Discussion Papers In Economics
And Business

Fiscal sustainability under physical and human capital
accumulation in an overlapping generations model
Takumi Motoyama
Discussion Paper 17-05

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANFiscal sustainability under physical and human capital
accumulation in an overlapping generations model
Takumi Motoyama
Discussion Paper 17-05

March 2017

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANFiscal sustainability under physical and human capital
accumulation in an overlapping generations model ∗Takumi Motoyama †Graduate School of Economics, Osaka University

Abstract
We consider fiscal sustainability by using an overlapping generations model with human capital accumulation (private and public education) and public debt. Based on this model, we explicitly show (i)the parameter region in which the economy cannot be fiscally sustainable for any initial endowment,and (ii) the threshold of initial endowment over (under) which the economy diverges (converges) tothe steady state. Importantly, the threshold is neutral to the level of initial human capital. Further, weshow the existence and uniqueness of the growth-maximizing level of each policy variable (i.e., thetax rate and public education/production ratio).
Keywords: Human capital accumulation, Public education, Public debt, Fiscal sustainability.
JEL Classification: E62; H52; H63; I28∗
I am grateful to Koichi Futagami, Akira Yakita, Kazutoshi Miyazawa, Koki Sugawara as well as the participants at the
Nagoya Macroeconomic Workshop on an earlier version of this paper. I especially thank Tatsuro Iwaisako for his invaluablesuggestions concerning this research. Needless to say, all remaining errors are my responsibility.
†
Address: 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail: pge028mt@student.econ.osaka-u.ac.jp.

1

Introduction

According to data from the International Monetary Fund,1 the public debt/GDP ratios in 2015 varied
across countries; for example, arranged in descending order, they were Japan (248%), Greece (177%),
the United States (105%), United Kingdom (89%), Germany (71%), China (43%), and so on. Then,the following question can be asked. How much of the primary deficit can the government control?
Or, in other words, how much public debt can the government issue? Such a problem is referred to asthat of “fiscal sustainability.
As Tirole (1985) implied,23 it has been known that in the overlapping generations (OLG) model
developed by Diamond (1965) without any policy alternatives, the economy can be fiscally sustainable if the long-run growth rate without public debt is higher than the interest rate and is equivalent
to the condition of dynamic ineﬃciency in his model, which we call the “debt-existing condition of
Tirolehereafter. Chalk (2000) showed that with a positive primary deficit in the OLG model, theeconomy can be fiscally sustainable when the primary deficit is suﬃciently low in addition to dynamic
ineﬃciency. In terms of endogenous growth, Bräuninger (2005) used the OLG model with AK production, as well as the constant government spending/GDP and deficit/GDP ratios, and showed thatthe economy becomes fiscally unsustainable when the deficit/GDP ratio is high. Arai (2011) considered fiscal sustainability with the OLG model in which the government invests into flow-government
productive spending, following Barro (1990), thus resulting in the AK model. In his model, it was
shown that the economy is fiscally sustainable when the government spending/production ratio ismoderate. Further, Yakita (2008) investigated fiscal sustainability with public capital, such as did Futagami et al. (1993) with the constant budget deficit/production ratio in the OLG model, and showedthat the threshold of initial public debt for the economy to be fiscally sustainable is increasing ininitial public capital.
On the other hand, since the advent of the papers written by Uzawa (1965) and Lucas (1988), it has
1
Source:International Monetary Fund, World Economic Outlook Database, October 2016 athttps://www.imf.org/external/pubs/ft/weo/2016/02/weodata/index.aspx. The data in the text are equivalent to thoseof “General government gross debt and Percent of GDP
in the database.
2
In fact, he considered the fiscal sustainability of asset bubbles instead of public debt by introducing useless assets inthe canonical OLG model. However, the result can be applied to the case of public debt by regarding the control variable(useless asset) as the state variable (public debt).
3
Grossman and Yanagawa (1993) showed the same debt-existing condition as that of Tirole (1985) by using the OLG
model with AK structure.

1

been considered that both physical and human capital accumulation are crucial for the growth engine;
hence, along the lines of the previous literature, it is important to consider fiscal sustainability by using
a two-sector endogenous growth model in which both physical and human capital accumulation exist,
that is, the Uzawa-Lucas model. By using this model, we can analyze the eﬀect of human capital on
fiscal sustainability. Further, in addition to private educational expenditure, it is important to consider
the eﬀect of public educational expenditure on economic performance, as studied in the previous
literature, such as Glomm and Ravikumar (1992), Blankenau and Simpson (2004), Blankenau et al.
(2007), and Agénor (2011). In fact, according to the report from the Organisation for Economic
Co-operation and Development (OECD) (2016), the OECD-average public education/GDP ratio in2013 was 4.8%, which is not a negligible value.4 Since the government must prepare funds to financepublic educational expenditure, it should aﬀect fiscal sustainability and, by using our model, we cananalyze the level of public education under which the economy can be fiscally sustainable besides thetax rate.
Therefore, we have extended the model developed by Boldrin and Montes (2005) by incorporating
public education and public debt. Based on this model, we obtain following three results. Firstly, we
explicitly show the parameter region in which the economy is fiscally unsustainable for any initialendowment. The condition is equivalent to that in which the economy has multiple steady states and,in line with the previous literature, an additional condition, besides the debt-existing condition of
Tirole, is needed to obtain a positive steady-state value of public debt. Secondly, we explicitly obtainthe threshold level of initial public debt over (under) which the economy diverges (converges) to thesteady state. Counterintuitive as it may be, the threshold is neutral to the initial human capital; that is,whether the economy initially has high education or not has nothing to do with fiscal sustainability.
Thirdly, we consider the eﬀect of a marginal increase in the tax rate and public education/production
ratio on the long-run growth rate. When the tax rate increases, private education decreases, since
the future income decreases, which is a marginal cost of an increase in the tax rate; primary deficit
improves; public debt decreases; and physical capital increases, which is a marginal benefit. Whenthe public education/production ratio increases, human capital directly increases, which is a marginal
4

See Table B4.2. on page 231 in OECD (2016). The public education/GDP ratio by country ranges from 3.3% in
Hungary to 7.3% in Norway.

2

benefit, while the primary deficit worsens, public debt increases, and physical capital is crowded out,
which is a marginal cost. Taking these into account, we can show the existence and uniqueness of thegrowth-maximizing tax rate and public education/production ratio, respectively.
Some literature exists that simultaneously treats public education and public debt in the growthmodel. Zhang (2003) considered public debt and subsidy for education with a constant public
debt/production ratio in the OLG model with altruism, and obtained the optimal policy and indirect
utility functions. Greiner (2008) considered the relationship between economic growth and public
debt with public education in the Ramsey model, and showed the steady states and their stability
through a calibration. Gottardi et al. (2015) considered the Ramsey model in which uninsurable
idiosyncratic shocks to human capital exist and the government can issue public debt, showing theexistence of a positive long-run, welfare-maximizing capital tax rate. Further, Ono and Uchida (2016)
simultaneously considered public education and public debt in the OLG model; they compared thewelfare of the only-tax regime as well as both tax and public debt regimes in terms of competitiveand political (middle-aged, welfare-maximizing policy) equilibriums. Nevertheless, these studies did
not focus on the fiscal sustainability problem, and, to my best knowledge, this is the first study thatconsiders fiscal sustainability under the model with both public debt and human capital accumulation.
The remainder of this paper is organized as follows. In Section 2, we describe the behavior ofa household, firm, and government, and obtain simultaneous diﬀerence equations that describe themodel’s equilibrium. In Section 3.1, we discuss the condition of fiscal sustainability; in Section 3.2,
we conduct comparative statics of the stable steady-state value with respect to each policy variable;
and in Section 3.3, the existence and uniqueness of the long-run, growth-maximizing tax rate andpublic education/production ratio are shown. Finally, Section 4 summarizes the study.

2 Model
2.1 Model settingThis model is a two-sector growth model consisting of final goods and human capital production. We
consider the three-period OLG model, in which three types of households coexist in each period (i.e.,
the young, middle-aged, and old).

3

Firstly, we consider the representative firm’s profit-maximizing problem. This firm produces final
goods by using physical and human capital, (kt , ht ) with Cobb-Douglas production technology:
Yt = F(kt , ht ) = Aktα h1−α
, A > 0, α ∈ (0, 1).
t
We assume that physical capital is fully depreciated after use;5 then, profit-maximizing conditions
become:1 + rt = Aαktα−1 h1−α,twt = A(1 − α)ktα h−α
t ,

(1)

where rt denotes the interest rate and wt denotes the wage rate per human capital.
Next, we turn to the household’s utility-maximizing problem. A t − 1 generation, born in periodt − 1, behaves as follows. In period t − 1, s/he is young and borrows et−1 from the middle-aged, andinvests it into education. Then, s/he obtains human capital described as follows:6ββ1−β −β221
Et−1
ht−1 1ht = Bet−1,

B > 0, β1 , β2 , 1 − β1 − β2 ∈ (0, 1).

(2)

In (2), Et−1 denotes the level of public education in period t − 1 and ht is positively aﬀected by theprevious level of human capital ht−1 , which captures the externality of human capital. Further, β1 (β2 )measures the productivity of private (public) educational expenditure. In period t, the t − 1 generation
becomes middle-aged and earns after-tax income (1 − τ)wt ht , where τ denotes the constant income
tax rate. S/he distributes her/his after-tax income into middle-age consumption cm
t , savings st , andrepayment of educational expenses to the old (1 + rt )et−1 . In period t + 1, the t − 1 generation becomes
old and consumes all of its income (1+rt+1 )st . The intertemporal utility function of the t−1 generationoois given by Ut−1 = ln cm
t + δ ln ct+1 , where δ ∈ (0, 1) is a discount factor and ct+1 is the consumption

level at old age. The t − 1 generation’s budget constraints are cm
t + st + (1 + rt )et−1 = (1 − τ)wt ht andcot+1 = (1 + rt+1 )st . Hence, the t − 1 generation’s utility-maximization problem is given as:[]β1
β2 1−β1 −β2max ln (1 − τ)wt Bet−1
Et−1
ht−1
− st − (1 + rt )et−1 + δ ln [(1 + rt+1 )st ] .

st ,et−1

5

If the yearly depreciation rate is 7%, after one period, say 25 years, about 84% of physical capital is depreciated, which
we can equate with full depreciation.
6
The functional form (2) is assumed in Blankenau and Simpson (2004).

4

The utility-maximizing conditions are given by:
et−1 = (1 − τ)β1
st =

wt ht
,
1 + rtδ[(1 − τ)wt ht − (1 + rt )et−1 ] .
1+δ

By substituting (1) into the above expressions, we obtain:et−1 = γkt ,(3)st = ηAktα h1−α
,
t

(4)

δ
where γ ≡ (1−τ)βα1 (1−α) denotes the propensity of private education and η ≡ 1+δ
(1 − τ)(1 − β1 )(1 − α) ∈(0, 1) denotes the savings rate.
Since the government can issue public debt, by defining bt as the stock of public debt in period t,
the government’s budget constraint is given by:∆bt+1 ≡ bt+1 − bt = rt bt + Et + Gt − τwt ht ,(5)where Gt denotes unproductive government expenditure. Equation (5) states that the sum of repayment of an interest of public debt rt dt and primary deficit Et + Gt − τwt ht is financed by issuing new
public debt ∆bt+1 . The government sets the level of public education Et as a fraction θ ∈ (0, 1) ofproduction Yt : 78
Et = θYt .

(6)

Similarly, the unproductive government expenditure Gt is set as a fraction d ∈ (0, 1) of production Yt ,as assumed in Chalk (2000) and Bräuninger (2005), that is, Gt = dYt .
Finally, savings are used as physical capital, public debt, and lending to the young generation;
thereby, the capital-market clearing condition becomes:
st = et + bt+1 + kt+1 .

(7)

7
In reality, θ in 2013 was 4.8% (OECD average) according to the OECD (2016). Further, that in 2005 was 4.7%, andthe ratio was relatively stable.
8
Instead of (6), even if the government sets the primary deficit as a fraction θ of production (i.e., Et + Gt − τwt ht = θYt ),the dynamics are almost the same as this model, since public education eventually becomes a constant fraction of production
under the constant tax rate and Gt = dYt .

5

2.2

Model solutionγ
Substituting kt = γ1 et−1 from (3) into (7) yields et−1 = 1+γ(st−1 − bt ); again, by substituting this and(4) into (7), we obtain the dynamics of kt , such as:
kt+1 =)
11 ((st − bt+1 ) =ηAktα h1−α
− bt+1 ,
t
1+γ
1+γ
(
)
bt+1
1
kt+1
α−1
=
ηAxt − yt
,
⇔
kt
1+γ
bt

(8)

where xt ≡ hktt denotes the physical/human capital ratio and yt ≡ bktt denotes the public debt/physicalcapital ratio. By combining (2) with (6) and (3), the dynamics of ht becomes:
β1

ht+1 = B(γkt+1 )1−β −β(θYt )β2 ht 1 2 = ψ

⇔

(

kt+1
kt

) β1

β +αβ2 1−β1 −αβ2
,
ht

kt 1

(
)β
ht+1
kt+1 1 β1 +αβ2
=ψ
,
xt
ht
kt

(9)

where ψ ≡ Bγβ1 θβ2 Aβ2 denotes a scale parameter of the long-run growth rate. Note that ψ is increasing in θ because the growth rate of human capital directly increases when a higher level of public
education is introduced, and that it is decreasing in τ, since a higher tax rate decreases the return ofprivate education; in turn, this decreases the propensity of private education γ. Finally, the dynamics
of bt are derived by substituting (1) and (6) into (5) as follows:
bt+1 = (1 + rt )bt + (θ + d)Yt − τwt ht = [αbt + (θ + d − τ(1 − α))kt ]Axtα−1 ,()()θ + d − τ(1 − α)
D
bt+1
⇔
= α+
Axtα−1 ≡ α +Axtα−1 ,
bt
yt
yt

(10)

t ht
where D ≡ θ + d − τ(1 − α) is a primary deficit/production ratio (i.e., Dt ≡ Et +GYt −τw= D), and wetassume D > 0, which means that there is a positive primary deficit. Then, by combining (8), (9), and(10), this economic system is characterized by the following two diﬀerence equations, with respect to(xt , yt ):αβxt+1 = ϕ(η − αyt − D)1−β1 xt 3 ,yt+1 = (1 + γ)

αyt + D≡ Ω(yt ),η − αyt − D

6

(11)
(12)

where ϕ ≡ ψ1()

A 1−β1and β3 ≡ 1 − β1 − β2 . Importantly, note that yt+1 depends only on yt .9 From1+γ(11), since xt must be a real number, yt must satisfy η > αyt + D in order for the economy to be fiscallysustainable. In other words, if yt > ȳ, where η = αȳ + D, the economy is fiscally unsustainable.
The debt-existing condition of Tirole is that the long-run growth rate without public debt is higher
than the interest rate. In our model, such a parameter region is given by:10
η − D > (1 + γ)α,(13)which is assumed hereafter. Note that the condition (13) implies η − D > 0; in turn, it also impliesȳ > 0.
In what follows, we consider the phase diagram of (xt , yt ). From (11), the ∆xt+1 ≥ 0 region is
given by:1−β1

1xt ≤ ϕ 1−αβ3 (η − αyt − D) 1−αβ3 .

(14)Moreover, from (12), by focusing on the region yt < ȳ, the ∆yt+1 ≥ 0 region is given by:Ω(yt ) ≥ yt , ⇔ αy2t − [η − D − (1 + γ)α]yt + (1 + γ)D ≡ F(yt ) ≥ 0.

(15)

Further the quadratic equation, F(y) = 0, has two distinct real roots if the following holds:[]
η − D − (1 + γ)α 2 ≡ A21 > 4α(1 + γ)D ≡ A2 .

(16)

Note that from (13), A1 is positive; thus, both roots are positive if both D > 0 and (16) are satisfied.
[]√
1
A1 ± A21 − A2 . By defining yH and yL as the roots ofThen, the roots of y are obtained as y = 2αF(y) = 0, satisfying yH > yL , the ∆yt+1 ≥ 0 region is given by:yt ≤ yL , and yt ≥ yH .

(17)

Note that yH is smaller than ȳ, and the economy is fiscally sustainable under both steady states, since
the diﬀerence equation of yt , Ω(y), goes to infinity as yt → ȳ (See Figure 3 in Appendix A2).
Now that we can draw the phase diagram in (xt , yt ) space, as in Figure 1, by using (14) and (17).
According to Figure 1, the steady state (xH , yL ) is locally stable, while the steady state (xL , yH ) is asaddle point. In summary, we can obtain the following proposition:
9
10

In Section 3.1, we consider the reason in detail.
See Appendix A1 for deriving (13).

7

‫ݔ‬௧

Ǽ ൌ Ͳ‫ݔ‬ுǼ‫ ݔ‬ൌ Ͳ
‫ݔ‬௅
௧

‫ݕ‬ത

‫ݕ‬ு

‫ݕ‬௅

Figure 1: Phase diagram under (16)Notes: The ∆x = 0 (∆y = 0) locus is Equation (14) ((17)) with equality. In the shaded area, yt > ȳ, and the economy isfiscally unsustainable. The steady state (xH , yL ) is locally stable and the economy with y0 < yH converges there. The steadystate (xH , yL ) is a saddle point and the saddle path is given by yt = yH .

Proposition 1 Under (13) and the positive primary deficit D > 0:
• if (16) holds, the dynamic system (11) and (12) has two steady states, (xH , yL ) and (xL , yH ),where xH > xL and ȳ > yH ≥ yL > 0, respectively; the former is locally stable, while the latter
is a saddle point, and• if (16) is violated, there are no steady states.
Proof. See Appendix A2 for the stability analysis. 
Finally, the long-run growth rate, which is equivalent to that of human capital, limt→∞ hht+1t , isobtained by substituting (8) into (9), and evaluating the resulting (9) at yt+1 = yt = yL and xt+1 = xt =xH 11 as follows:

(

A
g =ψ
1+γ
∗

) β1

α(1−β3 )(η − D − αyL )β1 xH

1

β1 +αβ2
1−β1= ψ 1−β1 xH

.

(18)

Since both xt and yt are state variables, only the stable steady state (xH , yL ) is economically meaningful and, thus, we
focus on this steady state.
118

The long-run growth rate g∗ is increasing in both the scale parameter ψ and physical/human capitalratio xH .

3 Policy implications
3.1 Fiscal sustainability
If (16) is satisfied, according to the phase diagram, the economy with the initial public debt/physicalcapital ratio y0 < yH converges to the stable steady state (xH , yL ), while that with y0 > yH eventuallycrosses the ȳ line (the vertical dotted line in Figure 1); thereafter the economy becomes fiscallyunsustainable. Hence, if there are two steady states, given the initial physical capital, the economy
with low initial public debt is fiscally sustainable, while that with high initial public debt becomes
fiscally unsustainable in some time. There are two policy implications as follows.
1. In addition to (13), (16) is required for the economy with low initial public debt to be fiscallysustainable. That is, the debt-existing condition of Tirole, (13), is necessary, since otherwise,yL becomes negative, but it is not suﬃcient for the debt-existing condition of this model. Since(16) is likely to be satisfied when D is small, the primary deficit/production ratio must be small
for the economy to be fiscally sustainable.12
2. The threshold line for whether the economy is eventually fiscally sustainable, that is, the saddle
path given by yt = yH , does not depend on x0 . This means that the initial human capitalis neutral to the threshold level. Hence, whether the economy has a high initial human capitallevel is irrelevant to whether it is fiscally sustainable. To explain the reason for this, we consider
why yt+1 depends only on yt , and for explanatory simplicity, let us assume that the productivityof private education β1 is zero, which also means that the propensity of private education γ iszero. Then, by dividing (7) by bt+1 , we obtain:
yt+1 =

kt+1
st
ηYt
=
−1=
− 1,
bt+1 bt+1
bt+1

(19)

12
If θ is small, D is also small, and (16) is likely to be satisfied, since there are no terms that include θ, except for D. If τ
is high, it decreases D, but at the same time, it decreases η and γ. Since an increase in γ increases (decreases) the right-hand(left-hand) side of (16), and we show that η − D is increasing in τ in Appendix A3, (16) tends to be satisfied when τ is high.
Therefore, (16) is likely to be satisfied when θ is small and/or τ is high.

9where the third equality holds from (4). The next level of public debt bt+1 is the sum of current
public debt including the interest payment and primary deficit. Then, when the production
Yt increases, savings increase by η, while debt obligations (1 + rt )bt increase by αyt becauseof an increase in the interest, and the primary deficit increases by D. Thus, from (19), an anincrease of Yt does not aﬀect yt+1 since it proportionally increases st and bt+1 , and the eﬀect iscompletely oﬀset. Since an increase of xt is equivalent to that of Yt , xt does not aﬀect yt+1 .
If (16) is violated, there are no steady states. In this case, F(yt ) is always positive since there
are no real roots of F(yt ) = 0; hence, ∆yt+1 is positive for any yt . Therefore, public debt continuously increases and the economy eventually crosses the ȳ line and becomes fiscally unsustainable
thereafter for any endowment. As mentioned above, the economy with high D tends to violate (16)and lapse into fiscal unsustainability. In summary, we obtain the following corollary regarding fiscal
sustainability.
Corollary 1 Under (13),• if (16) holds, the economy with y0 < yH is fiscally sustainability and that with y0 > yH eventually becomes fiscally unsustainable, where yH is given by
yH =

1 [
A1 +
2α

√

]
A21 − A2 ,and
• if (16) is violated, the economy eventually becomes fiscally unsustainable for any initial endowment.

3.2 Comparative statics of steady-state value
In this model, there are two policy variables—the tax rate τ and public education/production ratio
θ—and we consider the eﬀect of a marginal increase in each policy variable on the stable steady-statevalue (xH , yL ). At the same time, we also consider the comparative statics of yH , which is a thresholdlevel of the initial public debt/physical capital ratio, to consider whether the marginal increase ofeach policy variable widens or narrows the fiscally sustainable region. The results of the comparative
statics are given as the following proposition.
10

Proposition 2 The signs of the derivatives of (xH , yL , yH ), with respect to (τ, θ), become:
dxH
dyL
dyH
> 0,
< 0,
> 0,
dτ
dτ
dτ
dyL
dyH
dxH
< 0,
> 0,
< 0.
dθ
dθ
dθ

(20)

Proof. See Appendix A3. 
Because of its simplicity, we first consider the eﬀect of an increase in θ on yL . When θ increases,
the primary deficit increases since public education increases. Then, given the same level of tax,
public debt must increase to finance it; thus, yL increases. Considering this, xH is aﬀected by: (i) the
scale parameter of the long-run growth rate ψ increasing, which leads to a decrease of parameter ϕ,and (ii) yL increasing. Since (i) stems from an increase in the return of private education given byt(1−τ)wt dh
det , investment in education becomes attractive and human capital increases, and (ii) physicalcapital is crowded out as public debt increases according to the capital market clearing condition, boththe first and second eﬀects decrease xH . The sixth inequality, dyH /dθ < 0, means that the higher θ
is, the more the economy eventually becomes fiscally unsustainable. This is because a higher public
education/production ratio under the same tax rate worsens the primary deficit.
On the other hand, the eﬀect of an increase in τ on yL is threefold: (i) savings rate η decreases
since total income decreases, (ii) primary deficit DYt decreases since tax revenue increases, and (iii)
propensity of private education γ decreases since the return of private education decreases. The firsteﬀect decreases kt , while the second eﬀect increases kt , since pubic debt decreases; however, we
can show that the eﬀect of a decrease in D dominates that of η;13 Hence, the combination of firstand second eﬀects increases kt . The third eﬀect also increases kt since the household substitutes
investment into private education with savings, and the sum of three eﬀects increases kt , which, inturn, decreases yL , since the denominator increases. The eﬀect of an increase in τ on xH can be
categorized into three channels: (i) η − D increases, (ii) γ decreases, and (iii) yL decreases. All threeeﬀects reallocate the investment in human capital with that in physical capital; therefore, xH increases.
Finally, yH increases as τ increases since the primary deficit improves, which means that more of the13

In Appendix A3, we show that()
dδ(η − D) = (1 − α) 1 −(1 − β1 ) > 0.
dτ
1+δ

11

economy can be fiscally sustainable.

3.3

Growth-maximizing policy

In this subsection, we show the eﬀect of marginal increase in each policy variable on the long-run
growth rate g∗ given by (18), and consider whether there is a growth-maximizing level of policy
variables. Regarding the growth-maximizing policy, we obtain the following proposition:
Proposition 3
1. The growth-maximizing tax rate satisfies the following equation:()
β1 + αβ2ddyL(1 − α)β1dγ((1 − τ)(β1 + β2 ) − 1) ,(η − D) − α=
η − αyL − D dτdτγ(1 + γ)
dτ

(21)

d
where dτ(η − D) > 0, (1 − τ)(β1 + β2 ) < 1, and dγ
dτ < 0. There exists a unique tax rate thatsatisfies (21), τ∗ ∈ (τ̂, 1), where τ̂ is defined as A21 |τ=τ̂ = A2 |τ=τ̂ .
2. The growth-maximizing public education/production ratio satisfies the following equation:()
dyL
β2 (1 − α)β1 + αβ2
1+α=.
η − αyL − D
dθ
θ

(22)

There exists a unique public education/production ratio that satisfies (22), θ∗ ∈ (0, θ̂), where θ̂
is defined as A21 |θ=θ̂ = A2 |θ=θ̂ .
Proof. See Appendix A4. 
The eﬀect of an increase in tax rate τ on g∗ can be decomposed into: (i) γ decreases because of adecrease in the return of private education, which leads to a decrease in a scale parameter of the longrun growth rate ψ, and (ii) xH increases as shown in (20). The left-hand side (LHS) of (21) captures
the former marginal cost, while the right-hand side (RHS) captures the latter marginal benefit. In
Appendix A4, we show that the LHS of (21) is decreasing in τ, goes to infinity as τ → τ̂, where τ̂
satisfies A21 |τ=τ̂ = A2 |τ=τ̂ , and is positive when τ = 1, while the RHS of (21) is increasing in τ, ispositive when τ = τ̂, and goes to infinity as τ → 1. Taking these features into account, we obtainthe left panel of Figure 2 and show the existence and uniqueness of the growth-maximizing tax rateτ∗ ∈ (τ̂, 1).

12

RHS of (21)

RHS of (22)

LHS of (21)
LHS of (22)


̂

‫כ‬



ͳ

‫כ‬



ͳ

Figure 2: Growth-maximizing tax rate τ∗ (left panel) and public education/production ratio θ∗ (rightpanel)On the other hand, an increase in public education/production ratio θ aﬀects g∗ in the following
two ways: (i) ψ increases since higher θ directly increases hht+1t through higher public education, and(ii) xH decreases from a crowding-out eﬀect as shown in (20). The RHS of (22) captures the firstmarginal benefit, while the LHS captures the second marginal cost. Similar to the case of τ from
Appendix A4, we show that the LHS of (22) is increasing in τ, goes to infinity as θ → θ̂, where θ̂
satisfies A21 |θ=θ̂ = A2 |θ=θ̂ , and is positive when θ = 0, while the RHS of (21) is decreasing in τ, ispositive when θ = θ̂, and goes to infinity as θ → 0. Thus, we obtain the right panel of Figure 2 andshow that there uniquely exists θ∗ ∈ (0, θ̂), which maximizes the long-run growth rate. Note that if thegovernment can set τ = τ∗ and/or θ = θ∗ , A21 > A2 holds; therefore, (16) is satisfied and the economy
with y0 < yH is fiscally sustainable.14

4 Conclusion
In this study, we consider fiscal sustainability by using the OLG model with private and public education as well as public debt. Based on our model, if any, we obtain multiple steady states with respect
to physical/human capital ratio xt and public debt/physical capital ratio yt ; one of them characterized
14

However, when the government implements the growth-maximizing policy, the primary surplus may accrue (i.e.,
D < 0). In that case, the stable steady-state value of the public debt/physical capital ratio, yL , becomes negative andthe government becomes a saver in the long run.

13

by low public debt is locally stable, while the other characterized by high public debt is a saddle point.
The threshold level of public debt above which the economy eventually becomes fiscally unsustainable is given by the saddle path toward the saddle steady state, and we show that it is neutral to theinitial human capital level. This means that whether a country is educationally developed has nothing to do with fiscal sustainability. Moreover, the debt-existing condition of Tirole, that is, that thelong-run growth rate without public debt is higher than the interest rate, is a necessary, but suﬃcientcondition in our model, and we can analytically show the additional debt-existing condition given by(16), which tends to be satisfied when the primary deficit is low.
Further, we consider the eﬀect of a marginal increase in each policy variable on the long-run
growth rate in Section 3.3. There are two opposite eﬀects of raising each policy variable on therate: the improvement of primary deficit versus a decrease in private education in terms of the tax
rate, and an increase in public education versus the crowding-out eﬀect of physical capital in termsof the public education/production ratio. Taking these eﬀects into account, we show the existence
and uniqueness of the growth-maximizing tax rate and public education/production ratio, and that theeconomy satisfies (16) under this level. This means that if the government can set the tax rate and/or
public education/production ratio as the growth-maximizing level, the economy with a moderate,
initial public debt/physical capital ratio, which is given by y0 < yH , is fiscally sustainable.
Finally, in this paper, we consider the constant public education/production policy. Instead, if
the government implements a similar policy for budget deficit (i.e., ∆bt+1 = θYt ), as assumed inBräuninger (2005), Yakita (2008), and Arai (2011), the dynamics will change since the public education/production ratio does not become constant under the constant tax rate and unproductive government expenditure/production. It would be interesting to analyze how it changes the threshold of fiscal
sustainability and how the long-run growth rate is aﬀected by the budget deficit/production ratio. The
analysis is left to future studies.

14AppendicesA1. Derivation of (13)In this model, the debt-existing condition of Tirole is given by:15∂F(kt , ht )
− 1.
t→∞
y∗ =0
∂kt

g∗ |y∗ =0 − 1 > lim
1

β +αβ

∗ 11−β 2

t ,ht )g∗ |y∗ =0 is calculated from (18) as ψ 1−β1 x|y∗ =0 1 , while limt→∞ ∂F(kbecomes Aαx|∗α−1∂kty∗ =0 . Substiy∗ =0tuting these expressions into the above expression yields:
1−αβ∗ 1−β 3x|y∗ =0

1( ) 1−β1
1
1
.
> Aα
ψ

1−αβ

∗ 1−β 3

From (14), x|y∗ =0 1 becomes ϕ1−β1 (η − D), and by substituting this into the above expression and aftersome manipulation by using the definitions of ϕ and ψ, we obtain:
η − D > (1 + γ)α,which is equivalent to (13).

A2. Stability analysis of (11) and (12)Under (16), there exist two steady states, (xH , yL ) and (xL , yH ), where xH > xL and yH > yL , respectively, and we express these steady states by (x∗ , y∗ ). Then, by linearly approximating (11) and (12)around the steady state (x∗ , y∗ ), we obtain the following matrix form:

(
) 
)
α(1−β )x∗ (αβ3 η−αy∗1−D  dxt
dxt+1

= 
,
dyt+1
0
Ω′ (y∗ ) dyt
where dxt ≡ xt − x∗ and dyt ≡ yt − y∗ . Since the coeﬃcient matrix is a triangular matrix, characteristicroots are: λ1 = αβ3 ∈ (0, 1) and λ2 = Ω′ (y∗ ). To consider the value of Ω′ (y∗ ), note that Ω(yt ) is
increasing and convex for yt , since both Ω′ (yt ) and Ω′′ (yt ) are positive; hence, Ω′ (yL ) ∈ (0, 1) andΩ′ (yH ) > 1 must hold (see Figure 3). Therefore, the steady state (xH , yL ) is locally stable since bothλ1 and λ2 are in (0, 1), while the steady state (xL , yH ) is a saddle point since λ1 ∈ (0, 1) and λ2 > 1.
As xt and yt are state variables, only the stable steady state (xH , yL ) is economically meaningful.
15

Recall that we assume that the depreciation rate is one and define g∗ such as limt→∞ hht+1t , not as limt→∞ ht+1ht−ht .

15ߗሺ‫ݕ‬௧ ሻ‫ݕ‬௧ାଵ

‫ݕ‬௧ାଵ ൌ ‫ݕ‬௧ߗᇱ ሺ‫ݕ‬ு ሻ ൐ ͳߗᇱ ሺ‫ݕ‬௅ ሻ ൏ ͳ

‫ݕ‬௧
‫ݕ‬௅

‫ݕ‬ு

‫ݕ‬ത

Figure 3: Phase diagram of yt

A3. Proof of Proposition 2First, we consider the eﬀect of a marginal increase in τ. To show this, we calculate the derivative ofA1 ≡ η − D − (1 + γ)α, with respect to τ, as follows:dA1ddγ= (η − D) − α .
dτ
dτ
dτ
(
)
dA1
d
δ
Since dτ(η − D) = 1 − 1+δ
(1 − β1 ) (1 − α) > 0 and α dγdτ = −β1 (1 − α) < 0, dτ > 0. Similarly, thederivative of A2 ≡ 4α(1 + γ)D, with respect to τ, is:
[
]
dA2
d
dD
dγ
= 4α (1 + γ)D = 4α (1 + γ)
+D
.
dτ
dτ
dτ
dτ

16

dγ
dA2
Since dDdτ = −(1 − α) < 0 and dτ < 0, dτ < 0. Taking these into account, the sign of the derivative ofyL , with respect to τ, is given by:[)])− 12 (dyL
1 dA1 1 ( 2dA1 dA2=− A1 − A2
2A1
−
dτ
2α dτ
2
dτ
dτ
(
)− 12
(
[
)]
A21 − A2
1
1 dA2 /dτdA1
2=
(A1 − A2 ) 2 − A1 −2αdτ
2 dA1 /dτ


(
)− 21


2
(
)

A1 − A2
1
dA1  2
1 dA2 /dτ 
= (A − A2 ) 2 − A1 +
 < 0.
2αdτ | 1 {z
2 dA1 /dτ 
|{z}
} |
{z }
>0

<0

(23)

<0

2

We use the sign of ddτy2L in the proof of Proposition 3 in Appendix A4, and we show it as follows:[()]
d 2 yLdA1 1 dA2
d dA1
2
− 12
2α 2 =− (A1 − A2 )
A1
−
dτ dτ
dτ
2 dτ
dτ
(

(
)(
)
)
 dA1 2 1 d2 A2 
1 2
dA1 dA2
dA1 1 dA2
− 12 −1
2− 21 = (A1 − A2 )
2A1
−
A1
−
− (A1 − A2 ) 
−
2
dτ
dτ
dτ
2 dτ
dτ
2 dτ2 
(
(

)2
)2
2 A 


1
dA
dA
1
dA
1
d



1
1
2
2−− (A21 − A2 ) −= (A21 − A2 )− 2 −1  A1

2
dτ
2 dτ
dτ
2 dτ 




(
)
(
)
2
2
2

(A1 − A2 ) d2 A2 
dA1
dA1 dA2 1 dA2
2− 21 −1  > 0, (24)+ A2+
= (A1 − A2 )+−A1dτ |{z}dτ4 dτdτ2 } |{z}dτ2 
| {z

|{z}
>0

>0

<0

>0

where, note that:
dD dγ
d2 A2
= 8α
> 0.
2
dτ |{z}
dτ
dτ
|{z}
<0

<0

1−β1

1

From (14), xH is determined as xH = ϕ 1−αβ3 (η − αyL − D) 1−αβ3 , and taking logarithms in both sides
yields:ln xH =

1
1 − β1ln ϕ +ln(η − αyL − D).
1 − αβ31 − αβ3(25)Hence, the sign of the derivative of xH , with respect to τ, is:






 dyL d
1 dxH
1
dϕ
1 − β11−α=++ (η − D) > 0,xH dτϕ(1 − αβ3 ) |{z}dτ
1 − αβ3 η − αyL − D | {zdτ} |dτ {z }
>0

>0

17

>0

(26)

dγ
dψ
dϕ
where dϕdτ > 0 holds because dτ < 0 ⇒ dτ < 0 and dψ < 0. The sign of the derivative of yH , withrespect to τ, is directly given by:







1)− 2 
dyH
1  dA1 1 ( 2dA1dA2 

2A1 > 0.
=+ A1 − A2−dτ2α |{z}dτ2dτdτ 
|{z}
|{z}
>0

>0

(27)

<0

In what follows, similar to the case of τ, we derive the derivative of the stable steady state (xH , yL ),dA2
1
with respect to θ. By using dA
dθ = −1 and dθ = 4α(1 + γ), we obtain the derivative of yL , with respectto θ, as:[)])− 12 (dyL
1 dA1 1 ( 2dA1 dA2=− A1 − A2
2A1
−
dθ
2α dθ
2
dθ
dθ
[
]
1
(
)
−2
1
2
(−A1 − 2α(1 + γ))=−1 − A1 − A2
2α



1 

2
−

1
(
)2
(A1 − A2 ) 2 
2
A1 − A1 − A2 +2α(1 + γ) > 0.
=

|
2α
{z
}

(28)

>0

2

Note that the sign of ddθy2L is positive according to the following calculation:
]
[
(
)− 12
d 2 yLd 1
2(−A=−1 − A1 − A2
1 − 2α(1 + γ))
dθ 2α
dθ2
)− 1
1 d ( 2=A1 − A2 2 (A1 + 2α(1 + γ))
2α dθ
1
[
(
)
]
(A21 − A2 )− 2 −1 1
dA1 dA2
dA1
2
− 2A1
−
(A1 + 2α(1 + γ)) + (A1 − A2 )
=
2α
2
dθ
dθ
dθ
]
(A2 − A2 )− 2 −1 [= 1
(A1 + 2α(1 + γ))2 − (A21 − A2 ) > 0.
2α
1

(29)

Similar to (25), the sign of the derivative of xH , with respect to θ, becomes:
1 dxH
1
dϕ
1 − β1
α
dyL
=
−
< 0.
xH dθ
ϕ(1 − αβ3 ) |{z}dθ
1 − αβ3 η − αyL − D |{z}
dθ
<0

(30)

>0

Finally, the sign of the derivative of yH , with respect to θ, is simply given by:






)− 12 
dyH
1  dA1 1 ( 2
dA
dA
1
2
2A1 < 0. =
+ A1 − A2−dθ2α |{z}dθ2dθdθ 
|{z}
|{z}
<0

<0

18

>0

(31)

A4. Proof of Proposition 3
From (18), the derivative of g∗ , with respect to τ, is given by:ln g∗ =⇔

1
β1 + αβ3ln ψ +
ln xH ,
1 − β1
1 − β1

1 dψ β1 + αβ2 dxH
1 − β1 dg∗
=
+
.
g∗ dτ
ψ dτ
xH
dτ

(32)

Moreover, from the definition of xH , we obtain:
ln xH =

1
1 − β1ln ϕ +ln(η − αyL − D),
1 − αβ31 − αβ3⇔ (1 − αβ3 ) ln xH = − ln ψ + (1 − β1 ) ln A − (1 − β1 ) ln(1 + γ) + (1 − β1 ) ln(η − αyL − D),()
1 − αβ3 dxH
1 dψ 1 − β1 dγ
1 − β1
d
dyL
⇔
=−
−
+
(η − D) − α
.
(33)
xH
dτ
ψ dτ
1 + γ dτ η − αyL − D dτdτ
Substituting (33) into (32) yields:()()
1 dψ β1 + αβ2 1 dψ 1 − β1 dγ
β1 + αβ2 1 − β1ddyL
1 − β1 dg∗=−++(η − D) − α
.
g∗ dτ
ψ dτ
1 − αβ3 ψ dτ
1 + γ dτ1 − αβ3 η − αyL − D dτdτ
(34)
The sum of the first and second terms of the RHS of (34) becomes:()()
β1 + αβ2 1 dψ β1 + αβ2 1 − β1 dγ
1 − β1 1 − α dψ β1 + αβ2 dγ
1−
−
=
−
,
(35)
1 − αβ3 ψ dτ
1 − αβ3 1 + γ dτ 1 − αβ3ψ dτ
1 + γ dτ()
dψ
1 +αβ21)
where the second equality holds, since 1 − β1−αβ= (1−α)(1−β1−αβ3 . From the definition of ψ, dτ =3β1 ψ dγγ dτ , and by substituting this into (35), it becomes:

1 − β11dγ[β1 (1 − α)(1 + γ) − γ(β1 + αβ2 )] .
1 − αβ3 γ(1 + γ)dτ
Since:β1 (1 − α)(1 + γ) − γ(β1 + αβ2 ) = (1 − α)β1 − αγ(β1 + β2 )()αγ= (1 − α)β1 1 −(β1 + β2 ) = (1 − α)β1 (1 − (1 − τ)(β1 + β2 )) > 0,
(1 − α)β1
substituting this into (36) yields:dγ
1 − β1 (1 − α)β1
(1 − (1 − τ)(β1 + β2 )) ,1 − αβ3 γ(1 + γ)
dτ

19

(36)

and by substituting this into (34), we obtain:()
1 − β1 dg∗
dγ β1 + αβ2 1 − β1
1 − β1 (1 − α)β1ddyL
(1 − (1 − τ)(β1 + β2 ))=+(η − D) − α,g∗ dτ
1 − αβ3 γ(1 + γ)dτ
1 − αβ3 η − αyL − D dτdτ()
1 − αβ3 dg∗
β1 + αβ2ddyL(1 − α)β1dγ=(η − D) − α((1 − τ)(β1 + β2 ) − 1) .
⇔
−
∗
g
dτ
η − αyL − D dτdτγ(1 + γ)dτ∗The first-order condition of g∗ , with respect to τ, is given by dgdτ = 0 as:()
dβ1 + αβ2dyL(1 − α)β1dγ(η − D) − α=((1 − τ)(β1 + β2 ) − 1) ,
η − αyL − D dτdτγ(1 + γ)dτ(37)which is equivalent to (21). The LHS of (37) is decreasing in τ16 and goes to infinity as τ → τ̂,dyL
Lwhere A21 |τ=τ̂ = A2 |τ=τ̂ , since (A21 − A2 )− 2 → ∞ and dydτ < 0. Hence, dτ → −∞, and it is positive1

when τ = 1. On the other hand, the RHS of (37) is increasing in τ, and goes to infinity as τ → 1,since γ → 0. Therefore, γ−1 → ∞, while it is positive when τ = τ̂. This means that there is thegrowth-maximizing tax rate τ∗ ∈ (τ̂, 1) (see the left panel of Figure 2).
∗

Further, the derivative of g∗ , with respect to θ, is derived as follows; similar to (32), dg
dθ is calculated as:
1 − β1 dg∗
1 dψ β1 + αβ2 dxH
=
+
.
∗
g
dθ
ψ dθ
xH
dθ

(38)

By using a similar procedure to (33), we obtain dxdθH as:





 d

1 dψ 1 − β1 dγdy
1 − αβ3 dxH
1 − β1L (η − D) −α
 .
=−
−
+
xH
dθ
ψ dθ
1 + γ |{z}dθ
η − αyL − D |dθ {z }dθ 
=0

(39)

=−1

Combining (38) with (39) yields:[()]
1 − β1 dg∗
1 dψ β1 + αβ2 1 dψ
1 − β1
dyL
=
−
+
1+α
g∗ dθ
ψ dθ
1 − αβ3 ψ dθ η − αyL − Ddθ()β1 + αβ2dyL(1 − α)(1 − β1 ) 1 dψ
−
1+α
.
=
1 − αβ3ψ dθ η − αyL − Ddθβ2 ψ
By using dψdθ = θ , we obtain:()
1 − αβ3 dg∗ β2 (1 − α)β1 + αβ2
dyL
=
−
1+α
,
g∗
dθ
θ
η − αyL − Ddθ

16
From (24), we show that dydτL is increasing in τ. Considering this, the derivative of the LHS of (37), with respect to τ, ispositively proportional to()2
d 2 yLdyLd−α 2 (η − αyL − D) −(η − D) − α
,
dτ
dτ
dτ
which is negative; thus, the term is decreasing in τ.

20

∗and the first-order condition, dg
dθ = 0, is given by:()
β1 + αβ2dyL
β2 (1 − α)1+α=,
η − αyL − D
dθ
θ

(40)

which is equivalent to (22). The LHS of (40) is increasing in θ,17 And goes to infinity as θ → θ̂, wheredyL
L
A21 |θ=θ̂ = A2 |θ=θ̂ , since (A21 − A2 )− 2 → ∞ and dydθ > 0. Hence, dθ → ∞ and it is positive when θ = 0.
1

On the other hand, the RHS of (40) is decreasing in θ, goes to infinity as τ → 0, and is positive when
θ = θ̂. This means that there is growth-maximizing public education/production ratio θ∗ ∈ (0, θ̂) (seethe right panel of Figure 2). 

References[1] Agénor, P.-R., (2011) “Schooling and public capital in a model of endogenous growth”, Economica, 78, 108-132.
[2] Arai, R., (2011) “Productive government expenditure and fiscal sustainability”, FinanzArchiv:
Public Finance Analysis, 67, Number 4, 327-351.
[3] Barro, R. J., (1990) “Government spending in a simple model of endogenous growth”, Journalof Political Economics, 98, 103-125.
[4] Blankenau, R. W., and Simpson, B. N., (2004) “Public education expenditures and growth”,
Journal of Development Economics, 73, 583-605.
[5] Blankenau, R. W., Simpson, B. N., and Tomljanovich, M., (2007) “Public education expenditures, taxation, and growth: Linking data to theory”, American Economic Review, 97, No.2,
393-397.
[6] Boldrin, M., and Montes, A., (2005) “The Intergenerational State Education and Pensions”,
Review of Economic Studies, 72, 651-664.
[7] Bräuninger, M., (2005) “The Budget Deficit, Public Debt, and Endogenous Growth”, Journal of
Public Economic Theory, 7 (5), 827-840.
From (29), we show that dydθL is increasing in θ. Then, the denominator of the LHS of (40) decreases, while thenumerator increases as θ increases; thus, the term is increasing in θ.
1721

[8] Chalk, N, A., (2000) “The sustainability of bond-finance deficits: An overlapping generations
approach”, Journal of Monetary Economics, 45, 293-328.
[9] Diamond, P., (1965) “National debt in a neoclassical growth model”, American Economic Review, 55, 1126-1150.
[10] Futagami, K., Morita, Y., and Shibata, A., (1993) “Dynamic analysis of an endogenous growthmodel with public capital”, Scandinavia Journal of Economics, 95, 607-625.
[11] Greiner, A., (2008) “Human capital formation, public debt and economic growth”, Journal of
Macroeconomics, 30, 415-427.
[12] Glomm, G., and Ravikumar, B., (1992) “Public versus private investment in human capital:
Endogenous growth and income inequality”, Journal of Political Economy, 100, No.4, 818-834.
[13] Gottardi, P., Kajii, A., and Nakajima, T., (2015) “Optimal taxation and debt with uninsurable
risks to human capital accumulation”, American Economic Review, 105(11), 3443-3470.
[14] Grossman, G, M., and Yanagawa, N, (1993) “Asset bubbles and endogenous growth”, Journalof Monetary Economics, 31, 3-19.
[15] Lucas, R E., Jr., (1988) “On the mechanics of economics development”, Journal of Monetary
Economics, 22, July, 3-42.
[16] OECD, (2016) “Education at a glance 2016: OECD indicators”, OECD Publishing, Paris.
http://dx.doi.org/10.1787/eag-2016-en.
[17] Ono, T., and Uchida, Y., (2016) “Human Capital, Public debt, and Economic Growth: A Political Economy Analysis”, Discussion Papers In Economics and Business, Graduate School of
Economics and Osaka School of International Public Policy (OSIPP), Osaka University, 16-01,
August.
[18] Tirole, J., (1985) “Asset bubbles and overlapping generations”, Econometrica, 53, 1499-1528.
[19] Uzawa, H., (1965) “Optimal technical change in an aggregate model of economic growth”,
International Economic Review, 6, January, 18-31.
22[20] Yakita, A., (2008) “Sustainability of public debt, public capital formation, and endogenous
growth in an overlapping generations setting”, Journal of Public Economics, 92, 897-914.
[21] Zhang, J., (2003) “Optimal debt, endogenous fertilities, externalities in a model with altruistic
bequests”, Journal of Public Economics, 87, 1825-1835.

23