Budget Deficit, Life Expectancy, and Declining
Population Growth in an Endogenous Growth Modelwith Public Capital
Mitsuru Ueshina∗
August 17 2017

Abstract
This study constructs an endogenous growth model with public capital andpublic debt in a finite horizon economy. Most previous studies analyzing the eﬀects
of extension of life expectancy adopt an endogenous growth model with public
capital against the backdrop of an aging population. Referencing these works, ourmodel incorporates issues of public debt because some previous studies assume thebalanced budget rule. We show that debt-financed public investment decreases
public capital, possibly disrupting economic growth, particularly if life expectancy
is high. Next, we conduct a numerical simulation and demonstrate that a growthmaximizing tax rate decreases with an expanding fiscal deficit. We then extend ourmodel to examine the eﬀects of a declining birth rate and clarify that an extensionof life expectancy reduces the public debt-GDP ratio, while a declining birth rateincreases it. The model also shows that a declining birth rate does not aﬀect theeconomy when public investment is fully financed by fiscal deficit. In conclusion,
the eﬀects of life expectancy are more pronounced even when accompanied by asignificantly declining birth rate.
Keywords: Birthrate; Life Expectancy; Public Investment; Public Debt
JEL Classification: E62; H62; J13

1

Introduction

One of the most serious problems facing advanced countries is public debt accumulation.
For example, in 2014, the ratio of gross debt to GDP for the United States and Japanwas 105% and 249%, and the average ratio for Euro area countries was 82%.1 Further
the public debt is expected to accumulate in the United States and Japan, and many
researchers have therefore expressed concerns regarding the eﬀects of public debt and∗Graduate School of Economics, Kobe University, 2-1 Rokkoudai-cho, Nada-ku, Kobe 657-8501, Japan,
Tel +81-80-1900-4698, e2009014@gmail.com. I should like to express my deepest gratitude to Prof.
Tamotsu Nakamura, my supervisor, for his valuable comments and criticisms.
1
For more details, see World Economic Outlook Database, October 2016.

1

fiscal deficit on economic growth. Most theoretical and empirical studies show that excess
public debt accumulation adversely aﬀects economic growth.
For example, Checherita and Rother (2010) and Baum et al. (2012) use data forEuro area countries and empirically show an inverted U-shaped relationship between
economic growth rate and the debt-GDP ratio. Further, they state that the thresholdthat changes public debt eﬀects is about 90%, implying that an expansion of budget
deficit negatively aﬀects economic growth in advanced countries such as the United States,
Japan, and Euro area countries.2 Saint-Paul (1992), on the other hand, theoreticallyanalyzes the eﬀects of debt-GDP ratio on economic growth in an endogenous growthmodel and concludes that an increase in the ratio decreases the economic growth rate
through the crowding-out of private investment. In general, public debt is issued for
productive public investment because the resultant public capital benefits not only thecurrent but also the future generation3 . However, Saint-Paul (1992) fails to consider
government expenditure in his model. Nevertheless, many other studies have shown that
excess public debt hinders economic growth in the long run, even when considering public
investment that promotes economic growth.4 In sum, the abovementioned studies clarify
that excessive debt accumulation negatively influences economic growth, regardless of theextent of expenditures financed by fiscal deficit.
In addition to fiscal issues, we focus on the decreasing birth rate and increasing aging
population in advanced countries. In terms of aging population, the old-age dependency
ratio in G7 countries increased from 1950 to 2015 and is expected to continue growing in
the future. The aging population can be mainly attributed to the significantly expanding
life expectancy. As Tamai (2009) shows, population aging could reduce tax revenue and,thus, negatively influences public capital accumulation and economic growth. Drawing
on Blanchard (1985) and Yaari (1965), Tamai (2009) constructs a Futagami et al.-type(1993) endogenous growth model with public capital and a finite horizon and analyzes
the eﬀects of increasing life expectancy (i.e., aging population) on various key variables.
Tamai clarifies that the growth-maximizing income tax rate is lower than the elasticity of
public capital to output, which maximizes economic growth in an infinite horizon modelsuch as Futagami et al.’s (1993). Furthermore, in his model, the expansion of the aging
population increases the growth-maximizing income tax rate, implying the need for alarger government in the case of a growing aging population. Yakita (2008a) adopts asimilar framework for a two-period OLG model and shows that, to maximize economic
growth, a government must increase the tax rate in line with its aging population. However, these studies assume a balanced budget, that is, a tax-financed public investment,
and do not consider the eﬀect of a budget deficit. In recent years, such productive expenditures have been financed by fiscal deficit rather than tax revenue given the benefits
obtained across generations. Therefore, it is important to consider debt-financed public
investment to more realistically determine the eﬀects of aging. Recently, Kamiguchi andTamai (2017) constructed an endogenous growth model with debt-financed public capitalin a finite horizon, as in Blanchard (1985), and show that both the growth-maximizing tax
2

Reinhart et al. (2012) use the data for numerous advanced countries and show similar results interms of the threshold.
3
In recent years, social security expenses have also been financed by fiscal deficit given the shortageof tax revenue.
4
See, for example, Greiner and Semmler (2000), Futagami et al. (2008), Minea and Villieu (2009),
Greiner (2013).

2

CanadaFranceGermany
Italy
Japan
United Kingdom
United States

%
1.00

0.80

0.60

0.40

0.20

0

-0.20

-0.40

-0.60

-0.80
2015

2020

2025

2030

2035

2040

2045

2050

2055

2060

Figure 1: Changes in the population growth rate
Source. United Nations, Population Division (2015). World Population Prospect.

rate and the welfare-maximizing tax rate increase with an extension of life expectancy5 .
Another serious problem in many countries is the declining rates of birth and population growth. Figure 1 and 2 clearly show that the population growth rate continues
to decrease in G7 countries owing to the declining birth rate. This phenomenon can adversely aﬀect the fiscal position and has generated much interest in the eﬀect of a declining
population. Hansen (2013) introduces the role of a government that issues public debt
in a simple Solow model and shows that a declining population rate increases the debtGDP ratio. Kondo (2012) analyzes fiscal sustainability using an infinite horizon modeland demonstrates that sustainable public debt substantially decreases at the initial point
when the size or growth rate of population declines. Furthermore, he emphasizes thatwhile the quantitative eﬀect diﬀers, the reduction in both the size and growth rate of
population negatively aﬀects the sustainability of public debt. Their studies theoreticallyreveal that a decline in population worsens the fiscal situation even if the government
does not implement loose fiscal administration. However, Hansen (2013) assumes a fixedsaving rate, that is, he does not consider the optimization of households, while Kondo(2012) does not account for the role of private and public capital stock accumulation.
Referencing Yakita (2008b), Bokan et al. (2016) investigate the relationship between agrowth maximizing debt-GDP ratio and the population growth rate under the golden ruleof public finance (GRPF). As per the GRPF, the government can issue public debt tofinance only productive expenditures, where a certain proportion of public investment isfinanced by public debt, while the remaining is funded by tax revenue in our model. On
the other hand, Bokan et al. assume an extreme condition as a special case of the GRPF,
that is, public investment is fully financed by fiscal deficit. Therefore, it remains to be
determined if the growth-maximizing debt-GDP ratio (or tax rate) increases when fiscal
deficit expands.
This study constructs an endogenous growth model with public capital and debt to
examine the eﬀect of fiscal deficit on important variables, such as public capital accumu5

They also show that a growth-maximizing tax rate is not consistent with a welfare-maximizing tax
rate, although they do not take into account transitional dynamics, implying that the results of Barro(1990) and Futagami et al. (1993) do not hold in their model.

3births per 1,000
30.0

CanadaFrance25.0

Germany
Italy
Japan

20.0

United Kingdom
United States
15.0

10.0

5.0

0.0
1950

1955

1960

1965

1970

1975

1980

1985

1990

1995

2000

2005

2010

2015

2020

2025

2030

Figure 2: Changes in the birth rate
Source. United Nations, Population Division (2015). World Population Prospect.

lation and long-run economic growth rate, and mutual dependence among budget deficit,
declining birth rate and life expectancy under the GRPF. We draw on Tamai (2009),
who constructs an endogenous growth model with public capital in a finite horizon, andincorporate the role of public debt in our basic model. Although Kamiguchi and Tamai(2017) also consider the issue of aging population with debt-financed public investment
under the GRPF, they assume that public investment is fully financed by fiscal deficit
as in Bokan et al. (2016). Therefore, our analysis discusses the eﬀects of fiscal deficit
when life expectancy increases, and compares the results with those of the public debt
models in Minea and Villieu (2009) and Greiner (2013), who consider an infinite horizon.
In addition, we extend the present model to population growth to distinguish between
the eﬀects of expanding life expectancy and declining birth rate. In terms of a finite horizon, Tamai (2009) follows Blanchard’s (1985) standard model, implicitly assuming thatthe extension of life expectancy causes a consistently declining birth rate to maintain the
population growth rate at zero. Considering that, in reality, a declining birth rate is large
enough to decrease the population growth rate, it is important to clarify the net eﬀects.
Therefore, we focus on the eﬀects of not only an aging population but also falling birthrate that leads to declining population growth.
Our model oﬀers the following main results. In comparison with infinite horizon models
in Minea and Villieu (2009) and Greiner (2013), which show the negative eﬀects of fiscal
deficit, we demonstrate that an increase in fiscal deficit has complex eﬀects on public
capital accumulation and economic growth. As for the growth-maximizing income tax
rate, we numerically explain that an extension of life expectancy increases the tax rate,
while an increase in fiscal deficit decreases it. This result suggests that the government
should not necessarily increase the income tax rate under an expanding fiscal deficit
and life expectancy; this finding does not diﬀer from those for a balanced budget rule
(BBR), wherein a government does not issue public debt. Furthermore, we indicate thatthe extension of life expectancy reduces the debt-GDP ratio, while a declining birthrate increases it. We also show that the eﬀects of a declining population growth rate
disappear when a majority of public investment is financed by fiscal deficit, implying thatthe eﬀects of the extension of life expectancy, rather than those of a declining birth rate,4

are important. This suggests that, in this case, a declining birth rate does not aﬀect thegrowth-maximizing tax rate and debt ratio, which is in line with the results of Bokan et al.
(2016), who demonstrate that the growth-maximizing debt-GDP ratio does not depend on
the population growth rate in the absence of a subsidy for childrearing time. Our results
imply that changes in the population growth rate can aﬀect the growth-maximizing debtGDP ratio under the GRPF as long as a part of public investment is financed by tax
revenue.
The remainder of this paper is organized as follows. Section 2 presents the basic model.
Section 3 confirms the dynamic system and balanced growth path of the model. Section 4
conducts a comparative analysis and numerical simulation. Section 5 extends this modelto analyze a declining birth rate. Section 6 concludes the paper.

2

Basic model

We consider an endogenous growth model with public capital and public debt in a finite
horizon. In this section, our model focuses on whether government issues public debt. We
also elucidate how our model diﬀers from that of Tamai (2009).

2.1

Individuals

Following Blanchard (1985), we consider a representative individual who faces a constantmortality rate, θ, which is identical for all individuals. We assume that the birth rate isalso a constant, θ, such that the size of population Nt always equals 1 for any time t:∫ tθe−θ(t−v) dv = 1.
Nt =−∞Each individual inelastically supplies θ units labor. We compute the expected life time ofindividuals as follows:∫ ∞1tθe−θt dt = .
θ0Therefore, we regard a decrease in θ as the extension of life expectancy.
An individual born at time s maximizes the following expected utility at time t:∫ ∞Et U (s, t) = Et
[ln c(s, v)]e−ρ(v−t) dv∫ ∞t[ln c(s, v)]e−(θ+ρ)(v−t) dv,(1)=t

where c(s, t) denotes the individual consumption of generation s at time t, ρ is the rateof time preference, and Et is the expectation operator. In this model, we consider the
possibility that individuals with assets (or debt) suddenly exit the economy because they
always face a certain death rate. Thus, we also assume the existence of a perfect insurance
market. The rate of return of an individual’s assets at time t is the sum of the interest
rate r(t) and premium rate θ: r(t) + θ.
Each individual is subject to a budget constraint as follows:ȧ(s, v) = (1 − λ){[r(v) + θ]a(s, v) + w(v)} − c(s, v),
5(2)

where a(s, v) and w(v) denote total assets and wage income at time t for an individual
born at time s and λ ∈ (0, 1) is the constant income tax rate6 . We assume that newlyborn individuals do not have any assets: a(s, s) = 0. Solving the optimization of theindividual, we obtainċ(s, v)= (1 − λ)r(v) − λθ − ρ,c(s, v)(3)

lim e−(1−λ) t [r(z)+θ)]dz a(s, v) = 0.

(4)

∫v

v→∞

Equation (4) is the transversality condition. To obtain a consumption function for anindividual born at time s, therefore, we solve the diﬀerential equation (3) and substitutethe result into equation (2):c(s, t) = (ρ + θ)[a(s, t) + h(t)],(5)

where h(t) is the human wealth:∫ ∞∫vh(t) ≡ (1 − λ)w(t)e− t (1−λ)[r(z)+θ]dz dv.

(6)t

We also assume that the capital letter ∫of each variable represents the aggregate amounttin the economy, for example, P (t) ≡ −∞ p(s, t)θe−θ(t−s) ds. Thus, when we apply thisdefinition formula to equations (2), (5), and (6), the aggregate variables are denoted asfollows:
C(t) = (ρ + θ)[A(t) + H(t)],Ȧ(t) = (1 − λ)[r(t)A(t) + w(t)] − λθA(t) − C(t),Ḣ(t) = (1 − λ){[r(t) + θ]H(t) − w(t)}.

(7a)
(7b)
(7c)

Diﬀerentiating equation (7a) with respect to t and using equations (7b) and (7c), we
obtainA(t)Ċ(t)= (1 − λ)r(t) − λθ − ρ − θ(ρ + θ)
.
C(t)
C(t)

2.2

(8)

Firms

Following Glomm and Ravikumar (1994) and Greiner and Semmler (1999), we assume
the aggregate production function Y (t) as follows:
Y (t) = ξK(t)1−α [g(t)L(t)]α ,(9)where ξ, K(t), g(t), α ∈ (0, 1), and L(t)(= 1) are the technology parameter, private
capital stock, per capita public capital stock provided by the government, elasticity of
6

In our model, we assume that individuals earn the same wage income regardless of age. However,note that the model results remain unchanged even if we consider retirement; that is, an individual’s
wage income decreases with time, as in Blanchard (1985). For the analysis with retirement, see Appendix
A.

6

public capital to output, and labor input that equals population size, respectively.
Dividing both sides of equation (9) by L(t), we obtainy(t) = ξk(t)1−α g(t)α ,

7

(10)

where y(t) and k(t) are per capita output and per capita private capital: y(t) ≡ Y (t)/L(t)and k(t) ≡ K(t)/L(t). Under perfect competition, the interest rate and wage income are
given byr(t) = (1 − α)ξx(t)α ,w(t) = αξx(t)α−1 g(t),

(11a)
(11b)

where x(t) ≡ G(t)/K(t), and G(t) is aggregate public capital.

2.3

Government

A government levies a tax on income and issues bonds to finance government expenditure.
8
In addition, the government invests on public capital and makes the interest payments
on public debt as expenditure. Therefore, the government budget constraint isḊ(t) = r(t)D(t) + Ġ(t) − λ{[r(t) + θ]A(t) + w(t)L(t)},(12)

where D(t) and Ġ(t) are total government debt and public investment. Furthermore, we
assume that the government follows the GRPF, that is, the government issues bonds only
to finance productive expenditure such as public investment. Since public investment on
public capital benefits not only the current but also the future generation, under this rule,
all generations face a cost burden. In Japan, for example, a majority of public investment
is financed by fiscal deficit. Following Minea and Villieu (2009) and Greiner (2013), we
formulate this rule asḊ(t) = ψ Ġ(t),(13)

where ψ ∈ [0, 1) is the debt-financed ratio. Note that ψ = 0 implies that the government
always maintains a balanced budget, that is, the long-run results of ψ = 0 are consistent
with those of Tamai (2009).

3

Equilibrium

The equilibrium condition for the asset market is
A(t) = K(t) + D(t).

(14)

From equations (7b), (7c), (11a), (11b), (12), and (14), we also obtain the equilibrium
condition of a goods market:
Y (t) = C(t) + K̇(t) + Ġ(t).

(15)

7

Although equation (9) indicates a production function with a congestion eﬀect, it is consistent withthe standard Cobb-Douglas production function with public capital, as in Futagami et al. (2010), because
we do not assume population growth in this section. If we consider positive population growth, theproduction function ensures a consistent balanced growth path, where each per capita variable (i.e.,
production, private capital, and public capital) increases at the same rate in the long run. For the caseof population growth, see Section 4.
8
Tamai (2009) analyzes two types of tax rates: tax on output or income. Since the latter is more
realistic, we focus on the income tax rate.

7

3.1

Dynamic system

To describe the dynamic system for this model, we derive the growth rate for each economic variable. First, substituting equation (11a) and (14) into equation (8) gives us
]
[
Ċ(t)
1 + b(t)αγC ≡= (1 − λ)(1 − α)ξx(t) − λθ − ρ − θ(ρ + θ)
,
C(t)
q(t)

(16)

where q(t) ≡ C(t)/K(t) and b(t) ≡ D(t)/K(t). Dividing equation (15) by K(t), thegrowth rate for private capital is
γK ≡

K̇(t)
Ġ(t)
= ξx(t)α − q(t) −
x(t).
K(t)
G(t)

(17)

From equations (12) and (13), we obtain
γG ≡

Ġ(t)
[λ − (1 − α)(1 − λ)b(t)]ξx(t)α + λθ[1 + b(t)]=.
G(t)(1 − ψ)x(t)(18)

Finally, the growth rate for public debt is simply derived by dividing equation (13) byD(t) as follows:
γD ≡

3.2

Ḋ(t)
Ġ(t) x(t)
=ψ
.
D(t)
G(t) b(t)(19)

Balanced growth path

Here, we consider the balanced growth path (BGP) in which all variables in this economy
grow at a constant rate, γC = γG = γK = γD = γ ∗ . Then, equations (16) - (19) can be
rewritten as()
1 + b∗∗∗αγC = (1 − λ)(1 − α)ξx − λθ − ρ − θ(ρ + θ)
,
(20a)
q∗
α
∗
γK
= ξx∗ − q ∗ − γG∗ x∗ ,
(20b)
∗
∗α
∗
[λ − (1 − α)(1 − λ)b ]ξx + λθ(1 + b )
,
(20c)
γG∗ =
(1 − ψ)x∗
x∗
∗
(20d)
γD
= ψγG∗ ∗ .
bNote that all growth rates for aggregate variables have per capita growth rates because
we do not assume population growth in this section.
Equations (20b) and (20d) implyb∗ = ψx∗ ,αq ∗ = ξx∗ − (1 + x∗ )γG∗ .

(21a)
(21b)

When we use equations (21a) and (21b) to confirm the existence of BGP, equations (20a)and (20c) reduce to(1 − λ)(1 − α)ξx∗ =λθ
1 + ψx∗α−1
λAx∗ + ∗ + (1 − ψ)θ(ρ + θ)
+ (1 − ψ)ρ + λθ,xq(x∗ )α8(22)

q(x*)=0
Λ, Ω
Λ(x*)
E*

(1-ψ)ρ+λθ

Ω(x*)

0

x

x*

Ω(x*)

Figure 3: Existence of BGP

where q(x∗ ) comprises x∗ and other exogenous parameters from the abovementioned algebraic calculation. Let us define the left- and right-hand sides of the equation (22) as
Λ(x∗ ) and Ω(x∗ ), respectively. Accordingly, Figure 3 shows that Λ(x∗ ) slopes upward,
while Ω(x∗ ) is a discontinuous function at q(x∗ ) = 0 and slopes downward. As for the
existence and stability of BGP, we have the following proposition.
Proposition 1 For ψ = 0, there is a unique and stable BGP in the region where q ∗ > 0.
For ψ ∈ (0, 1), under plausible parameters, there is a unique and stable BGP in the regionq ∗ > 0 ( for the proof, see Appendix B.).

4

Characteristics of the steady state

To gain more detailed insight, we focus on the BGP, as E ∗ in Figure 3. First, we conductcomparative statics analysis on the income tax rate, life expectancy, and debt financeratio. Second, we derive the growth-maximizing income tax rate and then perform anumerical exercise that demonstrates how the tax rate should be changed in line with an
expanding life expectancy and fiscal deficit.

4.1

Comparative statics analysis

We mainly analyze the eﬀect on the ratio of public capital to private capital, x∗ . To obtain
the eﬀect of income tax rate on public capital ratio, we diﬀerentiate x∗ with respect tothe income tax rate λ in equation (22), obtaining9∗∗∂Ω(x ))− ∂Λ(xdx∗∂λ∂λ= − ∂Ω(x∗ ) ∂Λ(x∗ ) (> 0),
dλ
−
∗
∗
∂x

9

(23)

∂x

See Appendix B for the result of the denominator of equation (23).

9where∂Ω(x∗ ) ∂Λ(x∗ )(1 + ψx∗ )(1 + x∗ )θα−1−
= ξx∗ + ∗ + θ(ρ + θ)∂λ∂λxq(x∗ )2
[
]
θ
∗α−1
∗α
+ (1 − α)ψξx + ∗ + θψ (> 0).
· ξxx
Therefore, an increase in the income tax rate raises the public capital ratio. This result isthe same as those in obtained in Tamai (2009) and Greiner (2013). It eases the government
budget constraint such that available resources for public investment increase. In addition,
this result is used when deriving the growth-maximizing income tax rate in subsection
4.2.
Next, we show the eﬀects of an extending life expectancy on the public capital ratio,x∗ . Following the same method described above, we obtain the eﬀect of life expectancy
as follows:∗∗∂Ω(x ))− ∂Λ(xdx∗∂θ∂θ= − ∂Ω(x∗ ) ∂Λ(x∗ ) (> 0),
dθ
−
∗
∗
∂x

where

(24)

∂x

[
)]
(
∂Ω(x∗ ) ∂Λ(x∗ )1
θ(ρ + θ)(1 + ψx) 1−=λ ∗ +1++ ψ + 1 + ψx∂θ∂θxq(x∗ )2x∗(ρ + 2θ)(1 + ψx∗ )+(1 − ψ)(> 0).
q(x∗ )

This shows that the extension of life expectancy reduces the public capital ratio. It directly aﬀects individuals and the government10 . First, an extension of life expectancy
reduces government tax revenue, and thus, less available resources disrupt public investment. In addition, it reduces individuals’ marginal propensity to consume to prepare for
future consumption. Further, it boosts private investment, which in turn accumulates
private capital stock. These direct eﬀects reduce public capital and raise private capital, lowering the ratio of public to private capital. Although greater private investment
raises production, which has a stimulating eﬀect on public investment because it raises
tax revenue, the public capital ratio declines because the direct eﬀects exceed the indirect
eﬀects, as seen in equation (24).
We consider the eﬀects of fiscal deficit on the public capital ratio and economic growthrate. By diﬀerentiating x∗ with respect to ψ, we obtaindx∗
2⪌ 0 ⇔ θ(ρ + θ)Z ⪌ (1 − ψ)ρq ∗
dψ

(25)

where
Z⪌0⇔
[

]
1
α−1
2λθ ψ x + ∗ + (2 + ψ) + 2λξx∗
x
2 ∗2

10

Although this result is the same as that in Tamai (2009), note that it includes the eﬀect of not only
an extension of life expectancy but also a declining birth rate.

10⪌ [(1 − λ)(1 − α)(2ψ 2 + ψ + 1) + (1 − λ)(3ψ − 1) − 2ψ 2 ]ξx∗

α+1

+2ψ 2 (1 − λ)(1 − α)ξx∗

α+2+ [(1 + ψ)(1 − λ)(1 − α) + 1 − ψ − λ(1 + 3ψ)]ξx∗ .
αNote that Z can become negative depending on the parameters. If Z < 0, an increasein fiscal deficit reduces the public capital ratio. The result of equation (25) is diﬀerent
from that of an infinite horizon model as in Minea and Villieu (2009) and Greiner (2013)
who show negative eﬀects of fiscal deficit on public capital. In the short run, their modelshows that public investment financed by fiscal deficit accumulates public capital but
disturbs public capital accumulation because of the higher interest payment on public
debt in the long run. In our model, resources for public investment include not only
taxation on output (i.e., returns on private capital and wage income) but also those on
insurance premiums. An increase in fiscal deficit raises public debt accumulation, thus
leading to greater individual assets. Therefore, it does not necessarily reduce public
capital ratio in a finite horizon. Given equation (25) and the result of an infinite horizon
model, public investment financed by fiscal deficit reduces private capital in the long run
if life expectancy is suﬃciently high, that is, θ is suﬃciently low.
Proposition 2 Assume that life expectancy is suﬃciently high, that is, θ is suﬃciently
low. Then an increase in fiscal deficit reduces the ratio of public capital to private capital,x∗ , in the long run.
This suggests that debt-financed public investment, rather than tax-financed investment, is undesirable in developed countries where life expectancy continues to increase.11

4.2

Numerical simulation

In this section, we perform a numerical simulation to derive the growth-maximizing tax
rate and analyze the eﬀect of life expectancy and fiscal deficit on the tax rate. When we
diﬀerentiate the growth rate of public capital (20c) with respect to λ, we obtain
dγG∗
=0⇔
dλ
1
α
α
(ξx∗ + θ) + ψ[(1 − α)ξx∗ + θ] =
∗
x[
] ∗
λ∗
x∗α
∗
∗α −1 dx,[(1 − α)ξx + θ] + ψ(1 − ψ)(1 − λ )αξx
dλ
x∗2

(26)

where λ∗ is the growth-maximizing income tax rate. From equation (26), we can explicitly
derive the income tax rate λ∗ , but it has a complex form. Thus, we numerically computethe tax rate for a varying life expectancy and debt-finance ratio.
As for the parameter values, we choose them on the basis of previous studies. Following
Mourmouras and Lee (1999), we set the eﬀective time preference to θ + ρ = 0.08, which
11

Futagami et al. (2008) construct an endogenous growth model with public capital and debt under
the Maastricht Treaty and present similar implications. Although they assume an infinite horizon model,
they indicate that a rise in public debt increases the economic growth rate in the low-growth steady state
but decreases it in the high-growth steady state.

11

comprises θ = 0.05 and ρ = 0.03. While some studies set the elasticity of output topublic capital to a relatively high value, α = 0.4, we adopt α = 0.25, which is morecommonly used in the literature (e.g., Greiner and Semmler, 1999; Mourmouras and Lee,
1999; Greiner, 2013), and find that our results remain unchanged even if we adopt α =0.4. Finally, the debt-finance ratio ψ can take various values depending on the assumed
countries. In Japan, which is an extreme example, public investment is entirely financedby fiscal deficit, ψ → 1, whereas the government in Germany follows the BBR, ψ → 0.
Therefore, we consider various values of the debt-finance ratio as follows ψ = 0.1, 0.5, 0.9.
Figure 4 depicts the long-term relationship between the economic growth rate andincome tax rate for ψ = 0.1, 0.5, 0.9. For ψ = 0.1, 0.5, 0.9, the growth-maximizing tax
rate λ∗ is 18.8%, 17.5%, and 16.8%, respectively. Table 1 presents a combination of debtfinance ratio and the growth-maximizing tax rate, implying that the growth-maximizing
tax rate tends to decrease as the debt-finance ratio rises. This result considerably diﬀers
from that of an infinite horizon model, which shows that the growth-maximizing tax rateis equal to the elasticity of output to public capital. 12 The intuition behind this result
is as follows. An increase in fiscal deficit accumulates public debt, which has an eﬀect
on the optimal income tax rate. Increased public debt raises tax revenue because thegovernment taxes insurance premiums gained from individuals’ assets and this reduces
the optimal income tax rate. In this case, the government must reduce the income tax
rate to maximize the economic growth rate since public capital is large relative to private
capital.
Next, we analyze the eﬀects of an extension of life expectancy. Except for life expectancy and the debt-finance ratio, we use the same values as those in the above simulation. We set the debt-finance ratio to ψ = 0.5. Figure 5 illustrates the relationshipbetween the economic growth rate and income tax rate for diﬀerent values of life expectancy. For θ = 0.06, 0.05, 0.04 the growth-maximizing tax rate λ∗ is 16.6%, 17.5%,
and 18.6%, respectively. Furthermore, Table 2 shows the growth-maximizing tax rates
for θ = 0.03 − 0.07 and implies that an extension of life expectancy raises the growthmaximizing tax rate. Therefore, we obtain the same result as that in Tamai (2009). The
mechanism behind this is as follows: an extension of life expectancy reduces tax revenue
because the government taxes insurance premiums. The reduction in tax revenue lowers
public investment, in turn reducing the ratio of public capital to private capital. Thus,
the government must raise the tax rate to continue maximizing the economic growth rate
because the ratio of public to private capital is below the optimal level.
Tamai (2009) and Yakita (2008a) conclude that the size of the government should belarge for an increasingly aging population under the BBR. However, an increase in life
expectancy does not necessarily raise the growth-maximizing tax rate if the fiscal deficit
also increases simultaneously. This ambiguous result occurs because an increase in life
expectancy and a rising fiscal deficit exert opposite eﬀects on the growth-maximizing tax
rate. As in Japan, however, if public investment is entirely financed by fiscal deficit, i.e.,ψ → 1, there mainly exist the influence of life expectancy which continues to over time.
As has been discussed in the context of countries with high public debt, governments
must now consider fiscal consolidation. This suggests that the government must further
raise the tax rate to deal with an aging population.
12

Ueshina and Nakamura (2017) show an inverted U-shaped relationship between the economic growthrate and tax rate, whose absolute maximum is λ∗ = α.

12

0.05

ψ = 0.1
ψ = 0.5

0.04

ψ = 0.9

Economic growth rate

0.03

0.02

0.01

0
0

0.1

0.2

0.3

0.4

0.5

0.6

-0.01

-0.02

-0.03

Income tax rate

Figure 4: Economic growth and income tax rate for ψ = 0.1, 0.5, 0.9
Table 1: Debt-finance ratio and growth-maximizing tax rate
ψ
λ∗

0
19.0%

0.1
18.8%

0.3
18.0%

0.5
17.5%

0.7
17.1%

0.9
16.8%

0.05

θ = 0.06
θ = 0.05

0.04

θ = 0.04

Economic growth rate

0.03

0.02

0.01

0
0

0.1

0.2

0.3

0.4

0.5

0.6

-0.01

-0.02

-0.03

Income tax rate

Figure 5: Economic growth and income tax rate for θ = 0.04, 0.05, 0.06

Table 2: Life expectancy and growth-maximizing tax rate
θ
λ∗

0.03
19.8%

0.04
18.6%

0.05
17.5%

13

0.06
16.6%

0.07
15.8%

5

Extension

In this section, we incorporate population growth into the basic model. We distinguishbirth rate and death rate to analyze the eﬀects of a declining birth rate.

5.1

Declining population growth

Thus far, we assumed that the birth rate is equal to the mortality rate to maintain the
population growth rate at zero. Following Buiter (1988), we consider a positive population
growth13 . We denote birth rate and mortality rate as β and θ, respectively. Then, we
regard population growth rate n as n = β − θ and assume positive population growth:β > θ(> 0). Furthermore, we also assume that the size of the initial population is unity:
N0 = 1. Each individual inelastically supplies β units of labor. Therefore, the size of totallabor for any time t is given by∫ t−θt
Lt = βeeβs ds = ent .
−∞As for the GRPF, we rewrite equation (13) as follows:˙ = ψ ġ(t),
d(t)

(27)

where d(t) ≡ D(t)/L(t) and g(t) ≡ G(t)/L(t). In addition, we define other per capita variables such as k(t) ≡ K(t)/L(t) and c(t) ≡ C(t)/L(t). Note that an individual’s expectedutility and budget constraint do not change and, thus, equation (3) is the individual Euler
equation.
Based on the definition of per capita variables, rewriting dynamic systems (16) - (19)in per capita form gives us
γc ≡

ċ(t)
=(1 − λ)(1 − α)ξx(t)α − λθ − ρc(t)[]
1 + b(t)− β(ρ + θ)
q(t)

k̇(t)
ġ(t)
= ξx(t)α − q(t) −x(t) − n[1 + x(t)]
k(t)
g(t)
ġ(t) [λ − (1 − α)(1 − λ)b(t)]ξx(t)α + λθ[1 + b(t)]
γg ≡
=
g(t)
(1 − ψ)x(t)n[x(t) − b(t)−(1 − ψ)x(t)

γk ≡

γd ≡

˙
d(t)
ġ(t) x(t)
=ψ
.
d(t)
g(t) b(t)

(28)

(29)

(30)

(31)

13

Weil (1989) also considers positive population growth but assumes that individuals do not exit theeconomy.

14

Now we consider the BGP: γc∗ = γk∗ = γg∗ = γb∗ = γ ∗ > 0. Accordingly, equation (22)can be rewritten as(1 − λ)(1 − α)ξx∗ + (1 − ψ)n =λθ
1 + ψx∗α−1λξx∗ + ∗ + (1 − ψ)(ρ + θ)(θ + n)
+ (1 − ψ)ρ + λθ,xq(x∗ )
α(32)

where we again define the left- and right-hand sides of equation (32) as Λ(x∗ ) and Ω(x∗ ),respectively. As for the existence and stability of the BGP, we obtain the same results of
Proposition 1. 14 From equation (32), diﬀerentiating the ratio of public to private capitalwith respect to birth rate and life expectancy, respectively we obtain∗∗∂Ω(x ))− ∂Λ(xdx∗∂β∂β= − ∂Ω(x∗ ) ∂Λ(x∗ ) (< 0),
dβ
−
∗
∗
∗

dx
dθ

(33)

∂x
∂x
∂Ω(x∗ )
∂Λ(x∗ )− ∂θ∂θ(> 0),= − ∂Ω(x∗)∂Λ(x∗ )
−
∗
∗
∂x
∂x

(34)

where
∂Ω(x∗ ) ∂Λ(x∗ )(1 − ψ)(ρ + θ)h∗
−
=−
<0
∂β
∂β
q(x∗ )k ∗(35)(1 + ψx∗ )∂Ω(x∗ ) ∂Λ(x∗ )−= (1 − ψ)β∂θ∂θq(x∗ ))][(β(ρ + θ)(1 + ψx) 1
1+ ψ + 1 + ψx
> 0.
+λ ∗ +1+xq(x∗ )2
x∗

(36)

and

Note that the sign of the denominator of equation (33) and (34) is negative. Not surprisingly, we obtain the equation (24) by adding equations (35) and (36) and assuming thatthe birth rate is equal to the mortality rate, β = θ. Now, we consider the eﬀects of theseon the ratio of public debt to GDP. Since b∗ = ψx∗ , in the steady state, the eﬀects on
the debt-GDP ratio are proportional to those on the public capital ratio. Accordingly, we
obtain the following results:
db∗
db∗
< 0,
> 0.
dβ
dθ

(37)

This implies that life expectancy extension reduces the debt-GDP ratio, while a declining
birth rate increases the ratio.15 When these eﬀects simultaneously occur, such that the size
14

For proof, see Appendix B.
Although we consider the eﬀects on the ratio of public debt to private capital to analyze debt-GDP
ratio, note that both oﬀer the same results. In the BGP, the debt-GDP ratio can be rewritten using
equation (9) and b∗ as follows:
15

1−α
D
= cb∗ ,
Y

where c is a constant value.

15

of the population is constant, the net eﬀect of these is negative, that is, the debt-GDP
ratio declines. As we confirm in introduction, however, the rate of population growthtends to decline in many advanced countries. Therefore, the eﬀects of a declining birthrate can be quantitatively large, and thus, it is important to focus on the influence on
countries in which the population growth rate significantly declines. As for the declining
birth rate, the results can be interpreted as follows. Declining population growth as aresult of a decreasing birth rate has two eﬀects on public investment. First, it raises the
per capita burden of public debt but reduces the dilution eﬀect of public capital. Under
the golden rule, the ratio of public debt to private capital does not exceed public capitalratio: g ∗ ≥ b∗ . Therefore, with a declining population growth, the improvement in the
dilution eﬀect dominates the increase in the per capita burden of public debt, which isfurther accumulated through public investment under the GRPF.
The relationship between the eﬀects of a declining birth rate and debt-finance ratio
is highly complex, as can be seen in equations (33) and (35). Therefore, we consider thesituation in which most of the public investment is financed by fiscal deficit, as in the caseof Japan. This implies ψ → 1, and thus, equation (33) oﬀers the following result:db∗dx∗= 0, lim=0ψ→1 dβψ→1 dβlim

This demonstrates that a change in the birth rate does not influence those ratios if public
investment is entirely financed by fiscal deficit. This is because the improvement in the
dilution eﬀect is equal to the increase in the public debt burden, because both public debt
and public capital are of the same size. From the above results, we obtain the following
proposition.
Proposition 3 Slow population growth due to a decreasing birth rate raises the public
debt-GDP ratio, while an extension of life expectancy reduces it.
db∗
db∗
< 0,
> 0.
dβ
dθ
If a majority of public investment is financed by fiscal deficit and β and θ simultaneously
decrease, the eﬀects of a declining birth rate disappear, so the eﬀects of life expectancy
mainly remain.
This proposition indicates the need to also focus on a declining birth rate in the caseof a significantly declining population growth; however, more attention is needed for theextension of life expectancy when public investment is entirely financed by fiscal deficit.
In addition, our results are consistent with those of Bokan et al. (2016) when amajority of public investment is financed by fiscal deficit. Their study shows that thegrowth-maximizing debt-private capital ratio does not depend on birth rate under the
GRPF and in the absence of a subsidy for childrearing time16 . This suggests that theirresults can be obtained even in the special case of public debt being equal to public capital.

16

Bokan et al. (2016) assume that public investment is entirely financed by fiscal deficit and households
endogenously determine the number of children.

16

6Conclusion

In this study, we constructed an endogenous growth model with public capital and debt
in a finite horizon economy. Some previous studies analyze the eﬀects of the extensionof life expectancy under the balanced budget rule against the background of an aging
population. Therefore, we account for an aging population in our models to clarify theeﬀects of a fiscal deficit. We also focus on the influence of a declining birth rate given thetendency of population growth to significantly decline in many advanced countries.
Our main findings are as follows. First, under a finite horizon, debt-financed public
investment can increase or decrease the ratio of public to private capital, implying that
economic growth rises if it increases the ratio of public to private capital. However, debtfinanced public investment disturbs public capital accumulation by reduced tax revenue
if life expectancy is suﬃciently high as in the case of advanced countries. Second, as
for the growth-maximizing tax rate, we numerically show that an expanding fiscal deficit
reduces the tax rate, which is in contrast to the eﬀects of life expectancy. Although
this result renders policymaking a more complex process, it implies that the government
must further raise the tax rate when implementing fiscal consolidation to reduce excess
public debt accumulation. Third, we clarify the eﬀects of birth rate and life expectancy,
for which previous studies oﬀer mixed results. An extension of life expectancy reduces
the debt-GDP ratio, while a decrease in population growth owing to a declining birthrate raises the debt-GDP ratio. The eﬀects of a declining birth rate can be large whenpopulation growth is slow. In addition, when public investment is entirely financed by afiscal deficit, the eﬀects of birth rate disappear and that of life expectancy persists. Thissuggests the need to address the problem of a declining birth rate and aging population
depending on the fiscal situation. In particular, when the government excessively relies on
public debt issuance as public capital investment, it must ensure to implement measures
for an aging population rather than declining population growth.
Finally, further study is warranted on the following issues. This study does not consider
a welfare analysis on the current and future generations. When a government changes theincome tax rate to maximize the economic growth rate, it is important to reveal whether
the policy benefits various generations. In addition, an analysis of an aging society isworth considering in the context of unproductive government expenditures, such as social
security costs, which tend to be financed by fiscal deficit.

References
Barro, R. J. (1990) “Government Spending in a Simple Model of Endogenous Growth”,
Journal of Political Economy, 98(S5), S103-S126.
Baum, A., Checherita-Westphal, C., and Rother P. (2012) “Debt and growth: New evidence for the euro area,” ECB working paper series 1450, European Central Bank.
Bokan, N., Hallett, A., and Hougaard Jensen, S. (2016) “Growth-Maximizing Public Debt
under Changing Demographics,” Macroeconomics Dynamics, 20(6), 1640-1651.
Blanchard, O. (1985) “Debt, Deficit, and Finite Horizons,” The Journal of Political Economy, 92(2), 223-247.
17

Buiter, W. (1988). “Death, Birth, Productivity Growth and Debt Neutrality.” The Economic Journal, 98(391), 279-293. doi:10.2307/2233369.
Checherita-Westphal, C. and Rother, P. (2010) “The impact of high and growing debt
on economic growth. An empirical investigation for the euro area,” ECB working paper
series 1237, European Central Bank.
Futagami, K., Morita, Y., and Shibata, A. (1993) “Dynamic Analysis of an Endogenous
Growth Model with Public Capital,” The Scandinavian Journal of Economics, 95(4),604-625.
Futagami, K., Iwaisako, T., and Ohdoi, R. (2008) “Debt Policy Rule, Productive Government Spending, and Multiple Growth Paths,” Macroeconomics Dynamics, 12(4),445-462.
Futagami, K., T. Hori, and R. Ohdoi. (2010) “Debt Policy and Economic Growth in a Small Open Economy Model with Productive Government Spending,” ADBI Working Paper, 212. Tokyo:
Asian Development
Bank
Institute.
Available:
http://www.adbi.org/workingpaper/2010/04/15/3656.debt.policy.economic.growth.model/.
Glomm, G and Ravikumar, B. (1994) “Public investment in infrastructure in a simple
growth model,” Journal of Economic Dynamics and Control, Volume 18, Issue 6, Pages
1173-1187.
Greiner, A. (2013) “Debt and Growth: Is There a Non-Monotonic Relation?,” Economic
Bulletin, 33(1), 340-347.
Greiner, A. and Semmler, W. (1999) “An endogenous growth model with public capital and government borrowing,” Annals of Operations Research, 88: 65.
doi:10.1023/A:1018955002563.
Greiner, A. and Semmler, W. (2000) “Endogenous Growth, Government Debt and Budgetary Regimes,” Journal of Macroeconomics, 22(3), 363-384.
Hansen, A. (2013) “Economic Growth and Public Debt under a Declining Population,”
Tectum Verlag Marburg.
Kamiguchi, A. and Tamai, T. (2017) “Public investment and Golden Rule of Public
Finance in an Overlapping Generations Model,” KIER Discussion Paper, 971, Kyoto
University.
Kondo, A. (2012) “A Note on Public Debt Sustainability in an Economy with Declining
Fertility,” CRR Discussion Paper Series B, Shiga University.
Minea, A. and Villieu, P. (2009) “Borrowing to Finance Public Investment? The ‘Golden
Rule of Public Finance’ Reconsiderd in an Endogenous Growth Setting,” Fiscal Studies,30(1), 103-133.

18

Mourmouras, I. and Lee, J. (1999) “Government Spending on Infrastructure in an Endogenous Growth Model with Finite Horizons,” Journal of Economics and Business,
51, 395-407.
Reinhart, C. M., Reinhart, V. R., and Rogoﬀ, K. S. (2012) “Public Debt Overhangs:
Advanced Economy Episodes 1800,” Journal of Economic Perspectives, 26(3), 69-86.
Saint-Paul, J. (1992) “Fiscal Policy in an Endogenous Growth Model,” Quarterly Journalof Economics, 107(4), 1243-1259.
Tamai, T. (2009) “Public Capital, Taxation and Endogenous Growth in a Finite Horizons
Model,” Metroeconomica, 60(1), 179-196.
Ueshina, M. and Nakamura, T. (2017) “Inverted U-shaped relationship between public
debt and economic growth under the golden rule of public finance,” Mimeo, Kobe
University.
Weil, P. (1989) “Overlapping Families of Infinitely-lived Agents,” Journal of Public Economics, 38, 183-198.
Yaari, M. (1965) “Uncertain Lifetime, Life Insurance, and the Theory of the Consumer,”
The Review of Economic Studies, 32(2), 137-150.
Yakita, A. (2008a) “Ageing and public capital accumulation,” Int Tax Public Finance,
15(5), 582-598.
Yakita, A. (2008b) “Sustainability of Public Debt, Public Capital Formation, and Endogenous Growth in an Overlapping Generations Setting,” Journal of Public Economics, 92,
897-914.

Appendix A. The eﬀects of retirement
Drawing on Blanchard (1985) and Saint-Paul (1992), we focus on the eﬀects of retirement.17 We assume that an individual’s wage income declines at a constant rate, m > 0,
over time. An individual born at time s earns the following income wage at time t:w(s, t) = pw(t)em(s−t) ,(A.1)

where p is a constant value determined below. When aggregating all individual’ income
wage, the result must be equal to the wage income of the entire economy. This conditionis given by∫ tβeβs e−θt w(s, t)ds = w(t)L(t).
(A.2)
−∞

From equations (A.1) and (A.2), we obtain the value of p as follows:
p=
17

m+β
> 0.
m

We incorporate the role of retirement in the model presented in subsection 5.1.

19

We denote after-tax life time wages of an individual born at time s, evaluated at time t,as h(s, t). Using the above results to derive individual and aggregate human assets, we
obtain∫∫v(1 − λ)(m + β) m(s−t) ∞h(s, t) =e
w(v)e− t {(1−λ)[r(z)+θ]+m}dz dv,
β
t
∫ ∞
(A.3)
∫v
− t {(1−λ)[r(z)+θ]+m}dz
nt
H(t) = (1 − λ)e
w(v)e
dv.
t

The remaining procedure is identical to that in the previous section. In addition, notethat retirement influences only an individual’s behavior, so the Euler equation (28) canbe rewritten asċ(t)=(1 − λ)(1 − α)ξx(t)α − λθ − ρ + mc(t)[]
1 + b(t).
− (ρ + θ)(θ + m + n)
q(t)

(A.4)

Equations (29), (30), and (31) remain unchanged. In the BGP, therefore, using equations(29)-(31), and Equation (A.4), we obtain(1 − λ)(1 − α)ξx∗ + (1 − ψ)(n + m) =λθ
1 + ψx∗α−1λξx∗ + ∗ + (1 − ψ)(ρ + θ)(θ + n + m)
+ (1 − ψ)ρ + λθ.
xq(x∗ )α(A.5)

The diﬀerentiating the public capital ratio x∗ with respect to m to confirm the influenceof retirement gives us∗∗∂Ω(x ))− ∂Λ(xdx∗∂m∂m= − ∂Ω(x∗)∗ (< 0),dm− ∂Λ(x∗ )∗(A.6)∂Ω(x∗ ) ∂Λ(x∗ )(1 − ψ)(ρ + θ)h∗
−
=−
< 0.
∂m
∂m
q(x∗ )k ∗

(A.7)

∂x

∂x

where

Given the existence of retirement, the ratio of public capital (and public debt) to private
capital declines. The reason is simple: an individual increases savings for a future decline
in wage income, that is, retirement. Thus, public capital and debt are small relative
to private capital. In addition, the existence of retirement does not change our main
conclusions.
Saint-Paul (1992) analyzes the eﬀects of introducing unfunded social security on thebasis of Blanchard’s framework. Although he considers the role of public debt, he neglectspublic debt when considering unfunded social security in the analysis. Our model showsthe eﬀect of unfunded social security on the public debt ratio through a reduction in wage
income, m; that is, the wage income of relatively young individuals declines and that ofrelatively old ones increases.18 Given equation (A.6), the introduction of a social security
system increases the public debt ratio through a reduction of private capital.
18

Note that the wage income of the entire economy remains unchanged, and thus, an income transfer
occurs within the current generations.

20

Appendix B. Existence and stability of the BGP
Here, we confirm the existence and stability of the BGP using the model presented inAppendix A. This model is identical to that discussed in the Basic Model section if we
assume n = m = 0. As in section 3, we define the left- and right- hand sides of equation(A.5) as Λ(x∗ ) and Ω(x∗ ). Accordingly, these equations can be rewritten asΛ(x∗ ) = (1 − λ)(1 − α)ξx∗ + (1 − ψ)(m + n)αΩ(x∗ ) =λξx∗

α−1

+

(B.1)

λθ
+ (1 − ψ)ρ + λθ,x∗[]
1 + ψx∗.
+ (1 − ψ)(ρ + θ)(θ + m + n)q(x∗ )(B.2)

As in the case of equations (B.1) and (B.2), we obtain the following properties:∂Λ(x∗ )
∗α−1
=
α(1
−
λ)(1
−
α)ξx
>0
∂x∗

(B.3)

∂Ω(x∗ )λθ
α−2= − (1 − α)λξx∗ − ∗2
∗
∂x
x

[
]
∂ 1 + ψx∗
+ (1 − ψ)(ρ + θ)(θ + m + n) ∗< 0,∂xq(x∗ )

where

(B.4)

[
]
∂ 1 + ψx∗1
Γ(x∗ ) < 0,
=−
∗
∗
∂x
q(x )
(1 − ψ)q(x∗ )2Γ(x∗ ) = α(1 − λ)(1 − α)ψ 2 ξx∗

α+1+ (1 − α)[λψ + α(1 − λ)(1 + ψ)]ψξx∗α+ [2λψ(1 − α) + α(1 − λ)(1 − ψα)]ξx∗
λθ
2λθψ
α−2
+ λ(1 − α)ξx∗ + ∗2 ++ λθψ 2 (> 0).
∗
x
x
α−1

Therefore,∂Ω(x∗ ) ∂Λ(x∗ )
−
< 0.
∂x∗
∂x∗
Given limx∗ →0 Ω(x∗ ) = ∞ and limx∗ →∞ Ω(x∗ ) = (1 − ψ)ρ + λθ, from equation (B.2), we
can see that the balanced growth path (BGP) exists. However, in equation (B.2), Ω(x∗ )can be a discontinuous function at q(x∗ ) = 0. To confirm whether x∗ satisfies q(x∗ ) = 0,
we describe this equation as follows:q(x∗ ) = 0 ⇔λθ+ λθψx∗ + λθ + λθψx∗α+1α− (1 − λ)(1 − α)ψξx∗ − [1 − ψ + ψ(1 − λ)(1 − α) − λ]ξx∗ = 0.

f (x∗ ) ≡ λξx∗

α−1

+

21

(B.5)

f(x*)

0x*q(x*)=0

Figure 6: f (x∗ ) when ψ = 0

From equation (B.5), we obtain the following limit value for f (x∗ ): limx∗ →0 f (x∗ ) = ∞and limx∗ →∞ f (x∗ ) = −∞. Therefore, there is at least one x∗ that satisfies q(x∗ ) = 0
from the intermediate value theorem. Diﬀerentiating f (x∗ ) with respect to x∗ in order toconfirm this uniqueness leads to∂f (x∗ )λθαα−2= − (1 + α)(1 − λ)(1 − α)ψξx∗ − (1 − α)λξx∗ − ∗2
∗
∂x
x
∗α−1
− α[1 − ψ + ψ(1 − α)(1 − λ) − λ]ξx
+ λθψ.

(B.6)

We consider f (x∗ ) < 0 to show that x∗ uniquely satisfies f (x∗ ) = 0. Then, if λθψ = 0, thatis, if the constant term is suﬃciently small, f ′ (x∗ ) < 0 holds under f (x∗ ) < 0. Therefore,
there exists a unique x∗ , which implies a unique BGP where q ∗ > 0. On the other hand,
we obtain a negative value for equation (B.6) if we assume the BBR, ψ = 0. This result
is as follows:∂f (x∗ )λθα−2α−1= −(1 − α)λξx∗ − ∗2 − α(1 − λ)ξx∗
< 0.
∗
∂x
x
ψ=0

(B.7)

Figure 6 presents function f (x∗ ) when ψ = 0 and demonstrates that a unique x∗ thatsatisfies q(x∗ ) = 0 exists. As for ψ ∈ (0, 1), we obtain the same shape as the Figure 6 if
λθψ is suﬃciently small. Then, we obtain the unique BGP in region q ∗ > 0.
In the following, we consider the unique BGP and confirm its stability. From equations(29)-(31), and equation (A.4), we partially diﬀerentiate these equations with respect toeach endogenous variable as follows:{}∂ q̇(t)[1 + b(t)]= (β + m)(ρ + θ)
+ 1 q(t),
∂q(t)
q(t)2
{ (
)
}
[λ − (1 − λ)(1 − α)b(t)]nψ∂ q̇(t)α−1= − 1 − (1 − λ)(1 − α) −αξx(t)
−
q(t),
∂x(t)
1−ψ
1−ψ
22

∂ q̇(t)(β + m)(ρ + θ) [−(1 − λ)(1 − α)ξx(t)α + λθ + n]q(t)
=−
+
,
∂d(t)
q(t)
1−ψ
∂ ẋ(t)= x(t),∂q(t){}∂ ẋ(t)
1
nd(t)
α−1
=
−[λ − (1 − λ)(1 − α)b(t)](1 − α)ξx(t)−∂x(t) 1 − ψx(t){}λθ[1 + d(t)]1α−− {[λ − (1 − λ)(1 − α)d(t)] − 1} αξx(t) ,1−ψx(t)∂ ẋ(t)[1 + x(t)]=[−(1 − λ)(1 − α)ξx(t)α + n + λθ],
∂b(t)
1−ψ
∂ ḃ(t)= b(t),∂q(t)∂ ḃ(t) [b(t) + ψ]={[λ − (1 − λ)(1 − α)b(t)]αx(t)α−1 − n}
∂x(t)
1−ψ
− [αξx(t)α−1 + n]b(t),ψ∂ ḃ(t)=−{[λ − (1 − λ)(1 − α)b(t)]ξx(t)α + n[b(t) − x(t)] + λθ[1 + b(t)]}∂b(t)(1 − ψ)b(t)b(t) + ψ+[−(1 − ψ)(1 − α)ξx(t)α + n + λθ].
1−ψFirst, we assume ψ = 0 and evaluate the abovementioned results based on the BGP.
Then, the above equations can be rewritten as[](β + m)(ρ + θ)
∂ q̇(t)=+ 1 q ∗ (> 0),2∗∂q(t) ψ=0q
∂ q̇(t)= −(1 − λ)α2 ξx∗α−1 q ∗ (< 0),∂x(t) ψ=0[]
(β + m)(ρ + θ)
∂ q̇(t)∗α=− (1 − λ)(1 − α)ξx + λθ + n q ∗ ,∂b(t) ψ=0q∗
∂ ẋ(t)= x∗ (> 0),∂q(t) ψ=0
∂ ẋ(t)1αα+1= − ∗ [λ(1 − α)ξx∗ + λθ + α(1 − λ)ξx∗ ](< 0),∂x(t) ψ=0x
∂ ẋ(t)α= (1 + x∗ )[−(1 − λ)(1 − α)ξx∗ + n + λθ],∂b(t) ψ=0∂ ḃ(t)
∂q(t)

= 0,
ψ=0

23

∂ ḃ(t)∂x(t)∂ ḃ(t)
∂b(t)

= 0,
ψ=0

= −γk∗ .
ψ=0Then, we define Jacobi matrix J as
∂ q̇(t)
∂q(t)


ψ=0
 ∂ ẋ(t)J =  ∂q(t)ψ=0 ∂ b(t)
˙
∂q(t)

ψ=0

∂ q̇(t)∂x(t)∂ ẋ(t)∂x(t)∂ ḃ(t)
∂x(t)

ψ=0
ψ=0
ψ=0

∂ q̇(t)
∂b(t)





.
ψ=0 

∂ ḃ(t)ψ=0∂ ẋ(t)
∂b(t)
∂b(t)

ψ=0

We denote the eigenvalues of this matrix J as X. Given the above results, we can obtainthe sign of eigenvalues of J:

 [(
) (
)
]
∂
ḃ(t)
∂
q̇(t)
∂
ẋ(t)
∂
q̇(t)
∂
ẋ(t)
X −
· X −
· X−
−
·
= 0.
∂b(t)
∂q(t) ψ=0
∂x(t) ψ=0
∂x(t) ψ=0 ∂q(t) ψ=0
ψ=0This shows that one of the eigenvalues is negative: X1 = −γk∗ < 0. Then, computing adeterminant of J, we obtain)(∂ ẋ(t)∂ q̇(t)∂ ẋ(t)∂ q̇(t)∂ ḃ(t)
·
−
·
,
·
|J| =
∂b(t)
∂q(t) ψ=0 ∂x(t) ψ=0 ∂x(t) ψ=0 ∂q(t) ψ=0
ψ=0

where
∂ q̇(t)∂ ẋ(t)∂ q̇(t)∂ ẋ(t)·−·∂q(t) ψ=0 ∂x(t) ψ=0 ∂x(t) ψ=0 ∂q(t) ψ=0β(ρ + θ)αα+1[λ(1 − α)ξx∗ + (1 − λ)αξx∗ + λθ]
∗
∗
xq
}
q∗ {
α+1
− ∗ λθ + [1 − α(1 − λ)]αξx∗(< 0).
x=−

Since |J| > 0, it is clear that the product of the three eigenvalues is positive. Therefore, theproduct of the remaining two eigenvalues is negative since one of eigenvalues is negative.
Thus, the BGP is saddle-path stable for ψ = 0.
When ψ ∈ (0, 1), it is diﬃcult to analytically prove the stability of BGP. Following
previous studies, we numerically compute the eigenvalues. Table 3 shows that there exists
two negative eigenvalues and one positive eigenvalue under some plausible parameters. As
per the numerical calculation, the BGP can also be saddle-path stable when ψ ∈ (0, 1).

24

Table 3: Sign of eigenvalues

For n = 0 (β = θ = 0.05), ξ = 0.2
ρ = 0.03, λ = 0.3, m = 0.01
α = 0.2 α = 0.3 α = 0.4
0.1368 0.1383 0.1405
ψ = 0.3 -0.0448 -0.0442 -0.0435-0.1406 -0.1609 -0.1901
0.1310 0.1300 0.1286
ψ = 0.5 -0.0583 -0.0617 -0.0629-0.2409 -0.2663 -0.3095
0.1261 0.1223 0.1166
ψ = 0.7 -0.0996 -0.1186 -0.1317-0.4781 -0.5083 -0.5725
0.1219 0.1157 0.1065
ψ = 0.9 -0.3274 -0.4422 -0.5440-1.6724 -1.7119 -1.8501

For n = 0.01 (β = 0.06, θ = 0.05)
ξ = 0.25, ρ = 0.03, λ = 0.3, m = 0.01
α = 0.2 α = 0.3 α = 0.4
0.1452 0.1459
0.1470ψ = 0.3 -0.0590 -0.0554 -0.0520-0.1812 -0.1955 -0.2197
0.1383 0.1359
0.1324
ψ = 0.5 -0.0707 -0.0711 -0.0687-0.3036 -0.3211 -0.3579
0.1316 0.1254
0.1157
ψ = 0.7 -0.1085 -0.1240 -0.1316-0.5948 -0.6149 -0.6726
0.1250 0.1149
0.0997
ψ = 0.9 -0.3232 -0.4350 -0.5265-2.0645 -2.0866 -2.2239

For n = 0.02 (β = 0.07, θ = 0.05)
ξ = 0.25, ρ = 0.03, λ = 0.3, m = 0.01
α = 0.2 α = 0.3 α = 0.4
0.1498 0.1483
0.1467
ψ = 0.3 -0.0456 -0.0414 -0.0370-0.1773 -0.1905 -0.2129
0.1404 0.1346
0.1258
ψ = 0.5 -0.0511 -0.0484 -0.0415-0.2980 -0.3143 -0.3488
0.1309 0.1190
0.0990ψ = 0.7 -0.0753 -0.0822 -0.0768-0.5892 -0.6094 -0.6658
0.1204 0.1018
0.0726
ψ = 0.9 -0.2279 -0.3101 -0.3646-2.0674 -2.0998 -2.2457

25