ct /Yt in (42) do notchange. Therefore, for the optimality condition to hold, Ytθ−1 Pt1+γ must be constant.
Since Ṗt /Pt = g ∗ −gτ from (18), this condition requires (θ−1)g ∗ +(1+γ)(g ∗ −gτ ) = 0,or equivalently
g∗ =

1+γ
gτ .
θ+γ
21

(43)

Observe that, since θ > 1, condition (43) requires that the economy should growslower than the rate at which the environmental tax is increased.11 As we see insection 3, this occurs in equilibrium only when gτ > g max . In this case, the actual
value of g ∗ is determined by (30), which equals to (43) if and only if
gbτopt =(γ + θ)(1 − α − β)(B − δH − ρ).
θ(1 + γ)(1 − α − β) + β(γ + θ)This expression gives welfare-maximizing growth rate of environmental tax rate:gτ and g lim , the optimal rate of growth isSince bgτopt is positive and smaller than epositive and the dynamics is saddle stable.

To summarize, the main result that sustained growth is desirable does not changeeven 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, g max . This implies that, if pollution aﬀects the utility of agents
directly, the emission of pollutant should be eliminated in the long run even at thecost of accepting a slower (although positive) rate of economic growth.

5Concluding Remarks

The sustainability of economic growth has been analyzed in a two-sector model ofendogenous growth, taking into account the risk of natural disasters. Polluting inputs
are necessary for production, but they intensify the risk of natural disasters. In thissetting, we obtained following results.
First, the long-run economic growth can not be sustained if the private cost ofusing the polluting input is kept constant.12 Since, for simplicity, we do not consider the cost associated with extracting resources or the finiteness of those inputs,
11

Strictly speaking, condition (43) does not rule out g ∗ = gτ = 0. However, similarly to theprevious discussion, if the tax rate is kept constant, P ∗ is higher than the optimal amount ofpollutants (this should be even lower due to the disutility of pollution) unless the consumers discountthe future very heavily. Therefore, the optimal rate at which the tax rate is increased must be
positive.
12

When the damage to physical capital stock is much larger than that to human capital stock22

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 reasons, the environmental tax rate must be adjusted to absorb those changes. Moresubstantially, 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 isbeyond 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 environmental tax is initially slow growing, its acceleration will reduce the long-run level ofemission and the risk of natural disasters, which enhances the incentive to save andhence promotes economic growth. When the rate of environmental tax is alreadyfast growing, the amount of polluting input at the steady state is fairly small so thatfurther acceleration of environmental tax excessively impair the productivity of private firms, which works against economic growth. Therefore, the economic growth
can be maximized with choice of the most gradual increase in environmental tax ratethat 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 ceterisparibus induces private firms to use more polluting input, an appropriate environmental policy can lead firms to use more of human capital (e.g., by investing inalternative technology), which decreases their reliance on polluting inputs. The optimal 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.

(i.e., φ >> ψ), the steady-state value of P ∗ is rather large (see Equation 26). In this case, the speedof convergence to the steady state is slow, the economic growth declines gradually, and amount ofpollution increases during the transition to the steady state.

23

Appendix
Case 1First, we examine the transitional dynamics and the stability of the steady state inCase 1. From (14), (15), and (16), we obtain the dynamics of u as follows:(1 − α − β)(φ − ψ)u̇
= B(u − u∗ ) − (χ − χ∗ ) + β(z − z ∗ ) +(P − P ∗ ),
u
α

(44)

where z ∗ , χ∗ , u∗ , and P ∗ are the steady state value of z, χ, u, and P , respectively.
From (14) and (17), the dynamics of χ is given by:
θ−α(θ − 1)φχ̇= (χ − χ∗ ) −(z − z ∗ ) +(P − P ∗ ).
χ
θ
θ

(45)

From (11) and (44), the dynamics of z and of P are given by:(1 − α − β)(ψ − φ)ż(46)
= −(1 − α − β)(z − z ∗ ) +(P − P ∗ ),zα
α + β(1 − α − β)
(1 − 2α − β)φ − (1 − α − β)ψ
Ṗ
= −(χ − χ∗ ) +(z − z ∗ ) +(P − P ∗ ).
P
1−β
α
(47)
The Jacobi matrix of (44) - (47) is written as:

B −1
β


 0 1
− θ−α
θ
1

J =
 0 0 −(1 − α − β)
α+β(1−α−β)
0 −1
1−β

Λ
(θ−1)φ
θ

Λ
Ω

where

Λ≡

(1 − α − β)(φ − ψ)
,
α

and

Ω≡






,




(1 − 2α − β)φ − (1 − α − β)ψ
.
αDefine a matrix as follows:


1

− θ−α
θ



J ∗ =  0 −(1 − α − β)
α+β(1−α−β)
−1
1−β
24

(θ−1)φ
θ

Λ
Ω





.


The eigenvalues of J ∗ are the solutions of its characteristic equation:−λ3 + TrJ ∗ λ2 − BJ ∗ λ + DetJ ∗ = 0,

(48)

where
TrJ ∗ = Λ + (α + β − φ) > 0,
BJ ∗ =

1

− θ−α
θ

+

if α + β > φ−(1 − α − β) Λα+β(1−α−β)1−β0 −(1 − α − β)θ(1 − α − β)(φ − 1) − φ< 0,θψ(1 − α − β)
=
> 0.
θ

Ω

1

(θ−1)φ
θ

−1

Ω

+

,

=

DetJ ∗We determine the sign of the real parts of the roots of (48) based on Theorem 1 ofBenhabib and Perli (1994).
Theorem 1 (Benhabib-Perli ) The number of roots of the polynomial in (48) withpositive real parts is equal to the number of variations of sign in the scheme
−1

TrJ ∗

− BJ ∗ +DetJ ∗TrJ ∗DetJ ∗ .

∗> 0, and DetJ ∗ > 0, there can only be one
Since we have TrJ ∗ > 0, −BJ ∗ + DetJ
TrJ ∗positive real parts in the matrix J ∗ under the assumption α + β > φ. Hence, there
can be two positive eigenvalues and two eigenvalues with negative real parts in thematrix J 1 and the steady state is saddle path stable.
Case 2Turning to Case 2, the transitional dynamics of z, χ, u, and P are given by:
u̇
u
χ̇
χ
ż
z
Ṗ
P

β
1−α−β
(B + δK − δH ) − gτ ,
α
α
θ−α
θ−1
ρ θ−1
= χ−
z+
φP − +
δK ,
θ
θ
θ
θ
β
1−α−β
= −(1 − α − β)z + ΛP +(B + δK − δH ) − gτ ,ααα + (1 − α − β)β
1−α−β
α+β
= −χ +z + ΩP +
B−
gτ
1−β
α
α
(1 − 2α − β)δK − (1 − α − β)δH+.
α
= Bu − χ + βz + ΛP +

25

(49)
(50)
(51)

(52)

Linearizing from (49) to (52) around the steady state and substituting P ∗ = 0 andother steady state values into the linearization, we obtain:



 
u − u∗
u̇
J11 ∗
∗
∗



 



 
∗
 χ̇   0 J22 ∗∗  χ − χ 

,
'
J =



 
 ż   0
0 J33 ∗   z − z ∗ 



 
∗
P −P
0
0
0 J44
Ṗ
where

θ−1
θ−1
ρ
(B − δH ) −βgτ + ,
θθ(1 − α − β)θθ−α(θ − α)βρ=(B − δH ) −gτ + ,αθαθ(1 − α − β)
θ
1−α−β
β
1−α
= −
(B − δH ) + gτ +
δK ,
α
α
α
1
ρ θ(1 − α − β) + β=(B − δH ) − −gτ ,θθθ(1 − α − β)

J11 =
J22
J33
J44

Substituting g max into gτ of the above eigenvalues, we obtain:
J11 > 0,J22 > 0,J33 < 0,J44 = 0.

Moreover, substituting g lim into gτ , the above eigenvalues are:
J11 > 0,J22 > 0,J33 = 0,J44 < 0.

Thus, for gτ ∈ (g max , g lim ), there are two positive eigenvalues and two eigenvalues
with negative real parts in the matrix J and the steady state is saddle path stable.

26

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