Discussion Paper No. 619

HABIT FORMATION IN
AN INTERDEPENDENT WORLD ECONOMY

Shinsuke Ikeda
and
Ichiro Gombi

September 2004
Revised July 2008

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

Habit Formationin an Interdependent World Economy1
Shinsuke Ikeda2 and Ichiro Gombi3
Osaka University and Ritsumeikan University
September, 2004
Revised, July 2008

1 A shortened version of this paper is forthcoming in Macroeconomic Dynamics.

The authors are grateful for the helpful comments on the earlier versions to G.
Bauer, L. Cheng, C. Johnston, A. Kajii, K. Mino, A. Shibata, two anonymousreferees, and the participants of the 7th International Meeting of the Society forSocial Choice and Welfare (Osaka, Japan), the 79th WEAI Conference (Vancouver, Canada), the 57th IAEC meeting (Lisbon, Portugal), the JEA Spring meeting(Meiji-Gakuin University, Tokyo), the annual meeting of IEFS Japan (Kobe University, Kobe), and the seminars at Kyoto, Osaka, and Ritsumeikan Universities.
We appreciate financial supports for Ikeda from Grants-in-Aid for Scientific Research (C No. 1553021) from the Japan Society for the Promotion of Science andthe 21st COE Program from the Ministry of Education, Culture, Sports, Scienceand Technology, and for Gombi from Open Research Center Project for Private
Universities: MEXT, 2004-2008.
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 world economy, consumption-habit dynamics in one country are aﬀected, due to endogenous interest rate adjustments, by the othercountry’s habits and preferences. External indebtedness depends cruciallyon international diﬀerences in habit-adjusted net output less habitual living
standard. Interest rate adjustments enlarge the consumption impact of an
income shock. Consistently with the empirical facts, the habit parameterof a large country, therefore, would be underestimated, and the current account volatility overestimated, if estimated using a small-country model. An
increase in fiscal spending in one country can benefit the country and harm
the neighbor due to intertemporal terms-of-trade eﬀects.
JEL Classification Numbers: F41, D90.
Keywords: Habit formation, two country model, immiserizing growth.

1Introduction

An important stylized fact is that consumers’ habit forming behavior has asignificant eﬀect on intertemporal choices, and hence macroeconomic phenomena (e.g., Constantinides, 1990, Campbell and Cochrane, 1999, Carroll et.al., 2000, Díaz et.al., 2003). Accordingly, there have been numerous
attempts to better understand open-macroeconomic phenomena by incorporating habit formation. For example, by introducing consumers’ habitpersistent behavior to small-country models, Mansoorian (1993a, b) andIkeda and Gombi (1999) derive its implications for current account dynamics
and macroeconomic policies. Gruber (2002) provides empirical support to an“intertemporal current account model” with habit formation as explaining
the actual current account behavior of the G-7 countries.
However, in the existing studies, including those small-country analyses,
habit formation is examined within the representative-agent framework, inwhich consumers’ interactions due to pecuniary externalities are completely
neglected. When the interest rate is endogenously determined in a world
economy with two heterogeneous agents (countries), one consumer’s (or country’s) habit forming behavior is aﬀected by other consumers’ (or countries’)
habits and preferences. It has a particular importance to examine how thisproduces diﬀerent (or same) results from what the literature has, theoretically or empirically, predicted using small-country and/or representativeagent models; and to figure out what kind of bias would arise when a large
country’s consumption behavior is estimated by using a small-country model.
The two-country analysis would also enable us to investigate how habit formation in consumption aﬀects each consumer’s (country’s) long-run indebtedness and the transmission mechanism of various policies.
This paper presents a tractable two-country world economy model withheterogeneous habit forming consumers. The purposes of the paper are: (i) to
describe interactive habit-forming consumer behavior in a world economy; (ii)to examine long-run external indebtedness; (iii) to address how fiscal policies
in one country aﬀect the two countries’ consumptions, wealth holdings, andwelfare levels; and (iv) to figure out empirical implications. Issues (ii) and (iii)
are also examined by Devereux and Shi (1991) by using a variable discountrate model with capital accumulation.1 At the cost of assuming away capital1
The other literature concerning two-country dynamic models include Turnovsky andBianconi (1992), Kalayalcin (1996), and Bianconi and Turnovsky (1997), which focus
on supply side adjustments such as capital accumulation and/or leisure/labor choice. See1

accumulation, our main interest is to focus on habit formation as a more
empirically relevant component of the macroeconomic adjustment process.
Under habit formation, consumption dynamics depend crucially on intertemporal complementarities. For example, when the marginal utility of
consumption is strongly and positively related to habits, which Ryder and
Heal (1973) designate adjacent complementarity, optimum consumption istied tightly to and comoves positively with current habits. When there areheterogeneous consumers or countries, the behavior of individual consumers
is aﬀected by other consumers’ habits and preferences, as well as their own,
due to interest rate adjustments. Equilibrium intertemporal complementarity
in consumption can thus diﬀer from what is defined by individual consumers’
subjective preferences.
To examine habit forming consumer behavior in an interdependent world
economy, we propose to simplify the general equilibrium system by using
two aggregations. First, we construct a world felicity function from bothcountries’ felicity functions. One appealing aspect of this approach is theability to thereby characterize intertemporal complementarity in each country’s equilibrium consumption in a similar way as in the small-country case.
Even though habit preferences in one country display distant (resp. adjacent)complementarity, its equilibrium consumption behavior may exhibit adjacent(resp. distant) complementarity. A corollary is that even if one country has
no preference for habits, the time series of consumption rates will display
habit-persistent patterns.
Second, we define an aggregate habit capital as the sum of individual
countries’ habit capitals. When aggregate output is constant, the aggregate
habit capital is also constant since, from the market clearing condition, it
equals aggregate output. When aggregate output changes at a point in time
due to exogenous shocks, the aggregate habit formation aﬀects consumption
dynamics as an additional dynamic source. The resultant consumption behavior in each country is then aﬀected by the aggregate habit capital, as wellas its own habit capital.
We characterize long-run external indebtedness by defining the eﬀective
net output that can be allocated for saving under habit formation as surplus
income. It is shown that a country is more likely to be a long-run creditor
the larger surplus income it has. Surplus income depends on net output (i.e.,also Ikeda and Ono (1991) for an analysis of non-monotonic multi-country dynamics underheterogeneous consumer preferences.

2

output income less government spending), the initial stock of habit capital,
and the strength of adjacent complementarity. When a country’s net output
is larger (resp. smaller) than its habitual consumption level, we show thatthe country has larger (resp. smaller) surplus income and hence is more
likely to be a creditor (debtor), the stronger adjacent complementarity it
exhibits. This result is shown to be consistent with the usual proposition
that a country tends to be a long-run creditor the more patience it has. The
novelty of this paper is that which country is more patient depends on therelative strengths of adjacent complementarity and on how much greater net
output than the habitual consumption standard they have.2
We address macroeconomic adjustments in response to an increase incountry H’s fiscal spending financed with lump-sum taxes. As shown in theliterature (see, e.g., Mansoorian 1993a, b), in the small-country case, negative income shocks, such as an increase in fiscal spending, necessarily lowers
both initial and steady-state consumption levels; and decreases (resp. increases) net foreign assets under adjacent (resp. distant) complementarity.
By assuming that the initial holding of net foreign assets, and hence the income re-transfer eﬀect due to interest rate changes are negligible, we showthat an increase in a country’s fiscal spending (i) can increase the initialor steady-state consumption level of that country, with the other country’s
consumption being crowded out; (ii), as in the small country case, necessarily worsens (resp. improves) the steady-state external asset position if
the country’s preference displays adjacent (resp. distant) complementarity,
irrespective of the other country’s preferences; and (iii) has long-run andshort-run spill-over eﬀects on the other country’s consumption, depending
crucially on whether the policy implementing country displays adjacent or
distant complementarity.
Our two-country model helps to explain two puzzling facts that Gruber(2004) reports by using a small-country model. First, his GMM estimatesof the habit preference parameter for two large countries, United States andJapan, are smaller than those for smaller countries, e.g. Italy, Spain, andNetherlands. Second, in the case of the two large countries, his habit model
overpredicts the actual volatility of the current account, unlike for in thesmaller country case. To explain these, say, "large-country puzzles", we2
For the eﬀect of habit formation on wealth distribution, see Díaz et al. (2003). Theyshow that habit formation decreases wealth inequality by raising wealth-poor consumers’s
precautionary savings more than wealth-rich ones’.

3

show that, under adjacent complementarity, interest rate adjustments enlarge
the consumption eﬀect of an income shock and thereby mitigate its current
account eﬀect. As an important empirical implication, the habit parameter ofa large country would be underestimated, and the current account volatility
overestimated, if they were estimated, as in Gruber (2004), using a smallcountry model.
With non-negligible initial net foreign assets, an increase in fiscal spending induces additional income eﬀects by changing the world interest rate.
Even though fiscal spending is assumed to generate no direct utility, it isshown, as an important welfare implication, that due to the intertemporalterms-of-trade eﬀect, an increase in fiscal spending financed by lump-sum
taxation can benefit the policy implementing country and harm the neighboring country. This is an intertemporal version of (reversed) “immiserizing growth” discussed by, e.g., Bhagwati (1958) and Brecher and Bhagwati(1982).
The remainder of the paper is structured as follows: In Section 2, we
present a two-country model. In Section 3, we examine equilibrium dynamics
of consumption, the interest rate, and net foreign assets. Section 4 analyzes
the eﬀects of an increase in one country’s fiscal spending. In Section 5, welfare
enhancing government spending is discussed. Section 6 concludes the paper.

2

The Model

Consider a two-country world economy composed of home and foreign countries. Each country is populated with infinitely lived identical agents. The
representative agents in home and foreign countries are referred to as consumers H and F, respectively. They consume a single consumption good andhold wealth in the form of bonds. Both goods and bonds are assumed tobe costlessly traded in international markets. For brevity, the representative
agents H and F are assumed to be endowed with constant 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 =R t (∗)
α −∞ cs exp (−α (t − s)) ds, or equivalently
´
³
(∗)
(∗)
(∗)
żt = α ct − zt ,(1)
4

where ẋ represents the time derivative of variable x and α represents thediscount rate for past consumption rates. We assume that consumers in bothcountries have the common discount rate α for past consumption rates. This
enables us to obtain the tractable dynamics of a two-country equilibrium.
Consumers H and F have diﬀerent preferences over consumption andhabits. Consumer H’s preferences are specified asZ ∞U0 =u(ct , zt ) exp (−θt) dt,(2)0

where θ represents the subjective discount rate, which is assumed constant.
Following Ryder and Heal (1973), function u is assumed to satisfy the following regularity conditions: (C1) uc > 0; (C2) uz 5 0; (C3) uc (c, c)+uz (c, c) >0; (C4) u is concave in (c, z) ; (C5) limc→0 uc (c, z) = ∞ uniformly in z; and(C6) limc→0 [uc (c, c) + uz (c, c)] = ∞. Consumer F’s utility U0∗ is specified in
the same way.
Due to habit formation, consumer preferences are intertemporally dependent. Within the representative agent framework, Ryder and Heal (1973)
characterize the resulting intertemporal complementarity in consumption by
adjacent and distant complementarities: With adjacent complementarity, an
increase in today’s habits increases the marginal utility of today’s consumption more than it increases marginal disutility of habits, thereby enlarging
today’s optimal consumption. Under distant complementarity, an increase
in today’s habits increases marginal disutility of habits so much that it reduces today’s consumption rate. We apply this idea to characterize eachcountry’s preferences: Consumer H’s preference is said to display adjacentα(distant) complementarity when ucz (c, c) + θ+2αuzz (c, c) > (<)0. Similarly,
adjacent and distant complementarities in country F are defined in termsof u∗ . In the case of the representative agent economy, intertemporal complementarities displayed by consumers’ preferences definitely determine theequilibrium dynamics (see, e.g., Ryder and Heal 1973 and Ikeda and Gombi1999). In contrast, when the economy consists of heterogeneous agents, as in
the present case, pecuniary externalities through markets divert each agent’s
consumption saving behavior from what is predicted from each agent’s own
preference, as shown in the following section.
To ensure the steady state, let us assume thatθ = θ∗ .

5

Unlike in the single agent model, the interest rate r can deviate from thissubjective discount rate.
Let bt denote net foreign assets held by consumer H. The flow budget
constraint for consumer H is given byḃt = rt bt + y − ct − τ ,(3)where τ represents a lump-sum tax levied by the government in country H.
Given the initial values (b0 , z0 ) , consumer H chooses C0 = {ct , bt , zt }∞t=0 soas to maximize (2) subject to: (i) the flow budget constraint (3); (ii) theformation of consumption habits (1); and (iii) the transversality conditions.
Letting λt (= 0) be the shadow price of savings and ξ t (5 0) that of habit
formation, the optimal conditions are given byuc (c, z) = λt − αξ t ,(4)

λ̇t = (θ − rt ) λt ,(5)ξ̇ t = (θ + α) ξ t − uz (ct , zt ) ,(6)

together with (1), (3), and the transversality conditions for bt and zt . Consumer F’s behavior can be specified in exactly the same way.
The governments in the two countries follow the balanced budget principle. Their fiscal spendings g and g ∗ equal tax revenues from lump-sum taxes
τ and τ ∗ , respectively. By substituting this into (3) we obtain the balance of
payment equation,ḃt = rt bt + y − ct − g.
(7)
The model is closed by introducing the market clearing conditions:ct + c∗t = Y (≡ y + y ∗ − g − g ∗ ) ,(8)

bt + b∗t = 0,(9)where Y represents the disposable aggregate income. By Walras’ law, these
are not independent: (9) together with (7) and the corresponding constraint
for the foreign consumer imply (8). In sum, the equilibrium time path of(bt , b∗t , ct , c∗t , zt , zt∗ , rt , λt , λ∗t , ξ t , ξ ∗t ) is determined by equations (1), (4) through(7), the corresponding equations for F, and the market equilibrium condition(8) or (9).

6

3

Two-Country Equilibrium

3.1

The world felicity function

In this economy consumption dynamics in the two countries interact throughinternational market transactions. To ease the analysis, it is useful to define
aggregate habit capital Zt as
Zt ≡ zt + zt∗ .

(10)

Since α is assumed to be internationally identical, the dynamics of Zt can be
expressed from (1) and the market clearing condition (8) as
Żt = α (Y − Zt ) .

(11)

Note that this is an autonomous dynamic equation. If Zt and zt are given,zt∗ are determined from (10) as
zt∗ = Zt − zt .

(12)

By using (12), dynamics can be drastically simplified in the following
manner. Define σ as:λ
σ = ∗,
λ∗which is constant over time since λ̇t /λt = λ̇t /λ∗t from (5) (and the corresponding equation for F). We then construct aggregate indices for (u, u∗ )and (ξ, ξ ∗ ) as:v (c, z, Z) ≡ u (c, z) + σu∗ (Y − c, Z − z) ,ς ≡ ξ − σξ ∗ ,

(13)
(14)

where (8) and (12) are substituted.
The new felicity function v represents an aggregate utility function withweights being given by relative individual marginal utilities. We could call it
the world felicity function. The new shadow price ς is the weighted diﬀerencein the shadow prices of habits. From the definition of v, ς could be regarded
as net marginal gains of transferring consumption capital from consumer F
to H. With these definitions, we can reduce the equilibrium dynamics ofconsumption habits around a steady-state point as follows:7

Proposition 1: In equilibrium, consumer H’s habits around a steady-state
point are governed by
⎞⎛
⎛
⎞ ⎛
⎞
cz
− αv
ẑ
ż
vcc
vcz vcz +vcc vzz ⎠ ⎝⎝ ς̇ ⎠ = ⎝ Mς̂ ⎠ ;
(15)
vcc
Ż
Ẑ
0 0
−α
´
³
!
Ã
2
− vαcc−α 1 + vvcz
cc,M=
(vcz )2 −vcc vzz
α(vcc +vcz )
θ
+
vcc
vcc
where x̂ denotes a deviation of variable x from its steady-state value x̄: x̂ ≡xt − x̄.
Proof. See Appendix A.
Remark 1: The dynamic equation (15) ³
is block´ recursive, where coeﬃcientmatrix M for habit and its shadow price ẑ, ζ̂ is exactly the same as in thecase of the small-country model (e.g., Ikeda and Gombi 1999 and Mansoorian
1993a, b), except that v and ς represent the world average preferences, insteadof individual countries’ preferences.
Remark 2: As easily seen, Proposition 1 is an application of the second
theorem of welfare economics. Without any distortion, the Pareto optimal
resource allocation attained as a complete market equilibrium can be duplicated as a solution to a social welfare maximization problem:
max U0 + βU0∗ ,subject to Y = ct + c∗t , Zt = zt + zt∗ , żt = α (ct − zt ) ,and Żt = α (Y − Zt ) .

Indeed, the corresponding Hamiltonian function,
Ht = u(ct , zt ) + βu∗ (Y − ct , Zt − zt ) + ς t α (ct − zt ) + ιt α (Y − Zt ) ,produces the same optimal condition as in Proposition 1 when weight β isset equal to σ.
Since the dynamic equation (11) of aggregate habit Z is autonomous, it isnoteworthy that, if the economy is initially in steady-state equilibrium, and8

if the steady-state value of Z, which equals Y from (11), does not change, theaggregate habit stays at value Y . In other words, in the absence of aggregate
income shocks, the aggregate habit dynamics degenerate at Z̄(= Y ), so³ that´
the equilibrium dynamics in (15) are completely described by the ẑ, ζ̂dynamics, as in the small-country case. Note, however, that the transition
matrix here is defined in terms of the world felicity function.

3.2

Equilibrium dynamics

3.2.1

Habit and consumption

Dynamic system (15) has two stable roots:qθ − (θ + 2α)2 − 4α (θ + 2α) Ω(< 0) and − α,
(16)
ω≡
2
and one unstable root, which¡ is conjugate¢with ω, where ω is a characterα
istic root of M; and Ω ≡ − vcz + θ+2α
vzz /vcc . In parallel with intertemporal complementarities with respect to the individual consumers’ preferences, Ω captures equilibrium intertemporal complementarities: we refer tothe world felicity function as displaying adjacent (distant) complementarityαwhen vcz (c, c) + θ+2αvzz (c, c) > (<)0, i.e., Ω > (resp. <)0.
Let us introduce complementarity indices ΩH and ΩF for the individual
countries’ preferences, in parallel with Ω, as
¶
¶
µ
µ
α
α
H
F
∗
∗
Ω ≡ − ucz +/u∗cc ,
uzz /ucc and Ω ≡ − ucz +uθ + 2αθ + 2α zz

where, e.g., a positive ΩH represents adjacent (distant) complementarity for
H. The indices, ΩH and ΩF , capture the strength of habit formation of theindividual consumers H and F. When ΩH is larger than ΩF , for example,
complementarity between consumption and habit is stronger for consumer H
than for consumer F.
Definition: Consumer H is more (less) in the habit of consuming thanconsumer F when ΩH > (<)ΩF .
From the definition (13) of v, intertemporal complementarities with respect to the individual countries’ preferences can be related to those prevailing in equilibrium by:Ω = εΩH + (1 − ε) ΩF ;
(17)
9

ε≡

ucc
.
ucc + σu∗cc

Therefore, adjacent (distant) complementarity in both countries implies adjacent (distant) complementarity in the two-country equilibrium. When theindividual countries’ preferences display diﬀerent intertemporal complementarities, equilibrium intertemporal complementarity is determined by the relative strength of intertemporal complementarities regarding the individual
preferences.
As shown in Appendix B.1, the saddle plane governed by the two stableroots are expressed asż = ωẑ − (ω + α)(1 − δ)Ẑ,(18)

Ż = −αẐ,

(19)

where
δ ≡ εΩH /Ω.
By equating (18) to (1), the consumption dynamics are given by
µ
¶
´
ω+α ³
ĉ =
ẑ − (1 − δ)Ẑ .
α(20)

Diﬀerentiate this by t and substitute (1), (19), and (20) successively into
the result. Then, by taking (8) into account, we obtain the motion of eachcountry’s consumption asċ = ωĉ,(21)ċ∗ = ωĉ∗ .

(22)

Irrespective of the second-order habit dynamics of (18) and (19), therefore,
the equilibrium consumption dynamics are of the first order. Equations (1)and (21) jointly govern the equilibrium dynamics of (c, z); and equations (1)and (22) do dynamics of (c∗ , z ∗ ). The resulting phase diagrams are illustrated
in the (c, z) and (c∗ , z ∗ ) planes of Figure 1.
To understand the consumption-habit dynamics given by (20), consider
first the case without any aggregate income shocks, Ẑ = 0, in which case theequilibrium ĉ-ẑ relationship reduces to:
¶
¶
µ
µ
ω+α
ω+α ∗
∗
ẑt and ĉt =
ẑt .
ĉt =
(23)
α
α
10

From equations (16) through (23), when the world felicity function displays
adjacent complementarity (Ω > 0), ω + α is positive and thus the stablearms (23) are positively sloping, whereas under distant complementarity withrespect to v (Ω < 0) the trajectories have a negative slope with negative
ω + α. That is, in equilibrium, both countries experience positive or negative(∗)(∗)
comovements between consumption and habits, i.e., dct /dzt > (<) 0, as
the world felicity function displays adjacent or distant complementarity. This
property is the same as in the small-country case except that intertemporalcomplementarities here are defined in terms of the world felicity function,
instead of each country’s felicity function.
When aggregate habit capital Ẑ varies, the equilibrium consumption dynamics depend on Ẑ. The second term of (20) captures the eﬀect of anincrease in Ẑ for given ẑt . Alternatively, we substitute the definition of δ,(17), and (10) successively into (20) to obtainµ¶¢ω+α ¡ Hĉ =εΩ ẑ − (1 − ε) ΩF ẑ ∗ .
(24)
αΩSince (ω + α) /Ω is positive irrespective of whether the world felicity function
displays adjacent or distant complementarity,3 this reveals that the sign of ĉ
is determined by the relative magnitudes of εΩH ẑ and (1 − ε) ΩF ẑ ∗ . ĉ canthus be negative even when εΩH ẑ is positive if (1 − ε) ΩF ẑ ∗ is large enough.
Suppose that both the countries’ preferences display adjacent complementarity, ΩH , ΩF > 0. If ẑ and ẑ ∗ are positive, each country’s consumer plans
to consume a larger quantity than the steady-state level. Since a positiveĉ∗ implies a negative ĉ in equilibrium, the sign of equilibrium ĉ should be
negative when (1 − ε) ΩF ẑ ∗ is larger than εΩH ẑ. As will be shown later, themarket will be cleared by a rise in the interest rate.
It is important to note that even if one country has no preference for
habits, the time series of consumption rates will display seemingly habitpersistent patterns. For example, suppose that country H has no preference
for habits: uz = 0, uzz = 0, ucz = 0, and hence ΩH = 0. From (20), theequilibrium consumption of country H still depends on its own habit capital,
with root ω being determined solely by country F’s preferences for habits (see(16) and (17)). Given that c is related to c∗ by market equilibrium condition(8), and that z is related to z ∗ and Z by (10), the dependence of c∗ on z ∗3When Ω = 0, we have ω + α = 0. It can be shown from (16) that lim (ω + α)/ Ω = α.
Ω→011due to country F’s habit preferences induces the dependence of c on (z, Z)
even though c does not depend directly on z.
3.2.2

The interest rate

As is proven by Appendix B.2, the interest rate dynamics are given by
#
"
¡ F
¢
F
H
H F
¡ H
¢
−
Ω
Ω
κ
(1
−
ε)
Ω
Ω
Ω
r̂ = κ Ω − ΩF ẑ +
+
η Ẑ,
Ω
Ω

(25)

where κ and η are defined asκ≡αucc u∗cc
ωucc u∗cc (θ + 2α)(> 0) .
(>
0)
;
η
≡
−
λ∗ (α + θ − ω) (ucc + σu∗cc )λ∗ ucc + λu∗cc

Equation (25) reveals that the equilibrium interest rate is driven by country H’s habit capital ẑ and the aggregate habit capital Ẑ. Since ẑ − ẑ ∗ =2ẑ − Ẑ, a change in ẑ with a given Ẑ can be taken as a change in diﬀerence
between ẑ and ẑ ∗ . Equation (25) thus implies that the dynamics of r̂ rely on
the diﬀerence between, and the sum of, ẑ and ẑ ∗ . Given that the dynamics
of z and Z are jointly generated by (18) and (19), it is easy to show that theresulting dynamics of the interest rate can be non-monotonic. As seen from(25), r̂ also depends on the diﬀerence in ΩH and ΩF as well as on the averageΩ of the two.
It is too complicated to analyze r̂ by using (25) directly. Instead, we
focus on two special cases: that (i) the aggregate habit stock stays constantat the initial steady-state value, i.e., Ẑt = 0 for all t; and that (ii) the twocountries’ preferences exhibit identical degrees of adjacent complementarity,i.e., ΩH = ΩF = Ω.4
Case (i): Ẑt = 0 for all t. In this case, equation (25) reduces to:¡¢r̂ = κ ΩH − ΩF ẑ,
4(26)This case takes place (i) when the values of Ωi (i = H, F ) happen to be the samein steady state; (ii) when the felicity functions take quadratic forms with constant andinternationally identical second-order derivatives; or (iii) when the two countries are perfectly identical with respect to preferences, endowments, and the initial stocks of habit
capitals and net foreign assets. In case (iii), although the Ωi values are globally identical,
we cannot examine the case of non-zero b0 .

12

implying that r̂ positively (negatively) comoves with ẑ as consumer H is
more or less in the habit of consuming than consumer F, i.e., ΩH − ΩF ispositive or negative.5 To understand this, suppose that both consumers H
and F display adjacent complementarity (ΩH > 0, ΩF > 0); that ΩH > ΩF ;and that z (t) > z̄, and hence, from (12), z ∗ (t) < z̄ ∗ . Then, if r equaled θ,
consumer H would consume more now than in the future steady state, thereby
producing excess demand in the “present good” market, irrespective of that
consumer F would consume less now than in the future (note: ΩH > ΩF ).
The market clearing r should thus be higher than its steady-state level θ.
As shown later, the dynamics of ẑ are monotonically driven by stable rootω when Ẑt = 0. From (26), therefore, the interest rate in this case displays
monotonic dynamics,ṙ = ωr̂.
Case (ii): ΩH = ΩF = Ω. When the two countries’ preferences exhibit theidentical degrees of adjacent complementarity, the equilibrium interest rateis determined from (25) asr̂ = ηΩẐ,(27)

implying that Ẑ and Ω play crucial roles in this case. Suppose that Z (t) > Z̄.
Then, if consumers exhibit adjacent (distant) complementarity, i.e., Ω > 0(Ω < 0), ceteris paribus there prevails excess demand (supply) in the present
good market. This renders r higher (lower) than its steady-state value θ. The
equilibrium interest rate positively (negatively) comoves with the aggregate
habit stock, which exhibits monotonic motions with stable root −α (see (19)).
The resulting dynamics of r are given explicitly byṙ = −αr̂.
3.2.3

(28)

Net foreign assets

The transition dynamics of net foreign assets also depend on the property of
the world felicity function and international heterogeneity in habit formation.
5

If country H is identical to country F (ΩH = ΩF ), or if one of the two countries issmall in the sense that λ or λ∗ is infinitely large (λ or λ∗ → ∞), for which case κ is zero,the interest rate dynamics degenerate at θ as in the small-country case (e.g., Ikeda andGombi 1999).

13

As shown by Appendix B.3, by linearizing (7) and substituting (24) and (25)
into the result, we can obtain¶∙µ1(ω + α) (1 − ε) ΩFω+α− a1 b0 ẑt + −
b̂t =
θ−ω
α
α (θ − ω) Ω
¾¸
½
F
b0
a1 (1 − ε) Ω (ω + α)+(29)
− a2 Ẑt ,θ+α(θ − ω) Ω
where a1 and a2 represent the coeﬃcients of ẑ and Ẑ in (25):¡¢
a1 = κ ΩH − ΩF ,¡¢(1 − ε) ΩF ΩF − ΩH κ ΩH ΩF
a2 =
+
η,
Ω
Ω
respectively.
Two habit stocks ẑt and Ẑt aﬀect b̂t by changing consumption and theinterest rate. Changes in the habit stocks aﬀect consumption through (20)and thereby influence the accumulation of net foreign assets. This eﬀect
is expressed by the terms without b0 in (29), which depend on ω + α, i.e.,
whether the world felicity function displays distant or adjacent complementarity. The habit stocks also aﬀect the interest rate by (25). The resulting
change in interest income alters the time profile of net foreign assets. The
eﬀect is captured by the terms associated with b0 in (29). These income retransfer eﬀects through interest rate changes may well have various adverse
eﬀects. To avoid analytical complexity, we get rid of the eﬀects by assumingthat the initial amounts b0 of net foreign assets for both countries are negligible, as in Devereux and Shi (1991). The implications of a nonzero b0 will
be discussed briefly in Section 5 and in Appendix C.
Assumption 1: b∗0 ≈ 0.
With Assumption 1, equation (29) reduces to
´
ω+α ³
b̂t =
ẑt − (1 − δ) Ẑt .
α (θ − ω)Combining (20) with the above equation yields the saddle arm in the (b, c)
plane:1ĉ (t) .
(30)b̂ (t) =θ−ω
14Since from (21) the dynamics of ĉ (t) are monotonic, those of b̂ (t) are also
monotonic. Figure 1 depicts the saddle arm in the (b, c) plane as upward
sloping schedule DD0 .

3.3

Steady state

From (1), (5), (7), and (29), the steady-state equilibrium, (c̄, c̄∗ , z̄, z̄ ∗ , Z̄, b̄, b̄∗ , r̄),is determined by:z̄, c̄∗ = z̄ ∗ ,Y = Z̄,θ,c̄ + g,¢(ω + α) (1 − ε) ΩF ¡ω+αZ̄ − Z0 ,(z̄ − z0 ) +b̄ =α (θ − ω)α (θ − ω) Ω∗b̄ = −b̄ .

c̄c̄ + c̄∗r̄
r̄b̄ + y

=
=
=
=

(31)
(32)
(33)
(34)
(35)
(36)

together with (8), (9) and (12), where (35) comes from (29) evaluated at
t = 0.
By substituting (31) and (33) into (34), we obtainθb̄ + y = z̄ + g.

(37)

Combining (32) and (35) yields
ω+α(ω + α) (1 − ε) ΩF(z̄ − z0 ) +(Y − Z0 ) .
(38)
α (θ − ω)α (θ − ω) Ω¡ ¢Equations (37)and(38)
jointly
determineb̄, z̄ . Figure 2 depicts the deter¡ ¢0mination of b̄, z̄ : schedule BB depicted with a positive slope represents(37), whereas schedule SS 0 , which represents (38), can be either positively
or negatively sloping as the world felicity function displays either adjacentor distant complementarity. Even when schedule SS 0 is positively sloping,schedule
BB 0 is always steeper than SS 0 . The steady-state equilibrium point¡ ¢schedules. Given this,¡b̄, z̄∗ is∗ given¢ by the intersection point E of the two∗c̄, c̄ , z̄ , Z̄ is determined by (31) and (32); and b̄ by (36).
From linearity of (37) and (38), we can examine the determinants of thelong-run external asset distribution by solving the two equations for b̄ asω+αb̄ = −(I − I ∗ ) ,(39)
ω (θ + α) Ω
b̄ =

15

whereµ¶ω (θ + α)I = εΩ
y−g+z0 ,α (θ − ω)µ¶ω (θ + α) ∗
∗
F
∗
∗
z .
I = (1 − ε) Ω y − g +α (θ − ω) 0
H

Roughly, terms I and I ∗ represent net outputs y (∗) −g (∗) in excess of habitual(∗)
consumption levels ω(θ+α)
z , adjusted by the relative strengths of adjacentα(θ−ω) 0
complementarities εΩH and (1 − ε) ΩF . For short, we call I and I ∗ surplus
incomes.
Definition: I and I ∗ are referred to as surplus incomes.
Since (ω + α) /Ω is positive, (39) implies that b̄ is positively proportionate
to the international diﬀerence I − I ∗ in surplus income. This implies thefollowing proposition:
Proposition 2: With Assumption 1, a country is a creditor (resp. debtor)
in steady state when its surplus income is larger (resp. smaller) than theother country’s: b̄ T 0 ⇔ I T I ∗ .
To understand this result, suppose that a country’s net output exceeds the
habitual consumption level. Then, if country H’s preferences are of adjacent(resp. distant) complementarity, ceteris paribus it saves (resp. consumes) theexcessive income and thereby holds positive (resp. negative) external assets
in the long run. The same is true for the other country. In equilibrium,
the country with a larger (resp. smaller) surplus income accumulates (resp.
decumulates) external assets.
From Proposition 2, long-run indebtedness depends on three determinants: (i) net output; (ii) the strength of habit formation; (iii) the initial
habit stock. Net output is also emphasized as a key determinant by Frenkel
and Razin (1996) and Devereux and Shi (1991). Unlike in our model, in
the Frenkel and Razin model with intertemporally separable preferences, net
output aﬀects net foreign assets only when time preference diﬀers internationally; and in the Devereux and Shi model of endogenous discounting, larger
net output implies smaller net foreign assets.

16

Proposition 2 implies the three parts of Corollary 1 below. First, from(39), the eﬀects of increases in fiscal spending are obtained as
µ
¶
db̄
db̄∗
εΩH
ω+α
=−
=
,
(40)
dg
dg
(θ + α) ω
Ω
µ
¶
db̄∗
db̄
(1 − ε) ΩF ω + α,(41)=− ∗ =dg ∗dg(θ + α) ωΩimplying that an increase in a country’s fiscal spending reduces (resp. enlarges) its long-run external assets and hence enlarges (resp. reduces) theother country’s when consumer preferences in the policy implementing country display adjacent (resp. distant) complementarity. For example, an increase in g reduces or enlarges b̄ as ΩH is positive or negative. Irrespectiveof the two-country setting, note that this property is the same as in thesmall-country case.
Second, a permanent income transfer from country H to country F, −dy =dy ∗ > 0, aﬀects each country’s external asset position as¯¯db̄ ¯¯db̄∗ ¯¯
ω+α
.
(42)
=−
=−
¯
¯
dy dy=−dy∗
dy dy=−dy∗ω(α + θ)It follows that the income transfer reduces or enlarges country H’s net foreign
assets as the world felicity function exhibits adjacent or distant complementarity.
Third, the initial stock z0 of habit capital has qualitatively the same
implication for the external asset position b̄ as fiscal spending g. These
discussions can be summarized in the following corollary.
Corollary 1: With Assumption 1,(i) an increase in a country’s fiscal spending reduces (resp. enlarges) its longrun external assets and hence enlarges (resp. reduces) the other country’s
when consumer preferences in the policy implementing country display adjacent (resp. distant) complementarity: db̄/dg = −db̄∗ /dg S 0 ⇔ ΩH T 0;(ii) a permanent income transfer from country H to country F reduces (resp.
enlarges) H’s net foreign assets when the world¯ felicity function exhibits
adja¯
∗
¯
¯
cent (resp. distant) complementarity: db̄/dy dy=−dy∗ = − db̄ /dy dy=−dy∗ T

0 ⇔ Ω T 0; and17

(iii) the initial stocks of habit capitals have qualitatively the same implicationfor external asset positions as fiscal spending.
Remark 3: The existing literature (e.g. Devereux and Shi (1991), Frenkel
and Razin (1996), and Ikeda and Ono (1992)) shows that a country will tend
to be a long-run creditor the more patient it is and/or the greater is its net
output. Our results can be related to their discussion as follows. Startingfrom the initial situation that net output equals the habitual consumption
standard in each country, consider a marginal increase in country H’s net output, making its net output greater than the habitual consumption standard.
As conjectured from the small-country analysis (and as will be indeed shown
later), when country H’s preferences are more of adjacent complementarity
than country F’s, then, in response to the positive income shock, country H
is more reluctant to increase consumption in the short run and hence more
likely to be a creditor in the long run. In this case, country H can be regarded
as more patient than F in the short run and, as in the literature, the more patient country becomes a long-run creditor. The point is that which country ismore patient depends on the relative strengths of adjacent complementarityand on how much greater net output than the habitual consumption standard they have. To see this, consider, instead, a marginal decrease in country
H’s net output, in which case the new level of net output is lower than the
habitual consumption standard. When country H’s preferences are more ofadjacent complementarity than country F’s, country H is more reluctant todecrease consumption in response to the negative income shock and becomesa debtor in the long run. It is true that the less patient country is a long-run
debtor. But it is country H, in this case, that is the less patient.

3.4

WelfareFrom the linearized equilibrium dynamics obtained above, Appendix B.4
shows that the welfare levels of countries H and F are given by

U0 =u (z̄, z̄) (ω + α) uc + αuz+(z0 − z̄)θα (θ − ω)(ω + α) (1 − ε) ΩF[(θ + α) uc + αuz ] (Z0 − Y ) ,−α(θ + α) (θ − ω) Ω
18

(43)

U0∗ =u∗ (z̄ ∗ , z̄ ∗ ) (ω + α) u∗c + αu∗z ∗+(z0 − z̄ ∗ )θ
α (θ − ω)(ω + α) εΩH−[(θ + α) u∗c + αu∗z ] (Z0 − Y ) ,α(θ + α) (θ − ω) Ω

(44)respectively.

4Eﬀects of government spending

Let us next consider an increase in fiscal spending g in country H, financed
by lump-sum taxes.6

4.1

Consumption and net foreign assets

As shown by Corollary 1(i), an increase in g reduces or increases b̄ as thepolicy-implementing country H exhibits adjacent or distant complementarity,
as in the small-country case. From (8), (37) and (38), the eﬀects on steadystate consumption are obtained as follows:
µ
¶
dc̄
dz̄
db̄
εθΩH
ω+α
=
=θ −1=
− 1,
(45)
dg
dg
dg
(θ + α) ω
Ω
µ
¶
dc̄∗
εθΩH
ω+α
=−
T 0 ⇔ ΩH T 0.
(46)dg(θ + α) ωΩFrom (45), an increase in g decreases c̄ whenever ΩH is positive. Figure
2 illustrates the typical case that ΩH > 0; ΩF > 0; and Ω > 0, for which casethe slope of the SS 0 schedule is positive but gentler than that of BB 0 . An
6

We could consider three types of shocks: a shock that aﬀects both of the BB 0 and SS 0
schedules in Figure 2 (e.g., an increase in fiscal spending in country H); one that shifts the
BB 0 schedule, leaving the SS 0 schedule unchanged (e.g., an income transfer between thetwo counties, −dy = dy ∗ ); and one that shifts the SS 0 schedule, leaving BB 0 unchanged(e.g., productivity shocks dy ∗ in country F). For the eﬀects of income transfers, see Gombi
and Ikeda (2003). The eﬀects of productivity shocks dy ∗ in country F can be obtained
from the eﬀects of fiscal spending in country H by replacing country H’s variables withcountry F’s.

19

increase in g shifts schedule BB 0 to the left whereas SS 0 to the right, thereby
reducing b̄ and z̄ (and hence c̄).7
From (46), the spill-over eﬀect on country F’s consumption c̄∗ depends
crucially on whether country H’s preferences display adjacent or distant complementarity, ΩH T 0. This is because the eﬀects on b̄ and hence on b̄∗ rely
crucially on the sign of ΩH (see (40)). When country H’s preferences exhibit
adjacent complementarity (ΩH > 0), an increase in g decreases b̄, increasesb̄∗ and hence increases c̄∗ , and vice versa.
From (20) and (45) the impact eﬀect on country H’s consumption isderived asµ¶εΩH ω + αdc (0)
=
− 1.
(47)
dg
θ+α
Ω
This implies that, as in the small-country case, c (0) plumps less (resp. more)than its net output as the preferences exhibit adjacent (resp. distant) complementarity: dc (0) /dg ≷ −1 ⇔ ΩH ≷ 0.8
Given this, in turn, market-clearing condition (8) implies that c∗ (0) decreases or increases as ΩH is positive or negative. Formally, from (47) and(8), we obtain thatµ¶dc∗ (0)εΩH ω + α(48)=−
S 0 ⇔ ΩH T 0.
dgθ+αΩTherefore, when H’s preferences display adjacent (resp. distant) complementarity, country F’s consumption jumps down (or up) on impact. Comparing(46) with (48) yields

7³´
dc∗ (0)dc̄∗S0⇔
T 0 ⇔ ΩH T 0 ,dg
dg

When ΩH is negative, however, c̄ can increase due to an increase in interest incomeθb̄ in excess of the lump-sum tax increase if Ω = 0 and hence ΩF > 0. Formally, from (17)and (45), we can obtaindc̄ω (1 − ε) (θ + α) F
1−ε F
>0⇔−
Ω 5 ΩH <Ω .
dgεαε (θ − ω)
From this relation, it can be seen that necessary conditions for dc̄/dg to be positive are:T − 1−εΩH < 0, ΩF > 0, and Ω = 0 by noting that ω(1−ε)(θ+α)αε(θ−ω)ε ⇔ Ω T 0.
8
We cannot exclude the pathological case that equilibrium consumption c (0) increases
on impact when consumer H’s preferences display strong adjacent complementarity.

20implying that the impact eﬀect of an increase in g on c∗ is necessarily opposite
in signs to its steady-state eﬀect. This is because, without direct income
shocks in country F, any increases or decreases in c̄∗ should be financed byincreases or decreases in b̄∗ , which are only brought about by decreases orincreases in c∗ (0) to cause current account surplus or deficits in transition.
When consumer H’s preferences are of adjacent complementarity, ΩH >0, the transition dynamics are illustrated in Figure 3 by using the phase
diagrams introduced in Figure 1. From (40), an increase in g reduces b̄ andhence c̄ (see (45)). Although c (0) may decrease or increase (see (47)), savings
necessarily reduce to generate the current account deficits, as in the smallcountry case. The figure depicts the normal case that c (0) decreases, so thatsaddle arm DD0 in the (b, c) plane shifts downward. Net foreign assets b
gradually decumulate from b0 to b̄ along the new saddle arm, D1 D10 . Sinceb̄∗ (= −b̄) increases, c̄∗ increases (see (46)). To realize this, c∗ (0) necessarily
drops to induce the current account surplus (see (48)). As c∗ monotonicallyincreases toward c̄∗ , z ∗ sooner or later stops decreasing and starts rising
toward z̄ ∗ (= c̄∗ ).

4.2

The interest rate

The eﬀect on the interest rate can be computed by diﬀerentiating (25) withrespect to g and by substituting (45) into the result as©
αεu∗cc ΩH
dr (0)= ∗ucc (ω − θ) (θ + 2α) ΩH∗dgλ (θ + α) (α + θ − ω) (ucc + σucc ) Ωª− {α (ω + α) ucc + (θ + α) (α + θ − ω) σu∗cc } ΩF .
(49)

Instead of discussing the general properties of this complex result, we restrict
our attention to the special case (ii) discussed in Section 3.2.2.
Consider case (ii), in which consumer preferences exhibit identical degrees
of adjacent complementarity: ΩH = ΩF = Ω. Equation (49) then reduces to¯
dr (0) ¯¯= ηΩ T 0 ⇔ Ω T 0.
(50)
dg ¯ΩH =ΩF =Ω

This implies that an increase in g raises (or lowers) r (0) as the world felicity
function displays adjacent (or distant) complementarity. Such a policy aﬀects
the commodity market on both the supply and demand sides: it reduces the
supply of the good that should be consumed; and it decreases disposable21

income in country H. In the case of adjacent complementarity, the decreasein disposable income would lower c (0), but not so much as it would lowerc̄ if r (0) is constant at θ. This means that there would be excess demandin the time-zero good market. To clear the market, r (0) rises above θ,which, in turn, decreases aggregate consumption to the lowered aggregate
supply level. Alternatively, when the world felicity function exhibits distantcomplementarity, an increase in g produces excess supply in the present goodmarket, thereby lowering the interest rate. After the initial response, theinterest rate then monotonically converges toward θ as seen from (28).

4.3

WelfareIrrespective of how each country’s consumption responds over time, an increase in government spending g necessarily reduces country H’s welfare. To
see this, we substitute (8) into (43) and (44), diﬀerentiate the results by g,
and incorporate (45) and (46). The results are simply given by
dU0
λ
dU0∗
= − (< 0) and
= 0.
dg
θ
dg
The increase in g harms country H’s welfare by the amount equal to thepresent value of the marginal utility. The marginal change in g, however, does
not aﬀect country F’s welfare in this non-distortionary competitive equilibrium.9 However, these properties are limited to the case of b0 = 0, as shown
later in Section 5.

4.4

Empirical implications

By using a small-country model, Gruber (2004) shows that incorporating
habit formation improves the explanatory power of the intertemporal current
account model in advanced countries. However, we can find two puzzlingdiﬀerences between his result for two large countries, the United States andJapan, and that for smaller countries, such as Italy, Spain, and Netherlands.
First, the GMM estimates of the habit preference parameter for the twocountries are smaller than those for the smaller countries. Second, in the caseof the two large countries, the habit model overpredicts the actual volatility
9

The same result can be obtained from Bianconi and Turnovsky (1997, p.80) by assuming that g does not generate direct utility and that there is no initial external indebtedness.

22

of the current account, unlike in the smaller country case. We shall showthat these "large-country puzzles" could be attributed to biases that willarise when, with the small country assumption, the interest-rate adjustment
is ignored.
To consider the large-country puzzles in this deterministic framework,
we examine impacts of an income shock on consumption and on the current
account, instead of directly treating their volatilities. Consider an increase ing, a negative income shock, in two diﬀerent economies: (i) the two-country
world economy with the internationally same strength of habit formation,ΩH = ΩF = Ω > 0 and (ii) the small open economy (SOE) with the same Ω.
In the SOE, the consumption impact of an increase in g can be obtained10as¯
dc (0) ¯¯
α+ω
=
− 1,
(51)
¯
dg SOE
α+θ
whereas in the two-country economy, from (47), we obtain¯² (α + ω)
dc (0) ¯¯− 1,=¯dg ΩH =ΩF =Ω
α+θ

(52)

implying that the consumption impact is larger in the two-country world
economy than it would be in the SOE:
¯
¯ ¯ ¯
¯
¯
¯
¯ dc (0) ¯ ¯ ¯ dc (0) ¯
¯
¯ ¯<¯
¯
¯
(53)
¯ dg ¯ ¯ ¯ dg ¯ H F ¯ .
SOEΩ =Ω =ΩThis comes from the fact that in the two-country case an interest-rate hikeinduced by the increase in g (see (50)) enlarges the decrease of c (0). Empirically, inequality (53) implies that consumption is more volatile in large
countries than it would be in small countries if the habit parameter value isthe same.
From the viewpoint of estimating the habit parameter value, capturedby ω, from a given observed volatility of consumption, in turn, inequality(53) implies that the ω estimate is larger when it is estimated by using thelarge country model than by using the SOE model: Explicitly, from thecomparison of (51) and (52), the ω value required to explain a given value ofthe consumption change is larger in (52) than in (51) since ² ∈ (0, 1). The
10

See Ikeda and Gombi (1999).

23

habit parameter value of a large country would thus be underestimated if
it is estimated using a small country model. This consistently explains the
puzzle that the estimates of the habit parameters for the United States andJapan are smaller than the smaller countries.11
As for the current account eﬀect of an increase in g, inequality (53) implies
that the direct negative eﬀect of the g-increase on the current account is oﬀset
by a larger decrease in consumption in the two-country setting than in the
SOE setting. Formally, we can derive¯dḃ (0) ¯¯
α+ω
,
=−
¯
dg ¯
α+θ
SOE

whereas in the two-country case, from (7) and (47), we have¯dḃ (0) ¯¯² (α + ω)=−,¯
dg ¯ H Fα+θΩ =Ω =Ω

implying¯¯¯ dḃ (0) ¯¯¯¯¯¯ dg ¯SOE¯ ¯¯¯ ¯ dḃ (0) ¯
¯ ¯
¯
¯>¯
¯
¯ ¯ dg ¯ HΩ =ΩF =Ω

¯
¯
¯
¯.
¯

As a result, the current account volatility of a large country would be overestimated under the small country assumption since the interest-rate eﬀect
is ignored in the setting. This consistently explains Gruber’s overpredictionof the current account volatility for Japan and the United States.12
These empirical implications are summarized as follows:
Proposition 3: The habit parameter of a large country would be underestimated, and the current account volatility overestimated, if they were estimatedusing a small-country model.
The point is that, in a large country, the interest rate adjustment exaggerates consumption impact of an income shock, so that neglecting this eﬀect11

Gruber (2004) assumes linear habit formation in specifying felicity function, as we
1−ϕbelow in subsection 5.2, and thereby estimates γ as theshall define as u (c, z) = (c−γz)
1−ϕhabit parameter. It is easy to confirm that ω is positively related with γ.
12
Gruber (2004) gives the conjecture that the overpredictions of the current accountvolatility for United States and Japan could be attributed to the small-country assumption.
Our discussion provides a formal proof for it.

24

under the small country assumption brings upward bias to the impacts on
saving and on the current account for the large country and brings downward
bias to the estimated habit parameter of the large country.

5

Welfare-enhancing fiscal spending: An implication of b0 6= 0

5.1

Reversed immiserizing growth eﬀects

We have so far assumed that b0 equals zero by Assumption 1. When b0 6= 0,
however, an increase in government spending g has additional welfare eﬀects
by causing income re-transfers through interest rate changes, i.e., changes
in the intertemporal terms of trade. Even though government spending isassumed to provide no direct utility, we shall show that an increase in g canenhance country H’s welfare level due to favorable changes in the intertemporal terms of trade, i.e., rises (resp. falls) in the world interest rate if thecountry is a creditor (resp. debtor).
Let us now abandon Assumption 1 to allow b0 to take a non-zero value.
For expository purposes, we focus on case (ii), ΩH = ΩF = Ω. As shown inAppendix C, an increase in g aﬀects country H’s welfare by
¯
¶
µ
θ
λ
dU ¯¯
(54)
1−
ηΩb0 ,=−
dg ¯ΩH =ΩF =Ωθα+θθηΩb0 > 1.
implying that the increase in g improves country H’s welfare if α+θWith a positive b0 , the policy is welfare enhancing if Ω is a suﬃciently large
positive number, i.e., if consumers’ preferences are of suﬃciently strong adθjacent complementarity. Note that the term α+θηΩb0 represents an income
re-distribution eﬀect due to interest rate changes since, as derived in Appendix C, the eﬀect on the interest rate is given by¯
dr (0) ¯¯
T 0 ⇔ Ω T 0.
dg ¯ΩH =ΩF =Ω

With adjacent complementarity (Ω > 0), ceteris paribus an increase in gcauses excess demand for the present good, raises the interest rate, and enlarges the net output and the welfare level if b0 is positive and large. This is25

an intertemporal version of (reversed) immiserizing growth discussed by thetrade theory: a creditor country, i.e., an exporter of the present good, can bebetter oﬀ due to intertemporal terms-of-trade improvements by decreasing
the supply of the present good.
These changes in the intertemporal terms of trade have spill-over eﬀects
on country F. Appendix C derives¯
dU ∗ ¯¯
ηΩλ∗
=
−
b0 ,
dg ¯ H F
θ+αΩ =Ω =Ω

which reveals that the increase in g makes country F worse (resp. better) oﬀ
when Ωb0 > 0 (resp. Ωb0 < 0). With a positive b0 (= −b∗0 ), for example, anincrease in g induces a deterioration (resp. an improvement) in country F’s
intertemporal terms of trade, thereby harming (resp. benefiting) country F.
These results can be summarized as follows:
Proposition 4: Suppose that country H is as much in the habit of consuming
as country F: ΩH = ΩF = Ω. Then, due to intertemporal terms-of-tradeeﬀects, an increase in fiscal spending g in country Hθ(i) enhances the country’s own welfare if and only if α+θηΩb0 > 1; and(ii) harms country F if and only if Ωb0 > 0.

Remark 4: By using a two-country model, Bianconi (2003) shows the possibility that an increase in fiscal spending has a reversed immiserizing growtheﬀect due to changes in the intratemporal terms of trade. He shows that anincrease in a country’s fiscal spending financed by capital taxation contracts
the production frontier in the country, and thereby appreciates the real exchange rate, which can improve the asset position and the welfare level ofthe country. However, as pointed out in Bianconi (2003, p.26) this reversed
immiserizing growth eﬀect cannot take place when government spending isfinanced by a lump-sum tax, and/or when there is only one consumption
good. Proposition 4 shows that with habit formation, fiscal spending canmake the policy implementing country better oﬀ by improving the intertemporal terms of trade even when the policy is financed by lump-sum taxation
in a single good economy.

5.2

ExampleTo clarify the possibilities of the reversed immiserizing growth eﬀect causedby fiscal spending, assume that country H is the creditor: b0 > 0, and that the26

two countries have the same felicity functions specified, as in the literature(e.g., Constantinides (1990), Gruber (2004)), as:u (c, z) =(c − γz)1−ϕ
,
1−ϕ

(55)

where the habit parameter γ ∈ (0, 1) captures the strength of habit influence;HFand ϕ represents¡¢ a risk13aversion parameter. It is then valid that Ω = Ω =αγΩ = γ 1 − 2α+θ > 0. Therefore, from Proposition 4 (ii), an increase in gnecessarily harms the debtor country F.
As for Proposition 4 (i), the condition for the reversed immiserizing
θgrowth eﬀect, i.e., α+θηΩb0 > 1, reduces to
µ
¶
¶µ
αγ
αϕγ
b0
θ
¡
¢ 1−
> 1.
αγ
α + θ (1 − γ) 1 − α+θ
2α + θ
y + θb0 + z0∗

Note that, as γ → 1, the left hand side of this inequality diverges to infinity
for any parameter values. From Proposition 4 (i), this implies that an increase in g definitely causes the reversed immiserizing growth eﬀect under asuﬃciently large habit parameter γ.14
γ

ϕ0 .7 00 .7 5

0 .8 00 .8 50 .9 00 .9 5

z0∗

z0∗

z0∗

z0∗

z0∗

z0∗

0 .1

0 .5

0 .1

0 .5

0 .1

0 .5

0 .1

0 .5

0 .1

0 .5

0 .1

0 .5

2.0

2 .8 8

3 .92

1.92

2 .6 2

1 .2 1

1 .6 6

0 .7 00 .9 80 .3 4

0 .4 70 .1 1

0 .1 6

3.0

0 .1 1

2 .51

1.24

1 .7 00 .7 91 .0 90 .4 6

0 .6 4

0 .2 3

0 .3 20 .0 80 .1 14.0

1 .3 5

1 .84

0.92

1 .2 6

0 .5 90 .8 1

0 .3 5

0 .4 80 .1 70 .2 40 .0 60 .0 85.0

1 .0 7

1 .46

0.73

1 .0 00 .4 7

0 .6 50 .2 80 .3 80 .1 40 .1 90 .0 50 .0 7

Table 1: The net foreign assets-GDP ratio b0 required for the reversed
immiserizing growth eﬀect
13

Even if the risk³aversion parameter
ϕ diﬀers between the both countries, the relation,´αγΩH = ΩF = Ω = γ 1 − 2α+θ> 0, is retained. The discussions in this section, therefore,
can be extended without substantial changes to the heterogenous risk aversion cases.
14
Gruber (2004) obtains the significant estimates of γ as 0.789 to 0.920 from the current
account data of five advanced countries.

27

Corollary 2: Suppose that two countries have the same felicity function(55); and that country H is the creditor. Then, an increase in country H’s
fiscal spending(i) benefits country H under suﬃciently strong habit formation; and(ii) necessarily harms country F.
To evaluate tentatively how likely the reversed immiserizing growth eﬀect
occurs, we figure out the critical values of the net foreign assets-GDP ratio b0
under alternative values of the habit parameter γ, the risk aversion parameterϕ, and the initial habit stock z0∗ of country F, with fixing the parameters y, α,and θ to y = 1, α = 0.9, and θ = 0.05, respectively. Table 1 summarizes
the results, where each value represents the minimal b0 value required for theeﬀect under diﬀerent (γ, ϕ, z0∗ ) values. As seen from the table, the reversed
immiserizing growth eﬀect is not so implausible. For example, when (γ, ϕ) =(0.90, 3.0), the reversed immiserizing growth eﬀect takes place if the net
foreign assets-GDP ratio b0 is higher than 0.23 when z0∗ = 0.1 and 0.32 whenz0∗ = 0.5, both of which are smaller than the Japanese actual data (0.35).15
The critical condition can be even weaker if the risk aversion parameter ϕ islarger; and/or if the speed of habit formation α is larger.16

6Conclusions

We have examined macroeconomic adjustments in a tractable two-country
model of heterogeneous habit forming consumers, thereby figuring out implications for welfare and for the understanding of some empirical facts. Our
conclusions can be summarized as follows:
1. To focus on interactive consumption-habit dynamics, we have proposed
to construct the world felicity function and the aggregate habit capital,
which successfully help to characterize intertemporal complementarities
in the two-country equilibrium.
15

Although Gruber (2004) estimates Japan’s γ as 0.789, it is based on the small country
assumption. Compared with Gruber’s other estimates, the value 0.90 is not unrealisticallyhigh.
16When the felicity function is specified in the alternative form as u (c, z) =1−ϕ(c/z γ )/ (1 − ϕ) ; γ ∈ (0, 1) , however, the conditions for the reversed immiserizing eﬀect
are much more stict than in the case of (55).

28

2. External indebtedness depends crucially on international diﬀerence insurplus income, i.e., the eﬀective net output that can be allocated forsaving under habit formation. This is shown to be consistent with theexisting proposition (e.g., Devereux and Shi 1991) that a country tends
to be a long-run creditor the more patient it is. Our contribution isthat which country is more patient is determined endogenously from therelative strengths of adjacent complementarity and how much greater
net output than the habitual consumption standard they have.
3. When the initial holding of net foreign assets, and hence the income
re-transfer eﬀect due to interest rate changes are negligible, an increasein a country’s fiscal spending (i) can increase the initial or steady-state
consumption level of that country, with the other country’s consumption being crowded out; (ii), as in the small country case, necessarily
worsens (resp. improves) the steady-state external asset position if thecountry’s preference displays adjacent (resp. distant) complementarity, irrespective of the other country’s preferences; and (iii) has long-run
and short-run spill-over eﬀects on the other country’s consumption, depending crucially on whether the policy implementing country displays
adjacent or distant complementarity.
4. With non-negligible initial net foreign assets, an increase in fiscal spending in the creditor country can make the country better oﬀ and theneighbor debtor worse oﬀ due to the intertemporal terms-of-trade effect. With a specific felicity function, this necessarily occurs undersuﬃciently strong habit formation.
5. Interest rate adjustments enlarge the consumption eﬀect of an income
shock and thereby mitigate its current account eﬀect. The habit parameter of a large country, therefore, would be underestimated, and thecurrent account volatility overestimated, if they were measured usinga small-country model as in the literature.
For future research, two issues may be interesting. First, to evaluate
explicitly the eﬀect of habit formation on the current account volatility of alarge country, the analysis should be extended to a stochastic model. Second,
when a government knows that fiscal spending causes the reversed immiserizing growth eﬀect at the loss of the neighbor country’s welfare, some strategic
policy game naturally arises. It should be examined how the game changes
our basic results.
29

AppendicesA

Proof of Proposition 1∗

Note first that λ/λ∗ is constant because λ̇t /λt = λ̇t /λ∗t from (5). By eliminating c∗ and z ∗ using (8) and (12) from the foreign counterpart of (4),combining the resulting equation and (4) yields:uc (c, z) + αξλ∗ = ∗ = constant.
∗uc (Y − c, Z − z) + αξλBy totally diﬀerentiating this equation, we obtain:ĉ = −

∗
αλ
αλ∗
λu∗cz
λ∗ ucz + λu∗cz
ξ̂
+
ξ̂
ẑ
−
+
Ẑ.
λ∗ ucc + λu∗cc
λ∗ ucc + λu∗cc
λ∗ ucc + λu∗cc
λ∗ ucc + λu∗cc

We substitute this equation into (1), (6), and the foreign counterpart of(6) and eliminate c∗ and z ∗ using (8) and (12) from the resulting equation.
Then,from´ (11), the autonomous dynamic equation system with respect to³∗ẑ, ξ̂, ξ̂ , Ẑ is obtained as follows:
¶
µ ∗
∗
α2 λ
α2 λ∗
λ ucz + λu∗cz
ξ̂
+
ξ̂
+
1
ẑ
−
ż = −α∗∗∗λ ucc + λu∗cc
λ ucc + λu∗cc
λ ucc + λu∗cc
∗
αλucz
+ ∗
Ẑ,
λ ucc + λu∗cc¾µ¶½
αucz λ∗ucz (λ∗ ucz + λu∗cz )ξ̂− uzz ẑ + θ + α + ∗ξ̇ =λ∗ ucc + λu∗cc
λ ucc + λu∗cc∗
αucz λλucz u∗cz
− ∗
ξ̂
−
Ẑ,
(56)
λ ucc + λu∗cc
λ∗ ucc + λu∗cc¾½ ∗ ∗αu∗cz λ∗ucz (λ ucz + λu∗cz )
∗
∗
ẑ
−
−
u
ξ̂
ξ̇ = −zz
λ∗ ucc + λu∗cc
λ∗ ucc + λu∗cc
¶
¶
µ
µ
∗
λu∗2
αu∗cz λ∗cz∗ξ̂ +− uzz Ẑ,+ θ+α+ ∗λ ucc + λu∗cc
λ∗ ucc + λu∗cc
Ż = −αẐ.

From the definitions (13) and (14) of v and ς, respectively, this autonomous
system reduces to (15).
30

B

Equilibrium solutions

B.1

Dynamics of habit capital z: (18)The stable roots of dynamics (15) are given by ω and −α as in (16). Lettingm denote (z, ς, Z)0 , the general solution to (15) can thus be expressed asm̂ (t) = A1 exp (ωt) q + A2 exp (−αt) h,
where q ≡ (q1 , q2 , q3 )0 and h ≡ (h1 , h2 , h3 )0 represent the eigen vectors associated with stable roots ω and −α, respectively. From (15), it is easy toconfirm that q3 = 0. By eliminating A1 exp (ωt) and A2 exp (−αt) from thethree equations in the above vector equation, we obtainς̂ =q h2 − q2 h1
q2
ẑ + 1
Ẑ,
q1
q1 h3

(57)

where the coeﬃcients of ẑ and Ẑ can be obtained by exploiting the definitionof the eigenvectors q and h as(ω + α) vcc + αvcz
q2
= −
,
q1
α2
vcc (vcz vZZ + vcZ vzz )q1 h2 − q2 h1
= −
q1 h3
αvcc vzz + (2α + δ) vcc vcz{(ω + α) vcc + αvcz } {αvcc vZZ − (2α + δ) vcc vcZ }.
+α2 (αvcc vzz + (2α + δ) vcc vcz )Substituting (57) into the ż-equation in (15) yields (18).

B.2

The interest rate

From (1) and (4) through (6), the optimal consumption dynamics are givenby´
λ ³ċ = −r̂ − φ̂ ,uccwhere φ represents the rate of time preference,φ̂ =α (uzz + uzc )α (α + θ)
ẑ −
ξ̂.
λ
λ
31

Substitute (21) into the above Euler equation. The resulting equation canbe solved for r̂ as
r̂ = −α (uzz + ucz )α (α + θ)
ωucc
ĉ +
ẑ −
ξ̂.
λ
λ
λ

(58)

In the above, ξ̂ can be obtained from (14),(19), (56), and (57) asξ̂ = −

uzz
ucc ΩH
ĉ +
ẑ.
θ+α−ω
θ + 2α(59)

Substituting (59) and (20) successively into (58) yields (25).

B.3

Net foreign assets

Setb̂ = κ1 ẑ + κ2 Ẑ.

(60)

Diﬀerentiating (60) with respect to time t yieldsḃ = κ1 ż + κ2 Ż.
Since ḃ is given by (7), this equation implies³ ´ḃ = r̂b0 + rb̂ − ĉ = κ1 ż + κ2 Ż.

(61)

(62)

Substitute (18), (19), (20), (25), and (60) into (62). By comparing the coefficients of the resulting equation, we obtain
ω+α
,
µα
¶
ω+α
= a2 b0 +(1 − δ) .
α(ω − r) κ1 = a1 b0 −− (ω + α) (1 − δ) κ1 − (r + α) κ2This simultaneous equation can be solved for κ1 and κ2 as
κ1
κ2

¾
½
ω+α
1
− b0 a1 ,
=
r−ω
α
¾
½
b0
(1 − δ) (ω + α)
(1 − δ) (ω + α)+
a1 + a2 .
=α (r − ω)r+α(r − ω)Substituting (63) and (64) into (60) yields (29).
32

(63)
(64)

B.4

WelfareTo 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 obtainZ ∞U0 ={u (c̄, z̄) + uc ĉt + uz ẑt } exp (−θt) dt.
0

Since, from (21) and (1), ĉt and ẑt are solved as
¶
µ
αĉ0
αĉ0
ĉt = ĉ0 exp (ωt) and ẑt = ẑ0 −exp (−αt) +exp (ωt) ,ω+α
ω+αthe lifetime utility is expressed asZ ∞{u (c̄, z̄) + uc ĉ0 exp ((ω − θ) t) +
µ
¶
¾
αĉ0
αuz ĉ0+uz ẑ0 −exp ((ω − θ) t) dt
exp (− (α + θ) t) +
ω+α
ω+α
µ
¶
µ
¶
ĉ0
uz
αuz
αĉ0
u (c̄, z̄)
+
uc +
+
ẑ0 −
.
=
θ
ω−θ
ω+α
α+θ
ω+α

U0 =

0

The initial optimal consumption ĉ0 is obtained by setting t = 0 in (20).
Substituting it into the above equation yields (43). Equation (44) can beobtained in the same way.

C

Dynamics with non-zero b0: The identicaldegrees of adjacent complementarity

C.1

Solutions

By assuming that the two countries’ preferences exhibit the identical degrees
of adjacent complementarity, ΩH = ΩF = Ω, we shall derive equilibrium
solutions in the case of a nonzero b0 . In this case, from (18), (19), (20), (25),and (29), the equilibrium dynamics are generated by the following equationsż = ωẑ − (ω + α)(1 − δ)Ẑ,
33

(65)

ĉ =

µ

Ż = −αẐ,
¶
´
ω+α ³
ẑ − (1 − δ)Ẑ ,αċ = ωĉ,r̂ = ηΩẐ,½¾
ω+α(1 − δ) (ω + α)
b0 ηΩ
b̂ =
ẑ −
+
Ẑ.
α (θ − ω)α (θ − ω)
θ+α
U0 =

U0∗ =

(66)

u (z̄, z̄) (ω + α) uc + αuz+(z0 − z̄)θα (θ − ω)(ω + α) (1 − δ)−[(θ + α) uc + αuz ] (Z0 − Y ) ,α(θ + α) (θ − ω)u∗ (z̄ ∗ , z̄ ∗ ) (ω + α) u∗c + αu∗z ∗+(z0 − z̄ ∗ )θ
α (θ − ω)(ω + α) δ−[(θ + α) u∗c + αu∗z ] (Z0 − Y ) ,α(θ + α) (θ − ω)

(67)

(68)

(69)

respectively.
Define the eﬀective habit stock e as
e ≡ z − (1 − δ)Z.

(70)

By using the eﬀective habit stock, the equilibrium dynamics can be summarized as(ω + α)2ê,(71)ḃ = −αb̂ +α (θ − ω)ė = ωê.

(72)

Figure 4 illustrates the phase diagram in the (e, b) plane.
Note that (67) can be rewritten in terms of ê as
b̂ =

b0 ηΩ
ω+α
ê −
Ẑ.
α (θ − ω)
θ+α

34

(73)

The steady state equilibrium is determined by the following two schedules,
CC 0 and DD0 :CC 0 : b̄ − b0 =

0

ω+α
b0 ηΩ
(ē − e0 ) −(Y − Z0 ) ,α (θ − ω)θ+αDD0 : θb̄ = ē + (1 − ε)Y + g − y,

(74)
(75)

where schedule CC is obtained by evaluating (73) at t = 0; and DD0 is
obtained by substituting (70) for c̄ (= z̄) in (34). The two schedules areillustrated in Figure 4, where schedule CC 0 is positively- (resp. negatively) sloping when the world felicity function displays adjacent (resp. distant)complementarity, Ω > 0 (resp. Ω < 0). It can be easily shown that the slopeof schedule DD0 is larger than that of CC 0 ;17 and that, when Ω > 0, schedule
CC 0 is steeper than the ḃ = 0 schedule which is obtained by setting ḃ = 0 in
(71).18 The steady-state equilibrium is determined at the intersection of thetwo schedules.

C.2

The eﬀect of fiscal spending

The eﬀects of an increase in fiscal spending g on b̄ and ē are obtained from(74) and (75) asdb̄∗
δ (ω + α) αη (θ − ω) Ω
db̄
=−
=
−
b0 ,
dg
dg
ω (θ + α)ω (θ + α)2(76)αδ (θ − ω) αθη (θ − ω) Ω
dē
=
−
b0 .
dg
ω (θ + α)ω (θ + α)2(77)

As seen by comparing (40) and (76), the first terms on the right handsides in these equations represent the same eﬀects as are discussed in Section
4.2 whereas the second terms capture income eﬀects that arise under non-zerob0 due to interest rate changes. In fact, diﬀerentiating (50) by g yields¯
dr (0) ¯¯
= ηΩ,
(78)
dg ¯ΩH =ΩF =Ω
17

The slope of schedule DD0 , 1/θ, minus that of CC 0 , (ω + α) /α (θ − ω), equals−ω (θ + α) /αθ (θ − ω), which is positive.
18
The slope of schedule CC 0 , (ω + α) /α (θ − ω), minus that of the ḃ = 0 schedule,2(ω + α) /α2 (θ − ω), equals −ω (ω + α) /α2 (θ − ω), which is positive (resp. negative)
when ω + α > 0 (resp. < 0).

35

implying that, with adjacent (resp. distant) complementarity, Ω > (resp. <) 0,
ceteris paribus an increase in g causes excess demand (resp. supply) for thepresent good, and raises (resp. lowers) the interest rate. From (76), therefore, when country H is a creditor, b0 > 0, the increase (resp. decrease) in theinterest rate has an additional positive (resp. negative) eﬀect on steady-state
external asset holding of country H. Consequently, in contrast to the case ofa zero b0 , even under adjacent complementarity, an increase in g can enlarge
steady-state external asset holding of country H by increasing interest income
if b0 is positive and suﬃciently large.
Figure 5 illustrates such a case by assuming that Ω > 0 and b0 > 0. An
increase in fiscal spending shifts both the CC 0 and DD0 schedules upward.
Due to a dominant interest-income eﬀect, these shifts bring the steady-state
point from E0 to E1 with a higher b̄ and a lower ē. In transition the eﬀective
habit stock e decreases monotonically whereas net foreign assets increasenon-monotonically.
From (70) and (77), the eﬀects on consumptions and habit capitals are
given by
dz̄
dē
dY
dc̄
=
=
+ (1 − δ)
dg
dg
dg
dg
δθ (ω + α)αθη (θ − ω) Ω
b0 ,
=
−1−
ω (θ + α)ω (θ + α)2
dz̄ ∗
dc̄
dc̄∗
=
=1−
dg
dg
dg
δθ (ω + α) αθη (θ − ω) Ω
b0 ,
= −
+
ω (θ + α)ω (θ + α)2

(79)

(80)

in which the last terms on the right hand sides represent the income eﬀect
of interest rate changes. When Ωb0 > 0, an increase in g increases country
H’s interest income, thereby having a positive eﬀect on c̄ and a negative one
on c̄∗ .
The welfare eﬀects are obtained from (68), (69), (79) and (80) as
¯
¶
µ
dU ¯¯
θ
λ
1−
ηΩb0 ,=−
dg ¯ΩH =ΩF =Ωθα+θ¯dU ∗ ¯¯
ηΩλ∗
b0 ,
=
−
dg ¯ΩH =ΩF =Ω
θ+α36implying that

¯
dU ¯¯
θ
ηΩb0 T 1.
T0⇔¯
dg ΩH =ΩF =Ω
α+θ¯dU ∗ ¯¯
T 0 ⇔ Ωb0 S 0.
dg ¯ΩH =ΩF =Ω

(81)
(82)

Equations (78) and (81) reveal that an increase in g benefits country H when
the income eﬀect of interest rate changes is suﬃciently large. From (78) and(82), the spill-over eﬀect on country F’s welfare is positive or negative as theinterest rate revenue of country F is enlarged or reduced by the policy. For
example, when country H is a creditor (b0 > 0) and when preferences display
adjacent complementarity (Ω > 0), an increase in g raises the interest rateand causes income transfers from country F to H. This makes country F
worse oﬀ. If b0 and/or Ω are/is large enough, the increase in fiscal spending
makes country H better oﬀ by raising her interest revenues.

37

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39

c

c*
z& = 0b& = 0z& ∗ = 0DE

c& = 0E∗

c& ∗ = 0D'bz
Figure 1.

Equilibrium dynamics

z*

b

B
S
S ' (Ω > 0)

E

S

E′
B'

S ' (Ω < 0)z

Figure 2. Steady-state equilibrium and the effects ofan increase in fiscal spending

cc*z& = 0z& ∗ = 0Dc&∗ = 0c& = 0

D'

b
b0

z0

z

Figure 3. The effects of an increase in fiscal spending when Ω H > 0.

z0∗

z*

b

e& = 0C

D (Ω > 0)D
b& = 0D'D'(Ω < 0)C'e

Figure 4. Equilibrium dynamics

be& = 0

C1

C0

D1

D0
b& = 0

E1

E0

D1′

D0′

C1′

C0′
e

Figure 5. Effects of an increase in fiscal spending
when Ω > 0 and b0 > 0 .