ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E19-8

A Two-country Model of Public Infrastructure
Capital: Trade Patterns and Trade Gains in theLong Runby
Akihiko Yanase
Makoto Tawada

August 2019

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

A Two-country Model of Public Infrastructure Capital:
Trade Patterns and Trade Gains in the Long Run ∗Akihiko Yanase †Graduate School of Economics, Nagoya University

Makoto Tawada ‡Faculty of Economics, Aichi Gakuin University

Abstract
This study develops a two-country dynamic trade model with public infrastructure
capital that has a positive eﬀect on private sector productivity. Under the assumptions
that welfare-maximizing national governments determine the paths of production taxes
for financing public investment and that infrastructure has an “unpaid factor” property,
this study examines the economy’s trade pattern and the long-run eﬀects of trade. If
governments take world prices as given when they make public investments, a country
with a smaller labor endowment, a lower depreciation rate for infrastructure capitalstock, and/or a lower rate of time preference will become an exporter of a good thatis more dependent on public infrastructure and will unambiguously gain from trade,
whereas its trading partner may lose from trade. In the case of strategic governmentsthat recognize the eﬀect on the terms of trade when they determine their policy paths,
the country exporting (importing) the good that is more dependent on the stock of public
infrastructure will underaccumulate (overaccumulate) infrastructure in the long run.

Key Words: Public infrastructure capital; Two-country trade; Trade pattern;
Gains/losses from trade; Terms-of-trade eﬀect
JEL classification: F11; H41∗

We thank Ngo Van Long, Haiping Zhang, Mark Razhev, John Gilbert, Ryoji Ohdoi, Jota Ishikawa, Yan
Ma, Noritsugu Nakanishi, Yunfang Hu, Yoshifumi Ohkawa, and Yoshimasa Aoki for their helpful comments on
earlier drafts. This work has been financially supported by Japan Society for the Promotion of Science (JSPS)
Grant-in-Aid for Scientific Research (B) (No. 16H03612), (C) (No.17K03730), and Fund for the Promotion of
Joint International Research (No. 16KK0079).
†
Graduate School of Economics, Nagoya University. Furo-cho, Chikusa-ku, Nagoya 464-8601, JAPAN.
E-mail: yanase@soec.nagoya-u.ac.jp‡
mtawada2@dpc.agu.ac.jp1

1

Introduction

In economic activities, public infrastructure such as the electricity network, water supply,transportation, telecommunications, fundamental education, and the legal system plays a
key role in facilitating increased production and smooth economic transactions. In light of
the recent rapid globalization of the world economy, there is a growing need to consider therole of public infrastructure in connection with international trade, as documented by recent
empirical studies such as Limão and Venables (2001), Bougheas et al. (2003), and Francois
and Manchin (2013).
In international trade theory, productive public infrastructure has long been analyzed
by incorporating public intermediate goods in traditional (Ricardian or Heckscher–Ohlin–
Samuelson) trade models (Manning and McMillan, 1979; Tawada and Okamoto, 1983; Tawada
and Abe, 1984; Ishizawa, 1988; Abe, 1990; Altenburg, 1992; Suga and Tawada, 2007). There
are also studies on cost-reducing transport infrastructure and international trade (Martin andRogers, 1995; Bougheas et al., 1999; Bond, 2006; Mun and Nakagawa, 2010; Tsubuku, 2016).
However, most of these studies use a static framework. In reality, many public intermediate
goods have the characteristics of durable goods or capital stock; research and development
activities, national defense, and transportation and communication infrastructure are typicalexamples.
An exceptional study is McMillan (1978), who considers the stock eﬀect of a public intermediate good in an open economy. He considers a three-sector (two private goods and a public
intermediate good), one-factor (labor) small open economy with optimal supply of the public
intermediate good. It is shown that the stock of the public intermediate good determines theslope of the production possibility frontier, and thus determines the pattern of international
trade. Yanase and Tawada (2012) reexamine McMillan’s model and show the possibility ofmultiple steady states and history-dependent dynamic paths. Moreover, they discuss whether
trade is beneficial or not in McMillan’s model.
In both McMillan (1978) and Yanase and Tawada (2012), the public intermediate good isassumed to have an impact similar to the “creation of atmosphere”-type externality classified
by Meade (1952). That is, in their models, the technology in each private sector exhibits
constant returns to scale in primary inputs alone. In a one-factor model, this assumption
implies that for a given stock of the public intermediate good, the production possibility2frontier becomes linear, as in the standard Ricardian model, and thus the economy hardly
diversifies production.
There is another class of public intermediate good that can be interpreted as “unpaid
factors of production” according to Meade’s (1952) terminology. If the public good is of thistype, the production function of each private sector is characterized by constant returns toscale in all inputs, including the public intermediate good.1 In the case of an unpaid factor
type, congestion arises within an industry, although not among industries. For example, public
transportation systems like highways may be included in this type. The agriculture industry
uses the highway system in a certain season or region, while the manufacturing industry uses it
in a diﬀerent season or region. This highway system can be considered as public infrastructureof the unpaid factor type. Similarly, the communication system can be assumed to be ofthis type. In this study, we focus on this kind of public intermediate good and present a
dynamic trade model in which the stock of a public good has a positive eﬀect on private
sector production. Following McMillan (1978) and Yanase and Tawada (2012), we consider anopen economy in which two tradable private consumption goods and one nontradable public
intermediate good are produced by one primary input, that is, labor. However, in contrast
with McMillan (1978) and Yanase and Tawada (2012), we suppose that the private goods areproduced under constant returns to scale with respect to the stock of the public intermediate
good, as well as labor. Thus, the production possibility frontier becomes strictly concave tothe origin, even in the case of one primary factor. Therefore, in general, diversified production
is possible even after trade.
A dynamic trade model with public infrastructure of the unpaid factor type has alreadybeen analyzed by Yanase and Tawada (2017). However, their analysis is confined to a small
open economy in which commodity prices are constant. In addition, they focus on how thelabor endowment aﬀects a country’s comparative advantage and trade pattern. In the present
study, we consider a two-country model, and thus commodity prices are endogenously determined in the international market under free trade. We also examine other determinants oftrade patterns in addition to the labor endowment.
We begin with a dynamic two-country model in which the national governments, taking theprices of tradables as given, make infrastructure investments. The cost of public investment is
1

Tawada (1980) and Tawada and Okamoto (1983) used the alternative term “semi-public intermediategood” to describe this kind of good, compared with the “pure” public intermediate goods that have “creationof atmosphere”-type externalities.

3

financed by production taxes imposed on each private sector. We show that the competitive
equilibrium path with the optimal production taxes coincides with the optimal path determined by a social planner who directly controls the economy’s resource allocation. We also
show that there exists a unique saddle-point stable steady state if the economic fundamentalsof the two countries are slightly diﬀerent. We then proceed to analyze the trade pattern ofeach country. It is shown that if the economy is initially at the autarkic steady state, after
commencing trade, a country with a smaller (larger) labor endowment tends to be an exporterof a good that is more (less) dependent on the stock of public infrastructure. We also discuss
whether each country gains or loses from trade by comparing the steady-state welfare level
under free trade with that under autarky. It is shown that in comparison with the autarkic
steady state, free trade increases (reduces) the steady-state stock of public infrastructure andthereby expands (diminishes) the long-run production possibility in a country exporting agood that is more (less) dependent on public infrastructure. An interesting result is that asmaller country unambiguously gains from trade in the long run, whereas a larger country
may lose from trade.
We also consider an alternative scenario in which the governments strategically determine
their provision of public infrastructure, taking into consideration the eﬀect of each country’s
choice on the world market. If each country recognizes the eﬀect of its decisions on the world
price, namely the terms-of-trade eﬀect, the governments’ decision-making in relation to public
investment can involve, through the pursuit of national interests, an improvement of its termsof trade. Because of this strategic behavior undertaken by national governments, it is shownthat the steady-state stock of public infrastructure in each country diﬀers from that in the casewhere the governments have no such strategic incentives. That is, in comparison with the caseof nonstrategic governments, the country exporting (importing) a good that is more dependent
on the infrastructure stock would underaccumulate (overaccumulate) public infrastructure in
the long run.
The interdependence of public goods supply among countries in the presence of international trade has already been discussed in the literature. Devereux and Mansoorian (1992)
and Figuières et al. (2013) extend Barro’s (1990) model of endogenous growth driven by productive public expenditure to an open-economy context. Their main focus is on the strategic
distortions caused by the lack of cooperation in determining tax rates to finance the production of public goods, and the growth consequences of such noncooperative policy-making. The
4

ineﬃciency of noncooperative supply of infrastructure that reduces transportation costs in thepresence of trade is also discussed in Bougheas et al. (2003), who develop a static two-country,
two-good model. They also conduct an empirical analysis using data from 16 European countries over the period 1987–1995 and find evidence consistent with their theoretical predictions.
One shortcoming of these studies is that they assume that only one of two traded goods is
produced in each country. In other words, trade patterns are exogenously determined in thesestudies. By contrast, although we focus on the optimal policy for the provision of public
infrastructure from a single country’s viewpoint, we take a closer look at the basic theoremsin trade theory, namely the patterns of trade and the gains from trade, in a dynamic context.
The rest of the paper is organized as follows. Section 2 describes our two-country dynamic
model with public infrastructure. In Sections 3 and 4, we consider a situation in which
the government in each country makes a decision on public investment without taking into
account its eﬀect on world commodity prices and international trade. The properties of the
dynamic equilibrium path and the steady state are analyzed in Section 3, and each country’s
trade pattern and the possibility of trade gains are examined in Section 4. In Section 5, we
consider a situation in which the governments act strategically in the sense that they makepublic infrastructure investments while recognizing the eﬀect they have on the terms of tradeand compare the equilibrium conditions with those in the case of nonstrategic governments.
Section 6 concludes.

2

Model

We consider a world economy consisting of two countries, home and foreign, in which two
private and one public production sectors and one primary factor exist. The primary factor
is assumed to be labor. The public sector produces an investment good using nonincreasing
returns to scale technology with respect to labor. The investment good can be accumulated,
and its accumulated stock, public infrastructure, serves production in the private sectors as
a positive external eﬀect without congestion between sectors. The two private sectors aredenoted by sectors 1 and 2, where goods 1 and 2, respectively, are produced using constantreturns to scale technology with respect to labor and the stock of public infrastructure. Totallabor endowment is assumed to be constant over time. The private goods are traded between
countries, while the public good for infrastructure investment is assumed to be nontradable.

5

2.1

Production side

Let us focus on the home country. The foreign country, whose variables are denoted by anasterisk (∗ ), has a similar economic structure.
The production function of each private sector is assumed to take the following form:iYi = Rαi L1−α,i0 ≤ αi < 1,

i = 1, 2,(1)where Yi is the output of good i, R is the stock of public infrastructure, and Li is the labor
input in sector i. It is clear that the labor productivity in each private sector is given byi∂Yi /∂Li = (1 − αi )Rαi L−αand is dependent on the stock of public infrastructure R.
i

In the following analysis, we make the following assumption regarding the impact of thestock of public infrastructure on industries:
Assumption 1 α1 > α2 .
Because αi is the production elasticity of the infrastructure stock in sector i, i.e., αi =(∂Yi /∂R) · (R/Yi ), Assumption 1 can be interpreted as stating that sector 1 is more dependent
on the stock of public infrastructure than sector 2. At the same time, Assumption 1 implies
that sector 2 is more labor intensive than sector 1, as is usual in the standard Heckscher–
Ohlin–Samuelson model.
The production function of the investment good in the public sector is expressed as r =f (LR ), where r is the output and LR is the labor input in the public sector. Concerning f (LR ),we assume that this function is increasing and strictly concave (f ′ > 0 > f ′′ ), and f (0) = 0.
Given the initial stock R0 > 0, the infrastructure capital is assumed to accumulate over time
according to2
Ṙ = f (LR ) − βR,(2)where β > 0 is the depreciation rate of the stock of public infrastructure.
The full-employment condition of labor at each moment in time is given by
L1 + L2 + LR = L,

(3)

where L > 0 is labor endowment and is assumed to be constant over time.
2

A dot over a variable denotes a time derivative. To reduce the complexity of notation, we omit timearguments when no confusion is caused by doing so.

6

We consider a competitive economy and assume that the cost of public investment isfinanced by production taxes:
wLR = τ1 pY1 + τ2 Y2 ,(4)

where w is the wage rate, p is the world price of good 1 (good 2 is assumed to be numeraire),and τi is the production tax rate imposed in sector i = 1, 2.3 Then, a representative firm’s
profit in sector 1 is given by Π1 = (1 − τ1 )pY1 − wL1 , and that in sector 2 is given byΠ2 = (1 − τ2 )Y2 − wL2 . The first-order conditions for profit maximization of the respective
private firms are therefore given by
Y1
= w,
L1
Y2
= w.
(1 − τ2 )(1 − α2 )L2(1 − τ1 )(1 − α1 )p

(5)
(6)

The adjustment of wages in the labor market means that the full-employment condition oflabor (3) is satisfied. Therefore, the government’s budget constraint (4), profit maximization
conditions (5) and (6), production function (1), and market-clearing condition of labor (3)jointly determine the equilibrium values for Y1 , Y2 , L1 , L2 , LR , and w as a function of tax
rates τ1 and τ2 , as well as R, p, and L.
For the subsequent analysis, it is of interest to present the properties of Y1 , Y2 , and LR .
The comparative static results are shown in Appendix A.1, from which we can state that anincrease in the tax rate in each sector reduces output in that sector, i.e., ∂Yi /∂τi < 0, i = 1, 2,
and that the law of supply holds, i.e., ∂Y1 /∂p > 0 > ∂Y2 /∂p.

2.2

Consumption side

The consumption side of the economy is described by a representative household whose lifetime
utility is given by:∫ ∞U=

e−ρt [γ ln C1 + (1 − γ) ln C2 ] dt,(7)0

where Ci is consumption of good i (i = 1, 2), ρ is the rate of time preference, and γ ∈ (0, 1) isa parameter.

3

Kalaitzidakis and Tzouvelekas (2011) develop an endogenous growth model of a decentralized market
economy with N diﬀerent types of productive public capital, the investment costs of which are financed by
tax revenue. They show that the growth-maximizing tax rate and the growth-maximizing share of public
investment also maximize welfare.

7

Assuming that no borrowing or lending is permitted, the household’s utility maximization
behavior yields the optimal consumption for each good as C1 = γI/p and C2 = (1 − γ)I,
where I is the household’s income. I consists of firm profits Π1 and Π2 , and labor incomew(L1 + L2 + LR ). In light of (4), it is verified that I = pY1 + Y2 .
Substituting the household’s optimal consumption into the lifetime utility (7), the household’s indirect lifetime utility is derived as∫ ∞V =e−ρt {ln[pY1 (τ1 , τ2 , R, p, L) + Y2 (τ1 , τ2 , R, p, L)] − γ ln p + Γ} dt,(8)0where Γ ≡ γ ln γ + (1 − γ) ln(1 − γ).

3

Trading Equilibrium under Nonstrategic Behavior ofGovernments

In this and the next sections, we assume that governments in both countries take the world
price p as given when they determine the time paths of their stock of public infrastructure bycontrolling the tax rates.

3.1

Optimal policy

Let us focus on the home country. The home government’s dynamic optimization problem is to maximize the household’s indirect lifetime utility (8) subject to (2), where LR =LR (τ1 , τ2 , R, p, L), by choosing the time paths of τ1 and τ2 . The current-value Hamiltonian is
H = ln[pY1 (τ1 , τ2 , R, p, L) + Y2 (τ1 , τ2 , R, p, L)] − γ ln p + Γ + θ{f (LR (τ1 , τ2 , R, p, L)) − βR}.
The optimality conditions are derived as follows:()∂H
1∂Y1 ∂Y2∂LR=p++ θf ′ (LR )= 0,∂τ1
pY1 + Y2
∂τ1
∂τ1
∂τ1
(
)
∂H
1
∂Y1 ∂Y2∂LR=p++ θf ′ (LR )= 0,∂τ2
pY1 + Y2
∂τ2
∂τ2
∂τ2
{
(
}
)
∂LR
∂Y1 ∂Y2∂H
1′= θ ρ + β − f (LR )p+θ̇ = ρθ −−,∂R∂R
pY1 + Y2∂R∂R
lim e−ρt θ(t)R(t) = 0.
t→∞

(9)
(10)
(11)
(12)

Substituting the comparative statics results (A.4) in the Appendix into the first-orderconditions (9) and (10) and rearranging terms, it follows thatw.
τ1 = τ2 = 1 −(pY1 + Y2 )θf ′ (LR )8(13)Substituting (13) into the private firms’ profit maximization conditions (5) and (6) and rearranging terms, we have
(1 − α1 )pY1Y2= (1 − α2 )= (pY1 + Y2 )θf ′ (LR ).
L1
L2

(14)

In addition, substituting the comparative statics results (A.4) into the adjoint equation (11)and rearranging terms, we haveθ̇ = (ρ + β)θ −α1 pY1 + α2 Y2.
R(pY1 + Y2 )(15)

It is convenient to characterize the abovementioned set of equilibrium conditions with
optimal output taxes by using the GDP function defined by{
1
2
G(p, R, l) = max pRα1 L1−α
+ Rα2 L1−α
12
L1 ,L2}s.t. L1 + L2 = l ,where l ≡ L − LR is the sum of labor inputs in the private sectors. Applying the envelope
theorem to the GDP function, we obtain the following derivatives:4
Gp = Y1 ,GR =α1 pY1 + α2 Y2,R

Gl = (1 − α1 )pY1
Y2
= (1 − α2 ) .
L1
L2

(16)

Appendix A.2 provides other properties of the GDP function.
Lemma 1 Suppose that the government in each country determines the production taxes so as
to maximize national welfare, taking the world price as given, in a competitive world economy.
Then, the equilibrium path with optimal production taxes coincides with the optimal path in∫∞which a social planner determines the path of LR so as to maximize 0 e−ρt [ln G(p, R, L −LR ) − γ ln p + Γ]dt subject to (2).
Proof. Let us define the current-value Hamiltonian associated with the social planner’s
dynamic optimization problem asH̃(LR , R, θ) = ln[G(p, R, L − LR )] − γ ln p + Γ + θ̃ {f (LR ) − βR} .
Then, the optimal control must satisfy
∂ H̃
=0
∂LR
4

⇒

θ̃f ′ (LR ) =Gl (p, R, L − LR ).
G(p, R, L − LR )The subscripts denote partial derivatives: Gp = ∂G/∂p, and so on.

9

(17)

Moreover, the adjoint equation and the transversality condition are expressed, respectively, asθ̇ = ρθ̃ −∂ H̃
GR (p, R, L − LR )= (ρ + β)θ̃ −,∂R
G(p, R, L − LR )lim e−ρt θ̃(t)R(t) = 0.

t→∞

(18)
(19)

In light of (16), the optimality conditions (17), (18), and (19) are identical to (14), (15), and2(12), respectively, if we set θ̃ = θ.

The optimality conditions (17) and (18), which are equivalent to (14) and (15), respectively,
provide the following familiar interpretations. Eq.(17) indicates that the value of the marginal
product of labor should be equalized across the public and private sectors. Eq.(18) indicatesthat the sum of the capital gain (or loss if it is negative) of the infrastructure capital θ̇ and themarginal social benefit of an increase in infrastructure GR /G should be equal to the marginal
cost (ρ + β)θ, which is the sum of the intertemporal cost implied by the discount rate and thereplacement cost of depreciated infrastructure capital.5

3.2

Dynamic equilibrium pathAt each moment in time, the world market for each private good has to hold along an equilibrium path. We assume that the two countries share identical production technologies, meaningthat the GDP function in the foreign country is given by G(p, R∗ , L∗ − L∗R ), and have the same
felicity function, meaning that γ = γ ∗ . Then, the world market-clearing condition for good 1,
C1 + C1∗ = Y1 + Y1∗ , can be rewritten asγ{G(p, R, L − LR ) + G(p, R∗ , L∗ − L∗R )} = Gp (p, R, L − LR ) + Gp (p, R∗ , L∗ − L∗R ).
p(20)

The dynamic equilibrium of the world economy is characterized by the home country’s
static optimality condition (17) and dynamic equations (2) and (18), their foreign counterparts,and the world market-clearing condition (20).

5Needless to say, this is a dynamic version of Kaizuka’s condition for optimal resource allocation in an
economy with public intermediate goods (Kaizuka, 1965).

10

3.3

Steady state

At the steady state, it holds that Ṙ = Ṙ∗ = θ̇ = θ̇∗ = 0. In light of (2), (17), and (18),conditions for the free-trade, steady-state equilibrium are given byf (LR ) = βR,
GR (p, R, L − LR ) ′ρ+β =f (LR ),Gl (p, R, L − LR )f (L∗R ) = β ∗ R∗ ,GR (p, R∗ , L∗ − L∗R ) ′ ∗f (LR ),
ρ∗ + β ∗ =Gl (p, R∗ , L∗ − L∗R )

(21)
(22)
(23)
(24)

and (20).
From (21) and (22), the steady-state level of LR can be represented as a function of p(and the parameters β, ρ, and L): LR = ψ(p). Under Assumption 1, the following lemma is
obtained.
Lemma 2 (i) ψ ′ (p) > 0. (ii) There exist LR and LR such that 0 < LR < LR < L,
limp→0 ψ(p) = LR , and limp→∞ ψ(p) = LR .
Proof. See Appendix A.3.
From (21), the steady-state stock of public infrastructure is denoted as R = f (ψ(p))/β.
Analogously for the foreign country, from (23) and (24), we have L∗R = ψ ∗ (p), which hassimilar properties to Lemma 2, and R∗ = f (ψ ∗ (p))/β ∗ . Substituting these expressions into
(20), the steady-state, market-clearing condition is given by
{ (
)
(
)}
f (ψ(p))f (ψ ∗ (p)) ∗γ∗
G p,, L − ψ(p) + G p,
, L − ψ (p)
p
β
β∗
(
)
)
(
f (ψ(p))f (ψ ∗ (p)) ∗∗= Gp p,, L − ψ (p) .
, L − ψ(p) + Gp p,
β
β∗

(25)

Let us denote the solution of (25), i.e., the steady-state equilibrium solution for the relative
price of good 1 in the world market by pss . The steady-state stocks of public infrastructure in∗= f (ψ ∗ (pss ))/β ∗ ,the home and foreign country are then derived as Rss = f (ψ(pss ))/β and Rss

respectively. The existence, uniqueness, and stability of the steady state are characterized by
the following theorem.

11

Theorem 1 There exists at least one steady-state, free-trade equilibrium. If the diﬀerencesin discount rates, depreciation rates, and labor endowments between the two countries are
not very large, there exists a unique saddle-point stable steady state in which production isdiversified in both countries.
Proof. See Appendix A.4.

4

Trade Patterns and Trade Gains in the Long Run

4.1

Trade patterns

Let us focus on the home country’s excess demand for good 1 in the steady state:
(
)
(
)
γ
f (ψ(p))f (ψ(p))
ED(p) ≡ G p,
, L − ψ(p) − Gp p,, L − ψ(p) .
p
β
β

(26)

The foreign country’s excess demand in the steady state, ED∗ (p), can be analogously defined.
Let us denote the autarkic steady-state prices in the home country and foreign country by pa
and p∗a , respectively. Then, in the autarkic steady-state equilibrium, it holds that ED(pa ) =0 = ED∗ (p∗a ). It is verified that ED(p) and ED∗ (p) are downward sloping in the neighborhoodof the steady-state, autarkic equilibrium price.
Labor endowments Evaluating at the autarkic steady-state solutions, we obtain the shift
in excess demand in response to a change in the labor endowment L as follows:
(
)
(
) ′
)(
Gp Gl
Gp GR
f ∂ψ(p)∂ψ(p)∂ED(p)
=
− GpR
+
− Gpl
1−
,
∂L
G
β ∂L
G
∂L
p=pa
(

where
∂ψ
=(
GR GlR
∂L
−G
Gl

RR

)

GRl − GRGGl ll(f ′ )2
+
β

(

)

(27)

f′

)
>0
GlR − GRGGl ll f ′ − GR f ′′from (21) and (22).6 Using the calculations given in the Appendix, it can be verified thatthe sign of ∂ED(p)/∂L is equal to that of (α2 − α1 )f ′′ . Therefore, ∂ED(p)/∂L is positive
under Assumption 1. Suppose that the home country has a smaller labor endowment than
the foreign country: L < L∗ . Then, it holds that ED(p) < ED∗ (p) in the neighborhood of

12

Figure 1: Steady-state equilibrium prices in the case where ED(p) < ED∗ (p)autarkic equilibrium prices, as illustrated in Figure 1, and thus pa < p∗a . Hence, the homecountry exports good 1 in the steady state.
McMillan (1978) and Yanase and Tawada (2012) consider a dynamic model of a small open
economy with a stock of a public intermediate good in which the production technology of eachprivate good exhibits a Ricardian property, i.e., constant returns to scale with respect to labor.
They show that if the labor endowment is suﬃciently large (small), a small open country
specializes in a good whose productivity is more (less) sensitive to the public intermediate
good. This implies that after commencing trade, a country with a higher labor endowment
becomes an exporter of a good whose productivity is more sensitive to the public intermediate
good. However, in the present model with constant-returns technology with respect to laborand the public good stock, the result is reversed.
Intuitively, the diﬀerence between the present model and McMillan’s model can be interpreted as follows. In McMillan (1978) and Yanase and Tawada (2012), the private sectors’
production function is given by Yi = gi (R)Li , where gi (R) is increasing and concave in R,
and thus each country’s comparative advantage depends on the current stock of the public
6

See (A.22) in Appendix A.5 for the derivation.

13

intermediate good. The labor endowment aﬀects comparative advantage only indirectly, via
the accumulation of the public good, and a country with a higher labor endowment will havea higher R. Hence, a larger country tends to specialize in a good that is more dependent on R.
In the present model, by contrast, each country’s comparative advantage depends on both R
and L in a manner such that a higher R (L) increases Y1 to a greater (lesser) extent than Y2 .
Although a higher labor endowment implies a higher steady-state stock of R, the steady-state
relative factor endowment is decreasing in L. This implies that a country with a higher L has
comparative advantage in the labor-intensive good, which in turn is less dependent on R.
Depreciation rates The shift in excess demand in response to a change in the depreciation
rate of the public good stock β is derived as
(
)( ′
) (
)
∂ED(p)Gp GR
f ∂ψ(p)
f
Gp Gl
∂ψ(p)
=
− GpR
− 2 −
− Gpl
,
∂β
G
β ∂β
β
G
∂β
p=pa
(

where
∂ψ
=(
GR GlR∂β−GGl

GR GlR
− GRR
Gl

)

RR

(f ′ )2
+
β

(

)

(28)

ff′
− Gl
β2

)
.
GlR − GRGGl ll f ′ − GR f ′′Using the calculations given in the Appendix, it can be verified that ∂ED(p)/∂β > 0 under
Assumption 1. Suppose that the home country has a lower depreciation rate for its public
good stock than the foreign country: β < β ∗ . In this case, it holds that ED(p) < ED∗ (p) in
the neighborhood of autarkic equilibrium prices, and thus pa < p∗a . The home country exports
good 1 in the steady state. The intuition behind this result is straightforward. Other things
being equal, a smaller β implies a higher steady-state stock of public infrastructure, whichleads to a higher relative supply of good 1 under Assumption 1, and thus a lower relative priceof good 1 under autarky.
Discount rates The shift in excess demand in response to a change in the discount rate ρ
is derived as∂ED(p)
=
∂ρ
p=pa

(

Gp GR
− GpR
G

)

f ′ ∂ψ(p)
−
β ∂ρ

(

Gp Gl
− Gpl
G

)

∂ψ(p)
,
∂ρ

where
∂ψ
= −(
GR GlR
∂ρ
−G
Gl

)
RR

(f ′ )2
+
β

Gl
(
)
< 0.
GlR − GRGGl ll f ′ − GR f ′′

14

(29)

From (A.15) and (A.16) in the Appendix, it holds that Gp GR /G−GpR < 0 and Gp Gl /G−Gpl >0 under Assumption 1. Therefore, it follows that ∂ED(p)/∂ρ > 0, i.e., an increase in ρ
shifts ED(p) upwards. Suppose that the home consumer is more patient than the foreign
consumer: ρ < ρ∗ . In this case, it holds that ED(p) < ED∗ (p) in the neighborhood of
autarkic equilibrium prices, and thus pa < p∗a . The home country exports good 1 in the steadystate.
The intuition behind this result is as follows. Other things being equal, a smaller ρ implies
that people are more patient and put more weight on future consumption than on current consumption. Future consumption can be enhanced by the accumulation of public infrastructure,
which augments outputs of final goods in the future.
To summarize, we establish the following theorem regarding each country’s trade pattern.
Theorem 2 Other things being equal, (i) the country with a smaller (larger) labor endowment,(ii) the country with a lower (higher) depreciation rate for its public good stock, and/or (iii)the patient (impatient) country exports (imports) the good that is more dependent on the public
good stock in the steady state.

4.2

Gains from trade

Let us denote the autarkic steady-state stock of public infrastructure in the home and foreigncountries by Ra and Ra∗ , respectively. Suppose that the home country exports good 1 in
the free-trade, steady-state equilibrium, i.e., pa∗ > pss > pa . Then, from Lemma 2, it holds∗
that Rss > Ra and Rss< Ra∗ . That is, in comparison with the autarkic steady state, tradeliberalization increases the steady-state stock of public infrastructure in the country exportinga good that is more dependent on the public good stock, while it reduces the stock of public
infrastructure in the other country.
Theorem 3 The country exporting a good that is more dependent on the stock of public infrastructure unambiguously gains from trade in the long run, in the sense that the country
enjoys higher steady-state welfare under free trade than under autarky.
Proof. Let us define the expenditure function asE(p, u) = min {pC1 + C2
C1 ,C2s.t. γ ln C1 + (1 − γ) ln C2 ≥ u} .

15

It is easily verified that Eu > 0. Let us also denote the steady-state utility level under autarky
and free trade by ua and uss , respectively. In light of the budget constraint pY1 +Y2 = pC1 +C2 ,we have the following expression:E(pss , uss ) − E(pss , ua )= G(pss , Rss , lss ) − (pss Y1a + Y2a ) + (pss C1a + C2a ) − E(pss , ua ),(30)

where lss is the free-trade, steady-state level of l, and Yia and Cia are the autarkic steady-statelevels of output and consumption, respectively, of good i = 1, 2. From the definition of theexpenditure function, it holds that pss C1a + C2a ≥ E(pss , ua ). Using the GDP function, thesecond term on the right-hand side of (30) can be rewritten as G(pss , Ra , la ), where la is theautarkic steady-state level of l. Because the stead-state labor input in the public sector isrewritten as LR = f −1 (βR), the eﬀect of an increase in R on the maximized GDP for a given
price level can be derived asβ
Gl (p, R, L − f −1 (βR))
f′β= GR (p, R, L − f −1 (βR)) −GR (p, R, L − f −1 (βR))ρ+βρ−1
GR (p, R, L − f (βR)) > 0,=ρ+βGR (p, R, L − f −1 (βR)) −(31)

where we use (22). Eq.(31) implies that G(pss , Rss , lss ) > pss Y1a + Y2a . To summarize, itfollows that the sign of (30) is unambiguously positive, and thus uss > ua .

2

For the country importing a good that is more dependent on the stock of public infrastructure, international trade reduces the steady-state stock of public infrastructure, and thus
the steady-state national income evaluated at post-trade prices. If this reduction in national
income outweighs the eﬃciency gains from specialization and exchange, the country will suﬀer
steady-state losses from trade.
Note that the above discussion on gains/losses from trade is from a long-run viewpoint;
we have focused on a comparison of steady-state welfare levels. Let us now discuss the welfare
eﬀects of trade along the transition path. Suppose that the home country exports good 1 andthat both countries are initially under the autarkic steady state and open international trade.
Then, in the short run, both countries enjoy welfare improvement because their consumption
possibilities expand. However, along the transition path, the stock of public infrastructure
16increases over time in the home country and decreases in the foreign country. This implies
that the home country continues to improve welfare, but in the foreign country, instantaneous
welfare decreases over time and may be lower than autarkic welfare after a certain point in
time.

4.3

Possibility of loss from trade: an example

As indicated in the previous subsection, a country importing a good that is less dependent on
the stock of public infrastructure may suﬀer lower steady-state welfare than under autarky.
In this subsection, we demonstrate this possibility by considering a simplified version of themodel.
Let us assume that α1 = α > 0 = α2 , L = L∗ = 1, and β ∗ = λβ and ρ∗ = λρ, where λ > 1.
That is, we assume that only good 1 is dependent on the stock of public infrastructure andthe two countries are identical except for their rates of depreciation and time preferences. In
addition, let us specify the production function in the public sector as f (LR ) = LR − L2R /2.
From Theorem 2 (ii) and (iii), the home (foreign) country becomes an exporter of good 11−α1(good 2).7 This assertion is verified in this example. Let us define q ≡ (1 − α) α αp α . Then,as shown in Appendix A.6, we obtain8√(ρ + β){(1 + αγ)β + (1 − αγ)ρ},qa =1 − αγqa∗ = λqa ,

qss =

√
λqa ,

(32)

and thus qa < qss < qa∗ .
Remark Substituting q = qss into each country’s steady-state equilibrium level of labor
input in the public sector, we can conclude that both LR > 0 and L∗R > 0 hold only if(1 + αγ)β + (1 − αγ)ρ> λ > 1.
(1 − αγ)(ρ + β)(33)

In the following analysis, we assume that this condition is satisfied.

7

Alternatively, we can consider a situation in which the labor endowments diﬀer between the countries (e.g.,
L = 1 and L∗ = 1 + ϵ, where ϵ > 0. Theorem 2 (i) and 3 can be verified under this alternative specification,although the calculation becomes more complicated.
8
Because both α and γ are between 0 and 1, 1 − αγ > 0.

17

Denoting the steady-state welfare level by u ≡ γ ln C1 + (1 − γ) ln C2 , we obtain the welfare
comparison result in the home country:{}(λ − 1)[(1 + αγ)β + (1 − αγ)ρ]1 + αγuss − ua = ln 1 +
−
ln λ.
2β
2

(34)

Theorem 3 purports that the home country obtains unambiguously higher welfare under free
trade than under autarky. This assertion is also verified because uss > ua holds.9 However,for the foreign country, the following expression holds:{}(λ − 1)(1 − αγ)(β − ρ)
1 − αγ∗∗uss − ua = ln 1 +
−
ln λ,
2β
2

(35)

the sign of which is ambiguous. In particular, if ρ becomes higher (i.e., people are more
impatient), the sign of (35) is more likely to be negative.
Yanase and Tawada (2012) show that in McMillan’s (1978) model with a pure public
intermediate good, a country unambiguously gains from trade in the long run only if it has
a comparative advantage in a good with productivity that is more sensitive to the public
intermediate good; if the country has a comparative advantage in a good with productivity
that is less sensitive to the public intermediate good, the economy may lose from trade in
the long run. In the present model, we obtain a similar result. However, in their model, thecountry that gains (may lose) from trade is the larger (smaller) country, measured in terms oflabor endowments. By contrast, in the present model, as implied by Theorem 2, the country
that gains (may lose) from trade is the smaller (larger) country. In this sense, we can conclude
that the results in terms of gains/losses from trade obtained in Yanase and Tawada (2012) are
not robust, and are dependent on the property of the public intermediate good.

5

Trading Equilibrium under Dynamic Nash Provisionof Public Goods

So far, we have assumed that the national government in each country is a price taker in theworld commodity markets and public investment is undertaken by the government without
strategic motives to manipulate the country’s terms of trade. In this section, we consider asituation in which the governments act strategically in providing the public good, i.e., the9

The right-hand side of (34) is continuous in λ, becomes zero if λ = 1, and is strictly increasing in λ.

18

governments noncooperatively make an investment in public infrastructure, recognizing thatit can aﬀect international prices.10
As discussed in Section 2, the competitive equilibrium outputs of private goods are derived
from (1), (3), (4), (5), and (6), and the equilibrium consumption levels are C1 = γ(pY1 + Y2 )/pand C2 = (1−γ)(pY1 +Y2 ). Thus, the market-clearing condition for good 1 in the international
market under free trade is given by11
Y1 (τ1 , τ2 , R, p) + Y1 (τ1∗ , τ2∗ , R∗ , p)
γ {p [Y1 (τ1 , τ2 , R, p) + Y1 (τ1∗ , τ2∗ , R∗ , p)] + Y2 (τ1 , τ2 , R, p) + Y2 (τ1∗ , τ2∗ , R∗ , p)}
=
.
p

(36)

From (36), the equilibrium world price of good 1 can be derived as a function of tax rates andthe stock of public infrastructure in both countries, i.e., p = P (τ1 , τ1∗ , τ2 , τ2∗ , R, R∗ ), with thefollowing properties:
{
}
∂P
1
∂Y2
∂Y1
=
γ
− (1 − γ)p, x = τ1 , τ2 , R,
∂x
Ω
∂x
∂x
{
}
∂P
1
∂Y2∗
∂Y1∗
=
γ ∗ − (1 − γ)p ∗ , x∗ = τ1∗ , τ2∗ , R∗ ,
∂x∗
Ω
∂x
∂x
(

where
Ω ≡ (1 − γ)(Y1 + Y1∗ ) + (1 − γ)p∂Y1 ∂Y1∗
+
∂p
∂p

)

(
−γ

∂Y2 ∂Y2∗
+
∂p
∂p

(37a)
(37b)
)
>0

and the derivatives of outputs Y1 and Y2 with respect to τi , R, and p are given by (A.4).
The governments no longer take the world price p as given; rather, they take into accountthe eﬀects of the change in the tax rate at each moment in time and the stock of public
infrastructure on the world price. The home government’s problem is to maximize welfare (8)subject to the dynamics of domestic public infrastructure (2) and p = P (τ1 , τ1∗ , τ2 , τ2∗ , R, R∗ ).
Each government recognizes that the rival country’s choice of tax rates and stock of public
infrastructure also aﬀect the world price. In this sense, there exists dynamic and strategicinteraction, and thus the trading equilibrium is characterized as a diﬀerential game between
national governments that determine the appropriate levels of their domestic public infrastructure.
10

In a static model, Shimomura (2007) proves that if governments determine the level of public goods
noncooperatively, free trade is beneficial to all countries.
11
For simplicity of exposition, we omit the labor endowments L and L∗ from the arguments of Y1 (·) andY2 (·).

19

Two equilibrium concepts are frequently employed in dynamic game analysis in economics;
the open-loop Nash equilibrium, in which each player’s equilibrium strategy is a simple function independent of the current state of the system, and the Markov perfect Nash equilibrium,in which each player designs their optimal strategy as a feedback decision rule dependent only
on the state variable. Both concepts satisfy time consistency, but only the Markov perfectNash equilibrium satisfies subgame perfectness (Long, 2010). Because of its tractability, we
focus on the open-loop Nash equilibrium. This strategy concept requires that governments
commit themselves to particular strategy paths at the beginning of the game, and we simply assume that the commitment is credible. Formally, the open-loop Nash equilibrium forthis dynamic game is defined as a pair of time paths in relation to the home government’s∞∗∗
tax rates {τ1 (t), τ2 (t)}∞t=0 and the foreign government’s tax rates {τ1 (t), τ2 (t)}t=0 such that{τ1 (t), τ2 (t)}∞t=0 maximizes the home country’s national welfare subject to the dynamics of R,∞∗∗
taking {τ1∗ (t), τ2∗ (t)}∞t=0 as given, and {τ1 (t), τ2 (t)}t=0 maximizes the foreign country’s national

welfare subject to the dynamics of R∗ , taking {τ1 (t), τ2 (t)}∞t=0 as given.

5.1

The Nash equilibrium conditions

The current-value Hamiltonian for the home government’s problem is now defined as
H = ln [P (τ , R)Y1 (τ1 , τ2 , R, P (τ , R)) + Y2 (τ1 , τ2 , R, P (τ , R))]
− γ ln [P (τ , R)] + Γ + θ {f (LR (τ1 , τ2 , R, P (τ , R))) − βR} ,where τ ≡ (τ1 , τ1∗ , τ2 , τ2∗ ) and R ≡ (R, R∗ ). The optimality conditions are presented by
[(
)
]
∂H
1
∂Y1 ∂Y2 ∂P∂Y1 ∂Y2γ ∂P=
Y1 + p++p+−∂τ1
pY1 + Y2∂p∂p ∂τ1
∂τ1
∂τ1
p ∂τ1
(
)
∂LR ∂LR ∂P+ θf ′ (LR )
+
= 0,
(38)
∂τ1
∂p ∂τ1[(])∂H
1∂Y1 ∂Y2 ∂P∂Y1 ∂Y2γ ∂P=
Y1 + p++p+−∂τ2
pY1 + Y2∂p∂p ∂τ2
∂τ2
∂τ2
p ∂τ2
(
)
∂LR ∂LR ∂P+ θf ′ (LR )
+
= 0,
(39)
∂τ2
∂p ∂τ2{()}∂H∂LR ∂LR ∂Pγ ∂P′θ̇ = ρθ −= θ ρ + β − f (LR )
+
+
∂R
∂R
∂p ∂R
p ∂R
{(
)
}
∂Y1 ∂Y2 ∂P∂Y1 ∂Y2
1
Y1 + p
+
+p
+
,
(40)
−
pY1 + Y2
∂p
∂p ∂R
∂R
∂R

20

and the transversality condition. The optimality conditions for the foreign government’s problem can be analogously derived.
Since C1 = γ(pY1 + Y2 )/p, (38) and (39) can be rewritten as{}∂Y1 ∂Y2∂LR ∂P′
Y1 − C1 + p++ (pY1 + Y2 )θf (LR )∂p∂p∂p ∂τ1∂Y1 ∂Y2∂LR++ (pY1 + Y2 )θf ′ (LR )
= 0,
+p
∂τ1
∂τ1
∂τ
{
} 1
∂Y1 ∂Y2∂LR ∂P
Y1 − C1 + p++ (pY1 + Y2 )θf ′ (LR )∂p∂p∂p ∂τ2∂LR∂Y1 ∂Y2++ (pY1 + Y2 )θf ′ (LR )
= 0.
+p
∂τ2
∂τ2
∂τ2

(41)

(42)

From these equations, it holds that
1
R
2
p ∂Y
+ ∂Y
+ (pY1 + Y2 )θf ′ (LR ) ∂L
∂τ1
∂τ1
∂τ1

∂P
∂τ1

=

R
1
2
p ∂Y
+ ∂Y
+ (pY1 + Y2 )θf ′ (LR ) ∂L
∂τ2
∂τ2
∂τ2

∂P
∂τ2

,

which in light of (A.4) and (37) can be rewritten asγτ1 + (1 − γ)τ2 = 1 −θf ′ (Lw.
R )(pY1 + Y2 )(43)

Substituting (43) into the first-order condition (41) or (42) again and rearranging terms,12 we
obtainτ1 − τ2
1 − γτ1 − (1 − γ)τ2

{
}
∂Y1∗
∂Y2∗
∗
(1 − γ)(Y1 + Y1 ) + (1 − γ)p−γ= Y1 − C1 .
∂p
∂p

(44)

Eqs. (43) and (44) jointly determine the optimal tax rates for the home government, τ1 andτ2 , as a function of the tax rates in the foreign country τ1∗ and τ2∗ ,13 as well as the state andco-state variables in the home country, R and θ.
The foreign government’s optimality conditions can be analogously derived. That is, theoptimal tax rates for the foreign country, τ1∗ and τ2∗ , satisfy the following conditions:w∗,γτ1∗ + (1 − γ)τ2∗ = 1 − ∗ ′ ∗θ f (LR )(pY1∗ + Y2∗ ){}τ1∗ − τ2∗
∂Y1
∂Y2∗
(1 − γ)(Y1 + Y1 ) + (1 − γ)p−γ= Y1∗ − C1∗ .
1 − γτ1∗ − (1 − γ)τ2∗∂p∂pFrom (44) and (46), we immediately obtain the following proposition:
12
13

Either operation yields the same outcome.
Note that ∂Y1∗ /∂p and ∂Y2∗ /∂p depend on τ1∗ and τ2∗ .

21

(45)
(46)

Proposition 1 Suppose that the national governments determine their production taxes strategically, in the sense that they take into account the eﬀects on the international price. If thehome country exports good 1, which is more dependent on the stock of public infrastructure,and the foreign country imports that good, the home government chooses τ1 > τ2 , whereas theforeign government chooses τ1∗ < τ2∗ .
Proposition 1 highlights the national government’s incentive to improve its terms of tradein determining the tax rates for funding public investment. Higher production taxes add morecosts to firms, and thus dampen their outputs. The reduction in outputs leads to excess
demand for the good in the world market, which increases the relative price of that good. If
the home country exports good 1, its terms of trade improve when the relative price of good
1, p, increases, making the home country better oﬀ. Therefore, the home government has
an incentive to impose a higher tax rate on sector 1 than on sector 2, thereby reducing therelative output of good 1.
In light of (37), (41), (42), and the comparative statics results (A.4), the home government’s
adjoint equation (40) can be rewritten asθ̇ = (ρ + β)θ −(1 − τ1 )α1 pY1 + (1 − τ2 )α2 Y2.
[1 − γτ1 − (1 − γ)τ2 ](pY1 + Y2 )R

(47)

Eq.(47) can be interpreted in a similar manner to that in the case of nonstrategic governments;
it indicates that it is optimal for the home government to choose the resource allocation in
the home country that balances the sum of the capital gain/loss and the marginal benefitof an increase in public infrastructure with the marginal cost of providing the infrastructure.
However, the marginal social benefit of public infrastructure is distorted by the strategic useof production taxes. Comparing the second term on the right-hand side of (47) with that of(15) and making use of the fact that γ = p(Y1 + Y1∗ )/[p(Y1 + Y1∗ ) + Y2 + Y2∗ ] from the world
market-clearing condition (36), we obtain
α1 pY1 + α2 Y2(1 − τ1 )α1 pY1 + (1 − τ2 )α2 Y2−[1 − γτ1 − (1 − γ)τ2 ](pY1 + Y2 )R(pY1 + Y2 )R∗(τ1 − τ2 )p[α1 Y1 (Y2 + Y2 ) − α2 Y2 (Y1 + Y1∗ )].
=−R(pY1 + Y2 )[(1 − τ1 )p(Y1 + Y1∗ ) + (1 − τ2 )(Y2 + Y2∗ )](48)

If the home country exports good 1, Y1 (Y2 + Y2∗ ) > Y2 (Y1 + Y1∗ ) must hold.14 Given thisfact and Assumption 1, it follows that α1 Y1 (Y2 + Y2∗ ) > α2 Y2 (Y1 + Y1∗ ). In addition, from
14
Since C1 = γ(pY1 +Y2 )/p, it follows that Y1 −C1 = (1−γ)Y1 −γY2 /p = [Y1 (Y2 +Y2∗ )−Y2 (Y1 +Y1∗ )]/[p(Y1 +∗
Y1 ) + Y2 + Y2∗ ]. Therefore, the sign of Y1 − C1 is the same as that of Y1 (Y2 + Y2∗ ) − Y2 (Y1 + Y1∗ ).

22

Proposition 1, τ1 > τ2 holds in the case where the home country exports good 1. Therefore,
the sign of (48) is negative if the home country exports good 1. In the foreign country, which
imports good 1 and in which the government sets τ1∗ < τ2 , the opposite result occurs. Thus,
the following proposition can be established.
Proposition 2 Suppose that the national governments determine their production taxes strategically. Then, in comparison with the case where the governments act nonstrategically, themarginal social benefit of public infrastructure becomes smaller (larger) in a country that exports (imports) good 1, which is more dependent on the stock of public infrastructure.

5.2

Steady state

The steady state of the open-loop Nash equilibrium must satisfy (21), (23), and the following
two equations:(1 − τ1 )α1 pY1 + (1 − τ2 )α2 Y2 f ′ (LR )·,w
R(1 − τ1∗ )α1 pY1∗ + (1 − τ2∗ )α2 Y2∗ f ′ (L∗R )ρ∗ + β ∗ =
·
,
w∗
R∗
ρ+β =

(49)
(50)

where we made use of (43) and (47) in deriving (49) and the foreign counterpart (50) can beanalogously derived.
The left-hand sides of (49) and (50) are the sum of the discount and depreciation rates
in the home country and foreign country, respectively, and can be interpreted as the longrun marginal cost of providing the stock of public infrastructure in the respective countries.
The right-hand sides of these equations denote the shadow value of the public good stock
measured in terms of labor, and can be interpreted as the long-run marginal social benefitof public infrastructure in these countries. Other things being equal, the existence of theterms-of-trade eﬀect changes the marginal benefit. As demonstrated in Proposition 2, if thehome country exports good 1, which is more intensive to R, the marginal benefit will besmaller compared with the nonstrategic case. Because the marginal benefit is decreasing in
LR (see the proof of Lemma 2), this implies that in comparison with the nonstrategic case,the home government will have an incentive to underaccumulate the public good stock. It
is also verified in a similar manner that the foreign country will tend to overaccumulate thepublic good stock compared with the nonstrategic case if it imports good 1. This under- andoveraccumulation is the result of the terms-of-trade eﬀect; because an increase in the stock of23

public infrastructure augments the relative output of good 1, its relative price will fall. The
exporting country will suﬀer a deterioration in its terms of trade, and thus has an incentive
to reduce its public good provision to reduce the welfare loss.

6

Concluding Remarks

In this paper, we have developed a dynamic two-country model of international trade withproductive public infrastructure. In the case where the government in each country acts asa price taker in the world commodity market, we revealed that if the economy is initiallyat the autarky steady state, a country with a smaller (larger) labor endowment, a lower(higher) depreciation rate of its stock of infrastructure capital, and/or a lower (higher) rateof time preference exports (imports) a good that is more dependent on the stock of public
infrastructure. We also showed that the country exporting the good that is more dependent
on the stock of public infrastructure unambiguously gains from trade in the long run, but thecountry importing that good may lose from trade in the long run.
We also considered an alternative situation in which the governments strategically determine their provision of public infrastructure, taking into consideration the eﬀect of government
policy on the world commodity market. Because of the terms-of-trade eﬀect, in comparison
with the nonstrategic case, the country exporting (importing) the good that is more dependent
on public infrastructure would underaccumulate (overaccumulate) public infrastructure.
We should note that, with a few exceptions such as Bougheas et al. (2003) and Figuiéres etal. (2013) that consider spillover eﬀects of infrastructure across countries, most of the existingstudies on infrastructure and trade have focused on national public infrastructure, and ouranalysis is also confined to national public infrastructure. However, international public goods
are receiving increasing attention. For example, railway and highway construction is carriedout across national boundaries, and various communication systems are also available acrossboundaries. Thus, it is of increasing importance to accommodate infrastructure that has thecharacteristics of international public goods into the present model and to reexamine the trade
theorems. There is much work to be done once our attention is focused on this area.

24

Appendix
A.1

Comparative statics of the competitive equilibrium solutions

From (5) and (6), it holds that L1 = (1−τ1 )(1−α1 )pY1 /w and L2 = (1−τ2 )(1−α2 )Y2 /w. Substituting
these equations into (1) and (3), we obtain[](1 − τ1 )(1 − α1 )pY1 1−α1Y1 = R,w[](1 − τ2 )(1 − α2 )Y2 1−α2Y 2 = R α2,w
(1 − τ1 )(1 − α1 )pY1 + (1 − τ2 )(1 − α2 )Y2+ LR = L.
w
α1

(A.1)
(A.2)
(A.3)

Totally diﬀerentiating (4), (A.1), (A.2), and (A.3), and solving for dY1 , dY2 , dLR , and dw, we canderive the comparative statics results as follows:(1 − α1 )Y1 {α2 pY1 + [1 − α2 (1 − τ2 )]Y2 }
∂Y1
=−
,
∂τ1
∆(1 − τ1 )∂Y1(1 − α1 )Y1 Y2 [1 − α2 − α2 (1 − τ2 )]=,∂τ2∆(1 − τ2 )∂Y1Y1=
{α1 α2 [1 − α1 (1 − τ1 )]pY1 + [1 − α2 (1 − τ2 )][α1 − (1 − α1 )α2 ]Y2 },
∂R
∆R
∂Y1
(1 − α1 )Y1 Y2 [1 − α2 (1 − τ2 )]
=
,
∂p
∆p
(1 − α2 )pY1 Y2 [1 − α1 − α1 (1 − τ1 )]
∂Y2
=
,
∂τ1
∆(1 − τ1 )∂Y2
(1 − α2 )Y2 {[1 − α1 (1 − τ1 )]pY1 + α1 Y2 }=−,∂τ2∆(1 − τ2 )∂Y2
Y2={[1 − α1 (1 − τ1 )][α2 − α1 (1 − α2 )]pY1 + α1 α2 [1 − α2 (1 − τ2 )]Y2 },
∂R
∆R
∂Y2
(1 − α2 )Y1 Y2 [1 − α1 (1 − τ1 )]
=−
,
∂p
∆
{
[
] }
∂LR
(1 − α1 )pY1
(1 − α2 )(1 − τ2 )=α2 pY1 + τ2 +(α1 − τ1 ) Y2 ,∂τ1∆w
(1 − α1 )(1 − τ1 )
{[
]
}
∂LR
(1 − α2 )Y2(1 − α1 )(1 − τ1 )=
τ1 +(α2 − τ2 ) pY1 + α1 Y2 ,∂τ2∆w
(1 − α2 )(1 − τ2 )∂LR
(α1 − α2 )pY1 Y2=[(1 − α1 τ2 − α2 )τ1 − (1 − α1 − α2 τ1 )τ2 ],
∂R
∆wR
∂LR
Y1 Y2 {τ1 [1 − α2 (1 − τ2 )] − τ2 [1 − α1 (1 − τ1 )]}
=
,
∂p
∆w
∂LR
wpY1 α2 [1 − α1 − α1 (1 − τ1 )]=−,∂τ1∆(1 − τ1 )∂w
wY2 α1 [1 − α2 − α2 (1 − τ2 )]=−,∂τ2∆(1 − τ2 )25

(A.4a)(A.4b)(A.4c)(A.4d)(A.4e)(A.4f)(A.4g)(A.4h)(A.4i)(A.4j)(A.4k)(A.4l)(A.4m)(A.4n)

∂w
wα1 α2 {[1 − α1 (1 − τ1 )]pY1 + [1 − α2 (1 − τ2 )]Y2 }
=
,
∂R
∆R
wY1 α2 [1 − α1 (1 − τ1 )]
∂w
=
,
∂p
∆

(A.4o)(A.4p)

where ∆ ≡ α2 [1 − α1 (1 − τ1 )]pY1 + α1 [1 − α2 (1 − τ2 )]Y2 > 0.

A.2

Properties of the GDP function12+ Rα2 L1−α+ ω(l − L1 − L2 ). The first-orderLet us define the Lagrangian as follows: L = pRα1 L1−α12conditions are∂LY1
1= (1 − α1 )pRα1 L−α− ω = (1 − α1 )p− ω = 0,
1
∂L1
L1
∂L
Y2
2
= (1 − α2 )Rα2 L−α− ω = (1 − α2 )− ω = 0,
2
∂L2
L2
∂L
= l − L1 − L2 = 0.
∂ω

(A.5)
(A.6)
(A.7)

From (A.5) and (A.6), we obtain L1 = (1 − α1 )pY1 /ω and L2 = (1 − α2 )Y2 /ω, respectively. Substituting these into (A.7) and solving for ω, it follows that ω = [(1 − α1 )pY1 + (1 − α2 )Y2 ]/l. Therefore,(1) can be rewritten as]1−α1[(1 − α1 )pY1 l
α1
Y1 = R,(A.8)(1 − α1 )pY1 + (1 − α2 )Y2[]1−α2(1 − α2 )Y2 l
α2
Y2 = R.
(A.9)(1 − α1 )pY1 + (1 − α2 )Y2
Totally diﬀerentiating (A.8) and (A.9), and solving for dY1 and dY2 , we obtaindY1 ={(1 − α1 )α1 α2 pY1 + (1 − α2 )[α1 − (1 − α1 )α2 ]Y2 }Y1(1 − α1 )(1 − α2 )Y1 Y2
dp +
dR
∆′ p
∆′ R
(1 − α1 )α2 Y1 [(1 − α1 )pY1 + (1 − α2 )Y2 ]+dl,(A.10)∆′ l(1 − α1 )(1 − α2 )Y1 Y2{(1 − α1 )[α2 − α1 (1 − α2 )]pY1 + (1 − α2 )α1 α2 Y2 }Y2
dp +
dR
′
∆
∆′ R
α1 (1 − α2 )Y2 [(1 − α1 )pY1 + (1 − α2 )Y2 ]+dl,(A.11)∆′ l

dY2 = −

where ∆′ ≡ (1 − α1 )α2 pY1 + (1 − α2 )α1 Y2 .
Since G = pY1 + Y2 , (A.10) and (A.11) yield the partial derivatives of the GDP function as
follows:∂Y1 ∂Y2+= Y1 ,∂p∂p∂Y1 ∂Y2
α1 pY1 + α2 Y2GR = p
+
=
,
∂R
∂R
R
∂Y1 ∂Y2(1 − α1 )pY1 + (1 − α2 )Y2
Gl = p
+
=
= ω.
∂l
∂l
l
Gp = Y1 + p

26(A.12a)(A.12b)(A.12c)Thus, we have the properties of the GDP function (16). In addition, since Gpp = ∂Y1 /∂p, GpR =∂Y1 /∂R, and Gpl = ∂Y1 /∂l, (A.10) derives(1 − α1 )(1 − α2 )Y1 Y2
> 0,
∆′ p
{(1 − α1 )α1 α2 pY1 + (1 − α2 )[α1 − (1 − α1 )α2 ]Y2 }Y1
GpR =
,
∆′ R
(1 − α1 )α2 Y1 [(1 − α1 )pY1 + (1 − α2 )Y2 ]
Gpl =
> 0.
∆′ l
Moreover, from (A.12b) and (A.12c), it follows thatGpp =α1 α2 [(1 − α1 )pY1 + (1 − α2 )Y2 ]2
< 0,
∆′ R2
α1 α2 [(1 − α1 )pY1 + (1 − α2 )Y2 ]2< 0,
Gll = −∆′ l2
α1 α2 [(1 − α1 )pY1 + (1 − α2 )Y2 ]2
GRl =
> 0.
∆′ Rl
Finally, we present some results that are useful in the subsequent analysis:
GRR = −

(A.13a)(A.13b)(A.13c)(A.14a)(A.14b)(A.14c)Gp GR(α1 − α2 )[(1 − α1 )pY1 + (1 − α2 )Y2 ]Y1 Y2=,(A.15)G∆′ (pY1 + Y2 )R
Gp Gl(α1 − α2 )[(1 − α1 )pY1 + (1 − α2 )Y2 ]Y1 Y2
Gpl −
=−
,
(A.16)G∆′ (pY1 + Y2 )lGR Gl(α1 − α2 )2 [(1 − α1 )pY1 + (1 − α2 )Y2 ]pY1 Y2
GRl −
=−
< 0,
(A.17)G∆′ (pY1 + Y2 )Rl
GR Glp(α1 − α2 )Y1 Y2
GpR −
=
,
(A.18)Gl∆′ R
GR GRlα1 α2 [(1 − α1 )pY1 + (1 − α2 )Y2 ](pY1 + Y2 )
GRR −
=−
< 0,
(A.19)Gl∆′ R2
α1 α2 [(1 − α1 )pY1 + (1 − α2 )Y2 ](pY1 + Y2 )GR Gll=> 0.
(A.20)
GRl −
Gl
∆′ Rl
Under Assumption 1, the signs of (A.15) and (A.18) are positive, whereas the sign of (A.16) is
negative.
GpR −

A.3

Proof of Lemma 2(i) Substituting (21) into (22) yields
ρ+β =GR (p, f (LR )/β, L − LR ) ′f (LR ).
Gl (p, f (LR )/β, L − LR )Totally diﬀerentiating (A.21) and rearranging, we have{() ′ 2 ()}
GR GlR(f )GR Gll′′′− GRR+ GRl −
f − GR f dLR
Gl
β
Gl
)
(
)
(
GR Glp
GR Gllf ′ dp + GRl −f ′ dL= GRp −
Gl
Gl
{(
) ′
}
GR GlR
ff+− GRR− Gl dβ − Gl dρ.
Gl
β2

27

(A.21)(A.22)

From (A.19), (A.20), and f ′′ ≤ 0, the coeﬃcient of dLR in (A.22) is positive. In addition, from
(A.18), the coeﬃcient of dp is positive if α1 > α2 . Therefore, dLR /dp > 0 holds under Assumption
1.
(ii) In light of (16) and (A.12c), (22) can be rewritten as
ρ+β =

GR ′
(α1 pY1 + α2 Y2 )βf ′ (LR )(L − LR )f (LR ) =,Gl{(1 − α1 )pY1 + (1 − α2 )Y2 }f (LR )(A.23)

from which we have:GR ′α2 βf ′ (LR )(L − LR )f (LR ) =,p→0 Gl1 − α2f (LR )GR ′α1 βf ′ (LR )(L − LR )lim.
f (LR ) =p→∞ Gl1 − α1f (LR )
lim

Under Assumption 1, it holds that limp→0 GGRl f ′ (LR ) < limp→∞ GGRl f ′ (LR ). In addition, both limp→0 GGRl f ′ (LR )and limp→∞ GGRl f ′ (LR ) are decreasing in LR :d[βf ′ (LR )(L − LR )/f (LR )]β{[(L − LR )f ′′ − f ′ ]f − (L − LR )(f ′ )2 }
=
< 0.
dLR
f2
Therefore, the solution for LR that satisfies (A.23) in each limiting case (i.e., p → 0 and p → ∞) is
uniquely determined, as illustrated in Figure A1.
2

Figure A1: Determination of LR and LR

A.4

Proof of Theorem 1

Let us define the left-hand side of (25) by ζL (p) and the right-hand side by ζR (p). It is easily verified
that
lim ζL (p) = ∞, lim ζL (p) = 0,
p→∞
[
] α1
]α
[
f (L̄∗R ) 1 ∗f (L̄R )1−α1lim ζR (p) =(L − L̄∗R )1−α1 > 0.
(L − L̄R )
+
p→∞
β
β∗
p→0

lim ζR (p) = 0,

p→0

28

Because both ζL (p) and ζR (p) are continuous in p, it follows that there exists at least one solution for
p in (0, ∞). Diﬀerentiating ζL (p) and ζR (p), and rearranging in light of (A.21), we have, respectively,
{ (
)
}
γ
ρ
ρ∗ ∗ ∗′′′∗
ζL (p) = 2 p
Gl ψ + ∗ Gl ψ
− (Y2 + Y2 ) ,(A.24)
p
β
β
(
)
( ∗
)
∗ ∗
ρ + β Gl GpR
ρ + β ∗ Gl GpR
′
′
∗
′
∗
(A.25)
ζR (p) = Gpp + Gpp +− Gpl ψ +− Gpl ψ ∗ .
∗
∗
β
GR
β
GR
′ (p) > 0. Therefore, if ζ ′ (p) < ζ ′ (p) holds at
From (A.13a), (A.18), and Lemma 2, it follows that ζRLR
the equilibrium price such that ζL (p) = ζR (p), the equilibrium is unique. From (A.24) and (A.25),
we have
(
)
( ∗
)
∗ ∗
γρ
ρ + β Gl GpR
γ ρ ∗ ρ∗ + β ∗ Gl GpR
′
′
′
′
∗
G −
ζL (p) − ζR (p) =Gl −+ Gpl ψ ++ Gpl ψ ∗pββGRp β∗ l
β∗
G∗R

−

γ2
(Y2 + Y2∗ ) − (Gpp + G∗pp ).
p(A.26)Let us assume that the two countries are identical: L = L∗ , β = β ∗ , and ρ = ρ∗ . Then, the twocountries share an identical GDP function, and thus the world market-clearing condition ζL (p) =ζR (p) implies that γ/p = Gp /G. Substituting this into the first term on the right-hand side of (A.26),
we have
)
{ (
)
(
)}
(
Gl GpRρ Gp GpR
ρ + β Gl GpRγρGl −+ Gpl ψ ′ =−Gl + Gpl −
ψ′.
pβ
β
GR
β G
GR
GR
From (A.15), (A.18), and Lemma 2, the above expression is shown to be negative, as is the second′ (p), and thus there exists a unique solution p in the symmetricterm in (A.26). Therefore, ζL′ (p) < ζRsscase.
We next turn to the stability of the steady state. The dynamic system of the world economy isdescribed as
Ṙ = f (LR ) − βR,Ṙ∗ = f (L∗R ) − β ∗ R∗ ,GR (p, R, L − LR )θ̇ = (ρ + β)θ −,G(p, R, L − LR )GR (p, R∗ , L∗ − L∗R )θ̇∗ = (ρ∗ + β ∗ )θ∗ −,G(p, R∗ , L∗ − L∗R )γ
0 = {G(p, R, L − LR ) + G(p, R∗ , L∗ − L∗R )} − Gp (p, R, L − LR ) − Gp (p, R∗ , L∗ − L∗R ).
p
Note that LR is dependent on R, θ, and p, and given (17), the following derivatives are obtained:
GlR − GlGGR∂LR
GlR − θf ′ GR
=
=
> 0,
∂R
θGf ′′ − θf ′ Gl + Gll
θGf ′′ − θf ′ Gl + Gll
∂LR
f ′G
=−
> 0,
∂θ
θGf ′′ − θf ′ Gl + Gll
Glp − lG p
Glp − θf ′ Gp∂LR==
> 0 if∂p
θGf ′′ − θf ′ Gl + Gll
θGf ′′ − θf ′ Gl + Gll
GG

29

α1 > α 2 .

Assuming that the two countries are identical and linearizing the dynamic system around the symmetric steady state, we obtain   ′ ∂LRRR0f ′ ∂L0f ′ ∂L
f ∂R − βṘ
R − R̄
∂θ∂p
RR
R  R∗ − R̄∗ Ṙ∗  −β0f ′ ∂Lf ′ ∂L0f ′ ∂L

  

∂R
∂θ
∂p

 θ̇  = 
θ − θ̄ (A.27)

δR
0
δθ
0
δp 
  

,
 ∗
∗ 
 θ̇∗  
θ
−
θ̄
0
δR
0
δθ
δp 
0
0
χR
χR
χθ
χθ
χp
where
{
(
)
}
G2
GR Gl
∂LR
GRR − R +
− GRl
,
G
G
∂R
(
)
1 GR Gl∂LRδθ ≡ ρ + β −
− GRl
,
G
G
∂θ
{
(
)
}
GR Gp
GR Gl
∂LR
1
GRp −
+
− GRl
,
δp ≡ −
G
G
G
∂p
(
)
Gp GR
Gp Gl ∂LR
χR ≡− GpR + Gpl −
,
G
G
∂R
(
)
Gp Gl ∂LRχθ ≡ Gpl −
,
G
∂θ
{
(
)
}
Gp Gl ∂LR
Y1 Y2
χp ≡ 2 −− Gpp + Gpl −
.
pG
G
∂p
δR ≡ −1G

Let us denote the above matrix by J and the corresponding eigenvalues by z, which is determinedby the characteristic equation
1 0 0 0 0
0 1 0 0 0
 = 0.
001
00Ω(z) ≡ J − z 0 0 0 1 0
0 0 0 0 0
After some calculations, the characteristic equation can be rewritten as Ω(z) = 2Ω1 (z)Ω2 (z) = 0,
where
)
(
∂LR
2
′ ∂LR
− β δθ − f ′δR ,Ω1 (z) ≡ z − ρz + f
∂R
∂θ
{(
)
(
)}
Gp Gl ∂LR
Y1 Y2
Ω2 (z) ≡
Gpl −
−
+ Gpp
z2
G
∂p
pG
{(
)[
(
)
] (
) }
Gp Gl
∂LR
∂LR
Y1 Y2
′ l
+
Gpl −
f
+ β − δθ +δp ++ Gpp ρ z
G
∂p
R
∂θ
pG
(
)(
) (
)
Gp Gl
∂LR
∂LR
l
β
+ Gpl −
δp −
δθ f ′
+ ′
G
∂θ
∂p
R f
(
) [(
)
]
Y1 Y2
′ ∂LR
′ ∂LR
−
+ Gpp
f
− β δθ − f
δR .
pG
∂R
∂θ

30

The constant term of Ω1 (z) can be rewritten as
)
(
∂LR
′ ∂LR
− β δθ − f ′
δR
f
∂R
∂θ
(
(
)
)
f f ′ LR ∂LR R
GR Gl
1 ∂LR
′
=
f GRR − βGRl − ρ,− 1 (ρ + β) +R
f ∂R LRG ∂θ
G(A.28)

which is unambiguously negative because of the concavity of f . This indicates that the two solutions
to Ω1 (z) = 0 have opposite signs. Next, the z 2 term of Ω2 (z) is negative:()()
Gp Gl ∂LR
Y1 Y2
Gpl −
−
+ Gpp
G
∂p
pG
(
)
(
)
Gp Gl 2
1
Y1 Y2
=
Gpl −
−
+ Gpp < 0.
θGf ′′ − θf ′ Gl + Gll
G
pG
Moreover, since the sign of (A.28) is negative and
(
)(
)
Gp Gl
∂LR
∂LR
Gpl −
δp −
δθ
G
∂θ
∂p
){
(
)
}
(
GR Gp
Gp Gl
∂LR 1
∂LR
GRp −
+ (ρ + β)> 0,= Gpl −G∂θ GG∂p
the constant term of Ω2 (z) is positive. Therefore, the two solutions to Ω2 (z) = 0 also have opposite
signs. To summarize, there are two positive characteristic roots and two negative roots. Because
there are two state variables, R and R∗ , it follows that the steady state is a saddle point.
Finally, we can check that Ω(0) ̸= 0. It follows that the implicit function theorem ensuresthat even if the economic fundamentals of the two countries are slightly diﬀerent, the existence,uniqueness, and stability of the steady state are established (Chen et al., 2008).
2

A.5

Signs of ∂ED(p)/∂L and ∂ED(p)/∂β

Substituting the expression for ∂ψ/∂L into (27), we have)} ′ 2
)(
) (
)(
{(
Gp GR
Gp Gl
(f )
GR GlRGR Gll
−
G
G
−
−
G
+
−
G
RR
pR
Rl
pl
Gl
G
G
Gl
β
∂ED(p)() ′2 (
)
=
(f
)
G
G
G
G
∂L
R lR
R ll
p=pa
− GRR
f ′ − GR f ′′Gl
β + GlR − Gl
)
(
Gp Gl
′′
G − Gpl GR f
) ′2 ().
(A.29)−((f )GR GllGR GlR′ − G f ′′
−
G
+
G
−
f
RR
R
lR
Gl
β
Gl
However, from (A.15), (A.16), (A.19), and (A.20), it is( verified that) the first term in the above
Gp Gl
equation becomes zero. Moreover, from (A.16), the term
G − Gpl is positive under Assumption
1. Then, it follows that the sign of ∂ED(p)/∂L becomes nonnegative.

31

Substituting the expression for ∂ψ/∂β into (A.16), we have
{(
)} ′
)(
) (
)(
Gp GR
Gp Gl
ff
GR GllGR GlR
G
−
−
G
−
G
+
−
G
RR
pR
Rl
pl
Gl
G
G
Gl
∂ED(p)
β2) ′2 (()=−
GR GlR∂βp=pa− GRR (fβ) + GlR − GRGGl ll f ′ − GR f ′′
Gl
(
)}
(
)
{
′′
Gp GR
Gp Gl
f ′ Gp GR
−
G
−
−
G
G
+
−
G
GR fβf2
pR
pR
pl
l
G
β
G
G
(
) ′2 ()+.
(A.30)(f )GR GllGR GlR− GRR
f ′ − GR f ′′Gl
β + GlR − Gl
Again,(the first term
becomes zero. Moreover, from (A.15) and (A.16), the) in the( above equation
)
Gp GR
Gp Gl
− GpR are positive and negative, respectively, under Assumptionterms
G − Gpl andG
1. Then, it follows that the sign of ∂ED(p)/∂β becomes positive.

A.6

Analysis of the simplified model in Section 4.3

The GDP function in the simplified model is derived as{
G(p, R, l) ≡ max pRα L1−α
+ L2
1
L1 ,L2}1−α1L1 + L2 = l = (1 − α) α αp α R + l.

s.t.

(A.31)

The first- and second-order derivatives of the GDP function are
1−α

1−α

Gp = (1 − α) α p α R,
Gpp = (1 − α)

1−α
α

αp

1−α
α

R,

1−α

1

GR = (1 − α) α αp α ,GpR = (1 − α)

1−α
α

1
α

αp ,

Gl = 1,

Gpl = GRR = GRl = Gll = 0.

Using (A.31) and f (LR ) = LR − L2R /2, we obtain the steady-state conditions of the home country’s
optimal resource allocation:
L2R
= βR,
2
ρ + β = q(1 − LR ),
LR −

1−α

1

where we use the variable transformation q ≡ (1 − α) α αp α . Solving the above equations, we have
(
)(
)
ρ+β
1
ρ+β
ρ+β
, R=
1−
1+
.
(A.32)
LR = 1 −q
2βqq
Similarly for the foreign country, solving the optimal steady-state conditions
L∗2R= λβR∗ ,2λ(ρ + β) = q(1 − L∗R ),L∗R −we obtainλ(ρ + β),L∗R = 1 −
q

1
R =
2λβ
∗

{
}{
}
λ(ρ + β)λ(ρ + β)
1−
1+
.
q
q

The autarkic equilibrium condition in the home country isαγ(qR + 1 − LR ) = qR.

32(A.33)

Substituting (A.32) into the above equation and solving for q, we obtain√(ρ + β){(1 + αγ)β + (1 − αγ)ρ}qa =.
1 − αγ(A.34)

Similarly, in light of (A.33), the autarkic steady-state solution for q in the foreign country is derivedas√(ρ + β){(1 + αγ)β + (1 − αγ)ρ}qa∗ = λ.
(A.35)
1 − αγThe steady-state equilibrium condition under free trade, (25), is rewritten asαγ(qR + 1 − LR ) + αγ(qR∗ + 1 − L∗R ) = qR + qR∗ .
Substituting (A.32) and (A.33) into the above equation and solving for q, we obtain√λ(ρ + β){(1 + αγ)β + (1 − αγ)ρ}qss =.
1 − αγ(A.36)The steady-state welfare isu = γ ln C1 + (1 − γ) ln C2 = ln(qR + 1 − LR ) − αγ ln q + A,

(A.37)

where A ≡ γ{α ln α + (1 − α) ln(1 − α)} + γ ln γ + (1 − γ) ln(1 − γ). Substituting (A.32) and (A.34)
into (A.37), we obtain the autarkic steady-state welfare as follows:
ua =

ρ+β
1 + αγ1 − αγln−ln{(1 + αγ)β + (1 − αγ)ρ} + A.
2
1 − αγ2(A.38)

Substituting (A.32) and (A.36) into (A.37), we obtain the steady-state welfare under free trade as
follows:uss =1 − αγρ+β
1 + αγln−ln{λ[(1 + αγ)β + (1 − αγ)ρ]}2
1 − αγ2}{(λ − 1)[(1 + αγ)β + (1 − αγ)ρ]+ ln 1 ++ Γ.
2β(A.39)In a similar manner, from (A.33), (A.35), (A.36), and (A.37), the steady-state welfare under autarky
and free trade, respectively, in the foreign country is derived as
1 − αγρ+β
1 + αγln−ln{(1 + αγ)β + (1 − αγ)ρ} − αγ ln λ + A,
2
1 − αγ21 − αγρ+β
1 + αγu∗ss =ln−
ln{λ[(1 + αγ)β + (1 − αγ)ρ]}2
1 − αγ2{}(λ − 1)(1 − αγ)(β − ρ)+ ln 1 +
+ A.
2β
u∗a =

33

(A.40)(A.41)

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