The Optimum Quantity of Debt for an Aging Japan:
Welfare and Demographic Dynamics
Akira Okamoto (Okayama University)
September 13, 2022
Okayama University Discussion Paper No. I－113

1

The Optimum Quantity of Debt for an Aging Japan:
Welfare and Demographic Dynamics *Akira Okamoto**
Okayama University
September 13, 2022

Abstract
Japan’s government is heavily indebted, and the current net debt tends to increase. This
paper uses an extended lifecycle general equilibrium model with endogenous fertility toinvestigate an optimal size of government debt from two viewpoints: individual welfare
and future demographic dynamics. A simulation analysis finds that the level of net
government debt, which maximizes per-capita utility, is negative at −170% of grossdomestic product (GDP) for Japan. In contrast, it substantially decreases the total
population in the long run, compared to the baseline simulation with a debt-to-GDP ratioof 150%. Conversely, the level of net government debt, which produces the largest total
population for each year, is positive at 220% (or 230%) of GDP approximately from2045 to 2150; however, it severely deteriorates per-capita utility compared to thebaseline simulation.
Keywords: Government debt; welfare; demographic dynamics; Japanese economy;
simulation analysis
JEL classification: H30; C68*

An earlier version of this paper was presented at the 13th Irvine-Japan Conference on Public Policy held at the
University of California, Irvine in February in 2017, with the title of “The Optimum Quantity of Debt for a
Depopulating and Aging Japan: Welfare Level or Future Population?” I am grateful for the insightful comments
and suggestions of Professors Alan J. Auerbach, Ronald D. Lee, and Emmanuel Saez (The University of California,
Berkeley) and Professor Amihai Glazer (The University of California, Irvine). Additionally, I want to thank
Associate Professor Yoshinari Nomura (Okayama University), who helped me in writing the computer programused for the simulations. This research was assisted by a grant from the Abe Fellowship Program administered by
the Social Science Research Council in cooperation with and with funds provided by the Japan Foundation Center
for Global Partnership. Finally, I also wish to acknowledge the financial support from the Ministry of Education,
Culture, Sports, Science and Technology in Japan (Grant-in-Aid for Scientific Research (C) No. 20K01679).
**
Professor, Faculty of Economics, Okayama University, Japan. Tsushima-naka, Okayama 700-8530, Japan.
E-mail: okamot-a@okayama-u.ac.jp

2

1. Introduction
Many countries experienced sharp increases in outstanding government debt with the worldwide spreadof the new coronavirus. While governments implemented lockdowns to prevent the spread of the disease,
their budget deficits and outstanding debts rapidly increased to leverage the domestic economy. Figure 1,
based on data from the International Monetary Fund (2021), illustrates the net government debt-to-GDP(gross domestic product) ratio transition for six developed countries. Various countries’ economic
stimulus packages were expanded, and even Germany, which had been in relatively good fiscal
condition, increased its budget deficits sharply. In addition, the President of the United States (U.S.),
Biden, indicated that he would invest heavily in rebuilding the economy and expanding social welfare;thus, he expected spending to increase and the budget deficit to grow over the subsequent decade. As
Figure 1 shows, Japan’s net government debt tends to increase, and from 2016 to 2019, it was almost
constant at approximately 150% of its GDP. Furthermore, the net debt-to-GDP ratio in Japan is estimatedto balloon to 167.0% in 2020 and 171.5% in 2021, the highest among major developed countries.
The net government debt has recently skyrocketed in many countries worldwide, including Japan. In
light of this situation, exploring Japan’s desirable level of debt would be worthwhile. Few studies
investigated the preferable level of government debt for Japan; however, there is extensive literature on
the fiscal sustainability of Japan, including Sakuragawa and Hosono (2010), Doi et al. (2011), Hoshi andIto (2014), Hansen and İmrohoroğlu (2016), and Sakuragawa and Sakuragawa (2020) (see the following
literature review section for further details). Nakajima and Takahashi (2017) examined an optimal ratioof net government debt to GDP for Japan through a welfare analysis. Their analytical model is based onan Aiyagari (1994) style heterogeneous agent and incomplete market model with endogenous labor
supply, following Flodén (2001), who conducted a similar analysis using a model calibrated to match the
U.S. economy. Nakajima and Takahashi (2017) introduced idiosyncratic earnings risk in a model tocalculate an optimal government debt-to-GDP ratio for Japan, which can analyze the insurance effect ofgovernment debt. We examine an optimal level of net government debt for Japan using a different model
than Nakajima and Takahashi (2017).
Next, we describe our research method. Our model can evaluate a desirable government debt-toGDP ratio from two viewpoints: individual welfare and future demography. We use the lifecycle generalequilibrium simulation model of overlapping generations, developed by Auerbach and Kotlikoff (1983a,
1983b) and similarly applied in Auerbach and Kotlikoff (1987), Auerbach et al. (1989), Altig et al.
3(2001), Homma et al. (1987), Ihori et al. (2006, 2011), and Okamoto (2013, 2021). We investigate thequantitative effects of changes in the ratio of net government debt to GDP on per-capita welfare andfuture population using an extended Auerbach–Kotlikoff dynamic simulation model.
The simulation model in Okamoto (2020) introduced the number of children freely chosen byhouseholds into the utility function, thus incorporating endogenous fertility and future demographic
dynamics. Furthermore, in the extended framework with endogenous fertility, Okamoto (2022)
introduced the descendent link between a parent and children, providing the exogenous transition
probabilities from the parent’s income class to the same (or the other) income class to which theirchildren would belong. In other words, Okamoto (2022) introduced the descendent income inequality
from parents to their children into the simulation model with endogenous fertility. They incorporated two
representative households, the low-income class (high school graduates) and high-income classes(university graduates), into a cohort. Therefore, we can also evaluate the effect of different debt-to-GDP
ratios on the population ratio between the low-income and high-income classes.
This paper’s analytical model is based on Okamoto (2022). In the framework of Okamoto (2022),
we extended the model to freely change the government net debt-to-GDP ratio and analyze the impact ofchanges in the debt-to-GDP ratio on per-capita utility and future population dynamics. The significant
difference between Okamoto (2022) and our study is that our model extension investigates the impacts ofchanges in government net debt-to-GDP ratios. In contrast, an Okamoto (2022) model cannot analyze theeffects of changes in the size of government debt. This model extension from Okamoto (2022) allows us
to assess the impacts of alternative government net debt-to-GDP ratios. Based on data from IMF (2021),
we assume that the net government debt for Japan is 150% of GDP in the 2020 initial steady state. Sincea change in the net debt-GDP ratio would have a tremendous impact on the economy and severely disturb
an individual utility-maximizing behavior, our simulation avoids abrupt changes by setting the net debtto-GDP ratio to change smoothly over 10 years from 2021 to 2030.
From the above, it follows that we quantitatively analyze how the change in government debt forJapan impacts the future population levels and the welfare of all generations, including future generations
and the current generation. Concretely, we examine the effect of different net debt-to-GDP ratios on the
per-capita utility and the demographic dynamics for the transition process from 2020 to 2300. Thus, thispaper analyzes a long-run impact on economic growth, welfare, and population levels, assuming
alternative net debt-to-GDP ratios. This paper focuses primarily on the debt-to-GDP ratio that maximizes
4per-capita welfare in the long run and the debt-GDP ratio that provides the largest future population foreach year.
Finally, as shown in Okamoto (2022), this study introduces an additional government institution, theLump Sum Redistribution Authority (LSRA). Changes in the ratio of net government debt to GDP
generally improve the welfare of some generations but reduce that of others. If combined withredistribution from winning 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 andSmetters (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 pure efficiency gains or losses, we consider simulation cases via LSRA transfers where theratio of net government debt to GDP is increased/decreased. 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.
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 related to the level of government debt, especially in Japan. The
primary literature on the study is as follows.

First, we discuss two papers that analyzed an optimal level of net government debt—Flodén (2001) andNakajima and Takahashi (2017)—which are the most important for our analysis.
Government debt and redistributive taxation can help people to smooth consumption when facing
uninsurable individual-specific risks. Flodén (2001) examined the effects of variations in public debt andtransfers on risk sharing, efficiency, and the distribution of resources, determining that risk sharing canbe improved significantly by debt and transfers, but that debt has adverse effects on equity. Debt can5enhance welfare if transfers are lower than optimal when used in isolation; however, the beneficial
effects of public debt vanish if transfers are used optimally. Furthermore, the study also found that the
optimal level of government debt for the U.S. is 150% of its GDP.
Nakajima and Takahashi (2017) analyzed the effect of the large government debt for Japan on
welfare, using evidence based on macro-level and micro-level data. They used a heterogeneous agent, an
incomplete market model with idiosyncratic wage risk, a borrowing constraint, and endogenous labor
supply. They found that Japan’s optimal level of net government debt is −50% of its GDP. They also
showed that the welfare cost of keeping government debt to 130% of GDP, rather than the optimal levelof −50%, is 0.19% of consumption. Furthermore, according to their sensitivity analyses, if bothgovernment debt and public transfers can be set freely, then the optimal debt level is −120% of GDP.
From the above, Flodén (2001) and Nakajima and Takahashi (2017) analyzed the optimal level ofgovernment debt for the U.S. and Japan, respectively, using a similar welfare analysis. Their studies
obtained contrasting results. Flodén (2001) revealed that the optimal level of government debt for the
U.S. is positive, at 150% of GDP; conversely, Nakajima and Takahashi (2017) suggested that it isnegative, with −50% of GDP for Japan.
The basic model is the main difference between our study and Flodén (2001) and Nakajima and
Takahashi (2017); their studies are based on an Aiyagari (1994) style model, whereas our study is basedon Auerbach–Kotlikoff-type simulation model. Because those two studies introduce idiosyncratic
earnings risk in a model to calculate an optimal government debt-to-GDP ratio, they can analyze theinsurance effect of government debt. Conversely, our study can analyze a desirable government debt-toGDP ratio that produces the largest total population and a government debt-to-GDP ratio that maximizes
per-capita welfare. This is because our study extends the lifecycle model and incorporates endogenous
fertility, simulating variations in the future demographic dynamics induced by policy changes. Moreover,
Flodén (2001) and Nakajima and Takahashi (2017) focused on a stationary equilibrium where the debtto-GDP ratio is constant. In contrast, our paper analyzes the long-run impact of different debt-to-GDP
ratios on per-capita utility and the demographic dynamics for the transition process from 2020 to 2300.

Second, we discuss several papers that analyzed the issue of government debt using an overlapping
generations model, like our paper.
Arai and Ueda (2013) investigated the size of a primary deficit-to-GDP ratio that Japan’s
6

government can sustain. They used an overlapping generations model where multi-generational
households live, and the government maintains a constant ratio of the primary deficit to GDP. Theirresults numerically showed that the primary deficit could not be sustained unless the economic growthrate is unrealistically high, which, according to their settings, is more than five percent. They concluded
that Japan’s government needs to achieve a positive primary balance in the long run to avoid thedivergence of the public debt-to-GDP ratio.
Braun and Joines (2015) found that Japan’s aging population is already burdening government
finances and that the very high debt-GDP ratio constrains the country’s ability to confront the negative
fiscal implications of future aging. They found that Japan faces a severe fiscal crisis without imminent
remedial action, and they also analyzed alternative strategies for correcting Japan’s fiscal imbalances.
Kitao (2015) quantified the fiscal cost of Japan’s projected demographic transition over the next
several decades. That study analyzed the issue using a lifecycle model with endogenous saving,
consumption, and labor supply in both intensive and extensive margins. Kitao (2015) found that
preserving the current level of public transfers would require a significant increase in taxation.
Furthermore, using consumption taxes to balance the government budget, the tax rate was projected toreach the maximum value of 48% in the late 2070s. Finally, that study found that pension reform to
reduce benefits by 20% could result in a peak tax rate of 37%, which could be reduced to 28% bygradually raising the retirement age by 5 years.
İmrohoroğlu et al. (2016) built a micro data-based, large-scale overlapping generations model for
Japan, incorporating individuals’ ages, gender, employment type, income, asset holdings, and theJapanese pension rules. Using existing pension law, current fiscal policy, and medium variants ofdemographic projections, they produced future paths for government expenditures and tax revenues, with
implications for government debt and the public pension fund. Their study found that Japan’s fiscal
stability requires additional pension reform, a higher consumption tax, and higher female labor force
participation.

Finally, we look at several previous studies that analyzed Japan’s government debt issue.
Sakuragawa and Hosono (2010) investigated the sustainability of government debt by applying a
dynamic stochastic general equilibrium model of an exchange economy with infinitely lived agents to theJapanese economy. Introducing intermediation costs into the model helped successfully explain the7observed relationship between the interest and GDP growth rates, which is crucial in testingsustainability. Their study found that under the projected real growth rate of 2.5%, the debt-to-GDP ratio
gradually increases stochastically, resulting in unsustainable government debt. Furthermore, they foundthat the primary surplus must be 0.2% of GDP to recover sustainability.
Doi et al. (2011) constructed quarterly series of the revenues, expenditures, and outstanding debt forJapan from 1980 to 2010. They examined Japan’s fiscal sustainability, showing that the Japanese
government debt poses serious challenges. To stabilize the debt-to-GDP ratio, Japan must implement a
tax rate hike of an extraordinary magnitude. Such a dramatic tax increase for fiscal sustainability would
represent a drastic departure from the last 30 years of Japanese fiscal policy. If the government fails to
reduce the primary deficit by increasing taxes and reducing expenditures and transfer payments, Japan
would be forced to reduce the value of government debt through either inflation or outright default.
Through simulations under various scenarios, Hoshi and Ito (2014) showed that even if the Japanese
residents continue to invest their new savings into Japanese Government Bonds (JGB), Japan’s fiscal
situation is not sustainable. They found that if the Japanese government’s fiscal policy stance does notchange in the future, the amount of government debt will exceed the private sector financial assets
available for government debt purchase in the next 10 years. They also suggested that sufficiently
significant tax increases or expenditure cuts in the future would put the government debt on a sustainable
path. Thus, if the market believes that Japan will embark on such fiscal consolidation in the next 10
years, at most, the low JGB yields are justifiable. Conversely, if the expectation changes, a fiscal crisis
can be triggered even before the government debt hits the ceiling of the private sector financial assets.
Hansen and İmrohoroğlu (2016) used a standard growth model to measure the size of the Japanese
fiscal burden in the form of additional taxes required to finance projected expenditures and stabilize
government debt. They found that a massive fiscal adjustment is needed in 30–40% of total consumption
expenditures, requiring a distorting tax such as the consumption or labor income tax to rise tounprecedented highs. Therefore, they suggested the importance of considering alternatives that attenuate
the projected increases in public spending or enlarge the tax base.
Sakuragawa and Sakuragawa (2020) reconsidered Japan’s fiscal sustainability. They investigatedwhether the official projection is supported by a simulation conducted under the political constraint
imposed by a fiscal reaction function. First, Sakuragawa and Sakuragawa (2020) obtained Japan’s fiscal
reaction function by estimating the response of the primary surpluses to the past debt for a panel data set
8

of 23 OECD (Organization for Economic Co-operation and Development) countries. Then, theyevaluated the political feasibility of the official projection using their estimated reaction function. Thus,
they found that the Cabinet Office criterion for the debt-to-GDP ratio could realize fiscal sustainability,
attaining the government’s policy target of nonnegative fiscal surpluses. Notably, the negative growthadjusted bond yield and the high growth rate contribute to this finding.

3. Theoretical Framework
We calibrate the simulation of the Japanese economy by applying population data from 2017, 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 ofhouseholds, 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 q j +1| j be 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 bys −1

pst ( H ) =  q tj +1| j .
j =18

t

The probability q j +1| j is calculated from data estimated by the National Institute of Population and
9(1)

Social Security Research (2017). 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:Ut (H )

(

) 

1
105
1
1− 
1  ( H ) 40 t ( H )−( s −18)t ( H ) 1− (H )t (H )−( s −18)t ( H )  t ( H ) 1−=  ps (1 +  )ns+ (1 −  )  ps (1 +  )
Cs
ls

1
s =18
s =18

1 − 

(

)

)(



(2)
This utility function represents the lifetime utility of the agent born in year t . Cst (H ) , lst ( H ) andnst ( H ) are respectively consumption, leisure and the number of children to bear (only in the first 23

periods of the life) for an agent born in year t , of age s ;  (H ) is the utility weight of the number ofchildren relative to the consumption–leisure composite,  is the intertemporal elasticity of substitution,

 is the adjustment coefficient for discounting the future, and  is the consumption share parameterto leisure.
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 pay-as-yougo (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 a parent is assumed to obtain the utility from the number of children born ateach age.
As shown in Okamoto (2022), letting Ast ( H ) 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:)
Ast +( H= {1 + rt + s (1 −  r )} Ast ( H ) + (1 −  w −  tp+ s ) wt + s es( H ){1 − lst ( H ) − tcst (nst ( H ) )} + ast ( H ) − orst ( H )
1

(

)

c
t(H )
+ bst ( H ) {1 − lut ( H ) − tcut (nut ( H ) )}uRE− (1 − m)(1 +  tc+ s ) ts( H ) − m(1 +  tc+ s ) ts(U ) ,=20 − (1 +  t + s ) Cs

(3)
where rt is the pretax return to savings, and wt is the real wage at time t ;  w , rand  tc are thetax rates on labor income, capital income and consumption, respectively.  tp is the contribution rate to10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 thechildrearing cost for orphans. There are no liquidity constraints, and thus the assets As( H ) can be
negative. Terminal wealth must be zero. An individual’s earnings ability es( H ) is an exogenous functionof age.
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:(

H t ( H ) {1 − lut ( H ) − tcut (nut ( H ) )}uRE=20
bst ( H ) {1 − lut ( H ) − tcut (nut ( H ) )}uRE=20 = 
0

(

)

) (s  ST ) ,(4)( s  ST )

where

(

) RE1− 19  w e

H t ( H ) {1 − lut ( H ) − tcut (nut ( H ) )}uRE=20 =

RE

s =20

(H )
t +s s{1 − lst ( H ) − tcst (nut ( H ) )} .

(5)

The age at which a household born in year t starts to receive the public pension benefit is ST , theaverage annual labor income for the calculation of pension benefit for each agent is(), and the weight coefficient of the part proportional to H
H t ( H ) {1 − lut ( H ) − tcut (nut ( H ) )}uRE=20

t (H )is  .

The symbol bst ( H ) ({1 − lut ( H ) − tcut (nut ( H ) )}uRE=20 ) signifies that the amount of public pension benefit is afunction of the age profile of labor supply, {1 − lut ( H ) − tcut (nut ( H ) )}uRE=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, arerepresented by ts(H ) and ts(U ) , which are the cost for the children who will become high school
graduates and university graduates, respectively:

 s t(H )t(H )( s = 18,19,, 35)   (1 −  )nk
k =18 s
 ts( H ) =    t ( H ) (1 −  )nkt ( H ) ( s = 36, 37,, 40) ,k =s−17
 40 t ( H )t(H )( s = 41, 42,, 57 )
   (1 −  )nk
k =s−1711(6)ts( H ) = 0 ( s = 58, 59, , 105 ) ,(7) s t(H )(1 −  )nkt ( H ) ( s = 18,19,, 39) k =18,
 ts(U ) =  40 t ( H ) (1 −  )nkt ( H ) ( s = 40, 41,, 61)k = s − 21(8)

ts(U ) = 0( s = 62, 63,,105 ) ,(9) t ( H ) = NW t ( H ) ,(10)

where  t ( H ) is the childrearing cost for the parent born in year t ,  is the rate of government subsidy(including child allowances) to childrearing costs, and  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 thehigh-income class (i.e., university graduates) different from their parent, and 1 − m is the probability inwhich 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 , isrepresented by

tcst ( H ) = nst ( H ) ,(11)

where  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
all surviving low-income class households at time t . When BQt( H ) 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 byast ( H ) =(1 −  h ) BQt(+Hs ),Et(+Hs )

(12)

where
12

105

BQt( H ) =  ( N st −s −1( H ) − N st +−1s −1( H ) ) Ast +−1s −1( H ) .

(13)

s =18

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

N st ( H ) = pst ( H ) N 0t ( H ) .

(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 .
When ORt( H ) is the sum of childrearing costs incurred by the low-income class households at time t ,the childrearing cost for orphans for each low-income class household is defined byorst ( H ) =ORt(+Hs ),Et(+Hs )

(15)

where
57

61

s =18

s =18

ORt( H ) = (1 − m)  ( N st −−1s ( H ) − N st −s ( H ) ) ts−s ( H ) + m  ( N st −−1s ( H ) − N st −s ( H ) ) ts−s (U ) .

Therefore, the net amount of bequests is represented as a

(H )

(16)

− 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 )tt(H ) 0  l s  1 − tcs (ns ) (18  s  RE ).
 t (H )

l
=
1
(
RE
+
1

s

105
)
 s

(17)

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 p t ( H ) 
 t(H )1+ ,
Wst ( H ) =  st −( H1 )  Wr  s −1
 ps  1 + rt + s (1 −  ) 

(18)

13

Wst ( H ) =



(H )

1−

k

1



t (H )
s

(n

−

)

1



21


(1 +  tc+ s ) (1 − m) ts , g t ( H ) (1 −  ) + m ts , g t ( H ) (1 −  )
g =0
g =0



,

(19)

17

−1

where ts , 0 =1 for

g = 0, 

t
s, g

 g

=  {1 + rt + s−1+k (1 −  r )} .
 k =1


Similarly, that for the consumption–leisure composite is represented by pt ( H )  t (H )1+ ,
Vst ( H ) =  st −( H1 )  Vr  s −1
 ps  1 + rt + s (1 −  ) 





−

(20)

1

(1 −  ( H ) ) (Cst ( H ) ) (lst ( H ) )1−   (Cst ( H ) ) −1 (lst ( H ) )1−.
Vst ( H ) =1 +  tc

(21)

3.2 Firm Behavior
As shown in Okamoto (2022), 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. Capitalis homogeneous and depreciating, while labor differs only in efficiency. All forms of labor are perfectly
substitutable. Households with different 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 = K t L1t − ,(22)

where Yt is aggregate output (national income), K t is aggregate capital, Lt is aggregate labor supply
measured by the efficiency units, and  is capital’s share in production. Using the property subject to a
constant-returns-to-scale production function, we can obtain the following equation:
Yt = (rt +  k ) K t + wt Lt ,(23)

where  k is the depreciation rate.

3.3 Government Behavior
As shown in Okamoto (2022), at each time t , the government collects tax revenues and issues debt
( Dt +1 ) that it uses to finance government 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 apension sector, and a portion of revenues is transferred to the public pension sector. The public pension
14

system is assumed to be a simple PAYG style and consists only of earnings-related pension. Pension
account expenditure is financed 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 −  ) 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  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 + Bt + GSt ,(27)

where Gt includes transfers to the public pension sector ( Bt ) 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.
The total amount of government subsidies (including child allowances) to the childrearing cost in
year t is GSt :GSt = GSt( H ) + GSs(U ) ,

(28)

57
61


GSt( H ) =  (1 − m)  ( RCsa,(t H ) + RCsb,(t H ) + RCsc,(tH ) ) + m  ( RCsa,(tU ) + RCsb,(tU ) ) ,
s =18
s =18



(29)

15

s

a(H )
RC
=
N kt − s ( H ) t − s ( H ) nkt − s ( H ) ( s = 18,19,, 35)

s,t

k =18

s

b(H )
t −s ( H ) t −s ( H ) t −s ( H )nk( s = 36, 38,, 40) , RCs , t =  N k
k
=
s
−
17

40

c(H )
t −s ( H ) t −s ( H ) t −s ( H )nk( s = 41, 42,, 57 )
 RCs , t =  N k
k = s −17s
a (U )
RC=
N kt − s ( H ) t − s ( H ) nkt − s ( H ) ( s = 18,19,, 39)

s
,
t


k =18
,

40
t −s ( H ) t −s ( H ) t −s ( H ) RC b (U ) =( s = 40, 41,, 61)
 N k  nks,t
k = s − 21

(30)

(31)

where RCta ( H ) , RCtb ( H ) and RCtc ( H ) 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 RCta (U ) and RCtb (U ) are the costs when the children will belong to the high income class different
from their parent, namely, they will become university graduates.
61
57


GSt(U ) =  (1 − m)  ( RCsa,(tU ) + RCsb,(tU ) + RCsc,(tU ) ) + m  ( RCsa,(t H ) + RCsb,(t H ) ) ,s = 22s = 22



(29’)

s

a (U )
RC=
N kt − s (U ) t − s (U ) nkt − s (U ) ( s = 22, 23,, 40)s,tk = 2240

b (U )t − s (U ) t − s (U ) t − s (U ),nk
( s = 41, 42, 43) RCs , t =  N k
k
=
22

40

c (U )t − s (U ) t − s (U ) t − s (U )nk
( s = 44, 45,, 61) RCs , t =  N k
k = s − 21


(30’)

s

a(H )
RC
=
N kt − s (U ) t − s (U ) nkt − s (U ) ( s = 22, 23,, 39)

s
,
t


k = 22,40t − s (U ) t − s (U ) t − s (U ) RC b ( H ) =( s = 40, 41,, 57 )
 N k  nks,t
k = s −17(31’)

where RCta (U ) , RCtb (U ) and RCtc (U ) 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

Bt = Bt( H ) + Bt(U ) ,(32)

where Bt( H ) and Bt(U ) are the expenditure for the two income classes:
105

Bt( H ) =  N st − s ( H )bst − s ( H ) ,(33)

s = ST

16

105

(U )
t

B

=  N st − s (U )bst − s (U ) .

(33’)

s = ST

The total revenue from pension contributions and the total tax revenue are represented by
Rt =  p wt Lt ,(34)

Tt =  w wt Lt +  r rt AS t +  tc AC t +  h BQt ,(35)

where aggregate assets supplied by households, AS t , and aggregate consumption, ACt , are given by

AS t = AS t( H ) + AS t(U ) ,(36)

AC t = AC t( H ) + AC t(U ) .

(37)

For the low-income class, aggregate assets supplied by households, AS t( H ) , and aggregate
consumption, ACt( H ) , are given by105

AS t( H ) =  N st −s ( H ) Ast −s ( H ) ,

(38)

s =18
105

57

61

s =18

s =18

s =18

AC t( H ) =  N st −s ( H )Cst −s ( H ) + (1 − m)  ( RCsa,(t H ) + RCsb,(t H ) + RCsc,(tH ) ) + m  ( RCsa,(tU ) + RCsb,(tU ) ) ,(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).
For the high-income class, aggregate assets supplied by households, AS t(U ) , and aggregate
consumption, AC t(U ) , are given by105

AS t(U ) =  N st − s (U ) Ast − s (U ) ,(38’)

s = 22
105

61

57

s =22

s =22

s =22

AC t(U ) =  N st −s (U )Cst −s (U ) + (1 − m)  ( RCsa,(tU ) + RCsb,(tU ) + RCsc,(tU ) ) + m  ( RCsa,(t H ) + RCsb,(t H ) ) ,(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:

BQt = BQt( H ) + BQt(U ) ,(40)

ORt = ORt( H ) + ORt(U ) .

(41)

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 by17

Zt = Zt( H ) + Zt(U ) ,(42)

Et = Et( H ) + Et(U ) ,(43)

Ot = Ot( H ) + Ot(U ) .

(44)

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 / Z t . 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
105 40Z t( H ) =  N it − k −i ( H ) pkt − k ( H ) nit − k −i ( H ) ,(45)

k = 0 i =18
105 40Et( H ) =  N it −k −i ( H ) pkt −k ( H ) nit −k −i ( H ) ,(46)

k =18 i =18
105 40Ot( H ) =  N it −k −i ( H ) pkt −k ( H ) nit −k −i ( H ) .

(47)

k =65i =18

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
105 40Z(U )t=  N it − k −i (U ) pkt − k (U ) nit − k −i (U ) ,(45’)

k = 0 i = 22
105 40(U )tE=  N it −k −i (U ) pkt −k (U ) nit −k −i (U ) ,(46’)

k =22i =22
105 40(U )tO=  N it −k −i (U ) pkt −k (U ) nit −k −i (U ) .

(47’)

k =65i =22

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 governmentdebt,

AS t = 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,

Lt = L(t H ) + L(tU ) ,where L(t H ) =

RE

N

s =18

(49)
t −s ( H ) ( H )sse{1 − lst − s ( H ) − tcst (nst ( H ) )} ,
18

(50)

RE

(U )
t

L

=  N st − s (U ) es(U ) {1 − lst − s (U ) − tcst (nst (U ) )} .

(50’)

s = 22

3) Equilibrium condition for the goods market
Because aggregate production equals the sum of private consumption, private investment andgovernment expenditure,

Yt = AC t + {K t +1 − (1 −  k ) K t } + Gt .

(51)

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

4. Simulation Analysis
4.1. Method
The simulation model presented in the previous section is solved fundamentally, given the assumptionthat households have perfect foresight and correctly anticipate interest, wages, the tax and contribution
rates, and other factors such as the government net debt-to-GDP ratio. 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 the computation process).
Our study assumes the transitional economy of Japan from the initial steady state in 2020 to the final
steady state in 2300. Alternative scenarios with the different debt-to-GDP ratio are assumed to be
implemented at the end of 2020. For simplicity, 2020 is set as the starting year, and we simulate thedemography and the economy in the following years. For the generations that were alive in 2020 andhave survived in 2021, we need to pay attention to their formation of future expectations. In 2021, thesegenerations realized that their previous expectations no longer apply and thus again maximize theirremaining lifetime utility given perfect foresight. Based on the ex-post age profiles of the number ofchildren 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 change in government net debt-to-GDP
ratios just enough resources (possibly a negative 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 2020, at its age in 2021, 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 toeach future household that enters the economy after a change (from 2021 onward), at its age of 18 or
19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 2021 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 thechange 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 2021, satisfying the LSRA budget constraint (seeNishiyama and Smetters (2005) for further details).
4.2. Simulation cases
This study investigates the quantitative effects of different levels of net government debt in Japan on
individual welfare and future demographics, using an extended lifecycle general equilibrium model withendogenous fertility. The net debt-to-GDP ratio was constant at approximately 150% from 2016 to 2019,as illustrated in Figure 1. Accordingly, we assume that Japan’s net government debt is 150% of its GDP( d =1.5) in the 2020 initial steady state. The benchmark simulation assumes that the ratio remains 150%
annually until 2300. We consider alternative scenarios with the different ratios of net government debt toJapan’s GDP from −250% to 250% ( d = −2.5, −2.4, …, 2.4, 2.5). To avoid extra disturbance or
confusion from sudden changes in the net government debt-to-GDP ratio, we assume that the ratiochanges smoothly, interpolating over 10-year periods from 2021 to 2030.1 In addition, we consider thecase with LSRA transfers for each scenario with the different debt-to-GDP ratios. To distinguishpotential efficiency gains/losses from possibly offsetting changes in the welfare of different generations,
we introduce LSRA into the alternative simulation scenarios with different levels of net government
debt. The LSRA transfers produce a leveled and common welfare gain/loss for each future household in
both the low- and high-income classes.

1

For example, in the simulation case with net debt-to-GDP ratio of −150% ( d = −1.5), the ratio ( d ) is assumed tobe 1.5 in 2020, 1.2 in 2021, 0.9 in 2022, …, −1.2 in 2029, and −1.5 in 2030. After 2030, the debt-to-GDP ratio willremain constant at −150%.

20

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 1
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 2 presents the endogenous variables in the 2020 initialsteady state. Because the simulation results depend on the model setting and the given parameters, wemust 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.
Table 3 indicates the population ratio of individuals with different educational backgrounds in 2020,
estimated from the Basic Survey on Wage Structure (Chingin Sensasu) by the Ministry of Health,Labour, and Welfare (2021). The population share of high school graduates (including junior high school
graduates) and university graduates (including technical and junior college graduates) is 49.9% and50.1%, respectively. We estimated each population of high school graduates and university graduates
aged 0–105 in 2020, similar to Okamoto (2022). For the elderly, especially those of advanced age, highschool graduates significantly exceed university graduates, whereas the young and middle-aged number
is approximately the same. For those under 18 or 22 years old and undecided about becoming highschool or university graduates, we assume their population is the same, i.e., fifty–fifty, based on Kikkawa(2009).
Next, we describe how we assign parameter values for childrearing since our simulation model
incorporates endogenous fertility. The Cabinet Office (2010) indicated the average annual childrearing
costs for the first-born child to annual income for each age. Based on the survey in the Cabinet Office(2010), we assigned the parameter value of  (i.e., the ratio of childrearing costs to parental net lifetime
income) such that the ratio of the annual net childrearing costs to annual labor income for the individual
21

is, on average, close to 19.3%. Thus,  is assigned 0.0385 (the ratio is 20.38 % for the low-income
class and 18.90% for the high-income class).
The OECD (2022) presents public spending on family benefits in cash, services, and tax breaks forfamilies as a percentage of GDP in 2017. For Japan, public spending ratios on family benefits in cash,
services, and tax measures to GDP are 0.65%, 0.93%, and 0.20%, respectively.2 We assigned the valueof parameter  (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 1.14 %in the 2020 initial steady state.
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 time required forchildrearing; more children to bear, more time required for childrearing. The parameter determining thisrelation,  , is assigned under the simple assumption that one child required 1 h per day forchildrearing.3
Table 4 presents the scheduled number of children for young people aged 21 to 40 in Kikkawa(2018), which is based on a large-scale questionnaire survey (SSM2015). Accordingly, the scheduled
number of children for young high school-graduate couples is, on average, 1.14, whereas it is 0.875 foryoung university-graduate couples. The data were used to assign the parameter values that determine thedifference of fertility rates between the two income classes (  (H ) and  (U ) ). The parameter values
determining the fertility were chosen so that the total fertility rate (TFR) is 1.33 in the 2020 initial steadystate, reflecting that Japan’s actual TFR was 1.33 in 2020. Consequently, in the initial steady state, the
TFR is 1.45 for the low-income class and 1.15 for the high-income class.
4.3.3. 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 (2013–2022) for the 2012–2021 period. Figure 2 illustrates age–earnings profiles by education. The labor
2

In Japan, the ratio of total family benefits to GDP is only 1.79%, whereas it is, on average, 2.34% for the 37
OECD member countries. This shows that the level of governmental support for childrearing is considerably lower
in Japan than that in other countries.
3 Calibrating the value of parameter, μ, that determines the time cost in the model is difficult. In the 2020 initialsteady state, an average number of children to which a parent gives birth during the period from 18 or 22 to 40 is
0.0311 per year. We simply assume that a parent’s available time is 16 h per day and that the childrearing time costfor one child is 1 h per day.

22

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 )

(U )

and es , respectively, the following equationis constructed:
Qt = a0 + a1 At + a2 At2 ,(52)

where Q is the average monthly cash earnings for high school-graduate workers and universitygraduate workers, respectively, and A is the average age for each of the workers, including both males
and females. Because bonuses account for a large part of earnings in Japan, Q includes bonuses. 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 the university 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.
4.3.4. 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 ( Bt ) 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 2020 initial steady state (i.e., 13.13%). The ratio is held constant at 0.1 throughout theentire period.
4.3.5. 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 (  )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 (  ) for public pension benefits in Equation (4) isequal to 40%, following Braun et al. (2009).
23

The age at which households start to receive public pension benefits ( ST ) is constant during theentire period. The compulsory retirement age ( RE ) is the starting age of public pension benefits ( ST )minus 1. Thus, after households retire at the end of the year in which they reach compulsory retirement,
they immediately start to receive pension benefits from the beginning of the next year.
4.3.6. 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 as explained in the subsection simulation cases. After 2020, 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.
4.3.7. Share parameter on consumption in utility
The value of the consumption share parameter,  , in the utility function is assigned based on Altig et al.
(2001). Consequently, in the 2020 initial steady state, an individual devotes, on average, 59.0% for the
low-income class and 60.1% 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).
4.3.8. 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).

5. Simulation Results
Based on the simulation results in Japan, we first address the net government debt-to-GDP ratio—which
maximizes the per-capita utility—and discuss the mechanism behind the findings. Next, we address thenet government debt-to-GDP ratio—which produces the largest total population—and discuss themechanism behind the findings. Finally, we evaluate the effect of different net debt-to-GDP ratios on the
population ratio between low-income (high school graduates) and high-income classes (university
graduates).
24

The overall simulation analysis results reveal that the net debt-to-GDP ratio of −170% maximizes
the per-capita utility, substantially decreasing the long-run future population. The future population levelis maximized when the net debt-to-GDP ratio is 220% (or 230%) approximately from 2045 to 2150;
conversely, the net debt ratio of 220% (or 230%) significantly creates welfare loss compared to thebenchmark case with the ratio of 150%. In general, one policy instrument cannot achieve two policy
goals. Therefore, it may be better to improve per-capita utility by reducing Japan’s net debt-to-GDP ratio
and to maintain the future population level through other policy instruments, such as childcare support
measures or immigration policies.
5.1. Effect on individual welfareFirst, we evaluate the effect of alternative cases with different net debt-to-GDP ratios on individual
welfare. Figure 3 illustrates the leveled LSRA transfer value obtained by the simulation analysis for eachnet debt-GDP ratio, from −250% to 250% ( d = −2.5, −2.4, …, 2.4, 2.5). When the ratio is −170% ( d =−1.7), each individual’s leveled welfare gain is maximized, equivalent to 30.206 million Japanese yen(approximately 275,000 U.S. dollars in 2021), a considerable amount for each individual.4 Figure 3
shows that as the net debt-to-GDP ratio is lower or higher than −170%, the per-capita welfare is lower.
Therefore, these results show that the net debt-to-GDP ratio of −170% ( d = −1.7) is desirable from theviewpoint of per-capita welfare. In terms of efficiency, it is preferable to realize the net debt ratio of−170% of Japan’s GDP. This result means that the current high level of government debt (150% of GDP)is far from the optimal level of maximizing welfare, bringing about a considerable loss in economic
welfare.
Still, from the viewpoint of the future total population, the ratio of −170% ( d = −1.7) is notdesirable. Figure 4 shows that just after the reform from 2021, the population for the ratio ( d = −1.7)
slightly increased until 2027 compared to the baseline simulation ( d = 1.5); it is higher by 0.05% in 2023
and 2024. Yet, after 2027, this ratio’s population gradually decreases, standing apart from the baseline
simulation’s level; it is lower by 1.68% in 2050, 3.65% in 2070, and 8.10% in 2100. Therefore, the net
debt-to-GDP ratio of −170% ( d = −1.7) maximizes the per-capita welfare with a considerable equivalent
4

The Cabinet Office (2022) estimated that the GDP of Japan in 2020 was 528.23 trillion yen. Also, according todata from the Ministry of Internal Affairs and Communications (2022), the number of the people aged 20–64 years
was 69.37 million in 2020. We calculated the income per worker using these data and also derived the value fornational GDP in 2020 in our model, yielding a conversion rate between actual amounts of yen and values in themodel. Consequently, in 2020, unity in the model corresponded to 4.16966 million yen.

25

amount; however, it gradually decreases the total population, resulting in a substantial drop in the longrun. This is mainly because an increase in the wage rate improves the individual utility but raises theopportunity cost of raising children (see the following subsection for further details).
Next, we compare this result obtained in our analysis with that of previous studies. Flodén (2001)
found that the optimal net government debt-to-GDP ratio for the U.S. is 150%, indicating that the net
government debt maximizing welfare is a positive value (a budget deficit). In contrast, Nakajima and
Takahashi (2017) found that Japan’s optimal net government debt ratio, which maximizes welfare, is anegative value (a budget surplus). Our analysis also finds that Japan’s net government debt ratio, which
maximizes per-capita welfare, is negative (a budget surplus). This reveals that our simulation result isqualitatively the same as the previous study for Japan. Nakajima and Takahashi (2017) found that the
optimal net debt-to-GDP ratio for Japan is −50% under their standard parameter settings. Furthermore, in
their sensitivity analyses, when government debt and public transfers can be set freely, the optimal ratiois −120% of GDP. Thus, our paper’s desirable net government debt ratio for Japan, −170%, isquantitatively lower than the ratio obtained in Nakajima and Takahashi (2017).
Although the net government debt-to-GDP ratio of −170% maximizes per-capita utility, it seems to
be difficult, in reality, to achieve this ratio because it is currently approximately 150% in Japan and is
still increasing. As Figure 3 illustrates, the debt-to-GDP ratios of 100%, 50%, and 0% bring about theleveled welfare gains of 5.122 million yen (approximately 47,000 U.S. dollars in 2021), 10.174 million
yen (approximately 93,000 U.S. dollars in 2021), and 15.382 million yen (approximately 140,000 U.S.
dollars in 2021), respectively. Because a decrease in the debt-to-GDP ratio from 150% to 100% generates
such a considerable amount of per-capita welfare gain, the transition to the debt-to-GDP ratio of 100%
would be an immediate and realistic goal to enhance the efficiency in present Japan.

5.2. Mechanism behind the findings on individual welfare
Here, we consider why the net debt-to-GDP ratio of −170% ( d = −1.7) maximizes the per-capita welfare.
Large changes in the net debt-to-GDP ratio greatly influence the level of capital stock through Equations(25) and (48). A substantial decrease in the debt ratio from 150% ( d = 1.5) to −170% ( d = −1.7)
substantially increases the capital stock. For three scenarios with different net debt-to-GDP ratios ( d =−1.7, 0, and 2.2), Figures 5, 6, and 7 present the percent changes in national income, capital stock, andlabor supply, respectively, from the benchmark case ( d = −1.7). The capital stock for the net debt ratio of26

−170% ( d = −1.7) sharply increases and peaks at a 94.00% increase in 2030; the increase then graduallyshrinks over time. The labor supply for the net debt ratio of −170% ( d = −1.7) first drops by 7.05% in
2021 and sharply rises by 7.01% until 2030. From 2030, the labor supply will increase slightly andgradually decrease over time. Consequently, the national income for the ratio of −170% ( d = −1.7) also
rapidly increases and peaks at a 34.10% increase in 2030; the increase will gradually shrink over time.
For the three scenarios, Figures 8 and 9 illustrate the percent changes in interest rates and wage
rates, respectively, from the benchmark case. Reflecting a large capital stock, the interest rate for the net
debt ratio of −170% ( d = −1.7) drops sharply at the bottom by 4.725% in 2030; the decline will thengradually decrease over time. Conversely, the wage rate for the ratio of −170% ( d = −1.7) sharply
increases and peaks by 25.29% in 2030; the increase will then gradually decrease over time because thereduction in government debt increases real capital, resulting in a relative undersupply of labor.
Figure 10 shows percentage-point changes in consumption tax rates from the benchmark case forthree cases with different net debt-GDP ratios. The consumption tax rate for the net debt ratio of −170%( d = −1.7) tends to decrease gradually and drops at the bottom by 13.54% in 2073. Thereafter it slightly
increases and, from 2090, settles at a 12.13% decrease. Figure 10 reveals that under the net debt-to-GDP
ratio of −170% ( d = −1.7), the consumption tax rate is substantially lower throughout the entire period.
This means a lower tax burden for individuals, which is one of the main reasons for attaining the highest
utility for individuals in this simulation case ( d = −1.7).
Figure 11 illustrates percentage-point changes in contribution rates from the benchmark case forthree cases of the different net debt-GDP ratios. The contribution rate for the net debt ratio of −170% ( d= −1.7) sharply drops at the bottom by 3.92% in 2030 before beginning to increase. From 2061, thecontribution rate for this case ( d = −1.7) becomes higher than that in the benchmark case. After that, it
gradually increases and peaks at 3.46% in 2091, and, after 2091, again decreases gradually. A possible
reason for this observation is the following. After the reform started, the transition to the ratio of −170%
increased the capital stock and promoted economic growth, reducing contribution rates; however, thereform ( d = −1.7) is not desirable from the viewpoint of the future total population. As Figure 4
illustrates, although the reform slightly increases the total population until 2027, it decreases the total
population at an accelerated pace over time. Consequently, reducing the young working population
would increase contribution rates under a PAYG social security system.

27

5.3. Effect on future population
Next, we assess the impact of alternative cases with the different net debt-to-GDP ratios on the future
population. Figures 4 and 12 illustrate the percent changes of the total population for each year from thebenchmark case ( d = 1.5), concerning five cases of different net debt-GDP ratios ( d = −2.5, −1.7, −1,−0.5, and 0) and additional five cases ( d = 0, 0.5, 1, 2.2, and 2.5), respectively. Different net government
debt-to-GDP ratios bring about a different total population for each year. Therefore, the net debt-GDP
ratio that produces the largest total population depends on the year used for evaluation, from 2021 to2300. Figure 13 illustrates the transition of the net debt-GDP ratio that generates the largest total
population for each year. As Figure 13 shows, for 2021, the net government debt-to-GDP ratio of −150%( d = −1.5) creates the largest total population. After that, the net debt-GDP ratio that provides the largest
total population for each year sharply rises over time and reaches 230% ( d = 2.3) in 2048. From 2048 to
2154, it remains at 220% ( d = 2.2) or 230% ( d = 2.3). After that, the debt-to-GDP ratio gradually
decreases over time and reaches 150% ( d = 1.5) in 2258. After 2258, it settles at 150% ( d = 1.5). Thus,
the net debt-GDP ratio that generates the largest total population for each year is 220% ( d = 2.2) or 230%
( d = 2.3) approximately from 2045 to 2150. Before and after this period, the ratio is smaller, and in thevery long run, it settles at 150%, the current ratio for Japan.
Still, the case ( d = 2.2 or 2.3) is not desirable from the viewpoint of individual welfare. The leveled
welfare loss for each individual is equivalent to 6.8915 or 7.8763 million yen (approximately 63,000 or
72,000 U.S. dollars in 2021). Therefore, the net debt-to-GDP ratio of 220% ( d = 2.2) or 230% ( d = 2.3)
generates the largest total population approximately from 2045 to 2150; however, it substantiallydeteriorates the per-capita welfare compared to the benchmark case.

5.4. Mechanism behind the findings on future population
Here, we consider why the net debt-to-GDP ratio of 220% ( d = 2.2) or 230% ( d = 2.3) provides thelargest total population approximately from 2045 to 2150. As Figure 12 illustrates, a high net debt-GDP
ratio, such as 2.2 or 2.5, slightly decreases the total population in the beginning after the reformcompared to the benchmark case; however, after around 2040, the total population begins to increase. In
the case of the net debt-to-GDP ratio of 220% ( d = 2.2), the total population peaks at a 0.47% increase in
2126; the conversion from the benchmark 150% ratio case to the 220% ratio case increases the total
population from 32.210 to 32.365 million people. After that, it will gradually decline.
28

Changes in the net debt-to-GDP ratio affect the level of capital stock through Equations (25) and(48). An increase in the debt ratio from 150% ( d = 1.5) to 220% ( d = 2.2) decreases the capital stock
throughout the entire period, as illustrated in Figure 6. This brings about a rise in interest rates and a dropin wage rates, as shown in Figures 8 and 9. A decrease in wage rates diminishes the opportunity cost of
having children, resulting in more children being born and a larger total population. Thus, the case of the
net debt-to-GDP ratio of 220% produces the largest population approximately from 2045 to 2150.
Conversely, an increase in the debt ratio from 150% ( d = 1.5) to 220% ( d = 2.2) deteriorates theindividual utility, as mentioned above. Possible reasons for this are as follows. Figures 9 and 10 illustratethat the debt-ratio increase declines wage rates and increases the consumption tax rates; both of thesechanges would deteriorate individual welfare. Figure 11 shows that the contribution rates for the case of
the 220% debt-to-GDP ratio increase until 2061 compared to the benchmark case; after that, they begin
to decrease. This is because the negative effect induced by lower wage rates first exceeds the positiveeffect induced by a larger total (working) population; however, after 2061, the latter positive effectexceeds the former negative effect. Therefore, concerning contribution rates, increasing the debt ratio to220% deteriorates the individual utility until 2061, but after that, it improves utility.

5.5. Effect on the population ratio of each income class
Finally, for two scenarios of the net debt-to-GDP ratio, −170% ( d = −1.7) and 220% ( d = 2.2), we
evaluate the effect on the population ratio between low-income (high school graduates) and high-income
classes (university graduates). Table 5 presents the population ratio of the low-income class for threecases of the different net government debt-to-GDP ratios ( d = 1.5, −1.7, and 2.2). In the benchmark casewith the debt ratio of 150% ( d = 1.5), the population ratio of the low-income class is 69.670% in the2020 initial steady state. The ratio is substantially higher than that of the high-income class (30.330%)
because for the elderly, especially those of advanced age, there are many more high school graduates
than university graduates. For the young and the middle-aged, it is approximately fifty–fifty, and, forthose under 18 or 22 in 2020 (not yet decided to become high school or university graduates), we assume
that their population is the same, i.e., fifty–fifty, based on Kikkawa (2009). We also assume that theintergenerational mobility probability from a low-income class parent to high-income class children, orfrom a high-income class parent to low-income class children, is the standard of 0.3, i.e., the transitional
probability of reaching the same income class between a parent and children is 0.7. Therefore, the low29

income class’s population ratio gradually approaches 50% over time.
As Table 5 shows, in the benchmark case of the net debt-to-GDP ratio of 150% ( d = 1.5), thepopulation ratio of the low-income class gradually decreases over time and reaches 53.385% in 2300. In
the case of the debt ratio of −170% ( d = −1.7), where per-capita welfare is maximized, the low-income
class’s population ratio reaches 53.136% in 2300, slightly lower than that of the benchmark case. A
possible reason for this is that the debt ratio of −170% increases the capital stock and enhances the wage
rate, which increases opportunity costs for having children. This would decrease the number of children
born, especially for the low-income class with a higher preference for the number of children.
Concerning the net debt-to-GDP ratio of 220% ( d = 2.2), which produces the largest total population
approximately from 2045 to 2150, the population ratio of the low-income class reaches 53.431% in 2300,
slightly higher than that of the benchmark case. A possible reason for this is that the debt ratio of 220%
decreases the capital stock and diminishes the wage rate, reducing opportunity costs for having children.
This would increase the number of children born, especially for the low-income class with a higher
preference for the number of children.
Therefore, in the case of −170% ( d = −1.7), in which per-capita welfare is maximized, thepopulation ratio of the low-income class will be slightly lower in the very long run. Conversely, in thecase of the ratio of 220% ( d = 2.2), which generates the largest total population, the population ratio of
the low-income class will be slightly higher in the very long run.

6. Conclusions
This paper evaluated a desirable quantity of government debt for a model parameterized to mimic certain
features of the Japanese economy from two viewpoints: individual welfare and future demography.
Concretely, it examined the quantitative effects of different levels of the net government debt on percapita welfare and future population in an aging and depopulating Japan, using an extended lifecycle
general equilibrium model with endogenous fertility. The effects of alternative ratios of the government
net debt to GDP were quantitatively investigated during the transitional period, 2021–2300. An LSRA
was introduced to calculate the per-capita welfare and evaluate the pure efficiency gains or losses ofthese policy reforms.
The two main findings of our analysis are as follows.
First, we examined the net government debt-to-GDP ratio that maximizes the per-capita utility for30

all individuals, containing the future and current generations. We found that from the viewpoint of
economic efficiency, the optimal quantity of the net debt in Japan is −170% of its GDP because it
maximizes the per-capita welfare. This ratio is negative, showing that it is desirable to realize the fiscal
surplus in Japan from the viewpoint of efficiency. This result in our analysis is qualitatively the same as
Nakajima and Takahashi (2017), who found that the optimal quantity of net debt for Japan is negative(i.e., −50% of its GDP) under their standard parameter settings. In contrast, Flodén (2001) found that the
optimal net debt level is positive (i.e., 150% of GDP) for the U.S. In addition, we also found that the net
debt-to-GDP ratio of −170% produces a considerable per-capita welfare gain (30.206 million yen,approximately 275,000 U.S. dollars in 2021). Still, from the viewpoint of the future total population, thisratio of −170% is not desirable because, after 2027, the total population gradually decreases compared to
the level of the baseline simulation, resulting in an 8.10% decrease in 2100.
Second, we evaluated the net government debt-to-GDP ratio that provides the largest total
population. The debt ratio that generates the largest total population depends on the year used to evaluate
it. We found that the debt-to-GDP of 220% (or 230%) creates the largest total population approximatelyfrom 2045 to 2150. In 2126, the conversion from the benchmark 150% ratio to the 220% ratio increases
the total population by 0.47%, from 32.210 to 32.365 million people. Before and after this period, thedebt ratio that produces the largest total population is smaller, and in the very long run, it settles at 150%,
the current ratio for Japan. Therefore, from the viewpoint of the future population, Japan’s optimalquantity of net debt is 220% (or 230%) of its GDP; however, from the viewpoint of efficiency, this ratioof 220% (or 230%) is not preferable because it brings about the per-capita welfare loss equivalent to6.8915 (or 7.8763) million yen (approximately 63,000 or 72,000 U.S. dollars in 2021).

Finally, we discuss policy implications based on the simulation results. The results reveal that Japan’s
optimal net government debt-to-GDP ratio depends on the viewpoints. From the efficiency viewpoint, itis negative, with a ratio of −170%, whereas from the future population viewpoint, it is positive, with aratio of 220% (or 230%) approximately from 2045 to 2150. The net government debt should besubstantially decreased, to a negative ratio, to improve the per-capita welfare in Japan. In contrast, if it isessential to maintain the future population level in Japan, the net government debt would be allowed toincrease a little more; however, the debt currently tends to increase. In general, one policy instrument
cannot achieve two policy goals. Therefore, it may be better to improve per-capita welfare by reducing
31

Japan’s net debt-to-GDP ratio. For example, since a decrease in the debt-to-GDP ratio from 150% to100% generates considerable per-capita welfare gain (5.122 million yen, approximately 47,000 U.S.
dollars in 2021), the transition to a debt ratio of 100% may be an immediate and realistic goal to enhancethe efficiency in present Japan. From the viewpoint of maintaining the future population level, it may be
preferable to implement another policy instrument, such as childcare support measures or immigration
policies.

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 bys −1

pst (U ) =  q tj +1| j .

(1)’

j =22

Each agent who begins its economic life at age 22 chooses perfect-foresight consumption paths( Cst (U ) ), leisure paths ( lst (U ) ), and the number of born children ( nst (U ) ) to maximize a time-separable
utility function of the form:U t (U ) =()( ) 

1
105
1
1−

1− 1−  
1  (U ) 40 t (U )  ps (1 +  ) −( s−22) nst (U )  + (1 −  (U ) )  pst (U ) (1 +  ) −( s−22) Cst (U ) lst (U )


1
s
=
22
s
=
22


1−

(

)



(2)’
where Cst (U ) , lst (U ) and nst (U ) 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 .  (U ) is the utility weightof the number of children relative to the consumption–leisure composite.
Letting Ast (U ) 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:
Ast +(U1 ) = {1 + rt + s (1 −  r )} Ast (U ) + (1 −  w −  tp+ s ) wt + s es(U ) {1 − lst (U ) − tcst (U ) (nst (U ) )} + ast (U ) − orst (U )()+ bst (U ) {1 − lut (U ) − tcut (nut (U ) )}uRE=22 − (1 +  tc ) Cst (U ) − (1 − m)(1 +  tc ) ts(U ) − m(1 +  tc ) ts( H ) .

There are no liquidity constraints, and thus the assets can be negative. An individual’s earnings ability
32

(3)’

es(U ) is an exogenous function of age.
The pension benefit is assumed to comprise only an earnings-related pension:(

H t (U ) {1 − lut (U ) − tcut (nut (U ) )}uRE= 22
bst (U ) {1 − lut (U ) − tcut (nut (U ) )}uRE= 22 = 
0

(

)

) (s  ST ) ,(4)’( s  ST )

where

(

) RE 1− 21  w e {1 − lRE

H t (U ) {1 − lut (U ) − tcut (nut (U ) )}uRE=22 =

s =22

(U )
t +s s(t (U )
s− tcst (nut (U ) )} .

(5)’

)

The average annual labor income for each agent is H t (U ) {1 − lut (U ) − tcut (nut (U ) )}uRE=22 , and the weight
coefficient of the part proportional to Ht (U )is  . The symbol bst (U ) ({1 − lut (U ) − tcut (nut (U ) )}uRE=22 ) in

Equation (3)’ signifies that the amount of public pension benefit is a function of the age 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 thechildren when the parent born in year t is s years old are represented by ts(U ) and ts(H ) , which are
the cost for the children who will become university graduates and high school graduates, respectively:

 s t (U )t (U )( s = 22, 23,, 40)   (1 −  )nk
k =22

 40
t (U ),
 s =    t (U ) (1 −  )nkt (U ) ( s = 41, 42, 43)k=22 40 t (U )t (U )( s = 44, 45,, 61)   (1 −  )nk
k =s−21
ts(U ) = 0( s = 62, 63,, 105 ) ,

(6)’

(7)’

 s t (U )t (U )( s = 22, 23,, 39)   (1 −  )nk
k = 22t (H ),
 s =  40
t (U )t (U )  (1 −  )nk (s = 40, 41,, 57)k = s −17(8)’ts( H ) = 0(9)’( s = 58, 59,, 105 ) , t (U ) = NW t (U ) .

(10)’

The time cost for rearing the children when the parent born in year t is s years old is representedby33

tcst = nst (U ) .

(11)’

When BQt(U ) is the sum of bequests inherited by the high income class households at time t , thebequest to be inherited by each highast (U ) =income class household is defined by(1 −  h ) BQt(+Us),
Et(+Us)(12)’

where Et(U ) is the number of the high income class households conducting an economic activity
independently, aged 22 and above, and105

BQt(U ) =  ( N st −s−1(U )) − N st +−1s−1(U ) ) Ast +−1s−1(U ) .

(13)’

s =22

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

N st (U ) = pst (U ) N 0t (U ) .

(14)’

When ORt(U ) is the sum of childrearing costs incurred by the high income class households at timet , the childrearing cost for orphans for each high income class household is defined byorst (U ) =ORt(+Us),
Et(+Us)

(15)’

where
61

57

s =22

s =22

ORt(U ) = (1 − m)  ( N st −−1s (U ) − N st −s (U ) ) ts−s (U ) + m  ( N st −−1s (U ) − N st −s (U ) ) ts−s ( H ) .

(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 )tt (U )
 0  ls  1 − tcs (ns ) (22  s  RE ).
 t (U )( RE + 1  s  105 ) ls = 1(17)’

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 p t (U )  
 t (U )1+ ,
Wst (U ) =  st −(U1 )  Wr  s −1
 ps  1 + rt + s (1 −  ) 
1−

1

(18)’

−

1

 (U ) k  (nst (U ) ) 

Wst (U ) =(1 + 

c
t +s

21
17


) (1 − m) ts , g t (U ) (1 −  ) + m ts , g t (U ) (1 −  )
g =0
g =0



34

,

(19)’

−1

where ts,0

=1 for g = 0 , 

t
s, g

 g

=  {1 + rt + s−1+k (1 −  r )} .
 k =1


Similarly, that for the consumption–leisure composite is represented by p t (U )  
 t (U )1+ ,
Vst (U ) =  st −(U1 )  Vr  s −1
 ps  1 + rt + s (1 −  ) 





−

(20)’

1

(1 −  (U ) ) (Cst (U ) ) (lst (U ) )1−   (Cst (U ) ) −1 (lst (U ) )1−Vst (U ) =.
1 +  tc

(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 U t ( H ) in Equation (2) subject to Equations (3) and (17). Let theLagrange function be

105

Lt ( H ) = U t ( H ) +  ts( H ) − Ast +( H1 ) + {1 + rt + s (1 −  r )} Ast ( H ) + [1 −  w −  tp+ s ] wt + s es( H ) {1 − lst ( H ) − tcst (nst ( H ) )} + ast ( H )
s =18

(

)

− orst ( H ) + bst ( H ) {1 − lut ( H ) − tcut (nut ( H ) )}uRE=20 − (1 +  tc+s ) Cst ( H ) − (1 − m)(1 +  tc+s )ts( H ) − m(1 +  tc+s )ts(U )



RE



+  st ( H ) 1 − lst ( H ) − tcst (nst ( H ) ) ,



(B.1)

s =18

where ts( H ) and  st (H ) represent the Lagrange multiplier for Equations (3) and (17), respectively.
The first-order conditions on the number of children nst ( H ) , consumption Cst ( H ) , leisure lst ( H ) , andassets Ast +( 1H ) for s =18, 19, …, 105 can be expressed bypt (H )s(H )(1 +  )−( s −18)t (H )
s

(n

−

)

1



17

= ts( H ) [1 −  w −  tp+s ] wt +s es( H ) + (1 − m)(1 +  tc ) ts , g t ( H ) (1 −  )
g =0


21
105

w e ( H )+ m(1 +  tc ) ts , g  t ( H ) (1 −  ) +   tk( H ) t + s s + st ( H ) ,RE − 19
g =0
k = ST


(B.2)

−1

where ts,0

=1 for g =0, 

t
s, g

 g

=  {1 + rt + s−1+k (1 −  r )} , k =1  (C (1 −  )(C

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

−

1



−

1



35

t ( H )  −1s)(lst ( H ) )1− = ts( H ) (1 +  tc+s ) ,t (H ) s) (lst ( H ) ) −

(B.3)

=

t (H )s(1 −  −  )w e +  
105

w

p
t+s

(H )
t+s s

k = STt (H )kwt + s es( H )RE − 19+  st ( H )( s  RE ) ,(B.4)

ts( H ) = {1 + rt + s (1 −  r )}ts(+H1 ) ,(B.5)( s  RE ) ,st ( H ){1 − lst ( H ) − tcst (nst ( H ) )} = 0
1 − lst ( H ) = 0( s  RE ) ,

(B.6)
(B.7)

 st ( H )  0 .

(B.8)

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

n18t ( H ) , is given, the initial value, W18t ( H ) , can be derived from Equation (19). If the value, W18t ( H ) , isspecified, the value of each age, Wst ( H ) , can be derived from Equation (18), which generates the valueof each age, nst ( H ) . If the value, nst ( H ) , 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,

V18t ( H ) , is specified, the value of each age, Vst ( H ) , can be derived from Equation (20). If Vst ( H ) is
specified, the values of consumption, Cst ( H ) , 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:105 
tk( H ) wt + s es( H )  st ( H )  
w
p
(H )

(
1
−

−

)
w
e
+
+ t (H )  
 

t +s
t+s s
t (H )
RE−20
s   t ( H )k=STt (H )s
Cs = l .
c s(1 −  ) (1 +  t + s )





(B.9)

If the value of lst ( H ) is given under  st ( H ) =0, the value of Cst ( H ) can be obtained using a numericalmethod, and then the value of Vst ( H ) can be derived from Equation (21). The value of lst ( H ) is chosen
so that the value of Vst ( H ) obtained in the simulation is the closest to that calculated by evolution from

V18t ( H ) through Equation (20). If the value of lst ( H ) chosen is unity or higher, the value of Cst ( H ) is
obtained from Equation (21) under lst ( H ) =1. If it is less than unity, the value of Cst ( H ) is derived from
Equation (B.9).
For s =RE+1, RE+2, …, 105, the condition of lst ( H ) =1 leads to the following equation:

Vst ( H ) =(1 − 

) (C )
1 +  tc+ s(H )t (H )s− + −1.

(21)’’

36

The value of Cst ( H ) 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:
RE



s =18

(

105

t(H )
s

)

105

[1 −  w −  tp+ s ]wt + s es( H ) {1 − lst ( H ) − tcst (nst ( H ) )} +  st ( H )bst ( H ) {1 − lut ( H ) − tcut (nut ( H ) )}uRE=20 +  st ( H ) (ast ( H ) − orst ( H ) )s = ST

105

35

s =18

s

40

s

=  st ( H ) (1 +  tc+ s )Cst ( H ) + (1 − m)  st ( H ) (1 +  tc+ s ) t ( H ) (1 −  )nkt ( H ) + (1 − m)   st ( H ) (1 +  tc+ s ) t ( H ) (1 −  )nkt ( H )s =18

s =18 k =18

57

s =36 k = s −17

40

39

s

+ (1 − m)   st ( H ) (1 +  tc+ s ) t ( H ) (1 −  )nkt ( H ) + m  st ( H ) (1 +  tc+ s ) t ( H ) (1 −  )nkt ( H )s =41 k = s −1761

s =18 k =18

40+ m   st ( H ) (1 +  tc+ s ) t ( H ) (1 −  )nkt ( H ) ,(B.10)

s = 40 k = s − 21 s

 u =19

where 18t ( H ) =1 for s =18, st ( H ) =  {1 + rt +u (1 −  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 U t (U ) in Equation (2)’ subject to Equations (3)’ and (17)’. Let theLagrange function be

105

Lt (U ) = U t (U ) +  ts(U ) − Ast +(U1 ) + {1 + rt + s (1 −  r )} Ast (U ) + [1 −  w −  tp+ s ] wt + s es(U ) {1 − lst (U ) − tcst (nst (U ) )} + ast (U )
s =22

(

)

− orst (U ) + bst (U ) {1 − lut (U ) − tcut (nut (U ) )}uRE=22 − (1 +  tc+ s ) Cst (U ) − (1 − m)(1 +  tc )ts(U ) − m(1 +  tc )ts( H )
RE





+  st (U ) 1 − lst (U ) − tcst (nst (U ) ) ,

(C.1)

s =22

where ts(U ) and  st (U ) represent the Lagrange multiplier for Equations (3)’ and (17)’, respectively.
The first-order conditions on the number of children nst (U ) , consumption Cst (U ) , leisure lst (U ) , andassets Ast +(U1 ) for s =22, 23, …, 105 can be expressed by121−
pst (U ) (U ) (1 +  ) −( s−22) (nst (U ) )  = ts(U ) [1 −  w −  tp+s ] wt +s es(U ) + (1 − m)(1 +  tc ) ts , g t (U ) (1 −  )
g =0


17
105

w e(U )+ m(1 +  tc ) ts , g t (U ) (1 −  ) +   tk(U ) t + s s + st (U ) ,RE − 21g =0
k = ST


37

(C.2)



−1

where =1 for g =0, 

t
s,0

t
s, g

 g

=  {1 + rt + s−1+k (1 −  r )} , k =1  (C (1 −  )(C−

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

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





105

= ts(U ) (1 −  w −  tp+ s ) wt + s es(U ) +  tk(U )k = ST1−t (U )  −1s

) (lst (U ) )1− = ts (1 +  tc+ s ) ,1t (U ) swt + s es(U )RE − 21(C.3)+  st (U )) (lst (U ) )−( s  RE ) ,(C.4)

ts(U ) = {1 + rt + s (1 −  r )}ts(+U1 ) , st (U ) {1 − lst (U ) − tcst (nst (U ) )} = 0
1 − lst (U ) = 0(C.5)( s  RE ) ,(C.6)( s  RE ) ,(C.7)

 st (U )  0 .

(C.8)

The combination of Equations (C.2) and (C.5) produces Equations (18)’ and (19)’. If the initial value,t (U )t (U ), is given, the initial value, W22t (U ) , can be derived from Equation (19)’. If the value, W22 , isn22

specified, the value of each age, Wst (U ) , can be derived from Equation (18)’, which generates the valueof each age, nst (U ) . If the value, nst (U ) , 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 initialvalue, V22t (U ) , is specified, the value of each age, Vst (U ) , can be derived from equation (20)’. If Vst (U ) ist (U )

specified, the values of consumption, Cst (U ) , and leisure, ls, 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:105 
tk(U ) wt + s es(U )  st (U )  
w
p
(U )
+

 (1 −  −  t + s ) wt + s es +  t (U )RE − 21 ts(U )   t (U )k = ST st (U )


Cs =
l .

 s
(1 −  ) (1 +  tc+ s )





(C.9)

If the value of lst (U ) is given under  st = 0 , the value of Cst (U ) can be obtained using a numericalmethod, and then the value of Vst (U ) can be derived from Equation (21)’. The value of lst (U ) is chosen
so that the value of Vst (U ) obtained in the simulation is the closest to that calculated by evolution from

V22t (U ) through Equation (20)’. If the value of lst (U ) chosen is unity or higher, the value of Cst (U ) is38

obtained from Equation (21)’ under lst (U ) =1. If it is less than unity, the value of Cst (U ) is derived from
Equation (C.9).
For s =RE+1, RE+2, …, 105, the condition of lst (U ) =1 leads to the following equation:− + −1(1 −  (U ) ) (Cst (U ) ) 
Vst (U ) =1 +  tc+ s.

(21)’’’The value of Cst (U ) 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:REs = 22(105t (U )s)105[1 −  w −  tp+ s ]wt + s es(U ) {1 − lst (U ) − tcst (nst (U ) )} +  st (U ) bst (U ) {1 − lut (U ) − tcut (nut (U ) )}uRE= 22 +  st (U ) (ast (U ) − orst (U ) )s = ST105

40

s = 22s

4340=  st (U ) (1 +  tc+ s )C st (U ) + (1 − m)   st (U ) (1 +  tc+ s ) t (U ) (1 −  )nkt (U ) + (1 − m)   st (U ) (1 +  tc+ s ) t (U ) (1 −  )nkt (U )s = 22s = 22 k = 2261

s = 41k = 22

40

39

s

+ (1 − m)   st (U ) (1 +  tc+ s ) t (U ) (1 −  )nkt (U ) + m   st (U ) (1 +  tc+ s ) t (U ) (1 −  )nkt (U )s = 44 k = s − 21

57

s = 22 k = 2240

+ m   st (U ) (1 +  tc+ s ) t (U ) (1 −  )nkt (U ) ,(C.10)

s =40 k = s −17

where t (U )
22=1 for s =22, t (U )s s=  {1 + rt +u (1 −  r )}
 u =23

−1

for s =23, 24, …, 105.

Conflict of Interest: This research was assisted by a grant from the Abe Fellowship Programadministered by the Social Science Research Council in cooperation with and with funds provided
by the Japan Foundation Center for Global Partnership. The author also has received research grants
from the Ministry of Education, Culture, Sports, Science and Technology in Japan (Grant-in-Aid for
Scientific Research (C) No. 20K01679). However, the author declares that he has no conflict ofinterest.

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42

Figure 1 Net government debt-to-GDP ratio transition for six advanced countries
Source: International Monetary Fund (2021)

Figure 2 Age earnings profiles based on educational backgroundSource: The profiles are estimated from the Ministry of Health, Labour and Welfare (2013–2022) for the 2012–2021 period.

43

Figure 3 Leveled LSRA transfer value created by simulation analysis for each net debtGDP ratio

Figure 4 Changes in total population from the benchmark for five cases of different net
debt-GDP ratios from 2020 to 2050 (percent changes)

44

Figure 5 Changes in national income from the benchmark for three cases of different net
debt-GDP ratios (percent changes)

Figure 6 Changes in capital stock from the benchmark for three cases of different net debtGDP ratios (percent changes)

45

Figure 7 Changes in labor supply from the benchmark for three cases of different net debtGDP ratios (percent changes)

Figure 8 Changes in interest rates from the benchmark for three cases of different net
debt-GDP ratios (percentage-point changes)

46

Figure 9 Changes in wage rates from the benchmark for three cases of different net debtGDP ratios (percent changes)

Figure 10 Changes in consumption tax rates from the benchmark for three cases ofdifferent net debt-GDP ratios (percentage-point changes)

47

Figure 11 Changes in contribution rates from the benchmark for three cases of different
net debt-GDP ratios (percentage-point changes)

Figure 12 Changes in total population from the benchmark for five cases of different net
debt-GDP ratios from 2020 to 2200 (percent changes)

48

Figure 13 Net debt-GDP ratio that produces the largest total population for each year49

Table 1 Exogenous variables for the benchmark simulation
Parameter description

Parameter value

Data source = 0.5

Nishiyama & Smetters (2005):
 = 0.47

Share parameter for consumption
Weight parameter of the number of children to theconsumption–leisure composite in utility(H )= 0.03529
 (U )
 = 0.02629

 = 0.0001

Oguro et al. (2011):  = 0.01

Intertemporal substitution elasticity

 = 0.5İmrohoroğlu et al. (2017)

Ratio of government subsidies to childrearing costs

 = 0.1

Ratio of childrearing costs to net lifetime income

 = 0.0385

Time cost for childrearing =1.7234

Capital share in production

 = 0.3794

İmrohoroğlu et al. (2017)

Depreciation rate

 k = 0.0821

İmrohoroğlu et al. (2017)

Tax rate on labor income w = 0.065

Kato (1998):  w = 0.065

Tax rate on capital income

 = 0.4

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

Tax rate on inheritance h = 0.1

Kato (1998):  h = 0.1

Ratio of government expenditures to national
incomeg = 0.1

Ratio of the part financed by tax transfer to total
pension benefit

 = 0.25

Oguro & Takahata (2013):

Replacement ratio for public pension benefits

 =0.4

Braun et al. (2009):  =0.4

Ratio of net public debt to national income

d = 1.5

Rate of time preferencer

Oguro et al. (2011):

 = 0.1

 = 0.25

İmrohoroğlu et al. (2017),
Nakajima & Takahashi(2017) : d = 1.3

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 = 0.83252

Dependency ratio (i.e., aging rate)

O / Z = 0.28372

50

Table 2 Endogenous variables in the 2020 initial steady stateParameter description

Parameter value

Interest rate, r

0.0753

Wage rate, w

1.0625

Tax rate on consumption, Contribution rate, 

0.1313c

0.1499p

Capital–income ratio, K / Y

2.40961.3300Total fertility rate (TFR)
Ratio of net childrearing costs to annual labor income(low-income class 1.45; high-income class 1.15))
0.2038 (low-income class)0.1890 (high-income class)0.0114

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

Table 3 Population ratios among people with different educational backgrounds
Population (thousands)

Population share (%)

Junior high school graduates695.16

High school graduates

9,945.14

3.26Share (%)
49.94
46.68

Technical and junior college

2,149.95

10.09

graduates
University
graduates

8,515.62

39.97

Total (in year 2020)

21,305.87

50.06

Source: The Ministry of Health, Labour and Welfare (2021)

51

100
100

Table 4 Scheduled number of children for young people aged 21 to 40
Average

Number of children
Young female non-university graduates

1.32

Young male non-university graduates

0.96

Young female university graduates

0.91

Young male university graduates

0.84

1.14

0.875

Source: Kikkawa (2018)

Table 5 Population ratio of the low-income class for three cases of different net
government debt-to-GDP ratios
Net debt-to-GDP ratio

2020

2050

2100

2200

2300

150% ( d = 1.5)

69.670%60.489%55.942%53.495%53.385%−170% ( d = −1.7)

69.670%60.536%55.840%53.240%53.136%220% ( d = 2.2)

69.670%60.493%55.978%53.546%53.431%52