DBJ Discussion Paper Series, No. 1904

Trade, Resource Use and Pollution: A Synthesis

Akihiko Yanase（Nagoya University）
Gang Li（Toyo University）

May

2019

Discussion Papers are a series of preliminary materials in their draft form. No quotations,reproductions or circulations should be made without the written consent of the authors in order toprotect the tentative characters of these papers. Any opinions, findings, conclusions or
recommendations expressed in these papers are those of the authors and do not reflect the views of
the Institute.

Trade, Resource Use and Pollution: A Synthesis∗
Akihiko Yanase†

Gang Li‡

Nagoya University

Toyo University

May 2019

Abstract
We develop a two-sector dynamic model of trade and the environment, in which
both sectors are environmentally harmful and one sector is environmentally sensitivein the sense that its productivity depends on the environmental stock. The modelyields a rich set of insights regarding specialization pattern, environmental impacts,
and welfare effects of trade between countries subject to industrial pollution as well
as resource extraction. In the short run, labor allocations and world specialization
pattern are dependent on country size, technology, preference, and environmental
stocks. The environmental stocks evolve over time and, depending on the type ofeach trading country (revealed by whether the environmentally sensitive sector ismore environmentally harmful), the steady-state environmental stocks and world
specialization pattern are determined, deriving environmental and welfare consequences of trade. At least one country gains from trade in the long run if two
countries are of the same type. If two countries are of different types, however, both
may lose from trade by exporting their respective “dirtier” goods to one another.

Keywords: Renewable resource; pollution; environmental stock; long-run supply curve;
long-run PPF; world specialization pattern; long-run gains from trade
JEL classification: F18; Q27∗

We are grateful to Eric Bond, Ngo Van Long, and Yasuhiro Takarada for insightful discussions. We
also thank Kazumi Asako, Brian Copeland, Taiji Furusawa, Jota Ishikawa, David Jinkins, Hayato Kato,
Yan Ma, Carol McAusland, Hiroshi Mukunoki, Jun-ichi Nakamura, Takeshi Ogawa, Hiroshi Ohta, Gianmarco Ottaviano, Thomas Sampson, Yoichi Sugita, Katsuhisa Uchiyama, Morihiro Yomogida, and seminar and conference participants at the University of Bari, University of British Columbia, Copenhagen
Business School, Hitotsubashi University, Kobe University, London School of Economics, McGill University, Research Institute of Capital Formation of the Development Bank of Japan, Xiamen University,
Asia Pacific Trade Seminars, Australasian Trade Workshop, Japanese Society of International Economics
Meeting, IEFS Japan Annual Meeting, Summer Workshop of Economic Theory, and World Congress of
the Econometric Society for helpful comments. All remaining errors are ours. We gratefully acknowledge
the Japan Society for the Promotion of Science (JSPS) for the financial support (No.16H03617). We also
thank the Hitotsubashi Institute for Advanced Study (HIAS) for the support.
†
Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan.
E-mail: yanase@soec.nagoya-u.ac.jp‡
Department of Economics, Toyo University, 5-28-20 Hakusan, Bunkyo-ku, Tokyo 112-8606, Japan.
E-mail: ligang.hitu@gmail.com

1

1

Introduction

Although it has been a few decades since global environmental problems such as climatechange are recognized as one of the important issues to be worked on in the international
arena, local environmental problems are still significant, especially in emerging and developing economies. In comparison with developed countries, developing countries arestill reluctant to implement stringent environmental policies, claiming the right to pursue
economic growth at the expense of environmental protection. In addition, property rights
to natural and environmental resources are often less well-defined in these countries. For
these reasons, local-scale environmental problems, such as air and water pollution, soil
contamination, desertification, and deforestation, remain substantial in many developing
countries.1
Such local environmental problems vary by countries. In emerging economies, such as
China and India, pollution problems accompanied by rapid industrialization are standing
out, while less developed countries in Sub-Saharan Africa and Latin America rather face
the problem of overexploitation of natural resources, bringing about resource depletion as
well as pollution and loss of biodiversity from traditional agriculture (such as slash-andburn shifting cultivation). The causes of each environmental problem also vary. Take, forexample, deforestation, which is a result of excessive harvest of timber, but could also arise
from persistent acid precipitation caused by sulfur or nitrogen oxide emissions. Featuring
a wide range of the problem’s nature and causes, an economically significant consequenceof these problems is that, in addition to the possible risk to human health, primarygood industries that produce food and resource goods are vulnerable to environmental
deterioration.
In this study, we theoretically address the issue of trade and the environment withan eye on developing countries.2 These countries have increased their presence in the1

Take air pollution for instance. According to 2018 World Air Quality Report by IQAir AirVisual(available on https://www.airvisual.com/world-most-polluted-cities) Asian locations dominate the highest 100 average PM2.5 (particulate matter less than 2.5 microns in diameter) levels, with cities in India,
China, Pakistan and Bangladesh occupying the top 50 cities. The Report also notes that althoughsources of pollution vary by region and city, common contributors include vehicle exhaust, open cropand biomass burning, industrial emissions and coal burning.
2
The sensitivity of results on the relationship between trade openness and environmental quality todifferences between developing countries and OECD countries has been empirically analyzed by Managi
et al. (2009) and Tsurumi and Managi (2012). Managi et al. (2009) used panel data of sulfur dioxide(SO2) and carbon dioxide (CO2) emissions of 88 countries from 1973 to 2000, and showed that trade
openness increases SO2 and CO2 emissions in non-OECD countries, while it decreases them in OECD
countries. Using data on the annual rate of deforestation for 142 countries from 1990 to 2003, Tsurumi

2

world economy, as shown by the rapid growth in exports from developing countries andtrade between them.3 In view of the increased presence of developing countries in world
trade and the fact that the implementation of their environmental policies can be often
deficient, it is of great interest to investigate the extent to which trade liberalization
affects the environment and economic welfare in countries where governments fail to
implement appropriate environmental policies and environmental degradation harms the
productivity of sectors that produce environmentally sensitive, primary commodities.4
For the purpose, we develop a two-sector model in which both production sectors areenvironmentally harmful by hampering the recovery of the environment, measured by astock variable (environmental stock) that evolves over time. In addition, one sector among
the two is environmentally sensitive in the sense that its productivity depends on theenvironmental stock. This sector can be interpreted as an agricultural- or resource-good
sector, which can damage the environment via resource extraction or biomass burningand can also be affected by the environmental quality. The other sector can be considered
as a manufacturing sector, the production process of which emits pollution that has anegative effect on the accumulation of environmental stock. We explicitly formulate theidea that both resource extraction and industrial pollution can damage the environment,
and make an elaborate analysis that captures trade between, in particular, emerging andless developed countries. Our study contributes to the literature because existing studies
on trade and the environment with an environmentally-sensitive, resource-good sector
have considered resource use and pollution in different economic models.
The interaction between trade and renewable natural resources was explored by Brander and Taylor (1997b, 1998) in a dynamic Ricardian general equilibrium model with
open-access renewable resources. In their model, there are two final goods, the resource
good and the other good. Brander and Taylor (1997b, 1998) showed that while a resource
importer gains from trade, a diversified resource-exporting country necessarily suffers aand Managi (2012) found that an increase in trade openness increases deforestation for non-OECD
countries while slowing down deforestation for OECD countries.
3
According to UNCTAD (2013), the share of South–South trade in total world trade increased fromslightly less than 30 per cent in 1995 to slightly more than 40 per cent in 2012. Furthermore, the shareand value added of manufacturing in a developing country’s exports to other developing countries areusually much higher than exports to developed countries.
4
Pointing out that in many developed countries, the consumption of imported coffee, tea, sugar, textiles, fish, and other commodities causes a biodiversity footprint that is larger abroad (i.e., in developing
countries) than at home, Lenzen et al. (2012) found that 30 per cent of global species threats are due tointernational trade.

3

decline in the steady-state utility level.5 The Brander–Taylor model can also help understand the effects of trade involving economic activities both being environmentallysensitive and generating environmental impacts, such as timber industry that can lead to
deforestation and agriculture that uses slash-and-burn cultivation.6
To address the issue on trade and the environment with pollution emitted in an environmentally insensitive sector, Copeland and Taylor (1999) developed a two-sector Ricardian model in which the production of “smokestack” manufacturing generates pollution,
which lowers the productivity of an environmentally sensitive sector. Assuming a laissezfaire economy with pollution unregulated, they showed the extent to which international
trade may benefit both countries by spatially separating dirty and clean industries andthereby raising the world’s production possibilities.7
In terms of the model structure, we consider a hybrid model of the Brander–Taylor
and Copeland–Taylor models. In a study closely related to ours, Rus (2016) combined theBrander–Taylor model with the Copeland–Taylor model to analyze international tradepatterns and the effects of trade liberalization. His analysis is, however, confined to asmall-open economy. By contrast, we analyze a two-country model, which is a profoundstep in the complexity and in providing a formal tool to consider trade between countriesof different types. The first type is what we call the BT country where resource extractionis the main source of environmental deterioration. The second type is what we callthe CT country where industrial pollution is the main source. Trade between BT and
CT countries in our model thus represents typical contemporary South–South trade inreality; our hybrid model sheds light on the environmental and welfare effects of trade
5

The Brander–Taylor model has been extended in a number of directions. Brander and Taylor (1997b)
extended their original model by introducing countries that differ in their resource-management regimes(open access in one country and optimally regulated in the other country). Jinji (2007) analyzed thecase in which two countries may differ in both their resource-management regime and relative resource
abundance. By introducing land as another input and focusing on forest resources, Jinji (2006) examineda model with an endogenous carrying capacity of the resource. Copeland and Taylor (2009) developeda theory of resource management with an endogenous property-rights regime by combining the originalmodel with a simple model of moral hazard. Takarada et al. (2013) considered a model of fishery resources
shared by two countries.
6
Using a balanced panel of 732 municipalities in the Brazilian Amazon from 2000 to 2010, Faria andAlmeida (2016) empirically found that as openness to trade in the Amazon increases, deforestation also
increases, and that the production of soybeans and beef cattle drives deforestation.
7
There are a number of applications of the Copeland–Taylor model. Copeland and Taylor (1997)
considered government policy that controls pollution. Focusing on transboundary pollution, Unteroberdoerster (2001), Benarroch and Thille (2001), and Suga (2007) re-examined the spatial separation result.
In another direction, Kondoh (2006) and Beladi et al. (2000) applied the Copeland–Taylor model tointernational migration and international capital movement, respectively.

4between emerging economies, in which pollution emission is the often the main cause ofenvironmental problem, and less developed, resource-rich economies, in which extractionor the inappropriate use of natural resources is the main culprit.
Having set up the model, we begin with charactering the properties of supply side,
particularly those in the long run. The long-run (steady-state) supply curve is upwardsloping in a country of the BT type, yet downward-sloping in a country of the CT type.
It is this distinction that generates the contrasting dynamic responses of the respective
economies to trade liberalization.
In the analysis of trade between the two countries, we obtain a complete description of
the dynamic system in the trading equilibrium. For this purpose, we first derive how world
specialization patterns (which country specializes on which good) are distributed along
the comparative advantage index (depending on environmental stocks and technologies
in the two countries) and the relative effective size (depending on labor endowments andtechnologies).
In the long run, environmental stocks evolve over time, and so does the comparativeadvantage index. We scrutinize world specialization pattern, labor allocations, and environmental and welfare consequences of trade in the long run. Regarding welfare effects oftrade, at least one country gains from trade if two countries are of the same type. However, if the two trading countries are of different types, a pessimistic scenario emerges as apossibility; trade could result in environmental degradation in both countries, and consequently may harm the welfare of both countries. This scenario is of special interest since
it captures trade between emerging industrial nations and less developed resource-richcountries, and highlights that in the absence of appropriate regulation on environmentally
harmful economic activities, mutually harmful international trade might occur.
The possibility of mutually harmful trade in the presence of environmental resource
dynamics has been addressed in the literature. Karp et al. (2001) extended the North–South
trade model developed by Chichilnisky (1994) in which differences in property rights forenvironmental resources create a motive for trade among otherwise identical countries.
They showed that trade may aggravate the common property problem to the extent
that both countries lose from trade. The mechanisms driving the results in their model,
however, are different from ours. First, in their model, environmental distortion arises
from imperfect property rights, and environmental change matters by affecting extraction

5

cost of environmental services. Here, environmental distortion arises from market failure(among self-interest firms), and environmental change affects directly the productivityof final-good sector. Second, in their model, partial unemployment serves a key role incausing welfare losses under trade. In our model, full employment prevails; the driving
force behind welfare losses in both trading countries is the combination of a productivity
decline due to environmental degradation, and the possibility that both countries export
their respective “dirtier” goods to one another.
The rest of the paper is organized as follows. Section 2 presents the basic model.
Section 3 analyzes the supply side. Sections 4 and 5 deal with the autarkic equilibrium
and small open economy. Section 6 investigates two-country trade. Section 7 illustrates
the results with numerical examples. Section 8 concludes.

2

The basic model

There are two production sectors, resource-good and manufacturing, producing under
perfect competition with technologies:
Xf = A (S) Lf ,Xm = aLm ,(1)where the subscripts f and m denote respectively resource goods (food, forest or fisheryresources) and manufacturing goods; Xi and Li (i = f, m) are the output from and labor
allocated to the corresponding sector. Environmental stock S measures the capacity ofthe environment, which is interpreted as the stock of renewable resources in Brander
and Taylor (1997a, 1998), and as the quality of the environment in Copeland and Taylor (1999). The resource-good sector is environmentally sensitive in the sense that its
productivity increases with environmental stock: A0 (S) > 0. The productivity of the
manufacturing sector is fixed. The labor market is perfectly competitive with flexible
wages so that full employment prevails all the time:
Lf + Lm = L,6(2)

where L is labor endowment and is assumed to be constant over time. Environmental
stock evolves according toṠ = G (S) − E,(3)where G (S) is the natural growth of environmental stock and E the economic usage ofthe environment (the environmental impact of economic activities). To formulate the ideathat the maximum capacity of the environment is finite, impose the following assumption:
Assumption 1. There exists K > 0 such that G (K) = 0 and G0 (K) < 0.
If no economic activity occurs, E = 0 and the capacity of the environment at thesteady state is K, which is usually called the carrying capacity of the environment. The
economic usage of the environment depends on the scales of two sectors and can be
expressed by
E = lf Xf + lm Xm ,(4)where non-negative parameters lf and lm measure environmental impacts from per unit
output in the two sectors. We exclude the trivial case of lf = lm = 0 to prevent themodel from degenerating into a Ricardian model in the long run. Note that our modelturns into the Brander–Taylor model by letting lf = 1 and lm = 0, and becomes the
Copeland–Taylor model if lf = 0 and lm > 0.
The economic usage of the environment per unit labor is then lf A (S) in the resourcegood sector and lm a in the manufacturing sector. Their relative magnitude proves crucial
in determining the dynamic behavior of the economy.
Definition 1 (Sector type). A more (less) environmentally harmful, or “dirtier (cleaner)”,
sector is a sector with a higher (lower) economic usage of the environment per unit labor.
That is, the resource-good sector is “dirtier (cleaner)” if lf A (S) > lm a (< lm a).
Letting Sc defined by lf A (Sc ) = lm a, the resource-good (manufacturing) sector is “dirtier”
for any S > Sc (< Sc ). Figure 1 clearly illustrates this. Note that the Brander–Taylor
model, by assuming lm = 0, focuses on the special case that the resource-good sector is“dirtier” for any level of S. By contrast, the Copeland–Taylor model, by assuming lf = 0,
focuses on the special case of the manufacturing sector being always “dirtier”.
Assume finally the preference described by a representative household with the in-7

Figure 1: Environmental usage per unit labor. The resource (manufacturing) sector ismore environmentally harmful for any S > Sc (< Sc ).
stantaneous utility satisfyingu (Cf , Cm ) = b ln Cf + (1 − b) ln Cm ,(5)where parameter 0 < b < 1 indicates the share of income spent on resource goods.

3

Properties of the supply side

As a preliminary analysis, this section formally defines two types of countries, and derives some properties of the supply side, especially the supply curve and the production
possibility frontier (PPF) in the long run.

3.1

Short-run properties

It follows from (1) and (2) that at each point in time
Xf
Xm
+
= L.
A (S)
a

(6)

Firms under perfect competition make decisions by taking environmental stock S as given.
The opportunity cost of producing manufacturing goods measured in resource goods is

8

given by the marginal rate of transformation (MRT):
MRT =

A (S)
.
a

(7)

Note that (6) gives the expression of the short-run PPF, which is a straight line on the(Xm , Xf ) plane passing through point (aL, 0) with the absolute value of its slope equal
to the MRT. Clearly, the model behaves like a Ricardian economy in the short run.
In the long run, however, environmental stock S can change over time. A production
schedule in the short run is not necessarily feasible in the long run. To understand thelong-run properties of the supply side, it is convenient to define β ≡ Lf /L as the share oflabor allocated to the resource-good sector. The production functions can be rewritten
into
Xf = A (S) βL,

Xm = a (1 − β) L.

(8)Clearly, β will be endogenously determined in equilibrium. For the moment, however, we
focus on the supply side and treat β as exogenously given.
Let P denote the relative price of manufacturing goods to resource goods. Without
loss of generality, let the resource good be the numeraire. (Thus P also gives the priceof manufacturing goods.) Wages are equalized within the country since labor is freely
mobile across sectors. The necessary condition for both sectors to be active is therefore
w = A (S) = aP , or equivalently,
A (S)
= P.
a

(9)

It follows directly from (4) and (8) thatE = (lf A (S) β + lm a (1 − β)) L,(10)

which is increasing with S and bound by lf A (S) L and lm aL (since it is the linear combination of the two). Figure 2 shows that the locus of E lies in the shadowed area between
lf A (S) L and lm aL, and rotates counter-clockwise on (Sc , lm aL) as β increases.

9Figure 2: Economic usage of the environment. Given three different environmental growth
functions G1 (S), G2 (S), and G2 (S), environmental stocks at all steady states are lessthan, equal to, and greater than Sc .

3.2

Country type

Given labor allocation β, it follows from (3) and (10) that at the steady state
G (S) = (lf A (S) β + lm a (1 − β)) L.

(11)

The stability requires that G (S) intersects the E schedule from above, namely
G0 (S) < lf A0 (S) βL.

(12)

Let S∞ (β) denote the solution(s) of S to (11) which satisfies (12). For arbitrary parameters and functional forms, S∞ (β), namely the set of environmental stocks at the steadystate corresponding to labor allocation β, can be empty, multi-valued, or discontinuous.
For simplicity, we impose the following assumption.
Assumption 2. S∞ (·) is a positive-valued and continuous function in [0, 1].
The assumption is not as restrictive as it seems. It holds in the Copeland–Taylor
model, which assumes G (0) > lm a and G0 (S) < 0. It also holds in the Brander–Taylor
model, which assumes a logistic form of G (S) with the maximum greater than lm a.
Taking the total differential of (11) yields
0
S∞
(β) =

(lf A (S∞ (β)) − lm a) L.
G0 (S∞ (β)) − lf A0 (S∞ (β)) βL
10

(13)

The stability requires the denominator to be negative. Therefore, if the resource-good(manufacturing) sector is more environmentally harmful, a marginal increase in β lowers(raises) the steady-state level of environmental stock. In other words, an expansion(contraction) of the “dirtier” sector harms (enhances) the environment.
A question naturally arising is whether a change in labor allocation, β, can change
the steady-state level of environment stock to such an extent that the “dirtier” sector
becomes the “cleaner” one. The following lemma provides the answer.8
Lemma 1. The type of a sector remains unchanged at all steady states.
This suggests that which sector is “dirtier” at the steady state is an intrinsic natureof the supply side, only depending on parameter values and functional forms. Noteas well that Assumption 2 is the requirement for this result; otherwise a sector may
be the “dirtier” one at some steady states while the “cleaner” one at others. Figure
2 illustrates three different environmental growth functions. Endowed with G1 (S), we
have S∞ (β) < Sc for all β ∈ [0, 1] in the country, meaning that the manufacturing sector
is “dirtier” at all steady states. By contrast, S∞ (β) > Sc holds for all β ∈ [0, 1] in acountry endowed with G3 (S), ensuring that the resource-good sector is “dirtier” at all
steady states. If a country is endowed with G2 (S), which passes through (Sc , lm aL),S∞ (β) = Sc holds for all β ∈ [0, 1]. This knife-edge case is of no special interest since
it resembles the classic Ricardian model at the steady state, and is excluded from theanalysis. Lemma 1 suggests that countries can be categorized into two types as follows.
Definition 2 (Country type). A country is of the BT (CT) type if the resource-good(manufacturing) sector is “dirtier” at all steady states.
It follows immediately from the discussion above that
Proposition 1. A country is either of the BT type or the CT type. In a country of the0
BT (CT) type, S∞(β) < 0 (> 0) holds for all β ∈ [0, 1].9

Figure 2 helps infer that a country is more likely to be of the BT type if it has: smaller
labor endowment (lower L), faster growth of the environment (greater G (S) for givenS), higher productivity in the resource-good sector (greater A (S) for given S), lower
89

Appendix A provides formal proofs to lemmas and propositions.
0
In the knife-edge case, S∞(β) = 0 holds for all β ∈ [0, 1].

11

manufacturing productivity (lower a), higher per unit output environmental impact in
the resource-good sector (greater lf ), or lower per unit output environmental impact inthe manufacturing sector (greater lm ). Otherwise, it is more likely to be of the CT type.

3.3

The long-run supply curve

Consider the long-run supply curve of manufacturing goods, which draws the relationship between the output of manufacturing goods and its (relative) price at the steadystate. We focus on the manufacturing good because its productivity remains constant,which produces a clear-cut distinction in the shape of the long-run supply curve between
countries of different types.
The following result is useful regarding how the (relative) price of manufacturing
goods varies with the output.
Proposition 2. In a country of the BT (CT) type, the relative price of manufacturing
goods rises (falls) with the output at the steady state.
In a country of the BT type, a shift of labor into the “cleaner” manufacturing sectormeans a shift of labor out of the “dirtier” resource-good sector, which enhances the environment and consequently the productivity in the resource-good sector, resulting in ahigher relative price of manufacturing goods. In a country of the CT type, the manufacturing sector becomes the “cleaner” one, and a shift of labor into the manufacturingsector leads to the opposite.
Proposition 2 provides how the price varies with the output. To draw the long-run
supply curve, it is also required to characterize two endpoints: Xm = 0 (where β = 1)
and Xm = aL (where β = 0). When Xm = 0, the steady-state wage in the resourcegood sector is A (S∞ (1)) and the potential wage in the manufacturing sector is aP . A
necessary condition for all labor working in the resource-good sector is A (S∞ (1)) ≥ aP ,which yields P ≤ A (S∞ (1)) /a. Similarly, we can obtain P ≥ A (S∞ (0)) /a as a necessary
condition for Xm = aL. Figure 3 applies the arguments above and draws the long-run
supply curves (of manufacturing goods) for two types of countries.

12(a) BT type.

(b) CT type.

Figure 3: The long-run supply curve. The demand curve in autarky, which is a verticalline given the Cobb-Douglas preference, and the demand curve in a small open economy,
which is a horizontal line, are also drawn.

3.4

The long-run PPF

The long-run PPF is useful for two reasons. First, it illustrates the deviation in the
production cost between private firms and the whole economy. Firms take environmental
stock as given and the opportunity cost of producing manufacturing goods (measured inresource goods) equals the MRT in (7). This deviates from the social opportunity cost,or the social MRT (SMRT), which takes into account environmental changes, and can bemeasured by the slope of the long-run PPF. Second, it provides an intuitive tool for the
analysis of welfare effects of trade.
To derive the long-run PPF, substitute S∞ (β) into the first equation in (8) to obtain
Xf = A (S∞ (β)) βL. Noting that the second equation in (8) gives β = 1 − Xm /aL, thelong-run PPF can be then written as


 

Xm
Xm
Xf = T (Xm ) = A S∞ 1 −
L−
.
aL
a

(14)

The SMRT can be expressed by
SMRT = −dT (Xm )β0= A0 (S∞ (β)) S∞(β) + MRT,dXm
awhere β instead of Xm is used to save notation. It follows immediately that

13

(15)

Proposition 3. In a BT (CT) type country, the MRT is greater (less) than the SMRT
for all Xm > 0.
Intuitively, in a BT (CT) type country, the resource-good (manufacturing) sector is“dirtier” so that the cost of producing the good is, from the perspective of the whole
economy, underestimated by private firms (which have no concern over such sectoral
distinctions in environmental impacts).
To correct externalities, in a BT type country, a tax on resource goods (or a subsidy
on manufacturing goods) can be imposed to reduce the MRT that faces private firms.
For example, if the tax rate on the production of resource goods is τ , the MRT becomes
MRT(τ ) = (1 − τ ) A (S∞ (β)) /a. The optimal tax, if focusing on the steady state, can beachieved by equalizing MRT(τ ) to the SMRT. The opposite holds in a CT type country.
It follows from (6) that, on the (Xm , Xf ) plane, the straight line connecting (aL, 0)with a certain point on the long-run PPF is actually the short-run PPF corresponding to
the level of environmental stock that allows the economy to sustainably produce at thatpoint. The following summarizes two features of the long-run PPF.
Proposition 4. In a BT (CT) type country, the long-run PPF is strictly concave (convex)
around the Xm axis; a straight line passing through (aL, 0) intersects, if any, the long-run
PPF from above (below).
Panagariya (1981) considered a two-sector model with increasing returns to scales(IRS) in one industry and decreasing returns to scales (DRS) in the other. He showed thatthe economy’s PPF is strictly concave to the origin near the IRS axis and strictly convex
to the origin near the DRS axis, and that welfare maximization requires a permanent
tax/subsidy scheme encouraging the expansion of the IRS industry and the contractionof the DRS industry. Our findings summarized in Proposition 4 can be interpreted as avariant of Panagariya’s argument. In our model, only the resource-good sector is subject
to external diseconomies. However, depending on the relative magnitude of environmental
impacts (per unit labor) of two sectors, the resource-good sector can either be an “IRS”
sector or a “DRS” sector, which drives the difference in the shape of the long-run PPF.

14

(a) BT type.

(b) CT type.

Figure 4: The long-run PPF. Panel (a) draws the long-run PPF of a BT type country,
which is strictly concave around (aL, 0). Panel (b) draws that of a CT type country,
which is strictly convex around (aL, 0). In either case, the long-run PPF is not necessarily
entirely concave or entirely convex, as illustrated in both panels. Two short-run PPFs
corresponding to S = S∞ (1) and S = S∞ (β) (where 0 < β < 1) are also drawn.

4

Autarkic equilibrium

Having characterized the supply side, the demand side can be introduced to close themodel. In this section, we consider autarky as the benchmark to compare with trade.
In autarky, demand is fulfilled by domestic supply: Ci = Xi (i = f, m). The CobbDouglas preference ensures that both goods are produced in autarky, which requires the(relative) price of manufacturing goods satisfying P = A (S) /a and the wage satisfying
w = A (S) = aP . The income is then A (S) L. The maximization of utility requires thatthe share of income spent on the resource good is b, which gives
Cf = bA (S) L,

Cm = (1 − b) aL.

(16)

Hence the demand for manufacturing goods is a vertical line on the (Xm , P ) plane, as
illustrated in Figure 3. It follows from (1) that, in autarkic equilibrium,
β = b.

(17)

Environmental stock and the (relative) price of manufacturing goods at the autarkic steady state, denoted by SA and PA respectively, satisfy SA = S∞ (b) and PA =
15A (S∞ (b)) /a. Assumption 2 assures the uniqueness and stability of the steady state.10
Finally, the utility level at the autarkic steady state can be expressed by, using (16),
VA = B + ln L + b ln A (S∞ (b)) + (1 − b) ln a,(18)where B ≡ b ln b + (1 − b) ln (1 − b).

5

Small open economy

The main purpose of the present paper is to investigate free trade between two countries
and its long-run environmental and welfare consequences. In the section, however, weanalyze trade in a small economy to highlight the difference between two types of countries
in dynamic responses to trade liberalization, and illustrate the difference intuitively withthe long-run supply curve and the long-run PPF.
Let PW denote the world (relative) price of manufacturing goods. The comparativeadvantage can be revealed by comparing the MRT with the world price: if A (S) /a > PW(< PW ), the economy has a comparative advantage in resource (manufacturing) goods.
Firms take environmental stock as given, and the economy completely specializes in thegood in which it has a comparative advantage.
In the long run, environmental stock evolves over time, thereby affecting the MRT
and consequently trade pattern and specialization pattern. Geometrically, the tradesteady state in a small open economy can be obtained by the intersection of the long-run(steady-state) supply curve and the demand curve of manufacturing goods, as illustrated
in Figure 3.
The expression of the utility level at the trade steady state, denoted by VT , varies
with specialization patterns. To see this, note that when the economy specializes in the
resource good (β = 1), we have w = A (S∞ (1)) and therefore, Cf = bA (S∞ (1)) L and
Cm = (1 − b) A (S∞ (1)) L/PW . When the economy remains diversified (0 < β < 1),
we have w = A (S∞ (β)) = aPW and therefore, Cf = bA (S∞ (β)) L = baPW L and
Cm = (1 − b) A (S∞ (β)) L/PW = (1 − b) aL. When the economy specializes in the
manufacturing good (β = 0), we have w = aPW and therefore, Cf = baPW L and
10

Proposition 1 in Brander and Taylor (1997a) and Copeland and Taylor (1999) gives similar results.

16

Cm = (1 − b) aL. It then follows that, using (5),
B + ln L + ln A (S∞ (1)) − (1 − b) ln PWVT = B + ln L + b ln A (S∞ (β)) + (1 − b) ln a





B + ln L + b ln PW + ln aif β = 1,
if 0 < β < 1,(19)

if β = 0.

Focusing on the steady-state level of utility, we can compare VT with VA to assess whether
the small economy gains or loses from trade liberalization. The following propositionsummarizes the results regarding an economy of the BT type.
Proposition 5. A small open economy of the BT type at the trade steady state can becharacterized as follows.
(i) It exports manufacturing (resource) goods if the world relative price is higher(lower) than the autarkic price, and specializes if the world price is sufficiently high (low)in the sense that PW ≥ A (S∞ (0)) /a (≤ A (S∞ (1)) /a); otherwise, it remains diversified.
(ii) The environment improves (deteriorates) if exporting manufacturing (resource)
goods.
(iii) It gains from trade if exporting manufacturing goods, or if exporting resource
goods with the world relative price low enough:A (S∞ (1))
PW < P ≡a0A (S∞ (1))
A (S∞ (b))

b
 1−b

.

(20)

It loses from trade if exporting resource goods with PW > P 0 .11
Figure 5 illustrates welfare effects of trade in a small economy of the BT type. In bothpanels, BA is the budget line at the autarkic steady state (which is also the short-run PPF
given S = S∞ (b)). In panel (a), the economy faces a world price higher than the autarkic
price and exports manufacturing goods when opened to trade (yet not high enough forthe economy to specialize). The resulting budget line at the trade steady state, B1 , canbe obtained by rotating BA outward on (aL, 0), implying a higher steady-state utility
level compared to autarky.
In panel (b), the world price is lower than the autarkic price and the economy exports
resource goods under free trade (and low enough for the economy to specialize). The
Note that P 0 < A (S∞ (1)) /a. Given PW > P 0 , the small economy may specializes in resource goods(if PW ≤ A (S∞ (1)) /a) or remain diversified (if PW > A (S∞ (1)) /a)..
11

17

(a) Exports manufacturing goods and gains.

(b) Specializes in resource goods and beak-even.

Figure 5: Welfare effects of trade in a small BT type economy. In both panels, xA and BA
are the production (consumption) bundle and budget line at the autarkic steady state;uA is the corresponding indifference curve; the dashed line is parallel to BA . In panel (a),the economy exports manufacturing goods, with the production bundle x1 and budget
line B1 . In panel (b), the economy exports and specializes in resource goods, with the
production bundle x2 and budget line B2 (which is tangent to uA ).
resulting budget line at the trade steady state, B2 , happens to be tangent to the autarkic
indifference curve, uA . Hence, panel (b) illustrates the threshold case in which the resulting steady-state utility level is the same as in autarky. If the world price falls further, thebudget line rotates outward on (0, A (S∞ (1)) L) and therefore, the small economy gains
from trade; if the opposite occurs, it loses from trade.
One may have noticed that these results are similar to those in Brander and Taylor(1997a) and Rus (2016). This should not be a surprise as our model shares the similar
structure with Rus (2016) and, less obviously, an economy of the BT type in our model isindeed isomorphic to the economy formulated in Brander and Taylor (1997a). The present
paper, however, contributes an intuitive exposition of welfare effects of trade by using thelong-run PPF. Welfare effects of trade come from two sources: changes in the terms oftrade, called the TOT effect, and changes in the productivity of the resource-good sector,
called the green effect.12 In a small open economy, the terms of trade are exogenously
12

A specific explanation of the TOT and green
 effects is as follows. Let production (consumption)Abundle at the autarkic steady state denoted by Xm, XfA , and labor allocation, production bundle, and




T
T
consumption bundle at the trade steady state denoted respectively by βT , Xm
, XfT , and Cm, CfT .

18

given and thus, the TOT effect is a static effect and unambiguously positive.13 By contrast, the green effect is a dynamic effect, which comes from environmental changes, andcan be positive or negative. When the small BT type economy exports manufacturing
goods, the expansion of the “cleaner” manufacturing sector enhances gradually the environment, which raises the productivity in the resource-good sector. This produces apositive green effect, as illustrated by an outward shift of BA to the parallel dashed line
in panel (a) of Figure 5. The total welfare effect is therefore unambiguously positive.
When the economy exports resource goods, which is “dirtier” in the BT type economy,
however, there arises a negative green effect. The total welfare effect then depends onwhich effect, the positive TOT effect or the negative green effect, dominates. Panel (b)of Figure 5 shows the threshold case, in which the two opposing forces cancel each otherout and the economy remains unchanged in terms of the steady-state utility level.
A small economy of the CT type presents quit different responses to trade liberalization as well as long-run welfare gains from trade.
Proposition 6. A small open economy of the CT type at the trade steady state can becharacterized as follows.
(i) It exports and specializes in manufacturing (resource) goods if the world relative
price is higher (lower) than the autarkic price.
(ii) The environment deteriorates (improves) if exporting manufacturing (resource)
goods.
(iii) It gains unambiguously from trade.
The intuition about a small CT type economy always gaining from trade in the longrun comes by realizing two facts. First, a small CT type economy always specializes
The green effect arises from an income gap between trade and autarky (measured in autarkic price):TA∆Green = PA Xm+ XfT − PA Xm+ XfA = (A (S∞ (βT )) − A (S∞ (b))) βT L,
Twhere the second equality follows from PA = A (S∞ (b)) /a, Xm= a (1 − βT ) L, XfT = A (S∞ (βT )) βT L,
AXm= a (1 − b) L, and XfA = A (S∞ (b)) bL. This shows that the source of the green effect is productivity
change in the resource-good sector from A (S∞ (b)) to A (S∞ (βT )). The TOT effect, on the other hand,arises from the gap between expenditure and income under trade (also measured in autarkic price):


T
T
T
T
∆T OT = PA Cm+ CfT − PA Xm+ XfT = (PW − PA ) Xm
− Cm
,



T
T
where the second equality follows from trade balance condition PW Cm+ CfT = PW Xm+ XfT .

13
T
T
Note that ∆T OT = (PW − PA ) Xm
− Cm
is always positive in a small open economy. In a twocountry world, however, the TOT effect becomes a dynamic effect and can be negative.

19

(a) Specializes in manufacturing goods and gains.

(b) Specializes in resource goods and gains.

Figure 6: Welfare effects of trade in a small CT type economy. In both panels, xA and
BA are the production (consumption) bundle and budget line at the autarkic steadystate; uA is the corresponding indifference curve. In panel (a), the economy specializes inmanufacturing goods, with the production bundle x1 and budget line B1 . In panel (b),the economy specializes in resource goods, with the production bundle x2 and budget line
B2 ; the dashed line is parallel to BA .
at the trade steady state (since the diversified steady state is unstable). Second, theresource-good sector is the “cleaner” sector in a CT type economy. If the economy faces
a world relative price higher than the autarkic price, it specializes in the production of
manufacturing goods when opened to trade, which implies that (i) there is no green effect(since it produces no resource goods), and (ii) there is a positive TOT effect (by selling
manufacturing good at higher price). The total effect is therefore positive, as illustrated
in panel (a) of Figure 6, where the production bundle at the trade steady state is x1 andthe corresponding budget line is B1 (with the absolute value of the slope equal to theworld relative price) is steeper than the autarkic budget line BA .
By contrast, if the economy faces a world relative price lower than the autarkic price,
it specializes in resource goods and there is a positive green effect on top of the positive
TOT effect. The total welfare effect is still positive, as illustrated in panel (b) of Figure 6,
where the economy produces at x2 and the resulting budget line is B2 , which lies outward
against BA . The positive green effect comes from an upward shift of the budget line; thepositive TOT effect comes from a further counter-clockwise rotation. In either case, asmall economy of the CT type enjoys an outward shift of the budget line, resulting in a20

higher steady-state utility level compared to autarky.
An interesting phenomenon featured in a small CT type economy is that, if tradepattern is determined by some other forces instead of comparative advantage, the economy
could be locked in the “wrong” specialization pattern such that it loses from trade. To see
this, suppose the world relative price is lower than autarkic price: PW < A (S∞ (b)) /a.
According to comparative advantage, when opened to trade, the economy would export
resource goods and gain from trade. However, the government intends to stimulate
manufacturing by providing subsidies so that the economy exports manufacturing goods.
At the trade steady state, the economy specializes and the MRT facing private firms
becomes A (S∞ (0)) /a. As long as the world relative price satisfies PW ≥ A (S∞ (0)) /a,
the economy is better at producing manufacturing goods and remains specializing even
without subsidies from the government. The resulting steady-state utility level, however,is lower than autarky. This can be illustrated in panel (a) of Figure 6 by drawing abudget line passing through x1 (the production bundle) with the absolute value of theslope equal to the world relative price (thus lying below the autarkic budget line).

6

Two-country tradeThis section deals with our major concern in trade between two countries, called Homeand Foreign. Both are described by equations (1) through (5) with Foreign’s variables
and functions denoted by an asterisk in the superscript. For simplicity, assume identicalpreference among two countries.
Home’s environmental stock S and Foreign’s environmental stock S ∗ evolve over time,the dynamics of which can be described by, using (3) and (10),Ṡ = G (S) − (lf A (S) β + lm a (1 − β)) L,

(21)


∗ ∗
a (1 − β ∗ ) L∗ .
Ṡ ∗ = G∗ (S ∗ ) − lf∗ A∗ (S ∗ ) β ∗ + lm

(22)

To obtain a complete description of the dynamic system, it remains to reveal how labor
allocations β and β ∗ are dependent on environmental stocks S and S ∗ . The rest of
this section proceeds in two steps. First, we derive the correspondence between labor
allocations and environmental stocks. We then move on to the characterization of tradesteady state, transition dynamics, and environmental and welfare consequences of trade.
21

6.1

Labor allocations and environmental stocks

Consider how labor allocations are determined in the short run (given environmental
stocks S and S ∗ ). For the purpose, noting that labor allocation is bound from one side
if a country specializes (only able to shift out of the specialized sector), we shall discusstrade patterns and specialization patterns as well.
Featured with the short-run Ricardian structure (namely the MRT not varying withlabor allocation given environmental stock), trade pattern can be revealed by comparing
the MRTs, namely A (S) /a in Home and A∗ (S ∗ ) /a∗ in Foreign. It is convenient to define
the comparative advantage index byv≡

A∗ (S ∗ ) a.
A (S) a∗(23)

If v < 1 (> 1), Home (Foreign) has a comparative advantage in producing resource
goods, and exports the good under free trade. As long as v 6= 1, the short-run Ricardian
structure also ensures that at least one country completely specializes.14 If v = 1, neither
country has comparative advantage, and trade pattern is indeterminate in the short run.
The characterization of labor allocations is related to which country specializes inwhich good, called world specialization pattern. There are seven of them in total:15(f,d)

Home produces only resource goods; Foreign produces both.

(f,m)

Home produces only resource goods; Foreign produces only manufactures.

(d,m)

Home produces both goods; Foreign produces only manufactures.

(d,d)

Both countries produce both goods.

(d,f)

Home produces both goods; Foreign produces only resource goods.

(m,f)

Home produces only manufactures; Foreign produces only resource goods.

(m,d)

Home produces only manufactures; Foreign produces both.

14

Suppose for example v < 1. Under constant-returns technology and perfect competition, this meansthat when trade is opened, manufacturing firms in Foreign offer higher wages than resource-good firms
can, and the opposite occurs in Home. Consequently, the manufacturing sector in Foreign and theresource-good sector in Home expand. This continues until Foreign, or Home, or both specialize.
15
Neither pattern (m,m) nor pattern (f,f) arises for 0 < b < 1.

22

It is then of interest which world specialization pattern arises given parameters and environmental stocks at the moment. Three factors prove crucial: the comparative advantage
index v defined by (23), preference parameter b, and the relative effective size defined by
z≡

aL
.
a∗ L∗

(24)

The following proposition summarizes the results regarding the determination of world
specialization pattern and corresponding labor allocations.
Proposition 7. In the short run (given environmental stocks), labor allocations andworld specialization pattern can be characterized as follows.
(i) Pattern (f,d) arises if (1 − b) z/b < v < 1, where β = 1 and β ∗ = b − (1 − b) z/v.
(ii) Pattern (f,m) arises if v ≤ (1 − b) z/b ≤ 1 and v 6= 1, where β = 1 and β ∗ = 0.
(iii) Pattern (d,m) arises if v < 1 < (1 − b) z/b, where β = b (1 + 1/z) and β ∗ = 0.
(iv) Pattern (d,f ) arises if 1 < v < bz/ (1 − b), where β = b − (1 − b) v/z and β ∗ = 1.
(v) Pattern (m,f ) arises if 1 ≤ bz/ (1 − b) ≤ v and v 6= 1, where β = 0 and β ∗ = 1.
(vi) Pattern (m,d) arises if bz/ (1 − b) < 1 < v, where β = 0 and β ∗ = b (1 + z).
(vii) Pattern (d,d) arises only if v = 1. Given v = 1, however, other pattern may also
arise since labor allocations are indeterminate by satisfying
βz + β ∗ = b (z + 1) .

(25)

Intuitively, comparative advantage index v matters as it determines trade pattern
and indicates productivity difference between two countries. A greater v thus means that
Foreign (Home) becomes better (worse), in a relative sense, at producing resource goods,
which shifts labor into (out of) the resource-good sector in Foreign (Home).16 Preference
parameter b matters since it measures the share of income spent on resource goods. A
greater b thus requires more labor to be allocated to the resource-good sector, resulting in
16
Both β and β ∗ are increasing with preference parameter b, which is straightforward since a stronger
demand for resource goods induces more labor into the sector. On the other hand, both β and β ∗ aredecreasing with the effective relative size z for v < 1, and increasing with z for v > 1. Intuitively, whenv < 1, Home exports resource goods to Foreign; the portion of labor in Home for the production of
resource-good exports (which is actually equal to b/z) falls when Foreign becomes smaller compared toHome (namely greater z). This implies a smaller β. At the same time, Foreign exports manufacturing
goods to Home, and when Home becomes bigger compared to Foreign (greater z), the portion of laborin Foreign for the production of manufacturing exports (which is actually equal to (1 − b) z/v) rises,
resulting in a smaller β ∗ . The similar argument applies to v > 1.

23

(a) b > 1/2.

(b) b < 1/2.

Figure 7: Distribution of world specialization patterns. Panel (a) illustrates the distribution for b > 1/2 and panel (b) for b < 1/2.
greater β and β ∗ . Effective relative size z matters by indicating the relative size of Home
with respect to Foreign in terms of the capacity of manufacturing production. Suppose forexample that Home is much smaller than Foreign in the effective size. Home’s response totrade can be well described by a small open economy: it completely specializes (β = 1 or0) as long as v 6= 1. By contrast, Foreign will observe little change in its labor allocation
after trade liberalization (β ∗ ≈ b). From this thought experiment, we can infer that asmaller (greater) z implies a greater likelihood for Home (Foreign) to specialize and for
Foreign (Home) to remain diversified under free trade.
When there exists no comparative advantage among two countries (v = 1), labor
allocations β and β ∗ are indeterminate in the short run (taking environmental stocks asgiven) with one degree of freedom. As long as 0 < β < 1 and 0 < β ∗ < 1, we havepattern (d,d); otherwise pattern (f,d), (f,m), (d,m), (d,f), (m,f), or (m,d) arises. In thelong run, however, labor allocations are determinate as the steady-state condition comes
into effect in the long run as well.
Figure 7 provides a geometric illustration of Proposition 7 by showing how the sevenworld specialization patterns are distributed on the (z, v) plane. The distribution appears
somehow different for b > 1/2 and b < 1/2. In particular, when two countries are of24

similar size (around z = 1) and v 6= 1, one country must specialize in resource goods if
b > 1/2, but neither will specialize in resource goods if b < 1/2.17
Figure 7 is also useful for comparative static exercises regarding how world specialization pattern varies with labor endowments, production technologies, and environmental
stocks. For example, an increase in Home’s labor endowment L leads to an increase inz but no change in v. The resulting world specialization pattern can be obtained bymoving right horizontally on the plane. For another example, an increase in Home’s
manufacturing productivity a has two effects. First, it raises Home’s effective size andthus z, implying a horizontal movement to the right. Second, it reduces Home’s MRT andthus raises v, implying an upward vertical movement. The total effect is an upper-rightmovement on the plane, with v/z remaining unchanged.

6.2

Closing the dynamic system

Having derived how world specialization pattern and labor allocations are determined by
given environmental stocks, we can substitute these results into (21) and (22) to closethe dynamic system that governs this two-country world. Noting that labor allocations
are determined in different manners for v 6= 1 and for v = 1, we proceed by considering
the two cases one by one.
The dynamic system for v 6= 1. For any S and S ∗ satisfying v 6= 1, labor allocations
are uniquely determined as given in Proposition 7. The following summarizes the results
17

Intuitively, given b > 1/2, the world demand for resource goods is relatively strong such that the
supply would not be able to match the demand if no country were devoted to the production of resource
goods. If b < 1/2, however, the demand for resource goods is relatively weak, and there is no need forthe resource-exporting country to completely specialize.

25

for easy reference:18

β (v) =




1







b 1 + 1zb − (1 − b) vz





0

∗

β (v) =




0






b − (1 − b) zvb (1 + z)





1


if v < 1 and b 1 + z1 ≥ 1,
if v < 1 and b 1 + z1 < 1,(26)

if v > 1 and b − (1 − b) vz > 0,
if v > 1 and b − (1 − b) vz ≤ 0,
if v < 1 and b − (1 − b) vz ≤ 0,
if v < 1 and b − (1 − b) vz > 0,(27)

if v > 1 and b (1 + z) < 1,
if v > 1 and b (1 + z) ≥ 1.

We write β and β ∗ explicitly as the functions of v since b and z are exogenous parameters.
Plugging β (v) and β ∗ (v) into (21) and (22) for β and β ∗ yields, using the definition of v, ∗ ∗ A∗ (S ∗ ) a
A (S ) aṠ = G (S) − lf A (S) β+ lm a 1 − β
L,∗A (S) aA (S) a∗ ∗ ∗  ∗ ∗ A (S ) a
A (S ) a
∗
∗
∗
∗ ∗
∗
∗
∗ ∗
∗
Ṡ = G (S ) − lf A (S ) β+ lm a 1 − βL∗ ,∗A (S) aA (S) a∗




(28)
(29)

which gives a complete description of the dynamic system for v 6= 1. Noting that β (v)and β ∗ (v) are piecewise functions of v, the dynamic system (28) and (29) has multiple
regimes, which differ from each other in the rule that governs the dynamics.
The dynamic system for v = 1. The argument above, however, cannot apply to thecase of v = 1 (in which pattern (d,d) may arise) where labor allocations are indeterminate
by satisfying the world market-clearing condition (25). It follows from (26) and (27)that β (v) and β ∗ (v) jumps when crossing v = 1, meaning that the dynamic system is
discontinuous around v = 1.
In summary, the dynamic system has two features: (i) multiple dynamic regimes for
S and S ∗ satisfying v 6= 1, and (ii) discontinuity at v = 1. The next two subsections
18
The expressions of β (v) and β ∗ (v) can be written more specifically given specific range of preferenceb and effective relative size z, since the set of possible world specialization patterns (patterns that may
arise as v changes) varies with the domain of b and z, as shown in Figure 7. For example, given b = 2/3
and z = 3, patterns (d,m), (d,d), (d,f), and (m,f) arise as v increases. By contrast, if b = 1/3 and z = 1,
patterns (d,m), (d,d), and (m,d) arise as v increases. Appendix B discusses this in detail.

26

examine steady state and transition dynamics while dealing with the two difficulties.

6.3

Trade steady stateThis subsection characterizes the trade steady state in this two-country world, at which
environmental stocks are endogenously determined. We proceed by considering (i) the
trade steady state satisfying v 6= 1 and (ii) that satisfying v = 1, as the dynamic systembehaves differently in the two situations.
The steady state satisfying v 6= 1. In this case, the steady state can be derived byletting Ṡ = Ṡ ∗ = 0 in (28) and (29) and solving for S and S ∗ from the two equations.
Equivalently, notice that at the steady state  ∗ ∗ 
A (S ) a,
S = S∞ β
A (S) a∗

S

∗

∗
= S∞

  ∗ ∗ 
A (S ) aβ∗,
A (S) a∗(30)

from which we can solve for S and S ∗ at the trade steady state.
Instead of solving for environmental stocks directly, an alternative approach turns out
convenient by exploiting comparative advantage index. It follows from (23) that, at thetrade steady state,∗A∗ (S∞(β ∗ (v))) a= v,
A (S∞ (β (v))) a∗(31)

from which we can solve for the steady-state level of v. Labor allocations then follow byplugging the solution into β (v) and β ∗ (v), and environmental stocks can be obtained by∗substituting labor allocations into S∞ (β) and S∞(β ∗ ). For simple notation, defineg (v) ≡∗A∗ (S∞(β ∗ (v))) a,
A (S∞ (β (v))) a∗(32)

g (v) = v.

(33)then (31) can be written as

Technically, characterizing the trade steady state in such a manner facilitates the analysis
of under what condition a certain world specialization pattern arises at the trade steadystate by translating the problem into a relatively easier one: under what condition thesolution to (33) lies in the range corresponding to that world specialization pattern.

27

The steady state satisfying v = 1. In this case, we need to obtain first labor allocations at the trade steady state, and then derive the steady-state environmental stocks.
Specifically, at the trade steady state satisfying v 6= 1, it holds that∗(β ∗ )) aA∗ (S∞= 1,
A (S∞ (β)) a∗(34)

which together with the world market-clearing condition (25), namely βz +β ∗ = b (z + 1),
can be solved for labor allocations at the trade steady state. The steady-state environ∗mental stocks follow immediately by plugging them into S = S∞ (β) and S ∗ = S∞(β ∗ ).

In both situations, we can obtain environmental stocks, labor allocations, and world
specialization pattern at the trade steady state. The following proposition summarizes
the results regarding the existence and uniqueness of trade steady state with a specificworld specialization pattern arising. For simple notation, define

∗
A∗ S∞
b − (1 − b) vz a∆f d (v) ≡,A (S∞ (1)) a∗∗A∗ (S∞(1)) a1,∆df (v) ≡v∗
A S∞ b − (1 − b) z a v∗(β ∗ )) aA∗ (S∞∆dd (β, β ∗ ) ≡,A (S∞ (β)) a∗and let




1−b
1−b
sup
∆f d (v) :z < v ≤ 1 , ∆f d = sup ∆f d (v) :z<v≤1 ,
b
b




b
b
inf
sup
∆df = inf ∆df (v) : 1 ≤ v <z , ∆df = sup ∆df (v) : 1 ≤ v <
z ,
1−b
1−b

∆inf
f d = inf

sup
∗
∗
∗
∗
∆inf
dd = inf {∆dd (β, β ) : βz + β = b (z + 1)} , ∆dd = sup {∆dd (β, β ) : βz + β = b (z + 1)} .

Proposition 8. The conditions for the existence and uniqueness of trade steady state(s)with a certain world specialization pattern arising can be characterized as follows.
(i) There exists trade steady states(s) with pattern (f,d) if and only if
sup(1 − b) z/b < 1 and ∆inf
f d ≤ 1 ≤ ∆f d ,

(35)

sup
and, if ∆inf
f d = 1 or ∆f d = 1, there exists (1 − b) z/b < v1 ≤ 1 such that ∆f d (v1 ) = 1.

28

(ii) There exists a unique trade steady state with pattern (f,m) if and only if∗
A∗ (S∞(0)) a1−bz ≤ 1.
≤A (S∞ (1)) a∗b(36)(iii) There exists a unique trade steady states with pattern (d,m) if and only if∗
A∗ (S∞(0)) a1−b ≤ 1 <z.
1b
A S∞ b 1 + z
a∗

(37)

(iv) There exists trade steady state(s) with pattern (d,f ) if and only ifsupbz/ (1 − b) > 1 and ∆inf
df ≤ 1 ≤ ∆df ,

(38)

sup
and, if ∆inf
df = 1 or ∆df = 1, there exists 1 ≤ v < bz/ (1 − b) such that ∆df (v1 ) = 1.

(v) There exists a unique trade steady state with pattern (m,f ) if and only if∗bA∗ (S∞(1)) a.
1≤z≤
1−b
A (S∞ (0)) a∗(39)(vi) There exists a unique trade steady states with pattern (m,d) if and only if∗
A∗ (S∞(b (1 + z))) a
b
z<1≤
.
1−b
A (S∞ (0)) a∗(40)(vii) There exists trade steady state(s) with pattern (d,d) if and only if
sup∆inf
dd ≤ 1 ≤ ∆dd ,

(41)

sup
∗
∗
and, if ∆inf
dd = 1 or ∆dd = 1, there exist β1 , β1 6= 0, 1 satisfying (25) and ∆dd (β1 , β1 ) = 1.

The basic idea to prove these results is to check (i) for those steady states satisfying
v 6= 1, whether there exists solution(s) to (33) that lies in the range required by thecorresponding world specialization pattern, and (ii) for those steady states satisfying
v = 1, whether there exists feasible solution(s) of β and β ∗ to (25) and (34).
Several conditions in Proposition 8 can be further specified.
Corollary 1. If Foreign is of the BT type, there exists a unique trade steady state with29

pattern (f,d) if and only if (1 − b) z/b < 1 and∗∗A∗ (S∞(b − (1 − b) z)) aA∗ (S∞(0)) a≤1<∗
A (S∞ (1)) a
A (S∞ (1)) a∗



1−b
z
b

−1
.

(42)

If Home is of the BT type, there exists a unique trade steady state with pattern (d,f ) if
and only if bz/ (1 − b) > 1 and∗(1)) aA∗ (S∞A (S∞ (0)) a∗



b
z
1−b

−1
<1≤

∗
(1)) aA∗ (S∞ .
A S∞ b − 1−b
a∗
z

(43)

If both countries are of the same type, the trade steady state with pattern (d,d), if existing,is unique. The sufficient and necessary condition for the existence is given in Table 1.
Table 1: Conditions for pattern (d,d) to arise at the trade steady state.

z < min

min

 1−b
b

 1−b
b

b
, 1−b

b
, 1−b
< z < maxz > max

 1−b
b

b
, 1−b

 1−b
b

b
, 1−b

Both BT∗ (b(1+z)))a∗ (b−(1−b)z))aA∗ (S∞A∗ (S∞<
1<∗
A(S∞ (0))aA(S∞ (1))a∗

Both CT∗ (b−(1−b)z))a∗ (b(1+z)))aA∗ (S∞A∗ (S∞>
1>∗
A(S∞ (0))aA(S∞ (1))a∗

Both BT∗ (b−(1−b)z))a∗ (1))aA∗ (S∞A∗ (S∞<1<
1−b∗
A(S∞ (1))a∗A(S∞ (b− z ))aBoth CT∗ (1))a∗ (b−(1−b)z))aA∗ (S∞A∗ (S∞
>
1
>
∗
1−b
∗
A(S
∞ (1))aA(S∞ (b− z ))aBoth BT∗ (1))a∗∗A∗ (S∞< 1 < A SA (Sb ∞1+(0))a∗A(S∞ (b− 1−ba( ∞ ( ( z1 )))a∗z ))Both CT∗ (0))a∗ (1))aA∗ (S∞A∗ (S∞>1>
1−b∗
A(S∞ (b− z ))aA(S∞ (b(1+ z1 )))a∗

Notes: There are three intervals of z, with which the condition varies.

The following gives two examples of the application of Proposition 8 and Corollary 1.
∗
Example 1. If both countries are of the BT type and A∗ (S∞(0)) a/A (S∞ (0)) a∗ = 1,

neither country can specialize in manufacturing at the trade steady state.19
To see why, note that S∞ (1) < S∞ b 1 + z1



< S∞ (0) holds since Home is of the∗∗∗
BT type and S∞(1) < S∞(b (1 + z)) < S∞(0) holds since Foreign is of the BT type.
19

Example 1 suggests that part (i) of Proposition 4 in Brander and Taylor (1998), which says thatneither country can specialize in manufacturing, comes mainly from their assumption that A (S∞ (0)) =∗∗A∗ (S∞(0)) = αK and a = a∗ = 1 (which actually implies A∗ (S∞(0)) a/A (S∞ (0)) a∗ = 1).

30∗This together with A∗ (S∞(0)) a/A (S∞ (0)) a∗ = 1 yields∗∗∗
A∗ (S∞(0)) aA∗ (S∞(0)) aA∗ (S∞(0)) a

>
>
= 1,
1
A (S∞ (1)) a∗
A (S∞ (0)) a∗
A S∞ b 1 + za∗∗∗∗(1)) aA∗ (S∞(b (1 + z))) a(0)) aA∗ (S∞A∗ (S∞<<= 1.
∗∗
A (S∞ (0)) a
A (S∞ (0)) a
A (S∞ (0)) a∗

Note that inequalities in the first line violate (36) and (37) in Proposition 8, implyingthat patterns (f,m) and (d,m) cannot arise. Those in the second line violate (39) and(40), implying that (m,f) and (m,d) cannot arise.
∗
Example 2. If both countries are of the BT type and A∗ (S∞(1)) a/A (S∞ (1)) a∗ = 1,

pattern (d,d) arises at the trade steady state.
Given the constraint above, one can verify that the conditions in Table 1 necessarily
hold for any level of z, which ensures that pattern (d,d) arises at the trade steady state.
The following proposition focuses on the stability of trade steady state given that a
specific world specialization pattern has arisen.
Proposition 9. The stability of trade steady state can be characterized as follows.
(i) The steady state with one country specialized in manufacturing, either pattern(f,m), (d,m), (m,f ), or (m,d), if existing, is locally stable.
(ii) The steady state with pattern (f,d) or (d,f ), if existing, is locally stable if
g 0 (v) < 1.

(44)(iii) The steady state with pattern (d,d), if existing, is locally stable if both countries
are of the BT type, and unstable if both countries are of the CT type. If the two countries
are of different types, the steady state(s) with pattern (d,d) is locally stable if
∗0
z
S∞


∂v
∗0
∗ ∗0 ∗ ∗
0 ∂v
− S∞
G
−
l
A
β
L
(G0 − lf A0 βL) > 0,
f
∗
∂S
∂S
∂v
∗0
0 ∂v
S∞
z ∗ − S∞
< 0.
∂S
∂S

(45)
(46)

The following corollary is also useful.
Corollary 2. If both countries are of the BT type, there exists a unique, stable steadystate. If both countries are of the CT type, there may exist multiple stable steady states
31

(with the same or different world specialization patterns arising), at which at least one
country completely specializes.
The corollary suggests that there exists no stable steady state with pattern (d,d)
arising when two countries of the CT type are trading with each other. Intuitively, thecomparative advantage in a CT type economy is self-reinforcing in the sense that an
expansion of the exporting sector makes the CT type economy better at producing its
exports. This self-reinforcing process amplifies the short-run Ricardian structure so thatat least one country specializes at the steady state.

6.4

Transition dynamics

Transition from autarky toward the trade steady state can be analyzed intuitively withphase diagram, which draws the Ṡ = 0 and Ṡ ∗ = 0 curves on the (S, S ∗ ) plane. Appendix
C explains technical details about how to draw phase diagram. Depending on parameters and functional forms, a variety of steady-state world specialization patterns andtransition dynamics can arise. This subsection illustrates three example phase diagrams,
all satisfying: (i) demand for resource goods is relatively strong, namely b > 1/2, and(ii) two countries are similarly sized, namely (1 − b) /b < z < b/ (1 − b). Given this, the(S, S ∗ ) plane can be then divided into four regions, corresponding with patterns (f,m),(f,d), (d,f), and (m,f), respectively (see Appendix B for detail).
Figure 8 illustrates the first example phase diagram, in which both countries are of the
BT type. Panel (a) gives the whole picture of the dynamic system, and panel (b) provides∗∗
an enlarged view of the shadowed area [S∞ (1), S∞ (0)] × [S∞(1), S∞(0)] in panel (a) where

all possible steady states lie. At the autarkic steady state, environmental stocks are∗(S∞ (b) , S∞(b)), which lies in the region corresponding with pattern (f,m). This meansthat pattern (f,m) arises on the spot as two countries begin to trade with each other.
Since Home exports the “dirtier” resource goods and Foreign the “cleaner” manufacturing
goods, Home’s environment deteriorates gradually and Foreign’s improves. This yields∗
an upper-left trajectory from the autarkic steady state (S∞ (b) , S∞(b)) converging to thetrade steady state (S∞ (1) , ST∗ ), where pattern (f,d) arises.
It is worth noting that in this example the Ṡ = 0 and Ṡ ∗ = 0 curves are positively (yetnot necessarily strictly) sloped. This is because, as shown in Appendix C, both countries
are of the BT type. The uniqueness of trade steady state then follows immediately.
32

0
∗
𝑆̇ =

d)

𝑣=
Trade steady state

𝑆"∗

1−𝑏
𝑧
𝑏

Autarkic steady state𝑆̇ = 0

∗ 𝑏
𝑆$

∗ 1
𝑆$

(a) Phase diagram on the (S, S ∗ ) plane.

(f,

)
(f,m

Foreign’s environemtal stock 𝑆 ∗∗𝑆$
0 (d,f)

𝑆$ 1

𝑆$ 𝑏

Home’s environemtal stock 𝑆

𝑆$ 0(b) An enlarged view of the shadowed area.

Figure 8: An example phase diagram with both countries of the BT type and pattern (f,d)∗
arising at the trade steady state. Point (S∞ (b) , S∞(b)) indicates the levels of environmental stocks at the autarkic steady state, (S∞ (1) , ST∗ ) gives the levels of environmental
stocks at the trade steady state. Panel (b) enlarges the shadowed area in panel (a).
Moreover, world specialization pattern changes from (f,m) right after trade to (f,d) at
the steady state, suggesting that specialization pattern or even trade pattern could varyduring the transition.
Figure 9 gives other two example phase diagrams, in both of which Home is of the BT
type and Foreign of the CT type. The autarkic steady-state levels of environmental stocks
are located in the region corresponding to pattern (f,d), meaning that pattern (f,d) arises
right after trade is opened. By exporting resource goods, which are “dirtier” to a BT type
country, Home’s environment deteriorates gradually. By exporting manufacturing goods,
which are “dirtier” to a CT type country, Foreign’s environment deteriorates gradually,
too. Therefore, two countries of different types could export their respective “dirtier”
goods to each other, which worsens the environment in both countries.
In panel (a) of Figure 9, the Ṡ ∗ = 0 curve intersects with the Ṡ = 0 curve only once inthe (f,m) region, yielding a unique trade steady state with pattern (f,m) arising. In panel(b), the Ṡ ∗ = 0 curve in the (f,d) region is not negatively sloped everywhere such that itintersects with the Ṡ = 0 curve three times, meaning that there exist three trade steadystates: one with pattern (f,m) and the other two with pattern (f,d) arising. This verifies

33

(a) Unique trade steady state.

(b) Multiple trade steady states.

Figure 9: Two example phase diagrams with Home of the BT type and Foreign of the CT
type. In panel (a), there exists a unique trade steady state with pattern (f,m). In panel(b), three trade steady states exist, one with pattern (f,m) and two with pattern (f,d).
the existence of multiple trade steady states claimed in Proposition 8 and Corollary 1.
Moreover, it is easy to see in panel (b) that, among the two trade steady states withpattern (f,d), only (S∞ (1) , ST∗ 1 ) is stable. It then depends on other conditions to whichstable trade steady state (among the remaining two) the two-country world converges.

6.5

Environmental and welfare consequences of trade

Consider now environmental impacts and welfare gains from trade in the long run bycomparing the steady-state levels of environmental stocks and utility in autarky withthose under free trade. The following is concerned with environmental impacts of trade.
Proposition 10. Environmental impacts of trade can be characterized as follows.
(i) If two countries are of the same type, the environment improves in one country(by exporting “cleaner” goods) and deteriorates in the other (by exporting “dirtier” goods).
(ii) If two countries are of different types, the environment either improves in bothcountries (by both exporting their respective “cleaner” goods), or deteriorates in both countries (by both exporting their respective “dirtier” goods).
Intuitively, if two countries are of the same type and trading with each other, thesame good is either the “dirtier”, or the “cleaner” good in both countries. Hence, if one
34

country exports the “dirtier” good, the other must export the “cleaner” good. Suppose
for concreteness that both countries are of the BT type. Then the country exporting
resource goods are exporting the ‘dirtier” good. The other country then exports the“cleaner” manufacturing good. Driving the expansion of the exporting sectors, trade
harms the environment in the resource-exporting country but enhances that in the other.
By contrast, in a world with two countries of different types, the same good is “dirtier”
in one country yet “cleaner” in the other. Therefore, either both countries export theirrespective “dirtier” goods, or their respective “cleaner” goods to each other. In the former
case, trade improves the environment in both countries, and in the latter, as illustrated
in Figure 9, trade harms the environment in both countries.
As for welfare effects of trade in the long run, Home’s utility level at the trade steadystate and that of Foreign can be expressed by
B + ln L + ln A (S∞ (1)) − (1 − b) ln PT
if βT = 1,
VT = B + ln L + b ln A (S∞ (βT )) + (1 − b) ln a if 0 < βT < 1,





B + ln L + ln a + b ln PT
if βT = 0,∗B + ln L∗ + ln A∗ (S∞(1)) − (1 − b) ln PT
if βT∗ = 1,
∗VT∗ = B + ln L∗ + b ln A∗ (S∞(βT∗ )) + (1 − b) ln a∗ if 0 < βT∗ < 1,





B + ln L∗ + ln a∗ + b ln PT
if βT∗ = 0,
where βT and βT∗ denote labor allocations and PT the world relative price at the tradesteady state. Note that the world price at the steady state is endogenously determinedin this two-country world by general equilibrium and steady state conditions. In essence,
if there exists a country remaining diversified at the trade steady state, the world priceis determined by technological condition (the MRT) in the diversified country. If bothcountries specialize, the world price is then determined by technological conditions inboth countries as well as the balance of trade.
Based on the observation above, the following proposition summarizes the results
regarding welfare gains from trade between two countries of the BT type.
Proposition 11. If both countries are of the BT type, welfare effects of trade in the longrun are as follows.

35

(i) The country exporting “cleaner” manufactures gains unambiguously from trade.
(ii) The country exporting “dirtier” resource goods may gain or lose. It gains fromtrade if the world relative price is low enough in the sense that20


 b

 A(S∞ (1)) A(S∞ (1)) 1−baA(S∞ (b))
PT < P 00 ≡
b

 1−b
∗ (1))∗ (1))A∗ (S∞
 A∗ (S∞∗∗∗aif Home exports resource goods,(47)
if Foreign exports resource goods.

A (S∞ (b))

It loses if PT > P 00 .
The following result is concerned with welfare effects of trade if two trading countries
are of the CT type.
Proposition 12. If both countries are of the CT type, welfare effects of trade in the longrun are as follows.
(i) The country exporting “cleaner’ resource goods gains unambiguously from trade.
(ii) The country exporting “dirtier” manufactures may gain or lose. It gains from trade
if it specializes and the other country remains diversified, or if both countries specialize
with the world relative price high enough in the sense that21

PT > P 000 ≡∗ (b))
 A∗ (S∞∗if Foreign exports manufactures, A(S∞ (b))

if Home exports manufactures.

a
a

(48)

It loses from trade if both countries specialize with PT < P 000 , or if it remains diversified.
Deviating from a small open economy of the CT type (which necessarily specialize at
the trade steady state), a country of the CT type in a two-country world may remain diversified at the steady state because of the limited size of the world market. This together
with the fact that the world price is endogenously determined produces a possibility of anegative TOT effect in the two-country world.
To see this concretely, suppose Foreign exports manufacturing goods and remains
diversified at the steady state. Compared to autarky, environmental stock in Foreign
decreases, implying a productivity decline in the resource-good sector and consequently
a negative green effect (since Foreign also produces resource goods). This also produces a20
21With (47) holding, it must specialize in the production of resource goods.
The former includes patterns (m,d) and (d,m), and the latter includes patterns (m,f) and (f,m).

36

(dynamic) negative TOT effect (since the world price of manufacturing goods, Foreign’sexports, falls over time). The total welfare effect is therefore unambiguously negative.
It is worth emphasizing that the country exporting manufacturing goods may loss
from trade even if completely specializing. This is because that the CT type country canspecialize in manufacturing goods at the trade steady state while facing a lower relative
price compared to autarky. If this is the case, there is no green effect but a negative TOT
effect, and the resulting budget line lies below that in autarky, implying a welfare loss.
Thus far we consider two trading countries of the same type and show that at least
one country (the country exporting the “dirtier” good) gains from trade in the long run.
The results mimic those in Brander and Taylor (1998) and Copeland and Taylor (1999).22
The following proposition suggests that, however, if two countries are of different types,
it is possible for both to lose from trade in the long run.
Proposition 13. If two countries are of different types, welfare effects of trade in thelong run are as follows.
(i) Both countries gain unambiguously from trade if they export their respective “cleaner”
goods to one another (namely the BT type country exports manufactures and the CT type
country exports resource goods), or if trade pattern is reversed with (47) and (48) holding.
(ii) Both countries lose from trade if they export their respective “dirtier” goods to one
another with (47) and (48) holding reversely.
Part (ii) of the proposition highlights the possibility that trade can harm both trading
countries. For such a pessimistic scenario to arise it requires that trade liberalization
harms the environment in both countries, which can be satisfied only if (i) two countries
are of different types, and (ii) the BT type country exports resource goods and the CT
type country exports manufacturing goods. If this is the case, the openness of trade
shifts more labor into the “dirtier” sector in both countries, resulting in environmental
deterioration and a productivity decline of the resource-good sector in both countries.
In such a circumstance, it suffices for the CT type country to lose from trade byremaining diversified at the trade steady state, as illustrated in panel (a) of Figure 10.
22

As mentioned in the analysis of small open economy, a BT type country in our model is isomorphic
to the economy formulated in Brander and Taylor (1998), whereas a CT type country here is isomorphic
to the economy formulated in Copeland and Taylor (1999). Therefore, the two-country trade in Brander
and Taylor (1998) is presented in our model by a trade between two countries of the BT type, whereasthat in Copeland and Taylor (1999) is presented by a trade between two countries of the CT type.

37

First, it faces a negative green effect. Staying diversified, the resource-good sector still
contributes to the economy and a productivity decline in the sector reduces the total
income to the country. Second, it faces a negative TOT effect. Remaining diversified
means that the world price is determined by technological condition (the MRT) in thediversified country and thus, a productivity decline in the resource-good sector raises therelative price of imports to exports (recalling that the CT type country are now exporting
manufactures). The two negative effects together bring about a long-run welfare loss.
As mentioned above in the discussion of trade between two CT type countries, theCT type country could lose from trade even if specializing in manufacturing goods. Then
there is no green effect as it produces no resource goods. The total effect comes only fromthe TOT effect, which is not necessarily positive since the CT type country can specialize
in manufacturing while facing a relative price lower than autarky.
In the BT type country, a productivity decline in the resource-good sector (which isthe exporting sector now) also produces a negative green effect. On the other hand, theBT type country enjoys a positive TOT effect by facing a higher relative price of thecountry’s exports to imports compared to autarky (otherwise it will not export resource
goods). The BT type country loses from trade in the long run only if the negative green
effect dominates the positive TOT effect, which is the case illustrated in panel (b) of
Figure 10.23
23

The following concrete example may provide further intuition. Suppose energy is produced withcoal in one country and with hydro power in the other, so that energy-intensive manufacturing pollutes
in the coal country but not in the hydro-power country. Both countries have renewable resource stocks
whereas the hydro-power country is more vulnerable to excessive harvesting than the coal country is.
Then, if trade stimulates manufacturing in the coal country, two effects emerge. First, the resource-good
sector becomes less productive owing to environmental damage (which we call a negative green effect);second, resource goods become relatively more expensive compared to manufacturing goods, worsening
the trade condition (which we call a negative TOT effect). With the two negative effects, trade openness
unambiguously harms the coal country. On the other hand, by exporting resource goods, resource stockis depleted as well in the hydro-power country (a negative green effect). If for some reason, say, the hydropower country is relatively small, the prices are determined according to the coal country’s technologicalconditions, the hydro-power country would benefit from a rise in the relative price of resource goods (apositive TOT effect). Facing these two counter-effects, the hydro-power country may gain or lose fromtrade, and loses if the negative green effect dominates. We are grateful to an anonymous referee forkindly suggesting this intuitive example.

38

(a) Home (BT type).

(b) Foreign (CT type).

Figure 10: Both countries lose from trade with pattern (f,d) arising at the steady state.
Panel (a) draws the long-run PPF in Home, with xA and BA denoting the production(consumption) bundle and budget line at the autarkic steady state, uA the corresponding
indifference curve, xT and BT the production bundle and budget line at the trade steadystate; the dashed line parallel to BA . Panel (b) draws Foreign counterparts. Note that
BT and BT∗ are identically sloped (the absolute value equal to the world price PT ).

7

Numerical example

For illustration purpose, specify the general model analyzed in previous sections by letting
A (S) = αS, A∗ (S ∗ ) = αS ∗ ,(49a)

a = a∗ = 1,(49b)lf = lf∗ = 1,
G (S) =



δS

g
K,
if S ≤ δ+gg (K − S)g
if S > δ+g
K,G∗ (S ∗ ) =

(49c)


δ ∗ S ∗if S ∗ ≤ δ∗g+g∗ K ∗ ,g ∗ (K ∗ − S ∗ )if S ∗ > δ∗g+g∗ K ∗ ,

∗

∗

(49d)
where α, δ, g, δ ∗ , and g ∗ are parameters. Note that we assume that the growth functionsof the environment are “tent-shaped.” 24 Assumption 2 can be replaced by
24This tent-shaped environmental growth function approximates the widely used logistic function whileremaining highly tractable. See Benchekroun (2008) and Benchekroun and Long (2016) for applications.

39

Table 2: Numerical specification.
Specification

b

α

δ

g

lm

K

δ∗

g∗

∗
lm

K∗

I

0.55

1

9

1

1

2

9

1

1

2

II

0.55

1

9

1

1

2

9

1

1.5

3

Assumption 3. Home’s labor endowment and parameters satisfy
L≤

δg K
δ
and L ≤.
α
δ + g lm

Foreign’s labor endowment and parameters satisfy
L∗ ≤

7.1

δ∗
δ∗g∗ K ∗and L∗ ≤ ∗.
∗α
δ + g ∗ lm

Country type

Given (49), the steady-state environmental stocks given labor allocations, namely S∞ (β)∗(β ∗ ) in Foreign, can be expressed byin Home and S∞

gK − (1 − β) lm L
S∞ (β) =,g + αβL∗ ∗L
g ∗ K ∗ − (1 − β ∗ ) lm∗∗
S∞ (β ) =.
∗∗ ∗g + αβ L(50)

It then follows that0S∞(β) ≶ 0 if

K
1
L
≶
− ,
g
lm α

∗0
S∞
(β ∗ ) ≶ 0 if

L∗
K∗
1
≶
−
.
∗
g∗
lm
α

(51)

0
Recalling that Home is of the BT (CT) type if S∞< 0 (> 0), and Foreign is of the BT∗0(CT) type if S∞< 0 (> 0), equation (51) gives the condition for country types.

7.2

World specialization pattern at the steady state

Applying Propositions 7 and 8, we can derive the condition for a certain world specialization pattern to arise at the trade steady state, and Proposition 9 can be applied to checkthe stability of the steady state. To illustrate these results in a more specific manner,in what follows we present two numerical examples to visualize how world specialization40

patterns, and welfare gains from trade are distributed on the (L, L∗ ) plane.
Table 2 summarizes the numerical specifications. Specification I focuses on two symmetric countries (except for labor endowments) and specification II allows two countries
differing in environmental impacts from the manufacturing sector and in the carrying
capacity of the environment. For both specifications, Home is of the BT (CT) type if
L < 1 (> 1) and Foreign is of the BT (CT) type if L∗ < 1 (> 1).
Figure 11 draws the distribution of world specialization patterns for specification I,
and Figure 12 for specification II. In both figures, when there exists a country being of
the CT type (either L > 1 or L∗ > 1), there are areas corresponding to different world
specialization patterns overlapping, which means that, given any (L, L∗ ) within the area,
there exist multiple trade steady states, which correspond to these different patterns.
This verifies the existence of multiple trade steady states suggested by Proposition 8.

7.3

Welfare effects of trade in the long run

With the same numerical specification, we can assess the long-run welfare effects of tradeby showing which area on the (L, L∗ ) plane corresponds to (i) both countries gain fromtrade, (ii) Home gains and Foreign loses, (iii) Home loses and Foreign gains, (iv) both
lose from trade, and (v) both remain unchanged. Figure 13 illustrates the results for
specification I, and Figure 14 for specification II. It only arises in the latter specificationthat both countries lose from trade, where Foreign’s environment has greater carrying
capacity and thus, Foreign is more likely to remain diversified under free trade when it isof the CT type. The existence of the area of both losing verifies part (ii) of Proposition
13. Other areas in both figures are also consistent with Propositions 10 to 13.

7.4

Regime change

The discussion above also suggests that trade pattern is crucially dependent on the typesof countries involved in trade. A change in parameters may alter country types, thus
resulting in a dramatic change in trade pattern. Consider two ex-ante identical countries∗
described by (49) with L = L∗ , K = K ∗ , and lm = lm
. Suppose that initially labor en-

dowments are less than g (K/lm − 1/α), thereby both countries being of the BT type andthere is no trade between the two identical BT type countries even when trade is liberalized between them. Suppose then labor endowments in both countries increase gradually
41

1.5

1.5

1.5

1.0

1.0

0.5

0.5

(f,d)

,d

)

1.0

(d,f)

(d

0.5

0.0

0.0
0.0

0.5

1.0

1.5

0.0

0.0

(a) WSP right after trade opened.

0.5

1.0

1.5

0.0

(b) WSP at trade steady state.

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

0.0
0.0

0.5

1.0

1.5

0.5

1.0

1.5

0.0

(e) Pattern (d,m).

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

0.0
0.5

1.0

(g) Pattern (d,f).

1.5

0.5

1.0

1.5

(f) Pattern (d,d).

1.5

0.0

1.5

0.0
0.0

(d) Pattern (f,m).

0.0

1.0

(c) Pattern (f,d).

1.5

0.0

0.5

0.0
0.0

0.5

1.0

(h) Pattern (m,f).

1.5

0.0

0.5

1.0

1.5

(i) Pattern (m,d).

Figure 11: Distribution of world specialization pattern (WSP) with specification I in Table
2. The horizontal axis represents Home’s labor endowment and the vertical Foreign’s.
Home is of the BT (CT) type when its labor endowment less (greater) than one. The
same holds for Foreign. Panel (a) gives WSP distribution right after the openness of
trade, and panel (b) gives that at stable trade steady states. Panels (c) to (k) illustrate
respectively the distribution of each WSP at the trade steady state.

42

1.5

1.5

1.5

1.0

1.0

0.5

0.5

,d

)

(f,d)

(m

1.0

f
(m,

)

0.5

(d,f)
0.0

0.0
0.0

0.5

1.0

1.5

0.0

0.0

(a) WSP right after trade opened.

0.5

1.0

1.5

0.0

(b) WSP at trade steady state.

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

0.0
0.0

0.5

1.0

1.5

0.5

1.0

1.5

0.0

(e) Pattern (d,m).

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

0.0
0.5

1.0

(g) Pattern (d,f).

1.5

0.5

1.0

1.5

(f) Pattern (d,d).

1.5

0.0

1.5

0.0
0.0

(d) Pattern (f,m).

0.0

1.0

(c) Pattern (f,d).

1.5

0.0

0.5

0.0
0.0

0.5

1.0

(h) Pattern (m,f).

1.5

0.0

0.5

1.0

1.5

(i) Pattern (m,d).

Figure 12: Distribution of world specialization pattern (WSP) with specification II in
Table 2. The horizontal axis represents Home’s labor endowment and the vertical Foreign’s. Home is of the BT (CT) type when its labor endowment less (greater) than one.
The same holds for Foreign. Panel (a) gives WSP distribution right after the openness of
trade, and panel (b) gives WSP distribution at stable trade steady states. Panels (c) to(k) illustrate respectively the distribution of each WSP at the trade steady state. Notethat there exists no steady state with pattern (d,m).

43

1.5

1.5

1.0

1.0

Both gain
Home gains, Foreign loses
Home loses, Foregin gains
0.5

0.5

Both lose
Break even

0.0

0.0
0.0

0.5

1.0

1.5

0.0

(a) Welfare effects of trade.

0.5

1.0

1.5

(b) Welfare effects and WPS.

Figure 13: The long-run welfare effects of trade with specification I in Table 2. Panel (a)draws only the distribution of welfare effects; panel (b) draws the distribution of welfare
effects together with the distribution of world specialization patterns. Given specification
I, there is no labor endowments such that both countries lose from trade.

1.5

1.5

1.0

1.0

Both gain
Home gains, Foreign loses
Home loses, Foregin gains
0.5

0.5

Both lose
Break even

0.0

0.0
0.0

0.5

1.0

1.5

(a) Welfare effects of trade.

0.0

0.5

1.0

1.5

(b) Welfare effects and WPS.

Figure 14: The long-run welfare effects of trade with specification II in Table 2. Panel (a)draws only the distribution of welfare effects; panel (b) draws the distribution of welfare
effects together with the distribution of world specialization patterns.

44

by the same amount, which represents an upper-right movement along the ray L/L∗ = 1
in panel (b) of Figure 11. As long as labor endowment is less than g (K/lm − 1/α) (equal
to unity in the numerical examples), there remains no trade between the two countries.
When L and L∗ exceed that level, however, both countries turn into the CT type andconsequently trade volume between them soars. At the trade steady state, world specialization pattern is (f,d) or (d,f) as shown in the figure.

8

Conclusion

This study synthesized two kinds of models in the literature on trade and environment:
the model regarding renewable natural resources developed by Brander and Taylor (1997a,
1998), and that formulating inter-industrial externalities of industrial pollution developed
by Copeland and Taylor (1999). Our hybrid model captured trade between countries witha variety of sources of environmental deterioration. Yet involving regime changes anddiscontinuity in the dynamic system, a full treatment of the model provides insights into
specialization patterns among trading countries, and sheds light on environmental andwelfare consequences of trade between emerging economies (where industrial pollution
is the main source of environmental deterioration) and less developed economies (where
inappropriate use of natural resources is the primary factor of such damage).
A distinct feature of our model is the capability of formulating two types of countries(categorized by whether the resource-good sector, namely the environmental sensitive
sector, is more environmentally harmful). This provides a framework to analyze trade
between countries of different types, which in our opinion represents typical contemporary
trades between emerging economies and less developed (yet resource-rich) economies. We
showed that free trade between a country exporting manufacturing goods and a country
exporting resource goods may result in both countries losing from trade in the longrun. No formal analysis has thus far been undertaken regarding such a possibility inthe context of South–South trade without appropriate government policies to control
economic activities that harm the environment. This finding highlights the significanceof policy interventions for environmental preservation in developing countries, which tend
to be less stringent compared to developed countries.
In reality, various policies regulating pollution emissions and/or resource extraction

45

are implemented in developing economies. The analysis of trade and the environment
with explicit considerations of policy interventions is of great importance. Moreover, we
assumed in the present paper that factor endowment (labor) and manufacturing productivities are exogenous. Endogenising these factors would also have important implications
for considering the interaction between economic growth and the environment in open
economies. Extensions of our model in these directions are left for future research.

A

ProofsA.1

Proof of Lemma 1

Assume to the contrary that the sign of lf A (S∞ (β)) − lm a changes as β varies. From thecontinuity of S∞ (β) (Assumption 2), there exists β0 ∈ [0, 1] satisfying lf A (S∞ (β0 )) = lm a.
Two results follow: (i) S∞ (β0 ) = Sc since Sc is defined by lf A (Sc ) = lm a, and (ii) G (Sc ) =(lf A (Sc ) β0 + lm a (1 − β0 )) L = lm aL. The two results together imply that S∞ (β) = Sc for allβ ∈ [0, 1], and thus lf A (S∞ (β))−lm a = 0 holds for all β ∈ [0, 1], which leads to a contradiction.
To see why, assume to the contrary that there exists β1 6= β0 such that S∞ (β1 ) 6= Sc . It
follows immediately that G0 (Sc ) ≥ lf A0 (Sc ) β1 L; otherwise, we would rather have S∞ (β1 ) = Sc .
On the contrary, the stability at β = β0 requires G0 (Sc ) < lf A0 (Sc ) β0 L, which together with
G0 (Sc ) ≥ lf A0 (Sc ) β1 L gives β1 < β0 . Clearly, there exists β2 ∈ [β1 , β0 ) such that G0 (Sc ) =lf A0 (Sc ) β2 L. For any β ∈ (β2 , 1], the stability condition G0 (Sc ) < lf A0 (Sc ) βL holds and wehave S∞ (β) = Sc . For any β ∈ [0, β2 ], however, the stability condition fails to hold and we haveS∞ (β) 6= Sc . This gives limβ→β + S∞ (β) = Sc 6= S∞ (β2 ), meaning that S∞ (β) is discontinuous2
at β2 . This contradicts the continuity of S∞ (β). Hence, S∞ (β) = Sc for all β ∈ [0, 1].

A.2

Proof of Proposition 1

Since we have ignored the knife-edge case in which lf A (S∞ (β)) = lm a for all β ∈ [0, 1], according
to Lemma 1, either lf A (S∞ (β)) > lm a holds for all β ∈ [0, 1], or lf A (S∞ (β)) < lm a holds forall β ∈ [0, 1]. By the definition of country type, in the former case the country is of the BT type0 (β) < 0 (> 0)and in the latter case it is of the CT type. From (12) and (13), it follows that S∞
holds for all β ∈ [0, 1] in the country of the BT (CT) type.

A.3

Proof of Proposition 2

Plugging S∞ (β) into (9) gives A (S∞ (β)) /a = P . Taking the total differential yields, using (8),0 (β)
A0 (S∞ (β)) S∞
dP
=−
.
2
dXm
a L
0 (·) < 0 (> 0) holds in a BT (CT)
Recall that A0 (·) > 0 and, according to Proposition 1, S∞
type country. It then follows that dP/dXm > 0 (< 0) holds in a BT (CT) type country.

46

A.4

Proof of Proposition 30 (·) < 0 (> 0) in a country of the BT (CT) type, which together
According to Proposition 1, S∞
with (15) gives MRT > SMRT (< SMRT) for all β ∈ (0, 1].

A.5

Proof of Proposition 4

Consider a certain point (Xm1 , Xf 1 ) on the long-run PPF, which corresponds to the steady-state
environmental stock S∞ (β1 ). From (6), the straight line connecting (aL, 0) with (Xm1 , Xf 1 ) is
the short-run PPF given S = S∞ (β1 ). If the country is of the BT (CT) type, from Proposition
3, the MRT is greater (less) than the SMRT at (Xm1 , Xf 1 ) and the short-run PPF intersects thelong-run PPF from above (below), which implies that the long-run PPF is necessarily strictly
concave (convex) around (aL, 0).
To verify this, use (15) and β = 1 − Xm /aL to obtaindT (Xm ) =
dXm
∆
= 2 ,
a L
00



dT (Xm )
dXm



d
=
dβ



β A (S∞ (β))
00−A (S∞ (β)) S∞ (β) −aa0 (β))2 β + A0 (S (β)) S 00 (β) β + 2A0 (S (β)) S 0 (β). For the sakewhere ∆ ≡ A00 (S∞ (β)) (S∞∞∞∞∞of notation, henceforth use, say, A0 instead of A0 (S∞ (β)). It follows from (13) that
0 LΦ − Φ L (G00 S 0 − l A00 S 0 βL − l A0 L)
lf A0 S∞
2
1
f
f
∞
∞
Φ22



Φ1 L2
Φ1
00
00
0
=
l
A
βL
−
G
,
2l
A
+
f
f
Φ2
Φ22

00
S∞
(β) =

where Φ1 ≡ lf A − lm a, Φ2 ≡ G0 − lf A0 βL. Thus, we have
Φ1 L
00
A0 S ∞
β = 2 A0 βL
Φ2





Φ1
00
00
2lf A +
lf A βL − G
.
Φ2
0



02 β = Φ L/Φ2 Φ A00 βL and 2A0 S 0 = Φ L/Φ2 2Φ A0 . Thus,
Similarly, we can obtain A00 S∞
2
1
1
1
∞
2
2





Φ1 L
Φ1
0
0
00
00
00
0
∆ = 2 A βL 2lf A ++ Φ1 A βL + 2Φ2 A
lf A βL − G
Φ2
Φ2
 0
 
Φ1 L 02
G
Φ1
d G0
= 2 A
2 0−
βL
.
A
Φ2
dS A0
Φ2
It then follows that
T 00 (Xm ) =∆Φ1 A02=a2 L
a2 Φ22

 0
 
G
Φ1
d G0
2 0−
βL
.
A
Φ2
dS A0

It then follows that, noting that β → 0 as Xm → aL,
lim T 00 (Xm ) =Xm →aL

47

2Φ1 A0 G0.
a2 Φ22

The stability requires G0 − lf A0 βL < 0 holds for all β ∈ [0, 1], which implies G0 < 0 (sinceG0 − lf A0 βL = G0 as β = 0). This means that T 00 (Xm ) has the opposite sign to Φ1 as Xm → aL,
which is negative (positive) in a country of the BT (CT) type.

A.6

Proof of Proposition 5

We consider specialization pattern, then environmental consequences, and finally welfare effectsof trade, all focusing on the steady state.

Specialization pattern. Given that the world relative price satisfies
A (S∞ (1))
A (S∞ (0))< PW <,
aathere exists a unique steady state in a small economy of the BT type, as illustrated in panel (a)of Figure 3, at which the economy remain diversified. To see why, suppose that the economy
specializes in the resource good. The environmental stock then approaches S∞ (1) over time.
With PW > A (S∞ (1)) /a, PW > A (S) /a will hold at some point in time, meaning that theeconomy loses its comparative advantage in the resource good and thus, is unable to specialize
in it. A similar argument applies if the economy specialized in the manufacturing good. Clearly,
if the world price is sufficiently high (low) such that PW ≥ A (S∞ (0)) /a (≤ A (S∞ (1)) /a), theeconomy necessarily specialize in the manufacturing (resource) good at the steady state.

Environmental consequences. Let βT denote labor allocation at the trade steady state.
When exporting manufacturing (resource) good, either diversified or specialized, it holds that0 (·) < 0 holds in an economy of the BT type, the trade steady-state
βT < b (> b). Since S∞
level of environmental stock, which can be expressed by S∞ (βT ), satisfies S∞ (βT ) > S∞ (b)(< S∞ (b)), implying an increase (decline) in environmental stock compared to autarky.

Welfare effects. At the autarkic steady state, the utility level can be expressed by, according
to (18) and using PA = A (S∞ (b)) /a,
VA = B + ln L + ln a + b ln PA .
At the trade steady state, if the economy remains diversified, the utility level can be expressedby, according to (19),VT = B + ln L + b ln A (S∞ (βT )) + (1 − b) ln a,which can be rewritten into, noting that PW = A (S∞ (βT )) /a when remaining diversified,
VT = B + ln L + b ln PW + ln a.

48

If the economy specializes in the manufacturing good at the trade steady state, we have thesame expression as above. Compared with the autarkic state, the utility level changes by
VT − VA = b ln

PW
.
PA

(52)

This has two implications. First, the economy gains from trade (VT > VA ) if it exports manufacturing goods at the trade steady state since exporting manufacturing goods requires that
PW > PA holds. Second, the economy loses from trade (VT < VA ) if it exports resource goods
while remaining diversified, since this requires that PW < PA hold.
When the economy specializes in the resource good, from (19), we have
VT = B + ln L + ln A (S∞ (1)) − (1 − b) ln PW .
On the other hand, VA can be rewritten into
VA = B + ln L + ln A (S∞ (b)) − (1 − b) ln PA .
Therefore, when the economy specializes in the resource good at the trade steady state,
VT − VA = ln

A (S∞ (1))
PW
− (1 − b) ln.
A (S∞ (b))
PA

(53)

Note that, in an economy of the BT type, S∞ (1) < S∞ (b) and therefore, the first term on
the RHS of (53) is negative. On the other hand, specializing in the resource good requires
PW ≤ A (S∞ (1)) /a. Recalling that PA = A (S∞ (b)) /a, we have PW < PA . That is, the second
term on the RHS of (53) is positive. Clearly, there exists a threshold such that VT = VA if theworld relative price take the threshold value. Letting VT = VA in (53) gives the threshold
A (S∞ (1))
P =
a
0



A (S∞ (1))
A (S∞ (b))

 b

1−b

.

That is, a small economy of the BT type gains from trade if PW < P 0 .

A.7

Proof of Proposition 6

We consider specialization pattern, then environmental consequences, and finally welfare effectsof trade, all focusing on the steady state.

Specialization pattern. Given that the world relative price satisfies
A (S∞ (1))
A (S∞ (0))< PW <,
aathere exist three steady states, as illustrated in panel (b) of Figure 3, among which the two
specialized steady states are stable and the diversified one is unstable. However, if
PW ≥A (S∞ (0)),
a49the economy will stably specialize in manufacturing goods; if
PW ≤A (S∞ (1)),
ait will stably specialize in resource goods. In either case, a small economy of the CT type
always specializes at the steady state. Note that, right after trade liberalization, the economy
exports and specializes in the good that is relatively cheaper compared to the world. This tradepattern and specialization pattern remain at the trade steady state. To see this, suppose forexample that PW < PA holds before trade liberalization. When opened to trade, the economy
specializes in resource goods. Recalling that the resource good is less environmentally harmful ina CT economy, the environment improves gradually, which reinforces the economy’s comparativeadvantage in the resource good. The self-reinforcing process ensures that, in a small economyof the CT type, trade pattern and specialization pattern right after trade liberalization remainunchanged during the transition.

Environmental consequences. Let βT denote labor allocation at the trade steady state.
When specializing in the manufacturing (resource) good, βT = 0 < b (βT = 1 > b). Since
0 (·) > 0 holds in an economy of the CT type, S (β ) < S (b) (> S (b)), implying a
S∞
∞
∞
∞
T
decline (increase) in environmental stock compared to autarky.

Welfare effects. If the economy specializes in manufacturing goods, we have (52) holds.
Since PW > PA is required for the economy to export manufacturing right after trade liberalization, it follows immediately that VT > VA holds.
0 (·) > 0 in a CT
If the economy specializes in resource goods, (53) holds. Noting that S∞
economy, A (S∞ (1)) > A (S∞ (b)) holds. Moreover, PW < PA is required for the economy toexport resource goods right after trade liberalization. Hence, both terms on the RHS of (53)are positive, implying that VT > VA holds. In either case, the small economy enjoys a higher
steady-state level of utility under free trade compared to autarky.

A.8

Proof of Proposition 7

Here, we derive the conditions for respective world specialization patterns to arise in the shortrun, in which environmental stocks S and S ∗ are taken as given. As by-products, we also obtainlabor allocations β and β ∗ , and the world relative price P .

Pattern (f,d) in the short run. In pattern (f,d), Home produces only the resource good,∗ = a∗ (1 − β ∗ ) L∗ .
Xf = A (S) L, and Foreign produces both goods, Xf∗ = A∗ (S ∗ ) β ∗ L∗ and Xm
The world relative price of the manufacturing good with respect to the resource good satisfies

P =A∗ (S ∗ )
,
a∗

(54)

which depends only on Foreign’s condition. Recalling that the resource good is the numeraire,
P is also the world price of manufacturing good. Since the resource good is produced in bothcountries in pattern (f,d), the wage in Home is w = A (S) and that in Foreign is w∗ = A∗ (S ∗ ).

50

Since the manufacturing good is supplied only by Foreign, the market-clearing condition re∗ . This gives
quires, given the same preference in the two countries, (1 − b) (wL + w∗ L∗ ) = P Xm
Foreign’s labor allocation:z(55)
β ∗ = b − (1 − b) ,v
where as defined in (23) and (24),v≡

A∗ (S ∗ ) a,
A (S) a∗

z≡

aL
.
a∗ L∗

Since Foreign produces both goods, we have β ∗ > 0, which is equivalent to v > (1 − b) z/b.
Recalling that a necessary condition for (f,d), (f,m), and (d,m) is v ≤ 1, the necessary condition
for pattern (f,d) to arise is1−bz < v ≤ 1.
(56)b
A sufficient condition can be obtained by excluding v = 1:
1−bz < v < 1.
b

(57)

Pattern (f,m) in the short run. In pattern (f,m), Home produces only the resource good,∗ = a∗ L∗ . The wage in
Xf = A (S) L, and Foreign produces only the manufacturing good, Xm
Home is w = A (S), whereas that in Foreign is w∗ = a∗ P . Since both countries completely
specialize, the world (relative) price of the manufacturing good, P , is determined such thatworld supply equals world demand, b (wL + w∗ L∗ ) = Xf , which givesP =(1 − b) A (S) L.
ba∗ L∗(58)

For the two countries to completely specialize and Home to export the resource good, thenecessary condition is A∗ (S ∗ ) /a∗ ≤ P ≤ A (S) /a, which yieldsv≤1−bz ≤ 1.
b(59)

A sufficient condition can be obtained by excluding v = 1:v≤

1−bz ≤ 1 and v 6= 1.
b(60)

Pattern (d,m) in the short run. In pattern (d,m), Home produces both goods, Xf =∗ = a∗ L∗ .
A (S) βL and Xm = a(1 − β)L, and Foreign produces only the manufacturing good, Xm
The world manufacturing price depends only on Home’s condition:

P =

A (S)
.
a

(61)

The wage in Home is w = A (S), whereas that in Foreign is w∗ = a∗ P = A (S) a∗ /a. Since theresource good is supplied only by Home, the market-clearing condition requires b (wL + w∗ L∗ ) =51

Xf . This gives Home’s labor allocation:
1.
β =b 1+z(62)

Since Home produces both goods, we have β < 1 and therefore, using (62), (1 − b) z/b > 1.
Recalling that a necessary condition for (f,d), (f,m), and (d,m) is v ≥ 1, the necessary condition
for pattern (d,m) is
1−b
v≤1<
z.
(63)
b
A sufficient condition follows by excluding v = 1:
v<1<

1−b
z.
b

(64)

Pattern (d,f ) in the short run. In pattern (d,f), we have Xf = A (S) βL, Xm =a (1 − β) L, and Xf∗ = A∗ (S ∗ ) L∗ . The world price of the manufacturing good is given by(61). The wages are w = A (S) and w∗ = A∗ (S ∗ ). The market-clearing condition requires(1 − b) (wL + w∗ L∗ ) = P Xm , which gives Home’s labor allocation:v
β = b − (1 − b) .
z

(65)

It follows from β > 0 that v < bz/ (1 − b), which together with v ≥ 1 gives the necessary
condition for pattern (d,f):
b
1≤v<
z.
(66)
1−b
Again, excluding v = 1 yields a sufficient condition:
1<v<

b
z.
1−b

(67)

Pattern (m,f ) in the short run. In pattern (m,f), we have Xm = aL and Xf∗ =A∗ (S ∗ ) L∗ . The wages are w = aP and w∗ = A∗ (S ∗ ). The world price of the manufacturing good is determined by the market-clearing condition b (wL + w∗ L∗ ) = Xf , which givesP =(1 − b) A∗ (S ∗ ) L∗
.
baL

(68)

For the two countries to completely specialize and Foreign to export the resource good, thenecessary condition is A (S) /a ≤ P ≤ A∗ (S ∗ ) /a∗ , which yields
1≤bz ≤ v.
1−b(69)

A sufficient condition can be obtained by excluding v = 1:1≤bz ≤ v and v 6= 1.
1−b

52

(70)

Pattern (m,d) in the short run. In pattern (m,d), we have Xm = aL, Xf∗ = A∗ (S ∗ ) β ∗ L∗ ,∗ = a∗ (1 − β ∗ ) L∗ . The world price of the manufacturing good is given by (54). The wages
and Xm
are w = aP and w∗ = A∗ (S ∗ ). The market-clearing condition requires b (wL + w∗ L∗ ) = Xf∗ ,which gives Foreign’s labor allocation:β ∗ = b (1 + z) .

(71)

It follows from β ∗ < 1 that bz/ (1 − b) < 1, which together with v ≥ 1 gives the necessary
condition for pattern (m,d):bz < 1 ≤ v.
(72)
1−b
A sufficient condition follows by excluding v = 1:bz < 1 < v.
1−b

(73)

Pattern (d,d) in the short run. In pattern (d,d), both countries produce both goods.
A necessary condition for pattern (d,d) to arise is v = 1. The wages are w = A (S) andw∗ = A∗ (S ∗ ). The market-clearing condition requires b (wL + w∗ L∗ ) = A (S) βL+A∗ (S ∗ ) β ∗ L∗ ,which gives (25), namely
βz + β ∗ = b (z + 1) .
With (25) as the only constraint on the two variables, β and β ∗ are indeterminate (and thus
other patterns may arise).

A.9

Proof of Proposition 8

We derive the conditions for the existence and uniqueness of steady state(s) with a certain world
specialization pattern arising.

Pattern (f,d) at the steady state. Recall that a necessary condition for pattern (f,d)
to arise is (1 − b) z/b < 1. With this necessary condition holing, (i) pattern (f,d) arises at thesteady state if (1 − b) z/b < v < 1 holds at the trade steady state, and (ii) it may also arise evenif v = 1 at the steady state.
Consider (i) first. In this case, the steady state is determined by g (v) /v = 1, whereg (v) ≡∗ (β ∗ (v))) aA∗ (S∞.
A (S∞ (β (v))) a∗

In pattern (f,d), recalling that β ∗ = b − (1 − b) z/v (and β = 1), we have∗ b − (1 − b) zA∗ S∞vg (v) =A (S∞ (1)) a∗



a

.

The existence of steady state(s) with pattern (f,d) requires that there exists solution(s) of v

53

(satisfying (1 − b) z/b < v < 1) to∗ b − (1 − b) z
A∗ S ∞v
A (S∞ (1)) a∗



a1
= 1.
v

Consider (ii) then. The existence of steady state(s) with pattern (f,d) requires∗ (b − (1 − b) z)) aA∗ (S∞= 1.
A (S∞ (1)) a∗

Therefore, if we define∗ b − (1 − b) zA∗ S∞v
∆f d (v) ≡A (S∞ (1)) a∗



a1
,
v

the existence of trade steady state(s) with pattern (f,d) arising for both cases (i) and (ii) requires
there exits solution(s) satisfying (1 − b) z/b < v ≤ 1 to
∆f d (v) = 1.

(74)

Note that ∆f d (v) is not necessarily monotonic. Specifically, if Foreign is of the BT type,∗0
S∞ (·) < 0 and thus ∆f d (v) is decreasing with v. However, if Foreign is of the CT type,∗0 (·) > 0 and thus ∆ (v) is not necessarily monotonic. Let
S∞
fd


1−b
z
<
v
≤
1
,
∆inf
=
inf
∆
(v)
:
fd
fd
b


1−b
sup
∆f d = sup ∆f d (v) :z<v≤1 ,bthen the existence of solution(s) to (74) satisfying (1 − b) z/b < v ≤ 1 requires
sup∆inf
f d ≤ 1 ≤ ∆f d .

(75)

Note that (75) is a necessary condition, and the polar case may take place in which v = (1 − b) z/b
is the only solution to (74) (so that pattern (f,m) arises). The following condition helps to exclude
supthis polar case and obtain the sufficient and necessary condition: if ∆inf
f d = 1 or ∆f d = 1, there
exists v1 6= (1 − b) z/b such that ∆f d (v1 ) = 1.
Therefore, the sufficient and necessary condition for the existence of trade steady state(s)with pattern (f,d) is the combination of three conditions: (1 − b) z/b < 1, (75), and if ∆inf
fd = 1
supor ∆f d = 1, there exists v1 6= (1 − b) z/b so that ∆f d (v1 ) = 1.
Since ∆f d (v) is not necessarily monotonic, there may exist multiple solutions to (74), meaning that there may exist multiple steady states with pattern (f,d).

Pattern (f,m) at the steady state. Recall that a necessary condition for pattern (f,m)
to arise is (1 − b) z/b ≤ 1. With this necessary condition holing, (i) pattern (f,m) arises ifv ≤ (1 − b) z/b and v 6= 1 holds at the steady state, and (ii) it may also arise if v = (1 − b) z/b = 1
hold at the steady state.

54

Consider (i) first. In this case, the steady state can be obtained by solving g (v) /v = 1 withg (v) =

∗ (0)) aA∗ (S∞.
A (S∞ (1)) a∗

Thus the existence of steady state(s) with pattern (f,m) arising implies∗ (0)) a∗ (0)) aA∗ (S∞1−bA∗ (S∞
≤
z
and
6= 1.
A (S∞ (1)) a∗b
A (S∞ (1)) a∗

Consider (ii) then. In this case, the existence of steady state(s) with pattern (f,m) implies∗ (0)) aA∗ (S∞= 1.
A (S∞ (1)) a∗

Therefore, the sufficient and necessary condition for the existence of trade steady state(s)with pattern (f,m) is∗ (0)) aA∗ (S∞1−b≤z ≤ 1.
∗A (S∞ (1)) ab
Note that the steady-state environmental stocks with pattern (f,m) can be expressed by
S = S∞ (1) ,∗S ∗ = S∞(0) ,which are uniquely determined. This gives the uniqueness of the steady state.

Pattern (d,m) at the steady state. Recall that a necessary condition for pattern (d,m)
to arise is (1 − b) z/b > 1. With this necessary condition holing, (i) pattern (d,m) arises if v < 1holds at the steady state, and (ii) it may also arise even if v = 1 at the steady state.
Consider (i) first. In this case, the steady state can be obtained by solving g (v) /v = 1 withg (v) =

∗ (0)) aA∗ (S∞ .
a∗
A S∞ b 1 + z1

Thus, the existence of steady state(s) with pattern (d,m) arising implies∗ (0)) aA∗ (S∞< 1.
A S∞ b 1 + z1
a∗Consider (ii) then. The existence of steady state(s) with pattern (d,m) implies∗ (0)) aA∗ (S∞= 1.
A S∞ b 1 + z1
a∗Therefore, the sufficient and necessary condition for the existence of trade steady state(s)with pattern (d,m) is∗ (0)) aA∗ (S∞1−b≤1<z.
∗
A (S∞ (b (1 + 1/z))) ab

55

Note that the steady-state environmental stocks with pattern (d,m) can be expressed by
 

1
,
S = S∞ b 1 +z∗
S ∗ = S∞(0) ,which are uniquely determined. This gives the uniqueness of the steady state.

Pattern (d,f ) at the steady state. Recall that a necessary condition for pattern (d,f)
to arise is bz/ (1 − b) > 1. With this necessary condition holing, (i) pattern (d,f) arises at thesteady state if 1 < v < bz/ (1 − b) holds at the trade steady state, and (ii) it may also arise evenif v = 1 at the steady state.
Consider (i) first. In this case, the steady state is determined by g (v) /v = 1, withg (v) =

∗ (1)) aA∗ (S∞ .
A S∞ b − (1 − b) vz a∗

The existence of steady state(s) with pattern (d,f) requires that there exists solution(s) of v
(satisfying 1 < v < bz/ (1 − b)) to∗ (1)) a
1
A∗ (S∞
= 1.
v∗
A S∞ b − (1 − b) z a v

Consider (ii) then. The existence of steady state(s) with pattern (d,f) requires∗ (1)) aA∗ (S∞= 1.
A S∞ b − 1−ba∗z

Therefore, if we define∆df (v) ≡∗ (1)) aA∗ (S∞1,v∗
v
A S∞ b − (1 − b) z a

the existence of trade steady state(s) with pattern (d,f) arising for both cases (i) and (ii) requires
there exits solution(s) satisfying 1 ≤ v < bz/ (1 − b) to∆df (v) = 1.

(76)

Note that ∆df (v) is not necessarily monotonic. Specifically, if Home is of the BT type,0 < 0 and thus ∆ (v) is decreasing with v. However, if Home is of the CT type, S 0 > 0 andS∞
df∞thus ∆df (v) is not necessarily monotonic. Letbz ,∆df (v) : 1 ≤ v <
1−b


b
z
,
∆sup
=
sup
∆
(v)
:
1
≤
v
<
df
df
1−b
∆inf
df = infthen the existence of solution(s) to (76) satisfying 1 ≤ v < bz/ (1 − b) requires
sup∆inf
df ≤ 1 ≤ ∆df .

56

(77)

Note that (77) is a necessary condition, and the polar case may take place in which v = bz/ (1 − b)is the only solution to (76) (so that pattern (m,f) arises). The following condition helps to exclude
supthis polar case and obtain the sufficient and necessary condition: if ∆inf
df = 1 or ∆df = 1, there
exists v1 6= bz/ (1 − b) such that ∆df (v1 ) = 1.
Therefore, the sufficient and necessary condition for the existence of trade steady state(s)with pattern (d,f) is the combination of three conditions: bz/ (1 − b) > 1, (77), and if ∆inf
df = 1supor ∆df = 1, there exists v1 6= bz/ (1 − b) so that ∆df (v1 ) = 1.
Since ∆df (v) is not necessarily monotonic, there may exist multiple solutions to (76), meaning that there may exist multiple steady states with pattern (d,f).

Pattern (m,f ) at the steady state. Recall that a necessary condition for pattern (m,f)
to arise is bz/ (1 − b) ≥ 1. With this necessary condition holing, (i) pattern (m,f) arises ifv ≥ bz/ (1 − b) and v 6= 1 holds at the steady state, and (ii) it may also arise if v = bz/ (1 − b) = 1
hold at the steady state.
Consider (i) first. In this case, the steady state can be obtained by solving g (v) /v = 1 withg (v) =

∗ (1)) aA∗ (S∞.
A (S∞ (0)) a∗

Thus the existence of steady state(s) with pattern (m,f) arising implies∗ (1)) a∗ (1)) aA∗ (S∞bA∗ (S∞
≥
z
and
6= 1.
A (S∞ (0)) a∗1−b
A (S∞ (0)) a∗

Consider (ii) then. In this case, the existence of steady state(s) with pattern (m,f) implies∗ (1)) aA∗ (S∞= 1.
A (S∞ (0)) a∗

Therefore, the sufficient and necessary condition for the existence of trade steady state(s)with pattern (m,f) is∗ (1)) ab
A∗ (S∞1≤z≤.
1−b
A (S∞ (0)) a∗
Note that the steady-state environmental stocks with pattern (m,f) can be expressed by
S = S∞ (0) ,∗S ∗ = S∞(1) ,which are uniquely determined. This gives the uniqueness of the steady state.

Pattern (m,d) at the steady state. Recall that a necessary condition for pattern (m,d)
to arise is bz/ (1 − b) < 1. With this necessary condition holing, (i) pattern (m,d) arises if v > 1
holds at the steady state, and (ii) it may also arise even if v = 1 at the steady state.
Consider (i) first. In this case, the steady state can be obtained by solving g (v) /v = 1 withg (v) =∗ (b (1 + z))) aA∗ (S∞.
A (S∞ (0)) a∗57

Thus, the existence of steady state(s) with pattern (m,d) arising implies∗ (b (1 + z))) aA∗ (S∞> 1.
A (S∞ (0)) a∗

Consider (ii) then. The existence of steady state(s) with pattern (m,d) implies∗ (b (1 + z))) aA∗ (S∞= 1.
A (S∞ (0)) a∗

Therefore, the sufficient and necessary condition for the existence of trade steady state(s)with pattern (d,m) is∗ (b (1 + z))) ab
A∗ (S∞z<1≤.
1−b
A (S∞ (0)) a∗
Note that the steady-state environmental stocks with pattern (m,d) can be expressed by
S = S∞ (0) ,∗S ∗ = S∞(b (1 + z)) ,which are uniquely determined. This gives the uniqueness of the steady state.

Pattern (d,d) at the steady state. Recall that a necessary condition for pattern (d,d)
to arise is v = 1. Thus, the existence of trade steady state(s) with pattern (d,d) is equivalent to
the existence of 0 < β < 1 and 0 < β ∗ < 1 satisfying∗ (β ∗ )) aA∗ (S∞= 1,
A (S∞ (β)) a∗βz + β ∗ = b (z + 1) ,where the second equation comes from the world market-clearing condition (or, equivalently, thebalance of trade). Therefore, if we define∆dd (β, β ∗ ) ≡∗ (β ∗ )) aA∗ (S∞,A (S∞ (β)) a∗and let∗∗∆inf
dd = inf {∆dd (β, β ) : βz + β = b (z + 1)} ,∗∗∆supdd = sup {∆dd (β, β ) : βz + β = b (z + 1)} ,then the existence of trade steady state(s) with pattern (d,d) arising requires
sup∆inf
dd ≤ 1 ≤ ∆dd .

(78)

Note that (78) is a necessary condition since it cannot exclude the polar case in which either
β = 0, β = 1, β ∗ = 0, or β ∗ = 1 holds in all solution(s). The sufficient and necessary condition
supcan be obtained by imposing the following constraint: if ∆inf
dd = 1 or ∆dd = 1, there exist
β1 , β1∗ 6= 0, 1 so that β1 z + β1∗ = b (z + 1) and ∆dd (β1 , β1∗ ) = 1.
Note that, given that β and β ∗ satisfy βz + β ∗ = b (z + 1), ∆dd (β, β ∗ ) is monotonic as β58

(equivalently, β ∗ ) varies if both countries are of the same type, and therefore, the steady statewith pattern (d,d), if existing, is unique; otherwise, there may be multiple ones.

A.10

Proof of Corollary 1

If Foreign is of the BT type, ∆f d (v) is decreasing with v. It follows that
∗ (b − (1 − b) z)) aA∗ (S∞,A (S∞ (1)) a∗∗ (0)) aA∗ (S∞
1 − b −1
sup∆f d =z,
A (S∞ (1)) a∗
b

∆inf
fd =

which can be substituted into part (i) of Proposition 8 to obtain (42). Clearly, the solution is
unique, and so is the steady state with pattern (f,d).
If Home is of the BT type, ∆df (v) is decreasing with v. It follows that
∗ (1)) aA∗ (S∞
b
∆inf
=
z
df
∗
A (S∞ (0)) a
1−b∗ (1)) aA∗ (S∞
 ,
∆sup
df =
a∗
A S∞ b − 1−b
z



−1
,

which can be substituted into part (iv) of Proposition 8 to obtain (43).
0 and S ∗0 are of the same sign and therefore,
If both countries are of the same type, S∞∞∆dd (β, β ∗ ) is monotonic as β (or β ∗ ) varies (as long as β and β ∗ satisfy βz + β ∗ = b (z + 1)).
sup∗
We can then obtain ∆inf
dd and ∆dd using the corner values of β and β . Specifically, ifz < min



b
1−b
,
b
1−b

,

the corner values are (β, β ∗ ) = (0, b (1 + z)) and (β, β ∗ ) = (1, b − (1 − b) z). If

min

1−b
b
,
b
1−b




< z < max

1−b
b
,
b
1−b


,

the corner values are (β, β ∗ ) = (b − (1 − b) /z, 1) and (β, β ∗ ) = (1, b − (1 − b) z). Ifz > max

1−b
b
,
b
1−b


,

the corner values are (β, β ∗ ) = (b − (1 − b) /z, 1) and (β, β ∗ ) = (b (1 + 1/z) , 0).
Moreover, given that βz + β ∗ = b (z + 1), ∆dd (β, β ∗ ) is monotonic as β varies, and thus
sup∆inf
dd < 1 < ∆ddis the sufficient and necessary condition. Summarizing these results gives Table 1.
As for the uniqueness, note that the world market-clearing condition βz + β ∗ = b (z + 1)

59

represents a downward-sloping curve on the (β, β ∗ ) plane. On the other hand,∗ (β ∗ )) aA∗ (S∞= 1,
A (S∞ (β)) a∗0 and S ∗0 are
gives an upward-sloping curve if both countries are of the same type (since then S∞∞
of the same sign). This implies a unique solution to the two equations above, and thus a uniquesteady state with pattern (d,d). However, if the two countries are of different types, the later
0 and S ∗0 have the opposite signs).
equation represents a downward-sloping curve (since then S∞∞
The possibility remains of multiple solutions and thus multiple steady states.

A.11

Proof of Proposition 9

We first consider the stability of the steady state satisfying v 6= 1 and that satisfying v = 1.

The stability of the steady state satisfying v 6= 1. Given v 6= 1 at the trade steadystate, any pattern except for (d,d) may arise. As long as v 6= 1, the dynamic system is governedby (28) and (29), the Jacobian of which can be expressed by∂ Ṡ
∂S∂ Ṡ ∗∂S"∂v
G0 − lfA0 βL − (lf A− lm a) Lβ 0 ∂S

J=

=

#

"

∂ Ṡ
∂S ∗
∂ Ṡ ∗∂S ∗∗ a∗ L∗ β ∗0 ∂v− lf∗ A∗ − lm∂S∂v
− (lf A −lm a) Lβ 0 ∂S∗#∗ a∗ L∗ β ∗0 ∂v
G∗0 − lf∗ A∗0 β ∗ L∗ − lf∗ A∗ − lm
∂S ∗.

The stability around the steady state holds if trJ < 0 and detJ > 0, where trJ is the trace anddetJ is the determinant of J evaluated at the steady state.
As for trJ, we have

∂v
∂v
∗ ∗
trJ = G0 − lf A0 βL − (lf A (S) − lm a) Lβ 0a L∗ β ∗0 ∗+ G∗0 − lf∗ A∗0 β ∗ L∗ − lf∗ A∗ − lm
∂S
∂S






∂v
∂v
0
∗0 ∗0
= G0 − lf A0 βL 1 − S∞β0
+ G∗0 − lf∗ A∗0 β ∗ L∗ 1 − S∞
β
,
(79)
∂S
∂S ∗




0 and l∗ A∗ − l∗ a∗ L∗ = G∗0 − l∗ A∗0 β ∗ L∗ S ∗0
where (lf A (S) − lm a) L = (G0 − lf A0 βL) S∞m∞ff
can be obtained from (13) and its Foreign counterpart.
As for detJ, it is convenient to calculate(l A−lm a)L ∂v
1 − Gf0 −lf A0 βL β 0 ∂S
detJ

=
(l∗ A∗ −l∗ a)L∗ ∗0 ∂v− G∗0f −l∗ Am(G0 − lf A0 βL) G∗0 − lf∗ A∗0 β ∗ L∗
∗0 β ∗ L∗ β ∂S
f=0 β 0 ∂v
1 − S∞∂S∗0 β ∗0 ∂v
−S∞
∂S

0
= 1 − S∞β0

(l A−lm a)L ∂v− Gf0 −lf A0 βL β 0 ∂S∗∗ a∗ L∗(lf∗ A∗ −lm)∂v
1 − G∗0 −l∗ A∗0 β ∗ L∗ β ∗0 ∂S∗f

0 β 0 ∂v
−S∞
∂S ∗
∂v
∗0
∗0
1 − S∞ β ∂S∗∂v
∗0 ∗0 ∂v
− S∞
β
.
∂S
∂S ∗

(80)

Note that G0 − lf A0 βL < 0 and G∗0 − lf∗ A∗0 β ∗ L∗ < 0, which are required for stability when labor60allocations are given. Hence, detJ > 0 is equivalent to
0
S∞
β0

∂v
∗0 ∗0 ∂v
< 1.
+ S∞
β
∂S
∂S ∗

(81)

Now, we shall show that g 0 (v) < 1 is equivalent to detJ > 0 and trJ < 0. The equivalence
between g 0 (v) < 1 and detJ > 0 can be obtained as follows. With the definition of g (v):g (v) ≡∗ (β ∗ (v))) aA∗ (S∞,A (S∞ (β (v))) a∗

we have∗0 β ∗0 a
0 β0
A∗0 S∞A∗ aA0 S∞
−
Aa∗
A2 a∗
∗0
A0
0
∗0 ∗0 Aβ0v
= S∞
β v ∗ − S∞
A
A
0
0 ∂v
∗0 ∗0 ∂v.
= S∞ β+ S∞ β∂S∂S ∗g 0 (v) =It follows immediately from (81) that g 0 (v) < 1 is equivalent to detJ > 0
We then show that g 0 (v) < 1 is equivalent to trJ < 0 given that (81) holds. Noting that forany steady state satisfying that v 6= 1, at least one country specializes. If only Home specializes,
namely pattern (f,d) or (m,d) arises, we have β 0 = 0; if only Foreign specializes, namely pattern(d,f) or (d,m) arises, we have β ∗0 = 0; if both countries specialize, namely pattern (f,m) or (m,f)
0 β 0 ∂v/∂S + S ∗0 β ∗0 ∂v/∂S ∗ < 1 implies thatarises, we have β 0 = β ∗0 = 0. This together with S∞∞∗0 β ∗0 ∂v/∂S ∗ < 1 (if β 0 = 0), S 0 β 0 ∂v/∂S < 1 (if β ∗0 = 0), or S 0 β 0 ∂v/∂S = S ∗0 β ∗0 ∂v/∂S ∗ = 0
S∞∞∞∞
0 β 0 ∂v/∂S > 0 and 1 − S ∗0 β ∗0 ∂v/∂S ∗ > 0, which(if β 0 = β ∗0 = 0). In either case, we have 1 − S∞∞together with G0 − lf A0 βL < 0 and G∗0 − lf∗ A∗0 β ∗ L∗ < 0 yields trJ > 0.
The arguments above suggest that the condition for the local stability of the steady state(satisfying v 6= 1), namely detJ > 0 and trJ < 0, can be equivalently written as
g 0 (v) < 1.
In what follows, we apply the condition and check the stability for each specialization pattern.
(i) In a steady state with pattern (f,m) or (m,f), we have β 0 = 0 and β ∗0 = 0, which can besubstituted into (79) and (80) to obtain
trJ = G0 − lf A0 βL + G∗0 − lf∗ A∗0 β ∗ L∗ < 0,detJ = G0 − lf A0 βL G∗0 − lf∗ A∗0 β ∗ L∗ > 0.
This gives the local stability of a steady state with pattern (f,m) or (m,f).
(ii) In a steady state with pattern (f,d), β 0 = 0 and β ∗0 > 0. It follows that00

trJ = G − lf A βL + G∗0− lf∗ A∗0 β ∗ L∗

∗0 ∗0 ∂v
1 − S∞
β
∗



,
∂S


 ∗0

0
0
∗ ∗0 ∗ ∗∗0 ∗0 ∂v
detJ = G − lf A βL G − lf A β L
1 − S∞ β
.
∂S ∗

61

∗0 < 0 and
From the definition of v, we have ∂v/∂S ∗ > 0. If Foreign is of the BT type, S∞∗0 β ∗0 ∂v/∂S ∗ > 0, which implies trJ < 0 and detJ > 0. Therefore, a steady state
thus, 1 − S∞with pattern (f,d) is locally stable if Foreign is of the BT type; otherwise, it is locally stable
if g 0 (v) < 1. A similar argument applies to pattern (d,f). It follows that a steady state withpattern (d,f) is locally stable if Home is of the BT type; otherwise, it is locally stable if g 0 (v) < 1.
(iii) In a steady state with pattern (d,m) or (m,d), β 0 = 0 and β ∗0 = 0. Therefore,

trJ = G0 − lf A0 βL + G∗0 − lf∗ A∗0 β ∗ L∗ < 0,detJ = G0 − lf A0 βL G∗0 − lf∗ A∗0 β ∗ L∗ > 0.
This gives the local stability of a steady state with pattern (d,m) or (d,m).

The stability of the steady state satisfying v = 1. Given v = 1 at the trade steadystate, pattern (d,d) may arise and, as knife-edge situations, other patterns may arise as well.
For simplicity, here we ignore those knife-edge cases and focus on the stability of pattern (d,d).
Verifying the stability of the steady state with pattern (d,d) faces the difficulty that the
dynamic system (28) and (29) is discontinuous at v = 1.25 To get around this difficulty, redefine
β (v) and β ∗ (v) such that both are continuous around v = 1, while letting the redefined β (v)and β ∗ (v) have fluctuations sufficiently large around v = 1, that is, β 0  −1 and β ∗0  1.
This gives a continuous dynamic system that approximates the original discontinuous dynamic
system. Such an approximation allows us to use (79) and (80) to evaluate the stability.
Note that β (v) and β ∗ (v) are supposed to satisfy the equilibrium condition (25) aroundv = 1, which gives zdβ + dβ ∗ = 0 there. This implies, around v = 1,β ∗0 = −zβ 0 .

(82)

By substituting (82) into the expressions of trJ and detJ, we obtain






∗0
∗ ∗0 ∗ ∗
∗0
0 ∂v
0
0 ∂v
1 + S∞ zβ+ G − lf A β L
trJ = G − lf A βL 1 − S∞ β
∂S
∂S ∗


∂v
∂v
0
∗0
= −S∞
β0
G0 − lf A0 βL + S∞
zβ 0 ∗ G∗0 − lf∗ A∗0 β ∗ L∗
∂S
 ∂S



∂v
0
∗0
∗0
∗ ∗0 ∗ ∗0 ∂v
00= β S∞ z ∗ G − lf A β L − S∞
G − lf A βL ,
∂S
∂S
0

0

Note that β drops and β ∗ jumps as v crosses one. The magnitude of discontinuity varies withexogenous variables v and b, which can be categorized into four cases as in B. Specifically, in Case 1,
world specialization pattern changes from (f,d) to (m,d), and according to (101) in Appendix B, β drops
from 1 to 0 and β ∗ jumps from b − (1 − b) z to b (1 + z). In Case 2, the pattern changes from (d,m) to(d,f), and from (102), β drops from b 1 + z −1 to b − (1 − b) z −1 and β ∗ jumps from 0 to 1. In Case 3,
the pattern changes from (f,d) to (d,f), and from (103), β drops from 1 to b − (1 − b) z −1 and β ∗ jumps
from b − (1 − b) z to 1. In Case 4, the pattern changes from (d,m) to (m,d), and according to (104),
β drops from b 1 + z −1 to 0 and β ∗ jumps from 0 to b (1 + z). In all cases, the dynamic system is
discontinuous at v = 1.
25

62

and
∂v
detJ
∂v
0
∗0
 = 1 − S∞

β0
+ S∞
zβ 0 ∗∂S∂S(G0 − lf A0 βL) G∗0 − lf∗ A∗0 β ∗ L∗


∗0 ∂v
0 ∂v
z ∗ − S∞.
= β 0 S∞∂S
∂S
During the algebra above, we keep the terms relating to β 0 and drop others since β 0  −1. The
sufficient condition for the stability, trJ < 0 and detJ > 0, can expressed by
∗0
S∞
z



∂v
∗0
∗ ∗0 ∗ ∗
0 ∂v
0
0
G
−
l
A
β
L
−
S
G
−
l
A
βL
> 0,
f
∞
f
∂S ∗
∂S
0 ∂v
∗0 ∂v
S∞
z ∗ − S∞
< 0,
∂S
∂S

(83)
(84)

which are exactly (45) and (46). Note that ∂v/∂S < 0, ∂v/∂S ∗ > 0, G0 − lf A0 βL < 0, and
0 and S ∗0G∗0 − lf∗ A∗0 β ∗ L∗ < 0, whether the condition holds depends crucially on the signs of S∞∞(which are dependent on the type of Home and Foreign).
Now we shall apply the conditions above to check the stability of steady state with pattern
0 <0(d,d) for various combinations of country types. (i) If both countries are of the BT type, S∞∗0 < 0. This together gives (83) and (84). That is, the steady state with pattern (d,d) isand S∞
0 > 0 and S ∗0 > 0, which
locally stable. (ii) If both countries are of the CT type, we have S∞∞
gives the opposite signs to (83) and (84). That is, the steady state with pattern (d,d) is unstable.
0 and S ∗0 are differently signed. The stability(iii) If the two countries are of different types, S∞∞
holds if (83) and (84) hold.

A.12

Proof of Corollary 2

The idea of the proof is to check whether the intersection of the parameter sets (corresponding
to the condition required for each specialization pattern to arise) is empty. If the intersection isempty, then a unique world specialization pattern (among the seven) arises at the steady state;otherwise, there may exist multiple steady states which differ in specialization patterns. Hence,the proof proceeds by considering four cases (see Appendix B for detail), which differ in the setof possible world specialization patterns (obtained by varying environmental stocks S and S ∗ ).

Both countries of the BT type In Case 1, the set of possible world specialization patterns
includes (f,m), (f,d), (d,d), and (m,d). If both countries are of the BT type, according to
Proposition 8, the conditions for a certain pattern to arise at the steady state are (36) for
pattern (f,m), (42) for pattern (f,d), (40) for pattern (m,d), and Table 1 for pattern (d,d). It
is easy to verify that the intersection of the parameter sets corresponding to these conditions isempty, meaning that only one world specialization pattern among the four arises at the tradesteady state. One can verify that this also holds for Cases 2 to 4.
Moreover, it follows from Corollary 1 that, if both countries are of the BT type, the steadystate with a certain specialization pattern, if existing, is unique and, according to Proposition 9,
locally stable. In summary, if both countries are of the BT type the trade steady state is uniqueand locally stable.

63

Both countries of the CT type If both countries are of the CT type, in Case 1, thecorresponding conditions are, according to Proposition 8, (36) for pattern (f,m), (35) for pattern(f,d), (40) for pattern (m,d), and Table 1 for pattern (d,d). The intersection of the parameter
sets corresponding to these conditions is, however, not necessarily empty, implying that there
may exist multiple steady states corresponding to different specialization patterns. One canverify that this also holds for Cases 2 to 4.
Moreover, according to Corollary 1, if both countries are of the CT type, multiple stablesteady states with pattern (f,d) or (d,f) might exist, and according to Proposition 9, the steadystate with pattern (d,d), if any, is unstable. It follows from these results that, if both countries areof the CT type, multiple stable steady states may exist, either with the same world specialization
pattern or with distinct ones, and at least one country completely specializes at the steady state.

A.13

Proof of Proposition 10

Let βT and βT∗ denote labor allocations in Home and Foreign at the trade steady state. SupposingHome exports resource goods and Foreign exports manufacturing goods at the trade steady state,
we have
βT > b, βT∗ < b.
Consider two countries of the same type. If both countries are of the BT type,∗∗S∞(βT∗ ) > S∞(b) .

S∞ (βT ) < S∞ (b) ,If both countries are of the CT type, then∗∗S∞
(βT∗ ) < S∞(b) .

S∞ (βT ) > S∞ (b) ,In either case, environmental stock in one country increases and that in the other decreases at
the trade steady state (compared to the autarkic steady state). From similar arguments, thesame conclusion holds if trade pattern is reversed.
Consider now two countries of different types. If Home is the BT type and Foreign is of theCT type, then∗∗S∞ (βT ) < S∞ (b) , S∞
(βT∗ ) < S∞(b) .
If Home is the CT type and Foreign is of the BT type, the opposite holds:∗∗
S∞(βT∗ ) > S∞(b) .

S∞ (βT ) > S∞ (b) ,In either case, environmental stocks in two countries change in the same direction at the tradesteady state (compared to the autarkic steady state). Similarly, the same conclusion holds if
trade pattern is reversed.

A.14

Proof of Proposition 11

We first consider the situation in which Home exports resource goods at the trade steady state,
and then consider what if trade pattern is reversed.

64

Home exports resource goods.

Given that Home exports resource goods at the tradesteady state, Foreign must export manufacturing goods then, either remaining diversified (βT∗ >0) or specializing in manufacturing goods (βT∗ = 0). If Foreign remains diversified, the world
relative price at the trade steady state can be expressed byPT =∗ (β ∗ ))A∗ (S∞
T.
a∗

Since Foreign exports manufacturing goods (βT∗ < b) and is of the BT type, we have
PT > PA∗ =∗ (b))A∗ (S∞.
a∗(85)

On the other hand, the argument for deriving (52) in the proof of Proposition 5 applies, yieldingVT∗ − VA∗ = b ln

PT
.
PA∗

(86)

It follows from (85) and (86) that VT∗ > VA∗ holds.
If Foreign specializes in manufacturing goods at the trade steady state, it holds thatPT ≥∗ (0))A∗ (S∞,a∗(87)

from which we also obtain (85). Noting that (86) still holds, we have, again, VT∗ > VA∗ . In either
case, Foreign (the country exporting manufacturing goods) gains from trade in the long run.
Consider Home then. Home may remain diversified (βT < 1) or specialize in the resource
good (βT = 1) at the trade steady state. If Home remains diversified, thenPT =A (S∞ (βT ))
.
a

(88)

Since Home exports resource goods (βT > b) and is of the BT type,PT <A (S∞ (b))= PA .
a(89)

Similar as we obtain (52), it holds thatVT − VA = b ln

PT
.
PA

(90)

It follows from (89) and (90) that Home (the country exporting resource goods) loses from tradein the long run.
If Home specializes in resource goods at the trade steady state, thenPT ≤A (S∞ (1))
.
a

(91)

Now the argument for deriving (53) applies, which gives
VT − VA = ln

A (S∞ (1))
PT
− (1 − b) ln,
A (S∞ (b))
PA

65

(92)

the sign of which is indefinite given (91). Letting VT > VA in (92) yields
A (S∞ (1))
PT <
a



A (S∞ (1))
A (S∞ (b))

 b

1−b

.

(93)

Foreign exports resource goods. Given that Foreign exports resource goods at the tradesteady state, Home must export manufacturing goods, and similar arguments as above apply.
Specifically, Home (the country exporting manufacturing goods) necessarily gains from tradein terms of the steady-state level of utility. Foreign (the country exporting resource goods) loses
if it remains diversified at the trade steady state. If Foreign specializes, the argument for deriving(53) applies and we can obtainVT∗ − VA∗ = ln

∗ (1))A∗ (S∞PT− (1 − b) ln ∗ .
∗∗A (S∞ (b))
PA

Letting VT∗ > VA∗ gives∗ (1))A∗ (S∞PT <a∗

∗ (1))A∗ (S∞∗ (b))A∗ (S∞

(94)

 b

1−b

.

(95)

Combining (93) and (95) gives (47).

A.15

Proof of Proposition 12

We first consider the situation in which Home exports resource goods at the trade steady state,
and then consider what if trade pattern is reversed.

Home exports resource goods. Consider Home first. At the trade steady state, Homeeither remains diversified (βT < 1) or specializes in resource goods (βT = 1). If Home remains
diversified, as shown in the proof of Proposition 11, we have (88) and (90). Since Home exports
resource goods (βT > b) and is of the CT type, we have S∞ (βT ) > S∞ (b) and thus, by (88),
PT > PA . This together with (90) yields VT > VA .
If Home specializes in resource goods at the trade steady state, we have (91) and (92). Notethat
A (S∞ (1))
PT
A (S∞ (1))
> (1 − b) ln≥ (1 − b) ln,
ln
A (S∞ (b))
A (S∞ (b))
PAwhere the first inequality comes from S∞ (1) > S∞ (b) (since Home is of the CT type), and thesecond inequality from (91). This together with (92) gives VT > VA . In either case, Home (thecountry exporting resource goods) gains from trade in the long run.
Consider Foreign then. Since Home exports resource good, Foreign must export manufacturing goods at the trade steady state, either remaining diversified (βT∗ > 0) or specializing(βT∗ = 0). If Foreign remains diversified, (86) holds. Since Foreign exports manufacturing goods(βT∗ < b) and is of the CT type, we havePT =∗ (β ∗ ))∗ (b))A∗ (S∞A∗ (S∞T<= PA∗ ,a∗a∗which together with (86) gives VT∗ < VA∗ .

66

(96)

If Foreign specializes in manufacturing goods at the trade steady state, (86) holds as well.
Moreover, we have
∗ (0))∗ (b))A∗ (S∞A∗ (S∞PT ≥<= PA∗ ,
(97)a∗a∗from which the sign of (86) remains ambiguous. There are two possibilities.
The first possibility is that Home remains diversified at the trade steady state, which gives
PT > PA as obtained above. Note that trade pattern among two countries of the CT type isself-reinforced in the sense that trade pattern remains unchanged during dynamic transition (aslong as there is no exogenous shocks).26 This self-reinforcing feature ensures that, given that
Home exports resource goods at the trade steady state, Home also exports resource goods rightafter the openness of trade, which requires
PA ≥ PA∗ .
It then follows that PT > PA∗ and thus VT∗ > VA∗ .
The second possibility is that Home specializes in resource goods at the trade steady state,
which requires (91) to hold. However, with the constraints obtained above, the sign of (86) still
remains ambiguous. Letting VT∗ > VA∗ in (86) yields
PT > PA∗ =∗ (b))A∗ (S∞.
a∗(98)

Foreign exports resource goods. Given that Foreign exports resource goods at the tradesteady state, Home must export manufacturing goods then. Similar arguments as above apply.
Specifically, Foreign (the country exporting resource goods) necessarily gains from trade in
the long run. Home (the country exporting manufacturing goods) loses if it remains diversified.
If Home specializes, the argument for deriving (52) applies, which gives (90). Note thatPT ≥A (S∞ (0))
A (S∞ (b))<= PA ,aa

the sign of which remains ambiguous. There are also two possibilities. The first is that Foreign
remains diversified. Since Foreign exports resource goods (βT∗ > b) and is of the CT type, wehave∗ (β ∗ ))∗ (b))A∗ (S∞A∗ (S∞TPT =>= PA∗ .
a∗a∗
The self-reinforcing trade pattern (since both countries are of the CT type) ensures thatPA∗ ≥ PA ,which gives PT > PA and thus VT > VA . The second possibility is that Foreign specializes inmanufacturing goods, which requires (87) to hold. Yet with these constraints, the sign of (90)26

The environment in the country exporting resource goods will be enhanced, making it better at
producing its exports. By contrast, the country exporting manufacturing goods faces environment deterioration and thus gets better at producing the manufacturing good.

67

remains ambiguous. Letting VT > VA in (90) yields
PT >A (S∞ (b))
.
a

(99)

Combining (98) and (99) gives (48).

A.16

Proof of Proposition 13

Consider first what happens if Home is of the BT type (thus Foreign is of the CT type since twocountries are of different types), and consider then what if Foreign is of the BT type.

Home is of the BT type. As for part (i), consider first what if Home (the BT type country)
exports manufacturing goods (βT < b) at the trade steady state, which means that Foreign (theCT type country) exports resource goods (βT∗ > b). Home either remains diversified (βT > 0)or specializes (βT = 0). If Home remains diversified, thenPT =A (S∞ (βT ))> PA ,awhere the inequality comes from βT < b and that Home is of the BT type. If Home specializes,then
A (S∞ (0))> PA .
PT ≥a
In either case, PT > PA holds, which together with the expression of VT given by (90) (notingthat Home exports manufacturing goods) yields VT > VA . That is, Home (the BT type country
that exports manufacturing goods) gains from trade in the long run.
Similarly, Foreign either remains diversified (βT∗ < 1) or specializes in the production of
resource good (βT∗ = 1). If Foreign remains diversified, thenPT =∗ (β ∗ ))A∗ (S∞T> PA∗ ,a∗where the inequality comes from noting that βT∗ > b and that Foreign is of the CT type. Staying
diversified, the difference in Foreign’s steady-state utility level between trade and autarky canbe expressed by (86), which together with PT > PA∗ yields VT∗ > VA∗ . If Foreign specializes, thenPT ≤∗ (1))A∗ (S∞.
a∗(100)

The difference between the utility level at the trade steady state and that in autarky can be
expressed by (94). Note thatln

∗ (1))∗ (1))A∗ (S∞A∗ (S∞
PT
>
(1
−
b)
ln
≥ (1 − b) ln ∗ ,∗ (b))∗ (b))A∗ (S∞A∗ (S∞
PA∗ (1) >
where the first inequality comes from noting that Foreign is of the CT type and thus S∞∗ (b), and the second inequality from (100). This together with (94) yields, again, V ∗ > V ∗ .
S∞
TA
In either case, Foreign (the CT type country that exports resource goods) gains from trade.

68Consider then what if trade pattern is reversed, with Home (the BT type country) specializing
in resource goods (βT = 1) and Foreign (the CT type country) specializing in manufacturing
goods (βT∗ = 0) at the trade steady state. Similar arguments apply and we have (91) and (92)in Home, and (86) and (97) in Foreign. It then follows from Proposition 11 that Home gains
from trade if (47) holds and from Proposition 12 that Foreign gains from trade if (48) holds.
As for part (ii). If Home (the BT type country) exports resource goods and (47) holds
reversely, it may remain diversified or specialize, but necessarily loses from trade in the long run
as shown in Proposition 11. Similarly, if Foreign (the CT type country) exports manufacturing
goods and (48) holds reversely, it may remain diversified or specialize, but necessarily loses fromtrade in the long run as shown in Proposition 12.

Foreign is of the BT type. The arguments above remain true by switching Home (andHome related variables) for Foreign (and Foreign related variables).

B

Sets of possible world specialization patterns

Note that the sets of possible world specialization patterns, which can be obtained by varying
environmental stocks S and S ∗ , may change given different values of effective relative size z andpreference parameter b. Figure 7 suggests that there are four cases.27

Case 1: Relatively small Home. Consider the case that Home is relatively small:z < min

b
1−b
,
b
1−b


.

In this case, world specialization pattern changes from (f,m) to (f,d) and further to (d,d) and(m,d) as v increases. According to (55), (71), (60), (57), and (73), labor allocation in Home, β,and that in Foreign, β ∗ , depend on v and z as follows:


1−b




z,
0
if v ≤ 1−b
1
if
v
≤


b
b z


β (v) = 1 if 1−bβ ∗ (v) = b − (1 − b) vz if 1−b(101)b z < v < 1,b z < v < 1,






0 if v > 1,
b (1 + z)
if v > 1.
Note that β (v) and β ∗ (v) are continuous except at v = 1. When crossing v = 1, β (v) drops
from 1 to 0 and β ∗ (v) jumps from b − (1 − b) z to b (1 + z). For v = 1, it is likely that bothcountries produce both goods, leaving two variables, β and β ∗ , to be determined. However, only
one constraint, (25), can be obtained from the world market-clearing condition. As a result, β
and β ∗ are indeterminate at v = 1, with one degree of freedom.
27

We omit two knife-edge cases: z = (1 − b) /b and z = b/ (1 − b). Given that b ∈ (1/2, 1), if z =(1 − b) /b, world specialization pattern changes from (f,m) to (f,d) and further to (d,d) and (m,f); if
z = b/ (1 − b), the pattern changes from (f,m) to (d,d) and further to (d,f) and (m,f). Given thatb ∈ (0, 1/2), if z = (1 − b) /b, world specialization pattern changes from (f,m) to (d,d) and further to(m,d). If z = b/ (1 − b), the pattern changes from (d,m) to (d,d) and further to (m,f).

69

Case 2: Relatively small Foreign. Now consider the case that Foreign is relatively small:z > max

1−b
b
,
b
1−b


.

In this case, world specialization pattern changes from (d,m) to (d,d) and further to (d,f) and(m,f). According to (62), (65), (64), (67), and (70), we obtainb 1 + z1
β (v) = b − (1 − b) vz



0

if v < 1,b
if 1 < v < 1−bz,bif v ≥ 1−b
z,




0 if v < 1,bβ ∗ (v) = 1 if 1 < v < 1−bz,1 if v ≥ b z.
1−b

(102)

Again, β (v) and β ∗ (v) are continuous except at v = 1, where labor allocations become indeterminate: β can vary from b (1 + 1/z) to b − (1 − b) /z and β ∗ from 0 to 1 as long as the world
market-clearing condition (25) holds.

Case 3: Similar sized with strong demand for resource goods. Consider the casethat the demand for resource goods is relatively strong, namely b > 1/2, and two countries areof similar size:
b
1−b
≤z≤
.
b
1−b
In this case, world specialization pattern changes from (f,m) to (f,d) and further to (d,d), (d,f),and (m,f) as v increases. By (55), (65), (60), (57), (67) and (70), we haveβ (v) =




1




1


b − (1 − b) vz




0

if v ≤ 1−bb z,
1−b
if b z < v < 1,b
if 1 < v < 1−bz,bif v ≥ 1−bz,∗

β (v) =




0




b − (1 − b) z

v



1




1

if v ≤ 1−bb z,
if 1−bb z < v < 1,bz,
if 1 < v < 1−b

(103)

b
if v ≥ 1−bz.

Both β (v) and β ∗ (v) are continuous except at v = 1. For v = 1, the two countries canbe diversified and labor allocations become indeterminate: β can be any number from 1 tob − (1 − b) /z and β ∗ from b − (1 − b) z to 1 as long as they satisfy (25).

Case 4: Similar size with strong demand for manufacturing goods. Now consider the case that the demand for manufacturing goods is relatively weak, namely b < 1/2, andtwo countries are of similar size:
b
1−b
≤z≤
.
1−b
b
In this case, world specialization pattern changes from (d,m) to (d,d) and to (m,d). From (62),(71), (73), and (64), labor allocations areb 1 + 1  if v < 1,z
β (v) =
0
if v > 1,

0
if v < 1,β ∗ (v) =b (1 + z) if v > 1.

70

(104)

At v = 1, labor allocations become indeterminate: β can be any number from b (1 + 1/z) to 0
and β ∗ from 0 to b (1 + z) as long as they satisfy (25).

C

The Ṡ = 0 and Ṡ ∗ = 0 curves

Plugging the expressions of β (v) and β ∗ (v) in the corresponding case given in Appendix B into(28) and (29) gives the complete expression of the dynamic system that governs the two-country
world. The phase diagram, with the Ṡ = 0 and Ṡ ∗ = 0 curves on the (S, S ∗ ) plane, is useful inunderstanding the dynamics intuitively, which we explain how to draw in what follows.

Case 1: Relatively small Home. Consider the Ṡ = 0 and Ṡ ∗ = 0 curves in Case 1,
namely z < min {(1 − b) /b, b/ (1 − b)}. Plugging (101) into (28) and (29) yields the dynamic
system in Case 1. Figures 15a and 15b depict Home’s Ṡ = 0 curve, with the former corresponding
to the BT type and the latter to the CT type. Figures 15c and 15d show Foreign’s Ṡ ∗ = 0 curve,with the former corresponding to the BT type and the latter to the CT type. The following
explains how to draw the Ṡ = 0 and Ṡ ∗ = 0 curves in this case.28
First, patterns (f,m), (f,d), (d,d), and (m,d) arise in Case 1, given, respectively, v ≤(1 − b) z/b, (1 − b) z/b < v < 1, v = 1, and v > 1. We can draw two boundary curves,v = (1 − b) z/b = constant and v = 1, which divide the (S, S ∗ ) plane into three regions, corresponding to patterns (f,m), (f,d), and (m,d), respectively. Pattern (d,d), however, arises along
the v = 1 curve. Since v ≡ A∗ (S ∗ ) a/A (S) a∗ and both A∗ (·) and A (·) are strictly increasing
functions that satisfying A∗ (0) = A (0) = 0, both the v = (1 − b) z/b curve and the v = 1
curve start from the origin and have positive slopes on the (S, S ∗ ) plane.29 Moreover, because(1 − b) z/b < 1, the v = (1 − b) z/b curve lies below the v = 1 curve.
Now, move on to the characterization of the Ṡ = 0 and Ṡ ∗ = 0 curves in each region.
In the (f,m) region, that is, the area above the horizontal axis and below the v = (1 − b) z/b
curve, Home specializes in the resource good, that is, β = 1. Therefore, as long as S = S∞ (1),Ṡ = 0 holds regardless of the value of S ∗ . That is, the Ṡ = 0 curve is a vertical line given by
S = S∞ (1) in the (f,m) region, as shown in Figure 15a (for BT Home) and Figure 15b (for CT∗ (0), asHome). Similarly, the Ṡ ∗ = 0 curve of Foreign is a horizontal line given by S ∗ = S∞
shown in Figure 15c (for BT Foreign) and Figure 15d (for CT Foreign).
In the (f,d) region, that is, the area above the v = (1 − b) z/b curve and below the v = 1
curve, Home specializes in the resource good and therefore, the Ṡ = 0 curve remains being thevertical line; yet, Foreign becomes diversified and so the Ṡ ∗ = 0 curve is no longer horizontal.
To see this, assume to the contrary that the Ṡ ∗ = 0 curve is a horizontal line in the (f,d) region.
Moving left along this supposititious horizontal Ṡ ∗ = 0 curve, S decreases and S ∗ remains thesame, so the comparative advantage index, v, increases, leading to an increase in β ∗ according
to (101). This implies that the value of S ∗ changes when moving along the Ṡ ∗ = 0 curve, leading
to a contradiction to the assumption that the Ṡ ∗ = 0 curve is horizontal. Specifically, note that28

We draw the v = constant curves as straight lines in Figures 15 to 18, which in general is notnecessarily the case.
29
For a simple exposition, the v = (1 − b) z/b, v = 1, and v = bz/ (1 − b) curves appear as straightlines in the figure. In general, this is not the case since v is not necessarily a function of S ∗ /S.

71

∗ (β ∗ (v)) holds on the Ṡ ∗ = 0 curve. Some algebra yields thatS ∗ = S∞∗0 β ∗0 ∂v
S∞
dS ∗
∂S
=
,
∗0 β ∗0 ∂v
dS Ṡ ∗ =0 1 − S∞∂S ∗which suggests that, if Foreign is of the BT type, the Ṡ ∗ = 0 curve is positively sloped and lies
∗ (0) line, as shown in Figure 15c; if Foreign is of the CT type, the Ṡ ∗ = 0
below the S ∗ = S∞
∗ (0) and, however, the sign of the slope is indefinite. The Ṡ ∗ = 0
curve lies above the S ∗ = S∞
curve is drawn as negatively sloped in Figure 15d only for the purpose of simple illustration.
In the (d,d) region, which coincides with the v = 1 curve, both Home and Foreign are likely
to be diversified. For Home, β can vary in [b − (1 − b) /z, 1] as long as βz + β ∗ = b (z + 1) issatisfied, implying that the Ṡ = 0 and v = 1 curves overlap between S = S∞ (b − (1 − b) /z)
and S = S∞ (1), as illustrated in Figures 15a and 15b. Similarly, for Foreign, β ∗ can vary in[b − (1 − b) z, 1] as long as βz + β ∗ = b (z + 1) is satisfied. The Ṡ ∗ = 0 and v = 1 curves overlap∗ (b − (1 − b) z) and S ∗ = S ∗ (1), as shown in Figures 15c and 15d.
between S ∗ = S∞∞
Finally, in the (m,d) region, that is, the area above the v = 1 curve, Home specializes in the
manufacturing good, implying that the part of the Ṡ = 0 curve in the (m,d) region is the verticalline satisfying S = S∞ (0), as illustrated in Figures 15a and 15b. By contrast, Foreign remains
diversified; yet the part of the Ṡ ∗ = 0 curve in this region is now the horizontal line satisfying∗ (b (1 + z)) since β ∗ = b (1 + z) = constant in pattern (m,d) according to (101).
S ∗ = S∞

Case 2: Relatively small Foreign. Consider the Ṡ = 0 and Ṡ ∗ = 0 curves in Case 2,
namely z > max {(1 − b) /b, b/ (1 − b)}. The dynamics in Case 2 can be obtained by substituting(102) into (28) and (29). Figure 16 presents the Ṡ = 0 and Ṡ ∗ = 0 curves in Case 2. The following
explains how to draw the figure.
First, patterns (d,m), (d,d), (d,f), and (m,f) arise in Case 2, given, respectively, v < 1,
v = 1, 1 < v < bz/ (1 − b), and v ≥ bz/ (1 − b). We can draw two boundary curves, v = 1 andv = bz/ (1 − b) = constant, which divid the (S, S ∗ ) plane into three regions, corresponding to
specialization patterns (d,m), (d,f), and (m,f), whereas pattern (d,d) arises on the v = 1 curve.
Now, move on to the characterization of the Ṡ = 0 and Ṡ ∗ = 0 curves in each region. We haveobtained the properties of the two curves in (d,d) in Case 1, and we can apply the argument overthe (m,d) region in Case 1 to the (d,m) region here. In the following, we focus on the propertiesof the two curves within (d,f) and (m,f) regions.
In the (d,f) region, that is, the area above the v = 1 curve and below the v = bz/ (1 − b)
curve, Foreign specializes in the resource good and so the Ṡ ∗ = 0 curve is the horizontal line
∗ (1). At the same time, Home becomes diversified. Note that S = S (β (v))
satisfying S ∗ = S∞∞
holds on the Ṡ = 0 curve, from which it follows that0 β 0 ∂v
1 − S∞dS ∗∂S=.
0 β 0 ∂v
dS Ṡ=0S∞∗∂S

It is then clear that, if Home is of the BT type, the Ṡ = 0 curve is positively sloped and lies left
to the S = S∞ (0) line; if Home is of the BT type, the Ṡ = 0 curve lies right to the S = S∞ (0)line, and, however, the sign of the slope is indefinite.

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In the (m,f) region, that is, the area above the v = bz/ (1 − b) curve, Home specializes in
the manufacturing good. The Ṡ = 0 curve turns into the vertical line satisfying S = S∞ (0) in
the region, as illustrated in Figure 16a (for BT Home) and Figure 16b (for CT Home). Foreign
remains specialized in the resource good and the Ṡ ∗ = 0 curve remains a horizontal line, as
illustrated in Figure 16c (for BT Foreign) and Figure 16d (for CT Foreign).

Case 3: Similar sized with strong demand for resource goods. Consider the Ṡ = 0
and Ṡ ∗ = 0 curves in Case 3, namely b > 1/2 and (1 − b) /b ≤ z ≤ b/ (1 − b). The dynamics
in Case 3 can be obtained by substituting (103) into (28) and (29). Patterns (f,m), (f,d), (d,d),(d,f), and (m,f) arise in Case 3, given, respectively, v ≤ (1 − b) z/b, (1 − b) z/b < v < 1, v = 1,
1 < v < bz/ (1 − b), and v ≥ bz/ (1 − b). We can draw three boundary curves, which divid the(S, S ∗ ) into four regions, corresponding to specialization patterns (f,m), (f,d), (d,f), and (m,f).
Again, pattern (d,d) arises on the v = 1 curve. Since we have obtained the properties of theṠ = 0 and Ṡ ∗ = 0 curves within (f,m), (f,d), and (d,d) regions in Case 1, and those within (d,f)
and (m,f) regions in Case 2. Applying these results yields Figure 17, which presents Home’sṠ = 0 curve and Foreign’s Ṡ ∗ = 0 curve in Case 3.

Case 4: Similar sized with strong demand for manufacturing goods. Consider
the Ṡ = 0 and Ṡ ∗ = 0 curves in Case 4, namely b < 1/2 and b/ (1 − b) ≤ z ≤ (1 − b) /b. The
dynamics can be obtained by substituting (104) into (28) and (29). Applying the results above,
Figure 18 presents the Ṡ = 0 and Ṡ ∗ = 0 curves in Case 4.

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(a) Ṡ = 0 (BT Home).

(b) Ṡ = 0 (CT Home).

(c) Ṡ ∗ = 0 (BT Foreign).

(d) Ṡ ∗ = 0 (CT Foreign).

Figure 15: The Ṡ = 0 and Ṡ ∗ = 0 curves in Case 1. Panel (d) draws the Ṡ ∗ = 0 curve in(f,d) region as negatively sloped, which in general is not necessarily the case.

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(a) Ṡ = 0 (BT Home).

(b) Ṡ = 0 (CT Home).

(c) Ṡ ∗ = 0 (BT Foreign).

(d) Ṡ ∗ = 0 (CT Foreign).

Figure 16: The Ṡ = 0 and Ṡ ∗ = 0 curves in Case 2. Panel (b) draws the Ṡ = 0 curve in(d,f) region as negatively sloped, which in general is not necessarily the case.

75

(a) Ṡ = 0 (BT Home).

(b) Ṡ = 0 (CT Home).

(c) Ṡ ∗ = 0 (BT Foreign).

(d) Ṡ ∗ = 0 (CT Foreign).

Figure 17: The Ṡ = 0 and Ṡ ∗ = 0 curves in Case 3. Panel (b) draws the Ṡ = 0 curvein (d,f) region and panel (d) draws the Ṡ ∗ = 0 curve in (f,d) region as negatively sloped,
which in general are not necessarily the case.

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(a) Ṡ = 0 (BT Home).

(b) Ṡ = 0 (CT Home).

(c) Ṡ ∗ = 0 (BT Foreign).

(d) Ṡ ∗ = 0 (CT Foreign).

Figure 18: The Ṡ = 0 and Ṡ ∗ = 0 curves in Case 4.

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