Discussion Papers In Economics
And Business

Dynamic analysis of a renewable resource in a small
open economy: The role of environmental policies
for the environment
Koichi Futagami and Yasuhiro Nakamoto
Discussion Paper 10-21-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDynamic analysis of a renewable resource in a small
open economy: The role of environmental policies
for the environment
Koichi Futagami and Yasuhiro Nakamoto
Discussion Paper 10-21-Rev.
September 2014

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDynamic analysis of a renewable resource in a small
open economy: The role of environmental policies
for the environment
Koichi Futagami∗ and Yasuhiro Nakamoto†
September 5, 2014

Abstract
We examine the eﬀects of environmental policies such as a subsidy forreforestation and an export-income tax in a small open economy with a renewable resource. In the small economy, the harvested renewable resources
are exported to acquire foreign assets and consumers can invest in the natural
resource to preserve it. In the setup, we show how the environmental policies
aﬀect the natural resource and the domestic economy.
Key words: Renewable resources; The eﬀects of environmental policies.
JEL classification: F41; H21; Q28.

∗

Graduate School of Economics, Osaka University, 1–7 Machikaneyama, Toyonaka 650–0043

Japan. E-mail: futagami@econ.osaka-u.ac.jp†

Corresponding author : Faculty of Economics, Kyushu Sangyo University, 2-3-1 Matsukadai,

Higashi–ku, Fukuoka, 813–8503, Japan. E-mail: nakamoto@ip.kyusan-u.ac.jp Telephone number:+81-92-673-5228. Fax number: +81-92-673-5919.

1

1Introduction

Some low-income nations are facing a shortage of natural resources such as forestry,
fish and wildlife stocks. Governments in most countries have become more awareof such threats and the risk of natural resources depletion. However, there are no
stylized policies for preserving natural resources because government policies aﬀect
not only natural resources but also other economic variables such as consumption
and investment. In this paper, we examine the eﬀects of environmental policies on
renewable resources and on the domestic economy by using a small open economy
model.
The sustainability of major renewable resource stocks is a significant issue in many
countries, especially developing countries. For instance, there have been widely publicized claims that forests in countries such as Indonesia, Malaysia and the Philippines
have been harvested rapidly, with deforestation shifting from the temperate zone to
the tropics following World War II. Other renewable resources, including fish andwildlife stocks, are also under threat in many developing countries.
In these developing countries, most of the harvested resources are exported toacquire foreign assets, which enable these countries to achieve rapid growth of theireconomies. Repetto and Gillis (1988) mentioned that, by 1970, between 7 and 10
percent of the total forest area in Indonesia was being utilized to acquire foreign
assets, with the timber sector, specifically logs and plywood, providing a major
source of foreign exchange. Indeed, between 1970 and 1979, gross foreign exchangeearnings from the export of tropical hardwood grew from US $ 110 million to US$ 2.1 billion. Considering that gross domestic product in 1970 was about US $ 11
billion, the exports created by the deforestation were very valuable in Indonesia. In
the Philippines, the share of exports represented by logs and lumber in GDP rosefrom about 9 percent in the late 1940s to about 15 percent by 1970. In 1970, 16.6
million hectares, or 55 percent of the country’s land area, was made up of forests;
however, between 1971 and 1980, forest lands decreased by 1.7 million hectares. The
export of fish products in Iceland also contributed considerably to the nation’s foreign2

currency earnings, with the contribution of the fishing industry’s exports to GDP ataround 17 percent in 1978 and 7 percent in 2007.1
In many developing countries, the overuse of natural resources is continuing, sothat preservation of these resources is still required. While some developing countries have introduced short-term environmental policies, some of these environmental
policies have generated unintended negative eﬀects on the conservation of resources.
Two such environmental policies are regeneration policies, such as reforestation policy, and a specific income tax for harvesters.2 Repetto and Gillis (1988) focusedon the income tax policy in relation to logging investments in Indonesia and notedthat most timber companies had not paid income taxes at all from 1967 through to
1983. They concluded: More eﬀective income taxation would not have reduced therate of exploitation of the tropical forest, but would have clearly raised the benefits oftimber exploitation to the owner of the forest resource. In addition, some researchers
have examined the eﬀects of reforestation. In Indonesia, from 1946 through to 1983,
reforestation programs impacted on 2.3 million hectares of land. To examine the impacts of the subsidy on reforestation in Indonesia, Osgood (1994) made use of panel
data on 20 regions in Indonesia from 1972 to 1988. Interestingly, she concluded thatthe reforestation subsidy encouraged further deforestation. In addition, Repetto and
Gillis (1988) examined reforestation policies in Malaysia and the Philippines andfound that 21,000 hectares were replanted between 1974 and 1981 in Malaysia, whilemore than 78,000 hectares had been reforested between 1976 and 1983 in the Philippines. Shen and Contreras-Hermosilla (1995) noted that, in India, during 1985–89,
the total number of seedlings distributed under the farm forestry program was in theorder of 1.4–2.0 billion a year, which was enough to plant 560,000–800,000 hectares.

1

See the homepage of the Icelandic Ministry of Fisheries and Agriculture’s Icelandic Fisheries

Information Centre at: http://www.fisheries.is/iceland/
2

Further, Repetto and Gillis (1988) showed that a policy that bans the export of logs in Indonesia

has not necessarily promoted better forest conservation. Shen and Contreras-Hermosilla (1995)
mentioned that the logging ban in Thailand had a limited eﬀect on reducing the rate of deforestation,and instead may have caused illegal trade to increase.

3

Repetto and Gillis (1988) and Shen and Contreras-Hermosilla (1995) concluded thatthe subsidized forestation policies provided little or no incentive to plant trees sothat the reforestation policies aimed at regeneration have proved largely ineﬀective.
In the resource-dependent developing countries, environmental policies have alarge impact not only on these countries’ natural resource sectors but also on theirwhole economies. Hence, in this paper, we construct a dynamic general equilibrium
model of a small open economy with a renewable resource. The small economy in
our model has the following characteristics. The renewable resource is harvested to
produce exportable commodities, which are exported to acquire the foreign assets.
In addition, people can invest in the natural resource such as tree planting or fish
replenishment to avoid its depletion. Under this set-up, our focus is to examine thedynamic impact of temporary environmental policy changes on the resource as well ason the domestic economy because environmental policies are temporarily performedin actual economies such as Indonesia and Malaysia. To shed light on eﬀects ofsuch temporary policy changes, we compare them with eﬀects of permanent policychanges. Specifically, a government can use two policy instruments; a subsidy policy
for reforestation and an income tax policy against the harvesters. Our main finding is
the following: Permanent increases of these policy instruments can increase the levelof the natural resource; instead, temporary increases of these policy instruments
always decrease the level of the natural resource in the long run compared to its
original level, although the level of the natural resource increases in the short run.
Our study is closely related to some of the existing investigations in the environmental dynamic models with renewable resources and the international macrodynamic models (e.g., Elı́asson and Turnovsky, 2004., Fullerton and Kim, 2008 andSilva et al. 2013).3 Many environmental studies incorporate renewable natural re-3

Concretely, Elı́asson and Turnovsky (2004) construct an endogenous growth model with arenewable resource to examine the eﬀects of an increase in the productivity of the harvest sector
and the final output sector. Fullerton and Kim (2008) show that pollution tax revenues are notenough to pay for optimal public spending on abatement in the macrodynamic model with therenewable resource. Silva et al (2013) construct a more general equilibrium model with renewable4

sources into macrodynamic models; however, these existing studies do not focus on
the temporary eﬀect of environmental policies in a small open economy. Furthermore, even if the temporary policies are conducted in their closed economies, theeconomy would go back to the original steady state after the policy variables return to its original levels. As a result, the temporary environmental policies are notharmful for the natural resource in the long run.
Alternatively, the structure of our model is the same as Sen and Turnovsky (1989),
Turnovsky (1997) and Schubert and Turnovsky (2002) in the sense that when policy
temporarily changes under the assumption of perfect foresight, the small open economy does not return to its original steady-state equilibrium after the policy variables
returns to its original levels. This insight on public policies is completely diﬀerent
from the result of the closed macrodynamic models that the long-run equilibriumcoincides with the original steady state under temporary policy changes. However,
the existing studies do not include the natural resource sector in each model and furthermore, their attention is not to examine the temporary eﬀects of environmental
policies.
Finally, our motivation is closely related to that in Rondeau and Bulte (2007) inthe point that both the studies cast some doubts about the usefulness of environmental protection policies. Rondeau and Bulte (2007) make use of a single-country
partial equilibrium model with the interaction between habitat and open-access resource in Bulte and Horan (2003). They show that compensation schemes aimed at
reducing hunting mortality can actually decrease the wildlife stock. This is because
compensation distorts relative commodity prices, thereby being able to increase thereturns to agriculture and encourage agricultural expansion. Instead, we use a dynamic macroeconomic model where the natural resource sector is newly introduced,
especially, the harvested renewable resource is used to obtain the foreign assets.
The essential cause that the environmental policies may be harmful for the natural resource lies in the implementation term of environmental policies in our model.

and non-renewable resources to analyze the eﬀects of the emission tax on the economy.

5That is, the natural resource may be harmed by the temporary implementation ofenvironmental policies, not the permanent one.
The remainder of the paper is organized as follows. In the next section, we
present the model. Section 3 shows the existence and uniqueness of the steady state.
Section 4 examines the eﬀects of government policies on the economy. Section 5 gives
discussion. Our conclusions are summarized in Section 6.

2

The basic framework

This section presents a small open economy model with a renewable resource. The
population is constant over time and its size is normalized to be unity. This economy
consists of firms and a government as well as households.

2.1

The evolution of the renewable resource

The economy is endowed with a stock of a renewable resource nt , where t stands
for time. We assume that the reproduction of the renewable natural resource ischaracterized according to G(nt ). The reproduction function G(nt ) has an inverted
U-shape, with G′′ < 0 and G(0) = G(n̄) = 0, where n̄ represents the carrying capacityof the natural resource.4 This means that there is a level ñ that satisfies G′ (ñ) = 0.
In other words, ñ expresses the level of the renewable resource that provides the
maximum sustained yield. If the level of the natural resource is below the levelof ñ, then the marginal reproduction of the natural resource takes positive values,
whereas if it exceeds ñ, the marginal reproduction of the natural resource takes
negative values.5 In addition, we assume that, as the level of the natural resource

4

For example, in an economy with a renewable resource, Schaefer (1954), Bovenberg and

Smulders (1996), Brander and Taylor (1998), Ayong Le Kama (2001), Wirl (2004), Elı́asson andTurnovsky (2004), and López et al.(2007) made use of a reproduction function with a U-shape.
5

Due to the concavity of G(nt ), it is recognized that analysis of the stability is more complicated.

For instance, see Fullerton and Kim (2008).

6

approaches infinity, the change rate of the natural resource is given by G(∞) = −∞.6
The households can invest in reproduction of the natural resource.7 Thus, we assume
that the renewable resource held by a household evolves as follows:ṅt = Γ(at ) + G(nt ) − zt ,(1)

where Γ(·) represents the investment function for the natural resource, at is the investment in the natural resource, and zt is the harvested level of the natural resource
for use as an input into production.8 The investment function Γ(at ) : ℜ+ → ℜ+ is assumed to be twice continuously diﬀerentiable, strictly increasing, and strictly concavewith respect to the investment at ; furthermore, it satisfies the Inada conditions.9

2.2

The household and the firms

We describe the representative household’s preference as follows. The householdobtains utility from consumption and from the natural resource. In particular, we
assume that the household becomes happier as the quantity of the natural resource
6

Following a lot of existing papers (e.g., Brander and Taylor 1997, and Elı́asson and Turnovsky

2004), the following reproduction function satisfies the assumption:(nt )G(nt ) = ∆ × nt 1 −,n̄where ∆ is the intrinsic rate of growth of the resource. We will make use of this reproduction
function in numerical examples.
7

As for the investment at , López et al. (2007) commented that investments in natural resources

comprise tree planting, fish replenishment including aquaculture investments, protection or cleanupof ecosystems, soil protection including terracing drainage, and agricultural fallowing.
8

Unlike our model, the harvest of the natural resource is carried out according to the Schaefer

harvesting function in some papers (e.g., Brander and Taylor 1998a and 1998b).
9

Some studies introduce the natural resource sector including the investment in the natural

resource by households (e.g., Hoagland et al., 2003 and López et al., 2007). López et al. (2007)
assume that households can invest in the natural resource as in the current setting. Hoagland et al(2003) consider a more complicated natural resource sector, that is, the investment in the natural
resource by households indirectly increases the size of the natural fish stock through the expansionof the area devoted to aquaculture.

7

increases. For instance, we suppose that people feel happier when the number offish in own sea increases or when the number of trees in own forests and mountains
increases. In addition, the used preference follows Economides and Philippopoulos(2008). In this economy, there are two final goods: one is produced by physical capitaland labor according to the neoclassical production function per capita, f (k), where kstands for the capital stock per capita; the other is produced by harvesting the natural
resource according to the production function h(zt ). The production function h(·)
is strictly increasing, twice diﬀerentiable, and concave with respect to the harvested
natural resource; furthermore, it satisfies the Inada conditions. We assume that all of
the goods produced by the harvested renewable resource are exported. That is, thehousehold does not consume the goods produced by the harvested natural resource.10
Then, the preference of the household is expressed as follows:∫ ∞U [0] =[u(ct ) + v(nt )]e−ρt dt,(2)0

where u(·) and v(·) represent the instantaneous utility functions of private consumption ct and the natural resource nt , respectively, and ρ is the fixed rate of time
preference. The instantaneous utility functions, u(·) and v(·), are twice continuously diﬀerentiable, strictly increasing, and strictly concave in terms of ct , and nt ,respectively. In addition, these functions satisfy the Inada conditions.
In the small open economy, the world interest rate, r, is assumed to be constant.
As profit maximization of the firms ensures that r = f ′ (k), the domestic capital stock
takes a constant value, k, and the wage rate also becomes constant, w = f (k)−kf ′ (k).
By making use of the goods produced by the capital stock, the household has
the option of either consuming the goods or investing in reproduction of the natural

10

When the household consumes the export commodities, the preference is written as follows:∫ ∞U [0] =[u(ct ) + v(nt ) + ω(export goods)]e−ρt dt,
0where ω(·) represents the utility function of export goods consumption. Because the relative priceof consumption commodities is exogenously given in a small open economy, this generalization does
not change the essence of main finding obtained in this model.

8resource. Then, the accumulation of the foreign asset, bt is expressed as follows:ḃt = rbt + f (k) + (1 − τ y )ph(zt ) − ct − (1 − τ a )at − Z,(3)

where Z is the lump-sum transfer. Moreover, p shows the relative price of theexported good, measured by the price of the consumption good, which is exogenously
given because of the set-up in the small open economy.
In the accumulation equation for the foreign asset (3), we consider two typesof environmental policies. First, τ a represents the constant rate of the investment
subsidy. An increase in τ a implies that the relative price of the investment in thenatural resource decreases, which would further stimulate the investment. As thelevel of the natural resource is improved, the investment subsidy policy would be
regarded as a regeneration policy such as a reforestation policy. Second, τ y shows
the constant income tax rate imposed only on the exportable income. It is likely
to be one of the environmental protection policies targeting forest-based industry(or fisheries) because, when the rate of the export-income tax increases, the levelof the harvested natural resource would decrease, which could protect the natural
resource.11
Finally, the government is assumed to keep the following balanced budget:
Z = τ a at − τ y ph(zt ).

(4)

Note that when Z > 0, the government imposes a lump-sum tax, whereas whenZ < 0, the government makes transfers.
The representative household maximizes its lifelong utility (2) subject to theevolution of the renewable resource (1) and the budget constraint (3). To solve themaximization problem, we constitute the current value Hamiltonian as follows:H ≡ u(ct )+v(nt )+qt {rbt + f (k) + (1 − τ y )ph(zt ) − ct − (1 − τ a )at − Z}+ηt {Γ(at ) + G(nt ) − zt } ,(5)11Because we assume a small open economy, the result is the same even if the income tax is

imposed not only on the export income but also on the rest of the income.

9where ηt is the shadow value for the natural resource associated with (1) and qt is
the shadow value for the foreign assets associated with (3).
The first order conditions are:12u′ (ct ) = qt ,(6a)(1 − τ y )qt ph′ (zt ) = ηt ,(6b)

Γ′ (at )ηt = (1 − τ a )qt ,

(6c)

q̇t
r−ρ=− ,qt

(6d)v ′ (nt )η̇t+ G′ (nt ) − ρ = − .
ηt
ηt

(6e)

Finally, the transversality conditions for the foreign asset and the natural resourceare:lim qt bt e−ρt = 0, and lim ηt nt e−ρt = 0.

t→∞

2.3

t→∞

(6f)

Equilibrium dynamics

This subsection characterizes the equilibrium paths. Let us define a competitive
equilibrium as follows.
Definition. A competitive equilibrium is a sequence of allocation, {ct , bt , nt , zt , at , kt }∞t=0 ,such that, given the initial conditions b0 , k0 , and n0 , and a set of prices {w, r, p}, therepresentative household’s utility is maximized, the firm’s profits are maximized, thegovernmental budget constraint is balanced every time, and all markets are cleared.

12

A considerable number of existing papers assume that private agents do not take the motion ofnatural resources into account when optimizing their choices. Instead, to focus on the temporary
impact of the environmental policies, we assume that there is no environmental externality; however,
we confirmed that the main result in Proposition 2 is not changed even if the reproduction of therenewable resource, G(nt ) is used for the externality and is not internalized.

10

We now turn to deriving the dynamic equations of this economy. First, fromequations (6a) and (6d), we obtain:13ċtu′ (ct )= − ′′(r − ρ).
ct
ct u (ct )(7)Assuming that r = ρ in the small open economy, the level of consumption is constantover time. Hereafter, we omit the subscript t from consumption, and denote theconstant level of consumption by c̄.
Next, solving (6c) for ηt and substituting (6a) into it, we obtain the shadow pricefor the natural resource:ηt =(1 − τ a )u′ (c̄).
Γ′ (at )(8)

Diﬀerentiating (8) with respect to time and substituting this equation into (6e), we
obtain:ȧt
Γ′ (at )=−atat Γ′′ (at )()
v ′ (nt )Γ′ (at )′r − G (nt ) −≡ F (nt , at , ct , τ a ),(1 − τ a ) u′ (c̄)(9)

where we note that r = ρ.
From equations (6b) and (6c), we obtain:ph′ (zt ) =1 − τa.
(1 − τ y )Γ′ (at )(10)Solving (10) for the harvested resource zt yields:
zt = z(at , τ a , τ y ),(11a)where the derivatives are given by:
Γ′′ (at )h′ (zt )∂zth′ (zt )∂zth′ (zt )
∂zt
=− ′
< 0,
=−
> 0, and y =< 0.
∂at
Γ (at )h′′ (zt )∂τ a(1 − τ a )h′′ (zt )∂τ(1 − τ y )h′′ (zt )Furthermore, because the functions h(zt ) and Γ(at ) satisfy the Inada conditions,z(at , τ a , τ y ) has the following characteristics:
lim z(at , τ a , τ y ) = ∞, and

at →0
13

lim z(at , τ a , τ y ) = 0.

at →∞

(11b)

The additive separability of utility function is critical for the movement of consumption; there-

fore, in Section 5 we make use of the non-separable utility function so that the level of consumption
is not constant over time.

11Substituting equation (11a) into the evolution of the natural resource (1), we
obtain the dynamic equation of the natural resource as follows:ṅt = Γ(at ) + G(nt ) − z(at , τ a , τ y ) ≡ H(nt , at , τ a , τ y ).

(12)

Equations (9) and (12) constitute the dynamic system for (at , nt ) of the small open
economy, given a steady-state level of consumption and the rates of the investment
subsidy and the export-income tax.

3

Steady-state equilibrium

This section analyzes the existence and the stability of the steady-state equilibrium.14
With the rate of time preference and the interest rate both being exogenously given
constants in the small open economy, we require r = ρ for our system to have afinite interior steady-state value for the marginal utility of consumption. Under theassumption, the level of consumption is constant over time as seen in (7).
Let us assume that an asterisk indicates the steady-state levels of the variables.
Taking account of F (n∗ , a∗ , c̄, τ a ) = 0 and H(n∗ , a∗ , τ a , τ y ) = 0 in the steady state,
given consumption and the environmental policies we draw the phase diagram.
At first, we consider the shape of the ȧt = 0 locus. In this case, the slope of theȧt = 0 locus is:datFn= − (< 0),dnt ȧt =0
Fa

(13)

where the respective derivatives are:15()
Γ′ (a∗ )v ′′ (n∗ )(r − G′ (n∗ ))′′∗Fn = ′′ ∗ G (n ) +(> 0), Fa = r − G′ (n∗ )(> 0).
′∗Γ (a )v (n )In addition, as the level of the natural resource approaches infinity (zero), the level

14

As Fullerton and Kim (2008) point out, stability analysis of the steady state is complicated dueto the concavity of the reproduction function G(nt ).
15

As shown later, it always holds that r > G′ in the steady state.

12

of the investment in the natural resource approaches zero (infinity).16 As depicted
in Figure 1, ȧt > (<)0 in region above (under) the ȧt = 0 locus.
[ Figure 1 around here ]Next, making use of H(n∗ , a∗ , τ a , τ y ) = 0, we can depict the ṅt = 0 locus as the
U-shaped curve in Figure 1. This is because, by totally diﬀerentiating (23b), givenc̄∗ , τ a , and τ y , we obtain:
Hn
dat
=
.
dnt ṅt =0 Ha

(14)

where
Hn = G′ (n∗ ), Ha = Γ′ (a∗ ) −∂z(·)(> 0).
∂aThat is, it can be shown that the ṅt = 0 locus has the negative (positive) slope if
G′ (n∗ ) < (>)0. Furthermore, because G(0) = 0 as the level of the natural resource
approaches zero, the level of the investment in the natural resource approaches afinite level, which satisfies z(a∗ , τ a , τ y ) = Γ(a∗ ). Instead, because G(∞) = −∞ asthe level of the natural resource goes to infinity, the level of the investment in thenatural resource approaches infinity. It can be confirmed that ṅt > (<)0 in the region
above (under) the ṅt = 0 locus.
Linearization of the dynamic equations (9) and (12) around the steady stateyields:

  




ȧt
Fa Fn
at − a∗at − a∗
 =

≡M
.
∗
∗
ṅt
Ha Hn
nt − nnt − n(15)

The trace and the determinant of this system are given by:
Tr(M) ≡ Fa + Hn = r(> 0),)(
Hn F n−.
Det(M) ≡ Fa Hn − Fn Ha = Fa Ha
Ha
Fa

(16a)
(16b)

Because our dynamic system involves one jumpable variable, at , and one predetermined variable, nt , the economy has a saddlepoint property around the steady-state
16

′

∗

′

∗

)
(n )
= (1−τv a(nAs confirmed later, the equation (9) in the steady state is given by r−GΓ′ (a∗ ))u′ (c̄∗ )where we assume that r > G′ (0). Because the level of consumption is constant, we can confirm
that n∗ → ∞ as a∗ → 0 and n∗ → 0 as a∗ → ∞.

13

equilibrium if the sign of the determinant is negative. We can obtain the following
lemma.
Lemma 1. When ñ < n∗ , the economy satisfies the saddle-path stability. Instead,when ñ > n∗ , the steady state satisfies locally determinacy if the following inequality
is satisfied.
Hn
Fn
<
,
Ha
Fa

(17)

Proof. Suppose that ñ < n∗ so that Hn < 0. In this case, from (16b) the sign ofdeterminant is negative, showing that the steady state has a saddlepoint stability.
Next, consider that ñ > n∗ such that Hn > 0. Using the condition (18), the determinant has the negative sign. We assume that this inequality holds in the following
analysis. ■From (13) and (14), the condition (18) states that the slope of the ȧt = 0 locus
is steeper than that of the ṅt = 0 locus at the steady state. It is depicted that thedotted curve represented by the ȧt = 0 locus crosses the ṅt = 0 as shown in Figure
1. Specially, substituting each terms into this inequality, we can show that()
v ′′ (n∗ )Γ′ (a∗ )∂z(·)
v ′ (n∗ )Γ′ (a∗ )′ ∗+Γ (a ) −.
r<(1 − τ a )u′ (c̄∗ ) v ′ (n∗ )Γ′′ (a∗ )
∂a∗

(18)

From Lemma 1, the system has one stable root and one unstable root. We denotethese eigenvalues as λ1 < 0 and λ2 > 0. These eigenvalues are:√√
Tr(M) + (Tr(M))2 − 4Det(M)
Tr(M) − (Tr(M))2 − 4Det(M)< 0, λ2 ≡
> 0.
λ1 ≡
2
2
(19)
Thus, the solution of the linearized system can be written as follows:n t = n ∗ + D 1 e λ1 t + D 2 e λ2 t ,(20a)

at = a∗ + D1 Υ1 eλ1 t + D2 Υ2 eλ2 t ,(20b)where the entities of the eigenvectors, Υs (s = 1, 2), are given by:( ′′ ∗( ′′ ∗
v ′′ (n∗ )(r−G′ (n∗ )) )v ′′ (n∗ )(r−G′ (n∗ )) )Γ′ (a∗ ) G (n ) +Γ′ (a∗ ) G (n ) +v ′ (n∗ )v ′ (n∗ )Υ1 ≡ ′′ ∗< 0, and Υ2 ≡ ′′ ∗
.
′
∗
′
∗
Γ (a )
λ1 + G (n ) − rΓ (a )
λ2 + G (n ) − r
14Note that λ1 < 0 and λ2 > ρ = r. Because of the transversality condition, it must
hold that D2 = 0. From (20a), we can show that D1 = n0 − n∗ . Thus, the stableadjustment path is given by:
nt = n∗ + (n0 − n∗ )eλ1 t ,(21a)

at = a∗ + Υ1 (n0 − n∗ )eλ1 t .

(21b)

Next, let us derive the intertemporal solvency condition for this economy. Substituting (11a) into (3) yields:ḃt = rbt + f (k) + (1 − τ y )ph(zt (at , τ a , τ y )) − c̄ − (1 − τ a )at − Z.

(22a)Linearizing the budget constraint (22a) around the steady state under the balanced
budget of the government (4) yields:
)
(
∗
a
y
′ ∗ ∂z(a , τ , τ )− 1 (at − a∗ ) + r(bt − b∗ ).
ḃt = ph (z )
∗
∂a

(22b)

Substituting (21a) and (21b) into (22b), we obtain the linearized solution of bt for an
initial stock of the foreign asset b0 .
bt = b∗ + {b0 − b∗ + (n0 − n∗ )Ω1 Υ1 } ert − (n0 − n∗ )Ω1 Υ1 eλ1 t ,where Ω1 is given by:

(22c)

∗

ph′ (z ∗ ) ∂z∂a(·)
∗ − 1
Ω1 ≡
< 0.
ρ − λ1

Because of the intertemporal solvency condition of the economy, the following
equality must hold:b0 − b∗ = −Ω1 Υ1 (n0 − n∗ ).

(22d)Then, we obtain the stable path for bt as follows:bt = b∗ − Ω1 Υ1 (n0 − n∗ )eλ1 t .

(22e)Note that because both Ω1 and Υ1 take negative values, the level of the foreign asset
moves in the opposite direction to the level of the natural resource along the stable
path.
15

Finally, in order to express the steady-state equilibrium, we consider the economy
starting at time Tj and corresponding to a policy set τja and τjy . Moreover, the viable
steady state is associated with the initial levels of the natural resource and the foreign
asset nTj and bTj , respectively, at time Tj . Taking account of ṅt = ȧt = ḃt = 0 with
the intertemporal solvency condition, a viable steady state j given by τja and τjy , (c̄∗j ,a∗j , n∗j , b∗j ) is determined as follows:4r − G′ (n∗j )
1
=
,
∗
∗
a
v ′ (nj )Γ′ (aj )(1 − τj )u′ (c̄∗j )(23a)

G(n∗j ) = z(a∗j , τja , τjy ) − Γ(a∗j ),(23b)

rb∗j + f (k) + ph(zj (a∗j , τja , τjy )) = c̄∗j + a∗j ,(23c)bTj − b∗j = −Ω1 Υ1 (nTj − n∗j ).

(23d)

The eﬀects of government policy

In this section, we analyze how policy changes aﬀect the economy. It is well known
that in a closed economy, a temporary policy change influences the economy but,once the policy is removed, the system gradually returns to the original steady-state
equilibrium. However, in small open economy models such as the present analysis, it
is shown that the economy does not return to its original steady-state equilibrium.17
Our main result is that when the rate of investment subsidy or the export-income
tax temporarily increases, the level of the natural resource in the new steady-state
equilibrium always decreases relative to its original level.
Suppose that the economy is initially in the steady-state equilibrium in whichτja = τ0a and τjy = τ0y . We denote this original steady state with the subscript 0:
n0 = n∗0 = N (c̄∗0 , τ0a , τ0y ),Nc̄ > 0, Nτ a > 0, Nτ y > 0.

(24a)a0 = a∗0 = A(c̄∗0 , τ0a , τ0y ), Ac̄ > (<)0 if G′ < (>)0, Aτ a > 0 if G′ < 0, Aτ y < 0 if G′ > 0.
(24b)
17

See, for example, Turnovsky (1997) and Schubert and Turnovsky (2002).

16

b0 = b∗0 = B(c̄∗0 , τ0a , τ0y ),(24c)c̄0 = c̄∗0 = C(τ0a , τ0y , b0 , n0 ), Cτ a < 0, Cτ y < 0, CbT > 0, CbT > 0.

(24d)Here, we assume the following condition to obtain the signs of the derivative (24d):18
)
(
∂z(·)
(
)
′ ∗
r
Γ
(a
)
−
′
∗
′ ∗
∂a
v (n )Γ (a )Γ′ (a∗ ) v ′′ (n∗ )
′′
∗
r<
+
+ G (n ) . (25)
(1 − τ a )u′ (c̄∗ ) (r − G′ (n∗ ) − λ1 )(r − λ1 ) Γ′′ (a∗ ) v ′ (n∗ )Note that this condition (25) corresponds to the stability condition (18) when λ1
′

∗

(n ))
< 1, this condition is stricter
approaches zero. Furthermore, as (r−λ1r(r−G)(r−λ1 −G′ (n∗ ))than (18).

4.1

Permanent eﬀects

In this subsection, we examine the eﬀects of permanent changes in environmental
policies on the natural resource in the long run. Conducting a comparative statics
analysis, we obtain the following.
Proposition 1. A permanent increase in the rate of the investment subsidy increases(decreases) the steady-state level of the natural resource if the following inequality issatisfied:τa< (>)
1 − τa

(

)
1
c̄∗
−1
,
β
σph(z ∗ )1h′′ hu′′ (c̄)c̄− 1 ≡ − ′ 2 (> 0), σ ≡ − ′(> 0) (26)β(h )u (c̄)

where 0 < β < 1 and σ > 0.
A permanent increase in the rate of the export-income tax increases (decreases)
the steady-state level of the natural resource if the following inequality is satisfied:
1−η
c̄∗
τy
<
(>)
×
,
1 − τyσa∗1−η ≡−

Γ′′ a∗
(> 0)
Γ′

(27)

Proof. See Appendix B. ■18

Using the following procedure, we derive the equations (24a) − (24d), where Tj = 0. From(23a) and (23b), we obtain (24a) and (24b). Furthermore, substituting (24b) into (23c) yields (24c).
Finally, incorporating (24c) into (23d), we obtain (24d). We derive the equations explicitly inAppendix A.

17

We now explain the intuition behind this proposition.19 Figures 2 and 3 illustrate
the eﬀects of the investment subsidy and the export-income tax, respectively. Notethat Tem (or Per) in these figures show the cases that the environmental policies
change temporarily (or permanently).
First, we examine the eﬀects of the investment subsidy policy where E0 is theoriginal steady state. By using the equation (23a), we can show how the downward
sloping ȧt = 0 locus moves as a result of the permanent increase in the rate ofinvestment subsidy:




∂a∗j
a∗j 
1 
 σCτ a

=
+

.
a
∗
a
∂τj ȧt =0, n∗ =constant 1 − η  c̄j
1−
τ }
|
{z
j
| {z }
#1

(28a)

#2

Note that Cτ a has a negative sign, which represents the negative impact of theinvestment subsidy policy on private consumption; that is, the decrease in the cost of
the investment due to the investment subsidy reduces the demand for consumption
goods.20 The equation (28a) consists of the indirect eﬀect that occurs through achange in consumption, given by the term #1, and the direct eﬀect, given by theterm #2. The term #1 takes a negative value and the term #2 takes a positive
value. Therefore, if the eﬀect of the term #2 dominates the term #1, the ȧt = 0
locus moves upward. Figure 2 shows this case.
19

Since the conditions (26) and (27) are complicated, it would be hard for readers to understandthe values given in (26) and (27). To help the understanding of readers, we now use some specificfunctional forms and parameter values in numerical examples of Section 5.1: β = η = 0.5, p = 1.25
and σ = 1.75 so that pσ ≃ 2.2 and 1−η
σ ≃ 0.29. As a result, each condition is given by:
(26):

τa
c̄∗
τy
c̄∗
<
(>)
,
(27):
<
(>)
1 − τa2.2h(z ∗ )1 − τy0.29a∗

If the rate of investment subsidy or export-income tax is small enough, then we can show that thepermanent increase in each rate increases the level of natural resource in the long run. In particular,
the numerical examples will give the case where the permanent increase in the rate of investment
subsidy or export-income tax, from τ0a = 0 (or τ0y = 0) to τ1a = 0.2 (or τ1y = 0.2) leads to the larger
level of natural resource as in Figures 2 and 3.
20

See the equation (A.9a) in Appendix A.

18

On the other hand, the permanent increase in the rate of the investment subsidy
always shifts the ṅt = 0 locus upward:
∂z(·)

∂a∗j
∂τja
(> 0).
=∂τja ṅt =0, n∗ =constant Γ′ (a∗j ) − ∂z(·)
∗

(28b)

∂aj

j

Taking account of (28a) and (28b), as the ṅt = 0 locus shifts upward, the extent of
the change of the ȧt = 0 locus determines the direction of the impact on the natural
resource. When σ and τ a take suﬃciently small values, then the eﬀect of the term
#1 relatively becomes weak and, thus, the ȧt = 0 locus moves upward, as shownby Figure 2. Indeed, when σ and τ a take suﬃciently small values, the level of thenatural resource increases. In contrast, when σ and τ a take suﬃciently large values,
then the eﬀect of the term #1 relatively becomes strong and, thus, the ȧt = 0 locus
moves downward. This decreases the level of the natural resource.
Moreover, we characterize changes in the foreign asset over time. From (23d),
when the level of the natural resource in the new steady-state equilibrium is greaterthan that in the initial period, the level of the foreign asset in the steady-state
equilibrium is smaller than that in the initial period and vice versa. That is, ifn∗j > (<)n∗0 under a policy set τja and τjy , it always holds that b∗j < (>)b∗0 . This
negative relationship between the foreign asset and the natural resource is depicted
in Figure 2(b).21 As a result, Figure 2(b) shows that an increase in the rate of
the investment subsidy decreases the level of the foreign asset from b∗0 to b∗1 wheren∗1 > n∗0 . This is because the increase in the investment, as a result of the subsidy,
decreases exports and, thus, the level of the foreign asset.22
Second, we examine the eﬀect of a permanent increase in the rate of the exportincome tax on the natural resource in Figure 3. Unlike the case of the investment
21

The negative relationship also applies to the case of the export-income tax rate, as shown in

Figure 3(b).
22Instead, suppose that the ȧt = 0 locus moves downward where the ṅt = 0 locus always moves

upward. In this case, the level of the natural resource decreases over time as a result of thepermanent increase in the rate of the investment subsidy. However, we omit this case to focus on
the interesting case.

19

subsidy policy, there is no direct eﬀect on the ȧt = 0 locus produced by the exportincome tax. Thus, any permanent increase in the rate of the export-income tax
moves the ȧt = 0 locus downward:∂a∗j
σa∗j Cτ y< 0.
=∂τjy ȧt =0, n∗ =constant (1 − η)c̄∗j

(29a)

j

On the other hand, the ṅt = 0 locus moves downward:
∂z(·)

∂a∗j
∂τjy
=
(< 0).
∂τja ṅt =0, n∗ =constant Γ′ (a∗j ) − ∂z(·)
∗

(29b)

∂aj

j

From (29a), the downward shift of the ȧt = 0 locus is smaller as the values of σand η are smaller. Then, a permanent increase in the rate of the export-income tax
increases the level of the natural resource. As Figure 3(a) shows, after the investmentin the natural resource initially jumps from E0 to I, the level of the natural resource
increases toward the new steady state, E1 .23 In this case, from (23d), the level of theforeign asset decreases from E0 to E1 in Figure 3(b).

4.2

Temporary eﬀects

In the previous subsection, we examined the impacts of the permanent environmental
policies on the natural resource. However, where environmental policies have been
applied in practice, such as in Indonesia and Malaysia, all environmental policies
have been implemented temporarily. As a result, it is more important to investigate
the eﬀects of temporary changes of the environmental policies on the natural resource
over time, as well as the long-run eﬀects.
In the present model, as confirmed in (7), the consumption growth rate is zero;that is, after the initial jump of consumption, the level of consumption is constantover time as long as the given conditions do not unanticipatedly change. When thegovernment announces the initial changes of the policy instruments from the original
23

Because the ȧt = 0 and ṅt = 0 loci move downward in the initial period, it can be easily

confirmed that the initial jump of the investment in the natural resource is always downward, asconfirmed in Figure 3(a).

20levels to new levels, the household in the small economy changes its consumption
behavior only at the initial time. In the case of temporary policy changes, after someduration, [0, T ], has passed, the policy instruments permanently return to the originallevel. However, under the assumption of perfect foresight, the level of consumption
does not go back to its original level after its initial jump because the household
can initially anticipate the policy change at time T ; that is, the consumption levelstays there permanently so as to sustain the intertemporal solvency condition. As
a result, the temporary policy changes have long-run impacts on the level of thenatural resource.
The eﬀects of the temporary environmental policies on the natural resource canbe summarized as follows.
Proposition 2. The temporary increase in the rate of investment subsidy or theexport-income tax always lowers the level of the natural resource in the long run.
Proof. See Appendix C. ■Let us explain the intuition of this proposition why the temporary policies decrease the level of the natural resource in the long run. The key element is the
direction of the initial jump in consumption, showing that both policies have negative impact on consumption in the initial period. In detail, in the case of theinvestment subsidy policy, the increase in the relative price of consumption commodity compared with the investment in the natural resource lowers the demand in
consumption, whereas in the case of the export-income tax, the decrease in income
lowers that. As a result, the initial jump of consumption is downwards and thereafter
the consumption stays there in the long run.24
Remembering that the level of consumption decreases due to these policies, let
us suppose that the economy initially stays at the steady-state equilibrium. Suppose further that government unanticipatedly raises the rate of investment subsidy
or export-income tax at the initial time, leading to the decrease in consumption at
24

The negative relation between these policies and consumption is shown by (A.9a) and (A.9b)

in Appendix A.

21

the initial period. When only the negative impact on consumption is considered,equation (9) states that the investment growth rate becomes positive. Namely, thecurrent investment is substituted with the future investment, which delays the investment in the natural resource such as the tree planting. Consequently, because
only the negative impact of consumption stays in the long run after the temporary
environmental policy is completed, the level of the natural resource decreases relative
to the original level.
Finally, we want to confirm the whole economy’s evolution by using the phase
diagram. At first, we consider the case of investment subsidy policy. We can show
how the ṅt = 0 and the ȧt = 0 loci move when the rate of investment subsidy
increases. Using (28b), we can show that the ṅt = 0 locus moves upward to ṅt =0 (t < T ) and then returns to its original position if the rate of the investment subsidy
policy returns to its original level. From (28a), if the indirect impact is dominated by
the direct eﬀect, the ȧt = 0 locus moves upward to ȧt = 0 (t < T ) at the initial time
as shown in Figure 2(a). However, even if the government returns the investment
subsidy to its original level, the ȧt = 0 locus does not go back to its original position
because the level of consumption remains constant after the initial jump. Therefore,
the indirect impact of the investment subsidy shown by #1 in (28a) remains evenafter the government policy is removed. This indicates that the ȧt = 0 locus moves
downward relative to its original position in the long run, as shown in Figure 2(a).
When an increase in the investment subsidy is temporary, the upward jump of
the investment in the natural resource is smaller than the jump when the policychange is permanent. That is, the economy jumps from E0 to I ′ at the initial time(see Figure 2(a)).25 After the jump, the level of the natural resource increases toward25

The size of the initial jump is aﬀected by the expected duration of the temporary policy. For

instance, when the duration of the policy change is short, the initial jump of investment in thenatural resource can be below the level of â. In this case, the level of the natural resource decreases
along an unstable path; when the rate of investment subsidy returns to the original level at time T ,the level of the natural resource further decreases along a stable path as in Figure 2(a). However,
we omit this case to focus on the interesting case. Alternatively, if the initial point of jump I ′ were22

I ′′ along the unstable path. When the economy reaches I ′′ , the level of the natural
resource begins to decrease; the economy is still on the unstable path. The economy
reaches I ′′′ when the government returns to the original policy at time T . After thispolicy reversion, the economy follows the stable saddle path to reach the new steadystate, E2 . In regard to the changes in the foreign assets, we can show the following.
Considering that n∗2 < nT , from (23d) it holds that b∗2 > b∗T . Figure 2(b) shows thatthe level of the foreign asset decreases from E0 toward L and, thereafter, it increases
through L′ . After the subsidy policy is removed at the point L′ , the level of theforeign asset increases from L′ to E2 .26
Next, let us focus on the case of the export-income tax rate. As confirmed in
the last subsection, when the rate of the export-income tax increases, the ṅt = 0
and the ȧt = 0 loci move downward. After the rate of the export-income tax policy
returns to its original level, the ṅt = 0 locus returns to its original position, while
the ȧt = 0 locus will be located below its original position. Hence, after the initial
jump of the investment from E0 to I ′ , the level of the natural resource increases
along the unstable path. However, after the rate of the export-income tax returns to
its original level, the level of the natural resource decreases over time. To the end,
the level of the natural resource in the new steady-state equilibrium shown by E2 islower than its original level in E0 . In Figure 3, we can confirm the movement in theforeign assets which correspond to the changes in the natural resource.

5

Discussion

In the last section, we obtained Proposition 2, which would be interesting beyond our
expectation because the temporary environmental policies always harm the renew-

above I, the economy would move upper right, which means that the economy would move along
the unstable path at any time. Therefore, the economy cannot follow the stable path at the period
T , implying that the economy cannot arrive at the steady state.
26

Because the level of consumption decreases in the long run, from (B.1) it holds that b∗2 > b0

under G′ < 0.

23

able natural resource. In this section, we further support the finding of Proposition
2 in two ways. First, we make use of the numerical examples to see the quantitative
impacts of environmental policies. Second, using the non-separable utility function
rather than the separable one (2), we suppose the more complicated inter-connection
between consumption and the natural resource.

5.1

Numerical examples

The purpose to use the numerical method is mainly to see the quantitative impacts ofenvironmental policies. In addition, our interest is to provide numerical confirmationof our results in Proposition 2, and furthermore Figures 2 and 3 mathematicallycorrect.27
At first, we specify the production functions and the utility functions. The production of the single homogeneous good is represented by the production process
f (k) = ξ1 k α and the production function with the harvested natural resource ish(zt ) = ξ2 ztβ where ξ1 > 0, ξ2 > 0, 0 < α < 1 and 0 < β < 1. The utility functions
are given by u(ct ) = c1−σ/(1 − σ) and v(nt ) = χ log(nt ) where σ > 0 and χ > 0.
t
In the natural resource sector, following Brander and Taylor (1997) and Elı́asson
and Turnovsky (2004), we use the following reproduction function:(nt ), ∆ > 0.
G(nt ) = ∆nt 1 −
n̄

(30)

Finally, the investment function in the natural resource is specified as:
Γ(at ) = ξ3 aηt , ξ3 > 0, 0 < η < 1.

27

We make use of Matlab 2014b.

24

(31)

The parameters under our simulations are summarized as follows:
Production parameters: α = 0.3, β = 0.5, ξ1 = 2, ξ2 = 1.5.
Taste parameters: σ = 1.75, χ = 0.8,
Natural sectors parameters: n̄ = 1.5, ∆ = 0.5, η = 0.5, ξ3 = 0.2.
Price and tax rates: p = 1.25, τ0a = τ0y = 0, r = (ρ =)0.15.
Initial values: b0 = 1, n0 = 1.407.
By setting ∆ = 0.5 and n̄ = 1.5, we have G′ (0.75) = 0 and G′′ = −1/n̄. In particular,
we assume that the natural resources moves in the range that G′ < 0, which meansthat from Lemma1, the economy always satisfies the saddle-path stability. The pricep = 1.25 supposes the economy that the export good is more expensive than thedomestic one. Finally, the original rates of taxes are zero; instead, when the policychanges, the tax rate is assumed to be τ1a = 0.2 (or τ1y = 0.2).
Figures 4(a), 4(b) and 5 show the quantitative eﬀects of the investment subsidy
and the export-income tax. Figure 4(a) depicts the permanent eﬀect of the investment subsidy. Figure 4(b) depicts the temporary eﬀect of the investment subsidy.
Figure 5 depicts the permanent and the temporary eﬀects of the export-income tax.28
Furthermore, notice that the signs such as E0 and I in these figures are the same as
those in Figure 2 and 3; instead, the square shows the position of initial economy
and the sign (X) shows the end point of the economy.
First, it is shown that Figures 4(a), 4(b) and 5 are qualitatively the same as
Figures 2 and 3.
Second, we can confirm the main findings in Proposition 2, that is, the level ofnatural resource at the new steady state E2 is less than the original level at E0 , whichmeans that the temporary increase in the rates of investment subsidy and the exportincome tax harms the renewable natural resource in the long run. Furthermore, we
can confirm the key downward jump of consumption that leads to Proposition 2.
28

We divided into Figure 4(a) and 4(b) because the points E1 in Figure 4(a) and I ′′ in Figure4(b) are very close so that the readers may be confused. We used other parameter set; however,this relationship was always seen.

25

More concretely, the level of consumption jumps downward from c∗0 = 5.6907 toc∗2 = 5.6903 in the case of investment subsidy, and from c∗0 = 5.6907 to c∗2 = 5.6514
in the case of export-income tax.
Finally, we can see the quantitative impacts of environmental policies. Whenthe rate of investment subsidy increases to τ1a = 0.2, from Figures 4(a) and 4(b) we
can show that n∗1 = 1.443 and n∗2 = 1.406. In particular, the temporary impact ofinvestment subsidy on the natural resource would be quantitatively small, that is,the distance between n∗0 and n∗2 is very close, which leads to the similar relationshipof the foreign assets, b∗0 = 1 and b∗2 = 1.001. Even if the rate of investment subsidy
further increases to τ1a = 0.5, each level of natural resource is given by n∗0 = 1.407,n∗1 = 1.478 and n∗2 = 1.403. In the case of export-income tax, Figure 5 shows thatn∗0 = 1.407, n∗1 = 1.468 and n∗2 = 1.405 at τ1y = 0.2; instead, when τ1y = 0.5, we canshow that n∗1 = 1.499 and n∗2 = 1.395.

5.2

Non-separable utility function

In the baseline model, we make use of the additive separable utility function given in(2) for the tractability, which may be critical to obtain the main findings. Therefore,
in this subsection we use a more general utility function as follows:∫ ∞U [0] =u(ct , nt )e−ρt dt.

(32)

0

In this case, from the first-order condition of consumption we can derive thefollowing:ct = c(q̄, nt ).

(33)

Because the level of natural resource is not fixed over time, the level of consumption
is not constant unlike the baseline model. However, importantly, the shadow value
for the foreign assets is still constant over time, that is, q̇t /qt = 0 under r = ρ.
Therefore, the extended model is the same as the baseline model in the sense thatthe shadow value qt is constant along time.
We derive the dynamic equation of the investment in the natural resource as26

follows:ȧtΓ′ (at )Non-separable utility:=−atat Γ′′ (at )()
un (c(q̄, nt ), nt )Γ′ (at )′r − G (nt ) −.
(1 − τ a )q̄
(34a)

We can see the constant shadow value q̄ in (34a), which expects that when thetemporary shock arises, the new steady state does not coincide with the original
steady state where the dynamic equation of the renewable natural resource is identicalto (1). In particular, it may be useful to rewrite (9) as follows:()
ȧt
Γ′ (at )v ′ (nt )Γ′ (at )′Separable utility:=−r − G (nt ) −.
atat Γ′′ (at )(1 − τ a )q̄

(34b)

We can confirm that the diﬀerence between (34a) and (34b) is un (c(q̄, nt ), nt )under the non-separable utility and v ′ (nt ) under the separable utility. Therefore,
noting that v ′′ (nt ) < 0, the following assumption of decreasing marginal utility leads
to the qualitatively identical dynamic behavior of the investment in the natural
resource under the constant shadow value q̄:()
unc (·)2 unn (·)ucc (·)∂un (c(q̄, nt ), nt )=− 1 (< 0).
∂n
ucc (·)
unc (·)2(35)

Note that the concavity of utility function leads to the negative sign of (35). Therefore, our main findings in Lemma 1 and Proposition 2 still hold. For instance, Fn
has a positive sign under (35), which leads to Lemma 1.29 That is, when ñ < n∗ , thesteady-state equilibrium has the saddlepoint stability; instead, when ñ > n∗ , we cansee the saddle-path stability under (18). Furthermore, following the similar procedure in Appendix C, we can obtain Proposition 2 under (35), that is, the temporary
environmental policies always harm the renewable natural resource in the long run
if the marginal utility of the natural resource decreases in the natural resource.

29

dat
The slope of the ȧt = 0 locus is dn= − FFna :t ȧt =0Fn =Γ′ (a∗ )Γ′′ (a∗ )(G′′ (n∗ ) +∂un (c(q̄, n∗ ), n∗ ) (r − G′ (n∗ ))
∂n
q̄

27

)

, Fa = r − G′ (n∗ ),6Conclusion

In this paper, we have presented a dynamic model of a small open economy with arenewable resource. Our main results are that when the rate of an investment subsidy
or an export-income tax temporarily increases, the level of the natural resource
always decreases in the long run. On the other hand, when permanent increases inthese policies are realized, the long-run eﬀects of the environmental policies on thenatural resource can be positive. For instance, permanent increases in the investment
subsidy or export-income tax can increase the level of the natural resource when therate of investment subsidy or export-income tax is plausibly set.
We consider that the findings in this paper have some important policy implications in this field. Governments in developing countries are interested in conservationof renewable resources; instead, they have not found the stylized environmental policies. In particular, the environmental policies conducted in Indonesia, Malaysia andthe Philippines did not lead to great results. Based on our findings, the reason thatthese policies was not successful lies in implementation term of environmental policies; that is, the short-term implementation of environmental policies does not bringus any success in the long run; on the contrary, it harms the renewable resources. As
given in Proposition 1, when government sets the plausible rate of investment subsidy or the export-income tax and permanently conducts the environmental policies,
the long-run level of renewable resource would increase.
Acknowledgments
We are grateful to Jun-ichi Itaya, Noritaka Kudoh, Kazuo Mino, Yasuhiro Takarada,
Syunsuke Managi, and seminar participants at The Research Institute of Economy,
Trade, and Industry (RIETI) and the Otaru University of Commerce.

Appendices
Appendix A.

28

Derivation of (24a) − (24d):
Using equations (23a) − (23d), we derive (24a) − (24d).30 First, from (23b), we canshow that:a∗j = Π(n∗j , τja , τjy ),(A.1)

where the respective derivatives are:
Πn ≡

∂a∗j
G′ (n∗j )> (<)0,=−∂z ∗ (·)∂n∗jΓ′ (a∗ ) − j ∗jif G′ < (>)0,∂aj∂zj∗ (·)∗∂aj∂τjaΠτ a ≡ a =∂zj∗ (·)∂τjΓ′ (a∗j ) − ∂a> 0,∂zj∗ (·)

∂a∗j
∂τjy
y
< 0.
Πτ ≡ y =∂z ∗ (·)∂τjΓ′ (a∗ ) − j ∗j∂aj

Next, substituting (A.1) into (23a) yields:n∗j = N (c̄∗j , τja , τjy ),(A.2)

where we can show that:
Nc̄ ≡

∂n∗j
1 u′′ (c̄∗j )
=
> 0,
∂c̄∗j
W u′ (c̄∗j )∂n∗j
Γ′ (a∗j )1) > 0,(
Nτ a ≡ a = −∗∂τjW (1 − τ a ) Γ′ (a∗ ) − ∂zj (·)∗
jNτ y ≡

j

∂aj

∂n∗j
1 Γ′′ (a∗j )=−Π y > 0,∂τjy
W Γ′ (a∗j ) τj

where W is:
(
)]
[
Γ′′
v ′′ Γ′∂z(·)
G′′v ′ Γ′
′
) r−
− (
−
Γ −
< 0.
W ≡
r − G′ Γ′ Γ′ − ∂z(·)(1 − τja )u′ v ′ Γ′′
∂a∗

(A.3)

∂a

Note that, from (18), the sign of M takes a negative value.

30

We consider that the initial time is time T in Appendix A. Thus, taking account of the initialtime zero rather than T , equations derived in Appendix A correspond to (24a) − −(24d).

29

Now, substituting (A.2) into (A.1) yields:a∗j = Π(N (c̄∗j , τja , τjy ), τja , τjy ) ≡ A(c̄∗j , τja , τjy ),where:Ac̄ ≡∂a∗j= Πn Nc̄ > (<)0,
∂c̄∗j

(A.4)

if G′ < (>)0,∂a∗jAτ a ≡ a = Πn Nτ a + Πτ a > 0, if G′ < 0,∂τ∂a∗jAτ y ≡ y = Πn Nτ y + Πτ y < 0, if G′ > 0.
∂τHence, equation (23c) can be rewritten as:
rb∗j + f (k) + ph(zj∗ (A(c̄∗j , τja , τjy ), τja , τjy )) = c̄∗j + A(c̄∗j , τja , τjy ).
Then, we can obtain the following:b∗j = B(c̄∗j , τja , τjy ).

(A.5)

where each derivative is given by:∂b∗j1
Bc̄ ≡ ∗ =
∂c̄j
r

{

)
}
(
z ∗ (·)′ ∗ j
Ac̄ 1 − ph (zj ) ∗ + 1 ,
∂aj

{
(
)
}
∂zj∗ (·) ∂zj∗ (·)′ ∗−ph (zj ) Aτ a+
+ Aτ a ,
∂a∗j
∂τja
{
(
)
}
∂b∗j
∂zj∗ (·) ∂zj∗ (·)
1′ ∗−ph (zj ) Aτ y+
+ Aτ y .
Bτ y ≡ y =
∂τj
r
∂a∗j
∂τjy

∂b∗j
1
Bτ a ≡ a =∂τjr(A.6a)(A.6b)(A.6c)

Finally, substituting (A.5) and (A.2) into (23d) yields:}{
B(c̄∗j , τja , τjy ) − bTj = Ω1 Υ1 nTj − N (c̄∗j , τja , τjy ) .

(A.7)

Thus, consumption can be determined by:c̄∗j = C(τja , τjy , bTj , nTj ),

(A.8)

∂c̄∗j
Bτ a + Ω1 Υ1 Nτ a
< 0,
=−
a
∂τj
Bc̄ + Ω1 Υ1 Nc̄

(A.9a)

where we can show that:Cτ a ≡

30∂c̄∗jBτ y + Ω1 Υ1 Nτ y
Cτ y ≡ y = −< 0,∂τjBc̄ + Ω1 Υ1 Nc̄

(A.9b)

CbT ≡

∂c̄∗j
1
=
> 0,
∂bTj
Bc̄ + Ω1 Υ1 Nc̄

(A.9c)

CnT ≡

∂c̄∗j
Ω1 Υ1
=
> 0.
∂nTj
Bc̄ + Ω1 Υ1 Nc̄

(A.9d)The sign of the denominator in (A.9a) − (A.9d) is positive:
Bc̄ + Ω1 Υ1 Nc̄
)
(



∂z(·)
(
)
′


′ ∂z(·)
′
′′
′
r
Γ
−
1 − ph ∂a∂a1G′ − Γ G′′ + v (r − G ) ,=1 − Nc̄rΓ′′v′(r − G′ − λ1 )(r − λ1 ) Γ′ − ∂z(·)∂awhere Nc̄ > 0.
Finally, using (23a), we can show that:
Bc̄ + Ω1 Υ1 Nc̄
)
(



(
)
′ − ∂z(·)
∂z(·)


′
′
′
′
′′
′
r
Γ
1 − ph ∂a
∂a
1
v
Γ
Γ
v
(r
−
G
)
′′
 > 0,r −=1 − Nc̄−G+r
(1 − τ a )u′ Γ′′v′(r − G′ − λ1 )(r − λ1 ) Γ′ − ∂z(·)∂a(A.10)

where we make use of the condition (25).
Appendix B.
Taking account of the initial jump in consumption, diﬀerentiating (24a) withrespect to τ a yields:∂n∗= Nc̄ Cτ a + Nτ a ,∂τ a−Nc̄ Bτ a + Nτ a Bc̄
,=Bc̄ + Υ1 Ω1 Nc̄

(B.1)

where the sign of Bc̄ + Υ1 Ω1 Nc̄ is positive, as in (A.10). Arranging the numerator
on the right-hand side in (B.1) yields:
1()−Nc̄ Bτ a + Nτ a Bc̄ =∂zrW Γ′ − ∂a

{ ′′
}
u ∂z
Γ′
′ ′
(ph Γ − 1) −.
u′ ∂τ a1 − τaFinally, making use of (10) and τ y = 0, equation (B.2) can be rewritten as:()
Γ′ u′ h′′1
h′ u′′a)(τ − ′′ ′ .
−Nc̄ Bτ a + Nτ a Bc̄ =∂z h′′ u′u h(1 − τ a )rW Γ′ − ∂a
31

(B.2)

(B.3)

This shows the relationship between n0 and n∗1 . Substituting (10) into the parenthesisin (B.3) obtains the result in (26).
Next, we examine the eﬀects of an increase in the rate of the export-income tax
on the natural resource. Conducting the static comparative analysis in (23b), we
obtain:∂n∗= Nc̄ Cτ y + Nτ y ,∂τ y(−Nc̄ Bτ y + Nτ y Bc̄ )=.
Bc̄ + Υ1 Ω1 Nc̄
The numerator on the right-hand side in (B.4) can be rewritten as:{}
1∂z u′′
Γ′′′ ′()−Nc̄ Bτ y + Nτ y Bc̄ =(ph Γ − 1) − ′ .
∂z ∂τ yu′ΓrW Γ′ − ∂a

(B.4)

(B.5)

Then, substituting (10) and τ a = 0 into (B.5), we can show the eﬀect of a permanent
increase in the rate of the export-income tax on the natural resource as follows:
( y
)
∂z
τ
u′′
Γ′′ u′()(B.5) =
−
.
(B.6)
∂z
rW Γ′ − ∂au′ ∂τ y 1 − τ y Γ′ u′′Appendix C.
In this appendix, we show the temporary impacts of the environmental policies
on the natural resource. Let us suppose that the government announces changes ofthe policy instruments from the original levels τ0a and τ0y to τ1a and τ1y until time T ,which thereafter revert permanently to their original levels. Under the assumptionof perfect foresight, the households can initially anticipate the policy change at time
T . This implies that new information arrives only at time zero. Hence, consumption
jumps to the new steady state at the initial time zero and remains there permanently.
We divide the dynamics into two separate dynamics, Periods 1 and 2, as follows.
Period 1: 0 ≤ t < T
During Period 1, the economy moves along the unstable transitional path:
nt = n∗1 + D1 eλ1 t + D2 eλ2 t ,(C.1a)

at = a∗1 + D1 Υ1 eλ1 t + D2 Υ2 eλ2 t ,(C.1b)

bt = b∗1 − D1 Ω1 Υ1 eλ1 t − D2 Ω2 Υ2 eλ2 t ,(C.1c)32∂z ∗ (·)

where Ω2 is defined byph′ (z ∗ ) ∂a∗ −1.
ρ−λ2

In addition, the steady-state levels of eachvariable are determined by:n∗1 = N (c̄∗1 , τ1a , τ1y ) = N (C(τ1a , τ1y , b0 , n0 ), τ1a , τ1y ),(C.2a)a∗1 = A(c̄∗1 , τ1a , τ1y ) = A(C(τ1a , τ1y , b0 , n0 ), τ1a , τ1y ),(C.2b)b∗1 = B(c̄∗1 , τ1a , τ1y ) = B(C(τ1a , τ1y , b0 , n0 ), τ1a , τ1y ),(C.2c)c̄∗1 = C(τ1a , τ1y , b0 , n0 ).

(C.2d)

Notice that the initial stocks of the natural resource and the foreign asset are n0 and
b0 , respectively, and the rates of the investment subsidy and the export-income tax
are τ1a and τ1y , respectively. Furthermore, we must note that c0 ̸= c̄∗1 because thelevel of consumption jumps at the initial time.
Period 2: T ≤ t
During Period 2, the economy follows the stable path defined by:31′

nt = n∗2 + D1′ eλ1 t ,(C.3a)′

at = a∗2 + D1′ Υ1 eλ1 t ,(C.3b)′

bt = b∗2 − D1′ Ω′1 Υ′1 eλ1 t .

(C.3c)The steady-state levels of n∗2 , a∗2 and b∗2 are determined by:n∗2 = N (c̄∗2 , τ0a , τ0y ) = N (C(τ0a , τ0y , bT , nT ), τ0a , τ0y ),(C.4a)a∗2 = A(c̄∗2 , τ0a , τ0y ) = I(C(τ0a , τ0y , bT , nT ), τ0a , τ0y ),(C.4b)b∗2 = B(c̄∗2 , τ0a , τ0y ) = B(C(τ0a , τ0y , bT , nT ), τ0a , τ0y ),(C.4c)c̄∗2 = C(τ0a , τ0y , bT , nT ).

(C.4d)

Note that the level of consumption does not change c̄∗1 = c̄∗2 because the household
anticipates the removal of the policy under the assumption of perfect foresight.

31

Because the initial conditions are diﬀerent in Period 1 and Period 2, the values of D1 , Ω1 , Υ1and λ1 are diﬀerent from D1′ , Ω′1 , Υ′1 and λ′1 ; however, the fundamental forms and signs are the
same.

33

For simplicity, let us denote the policy changes as follows:τ1a − τ0a ≡ dτ a , τ1y − τ0y ≡ dτ y(C.5)

Furthermore, approximating the steady-state changes with the diﬀerentials, we canshow that:n∗2 − n∗1 ≡ N (c̄∗2 , τ0a , τ0y ) − N (c̄∗1 , τ1a , τ1y ) = −Nτ a dτ a − Nτ y dτ y ,(C.6a)n∗1 − n∗0 ≡ N (c̄∗1 , τ1a , τ1y ) − N (c̄∗0 , τ0a , τ0y ) = Nc̄ (Cτ a dτ a + Cτ y dτ y ) + Nτ a dτ a + Nτ y dτ y ,(C.6b)a∗2 − a∗1 ≡ A(c̄∗2 , τ0a , τ0y ) − I(c̄∗1 , τ1a , τ1y ) = −Aτ a dτ a − Aτ y dτ y ,(C.6c)a∗1 − a∗0 ≡ A(c̄∗1 , τ1a , τ1y ) − A(c̄∗0 , τ0a , τ0y ) = Ac̄ (Cτ a dτ a + Cτ y dτ y ) + Aτ a dτ a + Aτ y dτ y .
(C.6d)
Please note that the equality c̄∗1 = c̄∗2 holds.
Finally, from (C.6a) and (C.6b), we can characterize the eﬀects of a temporary
policy change on the natural resource as follows:n∗2 − n∗0 = Nc̄ Cτ a dτ a , n∗2 − n∗0 = Nc̄ Cτ y dτ y .

(C.7)

where Nc̄ > 0, Cτ i < 0, and Cτ y < 0. The level of the natural resource in the newsteady state is lower than the original level when the rate of investment subsidy or
the export-income tax temporarily increases. Because the results in Proposition 2
are derived by only the steady-state diﬀerences in (C.6a) and (C.6b), we must noticethat those are irrespective of the constants D1 , D2 and D1′ which aﬀect the movementof variables along time.32

32

We define the constants D1 , D2 and D1′ where we note that these constants are irrespective ofour main results in Proposition 2. By setting t = 0 in (C.1a) and (C.1b), these equations can be
rewritten as:−(n∗1 − n0 ) = D1 + D2 ,Next, using (C.1a), (C.1b), (C.3a), and (C.3b) at t = T , the matching conditions on the natural34

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37

Figure 1: Phase diagramatȧt = 0ȧt = 0ṅt = 0

a∗j

n̂

n∗j

nt

38

Figure 2: The eﬀects of an increase in the rate of the investment subsidy
atII′

Per: ṅt = 0(0 < t)Tem: ṅt = 0(t < T )

E1

a∗1

t>T

I ′′

t<T
ṅt = 0(t = 0)
Tem: ṅt = 0(t > T )

â
a∗0
a∗2
aT

E0
E2

Per: ȧt = 0(0 < t)Tem: ȧt = 0(t < T )
t<T
t>T

I ′′′
ȧt = 0(t = 0)

Tem: ȧt = 0(t > T )

(a)
bt
b∗2
b∗0

n∗2
E2

bT

n∗0 nT

nt

E0

L′

L

b∗1
(b)

n∗1

n∗2

n∗0 nT

E1

n∗1

39

nt

Figure 3: The eﬀects of an increase in the rate of the export-income tax
atṅt = 0(t = 0)
Tem: ṅt = 0(t > T )

E0

a∗0
a∗2
aT

I ′′

E2
I′

ȧt = 0(t = 0)
t>T

t<T

I

E1

a∗1

Per: ȧt = 0(0 < t)Tem: ȧt = 0(0 < t)

(a)
bt

n∗2

b∗2
b∗0

E2

n∗0

n∗1

nT

nt

E0

bT

L

b∗1

(b)

Per: ṅt = 0(0 < t)Tem: ṅt = 0(t < T )

E1

n∗2

n∗0

n∗1

nT

40

nt

Figure 4(a): The investment subsidy
(23a)

0.15
a

I

E1

a1
0.13
(23b)
(23b)

0.11
a0

E0
(23a)

1.4

1.42

1.43

1.44

E0

b

1

1.41

0.98
E1
0.96
1.4

n0 1.41

1.42

1.43

n
41

1.44 n1

Figure 4(b): The investment subsidy
I’
I’’
0.13
a

t<T T<t
(23b)

0.11
a2

E0

I’’’
(23a)

E2
(23a)

1.4

1.41

1.42

1.43

1.44

E2 E0

b

b2
1
b0

0.98

L’
L

0.96
1.4

(23b)

n2 n0 1.41 nT

1.42

1.43

n
42

1.44

Figure 5: The export-income tax
0.11

(23b)

E0
E2

I’’

a

I’

(23a)

0.09
(23a)

I
a1

E1
(23b)

1.4

1.42

1.44

1.46

1.48

b

b2
E0
b0 E2
0.96
L
E1

b1
1.4 n2 n0

1.42

nT

1.44

n
43

1.46

n1

1.48