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Aging, Housing, and Macroeconomic Inefficiency
Yasutaka Ogawa and Jiro Yoshida
Discussion Paper No. 2024-E-4

INSTITUTE FOR MONETARY AND ECONOMIC STUDIESBANK OF JAPAN
2-1-1 NIHONBASHI-HONGOKUCHOCHUO-KU, TOKYO 103-8660
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IMES Discussion Paper Series 2024-E-4
June 2024

Aging, Housing, and Macroeconomic Inefficiency

Yasutaka Ogawa* and Jiro Yoshida**
Abstract
This study quantifies the macroeconomic impact of population aging with a focus
on large houses owned by elderly households for bequest motives, although younger
generations may leave the inherited houses vacant. A quantitative overlapping
generations model incorporates age-specific mortality rates and bequest motives togenerate a hump-shaped age profile for consumption and an upward-sloping age
profile for housing and savings. When calibrated to the Japanese economy, themodel suggests that bequest-driven housing demand raises the output level but
reduces consumption, the natural rate of interest, capital allocation to the goods
sector, and housing affordability. These effects are more pronounced whenhouseholds intend to bequeath housing rather than financial assets and when morehouses become vacant upon inheritance.

Keywords: Aging; Natural Rate of Interest; Overlapping Generations Model;
Bequest Motives; Intergenerational Transfer of Housing; JapanJEL classification: D15, E22, J11, R21*Director and Senior Economist, Institute for Monetary and Economic Studies (currently,
Financial System and Bank Examination Department), Bank of Japan (E-mail:yasutaka.ogawa@boj.or.jp)**Pennsylvania State University and the University of Tokyo (E-mail: jiro@psu.edu)
The authors would like to thank Ippei Fujiwara, Stijn van Nieuwerburgh, Takatoshi Ito, Zheng
Liu, Nao Sudo, Brent Ambrose, Liang Peng, Eva Steiner, Makoto Nakajima, Shigeru Fujita,
Yasuo Terajima, Takeo Hoshi, Kiminori Matsuyama, Koji Nakamura, Mototsugu Shintani,
Mitsuru Katagiri, and seminar participants at the Bank of Japan, Penn State, Philadelphia Fed,Bank of Canada, Columbia, Princeton, University of Tokyo, AREUEA, and WEAI for valuablecomments. The views expressed in this paper are those of the authors and do not necessarilyreflect the official views of the Bank of Japan.

I.

Introduction

Population aging aﬀects a wide range of markets and macroeconomy (for example, seeWise, 1989, and subsequent publications from the NBER Program on the Economics ofAging). Our study focuses on the ownership of large housing capacities by elderly households,
part of which is underutilized and may eventually become vacant after being transferred to
the younger generation (Seko et al., 2023). Because housing consumption increases with age
in many countries (e.g., Yang, 2009; Cocco and Lopes, 2020), a large stock of accumulated
housing will become a common feature of an aging society.
Despite the accumulation of housing stock, the younger generation may still face housing
aﬀordability problems if the housing stock is not readily available to them due to excess
housing capacity held by the elderly and intergenerational mismatches in preferences. As aconsequence, an aging society will allocate more capital to the construction industry, which
has low productivity after decades of negative productivity growth in many countries (e.g.,
Goolsbee and Syverson, 2023; Garcia and Molloy, 2023). One possible consequence is low
marginal product of capital (interest rates) due to the slow growth of overall productivity
due to the composition eﬀect.
Insights into the macroeconomic impact of aging can be gained by studying Japan, which
is at the forefront of aging societies. Its potential support ratio (the number of working-age
people (15-64) per elderly person (65+)) is the lowest at 2.0 in 2021, compared with 3.9 for
the US, 5.3 for China, and 9.9 for India. In addition, Japan, like other countries, showsincreasing housing consumption with age. The homeownership rate increases monotonicallywith the age of the household head from 3% for households under 25, to 83% for those 85 andolder (2018 Housing and Land Survey). A similar upward-sloping age proﬁle is also observedfor housing wealth in Japan’s 2019 National Survey of Family Income, Consumption and
Wealth. Moreover, total factor productivity (TFP) in the construction and housing sector
also declined in the four decades leading up to the 2000s (ﬁgure 1), in stark contrast to
the growing productivity of other sectors. The Japanese economy is thus an ideal case for1our study of how the ownership of large housing capacities by elderly households aﬀects themacroeconomy.

Figure 1: Total factor productivity level for the construction and housing sector in Japan.
Source: Japan Industrial Productivity Database.
We quantify the eﬀect of elderly households’ accumulated housing by calibrating a quantitative, perfect-foresight, overlapping generations (OLG) model to the Japanese economy.
The model that incorporates bequest motives and the housing production sector replicates
well historical GDP growth rates, historical real interest rates, the current housing-toconsumption ratio, and the housing vacancy rate. It also replicates relatively well the age
proﬁle of consumption, housing assets, and savings. These replications and calibrations are
based on the exogenously speciﬁed trajectory of fertility rates, mortality rates, sector-speciﬁc
productivity, government spending, and government debt, in addition to constant parameters for age-speciﬁc labor productivity, the sectoral share of labor supply, and the pension
replacement rate.
We analyze the natural rate of interest, gross domestic product, total factor productivity,
capital allocation between two production sectors, and house prices between 1980 and 2050.
2

We ﬁrst decompose the baseline prediction of the equilibrium outcome into the six exogenous
factors: mortality, fertility, government debt, government spending, technology growth, andthe technology gap between the two sectors. To identify the contribution of each factor,
we “knock out” each factor at a time by keeping its 1980 value throughout the remaining
years as Bielecki et al. (2020) do.1 This decomposition analysis shows that the natural rateof interest will decline by 1.5-2.0 percentage points after 1980, mainly driven by stagnant
technology growth and aging. The GDP growth rate will also decline by 4 percentage points
for similar reasons. Housing value will increase from 2020, driven by the growing demand of
the elderly.
The model also identiﬁes the general-equilibrium eﬀect of bequest-driven ﬁnancial andhousing asset holdings, the balance between ﬁnancial and housing bequests, the eﬃciency
of intergenerational housing transfers, and inheritance tax rates. First, bequest motives
increase the level of GDP and the capital allocation to the housing sector but decrease
consumption and the natural rate of interest. Houses are less aﬀordable because prices are
inﬂated by both the extra demand and the less eﬃcient production of housing. Second,
the above eﬀects are exacerbated when housing bequests are preferred to ﬁnancial bequests
because these eﬀects are primarily driven by the extra demand for housing driven by bequest
motives. Third, the same eﬀects will be further exacerbated if more inherited houses become
vacant due to ineﬃcient intergenerational transfers. Finally, under the premise that the
government substitutes income taxes for inheritance taxes, changing the inheritance tax rate
has only a small impact on the economic outcome. Overall, the eﬃciency of the Japanese
economy can be improved by reducing the bequest-motivated housing demand of the elderly
and improving the eﬃciency of house matching.
Our model has several special features. First, we introduce age-speciﬁc mortality rates.
One challenge with a life-cycle model with a ﬁxed life span is that the model predicts a ﬂat
age proﬁle of consumption, whereas data show a hump-shaped age proﬁle. By incorporating
1

For the fertility rate, we use the average rate for 1980-1989 because the growth rate of the 20-year-old
population is negative between 1970 and 1980 due to WWII.

3

actual mortality rates into our model, we generate a hump-shaped age proﬁle of consumption.
However, the actual mortality rates lead to a consumption peak much later (after age 80) thanin the data (around age 50). Given that households take into account health-adjusted life
expectancy (HALE) when making consumption decisions, we adjust the empirical mortality
rates for healthy life expectancy. Our calibrated model shifts the consumption peak earlier,to around 70 years of age.
Second, we introduce motives for bequeathing housing. In the OLG model that includes
housing consumption, households’ ﬁrst-order conditions require that the age proﬁle of housing consumption be parallel to that of general consumption. However, the data show that
housing consumption, unlike non-housing consumption, increases monotonically with age.
Housing bequest motives in our model create an upward age proﬁle of housing consumption.
Third, we also introduce motives for bequeathing ﬁnancial assets. In an OLG model, savings usually peak just before retirement age and decline monotonically to zero as a household
ages. However, the data show that households do not dissave after retirement age (French
et al., 2023). Our model introduces intentional ﬁnancial bequests, which help us explain theslow decumulation of ﬁnancial assets after retirement.
However, our model does not include several factors. First, our perfect foresight model
does not allow for precautionary saving, which could help explain the persistently highlevel of savings after retirement. Second, our rational model does not allow for behavioral
biases, such as myopia and suboptimal housing choices. Behavioral biases could explain an
early peak in consumption and large housing capacities, but we intend to study the rational
benchmark. Third, our model does not take into account age-speciﬁc consumption goods.
For example, middle-aged parents typically spend a signiﬁcant amount on their children’s
education, resulting in a consumption peak well before the retirement age. Also, elderly
households may prefer home care and other goods and services associated with their current
home, which may explain the upward-sloping age proﬁle of housing consumption. Fourth,
our model omits ﬁnancial constraints, such as required down payments and foreclosure costs,

4and frictions beyond house vacancies. The data suggest that households borrow and increase
their housing wealth signiﬁcantly when the head of the household is in their 30s and 40s,
which is associated with ﬁrst-time homeownership. Furthermore, older homeowners may notsell their homes and downsize if the outstanding balance of long-term recourse mortgages
exceeds the market value of the house. One might think that the absence of downsizing is due
to transaction costs, but excess housing capacity cannot be explained solely by transaction
costs under perfect foresight because rational households will maintain undercapacity intheir middle years to avoid costly downsizing in the future. Last, our model does not allow
productivity to endogenously change by the resource allocated to each production sector.
Relaxing this restriction may lead to a larger eﬀect of housing choices of elderly households
on growth.
Our study contributes to the literature in four ways. First, we show that homeownershipis a new channel through which population aging aﬀects the economy. This channel is basedon the empirical regularity that the elderly are likely to own large housing capacities. Ourmodel features the bequest motive of rational households to generate additional housing
demand from elderly households, which leads to the allocation of capital to a less productive
sector. Although other factors, such as a behavioral bias, could generate a similar result, weemphasize that downsizing constraints alone cannot generate an upward-sloping age proﬁleof housing consumption if households are rational.
Second, our study provides quantitative insights into the impact of population aging on
the Japanese economy, such as the real interest rate, GDP growth, tax rate, productivity
growth, and house prices. By decomposing the changes in these variables between 1980 and
2050, we quantify the eﬀect of the expected changes in mortality and fertility rates. These
factors lead to lower interest rates and higher taxes, but they are almost neutral for GDP
and TFP growth.
Third, our study shows the general equilibrium eﬀect of age-speciﬁc housing demand on
macroeconomic variables, whereas existing studies typically show partial equilibrium eﬀects

5and age-independent housing choices. The increased housing capacity of the elderly due tobequest motives signiﬁcantly aﬀects consumption, output, productivity growth, and the natural rate of interest. Speciﬁcally, the additional demand for housing suppresses consumption
and real interest rates, while it expands the level of output.
Fourth, our model demonstrates that an OLG model can replicate a hump-shaped age
proﬁle for consumption and an upward-sloping age proﬁle for housing and savings by incorporating both age-speciﬁc mortality rates and bequest motives for ﬁnancial and housing
assets. It provides a benchmark modeling choice for future studies.
The remainder of this paper is organized as follows. Section II reviews the related literature, Section III describes our model, and Section IV explains the model calibration.
Sections V and VI presents the results, followed by discussions in Section VII. Section VIII
concludes the paper.

II.
A.

Literature Review

Population aging
Population aging can have a profound eﬀect on the economy. A series of conference

proceedings from the NBER Project on the Economics of Aging include a large numberof contributions since 1989 (Wise, 1989, 1990, 1992, 1994, 1996, 1998b,a, 2001, 2004, 2005,
2009, 2011, 2012, 2014, 2017). These studies show that aging has not only an obvious impact
on health care and the social security system but also a wide range of indirect eﬀects on
savings, consumption, and investment. Furthermore, Weil (1997) and Lee (2016) review theeﬀect of aging on the macro economy. The literature generally predicts a tendency toward
increased capital intensity, higher wages, and lower returns on capital. In an open economy
setting, ﬂows of capital and labor into the aging country are also expected to increase. The
eﬀect on output is generally expected to be negative (e.g., Sheiner et al., 2007; Sheiner,
2014; Börsch-Supan et al., 2014; Gagnon et al., 2021), but the literature does not completely
6agree on this prediction because capital deepening can overturn the eﬀect (e.g., Cutler et al.,1990). In a recent study, Maestas et al. (2023) further point out that population aging
slows economic growth by diminishing labor productivity growth in addition to employment
growth. These results critically depend on conditions such as labor supply at older ages, percapita consumption of the elderly relative to younger ages, strength of public pension andhealth care systems, and health and vitality of the elderly.

B.

Large-scale overlapping generations models
Following the seminal work by Auerbach and Kotlikoﬀ (1987), overlapping generations(OLG) models have been a leading tool to evaluate the macroeconomic eﬀect of demographic
change because they can incorporate empirical age distributions to replicate realistic lifecycle behavior. Recent OLG models predict a downward trend in the real interest rate
due to population aging (e.g., Bielecki et al., 2020; Papetti, 2021). Longevity can resultin more precautionary savings. In particular, Bielecki et al. (2020) is most closely related
to the present study. They develop a quantitative OLG model calibrated to match thelife-cycle proﬁles of the European economy. They particularly investigate the impact ofdemographics on the natural rate of interest (NRI) in the euro area, with a particular focus
on the role played by economic openness, migrations and pension system design. They showthat population aging explains approximately two thirds of the secular decline in the NRI
between 1985 and 2030. They also ﬁnd that an increase in the retirement age can reverse
the downward trend in the interest rate, but a fall in the replacement rate exacerbates thetrend. Papetti (2021) also conﬁrms that demographics account for a decrease in the naturalrate of interest in the euro area due to scarce labor input and more savings. However,
the results depend on substitutability between labor and capital, intertemporal elasticity of
substitution, higher productivity, and labor participation by older individuals. Gagnon et al.
(2021) predict that GDP growth and the natural rate of interest in the U.S. will remain low.
Carvalho et al. (2016) show that the natural rate of interest will be lower through potentially7

competing three channels: larger savings driven by longevity, larger capital per-worker, anda lower savings rate. Eggertsson et al. (2019) quantify the secular stagnation hypothesis
and predict a natural rate in the U.S. ranging from −1.5% to −2%. Sudo and Takizuka(2020) estimate that demographic changes account for approximately 270 basis points out of
the 640 basis-point decline in the natural rate of interest during the last 50 years in Japan.
Braun and Ikeda (2022) provide a quantitative theory of deﬂation and secular stagnation
using Japanese data.

C.

Macroeconomic models with bequests and housing
A bequest is a type of intergenerational transfer in addition to public pensions and ed-

ucation, and plays a central role in investment decisions. Thus, it critically aﬀects theconsequences of population aging. Artle and Varaiya (1978) is an early study that incorporates homeownership and bequest motives in the life cycle consumption model. However,they generally predict a monotonic age proﬁle of the optimal consumption, either sloping
downward or upward, depending on ﬁnancial constraints. Whether a bequest motive is a
critical factor for the economy has been debated. Hurd (1989) estimates a life cycle model
augmented by a bequest motive and ﬁnds that the bequest motive is statistically signiﬁcantbut economically trivial. Hurd (1987) argues that bequests are merely accidental resulting
from precautionary responses to imperfect insurance markets. However, Bernheim (1991)
ﬁnds that the typical household chooses to maintain a positive fraction of its resources inbequeathable forms even if insurance markets are perfect. Kopczuk and Lupton (2007) ﬁnd
that roughly three quarters of the elderly single population has a bequest motive and spends
less on consumption expenditure, doubling the amount of bequeathed net wealth. Intentional bequests can emerge either from altruism (Tomes, 1981) or self-interested exchangewith one’s heirs (Bernheim et al., 1985). Bequests are also incorporated in quantitative
macro models (e.g., Hurd, 1989; Hviding and Mérette, 1998; Fougère and Mérette, 1999;
Kraft and Munk, 2011).
8A model that features bequest motives in the utility function can also be seen as a versionof models featuring wealth in the utility function (Michaillat and Saez, 2021; Nakajima, 2003;
Barberis et al., 2001; Carroll, 1998; Dynan et al., 2002, 2004). Wealth in the utility function
may resolve macroeconomic puzzles related to the savings rate and consumption around thezero lower bound. Wealth in the utility function is often incorporated in partial-equilibrium
consumer models and dynamic stochastic general equilibrium (DSGE) models but not in
OLG models.
Housing plays a critical role in consumption choices and investment decisions becauseof its dual role as a consumption good and an investment and collateral asset. The role ofreal estate as collateral is a key insight provided by Kiyotaki and Moore (1997), Bernanke
et al. (1999), and subsequent studies on the eﬀect of ﬁnancial constraints on the ﬁnancial
crisis and monetary policy transmissions (e.g., Wong, 2021). Housing also aﬀects household
portfolio choices and equilibrium asset prices (e.g., Flavin and Yamashita, 2002; Piazzesi
et al., 2007). Iacoviello (2005) and Iacoviello and Neri (2010) incorporate housing in DSGE
models. They show that homeownership with mortgage ﬁnancing signiﬁcantly aﬀects how
demand and supply shocks propagate. Also, they ﬁnd that slow technological progress in the
housing sector is a major cause of house price appreciation. Furthermore, they show that
housing is increasingly important in explaining consumption. More recent DSGE models
also incorporate housing (Liu et al., 2013; Garriga et al., 2021; Adam and Woodford, 2021).
Housing is also incorporated in multi-sector OLG models (Galor, 1992; Farmer and Wendner, 2003; Yang, 2009; Kraft and Munk, 2011; Waters, 2020; Nakajima, 2020). In particular,
as in the present study, Nakajima (2020) focuses on the substitution between housing andnon-housing capital and ﬁnds that preferential tax treatment for owner-occupied housing
signiﬁcantly decreases the optimal capital income tax rate. A low capital income tax rate
improves welfare by narrowing a tax wedge between housing and non-housing capital. Nakajima and Telyukova (2020) study why retired homeowners dissave slowly. They show thatthe accumulated home equity during a housing boom cannot be easily cashed out. Cocco

9and Lopes (2020) study the role of housing wealth in retirement consumption and saving
decisions by the elderly. They ﬁnd that reverse mortgages combined with insurance against
a forced home sale is Pareto improving.

III.
A.

Model

The Model Setup
Consider a model for a closed-economy populated by overlapping generations (OLG) of

households that solve a standard life-cycle problem. There are four agents in this model
economy: households, two-sector ﬁrms (general goods and housing sectors), and the ﬁscal
authority. The model is mainly based on Bielecki et al. (2020) and Papetti (2021).

B.

Households
Each household consists of a single agent, who appears in our model at the age of 20,

indexed as j = 1. Agents can live up to 104 years (j = J = 85) but may die at year t
according to an age- and time-dependent mortality rate ωj,t . Hence, at each point in time,the model economy is populated by 85 cohorts of overlapping generations. The size of theage-j cohort is denoted by Nj,t .
The representative age-j household at time t maximizes its expected remaining lifetime
utility until the maximum age J:

Uj,t =J−j

β i Πi−1
k=0 (1 − ωj+k,t+k ) ×

i=0[(ln cj+i,t+i + χ ln hj+i,t+i ) + ωj+i,t+i ψ{ν ln bij+i,t+i + (1 − ν) ln hj+i,t+i }] .

(1)

where β denotes the subjective discount factor, and ωj,t ≡ 1 − Nj+1,t+1 /Nj,t denotes thetime-t mortality rate between age j and j + 1. Thus, the term Πi−1
k=0 (1 − ωj+k,t+k ) represents

10the probability of surviving for at least i more years. The per-period utility function has
two components. The ﬁrst component represents felicity as a function of the consumption
stream cj,t and housing services hj,t with the importance parameter χ. The second component
represents bequest motives conditional on death. Parameter ψ determines the importanceof bequest motives relative to felicity. Bequest motives are speciﬁed as the weighted averageof the log utilities from the intentional ﬁnancial bequest contribution bij,t and the housing
bequest hj,t . Note that housing services in the felicity function are a linear transformationof housing assets in the bequest function. Parameter ν denotes the utility weight for theintentional ﬁnancial bequest.
An age-j household has the following budget constraint:

cj,t + aj,t + bij,t + Pth hij,t = (1 + rt ) aj−1,t−1 + 1j≥JR pent+ (1 − 1j≥JR ) (1 − Tt ) {σjg wtg ξjg + 1 − σjg wth ξjh } + (1 − T a ) it − T h Pth iht . (2)Retirees, as indicated by 1j≥JR , receive pension beneﬁts pent that are independent of theirage. Participation in the pension system is mandatory, so that all workers have to make asocial security contribution that is levied as part of personal income tax at the rate of Tt .
Working-age households (1 − 1j≥JR ) inelastically supply labor until reaching the age of 64(j = JR = 45), either in the general goods production sector g or housing production sector
h. Labor income in sector s = {g, h} depends on the sector-speciﬁc gross wage rate wts andthe exogenously determined age-speciﬁc labor productivity ξjs . Working-age households also
inherit ﬁnancial assets it and housing assets iht evaluated at the market price of housing Pthafter paying inheritance taxes at rates T a and T h , respectively.
All households can smooth their consumption by adjusting liquid assets aj,t , which areclaims on productive capital and government debt. We allow households to borrow acrossdiﬀerent cohorts, so that asset holdings aj,t can be negative at any age j. Households invest
bij,t in the intentional ﬁnancial bequest account bj,t . At death, agents leave the accumulated

11

intentional ﬁnancial bequest, the outstanding liquid asset, and the outstanding housing asset
as bequests. Intentional bequest assets (bj,t ) are invested in productive capital and earn thereal interest rate rt , which is equivalent to the natural rate of interest in this real model
without short-term business cycles:

bj,t = (1 + rt ) bj−1,t−1 + bij,t ,(3)

b0,t = a0,t = aJ,t = 0.

(4)

Households can adjust housing with housing investment hij,t at the market price of housing Pth . The sign of hij,t can be negative if a household reduces the size of its dwelling. Thissmooth adjustment of housing by the representative household reﬂects the aggregate individual adjustments made by renters and homeowners. Although the direct interpretation of
the budget constraint is that a household is a homeowner who spends to purchase housing
and pays no periodic user costs, we do not assume that all households are homeowners. The
present speciﬁcation is equivalent to an alternative one that includes periodic user (rental)
costs but not purchase costs because there are no market imperfections such as asymmetric
information and incomplete contracts. The user cost rate equals the diﬀerence between the hhPt
total gross return to housing assets 1+rt +δh and the house price appreciation rate Pt+1
(See Appendix I). An advantage of the present speciﬁcation is that house prices are directly
speciﬁed.
Occupied and vacant houses (hj,t and vj,t ) are accumulated according to the following
equations, respectively.
hj,t = (1 − δh ) hj−1,t−1 + hij,t + φ (1 − 1j≥JR ) iht ,(5)

vj,t = (1 − δh ) vj−1,t−1 + (1 − φ) (1 − 1j≥JR ) iht ,(6)h0,t = v0,t = 0.

(7)

12

Housing assets depreciate at the rate of δh . Parameter φ denotes the intergenerational housematching eﬃciency, as deﬁned by the proportion of inherited housing iht used by younger
generations.
Inherited ﬁnancial assets and housing are deﬁned by:J
it = (1 + rt )j=1 (Nj,t−1 − Nj+1,t ) (aj,t−1 + bj,t−1 ),JR
j=1 Nj,t(8)

j=1 (Nj,t−1 − Nj+1,t ) hj,t−1,JR
j=1 Nj,t(9)J
iht = (1 − δh )where it and iht denote inherited ﬁnancial assets and housing, respectively. Bequeathedﬁnancial assets and housing are equally allocated to all working-age households.
In our model, demographic processes are governed by exogenous changes in the size of theyoungest cohort n1,t = N1,t /N1,t−1 − 1 and the mortality rate ωj,t . Then, the total numberof living agents Nt and the population growth rate nt are given by:

Nt =

J


Nj,t ,

(10)

Nt
− 1.
Nt−1

(11)

j=1

nt =

The number of agents in each cohort is given by:

Nj,t = (1 − ωj−1,t−1 ) Nj−1,t−1 .

(12)Table I summarizes the timing convention for each variable. The household’s ﬁrst-orderconditions are derived in Appendix A.

C.

Firms
Firms consist of general goods and housing sectors: s = {g, h}. A representative ﬁrm insector s hires capital and labor and produces non-diﬀerentiated output Yts according to the13tPeriodDate
Financial assets
Housing
Intentional bequests
CapitalGovernment debt
Population and cohort size
Consumption
Bequest investment
Housing investment
Mortality rate
Goods production
Housing production
Capital for goods sector
Capital for housing sector
Government spending
Inherited ﬁnancial assets
Inherited housing
Interest rate

t−1
(Beginning of period t)t(End of period t)

At−1 , aj−1,t−1
Ht−1 , hj−1,t−1
Bt−1 , bj−1,t−1
Kt−1
Dt−1

At , aj,t
Ht , hj,t
Bt , bj,tKt
Dt

Nt , Nj,tCt , cj,t
bij,t
hij,t
ωj,t
Ytg
Yth
Ktg
Kth
Gt
it
iht
rt

Table I: Timing Convention for Each Variable
following Cobb-Douglas production function:2
Yts = (Kts )αs (ζts Lst )1−αs ,(13)where ζts denotes labor-augmenting technology growing at the rate of gts :s.
ζts = (1 + gts ) ζt−1

(14)

Let θt denote the technology gap between the housing and general goods sectors:3
θt ≡ 1 −
2

ζth
.
ζtg

(15)

Inelastically-supplied land could be separately included in the housing production function, in whichcase decreasing returns to scale will contribute to lower productivity for larger production scale.
3
Labor-augmenting technologies can be directly compared across these sectors because they use the samekind of labor input.

14

Then, the housing-technology growth rate equals the general-technology growth rate adjusted
for the change in the technology gap:
1 + gth = (1 + gtg )1 − θt.
1 − θt−1(16)

The periodic proﬁt ﬂow is:
πts = Pts Yts − wts Lst − (rt + δk ) Kts ,(17)

where Ptg equals unity.

D.

Government
The government purchases general goods Gt in the market and spends on pensions pent .

It rolls over public debt by paying oﬀ the outstanding debt (1 + rt ) Dt−1 and issuing a new
debt Dt . It also levies taxes from ordinary income at a rate Tt , inherited housing at a rate
T h , and inherited ﬁnancial assets at a rate T a . The government’s budget constraint at time
t is:
JR−1



 g g g 
 h h  h h h JR−1ga
Tt Nj,t σj wt ξj + 1 − σj wt ξj + T Pt it + T it
Nj,t + Dt

j=1

j=1

= Gt + pentJ

Nj,t + (1 + rt ) Dt−1 (18)j=JRPensions per retired household are determined as the product of the replacement rate ρt
and the economy-wide average wage:

pent = ρt

wtg

JR−1
j=1




1 − σjg Nj,t ξjh
σjg Nj,t ξjg + wth JR−1
j=1
,
JR−1
j=1 Nj,t(19)

where σjg is the proportion of j-aged household choosing to work for the general goods sector.

15

The paths of public debt, government purchases, the replacement rate, and inheritancetax rates are exogenously determined, whereas the income tax rate Tt adjusts so that the
government budget constraint (18) is satisﬁed. Thus, for the government, three types oftaxes are perfect substitutes.

E.

Market clearing and other aggregate conditions
The ﬁnancial market clears with

Dt + Kt = At + Bt

(20)

Kt−1 = Ktg + Kth(21)

Labor market clears with:
Lgt =

J


σjg Nj,t ξjg ,

(22)

j=1

Lht =

J




1 − σjg Nj,t ξjh .

(23)

j=1

New housing market clears with:

Yth =

J


Nj,t hij,t .

(24)

Nj,t cj,t + Kt − (1 − δk ) Kt−1 + Gt .

(25)

j=1

The good market clears as:
Ytg =

J

j=1

16

The aggregate consumption, assets and housing are:

Ct =

J


Nj,t cj,t ,(26)

Nj,t aj,t ,(27)

Nj,t Bj,t ,(28)

Nj,t hj,t ,(29)

Nj,t vj,t .

(30)

j=1

At =

J

j=1

Bt =

J

j=1

Ht =

J

j=1

Vt =

J

j=1

F.

Exogenous process
The model economy is driven by several exogenous forces. Demographic processes arecharacterized by the growth rate of the 20-year-old cohort n1,t and the age-speciﬁc mortality
risk ωj,t (j = 1, ..., J). The ﬁscal authority exogenously determines its purchases of general
goods Gt , the replacement rate in the pensions system ρt , and the path of public debt Dt .
TFP is also exogenous for the general goods and housing sectors. The age-speciﬁc mortality
rate ωj,t (j = 1, ..., J) is adjusted for healthy life expectancy.
Table II shows the list of the exogenous variables.

G.

Equilibrium
For all periods t = 0, 1, ..., ∞, given exogenously given demographics {Nj,t }Jj=1 , Nt , nt
J

J

JR−1

and variables n1,t , ρt , ζtg , ζth , Gt , {ωj,t }Jj=1 , ξjg j=1 , ξjh j=1 , and σjg j=1 , the equilibriumof this economy is a sequence of aggregates Ct , At , Bt , Ht , Vt , prices wtg , wth , Pth , rt , transfers
Tt , pent , and quantities {cj,t , aj,t , bj,t , bij,t , hj,t , hij,t , vj,t }Jj=1 , it , iht , Kt , Ktg , Kth , Lgt , Lht , Ytg ,Yth , such that:4
4

The full set of equations that characterize the equilibrium is listed in Appendix B.

17

Variable

Description

Type

n1,t
ωj,t
ζtg
θt
1j≥JR
ρt
σjg
ξjg
ξjh
G/Y
D/Y

Fertility rate (the growth rate of 20-year-old cohort)Age-speciﬁc modiﬁed mortality rate
Labor-augumenting technology in general production sector
Technology gap between the housing and general production sectors
Retirement indicator
Pension replacement rate
Age-speciﬁc proportion for working in general production sector
Age-speciﬁc eﬀectiveness of labor in general production sector
Age-speciﬁc eﬀectiveness of labor in housing sector
Government spending ratio
Government debt ratio

periods x 1
periods x cohorts
periods x 1
periods x 1
periods x cohorts
periods x 1
cohorts x 1
cohorts x 1
cohorts x 1
periods x 1
periods x 1

Table II: List of Exogenous Variables
1. Demographics develops as (10)-(12) and the aggregation satisﬁes (26)-(30);
2. Households maximize the life-time utility (1) subject to constraints (2)-(9);3. Firms maximize proﬁts (17) given the production technology (13)-(16);
4. The government balances budget (18) and (19) by setting the income tax rate;
5. Factor markets (20)-(23), the housing market (24) and the general goods market (25)clear.

IV.
A.

Baseline Calibration

Exogenous variables and structural parameters
We calibrate the model to the Japanese economy using the data from 1925 to 2095. We

exogenously specify a demographic scenario, technological progress, and the ﬁscal conditionsof the government. The data source for calibration is summarized in Appendix C, the timeseries graph of each exogenous variable is shown in Appendix D , and the detailed solution
method is outlined in Appendix E.
For our demographic scenario, we use past data and projections on mortality rates andthe size of the 20-year-old cohort from the Population Projection for 2020 through 2050 from18

the National Institute of Population and Social Security Research. We smooth the mortality
rate and the cohort size by using the stochastic trend component from the Hodrick-Prescott(HP) ﬁlter. We further adjust the mortality rate ωj,t for the health-adjusted life expectancy(HALE) by using the Life Tables for 2019 and the Comprehensive Survey of Living Conditions
from the Ministry of Health, Labour and Welfare. In Japan, HALE is shorter than the actual
life expectancy by 8 to 12 years (ﬁgure 3). We assume that mortality rates are constant for
1925-1970 at the rate in 1970. Regarding the growth rate of the 20-year-old cohort n1,t ,we use the Population Estimates between 1925 and 2014 and the Population Projection
between 2015 and 2050 from the Statistics Bureau. After 2050, we assume that both themortality rate and the fertility rate stabilize at the 2050 level. Figure 2 contrasts the actual
age distribution of the population with the model-generated age proﬁle for 1980, 2019, and2050. The model replicates the actual demographics quite well.

Note: The health-adjusted life expectancy (HALE) is computed from the Life Tables for 2019 and theComprehensive Survey of Living Conditions by the Ministry of Health, Labour and Welfare.

Figure 2: Distribution of Population by Age in 1980, 2019 & 2050
We also estimate the age-speciﬁc labor productivity for the general goods sector ξjg andhousing sector ξjh and the age-speciﬁc proportion of workers in the general goods sector ρgj ,19

Note: Data on mortality rates and the size of the 20-year cohort are provided by the National Institute ofPopulation and Social Security Research. Mortality rates adjusted for health-adjusted life expectancy areheld constant at the 1970 level for 1925-1970 and at the 2050 level for 2050 and later.

Figure 3: Original and HALE-Based Survival Rates

20

using the 2020 Basic Survey on Wage Structure by the Ministry of Health, Labour andWelfare. Our empirical measure of labor productivity is the estimated hourly labor income
based on a regression on the constant, age, and age squared. The retirement age JR is 65
for all periods.
For technological progress, we estimate the labor-augmenting technology for the general
goods sector ζtg and housing sector ζth using the JIP Database between 1972 and 2018 fromthe Research Institute of Economy. We use the stochastic trend component out of the
Hodrick-Prescott ﬁlter. After 2019, we assume that the TFPs grow at the same rates as
2018.
For the ﬁscal condition of the government, we use the government spending ratio (ofGDP) and government debt ratio (of GDP) from the Japanese System of National Accounts(JSNA) data for 1980-2021. We assume that those are constant for 1925-1980 at the rate in1980 and for 2021-2095 at the rate in 2021. We also ﬁx the replacement rate at 45%, in linewith the literature.
Table III summarizes the parameters. The discount factor β, the capital depreciation
rate δg and the housing depreciation rate δh are based on the literature. The inheritancetax rates for ﬁnancial assets (T a ) and housing (T h ) capture the preferential tax treatment
for housing. Inherited ﬁnancial assets are heavily taxed up to the maximum tax rate of
55%, although there are some basic exemptions. In contrast, the assessed value of housing
is reduced by 80% up to a 330 m2 lot. Thus, many heirs who inherit only a house often
pay no inheritance taxes at all. Instead of specifying the complex inheritance tax codes,
we simply use a 40% eﬀective tax rate for inherited ﬁnancial assets and no eﬀective tax
for inherited housing. Other countries also have various forms of preferential treatment of
housing in estate taxes, such as a step-up in basis. Thus, this diﬀerence in the eﬀective tax
rate is relevant for many countries.
We determined the importance of housing consumption χ, the relative importance ofbequest motives ψ in the lifetime utility function, and the weight on ﬁnancial bequests ν21

relative to housing bequests by matching the following statistics with the 2019 data: (1) the
ratio of housing consumption to non-housing consumption (ph h/c) for ages 20-64 and 65-104
and (2) the average saving-to-consumption ratio ((a + b)/c) for 65-104.5
The eﬃciency parameter for intergenerational housing transfer φ is set to 0.46 to matchthe inverse vacancy rate (20% of the inherited houses are used by the younger generation).
The capital shares for general goods sector αg and for the housing sector αh are calibrated
to the JSNA data for 1994-2018.
Variable

Value

How to decide

β
χ
ψ
ν
δg
δh
φ
Ta
Th
αg
αh

0.996
0.385
22
0.6
0.15
0.03
0.46
0.4
0
0.37
0.31

Literature
Match the average P h h/c amongages 20-64 and 65-104, and theaverage (a + b) /c among ages 65-104
Litereture
Litereture
Match the average h/v
Litereture
Litereture
Calibrate with JSNA
Calibrate with JSNA

Discount factor
Importance of housing consumption
Importance of bequest motives
Importance of ﬁnancial bequests
Depreciation rate for capital
Depreciation rate for housing
Proportion of reused housing
Inheritance tax rate for ﬁnancial assets
Inheritance tax rate for housing
Capital share in general production sector
Capital share in housing sector

Table III: List of Structural Parameters

Table IV shows the parameter calibration results. The ratio of housing consumption to
non-housing consumption in the model (P h h/c) is identical to the data for both the 20-64
group (8.4) and the 65-104 group (13.3). However, these perfect matches and the realistic
age proﬁle of housing consumption come at the expense of a high saving-to-consumption
ratio ((a + b) /c = 15.2) compared to the actual average ratio (6.4) for 65-104. A lower
value of ψ can decrease the saving-to-consumption ratio but worsens the housing-to-nonhousing consumption ratio and the age proﬁle of saving and housing consumption. Because
the main focus of this study is on the eﬀect of housing consumption, we accept a highsaving-to-consumption ratio for the baseline. However, we perform a counterfactual analysis
by changing the value of ψ and ν. The inverse vacancy rate from the model (h/v = 5.0)
5In the long run, these parameters may change as a result of population aging and low fertility rates.
However, we assume constant parameters because reliable estimates are unavailable for potential futurechanges in bequest motives.

22matches the data. In fact, if we remove vacant rental units from the data, the empiricalinverse vacancy rate increases to 7.14.
Ratiosh

Average P h/c among age 20-64
Average P h h/c among age 65-104
Average (a + b) /c among age 65-104
Average h/v

Target Value

Model Value

8.4
13.3
6.4
5.0

8.4
13.3
15.2
5.0

Table IV: Calibration Results

B.

Baseline Calibration
Figure 4 illustrates the age proﬁle of consumption, housing, and savings in 2019. In

contrast to the basic model for the life-cycle and permanent income hypotheses (LCH/PIH),
our model generates a hump-shaped age proﬁle of per capita consumption due to variablemortality rates (panel 4a). The consumption growth for a household of age j between t andt + 1 provides an insight into this cross-sectional age proﬁle. Our model’s Euler equation is
as follows:
cj+1,t+1= β (1 + rt+1 ) (1 − ωj,t ) .
cj,t

(31)

This equation shows that the consumption growth rate is determined not only by the standard factors (the subjective discount factor and the interest rate) but also the survival rate
(1 − ωj,t ). Younger generations increase consumption each year based on the standard factors because changes in survival rates are negligible. However, the survival rate decreases
rapidly towards the end of the lifespan. As a result, consumption growth becomes less than
one when the survival rate is signiﬁcantly low. Panel 4a shows a hump-shaped age proﬁleof per capita consumption for both the model and data. One diﬀerence is that the peakoccurs at around 50 years old in the data, whereas it occurs at around 70 years old in themodel. One possible explanation for the discrepancy is the signiﬁcant amount of educational
expenses for children when the household head is around 50 years old. Our model does notaccount for age-speciﬁc consumption goods and services, which prevents it from replicating23

this early consumption peak. However, when consumption is weighted by the cohort size,
the age proﬁle of aggregate consumption is reasonably similar between the model and data
(see panel 4b).
Housing consumption, unlike per capita consumption, increases with age, as shown inpanel 4c. This upward-sloping age proﬁle of housing consumption is also observed in many
countries, including the U.S. (Yang, 2009). In the standard two-good model without bequest
motives, the ratio of housing to non-housing consumption should be independent of age
because both consumption and housing consumption have parallel age proﬁles (Yang, 2009).
Our model replicates the realistic age proﬁle of housing by incorporating bequest motives
and age-dependent mortality rate.
Housing consumption is determined by the following ﬁrst-order condition:(1 − δh ) hhj,t= [χ + ψ (1 − ν) ωj,t ] Pth −P
cj,t(1 + rt+1 ) t+1

−1

.

(32)

As in the standard model, the ratio of housing to non-housing consumption increases with theimportance of housing χ in the felicity function and decreases with housing costs. However,unlike the standard model, equation (32) includes the product of the mortality rate ωj,t , thebequest motive parameter ψ, and the housing bequest weight (1−ν) for retirees only (1j≥JR ).
Elderly households increase their housing consumption relative to non-housing consumption
as mortality increases due to their bequest motives. Note that housing consumption is alinear transformation of housing assets in the model. Thus, the age proﬁle shown in panel
4d is based on housing assets due to data availability. The model agrees well with the data,
except for a sharp increase for households in their 30s and 40s in the data. This sharp
increase is likely due to ﬁrst-time home purchases around child births, which the model does
not account for. However, cohort-weighted aggregate age proﬁle is quite similar between themodel and the data (panel 4d).
Savings exhibit a hump-shaped age proﬁle in the standard life-cycle model because house-24(a) Per-Capita Consumption(b) Aggregate Consumption(c) Per-Capita Housing Assets(d) Aggregate Housing Assets(e) Per-Capita Financial Assets(f) Aggregate Financial Assets

Figure 4: The Age Proﬁle of Consumption, Housing and Liquid Assets: Data and Model25

holds save during a high-income period and dissave during a low-income period to smoothconsumption. In our model, liquid asset savings aj,t has that age proﬁle. However, as French
et al. (2023) discuss, retired households deccumulate their assets very slowly in the US andother countries. Our data conﬁrm this tendency even for liquid assets (panel 4e); savings
peak at the retirement age and remain at the same level throughout the remaining life. Ourmodel addresses this issue by incorporating bequest motives. The second component of savings in the form of ﬁnancial bequests bj,t increases sharply with mortality rates because of
the following intra-temporal optimality condition:bij,t= ψνωj,t .
cj,t

(33)

Thus, the sum of liquid assets aj,t and ﬁnancial bequests bj,t exhibits a high level of savingsfor the elderly. However, the data show negative savings for ages between 30 and 50. Thisborrowing corresponds to a sharp increase in housing assets. Thus, this dip in savings for
middle-aged household heads likely corresponds to mortgages for ﬁrst-time home purchases
driven by the birth of children. Because our model does not account for such mortgage
ﬁnancing or a rapid change in the household size, it deviates from the data for these agegroups.
Figure 5 contrasts the model results with the data for the dynamics of key aggregate
variables. For annual real GDP growth rates (panel 5a), we use JSNA. We estimate thestochastic trend using the HP ﬁlter, setting parameter λ equal 100. The model shows adecreasing trend in GDP growth as in the data, but the predicted change is smaller than theactual change. However, both the model and the data show no growth around 2010. Ourmodel predicts negative real GDP growth rates after the 2000s.
For the real interest rate (panel 5b), we use the prime lending interest rate adjusted for
the GDP deﬂator, provided by the World Bank. This rate corresponds to the average rate
for safe government notes and risky borrowing by ﬁrms and households, because the model26

has a single interest rate for both lending and borrowing rates in the public and private
sectors. This interest rate represents the natural rate of interest in our model, although it ishigher than the rates estimated from the risk-free rates. The model predicts decreasing realinterest rates until the 2010s, which is consistent with the data, except for the period of the
Bank of Japan’s quantitative and qualitative easing regime since 2013. From the 2010s, thenatural rate of interest is stabilized around 2%.
The capital stock ratio (panel 5c) decreases between 1980 and 2020 in both the modeland the data. The model predicts continued decline in the ratio until 2050. For household
assets, we include land but exclude pension assets to make the data consistent with themodel.
The real interest rate can be characterized by both household’s Euler equation (equation(31)) and each production sector’s ﬁrst-order condition with respect to capital:
rt + δk = α g

Ktg
ζtg Lgt

αg −1= αhKthζth Lhtαh −1

Pth .

(34)

The secular decrease in the interest rate between the 1980s and 2010s inversely reﬂects the
dynamics of capital Ktg , which peaks in 2011. The interest rate also reﬂects the seculardecrease in productivity ζtg .
Furthermore, the model demonstrates that the outcomes exhibit heterogeneous dynamics
by age. The age-speciﬁc equilibrium dynamics in the baseline calibration are discussed inAppendix H.

V.

Factor Decomposition

We specify our exogenous variables including demographic ratios between 1925 and 2100
and obtain the model outcome between 1980 and 2050.
We obtain the model outcome for all equilibrium variables: aggregates {Ct , At , Bt , Ht ,

∞
∞
g
h
h
,
prices
w
,
w
,
P
,
r
,
transfers
{T
,
pen
}
,
and
quantities{cj,t , aj,t , bj,t , bij,t , hj,t ,V t }∞
t t=0
t
t t=0
t
t
t
t=0
27

(a) Real GDP Growth(b) Real Interest Rate(c) Capital to Asset Ratio(d) Housing Sector Share of GDP

Note: Panel (a): The annual real GDP growth rate from JSNA. Panel (b): The prime lending interest rateadjusted for the GDP deﬂator from the World Bank. Panel (c): Capital stock from JSNA. The ratio is
calculated as Kt /(At + Bt + (Ht + Vt )Pth ). Panel (d): Sector share of housing sector in GDP from JSNA

Figure 5: Dynamics of Baseline Estimation: Data and Model28

hij,t , vj,t }Jj=1 , it , iht , Ktg , Kth , Lgt , Lht , Ytg , Yth∞
t=0. However, for brevity, we discuss GDP growthrate, the natural rate of interest, income tax rate, housing price, per capita housing value,and per capita ﬁnancial assets.
To gain an insight into the dynamics of the equilibrium outcome, we decompose eachvariable into the eﬀects of six exogenous factors: mortality rates, fertility rates, government
debt, government spending, technology growth rates for the general goods sector, and technology gap between the housing and general goods sectors. We “knock out” each factor ata time by ﬁxing the exogenous factor value at the 1980 level and measuring the diﬀerence
between the baseline outcome and the outcome in the knock-out case. Residuals that cannot
be attributed to any single factor are considered as interactions.
Figure 6 shows the results. Panel 6a shows that the model’s GDP growth rate decreases
by more than three percentage points between 1980 and 2050, mainly because low fertility
rates reduce the working-age population. In addition, low technology growth rates in thegeneral goods sector reduce GDP growth.
Panel 6b shows that after the 2000s, the natural rate of interest is approximately 1.7
percentage points below its 1980 level. The main factors driving the decline in interest
rates are low technological growth and low mortality and fertility rates. These eﬀects are
consistent with those found in the literature (e.g., Kitao and Mikoshiba, 2020). After 2010,
however, rising government spending ﬁnanced by public debt will create demand for capitaland raise interest rates.
In panel 6c, income tax rates increase by approximately 37 percentage points by 2050,
mainly because of declining fertility rates and increasing government spending. Note that
we do not calibrate the model tax rate to the actual income tax rate because the model
does not include other sources of government revenue. Thus, this large increase in tax rates
should not be taken as a prediction, but as the necessary government revenue to ﬁnancethe exogenously speciﬁed government spending, pension replacement rate, and government
debt. If a 37 percentage point increase in the tax rate is unacceptable to voters, the Japanese29

(a) GDP Growth Rate(b) Natural Rate of Interest(c) Tax Rate(d) Housing Price(e) Per Capita Housing Value(f) Per Capita Financial Assets

Figure 6: The Decomposition of Equilibrium Dynamics
30

government would naturally reduce spending, debt, and the pension replacement rate. To
put this rate in perspective, the tax wedge for Japan in 2023 is 32.6%, which is signiﬁcantly
lower than Belgium’s 53.0%.6 Also, the national burden rate in 2020 is 47.9% for Japan,
which is lower than the rates for the Scandinavian countries (65.9% for Denmark, 59.7% for
Finland, 54.5% for Sweden, and 53.4% for Norway).
Panel 6d shows that the model housing price increases by 51% between 1980 and 2010, butthen levels oﬀ. The main source of the increase in housing prices is the gap in TFP growth,a supply-side factor. As ﬁgure 1 shows, housing production became increasingly less eﬃcientthan other sectors until 2010, when TFP growth rates converged in the 2010s. As a result,
the relative price of housing rose steadily for three decades. Another factor contributing toprice increases is declining fertility rates, which create housing demand through population
aging. However, our model does not replicate the rapid rise and fall of housing prices during
the so-called “bubble economy” following the Plaza Accord in 1985. The actual national
residential land price index showed a 266% premium over the 1980 value at its peak in 1991,
and then declined steadily to a 33% premium in 2010. Thus, our model replicates the longterm real price changes relatively well, although it does not incorporate overly optimistic
economic forecasts or monetary easing and tightening during this period.
Panel 6e shows that per capita housing value decreases by more than 10% between the
1990s and 2010s, but recovers by 2050.7 There are competing eﬀects on house values, which
are the product of price and quantity. On the one hand, government spending has a negativeeﬀect on house values by suppressing the demand for housing. On the other hand, declining
fertility rates, which cause population aging, have positive eﬀects by increasing housing
demand.
In panel 6f, the per capita value of ﬁnancial assets (the sum of liquid assets At andﬁnancial bequests Bt ) increases by approximately 41.7% by 2020, mainly due to government
6

See OECD data for tax wedge statistics:
https://www.oecd.org/tax/tax-policy/
taxing-wages-japan.pdf
7
The per capita values of housing and ﬁnancial assets in this ﬁgure do not include the eﬀect of trend
growth, but the trend adjustment does not change the dynamics signiﬁcantly.

31

debt. In equilibrium, household savings are used to ﬁnance government debt. However, as
the population declines mainly due to low fertility rates, total assets begin to decline after
2015.

VI.

Counterfactual Analysis

We conduct eight counterfactual simulations by changing the parameters as shown in
Table V. We obtain the counterfactual values for all of the endogenous variables but discuss
only the GDP level, the consumption level, the capital level, GDP growth rates, TFP growth
rates, the natural rate of interest, house prices and quantities, and the capital share of thegeneral goods sector.
Parameter
ψ
ν
φ
Ta
Th
β
χ
δg
δh
αg
αh

Importance of bequest motives
Importance of ﬁnancial bequests
Proportion of reused housing
Inheritance tax rate for ﬁnancial assets
Inheritance tax rate forhousing
Discount factor
Importance of housing consumption
Depreciation rate for Capital
Depreciation rate for housing
Capital share in generalproduction sector
Capital share in housing
sectorBaseline

CF1

CF2

CF3

CF4

CF5

CF6

CF7

CF8

22

0

40

-

-

-

-

-

-

0.6

-

-

0.3

0.9

-

-

-

-

0.46

-

-

-

-

0

1

-

-

0.4

-

-

-

-

-

-

0

0.4

0

-

-

-

-

-

-

0

0.4

0.996
0.385

-

-

-

-

-

-

-

-

0.15

-

-

-

-

-

-

-

-

0.03

-

-

-

-

-

-

-

-

0.37

-

-

-

-

-

-

-

-

0.31

-

-

-

-

-

-

-

-

Table V: Counterfactual Parameter Values
First, we change the parameter ψ, which represents the importance of bequest motives.
Starting with the baseline value of ψ = 22, we examine 0 (no motives) and 40 (stronger
motives). Second, we change the parameter ν, the weight on the ﬁnancial bequest motive
32

within the total bequest motives. We change the parameter from the baseline value of 0.6 to
0.3 and 0.9. Third, we change the parameter φ, which represents the matching eﬃciency forintergenerational housing transfers, from the baseline value of 0.46 to 0 and 1. When φ = 0,
all bequeathed houses become vacant because the younger generation’s housing demand does
not match the inherited houses due to geographical and taste gaps. In contrast, when φ = 1,
all inherited houses are fully utilized by younger generations; there is no vacant housing
in the economy. Matching eﬃciency can be improved by helping the elderly remodel theirhouses to match young households’ tastes. Last, we change the inheritance tax rates. The
baseline rates are 0 for housing and 0.4 for ﬁnancial assets. We change them to (0, 0) and(0.4, 0.4) by removing preferential treatment of housing assets in the inheritance tax code.

A.

Bequest Motives
Figure 7 shows the results when we change the importance of bequest motives. The

bold line (CF1) corresponds to the case with no bequest motives, and the dotted line (CF2)corresponds to the case with greater bequest motives. The removal of bequest motives forboth ﬁnancial and housing assets reduces GDP by 9% to 14% (panel 7a) and the capital stockby 16% to 25% (panel 7c) because the demand for the less productive housing production is
reduced. However, consumption increases by 9% to 13% (panel 7b) because a larger share ofhousehold income is spent on general consumption after each household optimizes housing
consumption only for its own ﬂow utility.
The growth rate of GDP (panel 7d) and average TFP (panel 7e) would have been 6 − 11percentage points higher until the 1990s if households did not have bequest motives. During
this period, technological growth in the construction sector consistently lagged behind thatof the general production sector. A lower demand for housing would have reduced the shareof the housing production sector in the economy and thus increased the overall productivityof the economy. However, because we assume that technological progress in the two sectors
will converge in the future, the eﬀect of reduced housing production has a negligible impact
33

(a) GDP Level(b) Consumption Level(c) Capital Level(d) GDP Growth(e) TFP Growth(f) Natural Rate of Interest(g) House Prices(h) Per Capita Housing Value(i) Goods Sector Capital Share

Note: The ﬁgure shows deviations from the baseline dynamics for counterfactuals with ψ = 0 (no bequest
motives, labeled CF1) and ψ = 40 (large bequest motives, labeled CF2). The baseline parameter for theimportance of bequest motives is ψ = 22.

Figure 7: The Eﬀect of Bequest Motives

34

on overall TFP growth after 2020. In contrast, GDP growth is depressed by the reduced
demand for housing in the absence of bequest motives.
In the absence of bequest motives, the natural rate of interest is 2.6 percentage points
higher than the benchmark rate in 2050 (panel 7f). Because the benchmark rate is projected
to decrease from 3.6% to 1.9% over the same period, this upward adjustment will keep
the natural rate of interest relatively constant. This upward shift corresponds to a higher
marginal product of capital (MPK) due to a smaller scale of production and higher aggregate
TFP.
House prices are 16% to 26% lower without bequest motives (panel 7g); that is, housing
becomes more aﬀordable as bequest-driven housing demand is removed. In other words, thecurrent equilibrium house prices are inﬂated by the additional demand of bequest motives.
Similarly, the equilibrium level of the housing stock is 11% to 15% smaller without bequest
motives (panel 7h). The reduction in both price and quantity also conﬁrms that this change
is caused by a negative shift in demand.
Furthermore, without bequest motives, 3% to 5% more capital is allocated to the general
goods production sector (panel 7i). A reduction in the capital share of the housing sector
makes the economy less exposed to the technological gap between the two sectors.

B.

Utility Weight on Financial Bequests
Figure 8 shows the results when we change the utility weight for ﬁnancial bequests as

opposed to housing bequests. The bold line (CF3) corresponds to a smaller weight on
ﬁnancial bequests, and the dotted line (CF4) corresponds to a larger weight.
The results are qualitatively similar to those for a change in the importance of the totalbequest motives. Thus, increasing the relative importance of ﬁnancial bequests has a smaller
but analogous eﬀect to the case of decreasing the bequest motive itself. However, the effect on the natural rate of interest is opposite to that of changing bequest motives (panel8f), although the magnitude is small because housing bequests and ﬁnancial bequests are
35

(a) GDP Level(b) Consumption Level(c) Capital Level(d) GDP Growth(e) TFP Growth(f) Natural Rate of Interest(g) House Prices(h) Per Capita Housing Value(i) Goods Sector Capital Share

Note: The ﬁgure shows deviations from the baseline dynamics for counterfactuals with ν = 0.3 (a small
weight on intentional ﬁnancial bequests, labeled CF3) and ν = 0.9 (a large weight on intentional ﬁnancial
bequests, labeled CF4). The baseline weight is ν = 0.6.

Figure 8: The Eﬀect of Relative Bequest Motives

36

substitutes in generating capital demand.

C.

Matching Eﬃciency of Intergenerational Housing Transfers
Figure 9 shows the eﬀect of changing the house matching eﬃciency–the rate of successfultransfers of bequeathed houses to younger generations. The bold line (CF5) corresponds to
a completely ineﬃcient transfer–all transferred houses become vacant, and the dotted line(CF6) corresponds to a completely eﬃcient transfer–no vacancies. Eﬃcient house matching
can also be interpreted as indicating an environmentally sustainable society, in which all of
the existing stock is reused. In such a sustainable economy, production activities are reduced.
Eﬃcient house matching reduces GDP by 1.6 − 2.9% (panel 9a) because less construction
is needed. It also reduces the total capital stock employed in the production sectors (panel9c). Thus, in a sustainable society, households can enjoy 2.4 − 4.7% more consumption(panel 9b) without expanding the size of the economy or increasing capital intensity. Most
importantly, eﬃcient house matching makes housing aﬀordable: house prices are 7 − 14%lower, while more housing units are available (panels 9g and 9h). The opposite directions ofprice and quantity changes suggests that eﬃcient house matching acts as a positive supply
shift in the housing market; i.e., housing is supplied without using additional labor andcapital. Also, a smaller presence of the less productive housing production sector would
have increased GDP growth (panel 9d) and average TFP growth (panel 9e) during theperiod of stagnant construction productivity.

D.

Inheritance Tax Rates
Figure 10 shows the eﬀect of changing inheritance tax rates. We explore two possibilities

of removing the preferential treatment for housing. The bold line (CF7) corresponds to noeﬀective inheritance taxes for both assets, and the dotted line (CF8) corresponds to thesame 40% eﬀective tax rate for both assets, possibly by removing the assessment reduction
for housing.
37

(a) GDP Level(b) Consumption Level(c) Capital Level(d) GDP Growth(e) TFP Growth(f) Natural Rate of Interest(g) House Prices(h) Per Capita Housing Value(i) Goods Sector Capital Share

Note: The ﬁgure shows deviations from the baseline dynamics for counterfactuals with φ = 0 (all bequeathedhousing becomes vacant, labeled CF5) and φ = 1.0 (all bequeathed housing is used, labeled CF6). The
baseline eﬃciency rate is φ = 0.46.

Figure 9: The Eﬀect of Matching Eﬃciency

38

(a) GDP Level(b) Consumption Level(c) Capital Level(d) GDP Growth(e) TFP Growth(f) Natural Rate of Interest(g) House Prices(h) Per Capita Housing Value(i) Goods Sector Capital Share

Note: The ﬁgure shows deviations from the baseline dynamics for counterfactuals with T a = T h = 0 (noinheritance tax for both ﬁnancial assets and housing, labeled CF7) and T a = T h = 0.4 (inheritance tax rates
are 0.4 for both ﬁnancial assets and housing, labeled CF8). The baseline tax rates are T a = 0.4, T h = 0.

Figure 10: The Eﬀect of Inheritance Tax

39

The results suggest that the eﬀects are negligible; for each variable, the eﬀect is less than
one percentage point. The reason is that these tax changes do not aﬀect the household’s
decision. In our model, households cannot decide to forgo inheritance when tax rates arehigh. Thus, a change in inheritance tax rates could aﬀect the macroeconomy only throughits eﬀect on households’ after-tax wealth. For the government, however, inheritance taxes
are a substitute for income taxes. As a result, the eﬀect of a change in inheritance taxes
is oﬀset by the opposite eﬀect of a change in income taxes on household wealth. Thus, if
our model allows for households to choose whether to accept inheritances, the eﬀect will belarger. Nevertheless, our analysis suggests that a reform of inheritance tax may have only asmall eﬀect compared with changes in household preferences.

VII.

Discussions

Bequest motives for housing create additional housing demand and increase GDP. Furthermore, capital demanded by the housing sector additionally increases general goods production. Thus, housing bequest motives increase aggregate demand. The eﬀect of bequest
motives continues to increase as society ages due to low fertility and mortality rates.
However, housing bequest motives have negative eﬀects on the eﬃciency of capital usage.
First, the growth rate of GDP and TFP is lower until the 2000s than in the counterfactual case of no bequest motives because the housing sector’s productivity growth lagged thegeneral goods sector’s. The negative eﬀect of bequest motives on TFP growth largely disappears after 2020 because we assume that technology growth rates converge between the two
sectors. Thus, future aggregate TFP growth rates depend more on technological progress in
the increasingly important housing sector. The eﬀect of bequest motives on GDP growth isgreater than the eﬀect on TFP growth because housing demand increases due to the growing
elderly population.
Second, with housing bequest motives, household assets are used less for productive

40

capital than in the case of no-bequest motives, while housing services provide only small
marginal utility. Without bequest motives, households hold more liquid assets, which are
eventually invested in productive capital.
Third, greater housing production suggests the lower marginal product of capital or alower natural rate of interest in two ways; the marginal product is diminishing with more
capital employed, and the marginal product is lower in the housing sector because of lower
productivity. Thus, a reduction in housing bequest motives will increase the marginal product of capital or interest rates.
Bequest motives for housing also suppress household consumption because households
spend more on housing. Nevertheless, the elderly are satisﬁed with bequeathing housing totheir heirs, despite suppressed consumption.
Bequest motives have an obvious direct eﬀect on the housing market. They create additional demand and thus increase both price and quantity of housing. In other words,
housing is less aﬀordable for new buyers because older generations occupy their housing
underutilized.
However, as long as bequeathed houses are eﬃciently transferred to the young generation,
the crowding-out issue is minimal. One problem of housing bequest is that bequeathed houses
do not perfectly match the young generation’s housing demand. The young generation
typically has diﬀerent tastes in housing and lives in diﬀerent cities from the old generation.
In the calibrated baseline model, we incorporate such ineﬃciencies by assuming that 40% ofbequeathed houses become vacant.
With such vacancies due to generational mismatch, usable housing stock does not accumulate as quickly as in the case of no wasted housing. Thus, the reduction of wasted
housing works as an additional housing supply without the need for new housing production. Thus, a more eﬃcient intergenerational transfer of housing will decrease house prices
while increasing the aggregate housing quantity. As long as transfers are eﬃcient, housing
bequest motives have a small eﬀect on aggregate housing aﬀordability. In reality, there will41

be inequality between those who inherit housing and those who do not. We do not model
household heterogeneity in housing assets.
Improving the matching eﬃciency can be achieved by improving house remodeling. Ineﬃciencies arise from taste and geographical gaps between generations. Thus, a better
remodeling of housing at the time of intergenerational transfer will solve taste gaps. However, a geographical mismatch cannot be resolved by remodeling. To address geographicalmismatch, it would be eﬀective to facilitate multi-party exchanges beyond within-family inheritance. For example, if taxes are exempted for the transaction of an inherited house to a
local buyer, housing assets will be better utilized.
The present study provides valuable insights, yet it omits several aspects of the economy. Future extensions will explore these additional aspects. First, the study only considers age heterogeneity, while omitting other forms of heterogeneity, such as income, wealth,and household size. As Mian et al. (2021) emphasize, the demand for saving by wealthy
households has signiﬁcant implications, particularly because age and wealth are positively
correlated. Second, the study does not account for geographical heterogeneity. Even whenintergenerational transfers appear to be eﬃcient at the aggregate level, the geographical concentration of particular generations and housing assets may render transfers impracticable inreality. For instance, housing shortages for younger generations in Tokyo will not be resolved
by houses in rural areas bequeathed by older generations. Third, the study does not consider the social cost of externalities due to vacant houses and the oversupply of housing that
becomes vacant eventually. Fourth, the study speciﬁes a log-linear utility function, whichsuggests a restrictive assumption on intra- and inter-temporal consumption substitutions.
More realistic utility functions, such as the CES-CRRA utility and the Epstein-Zin recursive
utility function, can suggest a more signiﬁcant role for housing in the economy. Finally, it
should be noted that the model is a closed economy, whereas in an open economy, ﬂows ofcapital and labor are expected to increase and mitigate some of the eﬀects.

42

VIII.

Conclusion

Japan is at the forefront of aging societies. Studying the macroeconomic eﬀects of aging
in Japan will provide valuable insights into the future of the global economy. This study
focuses on the ownership of large houses by elderly households, some of which are wasted as
vacant units after being transferred to the younger generation. Ownership of large houses by
the elderly is a common phenomenon in many countries, and the housing sector tends to beless productive than other sectors. A natural prediction is that an aging society will allocate
more capital to housing assets and a less productive housing sector. We quantify the impactof this prediction by calibrating a quantitative OLG model to the Japanese economy. The
model with housing bequest motives explains well the age proﬁle of housing assets.
The model shows that bequest driven demand for housing increases GDP, but decreases
GDP and TFP growth when technological growth is slow in the housing sector. Households
consume less and allocate less resources to productive capital. The natural rate of interest islower because a larger scale of production reduces the marginal product of capital. Houses
are less aﬀordable because prices are inﬂated by the additional demand. These eﬀects areexacerbated when inherited houses are more likely to become vacant and when the proportionof elderly people is higher. The eﬃciency of the Japanese economy can be increased by
reducing bequest demand for housing and improving the eﬃciency of housing allocation.

43

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48

Appendix A

Household First-Order Conditions

A representative j-aged household at time t maximizes its utility with respect to cj,t , aj,t ,bij,t , hj,t and hij,t . The Lagrangian for j = 1, ..., J is:

Lj,t =J−j

β i Πi−1
k=0 (1 − ωj+k,t+k ) ×

i=0{(ln cj+i,t+i + χ ln hj+i,t+i ) + ωj+i,t+i ψ [ν ln bij+i,t+i + (1 − ν) ln hj+i,t+i ]hh h{hij+i,t+i + (1 − 1j+i≥JR ) Tt+i
it+i }+ λj+i,t+i cj+i,t+i + aj+i,t+i + bij+i,t+i + Pt+i
− (1 + rt+i ) aj+i−1,t+i−1 − 1j+i≥JR pent+i



 h h


g
g
g
g
a
− (1 − 1j+i≥JR ) (1 − Tt+i ) {σj+i
wt+i
zj+i
+ 1 − σj+i
wt+i zj+i } + 1 − Tt+iit+i+λhj+i,t+i hj+i,t+i − (1 − δh ) hj+i−1,t+i−1 − hij+i,t+i − φ (1 − 1j+i≥JR ) iht+i .

FOCs with respect to cj+i,t+i , aj+i,t+i , bij+i,t+i , hj+i,t+i and hij+i,t+i are respectivelyβ Πi−1k=0 (1 − ωj+k,t+k )

⇐⇒



1

i

cj+i,t+i

λj+i,t+i = −+ λj+i,t+i=01
cj+i,t+i

(A.1)

i+1 i
β i Πi−1Πk=0 (1 − ωj+k,t+k ) λj+i+1,t+i+1 (1 + rt+i+1 ) = 0
k=0 (1 − ωj+k,t+k ) λj+i,t+i − β⇐⇒λj+i,t+i = β (1 − ωj+i,t+i ) λj+i+1,t+i+1 (1 + rt+i+1 )β i Πi−1
k=0 (1 − ωj+k,t+k ) ωj+i,t+i⇐⇒ψν
bij+i,t+iωj+i,t+i(A.2)+ β i Πi−1
k=0 (1 − ωj+k,t+k ) λj+i,t+i = 0ψν
bij+i,t+i49= −λj+i,t+i(A.3)

ψ (1 − ν)β Πi−1+ β i Πi−1
k=0 (1 − ωj+k,t+k ) ωj+i,t+i
k=0 (1 − ωj+k,t+k )hj+i,t+iχi

hj+i,t+i+ λhj+i,t+i− β i+1 Πik=0 (1 − ωj+k,t+k ) λhj+i+1,t+i+1 (1 − δh ) = 0⇐⇒[ωj+i,t+i ψ (1 − ν) + χ]

1
hj+i,t+i+λhj+i,t+i − β (1 − ωj+i,t+i ) λhj+i+1,t+i+1 (1 − δh ) = 0

(A.4)



h
h
=0
(1
−
ω
)
λ
P
−
λ
β i Πi−1
j+k,t+k
j+i,t+it+i
j+i,t+ik=0⇐⇒h= λhj+i,t+i
λj+i,t+i Pt+i

(A.5)

From A.1 and A.2, we have

cj+1,t+1 = β (1 + rt+1 ) (1 − ωj,t ) cj,t f or j = 1, ..., J − 1(A.6)

From A.1, A.2, A.4 and A.5, we have[ωj,t ψ (1 − ν) + χ] cj,t = Pth −(1 − δh ) hPhj,t(1 + rt+1 ) t+1

f or j = 1, ..., J − 1[ψ (1 − ν) + χ] cJ,t = Pth −(1 − δh ) hhJ,tP(1 + rt+1 ) t+1

(A.7)

(A.8)

From A.1 and A.3, we havebij,t = ωj,t ψνcj,t f or j = 1, ..., J − 1(A.9)

biJ,t = ψνcJ,t(A.10)

50

Appendix B

List of Model Equations

This section presents the full set of model equilibrium conditions that jointly determine
the evolution of real allocations and prices. Endogenous variables are {Nj,t , cj,t , aj,t , bj,t ,bij,t , hj,t , hij,t , vj,t }Jj=1 , a0,t , b0,t , h0,t , Nt , nt , Ct , At , Bt , Ht , Vt , wtg , wth , Pth , rt , Tt , pent , it ,iht , Kt , Ktg , Kth , Lgt , Lht , Ytg , Yth , which are 705 variables. Note that ωJ,t = 1.

A

Household

Budget Constraint:

cj,t + aj,t + bij,t + Pth hij,t = (1 + rt ) aj−1,t−1 + 1j≥JR pent+ (1 − 1j≥JR ) (1 − Tt ) {σjg wtg ξjg + 1 − σjg wth ξjh } + (1 − T a ) it − T h Pth iht
f or j = 1, ..., J. (B.1)

Bequeathed Assets Accumulation:bj,t = (1 + rt ) bj−1,t−1 + bij,t f or j = 1, ..., J.

(B.2)

hj,t = (1 − δh ) hj−1,t−1 + hij,t + φ (1 − 1j≥JR ) iht f or j = 1, ..., J.

(B.3)

Housing Accumulation:

Underutilized Housing Accumulation:
vj,t = (1 − δh ) vj−1,t−1 + (1 − φ) (1 − 1j≥JR ) iht f or j = 1, ..., J.

(B.4)

Initial Holdings of Assets and Housing:b0,t = a0,t = h0,t = v0,t = 0.
51

(B.5)

Terminal Asset Holdings:
aJ,t = 0.

(B.6)

cj+1,t+1 = β (1 + rt+1 ) (1 − ωj,t ) cj,t f or j = 1, ..., J − 1.

(B.7)

FOC with respect to consumption:

FOC with respect to housing:[ωj,t ψ (1 − ν) + χ] cj,t = Pth −(1 − δh ) hP
hj,t f or j = 1, ..., J.
(1 + rt+1 ) t+1(B.8)

FOC with respect to intentional bequest:bij,t = ωj,t ψνcj,t f or j = 1, ..., J.

B

(B.9)

Demography

Survival Rate:
Nj,tNj−1,t−1= (1 − ωj−1,t−1 ) f or j = 2, ..., J.

(B.10)

N1,t− 1.
N1,t−1(B.11)

Growth Rate of Initial Young:n1,t =
Total Population:
Nt =

J


Nj,t .

(B.12)

Nt
− 1.
Nt−1

(B.13)

j=1

Population Growth:
nt =

52

C

Aggregation over households

Aggregate Consumption:
Ct =

J


Nj,t cj,t .

(B.14)

Nj,t aj,t .

(B.15)

Nj,t bj,t .

(B.16)

Nj,t hj,t .

(B.17)

Nj,t vj,t .

(B.18)

j=1

Financial Assets:
At =

J

j=1

Bequest Assets:
Bt =

J

j=1

Housing Stock:
Ht =

J

j=

Underutilized Housing Stock:
Vt =

J

j=1

Financial Bequest:
J
it = (1 + rt )j=1 (Nj,t−1 − Nj+1,t ) (aj,t−1 + bj,t−1 ).
JR
j=1 Nj,t

Housing Bequest:

J
iht = (1 − δh )D

(B.19)

j=1 (Nj,t−1 − Nj+1,t ) hj,t−1.
JR
j=1 Nj,t(B.20)

Firms

Labor Demand by the Goods Sector:
wtg = (1 − αg ) ζtg

53



Ktg
ζtg Lgtα g.

(B.21)

Labor Demand by the Housing Sector:α hwth = (1 − αh ) ζthKth
ζth Lhtαg −1

Capital Demand:
rt + δ k = α g

Ktg
ζtg Lgt

= αh

Pth .

Kth
ζth Lht(B.22)

αh −1Pth .

(B.23)

General Goods Production:
Ytg = (Ktg )αg (ζtg Lgt )1−αg .

(B.24)

1−αh
 α 
Yth = Kth h ζth Lht.

(B.25)

Housing Production:

E

Government

Government Budget Constraint:
JR−1



 g g g 
 h h  h h h JR−1ga
Tt Nj,t σj wt ξj + 1 − σj wt ξj + T Pt it + T it
Nj,t + Dt

j=1

j=1

= Gt + pentJ

Nj,t + (1 + rt ) Dt−1 . (B.26)

j=JR

Pension Determination:pent = ρt

F

wtg

JR−1
j=1




1 − σjg Nj,t ξjh
σjg Nj,t ξjg + wth JR−1
j=1
.
JR−1
j=1 Nj,t(B.27)

Market Clearing

Capital Market:
Dt + Kt = At + Bt , Kt−1 = Ktg + Kth .

54(B.28)

Labor Market in the Goods Sector:
Lgt =

JR−1


σjg Nj,t ξjg .

(B.29)

1 − σjg Nj,t ξjh .

(B.30)

j=1

Labor Market in the Housing Sector:

Lht =

JR−1




j=1

New Housing Market:
J

Nj,t hij,t = Yth .

(B.31)

j=1

General Goods Market:
Ytg = Ct + Kt − (1 − δk ) Kt−1 + Gt .

55

(B.32)

Appendix C

Data Sources

This section summarizes the data source. Table A1 shows the exogenous variables andtheir data sources. The exogenous variables other than the fertility rate are ﬁltered by the
Hodric-Prscott ﬁlter with a smoothing parameter of 6.25.
Description

PeriodSources

Fertility rate

1925 to 20192020 to 2070
2071 to 2100
1925 to 1970
1971 to 20192020 to 2070
2071 to 2100
1925 to 2100
1925 to 1971
1971 to 20192020 to 2100
1925 to 1969
1970 to 20192020 to 2100
1925 to 2100
1925 to 2100
1925 to 2100
1925 to 2100
1925 to 1979
1980 to 20192020 to 2100
1925 to 1979
1980 to 20192020 to 2100

Population Projections (Actual)
Future Population Projections (2020 Projections)
As of 2070
As of 1971
Life Table (Actual)
Future Life Table (2020 Projections)
As of 2070
Comprehensive Survey of Living Conditions (2019)
As of 1972
JIP database (Actual)
Assumed to be Zero
Assumed to be ZeroJIP database (Actual)
As of 2019
1 (Older than 65)
Assumed to be 0.4
Basic Survey on Wage Structure (2019)
Basic Survey on Wage Structure (2019)
As of 1980
JSNA (Actual)
As of 2019
As of 1980
JSNA (Actual)
As of 2019

Mortality rate

(HALE-adjusting)Labor-augumenting technology

Technology gap

Retirement indicator
Replacement rate
Age-speciﬁc proportion for sectors
Age-speciﬁc labor productivity
Government spending ratio

Government debt ratio

Table A1: List of Exogenous Variables
The HALE-Based mortality rate is calculated as follows:

HALE Based M otarity Ratej,t =κj+1
ωj,t ,
κj(C.1)

where κj is the percentage of persons whose disability aﬀects their daily life in the age-jcohort.
The labor-augmenting technology in our Cobb-Douglas production function corresponds
to productivity per worker, unlike productivity per quality-adjusted hour worked in the JIP
56

database. Thus, we calculate the growth rate of labor-augmenting technology as follows:

ΔLabor Augmenting T echnology =[1 + ΔT F P + (1 − α) (ΔLabor Quality + ΔM an hour − ΔN umber of W orker)]1/(1−α) − 1,
(C.2)
where Δ indicates the annual rate of change for each period.
In our model calibration, we match the ratio of housing value to consumption. Housing
wealth in our model conceptually includes both rental and owneroccupied housing (moreprecisely, there is no distinction between owners and renters). However, the measured grosshousing wealth in the National Household Income and Expenditure Structure Survey is based
only on owner-occupied housing. Thus, we adjust the per capita housing value in the databy multiplying a factor of 1.58, the inverse proportion of owner-occupied housing from the
National Housing and Land Survey (2018). Table A2 shows the results.
The per capita value of each variable is calculated as follows to correct for the numberof household members:

V alue per capita =ValuablehP hc(a + b)v

V alue per household(N umber of people per household)1/2

.

(C.3)

Description

SourceCurrent residence/place of residence
Consumer spending
Net ﬁnancial assets (savings - debt)
Other than current residence/place of residence

National Household Income and Expenditure
Structure Survey (2019)

Table A2: List of Calibration Target

57

Appendix D

Exogenous Variables

Figures A1 to A4 show the graphs for each of the exogenous variables. The speciﬁcation
methodology is elaborated in IV.A.

(a) Fertility Rate (Growth Rate of 20-year-olds)(b) Mortality Rate (over the Next 1-Year)

Figure A1: Demographics(a) Technology Growth in General Goods Sector (b) Technology Gap Between the Housing and General Goods Sectors

Figure A2: Technology

58

(a) Government Spending(b) Government Debt

Figure A3: Fiscal Condition(a) Labor Properties(b) Total Labor

Figure A4: Labor

59

Appendix E

Solution Method

We follow Bielecki et al. (2020) to solve our model. We ﬁrst detrend the model (AppendixF) and estimate the steady state value for the initial (1925) and terminal (2095) states.
Although we use the scenarios of our demographic and other exogenous forces from 1925 to
2095 (consisting of artiﬁcial data, past data and projections), our focus is on developments
1980 to 2050.
A steady state is deﬁned as a stationary equilibrium where the exogenous demographic
variables assume a ﬁxed value forever. A steady state value is calculated by the following
algorithm.
1. We set the initial value for the aggregate variables as in Table A3.

Initial values

K̂ g
Lg

K̂ h
Lh

3

1.5

î
2.0

îh
4.0

pen
ˆ
0.2

Table A3: Intial Value of the Algorithm

2. Calculate the factor prices ŵg , ŵh , P h , r and T .
3. Calculate the sum of lifetime discount factors and calculate the NPV of lifetime labor
income and pension.
4. Calculate the initial consumption as in Appendix G and derive a sequence of otherJˆ j , ĥj , ĥij.
variables {Nj , ĉj , âj , bi
j=1

g

h

5. Evaluate the convergence of K̂, î, îh and pen.
ˆ If the convergence criteria are not
, K̂
Lg
Lh
met, guess the new values and repeat the process.
After deriving the initial and terminal states, we perform our deterministic simulation(with perfect foresight) for the entire period using MATLAB and Dynare.

60

Appendix F

Detrending

We detrend the model by deﬂating all variables by the TFP ζtg . We deﬁne x̂t ≡ xt /ζtg forx = cj , aj , bij , bj , hij , hj , vj , C, A, B, H, V, wg , wh , pen, i, ih , K, K g , K h , Y g , Y h . The model
equations listed in Appendix B are rewritten as follows.
gˆ j,t + P h ĥij,t = (1 + rt ) âj−1,t−1 + 1j≥JR pen
âj,t + biˆ tĉj,t + 1 + gt+1
t




+ (1 − 1j≥JR ) (1 − Tt ) {σjg ŵtg ξjg + 1 − σjg ŵth ξjh } + (1 − T a ) ît − T h Pth îht
f or j = 1, ..., J,gˆ j,t f or j = 1, ..., J,b̂j,t = (1 + rt ) b̂j−1,t−1 + bi
1 + gt+1gĥj,t = (1 − δh ) ĥj−1,t−1 + ĥij,t + φ (1 − 1j≥JR ) îht f or j = 1, ..., J,
1 + gt+1gv̂j,t = (1 − δh ) v̂j−1,t−1 + (1 − φ) (1 − 1j≥JR ) îht f or j = 1, ..., J,
1 + gt+1(F.4)

b̂0,t = â0,t = ĥ0,t = v̂0,t = 0,(F.5)

âJ,t = 0,gĉj+1,t+1 = β (1 + rt+1 ) (1 − ωj,t ) ĉj,t f or j = 1, ..., J − 1,
1 + gt+1(F.6)


(1 − δh ) h g
Pt+1 1 + gt+1ĥj,t f or j = 1, ..., J − 1,
(1 + rt+1 )(1 − δh ) h g[ψ (1 − ν) + χ] ĉJ,t = Pth −Pt+1 1 + gt+1ĥJ,t ,
(1 + rt+1 )[ωj,t ψ (1 − ν) + χ] ĉj,t = Pth −

(F.1)
(F.2)
(F.3)

(F.7)
(F.8)
(F.9)

ˆ j,t = ωj,t ψνĉj,t f or j = 1, ..., J − 1,bi

(F.10)

ˆ J,t = ψνĉJ,t ,bi(F.11)Nj,t= (1 − ωj−1,t−1 ) f or j = 2, ..., J,
Nj−1,t−1
N1,t− 1,
n1,t =
N1,t−1JNt =Nj,t ,(F.12)(F.13)(F.14)

j=1

nt =

Nt
− 1,
Nt−1

(F.15)

61

Ĉt =

J

Nj,t ĉj,t ,(F.16)

Nj,t âj,t ,(F.17)

Nj,t b̂j,t ,(F.18)

Nj,t ĥj,t ,(F.19)

Nj,t v̂j,t ,(F.20)

j=1

Ât =

J

j=1

B̂t =

J

j=1

Ĥt =

J

j=1

V̂t =

J

j=1



j=1 (Nj,t−1 − Nj+1,t ) âj,t−1 + b̂j,t−1ît = (1 + rt ),JR
j=1 Nj,tJ
j=1 (Nj,t−1 − Nj+1,t ) ĥj,t−1h,ît = (1 − δh )
JR
N
j,t
j=1
 αg

g
K̂t
,
ŵtg = (1 − αg )
Lgt

 αh
h
K̂
t
ŵth = (1 − αh ) (1 − θt )Pth ,(1 − θt ) Lhtαg −1αh −1K̂tg
K̂th
rt + δ k = α g
= αh
Pth ,
Lgt
(1 − θt ) Lht αggŶt = K̂tg
(Lgt )1−αg , αh 1−αhhŶt = K̂th,
(1 − θt ) Lht
JR−1


J

(F.21)(F.22)(F.23)(F.24)(F.25)(F.26)(F.27)

 JR−1 g g g 
 h h  h h h


g
g
a
Dt
Tt Nj,t σj ŵt ξj + 1 − σj ŵt ξj + T Pt ît + T ît
Nj,t + 1 + gt+1

j=1

j=1

= Gt + pen
ˆ t

J


Nj,t + (1 + rt ) Dt−1 ,(F.28)


Nj,t σjg ŵtg ξjg + 1 − σjg ŵth ξjh,JR−1
j=1 Nj,t(F.29)

j=JR

JR−1
pen
ˆ t = ρtj=1

Dt + K̂t = Ât + B̂t , K̂t−1 = K̂tg + K̂th ,(F.30)

62

Lgt =JR−1
σjg Nj,t ξjg ,(F.31)

j=1

Lht =

JR−1





1 − σjg Nj,t ξjh ,(F.32)

j=1
J

Nj,t ĥij,t = Ŷth ,(F.33)j=1gK̂t − (1 − δk ) K̂t−1 + Gt .
Ŷtg = Ĉt + 1 + gt+1

63(F.34)

Appendix GSet ŵj = (1 − 1j≥JR ) (1 − T )Initial Value Calculation
σjg ŵg zjg +1 − σjgŵh zjhahh h+ (1 − T ) î − P T î1+r
ˆ and 1 + rg = 1+g+1j≥JR pen
g . Then, the household budget constraint can be rewritten as:ˆ J + P h ĥiJ = ŵJ + (1 + r) âJ−1ĉJ + (1 + g g ) âJ + bi
ˆ J + P h (1 + g g ) ĥJ − (1 − δh ) ĥJ−1 − φ (1 − 1J≥JR ) îh⇐⇒ ĉJ + bi
ˆ J−1 − P h ĥiJ−1 + ŵJ−1 + (1 + r) âJ−2=ŵJ + (1 + rg) −ĉJ−1 − bi

ˆ J + (1 + rg) bi
ˆ J−1 + P h (1 + g g ) ĥJ − P h (1 − δh ) ĥJ−1⇐⇒ ĉJ + (1 + rg) ĉJ−1 + bi


h
g
h
+ (1 + rg) P (1 + g ) ĥJ−1 − (1 − δh ) ĥJ−2 − φ (1 − 1J−1≥JR ) î=ŵJ + (1 + rg) ŵJ−1 + P h φîh (1 − 1J≥JR ) + (1 + rg) (1 + r) aJ−2

ˆ J + (1 + rg) bi
ˆ J−1 + (1 + rg)2 b̂iJ−2⇐⇒ ĉJ + (1 + rg) ĉJ−1 + (1 + rg)2 ĉJ−2 + bi


2
h
g
+ P (1 + g ) ĥJ + (1 + rg) ĥJ−1 + (1 + rg) ĥJ−22h− P (1 − δh ) ĥJ−1 + (1 + rg) ĥJ−2 + (1 + rg) ĥJ−3=ŵJ + (1 + rg) ŵJ−1 + (1 + rg)2 ŵJ−2+ P h φîh (1 − 1J≥JR ) + (1 + rg) (1 − 1J−1≥JR ) + (1 + rg)2 (1 − 1J−2≥JR )+ (1 + rg)2 (1 + r) âJ−364⇐⇒ ĉJ + (1 + rg) ĉJ−1 + · · · + (1 + rg)J−1 ĉ1ˆ J + (1 + rg) bi
ˆ J−1 + · · · + (1 + rg)J−1 bi
ˆ1
+ bi


+ P h (1 + g g ) ĥJ + (1 + rg) ĥJ−1 + · · · + (1 + rg)J−1 ĥ1− P h (1 − δh ) ĥJ−1 + (1 + rg) ĥJ−2 + · · · + (1 + rg)J−1 ĥ0=ŵJ + (1 + rg) ŵJ−1 + · · · + (1 + rg)J−1 ŵ1


J−1
h h
+ P φî (1 − 1J≥JR ) + (1 + rg) (1 − 1J−1≥JR ) + · · · + (1 + rg)(1 − 11≥JR )+ (1 + rg)J−1 (1 + r) â0 .

Recalling that ĥj =(1+rg){ωj ψ(1−ν)+χ}ˆ j = ωj ψνĉjĉj and bi(r+δh )P h

f or j = 1, ..., J − 1, we haveĉJ + (1 + rg) ĉJ−1 + · · · + (1 + rg)J−1 ĉ1+ ψν ĉJ + (1 + rg) ωJ−1 ĉJ−1 + · · · + (1 + rg)J−1 ω1 ĉ11+r[{ψ (1 − ν) + χ} ĉJ + (1 + rg) {ωJ−1 ψ (1 − ν) + χ} ĉJ−1
r + δh
J−1+ · · · + (1 + rg){ω1 ψ (1 − ν) + χ} ĉ1
1 − δhJ−1−(1 + rg) {ωJ−1 ψ (1 − ν) + χ} ĉJ−1 + · · · + (1 + rg)(ω1 ψ (1 − ν) + χ} ĉ1r + δh+=ŵJ + (1 + rg) ŵJ−1 + · · · + (1 + rg)J−1 ŵ1


J−1
h h
+ P φî (1 − 1J≥JR ) + (1 + rg) (1 − 1J−1≥JR ) + · · · + (1 + rg)(1 − 11≥JR )


⇐⇒

1+ ν+

1+r
1+r
(1 − ν) ψ +χ ĉJr + δhr + δh+ (1 + rg) {1 + ωJ−1 ψ + χ} ĉJ−1 + · · · + (1 + rg)J−1 {1 + ω1 ψ + χ} ĉ1=ŵJ + (1 + rg) ŵJ−1 + · · · + (1 + rg)J−1 ŵ1+ P h φîh (1 − 1J≥JR ) + (1 + rg) (1 − 1J−1≥JR ) + · · · + (1 + rg)J−1 (1 − 11≥JR ) .

65

Further recalling that ĉj+1 = β (1 + rg) (1 − ωj ) ĉj f or j = 1, ..., J − 1, we have


1+r
1+r
ψ (1 − ν) +
χr + δhr + δh+β J−2 (1 − ωJ−2 ) · · · (1 − ω1 ) {1 + ωJ−1 ψ + χ} + · · · + β (1 − ω1 ) {1 + ω1 ψ + χ} ĉ1(1 + rg)

J−1

β

J−1

(1 − ωJ−1 ) (1 − ωJ−2 ) · · · (1 − ω1 ) 1 + ψν +=ŵJ + (1 + rg) ŵJ−1 + · · · + (1 + rg)J−1 ŵ1+ P h φîh (1 − 1J≥JR ) + (1 + rg) (1 − 1J−1≥JR ) + · · · + (1 + rg)J−1 (1 − 11≥JR )
⇐⇒ (1 + rg)J−1 β J−1
+

J−1

j=1

=

J



β j−1

 j−1


J−1

k=1


(1 − ωk )1+r
1+r
ψ (1 − ν) +
χr + δhr + δh

1 + ψν +

(1 − ωk ) (1 + ωj ψ + χ)

ĉ1

k=1



(1 + rg)J−j ŵj + P h φîh (1 − 1j≥JR ) .

j=1

The initial value of detrended consumption ĉ1 is written as a function of the steady state
prices.

66

Appendix H

Equilibrium Dynamics by Age

Figure A5 shows the age-speciﬁc dynamics of four detrended per-capita variables in thebaseline case: ﬁnancial assets (a), consumption (b), housing (c), and vacant housing (d). If
aggregate TFP growth is added back, non-zero TFP growth rates between 1980 and 2019 will
tilt the graphs. There is signiﬁcant cross-sectional variation by age (ages 25, 45, 65, and 85).
The 2019 values across ages correspond to ﬁgure 4, and the dynamics of population-weighted
average correspond to ﬁgure 6.
In panel (a), per capita ﬁnancial assets are negative for 25 years old throughout thesimulation period, consistent with the standard life-cycle model. In contrast, an 85 years old
person has the largest amount of ﬁnancial assets, partly for bequest motives.
In panel (b), per capita consumption consistently decreases after 1990 for 25, 45, and65 years old, whereas it increases for the 85 years old. In 1980, the peak of per capita
consumption was much earlier in life than in 2050. Thus, the 85 years old already reduced
consumption signiﬁcantly because of increasing mortality rates. In 2050, however, the 85
years old will not be as old as in 1980 because of longevity. As the consumption peak isdelayed, per capita consumption for 85 years old increases.
In panel (c), per capita housing consistently decreases after 1990 for 25, 45, and 65
years old because of lower real income. In contrast, housing for 85 years old is more stable
over time because older households maintain large housing for bequest motives. Thus, theupward-sloping age proﬁle of housing is exacerbated by population aging.
In panel (d), the largest amount of vacant housing is owned by 65 years old because a
proportion φ of the inherited housing accumulates continuously between ages 20 and 64.
Vacant housing increases rapidly from 2020 because inherited housing increases over time as
population ages.

67

(a) Per Capita Financial Assets(b) Per Capita Consumption(c) Per Capita Housing(d) Per Capita Vacant Housing

Figure A5: The Dynamics of Equilibrium Variables by Age

68Appendix I

The User Cost of Housing

The present model speciﬁcation is equivalent to an alternative one that includes periodic
user (rental) costs but not purchase costs because there are no market imperfections such asasymmetric information and incomplete contracts. The user cost rate equals the total gross
 h
h
Pt .
return to housing assets 1 + rt + δh less the house price appreciation rate Pt+1
Figure A6 shows the user cost rate as the diﬀerence between the total gross interest rate
(rt + δh ) and hosing price growth rate in the model. The implied user cost rate is approximately 8 − 10%, which is higher than the observed “cap rate” for residential properties.8
This is because our model omits the land component of housing. Yoshida (2020) estimatesthat land accounts for 48 − 87% of housing value in Tokyo and 28 − 70% outside Tokyo, depending on the age of the structure. If, for example, non-depreciating land accounts for 60%
of housing value, the implied depreciation rate of housing will be 4.5 percentage points lower.
Thus, our model predicts that the observed cap rate would be approximately 3.5 − 5.5%,
which is consistent with the data. The user cost was low in the late 1980s and early 1990s
because of the large house price appreciation, but it increased by 2020 because of the lower
house price appreciation rate.

8

The cap rate for real estate investment corresponds to the user cost rate in our model, which does notinclude maintenance, insurance, and property taxes. The cap rate equals the ratio of net operating income
to the property value.

69

Figure A6: Gross Interest Rate and House Appreciation Returns

70