DP2013-04

Ineq ualities and Patience for T om orrow*

Kazumichi IWASA
Laixun ZHAO

February 2, 2013

* The Discussion Papers are a series of research papers in their draft form, circulated to encourage
discussion and comment. Citation and use of such a paper should take account of its provisional character.
In some cases, a written consent of the author may be required.

Inequalities and Patience for Tomorrow
Kazumichi Iwasa
Institute of Economic Research, Kyoto University
Laixun Zhaoy
Research Institute of Economics and Business, Kobe University
February 2, 2013

Abstract
This paper examines how impatience interacts with inequalities in economic development. In a society of intrinsic inequality, we show that (i) poor households tend tobene…t more from positive shocks under decreasing marginal impatience (DMI) than under constant marginal impatience (CMI) and increasing marginal impatience (IMI); (ii)an unequal society may be preferable for poor households under DMI; (iii) urbanization
can increase the income inequality, while raising overall welfare; (iv) under DMI even if
all households are allowed to own assets, with di¤erent initial asset holdings, the economy
will not converge to the steady state where everybody is a capitalist.

1

Introduction

In fast growing economies such as China, urban and coastal regions experience rapid growth,
drastically increasing the income gap between these regions and remote areas. In 2009, theannual per-capita income in China was about $750 for rural residents, while that for urban
residents was about $2500. According to China Daily (May 23, 2012), the most a- uent 10
percent of the population makes 23 times more than the poorest 10 percent.1 Such income
inequality has caused workers to migrate in large scale from rural to urban areas and from inner
to coastal regions. Furthermore, even within city limits, the income inequality has been rising
rapidly, especially following the in‡ux of workers from rural areas. Meanwhile, many newly
constructed roads, bridges, railways, buildings and even food products are of poor or evenhazardous quality, causing fatal accidents; Air and water pollution has reached unbearable
Institute of Economic Research, Kyoto University, Kyoto 606-8501, Japan. Fax: +81-75-7537170. E-mail:iwasa@kier.kyoto-u.ac.jpyResearch Institute of Economic & Business, Kobe University, Kobe 657-8501, Japan. Fax/Tel: +81-78-8037006. E-mail: zhao@rieb.kobe-u.ac.jp
1
The inequality in India is slightly better. In the same year, the annual per-capita income in India was about$750 for rural but $1500 for urban residents.

1

levels. Public morality seems to be in a landslide, culminated by a recent death (October,2011) of a 2-year-old girl, run over twice by vehicles and subsequently ignored by 18 cyclists
and other passers-by (A rag collector eventually came to her aid). These phenomena havegenerated heated debates in the media, among policy makers and researchers alike. There aresoul searching cries in the popular press that China in its rush to modernity should slow down
the pace, in order to decrease man-made errors and potential disasters, to reduce ever increasing
pollution, and to make more e¢ cient use of its depleting resources. It is also argued that tosustain growth in the longer run, the government must adopt more patient policies and reduceincome and other inequalities.2
Similar to the popular press, in the academic literature quite a number of empirical studies…nd strong evidence that households are heterogeneous in terms of impatience (e.g., they discount the future at di¤erent rates), and that preference heterogeneity is an important factor in
explaining household income inequality, see for instance, Hausman (1979), Becker and Mulligan(1997), Samwick (1997), and Barsky et al. (1997). Lawrance (1991) and Warner and Pleeter(2001) …nd that more-educated households and individuals tend to have lower discount rates
than less-educated ones, thus heterogenous time preference may lead to income and wealthinequality through long-term investment and human capital accumulation. In fact, some studies have found that the marginal propensity to save is considerably higher among wealthierpeople.3
The present paper is motivated by the above heated debates and interesting empirical…ndings. We consider a society that does not have “equal opportunity” to begin with, as is afact in many developing countries with strong traditional systems (e.g., some Latin Americancountries, China and India, etc.).4 Speci…cally, there exist two types of households that aresymmetric in all aspects except that one type owns asset (e.g., capitalists), while the othertype (e.g., workers) is unable to own asset and hence consume all income at each point intime. We examine how such inequality evolves under globalization, in particular faced withtechnology improvements (such as the telecommunication and Internet revolution in the early
1990s), capital market ine¢ ciency, demand and other shocks. We are especially interested intransition economies where impatience can play important roles during the catch-up process tomodernization.
An important phenomenon we examine is the so-called “decreasing marginal impatience(DMI)”, under which rich households save and invest more when they become richer. The
patience of rich households enables them to invest more and accumulate more capital, generatinga trickle-down e¤ect to the poor in the long run. This mechanism thus increases the capitalstock, the productivity of workers, and the welfare of all households including the poor when2

The above phonomenon is closely related to the so-called "curse to the late comer", who in the rush to
become rich picks the easier way out by copying mature technologies and buying assembly lines from richcountries, instead of taking the more di¢ cult route of gradually developing institutional and political systems
conducive for innovation and sustainable growth. As such, current growth may be fast, but it is very costly andshort-lived, and inevitably it comes with various kinds of social inqualities due to the lack of a fair institution
or system.
3
Frederick et al. (2002) provides a detailed review of this literature.
4
For instance, although slowly being relaxed, the infamous family registration (hukou) system in China
determines whether one is a peasant, urban worker, or cadre, etc. at birth (!) following the mother. And India
is wellknown for the Caste system, which has largely broken down in cities, but persists in rural areas where
72% of India’s population resides. Several Latin American countries have the highest Gini index in the world,and such inqualities have been carried over from generation to generation, some even from the colonial period(Ferreira et al., 2004). As such the top 10% population is estimated to own 50% of the total income while thebottom 10% owns only 1.5%, compared to 30% and 2.5% for corresponding groups in developed countries.

2the rich becomes richer. Hence, inequality as a side-e¤ect on a country’s catching-up path may
not be so bad after all. In fact, we show that when the fraction of rich residents is su¢ ciently
small, subsidizing the rich and taxing the poor raises poor households’ welfare in the steadystate. These results match the experiences of many developing countries, in whose early yearsof development, widespread subsidies are provided to business owners, such as those appliedin Special Economic Zones (SEZ) where tax holidays, export, import and land subsidies arecommon.
However, the income gap between the rich and the poor may increase under certain conditions, especially in the short run. Such scenarios may justify government policies such as taxes
to redistribute income from the rich to the poor, to keep the income gap within boundaries,
otherwise social instability may arise.
A particularly interesting …nding arises under urbanization, which causes cities to annex
neighboring rural areas and the poor residents there. Such labor migration increases the productivity of capital and the income of the rich, which under DMI induces more capital accumulationand makes all households better o¤ in the long run, although it widens the income inequality.
This explains very intuitively and matches the fact that while fast growing economies such as
China and India are becoming richer on one hand, their income inequality rises on the other
hand.
Next, either a positive productivity shock or an increase in capital market e¢ ciency always
raises the income gap, because rich households bene…t in more ways or more directly from suchshocks while poor households only bene…t through changes in the wage rate. This is consistent
with Acemoglu (2002), who studies the impacts of skilled labor-biased technology improvement
and …nds it to be a major cause for income inequality in the twenty’s century. Nevertheless,in our model, poor households tend to bene…t more under DMI than CMI (constant marginal
impatience), because the former of which creates a long-run trickle down e¤ect that is absent
under CMI.
However, DMI can lead to counter-intuitive and discouraging results: the welfare of an
economy where everyone owns capital is lower than if only some people own capital! The logic
is, the lower the share of capitalists in the economy, the more they save and the more capital thecountry accumulates in the steady state. Hence, a country with higher inequality accumulates
higher capital stock and enjoys higher levels of welfare, ceteris paribus. On the contrary, if thesame amount of capital stock is owned by more capitalists, each capitalist saves less and thesteady-state welfare is lowered.
Moreover, even if both types of households own asset to begin with, the economy will
not converge to the steady state where all households have some positive level of asset (i.e.,everyone becoming a capitalist), as long as their initial asset holdings di¤er. Thus, asymmetryof households may be a natural consequence of endogenous time preferences with DMI.5
Finally on an international scale, patience may impact rich and poor countries di¤erently
in terms of income inequality. Developing countries use inferior technology and their capital
markets are less e¢ cient and less complete, leading to lower income. An increase in patience
may raise the income inequality because rich households tend to gain more asset income (higher
interest rates) when capital markets are less complete. Examples are again China and India,
where savings and investment are high, but the income inequality is increasing rapidly.6 In
5

In contrast, under IMI the economy converges to the steady state where all households have the same levelof asset, even when their initial asset holdings di¤er. See also Epstein (1987).
6
According to Li, Wei and Jing (2005), the Gini coe¢ cient of wealth inequality in China increased from 0.40
in 1995 to 0.55 in 2002. For India, Bardhan (2006) states that "in the 90’s Indian wealth distribution was much3

contrast in developed countries, the income inequality may fall when households become morepatient, for which case Luxumburg and Switzerland are perhaps good examples.7
In the theoretical literature, Krusell and Smith (1998) demonstrate that introducing time
preference heterogeneity can signi…cantly improve the Aiyagari (1994) model in explaining income inequality, and Hendricks (2007) incorporates preference heterogeneity into the life-cycle
model of Huggett (1996) to account for wealth inequality. Also, Epstein (1987), Das (2003),
Hirose and Ikeda (2008) and Chang (2009) investigate equilibrium stability and uniqueness issues under DMI. In Hirose and Ikeda (2012), the Harberger-Laursen-Metzler e¤ect is examined,
where at least one country has IMI in order to obtain saddle-point stability.
In contrast, the present paper focuses on the e¤ects of consumer preference and patience andtotal factor productivity. The interactions of two types of households generate interesting resultsthat are novel in the literature. Our conclusions imply that to attain sustainable economic
growth under DMI, developing countries on one hand can allow freer labor migration, improvethe e¢ ciency of capital markets, and even educate households to be more patient; on theother hand, they may levy income taxes and adopt lump-sum transfers to reduce inequality
across households. However, caution is needed since income transfers may hamper capital
accumulation, and eventually lower the steady-state level of welfare for all households.

2

The Basic Model

We consider an economy with two types of households, which are symmetric in all aspects
except that one type owns asset, while the other type is unable to own asset for some reason.8
That is, there exists intrinsic social inequality in this economy to begin with, as in many
developing countries with strong traditional systems. Our goal is to examine the relationshipbetween inequality and economic growth and how social inequality evolves in transitional andfast-growing economies such as the so-called BRIC countries (Brazil, Russia, India and China).
In the process of catching up, such economies inevitably face technology improvement, capital
market ine¢ ciency, labor migration, demand and other shocks. We analyze these issues andgovernment policies including tax reform that might be used to mitigate the existing andpossibly widening inequality.
There is only one good that can be consumed and saved as capital, whose output is givenby
Y = F (K; L);
where K and L are respectively capital stock and labor supply. Production exhibits constantreturns to scale technology, so that we havekKand f (k)
L

F

K
;1 :
L

more unequal than that in China."(p16) In addition, there exists more severe educational inequality in India.
7
A note of caution is in order though. The above results are obtained in the absence of any international
credit market. If however rich households can invest in foreign markets or assets, then the result that the welfareof poor households increases when the rich becomes richer should be weakened.
8
In Appendix 2, this is relaxed and we show that (i) even if all households are allowed to own assets withdi¤erent initial asset holdings, under DMI the economy will not converge to the steady state where everybody
is a capitalist; (ii) the steady state we shall examine in the main text is the same as the one when the borrowing
constraint is binding only for one type of households.

4

Then, the capital rental rate R and the wage rate w are respectively
R = f 0 (k) and w = f (k)kf 0 (k):The household’s inelastic labor endowment (and supply) is normalized to one, so that the
number of households is denoted by L; and k is the capital stock per household or simply
capital stock.

2.1

The Capital Market

Poor households consume all their current income so they do not participate in the capital
market. In contrast, rich households save a portion of their income as assets. Historically, amajor problem facing any economy during and post takeo¤ is how to accumulate capital tosustain economic development, because …nancial markets have not been developed and there
are limits for entrepreneurs to borrow from relatives and friends. Thus, we assume there
exists ine¢ ciency in the capital market that lowers the return on investments, perhaps due totransaction costs of some sort,r = e[R];where r is the interest rate, e 2 (0; 1] can be interpreted as the e¢ ciency of the capital market,
and is the depreciation rate on capital. Then the income for rich households (with asset a) is
given by w + ra; while that for poor households (without asset) is only w:
The government may levy a tax on the capital income of rich households (simply named‘the capital tax’), intended to reduce the income gap. The tax revenue is used as lump-sumtransfer, T; to poor households. Let 2 (0; 1] denote the exogenous share of households withasset.9 The post-tax income for rich households is thenI = w + (1)ra;while that for poor households becomes
I = w + T;
for which the government budget constraint ra = T (1 ) holds. Also, it is natural to assume
the income of rich households to be higher than that of poor households, I
I ; for which1
is required.
Each household owning asset maximizes the discounted sum of utility
Z 1u(c)Xdt;(1a)0

subject toa_ = w + (1
)ra
X_ =
(c)X;

c;

(1b)
(1c)

Rt
where for 8 c > 0; u(c) > 0; u0 (c) > 0; u00 (c) < 0: Also, X
exp[ 0 (c)ds] is the discountfactor at time t which depends on the past and present levels of consumption through thefunction ( ):9In the case of= 1, all households own the same level of asset.

5Following Das (2003) and Chang (2009), we assume that household preference exhibits DMI(diminishing marginal impatience) as follows,100(c) < 0 <00(c) for 8 c > 0; (0) < 1; and lim (c) = 0:
c!1

(2)

Intuitively, it says households with a higher income discount future less, since they can a¤ord
to defer consumption of additional income and wealth. As mentioned in the Introduction,
this assumption is supported by several empirical studies, such as Lawrance (1991), Samwick(1997), and Barsky et al. (1997), etc. We are interested in how the degree of impatience a¤ects
the evolvement of inequalities in the process of catching up and modernization, especially fordeveloping countries.
Further, expression (1c) implies the rate at which X decreases is (c): To be more precise,with an increase in consumption c, expression X decreases at a normal speed under CMI, but
at a higher speed under IMI and a lower speed under DMI.
The Hamiltonian associated with our optimization problem isHu(c)X + [w + (1

)ra

c]

(c)X;

and the necessary conditions for optimality are@H0= u0 (c) X(c)X = 0;
@c
@H
= (1
)r = _ ;@a@H= u(c)(c) = _ :
@X
Let Z

(3a)
(3b)
(3c)

=X to simplify notation: Then (3a) and (3b) can be rewritten as
Z = u0 (c)Z_ = Z[ (c)

0

(c);

(1

)r]:

Using the above, our dynamic general equilibrium system can be described asa_ = w(k) + (1Z_ = Z[ (c) (1_ = (c)
0 = u0 (c))r(k)a

u(c);
0
(c)

(4a)

c;

(4b)
(4c)
(4d)

)r(k)];
Z;

where w(k) f (k) kf 0 (k), r(k) e[f 0 (k) ]; and k = a from the market clearing condition
for asset, a L = K:11 Notice that w0 (k) = kf 00 (k) > 0; and r0 (k) = ef 00 (k) < 0; that is,10
In the case of IMI, we follow Uzawa’s assumption, i.e. (c)(u(c)) with (u) > 0; 0 (u) > 0; 00 (u) > 0;0(u)(u)u > 0 for all u > 0; and (u(0)) > 0: See also Ikeda and Hirose (2008) for assumptions on u andthat can be applied to both DMI and IMI.
11
Each household unable to own asset consumes all of his income at each point in time, and hence his
consumption is c = w(k) + r(k)a=(1): Then, the goods market clears when:(c + a)_ L + c (1)L + K = wL + RK

[(1

e)(R

)K];

where the last term on the right-hand side denotes the loss of output due to the ine¢ ciency in the capital
market.

6

when the capital stock rises, the wage rate increases but the interest rate decreases.12 Theinteractions of these two opposite e¤ects determine the changes of household income, which
will become clear soon.

3

The Steady StateWe de…ne the steady state of the model as when all variables for households with asset, i.e.,a (or k), Z; ; and c; are constant, and the consumption of households without asset is also
constant at c = i (k; ; ); wherei (k; ; )w(k) +r(k)k:1

Then, the steady state is a solution to the following system of equations13
1

c;

(5a)

0 = Z[ (c) (1)r(k)];
0 = (c) u(c);0Z = u0 (c)
(c):

(5b)
(5c)
(5d)

0 = w(k) +r(k)kThese conditions say that at the steady state, consumption must be equal to the post-tax
income (condition (5a)), the post-tax interest rate must be equal to the discount factor ofhouseholds with asset (5b), utility is constant ( _ = 0; (5c)), and (5d) equates the current valueof the shadow price to the marginal-utility increase. In what follows, we use “ ”to denote thesteady state value of each variable.
~ c~); whichMore speci…cally, conditions (5a) and (5b) give the steady state solution pair (k;can be rewritten as c = i(k; ; ) and k = (c; ); wherei(k; ; )(c; )w(k) +

1r(k)k;

(c)

r 1

:

1

When is small the income of rich households largely consists of asset income, and hence
it declines when the interest rate falls due to capital accumulation. However, when e is small
but is large, it depends also on the wage income and increases with capital accumulation.
Indeed, we have
Lemma 1 If < e0(1)e; capital accumulation can reduce the income of households
holding asset, even when the interest rate is positive:@i(k; ; )(=@ke0 )w0 (k)

12

+

(1

)r(k)

:

(6)

In standard models with = e = 1 and = 0; these two e¤ects are exactly cancelled out, and hencehouseholds’ income, given by f (k)k; always increases with capital accumulation, for any positive interest
rate.
13
Our model does not have a “satiated”steady state (Z being equal to zero) as discussed in Hirose and Ikeda(2008), since our assumption on u and ensures that the steady state value of Z must be positive.

7

The …rst term on the right-hand side of (6) can be negative, because the wage rate increases
but the interest rate decreases and the latter dominates if < e0 : Then, capital accumulation
may reduce the income of rich households, if the fraction of such households is too small.
Indeed, with < e0 ( > e0 ) the income of rich households is maximized at some level ofcapital stock that is lower (higher) than the level at which the capital rental is equal to thedepreciation rate. In the case of = e0 = 1; it is the same as the level given by the Golden
rule because it maximizes households’income. In contrast, capital accumulation increases theincome of poor households, i , given 2 [0; 1]:The capital stock that equates the interest rate to the discount factor, ; is increasing in cdue to DMI:0(c)
@ (c; )
=
> 0:
@c
(1
)r0 (k)Once the steady state c~ is determined, we see from (5c) and (5d) that
~=

u(~
c)
u0 (~
c) (~
c) u (~c) 0 (~
c)
and Z~ =
;
(~
c)
(~
c)

where ~ (increasing in c~) can be interpreted
as the steady state level of welfare in the sense
R1
that the discounted sum of utility, 0 u(c)Xdt; is equal to ~ when c(t) = c~ for 8 t 0:
Finally we examine the stability and uniqueness of the steady state. Let k1 and k2 be thevalues of the capital stock that equate the post-tax interest rate to (0) and zero respectively:k1

r 1

where k1 < k2 holds.14 Then, for any(0)

and k21

r 1 (0);(7)2 (0; 1];i(k1 ; ; ) = w(k1 ) +

(0)k1

> 0;

i(k2 ; ; ) = w(k2 ) < 1:
It is then apparent from @ =@c > 0; k1 = (0; ); and k2 = limc!1 (c; ) that there necessarily
exists an intersection of the two graphs, c = i(k; ; ) and k = (c; ); as in Figure 1.
In the rest of the paper, we assume15
Assumption 1: The intersection between c = i(k; ; ) and k = (c; ) is unique for any pair( ; ) with 2 (0; 1] and1:
Remark 1 This is true when i and are strictly concave respectively in k and c: An example
is provided in Section 4 (see Assumptions 2 and 3 below).
Under Assumption 1, at the intersection, the slope of c = i(k; ; ); i.e., @i=@k; must be
smaller than that of c = 1 ((1)r(k)); i.e., (@ =@c) 1 ; and hence the following inequality
14

The Inada conditions: limk!0 f 0 (k) = 1 and limk!1 f 0 (k) = 0 ensure the existence of k1 and k2 :If Assumption 1 is violated, there may be an odd number of steady states. (8) holds at the …rst steadystate, but fails at the second one, ..., and it also holds at the last one. One can verify from the proof of Lemma
2 (Appendix 9.1) that the steady states with (8) are saddle points, while the others are either unstable orindeterminacy arises around them. Assuming the uniqueness of the steady state, we will focus on a saddle
point.
15

8

holds:16(e0 )w0 (k) + (c) 0(c) > e0 f 00 (k):(8)

Combining this with the stability analysis in Appendix 9.1, we obtain:17
Lemma 2 Under Assumption 1, the steady state of the dynamic system is a saddle point andunique.
The above completes the basic setup of the model.

4Technology, Credit Constraints and Demand Shocks

It is not uncommon for transition economies to experience technology and demand shocks aswell as credit constraints, especially facing the current wave of globalization. In this section, we
examine how technology advancement, capital market ine¢ ciency, and demand shifts impact
the steady state, in the presence of DMI. Precisely due to the impact of DMI, any change thatmakes households with asset richer also has a positive e¤ect on the steady state income ofhouseholds without asset, as we shall clearly demonstrate below.
Rede…nef (k) = Af^(k) for 8 k and (c) = B^(c) for 8 c;
where A > 0 and B > 0 can be interpreted as respectively the total productivity and the levelof impatience.
We now consider the e¤ects of changes in A; B; and e on the steady state level of welfare forboth types of households,18 where changes in B can represent impatience changes that cause
a demand shift of all households. Totally di¤erentiating c = i(k; ; ) and k = (c; ) withrespect to A; B; e0 ; and to give
e0

=

~ 00
kf
e0 f 00

dk
dc

1
0

f^ + e0 k~f^0 0
e0 f^0
^

~
k
e0
e0

!
~

k
2

0

0

1
dA
B dB CB
C@ de0 A ;dwhere each element of the matrices is evaluated at the steady state. Then, we obtain:
16

Condition (8) holds at any intersection and hence the steady state is unique, if the following is met:1
min [ f 00 (k)](0) k2[k1 ;k2 ]0(0)
max 1 + (e0e0 k2[k1 ;k2 ])kf 00 (k):(0)

This inequality degenerates to the bounded slope assumption in Chang (2009), when e0 = = 1: Then thesteady state of his model (of endogenous time preference with DMI) is unique and is a saddle point.
17
Notice that with IMI or CMI, the intersection also exists, and under Assumption 1, (8) holds at the uniqueintersection (Assumption 1 is redundant in the case of CMI). In both cases, one can easily verify that Lemma
2 remains valid.
18
In the rest of the paper, we de…ne the steady state level of welfare for households without asset as u(~c )= (~c ),analogous to that for households with asset.

9Lemma 3 Under Assumption 1,0~ 0
f + [e0(e0) 0 k]f
@ k~
=
> 0;
@A
AD
@ k~
=
< 0;
@B
BD
0~
k)
(
@ k~
=
> 0;
@e0
De0
0~
@ k~
k
< 0;
=
@
D 2
@~
c
e0 ( f 0
f f 00 )
=
> 0;
@A
AD
~ 00 + ]
@~
c
[(e0
)kf
=
;
@B
BD
~ 00 )
@~
c
(
kf
> 0;
=
@e0
De0
~ 00
e0 kf
@~
c
< 0;
=
@
D 2

where
D

e0

~ 00 +
kf

0e0 f 00 > 0 and e0

(e0

(9a)
(9b)
(9c)
(9d)
(9e)
(9f)
(9g)
(9h)

) 0 k~ > 0

hold from (8).
From Lemma 3, one sees that the steady state capital stock increases when either technology
A or the e¢ ciency e of the capital market improves (or the tax rate falls), which must in turn~ ; ) and @i =@k > 0:
raise the steady state level of welfare for all households, since c~ = i (k;Observe a slight di¤erence here. Both labor and asset income change at once when a productivity shock arises, while only the latter changes immediately after shocks in the capital market
occur, because labor income gradually changes following the adjustment in the capital stock.
In contrast, when households become more patient for all consumption levels (i.e., a decreasein B), the steady state capital stock increases, but the steady state consumption for richhouseholds may decrease (see (9f)). To see this more precisely, from (6) and (7), we have(e0@i(k; ; )
=
@k
k=k2

)k2 f 00 (k2 )That is, under < e0 and Assumption 1, the steady state income for rich households decreases~in k~ when k~ is su¢ ciently close to k2 ; implying @~c=@B > 0 > @~c =@B since @ k=@B
< 0: Hencea decrease in impatience B lowers their consumption, while it raises that of poor households.
Intuitively, when rich households become more patient, they save more, increasing thecapital stock in the economy. If < e0 ; by Lemma 1, the income of rich households falls as
the capital stock rises when the interest rate is su¢ ciently small. Thus an increase in patiencetends to lower their consumption. In contrast, poor households’consumption increases due tothe rise in the wage rate.

10

4.1

Income Gaps

We have shown that when positive shocks occur, all households become better o¤ at the steadystate. A related issue is whether their income gap widens or not, which we investigate now.
The incomes of households with and without assets at the steady state are respectivelyc)k~~ + (~
I~ = w(k)~ +and I~ = w(k)

(~
c)k~
)(1

(1

)

:

We can de…ne the income gap in terms of both level and ratio di¤erences, respectively as:
1
g I~ I~ =
(~
c)k~
(1
)(1
)
or g^

I~
=
I~

~

(~
c)k
+ w(
~
k)

+ (1

)(1

)

~
(~
c)k
~
w(k)

:

Then the Gini coe¢ cient can be calculated as,
I~ (1)2 =2 + I~ (1) + I~ 2 =2
G=1
[I~ (1) + I~ ]=2=g^g^ + 1;

which gives:
Lemma 4 The Gini coe¢ cient G is increasing in the ratio income-gap g^:
Further, to obtain clear results, we assume
Assumption 2: The production technology takes the Cobb-Douglas form: f (k) = Ak ;(0; 1):

2

Next, using Lemma 3, we have
Lemma 5 Under Assumptions 1 and 2,i@ h( + e0 )
(1
) 0~
~
(~
c)k =
1+
k ;
@A
AD
i
2
@ h
(1
)e0
~
(~
c)k =
1
;
@B
BD
i
2
@ h
(1
)( + e0 ) 0 ~(~c)k~ =
1+
k ;
@e0
e0 D
e0
"
#
@
(~
c)k~
e0 [e0 + (1)( + e0 )] 0=
< 0;~
@A w(k)AD (1)( + e0 )
"
#
@
(~
c)k~
e0
=
> 0;
~
@B w(k)
BDf
"
#
"
2
0~
@
(~
c)k~
k
e0 + (1)=
1++~~ + e0 )@e0 w(k)
(1
)e0
Dk(
11

(10a)
(10b)
(10c)
(10d)
(10e)
0~

#

k ;

(10f)

~
where g^ (and hence G) is increasing in k=w
when

<1

:

The above conditions give rise to two contrasting results. (i). Under CMI and IMI, we get
GA

0

and
(11)

gA ; ge ; Ge > 0;

where to simplify notation, we have used gj ; g^j and Gj to denote @g=@j; @^g =@j and @G=@jrespectively, for j = A; B; e; and : These expressions imply that positive shocks in productivity A or e¢ ciency e in the capital market increases the level income-gap, and the latter also
raises the Gini coe¢ cient since it favorably a¤ects the income of rich households.
(ii). In contrast under DMI, poor households bene…t more from positive shocks becausesuch shocks make rich households more patient and hence accumulate more capital. It follows
that the Gini coe¢ cient is reduced by a rise in productivity increase: GA < 0:In addition, (11) is true for weak DMI (in the sense that Assumptions 2 and Assumption 3
below hold).19
Assumption 3: (c) =B(c + 1)

;

):20

2 (0; 1

Assumption 3 implies that as capital accumulates, the rental rate falls faster than the speed
at which rich households become patient (as their consumption levels rise), i.e.,dd(c + 1)
dR
<
c+1
R

dk
:
k

To sum up, the e¤ects of A and e on the Gini coe¢ cient can be restated as:
Proposition 1 Given 2 [0; 1) and Assumptions 1 and 2, (i). An increase in productivity
A reduces the Gini coe¢ cient G under DMI; (ii). An increase in capital-market e¢ ciency eraises G under CMI and IMI, and also under DMI given Assumption 3.
Proof. See Appendix 9.2.
Acemoglu (2002) argues that the widening of income gap is a result of technology improvement over the past century. Our results on the increase of A partly recon…rm his prediction:
It is only true for the gap in terms of income level (gA > 0), and with regards to the Ginicoe¢ cient G; only true under CMI and IMI, but not under DMI as shown in Proposition 1(GA < 0). Also, Proposition 1 says that an increase in capital market e¢ ciency widens theincome gap. Thus, under some mild DMI such as < 1; the bene…t of poor households
from the positive shocks is small, and the level income-gap is magni…ed by the shocks, as under
CMI or IMI. And during a …nancial crisis such as the current one, the income gap can actuallyfall, because when e falls the rich takes fewer risks and invests less.
Next, we turn to the impact of rich households’degree of impatience B:
19

See Appendix 9.2.
One can verify that Assumption 3 is consistent with (2), and that the assumptions on technology andpreference yield strictly concave functions i and in k and c; respectively; The steady state must be uniqueand a saddle point.
20

12

Proposition 2 Let 2 [0; 1) and Assumptions 1 and 2 hold. A decrease in impatience
B always reduces the Gini coe¢ cient, while it increases the level income-gap i¤ (~c) > (1)e0 = :
Proof. Straightforward from Lemma 5.
Proposition 2 gives interesting implications. Developing countries tend to have inferior
production technology (low A) and less e¢ cient capital market (low e), thus their steady-state
consumption c~ is low. A decrease in impatience B may widen the level income-gap in thesecountries. This arises because the interest rate tends to be high when the capital market is~ which in turn leads
incomplete, resulting in a low capital stock (recalling (~c) = (1)r(k)),
to a low wage income and a high interest income, widening the income gap between the haves
and the have-nots.21 For instance, in China and India, both savings and investment are atunbelievably high levels, and the income gap is also rapidly increasing. In contrast, in somedeveloped countries (usually with high A and e; and hence high c~), a decrease in impatience B
may reduce the level income-gap; Examples for this case are probably Luxemburg, Switzerland,etc., with low saving rates and low income gaps too.

5

Capital Tax and Poor Households

Since the income gap between rich and poor households increases when the economy grows(such as caused by a rise in A or e), the government may use taxes to reduce the inequality.
We now examine the e¤ects of on the level income gap, the Gini coe¢ cient and the steadystate welfare.

5.1

Capital Tax

Partial di¤erentiation gives the e¤ect of the capital tax
g =
g^ =

g
1
g^
1

1
(
I~

on the income gap as:
+

w
I~

+

ege
g
)

e^
ge
g^

;
;

~
where lim !0 [(I~w)= I~ ] = k=(1)w: From the above, we …nd the total e¤ects can bedivided into the sum of a direct e¤ect and an indirect e¤ect (as contained in the curled braces).
Notice that the indirect e¤ect can be negative under DMI, when a tax hike on rich households
reduces the capital stock by a large scale (i.e., under strong DMI), and hence the wage rate
falls sharply. Then it is possible for the income gap to widen when the elasticity of the gap
with respect to the capital e¢ ciency e is negative.
On the contrary, under mild DMI, the government can reduce the income gap by raising
the capital tax on rich households as follows.
Proposition 3 Let 2 [0; 1) and Assumptions 2 and 3 hold. An increase in
Gini coe¢ cient under DMI, as well as under CMI or IMI.
21

Since the di¤erence is given as [(1)= (1reduces the~ k;~ it is increasing in k~ if d[r(k)~ k]=d~ k~ = r0 (k)~ k+r(~~ > 0:)]r(k)
k)13

Proof. Straightforward from Proposition 1.
Proposition 3 to some extent justi…es the capital tax on rich households, in order to reducesocial inequality, as one might expect. However, it has a negative e¤ect on the steady state
capital stock, and hence it may reduce poor households’income including transfers. Then it isnecessary to look into the e¤ect of on poor households’steady state welfare.22
As a particularly interesting …nding, it is possible for the capital tax to lower the steadystate welfare of poor households, especially in an economy where almost all households are poor
and unable to own asset (e.g., a low ). In such a case, the government can increase the steadystate welfare for all households by setting to be negative, in e¤ect subsidizing the rich andtaxing the poor!
To be more speci…c, partially di¤erentiating c~ with respect to yields
k~
)(1

@~
c
=
@
(1

~
0 @k
+
+
w
)2
@
(1

)(1

@ k~
+
) @
(1

k~ 0
)(1

@~
c
;
) @

where the …rst term on the RHS denotes a positive direct e¤ect, and the rest is the sum of
several indirect e¤ects: the second and third terms are negative, which comes from the factthat an increase in reduces the capital stock, and hence both the wage rate and the amountof transfer fall, while the last term is positive due to a rise in the rental rate under DMI.
From Lemma 3 and @e0 =@ = e; we seen h@~c^ k~ (e0
=D
@
(
h
^
=D
(1
^
=D

h

(1

~ 00 +) 0 kf0

e0 f 000~k)

k~ 0 (e)(1) 0 k~2 f 00
i

+
e)(1

0 ~ 2 00k f

) 0 k~2 f 00 +

i

+ (1
o
~ 00 )
kf
0 ~ 2 00

k f +

)(1

~ 00 ()kf0~

k + (1

0~

k)

~ 00
)kf

e)(1

(12)
0~

k + (1^
where D
=D (1
)(1
)2 :
We are now in a position to state:

e)(1

~ 00
)kf

i

;

(13)

Proposition 4 Let Assumption 1 hold. (i). If the capital market is imperfect (e < 1) and< 1 e; then under DMI or CMI, there exists some " > 0 such that for 2 ( "; 1);
reducing the capital tax increases the steady state welfare of all households; (ii). If the capital
market is perfect (e = 1), a positive capital tax on rich households raises the welfare of the poor
under CMI and IMI, but may not do so under DMI: for a su¢ ciently small ; subsidizing therich and taxing the poor may raise all households’welfare in the steady state under DMI.
Proof. See Appendix 9.3.
In fact, under Assumptions 2 and 3 and e = 1; we …nd that for 2 ( ; 1); reducing thecapital tax or raising the capital subsidy increases the steady state welfare of poor households
22

Hamada (1967) considers the e¤ect of transfers between capitalists and workers on the latter’s income on
the equilibrium growth path with constant saving ratios, and he …nds the optimal tax rate is zero, = 0:14

in an economy with some ; and the welfare is maximized at some 2 ( ; 0);23 rather thanby an income transfer to them!
The results under DMI contrast sharply with those under CMI and IMI. Poor households’
steady state income is more likely to decrease under DMI than under CMI or IMI, when thegovernment raises the tax on rich households (the …rst and second terms in (13)). This canbe divided into two cases. First, if < 1 e; under DMI or CMI, the positive direct e¤ect isdominated by the indirect e¤ect which comes from the decrease in w(k) (see (12)), and from(13), @~
c =@ < 0 for
0; implying that subsidizing the rich and taxing the poor will raisepoor households’welfare in the steady state.
The above results remind us of the Chinese experience in the past 35 years. Until the early
1980s, most Chinese were very poor and owned almost zero assets. The government opened
Special Economic Zones to allow business owners (a tiny fraction of the population then) to dobusiness tax free, with land, import and export other subsidies. It is especially worth mentioningthat even North Korea opened such Special Economic Zones along the borders with China andSouth Korea.
Second, if > 1 e; which necessarily holds when e = 1; a positive capital tax on richhouseholds is preferable for poor households under CMI and IMI by increasing their present
and future incomes. But under DMI, it is possible that @~c =@ < 0 for
0 even with e = 1:
On the other hand, in a more mature economy with a higher (a higher fraction of population owning assets), then under Assumptions 2 and 3,24@~c
> 0 for any feasible@0;that is, some positive capital tax will maximize poor households’ steady state welfare. This
arises because (13) yields(!
)
0 ~ 2 00
@~
c
e
1
k
f
0
0
~ 00 +~ 00 ;
^
=D
kf
+ (1
)(1
)kf
e0 f 00 k~@
where, if is su¢ ciently close to 1, (i). the term in [:::] is positive from D > 0; (ii) > 0 k~2 f 00
holds under Assumptions 2 and 3, and hence the second term is also non-negative given
0;(iii) the …rst term in braces f:::g dominates the last term when goes to 1 (see footnote 24).
Intuitively, the above arises because when is su¢ ciently high, subsidizing the rich causes
an even higher tax burden on the poor. Thus, in present-day China, when the fraction of
the rich has reached a certain high level, taxing the rich may be a good policy to reduce theever-increasing income gap.

5.2

Unequal but Preferred

In this subsection we show that an economy with a small may be preferable for poor households, and such an economy comes with a su¢ ciently low Gini coe¢ cient.
As mentioned earlier, the decrease in raises the capital stock and the wage rate in thesteady state, which implies that some level of inequality may be preferable under DMI. Indeed,
we can show that an economy with an uneven distribution of income among households is
23

See the proof of Proposition 4 in Appendix 9.3.
~ ; ) = w(k)~ + r(k)~ k=(1~
Here, “feasible” means that i (k;right-hand side of which goes to zero when goes to 1.
24

15

)

0; i.e.,

(1

~
~ k;
~ the
)w(k)=r(k)

preferable for all households to an economy with each household having an identical level ofasset.
Suppose that all households own the same level of asset. Then, households’income is givenbyI(k)w(k) + r(k)k= f (k) (1 e)kf 0 (k)e k;which is increasing in k with k < k2 :25
I 0 (k) = e[f 0 (k)](1e)kf 00 (k):~Let k(1)and c~(1) be the steady state levels of capital and consumption in the economy with~
= 1: Then, k(1)< k2 holds, and hence the income (and consumption) in the steady statemust be smaller than I(k2 ) = w(k2 ) :~c~(1) = I(k(1))< w(k2 ):On the other hand, if there are two types of households and only few households own asset( is su¢ ciently small), capital accumulates at almost the same level as k2 ; because the verywealthy households have an extremely low discount rate due to DMI and they will accumulate
capital until the interest rate reaches almost zero. It can be veri…ed as follows. For k < k2 ;@i(k; ; )
=
@
and

(1

)r(k)k2< 0 and lim i(k; ; ) = 1;!0

does not depend on : Therefore,
lim k~ = k2 and lim c~ = 1:
!0

!0

(14)

Also, notice that
lim c~ = w(k2 );!0which implies that
lim c~ > c~(1):!0Thus we have
Proposition 5 Under Assumption 1, the steady state welfare in the economy without inequality(i.e., everyone becoming a capitalist with an identical level of asset), is lower than the poor
households’welfare in the steady state with a su¢ ciently small :The Proposition implies that under DMI, some level of inequality is preferable for the
economy, basically because rich households are more patient and save more, which lowers theinterest rate and in turn raises the wage income of poor households through production linkages.
On the contrary if the capital stock is spread over more capital owners (i.e., lowering inequality),
then each capitalist saves and invests less, resulting in lower welfare in the long run.26 The
25

Notice that k2 corresponds to the golden-rule level of capital (per household) when e = 1:
In the case of = 1; both types of households have the same level of income and the steady-state
consumption is given by the intersection between c = i(k; ; 1) and k = (c; 1); which must be smaller
than c~(1):26

16generated consequences from this Proposition are similar to those of the “Trickle Down Theory”(Aghion and Bolton, 1997), albeit via a starkly di¤erent mechanism.
Note that there is a non-monotonic relationship between and the Gini coe¢ cient G,
because G = 0 when either = 1; or = 0, i.e.,hi~ + (~
lim I~ = lim w(k)c)k~ = 0:!0!0Di¤erentiation gives,g^ + (1) g^
1( g^ + 1)2(1 2 )(^
g 1) + (1=( g^ + 1

G =

) g^
)2

[ (^
g

1)]2

;

where g^ < 0 by Proposition 8 below. Notice that G must be negative forg^ > 1; and it can be positive for a su¢ ciently small ; because G > 0 for any
lim !0 G = 0:27 Therefore we have

1=2; since
2 (0; 1) and

Proposition 6 Let 2 [0; 1); and Assumptions 2 and 3 hold. The Gini coe¢ cient fallsfor
1=2 but rises as the share of rich households increases when is su¢ ciently small.
The Proposition implies that the Gini coe¢ cient exhibits a sharp inverted-U shape, …rst
increasing then decreasing, following a rise in the percentage owning assets, . It again stems
from DMI and its trickle down e¤ect, as discussed extensively before. This relationship is similar
to the Kuznets’curve in the literature, and it may be good news for developing countries suchas China, India and Latin America, who are trying to catch up with developed countries. As
the fraction of population owning assets rises to above a certain level, inequality gradually falls,
even without any government intervention.

6

Urbanization

Rapid urbanization is a common feature of some newly emerging economies, especially China
and India. Inequality attracts large-scale migration of poor households from rural to urban
areas. In this section, we look into how such urbanization and domestic migration a¤ects the
economy in terms of inequality and welfare.
Consider an urban economy that has both rich and poor households. Urbanization enables
some rural poor people to become city residents, though still remaining poor without asset. By
Lemma 3, we have
Proposition 7 Given Assumption 1, under DMI, urbanization (i) raises the steady state welfare of all households in the urban area, if or is su¢ ciently small; (ii) lowers (raises) the
steady state welfare of poor (rich) households in the urban area, if is su¢ ciently close to 1
and is su¢ ciently close to 1
: In contrast, under CMI, it has no (a negative) e¤ect on
the steady state welfare of poor households when = 0 ( > 0), though it raises that of richhouseholds.
27

Indeed, one can verify that under Assumptions 2 and 3, lim !0 g^ =(^
g

17

1) =

(1 + ) 1 >1:

Proof. See Appendix 9.4.
To give an intuitive explanation for Proposition 7, we suppose that
economy has reached a steady state such thatKK
k~ = ; a~=; c~ = w
L
L

K
L

+

K
; and c~ = w
L= 0;28 and the urban
KL;

where = r(K=L):
As the city becomes bigger and annexes a neighboring rural region, the urban labor input L
rises to L0 (the ratio falls to 0 ), and hence the wage rate falls and the capital rental rises. The
whole economy thus accumulates capital and will eventually reach a new steady state, which
can be characterized by
0

0

K
K
; c~ = w
k~ = 0 ; a
~=
L
L

K0
L0

+

K0
; and c~ = w
L

K0
L0

;

where = r(K 0 =L0 ); and the rich households own more asset than in the old steady state.
Under CMI, K 0 =L0 = K=L must hold, and hence rich households’steady state consumption
will rise, while that of poor households does not change. As a consequence, existing poor
households are harmed by urbanization due to the short-term fall in the wage rate.
Under DMI, however, an increase in the capital labor ratio (K 0 =L0 > K=L) is accompanied
by an increase in the consumption level of rich households since they become more patient,and therefore poor households will be made better o¤ in the long run through the rich’s added
investment. In other words, DMI generates a further e¤ect above CMI, which in this case is apositive trickle-down, from the haves to the have-nots, similar to those predicted by the “Trickle
Down Theory,”even though our mechanism is di¤erent.
Next, consider the e¤ect of urbanization on the income gap among households. Sincea decrease in shifts the graph c = i(k; ; ) upward while leaving unchanged the graphk = (c; ); it increases the steady state levels of k and c along k = (c; ) (see Figure 1).
Notice that Assumptions 2 and 3 yield strictly concave functions for i in k and in c; andthat a decrease in lowers the amount of income transfer for a given k: Thus we have
Proposition 8 Let 2 [0; 1); and Assumptions 2 and 3 hold. Urbanization (lowering )widens both the level and the ratio income-gap under DMI as well as under CMI and IMI.
Notice that Propositions 7 (i) and 8 together imply that urbanization, which adds poor
people to the city, increases the steady-state welfare of all city households due to capital accumulation stimulated by labor migration, but it also widens the income gap between the richand the poor. These are consistent with observations in China and India in recent years, where
urban (and coastal) regions experience rapid growth in size, population and average income,but the income gap also rises drastically, as documented in the Introduction section.

7

Concluding Remarks

In an economy with intrinsic inequality to begin with, we have examined how endogenous time
preference a¤ects the social inequality, with special focus on DMI. Our analysis has shown that28

If is positive, a fall ink and :

implies a decrease in the amount of income transfer, T = r(k)k=(118); for …xed

(i) poor households tend to bene…t more from positive shocks under DMI than under CMI;(ii) positive shocks widen the income di¤erence between the rich and the poor when the e¤ectof DMI is small; (iii) reducing the inequality may not be preferable for all households; (iv)urbanization widens the income inequality while increasing welfare.
Also, we show in Appendix 2 that under DMI, even if both types of households own di¤erent
levels of asset to begin with, the economy will not converge to the steady state where allhouseholds have some positive levels of asset. That is, the steady state is the same as the one
when the borrowing constraint is binding only for one type of households. Thus, asymmetry ofhouseholds may be a natural consequence of endogenous time preferences under DMI.
Our result that expanding inequality (a fall in ) makes the poor households better o¤ isderived in the absence of international credit market. If international lending and borrowing
are available, this result may be altered, mainly because, a fall in implies a decrease in therich’s discount factor but an increase in the their consumption as well as the capital stock in
the whole economy, the last of which also raises the income of poor households. However, if
international credit markets are available, a fall in does not necessarily imply the last result.

8

References

Acemoglu, D., 2002, Technical change, inequality, and the labor market, Journal of Economic
Literature 40, 7–72.
Aghion, P. and P. Bolton, 1997, A theory of tricle-down growth and development, Review
of Economic Studies 64(2), 151-72.
Aiyagari, S.R., Uninsured Idiosyncratic Risk and Aggregate Saving, Quarterly Journal of
Economics 109, 1994, 659-684.
Barsky, R.B., F.T. Juster, M.S. Kimball, and M.D. Shapiro, 1997, Preference parameters
and behavioral heterogeneity: An experimental approach in the health and retirement study,
Quarterly Journal of Economics 112, 537-579.
Becker, G.S. and C.B. Mulligan, 1997, The endogenous determination of time preference,
Quarterly Journal of Economics 112, 729-758.
Chang, F.R., 2009, Optimal growth and impatience: A phase diagram analysis, International Journal of Economic Theory 5, 2, 245–255.
Das, M., 2003, Optimal growth with decreasing marginal impatience, Journal of Economic
Dynamics and Control 27, 1881-1898.
Epstein, Larry G., 1987, A simple dynamic general equilibrium model, Journal of Economic
Theory 41(1), 68-95.
Ferreira, F.H. et.al. Inequality in Latin America: Breaking with History? The World Bank,
Washington, D.C., 2004
Freeman, R.B., 1995. Are your wages set in Beijing? Journal of Economic Perspective 9(3),15-32.
Hamada, K. On the Optimal Transfer and Income Distribution in a Growing Economy,
Review of Economic Studies 34, 3, 1967, 295-299.
Hausman, J.A., Individual Discount Rates and the Purchase and Utilization of Energy-Using
Durables, Bell Journal of Economics 10, 1979, 33-54.
Hendricks, L. How Important is Discount Rate Heterogeneity for Wealth Inequality? Journalof Economic Dynamics & Control 31, 2007, 3042-3068.

19

Hirose, K.I. and S. Ikeda, 2008, On decreasing marginal impatience, Japanese Economic
Review 59, 3, 259-274.
Hirose, K.I. and S. Ikeda, 2012, Decreasing and increasing marginal impatience and theterms of trade in an interdependent world economy, Journal of Economic Dynamics and Control
36, 10, 1551-1565.
Huggett, M., Wealth Distribution in Life-Cycle Economies, Journal of Monetary Economics
38, 1996, 469-494.
Krugman, P. and Lawrence, R., 1994. Trade, jobs and wages. Scienti…c American, April
22-27.
Krusell, P. and A.A. Smith, Income and Wealth Heterogeneity in the Macroeconomy, Journal of Political Economy 106, 1998, 867-896.
Lawrance, E.C., 1991, Poverty and the rate of time preference, Evidence from panel data,
Journal of Political Economy 99, 54-77.
Samwick, A.A., 1997, Discount rate heterogeneity and social security reform, NBER Working Paper 6219.
Uzawa, H. (1968) ‘Time Preferences, the Consumption Function, and Optimal Asset Holdings,’ in J. N. Wolfe, ed., Value, Capital and Growth: Papers in Honour of Sir John Hicks,
University of Edinburgh Press: Edinburgh, 485–504.
Warner, J.T. and S. Pleeter, The Personal Discount Rate: Evidence from Military Downsizing Programs, American Economic Review 91, 2001, 33-53.
Wood, A., 1995. How trade hurt unskilled workers. Journal of Economic Perspectives 9(3),57-80.

9

Appendix 1

9.1

Proof of Lemma 2

We evaluate the elements of a Jacobian in the system (equations (4a)–(4d)) to study the local
dynamics around the steady state. Di¤erentiation gives,
J(x) = det[J26= det 64xI](e0 )w0 +
x
0
Z(1
)r
0
0

= (x; ; );where Mu00 +(x; ; )

00

0
x
0
1

0
0

1

x
0

3

Z 0 7
7
Z 5
M

> 029 and
M x3
Z[

M [(
0
(1

e0 )w0 + 2 ]x2 + fM [(e0 )w0 + ])r0 ]gx + Z[(e0 )w0 0 + 0(1)r0 ]:

Notice that (8) implies (0; ; ) > 0: This characteristic equation can be used to derive thelocal dynamics of the system in the neighborhood of the steady state.
29

In the case of IMI, we assume M > 0 to make the Hamiltonian H strictly concave in c:

20

It is clear from M > 0 and J(0) = (0; ; ) > 0 that J(x) = 0 has at least one negative
root, x1 : If (e0 )w0 + 2 = 0; thenJ(x) = M x3 + J 0 (0)x + J(0)x1 ) x2 + x1 x

= M (x

J(0)
:
M x1

Thus the other two roots, x2 and x3 ; satisfy x2 + x3 = x1 > 0 and x2 x3 = J(0)=M x1 > 0,
which implies that they have positive real parts.
Suppose that (e0 )w0 + 2 6= 0: Then, applying Routh’s (1905) theorem, the number ofthe roots of J(x) = 0 with positive real parts equals the number of changes in signs in thefollowing sequence:
M;

M [(

e0 )w0 + 2 ];(((e0 )w0 + 2 ; ; );e0 )w0 + 2(0; ; ):

(i). (
e0 )w0 + 2 > 0: Then, the number of changes in signs is two irrespective of the signof the third term.
(ii). (
e0 )w0 + 2 < 0: Then, (e0 )w0 + < 0 ande0 )w0 + 2 ; ; )(e0 )w0 + 2
M [(e0 )w0 + ][(((=e0 )w0 + 2 ] + Z (1(e0 )w0 + 2)r0 [(

e0 )w0 + ]

Z 2 0

< 0;
which implies that the number of changes is two.

9.2

Proof of Proposition 1

When 0 0; we see from (10a), (10c), and (10f) that gA ; ge ; and g^e are all positive. We willshow that this is also true under DMI with Assumption 3.
The sign of gA
We show that under Assumptions 2 and 3,
0

(~
c)k~
>
(~
c)

(1

(15)

)

holds for any pair of parameters, and hence gA must be positive due to (10a) and
Under Assumption 3, we have
0
(~
c)k~
k~
=
(~
c)
c~ + 1and from Lemma 3 and Assumption 2,!
@k~
k~ 1 f[ e0 + (1
) ] (~c)k~ + e0 g
=
< 0:
@A
c~ + 1D (~c + 1)2
Next we show that
lim

A!1

k~
c~ + 1

!
21

=

(1

)

:

<1

:
(16)

(17)

(18)

One can easily verify that limA!1 k~ = limA!1 c~ = 1: This implies~ = ;lim Af^0 (k)A!1because limc!1 (c) = 0 and (c) = e0 [Af^0 (k)] holds at any steady state. Then (18) holds
since
k~
1
=
:
~
~
c~ + 1(1)Af^0 (k)=+ e0 [Af^0 (k)]= + 1=k~From (16)–(18), we may conclude that (15) holds for any pair of parameters.
The signs of ge and Ge
Under Assumptions 2 and 3, (10f) becomes
#
"
"
2
k~
@
(~
c)k~
k~
=
1
~
~ + e0 )@e0 w(k)c~ + 1 (1) c~ + 1
Dk(
"
2
k~
+1
=
(1
)Ak~ +~ + e0 )(~Dk(
c + 1)(
2=
1 + k~ (1)Ak~ 1~ + e0 )(~Dk(
c + 1)
2=~ + e0 )(~Dk(
c + 1)

1+

1

k~
e0 (~
c + 1)
k~

#
k~
)

(1
+

+

e0

k~
e0
k~

1

1

)

c~ ;

1

which is positive due to 1
> : Therefore g^e and Ge must be positive underthat g^e > 0 and g^ > 1 together imply ge > 0:

9.3

#

<1

: NoteProof of Proposition 4

Let e < 1 and<1e: Then, for

2 [0; 1

);

@~
c
<0
@
clearly holds under 0 0: This proves Proposition 4 (i).
Next, suppose that e = 1: Then,
"
0~
@~
c
k
(1
=
2
2
@
D(1
)(1
)
Thus for

~ 00
)kf

#

:

(19)

0;
@~
c
>0
@

holds under 0 0:
We now show that if is su¢ ciently small, then the …rst term in square brackets of (19)dominates the second one, and hence we have@~c
< 0 for@22

2 [0; 1)(20)

under 0 < 0:First, we see from (14) that~ 00 (k)~ = k2 f 00 (k2 ) and lim (~lim kfc) = 0:!0!0Using L’Hôpital’s rule, we have
lim

(~
c)

!0

= lim 0
!0

= lim

!0

@~
c
@
~ 00
(1
)kf
2
~ 00 +
)kf
0 (1

(1

)f 00

(21)

:

~ + (~~ together imply
However, (14) and the fact that c~ = w(k)
c)k=
lim

(~
c)

!0

= 1:

(22)

From (21) and (22), we have
lim
If

(~
c) 0 (~
c)
2

!0

=

1:

(23)

is su¢ ciently small, then (20) holds, which proves the last part of Proposition 4.
Finally, under Assumptions 2 and 3, we have
"
#
0
~
~
(~
c)k~
k(1
)r(k)
lim
= lim
!0
!0
(~
c + 1)
= lim

!0

~
k(1
~ + k(1
~
w(k)

~
)r(k)
~ +
)r(k)
(24)

= :
Since (19) yields
@~
c
=
@

)(1

)2

we see from (22) and (24) that for any

>

D(1

"

(1

0~

k

~ 00 +)kf; there exists some!#;such that

@~
c
< 0:
@
Also, if
; then @~c =@ > 0: the value of that maximizes the steady state levels of
welfare for poor households must be greater than: This is because
"
#
~
0~
0
( 00 @@~c k~ + 0 @@k )
k
@
(~
c)k~
=
2
@
00
~ 00
( 0 )2 kf
e0e0 k~ (e0)k~ + 0
3
0
2
( )
D
~ 00
( 0 )2 kf=[( + 1)e0(e0)]e0 ( Ak~
D 3~ 00
e0 ( 0 )2 kf=[(1)Ak~ + k~ + 1]
D 2
> 0:

=

23

~
k)

e0 fAk~ [(1) + e0 ]e0 k~ + g

9.4

Proof of Proposition 7

It is apparent from (9h) in Lemma 3 that irrespective of the sign of 0 ; labor migration ofpoor households without asset from rural to urban areas raises the steady state welfare of richhouseholds.
For that of poor households, we have
0~
k
@ k~
@~
c
k~
+
+
)(1
) @
(1
)(1
) @
(1
)2 (1
)
!
0~
k ~ 00k~(1) 0~ 00 0 k~ +
kf
+
+1
e kf
1
(1
)2 (1
)
D 2
(
1
k~2 f 00 0 [(1)2 + (1)(1 e)]
+ k~
2
2
) (1
)D

@ I~
= w0 +
@
(1
=
=

1
D 2
(1

0
2

e0 f 00);

which is negative for su¢ ciently small and/or due to (23) under DMI.
Let = 1: Then,hi
0~ 00 +
D = (e 1)kf
ef 00 ;

and

@ I~
=
@
(1

nh
k~
(e
) 2D

~ 00 +
1)kf

@ I~
=
@
(1

k~
)2 (1

i

0

which is positive when is su¢ ciently close to 1 since D > 0:~
Under CMI, we have @ k=@= 0 and 0 = 0; and therefore)

2o
ef 00 ;;

the sign of which corresponds to that of :10

Appendix 2: Endogenous Di¤erence among Households

So far, we have examined the e¤ects of fundamental shocks, government policies and migrationof poor households, when ‘intrinsic’inequality exists in an economy under DMI. In this section,
we further show that the existence of these two types of households may be a natural outcome
based on the assumptions on preferences.
Speci…cally, we relax the model by allowing both types of households to own non-negative
asset (though they are not allowed to borrow from future wage incomes): They are symmetricin all aspects except for their initial asset holdings. We shall show that the steady state wehave examined in the previous sections is the one where the borrowing constraint is binding
only for one type of households, giving rise to an asymmetric steady state endogenously.
There does exist a symmetric steady state where both types of households own the samelevel of asset. However, there is no steady state where each type of households has a di¤erent
level of asset. And we will prove that under endogenous time preference with DMI, the symmetric one is unstable in the sense that the economy converges to it only if the two types of24households initially have an identical level of asset, i.e., there is only one type of households inthe economy.30

10.1

The Dynamic System

We add the following constraint on each household’s asset holdings to the original problem((1a)–(1c)), since both types can own asset now,a0; for 8 t > 0:

We assume away the income tax here by settingL= 0. The associated Lagrangian is

H + a:The necessary conditions for optimality are@L0= u0 (c) X(c)X = 0;@c@L= r + = _;@a@L
= u(c)(c) = _ ;@Xa 0;0; and a = 0:
Let z

(25a)
(25b)
(25c)
(25d)

=X: Then, (25b) can be rewritten as
Z_ = Z[ (c)r]z:

Our new dynamic general equilibrium model can be described asa_ = w(k) + r(k)a c;
a_ = w(k) + r(k)ac
Z_ = Z[ (c) r(k)] z;
Z_ = Z [ (c ) r(k)] z ;_ = (c) u(c);_ =(c ) u(c );0
0 = u0 (c)(c) Z;
00
0 = u (c )(c ) Z ;0 = az;
0=a z ;

(26a)
(26b)
(26c)
(26d)
(26e)
(26f)
(26g)
(26h)
(26i)
(26j)

where “ ”denotes the corresponding behavioral relations for the other type of households andk = a + a (1) from the market clearing condition for asset,
a L + a (130

)L = K:

Epstein (1987) proved that local stability fails if there are two (or more) individuals with DMI, in his central
planner problem with N > 1 agents.

25

10.2

The Steady States

The steady state is the solution for the system of equations
0 = w(k) + r(k)a c;
0 = w(k) + r(k)ac;0 = Z[ (c) r(k)] z;
0 = Z [ (c ) r(k)] z ;0 = (c) u(c);0=(c ) u(c );00
Z = u (c)(c);
0Z = u0 (c )(c );0 = az;
0=a z :

(27a)
(27b)
(27c)
(27d)
(27e)
(27f)
(27g)
(27h)
(27i)
(27j)

Suppose that a~ and a~ are positive. Then, z~ = z~ = 0 and c~ = c~ from (27c) and (27d), and~ from (27a) and (27b). Notice that under Assumption 1, (k;~ c~) ishence we have a~=a~ (= k)uniquely determined as the solution toc = i(k; 1; 0) and k = (c; 0):Thus, a symmetric steady state exists.
~ : The steady state pair (k;~ c~) is given
Next, let a~ > 0 and a~ = 0: Then, z~ = 0 and a~ = k=
as the solution toc = i(k; ; 0) and k = (c; 0):This corresponds to the steady state discussed in the previous sections.
Also, there is another asymmetric steady state with a~ = 0 and a~ > 0.
Thus, we have
Lemma 6 Under Assumption 1, there exist three steady states, one of which is symmetric andthe others are asymmetric. However, there is no steady state where each type of households has a~di¤erent level of asset. Furthermore, the symmetric equilibrium is characterized as a~=a
~ = k;
00~ + r(k)~ k;~ ~ = ~ = u(~c~ = c~ = w(k)c)= (~c); Z~ = Z~ = u (~c) ~ (~c); and z~ = z~ = 0; where~(k; c~) is determined as the solution toc = i(k; 1; 0) and k = (c; 0):~ ; a~ + r(k)~ k=~ ; c~ = w(k);~One of the two asymmetric steady state is a
~ = k=~ = 0; c~ = w(k)
0
0
0
0
~
~
~ = u(~c)= (~c); ~ = u(~c )= (~c ); Z = u (~c) ~ (~c); Z = u (~c ) ~ (~c ); z~ = 0; and~~z~ = Z [ (~c)(~c)] > 0; where (k; c~) is determined as the solution toc = i(k; ; 0) and k = (c; 0):The other asymmetric steady state can be analogously derived from the intersection ofc = i(k; 1; 0) and k = (c ; 0):

26

10.3

Stability of the Symmetric Steady StateDi¤erentiating equations (26a)–(26h) and setting z = z = 0; we obtain^J(x)= det[J^
2
6
6
6
6
6
= det 6
6
6
6
6
4
2

6
6
6
6
6
= det 6
6
6
6
6
4

xI]

3
1
0
(w0 +r0 a )(w0 +r 0 a )(1 )+r x
0
1 7
7
0
0
0
Zr
Zr (1
)
r x
Z
0 7
7
Z r0
Z r0 (1)
0r x
0 Z 0 7
7
0
0
0
0
x
Z
0 7
7
0
0
0
0
0
x 0
Z 7
7
0
0
0
1
0
0
G
0 5
0
0
0
0
1
0
0
G
3
r x
(w0 +r 0 a)(1 )
0
0
0
0
1
0
(r x)
0
0
(w +r a )(1 )+r x
0
0
0
0
0
1 7
7
1
0
0
0
Zr (1
)
r x
0
0
0
Z
0 7
7
0
Z r0 (1)
0r x
000 Z 0 7
7 : (28)
0
0
0
0
x
0
Z
0 7
7
0
0
0
0
0
x 0
Z 7
7
0
0
0
1
0
0
G
0 5
0
0
0
0
1
0
0
G

(w0 +r0 a) +r x

(w0 +r 0 a)(1

0
0

)

0
0
0

0
0
0
0

0
0
0
0
0

Let us evaluate (28) at the symmetric steady state. Then, we obtain the Jacobian J^ for the
dynamic system and the characteristic equation,
^
J(x)
= (x
where

(x)

M x2

M x

Z

0

) (x) (x; 1; 0);with M > 0: Then we have,

Proposition 9 Under Assumption 1, the symmetric steady state uniquely exists but is unstablein the sense that the economy does not converge to it if the initial asset endowments di¤er forthe two types of households.
Proof. As shown in the proof of Lemma 2, (x; 1; 0) = 0 has one negative root and tworoots with positive real parts. Clearly, (x) = 0 has two roots with positive real parts since^(0) > 0 due to DMI. And J(x)= 0 has only one root with a negative real part when it isevaluated at the symmetric steady state.
Remark 2 The Proposition implies the economy converges to a symmetric steady state only ifhouseholds’initial asset holdings are the same.
Remark 3 Assuming IMI, 0 > 0; we see that both (x; 1; 0) = 0 and (x) = 0 have one
negative root, and hence the symmetric steady state is a saddle point. Since rich households
become less impatient than poor households under IMI, the poor will eventually catch up withthe rich.
From Lemma 6 and Proposition 9, it is clear that in the economy where each household
is endowed with a di¤erent level of asset, no government policy can make all households own
positive asset in the long run, unless the government equalizes the level of households’initialasset endowments. Also, as shown in Proposition 5, an economy without income inequality
may not be desirable from a social point of view under DMI.
27

Figure 1ck   (c, )c

c  i (k , , )  w(k ) 

c 

1

r (k )kc  w(k )  r (k )kc  i (k , , )  w(k ) 

O



k1

k k2

1



r (k )kk