Discussion Papers In Economics
And Business

Growth Cycles in a Two-country Model of Innovation
Kunihiko KonishiDiscussion Paper 15-07

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANGrowth Cycles in a Two-country Model of Innovation
Kunihiko KonishiDiscussion Paper 15-07

May 2015

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANGrowth Cycles in a Two-country Model of Innovation∗
Kunihiko Konishi†, ‡Abstract
This study examines growth cycles in a simple discrete-time two-country model of innovation. In this setting, we ﬁnd that there are two key driving forces that give rise tocycles. They are perfect international capital mobility and perfect international knowledge
spillovers. In addition, this study shows that the opening of trade can create cycles in bothcountries, whereas pretrade equilibrium in each country initially jumps to the steady state.
That is, our results are characteristic of an open-economy framework.

Keywords: Two-country model, Cycles, Innovation
JEL classiﬁcation: E32, F44, O41∗

I am especially grateful to Koichi Futagami for helpful discussions and suggestions. I also thank participantsat the 2014 Autumn Annual Meeting of the Japanese Economic Association at the Seinan Gakuin University for
their useful comments. The ﬁnancial support from the Research Fellowship for Young Scientists of the Japan
Society for the Promotion of Science (JSPS) is gratefully acknowledged. Any errors are my responsibility.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, JAPAN;
E-mail: nge008kk@student.econ.osaka-u.ac.jp‡
Research Fellow of the Japan Society for the Promotion of Science (JSPS).

1

1

Introduction

It is well known that long-run growth in developed countries ﬂuctuates. Many studies argue that
endogenous cycles are a primary explanation for this fact. In R&D-based models, a numberof studies have undertaken theoretical investigations of the existence of endogenous cycles.1
However, these investigations are conﬁned to a closed-economy framework. In reality, developed
countries increase international globalization and market integration. Thus, the investigationof endogenous cycles should be conducted within an open-economy framework.
Few studies have considered endogenous cycles using R&D-based models in an open economy. Furukawa (2015) develops a two-country growth model that can capture how economic
leadership endogenously moves between countries along an equilibrium path. In the model, theknowledge accumulation in each country occurs via domestic innovation and spillovers from foreign innovation through foreign direct investment (FDI). Furukawa (2015) shows that knowledge
spillovers from foreign innovation are the key driving force behind growth cycles. Iwaisako andTanaka (2013) examine growth cycles in a North-South product-cycle model with overlapping
generations. They show that perpetual ﬂuctuations in the world growth rate are generated by
the interaction between innovation and imitation.
The objective of this study is to show that there are endogenous growth cycles even ina simple discrete-time two-country model, following Grossman and Helpman’s (1991) variety
expansion model. In this setting, we ﬁnd that there are two driving forces that give rise to cycles.
One is perfect international capital mobility and the other is perfect international knowledge
spillovers. First, perfect international capital mobility requires that the sum of the dividendrate and the rate of capital gains must be equalized between both countries. This equalization
generates cycles if the wage rates are unequalized between both countries along the equilibrium
path. Second, the domestic country enjoys knowledge spillovers from foreign innovation andcan achieve faster innovation. As a result, this can generate cycles, even if the wage rates areequalized between both countries along the equilibrium path.
With regard to another result, we show that the opening of trade can create cycles in bothcountries. In a closed-economy model, Haruyama (2009) shows that cycles can emerge evenin the simple R&D-based model, following Grossman and Helpman’s (1991) variety expansion
1
See Shleifer (1986), Gale (1996), Deissenberg and Nyssen (1998), Francois and Shi (1999), Freeman et al.
(1999), Matsuyama (1999), Francois and Lloyd-Ellis (2003, 2008, 2009), Wälde (2005), Furukawa (2007), andHaruyama (2009).

2

model. The important assumptions imposed in Haruyama (2009) are, ﬁrst, discrete time and,second, risk-neutral consumers (i.e., the utility function takes a linear form).2 However, in
this study, pretrade equilibrium in each country initially jumps to the steady state because theutility function takes a log form. Thus, our results are characteristic of an open economy. Ina related study, Nishimura et al. (2014) show a similar result to this study.3 They consider
a two-country, two-good, two-factor general equilibrium model with CRRA utility functions,asymmetric technologies across countries, and decreasing returns to scale in the production of
all goods. They show that opening to free trade can create persistent endogenous ﬂuctuations
at the global level, even if each country’s closed-economy equilibrium is saddle-point stable.4
However, their model is not an R&D-based model and the mechanism of cycles is diﬀerent fromours.
The rest of this paper is organized as follows. Section 2 establishes the model used in thisstudy. Section 3 derives the steady state and the dynamic system of the economy. Section 4
examines the local dynamics of the economy. Finally, Section 5 concludes the paper.

2

Model

Time is discrete. We follow the R&D-based endogenous growth model with expanding variety(Grossman and Helpman, 1991). The model consists of two countries, A and B. Each country
engages in two activities: (1) production of diﬀerentiated goods using workers and (2) R&D
conducted by workers. The two countries’ population sizes are LA and LB . Each individual
supplies one unit of labor inelastically in every period and has perfect foresight.

2.1

Consumers

All consumers in country i ∈ {A, B} maximize their lifetime utility:
Ui ≡

∞
∑

ρt log Cti ,t=0where Cti represents the temporary utility derived from the consumption of a composite goodand ρ ∈ (0, 1) is the discount factor. Each consumer in both countries consumes diﬀerentiated2

Furthermore, Haruyama (2009) shows that cycles can emerge if the utility function takes a constant relativerisk aversion (CRRA) form and the degree of relative risk aversion is suﬃciently small.
3
In addition, Nishimura et al. (2006, 2009) show similar results.
4
The closed economy model of Nishimura et al. (2014) has potential for endogenous ﬂuctuations. They showthat the restrictions of endogenous ﬂuctuations can be loosened through international trade.

3goods produced in both countries. Cti is given by
Cti =

[∫ nt

cit (j)

ε−1
ε

] ε

ε−1

dj

,

0

where cit (j) denotes the consumption of good j in country i. nit indicates the number of varieties
B
produced in country i and nt ≡ nA
t + nt . ε > 1 is the elasticity of substitution between any∫ntwo products. Denoting the expenditure of consumers in country i as Eti = 0 t pt (j)cit (j)dj, thedemand function for good j becomespt (j)−ε Eti
cit (j) = ∫ nt,
1−εdu
0 pt (u)where pt (j) is the price of good j. PD,t is the price index deﬁned as PD,t =

(∫ nt
0

) 1
pt (j)1−ε dj 1−ε .

Substituting the demand function into Cti , we obtain
Cti =

Eti
.
PD,t

We assume perfect international capital mobility, and thus, the rate of return on assets isequalized between both countries. The maximization problem for each consumer in country i
is as follows:max

Uisubject to Ait+1 = (1 + rt )Ait + wti − Eti ,where Ait , rt , and wti represent consumers’ asset holdings in country i, the rate of return on assets,
and the wage rate in country i, respectively. Solving the intertemporal utility maximization,
we obtain the Euler equation:
1 + rt+1 =

i
Et+1
.
ρEti

Following Grossman and Helpman (1991), we normalize Et = EtA LA + EtB LB = 1 for all t sothat 1 + rt+1 = 1/ρ holds.

2.2

Production

This economy has no transportation costs or tariﬀs. Labor is immobile across countries. We
assume that each diﬀerentiated good is produced by a single ﬁrm because the good is inﬁnitely
protected by a patent and the good must be produced in the country in which they weredeveloped. Furthermore, we assume that the production sector is monopolistically competitive4and one unit of labor input produces one unit of a diﬀerentiated good. The total demandA
B
B
function is xt (j) = cAt (j)L + ct (j)L . This implies that the ﬁrm manufacturing good jcharges the following price:pt (j) = pit =

ε
wi .
ε−1 t

Therefore, all goods produced in country i are priced equally. This pricing rule yields the total
demand function and the monopoly proﬁts of a ﬁrm as follows:( i )−εwt
ε−1ixt (j) = xt =)( )1−ε ,(1−εε nA wA+ nB w Bttπt (j) = πti =

2.3

t

1
w i xi .
ε−1 t t

(1)

t

(2)

R&D

Following Grossman and Helpman (1991), an entrepreneur in country i devotes a/Kti units oflabor at time t to develop a new variety of good. a is R&D productivity and Kti represents
the knowledge stock in country i. In empirical studies, Coe and Helpman (1995) and Coe etal. (2009) ﬁnd that both foreign and domestic knowledge spillovers have signiﬁcant impacts
on the level of total factor productivity. We adopt this result. Suppose that knowledge resultsfrom the R&D activities and knowledge moves freely and rapidly throughout the global researchcommunity. With perfect international knowledge spillovers, we assume Kti = nt . We let vti
denote the market value of a successful innovation in country i. The R&D sector is assumed tobe competitive and the free entry condition is as follows:
vti =

awti
nt

nit+1 − nit > 0.

if

(3)

i
i
The shareholders of the stocks earn dividends πt+1
and capital gains or losses vt+1
− vti . Under

the assumption of perfect international capital mobility, we obtain the following no-arbitrage
conditions:
1 + rt+1 =

i
πi
vt+1
1
= t+1
+
.
ρ
vti
vti

5

(4)

2.4

Market clearing condition

The labor in country i is used for production and R&D in country i. The labor market clearing
condition in country i becomes
nit xit + azti gti = Li ,(5)

where zti ≡ nit /nt is the share of goods manufactured in country i and gti ≡ (nit+1 − nit )/nit is thegrowth rate of diﬀerentiated goods in country i.

3Market equilibrium

3.1

Steady state

We ﬁrst study properties of the steady state. We let denote the world innovation rate bygt ≡ (nt+1 − nt )/nt . The steady state is deﬁned as the innovation rates in both countries being
constant and identical: (nit+1 − nit )/nit = g̃. Tildes represent variables in the steady state. We
show in Appendix A that such a steady state is a unique form of equilibrium if and only if
i(vt+1− vti )/vti = −gt /(1 + gt ). Furthermore, we show that steady state values are as follows:Li,LρL − a(ε − 1)(1 − ρ)g̃ =,a(ε − 1 + ρ)
ε−1+ρw̃ = w̃A = w̃B =,ε(L + a − ρa)z̃ i =(6)

where L ≡ LA + LB .

3.2

Dynamic system

In this subsection, we derive the dynamic system in the economy. (1), (3), and (5) imply( )−εzti wtiLi ε − 1i i≡ Λi (wtA , wtB , ztA ).
zt gt =−( )( )aεa z A wA 1−ε + z B wB 1−εtttNote that the deﬁnition of zti implies ztB = 1 − ztA . gt = ztA gtA + ztB gtB and (7) yield
( )−ε
( )−ε
+ ztB wtB
L ε − 1 ztA wtA
gt = −≡ Γ(wtA , wtB , ztA ).
( )
( )
a
εa z A wA 1−ε + z B wB 1−ε
t

t

t

(7)

t

(8)

t

From the deﬁnitions of ztA and gtA , we obtain
A
zt+1
=

ΛA (wtA , wtB , ztA ) + ztA≡ ∆(wtA , wtB , ztA ).
A
B
A
1 + Γ(wt , wt , zt )6

(9)

By using (1), (2), and (3), the no-arbitrage condition (4) becomes( i )1−ε
]
i
wt+1
wt+1
1[
1
1+Γ(wtA , wtB , ztA ) =
+
.
(
[
](
)
)
1−ε
1−ε
A
B
ρ
εawti ∆(wtA , wtB , ztA ) wt+1
wti
+ 1 − ∆(wtA , wtB , ztA ) wt+1
(10)Equations (9) and (10) formulate the autonomous dynamic system with respect to wtA , wtB , and
ztA .

4

Local dynamics

We examine the local dynamics around the steady state. In this study, we focus on the steadystate with a positive growth rate. Therefore, in the local dynamics analysis around the steadystate, the positive innovation rates in both countries are ensured; that is, (3) holds in bothcountries along the transitional dynamics. By using (6), (7), and (8), we approximate (9) and(10) linearly around the steady state. As shown in Appendix B, we obtain the following linear
system:   A
A
ŵt+1
J1 J2
0
ŵt
B  = Jŵt+1J4
0  ŵtB  ,
3
A
ẑt+1
J5 −J5 J6
ẑtA

(11)

where
) B
(
L
(ε − 1) La + 1 − ρ La
a +ε
}
{
,
χ
=
,
J1 = χ + J3 ,J2 = −
Lρ ερ − (ε − 1) aερ − (ε − 1) La() A(ε − 1) La + 1 − ρ La{
} ,J3 = −
J4 = χ + J2 ,ρ ερ − (ε − 1) Laερ − (ε − 1) La
(
) .
J6 =
ρ La + ε(ε − 1)(ρ + ε − 1)LA LB −2ε
J5 =
w̃ ,
ρ(L + εa)L2(ŵtA , ŵtB , ẑtA ) denotes a sequence of deviations from the steady state and Ji denote entities in
the Jacobian matrix of this system. The eigenvalues of the Jacobian matrix, J, are deﬁned as(λi (i = 1, 2, 3). Here, λ1 , λ2 , and λ3 are the roots of the characteristic equation, λ − χ − J2 −)()()
J3 λ − χ λ − J6 = 0. To check stability, we solve the characteristic equation and obtain its
roots as follows:
λ1 =

L
a +1

ρ

> 1,

λ2 =

L
a +ε

ερ − (ε − 1) La

,

λ3 =

ερ − (ε − 1) La
ρ( La + ε)

< 1.

λ1 is an unstable root. Then, we have to investigate the other characteristic roots. Figure
1 shows the relationship between λ2 and λ3 in (ε, L/a) space.5 Here, we assume ε > 2 for5

A detailed derivation is shown in Appendix C.

7

simplicity.6 From the steady state value g̃, the line L/a = (ε − 1)(1 − ρ)/ρ implies that thegrowth rate is zero at the steady state. Therefore, we focus on the region L/a > (ε − 1)(1 − ρ)/ρ
where the steady state growth rate is positive. Note that wtA and wtB are jump variables and ztA
is a predetermined variable. In region (A), λ2 > 1 and 0 < λ3 < 1 imply that the steady stateis stable and the equilibrium path is monotonic. In region (B), λ2 < −1 and −1 < λ3 < 0 imply
that the steady state is stable and the equilibrium path ﬂuctuates. In region (C), λ2 < −1 andλ3 < −1 imply that the steady state is unstable and the equilibrium path ﬂuctuates. In region(D), −1 < λ2 < 0 and λ3 < −1 imply that the steady state is stable and the equilibrium pathﬂuctuates.

Figure 1: The relationship between λ2 and λ3 in (ε, L/a) space.

As shown in Grandmont (2008), we can reduce a bifurcation analysis to a simple onedimensional invariant manifold. If a ﬂip bifurcation occurs, one eigenvalue λi goes through−1. In this study, there are two cases: if L/a = ε(1 + ρ)/(ε − 2) holds, λ2 is −1, and if
L/a = 2ερ/(ε − 1 − ρ) holds, λ3 is −1. In both cases, we observe a ﬂip bifurcation. In addition,
there generically exist stable or unstable two-period cycles on one side of the bifurcation points.
These results are summarized in the following proposition:

6

This assumption is in the range of empirically plausible parameters. A detailed discussion is provided inHaruyama (2009).

8

Proposition 1
A flip bifurcation will generically occur for L/a = ε(1 + ρ)/(ε − 2) or L/a = 2ερ/(ε − 1 − ρ).

Haruyama (2009) shows that endogenous cycles do not occur in a closed economy if theutility function is the log form. In this study, each country’s closed-economy equilibrium, inwhich capital mobility and knowledge spillover are not allowed, jumps to the steady state
initially. Hence, the opening of trade can create endogenous cycles in both countries.
We discuss the mechanism of cycles in the following subsections. The key driving forces
behind cycles are perfect international capital mobility and perfect international knowledge
spillovers.

4.1

Cycles generated by perfect international capital mobility

To understand the mechanism of cycles, it is helpful to consider the local dynamics converging
to the steady state. Solving the linear diﬀerence equation (11) yields7
 A

 
 

1
ŵt
0
1
ŵtB  = Θ1 (λ1 )t 1 + Θ2 (λ2 )t − LBA  + Θ3 (λ3 )t 0 ,
L
ẑtA
1
0
Ω

(12)

where Ω ≡ (LB −LA )J5 /(χ−J6 )LB and Θi are constants determined by the initial and terminal
conditions. λ1 > 1 implies Θ1 = 0.
For an economy in which (ε, L/a) is in region (D), we obtain Θ2 = (z0A − z̃ A )/Ω and Θ3 = 0.
(12) is rewritten as follows:
z0A − z̃ A(λ2 )t ,Ω(z A − z̃ A )LA
wtB = w̃ − 0(λ2 )t ,ΩLB
ztA = z̃ A + (z0A − z̃ A )(λ2 )t .
wtA = w̃ +From (3), the relationship between vtA and vtB is the same as that between wtA and wtB . In thisA
B , and v AB≶ wt+1case, wtA ≷ wtB implies that vtA ≷ vtB , wt+1
t+1 ≶ vt+1 hold, and thus, cycles

arise. The mechanism of the cycles is as follows. From (1), (2), (3), and (4), the unequal wage
i /v i and the rate of capital gains v i /v i must
rates imply that the sum of the dividend rate πt+1t
t+1 t
A /v A ≷ v B /v B implies π A /v A ≶ π B /v B .
be equalized between both countries; that is, vt+1
t
t+1 t
t+1 t
t+1 t

We show in Appendix E that this relationship generates cycles. Therefore, the cycles are
7

A detailed derivation is shown in Appendix D.

9generated by perfect international capital mobility if the wage rates are not equalized between
both countries.
We then consider why the cycles occur in region (D). By using (1) and (2), the monopoly
proﬁt isi
πt+1=

( i )1−εwt+11( A )1−ε()1−ε .
Aεnt (1 + gt ) zw+ z B wB
t+1

t+1

t+1

(13)

t+1

From Figure 1, region (D) implies that ε and L/a are suﬃciently high. A higher ε implies
that the monopoly price is lower and the monopoly proﬁt is lower. From (8), a higher L/a
implies that the growth rate, gt , is higher. In the variety expansion model, new goods displace
the monopoly proﬁts of old goods. Thus, a higher growth rate yields lower monopoly proﬁts.
Furthermore, from (13), the ﬂuctuation of the wage rate implies that the monopoly proﬁt also
ﬂuctuates. By using these results, if ε and L/a are suﬃciently high, the ﬂuctuation of themonopoly proﬁt caused by the ﬂuctuation of the wage rate becomes smaller. In addition, thisi /v i and v i /v i must
makes the ﬂuctuation of capital gains smaller because the sum of πt+1t
t+1 t

be equalized between both countries. As a result, a ﬂuctuating equilibrium path exists thatconverges to the steady state. On the other hand, the abovementioned ﬂuctuation becomeslarger if ε and L/a are suﬃciently low; that is, (ε, L/a) is in regions (A), (B), or (C). In thiscase, cycles caused by perfect international capital mobility are unstable (i.e., λ2 < −1). As
shown in the next subsection, the wage rates are equalized between both countries to satisfy
the equal rate of return on equity if λ2 < −1 holds.

4.2

Cycles generated by perfect international knowledge spillovers

We now consider the case in which (ε, L/a) is in regions (A) or (B). In this case, we obtain
Θ2 = 0 and Θ3 = z0A − z̃ A . (12) is rewritten as follows:
wtA = wtB = w̃,ztA = z̃ A + (z0A − z̃ A )(λ3 )t .
For an economy in which (ε, L/a) is in region (B), cycles arise. Because the wage rates areequalized between both countries along the equilibrium path, πtA = πtB and vtA = vtB hold for
all t, and thus, the source of the cycles is not perfect international capital mobility. The cycles
are generated by perfect international knowledge spillovers. The mechanism of the cycles is asfollows. Suppose that ε and L/a are suﬃciently high and ztA < z̃ A . ztA + ztB = 1 yields ztB > z̃ B .
10

A higher ε implies that the monopoly price is lower and the labor demand for production
is larger. A higher L/a implies that the eﬀective labor force becomes larger and innovation
activities are accelerated. Thus, from ztA < z̃ A , ztB > z̃ B , and the labor market clearing
condition, country B allocates more resources to production. Country A enjoys the knowledge
spillovers freely and can achieve faster innovation. gtA > gtB holds and the diﬀerence between gtA
A becomes larger. z A > z̃ A and z B <and gtB is suﬃciently large. As a result, an increase in zt+1
t+1
t+1

z̃ B hold. Then, at time t + 1, country A allocates more resources to production and country
A < z̃ A
B can achieve faster innovation from international knowledge spillovers. Therefore, zt+2
B > z̃ B hold. This process occurs repeatedly along the equilibrium path. However, if εand zt+2

and L/a are much higher, cycles caused by perfect international knowledge spillovers become
unstable; that is, λ3 < −1 holds.

5

Conclusion

We examined the existence of growth cycles using the simple discrete-time two-country modelwith log utility following Grossman and Helpman’s (1991) variety expansion model. Our main
result shows that if perfect international capital mobility and perfect international knowledge
spillovers are allowed, then endogenous cycles based on a ﬂip bifurcation can arise.
This study has potential to be extended in several directions. First, in order to keep theanalysis tractable, we chose as simple a model as possible. It would be interesting to investigate
how the presence of heterogeneous preferences and technologies between the two countries aﬀect
the existence of the conditions of growth cycles. In addition, we could introduce R&D subsidies
or patent breadth. If these policy variables were asymmetric, the symmetry between the twocountries would become broken. Second, it would be useful to examine how changes in theeﬃciency of international knowledge spillovers alter the existence of the conditions of growth
cycles. In this study, we assumed perfect international knowledge spillovers, as expressed by
Kti = nit + njt . We could loosen this assumption as follows: Kti = nit + ϕnjt , where ϕ ∈ [0, 1] is theeﬃciency of international knowledge spillovers. Future research should examine this problem.

11Appendix
A.

Condition of the steady state

In this Appendix, we show that if countries innovate at the same rate, (nit+1 − nit )/nit = gt , thenecessary and suﬃcient conditions for the steady state to be in a unique form of equilibriumiare (vt+1− vti )/vti = −gt /(1 + gt ). From (1), (3), and (5), we obtaingti =(vti )−εLi
ε−1−.
A 1−ε + nB (v B )1−ε
ε nA
azti
t (vt )tt

(A.1)(1), (2), (3), and (4) yieldi )1−ε
i
(vt+1
vt+1
1
1
=
−
.
A 1−ε + nB (v B )1−ε
ρ εvti nA
vti
t+1 (vt+1 )t+1 t+1(A.2)

i
We ﬁrst consider suﬃcient conditions. Suppose that (nit+1 −nit )/nit = gt and (vt+1−vti )/vti =−gt /(1 + gt ). By using (A.2), we obtain
vt = vtA = vtB .

(A.3)

Li
ε−1 1
.
−
i
ε nt vt
azt

(A.4)

Li
.
LA + LB(A.5)

From (A.1) and (A.3), we obtain
gt =(A.4) and ztA + ztB = 1 result in
zti =(A.2) and (A.3) imply
1
1(gt + 1 − ρ) =.
ρ
εnt vt(A.6)

By using (3), (A.4), (A.5), and (A.6), the innovation rate and the wage rate becomeg̃ =ρ(LA + LB ) − a(ε − 1)(1 − ρ),a(ε − 1 + ρ)
ε−1+ρ
w̃ =
.
A
ε(L + LB + a − ρa)Thus, we conﬁrm that the economy is in the steady state.
Next, we consider necessary conditions. Suppose that the economy is in the steady state.
By using (A.1), we obtain
g̃ =

ε−1
ε−1
LA
LB
ε
ε
−
=
−
.
( vtB )ε A A
az̃ A nA v A + ( vtA )ε nB v B
az̃ B
n v + nB v B
t

t

vtB

t

t

12

vtA

t

t

t

t

In the steady state, the respective second terms are constant. We deﬁne the constant values θAand θB as
θ

A

A
≡ nA
t vt +( A )ε
vt
B
nB
t vt
vtB

( B )εvt
A
B B
and θ ≡nAt vt + nt vt .
vtA
B

From these deﬁnitions, the relative ﬁrm values are as follows:
vtA = θvtB ,( A )1εθ.
where θ ≡θB

Thus, the growth rates of ﬁrm values in both countries are identical. As a result, we can conﬁrm
that (A.2) yields (A.3); that is, θA = θB . Finally, we derive the growth rates of ﬁrm values. By
using (3), we obtain
i
vt+1
− vti
nt
g̃
=
−1=−
.
i
nt+1
1 + g̃
vt

Hence, this proves the necessary condition.

B.

Derivation of the Jacobian matrix

By using (6), (7), and (8), we approximate (9) and (10) linearly in the neighborhood of thesteady state as follows: −ε  A  1w̃ − ψ1
ψ1
0
ŵt+1
ρ ψ2 + ψ3
1
−ε
B
 ψ2




w̃ − ψ2 0
ŵt+1 =
ρ ψ2
A
0
0
1
ẑt+1
ψ4

1
ρ ψ1

1
ρ ψ1 + ψ3−ψ4

  A
0ŵt
0ŵtB  ,
ẑtA
ψ5

where
ψ1 =
ψ3 =

L
a +ε

(ε − 1)( La + 1 − ρ)LB −εw̃ ,(ε − 1 + ρ)L

ε−1+ρw̃−ε ,

ψ4 =ψ2 =(ε − 1)( La + 1 − ρ)LA −εw̃ ,(ε − 1 + ρ)L(ε − 1)(ρ + ε − 1)LA LB −2εw̃ ,ρ(L + εa)L2The Jacobian matrix is given by
 1
 −ε
w̃ − ψ2−ψ10
ρ ψ2 + ψ3
1 −ψ2w̃−ε − ψ1 0   ρ1 ψ2
J=
ψ6
0
0
ψ6
ψ4

ψ5 =

1
ρ ψ1

1
ρ ψ1 + ψ3−ψ4ερ − (ε − 1) La() .
ρ La + ε

0
0 ,
ψ5

where ψ6 = (w̃−ε − ψ1 )(w̃−ε − ψ2 ) − ψ1 ψ2 .

C.

The relationship between λ2 and λ3

In this Appendix, we consider the relationship between λ2 and λ3 in (ε, La ) space to illustrate
Figure 1. We assume ε > 2 for simplicity and empirical plausibility. First, we investigate the13region in which the steady state growth rate is positive. From the steady state growth rate, theregion in which g̃ > 0 is as follows:(ε − 1)(1 − ρ)
L
>
.
a
ρ

(C.1)

We rewrite λ2 and λ3 as follows:
λ2 =

L
a +ε

,
L

λ3 =

ερ − (ε − 1) aερ − (ε − 1) La
ρ( La + ε)

< 1.

By using λ2 , we obtainλ2 ≷ 0⇔λ2 R 1⇔

λ2 R −1

⇔

L
ερ
≶
,
a
ε−1
L
R 1 − ρ,aLε(1 + ρ)
R
.
a
ε−2

(C.2)

From ε > 2, we obtain(ε − 1)(1 − ρ)
(1 − ρ)(ε − 1 − ρ)
− (1 − ρ) =
> 0.
ρ
ρ

(C.3)

(C.1), (C.2), and (C.3) imply that the region in which g̃ > 0 does not contain the region inwhich 0 < λ2 < 1. From these results, we can depict Figure A.1. We then examine the existenceof the region in which λ2 > 1. From Figure A.1, this condition is as follows:1−ρ2ρ2 + ρ − 1
=
> 0.
ρ
ρ
√
Therefore, 2ρ2 + ρ − 1 > 0 holds if ρ > (−1 + 5)/4. This parameter range is economically√plausible because (−1 + 5)/4 ≈ 0.309.
2ρ −By using λ3 , we obtainλ3 R 0⇔λ3 R −1

⇔

L
ερ
Q
,
a
ε−1
2ερ
L
Q
.
a
ε−1−ρ

From these results, we can depict Figure A.2.
Finally, in order to illustrate Figure 1, we consider whether the curve L/a = ε(1 + ρ)/(ε − 2)
is above the curve L/a = 2ερ/(ε − 1 − ρ) as follows:2ερ(1 − ρ)(ε − 1 + ρ)ε(1 + ρ)
−
=
> 0.
ε−2
ε−1−ρ
(ε − 2)(ε − 1 − ρ)
14

Figure A.1: The value of λ2 in (ε, L/a) space.

Figure A.2: The value of λ3 in (ε, L/a) space.

Thus, the curve L/a = ε(1 + ρ)/(ε − 2) is above the curve L/a = 2ερ/(ε − 1 − ρ).

D.

Solving the linear diﬀerence equations

We ﬁrst solve the linear diﬀerence equation (11) as follows:
 A
 
 
 
ŵt
ν11
ν21
ν31
ŵtB  = Θ1 (λ1 )t ν12  + Θ2 (λ2 )t ν22  + Θ3 (λ3 )t ν32  ,
ẑtA
ν13
ν23
ν33

(D.1)

where νij are the characteristic vectors and Θi are constants determined by the initial andterminal conditions. We derive the characteristic vectors. From the deﬁnition of characteristicroots and vectors, we obtain   
J1 − λ i
J2
0
νi1
0
 J3
J4 − λi0  νi2  = 0 .
J5−J5J6 − λi
νi3
0

(D.2)

Here, the calculation yields J1 − λ1 = −J2 and J1 − λ2 = J3 . By using J1 − λ1 = −J2 , we solve(D.2) as follows:ν11 = ν12and ν13 = 0.

By using J1 − λ2 = J3 , we solve (D.2) as follows:ν22 = −LA
ν21
LBand ν23 =(LB − LA )J5
ν21 .
(χ − J6 )LB

With regard to λ3 , we solve (D.2) as follows:(χ + J3 − J6 )ν31 + J2 ν32 = 0(D.3)

J3 ν31 + (χ + J2 − J6 )ν32 = 0

(D.4)

15

Rearranging (D.4) with respect to ν31 and substituting this into (D.3) yields(J6 − χ)(χ − J6 + J2 + J3 )ν32 = 0
J6 − χ ̸= 0 and χ − J6 + J2 + J3 ̸= 0 imply that ν32 = 0 holds. Thus, from (D.3) and ν32 = 0,
we obtain ν31 = 0. By setting ν11 = ν21 = ν33 = 1, we can rewrite (D.1) as follows:

 
 A
 

1
ŵt
1
0
ŵtB  = Θ1 (λ1 )t 1 + Θ2 (λ2 )t − LBA  + Θ3 (λ3 )t 0 ,
L
1
ẑtA
0
Ω
where Ω ≡ (LB − LA )J5 /(χ − J6 )LB .

E.

Cycles generated by perfect international capital mobility

i /v i and the rate
From (4), the unequal wage rates imply that the sum of the dividend rate πt+1t
i /v i must be equalized between both countries; that is, v A /v A ≷ v B /v Bof capital gains vt+1
t
t+1 t
t+1 t
A /v A ≶ π B /v B . Suppose that w A > w B . From (3), v A > v B holds. If v A /v A >implies πt+1
t
t
t
t
t
t+1 t
t+1 t
B /v B , the diﬀerence between v A and v B becomes larger. However, (3) and (6) imply that
vt+1
t
t
t

the steady state vti are equalized between both countries. In order to converge to the steadyA /v A < v B /v B holds. (1), (2), (3), (4), and v A /v A < v B /v B yieldstate, vt+1
t
t+1 t
t+1 t
t+1 t
B
A
πt+1
πt+1
>
vtA
vtB

⇔

( A )ε−1( B )ε−1
vtA vt+1< vtB vt+1.

A /v A < v B /v B imply v A
B
AB
Hence, vtA > vtB and vt+1t
t+1 t
t+1 < vt+1 . Similarly, wt < wt implies
A
B
that vtA < vtB and vt+1
> vt+1
hold. Summarizing these results, we can show that perfectinternational capital mobility certainly generates cycles if the wage rates are not equalized
between both countries.

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