Discussion Papers In Economics
And Business

Politics of Public Education and Pension with Endogenous
Fertility

YUKI UCHIDA, TETSUO ONO

Discussion Paper 24-07

May 2024

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANPolitics of Public Education and Pension with Endogenous
Fertility∗
Yuki Uchida†

Tetsuo Ono‡

Seikei University

The University of Osaka

May 9, 2024

Abstract
Implications of increased life expectancy on parental fertility decisions and subsequent
shifts in political inﬂuence between younger and older generations carry signiﬁcant consequences for government policies concerning education and pension. This study introduces
an overlapping generations growth model incorporating these eﬀects, qualitatively indicating that increased life expectancy correlates with lower fertility rates, decreased education
expenditure-GDP ratio, and increased pension beneﬁt-GDP ratio. A model simulation evaluates the impact of the projected increase in life expectancy until 2100 on four countrygroups: synthetic rich OECD, synthetic rich OECD Europe, Japan, and the United States.
The ﬁndings demonstrate similar trends as in the qualitative analysis, yet growth rates areprojected to vary signiﬁcantly across regions and countries due to diﬀering life expectancy
increases.
• Keywords: Fertility; Public Pension; Public Education; Probabilistic Voting; Overlapping Generations• JEL Classiﬁcation: D70, E62, H52, H55∗

The authors thank the seminar participants at Kobe University for their valuable comments. Additionally, theauthors acknowledge the ﬁnancial support received from the Japan Society for the Promotion of Science througha Grant-in-Aid for Early-Career Scientists (No.18K12802, Uchida) and a Grant-in-Aid for Scientiﬁc Research (C)(No.21K01539, Ono).
†
Yuki Uchida: Faculty of Economics, Seikei University, 3-3-1 Kichijoji-Kitamachi, Musashino, Tokyo 180-8633,
Japan. E-mail: yuchida@econ.seikei.ac.jp.
‡
Tetsuo Ono: Graduate School of Economics, The University of Osaka, 1-7, Machikaneyama, Toyonaka, Osaka
560-0043, Japan. E-mail: tono@econ.osaka-u.ac.jp.

1

Introduction
The declining birth rates and increasing life expectancy observed in most Organization for

Economic Cooperation and Development (OECD) member countries in recent decades haveincreased the share of older adults in the voting population (OECD, 2016). This trend is projected to continue over the next few decades (Rouzet et al., 2019). Accordingly, pension beneﬁts
for older adults are expected to increase (Gonzalez-Eiras and Niepelt, 2012), while government spending on schemes that may not directly beneﬁt older adults, such as public education(Poterba, 1997; Cattaneo and Wolter, 2009) could decrease. Simultaneously, an aging population
reduces the willingness of the working middle-aged population to pay higher taxes to meet the
government’s growing pension burden (Razin et al., 2002). However, older adults may not object
to education spending because of altruistic concerns for the younger generations or because suchspending may enhance productivity and ensure a higher level of tax revenues (Gradstein andKaganovich, 2004). Therefore, these opposing eﬀects lead to the following question: how does
the government allocate its limited budget to provide pension for older adults and education forthe younger generation in response to population aging?
Recent studies on this topic include Gonzalez-Eiras and Niepelt (2012), Lancia and Russo(2016), Ono and Uchida (2016), and Bishnu and Wang (2017). These studies have used probabilistic voting (Lindbeck and Weibull, 1987; Persson and Tabellini, 2000) to describe intergenerational conﬂicts in the allocation of government revenue among the public for pension andeducation. In this voting environment, each oﬃce-seeking candidate proposes a policy platform to maximize the probability of winning elections, resulting in the selection of policies thatmaximize an objective function that weights the utility of each generation by its share of the
population.
The aforementioned studies have assumed exogenous fertility and examined the impact of its
exogenous decline on equilibrium policy and resulting growth eﬀects across periods. However,fertility is endogenously determined from the optimizing behavior of households (Becker, 1991).
Particularly, an increase in life expectancy aﬀects fertility decisions of households (Ehrlich andLui, 1991; Zhang et al., 2001; Zhang and Zhang, 2005). This consequently aﬀects the population share or political weight of older adults during the subsequent period. Thus, as emphasized
by Gonzalez-Eiras and Niepelt (2012) and Bishnu and Wang (2017), the interaction of pension and education policy choices with fertility decisions is an important issue in analyzing thedetermination of ﬁscal policies and assessing its impact on economic growth.1
To demonstrate the interaction between the determination of education and pension policies
1

Gonzalez-Eiras and Niepelt (2012) say: “With endogenous fertility, the demographic structure would turninto an endogenous state variable, rendering an analytical solution of the policy game considered in the present
paper infeasible. ... We leave an analysis of these feedback eﬀects for future research.” Bishnu and Wang (2017)
add: “As the intergenerational distribution of political power is tied to the demographic change, which in turn isdetermined by the changing pattern of fertility and longevity, a natural extension of this study is to accommodate
individual choice of fertility and longevity. We leave this for future study.”1and parents’ decisions on fertility, we utilize the overlapping-generation model with physicaland human capital accumulation developed by Gonzalez-Eiras and Niepelt (2012) and Ono andUchida (2016). We follow de la Croix and Doepke (2004) and extend the model by introducing
the decisions regarding children, developed by Becker and Lewis (1973). Speciﬁcally, parents careabout consumption, the number of children, and the human capital of their children. Parents
vote for public education that aﬀects the formation of their children’s human capital and pension
provisions that beneﬁt retired older adults. Parents spend a part of their lives raising theirchildren. Given the education and pension policies, parents choose consumption, savings, andthe number of children, to maximize their lifetime utility.2
In this framework, our study uncovers the dynamic interplay between current fertility choices
and the determination of future pension beneﬁts. Firstly, current fertility choices inﬂuence thepolitical determination of future pension beneﬁts by altering the demographic composition of
future older adults. Additionally, these choices shape the labor supply of parents, impacting equilibrium market wages. Consequently, these wage ﬂuctuations propagate through theeconomy, aﬀecting savings and physical capital accumulation, thus inﬂuencing future pension
beneﬁts. Conversely, expected future pension beneﬁts reciprocally inﬂuence the fertility choicesof current parents.
By emphasizing the interrelationship between fertility choices and pension beneﬁts, which
has been underrepresented in the literature, we show that increased life expectancy directly
aﬀects parents’ fertility decisions for a given set of policy variables, as shown by Ehrlich and Lui(1991) and the literature that follows them. We also show that increased life expectancy has anindirect eﬀect through the political decisions regarding pension beneﬁts for older adults, which is
new to the literature. An indirect eﬀect occurs through the following four routes: First, increased
life expectancy raises the political weight of older adults, increasing pension beneﬁts. Second, anincrease in the number of older adults reduces pension beneﬁts per old-aged individuals. Third,
increased life expectancy incentivizes middle-aged individuals to save, leading to a lower labor
income tax rate and, in turn, lower pension beneﬁts. Fourth, increased savings lead to higher
future pension beneﬁts through physical capital accumulation. Thus, increased life expectancy
positively inﬂuences fertility through the ﬁrst and fourth routes but negatively impacts throughthe second and third. Overall, the net impact on fertility through public pension is negative inthe present framework.
We also demonstrate that increased life expectancy leads to an increase in the pension beneﬁtGDP ratio and a decrease in the education expenditure-GDP ratio. The political dimension ispivotal in this process: An increase in life expectancy and the resulting decline in fertility work
to raise the political weight of older adults. Consequently, the government prioritizes enhancing
pension beneﬁts for older adults over expenditure on education for the young. Furthermore,declining fertility boosts the capital equipment available for working middle-aged individuals,
2Life expectancy could be controllable through health investment (Grossman, 1972). However, in this study,
we assume it to be exogenous, and focus on the interaction between fertility and policy choices.

2

stimulating a higher growth rate.
To further investigate the eﬀects of the projected increase in life expectancy, we follow
Gonzalez-Eiras and Niepelt (2012) and employ a model simulation that incorporates life expectancy data up to 2100 for four distinct groups of countries: synthetic rich OECD (encompassing all included countries), synthetic rich OECD Europe (comprising exclusively European
countries), Japan, and the United States.3 Our simulation predicts a decline in the fertility
rate and the education expenditure-GDP ratio, accompanied by an increase in the pension
beneﬁt-GDP ratio. These projections are consistent with qualitative predictions derived fromthe model. However, the growth rate is expected to decrease for the rich OECD and experiencea slight decrease for the rich OECD Europe and Japan, with a temporary decline observed in the
United States. This ﬁnding contrasts with the comparative statics analysis mentioned earlier,suggesting that varying degrees of increase in life expectancy between periods lead to divergent
forecasts of the growth rate.
To evaluate the baseline case mentioned above, we develop and compare two alternative
scenarios. The ﬁrst involves a policy that maintains the pension-GDP ratio at the 2020 level forthe future, eﬀectively implying a reduction in pension expenditure. Similar to the baseline case,this scenario forecasts a decline in the fertility rate, albeit at a more rapid pace. Moreover, unlike
the baseline case, the ﬁxed pension beneﬁt-GDP ratio helps evade an increase in the tax burden,
thus indicating a tendency for the economic growth rate to increase. Therefore, implementing a
policy of ﬁxing the pension beneﬁt-GDP ratio entails a trade-oﬀ between fertility and growth.
The second scenario omits individual fertility choices from the model and incorporates projection data from the United Nations World Population Prospects. Like the baseline case, thepresent scenario indicates a trend toward lower fertility rates, higher pension beneﬁt-GDP ratios, and lower education expenditure-GDP ratios. However, these changes are rapid comparedto the baseline case. In addition, the present scenario reveals a signiﬁcant increase in the growthrate. The result implies that endogenous fertility plays an essential role in the pace of ﬁscal
policy changes and economic growth.
The remainder of this study is organized as follows. The next section reviews the related
literature. Section 3 describes the proposed model. Section 4 characterizes political equilibrium.
Section 5 presents a model-based simulation to predict changes in ﬁscal policies, fertility rates,and growth rates over time in response to projected improvements in life expectancy. Section 6
concludes with brief remarks. All the proofs are provided in the Appendix.

2

Related Literature
The literature on public education and pensions began with Pogue and Sgontz (1977), whoshow that pay-as-you-go (PAYG) social security incentivizes public investment in education.
3

We include the following countries into the set of the rich OECD countries: Australia, Canada, Denmark,France, Germany, Italy, Japan, New Zealand, Sweden, the United Kingdom, and the United States.

3

Such an incentive has also been indicated by Becker and Murphy (1988), who have demonstrated the role of PAYG social security in garnering political support from the current working
population for public investment in education. Subsequent studies by Cremer et al. (1992),
Kaganovich and Zilcha (1999), Pecchenino and Utendorf (1999), Boldrin and Montes (2005),
Poutvaara (2006), Cremer et al. (2011), and Andersen and Bhattacharya (2017) have focusedon how households behave when public education and pensions are provided by the government.
Therefore, decisions on these policies through voting are abstracted away from their analyses.
Early studies on the political economy of public education and pensions include those by
Bearse et al. (2001), Soares (2006), Iturbe-Ormaetxe and Valera (2012), Kaganovich and Meier(2012), Kaganovich and Zilcha (2012), and Naito (2012). A common feature of these studies isthat the two-dimensional voting aspect is reduced to one dimension to simplify the analysis. In
other words, they consider a vote on public education for a given pension beneﬁt, or a vote on
the allocation of tax revenue for a given tax rate. Therefore, these studies do not indicate how
the size of the government (i.e., the tax rate) and the allocation of government spending between
education and pensions are jointly determined through voting in the presence of generational
conﬂict.
This problem is resolved by introducing two-dimensional voting based on altruism (Tabellini,1991), party competition (Levy, 2005), issue-by-issue voting (Poutvaara, 2006), and reputation(Bellettini and Ceroni, 1999; Boldrin and Rustichini, 2000; Rangel, 2003). However, these studies have abstracted physical and/or human capital formation, and thus, have not examined theinteraction between policy and capital formation. The capital formation was introduced by
Kemnitz (2000), Gradstein and Kaganovich (2004), Holtz-Eakin et al. (2004), Tosun (2008),
and Bernasconi and Profeta (2012). These studies have assumed myopic voting, in which current voters consider future policy as a given. In other words, the forward-looking decisions ofvoters are absent in the analysis of these studies. Therefore, they abstract from the feedback
mechanism between current and future redistribution policies through physical and/or humancapital accumulation, which plays a crucial role in shaping ﬁscal policies.
The feedback mechanism is demonstrated by Beauchemin (1998), Forni (2005), Bassetto(2008), Mateos-Planas (2008), Gonzalez-Eiras and Niepelt (2012), Song (2011), Chen and Song(2014), and Arcalean (2018).4 In particular, the present study is closely related to GonzalezEiras and Niepelt (2012), Lancia and Russo (2016), Ono and Uchida (2016), and Bishnu andWang (2017), who have analyzed the politics of public education and pensions in the presenceof a feedback mechanism in the overlapping generations model. Among these studies, GonzalezEiras and Niepelt (2012) and Ono and Uchida (2016) have explored the eﬀects of exogenously
declining population growth rates on policy choices. Nonetheless, they have overlooked the
dynamic interplay between policies and fertility decisions triggered by shifts in life expectancy.
4

The studies of multiple policy instruments other than education spending in the presence of a feedback
mechanism include Hassler et al. (2003, 2005, 2007); Arawatari and Ono (2009, 2013); Song et al. (2012); Müller
et al. (2016); Röhrs (2016); Arai et al. (2018); and Uchida and Ono (2021).

4

This study makes a valuable contribution to the existing literature by emphasizing the pivotal
role of this interaction in evaluating the consequences of aging on policy determinations andeconomic growth.
This study contributes to the literature on aging and intergenerational conﬂict over policymaking through probabilistic voting, from a methodological perspective (Grossman and Helpman, 1998, Hassler et al., 2005, Gonzalez-Eiras and Niepelt, 2008, Song, 2011, Song et al., 2012,
Arai et al., 2018 and Uchida and Ono, 2021). The study is, to the best of our knowledge, theﬁrst to obtain a closed-form solution of policy functions in a dynamic setting with endogenous
fertility. de la Croix and Doepke (2009) and Kimura and Yasui (2009) have analyzed the politics of education when fertility is endogenous. However, their models are static in nature andthus, assume away an intertemporal interaction between fertility and policy choices via physical/human capital accumulation. The present study overcomes this limitation and demonstrates
the dynamic impact of fertility on policy decisions and the resulting resource allocation acrossgenerations.
This study also contributes to the literature on optimal pension and education policy. Andersen and Bhattacharya (2017), Bishnu et al. (2021, 2023) and Amol et al. (2022) have shown apension-education package that attains Pareto improving or optimal allocation in a dynamicallyeﬃcient economy. In particular, Amol et al. (2022) and Bishnu et al. (2023) have presented such a
policy package in an environment where fertility is endogenous. Our study shares a concern with
theirs in exploring pension and education policies in an endogenous fertility setting, but diﬀers
from theirs with respect to the time horizon of the government’s implementation of the policies.
Their analyses and results have relied on the implicit assumption that an inﬁnitely-lived government can calculate and implement a Pareto-improving or optimal redistribution of resources
across generations. This is a common assumption in analyses of decentralization in competitive
equilibrium. In contrast, our study, which belongs to the political economy literature, assumes
that while the current population can compute an intergenerational redistribution of resources,
there is no inﬁnitely lived government that can commit to such a redistribution. In other words,
an intergenerational resource reallocation can be performed only by a short-lived governmentthat represents successive generations living in a current period, and this government can only
reallocate resources in that period.

3

Model
The discrete-time economy starts in period 0 and consists of overlapping generations. Indi-

viduals are identical within a generation and live at most for the following three periods: young,middle, and old age. They face uncertainties in the third period of life. Let π ∈ [0, 1] denote thelife expectancy (i.e., the probability of living in old age). This is considered idiosyncratic for all
individuals and is constant across periods. Each middle-aged individual gives birth to a numberof children denoted by n. The number of middle-aged individuals in period t is represented by5

Nt , and the population grows at a rate of nt+1 following the equation Nt+1 = nt+1 Nt . The grosspopulation growth rate, nt+1 , is determined by the fertility decisions of the period-t middle-aged
individuals.

3.1

Individuals

Individuals exhibit the following economic behavior during their life cycle. In their youth,
individuals do not make any economic decisions and depend on their parents for their livelihood.
In middle age, individuals work, receive market wages, pay taxes, and make fertility and saving
decisions. In old age, they retire and consume returns from savings.
We consider middle-aged individuals in period t. Each of them is endowed with one unit oftime. Raising one child takes fraction ϕ ∈ (0, 1) of time. Each individual devotes ϕnt+1 unitsof time to raising children and supplies the remaining time, 1 − ϕnt+1 , to the labor market.
Each middle-aged individual obtains labor-income (1 − ϕnt+1 ) wt ht , where wt is the wage rate
per unit of labor and ht is the human capital endowment. After paying tax, τt wt ht (1 − ϕnt+1 ),where τt is the period t labor-income tax rate, the individual distributes the after-tax income
between consumption ct and savings held as an annuity and invested in physical capital, st .
Therefore, the period-t budget constraint for each middle-aged individual becomes
ct + st ≤ (1 − τt )wt ht (1 − ϕnt+1 ) .

(1)The period t + 1 budget constraint in old age is
dt+1 ≤

Rt+1
st + bt+1 ,π(2)where dt+1 is consumption, Rt+1 is the gross return from savings, and bt+1 is the pay-as-yougo public pension beneﬁt. If an individual dies at the end of the middle-age period, theirannuitized wealth is transferred via the annuity markets, to individuals who live throughout
their old age. Therefore, the return on savings becomes Rt+1 /π under the assumption of perfectannuity markets.
Children’s human capital over period t + 1, ht+1 , is a function of both ht and xt , where ht
represents parents’ human capital and xt denotes government expenditure on public education
per young individual. Particularly, ht+1 is formulated using the following equation:
ht+1 = h̄ (ht , xt ) ≡ D (ht )1−η (xt )η ,(3)

where D(> 0) is a scale parameter and η ∈ (0, 1) denotes the elasticity of education technology
with respect to education spending.
The following two remarks are in order. First, as in Gonzalez-Eiras and Niepelt (2012) and
Lancia and Russo (2016), we abstract private education and private old-age support away fromthe analysis; moreover, heterogeneity within a generation is abstracted from the analysis. This
simpliﬁcation enables us to demonstrate precisely how the results would change when fertility
6choice of households is introduced in their framework. Second, we do not distinguish between
spending on K-12 and higher education. Accordingly, we consider that x, an investment inpublic education, includes investments in both K-12 and higher education. In a real economy,
the beneﬁts of public education expenditure vary from person to person because some people
receive higher education while others do not. This model does not explicitly depict such intragenerational heterogeneity. Instead, we focus on a representative agent to demonstrate theextent to which each individual within a generation beneﬁts from public education investment
in K-12 and higher education levels on average.
Middle-aged individuals care about consumption, ct and dt+1 , their number of children,
nt+1 , and the human capital of children, ht+1 . The preferences of the middle-aged in period t
are speciﬁed by the following expected utility function à la de la Croix and Doepke (2003, 2004,2009):
ln ct + δ ln nt+1 ht+1 + βπ ln dt+1 ,(4)

where β ∈ (0, 1) is a discount factor, and δ(> 0) is the degree of preference for the children’squantity and quality.
We substitute the budget constraints (1) and (2) into the expected utility function in (4) toform the unconstrained maximization problem:
max ln ((1 − τt )wt ht (1 − ϕnt+1 ) − st ) + δ ln nt+1 ht+1 + βπ ln{st ,nt+1 }

(

)
Rt+1
st + bt+1 .
π

By solving this problem, we obtain the following fertility, savings, and consumption functions:bt+1(1 − τt )wt ht + Rt+1 /πδ1·,
nt+1 = n̄ (τt , bt+1 , wt ht ) ≡ ·ϕ 1 + δ + βπ
(1 − τt )wt ht
[
]
1+δ
bt+1
βπ
(1 − τt )wt ht −·,st = s̄ (τt , bt+1 , wt ht ) ≡1 + δ + βπ
βπ Rt+1 /π
[
]
1
bt+1
ct = c̄ (τt , bt+1 , wt ht ) ≡(1 − τt )wt ht +,1 + δ + βπ
Rt+1 /π
[
]
βRt+1
bt+1
dt+1 = d¯′ (τt , bt+1 , wt ht ) ≡(1 − τt )wt ht +,1 + δ + βπ
Rt+1 /π
′

(5)
(6)
(7)
(8)

where we drop the argument Rt+1 from the expressions of n̄′ (·), s̄(·), c̄(·), and d¯′ (·) because Rt+1
becomes constant, as demonstrated below. Superscript “′” in the expressions for n̄′ (·) and d¯′ (·)
denotes the next period.5

3.2

Firms

There exists a continuum of identical ﬁrms that are perfectly competitive proﬁt maximizers.
Each individual ﬁrm is indexed by i. The technology available to ﬁrm i is Yit = At (Kit )α (Lit )1−α5Equation (5) implies a negative relation between income and fertility, a standard ﬁnding in the existing
literature. However, Doepke et al. (2023) highlight the possibility that this relation may not hold recently indeveloped countries. This underscores the debatability of the assumption underlying the utility function in (4)and the household’s budget constraint in (1). Despite recognizing this aspect, by adhering to the standard utility
function and household’s budget constraint as expressed in (4) and (1), we can emphasize distinctions from priorstudies that presumed the fertility rate to be exogenous.

7

where Yit , Kit , and Lit stand for output, physical capital input, and labor input of ﬁrm i,
respectively. Here, At (> 0) represents the general level of factor productivity, given by theindividual ﬁrm, and α ∈ (0, 1) is a constant parameter representing the physical capital sharein production.
In each period, ﬁrm i chooses physical capital and labor to maximize its proﬁt, At (Kit )α (Lit )1−α −Rt Kit − wt Lit , where Rt is the gross return on physical capital and wt is the wage rate. Firms’
proﬁt maximization leads to
Kit : Rt = αAt (Kit )α−1 (Lit )1−α ,(9)Lit : wt = (1 − α)At (Kit )α (Lit )−α .

(10)Capital fully depreciates within a single period.
The productivity parameter At is assumed to be proportional to the per labor capital∑∑
At = Q · (Kt /Lt )1−α , where Kt = i Kit and Lt = i Lit represent the aggregate physicalcapital stock and labor, respectively, and Q(> 0) is constant. Thus, physical capital investment
involves a technological externality of the type often used in endogenous-growth theories. This
assumption, called the AK technology, results in a constant interest rate across periods, which isdemonstrated below. This approach helps us obtain a closed-form solution of the model. 6 Under
this assumption, the ﬁrst-order conditions in (9) and (10) are rewritten in aggregate terms asfollows:
Rt = R ≡ αQ,
wt = (1 − α)Q

3.3

(11)
Kt
.
Lt

(12)

Government Budget Constraint

Government expenditures include public investment in education and public pension payments. They are ﬁnanced by taxes on labor income. The government budget constraint in period
t is τt wt Lt = πNt−1 bt +xt Nt+1 , where τt wt Lt is the aggregate labor-income tax revenue, πNt−1 bt
is the public pension payment, and xt Nt+1 is the aggregate public expenditure on education.
Let kt ≡ Kt /Nt denote physical capital per middle-aged individual. By using (12) anddividing both sides of the constraint by Nt , we obtain the following expression of the government
budget constraint:τt (1 − α)Qkt =6

πbt+ nt+1 xt .
nt

(13)

The AK assumption, originally introduced by Romer (1986), has been subsequently employed in models
addressing endogenous fertility growth (Yip and Zhang, 1997; Blackburn and Cipriani, 1998; Chang et al., 2013).
Given the resemblance of results derived from the AK assumption to those in models of small open economies
where the interest rate is exogenously given, these ﬁndings may possess limited relevance for large, developed
countries. Nevertheless, certain investigations have undertaken the calibration of the model incorporating small
open economy assumptions to developed countries (Song et al., 2012; Lancia and Russo, 2016; Müller et al., 2016).
This study oﬀers novel insights into these preceding analyses.

8

3.4

Market Clearing

The market-clearing condition for physical capital is Kt+1 = Nt st , which expresses theequality of total savings by the middle-aged individuals in period t, Nt st , to the stock of aggregate
physical capital at the beginning of the period t + 1. We rewrite the physical capital market
clearing condition as
nt+1 kt+1 = s̄ (τt , bt+1 , wt ht ) ,(14)

where s̄(·) is deﬁned in (6).
The market-clearing condition for labor is
Lt = (1 − ϕnt+1 ) Nt ht ,(15)which expresses the equality of the aggregate labor demand, Lt , to the aggregate supply,(1 − ϕnt+1 ) Nt ht . Using (12) and (15), we deﬁne labor income as follows:
wt ht = w̄ (nt+1 , kt ) ≡(1 − α)Qkt .
1 − ϕnt+1(16)

Thus, the labor income, wt ht = w̄ (nt+1 , kt ), depends on nt+1 and kt , but is independent of ht .
Using (16), we can reformulate the fertility function in (5) as nt+1 = n̄′ (τt , bt+1 , w̄ (nt+1 , kt )),which implies that nt+1 aﬀects itself through the general equilibrium eﬀect in w̄ (nt+1 , kt ). Solving this expression for nt+1 leads to:t+1(1 − τt )(1 − α)Qkt + bR/πδ1·
nt+1 = n (τt , bt+1 , kt ) ≡ ·.
δ
ϕ 1 + δ + βπ (1 − τt )(1 − α)Qkt +· bt+1′

1+δ+βπ(17)

R/πEquation (17) suggests that an increase in the tax rate τt raises the fertility rate nt+1 by lowering
the relative price of birth. Additionally, (17) indicates that an increase in pension beneﬁts bt+1
leads to a rise in nt+1 through the income eﬀect.
Using (11), (16) and (17), we can also redeﬁne the saving function in (6) as[]βπ
1 bt+1st = s (τt , bt+1 , kt ) ≡(1 − τt )(1 − α)Qkt −·.
1 + βπβπ R/π(18)

Equation (18) suggests that an increase in the tax rate τt or pension beneﬁts bt+1 discourages
middle-aged individuals from saving. However, this eﬀect is oﬀset to some extent by the generalequilibrium eﬀect in wt resulting from a decrease in the fertility rate nt+1 . Nonetheless, theeventual decrease in savings occur due to the increase in τt or bt+1 .

4

Political Equilibrium
This section considers voting on ﬁscal policies. We employ probabilistic voting à la Lindbeck

and Weibull (1987), in which there is electoral competition between the two oﬃce-seeking candidates. Each candidate announces a set of ﬁscal policies subject to the government’s budget
constraints. As Persson and Tabellini (2000) have demonstrated, the two candidates’ platforms
9converge in the equilibrium to the same ﬁscal policy that maximizes the weighted average utilityof voters.
In the present framework, the young, middle-aged, and older adults have an incentive to vote.
While the young may beneﬁt from current public education expenditure through human capital
accumulation, we assume that their preferences are not considered by politicians. We impose
this assumption, which is often used in the literature (Saint-Paul and Verdier, 1993; Bernasconi
and Profeta, 2012; Lancia and Russo, 2016; Bishnu and Wang, 2017), for tractability reasons.
However, the assumption could be supported in part by the fact that a large number of theyoung are not allowed to participate in the voting process due to age restriction.
In period t, the oﬃce-seeking candidates consider the preferences of the middle-aged andolder adults in that period. Their preferences are represented by the indirect utility functions
VtM and VtO , respectively, and can be formulated as follows:(())
VtM (τt , xt , bt+1 , kt , ht ) = ln c̄ τt , bt+1 , w̄ n′ (τt , bt+1 , kt ) , kt + δ ln n′ (τt , bt+1 , kt ) · h̄ (ht , xt )))((+ βπ ln d¯′ τt , bt+1 , w̄ n′ (τt , bt+1 , kt ) , kt ,(19)
VtO (bt , kt , nt ) = ln d (bt , nt , kt ) ,(20)

where h̄(·), c̄(·), d¯′ (·), w̄(·), and n′ (·) are deﬁned in (3), (7), (8), (16), and (17), respectively, andd(·), representing consumption by older adults, is deﬁned as follows:d (bt , nt , kt ) ≡R
nt k t + bt .
π

(21)

In the probabilistic voting, the political objective in period t is the weighted sum of theutilities of older and middle-aged adults; this is given by πωVtO +nt (1 − ω) VtM , where ω ∈ (0, 1)
and 1−ω are the political weights placed on older and middle-aged adults, respectively. A larger
value of ω implies greater political power of older adults. We use the gross population growthrate nt to adjust the weight of the middle-aged, and life expectancy (i.e., the probability of living
in old age) π to adjust the weight of older adults, to reﬂect their population share. To obtain
the intuition behind this result, we divide the objective function by nt (1 − ω) and redeﬁne it,
denoted by Ωt , as follows:Ωt (τt , bt , xt , bt+1 , kt , ht , nt ) =πωV O (bt , kt , nt ) + VtM (τt , xt , bt+1 , kt , ht ) .
nt (1 − ω) t

(22)

The coeﬃcient πω/nt (1 − ω) of VtO represents the relative political weight of older adults.
The political objective function in (22) suggests that the current policy choice of (τt , bt , xt )aﬀects the future policy decisions through fertility choices and physical capital accumulation.
Speciﬁcally, the current choice of τt , bt , and xt aﬀects fertility decisions and the formation ofphysical capital in the next period, respectively. This inﬂuences political decision-making on
pension payments, bt+1 , in the next period. Conversely, as seen in (6) and (17), the level ofpension beneﬁts in the next period also aﬀects the economic decisions on savings and fertility
in the current period.
10

To demonstrate this mutual interaction between economic and political decisions, we employ
the Markov-perfect equilibrium concept, in which today’s ﬁscal policy depends on the current
payoﬀ-relevant state variables. In the current framework, the payoﬀ-relevant state variables inperiod t are the fertility rate, nt , and physical capital, kt ; the human capital, ht , is a payoﬀirrelevant state variable because of the speciﬁcation of the human capital formation function in(3) and the assumption of the logarithmic utility function in (4).7 Thus, we can drop ht from thearguments of VtM (·) and Ωt (·). Consequently, the expected provision of public pension in periodt + 1, bt+1 , could be given by the function of the period t + 1 payoﬀ-relevant state variables, kt+1
and nt+1 : bt+1 = B̄(kt+1 , nt+1 ).
Using the notation, with z ′ denoting the next period z, we can deﬁne a Markov-perfectpolitical equilibrium in the current framework as follows.
(
Deﬁnition 1 A Markov-perfect political equilibrium is a ﬁve-tuple ,)T̂ , B̂, X̂, Ŝ, N̂ , whereT̂ : ℜ+ × ℜ+ → [0, 1] is the tax rule, τ = T̂ (k, n); B̂ : ℜ+ × ℜ+ → ℜ+ is the pension
rule, b = B̂ (k, n); X̂ : ℜ+ × ℜ+ → ℜ+ is the education expenditure rule, x = X̂ (k, n);()
Ŝ : [0, 1]×ℜ+ → ℜ+ is the optimal private saving rule, s = Ŝ τ, k | B̂ ; and N̂ : [0, 1]×ℜ+ → ℜ+()
is the optimal private fertility rule, n′ = N̂ τ, k | B̂ , such that (i) for a given τ, k, and B̂, theoptimal private saving and fertility rules are the maps, Ŝ and N̂ , respectively, that solve
)
(
)) )
(
)
(
( (
Ŝ τ, k | B̂ = s τ, B̂ Ŝ τ, k | B̂ , N̂ τ, k | B̂ , k ,
(
( (
)
(
)) )
(
)
N̂ τ, k | B̂ = n′ τ, B̂ Ŝ τ, k | B̂ , N̂ τ, k | B̂ , k ,()()
where b′ = B̂ (k ′ , n′ ) with n′ = N̂ τ, k | B̂ and n′ k ′ = Ŝ τ, k | B̂ ; (ii) given the set of initialconditions, (k, n), and the political objective function
(
)
(
)
πω
Ω τ, b, x, k, n | B̂ ≡V O (b, k, n) + V M τ, x, b′ , k ,n(1 − ω)()()
where b′ = B̂ (k ′ , n′ ) with n′ = N̂ τ, k | B̂ and n′ k ′ = Ŝ τ, k | B̂ , the equilibrium ﬁscal
policies solve

(

)
(
)
T̂ (k, n) , B̂ (k, n) , X̂ (k, n) = arg max Ω τ, b, x, k, n | B̂
τ,b,x

subject to the government budget constraint, τ (1 − α) Qk = πb/n + n′ x, and the government
budget constraint satisﬁes:T̂ (k, n) (1 − α) Qk =()π B̂ (k, n)+ N̂ T̂ (k, n) , k | B̂ X̂ (k, n) .
nPart (i) deﬁnes the functional equations that map the current tax and physical capital()()
stock to achieve optimal private savings and fertility, s = Ŝ τ, k | B̂ and n′ = N̂ τ, k | B̂ .
This set of rules describes the private sector’s response to changes in τ under the expectation
7

Notice that ht solely appears within the h̄ (ht , xt ) term of (19). Given the Cobb-Douglas type speciﬁcation of
the function h̄ and the assumption of logarithmic utility function, ht does not inﬂuence the optimality conditions
with respect to τt , bt , and xt .

11

that future pensions will be set according to the equilibrium rule B̂ (k ′ , n′ ). Part (ii) describes
the government’s problem. In each period, the government sets ﬁscal policies subject to its
budget constraints and the private sector’s response, consistent with the expectation that future
governments will follow the Markov-perfect political equilibrium rule.

4.1

Characterization of Political Equilibrium

To obtain the set of policy functions in Deﬁnition 1, we conjecture the following policy
function of pension beneﬁts in the next period:πω)n′ (τ,b′ ,k)(1−ω) B + C R ′ (· n τ, b′ , k k ′πωπn′ (τ,b′ ,k)(1−ω) E + F
πω)n′ (τ,b′ ,k)(1−ω) B + C R (· s τ, b′ , k ,=πωπn′ (τ,b′ ,k)(1−ω) E + F

b′ =

(23)

where B, C, E, and F are constant parameters, and the equality in the the second line originatesfrom the capital-market-clearing condition in (14). Equation (23) implies that the amount of thepension beneﬁt, b′ , is set to match a certain proportion of the savings, n′ k ′ = s. The proportion
depends on the relative weight given to older adults, πω/n′ (1 − ω), in the the political objectivefunction.
Using the ﬁrst-order condition with respect to physical capital in (11), the conjecture in (23)is rewritten as follows:

πω

πb′ N
n′ (τ,b′ ,k)(1−ω) B + C
=
α.
πω
Y′
n′ (τ,b′ ,k)(1−ω) E + F

This shows that public pension payments are linearly related to GDP. The conjecture thatsatisﬁes this property is based on the results of Gonzalez-Eiras and Niepelt (2008) who show thelinear relation of the policy functions on GDP under an exogenous fertility rate. Our conjecture
here indicates that the same property holds for the endogenous fertility rate.
The conjecture in (23) suggests that b′ aﬀects itself through savings. Speciﬁcally, pension
beneﬁts b′ discourage middle-aged individuals from saving, which leads to a decrease in pension
beneﬁts. Solving (23) for b′ , we obtain

b′ =
1+

πω
B+C
n′ (τ,b′ ,k)(1−ω)πω
E+Fn′ (τ,b′ ,k)(1−ω)πω
B+Cn′ (τ,b′ ,k)(1−ω)πω
E+Fn′ (τ,b′ ,k)(1−ω)π
R· 1+βπ·βπ(1 − τ ) (1 − α) Qk.
1 + βπ(24)

Expressions in (17) and (24) suggest that a mutual interaction exists between pension beneﬁts, b′ , and the fertility rate, n′ . As shown in (17), an increase in pension beneﬁts b′ raises
the lifetime income of an individual, thereby improving fertility through the income eﬀect. Thisis the economic eﬀect of pension beneﬁts on the fertility rate. Conversely, an increase in thefertility rate leads to a decrease in the relative political weight of older adults, as seen in (24),which aﬀects the determination of pension beneﬁts. This is the political eﬀect of fertility on
12pension beneﬁts. Thus, pension beneﬁts and fertility interact with each other through economic
and political eﬀects.
Considering these two interactions, we substitute (17) into (24), and solve for b′ to obtainthe following equation:b′ = G · (1 − τ ) k,(25)

where G(> 0), a constant, is deﬁned in Appendix A.1. Equation (25) shows that a higher
labor-income tax rate is associated with a lower level of future expected pension beneﬁts. An
increase in the tax rate lowers the relative price of births. This incentivizes individuals toincrease fertility, which in turn aﬀects the level of pension beneﬁts through the political eﬀect of
fertility described above. Simultaneously, a rise in the tax rate diminishes savings by reducing
disposable income, thus lowering pension beneﬁts denoted as b′ . Consequently, the combined
impact of these eﬀects is negative.
We substitute the policy function of b′ from (25) into the fertility function presented in (17)and obtainn′ =G(1 − α)Q + R/π
1
δ
·
·
.
δ
G
ϕ 1 + δ + βπ (1 − α)Q + 1+δ+βπ
· R/π

(26)

Equation (26) reveals that the fertility rate remains constant across periods, independent ofvariations in the labor-income tax rate and state variables. This constancy implies that thedirect inﬂuence of taxes on fertility is counterbalanced by the indirect eﬀect of taxes throughpublic pension beneﬁts, denoted as b′ . It is important to note that this result is contingent upon
the assumption of a logarithmic utility function. If this assumption is relaxed, the two eﬀects
may not necessarily cancel each other out. Nonetheless, even under the current assumption,
there remains an endogenous determination of fertility through individuals’ utility-maximizing
behavior and their responses to changes in structural parameters.
Speciﬁcally, we focus on the impact of increased life expectancy on fertility rates. Given thepersistent increase in life expectancy, a prevalent trend in developed countries, each generation’s
weight within the political objective function undergoes transformation due to its impact on
fertility choices. This transformation, in turn, inﬂuences government policy decisions. Notably,
the dynamic interplay between life expectancy and policy formulation through fertility choices
sets our study apart from prior research (Gonzalez-Eiras and Niepelt, 2012; Lancia and Russo,2016). We will explore this unique aspect in Section 5.
Using the pension beneﬁts in (25) and the constant fertility rate in (26), we can reformulate
the political objective function in (22) as follows:Ω=πω
ln d (b, n, k) + ln c (τ, k) + δ ln n′ · h̄ (h, x) + βπ ln d′ (τ, k) ,n(1 − ω)where h̄(h, x) ≡ D (h)1−η (x)η as in (3), d(·) is deﬁned in (21), and c(·) and d′ (·) are sourced
from (7) and (8), respectively. Using (25) and (26), we can deﬁne the functions c(·) and d′ (·) as

13

follows:[]( ′ ) G · (1 − τ ) k1
c (τ, k) ≡· (1 − τ ) w̄ n , k +,1 + δ + βπ
R/π
[
]
(
) G · (1 − τ ) kβRd′ (τ, k) ≡· (1 − τ ) w̄ n′ , k +.
1 + δ + βπR/πGiven the government budget constraint in (13), we derive the ﬁrst-order conditions withrespect to τ , x, and b. They are summarized by a system of the following two functional
equations:d′τcτc + βπ d′db
πωn(1−ω) · d=,π(1 − α)Qk
n
′
d′τ
cτ
δ hhx′
c + βπ d′= ′ .
(1 − α)Qk
n

(27)
(28)

Expression in (27) captures the trade-oﬀ between the marginal cost of taxation and themarginal beneﬁt of pension. This trade-oﬀ reveals a conﬂict of interest between older and middleaged adults. Older adults want a higher tax rate to enjoy more pension beneﬁts. However, ahigher tax rate decreases middle-aged adults’ consumption (c and d′ ), because it decreases not
only their disposable income but also future pension beneﬁts, as shown in (25). Expression in(28) demonstrates the trade-oﬀ between the marginal cost of taxation and the marginal beneﬁt of
education expenditure. While education beneﬁts middle-aged adults through altruistic motives,
they bear the tax burden to ﬁnance it. The government determines education expenditure tobalance the aforementioned cost and beneﬁt.
Using the conditions in (27) and (28), alongside the government budget constraint in (13),we verify the conjecture in (23), and obtain the following result.
Proposition 1 Suppose that the following condition holds:{πω
1−α
for t = 0,n0 (1−ω) · α
1 + δη + βπ <πω
1−αfor t ≥ 1.
n′ (1−ω) · α(29)There is a Markov perfect political equilibrium such that the policy functions, τ , b, and x, areexpressed as follows:

]

[

πω
α
n(1−ω) + δη − (1 + βπ) 1−ατ=∈ [0, 1] ,πωn(1−ω) + (1 + δη + βπ)πω 1−αn(1−ω) α − (1 + δη + βπ) Rb=nk > 0,πωπn(1−ω) + (1 + δη + βπ)

x=

(31)

1
δη
· πω
Qk > 0,n′ n(1−ω)+ (1 + δη + βπ)

where n′ is the fertility rate given by
n′ =

(30)

− (1 + δ + αβπ) +√δ(1 + δ + αβπ)2 + 4αβ (1 + δη + βπ) 1−ωω ϕ
2αβ (1 + δη + βπ) 1−ω
ω
14

(32)

.

(33)

Proof. See Appendix A.1.
The condition (29) ensures b > 0. It indicates that life expectancy has three eﬀects on thegovernment’s provision of public pensions. First, the higher the life expectancy, the easier it isfor the government to provide pension, given the greater political weight of older adults. Thiseﬀect is captured by the term π on the right-hand side of (29). Second, the higher the life
expectancy, the higher the weight of consumption utility in old age. This provides an incentive
for the government to lower its labor-income tax rate to maintain the consumption level of theolder population. This eﬀect is captured by the term π on the left-hand side of the equation.
In period t = 0, if the eﬀect of the former exceeds that of the latter, pension beneﬁts will be
paid to older adults. From period t = 1 onward, there exists a third eﬀect in addition to the two
aforementioned eﬀects. An increase in life expectancy decreases fertility rates, as will be shownin Proposition 2 below. This increases relative political weight of older adults and strengthens
the government’s incentive to provide pension beneﬁts. This eﬀect is represented by the termn′ on the right-hand side of (29). Thus, because of this additional eﬀect, the increase in life
expectancy provides a stronger incentive for the government to provide pension beneﬁts fromperiod 1 onward compared to period 0.
Proposition 1 implies that the fertility and policy functions have the following features:First, the fertility rate remains constant and independent of physical capital over time. Thisstability arises from the previously mentioned factors: the direct inﬂuence of taxes on fertility
is oﬀset by the indirect eﬀects of taxes on public pension beneﬁts. Second, the levels of public
pension beneﬁts, b′ , and public education expenditure, x, are linear functions of output, Qk.
This property is necessary for generating a balanced growth path for the economy. Finally, thelabor-income tax rate is independent of the state variables and is constant across periods. This
property is necessary for the government’s budget to be balanced each period.

4.2

Eﬀects of Life Expectancy

The result in Proposition 1 suggests that increased life expectancy aﬀects the choice offertility and policy. These eﬀects extend to economic growth. This section focuses on an
unexpected and permanent increase in life expectancy and analyzes its eﬀects on fertility andsubsequently, on policies and growth.
First, we consider the eﬀect of life expectancy on the fertility rate in (33), which can besummarized as follows:
Proposition 2 A higher life expectancy is associated with a lower fertility rate: ∂n′ /∂π < 0.
Proof. See Appendix A.1.
To understand the mechanism behind the result in Proposition 2, recall the fertility functionin (26), showing the following two types of eﬀects of life expectancy on fertility. First is the
direct eﬀect on an individual’s decision regarding fertility for a given set of policy variables; the15other is the indirect eﬀect on fertility through political decisions regarding the level of pension
beneﬁts. In what follows, we examine various factors that contribute to these two eﬀects.
Firstly, we consider the direct eﬀects of the following two routes, named as “Eﬀect n i”
(i = 1, 2) for later references. First, an increase in life expectancy increases the weight of old-age
consumption utility. This strengthens the incentive for individuals to save, thereby increasing
the costs of raising children. This has a negative eﬀect on fertility (“Eﬀect n 1”). Second,an increase in life expectancy lowers the return on savings and thus reduces the incentive forindividuals to save. This has a positive eﬀect on fertility (“Eﬀect n 2”).
Next, we consider the indirect eﬀects of the following four routes, named as “Eﬀect n i”
(i = 3, 4, 5, 6). First, an increase in life expectancy increases the political weight of older adults.
This works to increase the level of pension beneﬁts through voting, having a positive eﬀect
on fertility (“Eﬀect n 3”). Second, an increase in life expectancy increases the weight of oldage consumption utility. This reduces the labor-income tax rate, which subsequently reduces
pension beneﬁts. This negatively aﬀects fertility (“Eﬀect n 4”). Third, an increase in the weightof old-age consumption utility increases the incentive for individuals to save. This leads to anincrease in the level of pension beneﬁts, as observed in the policy function of pension beneﬁts.
This has a positive eﬀect on fertility (“Eﬀect n 5”). Finally, the pension beneﬁts per old-aged
individual are reduced to maintain a constant pension beneﬁt-GDP ratio against an increase inlife expectancy. This negatively aﬀects fertility (“Eﬀect n 6”). Overall, there are six conﬂicting
eﬀects. However, in the current framework, the net eﬀect is negative.
As described above, life expectancy aﬀects the government’s policy choices. This implies
that life expectancy aﬀects the pension beneﬁt-GDP ratio, πbN− /QK, and the education
expenditure-GDP ratio, xN ′ /QK, where N− and N ′ denote the previous and next period N ,respectively. The following proposition shows the eﬀects of life expectancy on these ratios.
Proposition 3 An increase in life expectancy results in the following eﬀects: (i) an increasein the pension beneﬁt-GDP ratio; and (ii) a decrease in the education expenditure-GDP ratio:∂ (πbN− /QK) /∂π > 0 and ∂ (xN ′ /QK) /∂π < 0.
Proof. See Appendix A.2.
First, we consider the eﬀect of life expectancy on the pension beneﬁt-GDP ratio. As describedin the paragraphs following Proposition 1, there exist two positive and one negative eﬀects oflife expectancy on the pension beneﬁts per old-aged individual. Additionally, given the pension
beneﬁts per old-age individual, an increase in life expectancy leads to an increase in total pension
beneﬁts. Overall, there exist three positive and one negative eﬀects, and the former exceeds thelatter. Hence, an increase in life expectancy leads to an increase in the pension beneﬁt-GDP
ratio.
Next, we consider the education expenditure-GDP ratio. As life expectancy increases, thepolitical weight of older adults also increases. The decrease in fertility brought about by the16increase in life expectancy further increases the relative political weight of older adults. Theseeﬀects reduce education spending through voting. Furthermore, an increase in life expectancy
implies an increase in the weight of the utility of old-age consumption, which lowers the current
labor-income tax rate and decreases education expenditure. Considering these two negativeeﬀects, an increase in life expectancy leads to a decreased education expenditure-GDP ratio.
As discussed, life expectancy has three eﬀects on the labor-income tax rate, named “Eﬀect
τ i” (i = 1, 2, 3) for later reference. They can be summarized as follows. First, an increasein life expectancy increases the weight of utility from consumption in old age for middle-aged
individuals. This lowers the tax rate to maintain consumption or savings (“Eﬀect τ 1”). Second,an increase in life expectancy raises the political weight of older adults, incentivizing the government to increase pension beneﬁts. This has the eﬀect of raising the tax rate (“Eﬀect τ 2”).
Finally, because an increase in life expectancy leads to a decrease in fertility rate (Proposition2), the political weight of older adults further increases, reinforcing the second eﬀect (“Eﬀectτ 3”). In summary, life expectancy has two positive and one negative eﬀect on the labor-income
tax rate, and the net eﬀect is positive or negative, depending on the structural parameter values.
Finally, based on the results presented in Proposition 1, we derive the growth rate of theeconomy and investigate how it is aﬀected by increased life expectancy. The growth rate of theoutput per middle-aged individual is calculated as follows:
y′
Qk ′
s/n′
=
=
.
y
Qk
k

(34)

Equation (34) indicates that life expectancy aﬀects growth rate via the fertility rate, n′ , as well
as savings, s.
Proposition 4 An increase in life expectancy increases the growth rate of GDP per middle-aged
individual: ∂ (y ′ /y) /∂π > 0.
Proof. See Appendix A.3.
An increase in life expectancy leads to a decrease in the fertility rate (Proposition 2). As
observed in (34), this positively aﬀects the growth rate of GDP per middle-aged individual
because the capital equipment per middle-aged individual increases as the fertility rate declines.
To observe the growth eﬀect through savings, recall the savings function in (18), which is restatedas follows:

[
]
1 b′
βπ
(1 − τ )(1 − α)Qk − ·.
s=
1 + βπβ RLife expectancy aﬀects savings through the following three terms: a coeﬃcient βπ/(1 + βπ),
representing the propensity to save, the labor-income tax rate, τ , and pension beneﬁts, b′ .
As life expectancy increases, the weight on the utility of old-age consumption increases.
This strengthens the incentive for individuals to save, and positively aﬀects savings. Second anincrease in life expectancy aﬀects the labor income tax rate through Eﬀects τ 1, τ 2, and τ 3
mentioned above. Finally, an increase in life expectancy has three positive and one negative17

eﬀects on pension beneﬁt determination, as discussed above. Ultimately, the sum of the positiveeﬀects exceeds the sum of the negative eﬀects in the current framework. Thus, an increase inlife expectancy increases the growth rate.

4.3

Discussion

To assess the predictive capability of the model’s propositions outlined in Propositions 24, we examine their validity using OECD data. Figure 1 illustrates the correlation between
life expectancy and various factors for rich OECD countries. Panel (a) displays a negative
correlation with fertility, Panel (b) exhibits a positive correlation with the public pension beneﬁtGDP ratio, and Panel (c) portrays a negative correlation with the education expenditure-GDP
ratio. The model predictions of Propositions 2 and 3 align with the observation based oncross-country data in Panels (a), (b), and (c) of Figure 1. However, in Panel (d), a relatively
weak negative correlation between life expectancy and the growth rate is observed. This ﬁnding
contradicts Proposition 4, which suggests a positive correlation. The inconsistency arises because
our comparative statics analysis does not diﬀerentiate between current and subsequent period
life expectancies.
To gain a deeper understanding of this inconsistency, we derive the following expression for
the growth rate from the current period to the next period, utilizing the savings function from(18) and the policy functions from Proposition 1:
[
]
(
)
k′
1
βπ ′
1 G (π ′ , π ′′ )= ′ ′ ′′ ·(1 − α)Q − ·(1 − τ π, π ′ ),′kn (π , π ) 1 + βπ
β
R

(35)

where π, π ′ , and π ′′ represent the life expectancies for the current period, the next period, andthe period after next, respectively. Note that n, G, and τ are now represented as functions ofvariables π, π ′ , and/or π ′′ to observe how the timing of changes in life expectancy inﬂuencesthem.
It is noteworthy that the growth rate’s dependency on π, π ′ , and π ′′ is mediated through four
distinct terms in (35), denoted as n′ (π ′ , π ′′ ), βπ ′ / (1 + βπ ′ ), G (π ′ , π ′′ ), and τ (π, π ′ ). An increasein life expectancy for the current period, π, tends to reduce the growth rate by diminishing thesavings of middle-aged individuals in that period through an increased labor income tax rate
due to Eﬀects τ 2 and τ 3. In contrast, an increase in life expectancy for the next period, π ′ ,tends to stimulate the growth rate by reducing fertility, n′ (π ′ , π ′′ ) through Eﬀects n 1, n 2, n 3,
n 5 and n 6, and increasing the savings rate βπ ′ / (1 + βπ ′ ). In addition, life expectancy forthe next period inﬂuences pension beneﬁts, represented by G (π ′ , π ′′ ), and also aﬀects the labor
income tax rate through Eﬀects τ 1 and τ 3. Furthermore, life expectancy for the period after
next, π ′′ , negatively impacts the fertility rate via Eﬀect n 4 and potentially has an uncertain
eﬀect through pension beneﬁts. In summary, the growth eﬀects stemming from these factors
remain ambiguous.
Proposition 4 examined a scenario in which life expectancy in three successive periods increased uniformly from the same level. This analysis revealed a net positive eﬀect of increased
18

16
NZL

USA
2

Fer!lity rate (ave 95-15)

Public pension (% of GDP, ave 95-15)

2.2
FRA
GBR

AUS

SWE

1.8

CAN

1.6
DEU

1.4

ITA

1.2

JPN

y = -0.1018x + 9.8332
R² = 0.221

ITA

14
12

DEU

10
JPN

SWE

8

USA
GBR

6

NZL CAN
AUS

4
2
0

1
77

78

79

80

81

82

77

83

78

79

80

(a)
GDP per capita growth rate (% annual,
ave 95-15)

SWE
NZL
USA

6

FRA

5.5

CAN AUS

GBR

5

DEU
ITA

4.5
4
y = -0.3976x + 37.042
R² = 0.2251

3.5

JPN

3
77

78

79

82

83

(b)

7
6.5

81

Life expectancy (ave 95-15)

Life expectancy (ave 95-15)

Public educa!on (% of GDP, ave 95-15)

y = 0.569x - 37.812
R² = 0.0364

FRA

80

81

82

83

2.5
SWE
2

AUS
NZL
USA

GBR

1.5

CAN

DEU
FRA

1

JPN
y = -0.1573x + 13.894
R² = 0.1233

0.5

ITA

0
77

Life expectancy (ave 95-15)

78

79

80

81

82

83

Life expectancy (ave 95-15)

(c)

(d)

Figure 1: Association between life expectancy and fertility rates (Panel (a)), the public pension
beneﬁt-GDP ratio (Panel (b)), the public education expenditure-GDP ratio (Panel (c)), and the
GDP growth rate (Panel (d)) for rich OECD countries during 1995-2015.
Note. The set of rich OECD countries includes Australia, Canada, Denmark, France, Germany,
Italy, Japan, New Zealand, Sweden, the United Kingdom, and the United States.
Sources: Data on fertility rates are sourced from the United Nations Department of Economic
and Social Aﬀairs Population Division (2022) World Population Prospects 2022, Online Edition(https://population.un.org/wpp/) (accessed February 28, 2023). The sources of the otherdata are described in Appendix A.4.
life expectancy on the growth rate. However, in the real world, the rate of life expectancy increase varies among countries, and the impact of increased life expectancy in one period may
not align with that in another. Panel (d) of Figure 1 oﬀers evidence suggesting that the negativeeﬀect is more pronounced in Japan and Italy, while the positive eﬀect is stronger in Australia
and Sweden. These divergent patterns will be subject to more comprehensive investigation in
the model predictions and simulations detailed in Section 5.

5

Numerical Analysis
For numerical analysis, we calibrate the model economy in such a manner that the steady-state equilibrium matches the key statistics of the average synthetic rich OECD countries over19the time period 1995–2015. Following Gonzalez-Eiras and Niepelt (2012), we include the following countries into the set of the rich OECD countries: Australia, Canada, Denmark, France,
Germany, Italy, Japan, New Zealand, Sweden, the United Kingdom, and the United States.
Appendix A.4 describes the sources of the data used in calibration.

5.1

Calibration

We take one period in the model to correspond to 30 years in the data. This assumption isstandard in quantitative analyses of two- or three-period overlapping generations models (e.g.,
Gonzalez-Eiras and Niepelt, 2008; Lancia and Russo, 2016). The probability of living in old age,π, is derived based on the average life expectancy at birth. The average life expectancy in the richOECD countries is 79.989 years; therefore, individuals will, on average, live 19.989(= 79.989−60)
years into old age. In other words, individuals are expected to live 19.989/30 of their 30 yearsof old age, thus π = 0.666.
We ﬁx the share of capital at α = 0.3, which is considered a standard value in the literature(e.g., Gonzalez-Eiras and Niepelt, 2012). Our selection of R is 1.04 per year (e.g., Song et al.,2012; Lancia and Russo, 2016). The productivity parameter is Q = 9.730 because Q = R/α(=(1.04)30 /0.3). The opportunity cost of raising a child is approximately 20% of parents’ time.
(Kimura and Yasui, 2009). By assuming that the duration of parenthood is 18 years, ϕ is
0.2 × (18/30) = 0.12.
To determine the remaining four parameters, δ, η, ω, and β, we focus on the fertility rate,
the education expenditure-GDP ratio, the pension beneﬁt-GDP ratio, and the growth rate ofoutput per middle-aged individual. The fertility rate, n′ , is given by (33), and the two ratios,
xN ′ /Y and πbN− /Y, and the growth rate, y ′ /y, are as follows:
xN ′
δη
= πω
,
Y
n′ (1−ω) + (1 + δη + βπ)

(36)

πω
πbN−
n(1−ω) (1 − α) − (1 + δη + βπ) α=,πωYn(1−ω) + (1 + δη + βπ)

y′
k′
=
=(
y
k
1−

βπ
1+βπ (1 − α))αQω(1−ω)β +

(37)
(

)
.
δη
αn
1 + 1+βπ(38)

We use the data of the synthetic rich OECD average during 1995–2015 to solve the four
equations (33), (36), and (37) and (38) for δ, η, ω, and β. Annual population growth rate
data during the covered period, which is 1.0067, is taken as the target fertility rate. Therefore, the population growth rate over 30 years is considered (1.0067)30 .8 The average education
expenditure-GDP ratio is 0.0523, the average pension beneﬁt-GDP ratio is 0.0770, and the average output growth rate for 30 years is (1.0131)30 . We substitute these data and the values of
8
In this framework, the gross population growth rate and fertility rates are identical when π and n remain
constant. The population size in period t is calculated by adding πNt−1 , Nt , and Nt+1 , and that in period t + 1
is obtained by adding πNt , Nt+1 , and Nt+2 . Accordingly, the gross population growth rate from period t to
t + 1 is given by (πNt + Nt+1 + Nt+2 ) / (πNt−1 + Nt + Nt+1 ) = (π + nt+1 + nt+1 nt+2 ) / (π/nt + 1 + nt+1 ). Thisequation reduces to n when nt+1 = nt+2 = n.

20

Figure 2: Panel (a) The association between π and n. Panel (b) The association between life
expectancy and annual population growth rate.
Note. The life expectancy and annual population growth rate in Panel (b) are computed using
π × 30 + 60 and (n)1/30 , respectively.
α, ϕ, and π into (33), (36), (37), and (38), and solve them for δ, η, ω, and β. Accordingly, we
obtain (δ, η, ω, β) = (0.243, 0.573, 0.648, 0.780).
Based on this calibration, the impact of changes in life expectancy on fertility and theresulting population growth rate can be estimated, as demonstrated in Figure 2. Panel (a)illustrates the relationship between π, which represents the probability of living to old age, andn, which represents fertility rates, while Panel (b) illustrates the corresponding relationshipbetween life expectancy and the annual population growth rate. The ﬁndings in the ﬁgureshow that a higher life expectancy is associated with a lower fertility rate, as demonstrated in
Proposition 2.
Table 1 presents the calibration of life expectancy and its correlation with population growthin the following four country groups: the synthetic rich OECD countries, synthetic rich OECD
Europe (consisting exclusively of European countries), Japan, and the United States. Basedon the results presented in Table 1, an increase in life expectancy corresponds to a decrease in
the annual population growth rate. Particularly in the United States, where life expectancy iscomparatively low, a one-year increase in life expectancy results in a reduction of 211 individuals
per million. Conversely, in Japan, where life expectancy is higher than that in the United States,
a corresponding reduction is 203 individuals per million, which is lower than that in the UnitedStates. These ﬁndings suggest that the reduction in population growth due to an increase in life
expectancy is less signiﬁcant as life expectancy increases.

rich OECDJapan
the U.S.
Europe(1)
79.98982.02877.66479.937(2)-0.00020664-0.00020297-0.00021102-0.00020674(3)-206.644-202.971-211.022-206.740

Table 1: Column 1 displays the mean life expectancy between 1995 to 2015. Columns 2 and 3
demonstrate the predicted impact of a one-year increase in life expectancy on the rate of change
in the annual population growth (Column 2) and the number of population changes per million(Column 3).
21

5.2

Projected Changes in Fiscal Policies, Fertility, and Growth Rates

To facilitate our analysis, we utilize a model-based time series approach to predict the changes
in ﬁscal policies, fertility, and growth rates over time in response to projected improvements inlife expectancy. In Section 5.1, following Gonzalez-Eiras and Niepelt (2012), we have categorized the OECD countries included in our parameter estimation into four groups: synthetic
rich OECD (including all covered countries), synthetic rich OECD Europe (comprising only
European countries), Japan, and the United States. In this section we use life expectancy datafrom the United Nations World Population Prospects,9 and examine the impact of projected life
expectancy on ﬁscal policies, fertility, and economic growth over time, providing quantitative
estimates of the predicted eﬀects.
The analysis proceeds by computing three sequences of model predictions, each spanning a
30-year period. The ﬁrst sequence includes the years 1990, 2020, 2050, and 2080; the second
includes 2000, 2030, 2060, and 2090; and the third includes 2010, 2040, 2070, and 2100. To
present the time series predictions, these three sequences are merged into a single time series
following the procedure outlined by Gonzalez-Eiras and Niepelt (2012).
Figure 3 displays the predictions of life expectancy in our model for the four groups basedon the United Nations World Population Prospects. The other parameter values utilized in theanalysis are obtained from the calibration conducted in Section 5.1. It should be noted thatinstead of using average life expectancy between t − 1 and t, we utilize the life expectancy attime t, denoted by πt .10 Furthermore, πt for all groups, except the United States, is projected
to reach the upper limit of 1 after 2070 in Japan and after 2080 in the synthetic rich OECD andsynthetic rich OECD Europe. This is because, in the model, the upper limit of life expectancy
is set at 90 years, whereas the estimated life expectancy exceeds 90 years after 2070 or 2080.

Figure 3: The predictions of πt for the synthetic rich OECD, the synthetic rich OECD Europe,
Japan, and the United States.
Note: The ﬁgure plots measured values up to 2020 and projected values after 2030.
It is important to note that the data on πt of the previous year in each column, that is,
2080, 2090, and 2100, is solely used to calculate nt at t = 2050, 2060, and 2070, respectively. To
910

See Appendix A.4 for the source of the data.
In the model, πt represents the probability of survival into old age for middle-aged individuals in period t − 1.

22

Figure 4: Projected changes in fertility rate (Panel (a)), the labor-income tax rate (Panel (b)),
the pension beneﬁt-GDP ratio (Panel (c)), the education expenditure-GDP ratio (Panel (d)),
and the growth rate of physical capital per middle-aged individual (Panel (e)).
obtain the fertility rate nt and policy variables for the last three years of 2080, 2090, and 2100,
life expectancy πt data for 2110, 2120, and 2130, respectively, is required. However, since theavailable life expectancy estimates are limited to the year 2100, it is not possible to calculate thefertility rates and policy functions for the last three years. Therefore, we forecast fertility rate,
policy functions, and the growth rate for each column up to one period before the ﬁnal year.
Figure 4 illustrates the projections of fertility rates, ﬁscal policies, and growth rates for thefour groups of countries. The ﬁgure shows that the fertility rate and education expenditure-GDP
ratio will decrease, while the labor-income tax rate and pension beneﬁt-GDP ratio will increaseover time. This projection aligns with model-based qualitative predictions in Propositions 2
and 3. However, the growth rate of physical capital per middle-aged individual is projected todecrease for Japan and slightly decrease for the synthetic rich OECD and synthetic rich OECD
Europe, with a temporary decline also observed in the United States. This ﬁnding contrasts
with the result of Proposition 4, which states that the growth rate increases as life expectancy
increases, based on comparative statics that assume equal life expectancy increases between thecurrent and future periods. The numerical analysis indicates that the degree of increase in life
expectancy diﬀers between periods, leading to diﬀerent forecasts of economic growth rates.
To better understand the mechanism behind the diﬀerence in results, we refer to the equation for the growth rate of physical capital per middle-aged individual in (35). Equation (35)indicates that an increase in life expectancy aﬀects the growth rate through the following fourroutes: First, a decline in the fertility rate results in an increased physical capital equipment per
middle-aged individual. Second, middle-aged individuals are incentivized to save more becauseof their increased life expectancy. Third, increasing pension beneﬁts discourages individuals from23

saving. Fourth, an increase in the labor income tax rate reduces savings. Therefore, increased
life expectancy positively inﬂuences economic growth through the ﬁrst and second routes but
negatively impacts through the third and fourth.
The analysis provided above, based on equation (35), oﬀers insights into the disparity between the model prediction in Proposition 4 and the evidence depicted in Figure 1(d), as well as
the divergent projected changes across four country groups. In Japan, where life expectancy isexperiencing rapid growth, the adverse eﬀects surpass the positive ones, leading to a decline ineconomic growth. Conversely, in the United States, where life expectancy temporarily declined
between 2010 and 2020, the positive eﬀects outweigh the negative ones, resulting in a temporary
upsurge in economic growth from 2030 to 2040. Consequently, the impact of life expectancy on
economic growth hinges on the direction of change in life expectancy and the extent to which
this trend strengthens in the future.

5.3

Further Quantitative Implications

In this section, we explore two scenarios that diﬀer from the baseline analysis in Section
5.2 to understand their impact on the baseline results. In Section 5.3.1, we analyze the casewhere the pension beneﬁt-GDP ratio is ﬁxed, while in Section 5.3.2 we examine the scenario
where the fertility rates are determined by the United Nations World Population Prospects
projections. Both analyses are focused on the rich OECD case. We present only the results andinterpretations here, leaving the analysis details to Appendix A.5.
5.3.1

Fixed Pension GDP Ratio

The analysis in Section 5.2 predicts a decline in fertility, an increase in the pension beneﬁtGDP ratio, and a decline in economic growth against the expected trend of increasing life
expectancy in the rich OECD. Speciﬁcally, the increase in the pension beneﬁt-GDP ratio isalready observed in the rich OECD. To limit the projected further rise in this ratio, many
OECD countries are working to curb pension beneﬁts (OECD, 2019). We analyze the impact of
this eﬀort on fertility, policy variables, and economic growth. By comparing our analysis resultswith the baseline results, we can assess the eﬀect of the policy of ﬁxing the pension beneﬁt-GDP
ratio.
For the analysis, we assume that the pension beneﬁt-GDP ratio is ﬁxed at the 2020 level
after 2030. This policy is assumed to be introduced unexpectedly in 2030. In other words,
people do not expect the introduction of this policy before 2020. Figure 5 illustrates the eﬀects
of a ﬁxed pension beneﬁt-GDP ratio policy on fertility, tax rates, education expenditure-GDP
ratio, and economic growth. The results of the baseline analysis are also shown for comparison.
The scenario with a ﬁxed pension beneﬁt-GDP ratio exhibits the same declining trend in
the fertility rate and the education expenditure-GDP ratio as observed in the baseline case.
Speciﬁcally, while the fertility rate declines rapidly, the education expenditure-GDP ratio de-24

Figure 5: Projected changes in fertility rate (Panel (a)), the labor-income tax rate (Panel (b)),
the education expenditure-GDP ratio (Panel (c)), and the growth rate of physical capital per
middle-aged individual (Panel (d)).
creases slowly. One signiﬁcant diﬀerence from the baseline is that in the scenario where thepension beneﬁt-GDP ratio is ﬁxed, the labor income tax rate tends to decrease while the growthrate tends to increase. Consequently, reducing future pension beneﬁts by ﬁxing the pension
beneﬁt-GDP ratio entails a trade-oﬀ between lower fertility and higher economic growth. 11
The mechanism behind these results is as follows: When the pension beneﬁt-GDP ratio is
ﬁxed, decisions regarding x and τ do not impact the utility of older adults. Therefore, thegovernment determines x and τ by considering only the preferences of middle-aged individuals.
As life expectancy rises, middle-aged individuals prioritize their old-age consumption, leading
to a desire for a smaller tax burden and lower education expenditures. Consequently, the labor
income tax rate and the education expenditure-GDP ratio decline.
As noted earlier, a policy of ﬁxing the pension beneﬁt-GDP ratio implies a reduction in future
pension beneﬁts. This reduction diminishes the incentive to have and raise children, resulting
in a rapid decline in the fertility rate. Simultaneously, the decrease in pension beneﬁts and thelower labor income tax rate resulting from this policy promote physical capital accumulation.
Additionally, the decline in fertility increases the per capita capital equipment rate. These eﬀects
contribute to an increase in the rate of economic growth.

11This trade-oﬀ may be lessened by private old-age support. If individuals display altruistic behavior towards
their parents - a factor not considered in this study - then future generations might increase private transfers
to their parents when confronted with reduced pension beneﬁts. This could oﬀset pension reductions and thus
partially alleviate the negative eﬀect of pension cuts on fertility. It should be noted that our model does notincorporate these dynamics associated with private old-age support.

25

Figure 6: Projected changes in fertility rate (Panel (a)), the labor-income tax rate (Panel (b)),
the pension beneﬁt-GDP ratio (Panel (c)), the education expenditure-GDP ratio (Panel (d)),
and the growth rate of physical capital per middle-aged individual (Panel (e)).
5.3.2

Exogenous Fertility

The key property of the present study relative to Gonzalez-Eiras and Niepelt (2012) is that
fertility is endogenous. This property enables us to demonstrate the fertility decline due toincreased life expectancy and its eﬀects on policies and economic growth. In this section, we
consider an alternative case where the fertility rate is exogenous, and compare it with the baseline
case to see how the assumption of endogenous fertility aﬀects the results.
We utilize data from the United Nations World Population Prospects for the projected
fertility rate. For instance, n1990 represents the 1990 population/1960 population ratio. Using
this estimated fertility data, Figure 6 illustrates the predicted values of the policy variables andthe economic growth rate, comparing them with the predicted values in the baseline case. The
results in the ﬁgure suggest the following: Similar to the baseline case, the exogenous fertility
case demonstrates a trend toward lower fertility rates and higher labor income tax rates andpension beneﬁt-GDP ratios, along with lower education expenditure-GDP ratios. However, thepace of these changes is swifter in the exogenous fertility case compared to the baseline case.
Another signiﬁcant diﬀerence between the results of the two cases is observed in the economic
growth rate. While the baseline case shows minimal change in the growth rate, the exogenous
fertility case exhibits a substantial increase in the growth rate.
The mechanism driving these results is as follows: The data from the United Nations World
Population Prospects predict a faster decline in fertility compared to the endogenous fertility
baseline case. Consequently, the relative political inﬂuence of older adults in the exogenous
fertility case is greater than in the baseline case, leading to the government’s decision to implement higher labor income tax rates and pension beneﬁt-GDP ratios, along with lower education
26

expenditure-GDP ratios. These higher labor income tax rates and pension beneﬁt-GDP ratios
diminish the incentive for middle-aged individuals to save, thereby hindering physical capital accumulation and negatively impacting economic growth. However, the decline in the fertility rateincreases the physical capital equipment rate per middle-aged individual, which positively inﬂuences economic growth. As this positive impact outweighs the negative impact, the exogenous
fertility case predicts a higher rate of economic growth than the baseline case.

6

Conclusion
This study investigated the following two key questions: (1) How does the government al-

locate its limited budget between pension provisions for older adults and education for the
younger generation in response to population aging in each period? (2) What are the anticipated shifts in ﬁscal policies, fertility rates, and economic growth trajectories as a response
to projected advancements in life expectancy? To answer these questions, we employed an
overlapping-generation model that incorporates physical and human capital accumulation. We
further enhanced the model by introducing parental decisions regarding fertility. By endogenizing fertility choices, our approach oﬀers insights into the interplay between political decisions
on education and pension expenditures and parental fertility decisions.
Past studies on the political economy of education and pensions have typically treated the
population growth rate as an exogenous variable, focusing on the inﬂuence of changes in thisexternal factor (speciﬁcally, a decline in the population growth rate) on policy formulation. In
contrast, our study endogenously models households’ fertility decisions and demonstrates theirsubstantial impact on government policy choices and resource allocations. Additionally, ourstudy provides predictions regarding ﬁscal policies, fertility patterns, and economic growth overtime, considering anticipated changes in life expectancy.
While our analysis presents a novel perspective by incorporating fertility as an endogenous
variable, several unresolved issues require attention in this study. The ﬁrst pertains to the generalization of the functional form. The outcomes of this study heavily depend on the assumptions
underlying the logarithmic utility function and the AK production technology. Speciﬁcally,
altering our assumption to a utility function with a constant inter-temporal elasticity of substitution would impact the comparative statics result. Furthermore, relaxing the AK technology
assumption would enable us to depict the interplay among capital accumulation, interest rates,
and fertility, along with the corresponding endogenous fertility variability. While this exploration holds potential value, our preliminary investigation suggests that this extension would
render a closed-form solution impossible, necessitating further quantitative analysis.
The second issue concerns the ﬁnancial costs associated with child rearing. The current
framework concentrates on the opportunity costs of child rearing, sidestepping ﬁnancial costs
due to their lack of intrinsic diﬀerence under the logarithmic utility assumption. Introducing
ﬁnancial costs alongside a generalized utility function may yield several novel insights. The
27

third issue involves childcare subsidies, prevalent in many developed countries. This introduces
a conﬂict of interest between older and middle-aged adults and may exert additional inﬂuencewhen combined with the ﬁrst two extensions. While these extensions hold substantial potential
value, they are deferred for future exploration.

28

A

Appendices

A.1

Proofs of Propositions 1 and 2

We substitute the fertility function into (17) to (24) to obtain′′ϕ 1+δ+βπ
δ

b =

δ
b
(1−τ )(1−α)Qk+ 1+δ+βπ
· R/π
πω
′
b
1−ω B + C
(1−τ )(1−α)Qk+ R/πb′(1−τ )(1−α)Qk+ 1+δ+βπ · R/π πω
ϕ 1+δ+βπ′δ
1−ω E + F
(1−τ )(1−α)Qk+ b
δ

[
]
R
βπ
1
b′
· ·
· (1 − τ )(1 − α)Qk −·,π 1 + βπ
βπ R/π

R/π

(A.1)
or,

( )2
G1 · b′ + G2 (1 − τ )k · b′ + G3 · ((1 − τ )k)2 = 0,

(A.2)

where G1 , G2 , and G3 are deﬁned as:
[(
) (
)
]
δ
1
πω
F
πω
C
1
G1 ≡
·
·
E+
+
B+
,
1 + δ + βπ R/π
1−ω
ϕ
1−ω
ϕ 1 + βπ
{(
)
πω
δ
F
G2 ≡ (1 − α) QE+·1−ω
1 + δ + βπ ϕ
[
(
)
(
)]}
βπ
δ
πω
C
1
πω
δ
C
+
· −
·
B+
+
·
B+
·
,
1 + βπ1 + δ + βπ
1−ω
ϕ
βπ
1−ω
1 + δ + βπ ϕ
(
)
πω
δ
C
R
βπ
G3 ≡ (−1)
B+·· ·· [(1 − α) Q]2 .
1−ω
1 + δ + βπ ϕ
π 1 + βπAssuming G1 ̸= 0, we solve (A.2) for b′ and obtain√−G2 + (G2 )2 − 4G1 G3b′ = G(1 − τ )k =(1 − τ )k,2G1
where G is
G≡

−G2 +

√

(G2 )2 − 4G1 G3
2G1

(A.3)

.

We substitute VtM into (19) and VtO in (20) into Ωt in (22), and obtain
(
)
[
]
πω
R
b′
Ω≃
ln
nk + b + (1 + δ + βπ) ln (1 − τ )(1 − α)Qk +n(1 − ω)
π
R/π
[
]
′
δ
b
− δ ln (1 − τ )(1 − α)Qk ++ δη ln x.
·
1 + δ + βπ R/π(A.4)

We use the notation ≃ in (22) because irrelevant terms are omitted from the expressions forΩt . We further substitute (A.3) and the government budget constraint in (13) into the politicalobjective function of (A.4), and obtain
]
(
)
[
πω
R
πb
.
Ω≃
ln
nk + b + (1 + βπ) ln (1 − τ ) + δη ln τ (1 − α)Qk −n(1 − ω)
π
n

29

(A.5)

The ﬁrst-order conditions with respect to b and τ are:
b:

π
πω
1
n
· R
− δη
≤ 0,
n(1 − ω) π nk + b
τ (1 − α)Qk − πb
n

(A.6)

τ:

(−1) (1 + βπ)δη(1 − α)Qk
+
= 0.
1−τ
τ (1 − α)Qk − πb
n

(A.7)

Equation (A.7) is rewritten asδη(1 − α)Qk + (1 + βπ) πbn.
(1 + δη + βπ) (1 − α)Qk

τ=

(A.8)

Substitution of (A.8) into (A.6) leads to the policy function of pension beneﬁts:πω (1−α)− (1 + δη + βπ) Rn(1−ω) α
b=
nk,
πω
π
n(1−ω) + (1 + δη + βπ)(A.9)

verifying the conjecture of (23). Equation (A.9) indicates that a public pension is provided, thatis, b > 0 holds if the fertility rate falls below the following critical value:b > 0 ⇔ 1 + δη + βπ <

πω
1−α
·
.
n (1 − ω)α(A.10)We express this condition in (29).
We derive the policy function of τ by substituting (A.9) into (A.7):
]
[
πω
α
+
δη
βπ)
−
(1
+
1−α
n(1−ω)
] .
τ= [
πω
+
(1
+
δη
+
βπ)
n(1−ω)(A.11)

τ < 1 is obtained immediately from the expression in (A.11). τ > 0 holds if the sign of thenumerator in (A.11) is positive; that is, if[]πωα+ δη − (1 + βπ)> 0,n(1 − ω)1−αor,
1 + βπ − δη

1−α
πω
1−α
<
·
.
α
n (1 − ω)α(A.12)(A.12) holds if (A.10) (that is, (29)) holds true. Thus, we obtain τ ∈ [0, 1] if (29) holds.
Recall that the fertility rate for a given set of policy variables is (17). We must replace τand b′ in (17) with n and k using the policy functions in (A.9) and (A.11). Taking one period
lag of (A.9):b′πω 1−αz1 (n′ ) R ′ ′n′1−ω) α − (1 + δη + βπ) R ′ ′
=
n
k
=
nk,
πω
z0 (n′ ) πn′ (1−ω) + (1 + δη + βπ) π(A.13)

where z0 (·) and z1 (·) are deﬁned as:z0 (n′ ) ≡z1 (n′ ) ≡πωn′ (1 − ω)πωn′ (1 − ω)

+ (1 + δη + βπ) ,(A.14)1−α− (1 + δη + βπ) .
α(A.15)

30

The policy function b′ in (A.13) is reformulated as follows:
z1 (n′ ) Rsz0 (n′ ) π[]
z1 (n′ ) R βπ1b′=(1 − τ )(1 − α)Qk −·,z0 (n′ ) π 1 + βπ
βπ R/πb′ =where the equality in the ﬁrst line comes from (14), and the equality in the second line originatesfrom (6). By rearranging the terms, we have[]z1 (n′ ) R βπ 1 1
z1 (n′ ) R βπ′1+b=(1 − τ )(1 − α)Qk
z0 (n′ ) π 1 + βπ βπ R/πz0 (n′ ) π 1 + βπ
z1 (n′ ) R βπ
1 + βπ=(1 − α)Qk,
z0 (n′ ) π 1 + βπ (1 − α)z0 (n)where the equality in the second line is derived from (A.11). Thus, we obtainb′z1 (n′ ) R
βπz0 (n′ ) π (1−α)z0 (n)=(1 − α)Qk.
′)
1
1 + zz10 (n′(n ) 1+βπ(A.16)

We substitute (A.11) and (A.16) into the fertility function in (17) and obtain the following
equation that characterizes the equilibrium fertility rate,
z1 (n′ )1 + z0 (n′ )δ1·n′ = ·
ϕ 1 + δ + βπ 1 + z1 (n′′ ) · 1+δ
z0 (n ).

(A.17)

1+δ+βπWe reformulate the expression in (A.17) by substituting z0 (·) in (A.14) and z1 (·) in (A.15) andrearranging the terms to obtain
n′ =

1
δ
·
.
ϕ (1 + δ + αβπ) + αβπ (1 + δη + βπ) n′ (1−ω)(A.18)πωEquation (A.18) shows that there exists a unique n′ > 0 satisfying (A.18). Note that (A.18) isrewritten as a quadratic equation for n′ :αβ (1 + δη + βπ)

1 − ω ( ′ )2δn + (1 + δ + αβπ) n′ − = 0.
ωϕThus, we can solve this equation for n as√δ
− (1 + δ + αβπ) + (1 + δ + αβπ)2 + 4αβ (1 + δη + βπ) 1−ω
ω ϕ
′
n =
.
2αβ (1 + δη + βπ) 1−ωω(A.19)(A.20)The left-hand side of (A.19) increases in π and n′ , indicating that a higher π value is associatedwith a lower n′ in (A.19). Thus, we have ∂n′ /∂π < 0.
To obtain the policy function of x, we substitute the policy functions of b in (A.9) and τ in
(A.11) into the budget constraints in (13). Then, we have
x=

1
δη
· πω
Qk > 0.
′n n(1−ω) + (1 + δη + βπ)(A.21)■31

A.2

Proof of Proposition 3

The aggregate expenditure on public pensions is πbN− , and the aggregate GDP is QK.
Thus, the pension beneﬁt-GDP ratio is
πbN−
πb
=
QK
Qkn
πω
n(1−ω) (1 − α) − (1 + δη + βπ) α=,πωn(1−ω) + (1 + δη + βπ)

where the second equality comes from the policy function b in (31). After manipulation, we
obtain


[
]−1

1
1

(1 − α) − 1+δη β · n(1−ω) + 1
for t = 0,
πbN− +[ πω ω
]−1
=

QK

1
 (1 − α) − 1+δη1 β · n′ (1−ω)+1for t ≥ 1.
πω+ωAs ∂n′ /∂π < 0, we have ∂ [πbN− /QK] /∂π > 0.
The education expenditure-GDP ratio is
xn′
xN ′
=
QK
Qk
=

δη

,
πω
n(1−ω) + (1 + δη + βπ)

where the second equality comes from the policy function x in (32). Thus, we haveδη πω +(1+δη+βπ)
for t = 0,′xNn(1−ω)=δη πω +(1+δη+βπ)
QK
for t ≥ 1.
n′ (1−ω)As ∂n′ /∂π < 0, we have ∂ (xN ′ /QK) /∂π < 0.
■

A.3

Proof of Proposition 4

The aggregate output is QK; therefore, the output per middle-aged individual is QK/N =Qk. Thus, the gross growth rate of output is Qk ′ /Qk = k ′ /k. To compute k ′ /k, we recall thecapital market-clearing condition in (14), which we can rewrite as
[
]
1
βπ
b′
′ ′
(1 − τ )(1 − α)Qk −nk =·1 + βπβπ R/π
z1 (n′ ) Q βπ
1+βπβπ  1 + βπ11z0 (n′ ) π 1+βπ (1−α)z0 (n) (1 − α)Qk =(1 − α)Qk −··,′1 + βπ (1 − α)z0 (n)βπ R/π1 + z1 (n′ ) Q βπ 1 1
z0 (n ) π 1+βπ βπ R/πwhere the ﬁrst equality is derived using (6) and the second equality is derived by using (A.11)
and (A.16). We rearrange the terms in (A.20) and obtain
k′
=(
k
1−

βπ
1+βπ (1 − α))αQω(1−ω)β +

32

(

)
.
δη
1 + 1+βπ
αn′(A.22)

The ﬁrst term in the denominator on the right-hand side decreases in π, and the second termin the denominator also decreases in π because ∂n′ /∂π < 0. Thus, we obtain ∂ (k ′ /k) /∂π > 0.
■

A.4

Sources of Data

We source data on the average life expectancy and the average population growth rate in richOECD countries from the United Nations Department of Economic and Social Aﬀairs Population
Division (2022) World Population Prospects 2022, Online Edition (https://population.un.
org/wpp/) (accessed March 8, 2023). Data on the average education expenditure-GDP ratio aresourced from World Development Indicators (WDI) (https://datatopics.worldbank.org/
world-development-indicators/) (accessed March 2, 2023). Public education expenditure
includes expenditure funding through transfers from international sources to general government.
General government refers to local, regional, and central governments.
Data on the pension beneﬁt-GDP ratio are sourced from the OECD (2023), “Pension spending” (indicator), https://doi.org/10.1787/a041f4ef-en (accessed on March 2, 2023). Pension spending is deﬁned as all cash expenditure (including lump-sum payments) on old age andsurvivors’ pensions. Data on GDP per capita are from International Comparison Program,
World Bank | World Development Indicator Database, World Bank | Eurostat-OECD (https://datatopics.worldbank.org/world-development-indicators/) (accessed March 9, 2023).
GDP per capita is based on purchasing power parity (PPP).
■

A.5

Supplementary Explanation for Section 5.3

A.5.1

Fixed Pension GDP Ratio

To solve for the government problem under the ﬁxed pension-GDP ratio, we assume thatthe period-t pension-GDP ratio is ﬁxed at εt ∈ (0, 1]:πt Nt−1 bt= εt .
Yt
We reformulate this expression and obtain
nt
Qkt ,
πt
nt+1
bt+1 = εt+1
Qkt+1 .
πt+1
bt = ε t(A.23)(A.24)

With (A.24), the capital market clearing condition in (18) is reformulated as follows:[]
1 εt+1βπt+1(1 − τt ) (1 − α) Qkt −nt+1 kt+1 =
nt+1 kt+1 ,1 + βπt+1
βπt+1 αor,
kt+1 =

βπt+1
1·(1 − τt ) (1 − α) Qkt .
εt+1 ·
1
n
1
+
βπt+1
1 + 1+βπt+1 α
t+1
1

33

(A.25)

With (A.23), the government budget constraint in (13) is reformulated as follows:τt (1 − α) Qkt = εt Qkt + nt+1 xt .

(A.26)

Substitution of (A.25) into the fertility function in (17) leads to:
βπt+1
1
t+1
nt+1 + παQ·
εt+1 ·
1
1+βπt+1
1+ 1+βπ1δαt+1
nt+1 = ·
·
,
βπt+1
δ
1
ϕ 1 + δ + βπt+1 nt+1 +· πt+1 ·εt+1 ·
1
1+δ+βπt+1αQ

1+ 1+βπt+1

α1+βπt+1or,f (nt+1 ) ≡ (nt+1 )2 + Ḡ1 (πt+1 , εt+1 ) nt+1 − Ḡ2 (πt+1 , εt+1 ) ,(A.27)

where Ḡ1 (πt+1 , εt+1 ) and Ḡ2 (πt+1 , εt+1 ) are deﬁned by
(
)
δ
πt+1
1
βπt+1
1
·
·
−
Ḡ1 (πt+1 , εt+1 ) ≡,εt+1 ·
1
1 + δ + βπt+1
αQ 1 + 1+βπ
1 + βπt+1 ϕα
t+1Ḡ2 (πt+1 , εt+1 ) ≡

1
δ
πt+1
1
·
·
·
1
ϕ 1 + δ + βπt+1 αQ 1 + 1+βπ

βπt+1.
εt+1 ·
1 + βπt+1
t+1 αGiven the property of f (0) < 0, we can solve (A.27) for nt+1 to obtain√()2−Ḡ1 (πt+1 , εt+1 ) +Ḡ1 (πt+1 , εt+1 ) + 4Ḡ2 (πt+1 , εt+1 )
nt+1 =.
2(A.28)

Under the assumption in (A.23), the period-t old-age consumption is
dt =

R
πt
nt kt + εt Qkt .
πtnt(A.29)

Under the assumption in (A.24), the lifetime income becomes(1 − τt ) wt ht +

bt+1
nt+1
(1 − α) Q
πt+1= (1 − τt )
kt +
εt+1
Qkt+1
R/πt+1
1 − ϕnt+1
αQ
πt+1
)
(
1
εt+1
1
βπt+1
·
+
(1 − τt ) (1 − α) Qkt ,=εt+1 ·
1
1 − ϕnt+1α
1 + βπt+1
1 + 1+βπt+1 α(A.30)

where the equality in the ﬁrst line comes from (16) and the equality in the second line comesfrom (A.25). We substitute (A.30) into (7) and (8) to obtain
(
)
1
εt+1
1
1
βπt+1
+
(1 − τt ) (1 − α) Qkt ,ct =·εt+1 ·
1
1 + δ + βπt+1 1 − ϕnt+1α
1 + βπt+1
1 + 1+βπ
t+1 α
(
)
βR
1
εt+1
βπt+1
1
dt+1 =+(1 − τt ) (1 − α) Qkt .
·εt+1 ·
1
1 + δ + βπt+1 1 − ϕnt+1α
1 + βπt+1
1 + 1+βπαt+1Consequently, we haveln ct ≈ ln (1 − τt ) , ln dt+1 ≈ ln (1 − τt ) .

34(A.31)

The problem of the government in period t is to choose {τt , xt } to maximizeπt ωln dt + ln ct + δ ln nt+1 ht+1 + βπt+1 ln dt+1
nt (1 − ω)subject to the government budget constraint in (A.26), the fertility function in (A.28), andthe consumption functions in terms of utility in (A.31). Substituting the constraints into theobjective function, we can reformulate the problem asmax (1 + βπt+1 ) ln (1 − τt ) + δη ln (τt (1 − α) − εt ) .
τtThe ﬁrst-order condition with respect to τt results in the determination of τt :
τt =(1 + βπt+1 ) εt + δη (1 − α)
1
·
.
1−α
1 + βπt+1 + δη

(A.32)

Utilizing (A.26) and (A.32), we derive the solution for xt :xt =1
δη (1 − α − εt )·Qkt ,nt+1 1 + βπt+1 + δηand by substituting (A.32) into (A.25), we derive the growth rate of physical capital as:
kt+1
1
=
1
kt
1 + 1+βπ
A.5.2

1

·
εt+1 ·nt+1
t+1 α

βπt+1 (1 − α − εt )Q.
1 + βπt+1 + δη

Exogenous Fertility

When {nt+1 } is exogenously given, the individual’s utility maximization problem becomes()
Rt+1
max ln ((1 − τt ) wt ht (1 − ϕnt+1 ) − st ) + δ ln nt+1 ht+1 + βπt+1 ln
st + bt+1
st
πt+1
given nt+1 . The ﬁrst-order condition with respect to st leads to the following saving function:
(
)
βπt+1
1
bt+1
st =
(1 − τt ) wt ht (1 − ϕnt+1 ) −·,(A.33)
1 + βπt+1
βπt+1 Rt+1 /πt+1
and the associated consumption functions are as follows:
(
)
1
bt+1
ct =
(1 − τt ) wt ht (1 − ϕnt+1 ) +,1 + βπt+1
Rt+1 /πt+1
(
)
βRt+1
bt+1
dt+1 =(1 − τt ) wt ht (1 − ϕnt+1 ) +.
1 + βπt+1
Rt+1 /πt+1(A.34)(A.35)

Recall the conjecture of the pension beneﬁts in (23):πt+1 ω
R
nt+1 (1−ω) B̃ + C̃·
bt+1 = πt+1 ω
nt+1 kt+1 ,π
t+1
nt+1 (1−ω) Ẽ + F̃

where B̃, C̃, Ẽ, and F̃ are constant. Following the same manner as in the main analysis, theconjecture is reformulated as follows:

bt+1 =
1+

πt+1 ω
B̃+C̃
nt+1 (1−ω)πt+1 ω
Ẽ+F̃
nt+1 (1−ω)πt+1 ω
B̃+C̃ πnt+1 (1−ω)
t+1 /Rt+1
πt+1 ω
1+βπt+1
Ẽ+F̃
nt+1 (1−ω)

·

35

·

βπt+1
· (1 − τt ) (1 − α) Qkt .
1 + βπt+1(A.36)

Utilizing (16) and (A.36), we can write the lifetime income as follows:bt+1(1 − τt ) wt ht (1 − ϕnt+1 ) +Rt+1 /πt+1
πt+1 ω
B̃+C̃
nt+1 (1−ω)βπt+1·πt+1 ω
Ẽ+F̃ 1+βπt+11
nt+1 (1−ω)
·= 1 +
 · (1 − τt ) (1 − α) Qkt .
πt+1 ω
B̃+C̃
Rt+1 /πt+1
nt+1 (1−ω)
1
1 + πt+1 ω· 1+βπt+1
nt+1 (1−ω)(A.37)

Ẽ+F̃

Thus, with (A.34), (A.35), and (A.37), we obtainln ct ≈ ln (1 − τt ) , ln dt+1 ≈ ln (1 − τt ) .

(A.38)The political objective function Ωt now is given byπt ω
ln dt + ln ct + δ ln nt+1 ht+1 + βπt+1 ln dt+1
nt (1 − ω)
(
)
πt ω
Rt
≈
ln
nt kt + bt + (1 + βπt+1 ) ln (1 − τt ) + δη ln xt .
nt (1 − ω)
πt

Ωt =

This is identical to that under endogenous fertility (see (A.5)). Thus, policy functions remain
consistent regardless of whether fertility is exogenously or endogenously determined. Consequently, the comparative statics outcome outlined in Proposition 3, initially derived with endogenous fertility assumption, holds true even when fertility is assumed exogenous. However,quantitative comparative statics results are contingent upon fertility assumptions, as discussed
in Section 5.3.2.
■

36

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