Discussion Papers In Economics
And Business

Generational Conict and Education Politics:
Implications for Growth and Welfare

Yuki Uchida and Tetsuo Ono
Discussion Paper 18-05-Rev.2
April 2019

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANGenerational Conflict and Education Politics:
Implications for Growth and Welfare∗
Yuki Uchida†

Tetsuo Ono‡

Seikei University

Osaka University

Abstract
This study considers the politics of public education and its impact on economic
growth and welfare across generations. We employ probabilistic voting to demonstrate the generational conflict regarding taxes and spending and show that aging
results in a tax burden shift from the retired to the working generation, reduction
in public education spending, and ultimately in slowing down economic growth.
We subsequently consider a legal constraint that aims to boost education spending: a spending floor for education. This constraint stimulates economic growth
but creates a trade-oﬀ between current and future generations’ welfare. Finally, thequantitative implications of our results are explored by calibrating the model to theJapanese economy.
• Keywords: Public education, Economic growth, Capital income tax, Probabilistic voting• JEL Classification: D70, E24, H52.

∗

An earlier version of this study, titled “Capital Income Taxation, Economic Growth, and the Politics
of Public Education,” was presented at the Aoyamagakuin University Economic Theory and Policy Workshop held on January 26, 2018. We are grateful for the comments from the participants at that workshop.
The financial support from the Grant-in-Aid for Early-Career Scientists from the Japan Society for the
Promotion of Science (No. 18K12802, Uchida) and the Grant-in-Aid for Scientific Research (C) from theJapan Society for the Promotion of Science (No. 18K01650, Ono) is gratefully acknowledged.
†
Yuki Uchida: Faculty of Economics, Seikei University, 3-3-1 Kichijoji-Kitamachi, Musashino, Tokyo
180-8633, Japan. E-mail: yuchida@econ.seikei-u.ac.jp.
‡
Tetsuo Ono: Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka
560-0043, Japan. E-mail: tono@econ.osaka-u.ac.jp.

1

Introduction

Regarding how population aging aﬀects public education expenditure, median voter theory suggests that the composition of government spending is biased toward goods andservices that benefit the elderly because a decisive voter becomes older as population
ages. This implies that, with aging, the public education expenditures that provide lessbenefits to the elderly are expected to decrease. In fact, the Organisation for Economic
Co-operation and Development (OECD) reports that Japan, which has been experiencing
rapid population aging over the past two decades, showed the lowest ratio of public education expenditure to GDP among the 34 comparable OECD member countries in 2014
and 2015 (OECD, 2017, 2018). This negative association between aging and public education expenditure is also reported by Ohtake and Sano (2010), who analyze the eﬀects
of population aging on compulsory public education expenditure using prefectural panel
data from 1975 to 2005 on Japan.
A reduction in public education expenditure, stemming from population aging, reduces
human capital accumulation. In addition, population aging may induce politicians toshift the tax burden from the elderly to the middle-aged, thereby discouraging savings forthe working generation and reducing physical capital accumulation. These eﬀects imply anegative impact on economic growth via the choice of fiscal policy, as reported by Cattaneo
and Wolter (2009) and the references therein. One way to tackle this negative impact
is to introduce a spending floor constraint that supports spending on public education,
which is being or has been implemented in some East Asian countries such as Indonesia(OECD, 2010), Malaysia (OECD, 2016), South Korea, and Taiwan (Ho, 2004).1
Given this background, the present study proposes a political economy theory that
demonstrates the mechanism behind reductions in public education expenditure and economic growth. Specifically, we propose a three-period-lived overlapping-generations modelwith physical and human capital accumulation (e.g., Lambrecht, Michel, and Vidal, 2005;
Kunze, 2014; Ono and Uchida, 2016). Public education contributes to human capitalformation and is funded by taxing the labor income of the working generation (i.e., themiddle-aged) and capital income of the retired generation (i.e., the old). We employ probabilistic voting à la Lindbeck and Weibull (1987) to present the generational conflict overpublic education and taxation. Within this voting framework, the government, representing the middle-aged and old population, chooses expenditure and taxes to maximize
the weighted sum of the utility of the middle-aged and old. Based on this voting mechanism, we demonstrate the political determinant of both expenditures and taxes and theirimpacts on economic growth and welfare over time and across generations.
1

A similar constraint also existed in Brazil (Gordon and Vegas, 2005) and Connecticut, the UnitedStates (Bates and Santerre, 2003).

1

Within this framework, we consider a spending floor constraint on public educationand show that the constraint spurs education spending and promotes human capital accumulation. At the same time, it induces a rise in the labor income tax rate and thus
reduces physical capital accumulation. Overall, the spending floor promotes economic
growth, and thus benefits future generations. However, it forces the government to increase the capital income tax rate to finance its increased expenditure, and thus worsens
the welfare of the current old generation. Therefore, the constraint creates a trade-oﬀ
between current and future generations in terms of welfare.
We calibrate the model to the Japanese economy to quantitatively explore the implications of the spending floor constraint for Japan. The sample period for estimating
model parameters is 1995–2014. During this period, the average ratio of public education
expenditure to GDP is 0.0324. However, the average ratio during 1987–1989, or the ”bubble years”, is 0.0551, which is the highest over the last 40 years (see Figure 1). Thus, wetake the ratio 0.0551 as a target, and analyze how the growth and distribution of utility
over time and across generations would change if Japan were to increase the ratio andfollow the policy during the bubble years.
[Figure 1 here.]
For the growth and welfare evaluation of the constraint, we consider the presenceof a benevolent planner, who can commit to all his or her choices at the beginning of
the period, subject to the resource constraint. Assuming such a planner, we evaluate
the political equilibrium allocation in the presence of the constraint by comparing it to
the planner’s allocation and obtain the following results. The old are worse oﬀ, since thespending floor constraint increases the tax burden and thus crowds out their consumption.
The per capita output at the political equilibrium is higher than that in the planner’s
allocation over some periods.
We also show that some successive generations at political equilibrium are betteroﬀ relative to the planner’s allocation because the spending floor constraint works tostimulate human capital formation and thus increases income and consumption. These
results are obtained by assuming that the planner’s discount factor equals the individual
one. We also verify the robustness of the results by considering alternative discount factor
cases for the planner and show that the choice of the planner’s discount factor matters as
to whom benefits or loses from the spending floor constraint.
With these results, we can draw policy implications for Japan, which ranked lastamong the 34 comparable OECD member countries in public spending on educationin 2014 and 2015. The introduction of the spending floor is a way to increase public
education expenditure and, in turn, economic growth in Japan. However, at the sametime, such benefits accrue to future generations at the expense of the currently living old.
2

Thus, when policy makers decide on a spending floor or similar policies, they should keep
in mind that a trade-oﬀ between generations in terms of utility is created. The same
argument applies to the other countries with low education spending-to-GDP ratios andgrowth rates, such as Greece and Italy.
The rest of this paper is organized as follows. We first review the relevant literature
in Section 1.1. Thereafter, Section 2 presents the model and characterizes the economic
equilibrium. Section 3 characterizes the political equilibrium and investigates the eﬀects
of generational conflict on economic growth. Then, we introduce the spending floor for
education and evaluate it in terms of growth and welfare. Section 4 calibrates the modelto the Japanese economy and compares the political equilibrium in the presence of the
spending floor with the planner’s allocation. Section 5 provides concluding remarks.

1.1

Contribution to the LiteratureThis study is related to the literature on education politics and economic growth, initiated by Glomm and Ravikumar (1992) and Saint-Paul (1993). These studies, as wellas Kaganovich and Zilcha’s (2012), focus on intra-generational conflict over education
funding. The inter-generational conflict is inherent in their model formulation but isomitted from the analysis. Inter-generational conflict is demonstrated by Gradstein andKaganovich (2004), Naito (2012), Kunze (2014), Lancia and Russo (2016), Ono andUchida (2016), and Bishnu and Wang (2017), who focus on consumption or labor taxation
as a source of education funding. They assume away the capital taxation on the retired
generation and thus present no generational conflict over the tax burden. However, assuggested by Mateos-Planas (2010), Razin, Sadka, and Swagel (2004), and Razin and
Sadka (2007), demographic changes influence voters’ interests in taxing diﬀerent factors
as a political choice. On the other hand, Boldrin (2005) and Soares (2003, 2006) reflect
such influences but assume equal tax rates on labor and capital income. Instead, thisstudy assumes diﬀerent tax rates on capital and labor income and contributes to the literature by investigating the eﬀects of population aging on the distribution of tax burdens
between generations.
A second contribution to the literature is the focus on the spending floor for the publicfunding of education, an issue that was not addressed in the previous works, except for
Bates and Santerre (2003). As previously mentioned, Japan ranked at the bottom of
the OECD member countries in public spending for education in 2014 and 2015, owing
to the increased political pressure due to the elderly. Naturally, we expect the spending
floor would resolve this pressure, incentivizing politicians to increase public education
expenditure, and promote economic growth. Further, it would also benefit future generations through physical and human capital accumulation. However, such a constraint is3

also expected to increase the tax burden on currently living generations and thus worsen
their welfare. This study assesses the welfare costs and benefits of the spending floor by
calibrating the model to the Japanese economy.

2

Model

The discrete time economy starts with period 0 and consists of overlapping generations.
Individuals are identical within a generation and live for three periods: youth, middle, andelderly ages. Each middle-aged individual gives birth to 1 + n children. The middle-aged
population at period t is Nt and grows at a constant rate of n(> −1) : Nt+1 = (1 + n)Nt .

2.1

Individuals

Individuals display the following economic behavior over their lifecycles. During youth,
they make no economic decisions and receive public education financed by the government.
During the middle age, individuals work, receive market wages, and pay taxes. They use
after-tax income for consumption and savings. Individuals retire in their elderly years
and receive and consume returns from savings.
Consider an individual born during period t − 1. In period t, the individual is middleaged and endowed with ht units of human capital. He or she supplies them inelasticallyon the labor market and obtains labor income wt ht , where wt is the wage rate per eﬃcient
unit of labor in period t. After paying tax τt wt ht , where τt ∈ (0, 1) is the period t labor
income tax rate, the individual distributes the after-tax income between consumption ct
and savings invested in physical capital st . Therefore, the budget constraint for period tor the middle age becomes
ct + st ≤ (1 − τt )wt ht .
The budget constraint for period t + 1 or the elderly age is()
kdt+1 ≤ 1 − τt+1
Rt+1 st ,kis the period t + 1 capital income tax rate, Rt+1 (> 0) is
where dt+1 is consumption, τt+1
the gross return from investment in capital, and Rt+1 st is the return from savings.

Children’s human capital over period t + 1, ht+1 , is a function of government spending
on public education, xt , and parents’ human capital, ht . In particular, ht+1 is formulated
by using the following equation:
ht+1 = D (xt )η (ht )1−η ,(1)where D(> 0) is a scale factor and η ∈ (0, 1) denotes the elasticity of education technology
with respect to education spending.
4We note that private investment in education may also contribute to human capitalformation. For example, parents’ time (Glomm and Ravikumar, 1995, 2001, 2003; Glomm
and Kaganovich, 2008) or spending (Glomm, 2004; Lambrecht, Michel, and Vidal, 2005;
Kunze, 2014) devoted to education may complement public education. In this study,
we abstract private education from the main analysis to simplify the presentation of themodel and focus on the eﬀects of public education on growth and utility.
The preferences of an individual born in period t − 1 are specified by the following
utility function of logarithmic form:
Ut = ln ct + β ln dt+1 ,where β ∈ (0, 1) is a discount factor.2 We substitute the budget constraints into theutility function to form the following unconstrained maximization problem:
max ln [(1 − τt )wt ht − st ] + β ln Rt+1 st .
{st }By solving this problem, we obtain the following savings and consumption functions,
respectively:β(1 − τt )wt ht ,
1+β
1
ct =
(1 − τt )wt ht ,
1+β
(
)
k
β 1 − τt+1
Rt+1
dt+1 =(1 − τt )wt ht .
1+β
st =

2.2

Firms

Each period contains a continuum of identical firms that are perfectly competitive profit
maximizers. According to a Cobb–Douglas technology, they produce a final good Yt
using two inputs: aggregate physical capital Kt and aggregate human capital Ht ≡ Nt ht .
Aggregate output is given by
Yt = A (Kt )α (Ht )1−α ,where A(> 0) is a scale parameter and α ∈ (0, 1) denotes the capital share.
Let kt ≡ Kt /Ht denote the ratio of physical to human capital (interchangeably called“physical capital”). The first-order conditions for profit maximization with respect to Ht
and Kt are
wt = (1 − α)A (kt )α , and ρt = αA (kt )α−1 ,2(2)The results are qualitatively unchanged if we assume parents to be altruistic towards their children
and concerned about their income, wt+1 ht+1 (see Ono and Uchida, 2018).

5

where wt and ρt are the labor wage and rental price of physical capital, respectively. The
conditions state that firms hire human and physical capital until the marginal products
are equal to factor prices. We assume the full depreciation of physical capital.

2.3

Government Budget Constraint

Public education expenditure is financed by taxes on labor income and capital income.
The government budget constraint over period t is
τt wt ht Nt + τtk Rt st−1 Nt−1 = Nt+1 xt ,(3)

where τt wt ht Nt is the aggregate labor income tax revenue, τtk Rt st−1 Nt−1 the aggregate
capital income tax revenue, and Nt+1 xt the aggregate expenditure on public education.
By dividing both sides of the above expression by Nt , we obtain the per capita form of
the constraint:
τt wt ht +2.4

τtk Rt st−1
= (1 + n)xt .
1+n

(4)

Economic Equilibrium

The market-clearing condition for physical capital is Kt+1 = Nt st , which expresses theequality of total savings by the middle-aged population over period t, Nt st , to the stockof aggregate capital at the beginning of period t + 1, Kt+1 . By using kt+1 ≡ Kt+1 /Ht+1 ,
ht+1 = Ht+1 /Nt+1 , and the savings function, we can rewrite the condition as(1 + n)kt+1 ht+1 =β(1 − τt )wt ht .
1+β

(5)

The following defines the economic equilibrium in the proposed model.
{}∞
Definition 1. Given a sequence of policies, τt , τtk , xt t=0 , the economic equilibrium is a∞sequence of allocations {ct , dt , st , kt+1 , ht+1 }∞t=0 and prices {ρt , wt , Rt }t=0 with the ini()
tial conditions k0 (> 0) and h0 (> 0), so that (i) given wt , Rt+1 , τt , τtk , xt , (ct , dt+1 , st )solves the utility maximization problem; (ii) given (wt , ρt ), kt solves a firm’s profit)(
maximization problem; (iii) given (wt , ht , kt ) , τt , τtk , xt satisfies the government
budget constraint; (iv) the arbitrage condition holds: ρt = Rt ; and (v) the physicalcapital market clears (1 + n)kt+1 ht+1 = st .
At economic equilibrium, the indirect utility of the middle-aged over period t, VtM ,and that of the old over period t, Vto , can be expressed as functions of fiscal policy and6physical and human capital, as follows:()
k
VtM = V M xt , τtk , τt+1
, A (kt )α ht
(
)
(
) ( (
(
) ))
k
≡ (1 + β) ln Z xt , τtk , A (kt )α ht + β ln 1 − τt+1
R P xt , Z xt , τtk , A (kt )α ht , ht + C
(6)
( k
)
α
o
o
Vt = V τt , A (kt ) ht()
≡ ln 1 − τtk + ln αA (kt )α ht (1 + n),(7)
where Z(·, ·, ·), P (·, ·, ·), R(·), and C are respectively defined as follows:()
Z xt , τtk , A (kt )α ht ≡ (1 − α) A (kt )α ht − (1 + n)xt + ατtk A (kt )α ht ,
(8)
(

β
1+β

(
)
α
k
,
A
(k
)
h
Z
x
,
τ
, (9)
t
t
t
t
(1 + n)D (xt )η (ht )1−η
( (
(
) ))
( (
(
) ))α−1
R P xt , Z xt , τtk , A (kt )α ht , ht ≡ αA P xt , Z xt , τtk , A (kt )α ht , ht
, (10)β1+ β ln.
C ≡ ln1+β1+βP xt , Z() )xt , τtk , A (kt )α ht , ht ≡Functions Z, P,, and R represent the disposable income of the middle-aged, (1 − τt )wt ht ;physical capital to human capital ratio over the next period, kt+1 ; and gross interest rate,
Rt+1 . We use the government budget constraint in (4) to replace τt with τtk and xt inderiving Eq. (8), the capital market clearing condition in (5) and the human capitalformation function in (1) to derive Eq. (9), and the first-order condition with respect to
K in (2) to derive (10). The derivations of (6)–(10) are provided in Appendix A.1.

3

Politics

Here, we consider voting on fiscal policy. In particular, we employ probabilistic voting à
la Lindbeck and Weibull (1987). Under this voting scheme, there is electoral competition
between two oﬃce-seeking candidates. Each candidate announces a set of fiscal policies
subject to the government’s budget constraint. As demonstrated by Persson and Tabellini(2000), the two candidates’ platforms converge at equilibrium to the same fiscal policythat maximizes the weighted-average utility of voters.
In the present framework, the young, middle-aged, and elderly have an incentive tovote. While the young may benefit from public education expenditure in the future, we
assume they are unable to vote because they are below voting age. Thus, the politicalobjective is defined as the weighted sum of the utility of the middle-aged and old, given byΩ̃t ≡ ωVto + (1 + n)(1 − ω)VtM , where ω ∈ [0, 1] and 1 − ω are the political weights for theold and middle-aged in period t, respectively. The weight for the middle-aged is adjusted
7

by the gross population growth rate, (1 + n), to reflect their share of the population. For
simplicity, we divide Ω̃t by (1 + n)(1 − ω) and redefine the objective function as follows:Ωt =ω
Vto + VtM ,(1 + n)(1 − ω)

where the coeﬃcient ω/(1 + n)(1 − ω) of Vto represents the relative political weight of theold.
We substitute VtM in (6) and Vto in (7) into Ωt . By rearranging the terms, we obtain
(
)
(
)
ω
ln 1 − τtk + (1 + β) ln Z xt , τtk , A (kt )α htΩt ≃(1 + n)(1 − ω)
(
) ( (
(
) ))
k
+ β ln 1 − τt+1
R P xt , Z xt , τtk , A (kt )α ht , ht .
(11)
We use notation ≃ because the irrelevant terms are omitted from the expression of Ωt .
The political objective function in (11) suggests that the current policy choice aﬀects thedecision on future policy via physical and human capital accumulation. In particular,
period’s t choices of τtk and xt aﬀect the formation of physical and human capital for
period t + 1. This in turn influences the decision making on fiscal policy for period t + 1.
To demonstrate this intertemporal eﬀect, we employ the concept of the Markov-perfectequilibrium, under which the fiscal policy today depends on the current payoﬀ-relevant
state variables. In the present framework, the payoﬀ-relevant state variables are physicalcapital, kt , and human capital, ht . Thus, the expected rate of capital income tax for theknext period, τt+1
, is given by the function of the next-period stock of physical and humankcapital, τt+1 = T k (kt+1 , ht+1 ). By using recursive notation, with z ′ denoting the next
period z, we can define a Markov-perfect political equilibrium as follows.

Definition 2. A Markov-perfect political equilibrium is a set of functions, ⟨T, T k , X⟩,
where T : ℜ++ × ℜ++ → [0, 1] is the labor income tax rule, τ = T (k, h), T k : ℜ++ ×ℜ++ → [0, 1] the capital income tax rule, and τ k = T k (k, h), X : ℜ++ × ℜ++ → ℜ++
the public education expenditure rule, x = X(k, h), so that the following conditions
are satisfied:(i) The physical capital market clearsβ(1 − T (k, h)) (1 − α) A (k)α h;
(12)
1+β
⟨
⟩
(ii) Given k and h, T (k, h), T k (k, h), X(k, h) = arg max Ω subject to τ k′ = T k (k ′ , h′ ),the physical capital market-clearing condition in (12), the government budget con(1 + n)k ′ h′ =straint is
T (k, h) (1 − α) A (k)α h + T k (k, h)αA (k)α h = (1 + n)X(k, h),and the human capital formation function is h′ = D(h)1−η (X(k, h))η .
8

(13)

3.1

Characterization of Political Equilibrium

To obtain the set of functions in Definition 2, we conjecture that the capital income tax
rate over the next period, τ k′ , is independent of physical and human capital:
τ k′ = τ̄ k ,where τ̄ k ∈ (0, 1) is a constant parameter. Based on this conjecture, the first-orderconditions with respect to τ k and x areτkω
(1 + β) ∂Z
β ∂R′ ∂P
(1+n)(1−ω): (−1)++≤ 0,
1 − τkZ ∂τ k R′ ∂P ∂τ k

|

{z

(∗1)

x:

}

|

{z



(∗2)

}

|

{z



(∗5)

(14)

(∗3)

(1 + β) ∂Z
β  ∂R′ ∂Z ∂R′ + ′ ∂Z ∂x + ∂x  = 0.
Z∂xR
| {z }| {z } |{z}
(∗4)

}

(15)

(∗6)

A strict inequality holds for (14) if τ k = 0.
The first-order conditions in (14) and (15) state that the government chooses thecapital tax rate, τ k , and public education expenditure, x, to balance its marginal costs
and benefits. In particular, the condition in (14) indicates that there are three eﬀects ofτ k , as shown by terms (*1)–(*3). First, term (*1) implies that an increase in τ k lowers the
net return from savings for the old. Second, given x, the government can lower the labor
income tax rate, τ , by raising τ k . A reduction in τ leads to an increase in the disposable
income of the middle and, thus, an increase in their lifetime consumption, as shown byterm (*2). Third, the reduction in τ leads to an increase in savings and the next-period
physical-to-human capital ratio, which in turn lowers the marginal product of physicalcapital and thus the returns from savings, as shown by term (*3).
The condition in (15) indicates there are three eﬀects of x on the utility of the middleaged, as shown by terms (*4)–(*6). First, term (*4) implies that a rise in x leads toan increase in τ and, thus, a decrease in lifetime consumption. Second, an increase in τ
lowers saving and raises the return from savings, as shown by term (*5). Third, a risein x promotes human capital formation and, thus, lowers the physical-to-human capitalratio. This in turn increases the marginal product of physical capital and the returns
from savings, as shown by term (*6).
By using the conditions in (14) and (15), we can verify the conjecture and obtain thefollowing result.

9

Proposition 1. Assume that the following conditions hold:{}α
max 0,{1 + β (α + η (1 − α))} − βη
1−α
ω
α
≤
{1 + β (α + η (1 − α))} .
≤(1 + n)(1 − ω)
1−α

(16)

There is a Markov-perfect political equilibrium so that the policy functions are givenbyk≡1−
T k (k, h) = τun1
ω×∈ [0, 1) ,αΛ (1 + n)(1 − ω)Xun
A (k)α h,1+n1 + αβT (k, h) = τun ≡ 1 −∈ [0, 1) ,(1 − α)Λ

X(k, h) =whereω+ 1 + β (α + η (1 − α)) ,(1 + n)(1 − ω)βη(1 − α)
Xun ≡
.
Λ
Λ≡

Proof. See Appendix A.2.
k
The first and second inequalities in (16) imply τun≥ 0 and τun ≥ 0, respectively.
Subscript “un” implies that the choice of fiscal policy is unconstrained. In the next
section, we consider a case of a restriction on the education expenditure and compare itkwith the unconstrained case. The result in Proposition 1 suggests that tax rates, τunandτun , and public education expenditure, Xun , are aﬀected by the relative political weight ofthe old, ω/(1 + n)(1 − ω). A greater political power of the old leads to a larger weight of
the utility of consumption for this population. This incentivizes the government to shift
the tax burden from the old to the middle-aged and reduce public education expenditure.

3.2

Steady-State GrowthBased on the result in the previous subsection, we derive the steady-state growth rateof the economy and investigate how it is aﬀected by population aging. To this end, we
consider per capita output, yt , which is defined by yt ≡ Yt /Nt = A (kt )α ht . Then, thegrowth rate of per capita output is
A (k ′ )α h′y′=,yA (k)α hwhere z ′ denotes the next period, z(= k, h, y). In the steady state with k ′ = k, thegrowth rate of per capita output, y ′ /y, is equal to the growth rate of human capital, h′ /h.
Therefore, in the following, we focus on the steady-state growth rate of human capital.
10

To derive the steady-state growth rate of human capital, we recall the human capitalformation function, h′ = D(h)1−η (x)η . Given the policy function x presented in Proposition 1, we can reformulate the formation function as
(
)η
h′
Xun
α
=D
A (k)
.
(17)
h
1+n
By substituting this into the capital market-clearing condition in (12) and rearranging
the terms, we obtain the law of motion of physical capital as
′

k =

β
(1 − τun ) (1 − α)1+β( X )η (A (k)α )1−η .
un(1 + n)D 1+n(18)This equation implies that a unique and non-trivial steady state exists and, for any
initial condition k0 > 0, the sequence of k stably converges to the unique steady state.
By computing the steady-state value of k and substituting it into (17), we can express
the law of motion of steady-state human capital asαη[ β] 1−α(1−η)
1−α[ ()η ] 1−α(1−η)
′
′
(1
−
τ
)
(1
−
α)
η
un
Xun
h
h
1+β
=
≡ D
(A) 1−α(1−η) .
h
h un
1+n
1+n

(19)

This equation suggests that the steady-state growth rate is aﬀected by the relative
political weight of the old, ω/(1 + n)(1 − ω), through public education expenditure, Xun ,and the labor income tax rate, τun . As described above, an increase in the political weighton the old, ω/(1+n)(1−ω), results in a shift of the tax burden from the old to the middleaged and a reduction in public education expenditure. In other words, population aging
increases the labor income tax rate, which discourages savings by the middle-aged andthus reduces physical capital accumulation (i.e., lowers the steady-state level of k). In
addition, it lowers the ratio of public education expenditure to GDP, which also reduces
human capital accumulation. Therefore, aging has a negative impact on the steady-state
growth rate via the choice of fiscal policy.

3.3

Spending Floor for Public Education

The analysis in the previous subsection showed that population aging aﬀects fiscal policy
formation and that this, in turn, reduces the steady-state growth rate. This result suggests
that, given that populations grow older in most developed countries, these countries areunder pressure to act against the low growth rates. For this policy purpose, we here
consider a legal constraint: a spending floor for education to sustain economic growth.
Specifically, we consider the following constraint, which is introduced as an unchangeable
rule of law:Nt+1 xt≥ Xxc (> Xun ),
(20)
Yt
11

where Xxc ∈ (0, 1 − α) is an exogenously given lower bound of the ratio of public education
expenditure to GDP. The constraint, Xxc , is bounded above by 1 − α since Xxc = 1 − αis feasible, as long as the labor income tax rate is 100%, that is, τ = 1.
The government in the presence of the spending floor chooses a set of fiscal policies tomaximize the political objective function in (11) subject to the above constraint. Given
the assumption of Xxc > Xun , the constraint is binding at an optimum: (1 + n)x =Xxc · A(k)α h. The associated capital and labor income tax rates are given as follows.
Proposition 2. Assume that the ratio of public education expenditure to GDP is constrained by the spending floor constraint, as in (20). If the following conditions
hold,
(
)
1 − Xxcωα (1 + αβ)− 1 (1 + αβ) ≤
≤
,
(21)
1−α
(1 + n)(1 − ω)
(1 − α) − Xxc
there is a Markov-perfect political equilibrium so that the policy functions are given
by
[
]−1
1
−
X
1
+
αβ
xc
k
T k (k, h) = τxc≡1−1+∈ [0, 1) ,ωα(1+n)(1−ω)
X(k, h) =Xxc
A (k)α h,
1+n

[
]−1
ω
1 − Xxc(1+n)(1−ω)
T (k, h) = τxc ≡ 1 −1+∈ [0, 1) .
1−α
1 + αβProof. See Appendix A.3.
Subscript “xc” in the expressions of the policy functions in Proposition 2 means thatthe ratio of public education expenditure to GDP is binding at the spending floor conk
straint, Xxc . The first and second inequalities in (21) imply τxc≥ 0 and τxc ≥ 0,,
respectively.
Based on the characterization of the political equilibrium in Proposition 2, we compare
the cases in the presence and absence of the constraint in terms of tax rates, economic
growth, and welfare across generations. Hereafter, the old at the time of the introductionof the constraint are called the initial old.
Proposition 3. Consider the political equilibrium in the presence of the spending floor
constraint presented in Proposition 2.
(i) The growth rate and capital and labor income tax rates are higher at the politicalequilibrium in the presence of the constraint than at the political equilibrium in itskk, and τxc > τun .
> τunabsence: h′ /h|xc > h′ /h|un , τxc

12

(ii) The initial old are made worse oﬀ, whereas the steady-state generations are better oﬀ by the introduction of the constraint: V0o |xc < V0o |un and limt→∞ VtM xc >limt→∞ VtM un .
Proof. See Appendix A.4.
The introduction of the spending floor constraint forces the government to increase
public education expenditure. This action stimulates human capital accumulation andthus increases the growth rate. However, the increased expenditure incentivizes the government to raise the capital income tax rate. This lowers the welfare of the initial old
because they owe the tax burden but do not benefit from the fiscal policy. In addition, an
increased labor income tax rate lowers the disposable income of current and future generations, implying negative eﬀects on physical capital accumulation and economic growth.
Thus, there are two opposing eﬀects on economic growth and the result of Proposition
3(i) shows that the former positive eﬀect outweighs the latter negative eﬀect in the steadystate. This result implies that future generations benefit from increased income and arethus made better oﬀ at the expense of the initial old. As such, the introduction of the
spending floor constraint is non-Pareto-improving.

4

Planner’s Allocation

In the previous section, we used the Pareto criterion to evaluate the welfare consequencesof the spending floor constraint on public education expenditure. Here, we take an alternative approach by deriving an optimal allocation that maximizes an infinite discounted
sum of generational utilities for an arbitrary social discount factor (e.g., Bishnu, 2013). In
particular, we consider a benevolent planner, who can commit to all his or her choices atthe beginning of a period, subject to the resource constraint. Assuming such a planner,
we evaluate political equilibrium by comparing it with the planner’s allocation in termsof consumption, physical and human capital, and welfare across generations.

4.1

Characterization of the Planner’s Allocation

The planner is assumed to value the welfare of all generations. In particular, the objectiveof the planner is to maximize a discounted sum of the lifecycle utility of all current andfuture generations:∞∑SW =θt Ut ,
t=−1

under the resource constraint:
Nt ct + Nt−1 dt + Kt+1 + Nt+1 xt = A (Kt )α (Ht )1−α ,
13

or

1
dt + (1 + n)kt+1 ht+1 + (1 + n)xt = A (kt )α ht ,1+nwhere k0 and h0 are given. Parameter θ ∈ (0, 1) is the planner’s discount factor. Solving
the problem leads to the following planner’s allocation.
ct +Proposition 4. Given k0 and h0 , the planner’s allocation, {ct , dt , kt+1 , xt }∞t=0 , is characterized by
1
A (kt )α ht ,ϕ
(1 + n)βdt =A (kt )α ht ,
ϕθ
[
(
)
]
1
β
1
xt =
ϕ− 1+
− θϕ1
A (kt )α ht ,
1+n
θ
ϕ
(
)1−η
θϕ1
1
α
kt+1 =A (kt )
,
[ 1 {
(
)
}]η
ϕ
(1 + n)D 1+n
ϕ − 1 + βθ − θϕ1
ct =

where
1
ϕ≡
1 − αθ (1 − η)
ϕ1 ≡ αϕ.

[(

β
1+
θ

)

{

θ
+ η β + (1 + β)
1−θ

}]
,

Proof. See Appendix A.5.

4.2

Numerical Analysis

Based on the above results, we compare the planner’s allocation and the political equilibrium in the presence and absence of the spending floor constraint, respectively. Specifically, we calibrate the model in the absence of the constraint in Section 3 to the Japanese
economy. We first compute the education-to-GDP ratio, physical capital, and growth rateof per capita output in the steady state and then compare these in the planner’s allocation
to those in the political equilibrium. Subsequently, we plot the evolution of consumption,
physical and human capital, per capita output, and distribution of utility across generations and investigate the eﬀects of the constraint across periods and generations.
We fix the share of physical capital in production at α = 1/3 and assume each period lasts 30 years; these assumptions are standard in quantitative analyses of the twoor three-period overlapping-generations models (e.g., Gonzalez-Eiras and Niepelt, 2008,2012; Song, Storesletten, and Zilibotti, 2012; Lancia and Russo, 2016). We select β as0.99 per quarter, which is also standard in the literature (e.g., Kydland and Prescott,
1982; de la Croix and Doepke, 2003). Since the agents in the present model plan over14

generations that span 30 years, we discount the future by (0.99)120 . For η, Cardak (2004)
suggests that η is in the range of 0.1 − 0.3. Following this, we set η = 0.28 to attaininterior solutions for τ k and τ .
The sample period for estimating other model parameters is 1995–2014. During thissample period, the average rate of the gross population growth rate is 1.0232. To determine ω, we focus on the ratio of public education expenditure to GDP. In particular, theratio is given by
Nt+1 xtβη (1 − α)
= Xun ≡,
ωYt
+ 1 + β (α + η (1 − α))(1+n)(1−ω)
when we assume no spending constraint. Given α = 1/3, β = (0.99)120 , 1 + n = 1.0232,
and η = 0.28, we can solve this expression for ω by using the ratio observed in Japan.
The average ratio during 1995–2014 is 0.0324. We can determine ω by solving the above
expression for ω and obtain ω ≃ 0.368.
The productivity of human capital, D, is normalized to D = 1. For the productivityof final goods, we use data on a per capita GDP gross growth rate of 1.249 in Japan
during 1995–2014. We substitute this figure and the values of α, β, n, η, and D into
the expression of the per capita growth rate of human capital in Eq. (19) and solve theexpression for A to obtain A = 34.888. The data sources for the gross population growthrate, ratio of public education expenditure to GDP, and per capita growth rate are theWorld Development Indicators.3
The economy is assumed to be in a steady state in period 0. Initial physical capital
k0 is computed by solving Eq. (18) for k and is given by k0 ≃ 8.0318. The initial value ofhuman capital, h0 , is normalized at h0 = 1, allowing logarithmic utility to be positive.
The average ratio of public education expenditure to GDP is 0.0324 during the sample
period in Japan. However, the ratio is above 0.05 during the bubble years (1987–1989),
the highest over the past 40 years. We take the average ratio during 1987–1989, 0.0551,
as a target and set the spending floor constraint at Xxc = 0.0551. Thus, our analysis
may suggest how the growth and distribution of utility across time and generations wouldchange if Japan were to consider the ratio of Xxc = 0.0551.
4.2.1

Steady-State Comparison

Figure 2 illustrates the numerical results for the ratio of public education expenditure
to GDP (Panel (a)), steady-state physical capital (Panel (b)), and steady-state growth
rates (Panel (c)). In each panel, we compare the planner’s allocation with the politicalequilibrium in the presence and absence of the spending floor constraint by taking theplanner’s discount factor θ from 0 to 1 on the horizontal axis.
3

Source: https://data.worldbank.org/products/wdi (Accessed on August 26, 2017).

15

[Figure 2 here.]
Panel (a) shows that in the planner’s allocation, the ratio of public education expenditure to GDP increases as the planner’s discount factor θ increases. A higher θ implies
that the planner attaches a larger weight to future generations, thus having more incentives to invest in human capital through public education. Because of this incentive, theplanner’s allocation is more likely to attain a higher ratio of public education expenditure
to GDP than at the political equilibrium as his or her discount factor increases. However,when the spending floor constraint is introduced into political equilibrium, the planner’s
allocation is less likely to attain a higher ratio than that at the political equilibrium.
This result is straightforward, since the spending floor constraint works to spur the public
education expenditure.
Panel (b) plots the steady-state physical capital. In the planner’s allocation, steadystate physical capital is hump-shaped, peaking around θ = 0.75. This fact suggests
two opposing eﬀects of θ on physical capital accumulation: a positive eﬀect producedby the planner’s incentive to bequeath more physical capital to future generations anda negative eﬀect caused by the crowding out eﬀect of human capital investment. The
corresponding eﬀects also appear at the political equilibrium, but the negative eﬀect is
not as strong in the absence of the constraint. Thus, the political equilibrium in theabsence of the constraint attains higher steady-state physical capital than the planner’s
allocation. However, the negative eﬀect is strengthened by the spending floor constraint.
This implies that political equilibrium attains lower steady-state physical capital thanthat in the planner’s allocation for some range of θ. The presence of the constraint isthus crucial to determining the relative size of the steady-state physical capital stock in
the planner’s allocation and at political equilibrium.
Panel (c) plots the steady-state growth rate of per capita output. In the planner’s allocation, the steady-state growth rate increases as the planner’s discount factor increases.
A higher θ provides an incentive for the planner to invest more in education. In addition,
as argued above, a higher θ creates a positive eﬀect on physical capital accumulation,
which increases public education expenditure. Because of these two positive eﬀects on
education expenditure, the planner’s allocation attains a higher growth rate as his or her
discount factor increases. When the planner’s allocation is compared with the politicalequilibrium, the political equilibrium in the presence (absence) of the spending floor constraint attains a higher (lower) growth rate than the planner’s allocation for some rangeof θ. This is because, while the constraint has a negative eﬀect on physical capital accumulation through a rise in the labor income tax rate, it pushes up spending on public
education and thus promotes human capital accumulation. Therefore, the spending floor
constraint is crucial to determine the performance of economic growth in the political16

equilibrium relative to the planner’s allocation.
4.2.2

Comparison across Periods and Generations

As a baseline case, we assume θ = (0.99)120 , which is equal to the individual discountfactor β, and plot the movement of physical and human capital, output and consumption
over time, and the distribution of utility across generations in Figure 3. In this figure, wecall the old in period 0 “generation -1”, and the middle-aged in period j (= 0, 1, 2, · · · )“generation j”. In the next subsection, we analyze the outcomes by assuming diﬀerent
planner’s discount factors.
[Figure 3 here.]
We take a ratio of the variable at the political equilibrium to that in the planner’s
allocation for each period to compare them eﬀectively. Figure 3 plots the evolution ofphysical capital (Panel (a)), human capital (Panel (b)), per capita output (Panel (c)),
consumption in old age (Panel (d)), consumption in middle age (Panel (e)), and distribution of utility across generations (Panel (f)). The line denoted by un/pl (xc/pl) in
the figure implies the ratio of a concerned variable at the political equilibrium in theabsence (presence) of the spending floor constraint to that in the planner’s allocation.
For example, the line denoted by xc/pl shows the ratio of the variable in the presence ofthe constraint to that in the planner’s allocation. Each ratio implies that the politicalequilibrium outweighs the planner’s allocation when the ratio is above unity.
Comparison among political equilibria We first compare the political equilibrium
variables in the absence and presence of the spending floor constraint to facilitate theunderstanding of the result in Figure 3. The constraint has the following four eﬀects:(i) induces the politician to raise capital and labor income tax rates to finance increased
expenditure as presented in Proposition 2; (ii) increases human capital and thus theincome of successive generations as depicted in Panel (b) in Figure 3; (iii) lowers saving
and physical capital because of increased labor income tax burden as depicted in Panel (a)in Figure 3; and (iv) increases the marginal product of physical capital and thus returns
from savings because of the decreased physical capital. The per capita output depends
on the positive eﬀect of (ii) and the negative eﬀect of (iii), but the former outweighs thelatter as illustrated in Panel (c) of Figure 3.
The eﬀects on the consumption of the elderly and the middle are presented in Panels(d) and (e) of Figure 3, respectively. The eﬀects of (ii)–(iv) are irrelevant for the elderly
in period 0 (i.e., generation -1) because physical and human capitals are given as initialconditions; their consumption decreases as the lower bound of the constraint increases
17

owing to the negative eﬀect of (i). However, the eﬀects of (ii)–(iv) are relevant for theelderly from period 1 onward: their consumption increases as the lower bound increases
because the positive eﬀects of (ii) and (iv) outweigh the negative eﬀects of (i) and (iii).
The consumption of the middle-aged also depends on the eﬀects of (i)–(iv). In particular, period 0 consumption decreases when the spending floor constraint is introduced
because the negative eﬀects of (i) and (iii) outweigh the positive ones of (ii) and (iv).
However, the positive eﬀect of (ii) becomes larger over time, since human capital accumulates from one generation to the next. Therefore, the positive eﬀects of (ii) and (iv)come to outweigh the negative eﬀects of (i) and (iii) from middle-aged period 1 onward.
Based on the results above, we consider some welfare implications of the spending
floor constraint. First, the constraint crowds out the consumption of period 0 old and,thus, lowers their utility. Second, it increases human capital of the middle-aged fromgeneration 1 onward, which in turn increases income and consumption and, thus, improves
their utility. Finally, the constraint has two opposing eﬀects on the utility of the agents ingeneration 0: strengthening the constraint increases their old-age consumption, but lowers
their middle-age consumption. The agents in generation 0 are made better or worse oﬀ,
depending on the relative degree of the two opposing eﬀects on consumption.
Comparison between political equilibrium and planner’s allocation Next, we
compare the political equilibrium and planner’s allocation in terms of physical and humancapital, per capita output, middle- and old-age consumption, and utility. The physicalcapital is higher at the political equilibrium than in the planner’s allocation, as depicted
in Panel (a) of Figure 3. As observed in Panel (a) of Figure 2, the expenditure to GDP
ratio in the political equilibrium is lower than that in the planner’s allocation, regardlessof the presence or absence of the constraint. This implies that resources are devoted moreto physical capital formation at the political equilibrium than in the planner’s allocation.
Human capital in the political equilibrium is higher or lower than that in the planner’s
allocation. In particular, the relative strength depends on the following two opposing
eﬀects. First, the political equilibrium attains a lower education expenditure-to-GDP
ratio than the planner’s allocation. This implies that less resources are devoted to humancapital formation at the political equilibrium. Second, the political equilibrium realizes
a higher physical capital level. This in turn provides the agents with more resources foreducation expenditure and thus human capital formation. In the baseline case, the former
negative eﬀect outweighs the latter positive one, and so the human capital level is lower
in the political equilibrium than in the planner’s allocation, regardless of the presence orabsence of the constraint.
Consumption of the middle- and old-ages also depends on the degree of the spending
floor constraint as depicted in Panels (d) and (e) of Figure 3. Recall that the physical18

capital level in the political equilibrium is higher than that in the planner’s allocation.
This implies that politicians devote more resources for physical capital formation and lessfor consumption. This is a negative eﬀect on consumption at the political equilibrium.
However, the spending floor constraint stimulates human capital formation and, thus,
leaves more resources for consumption. Despite of this positive eﬀect, the net eﬀect on
the political equilibrium is negative for the old-age consumption of generations 1–7, as
depicted in Panel (d) of Figure 3. We observe the two opposing eﬀects for the middle-age
consumption at political equilibrium as well. The positive eﬀect outweighs the negative
one for generations 0–8 in the presence of the spending floor constraint. However, in
the absence of the constraint, the positive eﬀect overcomes the negative eﬀect only for
generations 0–2.
The per capita output at the political equilibrium becomes higher or lower than in theplanner’s allocation. Recall that the growth rate depends on physical and human capitalformation. As illustrated in Panel (a) of Figure 3, the physical capital is higher at thepolitical equilibrium than in the planner’s allocation. However, the education-to-GDP
ratio is lower at the political equilibrium than in the planner’s allocation. Therefore,
the political equilibrium attains a higher per capita output than the planner’s allocation
when the negative eﬀect through human capital formation is outweighed by the positiveeﬀect through physical capital formation. Such a case occurs across periods 1–6 when the
spending floor constraint is present; otherwise, the political equilibrium attains a lower
per capita output level than the planner’s allocation in the long run.
The results observed thus far allow us to draw some welfare conclusions, using theplanner’s allocation as benchmark. First, the period 0 old (i.e., agents in generation -1)
are made worse oﬀ by the political decision making, regardless of the presence or absenceof the spending floor constraint. Second, the agents in generation 0 are made better oﬀ
by the political decision making because the net eﬀect on their consumption is positive.
Finally, the agents from generation 1 onward are made better or worse oﬀ dependingon the presence or absence of the constraint. In particular, they are more likely to bebetter oﬀ in the presence of the constraint than in its absence because the lower boundworks to stimulate human capital formation and thus increases income and consumption.
Therefore, the spending floor could benefit some future generations at the cost of theperiod 0 old.
4.2.3

Sensitivity Analysis

The analysis in the previous subsection has been undertaken by assuming that the planner’s discount factor equals the individual one: β = θ = (0.99)120 . To check for therobustness of the results, we here perform a sensitivity analysis by considering alternative19

values of θ. In particular, we consider the following two scenarios: θ = (0.986)120 and(0.993)120 . The results are qualitatively unchanged when θ is below (0.986)120 or above(0.993)120 .
First, consider the scenario of θ = (0.986)120 , where the planner attaches lower weights
to future generations than in the baseline case. The results of this scenario are depicted
in Figure 4. In the present scenario, the planner has less incentive to invest in humancapital formation than in the baseline case, thereby resulting in a lower education-to-GDP
ratio in the planner’s allocation than at the political equilibrium. Therefore, the humancapital level is higher at political equilibrium than in the planner’s allocation. This inturn results in a higher physical capital level at political equilibrium than in the planner’s
allocation. Owing to these two positive eﬀects on political equilibrium, the consumption
and per capita output from period 1 onward and utility from generation 0 onward are
higher at the political equilibrium than in the planner’s allocation. However, the period 0
old are made worse oﬀ by the political decision making because less resources are devoted
to their consumption and more are devoted to education at political equilibrium.
[Figure 4 here.]
Next, consider scenario θ = (0.993)120 , where the planner attaches a higher weight
to future generations than in the baseline case. Figure 5 illustrates the results of this
scenario. We find the following three marked deviations from the baseline case. First,
human capital is higher in the planner’s allocation than at the political equilibrium. Thisis because the planner has now more incentive to invest in education than in the baseline
case.
[Figure 5 here.]
Second, the period 0 old’ s consumption is higher at the political equilibrium than in
the planner’s allocation. In other words, the initial old are made better oﬀ at the politicalequilibrium relative to the planner’s allocation. The reason for this result is that, given
a high weight to future generations, the planner now has more incentive to invest in
education as described above. This in turn results in less resources for consumption in
the planner’s allocation.
Third, the utility from generation 1 onward at the political equilibrium is lower thanin the planner’s allocation. That is, future generations are made worse oﬀ by politicaldecision making. This is because the planner that attaches a larger weight to future generations chooses a higher education-to-GDP ratio than the political equilibrium and thus
bequeaths more resources for future generation via human capital formation. Therefore,
current generations may benefit from the political decision making and the spending floor
constraint at the cost of future generations at the political equilibrium.
20Summarizing the above results, we can conclude that the political equilibrium generally diﬀers from the planner’s allocation and that the political decision making creates a
trade-oﬀ between generations in terms of welfare. In particular, the current generations
are made worse (better) oﬀ and the future generations are made better (worse) oﬀ by
the political decision making when the planner’s discount factor is low (high) relative
to the individual discount factor. This suggest that the choice of the planner’s discountfactor matters as to whom benefits and whom loses from the political decision making
and spending floor constraint.

5

Conclusions

This study presented a three-period-lived overlapping-generations model with physicaland human capital accumulation. Public education contributes to human capital formation and is funded by taxing the labor income of the working generation and capitalincome of the retired generation. Within this framework, we employed probabilistic voting
to demonstrate the political determinant of both taxes and expenditure and investigatedits impacts on economic growth. We showed that aging has a negative impact on economic
growth via fiscal policy choice.
To resolve the negative growth eﬀect, we proposed a spending floor for public education
expenditure, which is or was employed by some East Asian countries. We showed thatthe introduction of the spending floor stimulates economic growth and benefits future
generations at the expense of the current old. To further explore the implications of theresults, we calibrated the model to the Japanese economy and examined the growth andwelfare consequences of the spending floor constraint.
There are some caveats to the conclusions drawn in this study. First, it should be notedthat our analysis is based on the assumption that parents exhibit no altruism toward theirchildren. An alternative is to assume that parents are concerned about their children’s
income. In fact, earlier studies rely on altruism toward children to explain support for
public education policies (see, e.g., Glomm and Ravikumar, 1992, 1998; Saint-Paul andVerdier, 1993), and some recent studies also rely on this assumption (Kaganovich andMeier, 2012; Naito, 2012; and Lancia and Russo, 2016). Because the focus is on theanalysis of small open economies, the general equilibrium eﬀects of public funding of
education through interest rates are abstracted from these studies.
The public funding for education increases the return on saving and capital by improving the skills of the next generations. Thus, the present generation may wish to
support a publicly financed education system to enhance their future income. Soares(2003, 2006) shows that this non-altruistic incentive to support public education is quantitatively important. Based on this argument, the present study assumed away altruism
21toward children and focused on the general equilibrium eﬀect of education funding throughthe interest rate. An analysis in the presence of altruism is conducted by Ono and Uchida(2018), whose results are qualitatively unchanged under this alternative assumption.
Second, we assumed that the old do not enjoy public expenditure. This assumption
may be somewhat unrealistic since, in most advanced countries, the social expenditure on
the old, as a share of GDP, amounts to 20% (OECD, 2019), which is four times as large
as the public education expenditure (OECD, 2018). When the old are directly benefitingfrom the public expenditure such as public good provision, the spending floor constraint
would induce diﬀerent welfare eﬀects across generations through public good provision.
However, we show in Appendix B.1 that the intergenerational trade-oﬀ in Proposition 3
still exists: the old are made worse oﬀ, while future generations are made better oﬀ by
the introduction of the spending floor constraint.
Finally, private education was abstracted away from the main analysis. It would
be more reasonable to include private education into the model, since the latter partof this study focuses on the Japanese economy and the private education expenditure
share is quite high in Japan (OECD, 2018). In Appendix B.2, we demonstrate a model
including private education, and indicate that the presence of private education creates
two opposing eﬀects on the choice of public education. However, the extended model
suggests that there still arises an intergenerational trade-oﬀ in terms of welfare when the
spending floor constraint is introduced.

22

A

Proofs and Supplementary Explanations

A.1

Derivation of VtM and Vto

To derive VtM in (6), recall that the utility of the middle-aged is given by VtM = ln ct +β ln dt+1 . Given the consumption and savings functions and human capital formation
function in Section 2, this utility function is rewritten as()
k
β 1 − τt+1
Rt+1
1
M
Vt = ln(1 − τt )wt ht + β ln
(1 − τt )wt ht ,
1+β
1+β
or,

(
)
k
VtM = (1 + β) ln(1 − τt )wt ht + β ln 1 − τt+1
Rt+1 + C,

where
C ≡ ln(22)

1β+ β ln.
1+β1+βTerm (1 − τt )wt ht in (22) is rewritten as follows:()τtk Rt st−1α(1 − τt )wt ht = (1 − α)A (kt ) ht − (1 + n)xt −
1+n
(
)
α
k
= Z xt , τt , A (kt ) ht ≡ (1 − α)A (kt )α ht − (1 + n)xt + τtk αA (kt )α−1 kt ht ,(23)
where the first equality comes from the first-order conditions for profit maximization withrespect to Ht , wt = (1 − α)A (kt )α , and the government budget constraint in (4) and thesecond equality comes from the first-order conditions for profit maximization with respect
to Kt , ρt = αA (kt )α−1 , and the capital market-clearing condition, (1 + n) · kt ht = st−1 .
Term β ln Rt+1 in (22) is reformulated as follows:
β ln Rt+1 = β ln αA (kt+1 )α−1 ,(24)where the equality comes from the first-order conditions for profit maximization withrespect to Kt+1 and Ht+1 . Term kt+1 in (24) is reformulated by using the capital marketclearing condition as follows:st(1 + n)ht+1
β1=(1 − τt )wt ht
η
1−η ×
1+β
(1 + n) D (xt ) (ht )
) )
)
(
(
(
1
β
α
k
,
,
A
(k
)
h
= P xt , Z xt , τtk , A (kt )α ht , ht ≡
×
Z
x
,
τ
t
t
t
t
(1 + n) D (xt )η (ht )1−η 1 + β(25)

kt+1 =

where P (·, ·, ·), denoting the physical to human capital ratio, has the following properties:()∂P/∂xt = Px + Pz (∂Z/∂x) < 0 and ∂P/∂τtk = Pz ∂Z/∂τtk > 0, where Px and Pz denote23

the derivatives of P with respect to the first and second arguments, respectively. The
first line in (25) comes from the capital market-clearing condition, the second from thesavings function, and the third from the government budget constraint.
By using (23)–(25) and rearranging the terms, we can reformulate VtM in (22) as
follows:
(
)
(
) ( (
(
) ))
k
VtM = (1 + β) ln Z xt , τtk , A (kt )α ht +β ln 1 − τt+1
R P xt , Z xt , τtk , A (kt )α ht , ht +C.
We next derive Vto in (7). Recall that Vto is defined as Vto = ln dt + γ ln wt ht . Given
that st−1 = (1 + n)kt ht and Rt = αA (kt )α−1 , Vto is rewritten as follows:()
Vto = ln Rt 1 − τtk st−1()
= ln 1 − τtk + ln αA (kt )α ht (1 + n).
■A.2

Proof of Proposition 1

Assume that the first-order condition with respect to τ k holds with an equality. We
substitute (15) into (14). By rearranging the terms, we obtain((1 + n)x = βη(1 − α)ω
(1 + n)(1 − ω)

)−1

(
)
αA (k)α h 1 − τ k .

(26)

We substitute this into (14) and solve for τ k to obtainkτ k = τun
≡1−

ω
1
·
,
αΛ (1 + n)(1 − ω)

where Λ is defined as per Proposition 1. Thus, the conjecture is verified as long askτ k′ = τ̄ k = τun.
The corresponding public education expenditure becomes(1 + n)x =Xun
A (k)α h,1+nwhere Xun is defined as follows:Xun ≡βη(1 − α).
ΛWith the government budget constraint in (13), we can compute the labor income tax
rate as1 + αβτ = τun ≡ 1 −.
(1 − α)Λ
24

k
We find that the tax rates τunand τun are below one. They are greater than or equal
to zero if the following conditions hold:ωα≤{1 + β (α + η (1 − α))} ,(1 + n)(1 − ω)
1−α
ω
α
≥
{1 + β (α + η (1 − α))} − βη.
τun ≥ 0 ⇔(1 + n)(1 − ω)
1−α
k
τun
≥0⇔

k
Therefore, τun∈ [0, 1) and τun ∈ [0, 1) hold if the assumption in (16) holds.

■

A.3

Proof of Proposition 2

Conjecture that the constraint is binding: (1+n)x = Xxc ·A(k)α h. The first-order condition
with respect to τ k in (14) holds with an equality since the choice of τ k is unconstrained.
We assume an interior solution of τ k and substitute the conjecture (1+n)x = Xxc ·A(k)α h
into (14) to obtainω(1 + αβ) αA (k)α h(1+n)(1−ω)
(−1)
+
= 0.
1 − τk((1 − α) + ατ k ) A (k)α h − Xxc · A(k)α hkBy rearranging the terms, we obtain τ k = τxc.
k
We substitute (1 + n)x = Xxc · A(k)α h and τ k = τxc
into the first-order condition withrespect to x in (15) and rearrange the terms. Then, we obtain Xun ≤ Xxc . This condition
holds with a strict inequality by assumption. Thus, the conjecture is verified.
k
To derive the labor income tax rate, we substitute (1+n)x = Xxc ·A(k)α h and τ k = τxc

into the government budget constraint in (13) and obtain τ = τxc . These tax rates implythat
α (1 + αβ)ω≤,(1 + n)(1 − ω)
(1 − α) − Xxc()
1 − Xxcωτxc ≥ 0 ⇔
− 1 (1 + αβ) ≤.
1−α(1 + n)(1 − ω)
k
τxc
≥0⇔

■

A.4

Proof of Proposition 3

We first compare the growth rates. The growth rate in the absence of the constraint isgiven by (19). The growth rate in the presence of the constraint, denoted by h′ /h|xc , isgiven by replacing Xun and τun with Xxc and τxc , respectively. By direct comparison, we

25

have
h′
h′
≷
⇔ (Xun )1−α (1 − τun )α ≷ (Xxc )1−α (1 − τxc )α
h un
h xc
α

(
)1−α
1+αβ
Xxc
(1−α)Λ⇔
[]−1  ≷
ωXun(1+n)(1−ω)
1−Xxc· 1 + 1+αβ
1−α
]α
[
ω
1 + αβ + (1+n)(1−ω)⇔(Xun )1−α ≷ (1 − Xxc )α (Xxc )1−αΛ⇔ (1 − Xun )α (Xun )1−α ≷ (1 − Xxc )α (Xxc )1−α .
The right-hand side of the last expression, denoted by RHS, has the following properties:∂RHS(1 − α) − Xxc ∂RHS
∂RHS
= RHS ·
,
> 0,
= 0.
∂Xxc
(1 − Xxc ) Xxc ∂Xxc Xxc =Xun
∂Xxc Xxc =1−αThese properties imply thath′h′<∀Xxc ∈ (Xun , 1 − α) .
h un
h xc
We next compare capital income tax rates. Direct comparison leads to the following
result:]−1[1ω
1 − Xxc
1 + αβkkτun ≷ τxc ⇔ 1 −
·
≷1−
· 1+
ω
αΛ (1 + n)(1 − ω)α(1+n)(1−ω)⇔

1 − Xxc1≷ω+ 1 + αβΛ(1+n)(1−ω)βη (1 − α)≷ XxcΛ⇔ Xun ≷ Xxc .

⇔kkGiven the assumption of Xun < Xxc , we obtain τun< τxc.

The labor income tax rates are compared as follows:
1 − Xxc
1 + αβ (1 + γ)
1 + αβ (1 + γ)
≶1−·
(1 − α) Λ
1 − α Λ − β (1 + γ) η (1 − α)⇔ (1 − Xxc ) Λ ≶ Λ − β (1 + γ) η (1 − α)βη (1 − α)
⇔
≶ Xxc
Λ
⇔ Xun ≶ Xxc ,τun ≶ τxc ⇔ 1 −where the last line comes from the definition of Xun . Given the assumption of Xun < Xxc ,we obtain τun < τxc .
26

Finally, we compare the two cases in terms of the welfare of the initial old and steadystate generations. The initial old are made worse oﬀ by the introduction of the constraint
because the capital income tax rate increases. The welfare of the middle-aged in some
generation t is, from (6),[]VtM = (1 + αβ) ln (1 − α)A (kt )α ht − (1 + n)xt + τtk αA (kt )α ht()+ βη (1 − α) ln xt + β ln 1 − τ k + β (1 − α) ln D (ht )1−η + C.
Given that xt = Xj A (kj )α ht , j = un, xc, in the steady state, the above expression isreformulated as
[
]
M
Vt,j
≃ (1 + αβ) ln (1 − α) − Xj + τjk α A (kj )α()
Xj+ βη (1 − α) ln
A (kj )α + β ln 1 − τjk + (1 + β) ln htj .
1+nRecall that kj and τjk are constant stationary along the steady-state path. However,human capital htj increases along the steady-state path and the diﬀerence between ht,unMMand ht,xc also increases over time. Therefore, Vt,un< Vt,xc
holds in the steady state.

■

A.5

Proof of Proposition 4

In the present framework, state variable ht does not lie in a compact set because it
continues to grow along an optimal path. To reformulate the problem into one in which
the state variable lies in a compact set, we undertake the following normalization:c̃t ≡ ct /ht , d˜t ≡ dt /ht , and x̃t ≡ xt /ht .
Then, the above resource constraint is rewritten as
c̃t +

1 ˜
dt + (1 + n)kt+1 D (x̃t )η + (1 + n)x̃t = A (kt )α ,
1+n

(27)

and the utility function becomes
U−1 = β ln d˜0 + β ln h0 ,U0 = ln c̃0 + β ln d˜1 + ηβ ln x̃0 + (1 + β) ln h0 + β ln D,
Ut = ln c̃t + β ln d˜t+1 + βη ln x̃t + η (1 + β)t−1∑ln x̃jj=0+ (1 + β) ln h0 + {β + t (1 + β)} ln D, t ≥ 1.
The planner’s objective function is now given by
{
[
}
]
∞
∑
θ
β ˜
t
SW (k0 ) ≃θ ln c̃t + ln dt + η β + (1 + β)
ln x̃t ,
θ
1−θ
t=0
27

(28)

where the constant terms are omitted from the expression. Thus, we can express the
Bellman equation for the problem as follows:
}
}
{
{
θ
β ˜
′
V (k) = max
ln c̃ + ln d + η β + (1 + β)
ln x̃ + θV (k ) ,(29)
˜ ′ ,x̃}θ
1−θ{c̃,d,ksubject to (27), where k ′ denotes the next period stock of physical capital and V (·) the
optimal value function.
We substitute (27) into (29) to reformulate the problem as
]
{ [
1 ˜
α
′
η
d − (1 + n)k D(x̃) − (1 + n)x̃V (k) = max ln A (k) −˜ ′ ,x̃}
1+n
{d,k
{[
]
}
β ˜
θ
′
+ ln d + η β +(1 + β) ln x̃ + θ · V (k ) .
θ
1−θ

(30)

˜ k ′ , and x̃ are, respectively,
The first-order conditions with respect to d,β/θ
1/(1 + n)
=
,
(31)
d˜ :
c̃
d˜
(1 + n)D(x̃)η
= θ · V ′ (k ′ ),
(32)
k′ :
c̃
{
}
θ
η β + 1−θ
(1 + β)η(1 + n)k ′ D(x̃)η−1 + (1 + n)
x̃ :
=
.
(33)
c̃
x̃
We assume V (k ′ ) = ϕ0 + ϕ1 ln k ′ , where ϕ0 and ϕ1 are undetermined coeﬃcients. For
this assumption, (32) becomes(1 + n) · D(x̃)η · k ′ = θϕ1 · c̃.
From (33) and (34), we obtain

(34)

[

]
θ
(1 + n)x̃ = η · β +(1 + β) − θϕ1 · c̃.
1−θ

(35)

The substitution of (31), (34), and (35) into the resource constraint in (27) leads toc̃ =1
A (k)α ,
ϕ

(
}
)
{
β
θ
ϕ≡ 1+
(1 + β) .
+ θϕ1 (1 − η) + η β +θ1−θ˜ x̃, and k ′ become
The corresponding functions of d,
where

β 1
d˜ = (1 + n) · · A (k)α ,
θ ϕ
[
{(
)
}]
β
1
1
· ϕ−
1+
x̃ =
+ θϕ1
· A (k)α ,
1+n
θ
ϕ
(
)1−η
1
αβγ + θϕ1
α
′
k =
A (k)
.
)}]η ·(()[ 1 {ϕ· ϕ − 1 + βθ + θϕ1(1 + n)D 1+n
28

(36)
(37)

Substituting these policy functions into the Bellman equation gives
V (k) = Cons (ϕ0 , ϕ1 ) + αϕ ln k,where Cons (ϕ0 , ϕ1 ) includes constant terms. The guess is verified if ϕ0 = Cons (ϕ0 , ϕ1 )and αϕ = ϕ1 . Therefore, ϕ1 and ϕ0 are given by
[(
)
{
}]
α
β
θ
ϕ1 =
· 1+
+η β+
(1 + β) ,1 − αθ(1 − η)
θ
1−θ
[(
)
{
}]
1
β
θ
ϕ0 =
· 1+
+η β+
(1 + β) .
1 − αθ(1 − η)
θ
1−θ
■

29

B

Extensions

In the main text, we assume (i) no provision of public good that benefits the old and (ii)no private education as a complement to public education. In this appendix, we briefly
consider how the analysis and results would change if we relax each of these assumptions.

B.1

Public Good Provision

We introduce into the model the provision of a public good that benefits both the middleaged and the old. For analysis, consider the following utility function:
[ (
)
]
k
Ut = ln ct + θ ln gt + β · ln 1 − τt+1
Rt+1 st + θ ln gt+1 ,(B.1)
where g is a per capita public good provision and θ(> 0) is the weight on the utility of public good provision. The aggregate public good expenditure during period t is
(Nt + Nt−1 ) gt , so the government budget constraint in period t becomesτt wt ht Nt + τtk Rt st−1 Nt−1 = Nt+1 xt + (Nt + Nt−1 ) gt
or

τtk Rt st−1
2+n
= (1 + n)xt +
gt .
(B.2)
1+n
1+n
Following the same procedure as in the main text, we can express the indirect utility ofthe middle-aged during period t, VtM , and that of the old during period t, VtO , respectively,
as follows:
(
)
(
) ( (
(
) ))
k
VtM = (1 + β) ln Z̃ xt , τtk , gt , A (kt )α ht + β ln 1 − τt+1
R P̃ xt , Z̃ xt , τtk , gt , A (kt )α ht , ht
τt wt ht ++ θ · [ln gt + β ln gt+1 ] + C,()
VtO = ln 1 − τtk + ln αA (kt )α ht (1 + n) + θ ln gt ,

(B.3)
(B.4)

where C, including constant terms, is defined in Section 2, and Z̃ and P̃ are respectively
defined as follows:()
2+n
gt + ατtk A (kt )α ht ,Z̃ xt , τtk , gt , A (kt )α ht ≡ (1 − α) A (kt )α ht − (1 + n)xt −
1+n
β
)
(
)
)
(
(
1+β
α
k
P̃ xt , Z̃ xt , τtk , gt , A (kt )α ht , ht ≡η1−η Z̃ xt , τt , gt , A (kt ) ht .
(1 + n)D (xt ) (ht )From (B.3) and (B.4), we can reformulate the political objective function in (11) as
follows:
(
)
(
)
ω
Ω≃
ln 1 − τ k + (1 + β) ln Z̃ x, τ k , A (k)α h(1 + n)(1 − ω)
(
) ( (
(
) ))
+ β ln 1 − τ k′ R P̃ x, Z̃ x, τ k , A (k)α h , h[]ω+θ· 1+
ln g + βθ ln g ′ ,(B.5)(1 + n)(1 − ω)30

where the irrelevant terms are omitted from the expression of Ω.
We conjecture τ k′ and g ′ in (B.5) as follows:τ k′ = τ̄ k ,(B.6)

g ′ = G0 · A (k ′ ) h′ ,
α

(B.7)

where τ̄ k ∈ (0, 1) and G0 (> 0) are constant parameters. The conjecture in (B.7) leads to
[
]α
)]α
[
]1−α [ (β′· Z̃ x, τ k , A (k)α h.
(B.8)
g = G0 · A ·· D (h)1−η (x)η(1 + n)(1 + β)
Substituting (B.7), (B.8), and the definition of Z̃ into (B.5) and rearranging the terms,
we have()ω
ln 1 − τ k(1 + n)(1 − ω)
]
[
2+n
α
α
k
g + ατ A (k) h+ [1 + βα (1 + θ)] ln (1 − α) A (k) h − (1 + n)x −
1+n
[
]
(
)
ω
k
+ β ln 1 − τ̄ + βη (1 − α) (1 + θ) ln x + θ · 1 +
ln g.
(B.9)
(1 + n)(1 − ω)Ω≃The first-order condition, with respect to τ k , x, and g are as follows:τkω
[1 + βα (1 + θ)] αA (k)α h(1+n)(1−ω): (−1)+≤ 0,
1 − τkZ̃

[1 + βα (1 + θ)] (1 + n) βη (1 − α) (1 + θ)
= 0,
+
x
Z̃
]
[
ω
2+n
θ
·
1
+
[1 + βα (1 + θ)] 1+n(1+n)(1−ω)
g : (−1)
+
= 0.
g
Z̃

x : (−1)(B.10)(B.11)(B.12)

Suppose that τ k > 0 holds. Eq. (B.10) holds with an equality. Eqs. (B.10) and (B.11)
lead to the optimal relation between x and τ k , as follows:(1 + n)x =()(1 + n)(1 − ω)βη (1 − α) (1 + θ) αA (k)α h 1 − τ k .
ω

(B.13)In addition, Eqs. (B.10) and (B.12) lead to the optimal relation between g and τ k , asfollows:[]()(1 + n)(1 − ω)
ω
2+n
g=
θ 1+
αA (k)α h 1 − τ k .
(B.14)1+nω(1 + n)(1 − ω)The substitution of (B.13) and (B.14) into the first-order condition with respect to τ k ink, as follows:(B.10) leads to the policy function of τ k , denoted by τun
k
=1−
τun

1
ω
·
.
α (1 + θ) Λ (1 + n)(1 − ω)31

(B.15)

This shows that the conjecture in (B.6) is verified.
Next, we obtain the policy function of x, denoted by xun , by substituting (B.15) into
(B.13):(1 + n)xun = Xun A (k)α h.
(B.16)We also obtain the policy function of g, denoted by gun , by substituting (B.15) into (B.14):
2+ngun = Gun A (k)α h,
1+nwhere Gun is defined by
Gun ≡

[
]
ω
θ 1 + (1+n)(1−ω)(1 + θ) Λ(B.17).

Finally, substituting (B.15), (B.16), and (B.17) into the government budget constraint,
we obtain the policy function of τ , denoted by τun :τun = 1 −1 + βα (1 + θ).
(1 − α) (1 + θ) Λ

(B.18)

Next, we introduce the spending floor constraint and derive the policy functions in its
presence. Recall that the policy function of x in the absence of the constraint is (B.16).
This is equivalent to the function in the model without public good provision. Thus,
based on the analysis in Section 3.3, we introduce the following spending floor constraint:(1 + n)x≥ Xxc (> Xun ) .
A (k)α h(B.19)

Conjecture that this constraint is binding:(1 + n)x= Xxc .
A (k)α h(B.20)

The first-order conditions with respect to τ k in (B.10) and g in (B.12) lead to (B.14). We
assume an interior solution of τ k and substitute (B.14) and (B.20) into (B.10) to obtaink
the policy function of τ k , denoted by τxc
:
1
k
=1− ·
τxc

ω
(1 − Xxc )(1+n)(1−ω)α (1 + θ)

{[
]
}.
ω
1 + (1+n)(1−ω)+ βα(B.21)The substitution of (B.21) into (B.14) leads to the policy function of g, denoted by gxc :2+n
gxc = Gxc · A (k)α h,
1+nwhere Gxc is defined byGxc ≡θ
[1 + βα (1 + θ)] (1 − Xxc )
]
· [
.
ω
1+θ
1+
+ βα
(1+n)(1−ω)

32(B.22)

We substitute (B.20), (B.21), and (B.22) into the first-order condition with respect tox in (B.11) and rearrange the terms. Then, we obtain Xun ≤ Xxc . This condition holdswith a strict inequality by the assumption. Thus, the conjecture in (B.20) is verified. The
labor income tax rate, denoted by τxc , is obtained by substituting (B.20), (B.21), and(B.22) into the government budget constraint:(1 − Xxc )1(1+n)(1−ω){[]}.
τxc = 1 −·ω
1 − α (1 + θ) 1 ++ βαω

(B.23)(1+n)(1−ω)

Based on the characterization of the policy functions, we examine the welfare eﬀects
of the spending floor constraint across generations. For this, we first compare the indirect
utility functions of the initial old (i.e., the old in period 0) in the absence and presenceOOof the constraint, denoted by V0,unand V0,xc, as follows:

(
)
(
)
k
k
O
O
V0,un
≷ V0,xc⇔ ln 1 − τ0,un+ θ ln g0,un ≷ ln 1 − τ0,xc+ θ ln g0,xc
1 − Xxc
1 − Xxc⇔ 0 ≷ ln
+ θ ln.
1 − Xun
1 − Xun(B.24)

Given the assumption of Xun < Xxc , the sign of the right-hand side in (B.24) is negative.
The first term on the right-hand side shows the negative eﬀect of the increased capitalincome tax rate, while the second term shows the negative eﬀect of decreased public goodO
O
provision. Because of these two negative eﬀects, it holds that V0,un> V0,xc, indicating thatthe initial old are made worse oﬀ by the introduction of the spending floor constraint.
Next, consider the indirect utility of the middle-aged in period t, VtM , in (B.3). Substituting the policy functions into (B.3), we obtain

[
]
(
)
M
Vt,j
≃ (1 + β) ln (1 − α) − Xj − Gj + τjk α + β ln 1 − τjk+ θ [ln Gj + β ln Gj (Xj )η ] + (1 + β) (1 + θ) ln ht,j ,where j = un, xc and constant terms are omitted from the expression. The first threeterms are constant across periods, while the last term increases over time. Recall thatthe growth rate of human capital is given by (19). Given that Xun < Xxc and τun < τxc ,we have h′ /h|un < h′ /h|xc in the steady state. This implies that ht,un < ht,xc holds inMMholds from some future period t< Vt,xc
the long run. Thus, we can conclude that Vt,unonward. This suggests an intergenerational trade-oﬀ in terms of welfare.

B.2

Private Education as a Complement to Public Education

This subsection extends the model presented in Appendix B.1 by introducing private
education, denoted by et , that works as a complement to public education. The budget
constraint of the middle-aged is ct +st +et ≤ (1 − τt ) wt ht and the human capital formation33

is ht+1 = D (xt )η (et )δ (ht )1−η−δ , where δ ∈ (0, 1 − η) is the elasticity of human capital withrespect to et .
To create demand for private education, we assume that parents are altruistic toward
their children and concerned about their income in the middle age. In particular, thepreferences of the period-t middle-aged are specified by the following utility function oflogarithmic form:
Ut = ln ct + θ ln gt + β · [ln dt+1 + θ ln gt+1 + γ ln wt+1 ht+1 ] ,(B.25)

where γ(> 0) denotes the intergenerational degree of altruism.
We substitute the budget constraints and human capital formation function into (B.25)
to form the following unconstrained maximization problem:()
k
max ln [(1 − τt ) wt ht − st − et ] + β ln 1 − τt+1
Rt+1 st + βγ ln D (xt )η (et )δ (ht )1−η−δ .
{st ,et }By solving this problem, we respectively obtain the following consumption, saving, andprivate education functions:()
k
β 1 − τt+1
Rt+1
1
(1 − τt ) wt ht , dt+1 =(1 − τt ) wt ht ,ct =1 + β (1 + γδ)
1 + β (1 + γδ)
β
βγδ
st =
(1 − τt ) wt ht , et =(1 − τt ) wt ht .
1 + β (1 + γδ)
1 + β (1 + γδ)The substitution of these functions into (B.25) leads to the indirect utility function of
the period-t middle-aged:
))
(
)
(
) ( (
α
M
k
k
Vt = [1 + β (1 + γδ)] ln Z̃ xt , gt , τt , A (kt ) ht + β ln 1 − τt+1 R P̂ xt , Z̃ (·) , ht
+ θ [ln gt + β ln gt+1 ] + βγη ln xt + Ĉ (ht ) ,(B.26)

where Ĉ (ht ) includes the terms unrelated to political decisions, Z̃ is defined in Appendix
B.1, and P̂ is defined as follows:()

P̂ xt , Z̃ (·) , ht ≡[(1 + n)D(β
1+β(1+γδ)βγδ
1+β(1+γδ)

]δ

1−η−δ

·

)1−δ
Z̃ (·)(ht )(xt )η

.

The indirect utility function of the old is the same as in Eq. (B.4).
With the use of (B.4) and (B.26), we can formulate the political objective function as
follows:()()ω
ln 1 − τ k + (1 + β (1 + γδ)) ln Z̃ x, τ k , g, A (k)α hΩ≃(1 + n)(1 − ω)
))
(
) ( (
+ β ln 1 − τ k′ R P̂ x, Z̃ (·) , h[]ω+θ· 1+
ln g + βθ ln g ′ + βγη ln x.
(B.27)(1 + n)(1 − ω)34

Rearranging the terms, Eq. (B.27) is reduced to:()ω
ln 1 − τ k + βη [(1 + θ) (1 − α) + γ] ln x
(1 + n)(1 − ω)()
+ {1 + βα (1 + θ) + βδ [(1 + θ) (1 − α) + γ]} ln Z̃ x, τ k , g, A (k)α h[]ω+θ· 1+
ln g.
(1 + n)(1 − ω)Ω≃The main departure from the model in Appendix B.1 is that the altruism toward children, denoted by γ, and the elasticity of private education, denoted by δ, aﬀect politicaldecision making. The altruism induces policy makers to increase public education expenditure and private education induces the politician to attach more weight on the disposal
income of the middle-aged, Z̃ (·), thus reducing public education expenditure. Therefore,
there are two opposing eﬀects of private education on public education expenditure. However, the shape of the functions V O , V M , and Ω remains almost unchanged. Thus, we
can apply the analysis in Appendix B.1 and conclude that the spending floor constraint
creates an intergenerational trade-oﬀ in terms of welfare.

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Government expenditure on educa!on (% of GDP)
6
5.5
5
4.5
4
3.5
3

Figure 1: Evolution of the ratio of public education expenditure to GDP in Japan, 1971–
2015.
Note: Some observations are missing due to data unavailability.
Source: https://data.worldbank.org/products/wdi (Accessed on March 13, 2019).

39

Figure 2: Steady-state education-to-GDP ratio (Panel (a)), physical capital (Panel (b)),
and growth rate (Panel (c)). The horizontal axis shows the planner’s discount factor, θ.

40

Figure 3: Evolution of physical capital ratio (Panel (a)), human capital ratio (Panel (b)),
per capita output ratio (Panel (c)), consumption ratio in old age (Panel (d)), consumption
ratio in middle age (Panel (e)), and utility ratio (Panel (f)) when θ = (0.99)120 . Note:
Generation -1 represents the old in period 0; generation j (= 0, 1, 2, · · · ) represents the
middle-aged in period j.

41

Figure 4: Evolution of physical capital ratio (Panel (a)), human capital ratio (Panel (b)),
per capita output ratio (Panel (c)), consumption ratio in old age (Panel (d)), consumption
ratio in middle age (Panel (e)), and utility ratio (Panel (f)) when θ = (0.986)120 .

42

Figure 5: Evolution of physical capital ratio (Panel (a)), human capital ratio (Panel (b)),
per capita output ratio (Panel (c)), consumption ratio in old age (Panel (d)), consumption
ratio in middle age (Panel (e)), and utility ratio (Panel (f)) when θ = (0.993)120 .

43