Discussion Papers In Economics
And Business

Political Economy of Taxation, Debt Ceilings, and Growth

Tetsuo Ono and Yuki Uchida
Discussion Paper 18-22

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPolitical Economy of Taxation, Debt Ceilings, and Growth

Tetsuo Ono and Yuki Uchida
Discussion Paper 18-22

July 2018

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPolitical Economy of Taxation, Debt Ceilings, and
Growth∗Tetsuo Ono†

Yuki Uchida‡

Osaka University

Seikei University

Abstract
This study presents an overlapping-generations model with physical and humancapital accumulation and considers probabilistic voting over capital and labor taxes
and public debt to finance public education expenditure. Our analysis shows thatthe greater political power of the old induces the government to raise the labortax on the young and lower the capital tax on the old as well as issue debt. The
analysis also shows that the introduction of a debt ceiling rule calls for a rise in thelabor tax and thus lowers the welfare of the currently working generation. However,it increases the growth rate, and this growth eﬀect raises the welfare of future
generations. These benefits last for a long period even if the rule is imposed only
for a limited time.
• Keywords: Capital taxation, Public debt, Economic growth, Probabilistic voting, Overlapping-generations model• JEL Classification: D70, E24, H63∗

This work was supported by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid
for Scientific Research (C) Grant Number 18K01650.
†
Tetsuo Ono: Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka
560-0043, Japan. E-mail: tono@econ.osaka-u.ac.jp.
‡
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.

1

Introduction

How the government should distribute the tax burden between capital and labor income
has been a long-standing issue in the public finance literature since the seminal paperof Ramsey (1927). A large part of the literature has tackled this issue by assuming
a benevolent planner who can commit to future policies. For example, Judd (1985),
Chamley (1986), and Atkeson, Chari, and Kehoe (1999) show that the optimal tax rateon capital income is zero in the long run; expenditure should thus be financed solely bylabor income taxation. This conclusion is robust to whether the government can issue debt
or must run a balanced budget in each period (Ljungqvist and Sargent, 2012, Chapter
16); it is also robust in overlapping-generations models in which all households have finite
lifespans, whereas the planner has an infinite horizon (Erosa and Gervais, 2002; Garriga,2017).
However, this so-called Ramsey approach has the following two limitations. Firstly,
politicians, as representatives of the current population, are short-lived and unable tocommit to complete sequences of future policies in practice (Acemoglu, Golosov, andTsyvinski , 2011). In particular, these policymakers try to meet the conflicting preferencesof diﬀerent generations (Hassler et al., 2005; Bassetto, 2008). Secondly, older generations
are likely to pass the burden onto future generations by issuing public debt, a key sourceof government revenue (Song, Storesletten, and Zilibotti, 2012; Müller, Storesletten, andZilibotti, 2016). To control excessive debt issues, many countries have formulated nationalor supranational fiscal rules that ensure the sustainability of public finance (Schaechteret al., 2012).
Given this background, the following questions arise: (i) how does the conflict between
generations influence fiscal policymaking, (ii) how does policymaking in turn aﬀect growth
and welfare across generations, and (iii) does the introduction of a debt rule increase
growth and welfare? To answer these questions, we present an overlapping-generations
model with physical and human capital accumulation (e.g., Lambrecht, Michel, and Vidal,
2005; Kunze, 2014; Ono and Uchida, 2016, 2018b). Each generation comprises many identical individuals who live over three periods: young, middle, and old ages. Middle-aged
individuals (called parents) care about their children’s wage income. Public education
spending and parental human capital are inputs in the human capital production process,thereby contributing to children’s human capital formation and economic growth. Governments, as elected representatives, finance public education spending through taxes oncapital and labor income and public debt issues.
Under this framework, we consider the politics of fiscal policy formation. In particular,
we assume probabilistic voting à la Lindbeck and Weibull (1987) to demonstrate the extent
to which generations face conflict over such policies. In each period, middle-aged and old1individuals vote.1 The government in power maximizes the political objective function of
the weighted sum of the utilities of the middle-aged and old populations. In this voting
environment, the current policy choice aﬀects the decision on future policy via physicaland human capital accumulation. To demonstrate this intertemporal eﬀect, we employ
the concept of a Markov-perfect equilibrium under which fiscal policy today depends onthe current payoﬀ-relevant state variables, namely physical and human capital and public
debt. Given this process, we characterize the political equilibrium in the presence orabsence of a debt rule, which yields the following findings.
Firstly, the relative political power of the old is important for shaping tax rates. In
particular, the greater political power of the old incentivizes the government to shift the
tax burden from the old to the young. In addition, in the absence of any debt rule, theeducation expenditure-to-GDP ratio and debt-to-GDP ratio decrease with the greater political power of the old. The eﬀect on the education expenditure-to-GDP ratio is intuitivesince the old do not benefit from education expenditure. However, the eﬀect on the debtto-GDP ratio, which is the opposite to that suggested by Song, Storesletten, and Zilibotti(2012), is somewhat counterintuitive since the old may have an incentive to pass the fiscal
burden onto future generations by issuing public debt. This counterintuitive result arises
because the old want to lower the capital tax burden and public education expenditure as
their political weight increases. A decrease in capital tax revenue is compensated for by
an increase in debt issues, while a decrease in education expenditure is associated with adecrease in public debt issues. The analysis shows that the latter negative eﬀect on debt
outweighs the former positive eﬀect in the present framework.
Secondly, we consider an alternative scenario, called tax financing, in which the government is prohibited from issuing public debt; hence, its expenditure is financed solely
through taxation. This scenario, while is an extreme one, enables us to investigate in a
tractable way the eﬀect of controlling debt issues. We show that the labor (capital) tax
rate increases (decreases) if the government changes its instrument from debt financing
to tax financing. We then analyze the welfare consequences of this change and show a
trade-oﬀ in terms of utility across generations. The initial generation is worse oﬀ since its
labor tax burden increases to compensate for the loss of revenue from debt issues. However, tax financing removes the crowding-out eﬀect of public debt on physical capital andthus increases the growth rate. The initial generation cannot therefore benefit from thispositive eﬀect of increased growth, whereas future generations do benefit as it outweighs
the negative eﬀect of the increased tax burden.
1

The young may also have an incentive to vote since they benefit from public education in the future.
However, for the tractability of the analysis, we assume that politicians do not care about the young’s
preferences following Saint-Paul and Verdier (1993), Bernasconi and Profeta (2012), and Lancia andRusso (2016). This assumption is supported in part by the fact that a large number of the young are
below the voting age.

2

Thirdly, we consider a more realistic fiscal rule that sets an explicit ceiling for public
debt as a percentage of GDP, which has been widely introduced in developed countries(Schaechter et al., 2012). We investigate the eﬀect of such a fiscal tightening rule (i.e.,lowering the ceiling) based on a numerical analysis and obtain qualitatively similar growth
and welfare consequences as those obtained in the second analysis; in other words, thisfiscal tightening rule is growth-enhancing, but not Pareto-improving. We also find thateven if the rule is imposed only for limited periods, it has a long-lasting eﬀect on utility
across generations since increased human capital is bequeathed from generation to generation. This long-lasting eﬀect was absent from the study by Barseghyan and Battaglini(2016), who consider a temporary austerity program in the absence of human capital.
Our result suggests the importance of connecting seemingly unrelated subjects, namely
public education and public debt, to consider the impact of debt rules over time andacross generations.
The present study contributes to the following two strands of the literature. The
first is the literature on the politics of capital taxation in the overlapping-generations
framework. Earlier studies assumed no production sector and said nothing about the eﬀectof capital income (i.e., interest rate income) taxation on economic growth (Renström,
1996; Huﬀman, 1996; Dolmas and Huﬀman, 1997). In addition, these studies dropped
the conflict of interest between generations from their analyses by assuming that either
the young or the old generation is decisive in voting. Notable exceptions are Razin, Sadka,
and Swagel (2004) and Mateos-Planas (2010), who assume majority voting, and Bassetto(2008) and Ono and Uchida (2018a), who assume bargaining or competition between
generations. All these studies assume a balanced government budget and thus no budget
deficit. However, the current generation may have an incentive to pass the fiscal burden
onto future, unborn generations by issuing public debt. To address this issue, the present
study allows for deficit finance and investigates its impact on economic growth and welfare
across generations.
The second strand includes studies of the political economy of public debt (Battagliniand Coate, 2008; Barseghyan, Battaglini, and Coate, 2013; Debortoli and Nunes, 2013;
Azzimonti, Battaglini, and Coate, 2016; Barseghyan and Battaglini, 2016; Battagliniand Coate, 2016; Cunha and Ornelas, 2018). In particular, the present study is closely
related to Song, Storesletten, and Zilibotti (2012), Röhrs (2016), Müller, Storesletten,
and Zilibotti (2016), Song, Storesletten, and Zilibotti (2016), and Arawatari and Ono(2017), who demonstrate the conflict of interest between generations over public debt
policy. Arai, Naito, and Ono (2018), Ono (2018), and Ono and Uchida (2018b) extend
these analyses by including physical and/or human capital and present the eﬀects of thepolitics of public debt on economic growth and welfare across generations. However, all3

these studies have ignored capital taxation, which is an important policy instrument, andsaid nothing about the conflict of interest between generations over the distribution of tax
burdens. The present study overcomes this limitation by introducing capital taxation.
The organization of the remainder of this paper is as follows. Section 2 presents themodel. Section 3 describes the political equilibrium and Section 4 characterizes the steadystate equilibrium. Section 5 compares the debt- and tax-financing political equilibria; italso investigates the eﬀects of the debt ceiling rule. Section 6 presents concluding remarks.

2

Model

The discrete time economy starts in period 0 and consists of overlapping generations.
Individuals are identical within a generation and live for three periods: youth, middle,
and old ages. Each middle-aged individual gives birth to 1 + n children. The middle-aged
population for period t is Nt and the population 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.
In middle age, individuals work, receive market wages, and make tax payments. Theyuse after-tax income for consumption and savings. Individuals retire in their elderly years
and receive and consume returns from savings.
Consider an individual born in period t − 1. In period t, the individual is middleaged and endowed with ht units of human capital inherited from his or her parents. The
individual supplies them inelastically in 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 aftertax income between consumption ct and savings invested in physical capital st . Therefore,
the period t budget constraint for the middle age becomes
ct + st ≤ (1 − τt )wt ht .
The period t + 1 budget constraint in elderly age is)(k
Rt+1 st ,dt+1 ≤ 1 − τt+1
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. The
results are qualitatively unchanged if capital income tax is on the net return from saving

rather than the gross return from saving.
4

Period t middle-aged individuals care about their children’s income , wt+1 ht+1 . Children’s human capital in 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 using thefollowing 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.2
We assume that parents are altruistic toward their children and concerned about theirincome in middle age, wt+1 ht+1 . The preferences of an individual born in period t − 1 are
specified by the following expected utility function in the logarithmic form:
Ut = ln ct + β [ln dt+1 + γ ln wt+1 ht+1 ] ,where β ∈ (0, 1) is a discount factor and γ(> 0) denotes the intergenerational degree of
altruism. We substitute the budget constraints and human capital production function
into the utility function to form the following unconstrained maximization problem:[]
max ln [(1 − τt )wt ht − st ] + β ln Rt+1 st + γ ln wt+1 D (xt )η (ht )1−η .
{st }By solving this problem, we obtain the following savings and consumption functions:β· (1 − τt )wt ht ,
1+β
1
βRt+1
ct =
· (1 − τt )wt ht and dt+1 =· (1 − τt )wt ht .
1+β
1+β

st =

2.2

(2)

Firms

Each period contains a continuum of identical firms that are perfectly competitive profit
maximizers. According to 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.

2

Private investment in education may also contribute to human capital formation. 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 the present study, we abstract private education from the main analysis to simplify
the presentation of the model.

5Let kt ≡ Kt /Ht denote the ratio of physical to human capital. The first-order conditions for profit maximization with respect to Ht and Kt are
wt = (1 − α)A (kt )α ,(3)ρt = αA (kt )α−1 ,(4)where wt and ρt are labor wages and the rental price of capital, respectively. The conditions state that firms hire human and physical capital until the marginal products areequal to the factor prices. Capital is assumed to depreciate fully within each period.

2.3

Government Budget Constraint

Public education expenditure is financed by both taxes on capital and labor income andpublic bond issues. Let Bt denote aggregate inherited debt. The government budget
constraint in period t is
Bt+1 + τtk Rt st−1 Nt−1 + τt wt ht Nt = Nt+1 xt + Rt Bt ,where Bt+1 is newly issued public bonds, τtk Rt st−1 Nt−1 is aggregate capital tax revenue,
τt wt ht Nt is aggregate labor tax revenue, Nt+1 xt is aggregate expenditure on public education, and Rt Bt is debt repayment. We assume a one-period debt structure to derive
analytical solutions from the model as well as that the government in each period iscommitted to not repudiating the debt.
By dividing both sides of the above expression, we obtain a per-capita form of theconstraint:st−1+ τt wt ht = (1 + n)xt + Rt b̂t ,(5)(1 + n)b̂t+1 + τtk Rt
1+n
where b̂t ≡ Bt /Nt is per-capita public debt. We use the notation b̂t , rather than bt ,to distinguish per-capita public debt, b̂t ≡ Bt /Nt , from public debt per human capital,
bt ≡ Bt /Ht .

2.4

Economic Equilibrium

Public bonds are traded in the domestic capital market. The market-clearing condition
for capital is Bt+1 + Kt+1 = Nt st , which expresses the equality of total savings by the
middle-aged population in period t, Nt st , to the sum of the stocks of aggregate public
debt and aggregate physical capital at the beginning of period t+1, Bt+1 +Kt+1 . By using
kt+1 ≡ Kt+1 /Ht+1 , ht+1 = Ht+1 /Nt+1 , and the savings function in (2), we can rewrite thecondition as)(β(1 + n) kt+1 ht+1 + b̂t+1 =(1 − τt )wt ht .
(6)1+βThe following defines the economic equilibrium in the present model.
6{}∞Definition 1. Given a sequence of policies, τtk , τt , xt t=0 , an economic equilibrium is}∞{a sequence of allocations ct , dt , st , kt+1 , b̂t+1 , ht+1
and prices {ρt , wt , Rt }∞t=0 witht=0)(
the initial conditions k0 (> 0), b̂0 (≥ 0) and h0 (> 0), such that (i) given wt , Rt+1 , τtk , τt , xt ,( y o)
ct , ct+1 , st solves the utility maximization problem;(ii) given() ((wt , ρt ), kt solves) afirm’s profit maximization problem; (iii) given wt , ht , Rt , b̂t , τtk , τt , xt , b̂t+1 satisfies the government budget constraint; (iv)ρt = Rt holds;( an arbitrage condition)and (v) the capital market clears: (1 + n) · kt+1 ht+1 + b̂t+1 = st .
In the economic equilibrium, the indirect utility of the middle-aged population inperiod t, VtM , and that of the old population in period t, Vto , can be expressed as functionsof fiscal policy, physical and human capital, and public debt as follows:
VtM = (1 + β) ln(1 − α)A (kt )α ht (1 − τt ) + β [(α − 1) + γα] ln kt+1()
k
+ β ln 1 − τt+1
+ βγη ln xt + ϕM (ht ) ,
(7)
(
)
(
)
Vto = ln 1 − τtk + ϕO kt , ht , b̂t ,
(8)
(
)
M
O
where ϕ (ht ) and ϕ kt , ht , b̂t include policy-irrelevant and constant terms, and they
are defined by
)
(
β
1
1−η
M
+ β ln+ β (ln αA + γ ln (1 − α) A) ,ϕ (ht ) ≡ βγ ln D (ht )
+ ln
1+β
1+β
(
)
(
)
α−1
O
ϕ kt , ht , b̂t ≡ ln αA (kt )b̂t + kt ht (1 + n) + γ ln(1 − α)A (kt )α ht ,respectively.

3

Political Equilibrium

In this section, we consider voting on fiscal policy. In particular, we employ probabilistic
voting à la Lindbeck and Weibull (1987). In this voting scheme, there is electoral competition between two oﬃce-seeking candidates. Each candidate announces a set of fiscal
policies subject to the government budget constraint. As demonstrated in Persson andTabellini (2000), the two candidates’ platforms converge in the equilibrium to the same
fiscal policy that 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 current public education expenditure throughhuman capital accumulation, we assume that their preferences are not taken into accountby politicians. We impose this assumption, which is often used in the literature (e.g.,Saint-Paul and Verdier, 1993; Bernasconi and Profeta, 2012; Lancia and Russo, 2016),
for tractability reasons. However, the assumption could be supported in part by the factthat a large number of the young are below the voting age.
7

Thus, the political objective is defined as the weighted sum of the utility of the middleaged and old, given by Ω̃t ≡ ωVto + (1 + n)(1 − ω)VtM , where ω ∈ [0, 1] and 1 − ω are thepolitical weights placed on the old and middle-aged in period t, respectively. The weighton the middle-aged is adjusted by the gross population growth rate, (1 + n), to reflect
their share of the population. To gain the intuition, we divide Ω̃t by (1 + n)(1 − ω) andredefine the objective function as follows:Ωt =ω
V o + VtM ,(1 + n)(1 − ω) t

where the coeﬃcient ω/(1 + n)(1 − ω) of Vto represents the relative political weight on theold.
We substitute VtM in (7) and Vto in (8) into Ωt . By rearranging the terms, we obtain()ω
ln 1 − τtk + (1 + β) ln (1 − α) A (kt )α ht (1 − τt )(1 + n)(1 − ω)()
k
+ β [(α − 1) + γα] ln kt+1 + β ln 1 − τt+1
+ βγη ln xt .

Ωt ≃

(9)

We use the notation ≃ because irrelevant terms are omitted from the expression of Ωt .
We introduce the following notations to define the optimization problem with choicesfrom compact sets:b̂t+1
xt, b̃t+1 ≡,x̃t ≡αA (kt ) ht
A (kt )α ht
where x̃t and b̃t+1 denote the education expenditure-to-GDP ratio and government debtto-GDP ratio, respectively. The state and policy variables are assumed to be in compact
sets:
[ (
]
)
[
]
b̂t
∈ S ≡ 0, b̂/kh, x̃ ∈ Sx ≡ [0, x̃upper ] , b̃ ∈ Sb ≡ 0, b̃upper ,
kt ht
upper
[ k
]
k
τ ∈ Sτ k ≡ −τlower , 1 , and τ ∈ Sτ ≡ [−τlower , 1] .
()
Note that b̂/kh, x̃upper , and b̃upper are the upper limits of the corresponding variupperkables, while −τlower(< 0) and −τlower (< 0) are the lower limits of the corresponding

variables. These are exogenous and set suﬃciently high or low to ensure interior solutions. We allow for the possibility of negative tax rates, but will later derive the conditionsfor τ k ≥ 0 and τ ≥ 0.
With the use of (3)–(6), we can reformulate the expression in (9) as follows:

8

Ωt ≃

(
)
ω
ln 1 − τtk(1 + n)(1 − ω)
|
{z
}
(p.1)

[

)
]
b̂t
+ (1 + β) ln 1 −α+ 1 − (1 + n)x̃t + (1 + n)b̃t+1| {z }
k t ht
(p.2)
]
)
[
(
(
)
1
b̂
t
+ 1 − (1 + n)x̃t − (1 + n)b̃t+1+ β [(α − 1) + γα] ln 1 − 1 − τtk α|{z}
k t htβ(1 − τtk

)

(

(p.3)

(
)
k
+ β ln 1 − τt+1
+ βη (1 − α) (1 + γ) ln x̃t .
|{z}
|
{z
}
(p.4)

(10)

(p.5)

The terms (p.1)–(p.5) in (10) present the relative political weight on the old, the weight on
the young’s utility of lifetime consumption, the weight on the utility of interest rate income
plus the weight on the utility of children’s wage income, the weight on the marginal costof capital income taxation in terms of utility, and the weight on the utility of children’s
human capital, respectively. Appendix A.1 shows the derivation of (10).
The political objective function in (10) suggests that the current policy choice aﬀects
the decision on future policy via physical and human capital accumulation. In particular,
the period t choices of τtk , x̃t , and b̃t+1 (equivalently, τtk , xt , and b̂t+1 ) aﬀect the formationof physical and human capital in period t + 1. This in turn influences the decision making
on period-t + 1 fiscal policy. To demonstrate such an intertemporal eﬀect, we employ theconcept of a Markov-perfect equilibrium under which fiscal policy today depends on thecurrent payoﬀ-relevant state variables.
In the present framework, the payoﬀ-relevant state variables are the ratio of physical tohuman capital, kt , human capital, ht , and public debt, b̂t . These are summarized as b̂/kh
as observed in the political objective function in (10). Thus, the expected rate of capitalkincome tax for the next period, τt+1
, is given by the function (of the next period) stock ofkgovernment debt and physical and human capital, τt+1
= T k b̂t+1 /kt+1 ht+1 . By using
recursive notation with z ′ denoting the next period z, we can now define a Markov-perfectpolitical equilibrium in the present framework as follows.

Definition 2. A Markov-perfect political equilibrium is a set of functions, ⟨T, T k , X, B⟩,
where T : S → Sτ is a labor income tax rule, τ = T (b̂/kh), T k : S → Sτ k is a capitalincome tax rule, τ k = T k (b̂/kh), X : S → Sx is a public education expenditure rule,x̃ = X(b̂/kh), and B : S → Sb is a public debt rule, b̃′ = B(b̂/kh), such that thefollowing conditions are satisfied:(i) The capital market clears,((1 + n) b̃′ +k ′ h′
A (k)α h

)

9

=

β
(1 − τ ) (1 − α) ;
1+β

(11)

⟨
⟩
k
(ii) Given b̂/kh, T (b̂/kh), T (b̂/kh), X(b̂/kh), B(b̂/kh) = arg max Ω subject to τ k′ =()
T k b̂′ /k ′ h′ , the capital market-clearing condition in (11), the government budget
constraint,
(
)
b̂
b̂
(1 + n)b̃′ + τ (1 − α) + τ k α 1 += (1 + n)x̃ + α ,
(12)
kh
kh
(
)η
′
1−η
and the human capital formation function, h = D(h)X(b̂/kh) , where Ω isdefined by (10).

3.1

Characterization of the Political Equilibrium

To obtain the set of policy functions in Definition 2, we conjecture the following capitaltax rate in the next period:1kτ k′ = 1 − Tun,(13)′
1 + kb̂′ h′kwhere Tun(> 0) is constant. The subscript “un” means that public bond issuance is“unconstrained.” In the next section, we consider a case where public bond issuance is“constrained” by a constitutional rule and compare it with the unconstrained case.

The conjecture in (13) is reformulated by using (11) and (12) as follows:
)
(
) (
b̂
− (1 + n)x̃ − β1 (1 + n)b̃′
1 − 1 − τ k α 1 + kh
k′
k
(
)
.
τ = 1 − Tunb̂
1 − (1 − τ k ) α 1 + kh
− (1 + n)x̃ + (1 + n)b̃′(14)

The substitution of (14) into the political objective function in (10) leads to()ωΩ≃
ln 1 − τ k + ln Z 1 + αβ (1 + γ) ln Z 2 + βη (1 − α) (1 + γ) ln x̃, (15)
(1 + n)(1 − ω)
where

)
b̂
Z1 ≡ 1 − 1 − τ α 1 +− (1 + n)x̃ + (1 + n)b̃′ ,
kh
)
(
(
)
1
b̂
− (1 + n)x̃ − (1 + n)b̃′ .
Z2 ≡ 1 − 1 − τ k α 1 +
kh
β
(

)
k

(

The first-order conditions with respect to τ k , b̃′ , and x̃ are
(
(
)
)
b̂
b̂
α 1 + kh
αβ (1 + γ) α 1 + khω1
τk : −++= 0,(1 + n)(1 − ω) 1 − τ k
Z1Z2
1 + n α (1 + γ) (1 + n)
+
≤ 0,
b̃′ :
Z1
Z2
βη (1 − α) (1 + γ) 1 + n αβ (1 + γ) (1 + n)
x̃ :
−
−
= 0.
x̃
Z1
Z2
10

(16)
(17)
(18)

A strict inequality holds in (17) if b̃′ = 0. By using these conditions, we can verify the
conjecture in (13) and obtain the following result.
Proposition 1. There is a Markov-perfect political equilibrium distinguished by b̂′ = 0
if (1 − α) /α ≤ γ, and b̂′ > 0 otherwise. The corresponding policy functions of τ k ,x, τ , and b̂′ are as follows:kτ k = 1 − Tun

1b̂
1 + kh,(1 + n)x = Xun A (k)α h, 1 − 1 1+αβ(1+γ)
if 1−α≤ γ,ω+ψ1−α (1+n)(1−ω)
α
τ=
1+β
1
1−α
 1 − 1−α
if
> γ,
ω
+ψ
α
(1+n)(1−ω){
0
if 1−α≤ γ,
α(1 + n)b̂′ =Bun A (k)α h if 1−α> γ,αkwhere ψ, Tun, Xun , and Bun are defined byψ ≡ 1 + β (1 + γ) (α + η (1 − α)) ,ω(1+n)(1−ω),ω+ψα (1+n)(1−ω)

1
k
Tun
≡
Xun ≡

βη (1 − α) (1 + γ),ω+ψ(1+n)(1−ω)

Bun ≡β [1 − α (1 + γ)].
ω+ψ(1+n)(1−ω)

Proof. See Appendix A.2 .
The result in Proposition 1 suggests that the tax rates, τ k and τ , and public education
expenditure, x, are aﬀected by the weights, (p.1)–(p.3) and (p.5), of the political objectivefunction in (10). To understand the eﬀects, we reformulate the policy functions of τ k , τ ,and x in Proposition 1 to clearly specify the eﬀects of the weights on the policy functions
as follows:
{
}−1
1
1
(1
+
β)
+
[β
(α
−
1)
+
βγα]
+
βη
(1
−
α)
(1
+
γ)
×
τk = 1 −
1+
,
ω
b̂
α
1
+
(1+n)(1−ω)
kh

{
}
ω
−1
+βη(1−α)(1+γ)
1
1 + (1+n)(1−ω)
if 1−α≤ γ,
1 − 1−α(1+β)+[β(α−1)+βγα]
α
τ=
{
}
ω
−1
+[β(α−1)+βγα]+βη(1−α)(1+γ)
 1 − 1 1 + (1+n)(1−ω)
if 1−α
> γ,
1−α
(1+β)
α
{
}
−1
ω
+ (1 + β) + [β (α − 1) + βγα](1 + n)x(1+n)(1−ω)= 1+.
A (k)α hβη (1 − α) (1 + γ)
11The terms (p.1)–(p.3) and (p.5) in (10), which appear in the above equations, aﬀect
the tax rates and public education expenditure in the following ways. First, the term(p.1), representing the political weight on the old, implies that the greater political powerof the old leads to a larger weight on the utility of their consumption. This incentivizes thegovernment to shift the tax burden from the old to the young as well as to reduce public
education expenditure. Second, the term (p.2), representing the weight on the young’s
utility of lifetime consumption, implies that a larger weight on this utility provides thegovernment with an incentive to shift the tax burden from the young to the old as well as
to reduce the tax burden on the young by cutting public education expenditure. Third,
the term (p.5), representing the weight on the utility of children’s human capital, indicatesthat a larger weight on this utility implies greater altruism toward children. This in turnprovides the government with an incentive to increase public education expenditure byraising the capital and labor tax rates.
The term (p.3) includes the weight on the utility of the interest rate, represented by
β (α − 1), and the weight on the utility of children’s wage income, represented by βγα.
These two weights have opposing eﬀects on the tax rate and public education expenditure.
The first weight induces the government to reduce saving and capital to raise interest rateincome. For this purpose, the government shifts the tax burden from the old to the young;it also increases public education expenditure to raise the tax burden on the young. The
second weight has the opposite eﬀect since the young want to increase capital and thus
the wage income of their children. The net eﬀect depends on their relative magnitudes.
The policy function of b̂′ when b̂′ > 0 is reformulated as follows:
(p.3)

(p.2)

(p.4)

}|
{ z}|{z}|{ z(1 + n)b̂′β − [β (α − 1) + βγα] − β.
=ω+ψ
A (k)α h(1+n)(1−ω)
This indicates that the weights (p.2)–(p.4) are crucial to the government’s financial stance.
The term (p.2) implies that a greater weight on the young’s utility of consumption incentivizes the government to issue more public debt to finance the young’s increased
consumption. The term (p.3) implies that an increase in public bond issues crowds out
physical capital accumulation and thus increases the interest rate. However, at the sametime, it decreases children’s wage income, thereby giving the government an incentive to
cut public bond issues. The term (p.4) indicates that an increase in public bond issues
raises the capital tax burden on the young in their old-age period. This also incentivizes
the government to cut public bond issues. In sum, the government chooses b̂′ > 0 and,
therefore, borrows in the capital market if the eﬀect of (p.2) outweighs the sum of theeﬀects of (p.3) and (p.4), that is, if γ < (1 − α)/α.

12

3.2

Education Expenditure and Government Debt

The result established in Proposition 1 indicates that public education expenditure andgovernment debt are aﬀected by the two population factors, ω and n. To investigate theseeﬀects appropriately, we focus on the education expenditure-to-GDP ratio, Nt+1 xt /Yt , anddebt-to-GDP ratio, Bt+1 /Yt , and analyze the eﬀects of an increase in ω and a decrease inn on these ratios.
Proposition 2. Consider the Markov-perfect political equilibrium demonstrated in Proposition 1.
(i) The education expenditure-to-GDP ratio, Nt+1 xt /Yt , decreases with the greater political power of the old and a lower population growth rate: ∂ (Nt+1 xt /Yt ) /∂ω < 0 and∂ (Nt+1 xt /Yt ) /∂n > 0.
(ii) The government debt-to-GDP ratio, Bt+1 /Yt , decreases with the greater politicalpower of the old and a lower population growth rate: ∂ (Bt+1 /Yt ) /∂ω < 0 and∂ (Bt+1 /Yt ) /∂n > 0.
To confirm the statement in Proposition 2, we compute the public education expenditureto-GDP ratio and debt-to-GDP ratio as follows:
Nt+1 xtβη (1 − α) (1 + γ)
=
;
ω
Yt
+ψ
(1+n)(1−ω)
Bt+1
β [1 − α (1 + γ)]
=
.
ω
Yt
+ψ
(1+n)(1−ω)
These ratios depend on the two population factors, ω and n, through the term ω/(1 +n)(1 − ω), representing the relative political weight on the old. Higher ω and lower n
imply a larger political weight on the old in the political objective function. A larger
weight on the old in turn provides the government with an incentive to reduce public
education expenditure since the old benefit nothing from it. Therefore, the education
expenditure-to-GDP ratio decreases as ω increases and n decreases.
The debt-to-GDP ratio also decreases as ω increases and n decreases. This is somewhat
counterintuitive since the old may have an incentive to pass the fiscal burden onto future
generations. This incentive arises in the framework of Song, Storesletten, and Zilibotti(2012) in which the old are assumed to owe no tax burden but benefit from public good
provision. However, in the present framework, the old receive no benefit from public
expenditure, but owe a capital tax burden. This implies that the old want to lower thecapital tax rate and public education expenditure as their political weight increases. A
decrease in the capital tax rate is associated with increased public debt, whereas a decrease13

in public education expenditure is associated with decreased public debt. That is, there
are two opposing eﬀects on public debt, but the negative eﬀect outweighs the positive one
in the present framework. Thus, the debt-to-GDP ratio decreases as the political weighton the old increases.

4

Steady State

Having established the policy functions, we are now ready to demonstrate the accumulation of physical and human capital. We substitute the policy functions in Proposition 1
into the capital market-clearing condition in (11) and human capital formation functionin (1), and obtaink ′ = ΨK [A (k)α ] ,h′η= DΨH [A (k)α ] ,h1−ηwhere ΨK and ΨH are defined by{[ un ]η }−1 ΨK,b′ =0 ≡ β 1+αβ(1+γ)
if(1 + n)D X
ω
1+β (1+n)(1−ω)
+ψ
1+n
{
[
]
}
ΨK ≡
η −1
un
 ΨK,b′ >0 ≡ αβ(1+γ)(1 + n)D X
if
ω
+ψ
1+n
(1+n)(1−ω)

and

[

Xun
ΨH ≡
1+n

(19)
(20)

1−α
≤ γ,
α
1−α
> γ,
α

(21)

]η
,

(22)

respectively. Appendix A.3 shows the derivation of (19) and (20).
Given {k0 , h0 }, the sequence {kt , ht } is distinguished by the above two equations in(19) and (20). A steady state is defined as a political equilibrium with kt = kt+1 . In otherwords, the ratio of physical to human capital is constant in a steady state. Equation(19) implies that there is a unique, stable steady state. Human capital increases along
the steady-state path, as suggested in (20). The following proposition summarizes theargument thus far and identifies conditions for which tax rates are set within a range[0, 1).
Proposition 3. There is a unique, stable steady-state equilibrium distinguished by (i)b′ = 0, τ k ∈ [0, 1), and τ ∈ [0, 1) if (1 − α)/α ≤ γ and
α
βη
ω
α
ψ−
(1 + γ) (1 − α) ≤
≤
ψ,
1−α
1−α
(1 + n)(1 − ω)
1−α

(23)

and (ii) b′ > 0, τ k ∈ [0, 1), and τ ∈ [0, 1) if γ < (1 − α)/α and
1+β
ω
ψ
−ψ ≤
≤ .
1−α
(1 + n)(1 − ω)
γ
14

(24)

Proof. See Appendix A.4.
The result in Proposition 3 suggests that the relative political weight on the old,
represented by ω/(1 + n)(1 − ω), plays a crucial role in shaping the tax rates. Whenω/(1 + n)(1 − ω) is above the upper bound (below the lower bound) in (23) and (24),the relative political weight on the old is too high (low) to incentivize the government to
tax the old (young). The government would rather subsidize the old (young) by choosingτ k < 0 (τ < 0). We rule out this possibility by imposing the upper (lower) bound of
ω/(1 + n)(1 − ω).
Based on the result established thus far, we can define the growth rate of the economy
and investigate how it is aﬀected by the two population factors, ω and n. Consider the
output-to-human capital ratio defined by y ≡ Y /H = A (k)α h. Then, the growth rate ofy isy′A (k ′ )α h′=,y
A (k)α hwhere x′ denotes the next period x(= k, h, y). In the steady state with k ′ = k, the growthrate of y is equal to the growth rate of human capital, h′ /h. Therefore, in what follows,
we focus on the steady-state growth rate of human capital.
η

Recall the growth rate of human capital given by (20). From (19), the term (A (k)α )αηηηin the steady state becomes (A (k)α ) = (ΨK ) 1−α(1−η) (A) 1−α(1−η) . Thus, the steady-state
growth rate becomes
η
αη
h′
(25)
= DΨH (ΨK ) 1−α(1−η) (A) 1−α(1−η) .
h
The following proposition establishes the eﬀects of changes in ω and n on the growth rate
via the terms ΨH and ΨK .
Proposition 4. The steady-state growth rate increases as the population growth rate
decreases, but it decreases as the political weight on the old increases: ∂ (h′ /h) /∂n <0 and ∂ (h′ /h) /∂ω < 0.
The proof and interpretation of Proposition 4 is as follows. The steady-state growthrate in (25) is reformulated as follows:
′

h
=
h

{

1
1
ω
1 + n (1+n)(1−ω) + ψη} 1−α(1−η)× U R,

where U R includes the terms unrelated to n and ω. A decrease in ω leads to a larger
relative political weight on the old, as observed by the term ω/(1 + n)(1 − ω). This
provides the government with an incentive to cut public education expenditure, which
in turn reduces human capital accumulation. Thus, a greater political weight on the old
results in a lower steady-state growth rate.
15

A decrease in n has two opposing eﬀects on the steady-state growth rate. On the
one hand, it leads to a larger political weight on the old, as observed by the term ω/(1 +n)(1−ω). As in the case of increased ω, this provides the government with an incentive to
cut public education expenditure, which negatively aﬀects human capital accumulation.
On the other hand, a decrease in n results in an increase in per-capita public education
expenditure, as observed by the term 1/(1 + n). This works to enhance human capital
accumulation, and this positive eﬀect outweighs the negative eﬀect. Therefore, a lower
population growth rate results in a higher steady-state growth rate.

5

Fiscal Rules

In the previous section, we considered fiscal policy and economic growth in the absence ofconstraints on public bond issues except for the flow budget constraint. In other words,
we assumed no rule on public bond issues. However, in the real world, many countries
have introduced fiscal rules that control public bond issuance (Schaechter et al., 2012).
This brings us to the question of how fiscal rules shape the choice of fiscal policy andaﬀect economic growth over time.
To answer this question, in Section 5.1 we first consider the following alternative
scenario in which the government is prohibited from issuing public bonds and thus its
expenditure is financed solely through taxation. In this tax-financing case, we assume
γ < α/(1−α), which implies that in the absence of the tax-financing rule, the government
borrows in the capital market and issues public bonds. In other words, the government
wants to issue public bonds to finance its expenditure, but their issuance is prohibitedwhen the tax-financing rule is introduced. We then compare the tax rates, expenditure,and economic growth in the debt-financing case in the previous section with the taxfinancing case. We also investigate the welfare consequences of shifting from debt financing
to tax financing.
The requirement for tax financing is somewhat extreme because in reality the government is allowed to issue public bonds as long as their issuance is below some debt ceiling.
Hence, in Section 5.2, we overcome this shortcoming by considering an alternative fiscal rule for managing the debt issuance-to-GDP ratio widely introduced in developed
countries.

5.1

Tax Financing versus Debt Financing

The policy functions in the tax-financing case are those associated with b̂′ = 0 in Proposition 1. The corresponding steady-state growth rate is computed by setting b̂′ = 0 in(25). To investigate the diﬀerences between the tax-financing and debt-financing cases, we
16compare their tax rates, τ k and τ , public education expenditure-to-GDP ratio, (1+n)x/y,
and economic growth, h′ /h. the variables in the tax-financing and debt-financing cases
are denoted by the subscripts “tax” and “debt,”respectively.
Proposition 5. Suppose that γ < α/(1 − α) holds. Given the initial conditions k0 and
b0 , tax financing and debt financing are compared as follows:
τ0k tax = τ0k debt ; τtk tax < τtk debt for t ≥ 1; τ |tax > τ |debt ;(1 + n)x/y|tax = (1 + n)x/y|debt ; and h′ /h|tax > h′ /h|debt .
Proof. See Appendix A.5.
In the initial period, the government needs to finance the repayment of outstanding
public debt, b0 , regardless of the financing method. Thus, the capital tax rates are equalin the two financing cases in the initial period. However, from period 1 onward, thegovernment incurs no repayment costs in the tax-financing case, while it still incurs suchcosts in the debt-financing case. Because of this diﬀerence, the capital tax rate is lower
in the tax-financing case than in the debt-financing case from period 1.
By contrast, the labor tax rate is higher in the tax-financing case than in the debtfinancing case. When the tax-financing rule is introduced, the government needs to compensate for the loss of revenue from bond issues by raising the labor income tax rate.
An increase in revenue from the labor tax is oﬀset by a decrease in revenue from thecapital tax and public bond issues. Thus, the education expenditure-to-GDP ratio remains unchanged. However, the introduction of tax financing removes the crowding-out
eﬀect of public bonds. This positive eﬀect on physical capital enhances human capital
accumulation and economic growth.
The result in Proposition 6 suggests that the shift from debt financing to tax financing
increases the growth rate and benefits future generations, but may worsen the current
middle-aged population because of the increased labor tax burden. We investigate thiswelfare implication and obtain the following result.
Proposition 6. Suppose that γ < α/(1 − α) holds.
(i) The welfare of the initial old population is unaﬀected; generation 0 is made worse oﬀ
by shifting from debt financing to tax financing.
(ii) There is a critical period, denoted by t̂(> 1), such that generation t ≤ t̂ is made worseoﬀ, whereas generation t > t̂ is made better oﬀ by shifting from debt financing totax financing.
Proof. See Appendix A.6.
17

The welfare of the initial old population is unaﬀected by shifting to tax financing since
their tax burden is unchanged. However, the choice of tax financing has two opposing
eﬀects on current and future generations. Tax financing raises the tax burden of the
middle-aged population as demonstrated in Proposition 5. This lowers the lifetime incomeof the middle-aged and thus lowers their lifetime utility of consumption. This is thenegative eﬀect of tax financing. However, tax financing removes the crowding-out eﬀectof public bonds on capital and thus enhances human capital accumulation. This positiveeﬀect appears from generation 1 onward and accumulates over time, but generation 0
cannot enjoy this benefit. Therefore, generation 0 suﬀers from a negative eﬀect, whereas
distant future generations benefit from a positive eﬀect that outweighs the negative one.

5.2

Debt Rule

This section extends the analysis of the previous section by considering the following debt
rule:

Bt+1
≤ ū.
Yt
This rule resembles the debt rule that sets an explicit ceiling for public debt as a percentage
of GDP (Schaechter et al., 2012). This is reformulated as(1 + n)b̂t+1 ≤ ūA (kt )α ht ⇔ (1 + n)b̃′ ≤ ū,(26)

where ū is defined byū ≡ εBun , ε ∈ [0, 1) .
The rule resembles the tax-financing case in Section 5.1 when ε → 0 and the unconstrained
debt-financing case in Section 3 when ε → 1.
We retain the assumption of γ < (1 − α)/α, implying that the government chooses
debt financing in the absence of any debt rule. Debt issuance in the absence of the rulein (26) is given by (1 + n)b̂′ = Bun A (k)α h as demonstrated in Proposition 1. When thedebt rule in (26) is introduced, it is always binding since ε < 1. Thus, the issue of publicbonds in the absence of the rule in (26) is (1 + n)b̂′ = εBun A (k)α h, or(1 + n)b̃′ = εBun A.
With the use of (1 + n)b̃′ = εBun A, the government budget constraint in (12) isreformulated as
(
)
b̂
b̂
εBun + τ (1 − α) + τ k α 1 +(27)= (1 + n)x̃ + α ,khkh
and the capital market-clearing condition in (11) is rewritten as
εBun + (1 + n)

k ′ h′
β=(1 − τ ) (1 − α) .
αA (k) h
1+β
18

(28)

The political objective function in (10) is reformulated as()ω
ln 1 − τ k(1 + n)(1 − ω)
[
(
)
]
(
)
b̂
+ (1 + β) ln 1 − 1 − τ k α+ 1 − (1 + n)x̃ + εBun
kh
)
[
(
]
(
)
b̂
1
+ 1 − (1 + n)x̃ − εBun
+ β [(α − 1) + γα] ln 1 − 1 − τ k αkhβ()+ β ln 1 − τ k′ + βη (1 − α) (1 + γ) ln x̃.

Ω≃

(29)

Following the procedure described in Section 3, we conjecture the following capital tax
rate in the next period:1kτ k′ = 1 − Tcon,
1 + b̂′ /k ′ h′kwhere the subscript “con” of Tcon
implies that public bond issuance is “constrained” by
the rule in (26). From (26), (27), and (28), we can rewrite the above conjecture as follows:
(
)
b̂
b̂
k
εBun + (1 − α) + τ α 1 + kh − (1 + n)x̃ − α kh
− 1+β
εBun
β
k′
k
(
)
τ = 1 − Tcon
.
(30)
b̂
b̂
εBun + (1 − α) + τ k α 1 + kh
− (1 + n)x̃ − α kh

Appendix A.7 shows the derivation of (30).
We substitute (30) into the political objective function in (29) and rearrange the terms
to obtain
Ω≃
where

(
)
ω
ln 1 − τ k + ln Z̃ 1 + αβ (1 + γ) ln Z̃ 2 + βη (1 − α) (1 + γ) ln x̃, (31)
(1 + n)(1 − ω))b̂Z̃ 1 ≡ 1 − 1 − τ α 1 +− (1 + n)x̃ + εBun ,
kh
(
)
(
)
b̂
1
− (1 + n)x̃ − εBun .
Z̃ 2 ≡ 1 − 1 − τ k α 1 +
kh
β
(

)
k

(

The first-order conditions with respect to τ k and x̃ are()[]
αβ (1 + γ)
b̂
ω
1
1
k
+
+α 1+
= 0,
τ :−
(1 + n)(1 − ω) 1 − τ k
kh
Z̃ 1
Z̃ 2
[
]
βη (1 − α) (1 + γ)1
αβ (1 + γ)
x̃ :
−
+
= 0.
(1 + n)x̃Z̃ 1
Z̃ 2
By solving (34) and (35) for τ k and x̃, we obtain the following result.
19

(32)
(33)

(34)
(35)

Proposition 7. Suppose that γ < (1 − α)/α holds. In the presence of the debt rule in(26), a Markov-perfect political equilibrium is distinguished by the following policy
functions:kτ k = 1 − Tcon

1,
1 + b̂/kh(1 + n)x = Xcon A (k)α h,
[
(
)
]
ω
1
1
τ =1−
(1 + εBun ) − 1 +Xcon ,
1−α(1 + n)(1 − ω) βη (1 − α) (1 + γ)kwhere Tcon
and Xcon are defined as√
H − (H)2 − 4GI1ω1,
α (1 + n)(1 − ω) βη (1 − α) (1 + γ)
2G√
H − (H)2 − 4GI
Xcon ≡
,
2G
][
[
]
ω
1
ω
+ ψ > 0,G≡ 1+
(1 + n)(1 − ω) βη (1 − α) (1 + γ) (1 + n)(1 − ω)
[
][
(
)]
1
ω
εBun
H ≡ βη (1 − α) (1 + γ) 1 +(1 + εBun ) + 1 −(1 + n)(1 − ω) βη (1 − α) (1 + γ)
β
(
)
εBun
+ αβ (1 + γ) (1 + εBun ) + 1 −
β
> 0,
(
)
εBun
I ≡ βη (1 − α) (1 + γ) (1 + εBun ) 1 −
> 0.
β
k
Tcon
≡

Proof. See Appendix A.8.
Following the procedure described in Section 3, we show the existence and uniquenessof steady-state capital. Recall the capital market-clearing condition in (28), which isrewritten as follows:
′

k =
=

β
(1 − τ ) (1 − α) − εBun1+βA(k)α h(1 + n)D(h)1−η (x)ηβ
(1 − τ ) (1 − α) − εBun
1+β
( X )η
[A(k)α ]1−η ,con(1 + n)D 1+n(36)

where the first equality comes from the human capital formation function given by h′ =D(h)1−η (x)η and the second equality comes from the policy function of x presented in
Proposition 7. Equation (36) indicates that there is unique, stable steady-state capital.
In the next section, we focus on steady states and compare cases in the presence andabsence of the debt rule in (26) in terms of the education expenditure-to-GDP ratio,
capital and labor taxes, and growth rates. We also compare the cases in terms of utility
across generations.
20

5.3

Numerical Analysis

Our task here is to compare cases in the absence and presence of the debt rule in (26)based on numerical methods. Our strategy is to calibrate the model economy such thatthe steady-state equilibrium with b > 0 matches some key statistics of average OECD
countries during 1995–2014.3 We fix the share of capital at α = 1/3 following Song,Storesletten, and Zilibotti (2012) and Lancia and Russo (2016). Each period lasts 30
years; this assumption is standard in quantitative analyses of the two- or three-period
overlapping-generations model (e.g., Gonzalez-Eiras and Niepelt, 2008; Lancia and Russo,2016). Our selection of β is 0.99 per quarter, which is also standard in the literature (e.g.,
Kydland and Prescott, 1982; De la Croix and Doepke, 2002). Since agents in the present
model plan over generations that span 30 years, we discount the future by (0.99)120 .
We assume an annual population growth rate of 1.0059, which was the OECD averageduring 1995–2014. This assumption implies that the net population growth rate for 30
years is (1.0059)30 − 1. For η, the estimate in Cardak (2004) implies that the elasticity ofschool quality is in the range of 0.1 − −0.3. Based on this estimation, we set η = 0.2. We
also set ω to 0.45 to attain interior solutions for the capital and labor taxes.
For γ, we focus on the education expenditure-to-GDP ratio in the steady state:
Nt+1 xtβη (1 − α) (1 + γ)
= Xun =.
ωYt+ψ(1+n)(1−ω)
Given α = 1/3, β = (0.99)120 , 1 + n = (1.0059)30 , η = 0.2, and ω = 0.45, we can solve
this expression for γ by using the average ratio observed in OECD countries of 0.051 andobtain γ = 1.62.
To determine the two productivity parameters, A and D, we normalize the steady-state
wage, w, to unity. Thus, we have w = (1 − α)A(k)α = 1, or[ β(1 − τ ) (1 − α) − Bun
1+β( un )η
(1 − α)(1 + n) X
1+n]α/(1−α(1−η))(D)−α/(1−α(1−η)) (A)1/(1−α(1−η)) = 1. (37)We also use the data on the per-capita GDP gross growth rate of 1.02, which was theOECD average during 1995–2014. We substitute these data and the values of α, β, n, η,γ, and ω into the following equation expressing the per-capita GDP gross growth rate:
h′
=
h

(

Xun
1+n

)η [ β(1 − τ ) (1 − α) − Bun
1+β
(
)η
un
(1 + n) X
1+n]αη/(1−α(1−η))(D)(1−α)/(1−α(1−η)) (A)η/(1−α(1−η))= (1.02)30 .

(38)

3

Source: World Development Indicators, https://data.worldbank.org/products/wdi (Accessed on
March 3, 2018).

21

We solve the two equations, (37) and (38), for A and D, and obtain A = 3.37 and
D = 3.14.
The economy is assumed to be in a steady state in period 0. The initial capital
k0 is computed by solving equation (36) for k. The initial value of human capital, h0 ,is set at h0 = 2 to allow logarithmic utility to be positive for the tractability of theanalysis. From the result in Section 3, in the absence of any fiscal rule, the ratio b̂/khin the steady state is given by b̂/kh = [1 − α(1 + γ)] /α(1 + γ). Thus, we set b̂0 atb̂0 = {[1 − α(1 + γ)] /α(1 + γ)} k0 h0 and compare the cases with and without a debt rule
for the same initial conditions.
5.3.1

Comparative Statics

We study how the steady-state equilibrium responds to changes in the debt rule in (26). In
particular, we focus on ε. When ε = 1, the equilibrium policy functions and corresponding
economic growth rate coincide with those in the absence of the debt rule as in Section
3. When ε = 0, they coincide with those in the tax-financing case as in Section 5.1.
In the following, we consider a decrease in ε that aims to tighten fiscal discipline andinvestigate its impact on fiscal policy, economic growth, and welfare across generations.
Figure 1 plots the education expenditure-to-GDP ratio, capital and labor tax rates, ratioof physical to human capital, and per-capita growth rate in the steady state, taking ε on
the horizontal axis from 0 to 1.
[Figure 1 here.]Fiscal tightening (i.e., a decrease in ε) has the following eﬀects on the education
expenditure-to-GDP ratio. Firstly, the government raises the labor tax rate to compensate
for the loss of revenue from public bond issues, as depicted in Panel (a). This lowers thedisposable income of the middle-aged population and thus their lifetime consumption,
which in turn raises the marginal costs of public education expenditure in terms of utility.
This is a negative eﬀect of fiscal discipline that works through the term Z̃ 1 in (35).
Secondly, a decrease in the disposable income of the middle-aged population leads to lesssaving and thus a lower ratio of physical to human capital. However, at the same time,
fiscal tightening reduces the crowding-out eﬀect of public debt. The net eﬀect on theratio of physical to human capital is positive as shown in Panel (b). This eﬀect works
through the term Z̃ 2 in (35). In sum, the two opposing eﬀects on the choice of public
education expenditure produce an initial increase in the education expenditure-to-GDP
ratio followed by a decrease, as depicted in Panel (c).
Next, consider the eﬀect of fiscal tightening on the choice of the capital tax rate.
Firstly, this aﬀects the marginal benefit of capital taxation as shown by the term 1/Z̃ 1 +22

αβ (1 + γ) /Z̃ 2 . This eﬀect parallels the eﬀect on the choice of public education expenditure as described above. However, an additional eﬀect on the marginal benefit arises
through the term b̂/kh in (34). Fiscal tightening lowers the ratio b̂/kh and thus producesa negative eﬀect on the marginal benefit. This negative eﬀect outweighs the eﬀect via theterm 1/Z̃ 1 + αβ (1 + γ) /Z̃ 2 . Therefore, the government chooses a lower capital tax rate
to balance the marginal cost and benefit as ε decreases, as depicted in Panel (d). Finally,
fiscal tightening raises the per-capita growth rate as shown in Panel (e) because the positive eﬀects through the increased ratio of physical to human capital and decreased capitaltax rate outweigh the negative eﬀect through the increased labor tax and non-monotone
eﬀect through public education expenditure.
5.3.2

Comparative Dynamics

The comparative static analysis shows that the physical-to-human capital ratio and steadystate growth rate increase as ε decreases. This finding suggests that future generations
benefit from increased physical and human capital. However, are all generations madebetter oﬀ by fiscal tightening? To answer this question, Figure 2 plots the utility of generations from the initial old population for three scenarios, ε = 0.2, 0.5, and 0.8. Table
1 summarizes the results of the first four generations, indicating that the initial old population as well as generation 1 onward are made better oﬀ by the introduction of thedebt rule in (26), whereas generation 0 is made worse oﬀ (see also Figure 2). Thus, fiscal
tightening is not Pareto-improving.
[Figure 2 here.][Table 1 here.]
The mechanism behind the result is straightforward. Under the present assumption,
the government’s optimal choice of the debt-to-GDP ratio is Bun ; however, its choice islimited up to εBun (< Bun ) by the rule in (26). Because of this constraint, the government
in period 0 is unable to attain an “interior optimum.” In particular, the constraint hits the
middle-aged population in generation 0. Governments from period 1 are also constrainedby the rule, but they benefit from the higher levels of physical and human capital bequeathed from past generations. This benefit outweighs the cost of the constraint in (26).
Therefore, the introduction of the debt rule creates a trade-oﬀ between generations interms of utility.
The eﬀects of decreased ε on utility is monotone from generation 0 onward, as shownin Figure 2 and Table 1. However, the eﬀect is non-monotone as for the initial old
population. In particular, a decrease in ε from 1.0 to 0.5 improves their utility, but afurther decrease worsens it. This non-monotone eﬀect stems from the initial decrease
23

followed by an increase in the period 0 capital tax rate as depicted in Figure 3. The
U-shaped pattern of the period 0 capital tax rate parallels the hump-shaped pattern of
public education expenditure described above.
[Figure 3 here.]
Finally, we consider the case where the debt rule is imposed only for limited periods.
In particular, the debt rule in (26) is introduced in period 2, but terminated at theend of period 2, 3, or 4, meaning that successive governments from the termination
period onward are free to choose policies with no rule. Figure 4 illustrates the eﬀect of
this temporary implementation of the fiscal rule on fiscal policies, the ratio of physicalto human capital, the growth rate over time, and utility across generations when ε =0.5. As might be expected, the rule produces a temporary eﬀect on fiscal policies, thephysical-to-human capital ratio, and economic growth. However, it has a long-lasting
eﬀect on utility across generations owing to the increased human capital bequeathed fromgeneration to generation. This long-lasting eﬀect of the temporary rule, which was notshown in Barseghyan and Battaglini (2016), is caused by human capital accumulation
stimulated by debt-financed public education expenditure. Therefore, this result suggests
the importance of connecting seemingly unrelated subjects, public education and public
debt, to consider the impact of a debt rule over time and across generations.
[Figure 4 here.]

6

Conclusion

In this study, we developed an overlapping-generations model with physical and humancapital accumulation. In the model, public education and parental human capital wereused as inputs in the process of human capital formation. Public education spending
was financed through taxes on capital and labor income and public debt issues. Under
this framework, we demonstrated the endogenous policy formation and showed that thegreater political power of the old incentivizes the government to shift the tax burden fromthe old to the young, thereby resulting in a lower capital tax rate and a higher labor tax
rate. We also showed that the greater political power of the old leads to a lower debtto-GDP ratio. Moreover, we considered the role of a debt ceiling rule and found that
fiscal tightening increases the growth rate, but lowers the welfare of the currently working
generation. However, the growth eﬀect of the rule lasts for long periods via human capital
accumulation and benefits future generations even if it is imposed only for a limited time.
The presented model can be extended in several directions. For instance, one could
allow for the participation of the young in voting. Given that the young prefer more public education expenditure to allow their human capital to form, it would be reasonable24

to expect that their participation incentivizes the government to spend more on public
education by raising tax rates. One could also consider public good provision as an alternative mode of spending and investigate the conflict over the composition of expenditure.
One could go further and study alternative fiscal rules such as balanced budget rules,
expenditure rules, and revenue rules. Such extensions are left to future research.

25

A

ProofsA.1

Derivation of (10)Recall equation (9) in the main text:
(
)
ω
Ωt ≃
ln 1 − τtk + (1 + β) ln (1 − α) A (kt )α ht (1 − τt )|{z}
(1 + n)(1 − ω)
+ β [(α − 1) + γα] ln kt+1 + β ln
|{z}

(

k
1 − τt+1)(∗1)+ βγη ln xt .

(39)

(∗2)

The term (*1) is reformulated as follows:[(1 − α) A (kt ) ht (1 − τt ) = (1 − α) A (kt ) ht −
α

α

st−1
(1 + n)xt + Rt b̂t − (1 + n)b̂t+1 − τtk Rt

]

1+n
[
= (1 − α) A (kt )α ht − (1 + n)xt + αA (kt )α−1 b̂t − (1 + n)b̂t+1()]−τtk αA (kt )α−1 b̂t + kt ht ,(40)

where the first equality comes from (5) and the second equality comes from (3), (4), and(6). The term (*2) is reformulated as follows:[]β1
kt+1 =(1 − τt ) wt ht − (1 + n)b̂t+1(1 + n)ht+1 1 + β
{
(
)
(
)
β [
1
α
α−1
k
A (kt ) ht − 1 − τt αA (kt )=b̂t + kt ht(1 + n)D (ht )1−η (xt )η 1 + β]}−(1 + n)xt + (1 + n)b̂t+1 − (1 + n)b̂t+1
[
]
β
(
)
(
)
1
1+β
α
α−1
k
A (kt ) ht − 1 − τt αA (kt )b̂t + kt ht − (1 + n)xt − (1 + n)b̂t+1=
β(1 + n)D (ht )1−η (xt )η
(41)
where the first equality comes from (6) and the second equality comes from (3), (4),
and(6). From (40) and (41), Ωt in (39) is reformulated as()ω
ln 1 − τtk(1 + n)(1 − ω)
]
)
[
(
(
)
+ (1 + β) ln A (kt )α ht − 1 − τtk αA (kt )α−1 b̂t + kt ht − (1 + n)xt + (1 + n)b̂t+1
[
]
(
)
(
)
1
α
α−1
k
b̂t + kt ht − (1 + n)xt − (1 + n)b̂t+1+ β [(α − 1) + γα] ln A (kt ) ht − 1 − τt αA (kt )
β
)
(
k
(42)
+ β ln 1 − τt+1 + βη (1 − α) (1 + γ) ln xt .

Ωt ≃With the use of x̃t and b̃t+1 , we can further reformulate Ωt in (42) as in (10).
■
26

A.2

Proof of Proposition 1

Suppose that b̃′ = 0 holds. This leads to Z 1 = Z 2 . Thus, (17) implies that b̃′ = 0 if
(1 + n) − α (1 + γ) (1 + n) ≤ 0, that is, if (1 − α) /α ≤ γ.
When b̃′ = 0 holds, (16) and (18) are rewritten as
(
)
ω
1
1
b̂
= 1 α [1 + αβ (1 + γ)] 1 +,(1 + n)(1 − ω) 1 − τ k
Z
kh

(43)

and

1
1
= 1 (1 + n) [1 + αβ (1 + γ)] ,x̃Zrespectively. From (43) and (44), we obtain())(1 + n)(1 − ω)b̂ ((1 + n)x̃ =βη (1 − α) (1 + γ) α 1 +1 − τk .
ωkhβη (1 − α) (1 + γ)

(44)

(45)

We substitute (45) into (43) and rearrange the terms to obtain
1−τ =
k

ω
1
(1+n)(1−ω)().
ω+ ψ α 1 + b̂
(1+n)(1−ω)
kh

(46)

k
Thus, the guess in (13) is verified if Tun
is specified byω(1+n)(1−ω).
ωα (1+n)(1−ω)
+ψ

1
k
Tun
≡

By plugging (46) into (45), we obtain the policy function of X as in Proposition 1. We
can obtain the labor tax rate τ by substituting the policy functions of τ k , x, and b̃′ = 0
into the government budget constraint in (12).
Next, suppose that b̃′ > 0 holds; the first-order condition with respect to b̃′ in (17)holds with an equality. From (17) and (18), we obtain(1 + β) (1 + n)βη (1 − α) (1 + γ)
=
.
1
Z
x̃
From (16) and (18), we have
(
)
ω
)
b̂ (
(1+n)(1−ω)
k(1 + n)x̃.
α 1+
1−τ =
kh
βη (1 − α) (1 + γ)

(47)

(48)

We substitute (48) into the first-order condition with respect to b̃′ in (17). Afterrearranging the terms, we obtain the relation between x̃ and b̃′ :
[
(
)
] (
)
ω
1
(1+n)(1−ω)[1 − α (1 + γ)] 1 −+ 1 (1 + n)x̃ = α (1 + γ) +(1 + n)b̃′ .
βη (1 − α) (1 + γ)
β
(49)
27

We can obtain another relation between x̃ and b̃′ by substituting (48) into (47):
[
]
ω
1
+
β
(1+n)(1−ω)(1 + n)b̃′ =+1+(1 + n)x̃ − 1.
βη (1 − α) (1 + γ)βη (1 − α) (1 + γ)

(50)

By solving (49) and (50) for x̃ and b̃′ , we obtain the policy functions of x and b̂′ as in
Proposition 1. Note that b̂′ > 0 holds if and only if γ < (1 − α)/α.
We substitute the obtained policy function, X, into (48) to derive
1−τ =
k

ω
1
(1+n)(1−ω)().
ω+ ψ α 1 + b̂
(1+n)(1−ω)
khFinally, we can compute the labor tax rate by substituting the obtained τ k , x, and b̂′ into
the government budget constraint in (12).
■

A.3

Derivation of (19) and (20)Recall the human capital formation function, h′ = D(x)η (h)1−η . With the use of thepolicy function of x in Proposition 1, this function is rewritten as[]ηh′1 βη (1 − α) (1 + γ)η=D
[A (k)α ] ,ωh1 + n (1+n)(1−ω) + ψ

(51)

or,

h′
η
= DΨH [A (k)α ] ,hwhere ΨH is defined as in (22). The expression in (51) holds for both b̂′ = 0 and b̂′ > 0
because the policy functions of x are the same in these two cases.
Next, consider the capital market-clearing condition in (11). Suppose that b̂′ = 0 (i.e.,1−α≤ γ) holds. With the use of the policy functions in Proposition 1, (11) is rewrittenαas(1 + n)k ′ h′ =β 1 + αβ (1 + γ)
A (k)α h.
ω
1 + β (1+n)(1−ω) + ψ

By substituting (51) into (52) and rearranging the terms, we obtain{[]η }−1β 1 + αβ (1 + γ)
Xun
1−η
′
k =
(1 + n)D[A (k)α ] ,ω
1 + β (1+n)(1−ω) + ψ
1+n

(52)

(53)

where Xun is defined in Proposition 1.
Alternatively, suppose that b̂′ > 0 (i.e., 1−α> γ) holds. With the use of the policyαfunction of τ in Proposition 1, (11) is rewritten as(1 + n)k ′ h′ =αβ (1 + γ)
A (k)α h.
ω+ψ(1+n)(1−ω)
28

(54)

By substituting (51) into (54) and rearranging the terms, we obtain
αβ (1 + γ)k =ω+ψ(1+n)(1−ω)′{[]η }−1Xun1−η(1 + n)D[A (k)α ] .
1+n

(55)

(53) and (55) are summarized as in (19).
■A.4

Proof of Proposition 3

Equation (19) suggests that there is a unique, stable steady-state equilibrium level of thephysical-to-human capital ratio, k, for both (1 − α)/α ≤ γ and (1 − α)/α > γ.
Suppose that (1 − α)/α ≤ γ (i.e., b′ = 0) holds. The tax rates τ k and τ in Proposition
1 are
τ

k

ω
1
(1+n)(1−ω)
=1−
< 1,
ω
+ψα
(1+n)(1−ω)τ =1−

1 1 + αβ (1 + γ)
< 1.
ω1 − α (1+n)(1−ω)+ψ

Direct calculation leads toαω≤ψ,(1 + n)(1 − ω)
1−α
α
βη
ω
τ ≥0⇔
ψ−
(1 + γ) (1 − α) ≤
.
1−α
1−α
(1 + n)(1 − ω)τk ≥ 0 ⇔These are summarized in (23).
Next, suppose that (1−α)/α > γ (i.e., b′ > 0) holds. In the steady state, b̂/kh = b̂′ /k ′ h′holds, so
β[1−α(1+γ)]
A (k)α hω+ψ
1 − α (1 + γ)b̂(1+n)(1−ω)= αβ(1+γ)
=
,
(56)
α
kh
α (1 + γ)
A
(k)
h
ω
+ψ
(1+n)(1−ω)

where the first equality comes from the policy function of b̂′ in Proposition 1 and thecapital market-clearing condition in (54).
From (56), the tax rate on capital, τ k , when b′ > 0 becomes
τ

k

ω
1
(1+n)(1−ω)
(
),
=1−
ω
+ ψ α 1 + 1−α(1+γ)(1+n)(1−ω)α(1+γ)orτkω(1 + γ)(1+n)(1−ω)
< 1.
=1−
ω
+ψ
(1+n)(1−ω)

29Direct calculation leads toτk ≥ 0 ⇔ωψ≤ .
(1 + n)(1 − ω)γ(57)

The tax rate on labor, τ , is given byτ =1−
We obtain
τ ≥0⇔

1+β
1
< 1.
ω
1 − α (1+n)(1−ω) + ψ

1+β
ω
−ψ ≤
.
1−α
(1 + n)(1 − ω)(58)(57) and (58) are summarized in (24).
■

A.5

Proof of Proposition 5First, compare the capital tax rates. From Proposition 1, given k0 and b0 , we immediately
find that τ0k tax = τ0k debt holds. For t ≥ 1, τtk tax < τtk debt holds because b̂/kh = 0(> 0)holds in the tax-financing (debt-financing) case.
Next, compare the labor tax rates and growth rates. Direct comparison leads toτ |tax > τ |debt ⇔ γ <

1−α
,
α

and

1−α
.
α
The condition γ < α/(1 − α) holds by assumption. Finally, we immediately obtain(1 + n)x/y|tax = (1 + n)x/y|debt from Proposition 1.
h′ /h|tax > h′ /h|debt ⇔ γ <■A.6

Proof of Proposition 6(i) Recall the indirect utility function of the old given by (8). Because τ0k tax = τ0k debt
holds as demonstrated in Proposition 5, we immediately obtain V0o |tax = V0o |debt .
To consider the eﬀect on the utility of generation 0, V0M , recall the political objectivefunction in period 0:ωΩ0 =V o + V0M .
(1 + n)(1 − ω) 0
When γ < α/(1 − α) holds, Ω0 is maximized by choosing b̂′ > 0 (Proposition 1). However, the government choice is constrained when it is forced to finance its expendituresolely by taxes. That is, the government attains a lower value of its objective under tax
30

financing than under debt financing: Ω0 |tax < Ω0 |debt . This implies V0M tax < V0M debt
since V0o |tax = V0o |debt holds.
(ii) Recall the indirect utility function of the middle-aged in (7). Suppose that fromsome period t0 (≥ 1) onward, the economy is in a steady state regardless of the government’s financing method. The indirect utility function in (7) becomes( )α ()()
VtM j = (1 + β) ln(1 − α)A k|j
ht |j 1 − τ |j + β [(α − 1) + γα] ln k|j)(k+ βγη ln xt |j + ϕM (ht ) j ,+ β ln 1 − τt+1
j
(
)
(
)
k
≃ (1 + α (β + γ)) ln k|j + (1 + β) ln 1 − τ |j + β ln 1 − τt+1 j+ βγη ln xt |j + [(1 + β) + βγ (1 − η)] ln ht |j , j = tax, debt

(59)

where the constant terms are omitted from the expression.
With the use of the policy functions in Proposition 1, (59) is rewritten as follows:
(
)
1
(
)
VtM j ≃ (1 + α (β + γ)) ln k|j + (1 + β) ln 1 − τ |j + β ln
1 + b̂t+1 /kt+1 ht+1j( )α ()+ βγη ln A k|j
ht |j + [(1 + β) + βγ (1 − η)] ln ht |j ,or
VtM j ≃ [1 + α (β + γ + βγη)] ln k|j + (1 + β) ln(+ [1 + β (1 + γ)] ln h′ /h|j

)t−t0

(

1 − τ |j)ht0 |j , j = tax, debt.

The direct comparison of V1M tax and V1M debt leads toh′ /h|tax
h′ /h|debt
1 − τ |debtk|≷ [1 + α (β + γ + βγη)] ln debt + (1 + β) ln
k|tax
1 − τ |tax
ht |+ [1 + β (1 + γ)] ln 0 debt ,ht0 |tax

V1M tax ≷ V1M debt ⇔ (t − t0 ) [1 + β (1 + γ)] ln(60)

where the left-hand and right-hand sides of (60) are denoted by LHS and RHS, respectively. These satisfy the following properties:∂LHS/∂t > 0, LHS|t=t0 = 0, lim LHS = ∞, and ∂RHS/∂t = 0.
t→∞Therefore, there is a positive integer, denoted by t̂, such that LHS ≷ RHS ⇔ VtM tax ≷VtM debt for t ≷ t̂.
■
31

A.7

Derivation of (30)With the use of (26) and (28), we can rewrite b̂′ /k ′ h′ asb̂′=
k ′ h′=

β1
1+n 1+ββ
1+βThis leads to
1+

[

[

1
εBun A (k)α h1+n]α(1 − τ ) (1 − α) − 1+βεB
un A (k) hβεBun(1 − τ ) (1 − α) − 1+β
εBun
β

].

b̂′
(1 − τ ) (1 − α)
.
=
′
′
kh
(1 − τ ) (1 − α) − 1+β
εBun
β

(61)

1
k
From (61), the conjecture of τ k′ = 1 − Tcon
is rewritten as
1+b̂′ /k′ h′

τ

k′

k
= 1 − Tcon

(1 − τ ) (1 − α) − 1+βεBunβ(1 − τ ) (1 − α)

.

(62)

The substitution of the government budget constraint in (27) into (62) leads to (30).
■A.8

Proof of Proposition 7

The first-order conditions in (34) and (35) are summarized as
1 − τk =(1 + n)x̃1ω
().
(1 + n)(1 − ω) βη (1 − α) (1 + γ) α 1 + b̂/khWith the use of (63), Z̃ 1 and Z̃ 2 in (32) and (33) are rewritten as follows:

[
]
ω
1

Z̃ 1 = (1 + εBun ) − 1 + (1+n)(1−ω)(1 + n)x̃,
βη(1−α)(1+γ)[]ω
1 Z̃ 2 = (1 + εBun ) − 1 +(1 + n)x̃ − 1+βεBun .
(1+n)(1−ω) βη(1−α)(1+γ)
β

(63)

(64)

The substitution of (64) into the first-order condition with respect to x̃ in (35) leads toαβ (1 + γ)1
βη (1 − α) (1 + γ)+=.
(1 + n)x̃Z̃ 1 Z̃ 1 − 1+β
εB
un
β

(65)

Figure A.1 illustrates the graph of (65), taking (1 + n)x̃ on the horizontal axis. The
figure indicates that there are two candidates for a solution to (65). However, the larger
one is not feasible since Z̃ 1 < 0 holds. Therefore, the smaller one is the solution to (65).
[Figure A.1 here.]32

To derive the solution to (65), we reformulate (65) as follows:
εBun + αβ (1 + γ) Z̃ 1
Z̃ 1 − 1+ββη (1 − α) (1 + γ)
β
(
)
=
,
(1 + n)x̃Z̃ 1 Z̃ 1 − 1+β εB
β

un

or
f ((1 + n)x̃) ≡ G ((1 + n)x̃)2 − H(1 + n)x̃ + I = 0,(66)

where G, H, and I are defined in Proposition 7. Note that H > 0 and I > 0 hold because
εBun < β holds.
By solving (66) for (1 + n)x̃ and taking the smaller solution, we obtain√
H − (H)2 − 4GI(1 + n)x̃ = Xcon ≡
.
(67)
2G
The substitution of (67) into (63) yields
H−√(H)2 −4GI

1
ω
2G
(
),
1−τ =
(1 + n)(1 − ω) βη (1 − α) (1 + γ) α 1 + b̂/khk(68)

which verifies the initial guess. Finally, the labor tax rate is derived by substituting (67)and (68) into the government budget constraint in (27).
■33

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Figure 1: Eﬀects of decreased ε on the labor tax rate (Panel (a)), ratio of physical tohuman capital (Panel (b)), education expenditure-to-GDP ratio (Panel (c)), capital tax
rate (Panel (d)), and steady-state growth rate (Panel (e)).

37Figure 2: Utility of generations from the initial old population (denoted by -1) to generation 9.

38

Initial old population
Generation 0
Generation 1
Generation 2

ϵ = 0.2
1.30291.23462.33363.4144ϵ = 0.5
1.30311.23522.3201
3.3934ϵ = 0.8 ϵ = 1.0
1.3029 1.3026
1.2357 1.2359
2.3059 2.2961
3.3715 3.3563

Table 1: Utility of the first four generations

39

Figure 3: Period 0 capital tax rate.

40

Figure 4: Eﬀects of the temporary implementation of the fiscal rule in period 2. The
solid, dashed, and chain lines depict the cases where the rule is terminated at the end of
periods 2, 3, and 4, respectively. Note: Panel (g) plots the ratio of utility in the presenceof the temporary implementation to that in the absence of the implementation.

41

Figure A.1: Illustration of the left-hand side and right-hand side of equation (65).

42