Discussion Paper No. 536

ON DECREASING IMPATIENCE

Ken-ichi Hirose
andShinsuke Ikeda

April 2001

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

On Decreasing Impatience
Hirose, Ken-Ichi¤ and Shinsuke Ikeday
Ritsumeikan University and Osaka University
April 2001

Abstract
In the theory of endogenous time preference, one of the most common and mostcontroversial assumptions is that the degree of impatience, measured by the rateof time preference, is increasing in wealth. Although this empirically-unjusti…ed
assumption often helps ease dynamic analyses by ensuring stability, it has neverbeen discussed why decreasing impatience is theoretically problematic. We …rst
show that under certain conditions there exists no optimal solution when
impatience is decreasing. By considering problem-proof, well-behaved models, we
examine implications of decreasing impatience for three issues that have been
discussed in the literature: (i) long-run tax incidence of capital taxation; (ii) thee¤ect of in‡ation on growth; and (iii) wealth distribution
dynamics in the two-country world.

Keywords: Decreasing impatience, time preference, capital taxation, the Tobine¤ect, wealth distribution.
JEL classi…cation: D90, E00, F41.

¤

The Department of Economics, Ritsumeikan University, 1-1-1, Noji-Higashi, Kusatsu,
Shiga 525-8577, Japan. Telephone: +81(797)84-9633. E-mail: k1hirose@hotmail.com
yCorresponding author. 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

On Decreasing Impatience11Introduction

In the theory of endogenous time preference, one of the most common and,at the same time, the most controversial assumptions is that the degree of
impatience, measured by the rate of time preference, is increasing in wealth.
Although there is no empirical evidence to the assumption of increasing impatience,1 no research has so far been conducted on theoretical or economic
implications of decreasing impatience, notwithstanding the huge accumulation of research on time preference and/or recursive preferences. Why is theassumption of decreasing impatience theoretically problematic? Whether or
not can we consider well-behaved models of decreasing impatience that are
immune to the “theoretical problem”? And if such models are available,
what would be economic implications of decreasing impatience? All of these
problems are left unsolved. For example, it has often been stressed that theassumption of increasing impatience is required to ease dynamic analyses by
ensuring stability (see, e.g., Epstein (1987a,b), Lucas and Stokey (1984), andObstfeld (1990)). However, Svensson and Razin (1983) criticized this reasoning since unstable equilibrium time paths of the economy cannot be ruledout a priori. We should consider what would be problematic when deceasing
impatience destabilizes dynamics.
This paper tackles these problems regarding decreasing impatience. The
main aims are: (i) to clarify its theoretical di¢culty; and (ii) to examine
economic implications of decreasing impatience by considering well-behavedmodels. We …rst show that instability resulting from decreasing impatience
is inconsistent with optimality in a certain situation. In particular, withappropriate regularity conditions such as concavity, divergent consumption
paths under decreasing impatience are shown to make, sooner or later, thediscount rate negative, thereby violating the transversality condition for the
discount factor when the rate of interest is constant.
This di¢culty does not occur in two cases: (a) when there exists somedecreasing-return properties that stabilize economic dynamics; and/or (b)when there are upper bounds for consumption resources. As a typical example of case (a), we incorporate capital accumulation with the neoclassicalproduction function. The resultant well-behaved decreasing-impatience mod1

See Lawrance (1991), Becker and Mulligan (1997), and Samwick (1997). They commonly reported that the degree of impatience is decreasing in wealth or income. See
also Ogawa (1993), which reported that the shapes of the time preference schedule areinternationally di¤erent.

On Decreasing Impatience2

els are used to elucidate implications of decreasing impatience for two policy
issues which has been discussed in the literature of increasing impatience(e.g., Epstein and Hynes (1983) and Obstfeld (1990)): (i) the e¤ects of capital taxation; and (ii) the e¤ects of in‡ation on economic growth. As for(b), we consider a two-country model, in which consumption resources arebounded by a constant world output level. By using the model, we examine(iii) wealth distribution dynamics in a two-country world economy.
Main implications we show on issues (i) though (iii) are as follows. In the
neoclassical model with decreasing impatience, an increase in capital taxes
raises the long-run interest rate, so that the long-run tax incidence on capital are larger than in the cases of constant time preference and of increasing
impatience. When impatience is decreasing, a decrease in the capital stockcaused by capital taxation makes consumers less patient, which raises thelong-run interest rate irrespective of the capital tax increase, and thereby
causes a further reduction in the capital stock. This property is contrasted
sharply to what is obtained in the literature by assuming increasing impatience (e.g., Epstein and Hynes (1983)).
When money is introduced into the neoclassical model with decreasing
impatience, an increase in the rate of nominal money growth reduces thesteady state capital stock, that is the Tobin e¤ect does not hold. This implication, which again contrasts to the case of increasing impatience (e.g.,
Epstein and Hynes (1983)), is consistent with many empirical studies reporting of growth-harming in‡ation (e.g., Barro (1995) and Bruno and Easterly(1998)).
In the two-country context, decreasing impatience leads to various wealthdistribution dynamics. If impatience is decreasing in both of the two countries, the equilibrium time path of the world economy is unstable. When one
country displays decreasing impatience while the other displays increasing
impatience, the equilibrium time path can be either stable or unstable, depending on the relative degrees of the increasingness and decreasingness of
impatience. In the stable case, an upward shift of the subjective discount rateschedule in the country with increasing impatience lowers the steady-stateinterest rate. With decreasing impatience, there can be multiple steady-state
equilibria.
The rest of the paper is organized as follows. Section 2 presents an intertemporal utility maximization problem with decreasing impatience andpoints out the di¢culty of decreasing impatience. In Section 3, the e¤ects ofcapital taxation are examined. In Section 4, the e¤ects on capital accumula-On Decreasing Impatience

3tion of monetary growth are examined. In Section 5, wealth distribution ina two-country economy is investigated. Section 6 concludes the paper.

2

Di¢culty under decreasing impatience

2.1

Regular utility maximization problems

To clarify theoretical di¢culties of decreasing impatience, begin with checking a necessary and su¢cient condition for optimal consumption in thestandard model of endogenous time preference. Consider an in…nitely-lived
consumer who maximizes lifetime utility by choosing the time pro…le of n1
commodity consumptions fc (t)g1
t=0 = f(c1 (t) ; ¢ ¢ ¢ ; cn (t))gt=0 and total asset
holding fa (t)g1
t=0 . His or her problem is speci…ed as problem (P) in thefollowing:
(P)

max

Z 1

u (c (t)) exp(¡¢ (t))dt;(1)0subject to:a_ (t) = h(a (t) ; c (t)); a (0) = a0 (constant); a (t) ¸ 0;
_ (t) = ± (c (t)) ; ¢ (0) = 0;¢

where a dot represents the time derivative; u (c) represents the felicity function; ¢ denotes a cumulative discountrate with the instantaneous discountRt
rate being given by ± (c): ¢ (t) = 0 ± (c (¿ ))d¿ ; and h(a; c) is the law of
motion for a, which is assumed to be concave in (a; c).2
The regularity conditions to make problem (P) well-de…ned should betreated carefully since decreasing impatience may be inconsistent with thoseconditions. With the present-value Hamiltonian function de…ned by
H = u(c) exp(¡¢) + ¸p h(a; c) + ´ p ± (c) ;_ rewhere ¸p and ´ p represent the present-value shadow prices of a_ and ¢,3
spectively, we can prove the following proposition:
Proposition 1: Suppose that intertemporal utility maximization problem (P)satis…es:2

Asset holding a includes both …nancial wealth and human capital. We can rewrite thepositivity condition for a as the no-Ponzi game condition to bond holdings.
3
We abbreviate the notation of time when there is no risk of confusion.

On Decreasing Impatience4

1. u < 0 ;2. u is concave;
3. ¡u is log-convex; and4. ± is concave,then, the …rst-order condition:
@H
@H
p @H
= 0;
= ¡¸_ ;
= ¡´_ p ;@c@a@¢and the transversality conditions:lim a (t) ¸p (t) = 0; lim ¢ (t) ´ p (t) = 0;

t!1

t!1

are also su¢cient for the optimal consumption plan.
Proof. See Appendix A.
Remark 1: Under the conditions given in Proposition 1, the solution to
problem (P) may not be unique. The uniqueness of the solution is ensured
if conditions 2 through 4 are rewritten by using the “strict” versions of concavity and convexity.
The degree of impatience is increasing or decreasing in wealth as ± isincreasing or decreasing in consumption. In what follows, impatience is said
to marginally increasing (decreasing), or simply increasing (decreasing), when
± is increasing (decreasing) in c:
Throughout the paper, we assume that regularity conditions 1 through 4
in Proposition 1 are satis…ed. Note, however, that these conditions are notrelated to whether the subjective discount rate ± is increasing or decreasing.
What is required to the discount function ± is just concavity: it can be (i)monotonically increasing; (ii) monotonically decreasing; or (iii) hump-shaped
as illustrated in Figure 1, where the one-commodity case is depicted. Several
researchers (e.g., Fukao and Hamada (1991) and Jafarey and Park (1998))
consider U-shaped nonmonotonic discount-rate functions. Proposition 1 implies that with their discount-rate functions the usual …rst-order conditions
do not ensure optima.

On Decreasing Impatience

2.2

5Di¢culties of decreasing impatience

With only few exceptions, it has been assumed that impatience is increasingin the literature. One main reason is that decreasing impatience is likely
to destabilize dynamics. Svensson and Razin (1983) criticize this treatment,
pointing out that stability cannot be regarded a priori as more reasonable
than instability. As a result of the instability, however, there may be no
optimal solution when decreasing impatience, as we shall show now.
Under the regularity conditions given in Proposition 1, consider consumer
problem (P) with one-commodity by assuming:² (A1) impatience is decreasing: ±c (c) < 0; and² (A2) the rate of interest r is constant,where ±c (c) represents d± (c) =dc.
Let his or her ‡ow budget constraint be given by(2)

a_ = ra ¡ c:
When we de…ne the generating function g as
g (c; Á) = u (c) ¡ Á± (c) ;where Á (t) represents the lifetime utility U (t) from the consumption stream
after time t, g generates Á by the low of motion,
Á_ = ¡g (c; Á) s.t. lim Á (t) exp (¡¢ (t)) = 0:t!1

We assume that gc > 0 and gcc < 0.
Let ¸ represent the current-value shadow price for savings. With the
lifetime utility function (1), the optimal conditions are given by44gc (c; Á) = ¸;(3)¸_ = (± (c) ¡ r) ¸;(4)Letting ­ denote the discount facor ­ ´ exp (¡¢), the current-value shadow price¸ can be related to ¸p in Proposition 1 as ¸ = ¸p =­. The Á is the shadow price or the
costate variable for the evolution of the discount facor ­ : ­_ = ¡±­: It can be related tothe present-value costate variable ´p in Proposition 1 as ´p = ¡Á­.

On Decreasing Impatience6Á_ = ¡g (c; Á) ;(5)

lim exp (¡¢ (t)) ¸ (t) a (t) = 0;(6)

lim exp (¡¢ (t)) Á (t) = 0:

(7)

t!1

t!1

De…ne the rate of time preference ½ as ½ = ¡d ln ­ (t) =dtjc=0; where ­ ´_gc (c; Á) exp (¡¢) represents the present-value marginal utility of c. Then,from equation (3), equation (4) can be rewritten as
c_ = ¡

gc (c; Á)(r ¡ ½ (c; Á)) ;gcc (c; Á)

(8)

where
½ (c; Á) = ± (c) ¡g (c; Á)±c (c) :gc (c; Á)(9)The optimal path for (c; Á; a) must be jointly generated by equations (2),(5), and (8). In particular, local dynamics around steady state should bedetermined from:
0

1

0

10

1

r±c
0
0
c^
c_
gcc
B _ C
B
CB ^ C@ Á A = @ ¡gc r
0 A@ Á A;
a_
r
a
^
¡1 0

where the coe¢cient matrix is evaluated at the steady state; and a hat x^
denotes a deviation from the steady state value: x^ = x¡x¤ : The characteristicroots for this system are:
8
<

Ã

4r±c
r > 0; ! 0 ´ (1=2) :r + r2 ¡gcc gc8<Ã4r±c! ´ (1=2) :r ¡ r2 ¡gcc gc!1=2 9=;!1=2 9=> 0:;> 0; and(10)

Note that any time paths governed by the roots other than ! cannot
satisfy the transversality condition (6). The optimal solution, if it exists, isthus given by the eigenvector associated with smallest root !. As seen fromOn Decreasing Impatience

7(10), decreasing impatience implies a positive !:5 the resultant saddle pathgoverned by ! is unstable, which is consistent with what has been pointed
out in the literature. Given the concavity of ±, this unstable saddle pathcannot satisfy the TVC (7) if the initial wealth holding a0 is larger than thesteady state stock a¹, as we shall demonstrate just below.
0
Let c be any consumption level such that ± (c0 ) > 0: Due to concavity, ±0)satis…es that for c > c0 ¡ ±±(c0 ;c (c )± (c) < ± (c0 ) + ±c (c0 ) (c ¡ c0 )< 0;
implying that ± is negative for the consumption levels which are higher than a…nite critical value (see Figure 1). Suppose that a (0) is larger than a¤ . Then,if consumption evolves along the saddle trajectory governed by !, c grows
at that rate for ever. After some point in time, ± continues to take negativevalues. This consumption time-pro…le de…nitely violates the transversality
condition (7):
Proposition 2 (di¢culty of decreasing impatience): Under (A1) and (A2),
problem (P) has no optimal solution if a (0) > a¤ .
Remark 2: As an important corollary, decreasing impatience is not suitable for analyzing small country models since, in the constant interest-rate
economy, decreasing impatience may prevent us from having optimal solution. Intuitively, when consumption divergently grows, the subjective discount rate falls into the negative region, so that the transversality condition(7) is violated.
Remark 2 has an important implication for what Svensson and Razin(1983) discussed. They criticize Obstfeld’s result that the Harberger-LaursenMetzler (HLM) e¤ect cannot occur by claiming that it depends crucially on
the assumption of increasing impatience and showing that with decreasing
impatience the HLM e¤ect de…nitely takes place. However, their analysis islimited to a small country model, in which the assumption of decreasing impatience is inconsistent with the existence of the optimal solution, as shown
by Remark 2.
Proposition 2 does not imply that we cannot analyze decreasing impatience models in any situations. There are two cases in which we can treat
5

Root ! can be imaginary. In that case, the real part of ! is positive.

On Decreasing Impatience8

well-behaved models with decreasing impatience: that is, (i) when the models
are stabilized by some other decreasing properties in production technology
and/or preferences, for which case unstable time paths are excluded; and(ii) when there is some upper bound for consumption resources, which enables even divergent time paths to satisfy the transversality conditions. Asa typical example of (i), we incorporate capital accumulation with a usual
decreasing-return technology in Sections 3 and 4. The resultant well-behavedmodels are used to analyze the implications of decreasing impatience for thee¤ects of capital taxation and in‡ation. As for (ii), Section 5 considers a twocountry model, in which consumption resources are bounded by a constantworld output level.

3

The e¤ects of capital taxation

Let us examine the implication of decreasing impatience for the e¤ect of
capital taxation. Epstein and Hynes (1983) showed that with endogenous
time preference capital taxes reduce the steady state capital stock less thanunder constant time preference. However this result depends crucially on theassumption of increasing impatience. When impatience is decreasing, theresult is drastically changed, as we shall show below.
Consider the usual neoclassical model: there are two production factors,
labor and capital; there is a single multi-purpose commodity; it is produced
by using a constant-to-scale technology F ; and …rms are competitive. Consumers supply one unit of labor at each instant inelastically. Their preferences are speci…ed just as in the previous section. In particular, we assume
decreasing impatience, ±c (c) < 0. The government levies tax ¿ on capitalincome and pays back the revenue to consumers in the lump-sum manner.
Letting k represent the capital labor ratio and f a percapita production function satisfying fk > 0 and fkk < 0, we can easily obtain a reduced dynamic
system as follows:
c_ = ¡

gc (c; Á)((1 ¡ ¿ ) fk (k) ¡ ½ (c; Á)) ;gcc (c; Á)Á_ = ¡g (c; Á) ;k_ = f (k) ¡ c;(11)

On Decreasing Impatience

9

where the steady state equilibrium (c¤ ; k ¤ ) is jointly determined by:± (c¤ ) = (1 ¡ ¿ ) fk (k ¤ ) ;(12)

f (k ¤ ) = c¤ :(13)

Assuming the existence of a steady state equilibrium, consider the local
dynamics around the steady state. Then, we can prove the following saddlepoint stability condition:
Property 1: The equilibrium time path is uniquely given by the stable arm
if and only if:±c (c¤ ) fk (k ¤ ) > (1 ¡ ¿ ) fkk (k ¤ ) :Proof. See Appendix B.
Assumption 1: ±c (c) fk (k) > (1 ¡ ¿ ) fkk (k) :Remark 3: As shown in Appendix B, if Assumption 1 were not met, thethree characteristic roots would be all positive, so that the di¢culty shownby Proposition 2 would occur, insofar as the distortion ¿ is su¢ciently small(close to zero) or su¢ciently large (close to one).6
With Assumption 1, we can illustrate the determination of the steadystate capital stock k ¤ and the e¤ect of capital taxation on it by using Figure
2. From (12) and (13), k ¤ is determined by± (f (k ¤ )) = (1 ¡ ¿ ) fk (k ¤ ) :As shown in Figure 2, (1 ¡ ¿ ) fk (k ¤ ) can be depicted as a downward-sloping
schedule in the (r; k) plane. With decreasing impatience, ± (f (k ¤ )) can also
be expressed by a downward-sloping schedule, where, from Assumption 1, the(1 ¡ ¿ ) fk (k ¤ ) schedule is steeper than the ± (f (k ¤ )) schedule. The steadystate capital stock k ¤ is given at the intersection, say point E0 ; of the twoschedules.
6

When the distortion is neither su¢ciently small nor su¢ciently large, however, wecannot rule out a priori the (very little) possibility that without Assumption 1 two of
the three characteristic roots are negative. In that case, there exist multiple equilibria:
we have to be confronted with the indeterminacy problem, instead of the no-existence
problem.

On Decreasing Impatience10

Now suppose that capital tax ¿ is raised from ¿0 to ¿1 : It shifts the(1 ¡ ¿ ) fk (k ¤ ) schedule downward, thereby bringing the steady state pointfrom point E0 to E1 . Consequently, k ¤ decreases in response to the tax increase. Note that this reduction in k ¤ and hence the long-run tax incidence
on capital are larger than in the case of constant time preference: if ± wereconstant, the reduction would stop at k 0 : This property is contrasted sharply
to what Epstein and Hynes (1983) stressed by assuming increasing impatience. When impatience is decreasing, a decrease in k ¤ makes consumers
impatience, which raises the long-run interest rate irrespective of the capital
tax increase, and thereby causes a further reduction in k ¤ .
Implication 1: When impatience is decreasing, an increase in capital taxes
raises the long-run interest rate, so that the long-run tax incidence on capital is larger than in the cases of constant time preference and of increasing
impatience.

4In‡ation and growth with decreasing impatience

Many empirical studies have reported that in‡ation harms capital accumulation (e.g., Fischer (1993), De Gregorio (1992, 1993), Barro (1995), andBruno and Easterly (1998)), whereas theoretical models including Epsteinand Hynes (1983) often predict the Tobin e¤ect that in‡ation promotes
growth by shifting real money balances away in favor of consumption. As thesecond implication of decreasing impatience, we shall show that an increasein the rate of nominal money growth reduces the steady state capital stock.
Let us extend the model in section 3 by introducing money in the “moneyin-the-utility-function” framework. For brevity we specify the subjective discount rate as a function of the felicity level, ± = ± (u) as in Uzawa (1968),
where decreasing impatience is assumed by setting ± 0 (u) < 0. The consumer’s
problem is now given by
max

Z 1

u (c; m) exp(¡¢)dt;

(14)

0

subject to:_ = ± (u (c; m)) ;¢(15)

a_ = ra + w ¡ c ¡ (r + ¼) m + x;

(16)

On Decreasing Impatiencea = k + m;

11

(17)

where a represents total …nancial wealth; m real money holdings; w the wage
rate; ¼ the rate of in‡ation; and x the lump-sum transfer payments from thegovernment.
Assume that the regularity condition given in Proposition 1 is satis…ed.
Letting g (c; m; Á) be u (c; m) ¡ Á± (u (c; m)), the optimal consumption planshould satisfy:
gc (c; m; Á) = ¸;Â(c; m) ´um (c; m)= r + ¼;
uc (c; m)(18)¸_ = (± (u (c; m)) ¡ r) ¸;
Á_ = ¡g (c; m; Á) ;and the transversality conditions. In the same way to obtain (8), the …rstorder conditions can be reduced to:gc (c; m; Á)c_=¡(r ¡ ½ (c; m; Á)) ;c
cgcc (c; m; Á)m_gm (c; m; Á)=¡(r ¡ ½(c; m; Á));m
mgmm (c; m; Á)(19)where ½ (c; m; Á) is de…ned using g (c; m; Á) as in (9). We assume that gc > 0;
gm > 0; gcc < 0; gmm < 0; and gcm = 0.
Money is supplied by the government in the form of “helicopter money:”¹m = x;(20)

where ¹ denotes the growth rate of nominal money supply.
From the pro…t-maximizing behavior of …rms, r and w are given as
w = f (k) ¡ kfk (k) ; r = fk (k) :Substituting (17), (20), and (21) into (16) yields
k_ = f (k) ¡ c:(21)

On Decreasing Impatience

12m_
By combining m= ¹ ¡ ¼ with (18) and (19), we obtaingm(½(c; m; Á) ¡ fk (k)) = ¹ + fk (k) ¡ Â(c; m);mgmm
which can be solved for m as
m = À(c; Á; k; ¹):
By using this, the dynamic system is …nally reduced to:c_ = ¡

gc (c; À(c; Á; k; ¹); Á)(fk (k) ¡ ½ (c; À(c; Á; k; ¹); Á)) ;gcc (c; À(c; Á; k; ¹); Á)

Á_ = ¡g (c; À(c; Á; k; ¹); Á) ;k_ = f (k) ¡ c:
The steady state equilibrium (c¤ ; m¤ ; k ¤ ) is determined by± (u (c¤ ; m¤ )) = fk (k ¤ );Â(c¤ ; m¤ ) = fk (k ¤ ) + ¹;

(22)

f (k ¤ ) = c¤ :By analyzing the local dynamics around the steady state, we can show thefollowing:
Property 2: The equilibrium time path is saddle-point stable only if¤ ´ (1 ¡ÂÂÂc)±u uc fk ¡ fkk +±u uc fkk > 0:ÂmÂm

Proof. See Appendix C.
Assumption 2: ¤ > 0:
Remark 4: Assumption 2 is a necessary condition for the system to besaddle-point stable, under which the determinant of the coe¢cient matrix is
gm r±u um
negative. As shown in Appendix C, the trace of the matrix, 2r + mg;
mm Âm

On Decreasing Impatience13

should be positive for the saddle-point stability. Otherwise all of the threeroots might be negative, implying that there might be multiple equilibria.
Let us now examine the e¤ect of a permanent increase in the core rate ofin‡ation ¹. From (22), the steady state equilibrium is determined by± (u (f (k ¤ ) ; m¤ )) = fk (k ¤ );(23)

Â(f (k ¤ ) ; m¤ ) = fk (k ¤ ) + ¹:

(24)

Figure 3 depicts the determination, where schedule KK 0 represents (23)and M M 0 (24). The M M 0 schedule is positively sloping and, under Assumption 2, the slope of the KK 0 schedule is also positive and gentler thanthat of M M 0 . The steady state is determined at the intersection of the twoschedules. An increase in ¹ shifts the M M 0 schedule upward, bringing thesteady state point from E0 to E1 . As illustrated, the steady state capitalstock decreases in response to the rise in the in‡ation rate. In fact, from (23)and (24), we can derive
dk ¤
±u um
=¡
< 0:
d¹
Âm ¤
This result can be summarized as follows:(25)

Implication 2: When impatience is decreasing, an increase in the core rateof in‡ation harms capital accumulation, that is the Tobin e¤ect does not takeplace.
Remark 5: As seen from (25), whether an increase in ¹ increases or decreasesk ¤ depends crucially on the sign of @±=@m, but not on @±=@c, that is onwhether impatience with respect to m is increasing or decreasing, but not
on whether impatience with respect to c is increasing or decreasing. When ±depends on (c; m) independently of u, and hence when preferences are weakly
non-separable in the sense of Shi (1994), the same result as in implication 2
is valid even if impatience is increasing in c.

5

Wealth distribution dynamics in a two-country world
economyLet us …nally consider the implication of decreasing impatience in a twocountry model. Suppose that the world economy is composed of countries

On Decreasing Impatience14

1 and 2, each of which is populated with in…nitely-lived identical agents.
The representative agent in each country consumes a single consumption
good and holds wealth in the form of bonds. Both the goods and bonds aretraded freely in international markets. For brevity, the representative agentsin country i (i = 1; 2) are assumed to be endowed with constant amounts y i .
The budget constraint for the representative agent is given byb_ i = rbi + y i ¡ ci ;(26)

where bi represents net foreign assets held by country i.
By assuming the same preference structure as speci…ed in section 2.2, theoptimal conditions can be obtained in exactly the same way as in (3) through(7). They reduce to, for i = 1; 2;noc_i = ¾ i r ¡ ½i (ci ; Ái ) ;
³ ´

(27)

³ ´

i
Á_ = ± i ci Ái ¡ ui ci ;iand the transversality conditions, where ¾ i = ¡gci =gcc:
The market-clearing conditions are given byc1 (t) + c2 (t) = Y;(28)

b1 (t) + b2 (t) = 0;
where Y represents the aggregate output, y 1 + y 2 : The interest rate is endogenously determined by the market clearing condition: by di¤erentiating(28) and substituting (27) into the result, we can obtain
¾2
¾1
1 1
1
½ (c ; Á ) + 1½2 (c2 ; Á2 );
r= 1
2
2
¾ +¾
¾ +¾

(29)

implying that the equilibrium interest rate is determined as a weighted sumof the rates of time preference in the two countries.
From (28) and (29), the dynamic system (26) through (27) can be reduced
to:
c_1 =

o
¾1 ¾2 n 2
1
2
1 1
1
½
(Y
c
;
Á
)
½
(c
;
Á
)
;
¡
¡
¾1 + ¾2³ ´³ ´

1
Á_ = ± 1 c1 Á1 ¡ u1 c1 ;

On Decreasing Impatience
³

´

³

´

15

2
Á_ = ± 2 Y ¡ c1 Á2 ¡ u2 Y ¡ c1 ;

_1

b =

(

)

¾1
¾2
1 1
1
½
(c
;
Á
)
+
½2 (Y ¡ c1 ; Á2 ) b1 + y 1 ¡ c1 ;
1
2
1
2
¾ +¾
¾ +¾

the steady state of which is determined by
³

´

³

´

± 1 c1¤ = ± 2 Y ¡ c1¤ = r¤ ;
³

´

³

´

Á1¤ = u1 c1¤ =r¤ ; Á2¤ = u2 Y ¡ c1¤ =r¤ ;c1¤ = r¤ b1¤ + y 1 ; c2¤ = r¤ b2¤ + y 2 :By analyzing the local dynamics around the steady state, we can obtain thefollowing property:
Property 3: The steady-state point is locally saddle-point stable if and only
if
³

´

³

´

±c1 c1¤ + ±c2 Y ¡ c1¤ > 0:(30)

Proof. See Appendix D.
Remark 6: Unlike in the previous sections, stability (in the saddle-point
sense) is not necessary for an equilibrium path to exist. Even if (30) is notsatis…ed and hence even if the steady state is unstable, the time-path thatgoverned by the smallest root can be an equilibrium. In the previous sections,consumption is not bounded from the above, so that any divergent consumption paths, sooner or later, make the discount rate negative and thereby
violate the transversality condition. In the present two country setting, incontrast, consumption is bounded by the total quantity Y of world output,
which allows a divergent equilibrium consumption time-path. In particular,
this is the case when min (± 1 (Y ) ; ± 2 (Y )) > 0, for which case the discountrates remain positive for any divergent time-paths of the economy.
As implied by Property 3, if impatience is decreasing in both of the twocountries, the equilibrium time path is unstable. For a steady-state equilibrium to be saddle-point stable, at least one of the two countries should display

On Decreasing Impatience16

increasing impatience.7 Suppose that one of the two, say country 2, exhibits
increasing impatience, whereas the other’s, i.e., country 1’s impatience is decreasing. For a steady-state equilibrium allocation to be saddle-point stable,
a marginal transfer from country 1 to country 2 should make the recipient
country 2 less patient.8 Figures 4(a) and 4(b) illustrate the determination of
the steady-state equilibrium and equilibrium dynamics in the (c1 ; r)-plane in
the stable and unstable cases, respectively. The ± 1 and ± 2 are both depicted
as decreasing functions in c1 . The steady-state equilibrium is determined at
the intersection E of the two schedules. As is shown, steady state is stable
if and only if the ± 1 -schedule cuts the ± 2 -schedule from the below.9
Using Figure 4(a), consider the e¤ect of an upward shift of country 2’s
subjective discount rate schedule ± 2 . This brings the steady state point fromE to E 0 , lowering the steady state interest rate. The upward shift in ± 2 ;ceteris paribus, makes country 2 less patience than country 1, which induces
wealth movements from country 2 to 1 and thereby reduces the interest rate
along the ± 1 -schedule.
Implication 3: Suppose that a country exhibits decreasing impatience whereas
the other displays increasing impatience, and that saddle-point stability condition (30) is met. Then, an upward shift of the subjective discount rateschedule in the country with increasing impatience lowers the steady-stateinterest rate.
With decreasing impatience, there can be multiple steady-state equilibria. Figures 5(a) and 5(b) illustrate typical possibilities. First, consider thecase as illustrated in Figure 5(a). Only when the initial wealth distribution is su¢ciently equal, the economy will converge to the steady state point
7

For the analysis of the two-country world economy with increasing impatience, see
Devereux and Shi (1991).
8
The same condition is derived by Jafarey and Park (1998).
9
In Figure 4 (b), the divergent time paths towards (c1 ; c2 ) = (Y; 0) and (0; Y ) areequilibria, if
¡
¡ ¢
¡ ¢¢
gci ci ; ui ci =± i ci
= 0; i = 1; 2;
lim i i i i(c ; u (c ) =± i (ci ))ci !0 gcc
for which case points (Y; 0) and (0; Y ) are steady state equibria that are saddle-point stable.
It can be shown that a su¢cient condition for the above equation to be vaild is: (i) ± i is afunction of ui ; (ii) ui displays constant relative risk aversion; and (iii) Á±uu = (1 ¡ Á±u ) isbounded.

On Decreasing Impatience17

S with both countries consuming positive quantities of goods. If the initial wealth is distributed quite unequally, the economy will asymptoticallyconverge to the steady state point (c1 ; c2 ) = (Y; 0) or (0; Y ), at which only initially wealthy country has positive consumption. In this sense, co-prosperity
needs su¢ciently equal initial wealth distribution. Figure 5(b) illustrates another multiple-equilibrium case, where there is unstable steady state point U
with fairly equal wealth distribution, placed between two equilibria S1 andS2 with uneven distributions. In this case, unequal wealth distributions areboth stable and long-run wealth distribution is likely to be uneven. Thereis a threshold level for initial wealth holding that determines which unequal
distribution of S1 and S2 takes place.

6

Conclusions

This paper has examined theoretical as well as policy implications of decreasing impatience. We have …rst shown that in constant interest-rate economies
divergent consumption time-paths resulting from decreasing impatience areoften inconsistent with optimality. This di¢culty disappears (i) when there
are some deceasing-return properties built in the model; and/or (ii) when
there is some upper bound for consumption resources. Based on this observation, we have considered well-behaved decreasing-impatience models andinvestigated their policy implications.
In the neoclassical model with decreasing impatience, an increase in capital taxes raises the long-run interest rate and the long-run tax incidence on
capital are larger than in the case of constant time preference and of increasing impatience. When money is introduced into this model, the Tobin e¤ectdoes not hold and in‡ation harms capital accumulation, as many empiricalstudies have reported. In the two-country context, decreasing impatienceleads to various interesting multiple steady-state equilibria. We have shown
the existence of an equilibrium in which an upward shift of a country’s discount rate function lowers the world interest rate.

On Decreasing Impatience

A

18

Proof of Proposition 1:

Let us de…ne q(c; ¢) asq(c; ¢) = u(c) exp(¡¢);
and the Hamiltonian as
H = q(c; ¢) + ¸p h(a; c) + ´ p ± (c) :Suppose that (c¤ ; a¤ ; ¢¤ ) satis…es the …rst-order condition and the transversality condition of problem (P):Ã@H
= qci (c¤ ; ¢¤ ) + ¸p hci (a¤ ; c¤ ) + ´p ±ci (c¤ ) = 0; i = 1; ¢ ¢ ¢ ; n;(A.1)
@ci

!

Ã

@H
p
= ¸p ha (a¤ ; c¤ ) = ¡¸_ ;
@a

Ã

!

(A.2)

@H
= q¢ (c¤ ; ¢¤ ) = ¡´_ p ;
@¢

!

(A.3)

lim a¤ (t) ¸p (t) = 0;

(A.4)

lim ¢¤ (t) ´ p (t) = 0:

(A.5)

t!1

t!1

Then, the following lemmas hold.
Lemma A.1: q is concave in (c; ¢) if and only if (i) u < 0; (ii) u is concave,and (iii) ¡u is log-convex.
Proof. Because qci cj = uci cj exp(¡¢), qci ¢ = ¡uci exp(¡¢), q¢¢ = u exp(¡¢),
the necessary and su¢cient condition for the concavity of q (or the convexityof ¡q) is that01

¡uc1 c1 ¢ ¢ ¢ ¡ucn c1 uc1
B
..
..
.. C
...
B
.
.
. C
B
C
X´B
C
@ ¡uc1 cn ¢ ¢ ¢ ¡ucn cn ucn A
ucn
uc1
¢¢¢
¡u

On Decreasing Impatience19

is a positive semi-de…nite matrix, i.e., any determinant of the principal submatrix of X is non-negative:¯¯¯¯¯Xfi1 ;¢¢¢;im g ¯ ¸ 0; for any fi1 ; ¢ ¢ ¢ ; im g satisfying 1 · i1 < ¢ ¢ ¢ < im · n + 1;

where Xfi1 ;¢¢¢;im g denotes the principal
submatrix Ãof X which lies
in rowsÃ!!
¡uc1 c1 uc1. Then,i1 ; ¢ ¢ ¢ ; im and columns i1 ; ¢ ¢ ¢ ; im e.g., Xf1;n+1g =u c1¡uwe can ¯prove the¯ following properties:¯(a) ¯Xfn+1g ¯¯ ¸ 0 i¤ u < 0;¯¯(b) ¯¯Xfi1 ;¢¢¢;im g ¯¯ ¸ 0 for any fi1 ; ¢ ¢ ¢ ; im g satisfying 1 · i1 < ¢ ¢ ¢ < im · ni¤ u is ¯concave; and ¯(c) ¯¯Xfi1 ;¢¢¢;im ;n+1g ¯¯ ¸ 0 for any fi1 ; ¢ ¢ ¢ ; im g satisfying 1 · i1 < ¢ ¢ ¢ < im ·n i¤ ¡u is log-convex.
Since (a) and (b) are obvious, let us sketch the proof of (c). The necessary
and su¢cient condition for the log-convexity of ¡u is that01

uuc1 c1 ¡ uc1 uc1 ¢ ¢ ¢ uucn c1 ¡ ucn uc1
B
C
..
..
...
C
Z´B
.
.
@
A
uuc1 cn ¡ uc1 ucn ¢ ¢ ¢ uucn cn ¡ ucn ucn

is a positive semi-de…nite matrix, i.e., any determinant of the principal submatrix of Z is non-negative:¯¯¯¯¯Zfi1 ;¢¢¢;im g ¯ ¸ 0; for any fi1 ; ¢ ¢ ¢ ; im g satisfying 1 · i1 < ¢ ¢ ¢ < im · n:Now, we can show that

¯
¯
¯
¯
¯
¯
¯
¯
¯Xfi1 ;¢¢¢;im ;n+1g ¯ = (¡u)¡m+1 ¯Zfi1 ;¢¢¢;im g ¯ :For example,¯¯¯ ¡uc c ¢ ¢ ¢ ¡uc c
uc1 ¯¯
n 1
1 1
¯
¯
..
..
.. ¯¯
¯
¯
...
¯
¯
¯
.
.
. ¯
¯Xf1;¢¢¢;n;n+1g ¯ = ¯¯¯¯ ¡uc1 cn ¢ ¢ ¢ ¡ucn cn ucn ¯
¯
¯
¯ uc
u cn
¢¢¢
¡u ¯
1
¯
¯ uuc c¢ ¢ ¢ uucn c1
1 1
¯
¯
.
..
...
¯
..
.
= (¡u)¡n ¯¯¯ uuc1 cn ¢ ¢ ¢ uucn cn¯¯ ¡uuc ¢ ¢ ¢ ¡uuc1n¯

uc1 ¯¯.. ¯¯. ¯¯ucn ¯¯¡u ¯

On Decreasing Impatience20¯¯¯ uuc c ¡ uc uc¢ ¢ ¢ uucn c1 ¡ ucn uc1 uc1 ¯¯
1 1
1
1
¯
¯
..
..
.. ¯¯
...
¯
¡n ¯
.
.
. ¯
= (¡u) ¯¯¯ uuc1 cn ¡ uc1 ucn ¢ ¢ ¢ uucn cn ¡ ucn cn ucn ¯
¯
¯
¯
0
0
¢¢¢
¡u ¯
¯
¯
¯ uu
¯
c1 c1 ¡ uc1 uc1 ¢ ¢ ¢ uucn c1 ¡ ucn uc1 ¯
¯
¯
¯
..
..
...
¯
= (¡u)¡n+1 ¯¯
.
.
¯
¯
¯
¯ uuc1 cn ¡ uc1 ucn ¢ ¢ ¢ uucn cn ¡ ucn ucn ¯¯¯= (¡u)¡n+1 ¯¯Zf1;¢¢¢;ng ¯¯ :Properties (a), (b), and (c) imply lemma A.1.

Therefore, Proposition 1 follows from Lemma A.2 below:
Lemma A.2: Suppose that the following conditions are met:
1. q is concave in (c; ¢);2. ± is concave in c; and3. h is concave in (a; c).
Then, we have
D´

Z 1
0

fq (c¤ ; ¢¤ ) ¡ q (c; ¢)g dt ¸ 0

for any feasible paths (c; a; ¢).
Proof. D satis…es

D =

Z 1
0

¸
=

fq (c¤ ; ¢¤ ) ¡ q (c; ¢)g dt

Z 1 (Xn0i=1(c¤i ¡ ci ) qci (c¤ ; ¢¤ ) + (¢¤ ¡ ¢) q¢ (c¤ ; ¢¤ ) dt (the concavity of q)i=1(c¤i ¡ ci ) (¡¸p hci (a¤ ; c¤ ) ¡ ´p ±ci (c¤ )) + (¢¤ ¡ ¢) (¡´_ p )Z 1 (Xn0)= I1 + I2 ;)dt

On Decreasing Impatience

where I1 and I2 are:
I1 =
I2 =

Z 1 (Xn)21

0(c¤i ¡ ci ) (¡¸p hci (a¤ ; c¤ ))0(c¤i ¡ ci ) (¡´ p ±ci (c¤ )) + (¢¤ ¡ ¢) (¡´_ p )i=1
Z 1 (X
n
i=1

dt;
)

dt:

I1 and I2 are both positive as I shall show now. I1 satis…es:
I1 =
¸
=

Z 1 (X
n
0

Z 1n

i=1

(c¤i ¡ ci ) (¡¸p hci (a¤ ; c¤ )) + (a¤ ¡ a)³p¡¸ ha (a ; c ) ¡ ¸_
p

¤

¤

o

)
´

p
¡¸p (h (a¤ ; c¤ ) ¡ h (a; c)) ¡ (a¤ ¡ a) ¸_ dt (the concavity of h)

Z01 n

o

p
_ ¸p ¡ (a¤ ¡ a) ¸_ dt¡ (a_ ¤ ¡ a)0

= ¡ [(a¤ ¡ a) ¸p ]1
0p= lim a (t) ¸ (t) (a¤ (0) = a (0) = a0 )t!1

¸ 0 (¸p (t) ¸ 0):
As for I2 , we have:I2 ==Z 1 (X
n
0

Z 1

i=1

´

p

0

(

(c¤i ¡ ci ) (¡´p ±ci (c¤ )) +¤± (c ) ¡ ± (c) ¡

n
X
i=1

³

´

)

_¤¡¢
_ ´ p dt ¡ [(¢¤ ¡ ¢) ´ p ]1¢0)(c¤i ¡ ci ) (±ci (c¤ ))dt ¡ [(¢¤ ¡ ¢) ´ p ]10

¸ ¡ [(¢¤ ¡ ¢) ´ p ]1
0 (the concavity of ±)= lim ¢ (t) ´ p (t)t!1¸ 0 (´ p (t) ¸ 0 (See footnote 3 in the text.)).
This implies D ¸ 0:

B

Proof of Property 1

By linearizing system (11) around the steady state, the local dynamic system
can be obtained as
0

1

dt

0

10

1

)fkk
±±c
c^
c_
0
¡ gc (1¡¿
gcc
gcc
B _ C
B
CB ^ C@ Á A = @ ¡gc ±A@ Á A;
0
_k
k^
fk
¡1 0

On Decreasing Impatience

22

where the coe¢cient matrix is evaluated at the steady state. For this coe¢cient matrix,
trace = ± + fk > 0; det =±gc(±c fk ¡ (1 ¡ ¿ ) fkk ) :gcc

The linear system thus has two positive and one negative roots if and only±c fk ¡ (1 ¡ ¿ ) fkk > 0, as stated in Property 1.
When ±c fk ¡ (1 ¡ ¿ ) fkk < 0; the determinant of the coe¢cient matrix is
positive, implying that (i) the three roots are all positive; or that (ii) two
roots are negative and the other one is positive.. It is easy to show thatcase (ii) cannot occur insofar as the capital tax rate ¿ is su¢ciently small orsu¢ciently large, as in Remark 3. For example, when ±c fk ¡ (1 ¡ ¿ ) fkk < 0;
the three roots are all positive if
gc fkkgc(±c fk ¡ (1 ¡ ¿ ) fkk ) + ±fk ¡ ¿ ±c fk > 0:gcc
gcc
This inequality indeed holds valid when ¿ = 0 or 1:

C

Proof of Property 2

Noting thatÂm;Âc
r±u um= ¡;mgmm Âm
fkk
gm
= ¡
fkk ;
¡
Âm mgmm Âm

Àc = ¡
ÀÁÀk

the local dynamic system can be obtained as
0

1

0

1

c^
c_
B _ C
B ^ C
@ Á A = A@ Á A;
k_
k^
0

1

r±u uc0¡ ggccc fkk
gcc
B
C
Âm
fkk
gm
r±u umC
A=B@ ¡gc ¡ gm (¡ Âc ) r ¡ gm (¡ mgmm Âm ) ¡gm ( Âm + mgmm Âm fkk ) A ;0r¡1

On Decreasing Impatience

23

where coe¢cient matrix A is evaluated at the steady state. Then, the linear
system has two positive and one negative roots only if
"

#

gm
gc r
gm Âc
det (A) =r±u uc fkk +(1 ¡)r±u uc ¡ fkk < 0;gcc Âmgcc
gc Âmor alternatively stated, only if¤ = (1 ¡ ÂÂcÂ)r±u uc ¡ fkk +±u uc fkk > 0;ÂmÂm

as in Property 2. Note also that if ¤ > 0 and if(trace (A) =) 2r +gm r±u um> 0;mgmm Âm

the system indeed has two positive and one negative roots, and is saddle-point
stable, as demonstrated in Remark 4.

D

Proof of Property 3

By linearizing the dynamic system around the steady state, we obtain
0

1

0

1

c^1
c_1
C
B
Ã
!
B _1 C
B Á
^1 C
B Á C
B
O
C
B
3£1
B 2 C=B 2 C;
B _ C
C
B
^
C
r
@ Á A
@ Á A
^b
b_
0

0
¡
B
1
B=B
@ ¡gc rgc2
0³

¾ 1 ¾2¾ 1 +¾ 2

´

r±c1
gc1

³

0
r

¾1 ¾ 2¾ 1 +¾ 2

´

r±c2
gc2

1

C
C;
A

where the coe¢cient matrix is evaluated at the steady state. The roots ofthis linear system are thus r and three eigenvalues of B. Because trace(B) =2r > 0, B has two positive and one negative eigenvalues whendet (B) = ¡r2´¾1¾2 ³ 1
2
±
+
±
< 0:
c
c
¾1 + ¾2

Therefore, if ±c1 +±c2 > 0, the linear system has three positive and one negativeroots.

On Decreasing Impatience

24

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On Decreasing Impatience

25

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δ

δ(c)

′
c ′ − δ (c )
δ c (c ′)c′Figure 1. NonNon-monotonic impatience

c

r ,δ
(1 − τ ) f k (k )E1

E0

δ ( f (k ))kk′Figure 2. Capital taxation under decreasing impatience

k

M

K

E0
E1

K′

M′

m

Figure 3. Inflation and capital stock under decreasing impatience

r,δi

δ2 2
δ
δ1

E′

E

O1

c1

O2

Figure 4 (a). Two country equilibrium:
The case of saddle-point stability

r,δi

δ1

δ2

O1

c1
Figure 4 (b). Two country equilibrium:
The case of unstable steady state

O2

r,δi

δ2
S
U1

δ1
skd

δ1

U2
U1

O1

c1

O2

Figure 5 (a). Multiple steady-state equilibria

r,δi

δ1

δ1

U

δ2

S1

O1

S2

c1
Figure 5 (b). Multiple steady-state equilibria

δ2O2