Discussion Papers In Economics
And Business

Preference Structure and Volatility in a
Financially Integrated World
Kazuo Mino
Discussion Paper 08-05

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPreference Structure and Volatility in a
Financially Integrated World
Kazuo Mino
Discussion Paper 08-05

February 2008

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPreference Structure and Volatility in a
Financially Integrated World∗
Kazuo Mino†
Graduate School of Economics, Osaka University
February 2008

Abstract
This paper investigates a two-country model of capital accumulation with country-specific production externalities. The main concern of our discussion is to explore the presence of equilibrium indeterminacy in an open-economy setting. In contrast to the existing studies on equilibrium indeterminacy in small-open economies,
the present paper demonstrates that opening up international tradeand financial interactions between two counties does not necessarily enhance the possibility of indeterminacy of equilibrium. It isshown that the results depend heavily upon not only on the degreeof external increasing returns but also on the preference structures.
Keywords: Financial integration, Two-country growth model, Equilibrium determinacy, Preference structure
JEL classification: F43, O41∗

I gratefully acknowledge many years of fruitful discussion with Koji Shimomura on dynamic economic theory that has been very helpful in writing this paper. I thank Hideyuki
Adachi, Eric Bond, Ben Heijdra, Yoshiyasu Ono and Tamotsu Nakamura for their usefulcomments on earlier versions of Mino (2007) on which the present paper partially depends.
I also thank a referee of this paper for helpful comments.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, 5600043 Japan, e-mail: mino@econ.osaka-u.ac.jp1

1

Introduction

The equilibrium business cycle theory based on indeterminacy and sunspots
has claimed that economic fluctuations caused by extrinsic uncertainty tend
to more prominent in open economies than in closed economies. For example,
inspecting a small-open economy version of the two-sector model studied by
Benhabib and Farmer (1996), Weder (2001) concludes that the small-open
economy requires a lower degree of external increasing returns to hold indeterminacy than the closed-economy counterpart. Similarly, Aguiar-Conraria,
L. and Wen, Y. (2005), Lahiri (2001) and Meng and Velasco (2003, 2004)
also demonstrate that perfect capital mobility may enhance indeterminacyof equilibrium for small-open economies.1 The main reason for this results
is that in the small-open economies the interest rate is fixed in the world
financial market, so that consumption behaves as if the utility function werelinear. Since a high elasticity of intertemporal substitutability in consumption serves as a cause of indeterminacy, the small-open economies tend to
be more volatile than the closed economies with the same technologies andpreferences.
The purpose of this paper is to examine whether the volatility of open
economies emphasized by the existing studies on expectations-driven business cycles will hold in the world economy model as well. Since the world
economy is a closed economy with heterogeneous agents, the present paper
may be considered a study on the relationship between heterogeneity and indeterminacy.2 The analytical framework of this paper is a two-country modelof capital accumulation with external increasing returns. These two countries trade consumption goods. In addition, it is assumed that agents in bothcountries can access to the perfect international bond market. Given such asetting, we compare the indeterminacy conditions in the autarky equilibriumwith those in the world economy.3
1
Meng and Velasco (2003, 2004) study two-sector dependent economy models withsector-specific externalities in which investment goods are not traded. Lahiri (2001) examines a two-sector endogenous growth model with capital mobility. It is to be notedthat Nishimura and Shimomura (2002b) also consider equilibrium determinacy in a twosector small-open economy without factor mobility. Their argument follows the traditional
Heckscher-Ohlin modelling.
2
In a diﬀerent context, Ghiglino and Olszak-Duquenne (2005) also consider the relation
between heterogeneity of agents and equilibrium indeterminacy in a closed economy modelof economic growth with production externalities.
3
In the open-macroecomics literature, we have not seen any study on indeterminacy ina two-country model. It should be noted that in the context of the standard trade theory
context, Nishimura and Shimomura (2002b and 2006) investigate Heckscher-Ohlin models
with externalities that generate multiple equilibria. As for a general discussion concerning

2

Our main finding is that, as opposed to the results established in thesmall-open economy models, the global economy does not necessarily showa higher possibility of indeterminacy. In particular, if the preference structures of both countries satisfy additive separability between consumption
goods and labor, then opening up international transactions will not alterenhance the possibility of indeterminacy. Furthermore, even if we assume
a more genreal form of utility function, then we do not find unambiguous
results indicating that the world economy holds indeterminacy under weaker
restrictions on technology and preferences than the closed economies.
It is to be noted that the main concern of this paper is closely related
to a seminal contribution by Nishimura and Shimomura (2002b) who study
equilibrium indeterminacy of a world economy with free trade of commodities. Nishimura and Shimomura (2002b) utilize the standard Heckscher-Ohlin
setting where consumption as well as investment goods are freely traded but
there is no financial interactions between the home and the foreign countries.
They find that intermediacy conditions for a closed economy may generate
multiple equilibria in the world economy as well.4 Our study reconsiders thesame issue in the context of a two-country version of the real business cycle
model with production externalities and free capital mobility.
The rest of the paper is organized as follows. Section 2 sets up the basemodel. Section 3 assumes specific functional forms involved in the modeland explores indeterminacy conditions under alternative assumptions on theutility functions. Section 5 concludes the paper.

2

The Analytical Framework

2.1

Structure of the Model

There are two countries, country 1 and 2. Each country produces a countryspecific, single good. We assume that country 1 specializes in good x andcountry 2 specializes in good y. Each good can be either consumed or invested
for physical capital accumulation. We assume that the imported goods canbe consumed, but they cannot be used as investment goods. In addition,
agents in each country cannot access to the direct ownership of the foreign
capital stock. However, the agents in both counties may access to the perfect
the eﬀect of financial integration on macroeconomic volatility, see, for example, Evans andHanatkovska (2005).
4
Nishimura and Shimomura (2006) examines the possibility of equilibrium indeterminacy in a dynamic Heckscher-Ohlin model of a world economy in the absence of production
externalities. Indeterminacy in their model comes from the fact that intertemporal trade
between the home and foreign countries is prohibited,3

international bond market so that they can freely lend to or borrow fromeach other. Since the international bond market is assumed to be perfect,
the uncovered interest parity ensures that the nominal interest rates in bothcountries are equalized in each moment. 5 Although our assumption thatimported goods cannot be used for investment is restrictive, it is helpful todetermine real investment in each country without introducing additional
assumptions such as the presence of adjustment costs of investment.
Production
The production technology of each country is described by¡¢zi = f i ki , li , k̄i , ¯li , i = 1, 2,(1)where zi , ki and li respectively denote output, capital and labor input ofcountry i. In addition, k̄i and ¯li express external eﬀects associated with thesocial levels of capital and labor in country i. It is assumed that function
f i (.) is homogenous of degree one in ki and li and that it increases withk̄i and ¯li . This means that while the private technology under given levelof external eﬀects satisfies constant returns, the social technology exhibits
increasing returns to scale with respect to the aggregate levels of capital andlabor. We assume that those external eﬀects are country specific so that
there are no international spillovers of production technologies.
The commodity markets in both countries are competitive. Firms maximize their instantaneous profits under given levels of external eﬀects. Hence,letting ri and wi be the real interest rate and the real wage rate in country i,they respectively equal the net marginal product of capital and the marginal
product of labor:¡¢ri = fki ki , li , k̄i , ¯li − δi ,¡¢wi = f i ki , li , k̄i , ¯li ; i = 1, 2.
lwhere δ i ∈ (0, 1) denotes the depreciation rate of capital.

Households
The number of households in each country is normalized to one. The objective functional of the representative household in country i is a discounted
sum of utilities such thatZ ∞Ui =ui (xi , yi , li ) e−ρt dt, ρ > 0; i = 1, 2,
0

5

The structure of the base model is close to the two-country model examined by
Turnovsky (1997, Chapter 7). Since Turnovsky (1997) does not assume the presenceof external increasing returns, indeterminacy of equilibrium is not the issue in his argument. Baxter and Crucini (1995) use a similar model in their study on international realbusiness cycles.

4

where xi and yi denote consumption of x and y goods. By our assumption, y1
is exported from country 2 to country 1 and x2 is ex[rpted from country 1 tocountry 2. The instantaneous utility is assumed to be increasing in xi and yi ,and decreasing in labor li . The standard concavity assumption is imposed onui (.) . We assume that the households in both countries may have diﬀerent
utility functions but their discount rate is the same rate of ρ.6
The flow-budget constraint for the households in each country is given byω̇ i = ri ωi + wi li − mi , i = 1, 2,(2)

where ω i is the real asset holding and mi is real consumption expenditure.
For notational convenience, ω i , wi and mi are expressed in terms of the goodcountry i produces. Thus if p denote the price of good y in terms of good x,
the consumption spending in both countries are respectively determined by
m1 = x1 + py1 ,

m2 =

x2
+ y2 .
p

The asset consists of capital stock, ki , and the foreign bond holding, bi .
Therefore, we define
ωi = ki + bi , i = 1, 2.
where b1 and b2 are evaluated in terms of x good and y good, respectively.
The household maximizes Ui subject to (2) and the initial value of ω i bycontrolling consumption levels and labor supply. We impose the no-Ponzigame condition, and hence the following intertemporal budget constraint
holds as well:
µ Z t
¶
Z ∞
exp −
ri (s) ds wi (t) li (t) dt
ω i (0) +
0
0
µ Z t
¶
Z ∞
exp −
ri (s) ds mi (t) dt, i = 1, 2.
=00Market Equilibrium Conditions
Since physical capital stocks are not traded, the market equilibrium conditions for the commodity markets are:6

z1 = x1 + x2 + k̇1 + δ 1 k1 ,(3)

z2 = y1 + y2 + k̇2 + δ2 k2 .

(4)This assumption is introduced only for notational simplicity.

5The world financial market is assumed to be perfect. This means thatthe uncovered interest parity yieldsṗ
r1 = r2 + .
p(5)

The international borrowing and lending in the world economy should bebalanced in each moment, implying that the equilibrium condition for thebond market isb1 + pb2 = 0.
(6)Note that the homogeneity of production functions gives zi = ri ki + wi li(i = 1, 2). Hence, in view of (3) , (4) and ω i = ki + bi , the flow budget
constraints for the home and the foreign households present the dynamic
equations of b1 and b2 as follows:ḃ1 = r1 b1 + x2 − py1 ,

(7)

x2
.
p

(8)

ḃ2 = r2 b2 + y1 −Equations (7) and (8) respectively describe the current accounts of country
1 and 2.7
Perfect-Foresight Competitive Equilibrium
To sum up, the perfect-foresight competitive equilibrium (PFCE) of theworld economy is defined in the following manner:
Definition: The PFCE of the world economy is established if the following
conditions are satisfied:(i) The firms maximize instantaneous profits under given levels of external
eﬀects, k̄i . and ¯li .
(ii) The households maximize their discounted sum of utilities under given
sequences of prices, {ri (t) , wi (t) , p (t)}∞t=0 .
(iii) Commodity markets clear and the bonds market is in equilibrium.
(iv) The uncovered interest parity condition (5 ) is satisfied.
7

We assume that the solvency conditions for international lending and borrowing aresatisfied. Namely, the following hold:
µ Z t
¶
µ Z t
¶
Z ∞
Z ∞
b1 (0) +exp −r1 (s) ds x2 (t) dt =exp −r1 (s) ds p (t) y1 (t) dt,
0

0

0

0

µ Z t∙µ Z t∙
¸ ¶
¸ ¶
Z ∞
Z ∞
ṗ (s)ṗ (s)x2 (t)
b2 (0)+exp −exp −r2 (s) +r2 (s) +ds y1 (t) dt =
ds
dt.
p
(s)
p
(s)
p (t)
0
0
0
0

6

(v) External eﬀects satisfy consistency conditions, that is, k̄i = ki and¯li = li .
In what follows, we analyze the corresponding planning economy rather thanthe decentralized system displayed above.

2.2

A Pseudo-Planning Problem

In order to characterizes the equilibrium dynamics of the world economy, it
is convenient to consider the following pseudo-planning problem:
Z ∞
¤
£
max
μ1 u1 (x1 , y1 , l1 ) + μ2 u2 (x2 , y2 , l2 ) e−ρt dt, μi > 0, μ1 + μ2 = 1
0subject to¡¢k̇1 = f 1 k1 , l1 , k̄1 , ¯l1 − x1 − x2 − δ 1 k1 ,¡¢k̇2 = f 2 k2 , l2 , k̄2 , ¯l2 − y1 − y2 − δ 2 k2 ,

(9)
(10)

and given initial levels of k1 (0) and k2 (0) . A positive constant, μi , denotesai. Taking the sequences of external eﬀects,© weight on the utility in ªcountry∞k̄1 (t) , ¯l1 (t) , k̄2 (t) , ¯l2 (t) t=0 , as given, the planner solves this problem byselecting the optimal levels of li , xi and yi .
The Hamiltonian function for the planning problem can be set as
H = μ1 u1 (x1 , y1 , l1 ) + μ2 u2 (x2 , y2 , l2 )£ ¡¢¤+q1 f 1 k1 , l1 , k̄1 , ¯l1 − x1 − x2 − δ 1 k1
£ ¡¢¤+q2 f 2 k2 , l2 , k̄2 , ¯l2 − y1 − y2 − δ 2 k2 ,where qi represents the shadow value of ki . The necessary conditions for an
optimum are:μi uixi (xi , yi , li ) = q1 , i = 1, 2,(11)
μi uiyi (xi , yi , li ) = q2 , i = 1, 2,¡¢
μi uili (xi , yi , li ) = qi flii ki , li , k̄i , ¯li , i = 1, 2,£¡¢¤q̇i = qi ρ + δ i − f i ki , li , k̄i , ¯li , i = 1, 2,
ki

(12)
(13)
(14)

together with (9) , (10) and the transversality conditions:lim e−ρt qi ki = 0, i = 1, 2.

t→∞

(15)

Inserting k̄i = ki and ¯li = li into the optimal conditions, we find thenecessary conditions for an optimum of this pseudo-planning problem completely characterize PFEC conditions of the world economy defined above. .
7

Lemma 1 If μ1 and μ2 are appropriately selected, the necessary conditions
for an optimum for the planning problem and the PFEC in the world economy
are identical.
Proof. See Appendix 1.
Due to this fact, we can focus on the dynamic behavior of the pseudoplanning economy when examining the equilibrium dynamics of the world
economy.

2.3

Dynamical Systems

The necessary conditions for the planning problem may be summarized as
a complete dynamical system with respect to the state variables, k1 and k2 ,and the costate variables, q1 and q2 . The structure of the dynamic system
is sensitive to the specification of the preference structure. First, consider
the general case where the instantaneous utility functions are non separable
ones. In this case, by use of (11) and (12) , consumption demand xi and yi
may be expressed as functions of q1 . q2 and ki . Substituting those functions
into (13) and assuming that k̄i = ki and ¯li = li , we find that li depends on
q1 , q2 and ki . Therefore, the equilibrium levels of xi , yi and li are written as:
xi = xi (ki , q1 , q2 ) , yi = y i (ki , q1 , q2 ) , li = li (ki , q1 , q2 ) ; i = 1, 2.
Substituting those into (9) , (10) and (14) the complete dynamic system may
be written ask̇1 = z 1 (k1 , q1 , q2 ) − x1 (k1 , q1 , q2 ) − x2 (k2 , q1 , q2 ) − δ 1 k1 ,wherek̇2 = z 2 (k2 , q1 , q2 ) − y 1 (k1 , q1 , q2 ) − y 2 (k2 , q1 , q2 ) − δ 2 k2 ,¤£q̇i = qi ρ − ri (ki , q1 , q2 ) , i = 1, 2,¢¡
z i (ki , q1 , q2 ) ≡ f i ki , li (ki , q1 , q2 ) , ki , li (ki , q1 , q2 ) ,¡¢ri (ki , q1 , q2 ) ≡ fk ki , li (ki , q1 , q2 ) , ki , li (ki , q1 , q2 ) − δ i .

Next, assume that the utility function is additively separable between
consumption and labor in such a way thatui (xi , yi , li ) = φic (xi, yi ) + φiL (li ) .
Then it is easy to see that xi and yi depend only on q1 and q2 alone, whilez ι (.) and ri (.) involve only ki and qi
k̇1 = z 1 (k1 , q1 ) − x1 (q1, q2 ) − x2 (q1, q2 ) − δ1 k1 ,8

k̇2 = z 2 (k2 , q2 ) − y 1 (q1 , q2 ) − y 2 (q1 , q2 ) − δ 2 k2 ,¤£q̇i = qi ρ + δ i − ri (ki , qi ) , . i = 1, 2.

Finally, consider the case in which the utility function is additively separable for each argument, that is,ui (xi , yi , li ) = φiX (xi ) + φiY (yi ) + φiL (li ) .
As demonstrated by Turnovsky (1997, Chapter 7), this assumption simplifies
the dynamic system substantially: consumption demand for x goods depends
on q1 alone and that for y good depends on q2 alone. Thus the dynamic
system is described byk̇1 = z 1 (k1 , q1 ) − x1 (q1 ) − x2 (q1 ) − δ 1 k1 ,k̇2 = z 2 (k2 , q2 ) − y 1 (q2 ) − y 2 (q2 ) − δ 2 k2 ,¤£q̇i = qi ρ + δ i − ri (ki , qi ) , i = 1, 2.

Notice that this system consists of two independent sub-system with respect to (k1 , q1 ) and (k2 , q2 ) . Therefore, while the current account depends
on the foreign variables (see (7)equations (8)), consumption and investment
decisions in each country are not interdependent each other even in the financially integrated world. Since the most general case will not produce
tractable results in the base model, we focus on the second and the thirdcases when examining exogenous growth models in the next section. In section 4 we treats non-separable utility between consumption and labor in thecase of endogenous growth.

3

Equilibrium Dynamics

3.1

Specification

In order to obtain more concrete results about determinacy of equilibrium, we
now specify the production and the utility functions as the forms that havebeen frequently employed in the literature on indeterminacy and sunspots in
the real business cycle models. The production function of each country is
zi = kiαi li1−ai k̄iαi −ai ¯lβ i +ai −1 , 0 < ai < 1, αi , β i > 0, αi + β i > 1.
Accordingly, imposing k̄i = ki and ¯li = li , we obtain the social production
function in each country as follows:βzi = kiαi li i , i = 1, 2.
9(16)

In what follows we assume that 0 < αi < 1, so that capital externalities are
not so large that unbounded growth is possible. The rate of return to capitaland the real wage rate are respectively given byβri = fki − δ i = ai kiαi −1 li i − δ i , i = 1, 2,
β −1

wi = fli = (1 − ai ) kiαi li i , i = 1, 2.

(17)
(18)

The utility function is assumed to be additively separable between consumption and labor. More specifically, the instantaneous utility of the households in country i is given by
⎧ ¡
¢
1+γ
θi 1−θi 1−σi
⎪
x
y
−1
li i
⎪
i
i
⎪
⎨
−
, 0 < θi < 1, γ i > 0, σ i > 0,1 − σi1 + γiui (xi , yi , li ) =
1+γ
⎪
li i
⎪
⎪
ln
x
+
(1
−
θ
)
ln
y
−
, for σ i = 1.
θ
⎩ i
i
i
i
1 + γi

Given those specifications, the optimal conditions (11), (12), (13) and(14) may be written asθ (1−σ i )−1 (1−θi )(1−σi )yi= q1 ,

μi θi xi i

i = 1, 2,θ (1−σ i ) (1−θi )(1−σi )−1
yi= q2 ,μi (1 − θi ) xi iγ

i = 1, 2,

β −1μi li i = (1 − ai ) kiαi li i qi , i = 1, 2,³´
βq̇i = qi ρ + δ i − ai kiαi −1 li i , i = 1, 2.

3.2

(19)
(20)
(21)
(22)

Indeterminacy with Logarithmic Utility Functions

We first examine the case where the utility function has a log-additive form in
consumption. Ever since Benhabib and Farmer (1994), this specification has
been most popular in the studies on indeterminacy and sunspots in the realbusiness cycle settings. In this case σ 1 = σ 2 = 1, and hence the consumption
demand functions become
xi =

1
1
, yi =, i = 1, 2.
μi θi q1
μi (1 − θi ) q2

(23)

As pointed out in the previous section, the demand functions for each gooddepend on its own price alone. On the other hand, from (20) we obtain∙

1 − aili =μi1¸ γ +1−βiiαi
γ i +1−β i

ki

10

1
γ +1−β iqi i, i = 1, 2.

(24)

This states that the eﬀects of changes in capital and commodity price on
the equilibrium level of labor input depends on the sign of γ i , +1 − β i . If
the external eﬀect of labor is small enough to satisfy γ i + 1 > β i , then theemployment level is positively related to qi and ki . If γ i + 1 < β i , the labor
employment decreases with qi and ki .
In view of (9) , (10) , (22) , (23) and (24) , the complete dynamic systemwith logarithmic utility functions is given by the following:¸∙
α1 (γ 1 +1)
β1
111
γ 1 +1−β 1
γ 1 +1−β 1k̇1 = A1 k1
qi−+− δ1 k1 ,(25)
θ1 μ1 θ2 μ2 q1¸∙
α2 (γ 2 +1)
β2
1
11
γ 2 +1−β 2
γ 2 +1−β 2k̇2 = A2 k2
q2−+− δ 2 k2 ,(26)
(1 − θ1 ) μ1 (1 − θ2 ) μ2 q2∙¸
α1 (γ 1 +1)β1−1 γ +1−β
γ 1 +1−β 1
1
1
,
(27)
q̇1 = q1 ρ + δ 1 − a1 A1 k1
q1

where

∙
¸
α2 (γ 2 +1)β2−1 γ +1−β
γ 2 +1−β 222q̇2 = q2 ρ + δ 2 − a2 A2 k2,
q2∙

1 − aiAi =μi
In the steady state equilibrium,βi¸ γ +1−β
i

i

(28)

, i = 1, 2.

It is easy to see that the steady state equilibrium, if it exists, is uniquelydetermined. Letting ki∗ and qi∗ be the steady-state values of capital stocks
and prices., the dynamic system linearized at the steady state is written as
⎤ ⎡
⎤⎡
⎤
⎡
h11 0 h13 0
k1 − k1∗
k̇1⎢ k̇2 ⎥ ⎢ 0 h22 0 h24 ⎥ ⎢ k2 − k2∗ ⎥
⎥ ⎢
⎥⎢
⎥
⎢
⎣ q̇1 ⎦ = ⎣ h31 0 h33 0 ⎦ ⎣ q1 − q1∗ ⎦ ,0 h42 0 h44
q2 − q2∗
q̇2
where ∂ k̇1 /∂k1 , h13 = ∂ k̇1 /∂q1 , etc, all of which are evaluated by the steadystate values of ki∗ and qi∗ . Denote the eigenvalue of the coeﬃcient matrix
given above by λ. Then the characteristic equation of the coeﬃcient matrix
in the above linear system is written as
£ 2
¤
λ
−
(h
+
h
)
λ
+
h
h
−
h
h
11
33
11
33
13
31
£
¤
(29)
× λ2 − (h11 + h33 ) λ + h22 h44 − h24 h42 = 0.
11

αi (γ i +1)−1
γ i +1−β iNotice that in view of the steady-state condition, ai Ai ki
ρ + δ i , we obtain:hii

βi
γ i +1−β iqi=(γ 1 +1)β1−1 γ +1−βαi (γ i + 1) αγ 11+1−β
1
1
1
= A1
k1
qi
− δ1
γi + 1 − βi
∙
¸
δiβ i
γi + 1ρ+=, i = 1, 2,
γi + 1 − βiγi + 1

implying thatsign hii = sign (γ i + 1 − β i ) ,i = 1, 2.

We also see that when γ i + 1 − β i > 0, it holds that h13 , h31 , h24, h42 > 0
and h33 , h44 < 0. As a result, we show that if γ i + 1 − β i > 0, then (29) has
two positive and two negative real roots. The presence of two-dimensional
stable manifold around the steady state means that the initial levels of jump
variables, q1 and q2 can be uniquely determined. Therefore, we obtain thefollowing result:
Proposition 1 If the utility function is additively separable between domestic and imported good and production technologies of both countries satisfy
γ i + 1 − β i > 0 (i = 1, 2) , then the steady state of the world economy isdeterminate.
In contrast, if γ i +1 < β i (i = 1, 2), it holds that h11 , h22 , h13 , h24 < 0 and
h31, h33, h42 , h44 , > 0. Thus it is possible to have the following inequalities:
h11 + h33 < 0, h11 + h33 < 0,h11 h33 − h13 h31 > 0, h22 h44 − h24 h42 > 0
If this is the case, (29) have four roots with negative real parts, implyingthat there are a continuum of converging path near the steady state. Hence,the steady state of the world economy is at least locally indeterminate. If
γ 1 + 1 > β 1 and γ 2 + 1 < β 2 , then the steady state of country 2 will beindeterminate, but that of country 1 is determinate. Since fluctuations ofcountry 2 do not aﬀect investment and price in country 1.
In addition, volatility in country 2 does not fluctuate the current accountof country 1 either. To see this, we should note that, as shown in Appendix
A, the relative price p in the market economy is proportional to the ratio of
shadow values of capital, q2 /q1 . Given this fact, from (23) the current account12

of country 1 is described by
q2 μ2
y1 (q2 )q1 μ1
α1 (γ 1 +1)
β1
−1
1
μ2
− 2
.
= .a1 A1 k1γ1 +1−β1 q1γ 1 +1−β1 b1 +μ2 θ2 q1 μ1 (1 − θ1 ) q1ḃ1 = r1 (k1 , q1 ) b1 + x2 (q1 ) −What is noteworthy is that the current account of country 1 depends on k1
and q1 alone. Therefore, even if the equilibrium trajectory of country 2 isindeterminate so that q2 exhibits volatility, the imported good y in terms ofgood x is independent of the behavior of q2 .8 The current account of country 1
is aﬀected by sunspot-driven expectations only when the autarkic equilibriumof country 1 is indeterminate. In sum, we have shown:
Proposition 2 If the utility functions of both countries are logarithmic in
consumption goods, then the possibility of indeterminacy will not enhanced byinternational trade and financial integration. Indeterminacy in one country
will not spillover to the other country in any respect.
Obviously, this propoistion depends on the fact that the utility fuction isof a log-linear form. To see this, suppose that the utility function is given by
1+γ

ui (xi , yi , li ) =

yσ
l i
xσi
+ i − i,1 − σ 1 − σ 1 + γi

σ > 0, σ 6= 1.

Then= qi ,x−σ
iyi−σ = q

This means thatq2 μ2
y1 (q2 )q1 μ1
α1 (γ 1 +1)
β1
−1
1
μ2
− 2
.
= .a1 A1 k1γ1 +1−β1 q1γ 1 +1−β1 b1 +μ2 θ2 q1 μ1 (1 − θ1 ) q1ḃ1 = r1 (k1 , q1 ) b1 + x2 (q1 ) −3.3

Indeterminacy with CES Utility Functions

When σ i 6= 1 so that the utility functions are of CES forms in consumption,
equations (19) and (20) yield the following consumption demand functions:(1−σi )(1−θi )(1−σ́i )(1−θi )−1
−
σi
σi
i
q2
− δ 1 k1 ,xi = ∆x q1
8

i = 1, 2(30)

If this is the case, comparative dynamics analysis by Turnovsky (1997, Chapter 7)that assumes determinacy of equilibrium should be re-examined.

13

(1−σi )(1−θi )−1(1−σi )(1−θi )
+1 −
−1
σi
σi
i
yi = ∆y q1
q2− δ2 k2 ,i = 1, 2,

(31)

where
∆ix

1
=
μi θi

∆iy

1
=
μi θi

µ
µ

θi
1 − θi
θi
1 − θi¶(1−σi )(1−θi )> 0,¶(1−σi )(1−θi )+1> 0.

We see that the own price eﬀect is always negative, while the eﬀects of foreign
price on consumption demand for the home good depends on the sign of 1−σ i .
If σ i > 1, the substitution eﬀect dominate so that a rise in the foreign priceincreases in consumption demand for the home goods. Conversely, if σ i < 1,a higher foreign good price lowers demand for the home goods.
The dynamical equations in the case of non-separable utility between xi
and yi are thus displayed by
α1 (γ 1 +1)

β1k̇1 = A1 k1γ1 +1−β1 qiγ1 +1−β1 −
α2 (γ 2 +1)

β2k̇2 = A2 k2γ 2 +1−β2 q2γ2 +1−β2 −X(1−σ́i )(1−θ i )−1∆ix q1σi−

q2(1−σi )(1−θi )
σi

i=1,2

X

(1−σi )(1−θi )−1∆iy q1

σi

i=1,2

+1 −
q2

(1−σi )(1−θi )σi− δ 1 k1 ,
−1

(32)

− δ 2 k2 , (33)

together with (25) and (26).
Inspecting the dynamical system derived above, we find that if the closed
economies hold determinacy of equilibrium, opening up international trade
will not enhance volatility. More formally, we obtain the following result:
Proposition 3 Suppose that γ i +1−β i > 0 for i = 1 and 2 so that the steadystate in the autarkic equilibrium of both countries are locally determinate.
Then the world economy holds determinacy of equilibrium for any values of
σ i (> 0) .
Proof. See Appendix B.
In this model, the price of foreign good aﬀects capital formation of thehome country only through consumption demand. The above propositionindicates that such an interaction between the two countries may not enhance the possibility of indeterminacy. Therefore, in this case, globalization
would not produce fluctuations due to extrinsic uncertainty in self-fulfilling
expectations. On the contrary, globalization may be a ’stabilizing factor’, as
the following proposition demonstrates:
14

Proposition 4 Suppose that country 1 holds indeterminacy in the autarkic
equilibrium, while country 2 satisfies determinacy condition, i.e. γ 2 +1 > β 2 .
Then if the own price eﬀects in the consumption demand functions dominate
their cross price eﬀect, the world economy may establish determinacy of equilibrium.
Proof. See Appendix C.
The intuition behind this result may be stated as follows. From (21) we
obtainγ
μ1 l1 1β −1= (1 − a1 ) k1α1 l1 1 .
q1
Given the price, q1 , and capital, k1 , the left-hand side of the above expresses
labor supply curve and the right-hand side is the labor demand curve. If
country 1 is a closed economy, γ 1 + 1 < β 1 is a necessary condition for
local indeterminacy around the steady state. As discussed by Benhabib andFarmer (1994), in this case the labor demand curve has a positive slope andit is steeper than the labor supply curve. Now suppose that country 1 is in
the steady state initially. Then if the household anticipates a reduction in
price, q1 , the labor supply curve shifts up, implying that labor employment
and production increase. Since a lower price raises the domestic demand forx goods, such an increase in consumption demand may be consistent witha rise in production. This means that the initial anticipation of a lower
price will be self fulfilling. In the world economy, the mechanism that may
generate indeterminacy is essentially the same as that in the close economy.
If the households in the world anticipated a decrease in q1 , the consumption
demand for x goods increase. This again makes the labor supply curve incountry 1 shifts up, which raises the production level of x. However, inthe case of world economy a lower q1 increases the export for country 2 aswell as the domestic consumption demand. If such a demand eﬀect is large
enough, a rise in employment in country 1 is insuﬃcient to satisfy the newly
created consumption demand. In this case the initial anticipation cannot beself fulfilled. Therefore, the equilibrium is determinate even though labor
externalities are suﬃciently large to satisfy γ 1 + 1 < β 1 .

4Concluding Remarks

The central message of this paper is that the relation between indeterminacyof equilibrium and openness of the economy is sensitive not only to production technologies but also to preference structure. By use of a two-country
15

models with financial integration, we have examined whether or not financial
interactions and international trade enhance the possibility of indeterminacy.
We have found that, as opposed to the results shown by the existing studies on small-open economies, opening up international transactions does notnecessarily yield a higher possibility of indeterminacy. On the contrary, insome cases globalization may serve as a ’stabilizing factor’ in the sense thatthe equilibrium path of the world economy can be determinate even though
the autarky equilibrium of each country is indeterminate. Since our finding
depends heavily upon the model structure we use in the paper, it may not
be appropriate to claim that the global economy is in general less volatile
than closed economies. However, our analysis has suggested that the relation between the sunspot fluctuations and openness of the economy may not
be so straightforward as the small-open economy models demonstrate, if we
consider the general equilibrium of the world economy.

16Appendices
Appendix A: Proof of Lemma 1
First consider the optimization behavior of the households in country i.
The Hamiltonian function for the household’s problem is
Hi = ui (xi , yi , li ) + vi (ri Ωi + wi li − mi ) , i = 1, 2,
where m1 = x1 + py1 and m2 = x2 /p + y2 . The necessary conditions for the
optimization problems are summarized as follows:
u1x (x1 , y1 , l1 ) = v1 , u2x (x2 , y2 , l2 ) = v2 /p,(A1)

u1y (x1 , y1 , l1 ) = pv1 , u2y (x2 , y2 , l2 ) = v2 ,(A2)uil (xi , yi , li ) = vi wi , i = 1, 2,(A3)v̇i /vi = ρ − ri , i = 1, 2,(A4)

lim vi e−ρt ωi = 0, i = 1, 2.

t→∞

(A5)

Using the non-arbitrage condition (5) and (??), we obtain:ṗv̇2 v̇1− .
= r1 − r2 =p
v2 v1This shows that the relation between the relative price, p, and the marginalutility of capital, vi , must satisfy
p=μ

v2
,
v1

(A6)

where μ is a positive constant. From (??) μ satisfiesμ=

u1x (x2 , y2 , l2 ).
u2x (x1 , y1 , l1 )Thus if the economy involves a steady state in which each variable stays
constant over time, substituting the steady-state values of xi , yi and li into
the above equation determines the magnitude of μ.
From (A1), (A2) and (A6) we obtain
u2y
u1y
v2
=
,
y = p = μ
1
ux
ux
v1

17

u2y
u2x
v2
= μ.
=
=
1
1
ux
uy
v1 p

(A7)

Now remember that the optimal conditions (12) and (13) for the planningproblem yield:
u1y
u2y
u2y
q1
u2x
μ
=
= 2,
= 2 = 1.
(A8)
1
1
uy
q2
ux
uy
uy
μ2
Therefore, if we set q1 /q2 = p and μ1 /μ2 = μ, (A7) and (A8) receptively
correspond to (12) and (13) for the planning problem, so that the optimal
conditions for the consumption decisions in the market economy and thosein the planning economy are exactly the same. In addition, the optimal
conditions for labor supply (A3) is identical to (14) in the planning problem
and the commodity market equilibrium conditions (4) and (5) are describedby (10) and (11), repetitively.
Finally, the solvency conditions for international lending and borrowing
requires that
µ Z t
¶
µ Z t
¶
lim b1 (t) exp −r1 (t) ds = lim b2 (t) exp −r2 (s) ds = 0.
t→∞

t→∞

0

0

Thus the transversality conditions (A5) give limt→∞ e−ρt vit kit = 0 (i = 1, 2) .
Since we may set qi = μi vi , conditions in (A5) are equivalent to (16) for theplanning problem.
Appendix B: Proof of Proposition 3
The dynamic behavior of the closed economy is described byαi (γ i +1)βik̇i = kiγ i +1−βi qiγ i +1−βi −"1− δi ,qi

αi (γ i +1)−1
γ i +1−β iq̇i = qi ρ + δ i − ai ki

βi
γ i +1−β i

qi

(B1)
#

.

(B2)

Linearizing (B1) and (B2) at the steady-state equilibrium, it is easy to seethat the trace of the coeﬃcient matrix equals to ρ (> 0) . On the other hand,the coeﬃcient matrix of the linearized system of (32) , (33) , (25) and (26)isgiven by:¯¯
¯ λ − h110−h13−h14 ¯¯¯¯0λ − h22 −h23−h24 ¯¯
Λ (λ) = ¯¯
¯ = 0,
0
λ − h330¯ −h31¯¯
0−h42
0
λ − h44 ¯

where h13 = ∂x1 /∂q1 + ∂x2 /∂q1 , h14 = ∂x1 /∂q1 + ∂x2 /∂q1 , h23 = ∂y1 /∂q1 +∂y2 /∂q1 , .and h24 = ∂y1 /∂q2 + ∂y2 /∂q2 . All of the elements are evaluated at18

the steady state. The characteristic equation is thus written as
¤
£ 2
−
(h
+
h
)
λ
+
h
h
−
h
h
Λ
(λ)
=
λ
11
33
11
33
13
31
£
¤
× λ2 − (h22 + h44 ) λ + h22 h44 − h24 h42 − h14 h23 h31 h42 = 0.

(B3)

Given our assumptions, if h14 = h23 = 0, then Λ (λ) = 0 has two positive
and two negative roots. Thus if the number of negative roots of Λ (λ) = 0 is
higher than three for h14 6= 0 and h23 6= 0, then the steady state of the world
economy is locally indeterminate. Notice that
Λ (0) = (h11 h33 − h13 h31 ) (h22 h44 − h24 h42 ) − h14 h23 h31 h42 ,Λ0 (0) = − (h11 + h33 ) (h22 h44 − h24 h42 ) − (h22 + h44 ) (h11 h33 − h13 h31 ) .

As well as the closed system, if γ i + 1 > β i (i = 1, 2) holds, h11 + h33 > 0 andh22 +h44 > 0. In addition, the saddle-point properties of the closed economies
mean that h22 h44 − h24 h42 < 0 and h11 h33 − h13 h31 < 0. Hence, Λ0 (0) > 0
and Λ (0) > 0 for h14 = h23 = 0. This shows that Λ0 (λ) = 0 has two positive
and one negative roots, and thus when h14 = h23 = 0, the graph of thecharacteristic equation may be depicted as Figure A1. Therefore, regardlessof the sign of h14 h23 h31 h42 , the characteristic equation, Λ (λ) = 0, has at
most two negative roots.
Appendix C: Proof of Proposition 4
Since country 1 has an indeterminate steady state and country 2 has adeterminate one, (B3) involves three negative and one positive roots whenh14 = h23 = 0. Namely, Λ (0) in Appendix B has a negative value if h14 =h23 = 0. Observe that Λ (0) can be written asΛ (0) = h11 h22 h33 h44 − h11 h42 h24 h33 − h31 h13 h22 h44+h31 h42 (h13 h24 − h14 h23 ) .
Under our assumption, γ 1 +1 < β 1 and γ 2 +1 > β 2 , it holds that h31 > 0 and
h42 > 0. Therefore, if
h13 h24 − h14 h23 < 0, .
(C1)then Λ (0) may have a positive sign. As Figure A2 shows, if this is the case,the characteristic equation has two positive and two negative roots so thatthe steady state of the world economy is determinate. Condition (C1) isequivalent to
Ã
!Ã
! Ã
!Ã
!
X ∂xi
X ∂yi
X ∂xi
X ∂yi
−
> 0,
∂q1
∂q2
∂q2
∂q1
i=1,2
i=1,2
i=1,2
i=1,2
which states that the own price eﬀects on consumption demand dominates
the cross price eﬀects.
19

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21