Discussion Paper No. 773

GLOBAL HABITS, HABIT DIFFERENTIALS,AND
INTERNATIONAL MACROECONOMIC
ADJUSTMENT TO INCOME SHOCKS

Shinsuke Ikeda
Ichiro Gombi

April 2010

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Global Habits, Habit Diﬀerentials, andInternational Macroeconomic Adjustment toIncome Shocks1
Shinsuke Ikeda2 and Ichiro Gombi3
Osaka University and Ritsumeikan University
April 21, 2010

1 The authors are grateful for the helpful comments on earlier versions to Do-

minique Demougin, Masao Nakagawa, and the participants of the 69th International Atlantic Economic Conference in Prague, Czech. We appreciate financial
supports to Ikeda from Grants-in-Aid for Scientific Research (B No. 21330046)
from the Japan Society for the Promotion of Science and the 21st COE Programfrom the Ministry of Education, Culture, Sports, Science and Technology, and to
Gombi from Open Research Center Project for Private Universities: MEXT.
2 Corresponding author: S. Ikeda, The Institute of Social and Economic Research, Osaka University, Mihogaoka, Ibaraki, Osaka 567-0047, Japan. Telephone:
81-6-6879-8568, Facsimile: 81-6-6878-2766, Email: <ikeda@iser.osaka-u.ac.jp>.
3 I. Gombi, The Faculty of Economics, Ritsumeikan University, Kusatsu, Shiga
525-8577 Japan. Telephone: 81-77-561-4840, Email: <gombi@ec.ritsumei.ac.jp>.

Abstract
In a two-country model with habit formation, we focus on interdependent
macroeconomic adjustments to global and country-specific income shocks.
Global habits and habit diﬀerentials play key roles in the global equilibrium
dynamics, possibly nonmonotonic, and in the determination of international
asset distribution. A country’s steady-state holdings of net external assets
rely on (i) weighted income diﬀerence in excess of habit diﬀerentials and (ii)global income in excess of global habits. Local income shocks have greater
eﬀects on the international asset distribution than global income shocks.
With habit formation, positive income shocks lower the world interest rate,
thereby harming the creditor country and benefitting the debtor country due
to the intertemporal terms-of-trade eﬀect. In contrast to the case of trade
theory, this intertemporal immiserizing growth eﬀect is more likely to becaused by global income shocks than by country-specific income shocks.
JEL Classification Numbers: F41, D90.
Keywords: Global habits, habit diﬀerentials, immiserizing growth, twocountry model, global shock, country-specific shock.

1

Introduction

The purpose of this paper is to focus on habit formation as a key factor that
determines macroeconomic adjustments of an interdependent world economy.
In doing so, we describe two-country equilibrium dynamics in terms of theevolution of global consumption habits and international habit di¤erentials.
When consumers in both countries are rationally habit forming, the twocountries’ propensities to consume depend on their individual habits. The
global habits, de…ned as the sum of individual countries’habits, determine
the aggregate preference for “present” goods, and hence decide the world
equilibrium interest rate. The habit di¤erentials between the two countries
determine the di¤erence in the consumption propensities and hence a¤ect the
dynamics of the international asset distribution. The novelty of this paper
is to work out international interactions that these two forces produce.
We tackle three important issues: (i) the determinants of the equilibrium
external asset holdings, (ii) the di¤erences in the e¤ects of global and local
income shocks, and (iii) the welfare implications of habit-driven macroeconomic adjustments. We show that a country’s net foreign assets dependon habit di¤erentials and global habits. In particular, in steady state, acountry’s net external asset holdings rely on weighted income di¤erence inexcess of habit di¤erentials and global income in excess of global habits. One
important implication is that local income shocks have greater e¤ects thanglobal income shocks on the international income di¤erential and hence on
the international asset distribution.
With habit formation, positive income shocks lower the world interest
rate, thereby harming the creditor country and bene…tting the debtor country due to the intertemporal terms-of-trade e¤ect. This is an intertemporalversion of the immiserizing growth e¤ect developed by, e.g., Bhagwati (1958)
and Brecher and Bhagwati (1982). We show that this intertemporal immiserizing growth e¤ect is more likely to be caused by global income shocks
than by country-speci…c income shocks. This result contrasts to the case of
trade theory: in trade theory, positive supply shocks commonly occurring in
the individual countries’exporting sectors have smaller e¤ects on the static
terms of trade than positive supply shocks occurring locally in one of the twocountries. In our dynamic context, when a positive income shock is global,
the relative magnitudes of the harmful intertemporal terms-of-trade e¤ect
to the direct bene…cial income e¤ect is larger than they would be when theshock is country-speci…c.
1

The analytical model employs the two-county framework developed byGombi and Ikeda (2003) and Ikeda and Gombi (2009). Those papers wereconcerned with the e¤ects of …scal policies under preference heterogeneity inhabit formation, where the role of global habits in dynamic adjustment wereassumed away by setting the initial holdings of net foreign assets to be zero.
By assuming homogeneous preferences, the present research focuses on interactions among global habits, habit di¤erentials, and the consumption/saving
dynamics, and thereby examining their implications for international asset
distribution and the adjustments to global and local income shocks.1
The remainder of the paper is structured as follows: In Section 2, we
present a two-country model and examine equilibrium dynamics of consumption, the interest rate, and net foreign assets. In Section 3, we analyze thee¤ects of local and global income shocks on the economy and each countries’
welfare. Section 4 concludes the paper.

2

The Two-Country Model

2.1

The basic framework

Consider a two-country world economy composed of home and foreign countries. Each country is populated with in…nitely lived agents with homogeneous preferences. The representative agents in home and foreign countries
are referred to as consumers H and F, respectively. They consume a single
type of consumption goods and hold wealth in the form of bonds. Both goods
and bonds are assumed to be costlessly traded in international markets. For
brevity, the representative agents H and F are assumed to be endowed withconstant amounts of output y and y , respectively. Throughout the paper,
the foreign country’s variables are represented with superscript asterisks.
( )Consumption forms habits. Letting zt represent the time-t habit, we
( )
( )
( )
specify zt as the average of the past consumption rates cs ; s 5 t: zt =1

For small country models with habit formation, see, e.g., Mansoorian (1993a, b) andIkeda and Gombi (1999). Gruber (2002) provides empirical support to an “intertemporalcurrent account model” with habit formation. The literature concerning two-country
dynamic models includes Devereux and Shi (1991), Ikeda and Ono (1991), Bianconi andTurnovsky (1997), and Bianconi (2003).

2

Rt

( )
c exp (1 s(ts)) ds; or equivalently
( )

z_t

( )

=

ct

( )

zt

(1)

;

where x_ represents the time derivative of variable x and represents thediscount rate for past consumption rates. We assume that consumers in bothcountries have the common discount rate for past consumption rates. This
enables us to obtain the tractable dynamics of a two-country equilibrium.
Consumers H and F are assumed to have the same lifetime utility function
speci…ed as
Z 1( )( ) ( )U0 =u(ct ; zt ) exp ( t) dt;(2)0

where represents the subjective discount rate. To ensure the steady state,
the discount rate is the same in the two countries.
We focus on the typical e¤ect of habit formation in a simple way by
specifying the felicity function as
( )

( )

( )

u ct ; zt

=

( )

ct

1 '

zt
1

'

;

(3)

where the habit parameter 2 (0; 1) captures the strength of habit in‡uence;and ' represents a risk aversion parameter.2
With this felicity function, consumer preferences display adjacent complementarity, wherein an increase in today’s habits increases the marginalutility of today’s consumption more than it increases marginal disutility ofhabits, thereby, ceteris paribus, enlarging today’s optimal consumption. To
be formal, Ryder and Heal (1973) de…ne adjacent complementarity as the
felicity function satisfying ucz (c; c) + +2 uzz (c; c) > 0: To capture the intertemporal complementarity, de…ne as
ucz + +2 uzz =ucc . Then, withthe felicity function (3), the index can be computed as
=

1

2 +

> 0;

(4)

2
The felicity function satis…es the regularity conditions proposed by the Ryder and
Heal (1973). The same felicity function is assumed by, e.g., Constantinides (1990) andGruber (2004).

3

which implies that consumer preferences in the two countries commonly display adjacent complementarity.3
Let bt denote net foreign assets held by consumer H. The ‡ow budget
constraint for consumer H is given byb_ t = rt bt + y

(5)

ct :

Given the initial values (b0 ; z0 ) and the (perfectly predicted) time pro…le of1
the market interest rate frt g1
t=0 ; consumer H chooses C0 = fct ; bt ; zt gt=0 so
as to maximize (2) subject to: (i) the ‡ow budget constraint (5); (ii) theformation of consumption habits (1); and (iii) the transversality conditions.
Letting t (= 0) be the shadow price of savings and t (5 0) that of habit
formation, the optimal conditions are given byuc (ct ; zt ) =t_t = (

rt )_t = ( + ) tt;t;uz (ct ; zt ) ;

(6)
(7)
(8)

where uc (c; z) = (ctzt ) ' and uz (ct ; zt ) =(ctzt ) ' , together with(1), (5), and the transversality conditions for bt and zt : Consumer F’s behavior can be speci…ed in exactly the same way.
We assume away any role of the governments. The model is closed byintroducing the market clearing conditions:ct + ct = Y (y + y );

bt + bt = 0;

(9)
(10)

where Y represents the aggregate income. By Walras’law, these are not independent: (10) together with (5) and the corresponding constraint for the foreign consumer imply (9). In sum, the equilibrium time path of (bt ; bt ; ct ; ct ; zt ;zt ; rt ; t ; t ; t ; t ) is determined by equations (1), (5) through (8), the corresponding equations for F, and the market equilibrium condition (9) or (10).
3

Even if the risk aversion parameter ' di¤ers between the both countries, the relation,= F = =
1 2 + > 0, is retained. The discussions in this section, therefore,
can be extended without substantial changes to the heterogenous risk aversion cases.
H

4

2.2

Equilibrium dynamics

In this economy consumption dynamics in the two countries interact throughinternational market transactions. It is useful to de…ne global habit Zt as
Zt

(11)

zt + zt :Since is assumed to be common to the two countries, the dynamics of Zt
can be expressed from (1) and the market clearing condition (9) as
Z_ t =(Y(12)

Zt ) :

If Zt and zt are given, zt are determined from (11).
As in Ikeda and Gombi (2009), dynamics can be drastically simpli…ed in
the following manner. De…ne as:;=which is constant over time since _ t = t = _ t = t from (7) (and the corresponding equation for F). We then construct aggregate indices for (u; u )and ( ; ) as:v (c; z; Z)u (c; z) + u (Y
&

c; Z

z) ;

(13)
(14)

;

where (9) and (11) are substituted.
As shown in Appendix A, we can reduce the equilibrium dynamics ofconsumption habits around a steady-state point as follows:
1 0z_@ &_ A = B
@
Z_
0

2

1 + vvcz
cc(vcz )2 vcc vzz
vcc

0

+

vcc
(vcc +vcz )
vcc

0

vcz
vcc

10

zt
C
vcz vcz +vcc vzz A @ & t
vcc
Zt

1
z
& A;
Z

(15)
where x denotes a steady-state value of variable x: Dynamic system (15) has
two stable roots:q( + 2 )2 4 ( + 2 )!
(< 0) and
;
(16)
2
5

and one unstable root, which is conjugate with !; andis given by (4).
From the assumed property of adjacent complementarity, it can be shownthat ! + > 0.
As shown in Appendix B.1, the saddle plane governed by the two stableroots are expressed asz_ = ! (ztz)(! + )(1

Z_ =

Zt

where

) Zt

(17)

Z ;

(18)

Z ;

ucc
:
ucc + ucc

By equating (17) to (1), the consumption dynamics are given by
ct

c=

!+

(zt

z)

(1

) Zt

Z

:

(19)

Di¤erentiate this by t and substitute (1), (18), and (19) successively into
the result. Then, by taking (9) into account, we obtain the motion of eachcountry’s consumption asc_ = ! (ct c) ;(20)
c_ = ! (ct

c ):

(21)

Irrespective of the second-order habit dynamics of (17) and (18), therefore,
the equilibrium consumption dynamics are of the …rst order. Equations (1)and (20) jointly govern the equilibrium dynamics of (c; z); and equations (1)and (21) do dynamics of (c ; z ).
As is proven by Appendix B.2, the interest rate dynamics are given by
rt

=

Zt

Z ;

(22)

ucc ucc
where is de…ned as(> 0) ; implying that Zt plays a crucial
ucc + ucc
role. Suppose that Zt > Z. Then, due to adjacent complementarity, i.e., >0, ceteris paribus there prevails excess demand in the present good market.
This renders r higher than its steady-state value . The equilibrium interest
rate positively (negatively) comoves with the aggregate habit stock, which
exhibits monotonic motions with stable root(see (18)). The resulting
dynamics of r are given explicitly by

r_ =

(rt
6

):

(23)

The transition dynamics of net foreign assets also depend on the property of the world felicity function and international heterogeneity in habit
formation. As shown by Appendix B.3, by linearizing (5) and substituting(19) and (22) into the result, we can obtain
bt

b=

!+
(zt
(
!)

z)

(1

) (! + )
b0
+
(
!)
+

Zt

Z :

(24)

Two habit stocks zt and Zt a¤ect bt by changing consumption and theinterest rate. An increase in the habit stocks raises consumption from (19),which should be …nanced by net foreign assets greater than its steady-statelevel. This e¤ect is expressed by the terms without b0 in (24). An increasein the aggregate habit stock raises the interest rate by (22). The resulting
rise in interest income alters the time pro…le of net foreign assets. The e¤ect
is captured by the terms associated with b0 in (24).
In sum, the autonomous dynamics of (17) and (18) generate the evolutionof two habit stocks zt and Zt . Given the values of the two state variables,
consumption rates ct and ct , the interest rate rt , and net foreign asset bt
are determined by (19), (22), and (24), respectively. The dynamics can be
described more simply by introducing the weighted di¤erence of the twocountries’habit stocks:
et = zt (1) zt :(25)
We refer to et as weighted habit di¤erentials between countries H and F,
or simply habit di¤erentials. The equilibrium net foreign assets (24) can be
rewritten as
bt

b=

!+
(et
(
!)

e)

b0
+

Zt

Z ;

(26)

which implies the following results.
Proposition 1: Country H’s net foreign assets bt depend on habit di¤erentials et and global habits Zt in the following manners:(i) bt is larger as et is larger; and(ii) bt is larger as Zt is smaller (larger) if country H is the net creditor(debtor), b0 > (<)0.
With a habit di¤erential, the corresponding optimal consumption levelis high relatively to the other country’s, so that net foreign assets should7

be large enough to sustain it, as in (i) of Proposition 1. The e¤ect of global
habits is due to the interest rate adjustment. With adjacent complementarity,
a large Zt implies a large demand for present goods and hence a high interest
rate. The resulting interest revenue induces country H to accumulate net
foreign assets toward the long-run higher level. Equilibrium b thus negatively
relates to global habits Z, and (ii) follows.
By di¤erentiating (25) and (26), and substituting (11), (17), (18), (25),and (26) into the results, the equilibrium dynamic interactions of net foreign
assets and habit di¤erentials can be summarized as the autonomous systemof net foreign assets and habit di¤erentials:b_ =(! + )2
(et
b +
(
!)

bt

e_ = ! (et

(27)

e) ;

(28)

e) :

The phase diagram of (e; b) plane is depicted in Figure 1, where the b_ = 0
schedule is positively sloping and e_ = 0 schedule is in parallel to the verticalaxis.
As seen from (19) and (25), it is also noteworthy that the associatedconsumption dynamics are generated by
ct

c=

!+

(et

(29)

e) :

To obtain the welfare level of country H, linearize instantaneous utilitiesu (ct ; zt ) around a steady state and substitute the result into the lifetime
utility function (2) to obtain
Z 1
U0 =fu (c; z) + uc (ct c) + uz (zt z)g exp ( t) dt:
0

After substituting (29) into the above equation, substitute solutions to (17),(18), and (28) into the result. In this way, the lifetime utilities are obtainedasU0 =u (z; z)(! + ) uc + uz
(z0 z)(!)(! + ) (1)[( + ) uc + uz ] (Z0( + )(
!)

(30)

+

8

Y );

b

．

e=0

．

b=0

e
Figure 1. Equilibrium dynamics for relative habits and net foreign assets

9

U0 =

2.3

u (z ; z )(! + ) uc + uz
(z0 z )(!)(! + )[( + ) uc + uz ] (Z0( + )(
!)

(31)

+

Y );

Steady state

From (1), (5), (7), and (24), the steady-state equilibrium, (c; c ; z; z ; Z; b; b ; r),is determined by:b= z; c = z ;= Y = Z;= z (1
)z
= ;
= c;
!+
(z z0 )
b0 =
(
!)
b =
b:

(32)
(33)
(34)
(35)
(36)

c
c+c
e
r
rb + y(1) (! + )
b0
+
(
!)
+

Z

Z0 (37);(38)

where (37) comes from (24) evaluated at t = 0.
The steady state equilibrium conditions can be reduced to:
CC 0 : b

b0 =

!+
(e
(
!)

DD0 : b = e + (1

b0
+

e0 )
)Y

y;

(Y

Z0 ) ;(39)(40)

where the CC 0 schedule is obtained by evaluating (26) at t = 0; and DD0
schedule is obtained by substituting (32) through (35) into (36). Figure 2
illustrates the two schedules, where schedule DD0 is always steeper than CC 0 .
As shown by Figure 2, the steady-state equilibrium point e; b is givenby the intersection point E of the two schedules. Given this, c; c ; z; z ; Z
is determined by (32), (33) and (34); and b by (38).

10

b
D’

C’

E

b

C

D
e

e

Figure 2. Steady state equilibrium

11

From linearity of (39) and (40), we can examine the determinants of thelong-run external asset distribution by solving the two equations for b as
!+
(y
e0 )
b0 (Y Z0 ) ;(41)b=!( + )+
where y represents the weighted di¤erence of y and y :
y = y

(1

)y :

This implies that country H’s net foreign assets are determined as in Proposition 2:
Proposition 2: Country H’s steady-state holdings of external assets b arelarger as:
1. weighted income di¤erence in excess of relative depth of habits, yis larger and;2. global income in excess of global habits, Y
when b0 > 0 (resp. b0 < 0).

e0 ,Z0 is smaller (resp. larger)

To understand the …rst property, suppose that income di¤erentials y
exceeds habit di¤erentials e0 , y e0 . Then, with adjacent complementarity,ceteris paribus country H saves the excess income and thereby holds positive
external assets in the long run. We refer to this e¤ect as the relative surplusincome e¤ect. The second property represents the interest rate e¤ect: alarge global income in excess of global habits, Y Z0 , implies a low interest
rate, and when country H is a creditor b0 > 0, H’s interest rate income also
decreases thereby suppressing the long-term asset holdings. This e¤ect iscalled the interest rate e¤ect.

2.4

Income shocks

We consider global and local income shocks by …xing constant the magnitudeof the resulting increase dY in the aggregate output Y .4 Let us de…ne thetwo kinds of income shocks as follows:4This is just a simplifying assumption that eases comparison of the e¤ects of global andlocal income shocks. Even when we instead assume that increases in country H’s output
y are the same between the two shocks, and that, for local income shocks total output
increase dY equals dy, and for global income shocks dY equals dy=", our main results
below do not change.

12

global income shocks: dy = "dY and dy = (1") dY;

local income shocks in country H: dy = dY and dy = 0,
where " denotes the proposition of the associated shock dy to country H in
the total shock dY .
The e¤ects of the global and local income shocks on country H’s steadystate holdings of external assets b can be compared easily by using Proposition 2. Note …rst that the interest rate e¤ect, captured by the second term of(41), is the same for the global and local shocks. The relative surplus incomee¤ect, i.e., the …rst term of (41), is larger in the case of local income shocks
than in the case of global shocks because the local income shock has largere¤ect on income di¤erence between the two countries than the global shock.
The following corollary thus obtains:
Corollary 1: Local income shocks have greater e¤ects than global income
shocks on country H’s steady-state holdings of external assets b in the sense
that:
db
db
>
:
dY local shock
dY global shock3

Local and Global Income Shocks

Let us consider the e¤ects of positive local and global income shocks, dY > 0,in order. Before discussing each shock, note that the e¤ect on the interest
rate does not depend on whether the shock is local or global. This is because,by construction, the magnitudes of the resulting increase dY in the aggregate
output are the same between the two income shocks. Indeed, di¤erentiating(22) and (33) by Y yields
dr (0)
dr (0)==
dY local shock
dY global shock

< 0:

(42)

This implies that an increase in Y lowers r (0). With adjacent complementarity, the positive income shock, local or global, produces excess supply
in the time-zero good market, and hence lowers the equilibrium interest rate.
After the initial response, the interest rate then monotonically converges
toward as seen from (22).
1314

dr (0)
dY
dc̄
dY
dc̄∗
dY
dē
dY
db̄
db̄∗
=−
dY
dY
dU
dY
dU ∗
dY
−ηΩ < 0−ηΩ < 0δθ (ω + α) αθη (θ − ω) Ω+1−b0
ω (θ + α)ω (θ + α)2δθ (ω + α) αθη (θ − ω) Ω−b0
ω (θ + α)ω (θ + α)2αδ (θ − ω) αθη (θ − ω) Ω+−b0
ω (θ + α)ω (θ + α)2δ (ω + α) αη (θ − ω) Ω
−
+
2 b0
ω
(θ
+
α)
ω
(θ
+
α)
¶
µ
λ
θ
1−
ηΩb0
θ
α+θ
ηΩλ∗
b0 > 0θ+αε + (1 − ε − δ)θ (ω + α) αθη (θ − ω) Ω+b0
ω (θ + α)ω (θ + α)2θ (ω + α) αθη (θ − ω) Ω−
(1 − ε) − (1 − ε − δ)b0ω (θ + α)ω (θ + α)2α (θ − ω) αθη (θ − ω) Ω+
(1 − ε − δ)b0ω (θ + α)ω (θ + α)2(ω + α)αη (θ − ω) Ω
(1 − ε − δ)
+
2 b0
ω
(θ
+
α)
ω
(θ
+
α)
¶
µ
λ
θ
ε−
ηΩb0
θ µ α+θ¶θ
λ∗
1−ε+ηΩb0 > 0θα+θ(2) Global shocks
dy = εdY and dy ∗ = (1 − ε) dY(1) Local shocks
dy = dY and dy ∗ = 0

Table 1. The eﬀects of local and global income shocks0
α (1 − ε) (θ − ω)<0
ω (θ + α)α (1 − ε) (θ − ω)>0−
ω (θ + α)α (θ − ω)<0(1 − ε)ω (θ + α)(ω + α)(1 − ε)<0ω (θ + α)λ− (1 − ε) < 0θ
λ∗(1 − ε) > 0θEﬀect diﬀerences(2) − (1)

3.1

Local income shocks

We consider …rst the e¤ects of a positive local income shock, dy = dY > 0
and dy = 0. For simplicity, let us assume that b0 > 0 without further
notice. From (32) through (38), we obtain the steady-state e¤ect on each ofc; c ; e; b as in the second column of Table 1. They all are composed of thedirect income e¤ects, captured by the terms without b0 , and the interest ratee¤ects, represented by the terms with b0 . Without the interest rate e¤ect, asin the small country case (e.g., Ikeda and Gombi, 1999), the positive income
shock in country H increases savings, promotes interim asset accumulation
through current account surplus, and thereby increases c, e, and b while
decreasing b and c . The direct income e¤ects represent these in‡uences.
The interest rate e¤ects, caused by a fall in the interim market interest rate,
represent the countervailing negative income e¤ects on c, e, and b under apositive b0 , and positive e¤ects on b and c . Net e¤ects of the two haveambiguous signs.
Figure 3 illustrates the typical adjustment of the economy in the (e; b)and (e; c) plains. The local income shock shifts downward the CC 0 andDD0 schedules de…ned by (39) and (40) from C0 C00 to C1 C10 and D0 D00 to
DL DL0 , respectively, and thereby bringing the steady-state point from E0 to
EL , where the case in which b and e both increase is depicted. The interim
dynamics generated by (27) and (28) are illustrated as the arrowed pathfrom E0 to EL . Note that from a close look at Figure 1, the b dynamics canbe nonmonotonic: Country H can initially run the current account de…cit,and sooner or later it turns to surplus so as to generate a greater b. The
associated consumption dynamics are shown in the (e; c) plain, where thecase in which c increases from c0 to cL is depicted. After c jumps from F0 up
to FL01 instantaneously responding to the shock, it gradually rises up to FL
along the saddle trajectory (29).
The welfare e¤ects of the local income shock, dU /dY and dU /dY shownin column (1) of Table 1, are obtained by di¤erentiating (30) and (31) by
Y . Two properties are noteworthy: First, the positive local income shock
harms country H if and only if 1 <b0 =( + ). This represents theintertemporal version of immiserizing growth e¤ect that the associated fallin the interest rate, i.e., a deterioration in the intertemporal terms-of-tradefor creditor country H, harms it, irrespective of the direct bene…cial e¤ect of
the income increase. Second, this interest rate fall in turn de…nitely bene…ts
neighbor debtor country F.
15

b
D’0
C’0
D’L
C’1

D’G
bL

EL
E0

b0

bG

DL

EG
C0
C1
DG

D0

eG

e0

eL

FG

cG

FG01

c0

F0

FL01

FL

cL

c
Figure 3. Adjustment Processes

16eProposition 3 (the immiserizing growth e¤ect of local income shocks):
Suppose that country H is creditor b0 > 0 (resp. debtor b0 < 0). Then, due
to the intertemporal terms-of-trade e¤ects, a positive local income shock incountry H, (dy; dy ) = (dY; 0) ;1. reduces (resp. enhances) the country’s own welfare if and only if
1; and

+2. de…nitely bene…ts (resp. harms) country F.

3.2

Global income shocks

Let us next consider a positive global income shock, dy = "dY and dy =(1 ") dY , where the magnitude dY is the same as in the previous subsection. The steady-state e¤ect on c; c ; e; b is shown in the third column ofTable 1. As in the local shock case, the e¤ects comprise the direct incomee¤ects of domestic and foreign income increases, represented by the terms
without b0 , and the interest rate e¤ects, captured by the terms with b0 . Theinterest rate e¤ects are the same as in the local shock case. In contrast,
the direct income e¤ects are smaller for country H’s variables c; e; b; U andlarger for country F’s (c ; U ) than those in the local shock case. This is …rstly
because the associated increase in y is smaller under the global income shock,secondly because foreign income increase dy causes a negative crowding-out
e¤ect on c. As shown in the fourth column in the table, therefore the totale¤ects are smaller for country H’s variables c; e; b; U and larger for country
F’s (c ; U ).
Figure 3 depicts the typical dynamic adjustment to the global income
shock. The global income shock shifts the CC 0 schedule downward by thesame amount as in the local shock case, i.e., from C0 C00 to C1 C10 , whereas the
direction of the shift of the DD0 schedule depends on the relative magnitudesof " and 1(see (40)). Figure 3 illustrates a typical case in which " is smaller0
than 1, so that the DD0 schedule shift upward from D0 D00 to DG DG
withthe steady-state points e; b moving downward from points E0 to EG . Theinterim time path is indicated by arrows.5 The corresponding consumption
dynamics in the (e; c) plain are illustrated as the instantaneous downward
5

Although Figure 3 illustrates the transition dynamics of b are monotonic, they can be
nonmonotinic under certain parameter values.

17

b0 >

jump from F0 to FG01 , followed by over-time decreases along the saddle path
toward FG .
The welfare e¤ects of the global income shock, dU /dY and dU /dY shownin column (2) of Table 1, can be summarized as follows:
Proposition 4 (the immiserizing growth e¤ect of global income
shocks): Suppose that country H is creditor b0 > 0 (resp. debtor b0 < 0).
Then, due to intertemporal terms-of-trade e¤ects, a positive global income
shock, (dy; dy ) = ("dY; (1 ") dY ) ;1. reduces (resp. enhances) the country’s own welfare if and only if"; and

+2. de…nitely bene…ts (resp. harms) country F.
Note that the global income shock is more likely to cause the intertemporal immiserizing growth e¤ect than the local income shock does, because
compared to the local shock case, the direct income e¤ect (positive) is small
relatively to the intertemporal terms-of-trade e¤ect (negative). Indeed, comparing Propositions 3 and 4 implies the following corollary since " < 1.
Corollary 2: Intertemporal immiserizing growth e¤ects more likely takeplace due to global income shocks than due to local income shocks.
Remark 1: The result in the above corollary contrasts to that in the case ofstatic trade theory. In the typical static two-country trade model, where eachcountry specializes production in its export goods industry, global income
shocks that commonly occur in the export sectors of the two countries havesmaller e¤ects on the terms of trade, i.e., relative prices of two exporting
goods, than local income shocks of the same magnitude do, and hence are lesslikely to cause immiserizing growth e¤ects. In the present dynamic setting
the changes in the intertemporal term of trade, i.e., the interest rate, are thesame under global and local income shocks of the same magnitude, dY . Dueto the direct income e¤ects, global income shocks thus have greater e¤ects
on the two countries’welfare levels than local income shocks.6
6This reasoning is based on the simplifying assumption that increases in total output
Y are the same between global and local income shocks. Corollary 2 holds valid even

18

b0 >

4

Conclusions

In a two-country model with habit formation, we focus on interdependent
macroeconomic adjustments by considering global and country-speci…c income shocks on each country. The global habits and habit di¤erentials play
key roles in the global economic dynamics, possibly nonmonotonic, and in thedetermination of international asset distribution. A country’s net external
asset holdings rely on weighted income di¤erence in excess of habit di¤erentials and global income in excess of global habits. With habit formation,
positive income shocks lower the world interest rate, thereby harming thecreditor country and bene…tting the debtor country due to the intertemporalterms-of-trade e¤ect. In contrast to the case of trade theory, this intertemporal immiserizing growth is more likely to be brought about by global income
shocks than by country-speci…c income shocks.
For future research, two issues may be interesting. First, when a government can utilize the immiserizing growth e¤ect by means of …scal instruments, international strategic interactions would arise. It should be examined how the policy game changes our results. Secondly, extending empirical
study on habit formation to the interdependent world economy model would
be fruitful.

when we instead assume that increases in country H’s output y are the same between thetwo shocks, and that, for local income shocks total output increase dY equals dy, and for
global income shocks dY equals dy=". In that case, global income shocks have a larger
intertemporal terms-of-trade e¤ect than local income shocks, which results in Corollary 2
again. The point is that when a positive income shock is global, the relative magnitudesof the harmful intertemporal terms-of-trade e¤ect to the bene…cial direct income e¤ect islarger than they would be when the shock is country-speci…c.

19

AppendicesA

Deriving the Dynamic System (15)Note …rst that = is constant because _ t = t = _ t = t from (7). By eliminating c and z using (9) and (11) from the foreign counterpart of (6),combining the resulting equation and (6) yields:uc (c; z) +uc (Y c; Z z) +== constant.

By totally di¤erentiating this equation, we obtain:c^ =ucz + uczz^
ucc + ucc

ucc + ucc^+

ucc + ucc

^ +

ucz
^
Z:
ucc + ucc

We substitute this equation into (1), (8), and the foreign counterpart of(8) and eliminate c and z using (9) and (11) from the resulting equation.
Then, from (12), the autonomous dynamic equation system with respect toz^; ^; ^ ; Z^ is obtained as follows:22
ucz + ucz^+^
+ 1 z^
ucc + ucc
ucc + ucc
ucc + ucc
ucz
^
+
Z;
ucc + ucc
ucz ( ucz + ucz )
ucz
^
_ =
uzz z^ ++ +ucc + ucc
ucc + ucc
ucz
ucz ucz ^^Z;(43)
ucc + ucc
ucc + ucc
ucz ( ucz + ucz )
ucz
_ =
^
uzz z^
ucc + ucc
ucc + ucc
ucz
ucz2
^ +
^
+
+ +
uzz Z;
ucc + ucc
ucc + ucc
^
Z_ =
Z:

z_ =

From the de…nitions (13) and (14) of v and &, respectively, this autonomous
system reduces to (15).
20

B

Equilibrium Solutions

B.1

Dynamics of habit capital z: (17)The stable roots of dynamics (15) are given by ! andas in (16). Letting0
m denote (z; &; Z) , the general solution to (15) can thus be expressed asm^ (t) = A1 exp (!t) q + A2 exp (t) h;where q(q1 ; q2 ; q3 )0 and h(h1 ; h2 ; h3 )0 represent the eigen vectors associated with stable roots ! and, respectively. From (15), it is easy tocon…rm that q3 = 0. By eliminating A1 exp (!t) and A2 exp ( t) from thethree equations in the above vector equation, we obtain^& =q h2 q2 h1 ^q2z^ + 1
Z;
q1
q1 h3

(44)

where the coe¢ cients of z^ and Z^ can be obtained by exploiting the de…nitionof the eigenvectors q and h as
q2
=
q1
q2 h1
=

q1 h2
q1 h3

(! + ) vcc + vcz
2;vcc (vcz vZZ + vcZ vzz )vcc vzz + (2 + ) vcc vcz
f(! + ) vcc + vcz g f vcc vZZ (2 + ) vcc vcZ g:+
2 ( v v + (2 + ) v v )
cc zz
cc cz

Substituting (44) into the z-equation_in (15) yields (17).

B.2

The interest rate

From (1) and (6) through (8), the optimal consumption dynamics are givenbyc_ =r^ ^ ;
ucc
where represents the rate of time preference,^=(uzz + uzc )z^21( + )^:

Substitute (20) into the above Euler equation. The resulting equation canbe solved for r^ as
r^ =

!ucc

(uzz + ucz )c^ +( + )^:z^(45)

In the above, ^ can be obtained from (14),(18), (43), and (44) as
ucc
+

^=

H

!

uzz
z^:
+2

c^ +

(46)

Substituting (46) and (19) successively into (45) yields (22).

B.3

Net foreign assets

Set^b = {1 z^ + {2 Z:^(47)Di¤erentiating (47) with respect to time t yields_
b_ = {1 z_ + {2 Z:(48)

Since b_ is given by (5), this equation implies
b_ = r^b0 + r^b_c^ = {1 z_ + {2 Z:(49)

Substitute (17), (18), (19), (22), and (47) into (49). By comparing the coef…cients of the resulting equation, we obtain(!
(! + ) (1) {1r) {1 = a1 b0(r + ) {2 = a2 b0 +

!+

;

!+

(1

):

This simultaneous equation can be solved for {1 and {2 as
{1 =
{2 =

1
r

!+
!

(1

) (! + )
(r !)

(50)

b 0 a1 ;b0r+(1) (! + )a1 + a2 :(r !)Substituting (50) and (51) into (47) yields (24).
22

(51)

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24