Aging, Social Security Reform and Factor Pricein a Transition Economy∗
Tomoaki Yamada†First Draft: March 2007 - This Draft: November 2007

Abstract
This paper explores the macroeconomic and welfare implications of aging andsocial security reform in Japan. Based on the overlapping generations model with
idiosyncratic income risk, we demonstrate that aging induces capital deepening andwage increase in the transition path. We consider four social security reform plans;(1) the reduction of the replacement rate by half, (2) full privatization, (3) capitalincome tax, and (4) consumption tax. We compute the transition paths of eachcase, and find that the introduction of capital income tax improves the welfare of
the young and future households based on the cohorts’ viewpoint. On the contrary,
the introduction of consumption tax does not improves the social welfare of theeconomy because of intragenerational inequality and substitution effect. Further, weshow that under the transition, the earnings profiles become a more hump-shaped
and consumption profiles become steep with the changes in the replacement rate.
Keywords: Aging, Factor Price, Heterogeneity, Social Security Reform, Transition
JEL Classification: H55, D31, D33, E24, E25.

∗

The author acknowledges many helpful comments from Yukiko Abe, Jyun-ichi Itaya, Nori Kudo,
Kenji Miyazaki, Takashi Oshio, Makoto Saito, Takamasa Shirai and seminar participants at Hokkaido
University and 2007 Japanese Economic Association Autumn Meeting. This research is financially
supported by a Grant-in-Aid for Scientific Research. Of course, all remaining errors are my own. The
most recent version is downloadable at http://homepage2.nifty.com/~tyamada/research.html.
†
Faculty of Economics, Rissho University: 4-2-16, Osaki, Shinagawa-ku, Tokyo 141-8602. E-mail:tyamada@ris.ac.jp. Phone: +81-3-5487-3239.

1

1

Introduction

As the United Nations (2007) reports, Japan is one of the most aging societies in theworld. The reduction in the size of the of working population and the rise of old-age
dependency rate causes serious problems in social security reform. As far this issue
is concerned, in order to sustain the social security system, some reform should beconsidered. However, almost all social security reforms may induce well-known intergenerational conflicts. Moreover, the allocation of wealth and consumption are rather
complicated under the transition to an aging population. Therefore, we need to give
careful consideration to the macroeconomic and welfare implications of this transition.
In aging society, there are two possible scenarios. First, the capital-labor ratio decreases because a large burden of social security tax induces lower savings. Moreover,an increase in the number of retired households implies that many households decumulate their wealth for lifecycle reason. Second, in contrast, the capital-labor ratio
may increase when labor supply measured in terms of efficiency decreases, because thesocial security tax burden creates a distortion on labor supply decision. Based on theusual Cobb=Douglas-type production function, the capital-labor ratio determines factor
prices, the interest rate, and wage level. Our main focus in this paper is the dynamics of allocations under the transition to an aging society, with a particular focus on
the dynamics of factor prices and the welfare of each cohort. As Davila et al. (2006)
reveals, factor prices have different effects on the wealth rich and the poor. Moreover,the changes in factor prices have different effects on employees and retirees. Therefore,
to consider social security reform and evaluate the welfare of households, we need toinvestigate the dynamics of macroeconomic statistics and factor prices as well as thesocial security tax rate.
We consider the macroeconomic and welfare implications of aging and social security
reform under the transition by using the overlapping generations model with idiosyncratic income risks. This model features not only intergenerational heterogeneity but
also intragenerational heterogeneity because of the realization of idiosyncratic income
shocks. Thus, we consider the decisions of different asset and income level for eachage group. Our finding indicate that further capital deepening can be expected in thenear future in Japan. Further, if the capital-labor ratio increases, then based on the
Cobb=Douglas-type production function, the equilibrium wage increases by 6% and theinterest rate decreases by 1.5% in the transition path. Therefore, younger households
have a chance to obtain greater labor earnings. However, the output per capita decreases
by 20% because of the decrease in the aggregate capital and labor supply. If some typeof social security reform is implemented, such as the reduction of the replacement rate,
there will be further capital deepening. With the strong incentive toward savings for
retirement, households supply more labor and accumulate wealth. This may increase
the per capita GDP and welfare due to a significant reduction of the replacement rate.
Not surprisingly, these reforms induce a strong intergenerational conflict measured interms of the cohorts’ welfare. In addition, we investigate the other sources of finance2

for the social security system, namely consumption tax and capital income tax. We findthat the introduction of capital income tax flattens the factor prices path over time, andcapital income tax weakly improves the welfare of the young and future generations. On
the contrary, the consumption tax induces that young households consume more, andintragenerational consumption inequality of young generations increases. Therefore, atleast in our estimation, the consumption tax does not improves the welfare of the economy. With both of these new financing schemes, the payroll tax rate on labor income
decreases by 2% ∼ 4%.
Our paper is a branch of the traditional research that analyses transition dynamics
of aging and social security reform based on the overlapping generations model. Since
the initial research of Auerbach and Kotlikoff (1987), there have been many researches
on demographic structure and social security including Huang et al. (1997) and De
Nardi et al. (1999). Huggett (1996) and İmrohoroǧlu et al. (1995) extend these papers to consider not only intergenerational but also intragenerational heterogeneity byincluding the element of idiosyncratic income uncertainty. Huggett (1996) find that themodel with intragenerational heterogeneity generates a close-to-real wealth inequality
measured by the Gini coefficient. İmrohoroǧlu et al. (1995) demonstrates that the optimal replacement is zero when the discount rate is positive. Our approach is very similar
to those of Conesa and Krueger (1999) and Nishiyama and Smetters (2005b). Althoughthese works investigate the transition path of an aging economy with/without the social
security reform of replacement rate reduction, they do not consider financing reform
such as consumption tax and capital income tax. Therefore, we focus on the transition
dynamics with such social security reform in Japan.
The paper is organized as follows. In Section 2, we present our basic model andpresent some scenarios to be considered. Section 3 introduces the calibration parameters
for the Japanese economy. In Section 4, we report our main results on the transition
path. Section 5 discusses the implications of factor prices and the behavior of householdson the transition analysis. Section 6 presents the robustness of our analysis and Section
7 provides the concluding remarks.

2

The Model

2.1

Demographic Structure

We consider the overlapping generations model with a continuum of households.1 In themodel, time is discrete. The lifespan of the households is a maximum of J-years, but they
face mortality risks. The number of households aged j ∈ (0, . . . , 20, . . . , J) in period t isdenoted by µj,t . A fraction of households (1−φj,t ) exits the economy owing to death, andµj+1,t+1 = φj,t µj,t is the population of households aged j + 1 at period t + 1, where φj,t is1

Our model describes population dynamics and total factor productivity growth. Thus, we need to
distinguish between nominal and detrended variables to solve the model. For details, see the Appendix.

3

the survival probability. We assume that households begin economic activity at j = 20.
Because households are in their childhood at j = 0, 1, . . . , 19, they do not engage in
consumption or employment but are included in the population dynamics for computing
the future fertility rate. By assumption, µJ+1,t = φJ,t = 0. Let µt = (µ1,t , . . . , µJ,t )denote the population distribution in period t. Therefore, the population dynamics inour economy is expressed in the following matrix form:
1 + ψt 0 0
···
0
 φ
0 0
···
0 


1,t


0
φ2,t 0···0  µt ≡ Γt µt ,µt+1 = 


..
.. 
..
..

..
.

.
. 
.
.
0

···

0 φJ−1,t 0

where ψt is the population growth rate of age 0 from t to t + 1.2 New households enter
the economy in period t + 1 as µ0,t+1 = (1 + ψt )µ0,t , where the population growth rateis derived from the net fertility rate. The aggregate population including children at t
P
is Nt = Jj=0 µj,t . We denote the population growth rate from period t to t + 1 as nt ,i.e., Nt+1 = (1 + nt )Nt . Further, we assume that the aggregate population in period 0
is normalized to one, and the population growth rate is constant in the stationary stateP(i.e., Jj=0 µj,0 = N0 = 1, Nt = (1 + n)t N0 ). Although the population distribution isconstant over time at the stationary state (i.e., µj+1,t+1 /Nt+1 = µj,t /Nt ), the population
distribution varies under the transition path. In the following section, we consider boththe stationary and transition economy.

2.2
2.2.1

Households
Objective Function

A household born in period t has a lifespan of at most J periods, elastically supplies
labor until age jr , and faces idiosyncratic uncertainty with regard to its individual laborproductivity. The objective function of the household in period t is written as follows:


Ã j−1
!
J
X

Y
Ut = E20,t
β j−1
φi,t u(cj,t+j−20 , `¯ − `j,t+j−20 ) ,


j=20

i=20

where β > 0 is a discount factor. All households have labor endowment `¯ and supply¯ at j.
their labor, `j,t+j−20 ∈ [0, `],Since households of age j ∈ {20, . . . , jr } are of employable age, they can elasticallysupply labor. Thereafter, i.e., j ∈ {jr +1, . . . , J}, the households retire and receive social
security benefit from the government.

2

See also Rı́os-Rull (2001) for the details on the transition of population distribution.

4

2.2.2

Idiosyncratic Risk and Budget Constraint

All household face idiosyncratic income risk and also have deterministic labor productivity. The average earnings must be reflective of age-specific average labor productivity.
The average labor productivity grows when households are young and peaks at middleage around the 50s; in other words, the efficiency of a household has an hump-shaped
across age groups. We denote the deterministic productivity measured by hourly wager
as {ηj }jj=20.
Moreover, all households face idiosyncratic labor productivity risks when they arein employment. Following Storesletten, Telmer, and Yaron (2004), we assume thatthe idiosyncratic risk comprises three components; (1) transitory shocks, (2) persistentshocks, and (3) the fixed effect. The labor productivity process et is specified as follows.
ln et = α + zt + κt ,(1)

zt = ρzt−1 + εt .

(2)

The fixed effect is denoted by the variance of α, and the transitory shock as that of
κt . A persistent component of the idiosyncratic shock is represented by zt , which iscomposed of the persistence parameter ρ and the shock εt . Let s = (α, z, κ) denotes
a state of the idiosyncratic shocks for an individual household. We assume that allidiosyncratic shocks are discretized. The persistent component follows a Markov chain,and the transition probability is written as π(zj+1 |zj ) ∈ (0, 1). We assume that the
average efficiency profile and the stochastic productivity are independent of time t andacross agents. Thus, the before-tax labor earning of each age group is determined by
yj,t = wt ηj ej `j,t , where wt is the economy-wide wage level.
The government grants social security benefits through a constant payroll tax fromlabor earnings, and retired households receive the social security benefit. The constant
payroll tax rate is denoted as τtss . For a later analysis, we define the consumption tax
rate and linear capital income tax rate as τtc , and τta , respectively.3 After retirement, ahousehold receives social security benefit wt b(τt , Wg,t ), where b(τt , Wg,t ) is a replacement
rate determined from each tax rates τt ≡ (τtss , τtc , τta ) and social security funds Wg,t .
A household has some asset holdings aj,t ∈ A ⊆ R+ at age j and in period t. Sincehouseholds face mortality risk, some household may die with positive assets. We assume
that the existence of life insurance and that the accidental bequests are redistributed
over living households by an insurance company, which implies that capital income thatliving households can receives increases by 1/φj,t .4 Thus, the budget constraints for3

Conesa and Krueger (2005) consider an optimal progressive tax system on income flow wt ηj ej `j,t +rt aj,t . For our purpose, we need to divide the income sources because we will focus on incentives onsavings and consumption for each taxes.
4
In previous version of this paper, we have assumed that the accidental bequests are collected by thegovernment and redistributed for all households by lump sum manner. Those differences does not makelarge changes of macroeconomic variables in our analysis. Hansen and İmrohoroǧlu (2006) reveals thata lack of annuity markes affects large impacts on lifecycle consumption profile.

5

employees and retirees are(1 + τtc )cj,t + aj+1,t+1 ≤ (1 − τtss ) wt ηj ej `j,t + (1 + (1 − τta )rt /φj,t )aj,t , : employee(1 + τtc )cj,t + aj+1,t+1 ≤ wt b(τtss , Wg,t ) + (1 + (1 − τta )rt /φj,t )aj,t ,: retireewhere rt is the interest rate at t. We assume that households face a liquidity constraint,i.e., aj,t ≥ 0.

2.3

Behavior of Firms and the Factor Price

The aggregate production technology follows a Cobb=Douglas constant returns to scale
production function
Yt = At Ktθ L1−θ,twhere At denotes the total factor productivity (TFP) in period t, Kt is an aggregate
capital, and Lt is an aggregate labor supply measured by efficiency units. We assume thata sequence of the TFP is deterministic. Therefore, there are no aggregate uncertainties
in our economy, and the aggregate productivity and the population growth is perfectly1
forecastable. We denote the gross growth rate of the TFP as 1 + gt = ( AAt+1) 1−θ .
t
The asset holdings and labor supply of each households differs even in the same cohort and age group due to idiosyncratic income uncertainty. We denote a fraction ofhouseholds aged j with asset a and realized productivity s as Φt (aj , sj ).5 By construcRtion, dΦt (aj , ej ) = 1. The aggregate capital and labor supply are determined by thesums of each generation’s capital and labor, as follows.
Kt =

J
X

Z
µj,t

aj,t dΦt (aj , ej ) + Wg,t ,(3)

ηj ej `j,t dΦt (aj , ej ) .

(4)

j=20

Lt =

jr
X

Z
µj,t

j=20

The interest rate rt and wage wt at period t are determined as follows.
µrt = θAt

Kt
Lt

¶θ−1

µ
− δ, wt = (1 − θ)At

Kt
Lt

¶θ
,

where δ is the depreciation rate.

2.4

The Government and the Social Security System

We assume that the government collects tax to finance social security benefits and redistributes it to retired households in a lump-sum manner, and we do not consider othergovernment expenditures. In our setup, the social security system is basically governed
5

For details of the distribution function, see the Appendix.

6

by the Pay-as-you-go system although the social security system in Japan is conducted
as a “modified funded system”. Accordingly, the government collects payments from
employees and retirees using (τtss , τtc , τta ), and grants social security benefits throughb(τt , Wg,t ).6 We assume that the replacement rate is exogenously fixed and the corresponding tax rates are determiend endogenously. Because the old-age dependency rate
has been considerably lower in the past years, the government holds positive social security trust funds in Japan. We denote the social security payments by the payroll tax
as TtSS , by the consumption tax as TtC , by the capital income tax as TtA , and aggregatesocial security benefit as Bt . Therefore, if the contribution is greater than the benefit,
the government increases its fund, and vice verse.
The government should satisfy the following budget constraints.7
Wg,t+1 = (1 + rt )Wg,t + (TtSS + TtC + TtA ) − Bt ,
Z
jr
X
SS
Tt =
µj τtss wt ηj ej `j,t dΦt (aj , ej ) = wt τtss Lt ,

(5)

j=20

TtC =

J
X

Z
τtc cj,t dΦt (aj , ej ) = τtc Ct ,

µj

j=20

TtA =

J
X

Z
µj

τta aj,t dΦt (aj , ej ) = wt τta Kt ,

j=20

Bt =

J
X

µj,t wt b(τt , Wg,t )Lt = wt b(τt , Wg,t )Lt Ntret ,j=jr +1

where Ntret is a fraction of retired households over the total population who have alreadyentered the economy. We assume that the social security funds are invested into production and that they increase with the rise in the interest rate. Note that the average labor
income of all workers is wt Lt . Because the replacement rate b(τt , Wg,t ) is determined bya constant of the average labor income, wt b(τt , Wg,t )Lt is the social security benefit thatis exogenously determined by the government.
From the above, the aggregate states of the economy at period t is summarizedas Xt = (Kt , Lt , At , Φt , µt ). The government’s policy and funds are denoted as Ωt =6

In our model, we assume that the social security benefits are the same for all retirees and with thesame fixed effect α. We do not consider a too complicated social security system but a realistic one
because of the computational burden. For the redistributional effects of the social security system in theUS, see Huggett and Ventura (1999) and Storesletten et al. (1998).
7
At stationary state, output Yt also grows by the rate of population growth (1 + nt ) and the TFP
growth (1 + gt ). Thus, the government fund Wg,t has positive trend of (1 + nt ) (1 + gt ) in the stationary
1state. Normalized government budget constraints are determined by dividing At1−α Nt both sides ofequations.
1
W̃g,t+1 =[(1 + rt )W̃g,t + (TtSS + TtC + TtA ) − Bt ](1 + nt )(1 + gt )Note that the wage, the aggregate consumption and the aggregate capital grow by (1 + nt )(1 + gt ).

7(τt , Wg,t , b(τt , Wg,t )). Then, the Bellman equation of age j in period t is as follows.
©
ª¯ − `j,t ) + φj,t βEvt+1 (a0 , s0 ; Xt+1 , Ωt+1 ) ,vt (aj , sj ; Xt , Ωt ) = max
u(c
,
`
j,t
0
c,a

(6)

subject to(1 + τtc )cj,t + aj+1,t+1 ≤ (1 − τtss ) wt ηj ej `j,t + (1 + (1 − τta )rt /φj,t )aj,t ,(7)(1 + τtc )cj,t + aj+1,t+1 ≤ wt b(τt , Wg,t ) + (1 + (1 − τta )rt /φi,t )aj,t .

(8)

The intertemporal and intratemporal first order conditions are such that80uc (cj,t , `¯ − `j,t ) ≥ φj,t β(1 + (1 − τta )rt+1 /φj+1,t+1 )Ej vj+1
(a0 , e0 , j + 1; Xt+1 , Ωt+1 ),u` (cj,t , `¯ − `j,t ) ≥ uc (cj,t , `¯ − `j,t ) (1 − τ ss ) wt ηj ej .
t

2.5

Definition of a Competitive Equilibrium

Our concern in this paper is the stationary state and the transition paths of the economy. Therefore, we need three definitions of equilibrium in the stationary state and intransition.
Definition 1 (Recursive Competitive Equilibrium) Given the government’s policy {Ωt } and the population dynamics, the Recursive Competitive Equilibrium is a set ofvalue functions {vt }, policy functions {gc,t , g`,t , ga,t }, aggregate capital {Kt }, aggregate
labor {Lt }, factor prices {rt , wr }, payroll taxes {τt }, and government funds {Wg,t } thatsatisfy following conditions:(i) A Household’s Optimality: Given the factor prices {rt , wr } and payroll taxes {τt },the value function {vt } solves equation (6), and {gc,t , g`,t , ga,t } are the associated policy
functions. The value and policy functions are measurable.
(ii) A Firm’s Optimality: The factor prices are competitively determined as follows,
rt = θAt (Kt /Lt )θ−1 − δ, wt = (1 − θ) At (Kt /Lt )θ .
(iii) Market Clearing: The market clearing conditions of equations (3) and (4) aresatisfied.
(iv) the Government’s Budget: For each case of Wg,t > 0 and Wg,t = 0, the governments’ budget (5) clears.
(v) Transition Law: Φt+1 = H(Φt ).
Definition 2 (Stationary Recursive Competitive Equilibrium) The Stationary Recursive Competitive Equilibrium is a recursive competitive equilibrium with a stationarityof distribution Φj,t+1 = Φj,t (∀t) for each age group j.

8

For details, see the Appendix.

8

Definition 3 (Balanced Growth Path) A Balanced Growth Path is a recursive competitive equilibrium with Yt = ((1 + g)(1 + n))t Y0 , Kt = ((1 + g)(1 + n))t K0 , Wg,t =((1 + g)(1 + n))t Wg,0 , L1 = (1 + n)t L0 , rt = r0 , wt = (1 + g)t w0 , cj,t = (1 + g)t cj,0 ,aj,t = (1 + g)t aj,0 , and `j,t = `j,0 . In other words, without a labor supply, the aggregate
11variables are normalized by At1−θ Nt , and individual variables are normalized by At1−θ .
The final purposes of the paper to examine the macroeconomic and welfare implications of the competitive equilibrium on the transition path requires complex computational procedures. In this paper, we follow the method proposed by Conesa and Krueger(1999), who compute two stationary equilibria and the transition path by backward
induction. On this transition path, all the markets clear and the above definition ofcompetitive equilibrium is satisfied. To compute the transition, we need to calibrate
initial and final stationary states. We set an initial stationary state in the year 2000 anda final state in the year 2200. In an actual numerical procedure, we compute a detrended
path.

2.6

Policy Experiments

Under the transition path, there are mixed effects of allocation problems such as pureaging, increasing tax burden, and changing factor prices. In particular, the process of
aging in retired households accelerates, and in one of population projections, the fractionof the retirees over the employees exceeds 40%. Thus, the aging of society might at least
temporarily be considerably larger than in the final stationary equilibrium. Therefore,
we need to consider allocations in the transition path from 2000 to 2200.
As a benchmark, we use the medium variant of the population projection by the
National Institute of Population and Social Security Research (NIPSSR). Moreover, forpolicy experiments, we consider four social security reform plans. We compute thefollowing scenarios:9• As a benchmark, the replacement rate is targeted at 50%, i.e., b(τt , Wg,t ) = 0.5.
We investigate the effects of pure aging, namely changes in the payroll tax burdens
and factor price. We use the medium variant of the population projections andassume that the social security fund is zero.
• Social Security Reform I : Given zero social security fund, we consider a gradualcut of social security benefit to half; i.e., the final replacement rate is 25%.
• Social Security Reform II : Given a zero social security fund, we again consider agradual cut of social security benefit to nearly-zero. For computational reason, we
9Conesa and Krueger (1999) consider three cases; (1) a sudden cut in social security benefit, (2)agradual decrease in the replacement rate between 50 years, (3) cut in social security after 20 years.

9set the replacement rate as 0.1% and not as zero. This scenario implies the fullprivatization or transition to the funded social security system.
• Capital Income Tax : We consider the other source of finance to sustain the social
security system. First, we introduce s linear capital income tax to partly financethe social security benefits. The tax rate is set as 30%. In the tax rate, theremaining tax rate is approximately 5%, as mentioned later.
• Consumption Tax : Second, we introduce consumption tax to finance the remaining social security benefits. The consumption tax rate is set as 5%. We choose
the tax rates is such that the remaining payroll tax rate is almost the same asapproximately 5% of the tax rates.

Given these scenarios, we conduct a sensitivity analysis, considering the low and highvariants of the population projections, and the effect of a positive government fund in
the following section.

3

Calibration

3.1

Fundamental Parameters

We solve the model numerically because we have no closed-form solution. First, wecalibrate the fundamental parameters in the model. As a target of the initial stationary
state, we choose the Japanese economy in the year 2000.
The households enter into our economy at age 20, supply labor until age 65, and lives
till age 100 at most(i.e., J = 100, jr = 65). Although many workers compulsory retire
at 60 in Japan, for the year 2000, our model contains a voluntary labor supply even over
the age of 60. Even though we permit households to endogenously supply labor over age
60, a large fraction of the employees voluntary retires because of low productivity.
We assume that instantaneous utility function is of the Cobb=Douglas-type;10¡¢
u cj,t , `¯ − `j,t =hi1−γcσj,t (`¯ − `j,t )1−σ
1−γ.

Following Conesa and Krueger (1999), the relative risk aversion parameter is set at
γ = 2. This value is quite standard in the literature. Abe, Inakura, and Yamada(2007) estimate the fundamental parameters by structural estimation using Japanese
10

Heathcote, Storesletten, and Violante (2005) investigate the importance of insurance on income risks
when the utility function is separable and non-separable with regard to leisure. We use the non-separable
utility function that is used in broad macroeconomics literature because we consider growth economy.
Note that some empirical facts reveal that microeconomic behavior is consistent with the separable
utility function, although it contradicts with a growing economy.

10Panel Study of Consumers data compiled by the Institute of Household Economy, andfind that the relative risk aversion parameter ranges from 2 to 7. A share parameter for
consumption and leisure is set at σ = 0.38. This value is also close to the research by
Abe et al. (2007). Moreover the average hours worked in the model matches the actual
Japanese data using the parameter. Although Abe et al. (2007) estimate the discountfactor and find that the estimate value ranges over 0.95 ∼ 1.003, the rage is somewhat
broad to fix. Alternatively, as will be mentioned later, we use the capital-output ratio
in the model as a target to determine the discount factor, β = 0.984, which is close to
Hayashi and Prescott (2002) who calibrate the Japanese economy and set β = 0.976.
We set `¯ = 3.0 so as to aggregate the labor supply in the model close to one.
Lastly, we choose the parameters for the production function. The capital shareparameter, θ, is fixed at 0.312 to simultaneously match the capital-output ratio andinterest rate in the stationary equilibrium in the year 2000. The depreciation rate istaken from Hayashi and Prescott (2002), and the value is specified at δ = 0.089. Table
1 summarizes all the calibrated parameters.

3.2

Idiosyncratic Income Risk

It is difficult to estimate parameters for idiosyncratic income uncertainty that all households face because of the scarcity of micro data in Japan. Abe and Yamada (2006) isan exceptional work. Ohtake and Saito (1998) find that the logarithm of the varianceof income in Japan increases across age groups. Moreover, they show that the shape of
the age-variance profile is convex over age groups. To account for the convexity of thevariance profile, Abe and Yamada (2006) specify the labor income process and estimate
the parameters In this paper, we use the estimation shown in Appendix Table 2 in Abe
and Yamada (2006). For incorporating the nonlinearity of the income variances, we use
an age-dependent income variance shock.
As defined in Section 2.2.2, we assume that the idiosyncratic labor productivity process follows equation (1) and (2). Abe and Yamada (2006) report on the possibility ofρ ≥ 1 because of the convexity of the variance profile. However, incorporating ρ ≥ 1
makes the numerical computation far more difficult. Thus, we choose the persistence
parameter to be close to one and standard deviation of the persistence shock increases
across age groups(i.e., ρ = 0.98, σz20 = 0.25, σε20 = 0.0497, and 4σz = 0.001). Afterthe specification, we approximate the AR(1) process as 9-state Markov chain by using
Tauchen’s (1986) method.11,12 The initial value of the persistent shock, z20 , is assumed to
be one and the expected values for each age group are always one. The standard deviationof the transitory shock and the fixed effect are estimated to be σκ = σα = 0.135. We approximate the transitory shock which has three states, namely exp κ = {0.80, 1.07, 1.34}
11

Flodén (2007) demonstrates that a modified version of Tauchen’s method might be a better choice
for approximating an AR(1) process with high persistence.
12
For the details of the effect of the persistence parameter ρ on the variances of consumption andincome profile, see Yamada (2007).

11

with probabilities {0.35, 0.57, 0.07} respectively. In the benchmark model, we assume
that there are no fixed effect because of the computational burden.13 Moreover, we
assume that the structure of the income risk does not change over the transition path.
Figure 1 represents the cross sectional variances of the logarithms of consumption andearnings across age groups. Apparently, the simulated income profile traces the actual
variance profile in Japan. On the contrary, the simulated consumption variance profile
is much lower than the actual variance profile because we neglect the fixed effect.

3.3

Average Hourly Wage Profile

The efficiency unit of hourly wage for each age {ηj } determines the average wage profile.
We conduct the calculation following the method proposed by Braun et al. (2006) based
the Report on the Special Survey of the Labor Force Survey by the Statistics Bureau,
the Management and Coordination Agency, Government of Japan.14 Table 2 presents
the average hourly wage for each age group; we use a smoothed profile. The averagehourly wage of the data is taken from the before-tax earnings. Although we do notinclude the income tax explicitly, which implies that the disposable income is large, ourcalibration procedure does no affect our results.

3.4

Demographic Structure

We choose demographic parameters to replicate the actual and projected population
dynamics. The NIPSSR (2002) provides population projections from 2001 to 2050. We
set the survival probability of {φj,t }2050
t=2000 from the value estimated by the NIPSSR. The
data for the year 2000 are already realized, but those after 2001 are estimated values
based on the NIPSSR’s population projections. The fertility rate ψt is taken from eachof the three projections: the high, medium and low variants. Because the population
growth in our model is represented by the growth rate of 0-year old children, we use theratio of the projected population of new born people between period t and t + 1.
Since we need to compute two stationary states and the transition paths, we set theinitial stationary state in the year 2000. Although the population changes from 2000
to 2050 following the projection by the NIPSSR, the population growth rate converges
to zero after the transition. However, the convergence rate is slow and it takes approximately 100 years to reach a new stationary population distribution. Thus, we choose
the final stationary state in 2200. Following Braun et al. (2006), we assume that the
population growth rate converges to zero, ψt = 0, between 2050 and 2060.
One problem that arises here is how to choose an initial population distribution in
the initial stationary state. Apparently, the actual population distribution in 2000 does13

We confirm that the fixed effect does not affect macroeconomic variables significantly. Conesa andKrueger (1999) also show that such fixed effect have a minor role for investigating social security reforms.
14
For details on the calculation, see Hansen (1993) and the Appendix in Braun et al. (2006).

12

not seem to be stationary because of the existence of the baby-boomer generations as
depicted in Figure 2. However, to compute the initial stationary state, a population
distribution is required. Therefore, we assume that the households in our model believethat the actual population in 2000 is stationary.
The projection by the NIPSSR displays three variants: the high, medium and low
population projections. We plot a fraction of the child population (under the age of19), the working population (20 − 65), and retired households (66 − 100) in Figure 2.
The total population in 2000 is normalized as one. In each projection, the fraction ofretired households peaks in 2050, and thereafter, the rate converges to a new stationary
state. In the low variant, the fraction of the retired households reaches over 40% andthe working population sharply decreases with fewer birth.

3.5

Macroeconomic Variables as a Target

Finally, we choose the target macroeconomic variables to the calibrate Japanese economy.
In Japan, the social security trust funds have two components; the fund from (1) Kokumin Nenkin Hoken (9, 800 billion yen) and from (2) Kosei Nenkin (136, 900 billion yen).
Because Japan’s GDP in 2000 was approximately 502, 000 billion yen, the targeted ratio
for the social security fund in initial stationary state is set at Wg,t /Yt = 136.9/502.78.
In our model, the TFP growth rate is exogenously given and is perfectly forecastable.
From Hayashi and Prescott (2002), Chen et al. (2006a) and Braun et al. (2006) the³´ 1
At+1 1−θTFP growth is set at At= 1.01, and A0 is normalized as one.

4

Factor Prices and Social Welfare in a Transition Economy

4.1

Stationary State

To conduct our computational task, we need to know the stationary states of the initialand final periods. Before analyzing the transition path, we compare these stationary
states. Table 3 and 4 summarize the equilibrium factor prices, the aggregate capital andlabor, the payroll tax rate, the Gini coefficients and the macroeconomic statistics foreach cases. These tables contain several facts on the aging society and social security
reforms.
First, we confirm that the target macroeconomic variables of our model, the capitaloutput ratio and the equilibrium interest rate in the year 2000, are actually close to one
in Japan, as shown in Hayashi and Prescott (2002). The K over Y is 2.42 and the equilibrium capital return is approximately 4% in our model. Based on the medium variant13

of the population projection, there is further capital deepening in 2200. The capitaloutput ratio increases by 3.11% and the interest rate decreases by 39 basis points.15
In the benchmark case with the medium variant projection, only source for change ofmacro statistics is demographic structure, which also implies an increasing payroll tax
rate. From Table 3 and 4, we confirm that the capital deepening is caused by the
reduction of aggregate supply and not due to increased savings.
Second, the reduction of the replacement rate clearly promotes capital deepening
further because of the strong incentive to save for retirement. For comparison, we
compute a stationary equilibrium with the 25% and almost-0% replacement rate cases
for the year 2000. If the social security system does not exist (or is a fully funded system)
and all households must accumulate wealth for retirement, the capital-labor ratio is
considerably higher than that in the benchmark case, and the equilibrium interest rate
becomes 1.38%. Moreover, because capital deepening occurs in 2200, the equilibrium
net interest rate will fall below 1% when the replacement rate is 0.1%.
Two tax reforms that provide another source of finance for social security benefit
demonstrate different effects on households’ decisions. Note that the consumption tax
of 5% and the capital income tax of 30% implies that the remaining payroll tax is
approximately 5%. However these taxes have opposite effects on labor supply decisions.
When new financial resources are obtained from capital income tax, households increase
their labor supply. On the contrary, they choose not to work more when a source offinance is obtained through consumption tax. Moreover, the aggregate labor does not
move in the same direction as the number of hours worked, i.e. ch(L/N )6=ch(hours).
The aggregate labor does not increase as proportionately as the number of hours worked
because the less efficient or more aged workers increase their labor supply. Therefore,
even if consumption tax and capital income tax contribute the same to social security
fund, they have opposite effects on macro statistics. In other words, tax reforms changes
the labor profile and the substitution effect dominates the income effect.
Finally, we focus on wealth inequality because our model contains not only intergenerational but also intragenerational heterogeneity. The wealth Gini coefficient of total
population and that of 30-65 is close to the one observed in Japan. Takayama and Arita(1994) show that the Gini coefficient of the total wealth inequality (Shomi Shisan) was
estimated at 0.639 for the year 1989. Unfortunately, although the Gini coefficient ofwealth inequality for the total population is close to the Japanese data, that in the samegeneration is smaller, especially in middle-aged and old households. Thus, as comparedwith the actual data, intragenerational wealth dispersion is equal if we adjust the income
inequality based upon the estimation of Abe and Yamada (2006). A notable feature in
the intragenerational inequality is that the introduction of the consumption tax induceshigh wealth inequality in young generations.
15

The ch(•) in Table 3 and 4 represents percentage changes compared with the benchmark case in theyear 2000.

14

4.2

Benchmark Case

We will begin by considering the transition path of equilibrium factor prices, payrolltax, output per capita, capital, labor and the Gini coefficients in the benchmark case,illustrated in Figure 3. The transition path does not seem to be monotonic due tothe baby boomers and their children (baby boomer Jr.). As the economy has been
aging for 50 years, there is a sharp decrease in the equilibrium interest rate owing tocapital deepening. In contrast, the equilibrium wage increases up to 5% as comparedto the case in the initial stationary state. According to the decrease in the interest
rate, the Gini coefficient of the total population also reduces weakly over 50 years. The
wealth inequality moves close to the interest rate, because a high interest rate implies
a strong incentive to save. The high saving rate makes the wealthy households richer;however, unfortunately, the unlucky households who face a liquidity constraint cannot
save further. Therefore, the high interest rate leads to wealth inequality. We discuss
this point further in the following section.
To balance the government’s budget in equation (5) for the social security system, thepayroll tax rate must increase up to 18%. Although the output per capita decreases by
20% in the course of aging over 50 years and subsequently recovers weakly, there is morethan one reason for output decreases. The rise of the capital-output ratio stems from thereduction in the per capita labor supply and a weak increase in asset holdings. A sharp
rise in the payroll tax rate reduces the incentives to supply labor. On the contrary, theper capita asset increases because a fraction of near-retirement households increases as
depicted in Figure 2. Since asset holdings reach a peak in around the age of 60 years,
demographic changes affect the per capita capital. The labor supply of households
tends to monotonically decrease because the labor supply is a decreasing function ofasset holdings and due to the effect of aging. The wage increases also contribute toasset holdings for young households, which implies that such households prefer leisurein their old age. Therefore, in order to understand the pure effects of aging, we need toconsider the age profiles with respect to labor supply and asset holdings. We present
these profiles later.

4.3

Social Security Reform: Changes in the Replacement Rate

Sustaining current social security system causes a serious problem with respect to intergenerational equity. Specifically, the payroll tax creates a disincentive for labor supply
and the output per capita decreases. To ease the increasing burden of the payroll tax,
we consider two experimental cases: (1) the reduction of the replacement rate by half,and (2) a negligible replacement rate, i.e., 0.1%.16 The latter case implicitly assumes
the full privatization of the social security system or a transition to the funded system.

16

Nishiyama and Smetters (2005b) carefully investigate the relationship between the privatization ofsocial security system.and its efficiency gain.

15

First, we plot the transition paths of the macro variables with social security reform
I in Figure 4. Capital deepening persists and intensifies for the first 50 years and thewage level correspondingly increases by 10% over the transition path. Accordingly, thewealth inequality decreases. The payroll tax rate does not exceed 12%, and the peakof the payroll tax rate does not correspond to aging. The curve depicting the output
per capita is flatter than in the benchmark case since the per capita capital supply islarger by 15% and the aggregate labor does not decrease to great extent. Households
supply their labor primarily in the case of a relatively small payroll tax and a highwage level. Moreover, asset accumulation for retirement improves the aggregate capital
supply over a period of 40 years. In other words, future generations do not suffer as
much from aging when measured in terms of per capita GDP due to both the reductionof the payroll tax and the upward trend of economy-wide wage level. We investigate thewelfare implication of the social security reform in the following section.
The abolishment of the social security system results in substantial effects on thefactor prices as depicted Figure 5. On the transition path, the negative real return on
capital reduces further and the equilibrium wage sharply rises over 20%. As a result, thewealth inequality measured by the Gini coefficient for the total population also decrease
sharply corresponding with the interest rate. The payroll tax rate peaks at 10%.
Middle-aged and old households hold greater assets because of the sharp rise of theequilibrium wage level. Owing to the lower payroll tax, the after tax earnings do notdecrease even if the labor supply of young households is reduced. Not surprisingly, theper capita output increases for this transition path. Therefore, in this case, it can be said
that the future generation receives benefits from these social security reforms. Note that
unlike the young and future households at present, the middle-aged and old households
do not necessarily benefit from this reform in 2000. This is similar to the problem of oldhouseholds in the initial stationary sate in the usual overlapping generations model.

4.4

Capital Income Tax and Consumption Tax

For the last two scenarios, we consider the other financial schemes for social security
reform. The government collects social security tax not only by means of the payroll
tax but also by consumption or capital income tax. These financial sources of social
security benefits are different from that of the payroll tax on labor income because
all households including retired households are required to contribute to sustaining thesocial security system. As is already well known, the consumption tax does not distortthe intertemporal consumption choice after the introduction of the tax system. On
the contrary, the capital income tax affects disincentives for savings. From the above
definition of social security reform, inducing capital deepening may be beneficial foryoung households. Thus, the capital tax may not improve welfare of the economy,
especially of young households. However, recent research on capital tax reveals that16

good incentives for dissaving leads to increased labor supply and improves welfare.17
Moreover, by using a model with idiosyncratic income risk and precautionary saving,
Aiyagari (1995) shows that reducing the overaccumulation of capital could improves
welfare. Therefore, we consider the consumption and capital income taxes.
First, we examine the case where in the government introduced linear capital income
taxation in 2001, and thereafter, the linear capital tax rate is constant at 30%. All
macro statistics are summarized in Figure 6. Since a part of the social security benefits
is financed by the capital income tax, the maximum payroll tax rate does not exceed
16%, which is small as compared with the benchmark case. A notable feature of thecapital income tax is that it has a relatively small effect on the factor prices path. While
aging leads to capital deepening as depicted in Figure 3, the introduction of the capitaltaxation partially offset the capital deepening and corresponding factor prices becomerather flat over 50 years. Both the aggregate capital and labor simultaneously decrease
across the transition years and the Gini coefficient of wealth weakly increase over theyears. Since the introduction of capital income tax is immediately terminated, the precapita output sharply rises in 2001.
Next, we consider the case where the government introduces the consumption tax
in 2001 as a financial source of the social security system, and subsequently, the casewhere the consumption tax rate is set at 5%. The transition path of macroeconomic
statistics with consumption tax is close to the benchmark case as in Figure 7. The
maximum payroll tax does not exceed 14%, which is considerably smaller than that of
the benchmark case. Moreover, the payroll tax is lower than the case of capital income
tax. Further, the output per capita is higher than the benchmark case. Therefore, bothconsumption and capital income tax improves the output per capita although both havedifferent effects on factor prices.

5

Welfare Implications under the General and Partial Equilibrium Transition Paths

5.1

Evaluation of Welfare

For a comparison of the welfare implication of social security reform and aging, we needa criterion for evaluation the intergenerational and intracohort households. To evaluate
the welfare of households, following Aiyagari and McGrattan (1998), we employ expectedvalue of the initial period as follows:X
Evt (a20 , s20 ) =π(s)vt (0, s20 ).
(9)This criterion implies that we use a measure of the expected value of households whoenter the economy in period t at age 20. In other words, it is a life time discounted
17

For example, see Kocherlakota (2006).

17

value of each cohort before entering the economy. By assumption, the households have
no wealth, but they realize some labor productivity of the fixed effect. This welfare
measure also implies Rawls’s “veil of ignorance”. Note that the criterion does not actually measure households who have already entered the economy, i.e., the initial oldhouseholds. To partially avoid the problem partially, we compute the cohort’s welfare
before the year 2000. Moreover, following Conesa and Krueger (1999), we compute thefollowing welfare measureµ
EV (a20 , s20 ) =EvtReform (a20 , s20 )EvtBench (a20 , s20 )1¶ σ(1−γ),(10)which compare the consumption equivalent variation of cohorts between the benchmark
and some social security reform.
Figure 8 depicts the expected value of each cohort using the welfare criteria of equation (9) and (10). The horizontal axis represents the year that a household enter the
economy at age 20. In the benchmark case with the medium variant of the population
projection and no social security reform, the cohort’s welfare decreases for the aging period of 50 years and reaches the lowest point around 2050. A social security reform planwith replacement rate of 25% and 0.1% exacerbate the situation in new born households
that have already entered the economy, and improves that of the younger and future generations. Note that households who enter the economy in year 2000 do not benefit fromthe reform because the burden is doubled with the transition to the funded system. On
the contrary, the introduction of capital income tax weakly improves the welfare of thecurrent young and future generation households. Although capital income tax improves
the social welfare of the young and future cohorts, this does not imply that it results
in a Pareto improvement since the standard overlapping generations model contains oldhouseholds in the initial stationary state. A new financial scheme of social security financed by the consumption tax does not improves the welfare even of young and future
households. There are two reasons for not improving the welfare. First, the consumption
tax disperse the wealth inequality measured by Gini coefficients as described in Table
3. Such wealth inequality induces consumption inequality and declines the welfare ofcohorts. Another reason for this is the substitution effect of the consumption profile.
This is discussed in detail in later.

5.2

The Transition Path without Factor Price Adjustment

Before analyzing the details of the households behavior on the transition path, we needto consider the effect of factor price changes. Factor price dynamics constitute the key
to understanding the dynamics of social security reform in our paper. To analyze thepure effect of social security reform, we need to eliminate the factor price dynamics. We
investigate the transition dynamics with a fixed factor price path in a partial equilibrium

18

path or small-open economy.18 We fix the transition paths of factor prices to that of thebenchmark case. The replacement rate and each taxes are the same as in each reformcases.
Figure 9 is a revision of Figure 8 with a factor prices path that is the same as that in
the benchmark case. If there are no factor price changes, the welfare implication of thereforms change. If the replacement rate is reduced by the government, the old households
do not suffer as much damage by this reform. Moreover, there is a weak improvementin the welfare of future generations measured by equation (10). The social security
reform plan with capital income tax leads to a considerable improvement in the future
generations’ welfare relative to the case in a closed economy. The consumption equivalentvariations increase by approximately 2.5%. On the other hand, the consumption tax does
not improve their welfare even in this case though the reduction is small relative to thegeneral equilibrium model. Generally, the factor prices are affected by several factors, forexample, monetary policy, foreign capital flows, immigration, and changes in capital orlabor intensive technology. Therefore, the partial equilibrium path may be the possible
case.

5.3

Each Profiles under the Transition Path

Figure 10 − 12 plots a cross sectional profile of earnings, assets, and consumption. In
the benchmark case, although the earnings profiles do not change considerably over thetransition, middle-aged households consume more whereas retirees consume less. Thus,
the consumption profile acquires a more defined inverted u-shaped. In other words,
households substitute consumption at a young age for that at an old age. Further, theasset profile monotonically decreases over age.
Two social security reforms have the same qualitative features although their magnitude differs. When the replacement rate is reduced to 25% or 0.1%, two opposite effects
exist; income and substitution effects of labor supply. Under the transition path of thereform, middle-aged and old households would want to hold more assets around theirretirement. Because substitution effect dominates the income effect, young households
consume more; in fact they are almost hand-to-mouth consumers. Thus, they have almost zero assets. In their middle age, they work more and accumulate wealth rapidly.
Therefore, under the social security reform, the aggregate labor increases because themiddle-aged and old households increase their labor supply considerably. Because households prefer consuming in the earlier stage of their lives, their welfare increases withpositive discounting.
These figures also reveals why the consumption tax does not improve the households’
welfare. Both consumption tax and capital income tax weakly expand the labor sup18

We do not explicitly consider an open economy although international capital flow have strong effects
on capital return. Krueger and Ludwig (2006) and Attanasio, Kitao, and Violante (2007) investigate
the transition path of factor prices using a multi-country model.

19ply profile. However, consumption and asset profiles of those case differ each other. In
the transition path with capital income tax, the asset profiles monotonically decreases.
In comparison, in the transition path with consumption tax, a household prefers toaccumulate greater assets and consumes less. However, with lower interest rate, theconsumption profile with consumption tax is lower than that with capital income tax.
Moreover, the introduction of consumption tax leads the households consume more in
the early stage of lifecycle. As mentioned before, the consumption inequality of young
generation is larger in the case of consumption tax than that in the benchmark. Therefore, the expected value is lower in the case of consumption tax measured by equation(9). Nishiyama and Smetters (2003) also show that if there is no idiosyncratic risks,
introduction of the consumption tax as a new source of finance for the government expenditure improves welfare of the economy, on the contrary, if the shock is uninsurable,
such reform reduces efficiency.

6

Robustness Check

6.1

Various Estimations on the Population Projections

Our results are based on the medium variant of the population projection obtained
from the NIPSSR. Although the population projection follow a precise demographic
viewpoint, the actual population dynamics in recent years in Japan are rather close to
the low variant scenario. Thus, we perform a robustness check by reporting the transition
path with the low and high variant.
Figures 13 and 14 reveal the transition path with the low and high variants respectively. Not surprisingly, if the speed of aging in these variants is quicker than in themedium variant, there is greater capital deepening. If the low variant is realized, thepayroll tax rate exceeds 20%, which contradicts with the current system in Japan. Correspondingly, the output per capita decreases by over 20%. The governor in Japan claims
that the payroll tax will be fixed a maximum of 18.30%. Thus, some social security
reform is required.

6.2

Positive Government Funds

The social security system in Japan in year 2000 has positive government funds as statedin Section 2. In order to sustain the actual social security system in Japan, we cannot
escape from an increasing payroll tax or the reduction of the replacement rate. To
ensure the fairness of intergenerational allocation, a possible scenario is that where aweak reduction in the government’s fund relieves such intergenerational inequality. We
consider the case where the government cuts down the social security funds for the 50th
year of the aging period.

20

When the government gradually cut down the fund, the payroll tax over the transition
becomes smaller than that in the benchmark. Households allocate most of the increasein the disposable income to savings, which implies that the capital-output ratio increases
as compared with the benchmark as depicted in Figure 15. Therefore, even though there
is some government fund, it is not sufficient for the improvement of intergenerational
inequality.

7

Concluding Remarks

We have investigated the macroeconomic and welfare implications of aging in Japan.
When the low variant of population projection realized, some reform must be inescapable.
Therefore, we consider four social security reform plans: the reduction of the replacement rate to 25%, full privatization, capital income tax, and consumption tax. For ourcriteria of social welfare, we find that capital income tax weakly improves the young
and future cohorts’ welfare. Moreover, consumption tax should not necesarily improves
the welfare, though it increases the per capita output, because of intragenerational heterogeneity. On the other hand, it is not surprising that social security reform withchanges in the replacement rate leads to intergenerational conflicts between the old andfuture generations. As compared with the introduction of capital income tax, partial
privatization will improves the welfare of future cohorts.
An important issue to be considered in implementing these reforms is the change infactor prices and heterogeneity over the transition path. To a sizeable extent, a social
security reform should effect a change in the behavior of households with respect tosavings that will, in turn, change the capital-output ratio. Such effects may enlarge
or offset the effect of social security reform. In our experiments, if the factor priceadjustment is sufficiently effective, the introduction of capital income tax is a morefavorable option. To understand the effect of aging and social security reforms, we focus
on each profiles under transition. The key to understand this effect is the shape of
consumption profiles curves. If the changes in these factors are disregarded, it will leadto a misunderstanding of the aging effect and social security reform.

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24

A

Details of the Model and Numerical Procedures

A.1

Normalization

Because we consider growing economy with population dynamics, we need to detrend
aggregate and individual variables. The Bellman equation after normalization of population and TFP growth are as follows:no¯ − `j,t ) + φβ̃t Evt+1 (ã0 , e0 ; Xt+1 , Ωt+1 )u(c̃,`vt (ãj , sj ; Xt , Ωt ) = max
j,t
0
c̃,ã

subject to(1 + τtc )c̃j,t + (1 + gt )ãj+1,t+1 = (1 + (1 − τta )rt /φj,t )ãj,t + (1 − τtss ) wt ηj ej `j,t ,(1 + τtc )c̃j,t + (1 + gt )ãj+1,t+1 = (1 + (1 − τta )rt /φj,t )ãj,t + wt b(τt , Wg,t ),11where β̃t = β(1 + gt )σ(1−γ) , cj,t /(At1−θ Nt ) = c̃j,t and aj,t /(At1−θ Nt ) = ãj,t . Under thebalanced growth path, aggregate variables (Yt , Kt , Wg,t ) grow at the rate of (1 + gt )(1 +nt ), and the growth rate of aggregate labor Lt is (1 + nt ).

A.2

First-Order Conditions

From the first-order conditions of the Bellman equation, we obtainu0c (c̃j,t , `¯ − `j,t ) − λ(1 + τtc ) = 0,
(1 + gt )∂vt+1 (a0 , s0 )+ φj,t β̃t Ej≤ 0,c(1 + τt )∂a0(1 + (1 − τta )rt /φj,t ) 0∂vt (aj , sj )=uc (c̃j,t , `¯ − `j,t )∂a(1 + τtc )−u0 (c̃j,t , `¯ − `j,t ) + λ(1 − τ ss )wt ηj ej = 0,−u0c (c̃j,t , `¯ − `j,t )`twhere λ is a Lagrange multiplier on a budget constraint.
From the Envelope Theorem, intertemporal and intratemporal first-order conditions
are as follows;¡¢a )r
1 + (1 − τt+1
(1 + gt )t+1 /φj+1,t+10¯
uc (c̃j,t , ` − `j,t )= φj,t β̃t Ej
u0c (c̃j+1,t+1 , `¯− `j+1,t+1 ),c )(1 + τtc )(1 + τt+1[c̃σj,t (`¯ − `j,t )1−σ ]1−γ (1 + gt )c̃j,t
(1 + τtc )
)
¡
¢ ( σ
a )r
[c̃j+1,t+1 (`¯ − `j+1,t+1 )1−σ ]1−γ
1 + (1 − τt+1
t+1 /φj+1,t+1= φj,t β̃tEj,c )(1 + τt+1
c̃j+1,t+1⇒25

u0` (c̃j,t , `¯ − `j,t )u0c (c̃j,t , `¯ − `j,t )=,(1 − τtss )wt ηj ej(1 + τtc )[c̃σj,t (`¯ − `j,t )1−σ ]1−γ (1 − τtss )wt ηj ej[c̃σj,t (`¯ − `j,t )1−σ ]1−γ=σ.
⇒ (1 − σ)c̃j,t(1 + τtc )`¯ − `j,tThen, from the intratemporal first-order conditions, labor supply function is
·
¶
¸
µ
c)
(1
+
τ
1
−
σ
t
`j,t = max `¯ −c̃j,t , 0 ,(1 − τtss )wt ηj ejσ£ ¤where `j,t ∈ 0, `¯ .

A.3

Endogenous Gridpoint Method

Although there are many numerical methods for computing the policy function, we apply
the Endogenous Gridpoint Method by Carroll (2006) because it is a relatively safe andfaster method.19
Define the right hand side of the Bellman equation as
Γt (ãj , sj ) = φj,t β̃t Ej+1 vt+1 (ã0 , s0 ),¢¡a)rt+1 /φj+1,t+1 Ej+1 u0c (c̃j+1,t+1 , `¯ − `j+1,t+1 ),(1 + τtc )Γ0t (ãj , sj ) = φj,t β̃t 1 + (1 − τt+1
and take discretized grids on ã0 ∈ [a, a]. We set the number of grids to be 100. From
above, the intertemporal first-order condition is rewritten as follows.
Γ0t (ãj , sj )u0c (c̃j,t , `¯ − `j,t )=c ).
(1 + τtc )(1 + gt )(1 + τt+1

(11)

Thus, we can compute Γ0t for each discretized state (ãj , sj ). After taking inverse of theutility function, we obtain consumption c̃j for each state. From the Envelope Theorem
and marginal utility function, the first-order condition ish¡¢1−σ i1−γc̃σj+1 `¯ − `j+1
u0c (c̃j+1,t+1 , `¯ − `j+1,t+1 ) = σ.
c̃j+1
Suppose that next period’s consumption and labor supply function is already known as
cj+1,t+1 = gc,j+1 (ãj , sj ),`j+1,t+1 = g`,j+1 (ãj , sj ), if j ≤ jr ,`j+1,t+1 = 0, if j > jr .
wthen, by backward induction, we can compute the Γ0 (ãj , sj ) for each grids {ai0j,t }ni=1
foreach age.

19

For details on the endogeneous gridpoint method with endogenous labor supply, see appendix in
Krueger and Ludwig (2006).

26

A.4

Inverse of the Utility Function¯ j,t )
From the first order condition (11), by taking inverse of the utility function u0c (c̃j,t , `−`with respect to c̃j , we obtain c̃j for each choice variable ãj . Using the Euler equation
for leisure and removing `j,t , we have20µµ
u0c (c̃j,t , `¯ − `j,t ) = c̃−γ
j σ

1−σ
σ

¶

1 − τtc(1 − τtss )wt ηj ej¶(1−σ)(1−γ).

This equation is apparently invertible. Thus, we haveµ 0¶
Γt (ãj , sj )(1 + τtc )i−1c̃j = u ·c ) .
(1 + gt )(1 + τt+1
After that we can directly induce `ij . From the set of {c̃ij , `ij , ãij }, we have new cash on
hand x̃ij = (1 + gt )ãij + čij , where čij ≡ c̃ij + (1 − τtss )wt ηj ej (`¯ − `ij ).

A.5

Transition Function

From the policy function and transition probability of labor productivity π (z 0 |z), transition function Qt (·, ·) of states(a, s) and distribution function over the states can becomputed. Define the probability space as ((A × Z × J ), B ((A × Z × J )) , Φj ) where
B ((A × Z × J )) is a Borel σ-field and Φt (S) is a probability measure over S ∈ B ((A × Z × J )).
The probability measure is defined over individual state and also represents fraction ofhouseholds with state S ∈ B ((A × Z × J )). Because we assume that household ofage j = 0 have zero asset, Φ1 is equal to one on a1,t = 0. The transition function
Qj : (A × Z × J ) × B ((A × Z × J ))) → [0, 1] is defined as(X
π (z 0 |z) if ga,t (aj , sj ) ∈ S, for all j = 1, . . . , J.
Qj ((A × Z × J ), S) =
0
else
0
e ∈B

From the initial distribution Φ1,t , distribution function {Φj,t }Jj=1 for each j transit
by the following equation.
ZΦj+1,t+1 (S) = Qj ((A × Z × J ), S) dΦj,t , (∀B ∈ B ((A × Z × J ))) , j = 1, . . . , J,
Φt+1 = H(Φt )Population dynamics is adjusted by µt , and the growth of TFP is already included.
Thus, this distribution is purely a wealth distribution for each generation.
20

Without labor supply, we can compute the inverse of the utility function from the following equation:σ[c̃σj (`¯ − `˜j )1−σ ]1−γ
σ(1−γ)−1 ¯(1−σ)(1−γ)= σc̃j
`
.
c̃j

27

A.6

Computation of Steady State

Computation of the stationary state is the same as in Aiyagari (1994) model. Though
there are three markets, the factor prices are determined from capital-labor ration K/L.
Moreover, by the Walras law, we concentrate on K/L and governemnt budget celaringof τ ss .21
1. Given initial guess of (K 0 , L0 ), compute Y 0 . We also need initial guess of C 0 for
consumption tax. From calibration of the social security fund, compute Wg0 .
2. Given (r0 , w0 , K 0 , L0 , C 0 , Wg0 ) and exogenous (τ c , τ a ), compute the payroll tax
rate τ0ss from the government budget condition.
¡¢
3. Given r0 , w0 , τ ss,0 ; τ c , τ a , compute the policy function using the EGM and get
distribution function Φ0 for each age.
¡¢
4. Integrating the distribution function Φ0 , get the aggregate capital and labor K 1 , L1 .
¡¢5. If new K 1 , L1 and old (K 0 , L0 ) are sufficiently close to each other stop it, andwe have equilibrium prices for given τ ss,0 .
6. From new equilibrium condition (r1 , w1 , K 1 , L1 , C 1 , Wg1 ), re-compute a new payroll
tax τ ss,1 . Repeat step 3 − 5. If iteration error of τ ss is sufficiently small, stop it.
If criterion of step 5 and 6 are satisfied, we have an equilibrium.
1

Note that all computation above are already detrended by At1−α Nt .

A.7

Transition Dynamics

Aftre computation of the steady state, we compute transtion path of each equilibrium.
Basic idea here is the same as Conesa and Krueger (1999) and Nishiyama and Smetters(2004,2005a).
1. Guess future period T . We assume T = 201.
2. Set exogenous pair of tax rates (τtc , τta ).
3. Guess an equilibrium sequence of {rt , τtss , Lt , Wg,t , ξt ; b(τss,t , Wg,t ), At }Tt=1 among
stationary states.22 Note that we implicitly assume that benefit of social security
and the sequence of TFP {b(τtss , Wg,t ), At }Tt=1 is perfect foresight and exogenously
given. Thus, we have {rt , wt , τtss , b(τss,t , Wg,t ), ξt }Tt=1 which is required to computepolicy function.

21

We take 100 grids on asset a for computing policy function, and to compute the distriution we take5000 grids.
22
For simplicity, we start linear case.

28

4. Because already we know policy function of final stationary state in period T , given
sequences of step 2, compute the policy function from T to t = 1 by backward
induction.
5. Using the obtained policy function, compute the sequence {Kt , wt , Wg,t , b(τtss , Wg,t )}Tt=1
from t = 1 forwardly.
6. Check whether each market clearing conditions and government budget balances
are satisfied. If those are not in equilibrium, up-date the price sequences and repeas
step 3 − 5.23
7. If all markets clear in all periods, stop computation.

23

There are many efficient methods for update the price sequence. For example, Krueger and Ludwig(2006) and Ludwig (2006) uses a modified version of Gauss-Zeidel method for computing the transition
path.

29

B

Figures and Tables

β
0.985

γ
2.0

σ
0.38

θ
0.312

δ
0.089

J
100

jr
65

Table 1: Fundamental Parametersage
20 − 24
25 − 29
30 − 34
35 − 3940 − 44

hourly wage
1, 311
1, 7282, 1202, 5752, 919age
45 − 49
50 − 54
55 − 59
60 − 6465−

hourly wage
3, 0973, 0732, 9181, 9481, 655

Table 2: Average Efficiency for Each Age; yen

30

K/Y
K/L
ch(K/Y ): %r (%)
w
τ ss (%)
K/N
L/N
ch(L/N ): %ch(hours): %Y /NGini (20-100)
Gini (30-65)
Gini (20s)
Gini (30s)
Gini (40s)
Gini (50s)
Gini (60s)

Medium Variant
no fund with fund
2.42
2.48
3.61
3.75
−
2.69
4.01
3.67
1.03
1.04
10.17
9.04
3.50
3.65
0.97
0.97
−
0.25
−
0.41
1.45
1.47
0.596
0.609
0.531
0.545
0.586
0.587
0.589
0.580
0.393
0.417
0.263
0.274
0.303
0.316

Rep. Rate
25% 0.1%
2.63
3.03
4.07
5.02
8.72 25.53
2.97
1.38
1.07
1.14
5.09
0.02
4.10
5.36
1.01
1.07
3.78
9.97
4.35 11.52
1.56
1.76
0.590 0.583
0.549 0.565
0.591 0.605
0.586 0.589
0.420 0.443
0.254 0.232
0.238 0.171

Tax Reformτ a = 30%
2.45
2.24
3.69
3.22
1.54
−7.49
3.81
5.05
1.03
0.99
4.99
5.25
3.58
3.14
0.97
0.97
0.09
0.52
−0.04
0.74
1.46
1.40
0.605
0.611
0.543
0.548
0.643
0.588
0.634
0.580
0.409
0.424
0.267
0.276
0.302
0.314

τ c = 5%

Table 3: Stationary State Comparison in 2000

K/Y
K/L
ch(K/Y ): %r (%)
w
τ ss (%)
K/N
L/N
ch(L/N ): %ch(hours): %Y /NGini (20-100)
Gini (30-65)

MediumVariant
2.49
3.77
3.11
3.62
1.04
14.04
3.32
0.88
−9.31
1.31
1.33
0.608
0.555

Rep. Rate
25%
0.1%
2.76
3.28
4.39
5.62
13.96 35.59
2.42
0.62
1.09
1.18
7.02
0.03
3.99
5.44
0.91
0.97
−5.75 −0.14
5.98 13.69
1.45
1.66
0.589 0.589
0.592 0.592

Tax Reformτ a = 30%
2.54
2.32
3.87
3.40
4.90
−3.91
3.40
4.53
1.05
1.01
8.77
9.45
3.40
3.00
0.88
0.88
−9.28
−9.00
1.25
1.94
1.34
1.29
0.618
0.624
0.567
0.571

τ c = 5%

Table 4: Stationary State Comparison in 2200

31

Figure 1: Variance Profiles
Variance of Logarithm of Income Profile

cross cectional Variance

0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
20

25

30

35

40
age
data

45

50

55

60

55

60

simulation

Variance of Logarithm of Consumption Profile

cross cectional variance

0.30
0.25
0.20
0.15
0.10
0.05
0.00
20

25

30

35

40
age
data

45
simulation

50

Figure 2: Population Dynamics in Japan(a) Population Distribution in 2000(b) Population Dynamics: Low Variant

2500

0.7
0.6
population

population (1,000)

2000
1500
1000

0.5
0.4
0.3
0.2
0.1

500

0.0
2000 2020 2040 2060 2080 2100 2120 2140 2160 2180 2200
year

0
0

10

20

30

40

50
age

60

70

80

90

100

0-19

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
2000 2020 2040 2060 2080 2100 2120 2140 2160 2180 2200
year
0-19

20-65

66-10066-100(d)Population Dynamics: High Variant

population

population(c) Population Dynamics: Medium Variant

20-65

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
2000 2020 2040 2060 2080 2100 2120 2140 2160 2180 2200
year
0-19

20-65

66-100Figure 3: Benchmark Case (Medium Variant)
1.06

4.0%

1.05

3.5%

1.04

3.0%

1.03

2.5%

1.02

2.0%

1.01

1.5%

1.00

1.0%

0.99

0.5%

0.98

60%

25%

50%

20%

40%

replacement rate

4.5%

2000

2020

2040
year
interest rate

2060

10%

20%
5%

10%

0.97

0.0%

15%

30%

0%

0%
2000

2080

payroll tax rate

Social Security Systemwageinterest rate

Factor Prices

wage

2020

2040
year

2060

replacement rate

2080

payroll tax rate

Output Per Capita

Wealth Gini Coefficient
1.2
0.62
1.1

0.60
0.58
Gini

1.0
output

0.56
0.54
0.52

0.9
0.8

0.50
0.7

0.48
2000

2020

2040
year
Gini (20-100)

2060

2080

0.6
2000

2020

Gini (30-65)

Capital and Labor: Population Adjusted (K, L)

2040
year

2060

2080

2060

2080

Capital-Output Ratio：K/Y
3.00

1.1

2.50

0.95

1.0

0.90

0.9

0.85

0.8

0.80

0.7

2.00
K/Y

labor (L/N)

1.2

1.00
capital （K/N)

1.05

1.50
1.00

0.75

0.6
2000

2020

2040

2060

year
capital

2080

0.50
0.00
2000

labor

2020

2040
year

Figure 4: Social Security Reform I (25%)
1.12

4.0%

1.10

3.5%

1.08

3.0%

1.06

2.5%

1.04

2.0%

1.02

1.5%

1.00

1.0%

0.98

0.5%

0.96

0.0%

0.94
2000

2020

2040
year
interest rate

2060

60%

14%

50%

12%
10%

40%

replacement rate

4.5%

8%

30%
6%

20%

4%

10%

2%

0%

0%
2000

2080

payroll tax rate

Social Security Systemwageinterest rate

Factor Prices

2020

2040
year

2060

replacement rate

wage

2080

payroll tax rate

Output Per Capita

Wealth Gini Coefficient
1.2
0.62
1.1

0.60
0.58
Gini

1.0
output

0.56
0.54
0.52

0.9
0.8

0.50
0.7

0.48
2000

2020

2040
year
Gini (20-100)

2060

2080

0.6
2000

Capital and Labor: Population Adjusted (K, L)

1.10

1.0

2.50

1.05

0.9

1.00

0.8

0.95

0.7

1.00

0.6

0.50

0.90
2040

2060

year
capital

2080

K/Y

3.00
labor (L/N)

3.50

1.1

capital （K/N)

1.2

1.15

2020

2060

2080

2060

2080

2.00
1.50

0.00
2000

labor

2040
year

Capital-Output Ratio：K/Y

1.20

2000

2020

Gini (30-65)

2020

2040
year

Figure 5: Social Security Reform II (0.1%)
60%

12%

1.20

50%

10%

40%

8%

30%

6%

20%

4%

0.95

10%

2%

2080 0.90

0%

1.10
1.05

wage

1.15

1.00

2020

2040

2060

replacement rate

1.25

interest rate

4.5%
4.0%
3.5%
3.0%
2.5%
2.0%
1.5%
1.0%
0.5%
0.0%
-0.5% 2000

Social Security System

0%
2000

2020

2040
year

year
interest rate

payroll tax rate

Factor Prices

wage

2060

replacement rate

2080

payroll tax rate

Output Per Capita

Wealth Gini Coefficient
1.2
0.62
1.1

0.60

1.0

0.56
output

Gini

0.58

0.54
0.52

0.9
0.8

0.50
0.7

0.48
2000

2020

2040
year
Gini (20-100)

2060

2080

0.6
2000

Capital and Labor: Population Adjusted (K, L)

1.6

1.2

4.00

1.1

3.50

0.9

0.8
0.6

0.8

0.4
0.2
0.0
2060

year
capital

2080

2060

2080

2.00
1.50

0.7

1.00

0.6

0.50
0.00
2000

labor

2080

2.50
K/Y

1.0

2040

2060

3.00

1.0

1.2

labor (L/N)

capital （K/N)

1.4

2020

2040
year

Capital-Output Ratio：K/Y

1.8

2000

2020

Gini (30-65)

2020

2040
year

Figure 6: Capital Income Tax
Social Security System

1.05

3.5%

1.04

3.0%

1.03

2.5%

1.02

2.0%

1.01

1.5%

1.00

1.0%

0.99

0.5%

0.98

0.0%

0.97
2000

2020

2040
year

2060

interest rate

60%

16%
14%

50%

12%
40%

10%
8%

30%

6%

20%

payroll tax rate

1.06

4.0%

replacement rate

4.5%

wage

interest rate

Factor Prices

4%

10%

2%
0%

0%
2000

2080

wage

2020

2040
year

2060

replacement rate

2080

payroll tax rate

Output Per Capita

Wealth Gini Coefficient
1.05

0.64

1.00
0.62

0.95

0.60
Gini

0.90

0.58

output

0.85

0.56
0.54

0.75

0.52

0.70

0.80

0.50

0.65
2000

2020

2040
year
Gini (20-100)

2060

2080

0.60
2000

2020

Gini (30-65)

Capital and Labor: Population Adjusted (K, L)

2040
year

2060

2080

2060

2080

Capital-Output Ratio：K/Y
3.00

1.1

2.50

0.95

1.0

0.90

0.9

0.85

0.8

0.80

0.7

2.00
K/Y

labor (L/N)

1.2

1.00
capital （K/N)

1.05

1.50
1.00

0.75

0.6
2000

2020

2040

2060

year
capital

2080

0.50
0.00
2000

labor

2020

2040
year

Figure 7: Consumption Tax
Social Security System

3.5%
interest rate

3.0%
2.5%
2.0%
1.5%
1.0%
0.5%
0.0%
2000

2020

2040
year

2060

interest rate

60%

14%

50%

12%
10%

40%

replacement rate

4.0%

wage

1.06
1.05
1.04
1.03
1.02
1.01
1.00
0.99
0.98
0.97
0.96

4.5%

8%

30%
6%

20%

4%

10%

2%

0%

0%
2000

2080

payroll tax rate

Factor Prices

wage

2020

2040
year

2060

replacement rate

2080

payroll tax rate

Output Per Capita

Wealth Gini Coefficient
1.05
0.62

1.00
0.95

0.58

0.90

0.56

0.85

output

Gini

0.60

0.80

0.54

0.75

0.52

0.70

0.50

0.65
2000

2020

2040
year
Gini (20-100)

2060

2080

0.60
2000

2020

Gini (30-65)

Capital and Labor: Population Adjusted (K, L)

2040
year

2060

2080

2060

2080

Capital-Output Ratio：K/Y
3.00

1.1

2.50

0.95

1.0

0.90

0.9

0.85

0.8

0.80

0.7

2.00
K/Y

labor (L/N)

1.2

1.00
capital （K/N)

1.05

1.50
1.00

0.75

0.6
2000

2020

2040

2060

year
capital

2080

0.50
0.00
2000

labor

2020

2040
year

Figure 8: Welfare Comparison (Cohort at Age 20)
year-23.00-23.50 1950

1970

1990

2010

2030

2050

2070

2090

expected value-24.00-24.50-25.00-25.50-26.00-26.50EV

benchmarkss reform I

ss reform II

cap. taxcon. tax

1.20
1.15
1.10
1.05
1.00
0.95
0.90
0.85
0.80
1950

1970

benchmark

1990

2010

ss reform I

2030
year
ss reform II

2050

2070
cap. tax

2090
con. tax

Figure 9: Welfare Comparison (Partial Equilibrium)year-23.50
1950

1970

1990

2010

2030

2050

2070

2090

expected value-24.00-24.50-25.00-25.50-26.00EV

benchmarkss reform I

ss reform II

cap. taxcon. tax

1.20
1.15
1.10
1.05
1.00
0.95
0.90
0.85
0.80
1950

1970

benchmark

1990

2010

ss reform I

2030
year
ss reform II

2050

2070
cap. tax

2090
con. tax

Figure 10: Earnings Profiles
Medium Variant

2.5

Social Security Reform I

2.5

2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5
0.0

0.0
20

25

30
2000

35

40

45
2025

50

55

20

60 65
2050

25

30

35

40

2000

Social Security Reform II

45

50

55

2025

60

65

2050

Capital Income Tax

2.5

2.5

2.0

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0
20

25

30

35

40

2000

45

50

55

2025

60

65

2050

Consumptin Tax
2.5
2.0
1.5
1.0
0.5
0.0
20

25

30
2000

35

40

45
2025

50

55

60
2050

65

20

25 30
2000

35

40

45
2025

50

55

60
2050

65

Figure 11: Asset Profiles
Medium Variant

16.0

16.0

14.0
12.0
10.0
8.0

14.0
12.0
10.0
8.0

6.0
4.0
2.0
0.0

6.0
4.0
2.0
0.0
20 25 30 35 40 45 50 55 60 65 70 75 80
2000

16.0

2025

20 25 30 35 40 45 50 55 60 65 70 75 80

2050

Social Security Reform II

Social Security Reform I

2000

2050

Capital Income Tax

16.0
14.0
12.0

14.0
12.0
10.0

2025

10.0
8.0
6.0
4.0

8.0
6.0
4.0
2.0

2.0
0.0

0.0
20 25 30 35 40 45 50 55 60 65 70 75 80
2000

2025

2050

Consumptin Tax

16.0
14.0
12.0
10.0
8.0
6.0
4.0
2.0
0.0

20 25 30 35 40 45 50 55 60 65 70 75 80
2000

2025

2050

20 25 30 35 40 45 50 55 60 65 70 75 80
2000

2025

2050

Figure 12: Consumption Profiles
Medium Variant

1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0

1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0

20 25 30 35 40 45 50 55 60 65 70 75 80
2000
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0

2025

2025

2050

Consumptin Tax

1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0

20 25 30 35 40 45 50 55 60 65 70 75 80
2000

2025

2000

2050

2025

2050

Capital Income Tax

1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0

20 25 30 35 40 45 50 55 60 65 70 75 80
2000

20 25 30 35 40 45 50 55 60 65 70 75 80

2050

Social Security Reform II

Social Security Reform I

20 25 30 35 40 45 50 55 60 65 70 75 80
2000

2025

2050

Figure 13: Transition Path (Low Variant)
1.06

4.0%

1.05

3.5%

1.04

3.0%

1.03

2.5%

1.02

2.0%

1.01

1.5%

1.00

1.0%

0.99

0.5%

0.98

60%

25%

50%

20%

40%

replacement rate

4.5%

2000

2020

2040
year

2060

interest rate

10%

20%
5%

10%

0.97

0.0%

15%

30%

0%

0%
2000

2080

payroll tax rate

Social Security Systemwageinterest rate

Factor Prices

wage

2020

2040
year

2060

replacement rate

2080

payroll tax rate

Output Per Capita

Wealth Gini Coefficient
1.05
0.62

1.00

0.60

0.95

0.58
Gini

0.90

0.56

output

0.85

0.54
0.52

0.75

0.50

0.70

0.80

0.48

0.65
2000

2020

2040
year
Gini (20-100)

2060

2080

0.60
2000

2020

Gini (30-65)

Capital and Labor: Population Adjusted (K/N, L/N)
1.2

3.00

1.10

1.1

2.50

0.9

2080

0.95
0.8

2060

2080

2.00
K/Y

1.00

labor (L/N)

capital （K/N)

1.0

0.90

2060

Capital-Output Ratio: K/Y

1.15

1.05

2040
year

1.50
1.00

0.7

0.85
0.80

0.6
2000

2020

2040

2060

year
capital

2080

0.50
0.00
2000

labor

2020

2040
year

Figure 14: Transition Path (High Variant)Social Security System
1.05

4.0%

1.04

3.5%

1.03
40%

1.02

2.5%

1.01

2.0%

1.00

1.5%
1.0%

0.99

0.5%

0.98

0.0%

0.97
2000

2020

2040
year

2060

interest rate

20%
18%
16%
14%
12%
10%
8%
6%
4%
2%
0%

50%

wage

interest rate

3.0%

60%

replacement rate

4.5%

30%
20%
10%
0%
2000

2080

wage

2020

2040
year

2060

replacement rate

payroll tax rate

Factor Prices

2080

payroll tax rate

Output Per Capita

Wealth Gini Coefficient
1.05
0.62

1.00

0.60

0.95

0.58
Gini

0.90

0.56

output

0.85

0.54
0.52

0.75

0.50

0.70

0.80

0.48

0.65
2000

2020

2040
year
Gini (20-100)

2060

2080

0.60
2000

2020

Gini (30-65)

Capital and Labor: Populatoin Adjusted (K/N, L/N)

2060

2080

2060

2080

Capital-Output Ratio: K/Y
1.2

3.00

1.00

1.1

2.50

0.95

1.0

0.90

0.9

0.85

0.8

0.80

0.7

2.00
K/Y

labor (L/N)

1.05

capital （K/N)

2040
year

1.50
1.00

0.75

0.6
2000

2020

2040

2060

year
capital

2080

0.50
0.00
2000

labor

2020

2040
year

Figure 15: Transition Path with Government Fund
1.05

3.5%

1.04

3.0%

1.03

2.5%

1.02

2.0%

1.01

1.5%

1.00

1.0%

0.99

0.5%

0.98

60%

25%

50%

20%

40%

replacement rate

4.0%

2000

2020

2040
year

2060

interest rate

10%

20%
5%

10%

0.97

0.0%

15%

30%

0%

0%
2000

2080

payroll tax rate

Social Security Systemwageinterest rate

Factor Prices

wage

2020

2040
year

2060

replacement rate

2080

payroll tax rate

Output Per Capita

Wealth Gini Coefficient
1.05
0.62

1.00

0.60

0.95

0.58
Gini

0.90

0.56

output

0.85

0.54
0.52

0.75

0.50

0.70

0.80

0.48

0.65
2000

2020

2040
year
Gini (20-100)

2060

2080

0.60
2000

1.2

1.2

2.75

1.0

1.1

2.70

0.8

1.0

0.6

0.9

0.4

0.8

0.2

0.7

2.45

0.0

0.6

2.40

2020

2040

2060

year
capital

2080

2060

2080

2060

2080

2.65

K/Y

2.60
2.55
2.50

2.35
2000

labor

2040
year

Capital-Output Ratio: K/Ylabor (L/N)

capital （K/N)

Capital and Labor: Population Adjusted (K/N, L/N)

2000

2020

Gini (30-65)

2020

2040
year