Intertemporal elasticity of substitution and the transitional
dynamics and steady state of wealth distribution
Masakazu Emoto
Tamotsu Nakamura
January , 2021
Discussion Paper No.2101

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANIntertemporal elasticity of substitution and the transitional dynamics andsteady state of wealth distributionbyMasakazu Emoto and Tamotsu Nakamura ∗Graduate School of Economics, Kobe University
January 2021
Abstract
Although the steady state equilibrium is represented by a single point in the capital-consumption
plane in the standard Ramsey model, it is by a straight line in a Ramsey model withheterogeneous individuals. Taking advantage of this fact, this paper applies the backward
induction method to analyze the transitional dynamics of the Ramsey model with heterogeneous
individuals, and examines the role of heterogeneity in intertemporal elasticity of substitution(IES). When no heterogeneity exists in IES across individuals, then the wealth Gini declines ascapital accumulates, while the wealth gap expands. In contrast, with heterogeneity, various
dynamics of wealth distribution can emerge, including a U-shaped relationship between income
and inequality. It is also shown that an inverted U-shaped relationship, i.e., the Kuznets curve
can be explained by Stone-Geary preferences, which allow IES to change with wealth.
JEL classification: C63; D31; D90; E21; O41
Keywords: Ramsey model; intertemporal elasticity of substitution; wealth distribution;transitional dynamics; Stone-Geary preference
Address for Correspondence: Graduate School of Economics
Kobe University
2-1 Rokkodai, Nada-ku
Kobe 657-8501
JAPANnakamura@econ.kobe-u.ac.jp+81-78-803-6843∗

The authors would like to thank Hiroaki Sasaki and participants at the Economic Dynamics

Seminar at Kyoto University. The second author gratefully acknowledges Grants-in-Aid for
Scientific Research from the Japan Society for the Promotion of Science (18K01569) for
financial assistance. The usual disclaimer applies.

1. Introduction
Individual preferences, which are in essence characterized by time preference andintertemporal substitution, play a crucial role in determining the dynamics of wealthdistribution and the distribution in the steady state. The most well-known result about thewealth distribution in modern growth theory is probably the Ramsey conjecture, which is first
proposed by Ramsey (1928), and then confirmed by Becker (1980), and Mitra and Sorger(2013), among others, that the most patient individual, i.e., the individual whose timepreference rate is the lowest, will eventually have all the wealth in the economy. The
conjecture of course refers not to the dynamics of wealth distribution but to the long-run
phenomena. 1 To analyze the dynamics precisely, we cannot avoid the aggregation issue. 2
If the utility function is homogeneous with respect to the relevant arguments, then theaggregate behavior can be considered as generated by a single representative agent. 3 Timeseparable utility functions with a constant intertemporal elasticity of substitution (IES) that are
most widely used in contemporary growth theory have this homogeneity property. Since theaggregate behavior of the economy is independent of the individuals’ behavior, the analysis
becomes tractable. Taking advantage of this fact, Caselli and Ventura (2000) and Garcı´aPeñalosa and Turnovsky (2010), among others, analyze distributional issues to obtain theinsightful results.
To apply the above strategy, IES must be the same across individuals. However, a large
empirical literature, which includes Attanasio et al. (2002), Blundell et al. (2014) and
Browning et al. (1995), points out heterogeneity in IES. Nevertheless, probably due to theanalytical difficulty, enough attention may not have been paid to heterogeneity in IES in thedetermination of wealth distribution. Using a simple Ramsey model, this paper examines therole of heterogeneity on the wealth distribution not only in the steady state but also in thetransitional dynamics.
1

Heterogeneity in the time preference across individuals is important also in determining

transitional dynamics of wealth. See, for example, Fisher (2017).
2

Of course, the aggregation is crucial not only for the dynamics but also for the static analysis

when we build a macroeconomic model based on rigorous microfoundations.
3This important finding dates back to the seminal work by Gorman (1953).
1

The long-run steady state equilibrium is shown by a single point in the (k , c) plane in
the standard Ramsey model with homogeneous individuals, where k stands for per-capita
capital, while c for per-capita consumption. In contrast, as Nakamura (2020) shows, it isrepresented by a straight line in the (k , c) plane in the Ramsey model where individuals have acommon rate of time preference but different IES. Although the location and slope of the line
do not depend on the initial condition and heterogeneity in IES, the length depends crucially on
them. Putting it differently, although the initial condition and individuals’ IES are required todetermine the line precisely, the basic nature of the long-run steady state equilibrium can becharacterized independently of them. By utilizing that this fact allows us to apply the backward
induction method to analyze the transitional dynamics of the Ramsey model withheterogeneous individuals, this paper examines the role of heterogeneity in IES on the
dynamics of wealth distribution.
The empirical literature not only points out heterogeneity in IES but also finds that it
increases with wealth, i.e., the rich have high IES, while the poor have low IES. As Guvenen(2006) clearly explains, this finding is very important to reconcile conflicting evidence on IES
in business cycles. It is also crucial to understand the dynamics of wealth distribution and the
distribution in the long-run steady state. Here it should be noted that wealth is accumulated net
savings. The empirical finding that IES increases with wealth therefore implies that individuals
with high IES accumulate wealth more rapidly than individuals with low IES. As a result,
those with high IES have larger wealth than the others. This paper shows that this is true in the
dynamic general equilibrium. Moreover, the possibility is shown that the relatively poor
individuals in the initial period becomes relatively rich in the long-run steady state if their IES
are high enough. Hence, we have an inverted Kuznets curve, that is, a U-shaped relationshipbetween income and inequality.
To clarify the role of heterogeneity in IES, we treat it as exogenous variable in themost part of this paper. However, the individual IES may depend on wealth and/or
consumption, i.e., may change along the transition. For example, the introduction of minimum
consumption into individual preference allows IES to changes over the course of economic
growth. Alvarez-Pelaez and Diaz (2005) and Obiols-Homs and Urrutia (2005) analyze this2

situation to obtain interesting results about transitional dynamics in wealth distribution. As an
extension, we also analyze the Ramsey model with a Stone-Geary utility function to show thatan inverted U-shaped relationship between income and inequality, i.e., the Kuznets curve canemerge. The results obtained from the model with heterogeneity help intuitively understand themechanism behind the inverted U-shaped relationship.
The rest of the paper is organized as follows. Section 2 sets up the model and derives a
couple of important theoretical results that allow us to conduct the simulation analysis. Section
3 deals with numerical simulations on the dynamics of wealth distribution to clarify the role ofheterogeneity in IES. It also simulates the model with a Stone-Geary utility function to showthat the Kuznets curve can emerge. Section 4 concludes the paper.

2. The model
Consider an infinite horizon economy that consists of many infinitely-lived individuals andidentical firms. The individuals are divided into two types, P and R ; all individuals of eachtype are identical in terms of both wealth and preference, while both wealth and preference candiffer across the types. 4 The population of individuals is constant over time and normalized tounity. Denoting the population of type i individuals as λi ( 0 < λi < 1 ),λP + λ R =1.

(1)2-1. Technology and factor prices
Firms produce a homogeneous good according to the following standard constant-returns-toscale production function F (⋅) :Y (t ) = F ( K (t ), L(t ))(2)where Y (t ) is output, K (t ) is capital, and L(t ) is labor. The above can be rewritten as thefollowing per-capita production function:y (t ) = f (k (t )) with f '(k (t )) > 0 and f "(k (t )) < 0 ,(3)where y (t ) = Y (t ) L(t ) and k (t ) = K (t ) L(t ) . In equilibrium, the net rate of return on capital4

The main results remain unchanged if we assume more than two types of individuals.

3r (t ) and the wage rate w(t ) are as follows:=r (t ) r=(k (t )) f '(k (t )) − δ ,(4a)=w(t ) w=(k (t )) f (k (t )) − k (t ) f '(k (t )) ,(4b)

where δ is a constant capital depreciation rate.

2-2. Preferences, budget constraints and optimality conditions
The preference of type i individuals is expressed as follows:ci (t )1−1 εi − 1ρt
dt for i = P, R when ε i ≠ 1 ,
−
exp(
)
∫0
1−1 εi∞∞∫ exp(− ρt ) log c (t )dt for i = P, R when ε = 1 ,
i

0

i

(5a)
(5b)

where ρ > 0 stands for the rate of time preference, and ε i > 0 for the intertemporal elasticity
of substitution (IES). Assuming that each household supplies one unit of labor in each period,
the budget constraint of type i individuals is given byki (=t ) r (t )ki (t ) + w(t ) − ci (t ) with ki (0) = ki for i = P, R ,(6)

where ki (t ) is type i individuals’ capital stock, and k i is the initial level. The associated Euler
equations on consumption areci=(t ) ε i (r (t ) − ρ )ci=(t ) ε i ( f '(k (t )) − δ − ρ )ci (t ) for i = P, R ,(7)

and the transversality conditions are

lim e − ρ t ci (t ) −1 εi ki (t ) = 0 for i = P, R .
t →∞

(8)

2-3. Aggregate behavior of the economy
Since, as (1) shows, λi is a population weight of type i individuals, we have=k (t ) λP k P (t ) + λR k R (t ) ,=c(t ) λP cP (t ) + λR cR (t ) ,(9)where c(t ) represents the per-capita consumption. Taking (9) into account, the following law of
motion of the per-capita capital can be obtained from (4a), (4b) and (6):=k(t ) λP kP (t ) + λR kR (t )= r (t )[λP k P (t ) + λR k R (t )] + w(t )[λP + λR ] − [λP cP (t ) + λR cR (t )]= r (t )k (t ) + w(t ) − c(=t ) f (k (t )) − δ k (t ) − c(t ) .

4

(10)

The above shows the resource constraint of the economy, i.e., the goods market equilibrium.
Also, the following law of motion of the aggregate consumption can be obtained from (7):c(t ) = λP cP (t ) + λR cR (t ) = (r (k (t )) − ρ )[λPε P cP (t ) + λRε R cR (t )] .

(11)

If IES are the same across individuals, i.e., ε=ε=ε , then the above equation
PR
becomesc(t ) ε (r (k (t )) − ρ )c(t ) .
=(12)

The aggregate behavior of the economy is characterized by (10), (12), and the initial andtransversality conditions. In other words, as Caselli and Ventura (2000), among others, show, it
becomes the same as in the standard Ramsey model with homogeneous individuals althoughconsumption and capital can differ across individuals.
It is evident from (11) that the per-capita capital in the steady state is the same as themodified golden rule, k MGR , which is determined byf ' (k MGR ) − δ = r * = ρ .

(13)Once k MGR is given, the resource constraint (10) determines the per-capita consumption in thestationary state cMGR , which is given by=
cMGR f (k MGR ) − δ k MGR .

(14)

Proposition 1
Regardless of individual heterogeneity in the intertemporal elasticity of substitution, the percapita capital and consumption in the steady state are the same as those in the standard Ramseymodel with homogeneous individuals.

2-4. Wealth distribution in the steady state
Since ki (t ) = 0 in the steady state,ci* = r * ki* + w* = ρki* + w* ,(15)

where a variable with an asterisk denotes its steady state value. Since f (k ) = (r + δ )k + w
holds at any point in time, and hence f (k MGR ) =+(r * δ )kMGR + w* , (14) can be rewritten as
follows:

cMGR = (r * + δ )k MGR + w* − δk MGR = r * k MGR + w* ,5and therefore

cMGR = ρk MGR + w* or w* = cMGR − ρk MGR .

(16)

Substituting the above into (15),ci* = r *kMGR + (cMGR − ρ kMGR ) = ρ (ki* − kMGR ) + cMGR for i = P, R .

(17)

Since, as (13) shows, the slope of c = f (k ) − δk at (k MGR , cMGR ) is ρ , (17) implies that all
the points denoted by (k i* , ci* ) are on the tangent line to the curve of c = f (k ) − δk at(k MGR , cMGR ) , i.e., (ki* , ci* ) is on the line S P S R in Fig. 1. If ki* = 0 , then (17) becomes
ci* = − ρk MGR + cMGR . Substituting (16) into this equation gives ci* = w* , which shows that
consumption of an individual with no wealth in the steady state is equal to the wage. Thus, thelong-run steady state equilibrium is represented by a straight line in the (k , c) plane in the
Ramsey model with heterogeneity in IES, while it is shown as a single point in the (k , c) plane ,that is (k MGR , cMGR ) , in the standard Ramsey model with no heterogeneity.
[Fig. 1 is around here.]

Proposition 2 5
Suppose that the time-preference rate is the same across individuals, while the intertemporalelasticities of substitution and/or the initial wealth are different. Then, the long-run equilibrium
is represented by a tangent line to the curve of c = f (k ) − δk at (k MGR , cMGR ) in the (k , c)plane.

To derive the transition path to the long-run steady state, i.e., the dynamic generalequilibrium path, in the standard Ramsey model we use the backward induction method.
Without characterizing the transition path, we can determine the long-run steady state, which isshown by a single point in the (k , c) plane, such as (k MGR , cMGR ) in Fig. 1. Since the initialcondition is given only by the per-capita capital k (0) = k , we can find the unique per-capita
consumption in the initial period, c(0) , by using the backward induction method. The transition
is represented by a unique path that connects the stationary state represented by (k MGR , cMGR )with the initial state represented by (k (0), c(0)) .
5

The same proposition is put forward in a more general setting in Nakamura (2020).

6

In contrast, the long-run steady state equilibrium in our model is represented by astraight line, such as S P S R in Fig.1. Although the location and slope of the line are independentof the initial condition and heterogeneity in IES, the length depends crucially on them. Without
deriving the transition path, the long-run steady state equilibrium cannot be fully characterized.
In other words, the long-run equilibrium and the transition are simultaneously andinterdependently determined. However, assuming a steady state instead of determining it prior
to deriving the transition path allows us to use the backward induction method to derive thetransition path. The next section applies this strategy to the simulation analysis in order toanalyze changes in the wealth distribution over time and the effect of heterogeneity in IES on
the dynamics of wealth distribution.

3. The simulation analysis
This section conducts simulations to examine how the aggregate and individual wealth evolveas the economy grows. In the simulations, we will use the following Cobb-Douglass production
function:

Yt = AK tα L1t−α , i.e., yt = Aktα with α = 1 3 .
Also, we assume ρ = 0.05 and δ = 0.1 .
Taking advantage of the results in the previous section, we fix a straight line in the(k , c) plane that shows the long-run steady state equilibrium. In all of the following simulations,type R individuals are assumed to have 25 percent more capital than the average in the steadystate, while type P individuals have 25 percent less capital. In other words, the type R are
richer than the type P by two thirds in terms of wealth. We also assume that the population oftype R individuals is equal to that of type P individuals, i.e., λ=λ=0.5 . Hence, theRPwealth Gini coefficient in the steady state is 0.25 for all of the following simulations.

3-1. With no heterogeneity in IES
In this subsection we examine how the wealth distribution evolves during the transition in the
Ramsey model in which individuals are homogeneous in preference, i.e., they have the same7

IES in addition to the common rate of time preference. In concrete, we assume that ε=
1,
ε=
R
p
Panel (A) in Fig. 2 shows the phase diagram, in which the pink dashed line shows theequilibrium path of type R individuals, while the green dashed line shows the equilibrium pathof type P individuals. The aggregate behavior is represented by the blue solid line. If no initialdifference in wealth exists across individuals, our model becomes identical to therepresentative-agent Ramsey model. The blue line can therefore be interpreted as theequilibrium path in the standard Ramsey model. While Panel (B) shows the time paths of capitalfor the two types, Panel (C) shows the time path of the wealth Gini. 6[Fig. 2 is around here.]
The growth of the economy means increases in wealth of both types of individuals.
Since type R ’s wealth increases faster than type P ’s, the wealth gap increases through time.
At the same time, however, the total wealth also increases. As a result, the share of type R ’s intotal wealth decreases, while that of type P ’s increases. 7 In other words, the growth in theeconomy as a whole expands the wealth gap between the two types but reduces the wealth Gini.
One may suspect that this overall growth effect to reduce the wealth Gini depends crucially on
the speed of growth, i.e., IES. However, this reduction effect is valid for the relevant rage of IES
in our simple setting. For example, Fig. 3 shows the simulation results when ε=
0.5 .
ε=
R
p
Although the transition in Fig. 3 is longer compared to Fig. 2 because the growth is lower, thebasic properties of the dynamics are the same between Fig. 2 and Fig. 3. These observations
lead us to the following remark.
[Fig. 3 is around here.]6

See the appendix for the outline of computation code for the simulation analysis.

7

For example, type R’s wealth is 10 and type P’s is 5 in the initial period, while type R’s wealthis 20 and type P’s is 12 in the stationary state. The wealth gap is 5 and the wealth share of type
R’s is 2/3, i.e., about 0.67 in the initial period, while the wealth gap is 8 and the wealth share oftype R’s is 5/8, i.e. 0.625.

8

Remark 1
Suppose that the initial wealth differs across individuals in the Ramsey model with noheterogeneity in the intertemporal elasticity of substitution. Then, the wealth gap expands as the
economy grows, while the wealth Gini decreases over time.

3-2. Effects of heterogeneity in IES (1): Expanding inequality
Let us examine the effects of heterogeneity in IES. Since we have fixed the wealth distribution
in the steady state, at least two distinct transitions must be considered. One is with expanding
inequality, and the other is with shrinking inequality.
Even if the initial wealth is the same across individuals, the wealth can differ in thesteady state if IES are different. Individuals with high IES start with low consumption andaccumulate wealth rapidly. As a result, they have large wealth and consumption in the steadystate. Fig. 4 shows the simulation results when ε R = 1 and ε P = 0.5 . As Panel (A) shows, type

R individuals start with low consumption and end up with large capital and consumption in thesteady state, while type P individuals start with high consumption and end up with small
capital and consumption.
[Fig. 4 is around here]

Remark 2
Suppose that the initial wealth is the same across individuals in the Ramsey model withheterogeneity in the intertemporal elasticity of substitution (IES). Then, the individuals withhigher IES accumulate wealth rapidly and have a larger amount of wealth in the steady state.
As a result, the wealth Gini increases as the economy grows.

Many empirical studies such as Attanasio et al. (2002), Blundell et al. (2014) and
Browning (1995) have found that IES increases with wealth. Taking the fact into account that
wealth is accumulated net savings, the empirical finding can be interpreted conversely, i.e.,
individuals with high IES accumulate wealth more rapidly than individuals with low IES. As aresult, those with high IES have larger wealth than others. The above remark suggests it.

93-3. Effects of heterogeneity in IES (2): Shrinking inequality
Even if some individuals initially have a larger amount of wealth compared to others, therelative size is getting smaller in the transition if their IES are lower than the others. As
explained in the previous subsection, the individuals with low IES start with high consumption
and hence accumulate capital slowly. Fig. 5 shows the simulation results when ε R = 0.5 andε P = 1 . As Panel (A) shows, type R individuals accumulate wealth slowly by starting withhigh consumption, while type P individuals accumulate wealth rapidly by starting with low
consumption. As a result, the wealth gap is smaller in the steady state than in the initial period.
[Fig. 5 is around here]
Although the graphs in Fig. 5 look similar to those in Fig. 2 because the wealth Gini
starts at 0.5 and ends at 0.25 in the both figures, we should notice the difference in transition
period. It takes about 30 periods for the Gini to reach 0.25 in Fig. 2, while it takes about 40
periods in Fig. 4. Of course, this observation is consistent with our intuition. Since individuals
with high IES accumulate capital faster than those with low IES, the wealth gap shrinks more
slowly when the rich have lower IES ( ε R = 0.5 ) than the poor ( ε P = 1 ) compared to when bothtypes of individuals have the same high IES ( ε=
ε=
1 ).
R
P

3-4. U-shaped relationship with shoe on the other footIf type R individuals, who have by assumption a larger amount of wealth than type P
individuals in the steady state, have a smaller amount of wealth in the initial period, then thetype R must overtake the type P in terms of wealth during the transition. The previous results
have shown the possibility. If the type R have higher IES and start with smaller capital, thenthey can overtake the type P in terms of both capital and consumption before the economy
reaches the stationary state. Fig. 6 shows the simulation results when ε R = 1 , ε P = 0.5 , andtype R individuals have a smaller amount of capital in the initial period. The results can besummarized as the following remark.
[Fig. 6 is around here.]10Remark 3 (Inverted Kuznets Curve)
Suppose that individuals with high IES have small amounts of initial wealth, while individuals
with low IES’s have large amounts of initial wealth. Then, the possibility exists that the richindividuals in the initial period become relatively poor in the long run.

The above remark is important not only in the sense that it presents an interestingtransition of the wealth Gini such as the U-shaped, i.e. an inverted Kuznets curve, but also in thesense that it demonstrates the importance of transition period in determining the dynamics ofwealth distribution and the distribution in the steady state. The initial condition is crucial when
the transition period is short, while the preference plays a dominant role when the period is long.
As time passes, the effect of the initial condition gradually declines. Instead, that of thepreference gradually increases. As a result, the U-shaped relationship between inequality andincome can appear over time.

3-5. An Extension: Stone-Geary preferences
The above simulation method is not restricted to Ramsey models with constant IES. As an
example, let us examine the model with the following Stone-Geary utility function:∞∫ exp(− ρt )0(ci (t ) − c )1−(1 εi ) − 1
dt for i = P, R ,1 − (1 ε i )(18)where c stands for minimum consumption. As Alvarez-Pelaez and Diaz (2005) and ObiolsHoms and Urrutia (2005) correctly point out, IES change as capital accumulates even if ε i ’s
are constant through time. Denoting the instantaneous utility function by ui (ci (t )) , bydefinition, IES is expressed by∂ (ci (t ) ci (t ))∂ (ci (t ) ci (t ))for i = P, R .
IESi (t ) =−=−∂ (ui '(ci (t )) ui '(ci (t ))∂ (ui "(ci (t ))ci (t ) ui '(ci (t )))With the specification in (18),

IESi=
(t )


c 
= ε i 1 −
.
ci (t )
c
t
(
)
i



ε i (ci (t ) − c )The above expression leads to the following remark.

11

(19)

Remark 4
Suppose that individual preferences is give by a Stone-Geary utility function. Then, theintertemporal elasticity of substitution becomes larger as consumption becomes larger, i.e., aswealth accumulates.

Now the law of motion of k (t ) and c(t ) is given byk(t )= λP kP (t ) + λR kR (t )= f (k (t )) − δ k (t ) − c(t ) ,

(20a)

c=
(t ) λP cP (t ) + λR cR=(t ) (r (k (t )) − ρ )[λPε P (cP (t ) − c ) + λRε R (cR (t ) − c )] . (20b)
While (20a) is the same as (10), (20b) corresponds to (11). Although (20b) is different from(11), the both give the same modified golden rule level of capital accumulation in the steadystate, k MGR . Hence Propositions 1 and 2 remain true, and consequently we can apply the same
method as before to the model with Stone-Geary preferences.
Suppose, for example, ε=ε=
1 and c = 0.5 . If cR (0) = 1 and cP (0) = 0.8 , then,Rp
it is calculated from (19) that IES R (0) = 0.5 and IES p (0) = 0.375 . Thus, Stone-Geary
preferences lowers IES, with lowering the poor’s IES more than the rich’s. Since individuals
with high IES accumulate wealth rapidly, Stone-Geary preferences have an effect of expanding
the wealth gap.
[Fig. 7 is around here.]Fig. 7 shows the simulation results when ε=ε=1 and c = 0.5 . In the initial period,RP
the both types have a small amount of wealth although type R have slightly more than type P .
Since type R therefore have higher consumption than type P , type R ’s IES is higher thantype P ’s. As a result, type R accumulate wealth more rapidly than type P , as Panel (B) shows.
In other words, since the expanding effect by the Stone-Geary preference is dominant during theearly stage of transition, the wealth Gini increases, as Panel (C) shows. However, as Panel (A)shows, the gap in consumption between the two types shrinks as time goes. Since the gap in IES
also shrinks, the behavior of the model becomes similar to that of the Ramsey model with noheterogeneity in IES. We know from Remark 1 that the Gini decreases during the late stage oftransition, as is shown in Panel (C). Consequently, the inverted U-shaped relationship between
inequality and income is observed.

12Remark 5 (Kuznets Curve)
Introducing Stone-Geary preferences into a standard Ramsey model with no heterogeneity inthe intertemporal elasticity of substitution, the inverted U-shaped relationship between income
and inequality, i.e., the Kuznets Curve, can emerge.

4. Concluding Remarks
Both the initial conditions and individual preferences play a crucial role in the determination of
the dynamics of wealth distribution and the distribution in the steady state. Although mostprevious studies have paid much attention to time-preference rather than intertemporal
substitution, as this paper shows, the latter also plays an important role. To carefully examine
the wealth distribution in both the long run and short run, we must pay enough attention toindividual preferences, characterized by time preference and intertemporal substitution, as wellas to the initial conditions. In general, as the transition becomes longer, the effect of initialcondition becomes smaller, while that of individual preferences becomes larger. Our
simulation results have clearly shown that the roles of the intertemporal elasticity of
substitution (IES) in the transitional dynamics and steady state of wealth distribution cannot be
overlooked.
To clarify the role of heterogeneity in IES, we have only conducted the simulation
analysis. We believe that the backward induction method used in this paper can also be applied
to the calibration. It is hoped that our attempt stimulates future research.

13

Appendix: Pseudo CodeThe appendix gives the outline of computation code for the simulation, i.e., the pseudo code,in this paper.

1: procedure Calculation of Transition Dynamics
2:

Setting the parameters

3:

Calculate the steady state4:

 α A  1−α
k MGR = δ + ρ 

5:=
cMGR Ak MGRα − δ k MGR

−

1

6:

R
P
= 0.75k MGR
k MGR= 1.25kMGR , k MGR7:

PR= 0.75cMGR
cMGR= 1.25cMGR , cMGR

8:
9:Calculation of Coeﬃcient for Reverse Shooting Method

α −2
10:=
dw 0.5 Aα (α − 1)k MGR

α −1
11: =
dr 0.5 Aα (1 − α )k MGR

12:dr = dr + dw + ρ

13:
14:

for t = 0 : dt : T do

15:

i = i +1

16:

c 'R = cRT , c 'P = cPT

17:

k 'R = k RT , k 'R = k PT

18:

Calculation of cRt and cPt : Reverse Shooting Method (Judd(1998))

19:

for j = 1 : 4 do

20:
21:
22:
23:
24:

if T = 0 then(c =−0.5 ∗ ( dss + dss + 2ε (cα −2−0.5 ∗ dss 2 + dss 2 + 2ε R (cRMGR − c )(1 − α )α k MGR
cRt =
2

2

Pt

P

PMGR

else T ≠ 0 thencRt =−ε Rt (cRt − c )(α ktα −1 − ρ − δ )14

α −2− c )(1 − α )α kMGR))cPt =−ε Pt (cPt − c )(α ktα −1 − ρ − δ )

25:
26:

end if27:
28:

4th-order Runge-Kutta Methods

29:

dcRj = cRt dt

30:c='R cRt + 0.5cRj

31:

if j = 3 thenc '=cR + cRj
R

32:
33:

end if34:
35:

dcPj = cPt dt

36:c='P cPt + 0.5cPj

37:

if j = 3 thenc '=cP + cPj
P

38:
39:

end if40:

end for41:

Calculation of Consumption

42:

cR +1 =
cRt + 16 (dcR1 + 2dcR 2 + 2dcR 3 + dcR 4 )43:

cP +1 =
cPt + 16 (dcP1 + 2dcP 2 + 2dcP 3 + dcP 4 )44:=
ct +1 0.5cRt +1 + 0.5cPt +1

45:
46:

Calculation of kit47:

for j = 1 : 4 do

48:

kRt =−(rk Rt + w − cRt )49:kPt =−(rk Pt + w − cPt )

50:
51:

4th-order Runge-Kutta Methods

52:

dk Rj = kRt dt

53:

k '=
k R + 0.5kRjR15

if j = 3 then

54:

k '=
k R + kRj
R

55:
56:

end if57:
58:

dk Pj = kPt dt

59:

k '=
k P + 0.5kRjP60:

if j = 3 thenk '=k P + kPj
R

61:
62:

end if

63:

end for64:
65:

Calculation of Capital

66:

k R +1 =k Rt + 16 (dk R1 + 2dk R 2 + 2dk R 3 + dk R 4 )67:

k P +1 =k Pt + 16 (dk P1 + 2dk P 2 + 2dk P 3 + dk P 4 )68:=
kt +1 0.5k Rt +1 + 0.5k Pt +1

69:
70:

Calculation of Prices71:

rt +1 α Aktα+−11 − δ=72:

wt +1= (1 − α ) Aktα+1

73:

end for74: end procedure

16References
Alvarez-Pelaez, Maria J. and Diaz, Antonia, 2005, Minimum Consumption and Transitional
Dynamics in Wealth Distribution, Journal of Monetary Economics, Vol. 52(3), pp. 633-67.
Attanasio, Orazio P, Banks, James and Tanner, Sarah, 2002. Asset holding and consumption
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Blundell, Richard, Browning, Martin and Meghir, Costas, 1994. Consumer demand and thelife-cycle allocation of household expenditures, Review of Economic Studies, Vol. 61, 57–80.
Browning, Martin, Hansen, Lars Peter and Heckman, James J., 1999, Micro data and generalequilibrium models, in: Taylor, J., Woodford, M. (Eds.), Handbook of Macroeconomics, Vol.
1A. North-Holland, Amsterdam.
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Fisher, Thomas, 2017, “Thomas Piketty and the Rate of Time Preference,” Journal of
Economic Dynamics and Control, Vo1. 77, 111-33
Garcia-Penalosa, Cecilia; Turnovsky, Stephen J., 2006, Growth and income inequality: acanonical model, Economic Theory, Vol. 28, 25–49.
Gorman, William M., 1953. Community preference ﬁelds. Econometrica, Vol. 21, pp. 63–80.
Guvenen, Fatih, 2006, Reconciling conflicting evidence on the elasticity of intertemporal
substitution: A macroeconomic perspective, Journal of Monetary Economics, Vol. 53(7), 14511472.

17

Mitra, Tapan and Sorger, Gerhard, 2013, On Ramsey's Conjecture, Journal of Economic
Theory, vol. 148(5), pp. 1953-1976.
Nakamura, Tamotsu, 2020, On the long-run wealth distribution in a Ramsey model withheterogeneous households, Economic Modelling, vol. 84, 177-180.
Obiols-Homs, Francesc and Urrutia, Carlo, 2005, Transitional Dynamics and the Distributionof Assets, Economic Theory, Vo1. 25 (2), 381-400.

18

FiguresFig. 1

Fig. 1 Steady State Distribution

19

Fig. 2(A) Phase Diagram(B) Time Paths of Capital(C) Time Path of Wealth Gini
Fig. 2 Standard Ramsey Model When ε=
ε=
1
R
P

20

Fig. 3(A) Phase Diagram(B) Time Paths of Capital(C) Time Path of Wealth Gini
Fig. 3 Standard Ramsey Model When ε=
ε=
0.5
R
P

21

Fig. 4(A) Phase Diagram(B) Time Paths of Capital(C) Time Path of Wealth Gini
Fig. 4 Effects of Heterogeneity in IES (1) ( ε R = 1 and ε P = 0.5 )22

Fig. 5(A) Phase Diagram(B) Time Paths of Capital(C) Time Path of Wealth Gini
Fig. 5 Effects of Heterogeneity in IES (2) ( ε R = 0.5 and ε P = 1 )23

Fig. 6(A) Phase Diagram(B) Time Paths of Capital(C) Time Path of Wealth Gini
Fig. 6 U-shaped Relationship between Income and Inequality
24

Fig. 7(A) Phase Diagram(B) Time Paths of Capital(C) Time Path of Wealth Gini
Fig. 7 Inverted U-shaped Relationship under Stone-Geary Preferences25