Discussion Paper No. 882

DECREASING MARGINAL IMPATIENCEAND CAPITAL ACCUMULATIONIN A TWO-COUNTRY WORLD ECONOMY

Ken-Ichi Hirose
Shinsuke Ikeda

August 2013

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

Decreasing Marginal Impatience and Capital
Accumulation in a Two-country World
EconomyKen-Ichi Hirose and Shinsuke Ikeday
Otaru University of Commerce and Osaka University
August 2013

Abstract
This research is the …rst to examine dynamic general equilibriumin a growing two-country economy under decreasing marginal impatience (DMI). The stability condition is shown to be more restrictive
than in the case of an endowment economy and/or under increasing
marginal impatience (IMI). By analyzing global-economy adjustment
to time preference shocks, international transfers, and productivity
shocks, equilibrium dynamics in the presence of DMI di¤er drasticallyfrom what is obtained when the standard IMI model is used. For
example, in a country characterized by DMI, a positive productivity
shock improves the country’s welfare level and lowers its steady-statetime preference and, hence, the steady-state interest rate. This leads
to an increase in the neighboring country’s capital stock.
Keywords: decreasing marginal impatience, two-country economy,
capital accumulation.
JEL classi…cation: F41, F32, E00.
Corresponding author. The Faculty of Commerce, Otaru University of Commerce,
3-5-21 Midori, Otaru, Hokkaido 047-8501, Japan. Telephone: +81(134)27-5365. E-mail:hirose@res.otaru-uc.ac.jpy
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)6879-8583.
E-mail: ikeda@iser.osaka-u.ac.jp

1

Introduction

Although many theoretical studies on macroeconomic dynamics have been
conducted using endogenous time preferences (e.g., Epstein, 1987a, b; Lucas and Stokey, 1984; Obstfeld, 1990; Ikeda, 2006), it has usually been assumed that the degree of impatience— measured by the pure rate of time
preference— is marginally increasing in wealth, thereby ensuring stability.
When the degree of impatience is marginally decreasing in wealth, the wealthier are more patient and, ceteris paribus, become even wealthier over time.
However, many existing empirical studies often support the validity of decreasing marginal impatience (DMI) (e.g., Lawrance, 1991; Samwick, 1998;
Harrison et al., 2002; Ikeda et al., 2006). To develop realistic macroeconomic
models, it is necessary to examine dynamic implications of DMI in generalequilibrium settings.
Using a two-country capital-accumulation model, the purpose of this paper is to derive theoretical implications of DMI for dynamic stability, open
macroeconomic adjustments, and policy e¤ects. This is the …rst attempt tocharacterize solutions for a dynamic general equilibrium in a growing twocountry economy under DMI.
A seminal paper by Devereux and Shi (1991) examines a growing twocountry economy within the framework of endogenous time preferences. However, the analysis is limited to the case of increasing marginal impatience(IMI). For example, using the IMI model, they show that a positive productivity shock increases the steady-state interest rate slightly, and hence
decreases the neighboring country’s capital stock. However, under DMI, thesame productivity shock improves the country’s welfare level, thereby lowering its steady-state time preference and hence the steady-state interest rate;
this increases the neighboring country’s capital stock. Similarly, we show thatin the presence of DMI, the growing two-country economy exhibits dynamic
properties that drastically di¤er from what is obtained in the usual— but
empirically less relevant— IMI setting.
In the literature on DMI, Das (2003) …rst veri…es that DMI is compatible with dynamic stability in a production economy with capital. Hirose and
Ikeda (2008) show solution properties under DMI, including optimal satiatedconsumption. However, they do not discuss the general equilibrium implications of DMI under agent/country heterogeneity. Further, Hirose and Ikeda(2012a, b) solve two-country equilibria under DMI. However, their analysis
is restricted to the endowment economy model without capital.
1Supposing that one country has a DMI preference, for the steady-state
equilibrium to be saddle-point stable, it is necessary that the other country
displays IMI. Although the same property is valid in the case of an endowment economy, the saddle-point stability condition is more restrictive in acapital-accumulation economy than in an endowment economy.
Thus, to investigate the implications of DMI, we consider the case inwhich one country exhibits DMI while the other country exhibits IMI, withretaining the saddle-point stability of equilibrium dynamics. Then, an upward shift of the subjective discount rate schedule in the IMI country lowers
the steady-state interest rate and, hence, increases the steady-state capitalstock and output in both counties. This paradoxical result cannot be obtained in the case where both counties exhibit IMI, as in Devereux and Shi(1991).
Considering the saddle-path dynamics after international transfers in acapital-accumulation economy, the interest rate and capital stock sluggishly
deviate from their constant steady-state values initially and then returnto them. Moreover, although the convergence path of the IMI country’s
consumption is monotonic, that of the DMI country’s consumption is nonmonotonic. Such non-monotonic consumption adjustment after internationaltransfers never emerges if both counties exhibit IMI.
As the e¤ects of improvements in productivity, we obtain the following
results. First, the interest rate falls in the long run, although it rises inthe short run. Second, the steady-state capital stock and output increase inboth countries. Third, the steady-state consumption increases in the DMI
country, while it decreases in the IMI country. Fourth, an improvement in
the DMI country’s productivity typically increases its steady-state net foreign
assets, whereas an improvement in the IMI country’s productivity typicallydecreases its steady-state net foreign assets. These results are rather di¤erent
from those obtained in Devereux and Shi (1991).
The remainder of the paper is organized in the following manner. In Section 2, we construct a growing two-country economy model with endogenous
time preference, in which dynamic properties of the equilibrium solution are
discussed. In Section 3, we analyze global-economy adjustment to time preference shocks, international transfers, and productivity shocks. In Section 4,
we conclude the paper.

2

2

The model

Suppose that the world economy is composed of two countries 1 and 2, each ofwhich is populated with the same number of in…nitely-lived identical households. Without loss of generality, the population is assumed to be one. A
single type of goods, which can be either consumed or invested and is tradeable across countries, is produced by competitive …rms in both countries
using constant-returns-to-scale technologies with capital and labor. Equities
representing claims to the capital stock are traded in a perfect international…nancial market. Letting ai and k i (i = 1; 2) denote the asset holdings andthe capital stock in country i, respectively, country i’s net foreign assets bi
are expressed asb i = aiki:

(1)

In each country, one unit of labor is supplied inelastically by the representative household.

2.1

Households

The budget constraints for the representative households in country i (i =1; 2) are given by
a_ i = rai + wi ci ;(2)where ci denote consumption, wi the labor wage, and r the interest rate; anda dot represents the time derivative.
We specify the preferences by assuming variable time preferences as
Z 1
i
max
ui ci (t) exp((t))dt;(3)0where ui (ci ) (i = 1; 2) represent the instantaneous utility functions; and idenote cumulative
discount rates with instantaneous discount rates i (ci ):Rt i ii(t) = 0 (c ( ))d , or
_ i (t) =ici (t) ;i(0) = 0:(4)

For intertemporal preferences to be well-de…ned, we follow the literature(e.g., Epstein, 1987a; Obstfeld 1990; Hirose and Ikeda, 2008) in assumingthat the following standard regularity conditions are valid: (C1) ui < 03

(i = 1; 2); (C2) ui are strictly increasing and strictly concave in ci ; (C3) ui
are log-convex in ci ; and (C4) i are concave in ci . It is known, and will beshown later, that the degree of impatience, measured by the rate of time
preference, is marginally increasing or decreasing in wealth as the discountrate i is increasing or decreasing in ci . Since the regularity conditions,(C1)-(C4), are not related to the signs of the …rst derivatives of i (hereafter
d i =dci is expressed as ic , and so on), the degree of impatience can be either
marginally increasing or decreasing under the conditions.
Let i and i (i = 1; 2) represent the current-value shadow prices forisavings and the discount factor exp (), respectively, and the “generatingifunctions” are de…ned as
i

i

ci ;

i i

= ui cici :(5)Then, the optimal conditions to maximize lifetime utility function (3) are
given by
i
i
i
= i;
(6)
c c ;
_i =

i

_i =

ci

i

r

i

ci ;

i

(7)

;

(8)

;

lim expi(t)i(t) ai (t) = 0;

lim exp

i

(t)

i

(t)

t!1
t!1

i

(t) = 0;

(9)
(10)

As seen by solving di¤erential equation (8) under transversality condition(10), the optimal i (t) equals the lifetime utility obtained from the optimal
consumption stream after time t. We assume that the current value marginali i
1
utility of ci is positive: ic ci ; i (= uicc ) > 0.
Following Epstein (1987a) and Obstfeld (1990), we de…ne the rate of timeii
ii
preference i as i = d ln i (t) =dtjc_i =0 ; where i
exp (
)
c c ;
i
represents the present-value marginal utility of c . The time preference rateis computed as
i

ci ;

i

=

i

ci

i

ci ;

i

i
c

ci ;

i

i
c

ci :

(11)

In the following analyses, we assume away a possibility of satiation ( ic 5 0) which
may occur under decreasing marginal impatience ( ic < 0). See Hirose and Ikeda (2008)
for the implications of the satiated utility of decreasing marginal impatience consumers.
1

4

It is valid that i = i ic = ic around the steady state point in which i = 0(see (8)). This implies that the degree of impatience, measured by i , ismarginally increasing or decreasing in the utility index i as ic is positiveor negative. We refer to consumer preferences as increasing marginal impatience (IMI) when ic and hence i are positive, and as decreasing marginal
impatience (DMI) when ic and hence i are negative. From (6) and (7), theoptimal consumption dynamics can be written in terms of time preference as
c_i =

ci ;

i

r

i

ci ;

i

;

(12)

i
i
c = cc (> 0).

where

i

2.2

Firms

=

i

The per capita production functions for the representative …rms in countryi (i = 1; 2) are given by
y i = i f i (k i )(13)satisfying fki > 0, fkki < 0, where y i denote output, and i represent parameters of productivity. We assume that there is no capital depreciation andno adjustment cost of investment. As the results of pro…t maximization by…rms, we obtainifki (k i ) = r andif i (k i )ifki (k i )k i = wi :(14)

In our setup, instantaneous international capital movement may occur,
so that we should regard the world total capital stock K
k 1 + k 2 as astate variable. Noting that K = ( 1 fk1 ) 1 (r) + ( 2 fk2 ) 1 (r), where ( i fki ) 1
represents the inverse function of i fki , the interest rate r can be expressed
as a function of K:
r = (K); where

2.3

K (K) =

( 1 fkk1 )( 2 fkk2 )< 0:
1f 1 + 2f 2
kk
kk

(15)

The market equilibrium

The market-clearing conditions are given by
c1 + c2 + K_ = y 1 + y 2 and a1 + a2 = K:

5

(16)

Then, from (1), (2), (13), (14), and (16), overtime changes in net foreign
assets, or the current accounts, b_ i are expressed asb_ i = rbi + y i

k_ i :

ci(17)

From (2), (8), (12), (14), (15), and (16), we can derive the reduced dynamic system as follows:
c_1 =

1

c1 ;

1

(K)

1

c1 ;

1

;

(18)

c_2 =

2

c2 ;

2

(K)

i

ci ;

i

;

(19)

_1 =

1

c1

1

u1 c1 ;

(20)

_2 =

2

c2

2

u2 c2 ;

(21)

K_ =

1

f 1 (( 1 fk1 ) 1 ( (K))) +a_ 1 =1f 1 (( 1 fk1 ) 1 ( (K))) + (K)(a12f 2 (( 2 fk2 ) 1 ( (K)))

c1

c2 ;

(22)

( 1 fk1 ) 1 ( (K)))

c1 :

(23)

This dynamic system has two pre-determined state variables, K and a1 .

2.4

The steady-state equilibrium

From (1), (13), (14), (15), (16), (17), and (18) through (23), the steady-state
equilibrium is characterized by the following equations:
1
1

=

2

fk1 (k 1 ) =

2

c1

yi =
i

i

c2

= r (= (K ));(24)

fk2 (k 2 ) = r (= (K ));(25)

f i (k i ) (i = 1; 2) ;(26)

c1 + c2 = y 1 + y 2 ;(27)= ui c i=r (i = 1; 2) ;(28)

y i )=r (i = 1; 2) ;(29)

bi = (ciai = k i + bi (i = 1; 2) ;(30)

where an asterisk denotes the steady-state value. The values of c1 ; c2 ; k 1 ; k 2 ; y 1 ; y 2
and r (or K ) are jointly determined from (24) through (27). Then, i(i = 1; 2) are determined from (28), bi from (29), and ai from (30).
6By linearizing the dynamic system composed of (18) to (23) around thesteady state, we obtain
0 1 1
0 1
1
c_
c
c1
B c_2 CB c2 c2 C
B 1 C
B 1
C
B _ C
1 C
B
B
C
B
C
(31)
B _2 C = M B 1
2 C;
B
C
B
C
B
C
@ K K A
@ K_ A
a1 a1
a_ 1

with

0

0
B 0
B
1
B
c
M =B
B 0
B
@ 1
1

0
0
0
2
c

1
0

r 1c 1 = 1c
0
r
0
0
0

0
r 2c 2 = 2c
0
r
0
0

K
K

1
2

0
0
r
1
Kb

1
0
0 C
C
0 C
C;
0 C
C
0 A
r

where the coe¢ cient matrix is evaluated at the steady state.
Since this dynamic system has two pre-determined state variables, thesteady-state equilibrium is locally saddle-point stable if and only if M has
two negative eigenvalues. As proved in Appendix A, we thus obtain thefollowing property.2
Property 1: The steady-state equilibrium is locally saddle-point stable if
and only if
(r 1c
(r

2
K ) > 0; (r c
12
K )(r ccand( K )2 > 0:

K ) > 0;K)

(32)

Otherwise, the steady-state equilibrium is locally unstable.
Figure 1 displays the region of ( 1c ; 2c ) in which the saddle-point stability
conditions (32) are satis…ed. It makes clear the following property.
Property 2: For the steady-state equilibrium to be saddle-point stable, atleast one of the two countries needs to display IMI. If both counties exhibit
DMI, the steady-state equilibrium is necessarily unstable.
2

These stability conditions are consistent with the Appendix of Epstein (1987a).

7

Insert Figure 1.

The same property as Property 2 holds valid in the case of an endowment
two-country world economy (e.g., Jafarey and Park, 1998; Hirose and Ikeda,
2012a,b), in which the saddle-point stability condition is given by1c +2
c > 0:(33)

Note, however, that, as seen from Figure 1, the saddle-point stability condition in the endowment economy (33) is necessary, but not su¢ cient, for(32).3 This implies the following property.
Property 3: The saddle-point stability condition is more restrictive in ourcapital-accumulation economy than in the endowment economy.
In addition, the saddle-point stable region with either country having
DMI in Figure 1 shrinks as the absolute value of K becomes smaller. Notingr = (K), the following property is obtained.
Property 4: As r is less elastic to a change in K, the saddle-point stability
condition (32) is more restrictive.4
To be intuitive, the dynamic system is more easily adjustable in the endowment economy, in which the interest rate can be instantly adjustable
upon shocks, than in the capital-accumulation economy, in which the interest rate can change only sluggishly due to overtime changes in the aggregate
capital stock.5 That is why, as Property 3 shows, the stability condition
is more restrictive in our capital-accumation economy. As seen from this
discussion, the dynamic adjustablity in our capital-accumulation economy
becomes lower as the interest rate is less elastic to a change in the capitalstock, as Property 4 states.
In Figure 1, = 0 is tangent to 1c + 2c = 0 at ( 1c ; 2c ) = (0; 0).
Conversly, (32) approches to (33) as j K j ! 1 .
5
This intuition will be detailed in the analysis of Section 3.2.

3

4

8

3

The implications of DMI with capital accumulation

For investigating the implications of DMI in our two-country model with capital accumulation, hereafter, we assume the following saddle-point stability
in which one country displays a DMI preference.
Assumption 1: Country 1 exhibits DMI ( 1c < 0) while country 2 exhibits
IMI ( 2c > 0).
Assumption 2: The saddle-point stability conditions (32) hold valid.
Figure 2 illustrates the determination of the steady-state interest rate r
under these assumptions. By letting C
c1 + c2 and Y
y1 + y2 ,the "steady-state demand curve" represents the locus of (C ; r ) satisfying(24), while the "steady-state supply curve" represents the locus of (Y ; r )satisfying (25) and (26). Since C = Y (i.e., (27)) in equilibrium, r isdetermined at the intersection E of these two schedules.
Insert Figure 2.

The steady-state supply curve is negatively sloping, because a lower interest rate leads to capital accumulation and hence more output is supplied.
As derived in Appendix B (see Figure A1), the slope of the steady-state demand curve is also negative under Assumptions 1 and 2. If the steady-stateinterest rate r rises, the steady-state consumption demand in DMI country
1 c1 (= ( 1c ) 1 (r )) decreases while the steady-state consumption demand in
DMI country 2 c2 (= ( 2c ) 1 (r )) increases. Under Assumption 2, we have12
1 1
> ( 2c ) 1 , implying that c1 is more elasticc <c and hence ( c )than c2 in response to a change in r . Accordingly, a rise in r decreases thesteady-state world total consumption demand C .
Besides, the slope of the steady-state supply curve is steeper than thatof the steady-state demand curve. The slopes of the steady-state demand
1 2
dr
dr
and supply curves are given by= 1 c c 2 and= K , respectively.
dC
dY
r
+
c
c
1 2

Under Assumptions 1 and 2,> 0 in (32) implies that (0 >) 1 c c 2 > K .
r
c + c
9

3.1

Shifts in time preference

At …rst, we consider the e¤ects of upward shifts of subjective discount rateschedules. See Figure 3. When a upward shift in the subjective discount rateschedule takes place in DMI country 1, its result is quite natural. As shownin Appendix B (see Figure A2 (i)), a upward shift in the 1 -schedule shifts
the steady-state demand curve upward. In Figure 3, this upward shift inthe steady-state demand curve brings the intersection with the steady-state
supply curve from E to E 0 , implying that the steady-state interest rate r
rises and the capital stock are decumulated. This result is the same as in thecase where both two countries have the IMI preferences (e.g., Devereux andShi, 1991).
Insert Figure 3.

When a upward shift in the subjective discount rate schedule occurs in
IMI country 2, it leads to a paradoxical result. As shown in Appendix B (seeFigure A2 (ii)), an upward shift in the 2 -schedule shifts the steady-statedemand curve downward. In Figure 3, this downward shift in the steady-statedemand curve brings the intersection with the steady-state supply curve fromE to E 00 . Although the subjective discount rate schedule shifts upward, thesteady-state interest rate r lowers and capital accumulation is stimulated.
Proposition 1: Suppose that country 1 has DMI whereas country 2 has IMI(Assumption 1), and that the saddle-point stability conditions are met (Assumption 2). Then, an upward shift of the subjective discount rate schedule in
IMI country 2 paradoxically lowers the steady-state interest rate, and hence
increases the steady-state capital stock and output in both counties.

3.2

International transfers

Given the initial values of the state variables K(0) and a1 (0), the linearized
solution of the saddle-path dynamics is analytically calculated in Appendix
C.
Let us analyze the e¤ects of international transfers. Suppose that theworld economy is initially at the steady-state equilibrium and some wealth10

is transferred from one country to the other at time 0. By using a numericalexample given in Table 1, Figures 4 (a) and 4 (b) illustrate the saddle-path
dynamics in response to international transfers, where (a) represents a caseof a transfer from IMI country 2 to DMI country 1, and (b) represents a caseof a transfer from IMI country 1 to DMI country 2.
Insert Figures 4 (a) & 4 (b).
Insert Table 1.

The recipient country’s lifetime utility naturally improves while the donor
country’s lifetime utility becomes worse.6 In our model, however, international transfers have no long-run e¤ects, because the steady-state equilibrium
does not depend on the initial asset stocks (see (24) to (30)).
In our model with two state variables, there exists a possibility of nonmonotonic saddle-path dynamics. In fact, the saddle-path dynamics after
international transfers show the following non-monotonic characteristics, asseen in Figures 4 (a) and 4 (b).
Proposition 2: In our capital-accumulation economy under Assumptions 1
and 2, the saddle-path dynamics after international transfers have the following characteristics.
(i) The interest rate and the capital stock initially and sluggishly deviate
from their constant steady-state values and then return to them again.
(ii) A transfer from IMI country to DMI country causes the interest rate
deviate downwards, while a transfer from DMI country to IMI country causes
the interest rate deviate upwards.
(iii) The time path of DMI country’s consumption is non-monotonic,
whereas that of IMI country’s is monotonic.
For understanding these non-monotonic dynamics in our capital-accumulation
economy, it would be helpful to compare them with monotonic dynamics in
6

In these examples, we assume that initial net foreign assets are zero. With non-zero
initial net foreign assets, the transfer might harm the recipient country and bene…t thedonor country by changing the interest rate. This dynamic version of the transfer paradix
was pointed out by Epstein (1987a).

11

a endowment economy. Figures 5 (a) and 5 (b) depicts the e¤ects of international transfers in the case of the endowment economy.7
Insert Figures 5 (a) & 5 (b).

In Figure 5 (a), a transfer from IMI country 2 to DMI country 1 initiallymakes both countries patient. In Figure 5 (b), in contrast, a transfer fromDMI country 1 to IMI country 2 initially makes both countries impatient.
The saddle-point stability condition in the endowment economy (i.e., (33)),
which is met under Assumption 2, requires that the donor country (i.e.,country 2 in (a); country 1 in (b)) become more patient than the recipient
country (i.e., country 1 in (a); country 2 in (b)). In the endowment economy,
the interest rate r is determined as a weighted sum of the two countries’
rates of time preference 1 and 2 .8 The donor country, whose rate of time
preference is lower than r, accumulates wealth, whereas the recipient country,
whose rate of time preference is higher than r, decumulates wealth. As aresult, the world economy monotonically converges to the steady state.
In the capital-accumulation economy, the impact e¤ects of the international transfers on 1 and 2 are similar as those in the endowment economy,
but the interest rate cannot change instantaneously. Accordingly, r(0) >1(0) > 2 (0) is valid in Figure 4 (a), and r(0) < 1 (0) < 2 (0) in Figure 4(b). For stability, r must be adjusted …nally into the region between 1 and2, that is, the inequalities between r and 1 need to be reversed by overtimechanges in K, as illustrated in Figures 4 (a) and 4 (b). If the interest rate is
so inelastic that the inequalities between r and 1 keep eternally unchanged,
the economy cannot converge to the steady state.9
From (12), DMI country’s consumption c1 also shows non-monotonic adjustment, because the inequality relationships between r and 1 is reversed
midway on the saddle path. On the other hand, IMI country’s consumption
c2 monotonically converges to the steady-state level, since the inequality relationship between r and 2 never changes. If both countries exhibit IMI
7

The endowment is given by the steady-state output level in Table 1 (i.e., y 1 = y 2 = 20).

8

In the endowment economy, r(t) =

1

1
1
1+

2

(t)+

2
1+

2

(t) (see e.g., eq.(12) inHirose and Ikeda, 2012a).
9This argument details the intuitive reason for Properties 3 and 4 mentioned in theend of Section 2.4.

12(as in Devereux and Shi, 1991), non-monotonic consumption adjustment is
not brought about by such international transfers.10 We thus …nd that nonmonotonic consumption adjustment after international transfers is speci…conly to a DMI country in the capital-accumulation economy.

3.3

Improvements in productivity

Next, we investigate the e¤ects of improvements in productivity i (i = 1; 2).
The mathematical results of the steady-state e¤ects, derived from (24) to(30), are given by Table 2.
Insert Table 2.

3.3.1

An improvement in DMI country’s productivity

Column (I) of Table 3 displays the qualitative results of the steady-statee¤ects of an improvement in DMI country 1’s productivity 1 .
Insert Table 3.

An improvement in 1 lowers the steady-state interest rate r . As illustrated in Figure 6, it shifts the steady-state supply curve rightward and brings~ implying
the intersection with the steady-state demand curve from E to E,
that r falls. Therefore, not only in country 1 experiencing an improvement inproductivity but also in neighboring country 2, the steady-state capital stockand output increase. In DMI country 1, the steady-state consumption andlifetime utility increase. Because the increase in the steady-state consumption is larger than that in the steady-state output, country 1’s steady-state
10
In Devereux and Shi (1991), international transfers do not a¤ect the interest rate andthe capital stock even in the short run. This result is, however, due to a special assumption
for their lifetime utility function. When both country have IMI, generally speaking, thee¤ects of international transfers on the interest rate and the capital stock are ambiguous.

13

net foreign assets typically increase.11 In neighboring IMI country 2, thesteady-state consumption and lifetime utility decrease.
Insert Figure 6.

Concerning the impact e¤ects of an improvement in productivity, we can…nd analytically the following two facts. First, the impact e¤ects on theinterest rate is positive, though its steady-state e¤ect is negative. Notingthat K = k 1 + k 2 cannot change instantaneously, from (14) (or (15)), theinterest rate must rise immediately after a production shock, accompanied byinstantaneous capital movement to the country experiencing an improvementin productivity from the neighboring country.
Second, the impact e¤ect on the welfare of the country with a productivity
improvement is de…nitely positive and the impact e¤ect on the welfare of theneighboring country is approximately zero, when b1 (0) = b2 (0) = 0. From
Table 2 and Appendix C, we can obtain thatd i (0)=d i bi (0)=0

i i
cf

r

(> 0) (i = 1; 2);d j (0)= 0 (j 6= i):d i bi (0)=0
3.3.2

(34)

(35)

An improvement in IMI country’s productivity

Column (II) of Table 3 summarizes the qualitative results of the steady-statee¤ects of an improvement in IMI country 2’s productivity 2 .
An improvement in 2 lowers the steady-state interest rate r , similarly asin the previous case of an improvement in 1 (see again Figure 6), so that thesteady-state capital stock and output increase in both countries. However,the steady-state e¤ects on consumption and lifetime utility contrast to thosein the previous case. In the country with a productivity improvement, since it
is IMI country 2, the steady-state consumption and lifetime utility decrease,and therefore the steady-state net foreign assets also typically decrease. In
11

If country 1 has huge net foreign debt initially, it may be able to increase steady-state
net foreign debt, because the interest payment decreases due to a fall in r .

14

the neighboring DMI country 1, the steady-state consumption and lifetime
utility increase.
In this case as well as in the the previous case of an improvement in1, the impact e¤ects on the interest rate is positive in spite of the negative
steady-state e¤ect on it; the impact e¤ect on the welfare of the country with aproductivity improvement is positive; and the spill-over e¤ect on the welfareof the neighboring country ( 1 (0)) is negligible, when b1 (0) = b2 (0) = 0 (see(34) and (35)).
The e¤ects of improvements in productivity can be summarized as follows.
Proposition 3: In our setting under Assumptions 1 and 2, suppose that
productivity in one country improves. Then, we can obtain the following
results:(i) The interest rate falls in the long run, although it rises in the short
run after the production shock.
(ii) In both countries, the capital stock is accumulated and the steady-state
output increases.
(iii) Whichever country experiences an improvement in productivity, thesteady-state consumption increases in the DMI country whereas it decreases
in the IMI country.
Moreover, supposing that the initial net foreign assets are zero, we canalso say the followings:(iv) An improvement in productivity of one country improves the country’s
welfare and has no spill-over e¤ect on the neighboring country’s welfare.
(v) An improvement in the DMI country’s productivity increases its steadystate net foreign assets, while an improvement in the IMI country’s productivity decreases its steady-state net foreign assets.
3.3.3

Comparison with the case of two IMI countries

Finally, let us compare our results summarized as Proposition 3 with the one
in Devereux and Shi (1991) (hereafter D-S) where both two countries have
IMI preference. In Table 3, Column (III) summarizes the results in the D-Scase. Four points are especially deserving of notice.
First, the e¤ects on the steady-state interest rate are opposite to thosein the D-S setting. An improvement in productivity raises r in the D-S
setting whereas it reduces r in our setting, irrespective of in which country
the productivity improvement occurs (Proposition 3 (i)).
15Second, improvements in productivity commonly increase the world total
output. In the D-S setting, however, the output increases only in the country experiencing an improvement in productivity, while the output in theneighboring country decreases due to capital decumulation. In our setting,by contrast, the steady-state output increases in both countries (Proposition
3 (ii)).
Third, increases in the steady-state world total consumption are broughtby improvements in productivity. In the D-S setting, the steady-state consumption increases in both countries, implying that bene…ts of improvements
in productivity are shared by both countries in the long run. In our setting,whichever country experiences a productivity improvement, the steady-state
consumption increases only in the DMI country. It decreases in the IMI
country (Proposition 3 (iii)).
Fourth, the e¤ect on the steady-state net foreign assets in our setting
depends on whether an improvement in productivity happens in the DMI
country or in the IMI country (Proposition 3 (v)). This result is quite di¤erent from the one in the D-S setting, where the steady-state net foreign assetsof the country experiencing an improvement in productivity always decrease.
Consequently, the prediction in D-S that countries with higher (lower)
productivity should be long-run debtors (creditors) is not necessarily valid
in our setting. In the D-S setting, bene…ts of higher productivity are shared
between both countries in the long run, so that countries with higher productivity can diminish net foreign assets.
In our setting, the e¤ect on the interest rate a¤ects signi…cantly the international distribution of steady-state consumption, that is, under the lower
steady-state interest rate associated with higher productivity, steady-state
consumption tends to be biased toward the DMI country. Owing to thise¤ect, contrary to the D-S case, a country with higher productivity is more
likely to be a long-run creditor if it has DMI.

4

Conclusion

This paper has investigated the implications of DMI in a two-country world
economy with capital accumulation. Supposing that one country has a DMI
preference, for the steady-state equilibrium to be saddle-point stable, it isnecessary that the other country displays IMI. Although the same property isvalid in the case of an endowment economy, saddle-point stability condition is16more restrictive in our capital-accumulation economy than in the endowment
economy.
For investigating the implications of DMI, we thus consider the case inwhich one country exhibits DMI while the other country exhibits IMI, withretaining the saddle-point stability of the equilibrium dynamics. Then, an
upward shift of the subjective discount rate schedule in IMI country lowers
the steady-state interest rate, and hence increases the steady-state capitalstock and output in both counties. This paradoxical result cannot be obtained in the case where both counties display IMI as in Devereux and Shi(1991).
Considering the saddle-path dynamics after international transfers in ourcapital-accumulation economy, the interest rate and the capital stock initially and sluggishly deviate from their constant steady-state values and thenreturn to them again. Besides, although the convergence path of IMI country’s consumption is monotonic, that of DMI country’s consumption is nonmonotonic. Such non-monotonic consumption adjustment after internationaltransfers never emerges if both counties have IMI preference.
As the e¤ects of improvements in productivity, we obtain the following
results. First, the interest rate falls in the long run, although it rises inthe short run. Second, the steady-state capital stock and output increase inboth countries. Third, the steady-state consumption increases in the DMI
country while it decreases in the IMI country. Fourth, an improvement in
the DMI country’s productivity typically increases its steady-state net foreign
assets, whereas an improvement in the IMI country’s productivity typicallydecreases its steady-state net foreign assets. These results also quite di¤er
from those in the case of Devereux and Shi (1991).

17

Appendix A: The proof of Proposition 1
The characteristic equation of M is given bydet(MxI6 ) = (xr)2 (x) = 0;

where
(x) =
=
! =
=

1
r,
2

By letting x~ = x
x~4
Therefore,fx2 (x r)2 !x(x r) + g
x4 2rx3 + (r2 !)x2 + r!x + );2
1
+ (r 2c
(r 1c
K) ;
K)
(r2 1c 2c r 1c K r 2c K ) 1 2 :(x) = 0 is rewritten as

11
1 2(r + 2!)~x2 + r 4 + r 2 ! +
2
16
4

= 0:

o
p
1n 2
(2!)2 16;
r + 2!
4qp
1x~ =r2 + 2!
(2!)2 16 ;
2

x~2 =

and hence,

1
x=
2

r

q

r2 + 2!

p
(2!)2

16

:

1
2 2
2 1 2
Note that (2!)2 16 = 4f(r 1c
(r 2c
> 0.
K)
K ) g + 4( K )Let us denote
q
p
1
x1 =
r
r2 + 2! + (2!)2 16
;
2
q
p
1
r
x2 =
r2 + 2!
(2!)2 16
;
2
q
p
1
x3 =
r + r2 + 2!
(2!)2 16
;
2
q
p
1
x4 =
r + r2 + 2! + (2!)2 16:
2The signs of x1 , x2 , x3 , and x4 depend on the signs of ! and18

as Table A1.

Insert Table A1.

Therefore, the characteristic equation of M has two negative roots, implying that the steady-state equilibrium is locally saddle-point stable, if andonly if ! > 0 and > 0, or equivalently,
(r 1c
(r

2
K ) > 0; (r c2
1
K )(r ccand( K )2 > 0:

K ) > 0;K)

(32)

Otherwise, the characteristic equation of M has one or no negative root, sothat the steady-state equilibrium is unstable.

Appendix B: The derivation of the steady-statedemand curve
Figure A1 derives the negative slope of the steady-state demand curveunder Assumptions 1 and 2. When the steady-state world total consumption
demand is given by CM , for example, rM which satis…es 1 (c1 ) = 2 (c2 ) =rM and c1 + c2 = CM is given by the height of the intersection XM of1(c1 )-schedule and 2 (CM c1 )-schedule. The point (CM ; rM ) thus needs
to be on the steady-state demand curve. Similarly as (CM ; rM ), we canobtain (CL ; rL ), (CH ; rH ) and so on. The steady-state demand curve is thelocus of these points. Because country 1 has DMI ( 1c < 0) and country
2 has IMI ( 2c > 0), the 1 (c1 )-schedule and the 2 (Cc1 )-schedule are
21both negative sloping. From Corollary 1, c < c , that is, the slope of the2(Cc1 )-schedule is steeper than that of the 1 (c1 )-schedule. Then, we…nd that rL > rM > rH for CL < CM < CH , implying that the steady-statedemand curve is negatively sloping.
Insert Figure A1.

By using this derivation of the steady-state demand curve, Figure A2 (i)shows that it shifts upward when an upward shift in the 1 -schedule occurs,
whereas Figure A2 (ii) shows that it shifts downward when an upward shift
in the 2 -schedule takes place.

19

Insert Figures A2 (i) & (ii).

Appendix C: The linearized solution of the saddlepath dynamics
Eigenvectors of M corresponding x1 and x2 (x1 < x2 < 0) are1
1
00(r x1 )v11
(r x2 )v21
B (r x2 )v22 C
B (r x1 )v12 C
C
C
B
B
1 1
1 1
C
C
B
B
v
v
c 1
c 2
C and B
C;
B
2
2
2
2
C
C
B
B
v
v
1
2
c
c
C
C
B
B
A
A
@
@
1
1
1
1
v~1
v~2

respectively, where
v11

=

v21 =

1

K

x1 (r

x1 ) + r

v~11 = v11x1 (rx1 ) + r

2
1 1 ; v2 =x2 ) + r c
1
Kb; v~21 = v21
r x1

2

K

1

K

x2 (r

1 1;
c

v12 =

2 2 (= 1cv11 );(= 1v21 );2K

x2 (r x2 ) + r 2c 2
1
Kb
;
r x2

and xj (r xj ) + r 2c 2 6= 0 (j = 1; 2) are assumed.
The solution of (31) is thusc1 (t) = c1 + Z1 (rx1 )v11 exp(x1 t) + Z2 (rx2 )v21 exp(x2 t);(A.1)

c2 (t) = c2 + Z1 (rx1 )v12 exp(x1 t) + Z2 (rx2 )v22 exp(x2 t);

(A.2)

1
2

(t) =

1

+ Z1 1c v11 exp(x1 t) + Z2 1c v21 exp(x2 t);

(A.3)

(t) =

2

+ Z1 2c v12 exp(x1 t) + Z2 2c v22 exp(x2 t);(A.4)

K(t) = K + Z1 exp(x1 t) + Z2 exp(x2 t);(A.5)

a1 (t) = a1 + Z1 v~11 exp(x1 t) + Z2 v~21 exp(x2 t);

(A.6)

where
Z1 =

1
v~21

v~11

v~21 (K

K(0)) + (a120

a1 (0)) ;

Z2 =

1
v~21

v~11

v~11 (K

K(0))

(a1

a1 (0)) :Noting that r r = 1 fkk1 (k 1 k 1 ) = 2 fkk2 (k 2 (t)by linearizing (14) and (15), we obtain thatk2 ) =K (KK )K fZ1 exp(x1 t) + Z2 exp(x2 t)g;(A.7)

k 1 (t) = k 1 +K
fZ1 exp(x1 t) + Z2 exp(x2 t)g;
1f 1
kk

(A.8)

k 2 (t) = k 2 +K
fZ1 exp(x1 t) + Z2 exp(x2 t)g;
2f 2
kk

(A.9)

r(t) = r +and hence, from (1),b1 (t) = b1 +Z1 v~11

K
1f 1
kk

exp(x1 t)+Z2 v~21

21

K
1f 1
kk

exp(x2 t): (A.10)

References[1] Das, M, 2003. Optimal growth with decreasing marginal impatience.
Journal of Economic Dynamics & Control 27, 1881-1898.
[2] Devereux, M., Shi, S., 1991. Capital accumulation and the current account in a two-country model. Journal of International Economics 30,
1-25.
[3] Epstein, L.G., 1987a. A simple dynamic general equilibrium model.
Journal of Economic Theory 41, 68-95.
[4] Epstein, L.G., 1987b. The global stability of e¢ cient intertemporal allocations. Econometrica 55, 329-355.
[5] Epstein, L.G., Hynes, J.A., 1983. The rate of time preference and dynamic economic analysis. Journal of Political Economy 91, 611-625.
[6] Harrison, G.W., Lau, M.I., Williams, M.B., 2002. Estimating individual
discount rates in Denmark: A …eld experiment. American Economic
Review 92, 1606–1617.
[7] Hirose, K., Ikeda, S., 2008. On decreasing marginal impatience. Japanese
Economic Review 59, 259-274.
[8] Hirose, K., Ikeda, S., 2012a. Decreasing marginal impatience in a twocountry economy. Journal of Economics 105, 247-262.
[9] Hirose, K., Ikeda, S., 2012b. Decreasing and Increasing Marginal Impatience and the Terms of Trade in an Interdependent World Economy.
Journal of Economic Dynamics & Control 36, 1551-1565.
[10] Ikeda, S., 2001. Weakly non-separable preferences and the HarbergerLaursen-Metzler e¤ect. Canadian Journal of Economics 34, 290-307.
[11] Ikeda, S., 2006. Luxury and wealth. International Economic Review 47,
495-526.
[12] Ikeda, S., Ohtake, F., Tsutsui, Y., 2006. Time discounting, hyperbolic
discounting, and intertemporal choices: Evidences from experiments andquestionnaires. mimeo.
22[13] Jafarey, S., Park, H., 1998. The dynamics of optimal wealth distributions
with recursive utility. Economic Letters 61, 149-158.
[14] Lawrance, E.C., 1991. Poverty and the rate of time preference, Evidence
from panel data. Journal of Political Economy 99, 54-77.
[15] Lucas, R., Stokey, N., 1984. Optimal growth with many consumers.
Journal of Economic Theory 32, 139-171.
[16] Obstfeld, M., 1982. Aggregate spending and the terms of trade: Is therea Laursen-Metzler e¤ect?. Quarterly Journal of Economics 97, 251-270.
[17] Obstfeld, M., 1990. Intertemporal dependence, impatience, and dynamics. Journal of Monetary Economics 26, 45-76.
[18] Samwick, A.A., 1998. Discount rate heterogeneity and social security
reform. Journal of Development Economics 57, 117–146.
[19] Svensson, L.E.O., Razin, A., 1983. The terms of trade and the current account: The Harberger-Laursen-Metzler e¤ect. Journal of Political
Economy 91, 97-125.
[20] Uzawa, H., 1968. Time Preference, the consumption function and optimum asset holdings, in: Wolfe J.N. (Ed.), Value Capital and Growth:Papers in Honour of Sir John Hicks. Aldine, Chicago, pp.485-504.

23

Table 1: A numerical example under Assumptions 1 and 2
functionsubjective
discountrate
utilityfunction
production
functioncountry 1
country 2
1
DMI : c = 0:0005
IMI : 2c = 0:001
1 1
2 2(c ) ; 0:0005c1 + 0:06(c ) ; 0:001c2 + 0:03
1
around c = 20
around c2 = 20
ui (ci ) = (ci 50)2 ; ci 2 [0; 50)(i = 1;p2) i 0:25
i i if (k ) = 2 10(k )(i = 1; 2)

steady-state valueinterest rate
capital stock
outputconsumption
lifetime utility
net foreign assets

country 1
country 2
r = 0:05
1
k = 100
k 2 = 1001
y = 20y 2 = 20
c1 = 20c2 = 201
2= 18000= 18000
1b =0
b2 = 0Note:

10 4 ;

K =1:875= 3:4375

24

10 9

Table 2: The mathematical results of the steady-state e¤ects of improvements in productivity
country i : the country experiencing an improvement in productivity
country j : the neighboring countrydrrfki
r 1c 2c K
i
f
=
if i
d i
kk
i
dK
fki
rf
r 1c 2c
k
i
f
=
if i
if i
d i
kk
kk
1 2
i
1
r c c K
rfki
dk
i
= i i
f
fki
i
i
i
d
fkk
fkk
i
1
rf
dk j
r 1c 2c K
k
i
=
f
if i
jf j
d i
kk
kk
dy i
r
r 1c 2c K
rfki
i
=
+
1
f
if i
if i
d i
kk
kk
1
2
i
dy j
r
rf
r c c K
k
=
fi
if i
jf j
d i
kk
kk
j
i
i
dc
r c K
rfk
i
=
f
if i
d i
kk
i
j
i
dc
r c K
rf
k
i
=
f
if i
d i
kk
j
i
i
i
d
rf
K
c
c
k
i
=
f
if i
d i
kk
i
j
j
d
rfki
c K c
i
=
f
if i
d i
kk
1 2
dbi
1
r
1
c c K
=
fi
i
i
i
i
d
f
r
kk
c
!
1 2
dbj
1
r
rfki
c c K
i
=
f
j
if i
jf j
d i
kk
c
kk
1 2
i
i
rfk
da
r
1
c c K
=
fi
j
i
if i
d
K
kk
c
i
1 2
j
i
da
rfk
c K
c c K
i
=
f
fi
i
i
i
d
fkk

25

rfki
if i
kk
1 2
c c K

1 2
c c K

fi

1 2
fki
c c K
if i
kk
rfki
bj (0)
if i
kk

fi

rfki
if i
kk

rfki
if i
kk

bj (0)

fi

rfki
if i
kk

bi (0)bi (0)

Table 3: The qualitative results of the steady-state e¤ects of improvements in productivity
country i : the country experiencing an improvement in productivity
country j : the neighboring country

the e¤ects ofan improvement
in i
r
K
ki
yi
i
country i c ; i
bi
ai
kj
yj
country j cj ; j
bj
aj

(I)
country i : DMI
country j : IMI(i = 1; j = 2)(II)
country i : IMI
country j : DMI(i = 2; j = 1)

+
+
+
+
+
+
+
+

+
+
+

(III)
country i : IMI
country j : IMI(D-S, 1991)
+
?
?
+
+
?

+
+
+
+
+

+
+
+

Note: The e¤ects on bi , bj , ai , and aj are obtained when b1 (0) = b2 (0) = 0.

26

Table A1: The signs of xi (i = 1; 2; 3; 4)

!>0
!=0
!<0

>0
x1 < 0; x2 < 0;x3 > 0; x4 > 0
inconsistent with
(2!)2 16 > 0
x1 > 0; x2 > 0;x3 > 0; x4 > 0=0x1 < 0; x2 = 0;x3 > 0; x4 > 0
inconsistent with
(2!)2 16 > 0
x1 = 0; x2 > 0;x3 > 0; x4 > 0<0x1 < 0; x2 > 0;x3 > 0; x4 > 0
x1 < 0; x2 > 0;x3 > 0; x4 > 0
x1 < 0; x2 > 0;x3 > 0; x4 > 0

Note: xi > 0 (i = 1; 2; 3; 4) include the cases of imaginary roots withpositive real parts.

27

δ c2

saddle-point stable

γK
r

δ c1

O

unstableγKΨ=0rδ − γ K = 02
crrδ c1 − γ K = 0

δ c1 + δ c2 = 0

Figure 1. The saddle-point stability conditions

r

r∗

E
the steady-statedemand curve
the steady-state
supply curveO

Y ∗ (= C ∗ )C ∗ ,Y *Figure 2. The steady-state supply curve and the steady-state demand curve

r∗(i) r ↑

E′E∗(ii) r ↓(i)the steady-state

E ′′
(ii)

demand curve
the steady-state
supply curveC ∗ ,Y *O

Figure 3. The effects of shifts in subjective discount rate schedules(i) upward shift in δ -schedule
1(ii) upward shift in δ -schedule2

Figure 4. (a) The saddle-path dynamics in the capital-accumulation economy:
the case of transfer from IMI country 2 to DMI country 1

Figure 4. (b) The saddle-path dynamics in the capital-accumulation economy:
the case of transfer from DMI country 1 to IMI country 2

Figure 5. (a) The saddle-path dynamics in the endowment economy:
the case of transfer from IMI country 2 to DMI country 1

Figure 5. (b) The saddle-path dynamics in the endowment economy:
the case of transfer from DMI country 1 to IMI country 2

r(I), (II)
∗

r ↓

E

~
E
the steady-statedemand curve
the steady-state(I), (II)supply curveC ∗ ,Y *O

Figure 6. The effects of improvements in productivity(I) increase in α1(II) increase in α2r,δ 1 ,δ 2

δ 2 (C M∗ − c1 )XL

(C L∗ , rL∗ ) ∗ ∗(C M , rM )XM
XH

δ 1 (c 1 )c1

the steady-statedemand curveδ 2 (C H∗ − c1 )

δ 2 (C L∗ − c1 )O

(C H∗ , rH∗ )

C

∗
L

c2

C

∗
M

C

∗
H

C ∗ (= c1* + c 2∗ )Figure A1. The derivation of the steady-state demand curver,δ 1 ,δ 2

δ 2 (C M∗ − c1 )X M′

( C M∗ , rM′ ∗ )the steady-state

XM(C M∗ , rM∗ )δ (c )
1

1

O
c1

c

demand curveC ∗ (= c1* + c 2∗ )C∗
2 M

Figure A2. (i) An upward shift of the steady-state demand curveby an upward shift in δ -schedule1

r,δ 1 ,δ 2

δ (c )
1

1(C M∗ , rM∗ )XM

the steady-state

X M′′( C M∗ , rM′′ ∗ )δ 2 (C M∗ − c1 )
O
c1

c2

C

∗
M

demand curveC ∗ (= c1* + c 2∗ )Figure A2. (ii) A downward shift of the steady-state demand curveby an upward shift in δ -schedule2