Discussion Papers In Economics
And Business

Environmental discounting in a small open
economy with a renewable resource
Koichi Futagami and Yasuhiro Nakamoto
Discussion Paper 10-22

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEnvironmental discounting in a small open
economy with a renewable resource
Koichi Futagami and Yasuhiro Nakamoto
Discussion Paper 10-22

July 2010

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEnvironmental discounting in a small open
economy with a renewable resource
Koichi Futagami∗ and Yasuhiro Nakamoto†
July 16, 2010

Abstract: We construct a small open economy model with a renewable resource.
Households have an endogenous time preference rate that depends on the level of therenewable resource in the domestic economy. Although households know that thedegree of own patience depends on its resource, we assume that households believethat they cannot control the motion of the aggregate renewable resource. This isbecause they think that their impact is negligible so that there exists an externality
in the form of the patience of the households. Based on this framework, we analyze
the dynamic character of the steady state and show that the equilibrium path may
be indeterminate. We next examine the welfare eﬀects of tax policies. Finally, weinvestigate socially optimal tax policies.
Key words: Endogenous time preference rate; Indeterminacy; Renewable resources;
Optimal tax policy
JEL classiﬁcation: F41; H21; Q28∗

Faculty of Economics, Osaka University, 1–7 Machikaneyama, Toyonaka 650–0043 Japan. E-

mail address: futagami@econ.osaka-u.ac.jp; Tel: +81–6–6850–5266.
†

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

Higashi-ku, Fukuoka, 813-8503, Japan. E-mail address: nakamoto@ip.kyusan-u.ac.jp; Tel: +81–
92–673–5228.

1

1Introduction

Exportable renewable resources represented by timber and ﬁsheries are very valuable
for many economies. The International Trade Statistics Yearbook (2004) provides
data on these exportable resources. For example, in Finland, one of the most important exportable resources is lumber, which earned $1,838 million in 2004. In Chile,marine products are important exportable commodities that earned $2,000 million
in 2004, while the marine products exports of Thailand earned $2,600 million.1 The
existence of such renewable resources cannot be ignored because these resources occupy an important position in those economies. In fact, it is indispensable that small
economies harvest their natural resources to acquire foreign exchange. Table 1 shows
data on the exports of Finland, Portugal, and Sweden from their forestry sectors,
with gradual increases in the volume of their exports of wood fuel, roundwood, andindustrial roundwood.
Can the level of harvested natural resources in such economies continue to increase
indeﬁnitely? To answer this question, we have to consider how people evaluate thelevel of natural resources in the future. If people care about the present living more,
the level of harvested natural resources increases and then these natural resources
are further exported abroad. On the other hand, if they care about the future living
more, they may try to preserve the level of natural resources by refraining fromharvesting or by investing in those natural resources. Then, the exported quantitiesof these natural resources will decrease in the short term. This means that thedegree of their time preference, or patience, largely inﬂuences the level of harvestingof natural resources in the economy.2 Hence, we must pay attention to people’s time1

Elı́asson and Turnovsky (2004) use the examples of Iceland and New Zealand as follows: An

obvious example is that of Iceland where in the year 2000 the Fish and Fish Processing sector
was around 10% of GDP, having been 15% just a decade earlier...Another example is New Zealand,
which has developed a substantial forestry sector, with much of the timber being harvested for export.
2

For example, Field (1994) shows as an example that the present value of the future valuationis largely aﬀected by the discount rate. At an annual discount rate of 1%, the discounted present
value of obtaining $100 in a century’s time is $36.78, while at an annual discount rate of 6%, the2preference when considering the economic performance of an economy dependent on
natural resources.3
Conversely, natural resources can aﬀect people’s time preference. In the behavioral evidence, for instance, Viscusi et al. (2008) examine the relation between time
preference rate and water quality by using logit model and mixed logit model. They
conclude that visitors to water bodies have low rates of discount, while those who do
not visit have high rate of discount and low valuations of water quality. It concludes
that people who are familiar to natural resources have low rates of discount.
Alternatively, in the current paper, we intuitively consider that when the natural
resource endowment of the economy dwindles, people may begin to worry aboutsuch future decreases in their natural resources. In other words, people become less
impatient when the volume of their natural resources diminishes. Based on the idea,
households know that the degree of own patience depends on the aggregate natural
resource in the domestic country; however, we assume that households believe thatthey cannot control the movement of the aggregate natural resource because they
think that their impact is negligible.4 Because of the negligibility, households take it
as given when maximizing their behavior. It means that there exists an externality
produced by natural resources on the patience of the households.5 We call thischaracter a discount externality in the present paper.
present value is only $0.29
3Karp (2005) examines the role of hyperbolic discounting and concludes that the higher the rateof discount, the lower is the level of abatement.
4This argument is similar to those in many papers which incorporate consumption externalities

and capital ones (e.g., Futagami and Shibta (1997) and Liu and Turnovsky (2005)). For instance,
one example of this kind of problem is the consumption externality; a household’s utility function
depends on the average level of consumption in whole economy as well as private consumption.
Because households cannot control the aggregate level of consumption, a sequence of its averagelevel is exogenously given when the maximization of utility is carried out.
5

Lines (2005) incorporates an environmental externality into individuals’ patience where theenvironmental externality is a stock variable as in our setting (see page 353 in Lines (2005)). He
examines the existence and local stability of an equilibrium in an overlapping generations model.

3

We construct a small open economy model with a renewable natural resource byincorporating an endogenous time preference rate that depends on the whole level ofthe natural resource in the domestic economy. The evolution of the renewable natural
resource in the economy has the following characters. First, the renewable resourceis harvested in the small economy. Second, it is supposed that all of the harvested
natural resource is exported. Third, people can invest in the natural resource toincrease its level of output.
Based on this framework, we may summarize the main results of our analysis asfollows. First, we explore the dynamic characters of the equilibrium.6 It is shownthat there is a unique steady state. However, the equilibrium path converging to
the steady state is not uniquely determined. There can be multiple equilibrium
paths, that is, the stability of the steady state can exhibit indeterminacy.7 Second,
we analyze the eﬀects of tax policies on the welfare level. For the policy maker,
it is important to maximize welfare by imposing tax policies in an economy withexternalities. We cannot calculate these welfare eﬀects mathematically because theeconomy does not follow a unique stable transitional adjustment path. Hence, we
6

Incorporating the renewable resources in endogenously growth models, many researchers pay

attention to whether sustainable growth is achievable. For example, Agihon and Howitt (1998),
Ayong Le Kama (2001) and Wirl (2004) present simple growth models incorporating renewable
resources as an input in the production function. Elı́asson and Turnovsky (2004) examine impactsof some shocks such as technological improvement on economic performance in a small open economy
with a production externality. López, et al. (2007) show that there is no unique relationship between
the endowment of natural resources and the rate of economic growth when the engine of the growthis human capital accumulation.
7

So far, many researchers have examined the indeterminacy of the steady state in dynamic

general equilibrium models. For example, Meng (2006) shows how the indeterminacy of the steadystate arises in one sector model by incorporating an endogenous time preference rate that depends
on the average levels of consumption and the capital stock. Weder (2001) and Bian and Meng (2004)
examine the dynamic character of equilibrium at the steady state in a small open economy. Bianand Meng (2004) in particular make use of the endogenous time preference rate that depends on the
average level of consumption. Itaya (2007) shows that the indeterminacy arises in an endogenous
growth model with environmental externalities.

4make a comparison of the welfare levels between the steady states with diﬀerent
tax rates. We show that the welfare level of the steady state with a lower rateof a consumption tax is higher than that of the steady state with a higher rate of
consumption tax, whereas the welfare level of the steady state with a lower rate of atax on investment in the natural resource is lower than that of the steady state witha higher rate of tax on the investment in the natural resource. Finally, we examine
the socially optimal tax policies.
The remainder of the paper is structured as follows. Section 2 describes themodel. Section 3 examines the eﬀects of tax policies on the welfare level. Section 4
characterizes the socially optimal tax policies. Finally, Section 5 concludes.

2

The basic framework

2.1

The model

Suppose that there is a continuum of identical, inﬁnitely lived households in a small
open economy in which the total population, I is constant over time and its size
is normalized to unity. This economy consists of ﬁrms and a government as well
as households. The economy is endowed with a stock of a renewable resource. We
assume that the level of the renewable resource held by a household shows nt , while
the level of the renewable resource held by all households in the domestic economy
expresses Nt where subscript t stands for time. The renewable resource held by ahousehold evolves according to the following equation:ṅt = Γ(it ) + G(nt ) − zt ,(1)

where it is the investment in the natural resource, Γ(·) represents the eﬃciency function of investment, and zt is the rate of harvest for an input for production. We
suppose that G(nt ) denotes the reproduction function of the renewable natural resource held by the household.8 Following some existing papers, this function has an
8

Because all households are assumed to be identical, the investment function and the reproduc-tion function do not diﬀer among households.

5

inverted U shape with G(0) = G(n̄) = 0, where n̄ is the carrying capacity of the natural resource and is the level at which its growth ceases.9 Further, the reproduction
function G(nt ) is supposed to be a strictly concave function (i.e., G′′ < 0). Thus,
there is a unique value n̂ at which G′ (n̂) = 0, where n̂ expresses the level of therenewable resource providing the maximum sustained yield.
The level of the natural resource can be increased by investing a ﬁnal good according to the eﬃciency function Γ(it ). This function Γ(it ) : ℜ+ → ℜ+ is twice
continuously diﬀerentiable, strictly increasing, strictly concave with respect to theinvestment it , and satisﬁes the Inada conditions, that is, limit →0 Γ′ (it ) = ∞ andlimit →∞ Γ′ (it ) = 0. As for the investment it , López et al. (2007) mention thatinvestments in natural resources are supposed to involve tree planting, ﬁsh replenishment including aquaculture investments, protection or cleaning-up of ecosystems,
soil protection including terracing drainage, and agricultural fallowing.
In this economy, there are two ﬁnal goods. One is produced by physical capital and labor according to the neoclassical production function per capita, f (k).
The other is produced by using the harvested natural resource according to the production function h(zt ). The production function h(·) is strictly increasing, twice
continuously diﬀerentiable, concave with respect to the harvested natural resource,
and furthermore satisﬁes the Inada conditions. We assume that all production of thecommodity is exported. Thus, the households in this economy do not consume anyof their natural resources such as lumber or ﬁsheries.10 Then, the preference of the9See, for example, Ayong Le Kama (2001), Elı́asson and Turnovsky (2004), Koskela et al. (2002),

López et al. (2007), and Wirl (2004).
10

When households do consume export commodities, their preference is written as:∫ ∞U=[u(ct ) + v(nt ) + w(export goods)]e−Θt dt,
0where w(·) represents the utility function. In this case, lifelong utility of households is diﬀerent
from that in the present paper. However, even if we assume that the households consume some ofthe export commodities, the essence of this model is not changed. This is because, in a small open
economy, the relative price of consumption commodities can be exogenously given.

6households is given by:∫ +∞U=[u(ct ) + v(nt )] exp[−Θt ]dt,(2)0where u(·) and v(·) represent the instantaneous utility functions of private consumption ct and the natural resource nt , respectively. The households are assumed toobtain higher utility as the stock of the natural resources held by respective households increases. In other words, they feel happy when the level of own natural
resources held by the household increases. Furthermore, Θt stands for the integral
value of the instantaneous rate of time preference from the initial time to the current
time. 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.
Next, we consider the endogenous discount rate, which is deﬁned by:∫ tdΘt= ρ(Nt ), Θ0 = 0,Θt ≡ρ(Nv )dv,
dt
0

(3)

where ρ(Nt ) : ℜ+ → ℜ++ represents the instantaneous time preference rate. We
assume that this time preference rate depends on the aggregate level of the renewable
resource in the domestic economy. In addition, we consider that the households
become more patient as the natural resource stock decreases. This is because thehouseholds come to fear extinction of their natural resource in the future. The
function ρ(·) has the following properties:
ρ(Nt ) > 0,∂ρ(Nt )
∂ 2 ρ(Nt )
> 0,
> 0,
∂Nt
∂Nt2

lim ρ(Nt ) = ρ(N̄ ), and, lim ρ(Nt ) = ϵ,
Nt →0

Nt →N̄

(4a)(4b)

where ϵ and ρ(N̄ ) are the lower bound and the upper bound of the discount function,
respectively. Assumption (4a) shows that the discount function has the character ofincreasing marginal impatience with respect to the natural resource, implying thatas the level of the natural resource is smaller, the rate of the time preference is small(i.e., the households become more patient). Assumption (4b) shows that there exist
lower and upper bounds of the discount function so that the rate of time preference7

takes positive values, which implies that the optimization problem of the households
satisﬁes the transversality conditions.
The world interest rate r is exogenously given and is constant for a small open
economy. Perfect competition prevails in the economy. Because the total numberof the population in this economy is unity, the aggregate level of capital stock isshown by K. Thus, the proﬁt maximizing conditions are given by r = f ′ (k) andw = f (k) − kf ′ (k), where k is the level of the domestic capital stock per capita,
and w is the wage rate. Therefore, the level of the domestic capital stock k becomes
constant. Furthermore, we suppose that the renewable resource is harvested byhouseholds. Then, by making use of commodities produced by the domestic capitalstock f (k), the households in the economy have options for either consumption or
investing in the renewable resource. Then, the accumulation of the foreign assets
held by the domestic household ḃt is given by:ḃt = rbt + (1 − τy )(f (k) + ph(zt )) − (1 + τc )ct − (1 + τi )it + T,

(5)

where p is the relative price of the harvested natural resource measured by the priceof consumption goods. For the small open economy, the relative price p is assumed
to be constant.
The government has three kinds of tax policies. The government imposes a
consumption tax, an investment tax, and an income tax on the households. Let τcbe the tax rate on the consumption, τy the tax rate on the income, and τi the tax
rate on the investment in the natural resource. We assume that these tax rates areconstant over time. The government is assumed to keep to the following balanced
budget constraint:
I × T = I × {τy (f (k) + ph(zt )) + τi it + τc ct }.

(6)This equation means that the government collects tax revenue from domestic households and returns it to the households by lump-sum transfers where I shows the total
number of domestic population.
We now consider the maximizing problem of the representative household ina small open economy. In the present paper, we assume that households know
8that their time preference depends on the aggregate level of the renewable resource;
however, the households believe that they cannot control the motion of the aggregate
renewable resource, as they think that their impact is negligible.11 Hence, when thehouseholds optimize, they take it as given due to their negligibility. We call thisexternality a discount externality in this paper.12 Maximizing the discounted sumof the lifelong utility (2) subject to (1) and (5) leads to the following ﬁrst-orderconditions:u′ (ct ) = (1 + τc )qt ,(7a)(1 − τy )qt ph′ (zt ) = ηt ,(7b)

ηt Γ′ (it ) = (1 + τi )qt ,

(7c)

q̇t
+ Θ̇t ,
qt

(7d)

r=−

v ′ (nt )η̇t+ G′ (nt ) = − + Θ̇t ,
ηt
ηt

(7e)

where ηt and qt represent the costate variables associated with equations (1) and(5), respectively. In particular, note that the eﬀects of the natural resource on thetime preference rate are not included in the ﬁrst-order conditions due to the discountexternality.
Equation (7a) equates the marginal utility of consumption to the shadow valueof the foreign assets. Equation (7b) equates the marginal value of the harvested
natural resource zt to the shadow value of the natural resource ηt . Equation (7c)equates the marginal eﬃciency of investment in the natural resource to the cost of theinvestment. Equations (7d) and (7e) show the arbitrage conditions in this economy.
Equation (7d) is the arbitrage condition, which equates the rate of return on the11This is a similar argument in the existing papers with consumption externalities and capital

ones; a household’s utility is aﬀected by the average levels of consumption or capital as well asprivate consumption (e.g., Futagami and Shibata (1998) and Liu and Turnovsky (2005)). Thesepapers assume that households believe that they cannot inﬂuence the aggregate consumption or
the aggregate capital stock so that their average levels are exogenously given when they optimize.
12

Lines (2005) incorporates the environmental externality in the discount function where theenvironmental externality is a stock variable like that in our setting.

9foreign assets r to the rate of return on consumption. Equation (7e), which is the key
equation in the present paper, represents the arbitrage condition of the investment in
the natural resource. This equation (7e) equates the marginal beneﬁt of the natural
resource to the rate of return on investment in the natural resource. Please note thatthe households are assumed to make their decision without considering the degree
to which their patience depends on the whole level of the natural resource stock.
Finally, the next two equations are the transversality conditions for the foreign
assets and the natural resource, which are satisﬁed by assumption (4b):lim qt bt e−Θt = 0 and lim ηt nt e−Θt = 0.

t→∞

2.2

t→∞

(7f)

Equilibrium

This subsection shows that there exists a unique steady state and characterizes theequilibrium paths. In particular, we show that the equilibrium path of this economy
can be indeterminate, although the steady state is uniquely determined. Firstly,
because the total number of households is unity (i.e., I = 1), the renewable natural
resource is shown by nt = Nt . Hence, we can rewrite that the rate of the endogenous
time preference depends on nt . In this time, noting the discount externality, wedeﬁne a competitive equilibrium as follows.
Definition. A competitive equilibrium is a sequence of allocations, {ct , bt , nt , zt , it , kt }∞t=0 ,such that, given initial conditions b0 , k0 and n0 , a given set of exogenous prices{w, r, p} and the perfect foresight of the households, then the representative household’s utility is maximized, the ﬁrm’s proﬁts are maximized, the government budget
constraint is balanced at each time, and all markets are cleared.
As described in the above deﬁnition, due to the perfect foresight of the households, the rate of time preference moves along time as indicated by (3) where nt = Nt .
Making use of equations (6) and (7), we can summarize the macroeconomic equilibrium in terms of three dynamic equations for ct , nt , and it . First, substituting
equation (7a) into (7d), we obtain the well-known dynamic equation of consumption10

as follows:ċt1= (r − ρ(nt )),
ct
σc

(8)

where σc ≡ −u′′ (ct )ct /u′ (ct ) represents the elasticity of the intertemporal substitutionof consumption. Please note that the time preference rate depends on the level ofthe renewable resource.
Next, by using (7c) and diﬀerentiating ηt with respect to time, we obtain thefollowing equations:ηt =(1 + τi )q̇t (1 + τi )qt Γ′′ (it )i̇t
(1 + τi )qt
,
and,
η̇
=
−
.
t
Γ′ (it )Γ′ (it )(Γ′ (it ))2Note that the tax rate τi is constant over time. Substituting these equations andthe dynamic equation of consumption into equation (7e), we obtain the dynamic
equation of the investment it :
(
)
i̇t
1
1 + τc v ′ (nt )Γ′ (it )′=r − G (nt ) −,it
σi
1 + τi u′ (ct )(9)where σi ≡ −Γ′′ (it )it /Γ′ (it ).
Finally, substituting equation (7b) into (7c) yields:ph′ (zt )Γ′ (it ) =1 + τi.
(1 − τy )(10)Solving this equation for the harvested resources zt , we obtain zt as a function of it ,τy , and τi . We derive this function as follows:
zt = z(it , τy , τi ).

(11a)

As for this function, we can obtain the following derivatives:∂zt
Γ′′ (it )h′ (zt )
=− ′
< 0,
∂it
Γ (it )h′′ (zt )h′ (zt )
∂zt
=
< 0,
∂τy
(1 − τy )h′′ (zt )∂zth′ (zt )=< 0.
∂τi(1 + τi )h′′ (zt )

(11b)
(11c)
(11d)

Substituting function (11a) into the evolution of natural resource (1) yields:ṅt = G(nt ) + Γ(it ) − z(it , τy , τi ).
11

(12)

Let c∗ , n∗ , and i∗ denote the steady-state levels of consumption, the natural
resource, and the investment in the natural resource, respectively. Then, a stationary
solution (c∗ , n∗ , i∗ ) of the dynamic equations of (8), (9), and (12) is characterizedby ċt = ṅt = i̇t = 0 and is deﬁned as follows:r = ρ(n∗ ),(13a)G(n∗ ) = z(i∗ , τy , τi ) − Γ(i∗ ),(13b)

r − G′ (n∗ ) =1 + τc v ′ (n∗ )Γ′ (i∗ ).
1 + τiu′ (c∗ )(13c)¿From (13c), the sign of r − G′ (n∗ ) must be positive at the steady state. We assume
this in the following.
Then we obtain the following proposition with respect to the steady state in thiseconomy.
Proposition 1. Suppose that the world interest rate, r, is larger than the lower
bound of the discount rate, ϵ, and that the following inequality holds: r < ρ(n̄). Then
there exists a unique steady state in this economy.
Proof. See Appendix A.
Next, we examine the stability of the steady state. Let J denote the Jacobianmatrix of the dynamic equations linearized around the steady state. Then we obtain:
  


∗
′ ∗ ′
∗
′′ ∗
c −c
ċ
0 u (c )ρ (n )/u (c )0 t 

 t

  


∗
′
∗
′
∗
∗
(14)
ṅt  =  0
G (n )
Γ (i ) − ∂z(i , τy , τi )/∂i nt − n  ,
  


it − i∗
i̇t
Ω1
Ω2
Ω3
{z
}
|
J

where Ω1 , Ω2 , and Ω3 are given by:
1 + τc Γ′ (i∗ )2 v ′ (n∗ )u′′ (c∗ )< 0,
1 + τiΓ′′ (i∗ )u′ (c∗ )2()Γ′ (i∗ )1 + τc v ′′ (n∗ )Γ′ (i∗ )′′∗Ω2 = ′′ ∗ G (n ) +> 0,
Γ (i )
1 + τiu′ (c∗ )1 + τc Γ′ (i∗ )v ′ (n∗ )= r − G′ (n∗ ) > 0.
Ω3 =1 + τiu′ (c∗ )Ω1 = −

12

(15a)
(15b)
(15c)

Then the characteristic equation of the matrix J is given by:−λ3 + TrJλ2 − BJλ1 + DetJ = 0,(16)where the coeﬃcients are given by:
TrJ = r > 0,(17a)
) ′ ∗ ()
1 + τc v ′′ (n∗ )Γ′ (i∗ )∂z(·)Γ (i )
′
∗
′
∗
′ ∗
′′
∗
,
BJ = G (n )(r − G (n )) +− Γ (i )G (n ) +∂iΓ′′ (i∗ )1 + τiu′ (c∗ )
{z
}
|
(

(−)

(17b)
DetJ = Ω1u′ (c∗ )ρ′ (n∗ ) ′ ∗(Γ (i ) − ∂z(·)/∂i) > 0,u′′ (c∗ )(17c)where TrJ and DetJ indicate the trace and the determinant of this matrix, respectively. To examine stability, we make use of the application of ‘Ruth’s theorem’ to athird-order polynomial as shown in Benhabib and Perli (1994).
Theorem (Benhabib and Perli (1994)) The number of roots of the polynomialin (16) with positive real parts is equal to the number of variations of sign in thescheme:
−1,

TrJ,

−BJ +

DetJ
,
TrJ

DetJ.

(18)

First, both of the signs of determinant DetJ and trace TrJ are positive. Thus,
when the sign of BJ is negative, there exist two roots with positive real parts andone negative root. Because the economy has one stock variable nt and two unpredetermined variables, ct and it , there exist multiple equilibrium paths converging to
the steady state. In other words, the dynamic character of the steady state exhibits
indeterminacy. BJ takes a negative value if the following inequality holds:()()∂z(·) Γ′ (i∗ )1 + τc v ′′ (n∗ )Γ′ (i∗ )
′
∗
′
∗
′ ∗
′′
∗
G (n )(r − G (n )) < Γ (i ) −G (n ) +,∂iΓ′′ (i∗ )1 + τiu′ (c∗ )(19)
where the sign of the right-hand side is positive. Note that the sign of r − G′ (n∗ ) is
positive at the steady state.
13

We now summarize the above analysis in the next proposition.13
Proposition 2. In the small open country, the equilibrium path of this economy isindeterminate if inequality (19) is satisﬁed.
Considering the suﬃcient condition of indeterminacy, the character of the equilibrium at the steady state tends to be indeterminate as G′ (n∗ ) becomes smaller, whichmeans that the speed of reproduction at the steady state is close to its maximum
speed, G′ (n̂).

3Welfare comparison associated with tax policies

In this section, we examine the eﬀects of tax policies on the level of welfare. Because
the stability of the steady state of this economy exhibits indeterminacy, we focus on
a comparison of welfare between diﬀerent steady states associated with diﬀerent tax
rates.
We ﬁrst investigate the long-run eﬀects of tax policies on consumption and the13

We can present a simple example of the character of the equilibrium by using the speciﬁed

functions supposing that the rates of income tax, consumption tax, and investment tax are zero.
The discount function is given by ρ(nt ) = ϵ + dnt + gn2t , where ϵ represents the lower bound of
this function as the level of natural resources approaches to zero, and d and g are parameters ofcurvature. In particular, we restrict the range of the parameters d and g to satisfy the assumptionof the discount factor given by (4a) and (4b). Then, it is enough that the parameters of d and gtake positive values.
Next, the reproduction function of natural resources following Koskela et al. (2002) is given by
G(nt ) = A × nt − (1/2)b × n2t , where A and b are positive parameters. In particular, the growth rateof reproduction G′ (nt ) is given by (A − b × nt ), which implies that, when the stock of the renewable
resources provides the maximum sustained yield G′ (n̂) = 0, the level of the natural resource isn̂ = A/b.
Assume that the steady-state rate of time preference rate is 0.04. That is, ρ(n∗ ) = 0.04. Whenthe free parameters are given by ϵ = 10−6 , d = 0.05, g = 0.2, b = 1.5, and A = 0.5, we obtainn∗ = 0.34 and G′ (n∗ ) = −0.0090, which shows that the sign of BJ is negative. In this case, theequilibrium path is indeterminate.

14

natural resource. Making use of (13a)–(13c), we obtain the following lemma.
Lemma 1. (i) The consumption tax and the income tax have negative impacts
on consumption in the long run, whereas the investment tax has positive impacts on
consumption. That is, dc∗ /dτc < 0, dc∗ /dτy < 0, and dc∗ /dτi > 0. (ii) The consumption tax does not aﬀect investment in the natural resource, whereas the income tax
and the investment tax have negative impacts on investment. That is, di∗ /dτc = 0,di∗ /dτy < 0, and di∗ /dτi < 0. (iii) The consumption tax does not inﬂuence the harvesting of the natural resource, whereas the income tax and the investment tax havenegative impacts. That is, dz ∗ /dτc = 0, dz ∗ /dτy < 0, and dz ∗ /dτi < 0. (iv) The
tax policy does not aﬀect the renewable natural resources in the long run. That is,dn∗ /dτy = dn∗ /dτc = dn∗ /dτi = 0.
Proof. See Appendix B. 
We give explanations for the eﬀects of tax policies on consumption. When the rateof the consumption tax rises, the steady-state level of consumption decreases because
the price of consumption goods increases. When the rate of income tax increases,
the steady-state level of consumption decreases because overall output decreases. On
the other hand, the increase in the investment tax rate has two opposite eﬀects. One
is the substitution eﬀect of this tax. An increase in the investment tax decreases thelevel of investment in the natural resource, and thus raises the level of consumption.
The other is the indirect negative eﬀect of this tax on consumption. The increase in
the investment tax decreases the harvest level and thus lowers the level of income.
Although there are these positive and negative impacts of the investment tax on
consumption, we can calculate that the positive eﬀect of the investment tax on
consumption dominates the negative.14
Let us consider the eﬀects of tax policies on investment in the natural resource.
Equations (13a) and (13b) determine the steady-state level of investment in thenatural resource. Because these equations do not include the consumption tax, theconsumption tax does not have any impact on the long-run level of the investment as
14

The proof is given in Appendix B.

15

stated in Lemma 1 (ii). On the other hand, the income tax and the investment tax
have negative impacts on the long-run level of the investment in the natural resource.
When the rate of income tax rises, the decrease in total income reduces the level ofinvestment in the natural resource. Because an increase in the rate of investment
tax produces an increase in the cost of investing in the natural resource, it decreases
the level of investment in the natural resource in the long run.
Taking account of (11a), we can next examine the eﬀects of taxes on the harvestingof the natural resource. The consumption tax does not have any eﬀect on harvesting
because this tax does not aﬀect the level of investment in the natural resource. On
the other hand, the income tax and investment tax have two opposite impacts on
the harvesting of the natural resource. One is the direct negative eﬀect of the tax
policies, and the other is the indirect positive eﬀect through the negative impact
on investment in the natural resource. We can show that the direct negative eﬀectof the income tax or the investment tax on the harvesting of the natural resource
dominates the indirect positive eﬀect.15
Finally, let us consider the eﬀects of the income tax, the consumption tax, andthe investment tax on the natural resource. These tax policies do not have any eﬀect
on the natural resource. This is because the equality of the constant world interest
rate and the time preference rate uniquely determines the steady-state level of thenatural resource as shown by (13a).
Making use of these results, we can conduct a welfare comparison with respect
to the income tax, the consumption tax, and the investment tax.
Proposition 3. The welfare level at the steady state with a higher rate of the consumption tax or the income tax is lower than the welfare level at the steady state withlower rates of these taxes. On the other hand, the welfare level at the steady statewith a higher rate of the investment tax is higher than the welfare level at the steadystate with a lower rate of the investment tax.

15

The proof is given in Appendix B.

16

Proof. By diﬀerentiating the welfare level of the household, we obtain the following:∗dU
1′ ∗ dc=u(c), j = c, y, i.
dτjρ(n∗ )dτj(20)

Thus we can obtain Proposition 3.

4

The centrally planned economy

Because households neglect that the discount factor depends on the level of the natural resource stock, the macroeconomic equilibrium may not coincide with the social
optimum. In this section, we ﬁrst consider the ﬁrst best outcome, namely, when thesocial planner maximizes the intertemporal utility of the representative householdtaking account of this externality. First, we calculate the ﬁrst-order conditions derived by the maximizing problem of the social planner and investigate the distortions
caused by the discount externality. Next, we examine the socially optimal tax policies that enable the decentralized equilibrium to replicate the dynamic equations of
the centrally planned economy.

4.1

Comparison of steady-state equilibria

Let the variables with the upper bar denote those in the centrally planned economy.
Maximizing the intertemporal utility shown by equation (2) subject to (1), (3), (5),and (6) yields the ﬁrst-order conditions of the central planner as follows:u′ (c̄t ) = q̄t ,(21a)q̄t ph′ (z̄t ) = η̄t ,(21b)

η̄t Γ′ (īt ) = q̄t ,

(21c)

r=−

q̄˙t
+ ρ(n̄t ),
q̄t

(21d)

λ̄˙ t
u(c̄t ) + v(n̄t )= − + ρ(n̄t ),
λ̄t
λ̄t
′
v ′ (n̄t )λ̄η̄˙ tt ρ (n̄t )+ G′ (n̄t ) −= − + ρ(n̄t ),
η̄t
η̄t
η̄t
17

(21e)
(21f)

where η̄t , λ̄t , and q̄t , represent the shadow prices associated with (1), (3), and (5),respectively.
We compare these ﬁrst-order conditions with those in the decentralized economy,(7a)–(7e). First, because the social planner takes the discount externality into account, the dynamic equation of the shadow value of patience is given by equation(21e). Next, the third term of the left-hand side of equation (21f) shows the additional term derived by taking account of the discount externality. Noting that the
shadow value of patience is included in the additional term, we need to consider
the dynamic equation of the shadow value of the patience. Then there exist four
dynamic equations of the centrally planned economy, which are given by:1c̄˙t= (r − ρ(n̄t )),
c̄t
σ̄c

(22a)

n̄˙ t = G(n̄t ) + Γ(īt ) − z(īt ),
(
)
ī˙ t
1
v ′ (n̄t )Γ′ (īt )λ̄t ρ′ (n̄t )Γ′ (īt )′=r−− G (n̄t ) +
,
īt
σ̄i
u′ (c̄t )u′ (c̄t )λ̄˙ tu(c̄t ) + v(n̄t )=− ρ(n̄t ),
λ̄t
λ̄t

(22b)
(22c)
(22d)

where σ̄c ≡ −c̄t u′′ (c̄t )/u′ (c̄t )(> 0) and σ̄i ≡ −Γ′′ (īt )īt /Γ′ (īt )(> 0) represent the elasticity of the marginal utility and the elasticity of the marginal eﬃciency of the investment. In particular, the last term of the right-hand side of equation (22c) includes
the shadow value of patience λ̄t . Thus, the dynamic equation (22d) for this shadow
value must be taken into account.
Setting the rates of the income, consumption, and investment taxes in equations(9) and (12) to zero, we compare the dynamic equations of the decentralized economy
with those of the social optimum. Then, we obtain the following proposition.
Proposition 4. Suppose that both of the economies are at the steady state. The
steady-state levels of the natural resource and the investment in the natural resource
are the same in both economies. The steady-state level of consumption in the decentralized economy is smaller than that in the centrally planned economy.
Proof. When the tax rates are zero, we can obtain the same equation as (10) fromequations (21b) and (21c). From the dynamic equations of consumption and the18natural resource at the steady state with zero tax rates (i.e., equations (8), (12),(22a), and (22b)), the steady-state levels of the natural resource and investment in
the natural resource are the same in both economies, that is, n∗ = n̄∗ and i∗ = ī∗ .16
We next compare the levels of consumption. From the equations i̇t = 0 andī˙ t = 0, we obtain the following.
v ′ (n∗ )Γ(i∗ )+ G′ (n∗ ).
(23a)u′ (c∗ )v ′ (n̄∗ )Γ(ī∗ )λ̄∗ ρ′ (n̄∗ )Γ′ (ī∗ )′∗The centrally planned economy: r =
+
G
(n̄
)
−
. (23b)u′ (c̄∗ )u′ (c̄∗ )The decentralized economy: r =Because the steady-state levels of the natural resource and investment in thedecentralized economy are identical to those in the centrally planned economy, we
can subtract (23a) from (23b) and can obtain:()
1
λ̄∗ ρ′ (n̄∗ )Γ′ (ī∗ )1′∗∗v (n̄ )Γ(ī )−=−.
u′ (c∗ ) u′ (c̄∗ )u′ (c̄∗ )(24)

Because the right-hand side of (24) takes a negative value, the level of consumption
in the decentralized economy is smaller than that of the social optimum, that is,c∗ < c̄∗ . This proposition states that the steady-state levels of the natural resource andthe investment in the natural resource are the same in both economies. This isbecause the endogenous time preference rate determines the same level of the natural resource due to (8) and (22a). In contrast with this result, the steady-statelevel of consumption in the decentralized economy is smaller than that in the centrally planned economy. This is because the discount function has the characterof increasing marginal impatience with respect to the level of the natural resource(i.e., ρ′ (nt ) > 0). When the level of the natural resource is large enough, the social
planner largely discounts the future stream of utility compared with the householdsof the decentralized economy because they do not take account of this eﬀect. As
a result, the steady-state level of consumption in the centrally planned economy islarger than that of the decentralized economy. This indicates why the steady state16

The asterisks represent the steady-state level of the social optimum.

19

with the higher rate of investment tax has the larger welfare level (see Proposition
3). A higher rate of investment tax reduces the level of investment and thus raises
the level of consumption.

4.2

The socially optimal tax policies

In the last subsection, we have conﬁrmed that the discount externality creates a
distortion in resource allocation, which means that the government should develop
a tax policy to achieve the social optimum.
Let us consider a tax structure such that the decentralized economy replicates the
optimal path of the centrally planned economy over time. The equivalence between
the path in the decentralized economy and that in the centrally planned economy
means c̄t = ct , n̄t = nt , īt = it , and λ̄t = λt at every point in time. Thus, to achievethis, we allow the rates of the three kinds of tax, τy (t), τc (t), and τi (t) to be timevarying. Then we obtain the following lemma with respect to the consumption tax
rate.
Lemma 2. The rate of the consumption tax is constant over time to mimic thesocially optimal path in the centrally planned economy.
Proof. The replication of the social optimum requires q̇t /qt = q̄˙t /q̄t , which leads
to q̄t = αqt where α is an arbitrary constant. Thus, considering equations (7a)and (21a), the optimal rate of consumption tax is an arbitrary constant, that is,α = τc = constant. 
Taking account of this lemma, the rate of consumption tax must be constantover time, that is, τc (t) = τc = arbitrary constant.17 Hence, we set the tax rate of
consumption at zero through time, that is, τc (t) = 0.
Next, let us consider the optimal investment tax rate and the optimal income
tax to replicate the stable path in the centrally planned economy. Diﬀerentiating
17

Liu and Turnovsky (2005) show that, in a simple Ramsey model with consumption externalities,

the optimal tax rate of consumption needed to mimic the levels of a social optimum is arbitrary if
the labor supply is inelastic.

20

equation (7c) with respect to time, we obtain the dynamic equation of the investmentin the natural resource as follows:
(
)
i̇t
1
τ̇i (t)1v ′ (nt )Γ′ (it )′−.
=r − G (nt ) −it
σi
1 + τi (t) 1 + τi (t) u′ (ct )(25)

Thus, by comparing (25) with (22c), we obtain the optimal rate of the investment
tax τi (∞) at the steady state as follows:
τi (∞)λ̄∗ ρ′ (n̄∗ )= ′ ∗ =
1 + τi (∞)
v (n̄ )(u(c∗ )+1v(n∗ )

)

( ∗ ′

)

( ∗ ′

).

n ρ (n∗ )ρ(n∗ )n v (n∗ )v(n∗ )(26)

This equation implies that the government can drive the decentralized economy toreplicate the centralized economy if the following inequality holds at the steady state:( ∗ ′ ∗ )n ρ (n ))(ρ(n∗ )u(c∗ )() < 1.
(27)+
1n∗ v ′ (n∗ )v(n∗ )v(n∗ )

If the elasticity of the endogenous time preference rate is suﬃciently small, this inequality holds. Otherwise, the government cannot achieve a social optimal allocation
at the steady state by any tax policies.
Furthermore, the socially optimal rate of the investment tax must evolve according to the following dynamic equation:τ̇i (t)v ′ (n̄t )Γ′ (īt )=1 + τi (t)u′ (c̄t )(τi (t)
λ̄t ρ′ (n̄t )− ′1 + τi (t)v (n̄t )

)
.

(28)

Taking account of the rate of the investment tax at the steady-state level and on
the socially optimal transitional path, which are given by (26) and (28), we candetermine the initial rate of the investment tax.
Finally, we examine the optimal rate of the income tax through time to replicate
the harvested level of the natural resource of the centrally planned economy. Substituting (7b) into (7c), we obtain equation (10) where the income tax rate and theinvestment tax rate change through time. Similarly, substituting (21b) into (21c)yields:ph′ (z̄t )Γ′ (īt ) = 1.
21

(29)

Therefore, the comparison of (10) and (29) derives the optimal relationship between
the rate of the income tax and the rate of the investment tax as follows:τy (t) = −τi (t).

(30)

When the rate of the income tax is set according to (30), the harvested level of thenatural resource in the decentralized economy is the same as that in the centrally
planned economy. This results in ṅt = n̄˙ t . Then the tax policies deﬁned above enable
the decentralized economy to achieve the socially optimal path.
We can summarize these results as follows.
Proposition 5. Suppose that the inequality (27) holds. Then, the decentralized
economy can replicate the resource allocation in the centrally planned economy bysetting the rates of the income tax and the investment tax according to (26), (28),and (30), and by setting the rate of the consumption tax at an arbitrary constantlevel.

5

Conclusion

We have examined the eﬀects of tax policies in a small open economy that depends
on a renewable resource. In the present paper, it is supposed that the patience ofhouseholds depends on the whole level of the natural resource in the domestic economy. However, households believe that they cannot control its whole level because
they think that their inﬂuence is negligible. Hence, they do not consider this dependency when they optimize their consumption and investment decisions. In thepresent paper, we call this externality a discount externality. From this setting we
obtain the following results.
First, we have shown that there exists a unique steady state and that the equilibrium path can be indeterminate.
Second, we examined the welfare levels of two diﬀerent steady states corresponding to two diﬀerent tax rates. The welfare level at the steady state with a higher
rate of consumption or income taxes is lower than that at the steady state with lower
22

rates of these tax rates. Furthermore, the welfare level at the steady state with ahigher rate of the investment tax is higher than that at the steady state with a lower
rate of the investment tax.
Finally, we examined the optimal tax policy through time to replicate the transitional path in the centrally planned economy. It is possible for the decentralized
economy to replicate the optimal path when tax policies are mixed.

Acknowledgments
We are very grateful to Akira Momota, Yasutomo Murasawa, Yoshinao Sahashi, andseminar participants at Osaka Prefecture University for their valuable comments.
We also acknowledge the helpful advice about the data set provided by Tsunehiro
Otsuki.

Appendix
Appendix A
In this appendix, we show that the steady-state levels of consumption, the natural
resource, and the investment in the natural resource are uniquely determined.
Because ϵ < r < ρ(n̄) in (4b) and the discount rate is an increasing function of
the natural resource (4a), the steady-state level of the natural resource is uniquelydetermined by (13a).
We next consider the relationship between the harvested natural resource zt andthe level of investment in the natural resource it given the rates of the income and investment taxes. By diﬀerentiating the right-hand side of equation (13b) with respect
to it , we can show that the right-hand side of equation (13b) is a decreasing function of the investment in the natural resource, that is, ∂z(i∗ , τy , τi )/∂i − Γ′ (i∗ ) < 0.
When the level of the investment in the natural resource is approaching to zero, Γ′ (i)becomes inﬁnity, that is, limi→0 Γ′ (i) = ∞. Therefore, due to the Inada conditions
23

with respect to h(z), we obtain limi→0 z(i, τc , τi ) = ∞ (see equation (10)). Because
Γ(0) = 0, we can show:
lim[z(i, τc , τi ) − Γ(i)] = ∞.
i→0

(A.1)

On the contrary, when the level of the investment in the natural resource goes toinﬁnity, Γ′ (i) approaches to zero, that is, limi→∞ Γ′ (i) = 0. Therefore, we obtainlimi→0 z(i, τc , τi ) = 0. Consequently, we can show the following:lim [z(i, τc , τi ) − Γ(i)] < 0.

i→∞

(A.2)

Thus, equation (13b) uniquely determines a positive level of investment in the natural
resource in the steady state.
Finally, let us consider i̇t = 0 expressed by (13c) to determine the steady-statelevel of consumption given the tax rates. Because the marginal utility of consumptionu′ (·) is monotonically decreasing, the steady-state level of consumption is uniquelydetermined under the assumption of r > G′ (n∗ ).

Appendix B
In this appendix, we examine the eﬀects of increases in the rates of the income,consumption and investment taxes in the natural resource sector on consumption,and on the natural resource and investment in the natural resource. Totally diﬀerentiating the equations of ċt = ṅt = i̇t = 0 yields:   
′
∗
dc
0
0
0
dτ
0 −ρ (n )0  y
     
 0 G′ (n∗ ) Γ′ (i∗ ) − ∂z(·)/∂i dn = ∂z(·)/∂τy 0 ∂z(·)/∂τi   dτc    
di
0
Ω4
Ω5
dτi
Ω1
Ω2
Ω3
{z
}
|
#A

(B.1)

24

where the deﬁnition of Ωj (j = 1 − 5) is respectively given by:
1 + τc v ′ (n∗ )Γ′ (i∗ )u′′ (c∗ )< 0,
1 + τi(u′ (c∗ ))2
1 + τc v ′′ (n∗ )Γ′ (i∗ )Ω2 = −G′′ (n∗ ) −> 0,
1 + τiu′ (c∗ )1 + τc v ′ (n∗ )Γ′′ (i∗ )Ω3 = −> 0,
1 + τiu′ (c∗ )1 v ′ (n∗ )Γ′ (i∗ )
Ω4 =
> 0,
1 + τi u′ (c∗ )1 + τc v ′ (n∗ )Γ′ (i∗ )< 0.
Ω5 = −(1 + τi )2 u′ (c∗ )
Ω1 =

(B.2)
(B.3)
(B.4)
(B.5)
(B.6)

Thus, we obtain the following:
 


dc
G′ (n∗ )Ω3 − Ω2 (Γ′ (i∗ ) − ∂z(·)/∂i) ρ′ (n∗ )Ω3 −ρ′ (n∗ )(Γ′ (i∗ ) − ∂z(·)/∂i)
 

1 
 


′ ∗
dn =Ω1 (Γ (i ) − ∂z(·)/∂i)00  det 

′
∗
′
∗
di
−G (n )Ω1−ρ (n )Ω1
0

 
0
0
0
dτ

  y (B.7)
∂z(·)/∂τy 0 ∂z(·)/∂τi   dτc 

 
0
Ω4
Ω5
dτi
where det represents the positive determinant of the matrix #A, which is given bydet = −Ω1 ρ′ (n∗ )(Γ′ (i∗ ) − ∂z ∗ /∂i∗ ) > 0.
Noting that the signs of ∂z(·)/∂i, ∂z(·)/∂τy , and ∂z(·)/∂τi are negative, we canobtain the eﬀects of tax policies on consumption, the natural resource stock, and25

investment in the natural resource as follows:
dc
1 ′ ∗
∂z(·)
=
ρ (n )Ω3
< 0,
dτy
det
∂τy
dn
= 0,
dτy
1 ′ ∗
di
∂z(·)
=−
< 0,
ρ (n )Ω1
dτy
det
∂τy
(
)
dc
1 ′ ∗
∂z(·)
′ ∗
=−
ρ (n ) Γ (i ) −
< 0,
dτc
det
∂i
dn
= 0,
dτc
di
= 0,
dτc
[
(
) ]
dc
1
∂z(·)
∂z(·)
′
∗
′
∗
′ ∗
=
ρ (n )Ω3− ρ (n ) Γ (i ) −
Ω5
dτi
det
∂τi
∂i
1 ′ ∗ ′ ∗=−
Γ (i )ρ (n )Ω5 > 0,
det
dn
= 0,
dτi
di
1 ′ ∗
∂z(·)
=−
ρ (n )Ω1
< 0.
dτi
det
∂τi

(B.8a)(B.8b)(B.8c)(B.8d)(B.8e)(B.8f)(B.8g)(B.8h)(B.8i)

Next, let us consider the eﬀect of these tax policies on the harvested natural
resource. Diﬀerentiating (11a) with respect to the tax rates, the eﬀects of the income,consumption, and investment taxes on the harvested natural resource are given by:∂z ∗∂z ∗ ∂i∗
∂z ∗
= ∗
+
,
∂τy
∂i ∂τy ∂τy( ′′ ∗ ′ ∗)∂z ∗Γ (i )h (z )1=
− ′ ∗ ′′ ∗ ′ ∗+1 ,∂τyΓ (i )h (z ) Γ (i ) − ∂z ∗ /∂i∗Γ′ (i∗ )
∂z ∗
< 0.
=
∂τy Γ′ (i∗ ) − ∂z ∗ /∂i∗∂z ∗∂z ∗ ∂i∗
= ∗
< 0,
∂τc
∂i ∂τc∂z ∗ ∂i∗ ∂z ∗
∂z ∗
= ∗
+
,
∂τi
∂i ∂τi∂τi()
Γ′′ (n∗ )h′ (z ∗ )∂z ∗1− ′ ∗ ′′ ∗ ′ ∗
+1 ,
=
∂τi
Γ (i )h (z ) Γ (i ) − ∂z ∗ /∂i∗∂z ∗Γ′ (i∗ )=< 0.
∂τi Γ′ (i∗ ) − ∂z ∗ /∂i∗26

(B.9a)(B.9b)(B.9c)

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Table 1: Export Quantity of Forestry Sector (million cubic meters).
2000 2001 2002 2003 2004 2005
Finland

1.06

0.81

0.82

0.88

1.06

1.51

Portugal 1.14

1.62

1.64

2.04

2.02

2.56

Sweden

2.68

3.57

3.01

3.12

6.25

2.92

Data source: Food and Agriculture Organization of the United Nations.18

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We use data from the following homepage: http://www.fao.org/

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