ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E15-4

Public Capital Accumulation, Income Transfers,
and Fiscal Sustainability
byHikaru Ogawa
Akira Kamiguchi

May 2015

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Public Capital Accumulation, Income
Transfers, and Fiscal Sustainability¤
Hikaru Ogaway

Akira Kamiguchiz

Nagoya University

Nagoya University

Abstract
In this study, we present an endogenous growth model comprisingpublic investment and income transfer, with focus on ¯scal sustainability.
The main ¯nding of this paper is that the e®ects of an increase in the
public expenditure/GDP ratio on ¯scal sustainability evidently dependon how expenditures function. An increase in the income transfer/GDP
ratio may create a more sustainable economy, whereas, as previous studies
have shown, an increase in the public investment/GDP ratio creates a lesssustainable economy. The growth e®ects of changes in the two ratios arealso examined.

Keywords: Public capital accumulation; income transfers; debt sustainability
JEL classi¯cation: E62, H40, H63, O41¤ The research has been supported by grants from the Japan Society for the Promotion of
Science (25516007 and 15H03366).
y School of Economics, Nagoya University, Furocho, Chikusaku, Nagoya 4648601, Japan.
Email. ogawa@soec.nagoya-u.ac.jp
z School of Economics, Nagoya University, Furocho, Chikusaku, Nagoya 4648601, Japan.
Email. kamiguchi.akira@h.mbox.nagoya-u.ac.jp1

1

Introduction

The e®ect of government policy on ¯scal sustainability is an age-old topic; it not
only fascinates academic curiosity, but also attracts the attention of policymakers who cope with serious ¯scal sustainability problems. The classic analysis of
Domar (1944) presents the importance of economic growth for public debt accumulation sustainability, and Bohn (1998) proposes a new condition for ¯scal
sustainability associated with the primary surplus and debt-GDP ratios. Recent
theoretical studies have analyzed the e®ects of government policy on ¯scal sustainability by introducing practical policy rules. Using the model of exogenous
growth, Chalk (2000) and Rankin and Ro±a (2003) derive a ¯nite maximum
sustainable level of debt. Departing from the exogenous growth model and using an endogenous growth model, BrÄauninger (2005), Yakita (2008), and Arai(2011) also investigate the conditions for ¯scal sustainability, in which the government is supposed to spend a given pro! portion of GDP and to borrow a ¯xed
fraction of GDP. 1 The sustainability analysis under some exogenous policy rules
works for ¯nding out the e®ects of practical policies, such as Maastricht treaty
in EU and the roof constraint on public expenditure and °otation of government
bond introduced in countries being troubled over excessive public debt.
Our paper also concerns with the relationship between government policyand ¯scal sustainability in the endogenous growth model. Our paper, however, di®ers from precedent studies in breakdown of public spending and aims
to open up the option for government's policy instruments. While BrÄ
auninger(2005) assumes that government expenditures virtually play no role in the economy, Yakita (2008) and Arai (2011) study the ¯scal sustainability in a modelwith productivity-enhancing public expenditure. A critical di®erence between
Yakita (2008) and Arai (2011) is that the former assumes the investment inpublic capital formation based on a model of Futagami et al. (1993), and thelatter assumes the public expenditure based on Barro (1990)'s model in which
the public expenditure is treated as a °ow variable. Departing from the simple
setting of productivity-enhancing public expenditure, we incorporate the spending for income transfer policy. To ¯gure out the essence of our main argument,
we simply assume that the government allocates its ¯nancial resources between
productivity-enhancing investment and income transfer. The income transfer
1 Some other studies on the e®ects of debt policy on growth and welfare should be mentioned. For instance, Futagami et al. (2008) use an endogenous growth model with productive
public spending to prove that two balanced growth paths exist, one saddle-point stable, andthe other saddle-point stable, or asymptotically stable. They then show that a de¯cit-¯nanced
increase in public spending impacts the two balanced growth rates oppositely. Greiner (2008)
also uses an endogenous growth model with public capital to study the growth and welfaree®ects of debt policies. The critical di®erence of the two papers is the policy rule. Futagami etal. assume that government debt converges to a certain exogenously given debt ratio, whereas
Greiner (2008) follows Bohn's (1998) rule: debt policy is managed under the rule whereby the
primary surplus moves positively on public debt, which is justi¯ed by some empirical studies[Bohn (1998) and Greiner et al. (2007)]. Ghosh and Mourmouras (2004) analyze an economy in which the government provides productive government spending in a debt-¯nanced
economy from a welfare perspective. They show that the government's less strict budgetarystance reduces the welfare level.

2policies, in the form of social aid, child and social bene¯t, and social security
paycheck, account for a measurable share of government budget in modern welfare states. What the precedent studies pass over is the e®ects on the long-run¯scal sustainability of increase in the share of social welfare and the e®ects ofallocation between public investment and direct income transfers to households.
The aim of this paper is thus to redeem and generalize the work of BrÄ
auninger(2005), Yakita (2008), and Arai (2011) by incorporating an alternative policy
option of governments.
This extension can be evaluated from several standpoints. First, empiricalresearchers wishing to test the ¯scal sustainability desire to use model which
accounts not only productivity-enhancing policies but the policies with a highregard for income support. The results would be biased more than a bit if
we count out the e®ects of income redistribution policy in the modern welfare
states. This paper o®ers a simple but tractable model to distinguish the roleof public investment and income transfers. Second, researchers in applied workneed a model that clearly distinguishes two types of public expenditures since
the model with income transfers would be easily applied to various issues on
income redistribution. Although we do not explicitly formulate public pension,
welfare payments, and health-care expenditures, the income transfers policy
modeled in this paper explores an aspect of these practical policies.
The main argument of our paper is that the e®ects of increase in public
expenditure/GDP ratio on the ¯scal sustainability evidently depend on how expenditures work. Our analysis follows the capital accumulation model of Yakita(2008), which shows that the increase in expenditure/GDP ratio impacts negatively on the ¯scal sustainability since it increases the marginal productivityof private capital and thus induces interest-rate runup. An increase in interest rates is conductive to further dependence on government bond for interest
payment. From this standpoint, the decrease in public expenditure/GDP ratio is required for ¯scal sustainability. In contrast, in our model, the publicresources are spent not only for public capital accumulation but for income
transfers, which increases the private saving through income e®ect, and therefore it reduces the interest rate eventually. An interest-rate reduction works
positively on maintaining ¯scal sustainability. In fact, our analysis reveals thatthe decrease in public expenditure/GDP ratio has the negative e®ects on ¯scal
sustainability if public expenditures are used for something that increases theinterest rate, but it works positively on ¯scal sustainability if the expenditures
are made available for decreasing the interest rate.
Although the policy of income transfers to young generation is incorporated
in our study, we should not take income transfers to young in a restricted sense.
The point is to spare a thought of public spending that leads to encourage
younger generation's willingness to save, which results in a reduction in interest
rate. For instance, various policies for savings promotion, such as governmentled advantageous interest rate o®ers and preferred charge for service for young,fall into our argument. The employment-generating projects for young generation are other possible case, which will contribute enhancing savings of young
in a broad sense. Suppose that there are employed and unemployed workers in3

young generation. If unemployed succeeds in getting away from unemployment
pool, they are now able to save, which increase the total saving in the young
generation.2 After all, we can qualify public spending that enhance young generation to save as expenditure item which we add in our model.
The paper is organized as follows. Section 2 presents the model, and section
3 derives the sustainability condition. Section 4 is devoted to examining how
the changes in policy rules e®ect on the ¯scal sustainability. After con¯rming
the result of Yakita (2008), i.e., the increase in the ratio between public expenditure (investment) and GDP has negative e®ects on ¯scal sustainability, thissection shows that the increase in the ratio between public expenditure as aincome transfer and GDP might have positive impacts on ¯scal sustainability ifhouseholds have a low discount factor and/or the debt ¯nancing ratio is su±ciently small. The e®ects of changes in policy rule on the long-run growth rate
are discussed in section 5. Section 6 concludes the paper.

2

The Model

We follow Yakita (2008) and use a standard overlapping generations model,
which consists of ¯rms, a government, and households. Public capital acts as
the growth engine of this economy.

2.1

Households

Individuals live for two periods in our overlapping generations economy. Eachindividual supplies one unit of labor in the ¯rst period, and retires in the second
period. We assume that the population of each generation is constant over time.
The utility function of each individual belonging to generation t is given by
ut = (1 ¡ ±) ln cyt + ± ln cot+1 , where cyt and cot+1 represent the consumption in theyoung and old periods, respectively, and ± 2 (0; 1) denotes the time preference
rate. An individual in the young period allocates the sum of his/her disposable
income and a lump-sum transfer to consumption, cyt , and saving st . In theretirement period, the individual consumes using the net returns of savings.
The budget constraints of the individual in the ¯rst and second periods are
given by cyt + st = (1 ¡ ¿t )wt + pt and cot+1 = [1 + (1 ¡ ¿t+1 )rt+1 ]st , respectively,
where pt denotes the lump-sum income tran! sfer from the government. Fromthese equations, we derive the lifetime budget constraint:

cyt +

cot+1
= (1 ¡ ¿t )wt + pt :1 + (1 ¡ ¿t+1 )rt+1(1)

2 This argument con°ict with Yakita (2014, p.89)'s argument. In his model, all public
expenditures are ¯nanc! ed by debt issuance only. In this case, to ¯nance the lump-sumtransfers, the government must increase newly issued public bonds, which crowds out private
capital. This results in an increase the return rate to capital.

4

From the ¯rst-order conditions for utility maximization, the demand for consumption in the young and old periods and for savings are respectively derived
as
cyt

2.2

cot+1

=
=

st

=

(1 ¡ ±)[(1 ¡ ¿t )wt + pt ];±[1 + (1 ¡ ¿t+1 )rt+1 ][(1 ¡ ¿t )wt + pt ];±[(1 ¡ ¿t )wt + pt ]:

(2)
(3)
(4)

Firms

A representative ¯rm uses physical capital Kt and labor Lt to produce homogeneous goods. The production function is assumed as Yt = AKt® (Et Lt )1¡® ,where Et denotes labor e±ciency. To simplify the notation, we assume A = 1
in the following analysis. From the pro¯t-maximizing conditions of the ¯rm incompetitive markets, the interest rate and wage rate are respectively given as
rt = @Yt =@Kt = ®Yt =Kt and wt = @Yt =@Lt = (1 ¡ ®)Yt =Lt . We assume thatlabor e±ciency equals public capital stock per unit of labor, Et ´ Gt =Lt . Using
this assumption, we yield the aggregate production function as Yt = Kt® G1¡®
.
t
Assuming that factor markets are competitive, the interest rate, rt , and wage
rate for labor, wt , equal the marginal productivity:

rt
wt

2.3

= ®

µ

Gt
Kt

¶1¡®

= (1 ¡ ®)

µ

Gt
Kt

=®
¶¡®

Yt
;
Kt

(5)

Gt
Yt
= (1 ¡ ®) :
Lt
Lt

(6)

Government

The government levies a tax on income. We assume that the government spends
a given proportion µ 2 (0; 1) of GDP, µYt , on public investment, which contributes to public capital accumulation. The evolution of public capital is, thus,
speci¯ed as
Gt+1 = Gt + µYt :(7)

In addition to public investment, the government provides a lump-sum income
transfer to households. The income transfer amount, Pt , is constrained by
Pt = ´Yt , implying that the government transfers income at a constant fractionof GDP, ´ 2 (0; 1). We assume that µ + ´ < 1.
The government ¯nances a proportion of the expenditure, ¸ 2 (0; 1), by
issuing bonds, ¸(µ + ´)Yt . Thus, the dynamic equation of public debt, Dt , isgiven by5

Dt+1 = Dt + ¸(µ + ´)Yt :(8)

The government ¯nances its budget through public debt and tax revenues, andspends on interest payments of public debt, public capital investment, and lumpsum transfers to households:(Dt+1 ¡ Dt ) + ¿t (wt + rt st¡1 )N = rt Dt + (Gt+1 ¡ Gt ) + ´Yt ;(9)where N denotes the (constant) total population, ¿t is the income tax rate
in period t. Substituting (7) and (8) into (9), the budget constraint of thegovernment is rewritten as¸(µ + ´)Yt + ¿t (wt + rt st¡1 )N = rt Dt + (µ + ´)Yt :(10)

Here, using the individuals' budget constraints for the young and old periods
and the government budget constraint (10), we obtain the resource constraint
at period t as
Yt = cyt N + cot N + Gt+1 ¡ Gt + Kt+1 ¡ Kt ;where cyt and cot represent the consumption of young and old individuals at
period t, respectively.

2.4

Dynamics

The capital market in equilibrium satis¯es
Dt+1 + Kt+1 = st Lt :(11)

Using (4) and (6), (11) is rewritten as
Kt+1 = ±[(1 ¡ ¿t )(1 ¡ ®)Yt + Pt ] ¡ Dt+1 :(12)

Dividing both sides of (12) by Kt and using (8) and Pt = ´Yt , we obtain theevolution of private capital as
Kt+1= [±(1 ¡ ¿t )(1 ¡ ®) + ±´ ¡ ¸(µ + ´)]
Kt

µ

Gt
Kt

¶1¡®

¡

Dt
:
Kt

(13)

The evolution of public capital is derived by dividing both sides of (7) by Gt :Gt+1
Yt
= 1+µ :
Gt
Gt
6

(14)

The evolution of public debt is also obtained by dividing both sides of (8) by
Dt :
Dt+1
Yt
= 1 + ¸(µ + ´) :
Dt
Dt

(15)

Using (5), (6), and (10), we obtain the relationship between the tax rate andthe public debt/private capital ratio as
1 ¡ ¿t =1 ¡ (1 ¡ ¸)(µ + ´):
1 + ®(Dt =Kt )(16)

De¯ning gt ´ Gt =Kt and zt ´ Dt =Kt , and using (13)-(16), we depict the
dynamics of the economy using two dynamic equations:
gt+1
1 + µgt¡®
=;
gt³t (zt )gt1¡® ¡ zt

(17)

g 1¡®

1 + ¸(µ + ´) tzt
zt+1
=
;
zt
³t (zt )gt1¡® ¡ zt

(18)

where
³t (zt ) ´±(1 ¡ ®)[1 ¡ (1 ¡ ¸)(µ + ´)]+ ´(± ¡ ¸) ¡ ¸µ:
1 + ®zt

In the following analysis, to consider reasonable equilibria, we assume that
³t (zt ) > 0.
On the balanced growth path, we have gt+1 = gt and zt+1 = zt . From (17),(18), gt+1 = gt and zt+1 = zt , at the steady state, we have

gt =

3

µ
zt :
¸(µ + ´)

(19)

Debt Sustainability Condition

The steady-state equilibrium satis¯es (17), (18), gt+1 = gt and zt+1 = zt . Fora graphical representation of the steady-state equilibrium, we ¯rst use (17) andgt+1 = gt to obtain
1 + µgt¡® + zt = ³t (zt )gt1¡® :(20)

We de¯ne the left-hand side of (20) as ¯(gt ; zt ), and the right-hand side as"(gt ; zt ). Fig.1 depicts ¯(gt ; zt ) and "(gt ; zt ) for a given size of the public capital/private capital ratio, gt , which shows that the two curves labeled as ¯(gt ; zt )7and ²(gt ; zt ), intersect each other, and thus (20) is satis¯ed at zt = z1t . As the
public capital/private capital ratio increases to gt0 , the "(¢) curve shifts up andthe ¯(¢) curve shifts down, indicating that the debt/private capital ratio that0satis¯es (20) increases from z1t to z1t. Hence, equation (20) can be depictedusing the upward-sloping curve represented by GG in the gt ¡ zt plane in Fig.3.
This is formally con¯rmed using (20) to derive (dgt =dzt )jGG as follows.3
dgt
1 ¡ ³t0 (zt )gt1¡®
jGG =
> 0;
¡(1+®)
dzt[(1 ¡ ®)gt ³t (zt ) + ®µ]gt(21)

where ³t0 (zt ) ´ ¡®±(1 ¡ ®)[1 ¡ (1 ¡ ¸)(µ + ´)]=(1 + ®zt )2 < 0.
In a similar manner, using (18) and zt+1 = zt , we have
1 + ¸(µ + ´)gt1¡® zt¡1 + zt = ³t (zt )gt1¡® :(22)

De¯ning the left-hand side of (22) as Â(gt ; zt ), and because the right-hand sideof (22) is identical to "(gt ; zt ) de¯ned in (20), we can depict the two functions
as in Fig.2, which shows that there are two intersections. We denote z1t and z2t
as the public debt/private capital ratio levels that satisfy (22).4
An increase in the public capital/private capital ratio to gt0 shifts the Â(¢)
and "(¢) curves upward, with the shift of the "(¢) curve greater than that of theÂ(¢) curve.5 Thus, an increase in gt causes a reduction in the level of z1t andan increase in the level of z2t . Therefore, we can depict the ZZ curve, which
represents (22), on the gt ¡ zt plane in Fig.3 as a U-shaped ZZ curve.
Next, we con¯rm the slope of the curve ZZ. From (22), we have

dgt
jZZ =
dzt

h

®±(1¡®)(1¡(1¡¸)(µ+´))¡ ¸(µ+´)(1+®zt )2
zt2

i

gt1¡® + 1hi
? 0:(1 ¡ ®) ³t (zt ) ¡ ¸(µ+´)
gt¡®
zt

(23)

The sign of (23) depends on the sign of the numerator, because the sign of thedenominator is positive.6 The numerator in (23) tends to have a negative signwhen zt is small, but it is likely to have a positive sign when zt is large, implyingthat the ZZ curve has features of a U-shaped curve.
The dynamic equations on gt and zt are depicted in a phase diagram of Fig.3.
The intersections of the GG curve and ZZ curve in these ¯gures depict thesteady-state values of gt and zt . In Fig.3, there are two intersections, in which
3zt and gt have an upper and lower limit, respectively. See Appendix A.
4 To ensure at least one intersection, we must de¯ne the lower limit of g .
tÂt ! 1 + zt as
gt ! 0, and the interception of Â(gt ; zt ) as 1 when zt = 0. Hence, an interception of "(¢)
must be greater than 1 to ensure an intersection of the Â(¢) and "(¢) curves, indicating that
g 1¡®

g

the lower limit =t must satisfy 1 < [±(1 ¡ ®)(1 ¡ (1 ¡ ¸)(µ + ´)) + ´(± ¡ ¸) ¡ ¸µ] =t.
5 See Appendix B.
6 Equation (22) can be written as g 1¡® = z (1 + z )[z ³ (z ) ¡ ¸(µ + ´)]¡1 > 0. Hence, the
t
t
t t t
t
sign of the denominator in (23) is positive.

8

point A is a sink and point B is a saddle point (see Appendix C).7 The economy
with the initial public capital/private capital ratio g0 and public debt/private
capital ratio z0 starting from the point below the level of the stable arm, theeconomy converges to the equilibrium A. The economy with (g0 ; z0 ) startingfrom the point on the stable arm, the economy converges to the equilibrium
B. In contrast, the economy with (g0 ; z0 ) starting from the point higher thanthe level of the stable arm which converges to the saddle-point steady state, theeconomy cannot converge to the steady states and the economy is unsustainable
relative to public debt. Thus, some policy rule that sifts zB to the leftward andnarrows the range between zA and zB , such policies create an unsustainableeconomy.

4

Policy E®ects

We analyze the interaction between policy variable and debt sustainability.
First, we clarify how the GG and ZZ curves shift when the policy capturedby the variable ¸ changes. This a±rms the results presented by Yakita (2008).
From (20) and (22), we have

dgt
jGG
d¸

=

dgt
jZZ
d¸

=

h
i
1 ¡ ±(1¡®)(µ + ´)gt2
1+®zt

> 0;
(1 ¡ ®)gt ³t (zt ) + ®µhi
1
1 ¡ ±(1¡®)
1+®zt + zt (µ + ´)gt
hi > 0;(1 ¡ ®) ³t (zt ) ¡ ¸(µ+´)
zt

(24)

(25)

indicating that the GG and ZZ curves in Fig.3 shift upwards. Then, from (24)and (25), we ¯nd that the magnitude of the shift of the ZZ curve is greaterthan that of the GG curve;³´±(1¡®)
µ(µ
+
´)
1
¡
+ µ+´
1+®zt
zt [(1 ¡ ®)gt ³t (zt ) + ®µ]
dgt
dgt
h
i
jZZ ¡
jGG =
> 0:
(1¡®)
d¸
d¸
³t (zt ) ¡ ¸(µ+´)[(1 ¡ ®)gt ³t (zt ) + ®µ]gt
zt

Hence, an increase in ¸ is described by the shift of the GG and ZZ curves to
G0 G0 and Z 0 Z 0 , which shifts the upward-sloping stable arm which depicted as
the dotted curve leftward, as in Fig.3, implying that the economy is becoming
unsustainable. This is because if the economy with (g0 ; z0 ) starts from thepoint larger than the level of the stable arm that converges to the saddle-point
equilibrium B in Fig.3, the economy cannot converge to the steady states, andhence, the economy is not sustainable relative to public debt.
7 There is an exceptional case whereby only one equilibrium exists in the economy even if
both (18) and zt = zt+1 are satis¯ed.

9We now focus on the e®ects of a change in the ratio of spending for income
transfers, ´. The shifts of the GG and ZZ curves with a change in ´ can bechecked with the following equations:

dgt
jGG
d´

=

dgt
jZZ
d´

=

h
i
¡ ± ¡ ¸ ¡ ±(1¡®)(1¡¸)
gt2
1+®zt

7 0;
(1 ¡ ®)gt ³t (zt ) + ®µhi¸
¡ ± ¡ ¸ ¡ ±(1¡®)(1¡¸)
¡
1+®zt
zt gt
h
i
7 0:
(1 ¡ ®) ³t (zt ) ¡ ¸(µ+´)
zt

(26)

(27)

Furthermore, from (26) and (27), we havehihi
¡ ± ¡ ¸ ¡ ±(1¡®)(1¡¸)®µ + ¸(µ+´)(1¡®)
1+®zt
zt
dgt
dgt
h
i
jZZ ¡
jGG =
7 0:
(1¡®)
d´
d´
³ (z ) ¡ ¸(µ+´) [(1 ¡ ®)g ³ (z ) + ®µ]
t

gt

t

t t

zt

t

The levels of ± and ¸ are critical for determining how the GG and ZZ curves
shift with an increase in ´. We use three cases to study the e®ects of an increasein ´ on ¯scal sustainability.8
Case 1; Figure 4(a). ± <¸
1¡(1¡®)(1¡¸)
1+®zt. In this case, the GG and ZZ curves

shift upward and the magnitude of the shift in the ZZ curve is larger thanthat in the GG curve, indicating that an increase in ´ renders the economy
more unsustainable.
Case 2; Figure 4(b).

¸(1¡®)(1¡¸)
1¡ 1+®zt< ± <¸(1+ z1 )1¡t(1¡®)(1¡¸)
1+®zt. In this case, the GG

curve shifts downward and the ZZ curve shifts upward, directly making
the economy more unsustainable.
Case 3; Figure 4(c).

¸(1+ z1 )t1¡ (1¡®)(1¡¸)
1+®z< ±. In this case, GG curve and ZZ curvetshift downward, and the magnitude of the shift in the ZZ curve is larger
than that in the GG curve, indicating that an increase in ´ renders the
economy more sustainable.
In Cases 1 and 2, the increase in income transfer narrows the range between
z1t and z2t , and thus, such an income transfer policy renders the ¯scal budget
unsustainable. In a direction opposite to these cases, in Case 3, an increase in
spending for income transfers renders the ¯scal budgets sustainable. Based on
this arrangement, we conclude that if ± is su±ciently large and ¸ is su±ciently¸(1+ z1 )t
low to satisfy< ±, an increase in ´ leads to a more sustainable1¡ (1¡®)(1¡¸)
1+®zt
economy.
8 We exclude the case where the GG curve shifts upward and the ZZ curve shifts downward
because this never happens as ¸ < ¸(1 + zt¡1 ) holds for any given value of zt .

10

The intuition behind this result is simple. An increase in the income transfer
ratio ´ promotes further debt accumulation, which directly leads to an unsustainable economy. In contrast, an increase in ´ increases private savings throughthe income e®ect. When ± is su±ciently large, households put a high weight on
the consumption in the second period, and hence, from (4), private savings increase. Thus, an increase in ´ stimulates private savings in Case 3. The increasein savings lowers the interest rate in the capital market, which contributes to a
reduction in interest payments. In Case 3, the latter e®ect exceeds the former,
and thus the economy becomes more sustainable.

5E®ects on the Growth Rate

To study the e®ects of changes in ¸ and ´ on the economic growth rate, we ¯rstuse (13) to derive the growth rate in period t, 1 + ° = ³t (zt )gt1¡® ¡ zt . The
comparative statistics on ¸ are provided byd°®(µ + ´)gt¡®
=
d¸
H

½
·
¸¾
±(1 ¡ ®)
1 ¡ ³t0 (zt ) + µgt¡® 1 ¡;
1 + ®zt

(28)

where
¡(1+®)H ´ ¸(µ + ´)µ¡1 [1 ¡ ³t0 (zt )gt1¡® ] ¡ gt[(1 ¡ ®)gt ³t (zt ) + ®µ]:

(29)

Since the numerator in (28) is positive, the sign of (28) is determined by thesign of H. Because the slope of the GG curve at equilibrium B in Figs.4(a)(c) is steeper than ¸(µ + ´)=µ and gradual at equilibrium A, the sign of H ispositive at equilibrium B and negative at equilibrium A.9 We further ¯nd thatd°=d¸ > 0 holds at equilibrium B and d°=d¸ < 0 holds at equilibrium A, which
replicates the result of Yakita (2008).
The e®ects of an increase in income transfer ratio on the growth rate canalso be obtained as follows.
d°
®gt¡®
=
d´
H

½

¸[1 ¡ ³t0 (zt )gt1¡® ] ¡·¸¾µ±(1 ¡ ®)(1 ¡ ¸)
±¡
¡¸ :
gt®
1 + ®zt

H is positive at equilibrium B and negative at equilibrium A. The ¯rst term in
the angle bracket is positive. Hence, an increase in ´ increases the growth rateover the range of(1 + zt )®±· ¸;
1 + ®zt + ±(1 ¡ ®)at point B and reduces that at point A in Figures 4(a)-4(c).
9 See Appendix D.

11

(30)

The policy e®ects on the growth rate can be explained as follows. Let us focus
on the case of point B in Figures 4(a)-4(c), and thus on H > 0. A small increasein the income transfer, represented by ´, increases an individual's incentive tosave through the income e®ects. In contrast, an increase in ´ raises the tax rate,
given other variables [see (16)], which constitutes a limiting factor of individual
savings. The former e®ect tends to exceed the latter when the government
operates a bond-dependent budget, in other words, when ¸ is large. This isbecause the negative e®ects of tax increases on individual savings are relatively
small when the government heavily relies on debt-¯nancing and, thereby, the tax
rate is low. Thus, when ¸ is su±ciently large to satisfy (30), the positive e®ectof an increase in ´ on savings exceeds the negative e®ect, and an increase in
the transfer policy increases the savings and theref! ore accelerates the capital
accumulation and growth rate. In contrast, when the savings are contained
through a tax increase associated with an increase in spending for an income
transfer, an increase in ´ reduces the growth rate.

6

Conclusion

In this paper, we formulated an endogenous growth model comprising public
capital investment and income transfer, with focus on ¯scal sustainability. Whilepublic expenditure has had limited use on productivity-enhancing investmentin previous studies, the main focus of this paper is the e®ect of an expansionof public expenditures for income transfers on ¯scal sustainability. We ¯ndthat an increase in public expenditure for income transfers does not necessarily
create an unsustainable environment because income transfers increase private
savings and thereby reduce the interest rate, which mitigates interest payments.
A noteworthy feature of this paper is that in describing public expenditure, weformally incorporate two types of public expenditures, enabling us to examine
the various e®ects of expansion of public expenditure on ¯scal sustainability andlong-run economic growth.
The key to our argument is the public expenditure that promotes savingsin young generation. Although we take income transfers to young generation
for simple example to derive our main ¯nding, we should not restrict the policy
instrument in a narrow sense. Any public expenditure that enhances young
generation's savings fall into our model, and thus, an increase in the public expenditure/GDP ratio may create a more sustainable economy relative to public
debt.
Before concluding this paper, we discuss some problems that remain to besolved. First, because our main interest is to present a simple model examining the e®ects of public expenditure on ¯scal sustainability, we assume that allpolicy variables, µ, ´, and ¸ are exogenous: the current model considers neither
objective of the government. Incorporation of endogenous policy choices could
provide insightful information related to the implications of the optimal policies adopted. Second, a thorough empirical examination can be conducted on
the basis of this model. An important research topic from both academic and12practical perspectives would be to test how changes in public expenditure levela®ect the sustainability and growth paths. These are beyond of the scope of
this study and remain unresolved issues.

Appendices
Appendix A. Upper and Lower Limits of zt and gt .
Because the right-hand side of (21) is positive, zt has an upper limitz=±(1 ¡ ®)[1 ¡ (1 ¡ ¸)(µ + ´)] + ´(± ¡ ¸) ¡ ¸µ;®[¸µ ¡ ´(± ¡ ¸)]

whereas gt has a lower limit g when zt = 0, which satis¯es
1 + µg ¡® = [±(1 ¡ ®)(1 ¡ (1 ¡ ¸)(µ + ´)) + ´(± ¡ ¸) ¡ ¸µ]g 1¡® :Appendix B. Proof of dÂt =dgt < d"t =dgt .
The left- and right-hand sides of (22) are denoted by Ât and "t , respectively. Di®erentiation of Ât and "t with respect to gt provides
dÂt gtdgt 1 ¡ ®
d"t gtdgt 1 ¡ ®

=g 1¡®
¸(µ + ´) t
= Ât ¡ (1 + zt );zt=³t (zt )gt1¡® = "t :Using these equations, and evaluating at Ât = "t which holds under (22),
we have
gt
1¡®

µ

dÂt
d"t
¡
dgt
dgt

¶

= ¡(1 + zt ) < 0:Appendix C. Stability.
The linearized system of (17) and (18) around the stationary equilibrium
is
µ
¶ µ
¶µ
¶
gt+1 ¡ g¹@gt+1 =@gt @gt+1 =@zt
gt ¡ g¹=;
zt+1 ¡ z¹@zt+1 =@gt @zt+1 =@zt
zt ¡ z¹
where
gt+1

gt+1

zt+1

gt @ g t
gt @ gt
zt @ zt
@gt+1
@gt+1
@zt+1
=
+ 1;
=
;
=
;
@gt
@gt
@zt
@zt
@gt
@gt
zt @ zt+1
@zt+1
zt
and
=
+ 1:
@zt
@zt
13

These values are evaluated at the steady state. The characteristic polynomial is
µ
¶
µ
¶
@gt+1
@zt+1
@gt+1 @zt+1@gt+1 @zt+1
2
P (¹) = ¹ ¡
+
¹+
¡
:
(A.1)
@gt
@zt
@gt @zt@zt @gt
To analyze the local stability of each equilibrium, we con¯rm the sign ofP (¹) when ¹ = 0; 1. The values of P (0) and P (1) are
P (0) ==
P (1) ==

@gt+1 @zt+1@gt+1 @zt+1¡;@gt @zt@zt @gt(A.2)®±(1 ¡ ®) ®±(1¡®)(1¡(1¡¸)(µ+´))
gt1¡® zt + (® + (1 ¡ ®)zt )(1 + zt )(1+®zt )2(1 + µgt¡® )2
µ
¶ µ
¶
@gt+1
@zt+1
@gt+1 @zt+1@gt+1 @zt+1
1¡
+
+
¡
;
@gt
@zt
@gt @zt@zt @gtÃ gt+1 zt+1!
zt+1
@ gt @ zt@ gt+1
gt @ zt
gt zt
¡
? 0;
@gt @zt@zt @gt

where@ gt+1gt@gt@ gt+1gt@zt@ zt+1zt@gt@ zt+1
zt
@zt

=
=

=

=

¡

®µgt¡®¡1 + (1 ¡ ®)gt¡® ³t (zt )< 0;
1 + µgt¡®

1 + gt1¡® ®±(1¡®)(1¡(1¡¸)(µ+´))(1+®zt )2¡

1 + µgt¡®
hi(1 ¡ ®)gt¡® ³t (zt ) ¡ ¸(µ+´)
zt

gt1¡®

h

> 0;

< 0;

1 + µgt¡®

®±(1¡®)(1¡(1¡¸)(µ+´))¡ ¸(µ+´)(1+®zt )2
zt2¡®
1 + µgt

i

+1

? 0:

The sign of the last equation is positive (negative) when the sign of P (1)is negative (positive), implying thath both an unstable and a stablei equilib1¡® ®±(1¡®)(1¡(1¡¸)(µ+´))
rium exist. Since zt is large, gt¡ ¸(µ+´)+1 > 0(1+®zt )2
zt2
holds at equilibrium B. Further, as the slope of the ZZ curve is steeper
dgttthan that of the GG curve, dg
dzt jZZ > dzt jGG holds at equilibrium B. We
rewrite the slope of the ZZ and the GG curves as
@

zt+1
z

@

t
dgt
t
jZZ = ¡ @z
zt+1 ;
@ z
dzt
t

gt+1
g

t
dgt
t
jGG = ¡ @z
gt+1 :
@ g
dzt
t

@gt

@gt

dgt
t
Since dg
dzt jZZ > dzt jGG holds at equilibrium B, we derive
zt+1@ gt+1
gt @ zt

@gt

@zt
14

>

zt+1
@ gt+1g t @ zt

@zt

@gt

:

> 0;

(A.3)

Now, at equilibrium B, P (1) < 0 holds from (A.3). Thus, the values ofthe eigenvalues (¹i ) are 0 < ¹1 < 1 < ¹2 . Therefore, equilibrium B is asaddle point.
hi¸(µ+´)
By contrast, gt1¡® ®±(1¡®)(1¡(1¡¸)(µ+´))¡+ 1 < 0 holds at equi22(1+®zt )ztlibrium A, since zt is small at equilibrium A. Thus, P (1) is positive and thediscriminant of (A.1) is also positive, and P 0 (0) < 0 and P 0 (1) > 0 holdat equilibrium A. Therefore, we have real value eigenvalues 0 < ¹1 ; ¹2 < 1
and then, equilibrium A is a sink. However, we have imaginary value
solutions when the discriminant of (A.1) is negative.
Appendix D. Proof of sign of H.
The slope of the GG curve is given by (21). At point B in Figures 4(a)4(c), the slope of the GG curve is steeper than µ¸¡1 (µ + ´)¡1 ;dgt
1 ¡ ³t0 (zt )gt1¡®
µ
jGG =
>
:
¡(1+®)dzt¸(µ + ´)[(1 ¡ ®)gt ³t (zt ) + ®µ]gt
Rewriting this condition, we have¡(1+®)¸(µ + ´)µ¡1 [1 ¡ ³t0 (zt )gt1¡® ] ¡ gt[(1 ¡ ®)gt ³t (zt ) + ®µ] > 0:

Because the left-hand side is H [see (29)], H > 0 at point B in Figures
4(a)-4(c). The reverse is also true, that is, H < 0, at point A.

15

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16

¯; "

¯(gt ; zt )¯(gt0 ; zt )1 + zt"(gt0 ; zt )"(gt ; zt )

zt
z1t

0
z1t

Figure 1. Diagramatic representation of (20).

17

Â; "

Â(gt0 ; zt )1 + ztÂ(gt ; zt )"(gt0 ; zt )"(gt ; zt )

zt
0
z1t

z1t

z2t

0
z2t

Figure 2. Diagramatic representation of (22).

18

gt

µ
g = ¸(µ+´)
z

B

Z0Z0
ZZ
A
0

GG

0

GG

zA

0
zA

0
zB

zB

Figure 3. Phase diagram and e®ects of an increase in ¸ on the steady state.

19

zt

µ
g = ¸(µ+´)
z

g
B

ZZ
A

GG

zA

0
zA

0
zB

Figure 4(a). Case 1. ± <¸1¡ (1¡®)(1¡¸)
1+®z
t

20

zB

z

g

µ
g = ¸(µ+´)
z

B

ZZ

A

GG

0
zB

0
zA zA

Figure 4(b). Case 2.

¸(1¡®)(1¡¸)
1¡ 1+®z
t

21

<±<

zB

¸(1+ z1 )1¡t(1¡®)(1¡¸)
1+®zt

z

g

µ
g = ¸(µ+´)
z

B

ZZ
A
GG

0
zA
zA

0
zB zB

Figure 4(c). Case 3.

¸(1+ z1 )t1¡ (1¡®)(1¡¸)
1+®z
t

22

<±

z