Discussion Papers In Economics
And Business

Human Capital, Public Debt, and Economic Growth:
A Political Economy Analysis
Tetsuo Ono and Yuki Uchida
Discussion Paper 16-01-Rev.3

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANHuman Capital, Public Debt, and Economic Growth:
A Political Economy Analysis
Tetsuo Ono and Yuki Uchida
Discussion Paper 16-01-Rev.3

October 2017

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANHuman Capital, Public Debt, and Economic Growth:
A Political Economy Analysis∗
Tetsuo Ono†

Yuki Uchida‡

Osaka University

Seikei University

Abstract
This study considers the politics of public education policy in an overlappinggenerations model with physical and human capital accumulation. In particular, thisstudy examines how debt and tax financing diﬀer in terms of growth and welfare
across generations, as well as which fiscal stance voters support. The analysis showsthat the growth rate in debt financing is lower than that in tax financing, andthat debt financing creates a tradeoﬀ between the present and future generations.
The analysis also shows that debt financing attains slower economic growth thanthat realized by the choice of a social planner who cares about the welfare of all
generations.
• Keywords: Economic growth, Human capital, Public debt, Political equilibrium• JEL Classification: D70, E24, H63,∗

We are grateful for the comments from participants at the 2016 Japanese Economic Association
Spring Meeting and 2016 Asian Meeting of the Econometric Society. Tetsuo Ono acknowledges financial
support from Grant-in-Aid for Scientific Research (C) from the Ministry of Education, Culture, Sports,
Science and Technology of Japan No. 15K03509. Yuki Uchida acknowledges financial support fromGrant-in-Aid for JSPS Fellows of the Japan Society for the Promotion of Science No. 15J01375.
†
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

This paper considers the political determinants of fiscal policy and its impact on growth
and welfare across generations, with a focus on public education expenditures, which work
as both an inter-generational transfer from parents to children and an engine of economic
growth. Parents with altruistic concern for their children are likely to support public
education and bear the tax burden because they can benefit from highly educated children.
However, when the government has access to debt financing, parents may prefer debt totaxes because the former enables them to shift the fiscal burden to future generations.
The discussion above leads to the following questions: how do debt and tax financing diﬀer in terms of growth and welfare across generations and which fiscal stance dovoters adopt? To consider these questions, this study presents a three-period overlappinggenerations model with physical and human capital accumulation (see, e.g., Lambrecht,
Michel, and Vidal, 2005; Kunze, 2014). Parents care about their children’s wage income.
Public education spending and parental human capital are inputs in the human capitalformation process. Governments finance spending through tax on labor income and issue
bonds if they have access to debt financing.
Within this framework, we investigate the politics of fiscal policy formation. In eachperiod, three successive generations, the young, middle-aged, and elderly, can participate
in voting. However, the elderly are indiﬀerent between any two policies because they haveno tax burden and obtain no benefit from public education expenditure. The young may
benefit from public education in the future, but they are below the voting age. Therefore,
the government’s aim is to choose a fiscal policy that maximizes the utility of the middleaged. Given this process, we characterize the political equilibrium in both the tax- anddebt-financing cases, which yields the following findings.
First, the government’s choice of fiscal policy depends on the parents’ degree of altruism toward their children. Greater altruism encourages parents to leave higher wages totheir children, and they will do so by decreasing public bond issues and, thus, weakening
the crowding-out eﬀect on capital accumulation. In particular, there is a threshold levelof altruism such that the government finds it optimal (not) to issue public bonds whenaltruism is below (above) the threshold level. This result suggests that the degree of
altruism is a key explanation for the choice of fiscal policy.
Second, the growth rate in the debt-financing case is lower than that in the taxfinancing case. The debt overrides capital accumulation and adds the cost of debt repayment, reducing the government’s available resources and, thus, decreasing public education expenditure as an engine of economic growth. The peculiar negative impact on the
expenditure in the debt-financing case explains this result.
Third, debt financing has a tradeoﬀ between the present and future generations. When1

altruism is below the threshold level, the government in each period chooses debt financing
for the given physical and human capital levels. However, debt financing discourages
human capital accumulation and, thus, bequeaths less physical and human capital to
future generations. This implies that future governments must choose a fiscal policy
subject to lower levels of capital than those expected from tax financing. In other words,
the choice of the future governments is itself optimal for the given capital levels, but theyobtain less utility from debt financing than from tax financing.
Fourth, the political equilibrium generally attains under-accumulation of capital andunderinvestment of education, relative to the choice of the social planner who is assumed
to have the ability to commit to all of his or her choices at the beginning of a period,
subject to the resource constraint. The discrepancy between the choices of the plannerand politician arises because the planner values the welfare of all generations, whereas thepolitician cares only about existing generations. This shortsightedness of the politiciancauses less physical and human capital formation and, thus, slower economic growth than
do those realized by the planner’s choice.
The rest of this paper is organized as follows. We, first, present a literature survey in
Subsection 1.1. Thereafter, Section 2 presents the model. Section 3 describes the politicalequilibrium and Section 4 compares the debt- and tax-financing political equilibria. Section 5 characterizes the planner’s allocation and compares it to the political equilibrium.
Section 6 oﬀers some extensions to the basic model. Finally, Section 7 presents concluding
remarks.

1.1

Literature Review

Our study is related to research on public education and economic growth in terms ofparental altruism toward children. Examples include studies from Glomm and Kaganovich(2003, 2008) and Glomm and Ravikumar (1992, 2001, 2003), based in the competitive
equilibrium context and assuming that the central government can control policy processes
and outcomes across periods. Under this assumption, these researchers investigate howchanges in educational policy aﬀect growth and welfare across generations as well as the
optimal policy in terms of long-term growth and/or welfare.
Several studies have attempted to relax this assumption by introducing voting into
the policy-making process. These studies tend to focus on factors that aﬀect policy,
such as aging (Zhang, Zhang, and Lee, 2003; Gradstein and Kaganovich, 2004; Kunze,2014), inequality (Saint-Paul and Verdier, 1993), expectations of future policy (Glomm
and Ravikumar, 1995), private education as an alternative to public education (Gradsteinand Justman, 1996), social cohesion (Gradstein and Justman, 2002), and social security(Kemnitz, 2000; Poutvaara, 2006; Soares, 2006; Gonzalez-Eiras and Niepelt, 2012; Iturbe2

Ormaetxe and Valera, 2012; Kaganovich and Meier, 2012; Naito, 2012; Ono, 2015; Lancia
and Russo, 2016; Ono and Uchida, 2016). However, all of these studies assume financing
only from taxes and that the government budget constraint is balanced each period. In
other words, debt financing of public education is abstracted away from their analyses.
One exception is Zhang’s (2003) study that demonstrates the optimal policy for public
education when the government can issue public bonds. Thus, Zhang (2003) is based in thecompetitive equilibrium context, and mentions nothing about the political determinantsof debt-financing policy. As such, our study demonstrates the endogenous determinantsof debt-financed public education expenditures and their impact on growth and welfare
across generations. With this analysis, we can evaluate the relative performance of thetwo fiscal stances.
Our study is also related to the literature on the golden rule of public finance (see,e.g., Greiner and Semmler, 2000; Ghosh and Mourmouras, 2004a, 2004b; Yakita, 2008;
Minea and Villieu, 2009; Greiner, 2012). The rule allows the government to run a budget
deficit as long as it uses the deficit to finance productive public capital expenditures. In
the present framework, public education expenditures represent productive public capital
expenditures, as in Greiner (2008), who investigates the eﬀect of diﬀerent budgetary
regimes on growth and welfare. Yet, the present study diﬀers from Greiner (2008) in thatit incorporates the eﬀect of voting on fiscal policy, and considers its welfare eﬀects acrossgenerations.

2

Model

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

2.1

Individuals

Individuals display the following economic behavior over their life cycles. 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. 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 in period t − 1. In period t, the individual is middle-aged
and endowed with ht units of human capital. The individual supplies them inelastically3

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 income
tax rate, the individual distributes the after-tax income between consumption ct andsavings 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 the elderly age is:
dt+1 ≤ Rt+1 st ,where dt+1 is consumption, Rt+1 (> 0) is the gross return from investment in capital, and
Rt+1 st is the return from savings.
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 the parents’ human capital, ht . In particular, ht+1 is formulated using
the following equation:
ht+1 = D (xt )η (ht )1−η ,where D(> 0) is a scale factor and η ∈ (0, 1) denotes the elasticity of education technology
with respect to education spending.
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 the present
study, we abstract private education from the main analysis to simplify the presentationof the model and focus on the eﬀect of public education on growth and utility. Section
6.1 presents how the analysis and result would change when the model includes private
education.
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 of 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 }

4

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

(1)
(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 . The
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. 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 ,(3)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.

2.3

Government Budget Constraint

Public education expenditures are financed by both tax on labor income and public bondissues. Let Bt denote the aggregate inherited debt. A government budget constraint inperiod t is:
Bt+1 + τt wt ht Nt = Nt+1 xt + Rt Bt ,where Bt+1 is newly issued public bonds, τt wt ht Nt is the aggregate labor income tax
revenue, Nt+1 xt is the aggregate expenditure for public education, and Rt Bt is debt repayment. We assume a one-period debt structure to derive analytical solutions from themodel, and assume that the government in each period is committed to not repudiating
the debt.
By dividing both sides of the above expression, we obtain a per-capita form of theconstraint:(1 + n)b̂t+1 + τt wt ht = (1 + n)xt + Rt b̂t ,(4)where b̂t ≡ Bt /Nt is the per-capita public debt. We use the notation b̂t , rather than bt ,to distinguish the per-capita public debt, b̂t ≡ Bt /Nt , from the public debt per humancapital, bt ≡ Bt /Ht , which we introduce in the next section.
5

2.4

Economic Equilibrium

Public bonds are traded in a 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 . Using
kt+1 ≡ Kt+1 /Ht+1 , ht+1 = Ht+1 /Nt+1 , and the savings function in (1), we can rewrite thecondition as:()(1 + n) · kt+1 ht+1 + b̂t+1 =β· (1 − τt )wt ht .
1+β

(5)

The following defines the economic equilibrium in the present model.
∞
Definition 1. Given a sequenceof policies, {τt , x}t }t=0 , an economic equilibrium is a se{∞and prices {ρt , wt , Rt }∞t=0 with thet=0()
initial conditions k0 (> 0), b̂0 (≥ 0) and h0 (> 0), such that (i) given (wt , Rt+1 , τt , xt ) , cyt , cot+1 , stsolves the utility maximization problem;(ii) givent solves a firm’s profit() ((wt , ρt ), k)
maximization problem; (iii) given wt , ht , kt , b̂t , τt , xt , b̂t+1 satisfies the govern-

quence of allocations ct , dt , st , kt+1 , b̂t+1 , ht+1

ment budget
constraint; )(iv) ρt = Rt holds; and (v) the capital market clears:((1 + n) · kt+1 ht+1 + b̂t+1 = st .

3

Political Equilibrium

In this section, we consider voting on fiscal policy. In the present framework, the young,middle-aged, and elderly can participate in voting. However, the elderly are indiﬀerent
between any two policies because they bear no tax burden and obtain no benefits. The
young may benefit from the public education expenditure in the future, but we assume
that they are unable to participate in voting because they are below the voting age.
Under the assumption described above, the government’s aim in each period is tomaximize the indirect utility of the middle-aged, given by:
Vt = (1+β) ln {(1 − α)A (kt )α ht − τt wt ht }+{β(α − 1) + γα} ln kt+1 +γη ln xt +γ(1−η) ln ht +C0 ,(6)
where C0 includes constant terms and is defined by:
C0 ≡ ln1β+ β ln+ γ ln D + β ln αA + γ ln(1 − α)A.
1+β
1+β

The term {(1 − α)A (kt )α ht − τt wt ht } shows the after-tax income and, thus, represents
utility from consumption. The second term includes the utility from the return on savings,β(α − 1) ln kt+1 , and that from the next generation’s wage income, γα ln kt+1 . The thirdand fourth terms correspond to the utility from the next generation’s human capital, ht+1 .
6

Given the state variables kt , ht and b̂t , the period−t government chooses fiscal policy
to maximize Vt subject to the capital-market-clearing condition and government budget
constraint. In particular, we assume away government lending to the private sector, andfocus on the government borrowing case, b̂t+1 ≥ 0. Formally, Definition 2 describes thepolitical equilibrium in this framework.
Definition
2. A political equilibrium
is a sequence of policies {τt , xt }∞t=0 , allocations}∞{∞
ct , dt , st , kt+1 , b̂t+1 , ht+1
, and prices {ρt , wt , Rt }t=0 with the initial conditions
t=0

k0 (> 0), h0 (> 0) and b̂0 (≥ 0), such that (i) the conditions in Definition 1 (Economic
Equilibrium) are satisfied, and (ii) in period t(≥ 0), the government chooses xt (≥ 0)and b̂t+1 (≥ 0) to maximize Vt , subject to the capital market clearing condition andthe government budget constraint, given by kt , ht , and b̂t .
To formulate the government problem, we substitute the government budget constraintin (4) into Vt in (6) and obtain:()
Vt = (1+β) ln it − (1 + n)xt + (1 + n)b̂t+1 +{β(α − 1) + γα} ln kt+1 +γη ln xt +γ(1−η) ln ht +C0 ,(7)
where it is defined by:it ≡ (1 − α)A (kt )α ht − αA (kt )α−1 b̂t .
The term (1 − α)A (kt )α ht denotes labor income, and αA (kt )α−1 b̂t denotes debt repayment. Therefore, it presents the government’s available resources for its expenditures.
The government’s problem is to maximize Vt in (7) subject to the government budget
constraint in (4) and capital-market-clearing condition in (5). To more conveniently
reformulate the problem, we substitute the government budget constraint and humancapital formation function into the capital-market-clearing condition, and obtain:[]1β(1 + n)b̂t+1
kt+1 =·· it − (1 + n)xt −.
(8)(1 + n)D(xt )η (ht )1−η 1 + ββWe substitute this into Vt in (7) and rearrange the terms to obtain:[]
Vt = (1 + β) ln it − (1 + n)xt + (1 + n)b̂t+1[](1 + n)b̂t+1+ {β(α − 1) + γα} ln it − (1 + n)xt −β+ η(β + γ)(1 − α) ln xt + (1 − γ)(β + γ)(1 − α) ln ht + C1 ,where C1 includes the following constant terms:
C1 ≡ C0 + {β(α − 1) + γα} ln7β.
(1 + β) (1 + n)D(9)

The first three terms in (9) relate to policy-making. The first term represents theutility of lifetime consumption; the second term shows the utility of the interest income
and next generation’s wage income aﬀected by physical capital accumulation; and thethird term includes the three factors aﬀected by public education: the utility of the
return on savings, utility of the next generation’s wage income, and utility of the next
generation’s human capital. The government’s problem is to choose a pair of (xt , b̂t+1 )that maximizes Vt in (9) and the corresponding tax rate that satisfies the government
budget constraint. Solving the problem yields the following policy functions.
Lemma 1. The policy functions of xt , b̂t+1 , and τt in the political equilibrium are:xt = X0 · it ,b̂t+1 = B0 · it ,
τt = (1 + n) (X0 − B0 ) +αbt· {1 − (1 + n) (X0 − B0 )} · ,1−αktwhere:]−1(1 + β) + {β(α − 1) + γα}+1,X0 ≡ (1 + n) ·(β + γ)(1 − α)η
{}(1 + β) β − {β(α − 1) + γα}B0 ≡ max 0,.
(1 + n) [1 + (β + γ) {α + (1 − α)η}]−1[

Proof. See Appendix A.1.
To understand the property of the policy function of xt , let us focus on the three termsof X0 , 1 + β, β(α − 1) + γα, and (β + γ)(1 − α)η; they correspond to the coeﬃcients of thefirst, second, and third terms of Vt in (9), respectively. The term 1+β is the weight on theutility of consumption; a greater weight on it incentivizes the government to reduce public
education expenditures in order to increase the disposable income of the individuals and,thus, to raise consumption levels in their middle and elderly ages. The term β(α − 1) + γα
implies that an increase in public education expenditures reduces the physical-to-humancapital ratio, and this, in turn, creates two opposing eﬀects on the utility: an increasein the interest rate income, presented by β(α − 1), and a decrease in the children’s wage
income, presented by γα. Finally, the term (β + γ)(1 − α)η is the weight on the public
education expenditures; a greater weight enhances the children’s human capital formation
and, thus, improves parent’s utility. In summary, the terms β(α − 1) and (β + γ)(1 − α)η
work to increase public education expenditures, while the terms 1 + β and γα work to
decrease them.
The policy function of b̂t+1 indicates that (1 + β) β and {β(α − 1) + γα} are crucial
to the government’s financial state. They correspond to the coeﬃcients of the first andsecond terms of Vt in (9), respectively, so we can apply the abovementioned interpretation
8

for the policy function of xt . The term (1 + β) β implies that a greater weight on the utilityof consumption incentivizes the government to issue more public bonds in order to raise
consumption levels in the middle and elderly ages. The term {β(α − 1) + γα} implies
that an increase in public bond issues crowds out physical capital accumulation and,thus, increases the interest rate income, but decreases the children’s wage income. The
government chooses b̂t+1 > 0 and, therefore, borrows in the capital market if the eﬀectof the first term, (1 + β) β, outweighs the eﬀect of the second term, {β(α − 1) + γα};that is, if B0 > 0 ⇔ γ < β {(1 + β) + (1 − α)} /α. However, if B0 = 0, that is, γ ≥β {(1 + β) + (1 − α)} /α, the government finds it optimal to issue no public bonds.
The policy function of τt suggests an influence from both public education expendituresand public bond issues. The first term (1 + n) (X0 − B0 ) indicates that the government
attempts to increase the tax rate to finance public education expenditures, but can cut
the tax rate and finance a part of the expenditure by issuing public bonds. The second
term shows the eﬀect of debt repayment on the tax rate. In particular, the term “1” of{1 − (1 + n) (X0 − B0 )} indicates that the government raises the tax rate to finance debt
repayment. However, debt repayment pressure incentivizes the government to cut public
education expenditures and issue less public bonds. The former eﬀect, represented by(1 + n)X0 , works to decrease the tax rate, while the latter, represented by (1 + n)B0 ,works in the opposite direction.
Having established the policy functions, we are now ready to demonstrate the accumulation of physical and human capital as well as public debt. We substitute the policy
functions in Lemma 1 into the capital accumulation equation in (8), government budget
constraint in (4), and human capital formation function given by ht+1 = D (xt )η (ht )1−η ,and obtain:[]1−ηkt+1 = ΨK · (1 − α)A (kt )α − αA (kt )α−1 bt,[
αα−1 ]1−ηbt+1 = ΨB · (1 − α)A (kt ) − αA (kt )
bt
,
[
]η
ht+1
= D (X0 )η · (1 − α)A (kt )α − αA (kt )α−1 bt ,htwhere ΨK and ΨB are defined by:
1 − (1 + n)X0 − β B0
β
B0
·
ΨK ≡
and ΨB ≡,η1+β(1 + n)D (X0 )D (X0 )η1+nrespectively.
Given {k0 , b0 , h0 }, the sequence {kt , bt , ht } is distinguished by the above three equations, which we use to obtain Proposition 1.
Proposition 1. Suppose that the government can issue public bonds to finance its
expenditures. Given b0 /k0 ∈ (0, (1 − α)/α),9(i) there is a unique, stable, steady-state equilibrium with kt+1 > 0, bt+1 = 0, and τt ∈(0, 1) if γ ≥ β {(1 + β) + (1 − α)} /α;(ii) there is a unique, stable, steady-state equilibrium with kt+1 > 0, bt+1 > 0, and τt ∈(0, 1) if{}
β {(1 + β) + (1 − α)}β {(1 + β) + (1 − α)(1 − η)} β(1 + αβ)
max
,
<γ<
.
α + (1 − α)η
αα
Proof. See Appendix A.2.
The result in Proposition 1 suggests that the degree of altruism, denoted by γ, plays
a role in shaping fiscal policy. To gain the intuition for the γ assumption, let us, first,
consider the assumption in Proposition 1(i);γ≥β{(1 + β) + (1 − α)} ⇒ B0 = 0.
αAs mentioned above, the government finds it optimal to issue no public bond if the above
condition holds. However, if the condition is reversed, as expressed in the second inequality
condition in Proposition 1(ii), part of the expenditures is financed by borrowing in thecapital market.
Next, consider the role of the lower bound of γ. When γ is below the first lower boundgiven by β {(1 + β) + (1 − α)(1 − η)} / {α + (1 − α)η}, the degree of altruism is too low
to incentivize adults to provide higher levels of education and, thus, they would rather
support more public bond issues to finance public education expenditures. In this case,the government gains a surplus in revenue and, therefore, can refund it by subsidizing
households. We can rule out this possibility using the first lower bound of γ. In addition,
more public bond issues increases the debt repayment cost and, as such, results in a tax
rate higher than 100%. The second lower bound of γ guarantees that the tax rate is below
100% for period t ≥ 1, while the upper bound of b0 /k0 guarantees that the period 0 tax
rate is below 100%.
We should note that the present framework does not involve time inconsistency. In
period t, the government representing the period-t middle announces and implements
period-t policy. Such individuals are concerned about the period-t policy because it
creates costs and benefits in terms of their utility. However, they are indiﬀerent between
any two sets of fiscal policies in period t + 1 (i.e., in their old age) because they owe nocosts and receive no benefits from the period-t + 1 policy. Thus, they have no incentive
to act against any policy chosen by the next-period government.

10

4

Tax and Debt Financing

Public bond issues enable the current generation to pass costs on to future generations.
In fact, the public education expenditure, (1 + n)xt , plus the interest payment, (Rt − 1)b̂t ,is greater than the tax revenue, τt wt ht , along the steady-state debt-financing politicalequilibrium path with h′ /h > 1, where h′ denotes the next-period h. To confirm thisstatement, note that h′ /h > 1 implies (1+n)h′ /h > 1. We reformulate this as (1+n)h′ b >hb, or (1 + n)b̂′ > b̂ in the steady state. The government budget constraint in (4) implies
that (1 + n)b̂′ > b̂ leads to (1 + n)x < τ wh + (R − 1)b̂. Therefore, part of the fiscal burden
is passed on to future generations via public bond issues when adults have low altruism
toward children, γ < β {(1 + β) + (1 − α)} /α.
To consider the impact of debt financing in more detail, here, we assume the following
to focus on a situation in which the government chooses debt financing from the viewpointof maximizing its objective.
Assumption 1. γ < αβ {(1 + β) + (1 − α)} ⇔ B0 > 0.
To check the plausibility of this assumption, we calibrate the parameter γ to match theOrganisation for Economic Co-operation and Development (OECD) data during 2000–
2010 (see Appendix A.6). For this aim, we fix the share of capital at α = 1/3 following
Song, Storesletten, and Zilibotti (2012) and Lancia and Russo (2016). We set it atβ = (0.98)30 , implying that each period lasts 30 years, and that the period discountfactor is 0.98. In this setting, we obtain γ = 3.47 < β {(1 + β) + (1 − α)} /α = 3.62,
suggesting that Assumption 1 is empirically plausible.
Under Assumption 1, we consider a fiscal rule that prohibits debt financing and requires tax financing as an alternative. Here, we denote z|tax and z|debt the steady-statelevel of z(= k, b, τ, and h′ /h) for the tax- and debt-financing cases, respectively. We, first,
compare the aggregate expenditure-GDP ratio and the tax rate in the debt-financing casewith those in the tax-financing case, and obtain Proposition 2.
Proposition 2. Consider the aggregate expenditure-GDP ratio, Nt+1 xt /Yt and the tax
rate, τt . (i) For t = 0, N1 x0 /Y0 |tax = N1 x0 /Y0 |debt . For t ≥ 1, Nt+1 xt /Yt |tax >Nt+1 xt /Yt |debt . (ii) For t = 0, τ0 |tax > τ0 |debt . For t ≥ 1, τt |tax ≶ τt |debt if and only
if γ ≶ β(1 − α)/α + (1 + β)2 /(1 − α).
Proof. See Appendix A.3.
The first part of Proposition 2 states that both cases have equal aggregate expenditureGDP ratios in the initial period, but they diﬀer from period 1. To understand this result,recall the policy function xt , which indicates that the government uses a fraction, X0 , of11

its available resources, it , for public education expenditures in both cases. The government{}
has the same resources available in period 0, given by i0 = (1 − α)A (k0 )α − αA (k0 )α−1 b0 h0
for both cases. Therefore, the expenditure-GDP ratios are equal in period 0. However,they diﬀer from period 1. In the debt-financing case, the government must manage debt
repayment, which reduces its available resources. Therefore, the government attains alower expenditure-GDP ratio in the debt-financing case than in the tax-financing case.
The second part of Proposition 2 states that the tax rate in the tax-financing caseis higher than that in the debt-financing case in the initial period, but under a certain
condition, this relationship reverses in the next period. In the debt-financing case, thegovernment can implement a tax cut financed by public bond issues. Because of this tax
cut, the period 0 tax rate is lower in the debt-financing case than in the tax-financing case.
However, from period 1, the government has debt repayment costs in the debt-financing
case, and must then raise the tax rate. The second part of Proposition 2 shows that thistax-hike eﬀect is greater (less) than the tax-cut eﬀect if γ is below (above) the criticalvalue given by β(1 − α)/α + (1 + β)2 /(1 − α). Our calibration in Appendix A.6 suggeststhat γ is below the critical value. This indicates that the tax rate decreases from period
1 onward when debt financing is prohibited.
Next, we compare the steady-state growth rates for the two cases.
Proposition 3. The steady-state growth rate in the tax-financing political equilibrium
is higher than in the debt-financing political equilibrium; h′ /h|tax > h′ /h|debt .
Proof. See Appendix A.4.
The steady-state growth rates in the tax- and debt-financing cases are given by:
]η
[ (
) α
1
h′
β
1 − (1 + n)X0 1−α(1−η)= D · X0·{(1 − α)A} 1−α(1−η) ,h tax
1 + β (1 + n)D (X0 )η
η (α) 1−α(1−η)1) 1−α(1−η)(
1+n′
1 − (1 + n)X0 − β B0
h
β
ΨB
 ,
= D · X0·
(1 − α)A − αAηh debt1+β(1 + n)D (X0 )ΨKrespectively. The equation h′ /h|debt suggests that there are two negative eﬀects on thegrowth rate peculiar to the debt-financing case. First, the cost of debt repayment reduces
the government’s resources, which, in turn, decreases public education expenditures as anengine of economic growth. The term −(1 + n)B0 /β demonstrates this eﬀect. Second, thecost of debt repayment overwhelms capital accumulation and, thus, lowers the steadystate level of capital, resulting in a higher interest rate, which then further increases thedebt repayment cost. The term αAΨB /ΨK illustrates this eﬀect. These two negativeeﬀects from public debt result in a lower steady-state growth rate in the debt-financing
case than in the tax-financing case.
12Debt financing enables the government to cut the tax rate (Proposition 2(ii)). The
tax cut increases households’ disposable income and, thus, lifetime consumption for theperiod-0 middle-aged agents, while the public education expenditures remain equal in bothcases (Proposition 2(i)). The period-0 middle-aged agents find it optimal to choose debt
rather than tax financing to maximize their utility when γ < β(1−α)/α+(1 + β)2 /(1−α).
However, future generations would be worse oﬀ due to debt repayment and low growth
rates. This suggests the possibility of intergenerational utility tradeoﬀs.
To investigate the welfare consequences of debt financing across generations, here, we
compare the tax- and debt-financing cases in terms of utility, and obtain Proposition 4.
Proposition 4. Along the equilibrium path with h′ /h > 1, the steady-state utility ofthe middle-aged is higher in the tax-financing case than in the debt-financing case:
Vtax > Vdebt .
Proof. See Appendix A.5.
In each period, the government, which represents the middle-aged agents, finds it optimal to choose debt financing for the state variables k, b, and h. The choice of debt financing
discourages human capital accumulation (Proposition 3), which, in turn, bequeaths lessphysical and human capital to future generations. This implies that future governments
are forced to choose a fiscal policy subject to lower levels of capital. If the government
introduces tax financing and continues this for long periods, then future governments canenjoy higher capital and realize a higher utility for the middle-aged, although they are
constrained by the choice of fiscal policy. In other words, the choice of future governments
is itself optimal for given capital levels, but they obtain less utility for the middle-aged
from debt financing than from tax financing.

5

Planner’s Allocation

This section demonstrates an allocation chosen by a benevolent planner. The planner has
the ability to commit to all of his or her choices at the beginning of a period, subject to theresource constraint. We compare the planner’s allocation with the political equilibrium
allocation in terms of long-run capital, growth rates, and education-GDP ratios.
The planner is assumed to value the welfare of all generations. In particular, theobjective of the planner is to maximize a discounted sum of the life-cycle utility of all
current and future 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. The parameter θ ∈ (0, 1) is a planner’s discount factor.
ct +In the present framework, the 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, here, we undertake the following normalization:c̃t ≡ ct /ht , d˜t ≡ dt /ht , and x̃t ≡ xt /ht .
Then, the above resource constraint is rewritten as:
1 ˜
c̃t +
dt + (1 + n)kt+1 D (x̃t )η + (1 + n)x̃t = A (kt )α ,1+nand the utility function becomes:(10)

U−1 = β ln d˜0 + (β + γ) ln h0 + γα ln k0 + γ ln(1 − α)A,U0 = ln c̃0 + β ln d˜1 + γα ln k1 + (β + γ) η ln x̃0 + (1 + β + γ) ln h0+ γ ln(1 − α)A + (β + γ) ln D,
Ut = ln c̃t + β ln d˜t+1 + γα ln kt+1 + (β + γ) η ln x̃t + (1 + β + γ) ηt−1∑

ln x̃jj=0+ (1 + β + γ) ln h0 + γ ln(1 − α)A + {(β + γ) + t · (1 + β + γ)} ln D, t ≥ 1.
The planner’s objective function is now given by:
[
{
}
]
∞
∑
αγ
β ˜
θ
t
SW (k0 ) ≃ln k0 +
θ ln c̃t + ln dt + γα ln kt+1 + (β + γ) + (1 + β + γ)η ln x̃t ,
θ
θ
1
−
θ
t=0
where constant terms are omitted from the expression. Thus, we can express the Bellman
equation for the problem as follows:
{
}
}
{
θ
β ˜
′
′
SW (k) = max
ln c̃ + ln d + γα ln k + (β + γ) + (1 + β + γ)η ln x̃ + θSW (k ) ,˜ ′ ,x̃}θ
1−θ{c̃,d,k(11)
subject to (10), where k ′ denotes the next-period stock of capital.
Solving the problem in (11) leads to the policy functions of k ′ and x (see Appendix A.7).
We compute the corresponding steady-state capital level, growth rate, and education-GDP
ratio based on the calibration in Appendix A.6, and compare them to those in the political equilibrium, as depicted in Figure 1. The numerical result shows that the concernedvariables in the planner’s allocation realize larger values than those in the political equilibrium unless the planner’s discount factor is below 0.2. This result suggests that thepolitical equilibrium generally attains under-accumulation of capital and underinvestmentof education and, thus, suﬀers from a lower growth rate due to the shortsightedness ofpoliticians.
[Figure 1 here.]14

6

Extensions and Further Analysis

Up to this point, the analysis has assumed (i) no private education; (ii) no alternative
public spending, such as old-age benefits; (iii) no fertility choice; and (iv) no international
lending and borrowing. This section briefly considers how the analysis and results wouldchange if we relax each of these assumptions. A supplementary explanation for this section
is provided in Appendix B.

6.1

Private Education

This subsection introduces private education, denoted by et . The budget constraint of
the young is ct + st + (1 + n)et ≤ (1 − τt )wt ht , and human capital formation is ht+1 =D (xt )η (et )δ (ht )1−η−δ , where δ is the elasticity of human capital with respect to et . Solving
the utility maximization problem leads to the following indirect utility of the middle-aged:



 (
)
Vt = 1 + β + δ (γ + β (1 − α) − γα) ln it − (1 + n)xt + (1 + n)b̂t+1
{z
}
|


e.1


1 + γδ
+ {β(α − 1) + γα} ln it − (1 + n)xt − β ·(1 + n)b̂t+1  + (γ + β)(1 − α)η ln xt ,| {z }
e.2

where the terms (e.1) and (e.2) are relevant to private education. When δ = 0, thisexpression matches the indirect utility function in the absence of private education.
The term (e.1) includes the following eﬀects of private education. First, an increasein private education improves the human capital of the next generation, represented by
the term γ in parentheses. However, an improvement in the human capital decreases
the physical-to-human-capital ratio and, thus, lowers the next generation’s wage, butraises the interest rate. These eﬀects are represented by the terms β (1 − α) and −γα inparentheses, respectively. The net eﬀect of these three eﬀects on utility is positive since
γ + β (1 − α) − γα > 0. Therefore, the term (e.1) implies that the middle-aged attachlarger weights to the cost of public education as well as to the benefit of public bondissues when private education is available than in its absence.
The term (e.2) shows that private education increases the cost of public bond issues
because it lowers savings, which, in turn, strengthens the crowding-out eﬀect of publicbond issues. Overall, private education definitely decreases public education spending,while it has two opposing eﬀects on public bond issues. Direct calculation suggests thatthe net eﬀect is negative: the availability of private education lowers public bond issues.
To examine the long-run eﬀect of private education, here, we undertake the numerical analysis based on the calibration of the model to OECD countries, as presented in15

Appendix A.6. For analysis, we focus on the elasticity of private education, δ, and investigate its impact on steady-state debt, public and private education-GDP ratios, capital,
and growth rate, as illustrated in Figure 2. To see this, we consider an increase in δ,
accompanied by a one-to-one decrease in η, so that δ + η does not change and is fixed at
0.12.
[Figure 2 here.]
Panel (a) shows that steady-state debt decreases as δ increases. A higher δ implies ahigher elasticity of private education and, thus, incentivizes individuals to invest more inprivate education as a substitute for public education. In turn, this reduces government
expenditure on public education, so the government finds it optimal to reduce public bondissues. In particular, there is a threshold value of δ, such that the government issues no
public bonds and finances its expenditure solely by taxation when δ is above the thresholdvalue.
Panel (b) shows that public education expenditure decreases, while the private education expenditure increases as δ increases. This result is straightforward because ahigher elasticity of private education is associated with a lower elasticity of public education in the present assumption. The results in Panels (a) and (b) suggest that there
are two opposing eﬀects on capital accumulation: a positive eﬀect by the reduction of thecrowding-out eﬀect of public debt, and a negative eﬀect by the increase in private education associated with a decrease in savings. Below the threshold value of δ, the positiveeﬀect outweighs the negative eﬀect; beyond the threshold, the positive eﬀect disappears.
Thus, we can observe a hump-shaped pattern of the steady-state capital, as depicted inPanel (c).
Panel (d) shows an initial decrease in the growth rate, followed by an increase, in
response to an increase in the elasticity, δ. An increase in δ has the following three eﬀects
on the growth rate: (i) a decrease in public education expenditure; (ii) an increase in
private education expenditure; and (iii) an increase in capital in the presence of publicbond issues, and a decrease in capital in its absence. When δ is below the threshold level,such that the government issues public bonds, the positive eﬀects created by the last
two are outweighed by the first negative eﬀect. However, when δ is above the thresholdlevel, such that the government issues no public bonds, the positive eﬀect of the second
outweighs the negative eﬀects of the first and third. Therefore, the U-shaped growth
pattern is observed in the presence of private education.

6.2

Old-Age Benefits

This subsection introduces old-age benefits as an alternative mode of government spending, and investigates how they aﬀect public education expenditures. For this purpose, we
16modify the government budget constraint in (4) as:(1 + n)b̂t+1 + τt wt ht = (1 + n)xt +pt+ Rt b̂t ,1+nwhere pt denotes the per-capita benefits to the elderly living in period t. The budget
constraint of the elderly is dt ≤ Rt st−1 + pt . The elderly now have an incentive to control
the benefits through voting. We keep the assumption that the young are below votingage.
To reflect the preferences of the elderly and middle-aged in voting, here, we employ theprobabilistic voting mechanism according to Lindbeck and Weibull (1987). In particular,
the period-t political objective function is a weighted average of the elderly and middleaged:Ωt = ωVto + (1 + n)(1 − ω)Vtm ,where ω(> 0) is the relative political weight of the elderly, and Vto and Vtm are the indirect
utility functions of the elderly and middle-aged, respectively.
The problem of the government in period t is to choose a fiscal policy to maximize Ωtsubject to the government budget constraint and capital-market-clearing condition. The
first-order conditions with respect to pt and xt are summarized as follows:ωβ(1 − ω) (β + γ) (1 − α),pt =jt + 1+nxt
where the left-hand side shows the marginal benefit of p and the right-hand side shows
the marginal benefit of x. The government chooses x and p to equate these benefits. This
expression suggests that the government allocates its spending from public education toold-age benefits as the elderly’s political power, ω, increases.

6.3

Endogenous Fertility

This subsection introduces the fertility decisions of the individuals, and investigates how
this extension would aﬀect the result. Let nt+1 denote the population growth rate fromperiod t to period t + 1. Following de la Croix and Doepke (2004), we assume that parents
care about both child quantity, 1 + nt+1 , and quality, ht+1 . In particular, the problem of
the period-t middle is:max ln ct + β ln dt+1 + γ ln(1 + nt+1 )wt+1 ht+1
s.t. ct + st ≤ (1 − τt ) wt ht (1 − ϕ(1 + nt+1 ))dt+1 ≤ Rt+1 st ,where ϕ ∈ (0, 1) means that raising one child takes fraction ϕ of an adult’s time.
17

Solving the problem leads to:
1 + nt+1 =γ1·,
ϕ 1+β+γ

showing that the fertility rate is constant across periods as in de la Croix and Deopke(2004). Thus, the introduction of endogenous fertility does not qualitatively aﬀect theresult, as long as we keep the assumption of a logarithmic utility function.

6.4

Small Open Economy

In this subsection, we modify the model by assuming a small open economy, which allows
for international lending and borrowing for a given world interest rate, R. The profit maximization conditions imply that for a given R, per-capita capital k and wage w are uniquelydetermined and constant over time. This implies that there is no inter-temporal eﬀect of
current fiscal policy choice through capital accumulation. Within this specification, we
compare the tax- and debt-financing cases in terms of growth and utility.
We, first, consider the tax-financing case. We assume b̂t+1 = 0 and maximize Vt in(7), with respect to xt , to obtain:{
xt |tax =(γη
wh0 − Rb̂0(1+n)(1+β+γη)γη
w ht |tax(1+n)(1+β+γη))
for t ≥ 1.

This is the public education expenditure in the tax-financing case.
In the debt-financing case, the government’s objective function in (7) suggests thatsince k is constant, there is no crowding-out eﬀect of public bond issues. This incentivizes
the government to borrow as much as possible in the capital market. The borrowing
reaches its ceiling, that is, the natural debt limit, b̄t , which is defined by setting τt = 1
and xt = 0 in the government budget constraint. Thus, b̄t satisfies (1+n)b̄t +wht = 0+Rb̄t ,
or:
w
b̄t =
ht = b̄ · ht ,R − (1 + n)
where b̄ ≡ w/(R − (1 + n)). We assume R > 1 + n in the following.
Given b̂t+1 = b̄ · ht , the solution maximizing Vt in (7) is:(){γη
wh0 − Rb̂0 + (1 + n)b̄0(1+n)(1+β+γη)
xt | =
0for t ≥ 1.
In period 0, the government can finance its public education expenditure by borrowing in
the capital market. However, from period 1 onward, the government is unable to spend
its revenue for public education because it must use the revenue to repay debt. Therefore,
the choice of fiscal policy in the debt-financing case results in no human capital growth,18ht+1 /ht |debt = 0. This, in turn, leads to Vt |debt = −∞ from period 1 onward because of
the specification of the logarithmic utility function.
The result, thus far, is summarized as:
x0 |tax < x0 |debt ,xt |tax > xt |debt = 0 for t ≥ 1.
Given the property that growth and utility increase in response to an increase in public
education expenditure, we can conclude that in period 0, growth and utility are higher
in the debt-financing case than in the tax-financing case. However, the result is reversed
from period 1 onward because there is no growth as well as no available resources for
consumption and public education in the debt-financing case. This result is qualitatively
similar to that shown in the closed economy analysis of the main text.

7

Conclusion

This paper developed an overlapping-generations model with physical and human capital
accumulation using public education and parental human capital as inputs in the processof human capital formation. Public education spending is financed through tax on labor
income in the tax-financing case, and by both tax and public bond issues in the debtfinancing case. Within this framework, we demonstrated the endogenous policy formationand showed that the current generation will prefer debt financing to tax financing, since
the former enables the current generation to pass the debt repayment costs on to future
generations. However, the debt repayment costs induce successive generations to cut theexpenditure, making them worse oﬀ than in the tax-financing case. Our results provide
one possible explanation for why debt financing continues to dominate policy in many
developed countries, despite being expected to perform worse than tax financing.

19

A

ProofsA.1

Proof of Lemma 1

The first-order conditions with respect to xt and b̂t+1 are:xt :(1 + β)(1 + n)+{β (α − 1) + γα} (1 + n)

it − (1 + n)xt + (1 + n)b̂t+1
it − (1 + n)xt − 1+n
b̂t+1
β
1+n
{β (α − 1) + γα} β(1 + β)(1 + n)b̂t+1 :,≤b̂
it − (1 + n)xt + (1 + n)b̂t+1
it − (1 + n)xt − 1+n
t+1
β

=

(β + γ)(1 − α)η
, (12)
xt(13)

where an equality holds in (13) if b̂t+1 > 0, while an inequality holds if b̂t+1 = 0.
Assume that b̂t+1 > 0. We substitute (13) with an equality into (12), and obtain:{β (α − 1) + γα} 1+nβ
it − (1 + n)xt − 1+n
b̂t+1
β

+

{β (α − 1) + γα} (1 + n)
it − (1 + n)xt − 1+n
b̂t+1
β

=

(β + γ)(1 − α)η
,
xt

or
xt =

[
]
(β + γ)(1 − α)η
· βit − (1 + n)b̂t+1 . (14)
(1 + n) [{β (α − 1) + γα} (1 + β) + β(β + γ)(1 − α)η]We substitute (14) into (13). After rearranging the terms, we obtain the policy function of b̂t+1 as(1 + β)β − {β (α − 1) + γα}b̂t+1 =it .
(1 + n) [1 + (β + γ) (α + (1 − α) η)]
We then substitute this into (14) to obtain the policy function of xt as xt = X0 · it .
Therefore, we obtain
xt = X0 · it and b̂t+1 = B0 · it ,where X0 and B0 are defined in Lemma 1.
To obtain the policy function of τt , recall the government budget constraint, which isreformulated as (1 + n)b̂t+1 + τt (1 − α)A(kt )α ht = (1 + n)xt + αA(kt )α−1 b̂t . Plugging thepolicy function of xt and b̂t+1 into this constraint and using bt = b̂t ht , we have:τt (1 − α)A(kt )α ht = (1 + n)X0 it + αA(kt )α−1 bt ht − (1 + n)B0 it}{= (1 + n)(X0 − B0 ) (1 − α)A(kt )α ht − αA(kt )α−1 bt ht + αA(kt )α−1 bt ht= (1 + n)(X0 − B0 )(1 − α)A(kt )α ht + {1 − (1 + n)(X0 − B0 )}αA(kt )α−1 bt ht .
Dividing both sides by (1 − α)A(kt )α ht , we obtain:τt = (1 + n)(X0 − B0 ) +αbt{1 − (1 + n)(X0 − B0 )} .
1−α
kt
■
20

A.2

Proof of Proposition 1

First, suppose that γ ≥ β {(1 + β) + (1 − α)} /α (i.e., B0 = 0) holds. We obtain b̂t+1 = 0
from Lemma 1, and the corresponding tax rate as follows:{α(1 + n)X0 + 1−α{1 − (1 + n)X0 } kb00 for t = 0,τt =(1 + n)X0
for t ≥ 1.
The tax rate from period 1 onward satisfies τt ∈ (0, 1) because (1 + n)X0 ∈ (0, 1) holds;
and the tax rate in period 0 satisfies τ0 ∈ (0, 1) if
0 < (1 + n)X0 +

α
b0
< 1.
{1 − (1 + n)X0 }1−αk0

Given that 1 − (1 + n)X0 > 0, the first inequality holds if b0 /k0 > 0; the second inequality
holds if b0 /k0 < (1 − α)/α. Thus, τt ∈ (0, 1) holds if b0 /k0 ∈ (0, (1 − α)/α).
We can use the policy functions of xt and τt to reformulate the capital market clearing
condition in (5) as follows:{[]1−ηΨK · (1 − α)A (k0 )α − αA (k0 )α−1 b0
for t = 0,
kt+1 =α 1−ηΨK · [(1 − α)A (kt ) ]for t ≥ 1.
Given k0 (> 0) and b0 (> 0), a unique sequence of {kt } satisfies the above condition. The
condition for t ≥ 1 suggests that the sequence stably converges to the unique steady state.
Next, suppose that γ ≤ β {(1 + β) + (1 − α)} /α (i.e., B0 > 0) holds. We first derive
the conditions for which kt+1 > 0, bt+1 > 0, and τt ∈ (0, 1) hold along the equilibrium
path. Second, we show the existence and uniqueness of the steady-state equilibrium.
Finally, we show that the unique steady-state equilibrium is a sink.
Step 1.
Recall the capital and debt accumulation equations given by:
)1−η
it
,
ht
( )1−η
[itαα−1 ]1−η= ΨB ·,
bt+1 = ΨB · (1 − α)A (kt ) − αA (kt )
bt
ht

[
]1−η
kt+1 = ΨK · (1 − α)A (kt )α − αA (kt )α−1 bt= ΨK ·

(

(15)
(16)

respectively. kt+1 > 0 and bt+1 > 0 hold along the equilibrium path if it /ht > 0 for allt, ΨK > 0, and ΨB > 0.
In period 0, we reformulate the term i0 /h0 as:
[
]
i0
α
b0
α
= (1 − α)A (k0 ) · 1 −·.
h0
1 − α k0
Thus, i0 /h0 > 0 holds if

b0
1−α
<
.
k0
α
21

(17)

In period t ≥ 1, the term it /ht is rewritten as
[
]
it
α
bt
α
= (1 − α)A (kt ) · 1 −·ht
1 − α kt[]αΨBα= (1 − α)A (kt ) · 1 −·1 − α ΨK[{}]α(1+β)
1 − (1 + n)X0 − 1+n
B
−
(1
+
n)
B
0
0
β
(1−α)β= (1 − α)A (kt )α ·,1+n1 − (1 + n)X0 − β B0
where we obtain the second line using (15) and (16), and the third line using the definitionof ΨK and ΨB .
The expression above suggests that it /ht > 0 holds for t ≥ 1 if the following condition
holds:{}1+nα(1 + β)B0 > 0 and 1 − (1 + n)X0 −B0 − (1 + n)
B0 > 0.
(18)
β(1 − α)βIn addition, the definitions of ΨK and ΨB suggest that ΨK > 0 and ΨB > 0 hold if (18)
holds. After some manipulation, we have:β
B0 > 0 ⇔ γ < {(1 + β) + (1 − α)} ,
α
{
}
1+n
β
α(1 + β)1 − (1 + n)X0 −B0 − (1 + n)
B0 > 0 ⇔ (1 + αβ) < γ.
β(1 − α)β
α

(19)
(20)

Therefore, kt+1 > 0 and bt+1 > 0 for t ≥ 0 if (17), (19), and (20) hold.
Next, consider the tax rate:
τt = (1 + n) (X0 − B0 ) +btα· {1 − (1 + n) (X0 − B0 )} · ,
1−α
kt

where
1 − (1 + n) (X0 − B0 ) > 0.
Given b0 /k0 > 0, τ0 > 0 holds if X0 − B0 > 0, that is, if
β {(1 + β) + (1 − η)(1 − α)}< γ.
α + (1 − α)η

(21)

For t ≥ 1, τt > 0 holds if X0 − B0 > 0 and bt /kt > 0, that is, if (19), (20), and (21) hold.
To find the condition for τt < 1, let us first consider the period-0 tax rate. By direct
calculation, we have:b01−ατ0 < 1 ⇔<⇔ (17).
k0α
For period t ≥ 1, bt /kt = ΨB /ΨK holds. We thus have:τt < 1 ⇔αΨBβ(1 + αβ)·<1⇔< γ ⇔ (20).
1 − α ΨK
α22

The results established thus far suggest that kt+1 > 0, bt+1 > 0, and τt ∈ (0, 1) hold if(17), (19), (20), and (21) hold, that is, if:{}
β {(1 + β) + (1 − α)(1 − η)} β(1 + αβ)
β {(1 + β) + (1 − α)}
max
,
<γ<
,
α + (1 − α)η
α
α
and

b0
∈
k0

(
)
1−α
0,
.
α

Step 2.
The steady-state pair of (k, b) satisfies:[]1−η
k = ΨK · (1 − α)A (k)α − αA (k)α−1 b,[
αα−1 ]1−η.
b = ΨB · (1 − α)A (k) − αA (k)bThese equations lead to k/b = ΨK /ΨB , that is, b = (ΨB /ΨK ) · k. Plugging this into[]1−ηk = ΨK · (1 − α)A (k)α − αA (k)α−1 b, and rearranging the terms, we obtain a uniquevalue of k given by:[{}1−η ]1/(1−α(1−η))ΨB
k = ΨK · (1 − α)A − αA
.
ΨK

(22)

The corresponding value of b is also uniquely determined using b = (ΨB /ΨK ) k.
Step 3.
Recall the law of motions of capital and debt:[]1−η
kt+1 = ΨK · (1 − α)A (kt )α − αA (kt )α−1 bt,[
αα−1 ]1−ηbt+1 = ΨB · (1 − α)A (kt ) − αA (kt )bt.
Diﬀerentiating these with respect to k and b and evaluating them at the steady state, we
obtain:
[
]
[
]
dkt+1
dkt
=J
,
dbt+1
dbt
where

[
J≡

and

ΨK · ı̂(k, b) · α(1 − α)A(k)α−2 (k + b) −ΨK · ı̂(k, b) · αA(k)α−1
ΨB · ı̂(k, b) · α(1 − α)A(k)α−2 (k + b) −ΨB · ı̂(k, b) · αA(k)α−1[]−η· (1 − η) .
ı̂(k, b) ≡ (1 − α)A(k)α − αA(k)α−1 b

For any (k, b) ≫ 0, the trace and determinant of J are{
trJ = αA(k)α−1 · ı̂(k, b) · [(1 − α)ΨK − αΨB ] > 0,
detJ = 0.
23],

Here, the sign of the term (1 − α)ΨK − αΨB is(1 − α)ΨK − αΨB > 0 ⇔ γ >β(1 + αβ),αwhere γ > αβ (1 + αβ) is satisfied with the assumption in Proposition 5. Moreover, it istrue that:∆ ≡ (trJ)2 − 4detJ = (trJ)2 > 0.
Therefore, J has two distinct positive eigenvalues, denoted by λ1 and λ2 (> λ1 ), at thesteady state.
We obtain the eigenvalues of J by solving the following equation:p(λ) ≡ (λ)2 − (trJ) λ + detJ = 0.
The solution of p(λ) = 0 is
λ1 = 0 and λ2 = trJ(> 0).
The remaining task is to show λ2 < 1. For this purpose, recall the law of motion of cap[]1−ηital in the steady state, k = ΨK · (1 − α)A (k)α − αA (k)α−1 b. This is reformulated
as:
(
) −η
k 1−η
ı̂(k, b) = (1 − η)
.
(23)
ΨK
Using (22) and (23), we rewrite λ2 =trJ as follows:
λ2 = trJ(α−1= αA(k)= αA (ΨK )(1 − η)

−1
1−η

k
ΨK

−η
) 1−η

[(1 − α)ΨK − αΨB ]1(1 − η)ΨB(1 − α)A − αA Ψ
K

(

1
ΨK

−η
) 1−η

[(1 − α)ΨK − αΨB ]= α(1 − η) ∈ (0, 1),
where the first line comes from (23) and the second line is derived using (22).
■A.3

Proof of Proposition 2.

(i) In both cases, the policy function of x is xt = X0 ·it , where it ≡ {(1 − α)A(kt )α − αA(kt )α−1 } ht .
Given that Yt = A(kt )α ht Nt , we compute the ratio Nt+1 xt /Yt as follows:{}(1 + n)X0 (1 − α)A (kt )α − αA (kt )α−1 bt ht
Nt+1 xt=
Yt taxA (kt )α ht{(1 + n)X0 {(1 − α) − αb0 /k0 } for t = 0,=
(1 + n)X0 (1 − α) for t ≥ 1,
24

and

{
}
(1 + n)X0 (1 − α)A (kt )α − αA (kt )α−1 bt ht
Nt+1 xt=
Yt debt
A (kt )α ht= (1 + n)X0 {(1 − α) − α bt /kt |debt } ,where bt > kt holds for any equilibrium path. Therefore, we obtain the first part of
Proposition 2.
(ii) Recall that the tax rates in the tax-finance and debt-finance cases are given by:{
α b0(1 + n)X0 + (1 − (1 + n)X0 ) 1−αfor t = 0,k0
τt |tax =(1 + n)X0
for t ≥ 1,and{
τt |debt =α(1 + n) (X0 − B0 ) + 1−α· {1 − (1 + n) (X0 − B0 )} · kb00 for t = 0,ΨBα(1 + n) (X0 − B0 ) + 1−α· {1 − (1 + n) (X0 − B0 )} · Ψ
for t ≥ 1.
K

For t = 0, τ0 |tax and τ0 |debt are compared as follows:
τ0 |tax ≷ τ0 |debt ⇔

1−α
b0
≶
.
k0
α

Assuming kb00 < 1−αin Proposition 1, we obtain τ0 |tax > τ0 |debt .
αFor t ≥ 1, τt |tax and τt |debt are compared as follows:α1+β}{1 − (1 + n) (X0 − B0 )} {1+n1−α
β 1 − (1 + n)X0 − β B0
{
}
1+n
α
⇔ β 1 − (1 + n)X0 −B0 ≷(1 + β) {1 − (1 + n) (X0 − B0 )}β
1−ατt |tax ≷ τt |debt ⇔ 1 ≷⇔γ≷β (1 − α) (1 + β)2
+
,
α
1−α

where the second line comes from 1 − (1 + n)X0 − 1+n
B0 > 0, and the third line comesβfrom (β + γ) α/β − 1 > 0, which holds under the assumption in Proposition 1. Therefore,
we obtain the second part of Proposition 2.
■

A.4

Proof of Proposition 3

The policy function of x and the capital accumulation equation in the steady state forthe tax-finance case are given by
x = X0 · (1 − α)A(k)α h,()
k = ΨK |B0 =0 · {(1 − α)A(k)α }1−η .
25

Combining these equations, we obtain:α(1−η)() α
x = X0 (1 − α)A ΨK |B0 =0 1−α(1−η) {(1 − α)A} 1−α(1−η) h.

Substituting this into h′ = D(x)η (h)1−η leads to:[]ηα(1−η)() αh′ = D · X0 (1 − α)A ΨK |B0 =0 1−α(1−η) {(1 − α)A} 1−α(1−η) h (h)1−η ,or[() α ]η
1h′= D · X0 {(1 − α)A} 1−α(1−η) ΨK |B0 =0 1−α(1−η) .
h tax

(24)

Next, recall the set of the three equations for the debt-finance case and evaluate themat the steady state:[]1−η,k = ΨK · (1 − α)A (k)α − αA (k)α−1 b
[
]
1−η
,
b = ΨB · (1 − α)A (k)α − αA (k)α−1 b′[]ηh= D (X0 )η · (1 − α)A (k)α − αA (k)α−1 b .
h
The first two equations implyk = (ΨK )1
1−α(1−η)[ΨB(1 − α)A − αA
ΨK1−η
] 1−α(1−η).

Plugging this into the first equation, we obtain:η[] 1−α(1−η)
[
αη
Ψ
B
α
α−1 ]η
(1 − α)A (k) − αA (k)b = (ΨK ) 1−α(1−η) (1 − α)A − αA.
ΨK

We substitute this into the third equation and obtain:η[] 1−α(1−η)
αη
h′
Ψ
B
η
.
= D (X0 ) (ΨK ) 1−α(1−η) (1 − α)A − αA
h debtΨK(25)

A direct comparison of (24) and (25) leads to:
[
]
(
)α
h′
ΨB
h′
α
≷
⇔ ΨK |B0 =0 (1 − α)A ≷ (ΨK ) (1 − α)A − αA
h tax
h debtΨK[]αβ1 − (1 + n)X0⇔·
(1 − α)A
1 + β (1 + n)D (X0 )η
[]α []1 − (1 + n)X0 − 1+n
B0
β
ΨB
β
≷
·
(1 − α)A − αA1+β(1 + n)D (X0 )η
ΨK
[
]α [
]
1+n
ΨB
α
⇔ [1 − (1 + n)X0 ] (1 − α)A ≷ 1 − (1 + n)X0 −B0(1 − α)A − αA,βΨK
where we obtain the second line from the definitions of ΨK |B0 =0 and ΨK .
26

Given that B0 > 0 and X0 > 0, we have:
1+n
B0 ,
β
ΨB
(1 − α)A > (1 − α)A − αA.
ΨK

1 − (1 + n)X0 > 1 − (1 + n)X0 −′′Therefore, we find that hh tax > hh debt .
■A.5

Proof of Proposition 4

Recall the indirect utility function of the period-t middle-aged, (7). Given kt , bt , and ht ,we substitute the corresponding policy functions into (7) and obtain{}Vt,tax = {1 + (β + γ)(α + (1 − α)η)} ln (1 − α)A(kt )α − αA(kt )α−1 bt+ (1 − η)(β + γ)(1 − α) ln ht + Ctax{}
Vt,debt = {1 + (β + γ)(α + (1 − α)η)} ln (1 − α)A(kt )α − αA(kt )α−1 bt+ (1 − η)(β + γ)(1 − α) ln ht + Cdebt ,where
Ctax ≡ {1 + (β + γ)α} ln (1 − (1 + n)X0 ) + η(β + γ)(1 − α) ln X0 + C1 ,{}1+nCdebt ≡ (1 + β) ln [1 − (1 + n)X0 + (1 + n)B0 ] + + {β(α − 1) + γα} ln 1 − (1 + n)X0 −B0β+ (β + γ)(1 − α)η ln X0 + C1 ,and
Vt,tax ≶ Vt,debt ⇔ Ctax ≶ Cdebt .
Notice that Ctax < Cdebt holds because we assume the parametric condition that realizesb > 0, that is, Vt,tax < Vt,debt .
In the steady state, the expressions of Vt,tax and Vt,debt are reformulated as
Vtax = {1 + (β + γ)(α + (1 − α)η)} ln (1 − α)A(ktax )α
{z
}
|
)

(

(#1)

+ (1 − η)(β + γ)(1 − α) ln lim ht,tax + Ctax ,}| t→∞{z(#2){}
Vdebt = {1 + (β + γ)(α + (1 − α)η)} ln (1 − α)A(kdebt )α − αA(kdebt )α−1 b
{z
}
|
(

)

+ (1 − η)(β + γ)(1 − α) ln lim ht,debt + Cdebt ,| t→∞{z
}
(#4)

27

(#3)

where ktax and kdebt denote the steady-state capital in the tax-finance and the debt-financecases, respectively. While Ctax < Cdebt holds under the present parametric condition, theterm (#2) outweighs the term (#4) as demonstrated in Proposition 3. In addition, in thesteady state, we obtain()(1 − η)(β + γ)(1 − α) ln lim ht,tax + Ctaxt→∞()
> (1 − η)(β + γ)(1 − α) ln lim ht,debt + Cdebt ,t→∞because the diﬀerence between ht,tax and ht,debt becomes larger as time passes. Therefore,
Vtax > Vdebt holds in the steady state if (#1) is larger than (#3), that is, if ktax > kdebt .
To show ktax > kdebt , recall that ktax satisfies the capital-market-clearing condition,1−ηktax = ΨK |B0 =0 · {(1 − α)A (ktax )α } , that is,1( ) 1−α(1−η)
1−ηktax = Ψ̃K· [(1 − α)A] 1−α(1−η) ,and kdebt is given by
kdebt = (ΨK )1
1−α(1−η)[ΨB· (1 − α)A − αA
ΨK1−η
] 1−α(1−η),

from Eq. (22). Because ΨB > 0, ΨK > 0, and ΨK |B0 =0 > ΨK hold under the present
parametric condition, we obtain ktax > kdebt .
■A.6

Calibration of Parameters

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 inquantitative analyses of the two-period overlapping-generations model (see, for example,
Gonzaelz-Eiras and Niepelt, 2009; Lanica and Russo, 2016). Our selection of β is 0.98,
which is also standard in the literature. Since the agents in the present model plan overgenerations that span 30 years, we discount the future by (0.98)30 .
We assume an annual gross population growth rate of 1.007, which is the OECDaverage rate during 2000−2010.1 This assumption implies that the net population growthrate for 30 years is (1.007)30 − 1 ≃ 0.233. For η, the estimate in Card and Krueger (1992)
implies an elasticity of school quality of 0.12. In addition, recent simulation studies suggestthat η is in the rage of 0.1 − 0.3 (Cardak, 2004) and 0.05 − 0.15 (Glomm and Ravikumar,
1998). Following these earlier studies, we here set at η = 0.12.
1

Source: World Development Indicators (http://data.worldbank.org/products/wdi) (Accessed on May
24, 2016).

28

For γ, we focus on the income tax revenue-to-GDP ratio in the steady state, τ whN/Y =(1 − α)τ , or

[
]
α
ΨB
τ whN
= (1 − α) · (1 + n)(X0 − B0 ) +{1 − (1 + n)(X0 − B0 )}
.
Y
1−α
ΨK

Given α = 1/3, β = (0.98)30 , 1 + n = (1.007)30 , and η = 0.12, we can solve this expression
for γ by using the ratio τ whN/Y observed in OECD countries. In particular, we use thedata on “Tax on personal income,” defined as the taxes levied on the net income (grossincome minus allowable tax reliefs) and capital gains of individuals.2 The OECD averageduring 2000–2010 is 0.08665. We can determine γ by solving the above expression andobtain γ = 3.47.
The productivity of final goods, A, is normalized at A = 1. For the productivity ofhuman capital D, we use the data on the per capita GDP gross growth ratio of 1.072,
which is the OECD average during 2000 − 2010.3 We substitute this data and the valuesof α, β, n, η, γ, and A into the following equation that expresses the per capita growthrate in the debt-finance case:η{} 1−α(1−η)
αη
Ψ
h′
B
η
= D · (X0 ) · (ΨK ) 1−α(1−η) · (1 − α) − αA.
hΨK
We solve this expression for D and obtain D = 14.21.
■

A.7

Supplementary Explanation for Section 4

We substitute (10) into (11) to reformulate the problem as
]
[
1 ˜
α
′
η
d − (1 + n)k D(x̃) − (1 + n)x̃max ln A (k) −˜ ′ ,x̃}
1+n
{d,k
{
}
β ˜
θ
′
+ ln d + γα ln k + (β + γ) +(1 + β + γ) η ln x̃ + θ · SW (k ′ ).
θ1−θ˜ k ′ , and x̃ are
The first-order conditions with respect to d,
1/(1 + n)
β/θ
d˜ :
=
,
c̃
d˜
η
γα
′ (1 + n)D(x̃)= ′ + θ · SW ′ (k ′ ),
k :
c̃
k
{
}
θ
′
η−1
(β + γ) + 1−θ
(1 + β + γ) ηη(1 + n)k D(x̃)+ (1 + n)
x̃ :
=
.
c̃
x̃
2

(26)
(27)
(28)

Source: OECD, Tax on personal income (indicator). doi:10.1787/94af18d7-en (Accessed on June 6,2016).
3
Source: World Development Indicators (http://data.worldbank.org/products/wdi) (Accessed on May
24, 2016).

29

We make the guess SW (k ′ ) = ϕ0 + ϕ1 ln k ′ , where ϕ0 and ϕ1 are undetermined coeﬃcients. For this guess, (27) becomes(1 + n)D(x̃)η k ′ = (γα + θϕ1 ) c̃.
With (28) and (29), we obtain[(1 + n)x̃ = (β + γ) +]θ
(1 + β + γ) − (γα + θϕ1 ) ηc̃.
1−θ

(29)

(30)

Substitution of (26), (29), and (30) into the resource constraint in (10) leads to
c̃ =
where

1
A (k)α ,
ϕ

(
)
{
}
β
θ
ϕ≡ 1+
+ (γα + θϕ1 ) (1 − η) + (β + γ) +(1 + β + γ) η.
θ
1−θ

(31)

˜ x̃, and k ′ are
The policy functions of d,β 1d˜ = (1 + n) · · A (k)α ,
θ ϕ
[
{(
)
}]
1
β
1
x̃ =
· ϕ−
1+
+ (γα + θϕ1 )· A (k)α ,
1+n
θ
ϕ
(
)1−η
1
γα + θϕ1
α
′
A (k)
.
k =
((
)
)}]η ·[ 1 {ϕ· ϕ − 1 + βθ + (γα + θϕ1 )(1 + n)D 1+n

(32)
(33)

Substituting these policy functions into the Bellman equation gives
SW (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 is given by
[(
{
} ]
)
α
β
θ
ϕ1 =
· 1+
+ γα (1 − η) + (β + γ) +(1 + β + γ) η , (34)
1 − αθ(1 − η)
θ1−θand the corresponding policy function of x̃ is obtained by substituting (31) and (34)
into (32). We can also compute the steady-state capital level, growth rate given byh′ /h = D(x̃)η , and education-GDP ratio given by (1 + n)x/y, by using (31) – (34).
■

30

B

Supplementary Materials

B.1

Supplementary Explanation for Subsection 6.1

The utility maximization problem of the middle in the presence of private education is
max ln ct + β ln dt+1 + γ ln wt+1 ht+1
s.t. ct + st + (1 + n)et ≤ (1 − τt )wt ht ,dt+1 ≤ Rt+1 st ,ht+1 = D (xt )η (et )δ (ht )1−η−δ .
Solving this problem leads toβ1· (1 − τt )wt ht , ct =· (1 − τt )wt ht , and
1 + β + γδ
1 + β + γδγδ· (1 − τt )wt ht .
et =(1 + n) (1 + β + γδ)st =The capital-market-clearing condition is()(1 + n) · kt+1 ht+1 + b̂t+1 =(35)

β· (1 − τt )wt ht .
1 + β + γδThis is reformulated as
kt+1 =(

1it − (1 + n)xt + (1 + n)b̂t+1(1 + n)D (xt )
[
]
(
)
β
×
it − (1 + n)xt + (1 + n)b̂t+1 − (1 + n)b̂t+1 .
1 + β + γδηγδ·(1+n)(1+β+γδ)())δ(ht )1−η−δ
(36)

We substitute (35) and (36) into the utility function and obtain the indirect utility
function of the middle as follows:()
Vt = {1 + β + δ (γ + β (1 − α) − γα)} ln it − (1 + n)xt + (1 + n)b̂t+1()
1 + γδ
(1 + n)b̂t+1 + (β + γ)(1 − α)η ln xt ,+ {β(α − 1) + γα} ln it − (1 + n)xt −β
where irrelevant terms are omitted from the expression.
We solve the maximization problem of Vt and obtain the following policy functions:b̂t+1 = B̃0 · it ,xt = X̃0 · it ,
where

{
}
{(1+β)β−(β(α−1)+γα)}+δ·{β(β+γ)(1−α)−γ(β(α−1)+γα)}B̃0 ≡ max 0,,(1+n)(1+γδ)·[1+(β+γ){α+(η+δ)(1−α)}]η(β+γ)(1−α)
X̃0 ≡ (1+n)·[1+(β+γ){α+(η+δ)(1−α)}].

31

We compare X̃0 and X0 and obtainX̃0 < X0 ⇔ 0 < δ(β + γ)(1 − α),which holds for any positive parameter values. We also compare B̃0 and B0 and obtainB̃0 ≷ B0 ⇔ β{(β + γ)(1 − α) − γ(1 + β)} · [1 + (β + γ) (α + (1 − α)η)]
|
{z
}
(∗1)

≷ (1 + γδ)(β + γ)(1 − α)[(1 + β) β − (β (α − 1) + γα)].
|
{z
}
(∗2)

The term marked by (*1) is negative while the term marked by (*2) is positive under theassumption in Proposition 5. Therefore, we obtain B̃0 < B0 .
■B.2

Supplementary Explanation for Subsection 6.2

B.2.1

Utility Maximization

We consider the utility maximization of the middle-aged. The problem is
max ln ct + β ln dt+1 + γ ln wt+1 ht+1
s.t. ct + st ≤ (1 − τt ) wt ht
dt+1 ≤ Rt+1 st + pt+1 .
Solving this problem leads to the following saving function
[
]
pt+1
β
(1 − τt ) wt ht −
,
st =
1+β
βRt+1
and the indirect utility function of the middle-aged:
[
]
(
)
pt+1
1
β
m
Vt = (1 + β) ln (1 − τt ) wt ht ++β ln Rt+1 +γ ln wt+1 ht+1 + ln
+ β ln
.
Rt+1
1+β
1+β
(37)
The capital market clearing condition becomes
[
]
(
)
pt+1
β
· (1 − τt )wt ht −.
(38)
(1 + n) · kt+1 ht+1 + b̂t+1 =
1+β
βRt+1
We use (38), the factor market clearing conditions and the government budget constraint
to reformulate Vtm in (37) as
[
]
pt
pt+1
m
Vt ≃ (1 + β) ln it − (1 + n)xt −+ (1 + n)b̂t+1 +
1+nαA (kt+1 )α−1][
(1 + n)b̂t+1
pt+1
pt
−
−
+ {β(α − 1) + γα} ln it − (1 + n)xt −1+nββαA (kt+1 )α−1
+ (β + γ) (1 − α) η ln xt + (β + γ) (1 − α) (1 − η) ln ht ,
32

(39)

where irrelevant terms are omitted from the expression.
The indirect utility of the old is
Vto = β ln (Rt st−1 + pt ) + γ ln wt ht .
()
We substitute the capital market clearing condition, (1 + n) kt ht + b̂t = st−1 , and thefactor market clearing condition into Vto and obtain[]ptoVt = β ln jt ++ β ln(1 + n) + γ ln (1 − α) A (kt )α ht ,
(40)
1+n
(
)
where jt is defined as jt ≡ αA (kt )α−1 · kt ht + b̂t . By substituting (39) and (40) into
Ωt = ωVto + (1 + n)(1 − ω)Vtm and rearranging the terms, we obtain()
ptΩt ≃ ωβ ln jt +
1+n
[
]
pt
pt+1
+ (1 + n)(1 − ω) (1 + β) ln it − (1 + n)xt −+ (1 + n)b̂t+1 +
1+nαA (kt+1 )α−1[]pt
(1 + n)b̂t+1
pt+1+ (1 + n)(1 − ω) {β(α − 1) + γα} ln it − (1 + n)xt −
−
−
1+n
β
βαA (kt+1 )α−1
+ (1 + n)(1 − ω) (β + γ) (1 − α) η ln xt ,where irrelevant terms are omitted from the expression.
B.2.2

Solution to the Problem of Maximizing Ωt{}
The period-t government’s problem is to choose pt , xt , b̂t+1 that maximizes Ωt . However,
we should note that the next-period old-age benefit, pt+1 , which is chosen by the nextperiod government, is included in the period-t political objective function. We need to
speculate on the next-period government’s choice, which could be aﬀected by the current
policy choice. Based on the expression of Ωt , it is natural to conjecture that pt+1 =P (it+1 , jt+1 ). In particular, we conjecture a linear policy function, pt+1 = P0 it+1 + P1 jt+1 ,where P0 and P1 are constant.
Given this conjecture, the present value of the period-t + 1 old-age benefit in (39)becomes
)
(
(
)
pt+1
1−α
kt+1 ht+1 − b̂t+1 + P1 · kt+1 ht+1 + b̂t+1= P0 ·ααA (kt+1 )α−1
[
]
(
)
1−α
1
pt
(1 + n)b̂t+1
pt+1
β
= P0
+ P1
it − (1 + n)xt −−−α1+n1+β1+nββαA (kt+1 )α−1
+ (−P0 + P1 ) b̂t+1 ,33

where the expression in the second line is obtained by substituting (38) into the expression
in the first line. By rearranging the terms, we obtain]) 1 β [
( 1−α(1+n)b̂t+1
pt
+
P
i
−
(1
+
n)x
−
−
+ (−P0 + P1 ) b̂t+1
P
1 1+n 1+β
t
t
0 α
1+n
β
pt+1
(
)
.
=
1
1
1 + P0 1−α+PαA (kt+1 )α−11
α1+n 1+β(41)
We substitute (41) into Vtm in (39) to obtain()
ptΩt ≃ ωβ ln jt ++ (1 + n)(1 − ω) (1 + β) ln Z0t1+n(1 + n)(1 − ω) {β(α − 1) + γα} ln Z1t(1 + n)(1 − ω) (β + γ) (1 − α) η ln xt ,
where

{

(
)
}(
) {
}
1−α
1
pt
−P0 + P1
Z0t ≡ 1 + P0
+ P1
it − (1 + n)xt −+ 1+(1 + n)b̂t+1 ,
α
1+n
1+n
1+n
(
) {
}
pt
−P0 + P1
Z1t ≡ β it − (1 + n)xt −− 1+(1 + n)b̂t+1 .
1+n
1+n
0 +P1
= 0 and solve the problem of maximizing Ωt . After
We further guess at 1 + −P1+nsome calculation, we obtain the policy function of pt as follows:[](1 + n) it − (1+n)(1−ω)
{1
+
(β
+
γ)
(α
+
(1
−
α)η)}
j
t
ωβ
pt =
.
1 + (1+n)(1−ω)
{1
+
(β
+
γ)
(α
+
(1
−
α)η)}
ωβ

Thus, the guess is verified if P0 and P1 are given by(1 + n)P0 =,

1 + (1+n)(1−ω){1 + (β + γ) (α + (1 − α)η)}ωβ(−1) (1 + n) (1+n)(1−ω){1 + (β + γ) (α + (1 − α)η)}
ωβ
.
P1 =
1 + (1+n)(1−ω)
{1
+
(β
+
γ)
(α
+
(1
−
α)η)}
ωβ

0 +P1
These expressions imply that the guess 1 + −P1+n= 0 is also verified.

■

B.3

Supplementary Explanation for Subsection 6.4

In the tax-finance case, we have b̂t+1 = 0 for all t. Given kt+1 = k for all t, the solution of
maximizing Vt is
xt =

(
)
γη
wht − Rb̂t .
(1 + n)(1 + β + γη)34

Therefore, the growth rate is[) 1 ]η
(
ht+1
γη
=D·
w ht |tax − Rb̂t,
ht tax(1 + n)(1 + β + γη)ht |tax
and the indirect utility is
Vt |tax = (1 + β + γη) ln it |tax + {β(α − 1) + γα} ln k + γ(1 − η) ln ht |tax
1+βγη+ (1 + β) ln+ γη ln+ C0 .
1 + β + γη
(1 + n)(1 + β + γη)In the debt-finance case, the solution of maximizing Vt isb̂t+1 = b̄ · ht ,
xt =

(
)
γη
it + (1 + n)b̄t ,(1 + n)(1 + β + γη)where b̄ ≡ w/(R − (1 + n)). Therefore, the growth rate is
]η
[
(
) 1
ht+1
γη
it |tax + (1 + n)b̄t,=D·
ht debt(1 + n)(1 + β + γη)ht |debt
and the indirect utility function is()
Vt |debt = (1 + β + γη) ln it |debt + (1 + n)b̄t + {β(α − 1) + γα} ln k + γ(1 − η) ln ht |debtγη
1+β+ γη ln+ C0 .
+ (1 + β) ln
1 + β + γη
(1 + n)(1 + β + γη)These expressions suggest that in period 0, we haveh1h1<
and V0 |tax < V0 |debt .
h0 taxh0 debt
From period 1 onward, we have()γη
wht − Rb̄ht + (1 + n)b̄ht(1 + n)(1 + β + γη)()γη
R − (1 + n)
=
wht −
wht
(1 + n)(1 + β + γη)R − (1 + n)
= 0,

xt |debt =implying that ht+1 /ht |debt = 0 and Vt |debt = −∞ for t ≥ 1.
■

35

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37

8

×10-3 (a): steady-state capital(b): steady-state growth rate

11
10

6
9
4

8
7

2
Planner allocation
Political equilibrium

6

0

5
0

0.2

0.4

0.6

0.8

1

0

θ
0.8

0.2

0.4

0.6

0.8

1

θ

(c): steady-state education-GDP ratio

0.6
0.4
0.2
0
0

0.2

0.4

0.6

0.8

1

θ

Figure 1: Panel (a): steady-state capital level and θ. Panel (b): steady-state growth rateand θ. Panel (c): education-GDP ratio and θ.

38

2

×10-4 (a): steady-state debt

0.15(b): steady-state education-GDP ratio
public education
private education

1.5
0.1
1
0.05
0.5
0

0
0

3.2

0.02 0.04 0.06 0.08 0.1
δ0-3
×10 (c): steady-state capital(d): steady-state growth rate

8.6

3

8.4

2.8

8.2

2.6

8

2.4

7.8

2.2

0.02 0.04 0.06 0.08 0.1
δ

7.6
0

0.02 0.04 0.06 0.08 0.1
δ00.02 0.04 0.06 0.08 0.1
δFigure 2: Panel (a): steady-state debt and δ. Panel (b) steady-state education-GDP ratioand δ. Panel (c): steady-state capital and δ. Panel (d): steady-state growth rate and δ.

39