Discussion Paper No. 551

HETEROGENEOUS HABITS AND
THE TRANSFER PARADOX

Ichiro Gombi
andShinsuke Ikeda

August 2001

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

Heterogeneous Habits and The Transfer
ParadoxIchiro Gombi¤ and Shinsuke Ikeda yz
Ritsumeikan University and Osaka University
August 2001

By using a two-country model with habit-forming consumers, this paper showsthat the transfer paradox can take place in the free-trade, dynamically-stableworld economy. When the debtor is more habituated to consumption than thecreditor, an income transfer from the creditor to the debtor raises the interest
rate in transition through changes in time preference. With su¢ciently low
elasticities of intertemporal substitution and/or su¢ciently large stock of thecreditor’s assets, the intertemporal terms of trade e¤ect immiserizes the recipientand enriches the donor. Although the transfer paradox occurs only when theinternational bond market is “unstable” with respect to an ad hoc Walrasianadjustment process, the equilibrium dynamics are stable in the usual sense: giventhat the economy is always in the rational expectation equilibrium, the transfer
paradox generically occurs.
JEL Classi…cation Numbers: F00, D90.
Keywords: Habit, transfer paradox, two-country model, stability.

¤

I. Gombi, The Faculty of Economics, Ritsumeikan University, Kusatsu, Shiga 525-8577
Japan. Telephone: 81-77-561-4840, Email: <gombi@ec.ritsumei.ac.jp>.
yCorresponding author: S. Ikeda, The Institute of Social and Economic Research,
Osaka University, Mihogaoka, Ibaraki, Osaka 567-0047, Japan. Telephone: 81-6-68798568, Facsimile: 81-6-6878-2766, Email: <ikeda@iser.osaka-u.ac.jp>.
z
We are very grateful to F. Ohtake, R. Okumura, Y. Ono, S. Takagi, and the participants at the JEA Spring Meeting, Hiroshima Shudo University, and the Macro Research
Seminar for helpful comments and discussions on earlier versions of the paper.

Heterogeneous Habits and The Transfer Paradox

1

1

Introduction

Since Professor Paul Samuelson showed in his seminal work (Samuelson(1947, 1952)) that the transfer paradox, i.e., income transfers which enrichdonors and immiserize recipients, cannot take place in two-country Walrasstable equilibria insofar as there is no distortion, it has been the fundamental
result in the literature of the transfer problem. Starting from the result, thetrade theorists have made considerable contributions by (i) adding the third
country and/or by (ii) incorporating distortions.1
The contributions, however, have been limited to intrinsically static analysis and, except for several articles cited later, the intertemporal aspects of thetransfer problem have not been extensively examined. With international …nancial markets recently getting more and more integrated, international
income transfers may well induce variations in interest rates and in asset
prices, thereby bringing about rapid and huge endogenous re-transfers of resources. To incorporate this e¤ect using a dynamic-optimizing model would
help obtain a new insight into the transfer problem.
The purpose of this paper is to examine the transfer problem in a twocountry world economy with in…nitely-lived, heterogeneously habit-forming
consumers. Particularly emphasized is international heterogeneity in habituation to consumption. With the resultant heterogeneity in intertemporalconsumption complementarities, income transfers instantaneously a¤ect theworld interest rate, thereby causing the intertemporal terms of trade e¤ects
on both of the countries. By incorporating the e¤ects, we show that, without
any distortion, the dynamically-stable equilibrium of the two-country world
economy allows income transfers to immiserize the recipient and enrich thedonor.
As a stylized fact, habit formation has been often reported as playing important roles in determining actual consumer behavior (e.g., Braun, Constantinides, and Ferson (1993), Naik and Moore (1996), Campbell and Cochrane(1999), Carrol, Overland, and Weil (2000), and Boldrin, Lawrence, andFisher (2001)). However, without no exception, the existing models of con1

For the literature of (i), see, e.g., Gale (1974), Yano (1983), and Bhagwati, Brecher,and Hatta (1983). The literature of (ii) contains Brecher and Bhagwati (1982), Bhagwati,
Brecher, and Hatta (1985), and Yano and Nugent (1999). To be precise, the presenceof the third country can be regarded as a distortion insofar as the transferer and thetransferee do not jointly levy the optimal tari¤s against the third country. For this point,
see Bhagwati, Brecher, and Hatta (1983).

Heterogeneous Habits and The Transfer Paradox

2

sumption habits commonly rule out consumers’ interactions due to pecuniary
externalities: they focus on representative agent models (e.g., Ryder and Heal
(1973) and Carrol et al. (2000)) and/or small country models (Mansoorian(1993) and Ikeda and Gombi (1999)). When the interest rate is endogenously determined in heterogeneous consumer (or country) economies, one’s
consumption-habit behavior would be a¤ected by others’ through interestrate adjustments.
To characterize such internationally interactive habit dynamics, we construct a world felicity function from the two countries’ individual felicity functions. It captures pecuniary externalities caused by interest rate adjustments
helps describe the two-country equilibrium easily. Using the approach, thepossibility of the transfer paradox is shown to depend on: (i) internationaldi¤erences in the degree of habituation; (ii) initial asset distribution; and(iii) elasticities of intertemporal substitution. In particular, when the debtor
is more habituated to consumption than the creditor, an income transferfrom the creditor to the debtor raises the interest rate in transition throughchanges in time preference. With su¢ciently low elasticities of intertemporal substitution and/or su¢ciently large stock of the creditor’s assets, theresultant intertemporal terms of trade e¤ect immiserizes the recipient andenriches the donor.
A punch-line of the result is that it is seemingly inconsistent with Samuelson’s fundamental result: he claimed that the paradox never occurs in twocountry, free-trade Walras-stable equilibria. To reconcile the two propositions, we introduce an ad hoc Walras adjustment process, showing that ourtransfer paradox also takes place only when the international bond market
is “unstable” with respect to the ad hoc process. By doing it, however,
we do not intend to claim that the transfer paradox is implausible: theadjustment process is not a rational-expectation equilibrium. The rationalexpectation equilibrium we obtain is dynamically (saddle-point) stable in theusual sense. Our point is: given that the economy is always in the dynamic
rational expectation equilibrium, the transfer paradox generically occurs at
the neighborhoods of a certain steady state point.

1.1

Comparisons with the literature

There are several important contributions in the dynamic theory of the transfer problem. Among others Epstein (1987) is related most closely to thepresent research. In an application of his dynamic general equilibrium modelHeterogeneous Habits and The Transfer Paradox3with recursive preferences, he predicts that the transfer paradox can occur
under “turnpike stability” (Epstein (1987), footnote 8, p.91). In his model,
time preference, as a source of interest-rate adjustments, is endogenized byformalizing directly the discount rate as a function of consumption. In our
non-recursive preference model, variations in time preference are generated
endogenously by habit formation, in contrast. Furthermore, he does not
examine how his result can be reconciled to Samuelson’s result.2
Fries (1983) gives an example of the transfer paradox in a two-country,
two-commodity trade model with production uncertainty. The example isconstructed by assuming that equities are internationally-nontraded. On thecontrary, we emphasize the possibility that rapid interest-rate adjustments
in well-organized international capital markets bring about paradoxical intertemporal terms-of-trade e¤ects.
Galor and Polemarchakis (1987) examine the possibility of the transfer
paradox in the overlapping-generation framework. In their paper, dynamic
ine¢ciency arising from the overlapping-generation structure plays a critical role in producing the paradox. The analysis is, however, limited tothe steady-state e¤ect on welfare, neglecting utility gains and losses on thetransition path. We consider e¢cient equilibria in an in…nitely-lived agent
model, where welfare e¤ects are evaluated by taking into account the entire
time-pro…le of consumption. The punch-line lies in the …nding that without
allocative ine¢ciency, the intertemporal terms of trade e¤ect due to international heterogeneity in habits can cause the paradox.
Djajić, Lahiri, and Raimondos-Møller (1998) examine the intertemporal terms of trade e¤ect of income transfers by using a two-period model.
Although our attempt is similar to theirs in spirits, they rule out the possibility of the transfer paradox by adopting a time-additive utility function.
We incorporate time-nonseparability of preferences in the form of consumption habits, which has two e¤ects: First, it brings intrinsic dynamics into
the model. Second, it enables us to endogenize transitional shifting of time
preference, the dynamics of the interest rate, and the intertemporal terms oftrade e¤ect.3
2

As a minor di¤erence, in his model the interest rate can adjust only gradually over timeas the aggregate capital stock varies. In our model, in contrast, a transfer instantly a¤ects
the interest rate through the market clearing condition of the “present-good” market.
3
After we completed an earlier version of the paper, Professor S. Lahiri kindly sent us
the manuscript of Djajić, Lahiri, and Raimondos-Møller (2001). It focuses on consumption
habits through which a temporary foreign aid a¤ects the terms of trade and welfare ofHeterogeneous Habits and The Transfer Paradox

4

Our discussion is also related to the literature on the “correspondence
principle” initially proposed by Samuelson (1947): By imposing the stability
conditions on “disequilibrium” dynamics, one can derive “fruitful” results incomparative statics. The trade theorists including Samuelson himself applied
this principle to rule out the transfer paradox. As pointed out in a criticalsurvey by Kemp (1987), however, with re-emergence of dynamic theory ofmacroeconomics, economists have sought to consider (saddle-point) stabilityof intrinsically dynamic stock-‡ow processes by assuming that the economy
is always in dynamic equilibrium. For example, Burmeister and Long (1977)
reasonably proposed a new correspondence principle relating the saddle-point
stability and steady-state properties in dynamic optimization models, showing that some paradoxes can easily survive under the new criterion. Our
result also implies that the transfer paradox cannot be ruled out by the newcorrespondence principle.
The rest of the paper proceeds as follows: In Section 2, we present a twocountry model with consumption habits. In Section 3, equilibrium dynamics
are derived and characterized. In Section 4, the e¤ects of transfer payments
are examined, where the possibility of the transfer paradox is shown in themain proposition. Section 5 discusses the possibilities of immiserizing transfers more explicitly by considering quadratic felicity functions. Section 6summarizes the paper with concluding remarks.

2

The Two-Country Model

Consider a two-country world economy composed of home and foreign countries. Each country is populated with in…nitely-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 the goods and bonds are tradedfreely 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, foreign country’s variables are represented
with superscript asterisks.
(¤)
Consumption forms habits. Letting zt represent the time-t habit, we
the donor and recipient countries. In contrast to our setting, they introduce distortion by
assuming that there is no international asset trade; assume that habits are externalities;
and consider a two-period model.

Heterogeneous Habits and The Transfer Paradox
(¤)

(¤)

5
(¤)

specify zt as the average of past consumption rates cs ; s · t: zt =R t (¤)® ¡1 cs exp (¡® (t ¡ s)) ds; or equivalently
³
´
(¤)
(¤)
(¤)
(1)
z_t = ® ct ¡ zt ;where x_ represents the time derivative of variable x; and ® represents thediscount rate for past consumption rates. We assume that consumers in bothof the countries have the common discount rate ® for past consumption rates.
This enables us to have tractable dynamics of two-country equilibrium.
Consumers H and F have di¤erent preferences over consumption andhabits. Consumer H’s preferences are speci…ed as
Z 1(2)u(ct ; zt ) exp (¡µt) dt;
U0 =0

where µ 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 · 0; (C3) uc (c; c)+uz (c; c) >0; (C4) u is concave in (c; z) ; (C5) limc!0 uc (c; z) = 1 uniformly in z; and(C6) limc!0 [uc (c; c) + uz (c; c)] = 1: Consumer F’s utility U0¤ is speci…ed in
the same way.
Due to habit formation, consumer preferences are intertemporally dependent. Within the single-agent framework, Ryder and Heal (1973) characterize
the resulting intertemporal complementarities in consumption by adjacentand distant complementarities: With adjacent complementarity, an increase
in today’s habits increases the marginal utility of today’s consumption morethan it increases marginal disutility of habits, thereby enlarging the 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 their idea to characterize each country’s subjective or local preferences as follows:
De…nition: Intertemporal consumptions in country H are said to locallyor subjectively display adjacent (distant) complementarity, or simply display
local or subjective adjacent (distant) complementarity, LAC (LDC), if and®
only if (i¤) ucz (c; c) + µ+2®
uzz (c; c) > (<)0: Similarly LAC and LDC for
country F are de…ned by using u¤ .4
4Note that we are using term “local” for the meaning of “regional” or “country-speci…c”,
not the mathematical meaning of “neighborhoods”. Likewise we will later use “global” forthe geographical meaning.

Heterogeneous Habits and The Transfer Paradox

6

In the case of the single agent economy, these subjective complementarities de…nitely determine the equilibrium dynamics, as seen from the existing
literature (e.g., Ryder and Heal (1973) and Ikeda and Gombi (1999)). In contrast, when the economy consists of heterogeneous agents as in the present
case, some pecuniary externalities through markets divert their consumptionsaving behavior from what is predicted from the subjective intertemporalcomplementarities, as shown in the next section.
To ensure the steady state, let us assume thatµ = µ¤ :Unlike in the single agent model, however, the interest rate r can deviate
from this subjective discount rate.
Let bt denote net foreign assets held by country H. Given the initial values(b0 ; z0 ) ; consumer H chooses C0 = fct ; bt ; zt g1
t=0 so as to maximize (2) subject
to: (i) the ‡ow budget constraint,(3)
b_ t = rt bt + y ¡ ct ¡ ¿t ;where ¿t denotes lump-sum transfer payments from country H to F;5 (ii) theformation of consumption habits (1); and (iii) the transversality conditions.
Letting ¸t (¸ 0) be the shadow price of savings and »t (· 0) that of habit
formation, the optimal conditions are given byuc (c; z) = ¸t ¡ ®»t ;¸_ t = (µ ¡ rt ) ¸t ;»_ t = (µ + ®) »t ¡ uz (ct ; zt ) ;

(4)
(5)
(6)

together with (1), (3), and the transversality conditions for bt and zt :In exactly the same way, country F’s behavior can be speci…ed. The
model is closed by introducing the market clearing conditions:ct + c¤t = Y (´ y + y ¤ ) ;(7)

bt + b¤t = 0:(8)

By the Walras law, these are not independent: (8) together with (3) and thecorresponding constraint for the foreign consumer imply (7). In sum, theequilibrium time-path of (bt ; b¤t ; ct ; c¤t ; zt ; zt¤ ; rt ; ¸t ; ¸¤t ; »t ; »t¤ ) is determined byequations (1), (4) through (8), and the transversality conditions for (bt ; b¤t ; zt ; zt¤ ).
5Without loss of generality, we assume away the presence of governments in the bothcountries since this paper just focuses on the e¤ects of international transfer payments.

Heterogeneous Habits and The Transfer Paradox

3

7

Equilibrium Dynamics

3.1

The world felicity function

In this economy, consumption dynamics in the two countries interact throughinternational bond markets. To ease the analysis, it is useful to introduce“aggregate habits” Zt as
Zt ´ zt + zt¤ :Since ® is assumed to be internationally identical, the dynamics of Zt can be
expressed from (1) and the market clearing condition (7) as Z_ t = ® (ct + c¤t ¡ Zt ) =® (Y ¡ Zt ). Since Y is constant throughout the paper, this can be solved as
Zt = Y:
We can thus express zt¤ as a function of zt :6
zt¤ = Y ¡ zt :(9)

By using (9), dynamics can be drastically reduced in the following manner. De…ne ¾ as:
¾=

¸
;
¸¤

¤
which is constant since ¸_ t =¸t = ¸_ t =¸¤t from (5). We then construct aggregate
indices for (u; u¤ ) and (»; » ¤ ) as:v (c; z) ´ u (c; z) + ¾u¤ (Y ¡ c; Y ¡ z) ;(10)

& ´ » ¡ ¾» ¤ ;(11)where (7) and (9) are substituted.
The new felicity function v represents the aggregate utility function withweights being given by relative individual marginal utilities. We could call it
the world felicity function. The new shadow price & is the weighted di¤erencein the shadow prices of habits. From the de…nition of v; & could be regarded
as net marginal gains of transferring consumption capital from country F
6When aggregate consumable income Y varies due to …scal policies and/or supply
shocks, Zt evolves over time. For this case, see our companion paper, Ikeda and Gombi(2001).

Heterogeneous Habits and The Transfer Paradox

8

to H. With these de…nitions, we can reduce the equilibrium dynamics ofconsumption habits around a steady state point as follows:
Property 1: In equilibrium, consumer H’s habits around steady state aregoverned by
µ ¶
µ ¶
z_
z^
(12)
=M
;
&_
^&
M=

Ã

³
´
2
vcz
¡® 1 + vcc¡ v®cc(vcz )2 ¡vcc vzz
vccµ + ®(vccvcc+vcz )!

;where x^ denotes deviations of variable x from its steady-state value x¹: x^ ´
xt ¡ x¹.
Proof. See Appendix A.
This property can be easily understood by recalling the second theoremof public economics. Without any distortion, the Pareto-optimal resource
allocation attained in this equilibrium can be duplicated as a solution to asocial welfare maximization problem:7
max U0 + ¯U0¤subject to Y = ct + c¤t ; Y = zt + zt¤ ; and z_t = ® (ct ¡ zt ) :Indeed, the corresponding Hamiltonian function,
Ht = u(ct ; zt ) + ¯u¤ (Y ¡ ct ; Y ¡ zt ) + &t ® (ct ¡ zt ) ;produces the same optimal condition as in Property 1 when weight ¯ is set
equal to ¾:
The dynamic equation (12) for habits is exactly the same as in the caseof the small country model (e.g., Ikeda and Gombi (1999) and Mansoorian(1996)), except for that v and & represent the world average preferences,instead of individual consumers’ preferences. In parallel with intertemporalcomplementarities with respect to consumers’ individual preferences, we cointerms to characterize those with respect to the world felicity function:
7

The Pareto-optimality is shown in footnote 10.

Heterogeneous Habits and The Transfer Paradox

9

De…nition: Intertemporal consumptions are said to globally display distant(adjacent) complementarity, or simply display global adjacent (distant) com®
plementarity, GAC (GDC), i¤ vcz (c; c) + µ+2®
vzz (c; c) > (<)0:
From equation (12), we can obtain the characteristic equation: F ($) = 0,
where
µ
¶
® (µ + 2®)®
2
F ($) ´ $ ¡ µ$ ¡ ® (µ + ®) ¡vcz +vzz ; (13)vcc
µ + 2®
which has one stable and one unstable roots. Let ! be the stable root, which
is given in terms of the world felicity function asqµ ¡ (µ + 2®)2 ¡ 4® (µ + 2®) ­(14)!´(< 0) ;
2¡¢®
where ­ ´ ¡ vcz + µ+2®
vzz =vcc > (resp. <)0 i¤ GAC (resp. GDC) prevails.
This root speci…es a saddle trajectory for the equilibrium consumption¢

dynamics. In particular, since z^t = !^z stable arm
zt on the trajectory, the c^-^is obtained by substituting it into (1) as
µ
¶
!+®
(15)
c^t =
z^t ;
®
which, from the market clearing condition and (9), implies:
µ
¶
!+® ¤
¤
z^t :
c^t =
®

(16)

®
From equations (14) through (16), in the case of GAC (vcz (c; c)+ µ+2®
vzz (c; c) >0), ! + ® is negative and thus the stable arms (15) and (16) are positively®
sloping, whereas under GDC (vcz (c; c) + µ+2®
vzz (c; c) < 0) the trajectories
have a negative slope with negative ! + ®: These properties can be summarized as follows:

Proposition 1: In equilibrium, both of the countries experience positive(¤)(¤)(negative) co-movements between consumption and habits, i.e., dct =dzt >(<) 0; i¤ the world felicity function displays GAC (GDC).
This proposition implies that the equilibrium dynamics in each country
can di¤er from what is predicted by local habit preferences in individualHeterogeneous Habits and The Transfer Paradox

10

countries: For example, even if consumers in country H have the LDC preference, their consumption positively co-moves with habits in equilibrium if
the world felicity function displays GAC. Note, however, that, for this to be
the case, country F should display strong LAC since the world preference
captured by the world felicity function is an average of the two countries’ local preferences. More explicitly, by introducing indices ­H and ­F for local
preferences as
¶
¶
µ
µ
®
®
¤
¤
H
F
­ = ¡ ucz +uzz =ucc and ­ = ¡ ucz +u=u¤cc ;
µ + 2®
µ + 2® zz
where, e.g., ­H > (<)0 , LAC (LDC) for H, subjective intertemporal complementarities in individual countries can be related to intertemporal complementarity prevailing globally in equilibrium as follows:
Property 2: Global intertemporal complementarity is a weighted average ofthe individual countries’ local intertemporal complementarities in the sensethat:­ = ²­H + (1 ¡ ²) ­F ;where: ² = ucc = (ucc + ¾u¤cc ).
Therefore, LAC (LDC) in both countries implies GAC (GDC). When thelocal intertemporal complementarities di¤er internationally, global intertemporal complementarity is determined by their relative strength.
The exact meaning of the local complementarities can be explained asfollows. De…ne characteristic polynomial function F H ($) by replacing thesecond order derivatives vcc ; vcz ; and vzz in (13) with ucc ; ucz ; and uzz , respectively. F H ($) is a characteristic polynomial function for country H when r
is …xed to µ as if country H were a small country. The characteristic equation F H ($) = 0 has again one stable and one unstable roots. When ! H
represents the stable root, it is given in parallel with (14) asqµ ¡ (µ + 2®)2 ¡ 4® (µ + 2®) ­HH! ´(< 0) ;
2
implying that the sign of ! H + ® corresponds to local intertemporal complementarities in country H. The same discussion can be applied to theHeterogeneous Habits and The Transfer Paradox

11

case of country F by introducing the corresponding characteristic equation
F F ($) = 0 and its stable root ! F . Local intertemporal complementarities
can be characterized as follows:
Property 3:! i + ® > (<) 0 , LAC (LDC) in country i (i = H; F ).
From the de…nitions of !; ! H ; and ! F ; ! lies between ! H and ! F . Root! can be either larger or smaller than ! F , depending on whether ­H islarger or smaller than ­F . When ! H is larger than ! F , consumption adjustment in country H is slower due to stronger habituation to current consumption than in country F. We describe this relationship by using the following
terminology:H

De…nition: Country H is said to be more (less) habituated to consumption
than country F i¤ ! H > (<)! F and hence i¤ ­H > (<)­F :

3.2

Interest rate dynamics and pecuniary externalities

As shown in Proposition 1, equilibrium consumption dynamics are determined by globally formed habit preferences, which generally di¤er from local
preferences characterized by Property 3. This discrepancy is caused by pecuniary externalities of interest-rate adjustments, as shown in the next subsection. The interest-rate dynamics play an important role in global habit
formation and consumption behavior in each country. As is proven by Appendix B, the interest rate dynamics can be described by:¢¡(17)r^ = · ­H ¡ ­F z^;

where · is de…ned as·´!ucc u¤cc (µ + 2®)(> 0) :¸¤ (® + µ ¡ !) (ucc + ¾u¤cc )This reveals that how the interest rate co-moves with habits in country H
depends crucially on the signs of di¤erence ­H ¡ ­F , and hence on whether
country H is more or less habituated to consumption than country F:

Heterogeneous Habits and The Transfer Paradox

12

Property 4: The equilibrium interest rate positively (negatively) co-moves
with country H’s habits i¤ consumption in country H is more (less) habituated to consumption than in country F: in transition, we haver (t) ¡ µ
R 0 , ! H R ! F (i.e., ­H R ­F ):z (t) ¡ z¹
Remark: If country H is identical to country F (! H = ! F ), or if country F
is small in the sense that ¸¤ is in…nitely large (¸¤ ! 1), for which case · iszero, the interest rate dynamics degenerate at µ as in the small country case(e.g., Ikeda and Gombi (1999))
In the presence of international heterogeneity in the habit preference, theinterest rate r (t) in transition can deviate from µ. For example, supposethat country H’s preferences display LDC (! H + ® < 0) whereas country F
LAC (! F + ® > 0). Then, ! F is larger than ! H , so that r (t) negatively comoves with z (t). This can be explained intuitively as follows. Without lossof generality, suppose that z (t) < z¹, and hence, from (9), z ¤ (t) > z¹¤ . With
LDC, country H would consume more now than in the future steady state
if r equaled µ: Country F would also consume more now than in the future
due to LAC, which would give rise to excess demand in the “present good”
market. The market clearing r (t) should thus be higher than its steady statelevel µ, as shown in Figure 1. By the same reasoning, in the case of LAC
for country H with LDC prevailing in country F, r (t) is positively correlatedwith z (t).
These interest rate adjustments cause pecuniary externalities that divert
consumption dynamics from what is predicted from individual countries’ preferences, as pointed out in Proposition 1. Recall, for example, the case of LDC
for country H with GAC prevailing in equilibrium. When z (t) < z¹, a higherr (t) than µ induces country H to consume less now than in the future, sothat c (t) < c¹; as is predicted by GAC. This takes place when LAC in country
F is dominant. In contrast, if country H is dominant, a larger portion of theadjustment burden should be owed by country F’s consumption: a higherr (t) results in a lower c¤ (t) than c¹¤ , leaving c (t) larger than c¹ in country H,
so that GDC prevails re‡ecting H’s LDC.
The transition dynamics of net foreign assets also depends on the propertyof the world felicity function and international heterogeneity of habits. By

Heterogeneous Habits and The Transfer Paradox

13

linearizing (3) and substituting (15) and (17) into the result, we can obtain
^bt =

¢ ª
©
¡
1
! + ® ¡ ®· ­H ¡ ­F b0 z^t :® (µ ¡ !)(18)

Therefore, whether net foreign assets are positively or negatively correlatedwith consumption habits depends on: (i) whether the world felicity function
displays GDC or GAC (the signs of ! + ®); (ii) whether country H is moreor less habituated to consumption than country F (the signs of ­H ¡ ­F );and (iii) whether country H is initially a net creditor or debtor (the signs ofb0 ). Note that the last two factors capture the current-account e¤ect of theinterest-rate variation discussed in Property 4.

3.3

Steady state

From (1), (3), (4), (5), (6), (7), (8), (9), and (18), the steady state equilibrium, (¹c; c¹¤ ; z¹; z¹¤ ; ¹b; ¹b¤ ; ¹̧ ; ¹»; r¹), is determined by the following equations:c¹ = z¹;¹» =uc (¹c; z¹) =r¹ =r¹¹b + y =¹b ¡ b0 =z¹ + z¹¤ =c¹ + c¹¤ =¹b + ¹b¤ =1
c; z¹) ;uz (¹µ+®
¹̧ ¡ ®¹»;µ;c¹ + ¿;©
¡¢ ª1! + ® ¡ ®· ­H ¡ ­F b0 (¹z ¡ z0 ) ;® (µ ¡ !)
Y;
Y;
0;

(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)

where (19) represents z_ = 0 (see (1)); (20) represents »_ = 0 ((6)); (21)comes from the …rst order condition for consumption c (4); (22) represents¸_ = 0 ((5)); (23) represents b_ = 0 ((3)); (24) comes from (18) evaluated at
t = 0; (25) comes from (9); (26) and (27) represent the world market clearing
conditions for the good and bonds ((7) and (8)).
By substituting (19) and (22) into (23), we obtainµ¹b + y = z¹ + ¿;(28)

Heterogeneous Habits and The Transfer Paradox

14¡ ¢which, together with (24), determines ¹b; z¹ . Given this, c¹ is given by (19);¹» by (20); ¹̧ by (21); z¹¤ by (25); c¹¤ by (26); and ¹b¤ by (27). Figure 2 depicts¡ ¢the determination of ¹b; z¹ : schedule BB 0 depicted with a positive slope
represents (28), whereas schedule SS 0 represents (24).

4

The Transfer Paradox

4.1

The steady-state e¤ect of income transfer

Suppose that country H is a creditor: b0 > 0. Let us consider a permanent
income transfer from the creditor H to the debtor F: d¿ > 0:8 From (19),(24) and (28), we can obtain
¡
¢ ª¡1
d¹
d¹
z
c ©
=
= !(® + µ) ¡ ®µ· ­H ¡ ­F b0®(µ ¡ !);
d¿
d¿

(29)

©
¡
¢ ª d¹d¹b
1z=;! + ® ¡ ®· ­H ¡ ­F b0d¿d¿® (µ ¡ !)¡¢where it is assumed that !(® + µ) ¡ ®µ· ­H ¡ ­F b0 6= 0 for the existenceof the steady-state equilibrium.
The transfer payments have two e¤ects on country H: a direct e¤ect! (® + µ) in (29), due to a¡ decrease¢in output income, and an indirect effect represented by ¡®µ· ­H ¡ ­F b0 , which is caused by changes in theinterest rate. Steady state habits and consumption in country H are reducedby the transfer unless the indirect e¤ect is positive and dominant. We cannot, however, rule out a priori the pathological possibility that the transfer
payments increase the donor’s habits and consumption due to increases in
the interest rate. This causes the transfer paradox, as shown later. The e¤ect
on net foreign assets depends crucially on whether schedule SS 0 upward or
downward sloping.
These adjustments can be illustrated by using Figure 3. An increase in¿ shifts schedule BB 0 upward from B0 B00 to B1 B10 ; bringing the steady state
point from E0 to E1a , E1b , or E1c , corresponding to the SS 0 schedule Sa Sa0 ; Sb Sb0 ;8

Permanent transfer payments in the form of d¿, which are under consideration, is
economically and analytically the same as once-and-for-all transfers db0 of net foreign
assets since international bond markets are perfect.

Heterogeneous Habits and The Transfer Paradox

15or Sc Sc0 .9 Point E1c represents the paradoxical case which we shall focus on
next.

4.2

The main result

Let us now examine the welfare e¤ects of the transfer. First, linearize
the instantaneous utility u around the steady state to obtain: u (ct ; zt ) =u (¹c; z¹) + uc c^t + uz z^t . Next, substituting (15) and(!t) successivelyn z^t = z^0 expo
uc (!+®)
into this equation yields: u (ct ; zt ) = u (¹z ; z¹) ++ uz z^0 exp (!t). By®
substituting this into (2), the welfare at time zero is obtained as
U0 =

u (¹
z ; z¹) uc (! + ®) =® + uz+
(z0 ¡ z¹) :µµ¡!

From (29), the e¤ect of transfer d¿ (> 0) on donor H’s welfare is thus
given by©¡¢ª
! 1 + ®µ uc + uz d¹
dU0
z
= ¡
d¿
d¿
µ (µ ¡ !)½¾¡1¢(® + µ)(® + µ ¡ !)(® + µ) ! ¡ H
F(; 30)= ¡¡ ­ ¡ ­ b0®µ (2® + µ) (1 ¡ ²) ucc®µ·
whereas, from (9), the e¤ect on recipient F’s welfare is:©¡¢ª! 1 + ®µ u¤c + u¤z d¹
dU0¤
z
(31)
=
d¿
d¿
µ (µ ¡ !)½¾¡1¢(® + µ)(® + µ ¡ !)(® + µ) ! ¡ H
F=:¡ ­ ¡ ­ b0¾®µ (2® + µ) (1 ¡ ²) ucc®µ·

From the regularity assumption (C3), the …rst equations of (30) and (31)
imply that the sign of the welfare e¤ect is the same as that of d¹
z =d¿ for10
country H and opposite for country F.
9The above analysis can be applied to other income shocks with constant aggregate
income. For example, an increase in the subjective discount rate µ shifts the BB 0 schedule
upward or downward, depending on whether country H is initially a net debtor or creditor.
Given the direction of the shift, the same analysis as in Figure 3 can be applied.
10
Equations (30) and (31) imply:
dU0
dU ¤+ ¾ 0 = 0;d¿d¿meaning that the equilibrium dynamics we have obtained are indeed Pareto-e¢cient.

Heterogeneous Habits and The Transfer Paradox

16

From equations (30) and (31), the necessary and su¢cient condition for
the transfer paradox to occur is given as follows:theProposition 2: Transfer payments d¿ from country ¡H to F immiserize¢HF
recipient and enrich the donor i¤ (® + µ) != (®µ·) > ­ ¡ ­ b0 , i.e.,¢(® + µ) (® + µ ¡ !) (ucc + ¾u¤cc ) ¸¤ ¡ H
F
­
>
­
b0 :
¡
®µ (2® + µ) ucc u¤cc

(32)

Proposition 2 has the following four implications: First, since b0 is positive
by construction, the proposition implies that a necessary condition for thetransfer paradox to take place is that ­H < ­F : recipient country F is more
habituated to consumption than donor country H: ! F > ! H .
Second, the interest rate adjustment re‡ecting this habit heterogeneity
induces the intertemporal terms of trade e¤ect, thereby causing the transfer
paradox. To see this, obtain from equation (17) and (30)
¡
¢
½
¾¡1
¢
(µ ¡ !) ­H ¡ ­F
dr (0)(® + µ) ! ¡ H
F=¡:¡ ­ ¡ ­ b0d¿®µ·µThe transfer paradox thus takes place only when the interest rate instantaneously jumps up, and hence only when the intertemporal terms of trade forthe recipient deteriorates discretely.11 More speci…cally, when ­H < ­F , wecan easily show that whenever the interest rate r (0) goes up, the disposabledincome y + rb ¡ ¿ for the donor initially increases: d¿
fb0 r (0) ¡ ¿ g > 0:12 It
follows that the transfer payments bene…ts the donor only when the percentage rate of the initial rise in the interest rate is higher than the percentage
rate of the initial decrease in the total wealth b0 ¡ ¿ =r¤ :dr (0)(d¿ ) =¹
r
>
:
r¹
b0
11

Conversely, given that recipient country F is more habituated to consumption thandonor country H, the interest rate instantaneously jumps up only when the transfer paradox occurs.
12
We have b0 dr(0)¡ 1 > 0 if and only if
d¿
¢ ¡
¢ª
© ¡
! (® + µ) > ®·b0 min µ ­H ¡ ­F ; ! ­H ¡ ­F :Given that ­H < ­F , this is equivalent to (32).

Heterogeneous Habits and The Transfer Paradox

17

This situation is somewhat similar to the “debt-relief La¤er curve” e¤ect
discussed by Krugman (1989): by using a tentative model, he points out
the possibility that debt forgiveness actually increases expected payment on
the wrong side of the “debt-relief La¤er curve”. The discussion above showsthat this kind of the La¤er-curve e¤ect can be consistent with consumers’
dynamic utility maximizing behavior.13
Third, roughly speaking, given that recipient F is more habituated toconsumption than H, the larger the initial net foreign assets b0 held by donor
H, the more likely the paradox takes place. This is because a large b0 exaggerates the perverse terms-of-trade e¤ect. This statement is too rough,however, since an (exogenous) increase in b0 may well a¤ect the initial equilibrium resource allocation and hence the other parts of (32). A precise(¤)
implication of b0 is as follows. Let I0 denote the initial steady-state GNP:
(¤)
(¤)
(¤)
(¤)
I0 = µb0 + y (¤) , where note that I0 + I0¤ = Y and I0 = z0 : Suppose now
that
¢ z0
(® + µ) (® + µ ¡ !) (ucc + ¾u¤cc ) ¸¤ ¡ H> ­ ¡ ­F:¤®µ (2® + µ) ucc uccµ(33)

Then, for given (I0 ; I0¤ ), and hence for given (z0 ; z0¤ ), the larger b0 ; the more
likely the transfer paradox takes place.14
Fourth, ¡(32) depends ¢on the marginal
felicities ui ¢and¡ u¤i (i = c; ¢¢z) only
¡
¡
®
¸
®
®
¤
¤
¤
¤
¤
through ¸ = uc + ®+µ uz and ¾ = ¸¤ = uc + ®+µ uz = uc + ®+µ uz : Note³´(¤)
here that, as seen from the Euler equation (41) in Appendix B, ¡¸(¤) = c(¤) ucc
represent the elasticities of intertemporal substitution. Condition (32) thus
implies the following: Suppose that recipient
F is more habituatedto con¢¡sumption than donor H. Then, for given uij ; u¤ij ; ®; µ; ¾; b0 (i; j = c; z), thesmaller ¸ and ¸¤ ; and hence the smaller the elasticities of intertemporal substitution, the more likely the paradox occurs. This is natural again sincesmall elasticities of intertemporal substitution enlarge a response in the interest rate.15
13

In Krugman (1989), the “debt-relief La¤er curve” e¤ect takes place when debt forgiveness a¤ects the probability distribution of the nature in favor of the “good” state. Thee¤ect is thus Pareto-improving: it is also bene…cial to the debtor. In the present model,
in contrast, equilibrium is always Pareto e¢cient, so that the “debt-relief La¤er curve”
e¤ect is de…nitely harmful to the debtor.
14
Inequality (33) is necessary because b0 cannot exceed z0 =µ:15
From the same discussion, we can choose a set of parameter sets such that (32) holds
by choosing the …rst derivatives (uc ; ¡uz ; u¤c ; ¡u¤z ) arbitrarily small, with keeping ¾ and

Heterogeneous Habits and The Transfer Paradox

18

All these discussions can be summarized as follows:
Corollary:
1. The transfer paradox takes place only when recipient country F is more
habituated to consumption than donor country H.
2. The transfer paradox takes place only when the interest rate initiallyjumps up, thereby increasing the total wealth of the donor.
3. Suppose that recipient F is more habituated to consumption than donor
H. Then, the possibility of the paradox can be characterized as follows:(a) For given GNPs (I0 ; I0¤ ) of the two countries, the larger the net
foreign assets b0 held by the donor; the more likely the transfer
paradox occurs if (33) is met.
¡¢(b) For given uij ; u¤ij ; ®; µ; ¾; b0 (i; j = c; z), the smaller the elasticities of intertemporal substitution, the more likely the paradox
occurs.

4.3

The trick: relation to Samuelson

A striking aspect of Proposition 2 is that it appears to be inconsistent with
Samuelson’s fundamental result: we have shown that the transfer paradox
can occur even in a locally-stable, two-country equilibrium without any distortion.
The trick lies in a di¤erence in the stability concept: The fundamental
theorem requires the Walrasian stability, i.e., stability with respect to theWalrasian adjustment process, whereas the dynamics of our model are stablein the usual sense that the relevant characteristic root !, which characterizes
the equilibrium dynamics of the economy, is negative.
To see this, let us consider an ad hoc Walrasian adjustment process forthe time-zero international bond market as follows. Solve equation (24) forb0 and eliminate ¹b and z¹ from the resulting equation by using (28) and (17),respectively, where (17) is evaluated at t = +0. The resultant equation,
where b0 is expressed as a function of the time-zero interest rate r0 , could be
the other parameters constant. This point is discussed more explicitly in Section 5.2.

Heterogeneous Habits and The Transfer Paradox

19

regarded as a demand function for b0 . Letting D (r0 ) represent the demand
function, it is given by¡¢! (µ + ®) ¡ ®µ· ­H ¡ ­F b0y ¡ z0 ¡ ¿D (r0 ) ´
(r0 ¡ µ) ¡:HF®µ· (­ ¡ ­ ) (µ ¡ !)µAs in the usual rational-expectation models, we have so far implicitly
assumed that the international bond market is instantaneously cleared by r0
such that the demand D (r0 ) equals the existing stock b0 . Instead, imagine
for exposition that the market can be cleared only “sluggishly” according to:
dr0 (s)= k (b0 ¡ D (r0 (s))) ;
ds

(34)

where s represents some imaginary time; and k (> 0) is the adjustment speed.
Equation (34) represents an imaginary Walrasian adjustment process where
the interest rate rises “sluggishly” depending on the excess supply of thebond stock. With this process, the market could be said to be Walras-stablei¤¡¢! (µ + ®) ¡ ®µ· ­H ¡ ­F b0
0
(35)
> 0:
(D (r0 ) =)®µ· (­H ¡ ­F ) (µ ¡ !)From this observation, we can show that the Walrasian instability is a necessary condition for the transfer paradox to occur:
Proposition 3: The transfer paradox takes place only if the internationalbond market at time zero is Walras-unstable with respect to the ad hoc Walrasian adjustment process given by (34).
Proof. Suppose that inequality (32) holds true. This implies (i) the numerator of (35) is positive; and (ii) ­H ¡ ­F < 0; which together imply
D0 (r0 ) < 0:
We do not put Proposition 3 to show that the transfer paradox is unlikely
to occur, but to show the consistency of our result to Samuelson’s. Unlike
in his (and other trade theorists’) intrinsically-static models, however, we
incorporate dynamics in a natural way by considering consumers’ optimalsaving behavior, showing that the steady state equilibria are locally saddlepoint stable. Given the modelling, stability should reasonably be discussed

Heterogeneous Habits and The Transfer Paradox

20from the viewpoint of the equilibrium dynamics, as proposed by Burmeister and Long (1977). The local stability implies that the transfer paradox
generically occurs at the neighborhoods of the corresponding steady state
point. Epstein (1987) and Haaparanta (1989) also found that the transfer
paradox is compatible with saddle-point stability of equilibrium dynamics.16
All these results imply that, given that the economy is always in a dynamic
rational-expectation equilibrium, the transfer paradox cannot be ruled out
by the stability argument.17

5

Example: The case of quadratic felicity functions

5.1

Quadratic felicity functions

To discuss more explicitly the possibilities of the transfer paradox, let us
construct a simple example by using quadratic felicity functions, as in Becker
and Murphy (1988) and Obstfeld (1992):
(¤)2
(¤)2
¡
¢
(¤)
(¤) (¤)(¤) (¤) (¤)
(¤) c
(¤) z
u(¤) c(¤) ; z (¤) = Á(¤)
c
z
+Á
c
z
;(36)
¡Á
¡Á
¡Á
c
z
cz
cc
zz
2
2
¡
¢
for which case the partial derivatives of u(¤) c(¤) ; z (¤) evaluated at the initialsteady state are given by
¡ (¤)
¢ (¤)
(¤)
u(¤)
= Á(¤)cc ¡ Ácc ¡ Ácz z0 ;n¡ (¤)¢ (¤) o
(¤)
(¤)
u(¤)
=
Á
Á
Á
;
¡
¡
¡
z
z
cz
zz z0
(¤)
(¤)
(¤)
(¤)
u(¤)
= ¡Á(¤)
cc
cc ; ucz = Ácz ; uzz = ¡Ázz :For the regularity conditions to be satis…ed, we restrict the values of para(¤)(¤)
meters Ái and Áij as follows:
1. Ác > 0; Áz > 0; Ácc > 0; Ázz > 0; Ácz T 0;
(¤)

16

(¤)

(¤)

(¤)

(¤)

However, Epstein (1987) and Haaparanta (1989) do not elucidate how their results
can be consistent with Samuelson’s …nding. We can show that Epstein’s transfer paradox
also takes place only when the international capital market is unstable with respect to an
adhoc Walrasian adjustment process.
17
We do not intend to say that any “disequilibrium” dynamics are meaningless. Our
point is that the stability arguments based on ad hoc adjustment processes such as the
Walrasian process could not be a strong reason for ruling out a priori the possibilities of
the transfer paradox or other phenomena derived from rational agent models.

Heterogeneous Habits and The Transfer Paradox

21

³
´
³
´
(¤)
(¤)
(¤)
(¤)
(¤)
(¤)
(¤)
(¤)
(¤)
2. Ác ¡ Ácc ¡ Ácz z0 > 0; Áz ¡ Ácz ¡ Ázz z0 > 0 for uc > 0(¤)

and uz < 0;
³
´
n
³
´
o
(¤)
(¤)
(¤)
(¤)
(¤)
(¤)
(¤)
(¤)
(¤)
(¤)
3. Ác ¡ Ácc ¡ Ácz z0 ¡ Áz ¡ Ácz ¡ Ázz z0= Ác ¡ Áz ¡
³
´
(¤)
(¤)
(¤)
(¤)
(¤)
(¤)
Ácc ¡ 2Ácz + Ázz z0 > 0 for uc + uz > 0; and(¤) (¤)(¤)2

4. Ácc Ázz ¡ Ácz > 0 for concavity.

5.2

Possibilities of the transfer paradox

By using the felicity functions, we shall show that there actually exist parameter values that allow the transfer paradox. With felicity functions (36),the necessary and su¢cient condition (32) for the paradox to occur reduces
to
¡
where

¢
(® + µ) (® + µ ¡ !) Ácc + ¾Á¤cc ¤ ¡ H
F
¸
>
­
­
b0 ;
¡
®µ (2® + µ)Ácc Á¤ccµ¸ ´ u¤c +
¤

µ

¸
¾ ´ ¤
¸

¶

®
u¤
µ+® z

=

µ
­ = Ácz ¡H¶= Á¤c ¡ (Á¤cc ¡ Á¤cz ) z0¤ ¡(37)®
fÁ¤ ¡ (Á¤cz ¡ Á¤zz ) z0¤ g ;®+µ z®
Ác ¡ (Ácc ¡ Ácz ) z0 ¡ ®+µfÁz ¡ (Ácz ¡ Ázz ) z0 g;®
¤Á¤c ¡ (Á¤cc ¡ Á¤cz ) z0 ¡ ®+µ fÁ¤z ¡ (Á¤cz ¡ Á¤zz ) z0¤ g

(38)

µ
¶
¶
®
®
¤
¤
F
Ázz =Ácc ; ­ = Ácz ¡Ázz =Á¤cc :µ + 2®
µ + 2®

Start with choosing parameter values satisfying the regularity conditions,such that:² (a) country F is more habituated to consumption than H: ­F > ­H ;² (b) country H is initially a creditor: b0 > 0;
(¤)

(¤)

(¤)

(¤)

² (c) Ácc ¡ Ácz > 0; Ácz ¡ Ázz > 0:

Heterogeneous Habits and The Transfer Paradox

22

From two properties (a) and (b), the right hand side of inequality (37) is
negative. Since our purpose in this subsection is to verify the possibility ofthe paradox, we should start with some parameter set under which the righthand side of inequality (37) is larger than the left hand side.
But note that, by choosing (Ác ; Áz ; Á¤c ; Á¤z ) appropriately, we can make the(¤)(¤)
values of uc and uz and hence ¸¤ arbitrarily close to zero, with …xing thevalue of ¾ and other components of (37) at the values initially chosen. Thatis, we can take limits of ¸¤ # +0 by taking limits of (uc ; ¡uz ; u¤c ; ¡u¤z ) # +0,
which, in turn, can be taken, with keeping ¾ constant, by considering thelimit:(Ác ; Áz ; Á¤c ; Á¤z ) # ((Ácc ¡ Ácz ) z0 ; (Ácz ¡ Ázz ) z0 ; (Á¤cc ¡ Á¤cz ) z0¤ ; (Á¤cz ¡ Á¤zz ) z0¤ ) ;(39)
(¤)

(¤)

(¤)

which is positive from condition (c). For any (®; µ; Ácc ; Ácz ; Ázz ; z0 ; z0¤ ; b0 )that satis…es the regularity conditions and conditions (a) though (c) above,
this means that there always exists a continuum of the values for (Ác ; Áz ; Á¤c ; Á¤z )and hence for ¸¤ that validates inequality (37).
For example, limit (39) proposed³ above can be conducted´ as follows.
(¤)
(¤)
(¤)
First, for a given regular value of ®; µ; Ácc ; Ácz ; Ázz ; z0 ; z0¤ ; b0 , construct(Ác ; Áz ; Á¤c ; Á¤z ) as
Ác
Áz
Á¤c
Á¤z

=
=
=
=

(Ácc ¡ Ácz ) z0 + "c ;(Ácz ¡ Ázz ) z0 + "z ;(Á¤cc ¡ Á¤cz ) z0¤ + "¤c ;(Á¤cz ¡ Á¤zz ) z0¤ + "¤z ;(¤)

where the "i ’s are positive parameters. Let k and n be constant parameters(¤)
and n > 0. Then, by setting the "i ’s assuch that 0 < k < ®+µ®
"z = k"c ; "¤c =1k"c ; and "¤z = "c ;
n
n

(40)

we obtain from (38)
¾ = n;(¤)

which is a positive constant by construction, irrespective of the "i ’s values.
From (40), by taking limit "c ! 0, we can thus take limit (39) and hence ¸¤ #+0, with keeping ¾ and other components appearing in (37) constant. (Note:

Heterogeneous Habits and The Transfer Paradox

23

Components in (37) other than (¸¤ ; ¾) do not depend on (Ác ; Áz ; Á¤c ; Á¤z ) and(¤)
hence on the "i ’s.) In this way, by choosing an arbitrarily small "c , we makethe absolute value of the left hand side of (37) arbitrarily small, with keeping
the right hand side constant.

5.3

Numerical examples

In view of (37), let us now de…ne the paradox indicator function as:
T P X (®; µ; Ác ; Áz ; Ácc ; Ácz ; Ázz ; Á¤c ; Á¤z ; Á¤cc ; Á¤cz ; Á¤zz ; z0 ; z0¤ ; b0 )(® + µ) (® + µ ¡ !) Ácc + ¾Á¤cc ¤= ¡¸ ¡ (­H ¡ ­F ) b0®µ (2® + µ)Ácc Á¤cc½> 0 for the paradox case,· 0 for the normal case.
For the reference case, we consider the parameter set listed in Table 1
which satis…es all the regularity conditions and conditions (a)-(c):
®
1

µ Ác
Áz Ácc
1 15.2 0.1 16

Ácz
1

Ázz
1

Á¤c Á¤z Á¤cc
8.7 0.6 10Á¤cz Á¤zz
1.5 1

z0
1

z0¤
1

b0
0.4

Table 1: Reference Parameter Values

With these parameter values, the following properties are valid:
1. We have ­H = 4:1667 £ 10¡2 and ­F = 0:11667, implying that ­F >­H > 0; and hence ! F > ! H , that is country F is more habituated toconsumption. This is a necessary condition for the transfer paradox tooccur.
2. The stable root amounts to: ! = ¡0:92775; so that GAC prevails since! + ® > 0:
3. ¸ = ¸¤ = 0:15 and the weight applied to country F’s felicity in constructing function v is also unity: ¾ = 1.
4. However, the transfer paradox does not take place in the reference casesince T P X = ¡1:7576 £ 10¡2 < 0:

Heterogeneous Habits and The Transfer Paradox

24

As shown in Corollary, the transfer paradox likely occurs under a large
b0 : We can obtain the critical value ¯ for b0 required for the transfer paradox
to occur under the reference case by solving:
T P X (1; 1; 15:2; 0:1; 16; 1; 1; 8:7; 0:6; 10; 1:5; 1; 1; 1; ¯) = 0:This can be computed to obtain ¯ ' 0:64, which is smaller than z0 =µ = 1.
Therefore, if b0 is larger 0:64, transfer payments immiserize the recipient,insofar as b0 < 1: In fact, for example, when b0 = 0:7; T P X amounts to
4:92 £ 10¡3 , implying the paradox.
Let us next check item 3(b) in Corollary by examining the e¤ect of changesin (Á¤c ; Á¤z ). Under the reference values for the other parameters, (Á¤c ; Á¤z )should satisfy Á¤c > 8:5; Á¤z > 0:5; Á¤c ¡ Á¤z > 8:0 to meet the regularity conditions. A small Á¤c ¡ Á¤z ¡ 8:0 implies a small ¸¤ and hence a small elasticity
of intertemporal substitution of country F, where the elasticity of intertemporal substitution is given by ¸¤ =Á¤cc = ¸¤ =10 in the reference case. Table 2
summarizes the relationship among (Á¤c ; Á¤z ), elasticity of intertemporal substitution (EIS) ¸¤ =Á¤cc , and ¯. In the reference case (Á¤c ; Á¤z ) = (8:7; 0:6) ; EIS
amounts to 0:015, and a b0 larger than 0:64 results in the paradox. As is
expected, in case 1, in which EIS is much smaller than in the reference case,the paradox can occur for a smaller b0 (¯ = 0:25). In contrast, in case 2, EIS
equals 0:028, so that the paradox unlikely takes place since ¯ is almost equal
to one. In case 3, EIS is too large for the paradox to occur.18

reference case
case 1case 2case 3EIS (¸¤ =Á¤cc ) b0 for the paradox (¯)(Á¤c ; Á¤z )(8:7; 0:6)
0:015
0:64
(8:51; 0:509)
0:0006
0:25(8:93; 0:8)
0:028
0:98
(9; 1)
0:125;

Table 2: Elasticity of Intertemporal Substitution, External Assets, and thePossibility of the Transfer Paradox.
18

To be precise, these results are more than predicted by the analytical result 3(b) in
Corollary. When we compute the results in Table 2, all the e¤ects on parameters ¾ and !
are incoporated, whereas in statement 3(b) in Corollary, we …x these parameters.

Heterogeneous Habits and The Transfer Paradox

625

Concluding remarks

We have analyzed the transfer problem by using a one-good two-country
model with in…nitely-lived, heterogeneously habit-forming consumers. Due
to pecuniary externalities through endogenous interest-rate adjustments, theequilibrium consumption-habit dynamics are characterized by a world felicity
function constructed from the individual countries’ felicity functions, which
determines intertemporal consumption-complementarities prevailing in thetwo-country equilibrium. International transfer payments can harm the recipient and bene…t the donor, depending on: (i) international heterogeneity
in the preference for habits; (ii) initial asset distribution; and (iii) intertemporal consumption-complementarities in the world. In particular, when the
debtor is more habituated to consumption than the creditor, transfer from thecreditor to the debtor raises the interest rate in transition through changes intime preference. This intertemporal terms of trade e¤ect causes the paradox.
Our result implies that the transfer paradox cannot be ruled out by
the stability argument, given that the economy is always in the rationalexpectation dynamic equilibrium. We never intend to say that any “disequilibrium” analysis is meaningless; rather it must be examined whether
the paradoxical result can still survive when stable adjustment processes inmarket disequilibrium are properly modeled from the micro-founded rational
behavior of economic agents. It must be an important research topic in the
future.

Heterogeneous Habits and The Transfer Paradox

26

AppendicesA

Proof of Property 1¤
Note …rst that ¸=¸¤ which is constant because ¸_ t =¸t = ¸_ t =¸¤t from (5). Byeliminating c¤ and z ¤ using (7) and (9) from the foreign counterpart of (4),combining the resultant equation and (4) yields

uc (c; z) + ®»¸
= ¤ = constant.
¤¤uc (Y ¡ c; Y ¡ z) + ®»¸
By totally di¤erentiating this, we can obtainc^ = ¡®¸¤¸¤ ucz + ¸u¤cz®¸^» +^» ¤ :
z
^
¡
¤
¤
¤
¤
¤
¤
¸ ucc + ¸ucc
¸ ucc + ¸ucc
¸ ucc + ¸ucc

Substitute this equation into (1), (6), and the foreign counterpart of (6) andeliminate c¤ and z ¤ using (7) and (9) from the resulting equation. Then,we³ can¤ ´derive the autonomous dynamic equation system with respect toz^; ^»; ^» as
µ ¤
¶
®2 ¸
¸ ucz + ¸u¤cz®2 ¸¤^^» ¤ ;
z_ = ¡®
»
+
+
1
z
^
¡
¸¤ ucc + ¸u¤cc¸¤ ucc + ¸u¤cc¸¤ ucc + ¸u¤cc
½
¾
µ
¶
¤
¤
¤
(¸
u
+
¸u
)
®u
¸
®ucz ¸ ^¤
u
cz
cz
cz
cz
^» ¡
» ;
»_ =
¡ uzz z^ + µ + ® + ¤¤¤¤¤¸ ucc + ¸ucc
¸ ucc + ¸ucc
¸ ucc + ¸u¤cc
½ ¤ ¤
µ
¾
¶
¤
¤ ¤
¤ ¤
®u
u
®u
(¸
u
+
¸u
)
¸
¸
¤
cz
cz
cz
cz
cz
¤
^» + µ + ® +^» ¤ :»_ = ¡¡ uzz z^ ¡ ¤
¤
¤
¤
¤
¤
¸ ucc + ¸ucc
¸ ucc + ¸ucc
¸ ucc + ¸ucc
From the de…nitions (10) and (11) of v and &, respectively, this autonomous
system can be reduced to (12).

B

Derivation of (17)From (4), local consumption
dynamics can be expressed using time preference¯d ln(uc (c;z)+®») ¯
½´¡
¯ as
dt
c=0
_

c_ = ¡¸(^r ¡ ½^) ;
ucc

(41)

Heterogeneous Habits and The Transfer Paradox

27

where
^½ =

® (µ + ®) ^® (uzz + ucz )
z^ ¡
»:
¸
¸

Since c_ = !^c; (15) implies¶µ!+®
c_ = !
z^:®
Substituting the last two equations into (41), we obtain:r^ =® (µ + ®) ^¡ucc ! (! + ®) + ®2 (uzz + ucz )
z^ ¡
»:
®¸
¸

(42)

To eliminate ^» from the above, substitute »_ = !^» and (15) into (6) toobtain^» = (! + ®) ucz + ®uzz z^:® (µ + ® ¡ !)By substituting this equation into (42), we can obtain (17).

Heterogeneous Habits and The Transfer Paradox

28

References
Becker, G. and K. Murphy, 1988, A theory of rational addiction, Journal of Political Economy 96, 675-700.
Bhagwati, J.N., R.A. Brecher, and T. Hatta, 1983, The generalized
theory of transfers and welfare: Bilateral transfers in a multilateralworld, American Economic Review 73, 606-618.
Bhagwati, J.N., R.A. Brecher, and T. Hatta, 1985, The generalized theory of transfers and welfare: Exogenous (policy-imposed) and endogenous (transfer-induced) distortions, Quarterly Journal of Economics
100, 697-714.
Boldrin, M., L.J. Christiano, and J.D.M. Fisher, 2001, Habit persistence, asset returns, and the business cycle, American Economic Review 91, 149-166.
Braun, P., G. Constantinides, and W. Ferson, 1993, Time nonseparability in aggregate consumption: International evidence, European
Economic Review 37, 897-920.
Brecher, R.A. and J.N. Bhagwati, 1982, Immiserizing transfers fromabroad, Journal of International Economics 13, 353-364.
Burmeister, E. and Ngo Van Long, 1977, On some unresolved questions in capital theory: An application of Samuelson’s correspondence
principle, Quarterly Journal of Economics 91, 289-314.
Campbell, J. Y. and J. H. Cochrane, 1999, By force of habit: A
consumption-based explanation of aggregate stock market behavior,
Journal of Political Economy 107, 205-251.
Carrol, C. D., J. Overland, and D. N. Weil, 2000, Saving and growthwith habit formation, American Economic Review 90, 341-355.
Djajić, S., S. Lahiri, and P. Raimondos-Møller, 1998, The transfer problem and the intertemporal terms of trade, Canadian Journal of Economics 31, 427-436.
Djajić, S., S. Lahiri, and P. Raimondos-Møller, 2001, Logic of aid in aninternational setting, mimeo.

Heterogeneous Habits and The Transfer Paradox

29

Epstein, L.G., 1987, A simple dynamic general equilibrium model,
Journal of Economic Theory 41, 68-95.
Gale, D., 1974, Exchange equilibrium and coalition: An example, Journal of Mathematical Economics 1, 63-66.
Galor, O. and H.M. Polemarchakis, 1987, Intertemporal equilibrium
and the transfer paradox, Review of Economic Studies 54, 147-156.
Haaparanta, P., 1989, The intertemporal e¤ects of international transfers, Journal of International Economics 26,371-382.
Ikeda, S. and I. Gombi, 1999, Habits, costly investment, and current
account dynamics, Journal of International Economics 49, 363-384.
Ikeda, S. and I. Gombi, 2001, Global habit formation: A two-country
analysis, mimeo.
Kemp, M., 1987, Correspondence principle, in: J. Eatwell, M. Milgate,
and P. Newman eds., The New Palgrave: A Dictionary of Economics,
London: Macmillan Press.
Krugman, P., 1989, Market-based debt-reduction schemes, in: J.A.
Frenkel, M.P. Dooley, and P. Wickham eds., Analytical Issues in Debt,
International Monetary Fund, 258-278.
Mansoorian, A., 1993, Habit persistence and the Harberger-LaursenMetzler e¤ect in an in…nite horizon model, Journal of International
Economics 34, 153-166.
Naik, N. and M. Moore, 1996, Habit formation and intertemporal substitution in individual food consumption, Review of Economics andStatistics LXXVIII, 321-328.
Obstfeld, M., 1992, International adjustment with habit-forming consumption: A diagrammatic exposition, Review of International Economics 1, 32-48.
Ryder, H. E. and Heal, G. M., 1973, Optimal growth with intertemporally dependent preference, Review of Economic Studies 40, 1-33.

Heterogeneous Habits and The Transfer Paradox

30

Samuelson, P., 1947, Foundations of Economic Analysis, Cambridge:
Harvard University Press.
Samuelson, P., 1952, The transfer problem and transport costs: The
terms of trade when impediments are absent, Economic Journal 62,278-304.
Samuelson, P., 1954, The transfer problem and transport costs, 2:
Analysis of e¤ects of trade impediments, Economic Journal 64, 264289.
Yano, M., 1983, Welfare aspects of the transfer problem, Journal ofInternational Economics 15, 277-289.
Yano, M. and J.B. Nugent, 1999, Aid, nontraded goods, and transfer
paradox in small countries, American Economic Review 89, 431-449.

r

θ

r(z)

0

z (t )

Figure 1.

z

Equilibrium relationship between r and z:
The case of ω F > ω H .

z

b

B′

S′

b

S

B

0

z

z

Figure 2.
The case of

Steady state equilibrium:()ω + α − ακ Ω H − Ω F b01>.
θα (θ − ω )

b
S

a

Sc′
Sb

B1′
B0′

E1c

S a′

E1b
E1a

E0

Sa

B1
B0

Sb′
Sc

z

0

Figure 3. The effects of income transfers.