ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E1
No.E15
E15-2

On the Provision of International Public Goods
in a Dynamic Global Economy
byAkihiko Yanase

April 201
2015

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

On the Provision of International Public Goods in a Dynamic
Global Economy
Akihiko Yanase ∗March 29, 2015

Abstract
This paper develops a dynamic two-country model with an international public good, thestock of which has a positive eﬀect on the private-goods sectors’ productivity in each country
and the evolution of the stock is determined by each country’s voluntary contribution. It isshown that the country with a higher contribution technology becomes an exporter of thegood which is more dependent on the stock of international public good. It is also shownthat if the countries act cooperatively, the dynamics of the stock of international public
good and its shadow price under free trade coincide with those under autarky, although
the paths of each country’s contribution level are diﬀerent. Speciﬁcally, the contribution
level in the country with a lower contribution technology becomes smaller under free trade
than under autarky, and it is shown that this country unambiguously gains from trade. In
the noncooperative regime, free trade achieves a larger steady-state stock of international
public good than autarky.
Key Words: International public good; Two-country trade; Trade patterns; Cooperative solution; Open-loop Nash equilibrium
JEL classification: C73; F11; H54; H87∗
Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan; E-mail:yanase@soec.nagoya-u.ac.jp1

1

Introduction

The world economy is beneﬁted from various kind of public goods that are in the transnational
or global dimension. Because there is no coercive authority that can enforce an eﬃcient supplyof such international public goods by collecting tax from sovereign countries to ﬁnance the costof supply, the provision of such goods is implemented through voluntary contribution madeby countries that are beneﬁted from the goods in question. Examples of such international
public goods include transnational communication systems such as the Internet, internationalorganizations such as the United Nations, military alliances such as the North Atlantic Treaty
Organization, and reduction of greenhouse gas emissions to prevent global warming.
Public goods, regardless of local, national, or transnational ones, have external eﬀects that
beneﬁt the economy by raising consumers’ utility levels directly or make the economy betteroﬀ indirectly by augmenting private ﬁrms’ productivity. With respect to the latter kind of
public goods, i.e., public goods that have external eﬀects on productivity, these goods generally
have a characteristic of durable or capital goods; scientiﬁc knowledge and communication
infrastructure are typical examples. That is, it will be more reasonable to consider public
goods that can be accumulated over time when the stock of such goods has positive external
eﬀects the production side.
This paper considers a global economy consisting of two countries in the presence of aninternational public good, the stock of which has a positive external eﬀect on the private
sector’s productivity in each country and the evolution of the stock is determined by eachcountry’s voluntary contribution. The external eﬀect of the international public good can diﬀer
across sectors; for instance, a stock of scientiﬁc knowledge that is useful for reducing the riskof certain disease will beneﬁt medicinal chemical manufacturing, but has insigniﬁcant impact
on other sectors’ productivity. In addition, given that the provision of international public
goods is implemented through voluntary contribution by countries concerned, the eﬃciency or
productivity of contribution can diﬀer from one country to the next. These facts indicate thatthe provision of international public goods and the diﬀerence in the eﬃciency of contribution
across countries can be a source of comparative advantage. Moreover, in the presence ofinternational trade in private goods whose productivity depends on the stock of international
public good, the prices of private goods are endogenously determined in the world market.

2This means that under free trade, the countries will take into consideration the eﬀect of achange in the world price on the national welfare as well as the eﬀect of a change in the stock ofinternational public good and its shadow price. Then, it is of great interest whether and how thepresence of international economic transactions aﬀect each country’s contribution behavior.1
It is also of great interest how trade liberalization aﬀects the stock of international public
good and national welfare in each country. The purpose of this paper is to elucidate the tradepattern of each country and welfare eﬀects of trade in this world economy facing international
interdependence due to the spillover eﬀects of international public good and international trade.
Section 2 provides a brief review of the literature related to the present paper. Section
3 sets up the two country, two private-goods dynamic model with an international public
good. Section 4 considers an autarkic situation where there is no trade between countries.
Both cooperative solution, where the countries jointly determine their contribution levels forthe investment in the international public good in order to maximize the world welfare, andnoncooperative equilibrium, where each country determines its contribution level in order tomaximize it own national welfare, are derived. It is shown that under autarky the equilibrium
price of the good which is more dependent on the stock international public good becomes
lower in the country that has a higher contribution technology, regardless of whether the countries cooperatively or noncooperatively determine their contribution levels. Section 5 analyzes
outcomes under free trade. It is shown that if the countries act cooperatively, the dynamics of the stock of international public good and its shadow price under free trade coincidewith those under autarky, although the paths of each country’s contribution level are diﬀerent.
More speciﬁcally, trade liberalization increases the optimal contribution level in the country
with a higher contribution technology and reduces the contribution level in the country witha lower contribution technology. This implies that in the country with a lower contribution
technology, the total labor available in the production of private goods becomes larger underfree trade than under autarky, and it is shown that this country unambiguously gains fromtrade. By contrast, the country with a higher contribution technology may lose from trade.
1

In static trade models with public goods, some interesting results may emerge. Suga and Tawada (2007)
show that if the government of a country adopts the Lindahl pricing rule, the country may lose from trade. By
contrast, Shimomura (2007) shows that the classical gains-from-trade proposition is still valid if the governmentsin the trading world behave strategically with respect to the provision of public goods. Long and Shimomura(2007) show that the well-known neutrality theorem, i.e., the neutrality of small redistribution of wealth on the
Nash equilibrium allocation in the voluntary public-good contribution game, no longer holds if countries take
into account the eﬀect of their respective contributions on the world relative price of private goods.

3In the noncooperative regime, free trade achieves a larger steady-state stock of international
public good than autarky, and the gains-from-trade result still holds in the country with alower contribution technology.

2

Literature Review

As noted in the Introduction, this paper considers an economy consisting of sovereign countries,
each of which can make a decision independently in contributing to the investment on aninternational public good. Thus, in order to formulate the situation that this paper considers,
applying the models of dynamic voluntary or private provision of public good is suitable.
Fershtman and Nitzan (1991) develop a diﬀerential game model with a public good, thestock of which has a direct welfare eﬀect on private agents who make donations to the investment on the accumulation of the public good. In this dynamic framework, the authors
demonstrate the existence of a free-rider problem; compared to the Pareto optimal solution,
each agent has an incentive to contribute less in the Nash equilibrium, which thus brings the
economy about the lower stock of public good in the steady state. Moreover, the authors showthat the free-rider problem becomes worse, compared with the case where agents are able tocommit to a time path of contributions, when the agents use a strategy for their respective
contribution level that is linearly dependent on the stock of public good. The Fershtman–
Nitzan model is extended by Wirl (1996) and Itaya and Shimomura (2001). Wirl (1996) points
out that the ﬁnding of Fershtman and Nitzan (1991) crucially depends on the chosen set ofstrategies, i.e., the restriction to the linear strategy, and shows that the outcome can be better
when the agents use nonlinear strategies. Itaya and Shimomura (2001) consider a more general model than Fershtman and Nitzan (1991) and Wirl (1996), and elucidate the relationshipbetween the steady state equilibria of a dynamic contribution game and conjectural variations
equilibria of the corresponding static game.
A diﬀerent kind of diﬀerential game model of voluntary public-good provision is presented
by Shibata (2002), who develops an endogenous growth model with an infrastructure capitalthat is accumulated through voluntary investment by private agents, and analyzes how thepresence of strategic interactions between the agents aﬀects the long-run patterns of economic
growth. He shows that the equilibrium dynamics are heavily dependent on the commitment
behavior of agents; there exists a unique endogenous growth equilibrium if agents commit their4

announced paths of future actions, whereas multiple growth patterns emerge if agents condition
their actions on the stock of infrastructure capital. Moreover, in the latter case, some of thelong-run equilibria exhibit endogenous growth and others show no growth.
Although the above-mentioned studies do not explicitly take the international dimension
into consideration, the models may be interpreted as ones with international public goods,
where the economy consists of “countries” rather than individuals. However, such an interpretation seems hardly adequate because trade in goods between countries is assumed away inthese models.
The present paper is also closely related to the literature on international trade in the presence of public intermediate goods that have positive eﬀect on productivity in private production.
There have been a number of studies dealing with this issue (Manning and McMillan, 1979;
Tawada and Okamoto, 1983; Tawada and Abe, 1984; Ishizawa, 1988; Abe, 1990; Altenburg,
1992; Suga and Tawada, 2007). However, these studies are conﬁned to a static framework.
Dynamic models in which productivity eﬀects of the stock of a public intermediate good exists
and the national government determines the optimal path of the stock of public intermediate
good are analyzed by McMillan (1978) and Yanase and Tawada (2012, 2013).
McMillan (1978) shows that the stock of public intermediate good determines the slope of
the production possibility frontier and thus determines the pattern of international trade. By reexamining McMillan’s model, Yanase and Tawada (2012) show the possibility of multiple steadystates and history-dependent dynamic paths. In addition, Yanase and Tawada (2013) discuss
whether trade is gainful or not in McMillan’s model. In both McMillan (1978) and Yanase andTawada (2012), the stock of public intermediate good is assumed to have an impact similar to
the “creation of atmosphere” type externality classiﬁed by Meade (1952), where the technologyof each private sector exhibits constant returns to scale in primary factors of production only.
There is another class of public intermediate goods, which can be interpreted as “unpaid factorsof production,” again according to Meade’s terminology (1952), where the production functionof each private sector is characterized by constant returns to scale in all inputs, including the
public intermediate good. Yanase and Tawada (2013) develop a dynamic trade model with astock of public intermediate good of this type. However, these studies consider a small open
economy, where the price of goods are exogenously given, and take no account of international
public goods. In other words, interactions between countries have not been analyzed.

5

3

Model

I consider a world economy consisting of two countries, home and foreign, in which two private
goods, goods 1 and 2, are produced by using a single primary factor, labor. There is also aninternational public good, the stock of which has a positive external eﬀect on the productivityof these goods. The investment on the accumulation of international public good is made by
the government in each country, which takes the responses of the private sector and markets
into consideration. It is assumed that ﬁrms and households are price takers, and total laborendowment in each country is given and constant over time.

3.1

Production side

Let us focus on the home country. The foreign country, whose variables are denoted with 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 international public good, and Li is thelabor input in sector i. The parameter αi ∈ [0, 1) denotes the production elasticity of theinternational public good in each sector: αi = (∂Yi /∂R) · (R/Yi ).
In the following analysis, I make the following assumption regarding the impact of the stockof international public good to industries:

Assumption 1

α1 > α2 , i.e., sector 1 is more dependent on the stock of international public

good than sector 2.

Let us denote the labor contributed to the accumulation of international public good in thehome country by LR . Then, at each moment of time, the economy must face the following full
employment constraint on labor:
L1 + L2 + LR = L,(2)

where L > 0 is labor endowment and is assumed to be given and constant over time.
Let l ≡ L−LR is the total labor inputs in the private sectors. Letting good 2 be a numeraire,6the production side of the economy is characterized by the following GDP function:{2
G(p, R, l) = max pRα L1−α
+ Rα2 L1−α
12
L1 ,L2}s.t. L1 + L2 = l ,(3)where p is the price of good 1. By applying the envelope theorem to the GDP function, it
follows that the GDP function satisﬁes the following properties:2
Gp = Y 1 ,GR =α1 pY1 + α2 Y2,R

Gl = w,(4)

where w denotes the wage. Appendix A.1 gives additional properties of the GDP function.

3.2

Consumption SideThe 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,(5)0

where Ci is consumption of good i (i = 1, 2), ρ is the rate of time preference, and γ ∈ (0, 1) is aparameter. Let us denote the household’s total expenditure at each moment of time by E, andassume that no borrowing or lending is permitted. Then, the household’s optimal consumption
must satisfy C1 = γE/p and C2 = (1 − γ)E.
With no international borrowing and lending, national income must equal total expenditure
at all points in time: E = G(p, R, l). Substituting the household’s optimal consumption into
the lifetime utility (5), its indirect lifetime utility is derived as∫ ∞{}e−ρt ln[G(p, R, L − LR )] − γ ln p + ln[γ γ (1 − γ)1−γ ] dt.
V =

(6)

0

The national welfare can be measured by the lifetime indirect utility (6).

3.3International Public Good

The accumulation of the international public good is made by each country’s voluntary investment, which takes the form of producing a ﬂow of a homogeneous investment good. The
production of the investment good requires labor as an input, and I assume that the production technology is linear. Then, given the initial stock R0 > 0, the international public good
accumulates over time according to the following diﬀerential equation:3
Ṙ = ϕLR + ϕ∗ L∗R − δR,
2(7)

The subscripts denote partial derivatives: Gp = ∂G/∂p, and so on.
A dot over a variable denotes the time derivative. To avoid unnecessary complication in the notation, we
omit time arguments when no confusion arises from doing so.
37

where ϕ > 0 denotes the labor productivity of producing the ﬂow of international public good,which can be interpreted as the technology of contribution to the accumulation of international
public good, in the home country, ϕ∗ > 0 is the foreign country’s counterpart, and δ > 0 is thedepreciation rate of the stock of international public good.

4

AutarkyIn this section, I assume that the two countries do not trade the goods, and derive the cooperative and noncooperative solutions under autarky.

4.1

Market equilibrium

Under autarky, at each moment in time, the demand for for each private good must be equal
to the supply in each country. The market-clearing condition for good 1 in the home country,
C1 = Y1 , can be rewritten asγG(p, R, l) = Gp (p, R, l).
p(8)From (8), the autarkic equilibrium price of good 1 in the home country for a given pair of R
and l is derived as pa = P a (R, l), with the following properties:
PRa =γGR − pGpR,
(1 − γ)Gp + pGpp

Pla =γGl − pGpl.
(1 − γ)Gp + pGpp

(9)

I assume that the two countries share the identical preferences, meaning that ρ = ρ∗ and
γ = γ ∗ , and the identical technologies for producing private goods, i.e., αi = αi∗ , i = 1, 2.
These assumptions imply that the autarkic equilibrium price in the foreign country is derived
as Pa∗ = P a (R, l∗ ).

4.2

Cooperative Solution

In this subsection, I derive a cooperative solution where the governments in both countries
determine the paths of LR and L∗R in order to maximize the sum of these countries’ welfare
subject to the dynamics of international public good (7).
the current value Hamiltonian associated with the world-welfare maximization problem is8

deﬁned as
Hw = ln[G(P a (R, L − LR ), R, L − LR )] − γ ln[P a (R, L − LR )]+ ln[G(P a (R, L∗ − L∗R ), R, L∗ − L∗R )] − γ ln[P a (R, L∗ − L∗R )]+ 2 ln[γ γ (1 − γ)1−γ ] + θw (ϕLR + ϕ∗ L∗R − δR),
where θw can be interpreted as the shadow price of the international public good in the worldwelfare maximization problem. The ﬁrst-order conditions for maximizing Hw are∂Hw
Gl (P a (R, L − LR ), R, L − LR )=0 ⇒= θw ϕ,∂LR
G(P a (R, L − LR ), R, L − LR )Gl (P a (R, L∗ − L∗R ), R, L∗ − L∗R )
∂Hw
=
0
⇒
= θ w ϕ∗ ,∂L∗R
G(P a (R, L∗ − L∗R ), R, L∗ − L∗R )

(10)
(11)

where the market-clearing conditions in each country (8) are utilized. The adjoint equation isderived asθ̇w = ρθw −∂Hw∂R= (ρ + δ)θw −GR (pa , R, L − LR ) GR (p∗a , R, L∗ − L∗R )−,G(pa , R, L − LR )G(p∗a , R, L∗ − L∗R )(12)

where pa = P a (R, L − LR ) and p∗a = P a (R, L∗ − L∗R ), and the transversality condition is givenbylim e−ρt θw (t)R(t) = 0.

t→∞

(13)

From (10), the optimal contribution level in the home country can be expressed as LR =λa (θw ; ϕ, L), with the derivatives4λaθ = (

ϕG
)
,
Gl Gp
a + (Gl )2 − G
−
G
P
lp
ll
l
G
G

(14a)

λaϕ = (

θw G
)
,
Gl Gp
(Gl )2a
G − Glp Pl + G − Gll(14b)

λaL = 1.

(14c)

As shown in Appendix A.2, both λaϕ and λaL are positive, indicating that the optimal contribution level becomes larger in a country with higher contribution technology and/or with larger
labor endowment. Substituting these expressions into (7), the optimal path of the stock ofinternational public good is derived as
Ṙ = ϕλa (θw ; ϕ, L) + ϕ∗ λa (θw ; ϕ∗ , L∗ ) − δR.
4

It can be veriﬁed that the optimal level of LR is independent of R. See Appendix A.2.

9

(15)

The adjoint equation (12) is also rewritten as
GR (P a (R, L − λa (θw ; ϕ, L)), R, L − λa (θw ; ϕ, L))G(P a (R, L − λa (θw ; ϕ, L)), R, L − λa (θw ; ϕ, L))a
GR (P (R, L∗ − λa (θw ; ϕ∗ , L∗ )), R, L∗ − λa (θw ; ϕ∗ , L∗ ))−.
G(P a (R, L∗ − λa (θw ; ϕ∗ , L∗ )), R, L∗ − λa (θw ; ϕ∗ , L∗ ))θ̇w = (ρ + δ)θw −(16)

The evolution of the world economy under international cooperation is characterized by thesystem of diﬀerential equations (15) and (16).
Let us denote the autarkic steady-state solutions for R and θw under international cooperation by RaC and θaw . It can be veriﬁed that the steady state, if it exists, is unique andsaddle-point stable.5

4.3

Noncooperative Equilibrium

Let us turn to the situation where the governments in each country is self-interested and thus
determines the path of its contribution level in order to maximize its national welfare. Because
each country’s contribution level aﬀects the accumulation equation of the stock of international public good and this stock level in turn aﬀects each country’s welfare, this situation ischaracterized as a diﬀerential game.
There are two equilibrium concepts frequently employed in applications of diﬀerential gametheory in economics; one is 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 other is theMarkov perfect Nash equilibrium, in which each player designs its optimal strategy as a feedback
decision rule dependent only on the state variable. Both equilibrium concepts satisfy time
consistency, but the only the Markov perfect Nash equilibrium satisﬁes subgame perfectness(see, for example, Long, 2010). However, I focus on the open-loop Nash equilibrium becauseof its tractability. This strategy concept requires that governments can commit themselves
to particular strategy paths at the beginning of the game, and I simply assume that thecommitment is credible. Formally, the open-loop Nash equilibrium of this dynamic contribution∞
game is deﬁned as a pair of time paths {(LR (t), L∗R (t))}∞t=0 , such that {LR (t)}t=0 maximizes thehome country’s national welfare subject to the dynamics of R given by (7), taking {L∗R (t)}∞t=0 asgiven, and {L∗R (t)}∞t=0 maximizes the foreign country’s national welfare subject to the dynamics
of R, taking {LR (t)}∞t=0 as given.
5

See Appendix A.3.

10

Let us deﬁne the home country’s current value Hamiltonian as follows:
H = ln[G(P a (R, L − LR ), R, L − LR )] − γ ln[P a (R, L − LR )]+ ln[γ γ (1 − γ)1−γ ] + θ(ϕLR + ϕ∗ L∗R − δR),
where θ can be interpreted as the shadow price of the international public good in the homecountry. The optimality conditions are given by∂H
Gl (P a (R, L − LR ), R, L − LR )
=0 ⇒
= θϕ,
∂LR
G(P a (R, L − LR ), R, L − LR )∂H
GR (P a (R, L − LR ), R, L − LR )θ̇ = ρθ −= (ρ + δ)θ −,∂R
G(P a (R, L − LR ), R, L − LR )lim e−ρt θ(t)R(t) = 0.

t→∞

(17)
(18)
(19)

Notice that the ﬁrst-order condition (17) is the same as (10) except that θw in (10) is replacedby θ. Therefore, from (17), the home country’s optimal contribution level is expressed as
LR = λa (θ; ϕ, L). The foreign country’s optimality conditions can be derived analogously.
Substituting (17) and the foreign country’s counterpart into (7), the dynamic path of thestock of international public good is derived as
Ṙ = ϕλa (θ; ϕ, L) + ϕ∗ λa (θ∗ ; ϕ∗ , L∗ ) − δR.

(20)

The home country’s adjoint equation (12) and its foreign counterparts are also rewritten as
GR (P a (R, L − λa (θ; ϕ, L)), R, L − λa (θ; ϕ, L)),G(P a (R, L − λa (θ; ϕ, L)), R, L − λa (θ; ϕ, L))
GR (P a (R, L∗ − λa (θ∗ ; ϕ∗ , L∗ )), R, L∗ − λa (θ∗ ; ϕ∗ , L∗ ))θ̇∗ = (ρ + δ)θ∗ −.
G(P a (R, L∗ − λa (θ∗ ; ϕ∗ , L∗ )), R, L∗ − λa (θ∗ ; ϕ∗ , L∗ ))θ̇ = (ρ + δ)θ −

(21)
(22)

The evolution of the world economy in the open-loop Nash equilibrium is characterized by thesystem of diﬀerential equations (20), (21), and (22).
Let us denote the autarkic steady-state solutions for R, θ, and θ∗ in the open-loop Nashequilibrium by RaN , θa , and θa∗ . It can be veriﬁed that the steady state is a saddle point.6

4.4

Comparative Advantage

By substituting the optimal contribution levels in the case of international cooperation, theautarkic equilibrium prices of good 1 are pa = P a (R, L − λa (θw ; ϕ, L)) in the home country and6

See Appendix A.3.

11p∗a = P a (R, L − λa (θw ; ϕ∗ , L∗ )) in the foreign country. Analogously for the case of noncooperative equilibrium, the autarkic equilibrium prices are given by pa = P a (R, L − λa (θ; ϕ, L)) andp∗a = P a (R, L − λa (θ∗ ; ϕ∗ , L∗ )). The properties of autarkic equilibrium price in each country
are characterized by the following proposition.
Proposition 1 Suppose that there is no trade in goods between these countries and both countries choose positive contribution levels along the optimal path. Then, given the contribution
regime (cooperative or noncooperative), along the dynamic path, (i) the autarkic equilibrium
price of good 1 is lower in the country with a higher contribution technology, and (ii) laborendowment has no eﬀect on the autarkic equilibrium price.

Proof. Suppose ﬁrst the case of international cooperation. Along the optimal path, bothcountries faces the same R and θw , and thus the diﬀerences in the contribution technologies
and labor endowments determine the diﬀerence in the autarkic equilibrium prices. From (8),(9), and (14), it follows that()
G G
p · Gpl − pG l

∂pa
= −Pla λaϕ =λa ,∂ϕ(1 − γ)Gp + pGpp ϕ∂pa= Pla · (1 − λaL ) = 0.
∂L

(23)
(24)

From (A.12), it follows that ∂pa /∂ϕ < 0 under Assumption 1.
Next consider the case of noncooperative equilibrium. By diﬀerentiating (21) with respectto ϕ and using the derivatives of G derived in Appendices A.1 and A.2, it follows that
{(
)
}
λaϕ
GR Gp
∂ θ̇
GR Gl
a
=
GpR −
Pl + GRl −
∂ϕ
G
G
G
)(
(
)
GR Gp
GR Gp
GR Gl
p
− GpR
+ GRl −= GpR −
GG
(1 − γ)Gp + pGpp
G
= 0.

(25)

Moreover, in light of (14), it holds that ∂ θ̇/∂L = 0. Therefore, for a given R, any changes in ϕand L do not aﬀect the equilibrium path of θ. This implies that both countries face the same
paths of R and its shadow price. Then, as in the case of international cooperation, it follows
that along the equilibrium path pa < p∗a holds if ϕ > ϕ∗ and pa = p∗a if ϕ = ϕ∗ (even thoughL ̸= L∗ ).

2

12

Proposition 1 (i) indicates that along the optimal or equilibrium path, the country with ahigher contribution technology has a comparative advantage in a good that is more dependent
on the stock of international public good. Intuitively, a higher ϕ implies a larger contribution
level LR , and in turn, the total labor available in the private sectors l becomes smaller. A
decrease in l reduces both the output and consumption of good 1, but the reduction in C1 islarger than that in Y1 . This implies that for a given p the excess demand for good 1 becomes
smaller, and thus the autarkic equilibrium price of good 1 becomes lower.
Proposition 1 (ii) indicates that two countries that diﬀer only in their labor endowments
faces the same autarkic equilibrium price along the optimal or equilibrium path, and in otherwords, that the diﬀerence in labor endowments cannot be a source of comparative advantage.
This is because an increase in the labor endowment L increases LR by the same amount, andthus the total labor available in the private sectors does not change. Therefore, the diﬀerence
the labor endowment has no eﬀect on output or consumption of the private goods.
Notice that Proposition 1 holds for any parameter values for αi ∈ [0, 1), i = 1, 2, as longas both countries choose positive contribution levels. Therefore, in what follows, I put thefollowing assumption as a substitute for Assumption 1:
Assumption 1′

α1 = α > 0 = α2 .

Under Assumption 1′ , the GDP function is explicitly derived as
1−α1G(p, R, l) = (1 − α) α αp α R + l,

(26)

and using this, the autarkic equilibrium price is derived as[]αγlP a (R, l) =.
1−α
(1 − αγ)(1 − α) α R(27)

It follows that the optimal contribution levels under international cooperation are derived as7
λa (θw ; ϕ, L) = L −1 − αγ≡ λa (θw ),θw ϕ

λa (θw ; ϕ∗ , L∗ ) = L∗ −1 − αγ≡ λ∗a (θw ),θ w ϕ∗(28)and each country’s optimal contribution levels in the open-loop Nash equilibrium are λa (θ)
and λa (θ∗ ). The dynamic system of the cooperative solution path consisting of (15) and (16)7

Conditions under which the countries choose positive contribution levels are discussed in Appendix A.4.

13

is rewritten as2(1 − αγ)
− δR,
θw
2αγ
,
θ̇w = (ρ + δ)θw −R

Ṙ = ϕL + ϕ∗ L∗ −

(29)
(30)

and the steady-state stock of international public good is derived asRaC =αγ(ϕL + ϕ∗ L∗ ).
(1 − αγ)(ρ + δ) + αγδ(31)Analogously, the open-loop Nash equilibrium path consisting of (20), (21), and (22) is rewrittenas
1 − αγ 1 − αγ
−
− δR,
θ
θ∗
αγ
θ̇ = (ρ + δ)θ −,Rαγθ̇∗ = (ρ + δ)θ∗ −.
R

Ṙ = ϕL + ϕ∗ L∗ −

(32)
(33)
(34)

The steady-state stock of international public good is derived asRaN =αγ(ϕL + ϕ∗ L∗ ).
2(1 − αγ)(ρ + δ) + αγδ

(35)Comparing (31) and (35), it is easily veriﬁed that RaC > RaN . That is, under noncooperative
behavior, the international public good is under-supplied in the long run. This is because,
under noncooperative behavior, each country does not take account of the positive externalityof the international public good on the other country in determining the path of its contribution
level.
Substituting the steady-state solutions into P a (R, l), the autarkic steady-state equilibrium
prices of good 1 in each country are derived as
[
]α
[
]α
ρ
+
δ
ρ
+
δ
pC
, p∗C
1−α
1−α
a =
a =
2(1 − α) α αϕ2(1 − α) α αϕ∗in the case of cooperative solution and
]α
[
ρ
+
δ
,
pN
1−α
a =
(1 − α) α αϕ

[
p∗N
a =

ρ+δ
1−α

(1 − α) α αϕ∗

(36)

]α
(37)

in the case of noncooperative equilibrium, respectively. These expressions are consistent with
Proposition 1. Moreover, the following proposition is obtained.
Proposition 2 The autarkic steady-state equilibrium price of good 1 in each country is lower
under cooperative regime than under noncooperative regime.
14

Proposition 2 comes from the fact that the excess supply of good 1 is increasing in R (see themarket clearing condition (8)) and RaC > RaN . That is, the steady-state stock of international
public good is larger in the cooperative solution than in the noncooperative equilibrium, andthus the good 1, the output of which depends on the stock of international public good, canbe supplied more eﬃciently.

5Free Trade

5.1

Market Equilibrium

Under free trade, the total demand for for each private good must be equal to the total supply
in the world market at each moment in time. The market-clearing condition for good 1,
C1 + C1∗ = Y1 + Y1∗ , can be rewritten asγ[G(p, R, l) + G∗ (p, R, l∗ )] = Gp (p, R, l) + G∗p (p, R, l∗ ).
p(38)

In light of (26), the market-clearing condition (38) derives the equilibrium price of good 1 in
the world market as follows:
[
pf =

5.2

]α

γ(l + l∗ )1−α2(1 − αγ)(1 − α) α R≡ P f (R, l, l∗ ).

(39)

Cooperative Solution

The current value Hamiltonian associated with the world-welfare maximization problem isdeﬁned as
Hw = ln[G(P f (R, L − LR , L∗ − L∗R ), R, L − LR )] − 2γ ln[P f (R, L − LR , L∗ − L∗R )]+ ln[G(P f (R, L − LR , L∗ − L∗R ), R, L∗ − L∗R )] + 2 ln[γ γ (1 − γ)1−γ ]+ θw (ϕLR + ϕ∗ L∗R − δR).

15

In light of (26) and (27), the ﬁrst-order conditions are given by
2 − αγ∂Hw=−∂LR
αγ(L − LR + L∗ − L∗R ) + 2(1 − αγ)(L − LR )αγ−∗∗αγ(L − LR + L − LR ) + 2(1 − αγ)(L∗ − L∗R )2αγ++ θw ϕ = 0,
L − LR + L∗ − L∗R
∂Hw
αγ
=−
∗
∗
∗
∂LR
αγ(L − LR + L − LR ) + 2(1 − αγ)(L − LR )2 − αγ−αγ(L − LR + L∗ − L∗R ) + 2(1 − αγ)(L∗ − L∗R )2αγ++ θw ϕ∗ = 0.
L − LR + L∗ − L∗R

(40)

(41)

The adjoint equation and the transversality condition are shown to be the same as those underautarky.
The temporary optimal solutions for LR and L∗R can be derived by solving the nonlinear
equations (40) and (41). These equations derive a rather complicated solution pair for LR andL∗R , but adding up (40) multiplied by L − LR and (41) multiplied by L∗ − L∗R , it follows thatϕLR + ϕ∗ L∗R = ϕL + ϕ∗ L∗ − 2(1 − αγ)/θw . In light of (7), this means that the dynamics ofthe stock of international public good under free trade is given by (29), as in the cooperative
solution under autarky. Thus, the following proposition is established.
Proposition 3 Suppose that the governments in both countries act cooperatively and bothcountries choose positive contribution levels. Then, the optimal trajectories of the stock ofinternational public good and its shadow price under free trade coincides with those underautarky, and hence the steady-state stock of international public good under free trade, RfC ,coincides with RaC .
In contrast with the autarkic case, the welfare eﬀect of a change in the international priceof good 1 is asymmetric between countries. Suppose that ϕ > ϕ∗ and thus the home (foreign)country exports (imports) good 1 and that p increases. Then, other things being equal, theincrease in p makes the home country better oﬀ because of an improvement in its terms oftrade, whereas the foreign country becomes worse oﬀ. Nevertheless, in the cooperative solution
where both the home and foreign countries jointly maximize the world welfare, these positive
and negative welfare eﬀects induced by the price change are oﬀset, and thus the outcome willbe the same as that under autarky.
16

Although the paths of R and θw under free trade is the same as those under autarky,each country’s optimal contribution level under free trade is diﬀerent from that under autarky.
Evaluating the left-hand side terms of (40) and (41) at LR = λa (θw ) and L∗R = λ∗a (θw ) deﬁnedby (28), it follows that∂Hw
= θw ϕΦ(ϕ − ϕ∗ ) > 0,
∂LR LR =λa (θw ),L∗ =λ∗a (θw )R∂Hw= θw ϕ∗ Φ(ϕ∗ − ϕ) < 0,
∂L∗R LR =λa (θw ),L∗ =λ∗ (θw )
R

a

where
Φ≡

αγ[(2 − αγ)(ϕ2 + ϕ∗2 ) + 2αγϕϕ∗ ]> 0.
[(2 − αγ)ϕ + αγϕ∗ ][(2 − αγ)ϕ∗ + αγϕ](ϕ + ϕ∗ )From (40) and (41), the temporary optimal levels of LR and L∗R under free trade can be
expressed as λf (θw ) and λ∗f (θw ). Then, under assumption that ϕ > ϕ∗ , it holds that λf (θw ) >λa (θw ) and λ∗f (θw ) < λ∗a (θw ). That is, trade liberalization induces the home country, which
has a higher contribution technology, to contribute more to the investment on the international
public good, whereas the foreign country to contribute less.
Proposition 4 Suppose that the governments in both countries act cooperatively. Then, tradeliberalization increases the optimal contribution level in the country with a higher contribution
technology and reduces the contribution level in the country with a lower contribution technology.
Because labor endowment in each country is assumed to be constant over time, Proposition
4 implies that under free trade, less labor is available to private sectors in the home country
whereas the foreign country enjoys increased labor allocation in the private sectors. In otherwords, trade liberalization increases the foreign country’s level of “free ride” on the homecountry’s contribution eﬀort.
As shown in Proposition 1, under the assumption that ϕ > ϕ∗ , the home country has acomparative advantage in good 1. Under free trade, the home country actually exports good 1.
That is, denoting the equilibrium price of good 1 under autarky and free trade for a given pairof R and θw by (pa , p∗a ) and pf , respectively, it holds that pa < pf < p∗a . This can be veriﬁed
as follows. Since the paths of R and θw under free trade coincides with those under autarky,(27) and (39) imply the following:{}αpa2[L − λ(θaw )]=.
pf
L − λf (θw ) + L∗ − λ∗f (θw )17

As shown above, λf (θw ) > λa (θw ) and λ∗f (θw ) < λ∗a (θw ) hold. It was also shown that L −λa (θw ) < L∗ − λ∗a (θw ). Putting these inequalities together yields
L − λf (θw ) < L − λa (θw ) < L∗ − λ∗a (θw ) < λ∗f (θw ).

(42)

Moreover, since ϕλa (θw ) + ϕ∗ λ∗a (θw ) = ϕλf (θw ) + ϕ∗ λ∗f (θw ), the following inequality holds:
ϕ > ϕ∗⇒λf (θw ) − λa (θw ) < λ∗a (θw ) − λ∗f (θw ).

(43)

In light of (42) and (43), it holds that
2[L − λa (θw )]L − λa (θw ) + L∗ − λ∗a (θw )<L − λf (θw ) + L∗ − λ∗f (θw )L − λf (θw ) + L∗ − λ∗f (θw )=L + L∗ − [λa (θw ) + λ∗a (θw )]< 1,
L + L∗ − [λf (θw ) + λ∗f (θw )]and thus pa /pf < 1. Analogously, p∗a /pf > 1 can be veriﬁed.
As mentioned above, when the governments cooperatively determine their respective contribution levels, free trade achieves the same paths of the international public good and its shadow
price as autarky, but the contribution level in each country, and in turn, the total labor available in the production of private goods, diﬀers between autarky and free trade. Speciﬁcally,
under the assumption that ϕ > ϕ∗ , trade liberalization reduces L − LR and increases L∗ − L∗R
for a given stock level of R. This implies that the production possibility frontier for a given R
contracts in the home country and expands in the foreign country. Because the foreign country
enjoys larger GDP under free trade than under autarky, this country will gain from trade.
Proposition 5 Suppose that the governments in both countries act cooperatively. Then, thecountry with a lower contribution technology unambiguously enjoys the higher level of national
welfare under free trade than under autarky.
Proof. Let us deﬁne the expenditure function asE(p, u) = min {pC1 + C2
C1 ,C2s.t.

γ log C1 + (1 − γ) log C2 ≥ u} .

It is easily veriﬁed that Eu > 0. Let us also denote the foreign country’s utility levels underautarky and free trade for a given pair of R and θw by u∗a and u∗f , respectively. Because Yi = Ci
holds under autarky for i = 1, 2 and E(p, u) = G(p, R, L − LR ) holds under free trade, the18

following expression is obtained:E(pf , u∗f ) − E(pf , u∗a )∗∗= G(pf , R, L∗ − λ∗f (θw )) − G(pf , R, L∗ − λ∗a (θw )) + (pf C1a+ C2a) − E(pf , u∗a ),(44)∗ is foreign country’s autarkic consumption level of good i = 1, 2. From the deﬁnition
where Cia
∗ + C ∗ ≥ E(p , u∗ ). Moreover, from (26), (27),of the expenditure function, it holds that pf C1afa2a

and (39), it follows thatG(pf , R, L∗ − λ∗f (θw )) − G(pf , R, L∗ − λ∗a (θw )){}αγ[L − λf (θw )] + (2 − αγ)[L∗ − λ∗f (θw )]αγ[L − λf (θw ) + L∗ − λ∗f (θw )]=−+ L∗ − λ∗a (θaw )2(1 − αγ)2(1 − αγ)= λ∗a (θw ) − λ∗f (θw ).

(45)

As discussed above, λ∗a (θw ) > λ∗f (θw ) holds if ϕ > ϕ∗ . Then, it follows that the sign of (44) is
unambiguously positive, and thus u∗f > u∗a .

2

Contrary to the foreign country, the home country faces a reduction in its GDP levelevaluated at p = pf :G(pf , R, L − λf (θw )) − G(pf , R, L − λa (θw )) = λa (θw ) − λf (θw ) < 0.
Even though trade expands the consumption possibility, the home country’s national income
decreases, and if the negative eﬀect of the reduction in national income outweighs the positiveeﬀect of the improvement in the economy’s consumption possibility, the home country may
lose from trade.
The possibility of welfare loss in a country with a higher contribution technology can beillustrated in the following numerical example. As demonstrated in Proposition 3, the optimal
trajectories of R and θw under free trade coincide with those under autarky, which are givenby the system of nonlinear diﬀerential equations (29) and (30). I focus on the linearized systemof (29) and (30) around the steady state. For the parameter values, I choose α = 0.7, γ = 0.5,
ρ = 0.1, δ = 0.25, ϕ = 1.5, ϕ∗ = 1, L = 1000, L∗ = 1250, and R0 = 100. Figure 1 provides
a comparison between the path of contribution level under free trade (solid curve) and that
under autarky (dotted curve) in each country. It is shown that along the optimal path, thecontribution level in the home country, which has a higher contribution technology, is larger
19under free trade than under autarky for each moment in time, whereas the opposite holds in theforeign country. This ﬁnding is consistent with Proposition 4. With regard to the comparisonof welfare between autarky and free trade, I focus on the instantaneous welfare, which is given
by ln[G(p, R, L − LR )] − γ ln p + ln[γ γ (1 − γ)1−γ ]. As illustrated in Figure 2, in the foreigncountry the instantaneous welfare under free trade (solid curve) is unambiguously higher than
the instantaneous welfare under autarky (dotted curve) for each moment in time. This ﬁnding
is consistent with Proposition 5. Figure 2 also shows that in this numerical example, the homecountry’s instantaneous welfare is unambiguously lower under free trade than under autarky
for each moment in time.
600

600

500

500

400

400

LRHtL 300

LRHtL 300

200

200

100

100

0

0

5

10

15

0

20

0

5

10

time

15

20

time

(a) Home country(b) Foreign country

Figure 1: Paths of contribution levels in the cooperative solutions under autarky and free trade
7.5

7.5

7.0

7.0

6.5

6.5

WelfareWelfare
6.0

6.0

5.5

5.5

5.0

0

5

10

15

20

time

5.0

0

5

10

15

time

(a) Home country(b) Foreign country

Figure 2: Paths of instantaneous welfare in the cooperative solutions under autarky and free
trade

20

20

5.3

Noncooperative Equilibrium

The home country’s current value Hamiltonian is given by
H = ln[G(P f (R, L − LR , L∗ − L∗R ), R, L − LR )] − γ ln[P f (R, L − LR , L∗ − L∗R )]+ ln[γ γ (1 − γ)1−γ ] + θw (ϕLR + ϕ∗ L∗R − δR).
The ﬁrst-order condition is given by∂H
2 − αγ=−∗∂LR
αγ(L − LR + L − L∗R ) + 2(1 − αγ)(L − LR )αγ+
+ θϕ = 0.
L − LR + L∗ − L∗R(46)

The adjoint equation and the transversality condition are same as those under autarky. Analogously for the foreign country, deﬁning the current value Hamiltonian H∗ and deriving theﬁrst-order condition, it follows that2 − αγ
∂H∗
=−
∗
∗
∂LR
αγ(L − LR + L − L∗R ) + 2(1 − αγ)(L∗ − L∗R )αγ++ θ∗ ϕ∗ = 0.
L − LR + L∗ − L∗R(47)

Solving (46) and (47) for LR and L∗R , the Nash equilibrium contribution levels are derived,
which are dependent on θ and θ∗ . It is of interest whether trade liberalization increases theequilibrium contribution level in each country. As shown later, the equilibrium paths of R and
its shadow price in each country under free trade do not coincide with those under autarky.
Therefore, let us suppose that the stock of international public good is ﬁxed at the steady-statelevel under autarky: R = RaN and thus θa = θa∗ = αγ/[(ρ + δ)RaN ]. Evaluating the left-handside terms of (46) and (47) at the autarkic steady-state levels LR = λa (θa ) and L∗R = λ∗a (θa∗ ),it follows that under the assumption ϕ > ϕ∗ ,α2 γ 2 ϕ2 (ϕ − ϕ∗ )
∂H
=
> 0,
∂LR LR =λa (θa ),L∗ =λ∗a (θa∗ ) (ϕ + ϕ∗ )[αγϕ + (2 − αγ)ϕ∗ ](ρ + δ)RaN
R

α2 γ 2 ϕ∗2 (ϕ∗ − ϕ)
∂H∗
=
< 0.
∂L∗R LR =λa (θa ),L∗ =λ∗ (θ∗ ) (ϕ + ϕ∗ )[(2 − αγ)ϕ + αγϕ∗ ](ρ + δ)RaN
Raa

That is, as with the cooperative case, the home country has an incentive to contribute more
under free trade than under autarky, whereas the foreign country has an incentive to contributeless under free trade.

21

Adding up (46) multiplied by (L − LR )/θ and (47) multiplied by (L∗ − L∗R )/θ∗ , and substituting θ = θ∗ = αγ/[(ρ + δ)R], it follows thatϕLR + ϕ∗ L∗R = ϕL + ϕ∗ L∗ −(ρ + δ)R[(2 − αγ)Ψ − αγ],
αγ

where
Ψ≡

L∗ − L∗R
L − LR+.
(2 − αγ)(L − LR ) + αγ(L∗ − L∗R ) αγ(L − LR ) + (2 − αγ)(L∗ − L∗R )

Since
Ψ−1
=−

αγ(1 − αγ)[L − LR − (L∗ − L∗R )]2< 0,[(2 − αγ)(L − LR ) + αγ(L∗ − L∗R )][αγ(L − LR ) + (2 − αγ)(L∗ − L∗R )]it holds that (2 − αγ)Ψ − αγ < 2(1 − αγ). Then, comparing the steady-state condition underfree trade∗ ∗Ṙ = ϕL + ϕ L −{}[(2 − αγ)Ψ − αγ](ρ + δ)+δ R=0αγ

with the steady-state condition under autarky{}2(1 − αγ)(ρ + δ)∗ ∗Ṙ = ϕL + ϕ L −+ δ R = 0,αγit holds that the steady-state stock of international public good, RfN , is larger than the autarkic
level RaN (see also Figure 3).

Figure 3: Comparison of steady-state stocks of international public goodTo sum up, the following proposition is established.
22

Proposition 6 Suppose that the governments in both countries noncooperatively determine
their respective contribution levels by using the open-loop strategy. Then, the steady-state contribution level under free trade becomes larger in the country with a higher contribution technology, whereas it is smaller in the country with a lower contribution technology. The steady-state
stock of international public good under free trade is larger than the autarkic steady-state level.
In the absence of international cooperation over the provision of international public good,each country has a strategic incentive to control the contribution level so as to maximize its own
welfare. The optimality condition for national welfare maximization implies that the marginal
beneﬁt from the international public good should be equal to the marginal cost of contribution
in each country. From the assumption that ϕ > ϕ∗ , the home country increases its contribution
level under free trade compared to autarky, whereas the foreign country reduces its contribution
level. Moreover, ϕ > ϕ∗ implies that the home country has a superior contribution technology
compared to the foreign country, and thus the increase in LR caused by trade liberalization
outweighs the reduction in L∗R . Therefore, in comparison with autarky, there is a net increasein the sum of the change in each country’s contribution level multiplied by its contribution
technology, i.e., ϕLR + ϕ∗ L∗R , under free trade. Consequently, free trade achieves the larger
stock of international public good than the autarkic steady-state level.
With regard to the comparison of contribution levels and the stock of international public
good along the Nash equilibrium paths, I consider the following numerical example. I use
the same parameter values as those in the previous subsection, and obtained the equilibrium
solutions for LR (t), L∗R (t), and R(t), as illustrated in Figure 4 and Figure 5. The solid curvein each ﬁgure denotes the equilibrium solution under free trade, whereas the dotted curve
denotes that under autarky. Figure 4 shows that the home country, with a higher contribution
technology, allocates more labor for contributing to the accumulation of international public
good under free trade than under autarky, while the opposite holds for the foreign country.
Figure 5 shows that the Nash equilibrium trajectory of the stock of international public goodunder free trade is above the equilibrium trajectory under autarky.
In the cooperative regime, it was shown that even though the steady-state stock of international public good remains unchanged under free trade compared to autarky, the country
with a lower contribution technology unambiguously gains from trade in the long run. In thenoncooperative regime, RaN < RfN holds and the total labor available in the country with a
23

600

600

500

500

400

400

LRHtL 300

LRHtL 300

200

200

100

100

0

0

5

10

15

0

20

0

5

10

time

15

time

(a) Home country(b) Foreign country

Figure 4: Paths of contribution levels in the Nash equilibrium solutions under autarky and free
trade

2000

1500

RHtL 1000

500

0

0

2

4

6

8

10

time

Figure 5: Paths of R(t) in the Nash equilibrium solutions under autarky and free trade24

20

lower contribution technology increases under free trade compared to the autarkic steady state.
Therefore, in the noncooperative regime, the gains-from-trade result in the country with a lower
contribution technology is maintained.

6

Concluding Remarks

In this paper I developed a dynamic two-country model with an international public good, thestock of which has a positive eﬀect on the private sector’s productivity in each country andthe evolution of the stock is determined by each country’s voluntary contribution. I derived
both cooperative solution and noncooperative Nash equilibrium in open-loop strategies, underboth autarky and free trade. As far as I know, there are no studies that consider international
trade and the provision of international public goods in a uniﬁed dynamic trade model. I
showed that under autarky the equilibrium price of the good which is more dependent on thestock international public good becomes lower in the country that has a higher contribution
technology, regardless of whether the countries cooperatively or noncooperatively determine
their contribution levels. I also showed that if the countries act cooperatively, the dynamics
of the stock of international public good and its shadow price under free trade coincide withthose under autarky, although the paths of each country’s contribution level are diﬀerent.
More speciﬁcally, trade liberalization increases the optimal contribution level in the country
with a higher contribution technology and reduces the contribution level in the country witha lower contribution technology. This implies that in the country with a lower contribution
technology, the total labor available in the production of private goods becomes larger underfree trade than under autarky, and it was shown that this country unambiguously gains fromtrade. By contrast, the country with a higher contribution technology may lose from trade.
In the noncooperative regime, free trade achieves a larger steady-state stock of international
public good than autarky, and the gains-from-trade result is maintained in the country with alower contribution technology.
Throughout this paper I assumed that in the noncooperative regime, countries use openloop strategies to determine their respective contribution levels. However, the open-loop Nashequilibrium generally lacks subgame perfectness, and deriving the Markov perfect Nash equilibrium in this game would be more appropriate. Moreover, this paper assumed that theinternational public good is accumulated only through the public investment. However, for25example, the accelerated growth of the Internet is not only due to the eﬀorts of public sectors
but also private sectors. Therefore, it will also be interesting to extend the model to a case inwhich the international public good is accumulated through the contributions of private ﬁrms
as well as national governments. These issues are remained for future research.

Appendix
A.1

Properties of the GDP function

Let us deﬁne the Lagrangian:2
L(L1 , L2 , w, p, R, l) = pRα L1−α
+ Rα2 L1−α
+ w(l − L1 − L2 ),1
2

where the Lagrangian multiplier w can be interpreted as the wage. The ﬁrst-order conditions
for maximizing L are (1 − α1 )pY1 /L1 = w = (1 − α2 )Y2 /L2 and L1 + L2 = l. Given these
conditions and the production function (1), it follows that
]1−α1[
α1 (1 − α1 )pY1
Y1 = R,w]1−α2[α2 (1 − α2 )Y2
Y2 = R,w(1 − α1 )pY1 + (1 − α2 )Y2 = wl.
Totally diﬀerentiating eqs.(A.1), (A.1), and (A.3) gives  α1α1
1−α1  
1−α1
0
dY1
Y1
w
R dR + p dp
1−α2  α2α2.
dY2  = 
0
Y2
w
R dR
dw
−(1 − α1 )Y1 dp + wdl(1 − α1 )p 1 − α2 −l

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

(A.4)

From the solutions of (A.4), the second-order derivatives of the GDP function are obtained asfollows:(1 − α1 )(1 − α2 )Y1 Y2
∂Y1
=
> 0,
(A.5)
∂p
{(1 − α1 )α2 pY1 + (1 − α2 )α1 Y2 }p
(
)
∂Y2
1
αp ∂Y
+
α
2
∂R
∂R R − (αpY1 + α2 Y2 )α1 α2 (wl)2
GRR =
=
−
< 0,
R2
{(1 − α1 )α2 pY1 + (1 − α2 )α1 Y2 }R2
(A.6)
Gpp =

∂w
α1 α2 w 2
=−
< 0,
∂l
(1 − α1 )α2 pY1 + (1 − α2 )α1 Y2{(1 − α1 )α1 α2 pY1 + (1 − α2 )[α1 − (1 − α1 )α2 ]Y2 }Y1
∂Y1
=
,
GpR =
∂R
{(1 − α1 )α2 pY1 + (1 − α2 )α1 Y2 }R∂Y1
w(1 − α1 )α2 Y1
Gpl =
=
> 0,
∂l
(1 − α1 )α2 pY1 + (1 − α2 )α1 Y2
α1 α2 w 2 l
α1 ∂Y1 α2 ∂Y2
p
+
=
> 0.
GRl =
R ∂l
R ∂l
{(1 − α1 )α2 pY1 + (1 − α2 )α1 Y2 }R
Gll =

26

(A.7)
(A.8)
(A.9)
(A.10)

Moreover, the following calculation results, which are useful in the subsequent analysis, areobtained:
Gp GR(α1 − α2 )wlY1 Y2=,
G{(1 − α1 )α2 pY1 + (1 − α2 )α1 Y2 }(pY1 + Y2 )R
Gp Gl(α1 − α2 )wY1 Y2
Gpl −
=−
,
G
{(1 − α1 )α2 pY1 + (1 − α2 )α1 Y2 }(pY1 + Y2 )GR Gl
(α1 − α2 )2 wpY1 Y2
GRl −
=−
< 0.
G
{(1 − α1 )α2 pY1 + (1 − α2 )α1 Y2 }(pY1 + Y2 )R

GpR −(A.11)(A.12)(A.13)

If α1 > α2 , the sign of (A.11) is positive, whereas the sign of (A.12) is negative.

A.2

Derivatives of λa (θw ; ϕ, L)

In light of (8) and (9), the denominator of the derivatives in (14) is rewritten as
(
)
Gl Gp
(Gl )2− Glp Pla +
− Gll
G
G
)2
(
Gl Gp
p
(Gl )2− Glp+− Gll .
=
G
(1 − γ)Gp + pGpp
G

(A.14)

Since Gpp > 0 and Gll < 0 as derived in Appendix A.1, the sign of the above expression isunambiguously positive. Thus, λaθ and λaϕ have positive signs.
Suppose that the optimal level of LR that satisﬁes (10) depends on R, and solve for ∂λa /∂R.
The denominator of ∂λa /∂R is equal to (A.14), and its numerator is, in light of the derivativesof G derived in Appendix A.1, rewritten as
)
(
Gl Gp
Gl GR
− Glp PRa +
− GlR
G
G
(
)(
)
Gl Gp
Gp GR
p
Gl GR
=
− Glp
− GpR
+
− GlR
G
G
(1 − γ)Gp + pGppG
(α1 − α2 )2 wpY1 Y2 {[(1 − α1 )α2 pY1 + (1 − α2 )α1 Y2 ](pY1 + Y2 )[(1 − γ)Gp + pGpp ] − wlY1 Y2 }=.
{(1 − α1 )α2 pY1 + (1 − α2 )α1 Y2 }2 (pY1 + Y2 ){(1 − γ)Gp + pGpp }(A.15)
However, since8{(1 − α1 )α2 pY1 + (1 − α2 )α1 Y2 }(pY1 + Y2 ){(1 − γ)Gp + pGpp }}{(1 − α1 )(1 − α2 )(pY1 + Y2 )= {(1 − α1 )α2 pY1 + (1 − α2 )α1 Y2 }Y1 Y2 1 +(1 − α1 )α2 pY1 + (1 − α2 )α1 Y2= Y1 Y2 {(1 − α1 )pY1 + (1 − α2 )Y2 } = Y1 Y2 w(L1 + L2 ),the terms in the curly brackets in (A.15) become zero, and thus ∂λa /∂R = 0.
8
In the derivation of this equation, the ﬁrst-order conditions for maximizing the GDP and the market-clearing
condition (8) are utilized.

27

A.3

Uniqueness and stability of autarkic steady-state solutions

Cooperative solution

Let λa (θw ) ≡ λa (θw ; ϕ, L) and λ∗a (θw ) ≡ λa (θw ; ϕ∗ , L∗ ). From thesteady-state condition Ṙ = ϕλa (θw ) + ϕ∗ λ∗a (θw ) − δR = 0, it holds that
dθw
δ
> 0.
=
′
dR Ṙ=0 ϕλa + ϕ∗ λ∗′a(A.16)

From the steady-state condition θ̇w = 0,
∂ θ̇w /∂Rdθw=−.
dR θ̇w =0∂ θ̇w /∂θw

(A.17)

Straightforward calculations yield
)
)
(
( ∗ ∗
2
Gp GR
G∗2
Gp GR
a + GR − G
a∗
∗
∗
R
−
G
P
−
G
w
∗
pR
RR
R
pR PR + G∗ − GRR
G
G
G
∂ θ̇
=
+
∂R
G
G∗
)2
( ∗ ∗
)2
(
Gp GR
G G
∗
G2R
G∗2
∗
R
p pG R − GpR
p
−
G
pR
G∗
G − GRR
G∗ − GRR++=, (A.18)+{(1 − γ)Gp + pGpp }GG{(1 − γ)G∗p + pG∗pp }G∗G∗
which is unambiguously positive because Gpp > 0 and GRR < 0. Moreover, in light of (A.15)
it holds that
(
)
GR Gp
GR Gl
GRp −
Pla + GRl −
G
G
)
(
)(
GR Gp
Gl Gp
p
GR Gl
= GRp −− Glp
+ GRl −= 0,GG
(1 − γ)Gp + pGpp
G
and thus∂ θ̇w∂θw= ρ + δ + λ′a)(
G G
GRp − RG p Pla + GRl − GRGGl
G

+ λ∗′
a

(
)
G∗ G∗
G∗R G∗lG∗Rp − RG p Pla∗ + G∗Rl − G∗= ρ + δ > 0.

G∗
(A.19)

Therefore, the sign of (A.17) is unambiguously negative. It follows that the steady state, if itexists, is uniquely determined.
Linearizing the dynamic system (15) and (16) around the steady state (RaC , θaw ), it follows
that

[

] [
Ṙ
−δ
=
w
ΘC
θ̇
a

]
][
w
R − RaC
ϕλ′a (θaw ) + ϕ∗ λ∗′a (θ ),
ρ+δθw − θaw(A.20)w
C w
where ΘC
a is the value of (A.18) evaluated at (R, θ ) = (Ra , θa ). The determinant of theJacobian matrix in (A.20) is negative, and thus the dynamic system has one positive and one
negative eigenvalues. Since the dynamical system has one predetermined variable R, it follows
that the steady state is locally saddle-point stable.
28

Noncooperative equilibrium

The steady-state condition Ṙ = 0 is rewritten as R =[ϕλa (θ) + ϕ∗ λ∗a (θ∗ )]/δ. Substituting this into (21) and (22), it follows thatθ̇ = (ρ + δ)θ −
∗

∗

∗ ∗

∗

∗ ∗

∗

∗ ∗

∗

∗ ∗

∗

λa (θ )λa (θ )
, L − λa (θ)), ϕλa (θ)+ϕ, L − λa (θ))
GR (P a ( ϕλa (θ)+ϕδδθ̇ = (ρ + δ)θ −λa (θ )λa (θ )
G(P a ( ϕλa (θ)+ϕ, L − λa (θ)), ϕλa (θ)+ϕ, L − λa (θ))
δ
δ
∗ ∗

∗

∗ ∗

∗

∗ ∗

∗

∗ ∗

∗

,

λa (θ )λa (θ )GR (P a ( ϕλa (θ)+ϕ, L∗ − λ∗a (θ∗ )), ϕλa (θ)+ϕ, L∗ − λ∗a (θ∗ ))δδλa (θ )λa (θ )
G(P a ( ϕλa (θ)+ϕ, L∗ − λ∗a (θ∗ )), ϕλa (θ)+ϕ, L∗ − λ∗a (θ∗ ))δδ.

Then, it follows that}G2
PRa + GR − GRR
)
}
,
2
GR Gp
ϕ λa
dθ Ṙ=θ̇=0
a + GR − G
−
G
P
Rp
RR
R
δG
G
G
)
}
{( ∗ ∗
∗2
GR Gp
G
ϕλ′a
∗
∗a + R − G∗
−
G
P
∗
∗
∗
∗
Rp
R
RR
δG
G
G
dθ
{( ∗ ∗
)
},
=−
GR Gp
G∗2
ϕ∗ λ∗′dθ Ṙ=θ̇∗ =0
∗
∗a
∗
a
R
ρ + δ + δG∗
−
G
P
+
−
G
Rp
RR
R
G∗
G∗
dθ∗

′

a
ρ + δ + ϕλ
δG
{(
=−
∗ ∗′

{(

GR Gp
− GRp
G

)

(A.21)(A.22)and thus
dθ∗
dθ∗
<
< 0.
dθ Ṙ=θ̇∗ =0dθ Ṙ=θ̇=0Therefore, the steady state of the open-loop Nash equilibrium, if it exists, is uniquely determined.
Linearizing the dynamic system (20), (21), and (22) around the steady state (RaN , θa , θa∗ ),it follows that

  
Ṙ
−δ
 θ̇  =  ΘN
a
Θ∗N
θ̇∗
a



∗
ϕλ′a (θa ) ϕ∗ λ∗′
R − RaNa (θa )  θ − θa  ,
ρ+δ
0
0
ρ+δ
θ∗ − θa∗(A.23)∗N∗∗
where ΘNa and Θa are the values of ∂ θ̇/∂R and ∂ θ̇ /∂R, respectively, evaluated at (R, θ, θ ) =(RaN , θa , θa∗ ). The eigenvalues of the Jacobian matrix in (A.23) are ρ + δ and√′∗N ∗ ∗′{ρ ± ρ2 + 4[(ρ + δ)δ + ΘNa ϕλa + Θa ϕ λa ]}/2. Since the dynamic system has two positive
and one negative eigenvalues, it follows that the steady state is locally saddle-point stable.

A.4

Conditions for positive contribution levels

Consider the case in which the countries are under autarky and cooperatively determine theircontribution levels. Under Assumption 1′ , the interior solutions for the optimal contribution
levels are given by (28). Suppose that only the foreign country make a contribution to theinvestment in the accumulation of the international public good: λa (θw ) = 0 and λ∗a (θw ) > 0.
λa (θw ) = 0 holds if
L<

1 − αγ.
θw ϕ
29(A.24)

Substituting λa (θw ) = 0 and λ∗a (θw ) > 0 into (7), it follows that Ṙ = ϕ∗ L∗ − (1 − αγ)/θw − δR.
Given this and (30), the steady-state solution for θw is calculated as θaw = [(1 − αγ)ρ + (1 +αγ)δ]/[ρ + δ)ϕ∗ L∗ ]. Substituting this into (A.24), the condition under which only the foreigncountry make a contribution in the steady state is given byϕ∗ L∗(1 − αγ)ρ + (1 + αγ)δ
>
> 1.
ϕL
(1 − αγ)(ρ + δ)

(A.25)

Analogously, the condition under which only the home country make a contribution in thesteady state is derived asϕ∗ L∗(1 − αγ)(ρ + δ)
<
< 1.
ϕL
(1 − αγ)ρ + (1 + αγ)δ(A.26)

From (A.25) and (A.26), the condition under which both countries choose positive contribution
levels is(1 − αγ)(ρ + δ)ϕ∗ L∗
(1 − αγ)ρ + (1 + αγ)δ≤≤.
(1 − αγ)ρ + (1 + αγ)δϕL
(1 − αγ)(ρ + δ)

(A.27)

Then, it follows that if the diﬀerence between ϕL and ϕ∗ L∗ is not so large, both countries
choose positive contribution levels in the autarkic steady state under international cooperation,
as illustrated in Figure A.1.

Figure A.1: Conditions for positive contribution levels in the autarkic steady state

In the case where the countries noncooperatively choose their contribution levels, the condition under which both LR and L∗R are positive in the autarkic steady state is derived in a30

similar manner:(1 − αγ)(ρ + δ)ϕ∗ L∗
(1 − αγ)ρ + δ≤≤.
(1 − αγ)ρ + δ
ϕL(1 − αγ)(ρ + δ)

(A.28)

Comparing the above condition with (A.27), it follows that the region in which both LR > 0
and L∗R > 0 holds become narrower in the noncooperative equilibrium.
Acknowledgments The author would like to thank Steven Durlauf, Misuzu Azuma, Makoto
Yano, Koki Oikawa, Takahisa Yokoi, Kentaro Nakajima, Yuichi Furukawa, Nobuhito Suga,
Naoto Jinji, Takeshi Ogawa, Hiroshi Mukunoki, Hajime Kubota, Jota Ishikawa, Kojun Hamada,and seminar participants at Kyoto University, Academia Sinica, Tohoku University, Universityof Bari, and Konan University for their helpful comments on earlier manuscript drafts. The
author is also grateful to the Ministry of Education, Culture, Sports, Science and Technology
for its ﬁnancial support.

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32