Pension Reform for an Aging Japan:
Welfare and Demographic Dynamics
Akira Okamoto (Okayama University)
March 31, 2024
Okayama University Discussion Paper No. I－118

Pension Reform for an Aging Japan:
Welfare and Demographic Dynamics *Akira Okamoto**
Okayama University
March 31, 2024

Abstract
Life expectancy in Japan exceeds that in other developed countries; however, thestandard starting age for receiving public pension benefits remains at 65 years. This
paper uses an extended lifecycle general equilibrium model with endogenous fertility toinvestigate how a rise in the starting age for pension benefits will affect individual
welfare and future demographic dynamics. Our simulation analysis indicates that percapita welfare will increase if the starting age for pension benefits is raised to 68 or 70.
The higher the employment rate of individuals aged 65 and older, the higher the percapita welfare and the future population level. Conversely, when the employment rate
for elderly individuals remains at 50% (Japan’s current level), the total population would
decrease in the long run, compared to the benchmark case with the normal pensionable
age of 65. This paper demonstrates the importance of enhancing the employment rate ofelderly individuals while raising the standard pensionable age in Japan.
Keywords: Public pension; welfare; demographic dynamics; Japanese economy;
simulation analysis
JEL classification: H30; C68*

This paper was presented at the AGI (Asian Growth Research Institute) Seminar in March in 2024. I am gratefulfor the insightful comments and suggestions of Professors Tatsuo Hatta and Charles Yuji Horioka, and the otherseminar participants. I also want to thank Associate Professor Yoshinari Nomura (Okayama University), whohelped me in writing the computer program used for the simulations. Additionally, I wish to acknowledge thefinancial support from the Ministry of Education, Culture, Sports, Science and Technology in Japan (Grant-in-Aid
for Scientific Research (C) No. 20K01679, 23K01350).
**
Professor, Faculty of Economics, Okayama University, Japan. Tsushima-naka, Okayama 700-8530, Japan.
E-mail: okamot-a@okayama-u.ac.jp

11. Introduction
Many developed countries currently face problems related to aging populations. Many countries, such as
Germany, France, the United Kingdom, and the United States (U.S.), have already decided to raise thestandard age at which people begin receiving public pension benefits to accommodate longer life spans;however, although Japan has a longer life expectancy than other advanced countries, the standard age forpension benefits is still 65 years. Balancing public pension finance would necessitate raising the startingage for pension benefits in Japan from 65 to, e.g., 68 or 70 years. When the normal starting age rises, it
may be crucial for individuals aged 65 and above to secure employment. According to the Ministry ofInternal Affairs and Communications (2023a), the employment rate for people aged 65 to 69 in Japan was

50.8% in 2022, which has increased recently. An increasing number of seniors remain in Japan’s
workforce to compensate for the dwindling numbers of people in the working-age population. In Japan, a
new law (the amended Act on Stabilization of Employment of Elderly Persons) came into effect in April
2021, which prescribes employment until 70 years of age as one’s duty to make efforts. The measure
aims to compensate for the growing labor shortage in an aging and shrinking population by encouraging
elderly individuals to work longer and help support the social security system under demographicpressure.
Our model can evaluate public pension reforms from two viewpoints: individual welfare and future
demography. We use the lifecycle general equilibrium simulation model of overlapping generations,developed by Auerbach and Kotlikoff (1983a, 1983b) and applied in Auerbach and Kotlikoff (1987),
Auerbach et al. (1989), Altig et al. (2001), Homma et al. (1987), Ihori et al. (2006, 2011), and Okamoto(2013, 2020, 2021, 2022, 2024).1 Using an extended Auerbach–Kotlikoff dynamic simulation model, weinvestigate the quantitative effects of a rise in the normal starting age for public pension benefits on percapita welfare and future population.
The simulation model in Okamoto (2020, 2021) introduced the number of children freely chosen byhouseholds into the utility function, thus incorporating endogenous fertility and future demographic
dynamics. Furthermore, Okamoto (2022, 2024) incorporated two representative households into a cohort:the low-income class (high school graduates) and the high-income class (university graduates). Okamoto

1

The lifecycle model is considerably applicable to the Japanese economy. According to Horioka (2021), almost allof the available evidence suggests that the selfish lifecycle model applies, to some extent, in all countries and that
there is more consistent support for this model in Japan than in the United States and other countries.

2

(2022, 2024) introduced the descendent link between a parent and children in the extended frameworkwith endogenous fertility. This approach provided the exogenous transition probabilities from theparent’s income class to the same (or the other) income class to which their children would belong. In
other words, Okamoto’s (2022, 2024) simulation model with endogenous fertility incorporated thedescendent income inequality from parents to their children.
This paper’s analytical model is based on Okamoto (2022, 2024); we extended the model to freely
increase the standard starting age for receiving public pension benefits and to change the employment
rate for elderly individuals. Our study can analyze these impacts on per-capita utility and future
population dynamics. We quantitatively analyze how increasing the starting age for receiving public
pension benefits in Japan impacts the future population levels and the welfare of all generations,
including the future and the current generations. Specifically, we examine the effect of standard starting
ages and employment rates for elderly individuals on the per-capita utility and the demographic
dynamics for the transition process from 2022 to 2300. Thus, this paper analyzes the long-run impact on
economic growth, welfare, and population levels, assuming alternative standard starting ages on pensions
and employment rates for elderly individuals.
Finally, this study introduces an additional government institution, the Lump Sum Redistribution
Authority (LSRA). Rises of the standard starting age for receiving public pension benefits generally
improve the welfare of some generations but reduce that of others. If combined with redistribution fromwinning to losing generations, such changes may offer the prospect of Pareto improvements; however,
without implementing intergenerational redistribution, potential efficiency gains or losses cannot be
estimated. Therefore, like Auerbach and Kotlikoff (1987) and Nishiyama and Smetters (2005), we
introduce the LSRA as a hypothetical government institution that distinguishes potential efficiency
gains/losses from possible offsetting changes in the welfare of different generations. To isolate pureefficiency gains or losses, we consider simulation cases via LSRA transfers where the normal starting age
for receiving public pension benefits is raised to 68 and 70. The introduction of LSRA transfers enables
us to examine policy proposals from a long-term perspective, considering the welfare of current andfuture generations. Because of its ability to quantify alternative policies from a long-term perspective, we
can present concrete and valuable policy proposals.
To investigate the abovementioned issue, we first examine the projected trend of Japan’s aging
population and then explain Japan’s public pension system compared to other advanced countries.
3

Finally, we address the recent trend in the employment rate for elderly individuals in Japan.

1.1. Demographics in Japan
Figure 1 illustrates Japan’s population data and projections from 1990 to 2100. Actual results on the total
population until 2022 are based on the data from the Statistics Bureau of Japan (2023a). Projections after
2022 are based on data (by medium assumptions on fertility and mortality rates) from the National
Institute of Population and Social Security Research (2023). The population increased monotonically
until 2008, but the trend has reversed since 2008; the number of deaths exceeds births, and the total
population will continue to decrease throughout the rest of the century. According to the official
projection, by 2100, the population will reach 62.8 million, approximately half (50.2%) of the level in
2022 (124.9 million). Low fertility rates and a shrinking number of young females who can bear children
contribute to the ongoing decline in the Japanese population.
Simultaneously, Japan’s population is aging at an unprecedented speed for a developed nation.
Figure 2 presents historical and projected aging rates (defined as the ratio of the population aged 65 orolder against the total population) since 1990 for five major advanced countries. The data are based on
the United Nations (2023). Japan’s aging rate will rise sharply and reach 38.6% in 2065. The figure
shows that the population ratio of elderly individuals will rise remarkably compared to other advanced
countries, such as Germany, France, the United Kingdom, and the U.S. Thus, Japan’s speed andmagnitude of demographic aging are remarkable, even compared to other countries that face similar
challenges.
Figure 3 shows historical and projected total fertility rates since 1990. The rates first decreased,bottoming at 1.26 in 2005 before somewhat recovering and peaking at 1.45 in 2015. After 2015, fertility
rates again decreased, falling to 1.26 in 2022. This figure is the lowest since the start of the Current
Population Survey in 1947, indicating an accelerated birthrate decline. The National Institute ofPopulation and Social Security Research (2023) projects that the total fertility rate will recover, reaching
1.33 around 2035 and 1.36 by 2070.
Finally, we consider the effects of the COVID-19 pandemic (which affected the entire world) on an
aging and depopulating Japan. The spread of COVID-19 has had severe consequences in the form ofaccelerated population decline due to a sharp drop in births. Infection control measures included theprevalence of telework, travel, and fewer events; even if people could meet face-to-face, they wore4

masks and kept their distance (social distance), decisively reducing the opportunities for men and women
to meet. Figure 4 shows historical and projected births and actual marriage results since 2010. First, thenumber of births in Japan was in a critical situation, even before the COVID-19 outbreak, fallingsignificantly below the future population projection (medium projection) by the National Institute ofPopulation and Social Security Research (2017). From there, the number of births in Japan declined
significantly, partly because of the spread of COVID-19. The number of births in 2023 fell to 758,631(preliminary results) from 770,759 in 2022, the lowest since the start of the Current Population Survey in
1899, indicating an accelerated birthrate decline. Moreover, according to the Ministry of Health, Labourand Welfare (2024), the number of marriages (a prerequisite for childbearing in Japan) has decreased
sharply. The number of marriages declined to 489,281 couples (preliminary results) in 2023 from504,930 in 2022, the lowest in the postwar period.

1.2. Public pension in JapanThe current Japanese public pension program is operated fundamentally by a pay-as-you-go (PAYG)
style and is two-tiered: a basic flat pension and an amount proportional to each household’s averageannual labor income. In other words, the pension consists of two parts. The first is a basic pension (kisonenkin), provided for all retirees conditional on paying a required premium. The second is an
employment-based part (kosei nenkin), whose benefits are based on the premium contribution made byeach individual throughout their career. According to the data from the Organisation for Economic
Cooperation and Development (OECD) (2023a), total pension expenditures in Japan are 9.3% of thegross domestic product (GDP) in 2022, whereas it is, on average, 7.7% for the 38 OECD member
countries. Figure 2 illustrates that Japan’s aging rate is expected to rise rapidly compared to otherdeveloped countries. Therefore, Japan’s GDP ratio of total pension expenditures would be increasingly
higher than that of other developed countries.
The Japanese government implemented a significant pension reform in 2004. The purpose of thereform was to make the Japanese public pension system sustainable in the long run. The“macroeconomic slide” was introduced to control Japan’s rapid rise in benefit expenditures. The
mechanism automatically adjusts benefits downward with a rise in average life expectancy and a decline
in insured individuals; however, the macroeconomic slide has been triggered only five times (in 2015,2019, 2020, 2023, and 2024) since the reform was implemented in 2004 because the adjustment is
5

subject to enough inflation as the downward adjustment is restricted in nominal terms; the adjustment
does not exceed the (negative) inflation rate under a deflationary economy. The number of pension
recipients in Japan is currently increasing, and the average number of years of receiving public pensions
is simultaneously increasing as life expectancy continues to rise. The 2004 pension reform set the
timeline in which the contribution rate gradually increased by 0.354 percentage points yearly, from13.58% of earnings to 18.3% in 2017; since 2017, it has been fixed at 18.3%.
The normal retirement age is defined as the age at which individuals are entitled to receive pension
benefits in full, which is set by the policy. In contrast, the age at which individuals exit the labor force
depends on an individual’s choice. They do not have to coincide and differ for most individuals;
however, our simulation results revealed that individuals who maximize their utility work until thecompulsory retirement age (from 64 to 69 for our simulation cases). In the benchmark simulation, thenormal retirement age is assumed to be 65; individuals work up to 64 and start receiving pension benefits
from 65 throughout the entire period. In our model, after households retire at the end of the year in which
they reach compulsory retirement, they immediately start receiving full pension benefits from the
beginning of the following year. Additionally, our model does not assume accelerated or deferred pension
benefits.
The standard starting age for receiving basic and employment-based pensions is currently 65.
Individuals could start claiming benefits as early as 60 years old or delaying the initial take-up until aslate as 75. Early retirement at a reduced benefit is currently possible at 60 in basic and earnings-related
schemes. The benefit is reduced by 0.4% per month of early retirement, i.e., 4.8% per year. Lateretirement is currently possible between 65 and 75; deferral increases the pension benefit by 0.7% permonth, i.e., 8.4% per year; however, most individuals claim benefits at the normal retirement age (65),and very few people wait until after the normal retirement age to receive benefits. According to the
Ministry of Health, Labour and Welfare (2023b, Table 26), among those who started to receive national
pension benefits in 2022, 87.3% claimed at normal retirement age (65), and 10.8% and 2.0% claimed atages below and above the normal retirement age, respectively. This report also reveals that those who
claimed at ages below the normal retirement age tend to decrease, and those who claimed at the normal
retirement age and above tend to increase.
Table 1 compares the standard retirement age and life expectancy for five major developed
countries. The normal retirement age in Japan is 65 for the basic pension and employment-based part.
6

Many other countries have (or will have) the normal retirement age of 67 under current regulations, but
for Japan, the retirement age remains 65 under current regulations. Table 1 shows that Japan has (or willhave) the lowest normal retirement age among the five advanced countries without new regulations.
Given the life expectancy across these countries (OECD, 2023a), Japan has the longest expected durationof receiving public pension benefits (19.8 years), approximately 6.1 years longer than the average acrossthe OECD member countries (13.7 years). This report reveals that the expected duration of receiving
pension benefits is much longer in Japan than in other major advanced countries; however, the standard
starting age for receiving benefits in Japan remains lower than in other countries.

1.3. Employment for elderly individuals in JapanAs Japan’s population declines and its average age rises, the diminished working-age cohort faces
increasing challenges to support the ballooning numbers of seniors under a PAYG social security system.
According to Higo et al. (2016), Japan is far ahead of the rest of the world in terms of the aging
population curve and faces unprecedented pressure to delay workers’ retirement. In this national context,
workers’ retirement decisions and behaviors are primarily determined by the interplay between thegovernment and employers over reforming mandatory retirement corporate policies. The policies arethoroughly and uniformly institutionalized across the country’s workplaces, set at ages much younger
than in most developed countries worldwide. The primary micro-level determinants of retirement include
persistent gender roles, which still affect many female workers (currently aged 50 and older) as they areoften primary caregivers for children, aged parents, and the household.
Table 2 presents the employment rate by age groups in 2022 for male–female totals, based on datafrom the Statistics Bureau of Japan (2023b). The employment rate for the age group from 25 to 59 is high(more than 80%). It declines to 73.0% for the 60–64 age group and 50.8% significantly for the 65–69 age
group; however, the employment rate for both age groups has recently tended to increase.2 Figure 5,
based on data from the Ministry of Internal Affairs and Communications (2023a), illustrates thetransition of the employment rate for individuals aged 65–69. This figure suggests that the male
employment rate is higher than that of females, and both employment rates tend to increase.

2

For simplicity, our model assumes that the individual’s employment rate until 64 is 100% for all simulation
cases. This paper focuses on the analysis of rises in the standard starting age for receiving pension benefits under
different employment rates for individuals aged 65 and above; therefore, the assumption of the 100% employment
rate until 64 would not have essential effects on our main findings.

7

The remainder of this paper is organized as follows. Section 2 describes literature related to thisstudy, Section 3 identifies the basic model applied in the simulation analysis, Section 4 explains themethod and assumptions of simulation analysis, Section 5 evaluates the simulation findings, and Section
6 summarizes, concludes, and discusses policy implications.

2. Related LiteratureThis paper contributes to the literature on public pension reform in an aging society, especially in Japan.
We focus on five previous studies that analyzed Japan’s public pension issue; the primary literature is asfollows.
First, we discuss two papers that analyzed the public pension reform—Kitao (2015) and Kitao(2017)—which are the most important for our study.
Kitao (2015) quantified the fiscal cost of the demographic transition that Japan is projected toexperience over the next several decades. The author used a lifecycle model with endogenous saving,
consumption, and labor supply in both intensive and extensive margins. Retirement waves of baby-boom
generations, combined with a rise in longevity and low fertility rates, raised the old-age dependency ratio
to 85% by 2050, the highest among major developed countries. A significant budget imbalance occurredas the government faced rising public pension and health and long-term care insurance costs. Preserving
the current level of the transfers would require a significant increase in taxation. Kitao (2015) found thatthe tax rate reached the maximal value of 48% in the late 2070s, using consumption taxes to balance thegovernment budget. A pension reform to reduce benefits by 20% resulted in a peak tax rate of 37%,
which could be reduced further to 28% if the retirement age was also gradually raised by 5 years.
Kitao (2017) simulated pension reform to reduce the replacement rate by 20% and gradually raise
the retirement age by 3 years (from 65 to 68) over 30 years. That author considered three scenarios withdifferent points in time to initiate reform in 2020, 2030, and 2040, respectively, finding that a delay
would suppress economic activities, lowering output by up to 4% and raising the tax burden by morethan 8% of total consumption. Delaying reform implied a transfer of costs of demographic aging to theyoung and thus deteriorated the welfare of future generations by up to 3% in terms of consumption
equivalence.
Both papers considered the reduction of pension benefits by 20% and the rise of the normal
retirement age as pension reform measurements in Japan. In Kitao (2015), the standard retirement age
8was raised from 65 to 67 and 70, whereas in Kitao (2017), it gradually rose from 65 to 68 over 30 years.
Our study considers raising the normal retirement age from 65 to 68 and 70, but the transitional period
for the changes is not as long as the assumption in those papers (explained in Subsection 4.2).
We next examine three previous studies related to public pension reforms in Japan. Okamoto (2013)used a computable general equilibrium model with overlapping generations to explore the effects ofdifferent public pension schemes on economic welfare and intergenerational and intragenerational equity.
Besides the benchmark case based on the 2004 public pension reform, the author considered twoalternative reforms: financing the basic pension benefit through a consumption tax and eliminating theearnings-related pension benefit. The study’s results suggested that even the consumption-tax financing
of only the basic pension (i.e., the combination of both reforms) might not improve overall economic
welfare, although it increases economic output by inducing capital formation.
Braun and Joines (2015) modeled details of medical expenditures and the health insurance system,
examining the effects of public pension and health insurance reform in a general equilibrium lifecycle
model. They found that Japan’s aging population was already burdening government finances and thatthe very high debt-GDP ratio constrains its ability to confront the negative fiscal implications of futureaging. They also found that Japan faces a severe fiscal crisis without imminent remedial action andanalyzed alternative strategies for correcting Japan’s fiscal imbalances.
Maebayashi (2019) investigated how unfunded public pensions financed by value-added tax (VAT),
as discussed in Japan, affect economic growth, examining whether payroll tax (PT) or VAT is the more
growth-friendly tax structure for financing public pensions. The author studied these issues using
overlapping generations models with parental altruism, determining that a public pension systemfinanced by VAT may increase economic growth when bequests are operative. In contrast, public
pensions hinder growth when bequests are inoperative unless agents are sufficiently patient. The results
also indicated that public pensions financed by VAT are more growth-friendly than those financed by PT.

3. Theoretical Framework
We calibrate the simulation of the Japanese economy by applying population data from 2023, estimatedby the National Institute of Population and Social Security Research. The model includes 106
overlapping generations, corresponding to ages 0–105 years old. Three types of agents are incorporated:households, firms, and the government. The following subsections describe the basic structures of9households, firms, and the government, as well as the market equilibrium conditions.
Our model incorporates intergenerational mobility across income classes based on Kikkawa (2009)
who found that Japan’s income disparity stems fundamentally from different educational backgrounds
between high school and university graduates. On the basis of his study, our model introduces two typesof representative agents: the low-income class (i.e., (just) high school graduates) and the high-income
class (i.e., university graduates) into a cohort. In this section, we describe the behavior of the low-income
class household in the model (see Appendix A for the behavior of the high-income class).

3.1. Household behavior
The economy is populated by 106 overlapping generations that live with uncertainty, corresponding toages 0–105. Each agent is assumed to consist of a neutral individual because our model does notdistinguish by gender. Each agent enters the economy as a decision-making unit and starts to work at age
18 years, and lives to a maximum age of 105 years. Each household is assumed to consist of one adultand its children. The children aged 0–17 or 0–21 only consume, involving childrearing costs for their𝑡parent. Each household faces an age-dependent probability of death. Let 𝑞𝑗+1|𝑗be the conditional

probability that a household born in year 𝑡 lives from age 𝑗 to 𝑗 +1. Then the probability of ahousehold born in year 𝑡, surviving until s can be expressed by
𝑡(𝐻)

𝑝𝑠

𝑡
= ∏𝑠−1
𝑗=18 𝑞𝑗+1|𝑗 .

(1)𝑡
The probability 𝑞𝑗+1|𝑗
is calculated from data estimated by the National Institute of Population andSocial Security Research (2023). Since the survival probability is different among agents with different
birth year, agents born in different years have the different utility function.
Each agent who begins its economic life at age 18 chooses perfect-foresight consumption paths(𝐶𝑠𝑡 ), leisure paths (𝑙𝑠𝑡 ), and the number of born children ( nst ) to maximize a time-separable utility
function of the form:
𝑈

𝑡(𝐻)

=

1
1
1−𝛾

40

[𝛼

(𝐻)

𝑡(𝐻)

∑ 𝑝𝑠

1
𝑡(𝐻) 1−𝛾(1 + 𝛿)−(𝑠−18) (𝑛𝑠

)

+ (1

𝑠=18
1
1−
𝛾
𝑡(𝐻)
𝑡(𝐻) 𝜑 𝑡(𝐻) 1−𝜑−(𝑠−18)) ∑ 𝑝𝑠 (1 + 𝛿){(𝐶𝑠 ) (𝑙𝑠 )
} ].
𝑠=18
105

−𝛼

(𝐻)

𝑡(𝐻)
𝑡(𝐻)
This utility function represents the lifetime utility of the agent born in year 𝑡. 𝐶𝑠 , 𝑙𝑠
𝑡(𝐻)

𝑛𝑠

(2)
and

are respectively consumption, leisure and the number of children to bear (only in the first 23
10

periods of the life) for an agent born in year 𝑡, of age 𝑠; 𝛼 (𝐻) is the utility weight of the number ofchildren relative to the consumption–leisure composite, 𝛾 is the intertemporal elasticity of substitution,
𝛿 is the adjustment coefficient for discounting the future, and 𝜑 is the consumption share parameter toleisure.
Fertility choice in the model is only based on the direct utility that households obtain from theiroffspring, neglecting the investment element of children. The demand for children as investment goods
played an important role in traditional economies (and still does in developing countries), where transfersfrom the young to the old arise within the family. In modern advanced countries, however, a PAYG social
security scheme makes the investment aspect of children socialized, as Groezen et al. (2003) pointed out.
This creates the possibility for households to free-ride on the scheme by rearing fewer or no children,
still being entitled to a full pension benefit. Therefore, we treat children as “consumption goods” and aparent is assumed to obtain the utility from the number of children born at each age.
𝑡(𝐻)Letting 𝐴𝑠

be capital holdings for the agent born in year 𝑡, of age 𝑠, maximization of Equation(2) is subject to a lifetime budget constraint defined by the sequence:
𝑡(𝐻)

𝑡(𝐻)

𝑡(𝐻)

+ 𝑏𝑡(𝐻)− 𝑡𝑐𝑡𝑢 (𝑛𝑢({1 − 𝑙𝑡(𝐻)𝑠𝑢

𝐴𝑠+1 = {1 + 𝑟𝑡+𝑠 (1 − 𝜏 𝑟 )}𝐴𝑠

𝑡(𝐻)

𝑜𝑟𝑠

𝑡(𝑈)

𝜏𝑐𝑡+𝑠 )𝛷𝑠

(𝐻)

𝑝

𝑡(𝐻)

+ (1 − 𝜏 𝑤 − 𝜏𝑡+𝑠 )𝑤𝑡+𝑠 𝑒𝑠 {1 − 𝑙𝑠𝑡(𝐻)− 𝑡𝑐𝑠𝑡 (𝑛𝑠𝑡(𝐻))}𝛬𝑠 + 𝑎𝑠

𝑡(𝐻)

)}𝑅𝐸
− (1 − 𝑚)(1 + 𝜏𝑐𝑡+𝑠 )𝛷𝑠
) − (1 + 𝜏𝑐𝑡+𝑠 )𝐶𝑡(𝐻)
𝑠
𝑢=20

−

− 𝑚(1 +,

(3)

where 𝑟𝑡 is the pretax return to savings, and 𝑤𝑡 is the real wage at time 𝑡; 𝜏 𝑤 , 𝜏 𝑟 and 𝜏𝑡𝑐 are the tax
𝑝rates on labor income, capital income and consumption, respectively. 𝜏𝑡 is the contribution rate to the
public pension scheme at time 𝑡. All taxes and contributions are collected at the household level.
𝑡𝑐(𝑛(𝐻) ) is the time cost for childrearing. 𝑎(𝐻) is the bequest to be inherited, and 𝑜𝑟 (𝐻) is the(𝐻)childrearing cost for orphans. There are no liquidity constraints, and thus the assets 𝐴𝑠(𝐻)negative. Terminal wealth must be zero. An individual’s earnings ability 𝑒𝑠

can beis an exogenous functionof age, and 𝛬𝑠 denotes the employment rate of age 𝑠.
The public pension program is assumed to be a PAYG scheme similar to the current Japanesesystem. The program starts to collect contributions to the scheme from the age of 20, in accordance withthe law. The pension benefit is assumed to comprise only an earnings-related pension:
𝑡(𝐻)

𝑡(𝐻)

𝑏𝑠

𝑡(𝐻)

({1 − 𝑙𝑢𝑡(𝐻)− 𝑡𝑐𝑢𝑡 (𝑛𝑢)}𝑅𝐸
𝑢=20 ) = {𝜃𝐻𝑡(𝐻) ({1 − 𝑙𝑢

0

11

𝑡(𝐻)

− 𝑡𝑐𝑢𝑡 (𝑛𝑢)}𝑅𝐸
𝑢=20 )(𝑠 < 𝑆𝑇)(𝑠 ≥ 𝑆𝑇), (4)where𝑡(𝐻)

𝐻𝑡(𝐻) ({1 − 𝑙𝑢𝑡(𝐻)− 𝑡𝑐𝑢𝑡 (𝑛𝑢

1

(𝐻)

𝑡(𝐻)

𝑅𝐸
)}𝑅𝐸
𝑢=20 ) = 𝑅𝐸−19 ∑𝑠=20 𝑤𝑡+𝑠 𝑒𝑠 {1 − 𝑙𝑠𝑡(𝐻)− 𝑡𝑐𝑠𝑡 (𝑛𝑠)}𝛬𝑠 . (5)The age at which a household born in year 𝑡 starts to receive the public pension benefit is 𝑆𝑇, the𝑡(𝐻)average annual labor income for the calculation of pension benefit for each agent is 𝐻𝑡(𝐻) ({1 − 𝑙𝑢𝑡(𝐻)𝑡𝑐𝑢𝑡 (𝑛𝑢
𝑡(𝐻)

𝑏𝑠

−

𝑡(𝐻)
)}𝑅𝐸
is 𝜃. The symbol
𝑢=20 ), and the weight coefficient of the part proportional to 𝐻𝑡(𝐻)({1 − 𝑙𝑢𝑡(𝐻)− 𝑡𝑐𝑢𝑡 (𝑛𝑢)}𝑅𝐸
𝑢=20 ) signifies that the amount of public pension benefit is a function of𝑡(𝐻)the age profile of labor supply, {1 − 𝑙𝑢𝑡(𝐻)− 𝑡𝑐𝑢𝑡 (𝑛𝑢)}𝑅𝐸
𝑢=20 .

A parent is assumed to bear children with the upper limit of 40 years old, and expend for them until
they become independent of their parent, namely, during the period when children are from zero to 17 or
21 years old. Regarding the childrearing costs, the model takes account of both monetary and time costs.
Here note that the children aged below 18 or 22 years old do not conduct an economic activity
independently, and childrearing costs for their parent arise until they become independent of their parent.
The financial costs for rearing the children, for the parent born in year 𝑡 and 𝑠 years old, are𝑡(𝐻)represented by 𝛷𝑠𝑡(𝑈)and 𝛷𝑠, which are the cost for the children who will become high school

graduates and university graduates, respectively:𝑡(𝐻)∑𝑠𝑘=18 𝜉𝑡(𝐻) (1 − 𝜌)𝑛𝑘
𝑡(𝐻)
𝛷𝑠
=

𝑡(𝐻)
∑𝑠𝑘=𝑠−17 𝜉 𝑡(𝐻) (1 − 𝜌)𝑛𝑘𝑡(𝐻)40{∑𝑘=𝑠−17 𝜉 𝑡(𝐻) (1 − 𝜌)𝑛𝑘

(𝑠 = 18,19, ⋯ ,35)(𝑠 = 36,37, ⋯ ,40),(𝑠 = 41,42, ⋯ ,57)

𝑡(𝐻)

𝛷𝑠

= 0 (𝑠 = 58,59, ⋯ ,105),𝑡(𝐻)∑𝑠𝑘=18 𝜉𝑡(𝐻) (1 − 𝜌)𝑛𝑘
(𝑠 = 18,19, ⋯ ,39)
𝑡(𝑈)
𝛷𝑠
={
,
𝑡(𝐻)
𝑡(𝐻)
∑40
𝜉
(1
−
𝜌)𝑛
(𝑠
=
40,41,⋯,61)
𝑘=𝑠−21
𝑘
𝑡(𝑈)

𝛷𝑠

(6)

(7)
(8)
(9)

= 0 (𝑠 = 62,63, ⋯ ,105),

𝜉𝑡(𝐻) = 𝛽𝑁𝑊 𝑡(𝐻) ,(10)

where 𝜉𝑡(𝐻) is the childrearing cost for the parent born in year 𝑡, 𝜌 is the rate of government subsidy(including child allowances) to childrearing costs, and 𝛽 is the ratio of childrearing costs to the net
lifetime income, 𝑁𝑊 𝑡(𝐻) , for the parent born in year 𝑡.
The children who will become university graduates needs more monetary cost than the children who
will become high school graduates simply by the extra four-year (18–21) cost before the independence
from their parents. The mobility 𝑚 denotes the probability in which the children will belong to the highincome class (i.e., university graduates) different from their parent, and 1 − 𝑚 is the probability in
12which they will belong to the low-income class (i.e., high school graduates) same as their parent. The
number of children affects the whole available time for a parent, because of the time required forchildrearing. The time cost for rearing the children for the parent born in year 𝑡, of age 𝑠, is representedby𝑡(𝐻)𝑡𝑐𝑠𝑡 (𝑛𝑠𝑡(𝐻)) = 𝜇𝑛𝑠,(11)

where 𝜇 is the parameter that shows the relation between the number of children and the time requiredfor childrearing, which is simply assumed to be proportional to the number of born children. The time
cost is assumed to be same across the two types of children who will become high school graduates oruniversity graduates.
The model contains accidental bequests that result from uncertainty over length of life. The
bequests, which comprise assets previously held by deceased households, are distributed equally among(𝐻)

all surviving low-income class households at time 𝑡. When 𝐵𝑄𝑡

is the sum of bequests inherited by

the low-income class households at time 𝑡, the bequest to be inherited by each low-income household isdefined by
𝑡(𝐻)

𝑎𝑠

(𝐻)

=

(1−𝜏ℎ )𝐵𝑄𝑡+𝑠
(𝐻)

𝐸𝑡+𝑠

,

(12)

where
(𝐻)

𝐵𝑄𝑡

𝑡−𝑠−1(𝐻)= ∑105𝑠=18(𝑁𝑠𝑡−𝑠−1(𝐻)− 𝑁𝑠+1𝑡−𝑠−1(𝐻)

)𝐴𝑠+1

.

(13)

𝜏 ℎ is the tax rate on inheritances of bequests. The amount of inheritances received is linked to the age(𝐻)profile of assets for each household. 𝐸𝑡

is the number of the low-income class households conducting

an economic activity independently, aged 18 and older. The number of the generation with age 𝑠 years
born in year 𝑡 is represented by
𝑡(𝐻)

𝑁𝑠

𝑡(𝐻)

= 𝑝𝑠

𝑡(𝐻)

𝑁0

.

(14)

Total childrearing cost of the orphans, who are generated as a consequence of parents’ uncertainty
over length of life, is distributed equally among all surviving low-income class households at time 𝑡.
(𝐻)

When 𝑂𝑅𝑡

is the sum of childrearing costs incurred by the low-income class households at time 𝑡, thechildrearing cost for orphans for each low-income class household is defined by
𝑡(𝐻)

𝑜𝑟𝑠

(𝐻)

=

𝑂𝑅𝑡+𝑠
(𝐻)
𝐸𝑡+𝑠

,

(15)

where
(𝐻)

𝑂𝑅𝑡

𝑡−𝑠(𝐻)= (1 − 𝑚) ∑57
𝑠=18(𝑁𝑠−1

𝑡−𝑠(𝐻)− 𝑁𝑠

𝑡−𝑠(𝐻))𝛷𝑠

13

𝑡−𝑠(𝐻)+ 𝑚 ∑61𝑠=18(𝑁𝑠−1

𝑡−𝑠(𝐻)− 𝑁𝑠

𝑡−𝑠(𝑈))𝛷𝑠. (16)

Therefore, the net amount of bequests is represented as 𝑎(𝐻) − 𝑜𝑟 (𝐻) . When we consider the utility
maximization problem over time for each agent, besides the flow budget constraint represented by
Equation (3), the following constraint is imposed:
𝑡(𝐻)

{

𝑡(𝐻)

≤ 1 − 𝑡𝑐𝑠𝑡 (𝑛𝑠0 ≤ 𝑙𝑠)(18 ≤ 𝑠 ≤ 𝑅𝐸)
.

(17)

𝑡(𝐻)
𝑙𝑠
= 1 (𝑅𝐸 + 1 ≤ 𝑠 ≤ 105)

This is a constraint that labor supply is nonnegative, and that each household inevitably retires after passing
the compulsory retirement age, RE .
Let us consider the case where each agent maximizes expected lifetime utility under two constraints.
Each individual maximizes Equation (2) subject to Equations (3) and (17) (see Appendix B for further
details). From the utility maximization problem, the equation expressing the evolution of the number ofchildren over time for each individual is characterized by
𝑡(𝐻)

𝑊𝑠

𝑝

𝑡(𝐻)

= ( 𝑠−1𝑡(𝐻) ) [1+𝑟

1+𝛿

𝑡(𝐻)

𝑟
𝑡+𝑠 (1−𝜏 )𝑝𝑠] 𝑊𝑠−1 ,
1−

1

(18)
𝑡(𝐻) −

1𝛼 (𝐻) 𝑘 𝛾 (𝑛𝑠 ) 𝛾𝑡(𝐻)𝑊𝑠= (1+𝜏𝑐 )[(1−𝑚) ∑17 𝛺𝑡 𝜉𝑡(𝐻) (1−𝜌)𝑡𝑡(𝐻) (1−𝜌)],+𝑚 ∑21𝑔=0 𝑠,𝑔𝑔=0 𝛺𝑠,𝑔 𝜉
𝑡+𝑠

(19)

−1

𝑔

𝑡
𝑡
where 𝛺𝑠,0
=1 for 𝑔 = 0, 𝛺𝑠,𝑔
= (∏𝑘=1{1 + 𝑟𝑡+𝑠−1+𝑘 (1 − 𝜏 𝑟 )}) .

Similarly, that for the consumption–leisure composite is represented by
𝑡(𝐻)

𝑉𝑠

𝑡(𝐻)

𝑉𝑠

𝑝

𝑡(𝐻)

= ( 𝑠−1𝑡(𝐻) ) [
𝑝𝑠

1+𝛿
𝑡(𝐻)
,
]𝑉
1+𝑟𝑡+𝑠 (1−𝜏𝑟 ) 𝑠−1
𝑡(𝐻) 𝜑=(1−𝛼(𝐻) ){(𝐶𝑠1𝑡(𝐻) 1−𝜑 −𝛾

) (𝑙𝑠

)

}

(20)
𝑡(𝐻) 𝜑−1

𝜑(𝐶𝑠)𝑡(𝐻) 1−𝜑

(𝑙𝑠

1+𝜏𝑡𝑐

)

.

(21)

3.2. Firm behavior
The model has a single production sector that is assumed to behave competitively using capital and labor,
subject to a constant-returns-to-scale production function. Capital is homogeneous and depreciating,while labor differs only in efficiency. All forms of labor are perfectly substitutable. Households withdifferent income classes or different ages, however, supply different amounts of some standard measure
per unit of labor input.
The aggregate production technology is the standard Cobb-Douglas form:
𝑌𝑡 = 𝐾𝑡𝜀 𝐿1−𝜀
𝑡 ,

(22)

where Yt is aggregate output (national income), K t is aggregate capital, Lt is aggregate labor supply
measured by the efficiency units, and 𝜀 is capital’s share in production. Using the property subject to a
constant-returns-to-scale production function, we can obtain the following equation:14

𝑌𝑡 = (𝑟𝑡 + 𝛿 𝑘 )𝐾𝑡 + 𝑤𝑡 𝐿𝑡 ,(23)

where 𝛿 𝑘 is the depreciation rate.

3.3. Government behavior
At each time t , the government collects tax revenues and issues debt (𝐷𝑡+1 ) that it uses to financegovernment purchases of goods and services (𝐺𝑡 ) and interest payments on the inherited stock of debt
(𝐷𝑡 ). The government sector consists of a narrow government sector and a pension sector, and a portionof revenues is transferred to the public pension sector. The public pension system is assumed to be asimple PAYG style and consists only of earnings-related pension. Pension account expenditure isfinanced by both contributions and a transfer from the general account.
The budget constraint of the narrower government sector at time t is given by
𝐷𝑡+1 = (1 + 𝑟𝑡 )𝐷𝑡 + 𝐺𝑡 − 𝑇𝑡 ,(24)

where Gt is total government spending on goods and services, Tt is total tax revenue from labor
income, capital income, consumption and inheritances, and 𝐷𝑡 is the net government debt at the
beginning of year t . 𝐷𝑡 is gross public debt minus the accumulated pension fund because the model
abstracts the public pension fund, which is represented as a ratio to national income:𝐷𝑡 = 𝑑𝑌𝑡 ,(25)

where 𝑑 is the ratio of net public debt to national income.
The public pension system is assumed to be a simple PAYG style. The budget constraint of pension
sector at time t is represented by
𝑅𝑡 = (1 − 𝜋)𝐵𝑡 ,(26)

where Rt is total revenue from contributions to the pension program, 𝐵𝑡 is total spending on thepension benefit to generations of age 𝑆𝑇 and above, and 𝜋 is the ratio of the part financed by the tax
transfer from the general account.
The total government spending on goods and service is defined by
𝐺𝑡 = 𝑔𝑌𝑡 + 𝜋𝐵𝑡 + 𝐺𝑆𝑡 ,(27)

where Gt includes transfers to the public pension sector (𝜋𝐵𝑡 ) and the government subsidies to childrearing (𝐺𝑆𝑡 ). The government spending except for the transfers and the subsidies is 𝑔𝑌𝑡 , which isassumed to be represented as a constant ratio (𝑔) of national income. The spending is assumed to either
generate no utility to households or enter household utility functions in a separable fashion.
15

The total amount of government subsidies (including child allowances) to the childrearing cost in
year 𝑡 is GSt :(𝐻)

𝐺𝑆𝑡 = 𝐺𝑆𝑡
(𝐻)(𝑈)

+ 𝐺𝑆𝑠 ,

(28)𝑎(𝐻)= 𝜌 [(1 − 𝑚) ∑57
𝑠=18(𝑅𝐶𝑠,𝑡

𝐺𝑆𝑡

𝑎(𝐻)

𝑏(𝐻)

+ 𝑅𝐶𝑠,𝑡𝑡−𝑠(𝐻) 𝑡−𝑠(𝐻) 𝑡−𝑠(𝐻)
𝜉
𝑛𝑘
𝑏(𝐻)
𝑡−𝑠(𝐻)𝑡−𝑠(𝐻)
𝑅𝐶𝑠,𝑡 = ∑𝑠𝑘=𝑠−17 𝑁𝑘
𝜉𝑡−𝑠(𝐻) 𝑛𝑘𝑐(𝐻)𝑡−𝑠(𝐻) 𝑡−𝑠(𝐻) 𝑡−𝑠(𝐻)40𝜉𝑛𝑘{ 𝑅𝐶𝑠,𝑡 = ∑𝑘=𝑠−17 𝑁𝑘

𝑅𝐶𝑠,𝑡= ∑𝑠𝑘=18 𝑁𝑘𝑎(𝑈)𝑡−𝑠(𝐻)𝑡−𝑠(𝐻)

𝑅𝐶𝑠,𝑡 = ∑𝑠𝑘=18 𝑁𝑘
𝜉𝑡−𝑠(𝐻) 𝑛𝑘{ 𝑏(𝑈)𝑡−𝑠(𝐻) 𝑡−𝑠(𝐻) 𝑡−𝑠(𝐻)𝑅𝐶𝑠,𝑡 = ∑40𝜉𝑛𝑘
𝑘=𝑠−21 𝑁𝑘𝑎(𝐻)where 𝑅𝐶𝑡

𝑏(𝐻)

, 𝑅𝐶𝑡

𝑐(𝐻)

and 𝑅𝐶𝑡

𝑐(𝐻)

𝑎(𝑈)

+ 𝑅𝐶𝑠,𝑡 ) + 𝑚 ∑61
𝑠=18(𝑅𝐶𝑠,𝑡𝑏(𝑈)+ 𝑅𝐶𝑠,𝑡 )], (29)(𝑠 = 18,19, ⋯ ,35)(𝑠 = 36,38, ⋯ ,40) ,(30)(𝑠 = 41,42, ⋯ ,57)(𝑠 = 18,19, ⋯ ,39),(𝑠 = 40,41, ⋯ ,61)

(31)

are monetary costs for childrearing when the children will belong

to the low-income class same as their parent, namely, they will become high school graduates, and
𝑎(𝑈)

𝑏(𝑈)

and 𝑅𝐶𝑡

𝑅𝐶𝑡

are the costs when the children will belong to the high income class different fromtheir parent, namely, they will become university graduates.
(𝑈)

𝐺𝑆𝑡

𝑎(𝑈)

= 𝜌 [(1 − 𝑚) ∑61
𝑠=22(𝑅𝐶𝑠,𝑡

𝑎(𝑈)

𝑏(𝑈)

+ 𝑅𝐶𝑠,𝑡𝑡−𝑠(𝑈) 𝑡−𝑠(𝑈) 𝑡−𝑠(𝑈)
𝜉
𝑛𝑘
𝑏(𝑈)
𝑡−𝑠(𝑈) 𝑡−𝑠(𝑈) 𝑡−𝑠(𝑈)𝑅𝐶𝑠,𝑡 = ∑40𝜉𝑛𝑘
𝑘=22 𝑁𝑘𝑐(𝑈)𝑡−𝑠(𝑈)𝑡−𝑠(𝑈)40
𝜉𝑡−𝑠(𝑈) 𝑛𝑘{ 𝑅𝐶𝑠,𝑡 = ∑𝑘=𝑠−21 𝑁𝑘

𝑅𝐶𝑠,𝑡= ∑𝑠𝑘=22 𝑁𝑘𝑎(𝐻)𝑡−𝑠(𝑈)𝑡−𝑠(𝑈)

𝑅𝐶𝑠,𝑡 = ∑𝑠𝑘=22 𝑁𝑘
𝜉𝑡−𝑠(𝑈) 𝑛𝑘{ 𝑏(𝐻)𝑡−𝑠(𝑈) 𝑡−𝑠(𝑈) 𝑡−𝑠(𝑈)𝑅𝐶𝑠,𝑡 = ∑40𝜉𝑛𝑘
𝑘=𝑠−17 𝑁𝑘𝑎(𝑈)where 𝑅𝐶𝑡

𝑏(𝑈)

, 𝑅𝐶𝑡

𝑐(𝑈)

and 𝑅𝐶𝑡

𝑐(𝑈)

𝑎(𝐻)

+ 𝑅𝐶𝑠,𝑡 ) + 𝑚 ∑57
𝑠=22(𝑅𝐶𝑠,𝑡𝑏(𝐻)+ 𝑅𝐶𝑠,𝑡 )], (29’)(𝑠 = 22,23, ⋯ ,40)(𝑠 = 41,42,43)

,

(30’)

(𝑠 = 44,45, ⋯ ,61)(𝑠 = 22,23, ⋯ ,39),(𝑠 = 40,41, ⋯ ,57)

(31’)

are financial costs for childrearing when the parent is 22 to 61

years old. Once the parent becomes 62 years old, the cost does not exist because all children are
independent from their parent.
The total spending on the pension benefit to generations of age ST and above is represented by(𝐻)

𝐵𝑡 = 𝐵𝑡(𝐻)

where 𝐵𝑡

(𝑈)

+ 𝐵𝑡 ,(𝑈)

and 𝐵𝑡(𝐻)= ∑105𝑠=𝑆𝑇 𝑁𝑠(𝑈)= ∑105𝑠=𝑆𝑇 𝑁𝑠

𝐵𝑡
𝐵𝑡

(32)
are the expenditure for the two income classes:𝑡−𝑠(𝐻) 𝑡−𝑠(𝐻)
𝑏𝑠
,

(33)

𝑡−𝑠(𝑈) 𝑡−𝑠(𝑈)
𝑏𝑠
.

(33’)

The total revenue from pension contributions and the total tax revenue are represented by
16

𝑅𝑡 = 𝜏 𝑝 𝑤𝑡 𝐿𝑡 ,(34)

𝑇𝑡 = 𝜏 𝑤 𝑤𝑡 𝐿𝑡 + 𝜏 𝑟 𝑟𝑡 𝐴𝑆𝑡 + 𝜏𝑡𝑐 𝐴𝐶𝑡 + 𝜏 ℎ 𝐵𝑄𝑡 ,(35)

where aggregate assets supplied by households, AS t , and aggregate consumption, ACt , are given by(𝐻)+ 𝐴𝑆𝑡 ,

(𝑈)

(36)

(𝐻)

+ 𝐴𝐶𝑡 .

(𝑈)(37)

𝐴𝑆𝑡 = 𝐴𝑆𝑡

𝐴𝐶𝑡 = 𝐴𝐶𝑡

(𝐻)

For the low-income class, aggregate assets supplied by households, 𝐴𝑆𝑡 , and aggregate(𝐻)consumption, 𝐴𝐶𝑡 , are given by(𝐻)

𝑡−𝑠(𝐻) 𝑡−𝑠(𝐻)= ∑105𝐴𝑠,
𝑠=18 𝑁𝑠(𝐻)
𝑡−𝑠(𝐻)𝑡−𝑠(𝐻)
𝑎(𝐻)
𝑏(𝐻)
𝑐(𝐻)
𝐴𝐶𝑡 = ∑105𝐶𝑠
+ (1 − 𝑚) ∑57+ 𝑅𝐶𝑠,𝑡 + 𝑅𝐶𝑠,𝑡 ) +𝑠=18 𝑁𝑠𝑠=18(𝑅𝐶𝑠,𝑡
𝐴𝑆𝑡

𝑎(𝑈)

𝑚 ∑61
𝑠=18(𝑅𝐶𝑠,𝑡

(38)

𝑏(𝑈)

+ 𝑅𝐶𝑠,𝑡 ),(39)

where aggregate consumption consists of adult’s consumption (at age 18–105 years old) and children’s
consumption or cost (at age zero to 17 or 21 years old).
(𝑈)

For the high-income class, aggregate assets supplied by households, 𝐴𝑆𝑡 , and aggregate(𝑈)consumption, 𝐴𝐶𝑡 , are given by(𝑈)= ∑105𝑠=22 𝑁𝑠(𝑈)= ∑105𝑠=22 𝑁𝑠

𝐴𝑆𝑡

𝐴𝐶𝑡𝑡−𝑠(𝑈) 𝑡−𝑠(𝑈)𝐴𝑠, (38’)𝑡−𝑠(𝑈) 𝑡−𝑠(𝑈)
𝑎(𝑈)
𝑏(𝑈)
𝑐(𝑈)
𝐶𝑠
+ (1 − 𝑚) ∑61+ 𝑅𝐶𝑠,𝑡 + 𝑅𝐶𝑠,𝑡 ) +𝑠=22(𝑅𝐶𝑠,𝑡

𝑎(𝐻)

𝑚 ∑57
𝑠=22(𝑅𝐶𝑠,𝑡𝑏(𝐻)+ 𝑅𝐶𝑠,𝑡 ),(39’)

where aggregate consumption consists of adult’s consumption (at age 22–105 years old) and children’s
consumption or cost (at age zero to 21 or 17 years old).
The total sum of bequests inherited by the households and the total childrearing cost of the orphans
at time t are as follows:(𝐻)+ 𝐵𝑄𝑡 ,

(𝑈)

(40)

(𝐻)

+ 𝑂𝑅𝑡 .

(𝑈)(41)

𝐵𝑄𝑡 = 𝐵𝑄𝑡
𝑂𝑅𝑡 = 𝑂𝑅𝑡

Total population (i.e., the population aged zero to 105), the population aged 18 or 22 to 105 (i.e.,
independents financially), and the population aged 65 to 105 (i.e., retirees) in year t are respectively
represented by(𝐻)+ 𝑍𝑡 ,

(𝑈)

(42)

(𝐻)

+ 𝐸𝑡 ,

(𝑈)

(43)

(𝐻)

+ 𝑂𝑡 .

(𝑈)(44)

𝑍𝑡 = 𝑍𝑡

𝐸𝑡 = 𝐸𝑡

𝑂𝑡 = 𝑂𝑡

17

The aging rate (i.e., the old-age dependency ratio), the ratio of the population aged 65 and above to thetotal population, is given by 𝑂𝑡 /𝑍𝑡 . For the low-income class, the total population, the population aged
18 to 105, and the population aged 65 to 105 in year t are respectively represented by(𝐻)40= ∑105𝑘=0 ∑𝑖=18 𝑁𝑖(𝐻)40= ∑105𝑘=18 ∑𝑖=18 𝑁𝑖(𝐻)40= ∑105𝑘=65 ∑𝑖=18 𝑁𝑖

𝑍𝑡

𝐸𝑡

𝑂𝑡

𝑡−𝑘−𝑖(𝐻) 𝑡−𝑘(𝐻) 𝑡−𝑘−𝑖(𝐻)
𝑝𝑘
𝑛𝑖
,

(45)

𝑡−𝑘−𝑖(𝐻) 𝑡−𝑘(𝐻) 𝑡−𝑘−𝑖(𝐻)
𝑝𝑘
𝑛𝑖
,

(46)

𝑡−𝑘−𝑖(𝐻) 𝑡−𝑘(𝐻) 𝑡−𝑘−𝑖(𝐻)
𝑝𝑘
𝑛𝑖
.

(47)

For the high-income class, the total population, the population aged 22 to 105, and the population
aged 65 to 105 in year 𝑡 are respectively represented by(𝑈)40= ∑105𝑘=0 ∑𝑖=22 𝑁𝑖(𝑈)40= ∑105𝑘=22 ∑𝑖=22 𝑁𝑖(𝑈)40= ∑105𝑘=65 ∑𝑖=22 𝑁𝑖

𝑍𝑡

𝐸𝑡

𝑂𝑡

𝑡−𝑘−𝑖(𝑈) 𝑡−𝑘(𝑈) 𝑡−𝑘−𝑖(𝑈)
𝑝𝑘
𝑛𝑖
,

(45’)

𝑡−𝑘−𝑖(𝑈) 𝑡−𝑘(𝑈) 𝑡−𝑘−𝑖(𝑈)
𝑝𝑘
𝑛𝑖
,

(46’)

𝑡−𝑘−𝑖(𝑈) 𝑡−𝑘(𝑈) 𝑡−𝑘−𝑖(𝑈)
𝑝𝑘
𝑛𝑖
.

(47’)

3.4. Market equilibrium
Finally, equilibrium conditions for the capital, labor and goods markets are described.
1) Equilibrium condition for the capital market
Because aggregate assets supplied by households equal the sum of real capital and net government
debt,
𝐴𝑆𝑡 = 𝐾𝑡 + 𝐷𝑡 .

(48)2) Equilibrium condition for the labor market
Measured in efficiency units, because aggregate labor demand by firms equals aggregate labor supply
by households,(𝐻)

𝐿𝑡 = 𝐿𝑡(𝑈)+ 𝐿𝑡 ,

(49)

(𝐻)

= ∑𝑅𝐸
𝑠=18 𝑁𝑠(𝑈)= ∑𝑅𝐸
𝑠=22 𝑁𝑠

where 𝐿𝑡

𝐿𝑡𝑡−𝑠(𝐻) (𝐻)𝑡−𝑠(𝐻)𝑡(𝐻)𝑒𝑠 {1 − 𝑙𝑠
− 𝑡𝑐𝑠𝑡 (𝑛𝑠 )},(50)

𝑡−𝑠(𝑈) (𝑈)𝑡−𝑠(𝑈)𝑡(𝑈)𝑒𝑠 {1 − 𝑙𝑠
− 𝑡𝑐𝑠𝑡 (𝑛𝑠 )}.

(50’)3) Equilibrium condition for the goods market
Because aggregate production equals the sum of private consumption, private investment andgovernment expenditure,
𝑌𝑡 = 𝐴𝐶𝑡 + {𝐾𝑡+1 − (1 − 𝛿 𝑘 )𝐾𝑡 } + 𝐺𝑡 .

(51)

An iterative program is performed to obtain the equilibrium values of the above equations.

18

4. Simulation Analysis
4.1. Method
The simulation model presented in the previous section is solved, given the assumption that households
have fundamentally perfect foresight and correctly anticipate interest, wages, the tax and contribution
rates, and other factors such as the normal retirement age and the employment rate for the elderly. If the
tax and social security systems and other elements are determined, then the model can be solved using
the Gauss–Seidel method (see Auerbach and Kotlikoff (1987) and Heer and Maußner (2005) for thecomputation process).
Our study assumes the transitional economy of Japan from the initial steady state in 2022 to the final
steady state in 2300. Alternative scenarios with the different normal retirement age and the different
employment rate for the elderly are assumed to be implemented at the end of 2022. For simplicity, 2022
is set as the starting year, and we simulate the demography and the economy in the following years. For
the generations that were alive in 2022 and have survived in 2023, we need to pay attention to theirformation of future expectations. In 2023, these generations realized that their previous expectations nolonger apply and thus again maximize their remaining lifetime utility given perfect foresight. Based on
the ex-post age profiles of the number of children to bear, consumption, and leisure for these generations,
we calculated their lifetime utility at 18 and 22 years for the low- and high-income classes, respectively.
The LSRA first transfers to each household affected by the reform just enough resources (possibly anegative amount) to return its expected remaining lifetime utility to its pre-change level in the benchmark
simulation. For each household that is alive when a change occurs at the end of 2022, at its age in 2023,
the LSRA makes a lump sum transfer, to return its expected remaining lifetime utility to its pre-change
utility level. The LSRA also makes a lump-sum transfer to each future household that enters the economy
after a change (from 2023 onward), at its age of 18 or 22 years, to return its expected entire lifetime
utility back to its pre-change level.
Note that the net present value of these transfers in 2023 across living and future households will
generally not sum to 0. Thus, the LSRA makes an additional lump sum transfer to each future householdso that the net present value across all transfers is 0. To illustrate, let us assume that these additionaltransfers are uniform across all future generations, including the low- and high-income classes. If thetransfer is positive, then the change has produced extra resources after the expected remaining lifetime
utility of each household has been restored to its pre-change level. In this case, we can interpret that the19change has created efficiency gains, i.e., Pareto improvements. Conversely, if the transfer is negative,then the change has generated an efficiency loss. Thus, the total net present value of all lump sum
transfers to current and future generations sums to 0 in 2023, satisfying the LSRA budget constraint (seeNishiyama and Smetters (2005) for further details).
4.2. Simulation cases
This study uses an extended lifecycle general equilibrium model with endogenous fertility to investigate
the quantitative effects of a rise of the normal starting age for receiving pension benefits in Japan on
individual welfare and future demographics. We assume the starting age for pension benefits is 65 in the2022 initial steady state. Table 3 presents the simulation cases considered in our analysis. The benchmark
simulation assumes that the starting age remains 65 throughout the entire period. We consider two typesof scenarios for the six alternative cases where the starting ages for receiving pension benefits rise to 68
and 70. For each scenario, we also consider three kinds of experiments in which the employment rates
for individuals aged 65 and above are 50%, 75%, and 100%, respectively.
We assume that the changes are gradually performed to avoid extra disturbance or confusion fromsudden rises in the standard starting age for receiving pension benefits. The starting age is 66 for people
born in 1958, 67 in 1959, 68 in 1960, 69 in 1961, and 70 in 1962. For the scenarios with the 68-pension
starting age, it is 68 for the people born after 1960. For the scenarios with the 70-pension starting age, it
is 70 for the people born after 1962. Additionally, we will consider the case with LSRA transfers for eachscenario. To distinguish potential efficiency gains/losses from possibly offsetting changes in the welfare
of different generations, we introduce LSRA into the alternative six scenarios, Cases 68-50%, 68-75%,
68-100%, 70-50%, 70-75%, and 70-100% in Table 3. The LSRA transfers produce a leveled andcommon welfare gain/loss for each future household, including the low- and high-income classes.
4.3. Specification of the parameters
We chose realistic parameter values for the Japanese economy based on the literature (Nishiyama andSmetters, 2005; Oguro et al., 2011; İmrohoroğlu et al., 2017; Kitao and Mikoshiba, 2020). Table 4
displays the parameter values assigned in the baseline simulation, and the data source used in thecalibration. Parameter values were chosen such that the calculated values of the model’s endogenous
variables approached the actual data values. Table 5 presents the endogenous variables in the 2022 initialsteady state. Because the simulation results depend on the model setting and the given parameters, we20

must be careful about the effects of any parameter changes.
4.3.1. Demography
Japan’s population is aging at an unprecedented speed for a developed nation; simultaneously, thepopulation is decreasing, which has become one of Japan’s most important problems. Japan’s speed andmagnitude of demographic aging are remarkable, even compared to other advanced countries facing
similar challenges. Our extended lifecycle general equilibrium simulation model with endogenous
fertility rigorously reflects such demographic dynamics in Japan.
t

The age-specific survival probability q j +1| j is calculated from data estimated by the National
Institute of Population and Social Security Research (2023). We used the average values for males andfemales on future life tables by age from 2020 until the last year for which official projections areavailable, 2070; after 2070, we used the 2070 life table data. For simplicity, the survival rate for the lowincome class (i.e., high school graduates) is unity at 18 years old, and that for the high-income class (i.e.,
university graduates) is unity at 22.
Table 6 indicates the population ratio of individuals with different educational backgrounds in 2022,
estimated from the Basic Survey on Wage Structure (Chingin Sensasu) by the Ministry of Health, Labourand Welfare (2023a). The population share of high school graduates (including junior high school
graduates) and university graduates (including technical and junior college graduates) is 49.4% and50.6%, respectively.
Figure 6 illustrates the age–population distribution in 2022 based on data from the Statistics Bureau
of Japan (2023a), denoting the population of high school and university graduates, respectively, for eachage. We estimated each population of high school graduates and university graduates aged 0–105 in
2022, similar to Okamoto (2022, 2024). For the elderly, especially those of advanced age, the number ofhigh school graduates exceeds that of the university graduates, whereas for the young and the middleaged, it is approximately fifty–fifty. For those who are under 18 or 22 years old and undecided to become
high school or university graduates, we assume that their population is the same i.e., fifty–fifty on thebasis of Kikkawa (2009).
4.3.2. Preference parameter on the number of children
Regarding the preference parameter for children in the utility function of households, the parameter value
in the benchmark simulation is the same between the low-income class (i.e., high school graduates) and21

the high-income class (i.e., university graduates). In other words, the utility weight of the number ofchildren relative to the consumption–leisure composite in Equations (2) and (2)’ is the same between thetwo income classes (  (H ) =  (U ) = 0.028148). This parameter setting is implemented after
comprehensively considering several empirical studies, such as Kikkawa (2018) and Adsera (2017).
(2022).
Initially, Kikkawa (2018) suggested that the low-income class tends to have more children than thehigh-income class. That study presents the scheduled number of children for young people aged 21 to 40,
which is based on a large-scale questionnaire survey (SSM2015). Accordingly, on average, the scheduled
number of children for young high school-graduate couples is 1.14, whereas it is 0.875 for young
university-graduate couples. The data revealed that the low-income class has more children than thehigh-income class.
Conversely, some previous studies, such as Adsera (2017), revealed that such tendencies have
weakened recently. Adsera (2017) investigated the effects of a possible increase in the employment andincome gaps between highly educated and low-educated workers on their fertility. They suggested that
educational attainment’s negative fertility gradient has recently weakened in developed countries, and thegap in the number of children born between more-educated and less-educated women has shrunk. Theirstudy also suggested that rising inequality is one mechanism that could underlie this apparent fertility
convergence. As some middle-income jobs seem to disappear, polarization in the labor market has
increased. This change in the labor market could exert downward pressure on the fertility of mediumand less-educated couples and further flatten the educational gradient.
The empirical data derived by Kikkawa (2018) show that the fertility rate of the low-income class ishigher than that of the high-income class, which is based on a reasonable rationale and has a certain
validity; however, the findings of Adsera (2017) suggest that the fertility rate differences between the two
income classes have recently decreased. Based on the above considerations, this paper’s benchmark
assumed that the parameter (𝛼) related to the preference for the number of children in the households’
utility function was set to the same value between the two income classes.
The parameter value determining the fertility was chosen so that the total fertility rate (TFR) is 1.26
in the 2022 initial steady state, reflecting that Japan’s actual TFR was 1.26 in 2022. Consequently, theparameter value (  (H ) ,  (U ) ) is set to 0.028148; In the initial steady state, the TFR is 1.33 for the lowincome class and 1.15 for the high-income class, indicating that the TFR is higher for the low-income22class. A possible reason is that the utility obtained from the number of children is relatively higher for the
low-income class than the high-income class because of the lower wage income per unit of labor.
4.3.3. Childrearing costs
Next, we describe how we assign parameter values for childrearing since our simulation model
incorporates endogenous fertility. Based on empirical data, such as Kikkawa (2009), in our model, 70%of children from the high-income class will become high-income class households, and 70% of children
from the low-income class will become low-income. In Japan, the high-income class spends more on
educating their children than the low-income class because private education has a higher weight. This
fact justifies the model setting that childrearing costs are proportional to the parent’s lifetime income.
The Cabinet Office (2010) indicated the average annual childrearing costs for the first-born child toannual income for each age. Based on the survey in the Cabinet Office (2010), we assigned the parametervalue of 𝛽 (i.e., the ratio of childrearing costs to parental net lifetime income) such that the ratio of theannual net childrearing costs to annual labor income for the individual is, on average, close to 19.3%.
Thus, 𝛽 is assigned 0.0385 (the ratio is 19.2 % for the low-income class and 19.1% for the high-income
class).
The OECD (2023b) presents public spending on family benefits in cash, services, and tax breaks forfamilies as a percentage of GDP in 2019. For Japan, public spending ratios on family benefits in cash,
services, and tax measures to GDP are 0.66%, 1.08%, and 0.20%, respectively.3 We assigned the valueof parameter 𝜌 (government childcare subsidies divided by childrearing cost) to 0.1 in the model, as in
Oguro et al. (2011). Consequently, the ratio of total government subsidies to national income was 1.11 %in the 2022 initial steady state.
Additionally, our model incorporated not only the monetary costs of childrearing but also the time
costs. Increases in the number of children diminish the parent’s available time, because of the timerequired for childrearing; more children to bear, more time required for childrearing. The parameter
determining this relation, 𝜇, is assigned under the simple assumption that one child required 1 h per day
for childrearing.4
3

In Japan, the ratio of total family benefits to GDP is only 1.95%, whereas it is, on average, 2.29% for the 38
OECD member countries. This shows that the level of governmental support for childrearing is considerably lower
in Japan than that in other countries.
4 Calibrating the value of parameter, μ, that determines the time cost in the model is difficult. In the 2022 initial23

4.3.4. Age profile of labor efficiency
The age profiles of earning ability for the two income classes were estimated with data from the Basic
Survey on Wage Structure (Chingin Sensasu) by the Ministry of Health, Labour and Welfare (2014–
2023a) for the 2013–2022 period. Figure 7 illustrates age–earnings profiles by education. The labor
efficiency profiles are constructed from the Japanese data on employment, wages, and monthly work
hours.
To estimate the age profiles of earnings ability, 𝑒𝑠 (𝐻) and 𝑒𝑠 (𝑈) , respectively, the following
equation is constructed:
𝑄𝑡 = 𝑎0 + 𝑎1 𝐴𝑡 + 𝑎2 𝐴2𝑡 , (52)where 𝑄 is the average monthly cash earnings for high school-graduate workers and university-graduate
workers, respectively, and 𝐴 is the average age for each of the workers, including both males andfemales. Because bonuses account for a large part of earnings in Japan, 𝑄 includes bonuses. Using theabove data, we use the ordinary least squares (OLS) method to perform estimation. Figure 7 presents theresults, illustrating age–earnings profiles by educational background. For the high school graduates, they
start to work earlier (18 years old), but their age profile of earnings is flatter with a lower level than theuniversity graduates. For the university graduates, they start to work later (22 years old), but their age
profile of earnings is steeper with a higher level.
Figure 7 shows age–earnings profiles for two representative agents (high school graduates anduniversity graduates) with different employment rates (50%, 75%, and 100%) for individuals aged 65
and above. For Cases 68-100% and 70-100%, the estimated age profiles of earnings ability from 65 to 67
and 69, respectively, are used as they are for the two representative agents. For Cases 68-75% and 7075%, the estimated age profiles of earnings ability from 65 to 67 and 69, respectively, are weighted by
75% for both agents. For Cases 68-50% and 70-50%, the age profiles from 65 to 67 and 69, respectively,
are weighted by 50% for both agents. In other words, 𝛬𝑠 (the employment rate of age 𝑠) in Equation (3)are all unity before the starting age of pension benefits for Cases 68-100% and 70-100%. For Cases 6875% and 70-75%, 𝛬𝑠 are all unity until 64 and 0.75 for ages 65–67 and 65–69, respectively. For Cases
68-50% and 70-50%, 𝛬𝑠 are all unity until 64 and 0.5 for ages 65–67 and 65–69, respectively.
steady state, an average number of children to which a parent of the low-income class gives birth during the period
from 18 to 40 is 0.0290 per year. For a parent of the high-income class, it is 0.0302 per year during the period from22 to 40. We simply assume that a parent’s available time is 16 h per day and that the childrearing time cost for one
child is 1 h per day.

24

4.3.5. Taxes and expenditures
Tax rates on labor income, capital income, and inheritances are assumed to be fixed at the current levels(6.5%, 40%, and 10%, respectively) during the entire period until 2300. Tax rates on consumption are
endogenously determined to satisfy Equations (24) and (35). General government expenditures, except
for transfers to the public pension sector (𝜋𝐵𝑡 ) and government subsidies to childrearing (𝐺𝑆𝑡 ), areproportional to national income (𝑌𝑡 ), as indicated in Equation (27). The ratio of general expenditure tonational income, 𝑔, is assigned 0.1 such that the endogenous tax rate on consumption is realistic andplausible in the 2022 initial steady state (i.e., 13.08%). The ratio is held constant at 0.1 throughout theentire period.
4.3.6. The public pension system
The public pension program is assumed to be a simple PAYG system similar to the current Japanesesystem. The benefit is assumed to comprise an earnings-related pension, although Japan’s actual public
pension system is two-tiered: a basic flat pension and an amount proportional to the average annual labor
income for each household. General tax revenue finances half of the flat part, whereas contributions to
the pension system fund both the remaining half and the entire proportional part. We assign the ratio (𝜋)of the part financed by the tax transfer from the general account in Equation (26) as 0.25, taken from
Oguro and Takahata (2013). The replacement ratio (𝜃) for public pension benefits in Equation (4) isequal to 40%, following Braun et al. (2009).
The age at which households start to receive public pension benefits (𝑆𝑇) is constant at 65 during
the entire period for the baseline simulation. For six alternative simulation cases with a pension reform, itvaries for each simulation case, as explained in Subsection 4.2. The compulsory retirement age (𝑅𝐸) isthe starting age of public pension benefits (𝑆𝑇) minus 1. Thus, after households retire at the end of theyear in which they reach compulsory retirement, they immediately start to receive pension benefits fromthe beginning of the next year.
4.3.7. Government deficits
Net government debt (𝐷𝑡 ) is assumed to be proportional to national income to make our simulation
feasible. The value of parameter 𝑑, which is the ratio of net public debt to national income as given in
Equation (25), is assigned based on data from the Ministry of Finance (2023) and the Cabinet Office25

(2023). After 2022, Japan’s national income is expected to decrease as the population declines.
Therefore, the assumption that net government debt is proportional to national income during the entire
period implicitly implies that the government will successfully reduce future government deficits.
4.3.8. Intertemporal elasticity of substitution
Following İmrohoroğlu et al. (2017), the intertemporal elasticity of substitution (  ) in the individual
utility function is set to 0.5. Our model also set the same value between the number of children and theconsumption-leisure composite parameter, as in previous studies, such as Oguro, Takahata, andShimasawa (2011) and Oguro and Takahata (2013).
4.3.9. Share parameter on consumption in utility
The value of the consumption share parameter,  , in the utility function is assigned based on Nishiyama
and Smetters (2005). Referring to Nishiyama and Smetters (2005), where  = 0.47, we set  = 0.5 inthis paper. Consequently, in the 2022 initial steady state, an individual devotes, on average, 58.9% for the
low-income class and 60.7% for the high-income class, of the available time endowment (of 16 h perday) to labor during their working years (ages 18–64 or 22–64 years).
4.3.10. Adjustment coefficient for discounting the future
The adjustment coefficient for discounting the future,  , is set such that the capital–income ratio (K/Y)
in the model, that is 2.46, approaches its plausible value, 2.5 which is estimated by Hansen andİmrohoroğlu (2016).
4.3.11. Technological progress
The technological progress of private production is significant because it greatly influences economic
growth. Thus, careful attention should be paid to our assumptions. Technological progress is assumed to
be 0 in the simulation, reflecting Japan’s experience during the past two or three decades (see Ihori et al.,2006).

26

5. Simulation Results
We analyze alternative public pension reforms that raise the normal retirement age to 68 and 70 under
three scenarios of the employment rate for elderly individuals in Japan. Based on the simulation results,
we first address the effect of a rise in the normal retirement age on per-capita utility and the future
population level. We then discuss the mechanism behind those findings.
5.1. Effect of a rise in the starting age for pension benefits on individual welfareFirst, we evaluate the effect of a rise in the starting age for receiving pension benefits on individual
welfare. Table 3 presents six alternative cases, which have two types of scenarios of the starting age (68and 70) and three kinds of experiments of elderly individual’s employment rate (50%, 75%, and 100%).
Table 7 shows leveled welfare gains for each individual based on the simulation results. The table showsthat a rise in the starting age for receiving pension benefits enhances the welfare gains, indicating that the
70-starting age cases have a larger welfare gain than the 68-starting age cases. It also reveals that thehigher the employment rate, the more the individual welfare improves. Consequently, Case 70-100%
attains the highest individual’s leveled welfare gain, equivalent to 25.377 million Japanese yen (JPY)(approximately193,000 U.S. dollars [USD] in 2022), a considerable amount for each individual.5 In
contrast, the welfare gain for Case 68-50% is only 10.265 million JPY (approximately 78,000 USD in2022). These results suggest that the higher the starting age for receiving pensions and the higher theemployment rate for elderly individuals, the more per-capita welfare improves. Therefore, we find thatthe case of 70-100% is the most desirable from the viewpoint of per-capita welfare.
5.2. Effect of a rise in the starting age for pension benefits on future population
Next, we examine the effect of a rise in the starting age for receiving public pensions on the future
demography. Figures 8 and 9 illustrate changes in the total population for three cases of different
employment rates for elderly individuals (50%, 75%, and 100%) under the pension starting ages of 68
and 70, respectively, from the benchmark case with the 65-pension starting age. The higher theemployment rate for elderly individuals, the larger the total population in the future. The qualitative

5

The Cabinet Office (2023) estimated that the GDP of Japan in 2022 was 551.81 trillion yen. According to datafrom the Ministry of Internal Affairs and Communications (2023b), the number of the people aged 20–64 years was
58.66 million in 2022. We calculated the income per worker using these data and also derived the value for national
GDP in 2022 in our model, yielding a conversion rate between actual amounts of yen and values in the model.
Consequently, in 2022, unity in the model corresponded to 4.90035 million yen.

27

result is the same between the cases with the 68-starting age for public pensions and the 70-starting age,
but the magnitude of the effect is greater for the case with the 70-starting age. Figure 8 illustrates that the
total population in Case 68-100% continues to increase steadily, resulting in a 6.29% increase in 2200.
The total population in Case 68-75% first gradually increases and peaks at a 1.50% increase around2154; the increase then slightly shrinks over time. The total population in Case 68-50% first slightly
increases but is almost flat; however, from 2110 onward, it starts to decrease, compared to thebenchmark case with the 65-starting age. Figure 9 reveals that the total population in Case 70-100%
increases by 9.47% in 2200. The total population in Case 70-75% peaks at a 2.05% increase around2146; the increase slightly shrinks over time. The total population in Case 70-50% first slightly increases
but is almost flat; however, from 2112 onward, it starts to decrease, compared to the benchmark casewith the 65-starting age.
5.3. Findings and mechanism behind them
Based on the above simulation results, our findings are as follows. A rise in the starting age for receiving
pension benefits enhances individual welfare, and the magnitude of the effect is greater in the cases withthe 70-pension starting age. Furthermore, a higher employment rate for elderly individuals improves
individual welfare. The effect of a rise in the starting age on future demography depends significantly on
the employment rate of elderly individuals. In the cases with the 100%-employment rate, the future
population will steadily increase; however, in the cases with the 50%-employment rate, the current level
in Japan, the future population starts to decrease after around 2110. Therefore, these results suggest that arise in the starting age for pension benefits is desirable from the viewpoint of individual welfare, butfrom the viewpoint of the future population, it depends significantly on the employment rate of elderly
individuals. If Japan’s employment rate remains at 50%, it will harm the total population in the long run.
Therefore, when the starting age for pension benefits is raised, it is necessary to simultaneously increase
the employment rate of elderly individuals as much as possible.
Figures 10 and 11 illustrate the percent changes in national income for three cases of different
employment rates (50%, 75%, and 100%) for two scenarios of starting ages of 68 and 70, respectively,
for pension benefits. Until 2200, the level of national income for the six simulation cases is higher
throughout the entire period than that of the benchmark case with the 65-pension starting age. The higher
the employment rate, the higher the level of national income. Case 68-100% attains a 15.4% increase in28

the national income in 2200; furthermore, Case 70-100% achieves a 24.3% increase in 2200. Conversely,
the national income for Case 68-50% gradually increases and peaks at a 3.9% increase in 2083; theincrease gradually shrinks over time. The national income for Case 70-50% gradually increases andpeaks at a 6.2% increase in 2084; the increase gradually shrinks over time.
Figures 12 and 13 show the percent changes in capital stock and labor supply from the benchmark
case under the scenario with the 70-pension starting age. The higher the employment rate of elderly
individuals, the higher the level of capital stock. A rise in the starting age for receiving pension benefitsmeans a reduction of public pension, which stimulates individual savings and increases capital stock.
Case 70-100% achieves a 28.5% increase in 2200. Conversely, the capital stock for Case 70-50%
gradually increases and peaks at 11.8% in 2101; the increase gradually shrinks over time. The higher theemployment rate of elderly individuals, the higher the level of labor supply. This result suggests that ahigh employment rate means more labor supply, as measured by the efficiency units, because ourmodel’s employment rate represents the labor efficiency level. Case 70-100% attains a 21.9% increase in
2200. Conversely, the labor supply for Case 70-50% gradually increases and peaks at a 3.4% increase in
2082; the increase gradually shrinks over time.
Figures 14 and 15 illustrate the changes in interest rates and wage rates, respectively, from thebenchmark case under the scenario with the 70-pension starting age (percentage-point changes for Figure
14; percent changes for Figure 15). Figure 14 reveals that the interest rate for Case 70-100% is firsthigher than that of the benchmark case but decreases after 2052. In contrast, Figure 15 denotes that thewage rate for Case 70-100% is first lower than that of the benchmark case but increases after 2052.
These results suggest that the reform of Case 70-100% creates an additional considerable labor supply,
which reduces the wage rate and increases the interest rate. As mentioned, Case 70-100% increases thecapital stock over time and reduces the interest rate. After 2052, this effect of lowering interest rates
through increases in capital stock might have exceeded the abovementioned effect of raising interest rates
through increases in the labor supply. A possible reason for this is as follows. Initially, after the pension
reform, many middle-aged and older generations suddenly faced a raised starting age for receiving
pension benefits and a longer working period. Therefore, those generations might use the labor income
from the additional working period mainly for consumption during their remaining lifetime. As time
passes, more generations can gradually maximize their utility over the entire lifetime, including a longer
working period, at younger ages. This situation may increase the portion of the additional labor income
29put into individual savings, which gradually promotes capital accumulation and significantly lowers
interest rates.
For Case 70-50%, the additional labor supply measured by the efficiency units is not as large as in
the other cases. In the 70-50% case, the effect of increasing interest rates through increases in the labor
supply is weak; thus, decreasing interest rates through increases in the capital stock is dominant.
Consequently, the latter effect might have exceeded the former after 2030, earlier than in Case 70-100%.
Additionally, Case 70-50% (with a low employment rate for elderly individuals) enhances the wage rate,
thereby increasing opportunity costs for having children. As a result, the total population for Case 7050% is increasingly smaller over time than in the other simulation cases, as illustrated in Figure 9.
Figure 16 denotes percentage-point changes in consumption tax rates from the benchmark case forCases 70-50%, 70-75%, and 70-100%. The consumption tax rate for the three cases sharply drops until
around 2080. From 2045 onward, the consumption tax rate for Case 70-100% is the lowest while the rate
for Case 70-50% is the highest. Case 70-100% substantially increases the total population and national
income, resulting in a higher tax revenue from labor income. This situation brings about lower
consumption tax rates, which are endogenous variables in our model. This element relatively improves
individual welfare in Case 70-100%. Conversely, in Case 70-50%, the total population is smaller than thebenchmark case after 2111 (Figure 9), resulting in a relatively low national income (Figure 11). Thissituation leads to a lower tax revenue from labor income, resulting in higher consumption tax rates. Thisfactor relatively deteriorates the individual welfare in Case 70-50%.
Figure 17 illustrates percentage-point changes in contribution rates from the benchmark case forCases 70-50%, 70-75%, and 70-100%. The contribution rate for the three cases sharply drops until
around 2065. After 2060, the contribution rate for Case 70-100% is the lowest, while that for Case 7050% is the highest. Figure 9 shows that Case 70-100% substantially increases the total population, which
increases the young working population and reduces contribution rates under a PAYG social security
system. This element also relatively improves individual welfare in Case 70-100%. Conversely, Figure 9shows that the total population is relatively smaller in Case 70-50%. Consequently, reducing the young
working population increases contribution rates under a PAYG social security system, which is also one
factor that relatively deteriorates the individual welfare in Case 70-50%.

30

6. Conclusions
This paper evaluated the effects of a rise in the standard starting age of receiving pension benefits for amodel parameterized to mimic certain features of the Japanese economy. We examined this issue fromtwo viewpoints: individual welfare and future demography. Concretely, we examined the quantitative
effects of a rise of the normal starting age from 65 to 68 and 70 on per-capita welfare and future
population in an aging and depopulating Japan, using an extended lifecycle general equilibrium modelwith endogenous fertility. Our study focuses primarily on the difference in the employment rate forindividuals aged 65 and above. The effects of two types of scenarios with the rising starting age (68 and70) were quantitatively investigated under three kinds of experiments with different employment rates(50%, 75%, and 100%) for elderly individuals during the transitional period, 2022–2300. We introduced
an LSRA to calculate the per-capita welfare and evaluate the pure efficiency gains or losses of thesepension reforms.
The two main findings of our analysis are as follows. First, a rise in the standard starting age forreceiving pension benefits enhances individual welfare, and the magnitude of this effect is greater for thecase with the 70-pension starting age than that with the 68-starting age. Furthermore, the higher theemployment rate for individuals aged 65 and above, the more the individual welfare improves. Possible
reasons for this result are as follows. A rise in the starting age for pension benefits means public pensions
are reduced, which stimulates individual savings. This promotes capital formation, increases wage rates,
and improves individual welfare. Additionally, a higher employment rate for elderly individuals means
more labor supply, as measured by the efficiency units, which brings about a higher labor income andimproves individual welfare.
Second, the effect of a rise in the normal starting age on future demography depends significantly on
the employment rate of individuals aged 65 and above, especially in the long run. With the 100%
employment rate for elderly individuals, the future population steadily increases. In the case of thecurrent 50% employment rate, however, the future population will start to decrease after approximately
2110, compared to the benchmark case with the 65-pension starting age. A possible reason for this result
is that a low employment rate for elderly individuals means a lower labor supply, as measured by theefficiency units, which increases the wage rates and enhances opportunity costs for having children.
Finally, we discuss policy implications based on the simulation results. The above main findings
suggest that a rise in the standard starting age for pension benefits is desirable from the viewpoint of31

individual welfare; however, from the viewpoint of future population, it depends significantly on theemployment of individuals aged 65 and above. A high employment rate for elderly individuals improves
welfare and the future population; however, when Japan’s employment rate remains at 50%, it would
harm the total population level in the long run, compared to the benchmark case with the 65-starting age
for pension benefits. Therefore, if the starting age for pension benefits increases, it would be desirable tosimultaneously increase the employment rate of elderly individuals, for example, up to 75%.

Appendix A: Model for the High-Income Class (University Graduates)
Here, we describe the household behavior of the high-income class household (i.e., university graduates).

A.1 Household behavior
Each agent enters the economy as a decision-making unit and starts to work at age 22 years, and lives toa maximum age of 105 years with uncertainty of death. The children aged 0–17 or 0–21 only consume,
involving childrearing costs for their parent. The probability of a household born in year 𝑡, surviving
until s , can be expressed by
𝑡(𝑈)

𝑝𝑠

𝑡
= ∏𝑠−1𝑗=22 𝑞𝑗+1|𝑗 .

(1)’Each agent who begins its economic life at age 22 chooses perfect-foresight consumption paths
𝑡(𝑈)

(𝐶𝑠

𝑡(𝑈)

), leisure paths (𝑙𝑠), and the number of born children ( nst (U ) ) to maximize a time-separable utility

function of the form:
1

𝑈

𝑡(𝑈)

=

1
1−

𝑡(𝑈)
𝑡(𝑈) 1−𝛾𝑡(𝑈)(𝑈) ∑40(1 + 𝛿)−(𝑠−22) (𝑛𝑠 )+ (1 − 𝛼 (𝑈) ) ∑105
(1 +
1 [𝛼
𝑠=22 𝑝𝑠
𝑠=22 𝑝𝑠𝛾1−(𝑠−22)𝛿)

1−
𝛾𝑡(𝑈) 𝜑 𝑡(𝑈) 1−𝜑{(𝐶𝑠 ) (𝑙𝑠 )
} ].

𝑡(𝑈)

where 𝐶𝑠

𝑡(𝑈)

, 𝑙𝑠

𝑡(𝑈)

and 𝑛𝑠(2)’

are respectively consumption, leisure and the number of children to bear

(only in the first 19 periods of the life) for an agent born in year 𝑡, of age 𝑠. 𝛼 (𝑈) is the utility weightof the number of children relative to the consumption–leisure composite.
𝑡(𝑈)Letting 𝐴𝑠

be capital holdings for the agent born in year 𝑡, of age 𝑠, maximization of Equation(2)’ is subject to a lifetime budget constraint defined by the sequence:
𝑡(𝑈)

𝑡(𝑈)

𝐴𝑠+1 = {1 + 𝑟𝑡+𝑠 (1 − 𝜏 𝑟 )}𝐴𝑠
𝑡(𝑈)

𝑜𝑟𝑠

(𝑈)

𝑝

𝑡(𝑈)

+ (1 − 𝜏 𝑤 − 𝜏𝑡+𝑠 )𝑤𝑡+𝑠 𝑒𝑠 {1 − 𝑙𝑠

𝑡(𝑈)

𝑡(𝑈)

− 𝑡𝑐𝑠

𝑡(𝑈)

+ 𝑏𝑡(𝑈)− 𝑡𝑐𝑡𝑢 (𝑛𝑢 )}𝑅𝐸
− (1 − 𝑚)(1 + 𝜏𝑐𝑡 )𝛷𝑠({1 − 𝑙𝑡(𝑈)) − (1 + 𝜏𝑐𝑡 )𝐶𝑡(𝑈)
𝑠
𝑢
𝑠
𝑢=22

(3)’
32

𝑡(𝑈)

(𝑛𝑠

𝑡(𝑈)

)}𝛬𝑠 + 𝑎𝑠−𝑡(𝐻)− 𝑚(1 + 𝜏𝑐𝑡 )𝛷𝑠.

There are no liquidity constraints, and thus the assets can be negative. An individual’s earnings ability(𝑈)𝑒𝑠

is an exogenous function of age, and 𝛬𝑠 denotes the employment rate of age 𝑠.
The pension benefit is assumed to comprise only an earnings-related pension:
𝑡(𝑈)

𝑡(𝑈)

𝑏𝑠

𝑡(𝑈)

({1 − 𝑙𝑢𝑡(𝑈)− 𝑡𝑐𝑢𝑡 (𝑛𝑢)}𝑅𝐸
𝑢=22 ) = {𝜃𝐻𝑡(𝑈) ({1 − 𝑙𝑢𝑡(𝑈)− 𝑡𝑐𝑢𝑡 (𝑛𝑢0)}𝑅𝐸
𝑢=22 )(𝑠 ≥ 𝑆𝑇), (4)’(𝑠 < 𝑆𝑇)

where
𝑡(𝑈)

𝐻𝑡(𝑈) ({1 − 𝑙𝑢𝑡(𝑈)− 𝑡𝑐𝑢𝑡 (𝑛𝑢)}𝑅𝐸
𝑢=22 ) =

1
(𝑈)
𝑡(𝑈)
𝑡(𝑈)
∑𝑅𝐸 𝑤 𝑒 {1 − 𝑙𝑠 − 𝑡𝑐𝑠𝑡 (𝑛𝑠 )} 𝛬𝑠 .
𝑅𝐸−21 𝑠=22 𝑡+𝑠 𝑠𝑡(𝑈)The average annual labor income for each agent is 𝐻𝑡(𝑈) ({1 − 𝑙𝑢𝑡(𝑈)− 𝑡𝑐𝑢𝑡 (𝑛𝑢𝑡(𝑈)weight coefficient of the part proportional to 𝐻𝑡(𝑈) is 𝜃. The symbol 𝑏𝑠𝑡(𝑈)𝑡𝑐𝑢𝑡 (𝑛𝑢

(5)’

)}𝑅𝐸
𝑢=22 ), and the𝑡(𝑈)({1 − 𝑙𝑢−)}𝑅𝐸
𝑢=22 ) in Equation (3)’ signifies that the amount of public pension benefit is a function of theage profile of labor supply, {1 − lut (U )− tcut (nut (U ) )}uRE=22 .

A parent is assumed to bear children and expend for them until they become independent of theirparent, namely, during the period when they are from zero to 21 years old. Here, note that the children
aged below 22 years old do not conduct an economic activity independently, and only childrearing costfor their parent arises until they become independent of their parent. The financial costs for rearing the𝑡(𝑈)children when the parent born in year 𝑡 is 𝑠 years old are represented by 𝛷𝑠𝑡(𝐻)and 𝛷𝑠, which are

the cost for the children who will become university graduates and high school graduates, respectively:
𝑡(𝑈)

𝑡(𝑈)
𝛷𝑠
=

∑𝑠𝑘=22 𝜉𝑡(𝑈) (1 − 𝜌)𝑛𝑘

(𝑠 = 22,23, ⋯ ,40)

𝑡(𝑈)
𝑡(𝑈)
∑40
(1 − 𝜌)𝑛𝑘
𝑘=22 𝜉
𝑡(𝑈)40{∑𝑘=𝑠−21 𝜉 𝑡(𝑈) (1 − 𝜌)𝑛𝑘

(𝑠 = 41,42,43),(𝑠 = 44,45, ⋯ ,61)

𝑡(𝑈)

𝛷𝑠

= 0 (𝑠 = 62,63, ⋯ ,105),𝑡(𝑈)∑𝑠𝑘=22 𝜉𝑡(𝑈) (1 − 𝜌)𝑛𝑘
(𝑠 = 22,23, ⋯ ,39)
𝑡(𝐻)
𝛷𝑠
={
,
𝑡(𝑈)
40
𝑡(𝑈)
∑𝑘=𝑠−17 𝜉(1 − 𝜌)𝑛𝑘(𝑠 = 40,41, ⋯ ,57)
𝑡(𝐻)

𝛷𝑠

(6)’

(7)’
(8)’
(9)’

= 0 (𝑠 = 58,59, ⋯ ,105),

𝜉𝑡(𝑈) = 𝛽𝑁𝑊 𝑡(𝑈) .

(10)’

The time cost for rearing the children when the parent born in year 𝑡 is s years old is representedby𝑡(𝑈)𝑡𝑐𝑠𝑡 (𝑛𝑠𝑡(𝑈)) = 𝜇𝑛𝑠(𝑈)

When 𝐵𝑄𝑡.

(11)’

is the sum of bequests inherited by the high-income class households at time 𝑡, thebequest to be inherited by each high-income class household is defined by
33

𝑡(𝑈)

𝑎𝑠

(𝑈)

=

(𝑈)

where 𝐸𝑡

(1−𝜏ℎ )𝐵𝑄𝑡+𝑠
(𝑈)
𝐸𝑡+𝑠

,

(12)’

is the number of the high-income class households conducting an economic activity

independently, aged 22 and above, and
(𝑈)

𝐵𝑄𝑡

𝑡−𝑠−1(𝑈))= ∑105𝑠=22(𝑁𝑠𝑡−𝑠−1(𝑈)− 𝑁𝑠+1𝑡−𝑠−1(𝑈)

)𝐴𝑠+1

.

(13)’

The number of the generation born in year 𝑡, of age s , is represented by
𝑡(𝑈)

𝑁𝑠

𝑡(𝑈)

𝑁0

(𝑈)

is the sum of childrearing costs incurred by the high income class households at time

= 𝑝𝑠

When 𝑂𝑅𝑡

𝑡(𝑈)

.

(14)’

𝑡, the childrearing cost for orphans for each high income class household is defined by
𝑡(𝑈)

𝑜𝑟𝑠

(𝑈)

=

𝑂𝑅𝑡+𝑠
(𝑈)
𝐸𝑡+𝑠

,

(15)’

where
(𝑈)

𝑂𝑅𝑡

𝑡−𝑠(𝑈)= (1 − 𝑚) ∑61
𝑠=22(𝑁𝑠−1

𝑡−𝑠(𝑈)− 𝑁𝑠

𝑡−𝑠(𝑈))𝛷𝑠𝑡−𝑠(𝑈)+ 𝑚 ∑57
𝑠=22(𝑁𝑠−1

𝑡−𝑠(𝑈)− 𝑁𝑠

𝑡−𝑠(𝐻))𝛷𝑠. (16)’

When we consider the utility maximization problem over time for each agent, besides the flow
budget constraint represented by Equation (3)’, the following constraint is imposed:
𝑡(𝑈)

{

𝑡(𝑈)

≤ 1 − 𝑡𝑐𝑠𝑡 (𝑛𝑠0 ≤ 𝑙𝑠)(22 ≤ 𝑠 ≤ 𝑅𝐸)
.

(17)’

𝑡(𝑈)
𝑙𝑠 = 1 (𝑅𝐸 + 1 ≤ 𝑠 ≤ 105)

Each individual maximizes Equation (2)’ subject to Equations (3)’ and (17)’ (see Appendix C for
further details). From the utility maximization problem, the equation expressing the evolution of thenumber of children over time for each individual is characterized by
𝑡(𝑈)

𝑊𝑠

𝑝

𝑡(𝑈)

= ( 𝑠−1𝑡(𝑈) ) [
𝑝𝑠

1+𝛿
𝑡(𝑈)
] 𝑊𝑠−1 ,1+𝑟𝑡+𝑠 (1−𝜏𝑟 )
1−

1

(18)’
𝑡(𝑈) −

1

𝛼 (𝑈) 𝑘 𝛾 (𝑛𝑠 ) 𝛾𝑡(𝑈)𝑊𝑠= (1+𝜏𝑐 )[(1−𝑚) ∑21 𝛺𝑡 𝜉𝑡(𝑈) (1−𝜌)𝑡𝑡(𝑈) (1−𝜌)],+𝑚 ∑17𝑔=0 𝑠,𝑔𝑔=0 𝛺𝑠,𝑔 𝜉
𝑡+𝑠

(19)’

−1

𝑔

𝑡
𝑡
where 𝛺𝑠,0
=1 for 𝑔 = 0, 𝛺𝑠,𝑔
= (∏𝑘=1{1 + 𝑟𝑡+𝑠−1+𝑘 (1 − 𝜏 𝑟 )}) .

Similarly, that for the consumption–leisure composite is represented by
𝑡(𝑈)

𝑉𝑠

𝑝

𝑡(𝑈)

1+𝛿
𝑡(𝑈)
𝑟 )] 𝑉𝑠−1 ,
(1−𝜏
𝑡+𝑠

(20)’

= ( 𝑠−1𝑡(𝑈) ) [1+𝑟
𝑝𝑠

(1−𝛼
𝑡(𝑈)
𝑉𝑠
=

1
−
(𝑈) ){(𝐶 𝑡(𝑈) )𝜑 (𝑙𝑡(𝑈) )1−𝜑 } 𝛾 𝜑(𝐶 𝑡(𝑈) )𝜑−1 (𝑙 𝑡(𝑈) )1−𝜑
𝑠
𝑠
𝑠
𝑠
1+𝜏𝑡𝑐

34

.

(21)’

Appendix B: The Utility Maximization Problem for the Low-Income Class
The utility maximization problem over time for each low-income class household in Section 2 isregarded as the maximization of 𝑈𝑡(𝐻) in Equation (2) subject to Equations (3) and (17). Let theLagrange function be𝑡(𝐻)𝐿𝑡(𝐻) = 𝑈𝑡(𝐻) + ∑105
𝑠=18 𝜆𝑠
𝑡(𝐻)

𝑙𝑠

𝑡(𝐻)

− 𝑡𝑐𝑠𝑡 (𝑛𝑠
𝑡(𝐻)

𝑐
𝜏𝑡+𝑠
)𝐶𝑠

𝑡(𝐻)

𝑡(𝐻)

+ [1 − 𝜏 𝑤 − 𝜏𝑡+𝑠 ]𝑤𝑡+𝑠 𝑒𝑠 {1 −𝑡(𝐻))} 𝛬𝑠 + 𝑎𝑠

𝑡(𝐻)

− 𝑜𝑟𝑠

𝑡(𝐻)

𝑐
− (1 − 𝑚)(1 + 𝜏𝑡+𝑠
)𝛷𝑠

𝑡(𝐻)

+ 𝑏𝑠

𝑡(𝐻)

− 𝑡𝑐𝑢𝑡 (𝑛𝑢({1 − 𝑙𝑢𝑡(𝑈)𝑐− 𝑚(1 + 𝜏𝑡+𝑠
)𝛷𝑠

(𝐻)

𝑝

[−𝐴𝑠+1 + {1 + 𝑟𝑡+𝑠 (1 − 𝜏 𝑟 )}𝐴𝑠

𝑡(𝐻)

)}𝑅𝐸
𝑢=20 ) − (1 +

𝑡(𝐻)
𝑡
𝑡(𝐻)
{1 − 𝑙𝑡(𝐻)− 𝑡𝑐𝑠 (𝑛𝑠 )}] + ∑𝑅𝐸
𝑠=18 𝜂𝑠
𝑠

(B.1)
𝑡(𝐻)

where 𝜆𝑠𝑡(𝐻)and 𝜂𝑠represent the Lagrange multiplier for Equations (3) and (17), respectively.
𝑡(𝐻)The first-order conditions on the number of children 𝑛𝑠𝑡(𝐻), consumption 𝐶𝑠𝑡(𝐻), leisure 𝑙𝑠

, and

𝑡(𝐻)

assets 𝐴𝑠+1 for s =18, 19, …, 105 can be expressed by𝑡(𝐻) (𝐻)𝑝𝑠𝛼1𝑡(𝐻) −𝛾(1 + 𝛿)−(𝑠−18) (𝑛𝑠

)

17
𝑡(𝐻)
(𝐻)
𝑝
𝑐
𝑤
𝑡
= 𝜆𝑠 {𝜇[1 − 𝜏 − 𝜏𝑡+𝑠 ]𝑤𝑡+𝑠 𝑒𝑠 𝛬𝑠 + (1 − 𝑚)(1 + 𝜏𝑡+𝑠 ) ∑ 𝛺𝑠,𝑔𝜉𝑡(𝐻) (1 − 𝜌)𝑔=0
(𝐻)𝑡(𝐻) 𝜃𝑤𝑡+𝑠 𝑒𝑠 𝛬𝑠𝑡(𝐻)+ 𝜇𝜂𝑠 , (B.2)
𝑅𝐸−19

𝑐
𝑡
𝑡(𝐻)
+𝑚(1 + 𝜏𝑡+𝑠
) ∑21
(1 − 𝜌)} + 𝜇 ∑105
𝑔=0 𝛺𝑠,𝑔 𝜉
𝑘=𝑆𝑇 𝜆𝑘−1𝑔𝑡𝑡where 𝛺𝑠,0
=1 for 𝑔 =0, 𝛺𝑠,𝑔
= (∏𝑘=1{1 + 𝑟𝑡+𝑠−1+𝑘 (1 − 𝜏 𝑟 )}) ,
1

−
𝑡(𝐻)
𝑡(𝐻)
𝑡(𝐻)
𝑡(𝐻)
𝑡(𝐻)
𝑡(𝐻)
𝑐
𝑝𝑠 (1 − 𝛼 (𝐻) )(1 + 𝛿)−(𝑠−18) {(𝐶𝑠 )𝜑 (𝑙𝑠 )1−𝜑 } 𝛾 𝜑(𝐶𝑠 )𝜑−1 (𝑙𝑠 )1−𝜑 = 𝜆𝑠 (1 + 𝜏𝑡+𝑠),(B.3)

pst ( H ) (1 −  ( H ) )(1 +  ) −( s −18 ) (Cst ( H ) ) (lst ( H ) )1−
𝑡(𝐻)

= 𝜆𝑠

𝑡(𝐻)

{1 − 𝑙𝑠

𝑡(𝐻)

𝑡(𝐻)

1 − 𝑙𝑠
𝑡(𝐻)

𝜂𝑠

1



𝑡(𝐻)

− 𝑡𝑐𝑠𝑡 (𝑛𝑠t (H ) s) (lst ( H ) ) −(𝐻)𝑡(𝐻) 𝜃𝑤𝑡+𝑠 𝑒𝑠 𝛬𝑠
𝑡(𝐻)
+ 𝜂𝑠
𝑅𝐸−19

(𝐻)

𝑝

= {1 + 𝑟𝑡+𝑠 (1 − 𝜏 𝑟 )}𝜆𝑠+1 ,𝜂𝑠−{(1 − 𝜏 𝑤 − 𝜏𝑡+𝑠 )𝑤𝑡+𝑠 𝑒𝑠 𝛬𝑠 } + ∑105
𝑘=𝑆𝑇 𝜆𝑘𝑡(𝐻)𝜆𝑠 (1 −  )(C(B.4)(𝑠 ≤ 𝑅𝐸),

𝑡(𝐻)

(B.5)

)} = 0 (𝑠 ≤ 𝑅𝐸),

(B.6)
(B.7)

= 0 (𝑠 > 𝑅𝐸),

(B.8)

≥ 0.

The combination of Equations (B.2) and (B.5) produces Equations (18) and (19). If the initial value,
𝑡(𝐻)

𝑡(𝐻)

𝑛18 , is given, the initial value, 𝑊18𝑡(𝐻), can be derived from Equation (19). If the value, 𝑊18
35

, is

𝑡(𝐻)

specified, the value of each age, 𝑊𝑠𝑡(𝐻)each age, 𝑛𝑠𝑡(𝐻). If the value, 𝑛𝑠, can be derived from Equation (18), which generates the value of, is specified, the child rearing cost for lifetime is calculated, which

gives the lifetime budget constraint represented by Equation (B.10).
The combination of Equations (B.3) and (B.5) produces Equations (20) and (21). If the initial value,
𝑡(𝐻)

𝑉18

𝑡(𝐻)

, is specified, the value of each age, 𝑉𝑠𝑡(𝐻)specified, the values of consumption, 𝐶𝑠𝑡(𝐻), can be derived from Equation (20). If 𝑉𝑠

is, and leisure, lst ( H ) , at each age are obtained in the methodthat follows.
For 𝑠 =18, 19, …, RE, the combination of Equations (B.3) and (B.4) yields the following
expression:(𝐻)𝑝𝜑{(1−𝜏𝑤 −𝜏𝑡+𝑠 )𝑤𝑡+𝑠 𝑒𝑠

𝑡(𝐻)

𝐶𝑠

=[

𝜆

𝜆𝑠

(𝐻)
𝑡(𝐻)
𝜃𝑤𝑡+𝑠 𝑒𝑠 𝛬𝑠 𝜂𝑠+ 𝑡(𝐻) }
𝑅𝐸−19
𝜆𝑠

𝑡(𝐻)

𝑡(𝐻)

is given under 𝜂𝑠
𝑡(𝐻)

𝑡(𝐻)

𝑡(𝐻)

=0, the value of 𝐶𝑠

can be obtained using a numerical

can be derived from Equation (21). The value of 𝑙𝑠

is chosen so

obtained in the simulation is the closest to that calculated by evolution from𝑡(𝐻)through Equation (20). If the value of 𝑙𝑠

𝑉18

(B.9)

.

𝑡(𝐻)

method, and then the value of 𝑉𝑠
that the value of 𝑉𝑠

𝑡(𝐻)

] 𝑙𝑠

𝑐 )
(1−𝜑)(1+𝜏𝑡+𝑠

If the value of 𝑙𝑠

𝑡(𝐻)

𝑡(𝐻)

𝑘
𝛬𝑠 +∑105
𝑘=𝑆𝑇 𝑡(𝐻)𝑡(𝐻)obtained from Equation (21) under 𝑙𝑠𝑡(𝐻)chosen is unity or higher, the value of 𝐶𝑠𝑡(𝐻)=1. If it is less than unity, the value of 𝐶𝑠

isis derived from

Equation (B.9).
𝑡(𝐻)For 𝑠 =RE+1, RE+2, …, 105, the condition of 𝑙𝑠
𝑡(𝐻)

𝑉𝑠

=

𝜑
𝑡(𝐻) − +𝜑−1
(1−𝛼(𝐻) )𝜑(𝐶𝑠 ) 𝛾𝑐1+𝜏𝑡+𝑠𝑡(𝐻)The value of 𝐶𝑠=1 leads to the following equation:.

(21)’’

is chosen to satisfy this equation.
𝑡(𝐻)

𝑡(𝐻)

From Equation (3) and the terminal condition 𝐴18 =𝐴106 =0, the lifetime budget constraint for anindividual (= NW t ( H ) ) is derived:
𝑡(𝐻)
(𝐻)
𝑡(𝐻)
𝑡(𝐻)
𝑡(𝐻) 𝑡(𝐻)
𝑡(𝐻)
𝑝
∑𝑅𝐸
[1 − 𝜏 𝑤 − 𝜏𝑡+𝑠 ]𝑤𝑡+𝑠 𝑒𝑠 {1 − 𝑙𝑠 − 𝑡𝑐𝑠𝑡 (𝑛𝑠 )}𝛬𝑠 + ∑105𝑏𝑠 ({1 − 𝑙𝑢 −𝑠=18 𝛹𝑠
𝑠=𝑆𝑇 𝛹𝑠𝑡(𝐻)𝑡𝑐𝑢𝑡 (𝑛𝑢

𝑡(𝐻)

105
)}𝑅𝐸
𝑢=20 ) + ∑𝑠=18 𝛹𝑠𝑡(𝐻)𝑠𝑚) ∑35𝑠=18 ∑𝑘=18 𝛹𝑠

𝑡(𝐻)

𝑠
𝑚 ∑39
𝑠=18 ∑𝑘=18 𝛹𝑠

𝑡(𝐻)

− 𝑜𝑟𝑠

𝑡(𝐻)

) = ∑105
𝑠=18 𝛹𝑠𝑡(𝐻)𝑐(1 + 𝜏𝑡+𝑠)𝜉𝑡(𝐻) (1 − 𝜌)𝑛𝑘

𝑡(𝐻)

𝑐
𝜏𝑡+𝑠
)𝜉𝑡(𝐻) (1 − 𝜌)𝑛𝑘

𝑡(𝐻)

(𝑎𝑠

𝑡(𝐻)

𝑡(𝐻)

+ (1 −𝑡(𝐻)𝑠+ (1 − 𝑚) ∑40
𝑠=36 ∑𝑘=𝑠−17 𝛹𝑠40+ (1 − 𝑚) ∑57
𝑠=41 ∑𝑘=𝑠−17 𝛹𝑠(1 + 𝜏𝑐𝑡+𝑠 )𝜉𝑡(𝐻) (1 − 𝜌)𝑛𝑘𝑡(𝐻)𝑐
(1 + 𝜏𝑡+𝑠
)𝐶𝑠

𝑡(𝐻)

(1 + 𝜏𝑐𝑡+𝑠 )𝜉𝑡(𝐻) (1 − 𝜌)𝑛𝑘𝑡(𝐻)40
+ 𝑚 ∑61𝑠=40 ∑𝑘=𝑠−21 𝛹𝑠

(1 +

+
𝑡(𝐻)

(1 + 𝜏𝑐𝑡+𝑠 )𝜉𝑡(𝐻) (1 − 𝜌)𝑛𝑘,

(B.10)
36

𝑡(𝐻)

𝑡(𝐻)

where 𝛹18 =1 for 𝑠 =18, 𝛹𝑠= (∏𝑠𝑢=19{1 + 𝑟𝑡+𝑢 (1 − 𝜏 𝑟 )})−1 for 𝑠 =19, 20, …, 105.

Appendix C: The Utility Maximization Problem for the High-Income Class
The utility maximization problem over time for each high-income class household in Appendix A isregarded as the maximization of 𝑈𝑡(𝑈) in Equation (2)’ subject to Equations (3)’ and (17)’. Let theLagrange function be𝑡(𝑈)𝐿𝑡(𝑈) = 𝑈𝑡(𝑈) + ∑105
𝑠=22 𝜆𝑠
𝑡(𝑈)

𝑙𝑠

𝑡(𝑈)

− 𝑡𝑐𝑠𝑡 (𝑛𝑠

𝑡(𝑈)

𝑡(𝑈)

𝑡(𝑈)

)} 𝛬𝑠 + 𝑎𝑠(𝑈)𝑝[−𝐴𝑠+1 + {1 + 𝑟𝑡+𝑠 (1 − 𝜏 𝑟 )}𝐴𝑠

+ [1 − 𝜏 𝑤 − 𝜏𝑡+𝑠 ]𝑤𝑡+𝑠 𝑒𝑠 {1 −− 𝑜𝑟𝑠𝑡(𝑈) + 𝑏𝑠𝑡(𝑈) ({1 − 𝑙𝑢𝑡(𝑈) − 𝑡𝑐𝑢𝑡 (𝑛𝑢𝑡(𝑈) )}𝑅𝐸
𝑢=22 ) − (1 +𝑐 )𝐶 𝑡(𝑈)𝑐 )𝛷 𝑡(𝑈)𝑐 )𝛷 𝑡(𝐻)𝑡 (𝑈 )
𝜏𝑡+𝑠− (1 − 𝑚)(1 + 𝜏𝑡+𝑠
− 𝑚(1 + 𝜏𝑡+𝑠
] + ∑𝑅𝐸{1 − 𝑙𝑡𝑠(𝑈) − 𝑡𝑐𝑡𝑠 (𝑛𝑡𝑠(𝑈) )}, (C.1)
𝑠=22 𝜂𝑠𝑠𝑠𝑠𝑡(𝑈)where 𝜆𝑠𝑡(𝑈)and 𝜂𝑠represent the Lagrange multiplier for Equations (3)’ and (17)’, respectively.
𝑡(𝑈)The first-order conditions on the number of children 𝑛𝑠𝑡(𝑈), consumption 𝐶𝑠𝑡(𝑈), leisure 𝑙𝑠

, and

𝑡(𝑈)

assets 𝐴𝑠+1 for s =22, 23, …, 105 can be expressed by𝑡(𝑈) (𝑈)𝑝𝑠𝛼1𝑡(𝑈) −𝛾(1 + 𝛿)−(𝑠−22) (𝑛𝑠

)

21
𝑡(𝑈)
(𝑈)
𝑝
𝑐
𝑤
𝑡
= 𝜆𝑠 {𝜇[1 − 𝜏 − 𝜏𝑡+𝑠 ]𝑤𝑡+𝑠 𝑒𝑠 𝛬𝑠 + (1 − 𝑚)(1 + 𝜏𝑡+𝑠 ) ∑ 𝛺𝑠,𝑔𝜉𝑡(𝑈) (1 − 𝜌)
𝑔=0(𝑈)𝑡(𝑈) 𝜃𝑤𝑡+𝑠 𝑒𝑠 𝛬𝑠𝑡(𝑈)+ 𝜇𝜂𝑠 , (C.2)
𝑅𝐸−21

𝑐
𝑡
𝑡(𝑈)
+𝑚(1 + 𝜏𝑡+𝑠
) ∑17
(1 − 𝜌)} + 𝜇 ∑105
𝑔=0 𝛺𝑠,𝑔 𝜉
𝑘=𝑆𝑇 𝜆𝑘−1𝑔𝑡𝑡where 𝛺𝑠,0
=1 for 𝑔 =0, 𝛺𝑠,𝑔
= (∏𝑘=1{1 + 𝑟𝑡+𝑠−1+𝑘 (1 − 𝜏 𝑟 )}) ,
1

𝑡(𝑈)

𝑝𝑠

𝑡(𝑈) 𝜑(1 − 𝛼 (𝑈) )(1 + 𝛿)−(𝑠−22) {(𝐶𝑠𝑡(𝑈) 1−𝜑 −𝛾

) (𝑙𝑠



pst (U ) (1 −  (U ) )(1 +  ) −( s − 22 ) (Cst (U ) ) (lst (U ) )1−
𝑡(𝑈)

= 𝜆𝑠
𝑡(𝑈)

}

−

1


)

𝑡(𝑈) 1−𝜑(𝑙𝑠)𝑐= 𝜆𝑡𝑠 (1 + 𝜏𝑡+𝑠), (C.3)t (U ) s) (lst (U ) ) −(𝑈)𝑡(𝑈) 𝜃𝑤𝑡+𝑠 𝑒𝑠 𝛬𝑠
𝑡(𝑈)
+ 𝜂𝑠
𝑅𝐸−21

(𝑈)

𝑝

𝑡(𝑈) 𝜑−1

𝜑(𝐶𝑠 (1 −  )(C{(1 − 𝜏 𝑤 − 𝜏𝑡+𝑠 )𝑤𝑡+𝑠 𝑒𝑠 𝛬𝑠 } + ∑105
𝑘=𝑆𝑇 𝜆𝑘(𝑠 ≤ 𝑅𝐸),

(C.4)

𝑡(𝑈)

𝜆𝑠

= {1 + 𝑟𝑡+𝑠 (1 − 𝜏 𝑟 )}𝜆𝑠+1 , (C.5)

𝑡(𝑈){1 − 𝑙𝑠

𝜂𝑠

)

𝑡(𝑈)

𝑡(𝑈)

1 − 𝑙𝑠
𝑡(𝑈)

𝜂𝑠

𝑡(𝑈)

− 𝑡𝑐𝑠𝑡 (𝑛𝑠(C.6))} = 0 (𝑠 ≤ 𝑅𝐸),

(C.7)

= 0 (𝑠 > 𝑅𝐸),

(C.8)

≥ 0.

The combination of Equations (C.2) and (C.5) produces Equations (18)’ and (19)’. If the initial value,
37

𝑡(𝑈)

𝑡(𝑈)

𝑡(𝑈)

𝑛22 , is given, the initial value, 𝑊22 , can be derived from Equation (19)’. If the value, 𝑊22 , is𝑡(𝑈)specified, the value of each age, 𝑊𝑠𝑡(𝑈)of each age, 𝑛𝑠𝑡(𝑈). If the value, 𝑛𝑠, can be derived from Equation (18)’, which generates the value, is specified, the child rearing cost for lifetime is calculated, which

gives the lifetime budget constraint represented by Equation (C.10).
The combination of Equations (C.3) and (C.5) produces Equations (20)’ and (21)’. If the initial
𝑡(𝑈)

𝑡(𝑈)

value, 𝑉22 , is specified, the value of each age, 𝑉𝑠𝑡(𝑈)specified, the values of consumption, 𝐶𝑠𝑡(𝑈), can be derived from equation (20)’. If 𝑉𝑠t (U ), and leisure, ls

is, at each age are obtained in the methodthat follows.
For 𝑠 =22, 23, …, RE, the combination of Equations (C.3) and (C.4) yields the following
expression:
(𝑈)

𝑝

𝑡(𝑈)

𝐶𝑠

𝜑{(1−𝜏𝑤 −𝜏𝑡+𝑠 )𝑤𝑡+𝑠 𝑒𝑠

=[

𝑡(𝑈)

𝜆𝑠

(𝑈)
𝑡(𝑈)
𝜃𝑤𝑡+𝑠 𝑒𝑠 𝛬𝑠 𝜂𝑠+ 𝑡(𝑈) }
𝑅𝐸−21
𝜆𝑠

𝑡(𝑈)

𝑡(𝑈)

is given under 𝜂𝑠𝑡 = 0, the value of 𝐶𝑠𝑡(𝑈)method, and then the value of 𝑉𝑠𝑡(𝑈)so that the value of 𝑉𝑠

(C.9)

.

can be obtained using a numerical𝑡(𝑈)can be derived from Equation (21)’. The value of 𝑙𝑠

is chosen

obtained in the simulation is the closest to that calculated by evolution from𝑡(𝑈)through Equation (20)’. If the value of 𝑙𝑠

𝑉22

𝑡(𝑈)

] 𝑙𝑠

𝑐 )
(1−𝜑)(1+𝜏𝑡+𝑠

If the value of 𝑙𝑠

𝑡(𝑈)

𝜆

𝑘
𝛬𝑠 +∑105
𝑘=𝑆𝑇 𝑡(𝑈)𝑡(𝑈)obtained from Equation (21)’ under 𝑙𝑠𝑡(𝑈)chosen is unity or higher, the value of 𝐶𝑠𝑡(𝑈)=1. If it is less than unity, the value of 𝐶𝑠

isis derived from

Equation (C.9).
𝑡(𝑈)For 𝑠 =RE+1, RE+2, …, 105, the condition of 𝑙𝑠
𝑡(𝑈)

𝑉𝑠

=

𝜑
𝑡(𝑈) − +𝜑−1
(1−𝛼(𝑈) )𝜑(𝐶𝑠 ) 𝛾𝑐1+𝜏𝑡+𝑠𝑡(𝑈)The value of 𝐶𝑠=1 leads to the following equation:. (21)’’’

is chosen to satisfy this equation.
𝑡(𝑈)

𝑡(𝑈)

From Equation (3)’ and the terminal condition 𝐴22 =𝐴106 =0, the lifetime budget constraint for anindividual (= NW t (U ) ) is derived:

38

𝑅𝐸
𝑡(𝑈)

∑ 𝛹𝑠

(𝑈)

𝑝

[1 − 𝜏 𝑤 − 𝜏𝑡+𝑠 ]𝑤𝑡+𝑠 𝑒𝑠𝑡(𝑈){1 − 𝑙𝑠𝑡(𝑈)− 𝑡𝑐𝑠𝑡 (𝑛𝑠

)}𝛬𝑠

𝑠=22
105

105
𝑡(𝑈) 𝑡(𝑈)𝑡(𝑈)𝑡(𝑈) 𝑅𝐸𝑡(𝑈) 𝑡(𝑈)𝑡(𝑈)𝑡+ ∑ 𝛹𝑠 𝑏𝑠 ({1 − 𝑙𝑢 − 𝑡𝑐𝑢 (𝑛𝑢 )}𝑢=22 ) + ∑ 𝛹𝑠 (𝑎𝑠 − 𝑜𝑟𝑠 )
𝑠=𝑆𝑇
𝑠=22
105
𝑡(𝑈)

= ∑ 𝛹𝑠𝑡(𝑈)𝑐(1 + 𝜏𝑡+𝑠
)𝐶𝑠

+ (1

𝑠=22
40

𝑠
𝑡(𝑈)

𝑐
(1 + 𝜏𝑡+𝑠)𝜉 𝑡(𝑈) (1 − 𝜌)𝑛𝑘𝑡(𝑈)𝑐
(1 + 𝜏𝑡+𝑠)𝜉 𝑡(𝑈) (1 − 𝜌)𝑛𝑘

− 𝑚) ∑ ∑ 𝛹𝑠

𝑡(𝑈)

+ (1

𝑠=22 𝑘=224340− 𝑚) ∑ ∑ 𝛹𝑠𝑡(𝑈)𝑠=41 𝑘=22𝑡(𝑈)40+(1 − 𝑚) ∑61
𝑠=44 ∑𝑘=𝑠−21 𝛹𝑠
𝑡(𝑈)

𝜌)𝑛𝑘

𝑡(𝑈)

40
+ 𝑚 ∑57
𝑠=40 ∑𝑘=𝑠−17 𝛹𝑠

𝑡(𝑈)

𝑡(𝑈)

𝑐
(1 + 𝜏𝑡+𝑠)𝜉𝑡(𝑈) (1 − 𝜌)𝑛𝑘𝑡(𝑈)where 𝛹22 =1 for 𝑠 =22, 𝛹𝑠𝑡(𝑈)𝑐(1 + 𝜏𝑡+𝑠)𝜉𝑡(𝑈) (1 − 𝜌)𝑛𝑘𝑡(𝑈)𝑠+ 𝑚 ∑39𝑠=22 ∑𝑘=22 𝛹𝑠𝑐(1 + 𝜏𝑡+𝑠)𝜉𝑡(𝑈) (1 −, (C.10)= (∏𝑠𝑢=23{1 + 𝑟𝑡+𝑢 (1 − 𝜏 𝑟 )})−1 for 𝑠 =23, 24, …, 105.

39

References
Adsera, A. (2017) “Education and Fertility in the Context of Rising Inequality,” Vienna Yearbook of
Population Research 15, 63–92.
Aiyagari, S. R. (1994) “Uninsured Idiosyncratic Risk and Aggregate Saving,” The Quarterly Journal of
Economics, 109 (3), 659–684.
Altig, D., A. J. Auerbach, L. J. Kotlikoff, Smetters, K. A., and Walliser, J. (2001) “Simulating Fundamental
Tax Reform in the United States,” American Economic Review 91 (3), 574–595.
Arai, R. and J. Ueda (2013) “A Numerical Evaluation of the Sustainable Size of the Primary Deficit in Japan,”
Journal of the Japanese and International Economies 30, 59–75.
Auerbach, A. J. and Kotlikoff, L. J. (1983a) “National Savings, Economic Welfare, and the Structure of
Taxation” in Feldstein, M. ed., Behavioral Simulation Methods in Tax Policy Analysis, Chicago:
University of Chicago Press, 459–493.
Auerbach, A. J. and Kotlikoff, L. J. (1983b) “An Examination of Empirical Tests of Social Security and
Savings” in Helpman, E., Razin, A. and Sadka, E. eds., Social Policy Evaluation: An Economic
Perspective, New York: Academic, 161–179.
Auerbach, A. J. and L. J. Kotlikoff (1987) Dynamic Fiscal Policy, Cambridge: Cambridge University Press.
Auerbach, A. J., Kotlikoff, L. J., Hagemann, R. P., and Nicoletti, G. (1989) “The Economic Dynamics of an
Aging Population: The Case of Four OECD Countries,” OECD Economic Studies, 97–130.
Braun, A. and Joines, D.H. (2015) “The Implications of a Graying Japan for Government Policy,” Journal of
Economic Dynamics and Control 57, 1–23.
Braun, R. A., Ikeda, D., and Joines, D. H. (2009) “The Saving Rate in Japan: Why It Has Fallen and Why It
Will Remain Low,” International Economic Review 50 (1), 291–321.
Hayashi, F. and Prescott, E. C. (2002) “The 1990s in Japan: A Lost Decade,” Review of Economic Dynamics 5(1), 206–235.
Heer, B. and Maußner, A. (2005) Dynamic General Equilibrium Modelling, Springer.
Higo, M., Schröder, H., and Yamada, A. (2016) “Japan: Determinants of Retirement in a Hyper-Aged Society”
in Hofäcker, D., Hess, M., and König, S. eds., Delaying Retirement, Palgrave Macmillan London, 241–
268.
Homma, M., Atoda, N., Iwamoto, Y., and Ohtake, F. (1987) “Pension: Aging Society and Pension Plan” inHamada, K., Horiuchi, A., and Kuroda, M. eds., Macroeconomic Analysis of the Japanese Economy,
University of Tokyo Press, 149–175 (in Japanese).
Horioka, C. Y. (2021) “Is the Selfish Life-Cycle Model More Applicable in Japan, and If So, Why? A
Literature Survey,” Review of Economics of the Household 19 (1), 157–187.
Ihori, T., Kato, R. R., Kawade, M., and Bessho, S. (2006) “Public Debt and Economic Growth in an Aging
Japan,” in Kaizuka, K. and Krueger, A. O. eds., Tackling Japan's Fiscal Challenges: Strategies to Cope40

with High Public Debt and Population Aging, Palgrave Macmillan, 30–68.
Ihori, T., Kato, R. R., Kawade, M., and Bessho, S. (2011) “Health Insurance Reform and Economic Growth:
Simulation analysis in Japan,” Japan and the World Economy 23 (4) 227–239.
İmrohoroğlu, S., S. Kitao and T. Yamada (2016) “Achieving Fiscal Balance in Japan,” International Economic
Review 57 (1), 117–154.
İmrohoroğlu, S., S. Kitao and T. Yamada (2017) “Can Guest Workers Solve Japan’s Fiscal Problems?,”
Economic Inquiry 55 (3), 1287–1307.
Kato, R. (1998) “Transition to an Aging Japan: Public Pension, Savings, and Capital Taxation,” Journal of theJapanese and International Economies 12 (3), 204–231.
Kikkawa, T. (2009) The Society Divided by Educational Backgrounds, Chikumashobo (in Japanese).
Kikkawa, T. (2018) The Japanese Fragmentation: Detached Non-University Graduate Youngsters, Kobunsha(in Japanese).
Kitao, S. (2015) “Fiscal cost of demographic transition in Japan,” Journal of Economic Dynamics and Control
54, 37-58.
Kitao, S. (2017) “When do we Start? Pension reform in ageing Japan,” The Japanese Economic Review 68(1), 26–47.
Kitao, S. and M. Mikoshiba (2020) “Females, the elderly, and also males: Demographic aging andmacroeconomy in Japan,” Journal of the Japanese and International Economies 56, 101064.
Maebayashi, N. (2019) “Is an unfunded social security system good or bad for growth? A theoretical analysis
of social security systems financed by VAT,” Journal of Public Economic Theory 22 (4), 1069–1104.
Nishiyama, S. and K. Smetters (2005) “Consumption Taxes and Economic Efficiency with Idiosyncratic Wage
Shocks,” Journal of Political Economy 113 (5), 1088–1115.
Oguro, K. and J. Takahata (2013) “Child Benefits and Macroeconomic Simulation Analyses: An OverlappingGenerations Model with Endogenous Fertility,” Public Policy Review 9 (4), 633–659.
Oguro, K., J. Takahata, and M. Shimasawa (2011) “Child Benefit and Fiscal Burden: OLG Model with
Endogenous Fertility,” Modern Economy 2 (4), 602–613.
Okamoto, A. (2013) “Welfare Analysis of Pension Reforms in an Ageing Japan,” The Japanese Economic
Review 64 (4), 452–483.
Okamoto, A. (2020) “Childcare Allowances and Public Pensions: Welfare and Demographic Effects in an
Aging Japan,” The B.E. Journal of Economic Analysis and Policy 20 (2), Article 20190067.
Okamoto, A. (2021) “Immigration Policy and Demographic Dynamics: Welfare Analysis of an Aging Japan,”
Journal of the Japanese and International Economies 62, Article 101168.
Okamoto, A. (2022) “Intergenerational Earnings Mobility and Demographic Dynamics: Welfare Analysis of
an Aging Japan,” Economic Analysis and Policy 74, 76–104.
Okamoto, A. (2024) “The Optimum Quantity of Debt for an Aging Japan: Welfare and Demographic
41

Dynamics,” The Japanese Economic Review 75 (3) (in press).
Sakuragawa, M. and K. Hosono (2010) “Fiscal Sustainability of Japan: A Dynamic Stochastic General
Equilibrium Approach,” The Japanese Economic Review 61, 517–37.
van Groezen, B., T. Leers and L. Meijdam (2003) “Social Security and Endogenous Fertility: Pensions andChild Allowances as Siamese Twins,” Journal of Public Economics 87, 233–251.

Data
Cabinet Office (2010) “The Questionnaire Survey through the Internet for the Child-rearing Cost,” Cabinet
Office Director-general for Policies on Cohesive Society, Japan (in Japanese).
Cabinet Office (2023) “National Accounts Statistics: GDP Statistics,”
https://www.esri.cao.go.jp/jp/sna/menu.html (in Japanese).
Ministry of Finance (2023) “FY2022 budget,” https://www.mof.go.jp/english/budget/budget/index.html
Ministry of Health, Labour and Welfare (1991–2024) “Demographic Survey.” (in Japanese)
https://www.mhlw.go.jp/toukei/list/81-1a.html
Ministry of Health, Labour and Welfare (2014–2023a) “Basic Survey on Wage Structure (Chingin Sensasu),”
https://www.e-stat.go.jp/ (in Japanese).
Ministry of Health, Labour and Welfare (2023b) “Overview of the employees’ pension and national

pension system.” (in Japanese) https://www.mhlw.go.jp/stf/seisakunitsuite/bunya/0000106808_1.html
Ministry of Internal Affairs and Communications (2023a) “Employment of the Elderly.” (in Japanese)
https://www.stat.go.jp/data/topics/topi1380.html
Ministry of Internal Affairs and Communications (2023b) “Labour force and labour force participation rate by
age group: Whole Japan,” https://www.stat.go.jp/data/roudou/longtime/03roudou.htm (in Japanese).
National Institute of Population and Social Security Research (2017) Population Projections for Japan: 2016–
2115 (Estimation in April, 2017), Japan (in Japanese).
National Institute of Population and Social Security Research (2023) Population Projections for Japan: 2021–
2120 (Estimation in 2023), Japan (in Japanese).
OECD (2023a) “Pensions at a Glance 2023: OECD and G20 Indicators.”
https://www.oecd.org/publications/oecd-pensions-at-a-glance-19991363.htm
OECD (2023b) OECD Family Database, OECD, Paris. http://www.oecd.org/social/family/database.htm.
Statistics Bureau of Japan (2023a) “Population Estimation,” (in Japanese)
https://www.stat.go.jp/data/jinsui/index2.html#suikei
Statistics Bureau of Japan (2023b) “Labor Force Survey,” (in Japanese)
https://www.stat.go.jp/data/roudou/sokuhou/nen/ft/pdf/index1.pdf
United Nations (2023) World Population Prospects 2022, Population Division: Department of Economic andSocial Affairs.
42Figure 1 Total population in Japan: Actual results and projections
Source: Statistics Bureau of Japan (2023a) for actual results until 2022 on the total population; National
Institute of Population and Social Security Research (2023) for government projections after 2022

Figure 2 Aging rate transition for five advanced countries
Source: United Nations (2023)
Note: Aging rates indicate the ratios of the people aged 65 and above to the total population.

43

Figure 3 Total fertility rate in Japan: Actual results and projections
Source: Ministry of Health, Labour and Welfare (1991–2023a) for actual results until 2022 on the total
fertility rate; National Institute of Population and Social Security Research (2023) for government
projections after 2022

Figure 4 Number of Births and Marriages in Japan
Source: Ministry of Health, Labour and Welfare (2011–2024) for actual results until 2023 on births andmarriages; National Institute of Population and Social Security Research (2023) for government
projections on births after 2023

44

Figure 5 Employment rate for the elderly aged 65–69
Source: Ministry of Internal Affairs and Communications (2023)

Figure 6 Age–population distribution in the 2022 initial steady state
Notes: The vertical gap between the total population and the number of university graduates is thenumber of high school graduates for each age. For young people unsure if they will be (just) highschool graduates or university graduates, we assume 50/50.
Source: Statistics Bureau of Japan (2023a)

45

Figure 7 Age earnings profiles for two representative agents with different
employment rates (50%, 75%, and 100%)
Note: Age earnings profiles consider different employment rates after age 64 for each simulation case.
Source: The profiles are estimated from the Ministry of Health, Labour and Welfare (2014–2023b) for the
2013–2022 period.

Figure 8 Changes in total population for three cases of different employment rates(50%, 75%, and 100%) under the 68-starting age for pension benefits
Note: The figure shows changes in the total population from the benchmark case with the 65-starting age
for pension benefits.

46

Figure 9 Changes in total population for three cases of different employment rates(50%, 75%, and 100%) under the 70-starting age for pension benefits
Note: The figure shows changes in the total population from the benchmark case with the 65-starting age
for pension benefits.

Figure 10 Changes in natinal income for three cases of different employment rates(50%, 75%, and 100%) under the 68-starting age for pension benefits
Note: The figure shows changes in the national income from the benchmark case with the 65-starting age
for pension benefits.

47

Figure 11 Changes in national income for three cases of different employment rates(50%, 75%, and 100%) under the 70-starting age for pension benefits
Note: The figure shows changes in the national income from the benchmark case with the 65-starting age
for pension benefits.

Figure 12 Changes in capital stock for three cases of different employment rates(50%, 75%, and 100%) under the 70-starting age for pension benefits
Note: The figure shows changes in the capital stock from the benchmark case with the 65-starting age forpension benefits.

48

Figure 13 Changes in labor supply for three cases of different employment rates(50%, 75%, and 100%) under the 70-starting age for pension benefits
Note: The figure shows changes in the labor supply from the benchmark case with the 65-starting age forpension benefits.

Figure 14 Changes in interest rates for three cases of different employment rates(50%, 75%, and 100%) under the 70-starting age for pension benefits
Note: The figure shows changes in interest rates from the benchmark case with the 65-starting age forpension benefits.

49

Figure 15 Changes in wage rates for three cases of different employment rates(50%, 75%, and 100%) under the 70-starting age for pension benefits
Note: The figure shows changes in wage rates from the benchmark case with the 65-starting age forpension benefits.

Figure 16 Changes in consumption tax rates for three cases of different
employment rates (50%, 75%, and 100%) under the 70-starting age for pension
benefits
Note: The figure shows changes in consumption tax rates from the benchmark case with the 65-startingage for pension benefits.

50

Figure 17 Changes in contribution rates for three cases of different employment
rates (50%, 75% and 100%) under the 70-starting age for pension benefits
Note: The figure shows changes in contribution rates from the benchmark case with the 65-starting age
for pension benefits.

51

Table 1 Normal retirement age of the public pension system and life expectancy
Normal retirement age

Life expectancy

Difference

Japan

65

84.8

19.8

Germany

67

81.0

14.0

France

67

83.2

16.2

United Kingdom

67

82.1

15.1

United States

67

78.2

11.2

Average (OECD)

67

80.7

13.7

Source: OECD (2023a)

Table 2

Employment rate by age groups in 2022 for male–female totals

Age group

15–24

25–34

35–44

45–54

55–59

60–64

65–69

70–74

75–

Employment rate (%)

46.6

86.6

86.2

86.6

82.8

73.0

50.8

33.5

11.0

Source: Statistics Bureau of Japan (2023b)

Table 3 Simulation cases
Case

Starting age for pension benefits

Benchmark65

Employment rate for individuals aged 65 and above

68-50%
68-75%

50%
75%

68

68-100%100%70-50%

50%

70-75%

75%

70

70-100%100%

52

Table 4 Exogenous variables for the benchmark simulation
Parameter description
Share parameter for consumption
Weight parameter of the number of children to
the consumption–leisure composite in utility

Parameter value

Data source

𝜑 = 0.5

Nishiyama & Smetters(2005): 𝜑 = 0.47

𝛼 (𝐻) =𝛼 (𝑈) = 0.028148

𝛿 = 0.0001

Oguro et al. (2011): 𝛿 =0.01

Intertemporal substitution elasticity

𝛾 = 0.5İmrohoroğlu et al. (2017)

Ratio of government subsidies to childrearingcosts

𝜌 = 0.1

Oguro et al. (2011): 𝜌 =0.1

Ratio of childrearing costs to net lifetime income

𝛽 = 0.0385

Time cost for childrearing

𝜇 =1.7234

Capital share in production

𝜀 = 0.3794

İmrohoroğlu et al. (2017)

Depreciation rate

𝛿 𝑘 = 0.0821

İmrohoroğlu et al. (2017)

Tax rate on labor income𝜏 𝑤 = 0.065

Kato (1998): 𝜏 𝑤 = 0.065

Tax rate on capital income𝜏 𝑟 = 0.4

Hayashi & Prescott(2002): 𝜏 𝑟 = 0.48;
İmrohoroğlu et al. (2017):
𝜏 𝑟 = 0.35

Tax rate on inheritance𝜏 ℎ = 0.1

Kato (1998): 𝜏 ℎ = 0.1

Rate of time preference

Ratio of government expenditures to national
incomeRatio of the part financed by tax transfer to total
pension benefit

𝑔 = 0.1
𝜋 = 0.25

Replacement ratio for public pension benefits

𝜃 = 0.4

Ratio of net public debt to national income

𝑑 = 1.5

Compulsory retirement age

𝑅𝐸 = 64

Starting age for receiving public pension benefits

𝑆𝑇 = 65

Ratio of people aged 18 (or 22) and above to thetotal population

𝐸/𝑍 = 0.84904

Dependency ratio (i.e., aging rate)

𝑂/𝑍 = 0.31209
53

Oguro & Takahata (2013):
𝜋 = 0.25
Braun et al. (2009): 𝜃=0.4İmrohoroğlu et al. (2017),
Nakajima & Takahashi(2017) : 𝑑 = 1.3

Table 5

Endogenous variables in the 2022 initial steady stateParameter description

Parameter value

Interest rate, 𝑟

0.0722

Wage rate, 𝑤

1.0758

Tax rate on consumption, 𝜏 𝑐0.1308Contribution rate, 𝜏 𝑝

0.1602

Capital–income ratio, 𝐾/𝑌

2.45951.2600Total fertility rate (TFR)
Ratio of net childrearing costs to annual labor income(low-income class 1.334; high-income class 1.147))
0.1919 (low-income class)0.1908 (high-income class)

0.0108

Ratio of government childcare subsidies to national income, 𝐺𝑆/𝑌

Table 6

Population ratios among people with different educational backgrounds
Population (thousands)

Population share (%)

Junior high school graduates

689.84High school graduates

10,000.91

3.19Share (%)
49.43
46.24

Technical and junior college

2,146.77

9.93

graduates
University
graduates

8,791.18

40.65

Total (in year 2022)

21,629.35

50.57

Source: The Ministry of Health, Labour and Welfare (2023b)

Table 7

Leveled welfare gains for each individual for six scenarios

Case

Welfare gains in yen (million)

68-50%

10.2653

68-75%

13.3978

68-100%17.2238

70-50%

16.481070-75%

20.5372

70-100%25.376654

100
100