Discussion Paper No. 503

WEAKLY NONSEPARABLE PREFERENCE
AND THE CURRENT ACCOUNT:YES, THERE IS
A HARBERGER–LAURSEN–METZLER EFFECT

Shinsuke Ikeda

March 2000

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

Weakly Nonseparable Preference and theCurrent Account: Yes, There is a HarbergerLaursen-Metzler Eﬀect
Shinsuke IKEDA∗, †The Institute of Social and Economic Research, Osaka University
March 2000

Abstract
We examine the current account eﬀect of a terms-of-trade deterioration for asmall country model, incorporating weakly nonseparable preference à la Shi(1994) under endogenous time preference. This enables us to emphasize a welfarechange as an important determinant of the current account. Unlike in theliterature, even with increasing marginal impatience, the Harberger-LaursenMetzler eﬀect occurs if consumers’ preference toward imports is wealth-enhanced
enough: in that case a terms-of-trade deterioration must reduce steady-state
welfare so as to shift preference away from imports to exports. Several empiricalimplications are also derived.

Keywords: The Harberger-Laursen-Metzler eﬀect; time preference; weakly
nonseparable preference; current account
JEL classification: F41; F32∗

Correspondence. The Institute of Social and Economic Research, Osaka University,
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan. Telephone: +81(6)6879-8568. Fax:
+81(6)6878-2766. E-mail: ikeda@iser.osaka-u.ac.jp†
The author is grateful to N. Fujimoto, D. R. Hallett, Y. Iwamoto, K. Mino, M. Saito,
Y. Tsutsui, T. Ono, Y. Ono, and the participants in the Macro-Research Workshop, theMEW meeting, and the IEFS Japan 4th meeting for helpful discussions on earlier versionsof the paper. This research is financially supported by the Japan Economic Research
Foundation.

Weakly Nonseparable Preference and the Current Account1

1

Introduction

Since Harberger (1950) and Laursen and Metzler (1950) developed the Keynesian view that a terms-of-trade deterioration decreases savings by reducing
real income, this eﬀect — the Harberger-Laursen-Metzler eﬀect — has so far
attracted much attention in various contexts in macroeconomics. Since theeﬀect implies that the current account will deteriorate in response to a termsof-trade worsening if changes in investment and the government deficit are
negligible, it is usually discussed in the context of current-account dynamics (e.g., Obstfeld (1982), Svensson and Razin (1983), Persson and Svensson(1985), Sen and Turnovsky (1989), Mansoorian (1993), Backus (1993), andBackus et al. (1994)).1
Obstfeld (1982) is the first to question whether or not there can be a
Harberger-Laursen-Metzler eﬀect in a modern intertemporal utility-maximizing
framework. Using a small country model with Uzawa’s (1968) time preference, he gives a negative answer to this question, showing that a permanentterms-of-trade deterioration causes real expenditures to decrease sharply, andthereby improves the current account. Svensson and Razin (1983) attributehis result to Uzawa’s assumption that the degree of impatience measured by
the rate of time preference is increasing in current welfare. In that case, theequilibrium dynamics are stable and, at the same time, given the UzawaObstfeld setting, there is a unique target level of steady-state welfare determined by the constant world interest rate. Thus, in response to a terms-oftrade deterioration and to the resultant real-income reduction, the current
account should run a surplus in the short run to enable consumers to attainthe steady-state target expenditures.
However, fixed steady-state welfare is not of logical necessity of recursive
preference, although many international finance models including Obstfeld’shave this property. This comes from the implicit assumption that consumers’
preference is weakly separable. In that case, steady-state time preference depends solely on welfare. Since steady-state time preference must equal theworld interest rate, steady-state welfare is then fixed by the world interest
rate. Instead, if preference is weakly nonseparable, steady-state welfare canadjust freely to a terms-of-trade deterioration, so that even with increasing
impatience the current account could deteriorate so as to generate the wel1

For surveys of discussions before 1980’s on the Harberger-Laursen-Metzler eﬀect, see
Obstfeld (1982) and Svensson and Razin (1983).

Weakly Nonseparable Preference and the Current Account2

fare adjustment. Actually, using a two-period model with a general (possibly,
weakly nonseparable) preference structure, Svensson and Razin (1983) showthat the current-account eﬀect of a terms-of-trade deterioration is of ambiguous sign owing to three diﬀerent eﬀects: (i) a direct eﬀect, caused by a revaluation of the net export vector; (ii) a wealth or welfare eﬀect, due to a welfarechange; and (iii) a pure substitution eﬀect, which results from compensatedchanges in consumption caused by the relative-price change. However, their
discussion based on general preference is limited to the two-period frameworkand it is unclear how it can be extended to dynamic settings.
The purpose of this paper is to re-examine the eﬀect of a terms-of-trade
deterioration on a small open economy using an infinite-horizon model withweakly nonseparable preference.2 Our model owes much to the procedures
used by Shi (1994) to specify weakly nonseparable preference by extending the familiar recursive preference model à la Uzawa (1968) and Epsteinand Hynes (1983).3 Weak nonseparability is incorporated in such a waythat the intratemporal marginal rate of substitution between exportable andimportable goods depends on future consumption of these goods throughcurrent welfare. Relative preferences toward the two goods then depend on
current welfare and hence current wealth. When wealth increases, preference
toward one good can be more or less enhanced than that toward the othergood. With such a preference bias, a permanent deterioration in the termsof trade induces a change in steady-state welfare and wealth so as to cause
preference to shift away from imports in favor of exports. This will aﬀect
current-account dynamics.
As a main result, we indeed find that, when preference toward imports is
suﬃciently wealth-enhanced, a terms-of-trade deterioration worsens largely
steady-state welfare, so that, even under increasing impatience, the current
2

We assume away any adjustments on the production side. For this respect, see Persson
and Svensson (1985) and Sen and Turnovsky (1989). By integrating saving and investment dynamics in an overlapping-generation model, Persson and Svensson show that theHarberger-Laursen-Metzler eﬀect can have any sign for plausible parameter values. Sen
and Turnovsky discuss the eﬀect by incorporating the labor-leisure choice and costly investment. For analyses using models of a stochastic exchange economies, see Backus (1993)
and Backus et al. (1994).
3
Weakly nonseparable preference under endogenous time-preference formation is considered by Lucas and Stokey (1984), Judd (1985), Epstein, Ham, and Zin (1988), andShi (1994). Shi (1994) conducts the most systematic analysis on this topic. He characterizes the consumption-leisure choice under weakly nonseparable preference to discuss
implications of distortionary taxation on capital and/or labor.

Weakly Nonseparable Preference and the Current Account3

account will deteriorate to support the welfare change. In so doing, we reformulate the three eﬀects which Svensson and Razin decompose using ourdescription of the steady-state equilibrium. In this sense, the present attempt
could be regarded as a dynamic extension of their analysis or a synthesis of
theirs and Obstfeld’s (1982).
The main result is consistent with recent empirical research. For example,
Backus et al. (1994) report that signs of correlation between changes in theterms of trade and those in the trade account diﬀer internationally. Our
proposition implies that even with the endogenous time-preference model we
could accommodate these mixed results consistently by incorporating weak
nonseparability of preference. Another empirical implication is also derived
by relating weak nonseparability to the notion of luxury goods.
The rest of the paper is organized as follows. Section 2 presents theanalytical framework. Section 3 derives the equilibrium dynamics. In section
4, the eﬀect of a permanent deterioration in the terms of trade is examined.
In section 5.1, we relate our discussion to Svensson and Razin’s (1983). In
section 5.2, empirical implications are discussed. Section 6 concludes thepaper.

2

The model

Consider a small open economy populated with infinitely-lived identical agents.
They consume domestic goods d and foreign goods f. To abstract from thelink between domestic investment and the current account, the economy isassumed to be endowed with constant units y d of the domestic good andy f of the foreign good at each instant. y d is suﬃciently large whereas y f issmall, so that a part of y d is exported and a part of f imported.4 By thesmall-country assumption, the relative price p of the foreign good in termsof the domestic one is exogenously given. The representative agent holds
non-human wealth in the form of net foreign assets b. They can be tradedfreely at a constant interest rate r in perfect international capital markets.
Let u(d, f ) denote the instantaneous utility function of the representative consumer, which is assumed to satisfy the Inada condition. He or she
4 fy could be assumed to be zero for brevity. As will be shown below (remark 2),
however, a nonzero y f enables us to prove easily that there exists a set of parametervalues such that the Harberger-Laursen-Metzler eﬀect takes place.

Weakly Nonseparable Preference and the Current Accountmaximizes the following lifetime utility:
Z ∞U (0) ≡u(d (t) , f (t)) exp (−∆ (t)) dt,

4

(1)

0

where
∆ (t) =Z t

δ (d (s) , f (s)) ds,(2)0with δ (·, ·) (> 0) representing the instantaneous subjective discount rate. As
in the literature (e.g., Obstfeld (1982) and Devereux and Shi (1991)), increasing marginal impatience is assumed: δd > 0 and δf > 0, where δd = ∂δ/∂d,etc.5
To specify preference more clearly, let us construct the generating functiong,g (d, f, φ) = u (d, f ) − φδ (d, f ) ,(3)

where φ (t) represents the lifetime utility U (t) from the consumption stream
after time t. g generates φ according to the diﬀerential equation,φ̇ = −g (d, f, φ) s.t. lim φ (t) exp (−∆ (t)) = 0,
t→∞

(4)

where a dot represents the time derivative, i.e., φ̇ (t) =dφ (t) /dt. The firstorder partial derivatives gd and gf equal the current-value marginal utilitiesof d (t) and f (t) defined in terms of the Voltera derivative, e.g.,∂U (0)= gd (t) exp (−∆ (t)) .
∂d (t)
We assume their positivity: gd > 0 and gf > 0.
The marginal rate of substitution (MRS) between d and f at time t isgiven by the ratio of gf and gd at that time: dd/df |U (0)=const. = gf /gd .
The preference is thus weakly nonseparable when gf /gd depends on current
welfare φ since φ depends on the future consumption stream. To characterize
5Formally, impatience is defined as increasing (decreasing) if the rate of time preference
is increasing (decreasing) in current welfare. It is well-known that impatience is increasing(decreasing) if and only if the subjective discount rate is increasing (decreasing) in consumption (e.g., see Lucas and Stokey (1984) and Epstein (1987a)). This property holds
valid also in the present model, as can easily be seen from (13) below.

Weakly Nonseparable Preference and the Current Account5

the weak nonseparability, we follow Shi (1994) in introducing the following
nonseparability index ξ:ξ (d, f, φ) ≡1 ∂ (gf /gd )
δd δf
− ,
=
gf /gd
∂φ
gd gf

(5)

where we assume gdf = 0 and
gdd < 0, gf f < 0,

δgdd
δgf f
<
ξ
<
−
.
gd2
gf2

(6)

These inequalities ensure the local concavity of the preference.6 Furthermore,
it is easy to show that (6) and the positivity of gd and gf imply the following
technical property:
Property: ξ satisfies: 7rδd gf f;
gd gf2rδf gdd(ii) ξ < 0 ⇒ ξ 2 < −.
gf gd2(i) ξ > 0 ⇒ ξ 2 < −(7)

Weak nonseparability is captured by a nonzero ξ. A positive (negative) ξmeans that a rise in lifetime utility shifts preference away from consumption
d to f (f to d). For this reason, when ξ is positive (negative), preference
toward foreign goods f is referred to as more (less) wealth-enhanced thanthat toward d.8 From (5), for any (d, f, φ) such that preference toward f
6

See Shi (1994, appendix).
We have ξ < gδdd from the definition of ξ and the positivity of δf /gf . From the third
inequality in (6), it follows that when ξ > 0, we have
7

ξ2 <

δd
δd δgf f
ξ<−,gd
gd gf2

which is (i) in (7). (ii) can be shown in the same way.
8
Shi (1994) uses a diﬀerent terminology, as discussed at the end of this section. We
newly coin the terminology “more wealth-enhanced” because we can thereby see in which
direction a welfare- (or wealth-) increase aﬀects relative preferences toward the two goods,
depending on signs of ξ. As conjectured from what our terminology means, we can relate
index ξ to the notion of luxury goods. This will be discussed in section 5.2.

Weakly Nonseparable Preference and the Current Account6

is more (less) wealth-enhanced than toward d, the MRS for f along thecorresponding indiﬀerence curve, gf /gd , is larger (smaller) than that along
the corresponding discount-rate, δf /δd , implying that consumers are more(less) willing to sacrifice f for d to ¯keep the discount rate
¯ constant than
dd ¯
dd ¯
to keep lifetime utility constant: df ¯
> (<) df ¯
. Since, as
U (0)=const

δ=constshown later, the steady-state discount rate is fixed by the world interest
rate, these discrepancies in the MRS’s lead a demand shift between d and f,
caused by an exogenous perturbation (e.g., a terms-of-trade change), to aﬀect
welfare in steady states. By contrast, Obstfeld (1982) follows Uzawa (1968)
in specifying the discount rate δ as a function of felicity u, for which caseboth of gf /gd and δf /δd equal uf /ud , so that ξ equals zero (i.e., preferenceis weakly separable) and steady-state welfare is fixed by the world interest
rate.9The utility maximization is conducted subject to the following four constraints: (i) the flow budget constraint,ḃ (t) = rb (t) + y d + p (t) y f − d (t) − p (t) f (t) ;(8)(ii) the law of motion (2) for the discount factor; (iii) the initial condition,
b0 =given; and (iv) the no-Ponzi game condition, lim exp (−rt) b (t) = 0.
t→∞
Letting λ denote the current-value shadow price of savings, the first-orderconditions are given by:
gd (d, φ) (≡ ud (d, f ) − φδd (d, f )) = λ,(9)gf (f, φ) /gd (d, φ) = p,

(10)λ̇ = (δ (d, f) − r) λ.

(11)

The optimal dynamics for (b, d, f, φ, λ) are given by a time-profile whichis generated by five equations (4) and (8) through (11) under the initial andno-Ponzi game conditions. We can reduce the system by expressing the Euler
condition (11) in terms of the rate of domestic-good consumption. Define therate of time preference ρd with respect to d as¯
dgd (d (t) , φ (t)) exp(−∆ (t)) ¯¯dρ ≡ − ln
¯˙ ,
dt
d=0
9

Similarly, preference is weakly separable when δ is constant (time-additive preference),or when u is constant (e.g., Epstein and Hynes (1983)).

Weakly Nonseparable Preference and the Current Account7

where gd exp(−∆) represents the present-value marginal utility of d. Then,from (9), the Euler condition (11) can be rewritten as¢
gd ¡d˙ = −r − ρd (d, f, φ) ,
(12)
gdd
where

ρd (d, f, φ) = δ (d, f ) −δd (d, f )g (d, f, φ) .
gd (d, φ)(13)

In exactly the same way, the optimal dynamics of foreign-good consumption
can be obtained by defining the rate of time preference ρf with respect to f .
From (10), the dynamics are not independent of (12).
By comparing ρd and ρf , we can characterize ξ from a dynamic viewpoint.
In particular, from (5) and the definitions of ρd and ρf , we have around thesteady state
ξR0⇔

∂ρd
∂ρf
R
,
∂φ
∂φ

implying that impatience with respect to d is more increasing in φ than withrespect to f when ξ > 0, i.e., when preference toward f is more wealthenhanced than toward d and vice versa. Since increasing impatience has astabilizing eﬀect on dynamics, we can follow Shi (1994) in saying d as more(less) welfare-stabilizing than f if ξ > (<)0.10

3

Steady state and equilibrium dynamics

The equilibrium dynamics for (b, d, f, φ) are described completely by (4),(8), (10), and (12) together with the no-Ponzi game condition and the initialcondition. The steady-state equilibrium, (b∗ , d∗ , f ∗ , φ∗ ), is determined by thefollowing equations:¡ d ∗ ∗ ∗ ¢ρ (d , f , φ ) = δ (d∗ , f ∗ ) = r,
(14)φ∗ = u (d∗ , f ∗ ) /r,

10

(15)

Alternatively speaking, when lifetime utility φ displays an increasing time path (i.e.,φ̇ = −g > 0), ρd is larger (smaller) than ρf if ξ > (<)0: the increase in φ is supported
by a slow (rapid) increase in d and a rapid (slow) increase in f . In contrast, if φ is on adecreasing path, a positive (negative) ξ implies that ρf > (<)ρd , so that the decrease in
φ is generated by a slow decrease in d and a rapid decrease in f .

Weakly Nonseparable Preference and the Current Account8

gf (f ∗ , u (d∗ , f ∗ ) /δ (d∗ , f ∗ ))= p,
gd (d∗ , u (d∗ , f ∗ ) /δ (d∗ , f ∗ ))(16)

rb∗ + y d + py f = d∗ + pf ∗ .

(17)

In the above, (14) represents d˙ = 0 where (13) and g ∗ = 0 are substituted
successively into (12). (15) comes from (3) and (14) with g ∗ = 0. (16) isobtained from (10) and (15). (17) represents the external balance condition,ḃ = 0 (see (8)).
Given the terms of trade p, (14) and (16) jointly determine consumptionsd∗ and f ∗ . Steady-state welfare φ∗ and net foreign assets b∗ are then givenby (15) and (17), respectively.
Figure 1 illustrates the determination of the steady-state equilibrium.
Schedules RR0 and P P 0 represent (14) and (16), respectively. RR0 represents
the locus of (d∗ , f ∗ ) which equalizes the steady-state rate of time preference
to the world interest rate. P P 0 is the locus along which the intratemporal
marginal rate of substitution equals the terms of trade. The RR0 -schedule,
which could be referred to as the steady-state time preference curve, necessarily has a negative slope, whereas the slope of P P 0 can take either sign.
The figure assumes a normal case in which P P 0 is positively sloping. The
steady-state consumption basket (d∗ , f ∗ ) is determined by the intersection
point E of schedules RR0 and P P 0 . Schedule BB 0 , which goes through point
E with slope −1/p, represents the external balance condition, (17). The
horizontal intercept, i.e., point B, equals total income in terms of exportable
goods, rb∗ + y d + py f . From (16), if we define a steady-state indiﬀerence
curve which corresponds to the steady-state utility level at point E as½¾
u (d∗ , f ∗ )∗∗I (E) ≡ (d , f ) |= utility at point E ,δ (d∗ , f ∗ )the curve should be tangent to schedule BB 0 at point E.11 Schedule BB 0 , asa budget line, determines steady-state asset holdings required for the equilibrium consumption basket at E. At point E, the slope −1/p of the externalbalance schedule BB 0 and hence the gradient −gd /gf of the steady-state indiﬀerence curve are smaller or larger in magnitude than the gradient −δd /δfof the steady-state time preference curve RR0 in accordance to whether nonseparability index ξ is positive or negative. In Figure 1, ξ is assumed to be
11

The gradient of the long-run indiﬀerence curve equals −gd /gf , which is (the negativeof) the left hand side of (16).

Weakly Nonseparable Preference and the Current Account9

positive. As will be shown later, the relative magnitudes of the gradients ofthe two schedules, RR0 and BB 0 , play an important role in determining thesteady-state eﬀects of a permanent terms-of-trade deterioration on welfare
and net foreign assets.
Given the steady-state equilibrium, the local dynamic system for n ≡(d, φ, b) can be obtained by substituting (10) into (4), (8), and (12) and bylinearizing the resulting system around the steady state as ṅ (t) = An̂ (t);³ g2 ξ 2
´


ξgf2
gd
rδd
f
+
−
0


³ gf f´ gdd gf f ξg2 gd


f
(18)
A ≡  −gd 1 + p2 ggdd
0
,
r
+
g
f f´
ff


³
ξg
− 1 + p2 ggfddf
r
p gf ff

where the hats placed above the variables denote deviations from their steadystate values, e.g., dˆ(t) = d (t) − d∗ ; and the coeﬃcient matrix is evaluated at
the steady-state point.
We can show easily that the linear system has two positive and one negative roots, where the negative root is given by
r
ω≡

r−

4g 2 g 2

r2 + gdddgfff Ψ
2

,

where

½
µ
¶¾
rδd δf gdd
gf f2
Ψ≡− ξ ++,
gd gf δd gd δf gf(19)

which can be shown to be positive.12 The other roots are r and the costateroot of ω. As can easily be seen, either unstable root is not smaller than r,
so that any other paths than the saddle path governed by ω cannot satisfy
the no-Ponzi game condition. This implies that the relevant dynamic system
exhibits saddle point stability.
12

When ξ > 0, we can obtain from property (i) in (7)µ¶rδd δfgf f
gddrδf gddrδd gf f
−
+
> 0.
=− 2
Ψ>
2
gd gf
gd gf gd δd gf δf
gd gf

In the same way, we can show the positivity of Ψ for a negative ξ by using property (ii)in (7).

Weakly Nonseparable Preference and the Current Account10

The saddle dynamics can be derived from the eigen vector associated with(18) as follows:ḃ (t) = ω b̂ (t) , b̂ (0) = b0 − b∗ ,
(
)
gf2
g
(r
−
ω)
ff
dˆ(t) =
+ ξ b̂ (t) ,(gf f + p2 gdd )
gf2
gf gd
fˆ (t) =(gf f + p2 gdd )½¾
gdd (r − ω)− ξ b̂ (t) ,gd2

φ̂ (t) = gd b̂ (t) .

(20)
(21)

(22)
(23)

Equation (20) gives monotonic saddle dynamics for the state variable, b.
Given the dynamics, transitional paths for consumptions d and f and welfare
φ are determined on stable arms (21), (22), and (23), respectively.13 The
stable arms are all positively-sloping:14 when an unanticipated permanent
deterioration in the terms of trade takes place, consumptions d and f as wellas lifetime utility φ co-move with net ³
foreign assets´ b on the transitional path.
ˆˆ
In Figure 1, the saddle trajectory for d (t) , f (t) , which is a linear subspaceobtained by eliminating b̂ from (21) and (22), is depicted as a positivelysloping trajectory SS 0 . Since this schedule depicts positive co-movements ofd and f generated by (endogenous) wealth variation, it can be regarded as
the wealth-consumption curve or the Engel curve defined with permanent
income.

4

The eﬀects of a terms-of-trade deterioration

Let us now examine the eﬀects of an unanticipated, permanent increase in
the terms of trade at the initial point in time, t = 0. Since the equilibrium
dynamics along the saddle arm are monotonic, short-run eﬀects on the current account are qualitatively the same as steady-state eﬀects on external
13

Equation (23) is easy to understand. Letting J (b, p) denote the indirect utility function, we have φ = J (b, p), implying φ̂ = Jb b̂ in the case of a permanent change in p. (23)follows from this, since by the envelope theorem the shadow price Jb of wealth equals themarginal utility gd of domestic-good consumption.
14
The positivity of the slopes of (21) and (22) comes from (6).

Weakly Nonseparable Preference and the Current Account11

asset holdings. For example, a reduction in b∗ implies a current accountdeterioration in the interim run.
From (14) and (15), a permanent rise in the terms of trade p aﬀects thesteady-state consumption as follows:
dd∗
rδf
df ∗
rδd
= 2 > 0,= − 2 < 0,
dp
gf Ψ
dp
gf Ψ

(24)

where Ψ(> 0) is given by (19). Thus, an increase in the foreign-good price p
shifts consumption from f ∗ in favor of d∗ .
By combining these results with (15) and (17), the steady-state eﬀects on
welfare and net foreign assets can be obtained asdφ∗ξ
=−
T 0 as ξ S 0,
dp
pΨ

(25)

db∗
f ∗ − yfξf ∗ − yfξ=−T 0 as
T
.
dp
r
gf Ψ
r
gf Ψ

(26)

From (25), the steady-state eﬀect on welfare can be either positive or
negative in accordance with whether the nonseparability index ξ is negativeor positive. This is because when ξ <(>)0, i.e., when preference toward d ismore (less) wealth-enhanced than that toward f , consumers are more (less)willing to increase d for f to maintain the discount rate than to keep lifetime
utility. Therefore, since any changes in (d∗ , f ∗ ) should take place to keep δ
fixed at r (see (14)), the positive eﬀect of the increase in d∗ given by the firstequation of (24) on steady-state welfare u∗ /δ ∗ dominates (is dominated by)
the negative eﬀect of the decrease in f ∗ shown by the second equation of (24).
Intuitively, steady-state welfare adjusts to a terms-of-trade deterioration insuch a way that it promotes consumption substitution from f ∗ to d∗ : whenpreference toward d∗ is more (less) wealth- and hence welfare-enhanced than
toward f ∗ , steady-state welfare increases (decreases) so as to enhance thepreference toward d∗ .
When a terms-of-trade deterioration worsens steady-state welfare largely,
the stock of net foreign assets may decrease correspondingly in the steadystate and, hence, the current account may deteriorate in the interim run.
Indeed, (26) shows that the eﬀect on net foreign assets b∗ depends crucially∗fon the relative magnitudes of f −y
and gfξΨ , which can be interpreted asrfollows: First of all, in response to a rise in the import price, net foreign assets

Weakly Nonseparable Preference and the Current Account12

and hence the interest revenue should increase to maintain the initial living∗fstandard. f −yrepresents this income-compensating eﬀect. Secondly, asrshown by (25), since steady-state welfare is aﬀected, financial wealth shouldchange so as to support the welfare change. When ξ is positive (negative),ceteris paribus, net foreign assets should decumulate (accumulate) so as torealize a deterioration (an improvement) in steady-state welfare. This could
be referred to as the welfare-supporting eﬀect. The income-compensatingeﬀect is always positive, whereas the welfare-supporting eﬀect can take either
sign.15 When ξ is negative, a permanent rise in the terms of trade necessarily
improves the steady-state position of external assets, whereas in the case of
a positive ξ, the terms-of-trade deterioration can reduce the steady-state net
foreign assets due to a dominant negative welfare-supporting eﬀect, thereby
causing a current-account deficit, as in the case of the Harberger-LaursenMetzler eﬀect.
Proposition 1 Consider the model with weakly nonseparable preference insection 2. Then, the Harberger-Laursen-Metzler eﬀect takes place if andonly if preference for imports is so wealth-enhanced that the negative welfaresupporting
eﬀect,¡¢ −ξ/(gf Ψ), dominates the positive income-compensating effect, f ∗ − y f /r.

Remark 1 When preference is weakly separable (ξ = 0), the welfare-supporting
eﬀect degenerates, so that the current account is necessarily improved by
the income-compensating eﬀect, as Obstfeld (1982) and Svensson and Razin(1983) show.
d fRemark 2 We can indeed choose values for exogenous variables (p,¡ ∗r, y ,fy¢ )such that the Harberger-Laursen-Metzler eﬀect occurs (ξ/(gf Ψ) > f − y /r).
From (14) and (16), (d∗ , f ∗ ) and hence ξ/(gf Ψ) do not depend on (y d , y f )but solely on (p, r). Therefore, starting from an arbitrary initial steadystate equilibrium¡ such thatξ > 0 and f ∗ − y f > 0, where ξ/(gf Ψ) may¢∗fbe smaller than¡ ∗ f f−¢ y /r, we can construct another equilibrium satisfying
ξ/(gf Ψ) > f − y /r by choosing a suﬃciently large value for y f on one
hand and fixing (p, r) at the arbitrarily-chosen, initial values on the other.
15

As in footnote 13, we can clarify the meanings of the two eﬀects by using the indirect utility function J (b; p). Diﬀerentiating φ∗ = J (b∗ ; p) and rearranging the result
yield db∗ = (1/Jb )dφ∗ − (Jp /Jb )dp, where the first term can be regarded as the welfaresupporting eﬀect and the second as the income-compensating eﬀect.

Weakly Nonseparable Preference and the Current Account

13

Proposition 1 can be illustrated by using Figure 2, where p0 and p1 represent initial and new import-prices, respectively. For brevity, y f is assumed
to be zero, with production income being given by point Y on the d-axis.
With initial steady-state consumption point E0 , the initial stock of net foreign assets (multiplied by r) is represented by B0 Y . Recall that the signs ofξ and hence of the welfare eﬀect are determined by the relative gradients ofthe steady-state time preference curve RR0 and the steady-state indiﬀerence
curve I(E0 ) (see section 3). To depict the Harberger-Laursen-Metzler eﬀect,
Figure 2 focuses on the case of a suﬃciently large ξ. As illustrated by theshift to schedule P1 P10 , a terms-of-trade deterioration shifts the P P 0 -scheduleto the right, which brings the steady-state consumption point from E0 to
E1 , and correspondingly the steady-state external asset position from Y B0
to Y B1 . The consumption shift can be decomposed into two parts: the shifts
from E0 to E01 and from E01 to E1 . The former represents a compensatedconsumption change. The gradient of the steady-state indiﬀerence curve
I(E0 ) at point E01 equals −1/p1 . Segment B00 Y on the horizontal axis represents net foreign assets required for the compensated consumption underprice p1 . The change from B0 to B00 thus represents the income-compensatingeﬀect. Since the steady-state indiﬀerence curve I (E0 ) is flatter than thesteady-state time preference curve RR0 in the present case, the consumptionchange from point E01 to E1 causes a long-run deterioration in welfare. The
resulting negative income change from point B00 to B1 captures the welfaresupporting eﬀect on net foreign assets. In Figure 2, the steady-state asset
position deteriorates due to the dominant negative welfare-supporting eﬀect.
In the case of weakly separable preference (ξ = 0), the intratemporal
marginal rate of substitution, gf /gd , always equals the marginal rate of substitution along the steady-state time preference curve, δf /δd . In Figure 2,
this means that the steady-state time preference curve RR0 coincides with thesteady-state indiﬀerence curve I (E0 ), for which case the welfare-supporting
eﬀect, B00 → B1 , degenerates. As stated in remark 1, therefore, a termsof-trade deterioration increases the steady-state position of external assets
through the income-compensating eﬀect, B0 → B00 .
In sum, when preference is weakly separable, there is a unique level ofliving standard which should be maintained in the steady state, so that,in response to a terms-of-trade deterioration, the current account should
improve to attain the target living standard. In contrast, when preferenceis weakly nonseparable, the steady-state welfare level is variable, and thecurrent account will adjust so as to support the welfare change. In particular,Weakly Nonseparable Preference and the Current Account14

if a terms-of-trade deterioration is suﬃciently harmful in the steady state,
the current account should be worsened.
Transitional paths in the interim run are determined by the stable arms
given by (20) through (23). In particular, as for the initial response of consumption, when a terms-of-trade deterioration increases steady-state external
assets, which is the case when preference toward imports is not suﬃciently
wealth-enhanced, domestic-good consumption d instantly jumps upward byless than in the long run whereas foreign-good consumption f jumps downward by more than in the long run. At the same time, the rates of time preference, ρd and ρf , decrease instantly below the interest rate, which causesd (t) and f (t) to approach increasingly the steady-state levels over time.16 In
contrast, when the steady-state asset position is deteriorated, the initial upward jump in d overshoots its long-run increase while the initial discrete fallin f comes short of its long-run reduction. With ρd and ρf rising instantly
above r, d and f after that gradually decrease toward the steady-state levels
over time.

5

Discussions

5.1

Direct, welfare, and pure substitution eﬀects

Within a two-period model, Svensson and Razin (1983) decompose the eﬀectof a terms-of-trade deterioration into three diﬀerent eﬀects: (i) a direct eﬀect,caused by a revaluation of the net export vector; (ii) a wealth or welfare
eﬀect, due to a welfare change; and (iii) a pure substitution eﬀect, which
results from compensated changes in consumption caused by the relativeprice change. Their discussions can be recast using our description of thesteady-state equilibrium.
16

For t ≥ 0, we obtain from (13)
(
)
2
g
ω
g
g
(r
−
ω)
dd
f
f
f
+ ξ b̂ (t) ,ρ̂d (t) =gd (gf f + p2 gdd )gf2
gd gf f ωρ̂ (t) =gf f + p2 gdd
f½¾
gdd (r − ω)− ξ b̂ (t) ,gd2

implying that ρ̂d (t) b̂ (t) > 0 and ρ̂f (t) b̂ (t) > 0.

Weakly Nonseparable Preference and the Current Account15

From (14) through (16), the steady-state consumptions satisfyu (d∗ , f ∗ )gf (f ∗ , φ∗ )
∗
and
=
φ
= p.
δ (d∗ , f ∗ )gd (d∗ , φ∗ )They are thus aﬀected by changes in p and φ∗ asµ pµ gf ¶µ ∗ ¶¶g(p − δf ) − gffd
1
1
dd
∗
r
r
=
dφ +
dp,
pg
g
1
df ∗
(δf − pδd ) − gfdfΩ
Ω − rdr(27)

where Ω = − (gf f + p2 gdd ) /r (> 0). On the right hand side of (27), thefirst and second terms represent welfare and pure substitution eﬀects on
consumptions, respectively. The eﬀect on steady-state spending z ∗ (≡ d∗ +pf ∗ ) is obtained from (27) as
µ p
¶
gf f
(p
−
δ
)
−
1
f
∗
∗
gd
dz = f dp + (p, 1) 1 r
dφ∗ .
pgf f
(δ
−
pδ
)
−
Ω
f
d
r
gd
In the above, the first term on the right hand side represents what Svensson and Razin call a direct eﬀect and the second, a welfare or wealth effect. The direct and wealth eﬀects thus equal our income-compensating, andwelfare-supporting eﬀects, respectively. Note, however, that pure substitution eﬀects on both spending and the current account are always zero since,
as can be seen from (16) and the second term of (27), pure substitution eﬀects
on d∗ and f ∗ just oﬀset each other. Proposition 1 implies that the HarbergerLaursen-Metzler eﬀect takes place when the “wealth eﬀect” on steady-state
spending is dominantly negative.
In Figure 2, pure substitution and wealth eﬀects on consumption are
depicted by the changes from points E0 to E01 and from points E01 to E1 , respectively. The direct and wealth eﬀects on spending and the current accountare illustrated by the changes from B0 to B00 (i.e., the income-compensatingeﬀect) and from point B00 to B1 (i.e., the welfare-supporting eﬀect), respectively.17
17
In Figure 2, the property that the pure substitution eﬀect on spending is zero can beseen by noting that the intercept on the d-axis of the line segment from point E01 withthe slope of −1/p0 is nearly identical to point B0 , which represents initial spending.

Weakly Nonseparable Preference and the Current Account

5.2

16

Empirical implications

Regarding empirical implications, it is worthwhile noting the following threepoints. First, Backus et al. (1994) report that signs of correlation betweenchanges in the terms of trade and those in the trade account diﬀer internationally. Proposition 1 shows that even with the assumption of increasing
impatience we could account for this mixed result consistently by incorporating weak nonseparability.
Secondly, to obtain a testable prediction from our result, it is useful torecognize the relation between the notion of luxury goods and the sign of nond +dξ/(r−ω)
separability index ξ. From (21) and (22), we have dlnpf= γγf −pfalongξ/(r−ω)dlnd
0dthe Engel curve (i.e., schedule SS in Figure 1), where γ = −dgdd /gd andγ f = −pf gf f /gf represent measures of the desire to smooth consumption.18
This and the identity dz dlnd
+ pfz dlnpf= 1 imply
dlnz
dlnz
dlnpf
=
dlnz

µ

pf
γ f − pf ξ/ (r − ω)+ dd
γ + dξ/ (r − ω)

¶−1 µ

¶
pf
+1 .
d

Whether the consumption share of f in total spending z is increasing or
decreasing in wealth is thus determined as:¡¢
d (pf /z)ξz
R0⇔
+ γ d − γ f R 0.
db
r−ω

(28)

Therefore, if a diﬀerence between γ d and γ f is negligible, the sign of ξ determines which good increases its share in total spending when wealth increases,that is, which good is a luxury. When preference toward f is more wealthenhanced (ξ > 0), f is a luxury, and vice versa. Proposition 1 predicts thatthe Harberger-Laursen-Metzler eﬀect likely holds valid if imports are luxuries. In the first step, the empirical validity of our proposition could bechecked by examining how correlations between the current account and theterms of trade depend on the shares of luxury-goods in imports and exports.
Conversely, provided that our model successfully describes the real world,
we could predict the steady-state welfare eﬀect from empirical information
about correlations between the current account and the terms of trade. For
example, in Japan and many European countries like U.K. and France, it
is reported that net exports and the terms of trade display strong negative
correlations with each other (see Backus et al. (1994)). From proposition 1,
18

Epstein (p.76, 1987) uses a similar terminology.

Weakly Nonseparable Preference and the Current Account17

this means that ξ must be large enough, implying in turn from (15) that anincrease in the terms of trade will have a substantial detrimental eﬀect on
steady-state welfare in these countries.

6

Concluding remarks

To give a new insight into current-account dynamics under endogenous time
preference, we have incorporated weakly nonseparable preference. Due toadjustment through steady-state welfare changes, the Harberger-LaursenMetzler eﬀect can occur even if impatience is increasing in welfare.
As can easily be seen from (19), decreasing impatience would destabilize the equilibrium dynamics on the saddle path, thereby inducing various
technical diﬃculties. To avoid this, we must incorporate some stabilizing
factors into the model. For example, we can show that with capital market
imperfection the equilibrium dynamics can be stable even under decreasing
impatience. By using such models, it would be interesting to examine theimplications of decreasing impatience for the eﬀect of a terms-of-trade deterioration.
As a stylized fact which we have not discussed here, Backus et al. (1994)
report some non-monotonic adjustments of the current account, such as the
J-curve dynamics. To explain the fact, there may be several ways of extension. The most natural and direct one is to consider temporary or/and
anticipated changes in the terms of trade, as in Persson and Svensson (1985).
As conjectured from Obstfeld (1990), it will produce some nonmonotonic dynamics of the current account.

Weakly Nonseparable Preference and the Current Account18

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19

Mansoorian, A., 1993, Habit persistence and the Harberger-LaursenMetzler eﬀect in an infinite horizon model, Journal of International
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f

R

P
S

B’

E

I(E)
P’
-1/p
S’

R’
R’

d

O
B

Figure 1. Steady-state equilibrium:ξ
ξ> 0.

f

P1
R
p↑
E0
E01
I(E1)
E1
I(E0)
-1/p1
P1’

R’
-1/p0

O

Y

B1 B0

B0’

Figure 2. The effects of a terms-of-trade deterioration:ξ/(gfΨ)>(f* - yf)/r .

d