GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.107

Trade Structure and Equilibrium Indeterminacy in a
Two-Country Model

Yunfang Hu and Kazuo Mino

December 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Trade Structure and Equilibrium Indeterminacy in a
Two-Country Model∗
Yunfang Hu†and Kazuo Mino‡
December 2009

Abstract
This paper explores a dynamic two-country model with production externalities inwhich capital goods are not traded and international lending and borrowing are allowed.
Unlike the integrated world economy model based on the Heckscher-Ohlin setting, ourmodel yields indeterminacy of equilibrium under a wider set of parameter values than in
the corresponding closed economy model. Our finding demonstrates that the assumption
on trade structure would be a relevant determinant in considering the relation between
globalization and economic volatility.
Keywords: two-country model, non-traded goods, equilibrium indeterminacy, social constant returns
JEL classification: F43, O41∗

We thank Naohito Abe, Kazumi Asako, Costas Azariadis, Volker Böhm, Masaya Sakuragawa, Makoto

Saito and Etsuro Shioji for their helpful comments on earlier versions of this paper. Our research has been
financially supported by the Grant-in-Aid for Scientific Research.
†
Graduate School of International Cultural Studies, Tohoku University, 41 Kawauchi, Aoba-ku, Sendai,
980-8576 Japan‡
Corresponding Author: Institute of Economic Research, Kyoto University, Yoshida Honmachi, Sakyo-ku,
Kyoto, 606-8051 Japan

1

Introduction

Does globalization enhance economic volatility? The equilibrium business cycle theory basedon indeterminacy and sunspots has presented two diﬀerent answers to this question. On
the one hand, authors such as Meng (2003), Meng and Velasco (2003 and 2004) and Weder(2001) show that small-open economies with production externalities produce indeterminacyof equilibrium under a wider set of parameter values than in the corresponding closed economy
model. Hence, according to these studies, internationalization of an economy may increase
economic volatility.1 Nishimura and Shimomura (2002a), on the other hand, reveal that aworld economy consisting of two symmetric countries with production externalities holds thesame stability conditions as those for a closed economy counterpart. In addition, Sim and Ho(2007a) find that if one of the two counties has no production externalities in Nishimura andShimomura’s model, then the equilibrium path of the world economy would be determinate
even though the country with production externalities exhibits autarkic indeterminacy. Thesestudies indicate that opening up international trade does not necessarily enhance economic
fluctuations.
These opposite results seemingly stem from the diﬀerence in the analytical frameworks
used by the foregoing studies. The small-open economy models studied by Meng (2003) andothers are based on the partial equilibrium analysis in which behavior of the rest of the world isexogenously given. In contrast, the models of world economy employ the general equilibrium
approach that treats the world economic system as a closed economy consisting of multiple
countries. The world economy models thus consider more complex interdependency between
the countries than that assumed in the small-open economy models. One may conjecturethat such a diﬀerence would generate the contrasting views as to the destabilizing eﬀect of
globalization.
The purpose of this paper is to reveal that the diﬀerence in conclusions mentioned above
mainly comes from the assumptions on trade structures rather than from the modelling
strategies. To confirm this fact, we modify the model studied by Nishimura and Shimomura
1

Lahiri (2001) also examines indeterminacy in a small-open economy model. Since he uses a somewhat

non-standard framework, the model needs a high degree of external increasing returns to yield indeterminacy.
Yong and Meng (2004) and Zhang (2008) also discuss equilibrium indeterminacy in small-open economies.

2

(2002a) by introducing non-traded goods and international financial transactions. Nishimura
and Shimomura (2002a) use the standard Heckscher-Ohlin framework where both investment
and consumption goods are freely traded but there is no intertemporal trade between the twocountries. We assume that consumption goods are internationally traded but investment
goods are non-tradables. Instead, it is assumed that international lending and borrowing
are possible. Unlike the Heckscher-Ohlin setting, in the presence of non-traded goods, thefactor price equalization fails to hold in our model. As a result, in our modified framework
the factor intensities of production sectors in the home and foreign countries may diﬀer fromeach other. This means that the dynamic behavior of our model out of the steady state will
not be the same as that of a corresponding closed economy. Such a diﬀerence in transition
dynamics generates the divergence of determinacy conditions between the closed economy
and the integrated world economy consisting of symmetric countries.
Our main finding is that the equilibrium determinacy/indeterminacy conditions for theworld economy with non-traded goods and financial transactions are similar to the stability
conditions for the small-open economy models. More specifically, we show that our model
may exhibit indeterminacy regardless of the restrictions on the preference structure. The
closed-economy version of our model, which is essentially the same as the integrated world
economy model of Nishimura and Shimomura (2002a), needs a high elasticity of intertemporal
substitution in consumption to hold indeterminacy. It is to be noted that most of the smallopen economy models with equilibrium indeterminacy assume the presence of international
lending and borrowing.2 Our study, therefore, demonstrates that even though the countries
in the world economy have identical technologies and preferences, the presence of non-traded
final goods and financial capital mobility would generate a divergence in dynamic behaviorof the integrated world economy and a closed, single country. In this sense, the structure ofinternational trade would be a relevant determinant for the relation between globalizationand economic volatility.
The rest of the paper is organized as follows. The next section presents the basic assumptions for the following discussion. Section 3 reformulates the model of Nishimura andShimomura (2002a) as a pseudo-planning problem and summarizes their conclusions. Section
2This is not the case for Nishimura and Shimomura (2002b) who explore the small-country version of the

dynamic Heckscher-Ohlin model.

34 modifies the planning problem in Section 3 in order to consider the presence of non-tradedcapital goods and intertemporal trade. This section displays our main findings. Section 5gives economic implications of our finding and Section 6 presents concluding remarks. Finally, the Appendix discusses the equivalence between the optimal solution of the planningproblem and the competitive equilibrium of our economy.

2

Baseline Setting

Consider a world economy consisting of two countries, home and foreign. Both countries
have the same production technologies. In each country there is a continuum of identical,
infinitely-lived households. All the agents in both countries have an identical time discountrate and the same form of instantaneous felicity functions. The consumption-saving decision
is made by the representative agent whose objective is to select the sequences of consumption
to maximize a discounted sum of utilities over an infinite horizon. We assume that labor
supply is fixed and each household supplies one unit of labor in each moment.
As for the production side of the model, it is assumed that there are two production
sectors in each country. The first sector (i = 1) produces investment goods and the second
sector (i = 2) produces pure consumption goods. The production function of i-th sector in
the home country is specified as
Yi = Ai Kiai Lbi i X̄i ,ai > 0, bi > 0, 0 < ai + bi < 1,

i = 1, 2

where Yi , Ki and Li are i-th sector’s output, capital and labor input, respectively. Here X̄idenotes the sector and country-specific production externalities. We define:
1 −bi,X̄i = K̄iαi −ai L̄1−αiαi > ai ,1 − αi > bi

i = 1, 2.

If we normalizes the number of producers to one, then it holds that K̄i = Ki and L̄i =
Li (i = 1, 2) in equilibrium.3 This means that the i-th sector’s social production technologies
3

As shown by Mino (2001), the main argument of this paper holds for a more general production function

specified asYi = f i (Ki , Li ) E i K̄i , L̄i ,i = 1, 2,

where function f i (.) is homogeneous of degree γ ∈ (0, 1) in Ki and Li , while function E i (.) is homogeneousof degree 1 − γ.

4that internalize the external eﬀects are:i,Yi = Ai Kiαi L1−αii = 1, 2.

Hence, the social technology satisfies constant returns to scale, while the private technology
exhibits decreasing returns to scale.4
We also assume that capital and labor are perfectly shiftable between the production
sectors within a country, but they cannot move across the border. Therefore, the fullemployment conditions for production factors are given by
K = K1 + K2 ,1 = L1 + L2 ,where the total labor force is assumed to be unity.
As was assumed, the foreign country has the same production technologies as those of thehome country. It is also assumed that the labor force in the foreign country is normalized
to unity as well. Thus the home and foreign countries diﬀer only in their initial holdings ofcapital stocks.

3

A Dynamic Heckscher-Ohlin Model

To emphasize the role of trade structure in dynamic world economy models, we first summarize the dynamic properties of the Hecksher-Ohlin model of the integrated world economy
with sector as well as country specific production externalities. For this purpose, we consider
a pseudo-planning problem whose solution mimics the competitive equilibrium of the world
economy. This approach simplifies model manipulation and helps to clarify the diﬀerence
between the Heckscher-Ohlin setting and our model with non-traded goods and international
financial transactions. The market economy version of the model in this section is discussed
in detail by Nishimura and Shimomura (2002a) and Sim and Ho (2007a).5
4

Since the private technologies exhibit decreasing returns to scale, there exist positive profits in bothproduction sectors. According to Benhabib and Nishimura (1998), we implicitly assume that there is an entry
barrier in each industry to generate positive profits in each production sector.
5
Nishimura and Shimomura’s study is based on the dynamic Hechscher-Ohlin models examined by, forexample, Chen (1992) and Stiglitz (1970).

5

3.1

A Pseudo-Planning Problem

In the standard Heckscher-Ohlin framework, it is assumed that both consumption and investment goods are tradables, but international lending and borrowing are impossible. In
this setting, the representative agent in the home country solves the following problem:
Z ∞C 1−σ − 1dt, σ > 0, ρ > 0
e−ρt
max
1−σ
0
subject toK̇ = Y1 + p̂Y2 − p̂C − δK, K0 = given (> 0) ,where C is consumption, p̂ denotes the world price of consumption good in terms of the investment good and δ ∈ (0, 1) is the rate of capital depreciation. Similarly, the the representative
household of the foreign country solves
Z ∞C ∗1−σ − 1
dt
e−ρt
max
1−σ
0
subject toK̇ ∗ = Y1∗ + p̂Y2∗ − p̂C ∗ − δK ∗ , K0∗ = given (> 0) ,where asterisks denote corresponding foreign variables. The world market equilibrium conditions for investment and consumption goods are respectively given byK̇ + K̇ ∗ = Y1 + Y1∗ + δK + δK ∗ ,(1)

C + C ∗ = Y2 + Y2∗ .

(2)

In the pseudo-planning formulation that corresponds to the market economy describedabove, the planner is assumed to solve the following problem:¸
Z ∞ ∙ 1−σ
∗1−σ − 1C−1∗C+μe−ρt dt
max
1
−
σ
1
−
σ
0
subject to∗1
K̇w = A1 K1a1 Lb11 X̄1 + A1 K1∗a1 L∗b
1 X̄1 − δKw ,

(3)

∗
2
C + C ∗ = A2 K2a2 Lb22 X̄2 + A2 K2∗a2 L∗b
2 X̄2 ,

(4)

K = K1 + K2 ,K ∗ = K1∗ + K2∗ ,(5)

1 = L1 + L2 ,

1 = L∗1 + L∗2 ,(6)

6

Kw = K + K ∗ ,(7)

together with the given initial levels of capital stocks, K0 and K0∗ . Here, Kw stands forthe aggregate capital stock in the world economy at large. In addition, μ∗ in the objectivefunction denotes a constant welfare weight on the instantaneous felicity of the foreign agents
relative to the felicity in the home country. This value should be selected to make theplanning solution equivalent to the competitive equilibrium. Constraints (3) and (4) are theequilibrium conditions for investment and consumption goods, respectively. Equations (5)and (6) represent the resource constraints in each country. Following Kehoe et al. (1992),
we assume that in solving this problem the planner takes the sequences of external eﬀects,
ª∞
©
ª∞
©
X̄i (t) t=0 and X̄i∗ (t) t=0 (i = 1, 2) , as given.
In what follows, we focus on an interior solution. Set up the current value Hamiltonian

function:H =C ∗1−σ − 1
C 1−σ − 1∗1
+ μ∗
+ q(A1 K1a1 Lb11 X̄1 + A1 K1∗a1 L∗b
1 X̄1 − δKw )1 h− σ1−σ+λ A2 (K − K1 )a2 (1 − L1 )b2 X̄2 + A2 (K ∗ − K1∗ )a2 (1 − L∗1 )b2 X̄2∗
−C − C ∗ ] + φ (Kw − K − K ∗ ) .

In the above, q denotes the implicit price the aggregate capital, Kw , and λ and φ are Lagrangian multipliers. It is easy to see that q/λ corresponds to 1/p̂, that is, the price ofinvestment good in terms of consumption good in the decentralized world economy.6 The
necessary conditions for an optimum include the following:
C −σ = λ,μ∗ C ∗−σ = λ,qa1 A1 K1a1 −1 Lb11 X̄1 − λa2 A2 K2a2 −1 Lb22 X̄2 = 0,

(9)

qb1 A1 K1a1 Lb11 −1 X̄1 − λb2 A2 K2a L2b2 −1 X̄2 = 0,(10)

∗a2 −1 ∗b2 ∗∗1
L2 X̄2 = 0,
qa1 A1 K1∗a1 −1 L∗b
1 X̄1 − λa2 A2 K2

(11)

1 −1
2 −1
qb1 A1 K1∗a1 L∗b
X̄1∗ − λb2 A2 K2∗a2 L∗b
X̄2∗ = 0,
1
2

(12)

λa2 A2 K2 a2 −1 Lb22 X̄2 − φ = 0,
6

(8)

∗
2
λa2 A2 K2∗a2 −1 L∗b
2 X̄2 − φ = 0,

(13)

Notice that λ equals the marginal utility of consumption and q equals the marginal value of capital interms of utility. Therefore, q/λ denotes the value of investment good in terms of consumption good.

7q̇ = q (ρ + δ) − φ,(14)

lim qe−ρt Kw = 0.

(15)

t→∞

Equations (8) through (13) display the first-order conditions for maximizing the Hamiltonian
function with respect to the control variables, C, C ∗ , K1 , L1 , K1∗ , L∗1 , K and K ∗ under given
levels of X̄i and X̄i∗ (i = 1, 2) . Equation (14) is the canonical equation of the costate variable
for the aggregate capital, Kw , and (15) is the transversality condition.

3.2

Equilibrium Dynamics of the Integrated Economy

First, by use of (9) through (12) , we obtain the following relations:µµ¶¶
a2 b1 K1
a2 b1 K1∗
K2
K2∗
=
,
=
.
L2
a1 b2 L1
L∗2
a1 b2 L∗1

(16)

From the equilibrium conditions, K̄i = Ki , L̄i = Li , K̄i∗ = Ki∗ ,and L̄∗i = L∗i , we find that (9)and (11) present:12= λa2 A2 K2α2 −1 L1−α,
qa1 A1 K1α1 −1 L1−α
1
2

(17)

1
2
= λa2 A2 K2∗α2 −1 L∗1−α.
qa1 A1 K1∗α1 −1 L∗1−α
1
2

(18)

Using (16) , (17) and (18) , we obtain
q
λ

=

A2
A1

=

A2
A1

µ
µ

a2
a1
a2
a1

¶α2 µ
¶α2 µ

b2
b1
b2
b1

¶α2 −1 µ¶α2 −1 µ

K1
L1
K1∗
L∗1

¶α2 −α1¶α2 −α1.

(19)

As shown by the above conditions, because of the symmetry of the two countries, the factor
intensities of the social technology in both countries are the same: Ki /Li = Ki∗ /L∗i (i = 1, 2) .
Denoting q/λ ≡ p, from (19) we can express the capital intensities in the following manner:
Ki /Li = Ki∗ /L∗i = ki (p) ,i = 1, 2.

The full-employment conditions in each country (5) and (6) are respectively summarizedas
L1 k1 (p) + (1 − L1 ) k2 (p) = K,L∗1 k1 (p) + (1 − L∗1 ) k2 (p) = K ∗ .
8

In view of these full-employment conditions, we may express the social level of investment
good output in each country as follows:
Y1 = L1 A1 k1 (p)α1 =K − k2 (p)A1 k1 (p)α1 ,
k1 (p) − k2 (p)(20)

Y1∗ = L∗1 A1 k1 (p)α1 =K ∗ − k2 (p)A1 k1 (p)α1 .
k1 (p) − k2 (p)(21)

From (1) , (20) and (21), we see that the dynamic equation for the aggregate capital of theworld economy is given byK̇w =Kw − 2k2 (p)A1 k1 (p)α1 − δKw .
k1 (p) − k2 (p)(22)

Equations (9) , (13) and (14) yield the dynamic behavior of the shadow value of Kw :ihq̇ = q ρ + δ − a1 A1 k1 (p)α1 −1 .

(23)Equations in (8) mean that C ∗ /C = μ∗−1/σ ≡ m̄ for all t ≥ 0. Since the households in bothcounties have an identical form of homothetic utility function, the relative level of optimal
consumption stays constant over time. Thus, considering that Y2 = (1 − L1 ) A2 k2α2 andY2∗ = (1 − L∗1 ) A2 k2α , the world market equilibrium condition for consumption goods (4) isexpressed as
1(1 + m̄) λ− σ =2k1 (p) − Kw
A2 k2 (p)α2 .
k1 (p) − k2 (p)(24)

This equation shows that the equilibrium level of λ can be expressed as λ = λ (Kw , p; m̄) .
As a result, wep=q≡ π (Kw , q; m̄) .
λ (Kw , p, m̄)(25)

Plugging (25) into (22) and (23) yields a complete dynamics system of the integrated world
economy with respect to Kw and q.
Inspecting dynamic system (22) and (23) , Nishimura and Shimomura (2002a) confirm
that the steady state of the world economy where both countries imperfectly specialize isuniquely given under weak restrictions on parameter values. Then they show the following
proposition:

9

Proposition 1 (Nishimura and Shimomura 2002a) The steady-state equilibrium of the world
is locally indeterminate if (i) the investment good sector is more capital intensive than the
consumption good sector from the social perspective but it is less capital intensive from theprivate perspective, and (ii) the elasticity of intertemproal substitution in consumption, 1/σ,
is suﬃciently high.7
Given the conditions shown above, the steady state of the aggregate dynamic system is asink so that there is a continuum of converging paths towards the steady-state equilibrium.
Either if the social and private factor intensity rankings are the same or if the elasticity ofintertemporal substitution in consumption is low enough, the dynamic system of the integrated world economy exhibits saddlepoint stability and, hence, the competitive equilibrium
is at least locally determinate. As pointed out by Sim and Ho (2007b), the Heckscher-Ohlinmodel of two symmetric countries with constant-returns-to-scale technologies and homothetic
preferences has the same dynamic properties as those of the corresponding closed economy.
Therefore, the suﬃcient conditions for holding equilibrium indeterminacy shown above are
essentially the same conditions for the closed economy with sector-specific externalities andsocial constant returns examined by Benhabib and Nishimura (1998).8 This result demonstrates that in the standard Heckscher-Ohlin world with symmetric countries, opening up
international trade neither enhances nor diminishes economic volatility of each country.
When we consider the distributional dynamics between the two countries, it should be
noted that the equilibrium trajectory of the world economy depends on m̄: see equation (25) .
Nishimura and Shimomura (2002a) show that if the competitive equilibrium is indeterminate,¢¡
the value of m̄ = μ∗−1/σ cannot be pinned down by the initial distribution of capital stocks,
K0 and K0∗ , alone. In the dynamic Heckscher-Ohlin world, the steady-state levels of K and
7

More precisely, σ should satisfy(1 − α1 )a2 b1 (ρ + δ) + α1 a1 [ρb2 + δb1 a2 + (1 − a1 )b2 δ]1> max 1,σ
(a2 b1 − a1 b2 ) (α1 − α2 ) [ρ + δ(1 − a1 )]to establish local indeterminacy in the steady-state equilibrium.
8
In discussing two-sector closed economy model, Benhabib and Nishimura (1998) assume that the instantaneous utility function is linear in consumption (i.e. σ = 0) . Hence, their model exhibits indeterminacy if
condition (i) in Proposition 1 is satisfied. In the two-sector endogenous growth model with physical and humancapital, condition (i) in Proposition 1 is suﬃcient for establishing indeterminacy: see Benhabib et al. (2000)
and Mino (2001).

10K ∗ are path dependent and they are determined by the initial values of K0 and K0∗ , if theconverging path is determinate. If indeterminacy holds, then the level of m̄ is indeterminate
as well, and thus the terminal distribution of capital stocks in the steady state equilibrium isalso indeterminate. As a result, the steady-state levels of relative factor endowment (so thesteady-state patterns of international trade) may be aﬀected by sunspot-driven fluctuations.9

4

A Model with Non-Tradable Capital

We now assume that consumption goods are internationally traded but investment goods are
non-tradables. Instead, we assume that international lending and borrowing are allowed. In
our modelling, the international transaction of financial asset means that households in both
home and foreign countries can trade ownership of their capital stocks, while neither installedphysical capital nor final goods for new investment can cross the border.10 Although suchan assumption is restrictive one, it helps to elucidate the eﬀect of the presence of non-traded
goods in comparison with the case of free trade of final goods in the Heckscher-Ohlin model
discussed in the previous section. Additionally, since a large portion of investment goods
includes construction and structures, the investment goods sector shares a larger part ofnontradables than the consumption good sector.11

4.1

Decentralized Economy

Suppose that both home and foreign countries produce investment as well as pure consumption goods. Consumption goods produced in each country is homogeneous and they areinternationally traded. Investment goods are traded in the domestic market alone. Althoughinstalled physical capital are not shiftable internationally, households in each country can
9

See also Nishimura and Shimomura (2006) for further investigation on equilibrium indeterminacy in the

dynamic Heckscher-Ohlin model.
10
The structure of our model is one of the dependent economy models discussed in open-economy macroeconomics literature. Sen and Turnovsky (1995) treat a small-open economy model with non-tradable capitaland Turnovsky (1996, Chapter 7) studies a neoclassical two-country, two-sector model in which capital goods
are not traded. See also Chapter 5 in Turnousky (2009) for a brief review of dependent economy models.
11
Bems (2008) finds that the share of investment expenditure on non-traded goods is about 60%. and that
this figure has been considerably stable over the last 50 years both in developed and developing countries.

11

own capital stock in the other country.12 Let us denote the capital stocks domiciled in thehome (resp. foreign) country owned by the domestic and foreign households by Kh and Kf(resp. Kh∗ and Kf∗ ). Then it holds that
K = Kh + Kf ,K ∗ = Kh∗ + Kf∗ .
Thus the net foreign asset position, i.e. the net stock of traded bonds, held by the home andforeign households are respectively defined as
B = p∗ Kh∗ − pKf ,B ∗ = −B = pKf − p∗ Kh∗
where p and p∗ respectively denote price of investment good in terms of the consumption
goods in the home and foreign countries. (It is to be noted that since the investment goods
are nontradables, the price of investment goods in the home country, p, may not be thesame as p∗ determined in the foreign country.) Here, B and B ∗ are measured in terms of the(homogeneous) consumption goods. Thus the net wealth (in terms of consumption goods)
held by the home and foreign households are given byΩ = pK + B,Ω∗ = p∗ K ∗ + B ∗Since B and B ∗ are measured by consumption goods, the non-arbitrage conditions between capital and bond in each country are given by
r+

ṗ∗
ṗ
= R = r∗ + ∗ ,
p
p

(26)

where R is the interest rate on bonds, and r and r∗ respectively denote denote the net rateof return to capital in the home and foreign countries.
Given the above setting, the optimization problem for the representative household in thehome country is described by the following:Z ∞ 1−σC− 1 −ρt
e dt
max
1−σ
0
12

In this paper we assume a simple form of asset structure. For the relevance of asset structure of foreign

trade in the real business cycle studies, see Baxter and Crucini (1995).

12

subject to the flow budget constraintΩ̇ = RΩ + w + π 1 + π 2 − C,

(27)

µ Z ∞
¶
Rs ds Ωt ≥ 0
lim exp −(28)

and the no-Ponzi-game schemet→∞ttogether with the initial capital holdings of Ω0 . In the above, wt the real wage in terms of
consumption goods, and π 1 and π 2 are the excess profits generated by the investment andconsumption goods production.13
Similarly, the foreign households solve the following problem:
Z ∞ ∗1−σC− 1 −ρt
e dt
max
1
−
σ
0
subject toΩ̇∗ = RΩ∗ + w∗ + π ∗1 + π ∗2 − C ∗ ,µ Z ∞¶lim exp −Rs ds Ω∗t ≥ 0
t→∞

(29)

t

and the initial conditions. The market equilibrium conditions for investment goods in thehome and foreign countries are
I = Y1 ,I ∗ = Y1∗ .

(30)

The international market equilibrium condition for consumption goods and bonds are respectively given by
C + C ∗ = Y2 + Y2∗ ,(31)

B + B ∗ = 0.

(32)

In addition, as shown in the Appendix, the flow budget constraint for the households andthe market equilibrium conditions for investment goods, the change in net asset position, i.e.
the current account, of each country is
Ḃ = RB + Y2 − C,
13

(33a)

It is assumed that the profits earned by the firms are distributed back to the households. Since we haveassumed that a part of domestic capital may be owned by the foreign households, it is rather arbitrary toassume that the profits of domestic industries are entirely owned by the domestic households. However, thepattern of profit distribution does not aﬀect the optimal consumption/savings decision of the households, sothat we ignore the relation between the ownership of capital and international profit distribution.

13

Ḃ ∗ = RB ∗ + Y2∗ − C ∗ .

(33b)

The lending-borrowing relation between the two countries also imposes the following nonPonzi-game conditions:µ Z ∞¶Rs ds Bt ≥ 0,
lim exp −

(33c)

µ Z ∞
¶
lim exp −Rs ds Bt∗ ≥ 0.

(34a)

t→∞

t→∞

t

t

Finally, the production side of the economy is the same as that of the Hechscher-Ohlinmodel in the previous section. Profit maximization of both sectors equates the private marginal productivity of each factor and the factor prices, so that we obtain the following conditions:r = pa1 A1 k1α1 −1 − δ = a2 A2 k2α2 −1 − δ,(35a)w = pb1 A1 k1α1 = b2 A2 k2α2 ,(35b)r∗ = p∗ a1 A1 k1∗α1 −1 − δ = a2 A2 k2∗α2 −1 − δ,(35c)w∗ = p∗ b1 A1 k1∗α1 = b2 A2 k2∗α2(35d)

Again, we assume that both factor inputs are not traded so that the full employment conditions in both countries are:4.2

K = K1 + K2 ,K ∗ = K1∗ + K2∗

1 = L1 + L2 ,

1 = L∗1 + L∗2 .

A Pseudo-Planning Problem

As shown in the Appendix of the main text, the competitive equilibrium of the world economy can be characterized by the solution of the following pseudo-planning problem. In thisproblem the planner is assumed to solve the following:¸
Z ∞ ∙ 1−σ
∗1−σ − 1−1C∗C+μe−ρt dt
max
1−σ
1−σ
0
subject toK̇ = A1 K1a1 Lb11 X̄1 − δK,
14

∗
∗
1
K̇ ∗ = A1 K1∗a1 L∗b
1 X̄1 − δK∗
2
C + C ∗ = A2 K2a2 Lb22 X̄2 + A2 K2∗a2 L∗b
2 X̄2 ,

K ∗ = K1∗ + K2∗ ,K = K1 + K2 ,1 = L∗1 + L∗2 ,

1 = L1 + L2 ,as well as to the initial levels of K0 and K0∗ . The diﬀerence between the planning problem
given above and one discussed in the previous section is that in the present regime eachcountry has its own capital accumulation equation due to the assumption that investment
goods are not internationally traded.
The current-value Hamiltonian function is given byH =C ∗1−σ − 1
C 1−σ − 1
+ μ∗
1−σ
1−σ

³
´
∗
∗
1
−
δK
X̄
+q(A1 K1a1 Lb11 X̄1 − δK) + q ∗ A1 K1∗a1 L∗b1
1hi+λ A2 (K − K1 )a2 (1 − L1 )b2 X̄2 + A2 (K ∗ − K1∗ )a2 (1 − L∗1 )b2 X̄2∗ − C − C ∗ ,where q and q ∗ are the shadow values of capital stock in the home and foreign country,
respectively. In what follows, we focus on the interior solution in which both countries
imperfectly specialize in producing consumption and investment goods. The control variablesin this problem are C, C ∗ , K1 , L1 , K1∗ and L∗1 , while the state variables are K and K ∗ . In
parallel with the optimization in the previous section, we find that the necessary conditions
for au optimum include the following :C −σ = λ,μ∗ C ∗−σ = λ,(36)

qa1 A1 K1a1 −1 Lb11 X̄1 − λa2 A2 K2a2 −1 Lb22 X̄2 = 0,(37)

q ∗ b1 A1 K1a1 L1b1 −1 X̄1 − λb2 A2 K2a Lb22 −1 X̄2 = 0,(38)

∗a2 −1 ∗b2 ∗∗1
L2 X̄2 = 0,
qa1 A1 K1∗a1 −1 L∗b
1 X̄1 − λa2 A2 K2

(39)

1 −1
2 −1
q ∗ b1 A1 K1∗a1 L∗b
X̄1∗ − λb2 A2 K2∗a L∗b
X̄2∗ = 0,
1
2

(40)

q̇ = q (ρ + δ) − λa2 A2 K2a2 −1 Lb22 X̄2 ,

(41)

∗
2
q̇ ∗ = q ∗ (ρ + δ) − λa2 A2 K2∗a2 −1 L∗b
2 X̄2 ,

(42)

lim e−ρt qK = 0,

(43)

t→∞

lim e−ρt q ∗ K ∗ = 0.

t→∞

15

4.3

Dynamic System

Again, we define q/λ ≡ p and q ∗ /λ ≡ p∗ , which represent the prices of consumption goods interms of investment goods in the home and foreign countries, respectively. Then we replace(19) in the Heckscher-Ohlin model with the following:µ ¶ µ ¶µ ¶A2 a2 α2 b2 1−α2 K1 α2 −α1
= p,
A1 a1
b1
L1
µ ¶ µ ¶µ¶
A2 a2 α2 b2 1−α2 K1∗ α2 −α1= p∗ .
A1 a1
b1
L∗1
These conditions, together with (16) , yield the following:

K1∗
=
L∗1

¶

µ

A1
A2

µ

¶

K1
=
L1

A1
A2

µ

a1
a2

µ

¶

1
α2 −α11α2 −α1

a1
a2

¶

µ

b1
b2

¶ α2 −1p α2 −α1 ≡ k1 (p) ,µ¶ α2 −1p α2 −α1 ≡ k1 (p∗ ) .

α2α2 −α1α2α2 −α1b1b2

α1 −α2α1 −α2

1

∗

(44a)

1

(44b)

Hence, from (16) the capital intensity in the consumption good sectors are given by:
µ

A1
A2

µ

¶

K2
=
L2
K2∗
=
L∗2

A1
A2

¶

µ

a1
a2

µ

¶

1
α2 −α11α2 −α1

a1
a2

¶

µ

b1
b2

¶ α1 −1p α2 −α1 ≡ k2 (p) ,µ¶ α1 −1p α2 −α1 ≡ k2 (p∗ ) .

α1α2 −α1α1α2 −α1b1b2

α1 −α2α1 −α2

1∗

1

These expressions show thatsign ki0 (p) = sign ki0 (p∗ ) = sign (α2 − α1 ) , i = 1, 2.

(45)

Here, the sign of∆p = α1 − α2represents the factor intensity ranking from the social perspective. When ∆p is positive(negative), the aggregate technology of investment good sector is more (less) capital intensive
than that of the consumption good sector.
Note that we have restricted our attention to the interior equilibrium in which bothcountries imperfectly specialize in producing consumption and investment goods. To ensure
this restriction, we assume that relative price in each country satisfies the following condition:
0 < L1 =K − k2 (p)< 1,
k1 (p) − k2 (p)
16

(46a)

0 < L∗1 =K ∗ − k2 (p∗ )< 1.
k1 (p∗ ) − k2 (p∗ )(46b)Using functions k1 (p) and k2 (p) . we see that capital accumulation equation in each country
is written as
K̇ = y1 (K, p) − δK,(47)

K̇ ∗ = y1 (K ∗ , p∗ ) − δK ∗ ,

(48)

where
y 1 (K, p) ≡K − k2 (p)A1 k1 (p)α1 ,
k1 (p) − k2 (p)K ∗ − k2 (p∗ )A1 k1 (p∗ )α1 .
∗∗
k1 (p ) − k2 (p )y 1 (K ∗ , p∗ ) ≡It is easy to see that these supply functions of investment goods satisfy:
µ
¶
a1 a2
1
1
∗ ∗
−
,
sign yK (K, p) = sign yK ∗ (K , p ) = sign
b1
b2
¶
µ
a1 a2
1
1
∗ ∗
sing yp (K, p) = sing yp∗ (K , p ) = sign
(α1 − α2 )
−
b1
b2

(49)
(50)

(51a)

(51b)

Notice that the sign of
∆s =

a1 a2
−
b1
b2

shows the factor intensity ranking from the private perspective.
The shadow values of capital in both countries change according toq̇ = q[ρ + δ − r (p)],(52)

q̇ ∗ = q ∗ [ρ + δ − r (p∗ )] ,(53)

where r (p) ≡ a1 A1 k1 (p)α1 −1 and r (p∗ ) ≡ a1 A1 k1 (p∗ )α1 −1 . Dynamic equations (47) , (48) , (52)and (53) depict behaviors of capital stocks and implicit prices of capital in the home and foreign countries.
To derive a complete dynamic system, we should relate p and p∗ to K, K ∗ , q and q∗ . The
world market equilibrium condition for the consumption good in the Heckscher-Ohlin world(equation (24)) is now replaced with1

(1 + m̄) λ− σ = y 2 (K, p) + y 2 (K ∗ , p∗ ) ,

17

(54)

where m̄ = μ∗−1/σ andy 2 (K, p) =y 2 (K ∗ , p∗ ) =k1 (p) − K
A2 k2 (p)α2 ,k1 (p) − k2 (p)k1 (p∗ ) − K ∗A2 k2 (p∗ )α2 .
k1 (p∗ ) − k2 (p∗ )The supply functions of consumption goods satisfy the following:
µ
¶
a1 a2
2
2
∗ ∗
−
,
sign yK (K, p) = sign yK ∗ (K , p ) = −sign
b1
b2
¶
µ
a1 a2
sign yp2 (K, p) = sign yp2∗ (K ∗ , p∗ ) = −sign(α1 − α2 ) .
−
b1
b2

(55)
(56)

(57a)

(57b)

In view of (54) , we see that λ is expressed as a function of capital stocks, prices and m̄ :λ = (1 + m̄)σ [y2 (K, p) + y 2 (K ∗ , p∗ )]σ≡ λ (K, K ∗ , p, p∗ ; m̄) .

(58)

Thus by the definitions of p and p we obtainp=q,
λ (K, K ∗ , p, p∗ ; m̄)

p∗ =

q∗
.
λ (K, K ∗ , p, p∗ ; m̄)Solving these equations with respect to p and p∗ yields the following expressions:p = π (K, K ∗ , q, q∗ ; m̄) ,(59)p∗ = π ∗ (K, K ∗ , q, q∗ ; m̄) .

(60)

Substituting (59) and (60) into (47) , (48) , (52) and (53) , we obtain a complete dynamic
system that depicts the behaviors of K, K ∗ , q and q∗ .

4.4

Equilibrium IndeterminacyFirst, let us characterize the stationary equilibrium of the world economy. The steady stateof the dynamic system derived above is established when K̇ = K̇ ∗ = q̇ = q̇ ∗ = 0. From (59)and (60) the relative price in the home and foreign countries, p and p∗ , also stay constant in
the steady-state equilibrium. As for the existence of a feasible steady state, we can confirm
the following:18

Proposition 2 There exists a unique steady state in which both countries imperfectly specialize.
Proof. When q̇ = q̇ ∗ = 0 in (52) and (53) , it holds that
a1 A1 k1 (p)α1 −1 = a1 A1 k1 (p∗ )α1 −1 = ρ + δ.
Thus by use of (44a) and (44b) , we find that
∗

p=p =

µ

A2
A1

¶µ

a2
a1

¶α2 µ

b2
b1

¶1−α2 µρ+δa1 A1

¶ α2 −α1
α1 −1.

Thus the steady-state levels of p and p∗ are uniquely given and it holds that p = p∗ in thesteady state. The steady-state levels of capital stocks satisfying K̇ = K̇ ∗ = 0 in (47) and(48) are determined by the following conditions:K − k2 (p)A1 k1 (p)α1 = δK,
k1 (p) − k2 (p)K ∗ − k2 (p∗ )A1 k1 (p∗ )α1 = δK ∗k1 (p∗ ) − k2 (p∗ )Using the conditions for ṗ = ṗ∗ = 0 and the fact that p = p∗ holds in the steady state, wefind that the steady-state level of capital stock in each county has the same value, which is
given byα1

1

(aA1 ) 1−α1 (ρ + δ) α1 −1
´
³
K=K =
ρ + δ 1 − δ + ab22b1
∗

µ

a2 b1
a1 b2

¶

,

which has a positive value. In view of the steady-state levels of p and K derived above, thesteady-state values of labor allocation to the investment good sector are:´³
a1 δ aa21 bb12³
´ ∈ (0, 1) .
L1 = L∗1 =ρ + (1 − a1 )δ + a1 δ aa21 bb12

Hence, (46a) and (46b) are fulfilled so that both countries imperfectly specialize. In addition,
when p, p∗ , K and K ∗ are given, from (54) the steady-state value of λ is uniquely determined
as well, implying that q = pλ and q ∗ = p∗ λ are also uniquely given in the steady state
equilibrium.
In order to inspect local stability of the steady state, the following facts are useful:19Lemma 1

In the symmetric steady state where K = K ∗ and q = q∗ , it holds the following

relations:ii∗ ∗(K, p) = yK
yK
∗ (K , p ) ,i = 1, 2,

ypi (K, p) = ypi ∗ (K ∗ , p∗ ) ,i = 1, 2,

π K (K, K ∗ , q, q ∗ ) = π ∗K (K, K ∗ , q, q∗ ) = π K ∗ (K, K ∗ , q, q ∗ ) = π ∗K ∗ (K, K ∗ , q, q ∗ ) ,π q (K, K ∗ , q, q∗ ) = π ∗q∗ (K, K ∗ , q, q ∗ ) ,π q∗ (K, K ∗ , q, q ∗ ) = π ∗q (K, K ∗ , q, q∗ ) .
Proof. By the functional forms of yji (·) (i = 1, 2, j = K, K ∗ , p, p∗ ), it is easy to seei (K, p) = y i (K ∗ , p∗ ) and y i (K, p) = y i (K ∗ , p∗ ) are established when p = p∗ andthat yKpp∗
K∗

K = K ∗ .As for the rest of the results, we may use pλ (·) = q and p∗ λ (·) = q∗ to drive thefollowing:λK∂p= πK =,∂K
λ + pλP∂p
λK ∗= πH ∗ =,∗∂K
λ + pλP

(61a)

λK
∂p∗
= π ∗K =,∂K
λ + p∗ λP ∗

∂p∗
λK ∗
= π ∗K ∗ =,∗∂K
λ + p∗ λP ∗

(61b)

pλp
∂p
,
= πq∗ = −∂q ∗λ(λ + 2pλp )(61c)

λ + pλp∂p= πq =,∂qλ(λ + 2pλp )p∗ λp∗∂p∗= π ∗q = −,∂q
λ(λ + 2p∗ λp∗ )λ + p∗ λp∗
∂p∗
∗
.
=
π
∗ =
q
∂q ∗
λ(λ + 2p∗ λp∗ )(61d)Since λK (·) = λK ∗ (·) and λp (·) = λp∗ (·) in the steady state where K = K ∗ and p = p∗ , weobtain π K = π ∗K = π K ∗ = π ∗ , π q = π ∗q∗ and π q∗ = π ∗q .
We now inspect the dynamic behavior of our economy. As for local determinacy of thesteady state, we find the following:
Proposition 3 The steady-state equilibrium in the model with non-tradable capital is locallyindeterminate, if the investment good sector is more capital intensive than the consumption
good sector from the social perspective but it is less capital intensive from the private perspective.

20

Proof. Let us linealize the dynamic system of (47) , (48) , (52) and (53) at the steadystate. The coeﬃcient matrix of the linealized system is given by⎤⎡
1 − δ + y1π
1π ∗
1π
1π ∗
yK
y
y
y
q
q
K
K
p
p
p
p
⎥
⎢
⎥
⎢
⎥
⎢
1 − δ + y1 π∗ ∗
1 π∗
1 π∗
⎢
yp1∗ π ∗K
yK
y
y
∗
p∗ K
p∗ q
p∗ q ∗ ⎥
⎥
⎢
J =⎢
⎥.
⎥
⎢
0
0
0
0
⎢
−qr π K−qr π K ∗−qr π q −qr π q∗ ⎥
⎥
⎢
⎦
⎣
0
∗
0
∗
0
∗
0
∗
−qr π K−qr π K ∗−qr π q −qr π q∗
In view of Lemma 1, the characteristic equation of J is written asΓ (η) = det [ηI − J]
⎡
⎤
1
1
1
1
1
η − (yK − δ + yp π K )−yp π K−yp π q−yp π q∗
⎢
⎥
⎢
⎥
⎢
⎥
1 − δ + y1 π )
1π ∗
1π
⎢
⎥
−yp1 π K
η − (yK
−y
−y
p K
p q
p q
⎢
⎥
= det ⎢
⎥
⎢
⎥
0
0
0
0
⎢
qr π Kqr π K
η + qr π q
qr π q∗ ⎥⎢⎥⎣⎦qr0 π Kqr0 π Kqr0 π q
η + qr0 π q
⎡
⎤
¡ 1
¢
η − yK − δ
0
η
0
⎢
⎥
⎢
⎥
⎢
⎥
1 − δ)⎢⎥0η − (yK0η⎢⎥= det ⎢
⎥
⎢
⎥
0
0
0
0
⎢
qr π Kqr π K
η + qr π q
qr π q∗ ⎥⎢⎥⎣⎦qr0 π Kqr0 π Kqr0 π q∗
η + qr0 π q
£
¡ 1
¢¤ £
¤
= η − yK− δ η + qr0 (π q − π q∗ ) ξ (η) .

where η denotes the characteristic root of J and

£
¡ 1
¢
¢
¤
¡ 1
− δ − 2yp1 π K η − qr0 yK− δ (π q + π q∗ ) .
ξ (η) ≡ η2 + qr0 (π q + π q∗ ) − yK

Our assumptions mean that ab11 − ab22 < 0 and α1 − α2 > 0. Thus from (57a) we see that1 − δ < 0. In addition, note that from (61c) it holds that π − π ∗ = 1/λ (> 0) . Hence,yKqqusing r (p) ≡ a1 A1 k1 (p)α1 −1 , we obtain:r0 (π q − π q∗ ) = a1 (a1 − 1) A1 (k1 (p))a1 −2

k10 (p)> 0.
λAs a consequence, at least two roots of Γ (η) = 0 have negative real parts. Equations in (61c)also show
π q + π q∗ =1,
λ + 2pλp21

where¤− 1
1 £∂(1 + m̄) σ y2 (K, p) + y2 (K ∗ , p∗ ) σ∂p¤− 1 −1
yp2
1 £= − (1 + m̄) σ y2 (K, p) + y2 (K ∗ , p∗ ) σ < 0.
σ

λp =Therefore, in the steady state equilibrium. the following holds:"#
pyp2 (K, p)1
λ + 2pλp =σ− 2σy (K, p)

Notice that under our assumptions, it holds that yp2 (K, p) > 0. Suppose that σ is smallenough to satisfy σ < pyp2 /y2 . Then λp + 2pλp > 0 so that π q + π q∗ < 0, which leads to¢¡ 1− δ (π q + π q∗ ) < 0.
−qr0 yK

This means that ξ (η) = 0 has one positive and one negative roots. As a result, Γ (η) = 0
has three stable roots. Hence, if σ is smaller than the price elasticity of supply function of
consumption goods, then there locally exists a continuum of equilibrium paths converging to
the steady state.
Now suppose that σ is larger than pyp2 /y 2 . Then we obtain π q + π q∗ > 0. Furthermore, itholds that
−2yp1 π K¶
pλK=
λ + 2pλp"#σ−1 ¡¢−σ−1 −1
2pyp1
(1
+
m̄)
2
= −
yK
> 0,
2y 2
λ + 2pλp
σ
−2yp1

µ
−

2 > 0 under our assumptions. Consequently, the following inequalities
because yp1 < 0 and yK

are established:¡ 1¢− δ (π q + π q∗ ) > 0,
−qr0 yK¡ 1¢1− δ − 2yK
π K > 0.
qr0 (π q + π q∗ ) − yK

These conditions mean that ξ (η) = 0 has two roots with negative real parts and, hence, all
the roots of Γ (η) = 0 are stable ones. In sum, if ∆p = ab11 − ab22 < 0 and ∆s = α1 − α2 > 0,then the characteristic equation of the linearlized system involves at least three stable roots,meaning that the converging path towards the steady state is locally indeterminate.

22

It is to be emphasized that, as the above proposition shows, in our setting indeterminacy may emerge regardless of the magnitude of σ. This is in contrast to the conclusion in
Nishimura and Shimomura (2002a) showing that the indeterminacy conditions involve a highelasticity of substitution in consumption, 1/σ. Since the closed economy version of our modelis the same as that of Nishimura and Shimomura (2002a), we need the same condition forholding indeterminacy if our model economy is closed. Hence, our result shows that the financially integrated world with non-tradable capital goods tends to produce indeterminacy undera wider range of parameter spaces than in the closed economy counterpart. In this sense,
our model claims that internationalization may enhance the possibility of sunspot-deriven
economic fluctuations.
Finally, to complete our stability analysis, we summarize the findings for the other cases.
Proposition 4 (i) If the private and the social factor-intensity rankings are the same, thenthe steady-state equilibrium is locally determinate, and (ii) if the capital good sector is more
capital intensive than the consumption good sector from the private perspective but it is lesscapital intensive from the social perspective, then the steady state is unstable.
hiProof. (i) Note that sign r0 (p) = sign (a1 − 1) A1 k1a1 −2 k10 (p) = sign (α1 − α2 ) . Thus´³
if ab11 − ab22 (α1 − α2 ) > 0, thenIn addition, when

³

a1
a2
b1 − b2¢£¤¡ 1− δ −qr0 (π q − π q∗ ) < 0.
sign yK´(α1 − α2 ) > 0,we obtain1
sign [−qr0 (yK− δ)(π q + π q ∗)] = sign −1 − δ)r0 (yK< 0,
λ + pλp¢¡ 1− δ > 0. As a results, Γ (η) = 0 has two stablebecause sign λp = −sign yp2 > 0 and r0 yK
and two unstable roots, so that there is a unique converging path around the steady state.

(ii) If ab11 − ab22 > 0 and α1 − α2 < 0, then λp < 0. Hence, in this case the sign of π q + π q∗ is notdetermined without imposing further restrictions. In the case of π q + π q∗ > 0, we have two
1 − δ > 0. On the other hand, if it holds that
positive eigenvalues, r0 (π q + π q∗ ) > 0 and yK¢¡ 1− δ (π q + π q∗ ) < 0,
−qr0 yK23

then ξ (η) = 0 has one positive and one negative root. If π q + π q∗ > 0, we see that1 − δ > 0. In addition, if
r0 (π q + π q∗ ) < 0 and yK¢¡ 1− δ (π q + π q∗ ) < 0,
−qr0 yK1−(yK− δ) < 0,

qr0 (π q + π q∗ ) < 0,
−2yp1 π K < 0,then ξ (η) = 0 has two positive roots. Therefore, regardless of the sign of π q + π q∗ , Γ (η) = 0
has only one stable root and thus the steady state equilibrium is locally unstable.If ab11 − ab22 > 0
1 − δ > 0. Additionally, it is seen thatand α1 − α2 < 0, then r0 (π q + π q∗ ) > 0 and yK¢¡ 1− δ (π q + π q∗ ) < 0,
−qr0 yKso that ξ (η) = 0 has one positive and one negative root. This reveals that Γ (η) = 0 has only
one stable root and thus the steady-state equilibrium is locally unstable.
These results are also close to the stability conditions for the small open economy models
with capital mobility examined by Meng and Velasco (2003 and 2004). This proposition
gain emphasizes that the dynamic behavior of the financially integrated world economy withsymmetric countries and non-traded capital goods is closer to the behavior of corresponding
small-open economy rather than to the closed economy counterpart.

5

Discussion

5.1

The Steady-State Characterization and Equilibrium Determinacy

As was stated in Section 3.2, if the perfect-foresight competitive equilibrium is indeterminate
in the dynamic Hechscher-Ohlin model, the steady-state capital distribution between thetwo countries cannot be selected by the initial distribution of capital alone. Hence, sunspotderiven changes in expectations may aﬀect the equilibrium path towards the steady state,
which means that the long-run pattern of trade also depends on expectations formationof agents in the world market. In contrast, since the final goods for investment are notinternationally traded in our model, the steady-state level of physical capital in each country
is uniquely determined regardless of the presence of equilibrium indeterminacy.

24

It is to be noted that in our economy the long-run level of financial asset holding ineach country would be aﬀected by determinacy/indeterminacy of equilibrium. This result issummarized as follows:
Proposition 5 If the steady-state equilibrium of the world economy is locally determinate(indeterminate), then the steady-state level of asset position of each country is determinate(indeterminate).
Proof. From (54) in which p = p∗ and K = K ∗ in the steady state, the equilibrium
condition is written as(1 + m̄) C = 2y2 (K, p) .

(62)

Since the steady-sate levels of p and K are uniquely determined, the magnitude of total
consumption demand, (1 + m̄) C, is uniquely given as well. When the equilibrium path isdeterminate, there at least locally exists a two-dimensional stable manifold on which the
implicit prices of capital stocks are uniquely expressed by the following functions:qt = q (Kt , Kt∗ , m̄) ;qt∗ = q ∗ (Kt , Kt∗ , m̄) .

Thus (59) and (60) show that in the initial period p and p∗ are written as p0 = π̂ (K0 , K0∗ , m̄)and p∗0 = π̂ ∗ (K0 , K0∗ , m̄) . Using these functions and (58) , we may express the initial value of
λ in the following manner:
λ0 = λ (K0 , K0∗ , π̂ (K0 , K0∗ , m̄) , π̂ ∗ (K0 , K0∗ , m̄) ; m̄) .
As a result, (36) and (62) yield:(1 + m̄) [λ (K0 , K0∗ , π̂ (K0 , K0∗ , m̄) , π̂ ∗ (K0 , K0∗ , m̄) ; m̄)]−1/σ = 2y2 (K, p) .
¡¢This equation may determine the level of m̄ = μ∗−1/σ . If there is a unique level of m̄satisfying the above, it depends on the initial capital distribution (K0 , K0∗ ) as well as on
the steady-state levels of p and K. Once m̄ is uniquely selected, then the values of C andC ∗ (= m̄C) in the steady state are also uniquely given.14
In contrast, if the steady state of the world economy is locally indeterminate, the stablemanifold has at least three dimensions so that the implicit prices of capital cannot be functions
14

See also Appendix for determination of m̄.

25

of Kt , Kt∗ and m̄. If this is the case, m̄ cannot be uniquely determined by K0 , K0∗ and thesteady-state values of p and K. This means that the stead-state levels of C and C ∗ areindeterminate. Note that the from the flow budget constraints for the households in thedecentralized economy, in the steady state it holds that RΩ+w−C = 0 and ΩR+w∗ −C ∗ = 0.
Therefore, the steady state level of net asset positions are:B=(m̄ − 1)y 2 (K, p) − C=
y2 (K, p) ,ρ+δ
(1 + m̄) (ρ + δ)

(63a)

B∗ =

(1 − m̄)y 2 (K, p) − m̄C=y2 (K, p) .
ρ+δ
(1 + m̄) (ρ + δ)

(63b)

Since m̄ cannot be uniquely given in the case of presence of equilibrium indeterminacy, thelong-run levels of net asset position are indeterminate as well.
Equations (63a) and (63b) demonstrate that the net asset position of each country inthe steady state entirely depends upon the level of m̄. If the equilibrium is determinate, m̄
is uniquely determined at the initial period,. Hence, for example, if the home country is acreditor at the outset, we tend to have m̄ > 1, so that the home country will be a creditor
in the long-run equilibrium as well. In the equilibrium is indeterminate, then the level ofm̄ may not reflect the asset positions in the initial period. This implies that the long-run
asset positions cannot be predicted without specifying expectations of the households in bothcountries.

5.2

The Case of Small-Open Economy

When we examine the small-country counterpart of our model, we simply solve the representative household in the home country by fixing the interest rate on net wealth: R = R̄for all t ≥ 0, where R̄ denotes an exogenously given world interest rate. In this case, thenon-arbitrage condition between holding bonds and capital becomesṗ= R̄ − r = R̄ − αAk1 (p)α−1 .
pTherefore, the behavior of the relative price is independent of quantity side of the economy,so that from (45) the stability of relative price depends only on the sign of k10 (p), i.e. thesocial factor intensity ranking, ∆p = α1 −α2 .. In addition to this price equation, the quantity

26

system is given by the equilibrium condition for the investment goods market:
K̇ = y1 (K, p) − δK.
Meng and Velasco (2004) demonstrate that as for the dynamic system given above, the factorintensity ranking conditions presented in Proposition 3 are necessary and suﬃcient for holding
local indeterminacy in the small-open economy with non-traded capital. Remember that
Proposition 3 means that the factor-intensity ranking condition is not necessary but suﬃcientfor generating indeterminacy, implying that the possibility of equilibrium indeterminacy inthe world economy is higher than that in the small-open economy. The main reasons forthis result is that the interest rate, R, in an endogenous variable in the two-country model.
The world interest rate depends on the both prices and capital stocks of two counties, which
enhances the range of parameter values under which the steady-state equilibrium is locallyindeterminate.

6

Conclusion

This paper has investigated the relation between trade structure and equilibrium indeterminacy in a two country world. We have introduced non-traded capital goods and international
financial transactions into the dynamic Heckscher-Ohlin model with production externalities
examined by Nishimura and Shimomura (2002a). Our extension has demonstrated that the
introduction of non-traded goods and financial asset mobility enhances the range of parameter values under which the perfect-foresight competitive equilibrium of the world economy
is indeterminate. Since the standard Heckscher-Ohlin setting used by Nishimura and Shimomura (2002a) establishes the same stability conditions as these held in the corresponding
closed economy, our finding indicates that the assumptions of trade structure of the world
economy would be a critical determinant in considering relation between globalization andeconomic volatility.
The world economy as a whole is a closed economy in which there are heterogenous
countries. Therefore, its model structure is similar to that of a closed, single economy modelwith heterogenous agents. In particular, if consumption and saving decisions are made by the
representative household in each country, the behavior of the world economy model is closely27connected to that of the closed economy model with heterogenous households. There is,however, a key diﬀerence between the world economy and the single country settings: when
dealing with the world economy model, we should specify the transaction structure between
the countries. Both of the Heckscher-Ohlin theory and the discussion in this paper assume
specific structures of international trade. It is worth investigating how our conclusion would
be modified under alternative forms of international trade.15
In the literature on indeterminacy and sunspots, some authors have explored how thepresence of heterogenous households may alter the determinacy/indeterminacy conditions in
the real business cycle models with market distortions. These studies have shown that theheterogeneity of agents often aﬀects stability condition in a critical manner.16 As mentioned
in Section 1, Sim and Ho (2007a) reveal that the introduction of technological heterogeneity
into the Nishimura-Shimomura model may produce a substantial change in equilibrium indeterminacy results. Those existing findings suggest that it is worth extending our model byconsidering further heterogeneity between the two countries in order to consider the impact
of globalization on aggregate stability in a more general framework than the present paper.

Appendix
In this appendix we show that the pseudo-planning problem discussed in the main text
characterizes the competitive equilibrium of the decentralized world economy.17 For thispurpose, we first derive the optimization conditions of the households and firms in bothcountries.
Set up the Hamiltonian function for the households in the home country in such a waythatH=

C 1−σ − 1+ ζ (RΩ + w + π 1 + π 2 − C) ,1−σwhere ζ denotes the implicit value of net wealth. The necessary conditions for an optimum
15

In the trade theory literature, the relation between equilibrium characterization of the world economy andtrade structures have been discussed extensively: see, for example, Ethier and Svensson (1986) and Cremers(1997). We may use the results obtained in those studies to extend our argument.
16
See, for example, see Ghiglino and Olszak-Duquenne (2005).
17
See Hu and Mino (2009) for a detailed analysis of the market economy version of the model,

28

include the following:
C −σ = ζ,(A1)ζ̇ = ζ (ρ − R) ,(A2)

together with the transversality conditions: limt→∞ e−ρt ζ t Ωt = 0. Note that the transversality condition means that the non-Ponzi-game restriction holds with an equality. Profit
maximization conditions in (??) yieldr = pa1 A1 K1a1 −1 Lb11 X̄1 − δ = a2 A2 K2a2 −1 Lb22 X̄2 − δ,(A3)

w = pb1 A1 K1a1 L1b1 −1 X̄1 = b2 A2 K2a2 L2b2 −1 X̄2 .

(A4)From the non-arbitrage condition (26) we obtain
R=r+

ṗ
ṗ
1
= a1 A1 K1α1 −1 L1−α
−δ+ .
1
p
p

(A5)

In the same vein, we obtain the conditions for the foreign country corresponding to theabove as follows:C ∗−σ = ζ ∗(A6)ζ̇ = ζ ∗ (ρ − R)

(A7)

∗a2 −1 ∗b2 ∗∗1r∗ = p∗ a1 A1 K1∗a1 −1 L∗b
L2 X̄2
1 X̄1 = a2 A2 K2

(A8)

1 −1
w∗ = p∗ b1 A1 K1∗a1 L∗b
X̄1∗ = b2 A2 K2∗a2 L2∗b2 −1 X̄2∗
1

(A9)

∗

R = r∗ +ṗ∗ṗ∗∗α1 −1 ∗1−α1
=
a
A
K
L
−
δ
+
.
1
1
1
1
p∗
p∗

(A10)

It is seen that if we set p = q/λ and p∗ = q ∗ /λ, then (A3), (A4), (A8) and (A9) respectively
correspond to (37) through (40) in the planning problem. Furthermore, by use of (A5), (A10),
i −bi
i −bi
and X̄i∗ = Ki∗αi −ai L∗1−α, we findX̄i = Kiαi −ai L1−αiiṗ∗ ṗq̇ ∗ q̇1
1=− = a1 A1 K1α1 −1 L1−α−
− a1 A1 K1∗α1 −1 L∗1−α
.
1
1
p∗ p
q
q
This relation can be obtained fromq̇1
= ρ + δ − a1 A1 K1α1 −1 L1−α
,
1
q
q̇ ∗
1
= ρ + δ − a1 A1 K1∗α1 −1 L∗1−α
,
1
q∗
29

which respectively correspond to (41) and (42).
To examine the relation between the transversality conditions for the market economy
and those in the planning problem, it is to be noted that (A2) and (A7) mean that ζ̇/ζ =∗ζ̇ /ζ ∗ = ρ − R. Therefore, in view of (A1) and (A6), we see that C ∗ /C = (ζ ∗ /ζ)−1/σ stays

constant over time. Thus we may set ζ ∗ /ζ = μ∗ , i.e. the relative welfare weight on the foreign¡R ∞¢households in the planning problem. In addition, (A2) gives ζ t = ζ 0 exp t (ρ − Rs )ds .

Therefore, from the definitions of Ωt = B + pK and p = q/λ, the non-Ponzi game condition,
together with the transversality condition, for the household in the home country becomes
µ
¶
µ Z t
¶
qt
−ρt
Rs ds = ζ 0 lim e ζ t Bt + Kt = 0.
lim Ωt exp −tλt
0Hence, the non-Ponzi game scheme the economy as a whole (condition (33c)) implies that
lim e−ρt ζ t λqtt Kt = 0, so that the transversality condition for the planing problem, limt→∞ e−ρt qt Kt =0, is established by setting ζ t = λt . Since ζ ∗t = μ∗ ζ t , the non-Ponzi game conditions for theforeign households yields
µ
¶
µ Z t
¶
qt∗ ∗
∗
−ρt
∗
Rs ds = μ ζ 0 lim e ζ t Bt + Kt = 0.
lim Ωt exp −tλt
0This and (34a) ensure the transversality, condition limt→∞ e−ρt qt∗ Kt∗ = 0, in the planningproblem.
To select the value of μ∗ in the planning problem, consider the intertemporal budget
constraints for the households in both countries. Due to the transversality as well as nonPonzi game conditions, they are given by
µ Z t
¶
µ Z t
¶
Z ∞
Z ∞
exp −
Rs ds Ct dt =exp −Rs ds (wt + π 1,t + π 2,t ) dt + Ω0 ,
0

Z ∞
0

0

0

0

µ Z t
¶
µ Z t
¶
Z ∞
¡
¢
∗
exp −
Rs ds Ct dt =exp −Rs ds wt∗ + π ∗1,t + π ∗2,t dt + Ω∗0 .
0

00Using Ct∗ = m̄Ct , we thus obtain

³ R
´¡
¢
t
exp
−
R
ds
wt∗ + π ∗1,t + π ∗2,t dt + Ω∗0
s
0
0
³ R
´
.
m̄ = μ∗−1/σ = R
∞
t
exp
−
R
ds
(w
+
π
+
π
)
dt
+
Ω
s
t
1,t
2,t
0
0
0
R∞

Therefore, if the converging path is uniquely given, the entire sequences of wages, profits andinterest rate are determinate, and hence m̄ is also uniquely selected under given levels of Ω030and Ω∗0 . In contrast, if there is a continuum of converging paths, the sequences of wages,
profits and interest rate are indeterminate, which generates indeterminacy of m̄.
Finally, let us check the Walras law in the market economy. First, note that Ω̇ + Ω̇∗ =pK̇ + p∗ K̇ ∗ + ṗK + ṗ∗ K ∗ . Thus adding up the flow budget constraint for the households ineach country gives
pK̇ + p∗ K̇ ∗ + ṗK + ṗ∗ K ∗= R (pK + p∗ K) + w + w∗ − C − C ∗
¶
µ
¶
µ
ṗ∗
ṗ
pK + r + ∗ p∗ K ∗ + w + w∗ − C − C ∗ .
=
r+
p
p

(A11)

By use of the full-employment conditions, K = K1 + K2 and 1 = L1 + L2 , we obtain¶µw
rpK + w + π 1 + π 2 = p rK1 + L1 + π 1 + prK2 + wL2 + π 2
p= pY1 + Y2 − δK.
(A12)Similarly, it holds thatr∗ p∗ K ∗ + w∗ + π ∗1 + π ∗2 = p∗ Y1∗ + Y2∗ − δK ∗ .

(A13)

Substituting (A12) and (A13) into (A11) and using Y1 = K̇ + δK and Y1∗ = K̇ ∗ + δK ∗ , weobtain the world market equilibrium condition of the consumption goods: Y2 + Y2∗ = C + C ∗ .

31

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34