Discussion Papers In Economics
And Business

Sustainability of the public debt and wealth inequality
in a general equilibrium model
Noritaka Maebayashi and Kunihiko KonishiDiscussion Paper 16-33

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANSustainability of the public debt and wealth inequality
in a general equilibrium model
Noritaka Maebayashi and Kunihiko KonishiDiscussion Paper 16-33

December 2016

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANSustainability of the public debt and wealth inequality
in a general equilibrium model∗
Noritaka Maebayashi†

Kunihiko Konishi‡§

Abstract
This study investigates the correlation between sustainability of the public debt
and wealth inequality in an endogenous growth model with heterogeneous agents.
We show that the threshold for the sustainability of public debt can be related to not
only relative size of public debt but also wealth inequality. In addition, this study
examines the eﬀects of budget deficit and redistributive policies on the sustainabilityof the public debt and wealth inequality. We show that an increase in the deficit
ratio or the redistributive tax makes the public debt less sustainable. If the economy
falls into the unsustainable region as a result of the policy change, both public debt
and wealth inequality continue to increase.

JEL classification: H62, H63
Keywords: Fiscal sustainability, Public debt, Wealth inequality, Redistributive policy,
Endogenous growth∗

We thank the seminar participants at the Asian Growth Research Institute, Kyushu University, Hiroshima University, Otaru University of Commerce, and Tokai University for their comments. Any errors
are our responsibility.
†
Faculty of Economics and Business Administration, The University of Kitakyushu, 4-2-1 Kitagata,
Kokura Minami-ku, Kitakyushu, Fukuoka 802-8577, JAPAN; e-mail: non818mn@kitakyu-u.ac.jp‡
Institute of Economic Research, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501,
JAPAN; e-mail: k.konishi.econ@gmail.com§
Research Fellow of the Japan Society for the Promotion of Science (JSPS).

1

1

Introduction

Since the recent default risk on Greek government debt exposed by the 2008–2009 worldcrisis, the concern over whether a government deficit and debt are sustainable has been
growing among countries whose public debt is very large. In addition to increased public
debt, the expansion of wealth and income inequality is also observed in many developed
countries. According to Cagetti and Nardi (2008), the concentration of wealth is more
serious than that of income. Because the sustainability of the government debt depends
on the amount of wealth in its country, the accumulation of the government debt andwealth inequality among households can be closely related to each other.1 Therefore,
investigating the relationship between the sustainability of the public debt and wealthinequality is an important issue.
Some recent studies examine fiscal sustainability using the overlapping-generations(OLG) models (life-cycle models) developed by Diamond (1965). In the OLG models,
fiscal sustainability means that the ratio of public debt to GDP (or capital) converges to astable level in the long run. Chalk (2000) shows that under a constant primary deficit rule,
if the initial public debt is very large, public debt does not converge to a stable level but
explodes, and thus, public debt is not sustainable. More recent studies (e.g., Bräuninger
2005; Yakita 2008; Arai 2011; Teles and Mussolini 2014; Agénor and Yilmaz 2016) extend
the analysis to the OLG model with endogenous growth structure and obtain a similar
result to Chalk (2000).2 Nevertheless, these studies ignore the relationship between thesustainability of the public debt and wealth inequality.
To our knowledge, there are some studies that investigate how the public debt aﬀects
inequality or wealth distribution (e.g., Mankiw 2000; Michel and Pestieau 2005; Pestieau
and Thibaut 2012).3 However, these previous studies do not pay attention to the sustainability of the public debt but focus on the steady state in which the public debt converges
1

For example, as shown in Table 1 in Hoshi and Ito (2014), more than 90% of Japanese government
debt is funded domestically, which ensures the Japanese government’s solvency even though the debt isvery large.
2
In these endogenous growth models, a huge outstanding public debt can induce higher growth of
public debt than growth of private capital and output and can make fiscal policy unsustainable.
3
There is no consensus on how the public debt aﬀects inequality or wealth distribution among thesestudies. Mankiw (2000) and Michel and Pestieau (2005) show that public debt increases steady-state
inequality whereas Pestieau and Thibaut (2012) show that public debt redistributes wealth from the top
wealthy to others.

2to a stable level.
Accordingly, this study addresses the following open questions. (i) How are the sustainability of public debt and wealth inequality co-related? (ii) How does public deficit policy
aﬀect the sustainability of public debt and wealth inequality? (iii) How do redistributive
policies aiming for a reduction in wealth inequality aﬀect the sustainability of public debt?
To tackle these problems, we develop a tractable OLG model in which the transition pathof public debt and that of wealth inequality are jointly determined.
The present model is in line with Bräuninger (2005), which explores the sustainabilityof public deficit policy under a constant deficit/GDP rule in an AK model.4 We incorporatea mechanism generating endogenous transmission of wealth inequality into Bräuninger’s(2005) model. A key mechanism generating wealth inequality in this model is composed of(i) the heterogeneity in the agents’ subjective discount factors based on Becker (1980) and(ii) joy-of-giving bequest motives (e.g., Abel and Warshawsky 1988; Andreoni 1989, 1990).
Our model splits the population into two classes, the rich and poor, and assumes that therich have higher exogenous subjective discount factor than the poor (i.e., the rich are morepatient than the poor).5 In addition, we assume that agents have joy-of-giving bequest
motives independently of whether they belong to the rich or poor. Under these assumptions, the rich save more and bequeath more wealth to their children, which becomes thesource of wealth inequality. Some empirical studies consider that such intergenerational
linkages in saving behavior and wealth accumulation generate wealth inequality. For example, Dynan et al. (2004), Bozio et al. (2013), Alan et al. (2014), and Néstor (2015)
show that the rich have a higher savings rate than do the poor. Other empirical evidence
supports that bequests are one of the major causes of wealth inequality (e.g., Kotlikoﬀ
and Summers 1981; Gale and Scholz 1994).
Using this framework, we illustrate that the transition paths of both public debt andwealth inequality are determined in a two-dimensional phase diagram and we obtain thefollowing results.
4

The constant deficit/GDP rule employed by Bräuninger (2005), Yakita (2008), Arai (2011), and Teles
and Mussolini (2014) follows the criterion of the Maastricht Treaty.
5
As for such the classification of agents’ types, Borissov and Lambrecht (2009, pp.99) point out: “In theterminology of Mankiw (2000), this classes might be called savers and spenders.” Some empirical studies
show that the rich are more likely to be patient than are the poor (see, e.g., Lawrence 1991; Harrison etal. 2002).

3

(i) There is a threshold of public debt for each level of wealth inequality in order for thegovernment to sustain the fiscal policy, and the threshold of public debt is increasing
in wealth inequality. When the initial public debt is small, the economy can reacha stable equilibrium in which both public debt and wealth inequality converge tothe stable level. When the initial public debt is very large, the economy with higher
wealth inequality can converge to a stable equilibrium whereas the economy withlow wealth inequality cannot converge to any stable equilibrium. If the economy
is in the unsustainable region, both public debt and wealth inequality continue toincrease, and the economy goes bankrupt in the long run.
(ii) An increase in the public deficit ratio makes the public debt less sustainable. Therefore, if the economy with large public debt falls into the unsustainable region as aresult of expanding the public deficit ratio, wealth inequality increases as public debt
grows during the bankruptcy path.
(iii) A redistributive policy that attempts to reduce wealth inequality aﬀects the sustainability condition. If the government taxes bequests of the rich and redistributes therevenue to the poor, the economy is more likely to fall into the region in which the
public debt is not sustainable and wealth inequality continues to increase. Thus, inthe economy with large public debt, introducing such a redistributive policy might
be very risky.
(iv) The policy eﬀects in the stable steady state are as follows. A rise in the public deficit
ratio enlarges wealth inequality and decreases the growth rate. An increase in theredistributive tax reduces both wealth inequality and the growth rate, and hence,leads to the trade-oﬀ between equality and growth.
The remainder of this paper is organized as follows. Section 2 establishes the basemodel whose objective is to derive the relationship between sustainability of the public
debt and wealth inequality. To achieve this as simply as possible, this base model excludes
the redistributive public policy. Sections 3–5 explore this base model as follows. Section 3
derives the equilibrium condition and dynamic system of the economy. Section 4 derives
the transition dynamics and the relationship between wealth inequality and sustainability4

of public debt. Section 5 analyzes the eﬀects of changes in the public debt finance ratioon the sustainability of the public debt and wealth inequality. Section 6 introduces theredistributive public policy into the base model and examines its eﬀect on the sustainabilityof the public debt and wealth inequality. Section 7 concludes.

2

Model

2.1

Individuals

We consider a two-period OLG model following Diamond (1965). An individual lives fortwo periods and a cohort born in period t is called generation t. Therefore, two generations
exist in period t; that is, generation t (the young generation) and generation t − 1 (the oldgeneration). In each period, the size of the newly born cohort is given by N . There are
two groups of families, “rich” and “poor,” denoted by R and P , respectively. We assume
that a constant fraction δ ∈ (0, 1) of individuals is the rich and a constant fraction 1 − δ ofindividuals is the poor. Each individual supplies one unit of labor inelastically and earns
labor income in their young period, and then the total labor supply is Lt = N . In theold period, they are retired, consume their savings, and leave bequests to their children.
Individuals have perfect foresight.
Each individual i ∈ {R, P } born at period t maximizes utility,[]
2ii
Uti = (1 − αi ) log c1i
t + αi (1 − β) log ct+1 + β log bt+1 ,(1)iwhere ct1i is the consumption when young, c2i
t+1 is the consumption when old, and bt+1 isthe bequest passed on to the child. Note that the utility depends on the amount bit+1 .
This reflects a “joy of giving” savings motive. αi ∈ (0, 1) is the intertemporal preference
parameter and β ∈ (0, 1) is the relative importance parameter of the consumption when
old and the bequest. We assume β does not diﬀer between the rich and poor following
Bossmann et al. (2007). In addition, we assume that αR > αP based on Becker (1980)and some empirical evidence (e.g., Lawrence 1991; Harrison et al. 2002). This assumption
generates the mechanism by which the rich save a larger proportion of their income than
do the poor, which is empirically supported by Dynan et al. (2004) for the US, Bozio et al.
5(2013) for the UK, Alan et al. (2014) for Canada, and Néstor (2015) for Latin America. In
addition, we assume that the wealth endowment of the rich old generation at initial period(t = 0) is larger than that of the poor old generation. As a result, the rich bequeath larger
wealth than the poor do. Let sit be savings in youth. The budget constraint of generationt can be written as follows:iic1i
t = (1 − τt )wt − st + bt ,

(2)

i
i
c2i
t+1 = [1 + (1 − τt+1 )rt+1 ] st − bt+1 ,(3)

where wt , rt+1 , and τt represent the wage rate, interest rate, and tax on wage and interest
income. By solving the intertemporal utility maximization, we obtain the following optimal
conditions:
[
]
i
c1i
=
(1
−
α
)
(1
−
τ
)w
+
b
i
t
t
t
t ,
[
]
sit = αi (1 − τt )wt + bit ,

(4a)
(4b)

i
c2i
t+1 = (1 − β) [1 + (1 − τt+1 )rt+1 ] st ,(4c)

bit+1 = β [1 + (1 − τt+1 )rt+1 ] sit .

(4d)

From (4b) and (4d), savings are determined as follows:






sit = αi (1 − τt )wt + β{1 + (1 − τt )rt }sit−1  .
|
{z
}

(5)

the bequest from parents

Equation (5) indicates the following. First, the savings of the current generation sit arelinked to the savings of parents sit−1 . This is because the bequest from parents depends ontheir wealth income [1 + (1 − τt )rt ]sit−1 from (4d). Second, from assumption αR > αP , therich save more than the poor do, and leave more wealth to their oﬀspring, who in turn,tend to do the same. This means that the rich tend to accumulate more wealth than thepoor do.
P
The total assets (savings) held by young agents in period t, At ≡ δsRt N + (1 − δ)st N ,are composed of public bonds, Dt+1 , and private capital, Kt+1 . Hence, the asset market

6clears as follows:
Kt+1 + Dt+1 = At .

2.2

(6)

Production

There is a large number of identical firms denoted by j. Firm j produces a single finalγ
good using the production technology given by Yj,t = ΓKj,t(at Lj,t )1−γ (0 < γ < 1), where

Yj,t , Kj,t , and Lj,t represent the output level, private capital, and labor input of firm j,
respectively. at is the labor eﬃciency at time t. From profit maximization in competitive markets, factor prices become equal to the marginal products: rt = ∂Yj,t /∂Kj,t =γΓ(Kj,t /Lj,t )γ−1 a1−γ
and wt = ∂Yj,t /∂Lj,t = (1 − γ)Γ(Kj,t /Lj,t )γ a1−γ.
ttFollowing Romer (1986), we assume that the average capital per worker has positive
external eﬀects on labor productivity, and specify at = Kt /Lt , where Kt is the averagestock of private capital and Lt is the average labor input in the economy. In equilibrium,
Kj,t = Kt and Lj,t = Lt hold for all j, and thus, the factor prices and aggregate output,
Yt , in period t can be written as follows:
wt = Γ(1 − γ)

2.3

Kt
,
Lt

(7a)

rt = Γγ,(7b)

Yt = ΓKt .

(7c)

Government

The government in period t imposes a tax on income, (wt Lt + rt At−1 ), and issues bonds,
Dt+1 − Dt , to finance public spending, Gt and interest payment for the public debt, rt Dt .
The tax revenue in period t, τt (wt Lt + rt At−1 ), is rewritten by τt (Yt + rt Dt ) using (6), (7a),(7b), and (7c). Thus, the budget constraint of the government is
Dt+1 − Dt + τt (Yt + rt Dt ) = Gt + rt Dt ,7(8)

Following Bräuninger (2005), we assume that a constant proportion, g ∈ (0, 1), of national
income, Yt , is used for public expenditure: Gt = gYt . In addition, the government borrows
a constant proportion, λ ∈ (0, 1), of GDP, that is, the government fixes the deficit ratio
as follows:
Dt+1 − Dt = λYt .

(9)As in Bräuninger (2005), Yakita (2008), Arai (2011), and Teles and Mussolini (2014), wheng and λ are kept constant, the government must adjust the income tax rate, τt , so as tosatisfy the budget constraint (8). By using (7b), (7c), (8), and (9), we obtainτt = 1 −1+λ−g,
1 + γxt(10)

where xt ≡ Dt /Kt . A higher level of public debt means that a higher level of income
taxation must be used to pay for the interest payments on the debt. Therefore, an increasein the ratio of public debt to private capital x raises income tax rate: dτt /dxt > 0 as inBräuninger (2005), Yakita (2008), Arai (2011), and Teles and Mussolini (2014). We callthis the tax burden eﬀect.

3

Equilibrium condition and dynamic system

From individuals’ budget constraints: (2), (3), distribution of output: Yt = rt Kt + wt Lt ,the government’s budget constraint: (8), and the asset market-clearing condition: (6), the1P
2R
market equilibrium satisfies the resource constraint: Yt = δc1R
t N + (1 − δ)ct N + δct N +(1 − δ)c2P
t N + Kt+1 − Kt + Gt .
We then characterize the equilibrium paths in this economy. Substituting (5) intoP
At = δsRt N + (1 − δ)st N , we obtain][P
At = ᾱ(1 − τt )wt N + β[1 + (1 − τt )rt ] αR sR
t−1 δN + αP st−1 (1 − δ)N ,

8

(11)

where ᾱ ≡ δαR + (1 − δ)αP . By using (6) and the definition of xt , we obtain
Kt = (1 + xt )−1 At−1 .

(12)This equation (12) indicates that an increase in xt reduces investment in private capital
because public bonds account for a larger proportion of aggregate assets, At−1 . We callthis the crowding-out eﬀect.
Dividing (11) by At−1 and substituting (7a), (7b), (10), and (12) into (11), we obtain
the growth of aggregate savings, A, as follows:()
Atᾱ(1 − γ)µ1
γµ1=[(αR − αP )φt−1 + αP ]
+β 1+
At−1
(1 + xt )(1 + γxt )1 + γxt≡ GA (xt , φt−1 ),where φt ≡

δsR
t N
At

(13)

and µ1 ≡ Γ(1 + λ − g). Note that 1 − φt =(1−δ)sPt N
Atholds from thedefinition of φt and At . In this study, because φt represents the ratio of total savings ofthe rich to aggregate savings, φt serves as a convenient measure of wealth inequality.
AA(xt ,φt−1 )t−1 )< 0 and (b) ∂G ∂φ> 0. The
From (13), GA (xt , φt−1 ) satisfies (a) ∂G (x∂xt ,φtt−1

former, (a), comes from the following two channels. First, an increase in xt reduces thewage rate through the crowding-out eﬀect (see (12)). Second, an increase in xt decreases
disposable wage and bequest incomes through the tax burden eﬀect (see (10)). Bothhave negative eﬀects on the growth of aggregate savings. The latter, (b), indicates thefollowing. The rich accumulate more wealth than the poor do and hold a larger proportionof aggregate wealth (see (5)). Then, wealth inequality φ driven by the rich contributes to
the growth of aggregate savings GA (·).
By using (5), (7a), (7b), (10), (12), and the definition of φt , we obtain the growth of
the rich’s savings as
[
(
)
]
αR
δ(1 − γ)µ1
γµ1
δsR
t N
=
+β 1+
φt−1
φt−1 (1 + xt )(1 + γxt )1 + γxtδsR
t−1 N≡ GR (xt , φt−1 ).

(14)

9

Substituting (7c) into (9) and using Dt /Kt ≡ xt yields the growth of public debt as
Dt+1
λΓ
=1+
≡ GD (xt ).
Dt
xt

(15)

From (6) and (9), we obtain Kt+1 + λYt + Dt = At . Dividing both sides of it by Kt andusing (7c) yield Kt+1 /Kt = (At /At−1 )(At−1 /Kt ) − (xt + λΓ). Substituting (12) and (13)into it, we obtain
Kt+1= (1 + xt )GA (xt , φt−1 ) − (xt + λΓ) ≡ GK (xt , φt−1 ).
Kt

(16)

The growth of private capital GK (·) is linked positively with GA (·) through the asset
market-clearing condition (see (12)). From (13), (14), and the definition of φt , we obtain
δsR
t N

φt
GR (xt , φt−1 )
δsR N
= t−1
.
=
At
φt−1
GA (xt , φt−1 )
At−1

(17)

The growth of wealth inequality decreases (increases) when GA (·) is relatively larger(smaller) than GR (·). This is attributed to the definition of φ. From (15) and (16),
we obtain
xt+1
GD (xt )1 + λΓ/xt
= K
=
.
A
xt
G (xt , φt−1 )(1 + xt )G (xt , φt−1 ) − (xt + λΓ)(18)

The above two diﬀerence equations (17) and (18) together with the initial values φ−1 andx0 characterize the dynamics of the economy. Note that both xt and φt−1 in period t arepredetermined variables.

4

Transition dynamics of wealth inequality and the
public debt/private capital ratio

In this section, we derive global transition dynamics of the economy and investigate how
the accumulation of public debt and wealth inequality correlate to each other.
We begin with the derivation of φt = φt−1 locus on the (xt , φt−1 ) plane. Setting

10R(xt ,φt−1 )φt = φt−1 in (17), that is, 1 = G
yieldsGA (xt ,φt−1 )[]β(1 + xt ) µ−1
1 (1 + γxt ) + γ =1−γ(αR − αP )(1 − φt−1 )

)
(
δαR
ᾱ −
.
φt−1

(19)

Let us define the left- and right-hand side of (19) as ε(xt ) and η(φt−1 ), respectively. Examining (19), we arrive at the following.
Lemma 1(i) φt = φt−1 locus is an upward-sloping curve on the (xt , φt−1 ) plane.
(ii) φt = φt−1 locus has an asymptote φt−1 = 1 when xt → ∞ and takes a lower limit(0, φ̃) on the (xt , φt−1 ) plane. φ̃ is defined in Appendix A.
Proof: See Appendix A.
Next, we derive xt+1 = xt locus on the (xt , φt−1 ) plane. Setting xt+1 = xt in (18), that is,1+λΓ/xt
1 = (1+xt )GA (x, leads tot ,φt−1 )−(xt +λΓ)φt−1 =whereζ(xt ) ≡ζ(xt ),
αR − αP(20)()(1 + xt )(1 + γxt ) 1 + λΓ
− ᾱ(1 − γ)µ1xt
β(1 + xt ) [1 + γ(xt + µ1 )]− αP .

Examining (20), we arrive at the following.
Lemma 2 Suppose that γ(1 − g) > λ.6(i) xt+1 = xt locus is a U-shaped curve on the (xt , φt−1 ) plane.
−1−αP(ii) xt+1 = xt locus has asymptotes: limxt →∞ φt−1 = βαR −α> 1 and limxt →0 φt−1 = +∞P

on the (xt , φt−1 ) plane.
Proof: See Appendix B.
6This is satisfied if λ is not so large and (γ, g) takes conventional parameters used in the literature, suchas Bräuninger (2005) and Michel et al. (2010). For example, taking a parameter set (γ, g) = (0.2, 0.2),
this condition is satisfied under λ < 0.16.

11Hence, we can depict φt = φt−1 and xt+1 = xt loci on the (xt , φt−1 ) plane in Figure 1.
Finally, we examine the regions in which Kt+1 /Kt ≥ 0, that is, (1 + xt )GA (xt , φt−1 ) −(xt + λΓ) ≥ 0 from (16) so far.7 This condition can be rewritten as
φt−1 ≥()(1 + γxt ) 1 + λΓxt

1ζ(xt ) − ≡ Λ(xt ) .
αR − αPβ(1 + xt )[1 + γ(xt + µ1 )]αR − αP(21)

Thus, we can recognize that Kt+1 /Kt > 0 is satisfied above Kt+1 /Kt = 0 locus. Examining(21), we arrive at the following. From (20) and (21), Kt+1 /Kt = 0 locus is always below
xt+1 = xt locus. In addition, Appendix C shows that Kt+1 /Kt = 0 locus is an upward−1−αP
sloping curve if ᾱ(1 − γ)(1 + λ − g) > λ and has an asymptote limxt →∞ φt−1 = βαR −α>1Pon the (xt , φt−1 ) plane.8 Then, we can depict Kt+1 /Kt = 0 locus as the broken curve in
Figure 1.
[Figure 1]
Now we investigate the steady states of the economy wherein both xt and φt−1 areconstant. In this study, we use an asterisk to represent variables in the steady states (i.e.,
xt = xt+1 = x∗ and φt−1 = φt = φ∗ ). As shown in Figure 1, the steady state values of(x∗ , φ∗ ) are determined by the intersections of the curves, φt = φt−1 and xt+1 = xt loci,on the (xt , φt−1 ) plane. From Lemmas 1 and 2, there are either two long-run equilibria
or none. A case of two steady states is represented in Figure 1, and is obtained if C1 issatisfied.9C1:ε(x̄) > η(φ̄), where φ̄ ≡ ζ(x̄).

In addition, C2 allows us to focus on the economy in which wealth of the rich is larger
7

In the region where Kt+1 /Kt < 0, the asset market-clearing condition (6) indicates that Dt+1 becomeslarger than At . In this situation, the public debt cannot be absorbed by aggregate savings and then no
capital is installed in the production sector.
8
Similarly to the assumption of γ(1 − g) > λ in Lemma 2, ᾱ(1 − γ)(1 + λ − g) > λ is satisfied if λ is
not so large and (γ, g, ᾱ) takes conventional parameters used in the literature, such as Bräuninger (2005)
and Michel et al. (2010). For example, taking a parameter set (γ, ᾱ, g) = (0.2, 0.35, 0.2), this condition issatisfied under λ < 0.31.
9
A case of no steady state is realized and the fiscal policy is always unsustainable when the public
debt finance ratio, λ, is very large, as in Bräuninger (2005). We rule out this case because of the same
discussion by Bräuninger (2005).

12

than that of the poor (i.e., φt−1 > 0.5 for all t).10φ̃ ≡ η −1 (ε(0)) > 0.5.

C2:

Let us refer to (x∗k , φ∗k ) as the steady state k (k ∈ {S, U }). We obtain x∗S < x∗U and φ∗S < φ∗Ubecause φt = φt−1 locus is increasing in an upward-sloping curve. From (7c), (15), (16),and constant x∗k , the steady state growth rate at each state is as follows:(Ŷk∗ ≡

Yt+1
Yt

)∗

(
=

Kt+1
Kt

)∗

(
=

At
At−1

)∗

(
=

Dt+1
Dt

)∗
=1+

λΓ
x∗k

f or k ∈ {S, U }. (22)Therefore, x∗S < x∗U implies that the growth rate of the steady state S is higher thanthat of the steady state U . As shown in Appendix D, we can illustrate a phase diagramof this economy in Figure 1. As we can observe from the phase diagram, the steadystate S is stable and the steady state U is saddle-point stable.11 A dotted line HH in
Figure 1 represents the stable arm converging to the steady state U . Because xt and φt−1
are predetermined variables at time t, as we mention before, and we need to note thefollowing two points. First, the initial state of the economy is given by a point (x0 , φ−1 )on the (xt , φt−1 ) plane. Second, the saddle arm, HH, is a knife-edge.
These facts lead to the following two cases. When an economy starts at the initialstate (x0 , φ−1 ) in the upper-left of the saddle arm: HH, it converges to the steady state
S. At the steady state S, both xt and φt−1 are constant, and the government can run theconstant budget deficit policy permanently because private capital grows at the constantrate ŶS∗ . By contrast, when (x0 , φ−1 ) is in the lower-right of the saddle arm, HH, aneconomy will not converge to any steady states. In this case, the public debt grows morethan private capital does: GD (·) > GK (·); and finally the economy falls into Kt+1 /Kt < 0region in which the public debt/private capital ratio, xt , becomes too large to sustain theinvestment in private capital. Therefore, the dotted line HH in Figure 1 represents thethreshold of public debt for each level of wealth inequality in order for the government to10

φ̃ ≡ η −1 (ε(0)) > 0.5 holds if and only if()
2(1 − γ)[δαR − (1 − δ)αP ]ε(0) > η(0.5) ⇔ β µ−1.
1 +γ >−
αR − αP11In Appendix E, we consider the local stability at each steady state by using numerical examples.

13

sustain the fiscal policy. This threshold level of public debt has a positive relationshipwith wealth inequality. This is a noticeable result that departs from previous studies that
do not include wealth inequality (e.g., Bräuninger 2005; Yakita 2008; Arai 2011; Teles andMussolini 2014).
In summary, we can state the following proposition:
Proposition 1. Under a fiscal deficit policy, there is a threshold of public debt for eachlevel of wealth inequality in order for the government to sustain its policy. The thresholdof public debt is increasing in wealth inequality.
In order to consider the intuition behind Proposition 1, let us begin with the case inwhich the initial public debt/private capital ratio, x, is suﬃciently small. In this case,both the tax burden eﬀect and the crowding-out eﬀect are small from (10) and (12), andthen wage and bequest incomes are relatively large. Under such an environment, aggregate
savings can grow strongly, which avoids rapid growth of φ from (17) and drives private
capital to grow strongly from (16). Therefore, we find that a suﬃciently large growth ofaggregate savings, GA (·), helps both the public debt/private capital ratio, x, and wealthinequality, φ, converge to the steady state S.
Next, we move to the case when the initial xt is large enough to be near the saddle arm,
HH, for example, x0 = xH in Figure 1. In this case, both the tax burden eﬀect and thecrowding-out eﬀect are so large that wage and bequest incomes become very small. Thus,
the growth rates of both aggregate savings, GA (·), and private capital, GK (·), are small.
Even under such an environment, if the initial wealth of the rich is relatively larger thanthat of the poor, wealth accumulation of the rich can be strong enough to reinforce thegrowth of aggregate savings, GA (·) (see (13)). Therefore, the economy with high wealthinequality as represented by Q2 can sustain its public debt.
By contrast, in the economy with low wealth inequality, as represented by Q1 , thegrowth of aggregate savings, GA (·), tends to remain small. In this case, public debt grows
more than private capital does, and then, the economy cannot sustain its public debt.
It is of great interest that wealth inequality increases as public debt grows during thebankruptcy path. An intuitive reason for this is as follows. As public debt grows, thegrowth of the wage rate keeps decreasing through both the tax burden eﬀect and the14crowding-out eﬀect. In this situation, bequest income plays a more important role inwealth accumulation than does the wage income. Because the rich leave more wealth totheir oﬀspring than do the poor, wealth inequality increases and absorption of larger public
debt tends to rely more on wealth accumulation by the rich. However, this situation does
not last long and the economy goes bankrupt in the long run.

5

Changes in public debt finance ratio λA few studies investigate the relationship between public debt and inequality (e.g., Mankiw
2000; Michel and Pestieau 2005) and show that increases in public debt finance raise
wealth (or income) inequality. However, these studies focus only on the steady state,
which corresponds to the stable steady state S in our study. Therefore, previous studies
neglect the possibility of the unstable (unsustainable) paths. The main objective here is
to study how increases in public debt finance aﬀect wealth inequality and sustainabilityof the public debt.
We begin with the eﬀects of an increase in public debt finance ratio λ on the φt = φt−1
locus. Taking the total diﬀerentials of (19) yields
βΓµ−2
dφt−1
1 (αR − αP )(1 + xt )(1 + γxt )(1 − φt−1 )=−< 0.
dλ φt =φt−1
η(φt−1 )(αR − αP ) + (1 − γ)δαR φ−2
t−1

(23)

Note that 0 < φt−1 < 1 holds from the definition of φt−1 . We next investigate the eﬀectof changes in λ on xt+1 = xt locus. Diﬀerentiating (20) with respect to λ, we obtain[()]Γ(1 + γxt ) 1 + x−1(1 + γµ1 − γλΓ) − (1 − γ)ᾱ
dφt−1
t
=
.
dλ xt+1 =xt(αR − αP )β(1 + xt ) [1 + γ(xt + µ1 )]2(24)

Because of 1+γµ1 −γλΓ = 1+γΓ(1−g) > 1 (from the definition of µ1 ) and (1−γ)ᾱ ∈ (0, 1),
dφt−1> 0 holds. Thus, we obtain the following.
dλ xt+1 =xt

Lemma 3
When the government increases the public debt finance ratio λ, (i) φt = φt−1 locus shifts
downward from (23), and (ii) xt+1 = xt locus shifts upward from (24). These are represented in Figure 2.
15

[Figure 2]
From Lemma 3, we can observe that the steady state U shifts left-and-downward andtherefore, the saddle arm, HH, also shifts left-and-downward, as depicted in Figure 2. Thatis, the threshold of public debt for each level of inequality in order for the government tosustain the fiscal policy becomes lower. This result implies that the range in which thedeficit policy is sustainable is shrunk by an increase in λ. Thus, we obtain the following
proposition.
Proposition 2. An increase in the deficit ratio reduces the range of sustainable initialpublic debt.
A larger budget deficit reinforces both the tax burden eﬀect and the crowding-out eﬀect,
which leads to a decline in aggregate savings. Therefore, even if the initial level of wealthinequality is somewhat high, the economy is more likely to fall into an unsustainable
path. This result leads to the following policy implication. In the economy with a large
public debt, an increase in the public debt finance ratio not only makes public debt lesssustainable but also can induce wealth inequality to increase persistently.
In the rest of this section, we shed light on the steady state S and investigate the eﬀects
of changes in λ on the public debt/private capital ratio, wealth inequality, and the longrun growth rate. Lemma 3 shows that at the steady state S, an increase in λ raises the
public debt/private capital ratio, x∗S , but the eﬀect on wealth inequality, φ∗S , is ambiguous.
Moreover, the eﬀect of λ on the growth rate at the steady state S is ambiguous. Then, we
conduct a numerical analysis. We adopt the following benchmark parameters: γ = 0.2,
Γ = 12, g = 0.2, δ = 0.5, β = 0.3, αR = 0.45, and αP = 0.25.12 Table 1 shows the steadystate values for each deficit ratio and leads to the following result.
Result 1. In the steady state S, an increase in λ (i) raises wealth inequality and (ii)reduces the long-run growth rate.
12

The values of γ, Γ, and g follow the methodology in Bräuninger (2005). γ = 0.2 and Γ = 12 representr = γΓ = 2.4. If we assume that the time period is about 30 years, r = 2.4 implies that the annualinterest rate is 4.2%. We employ αR = 0.45 and αP = 0.25 so as to satisfy the conventional value ofᾱ = δαR + (1 − δ)αP = 0.35. The value of δ follows the methodology in Galı́ et al. (2007). The resultspresented here are robust to other parameter values, δ. We conduct robustness checks in the technicalappendix, which is available on request.

16

[Table 1]
The result (i) is similar to that of Mankiw (2000) and Michel and Pestieau (2005).
As mentioned in the paragraph below Proposition 2, a higher budget deficit decreases
aggregate savings, which reduces the growth of the wage rate because a decline in aggregate
savings reduces the investment in private capital (see (7a) and (12)). In this study, from(5), the income of the young consists of the wage and bequest incomes. Lower wage
income indicates that the bequest income becomes more important for the accumulationof wealth. The rich tend to hold a larger proportion of total wealth, and then wealthinequality increases. Furthermore, result (ii) is similar to that of Bräuninger (2005). A
higher budget deficit implies that an increase in the level of public debt raises the interest
payment of the government. To satisfy the government budget constraint, the income tax
rate increases. This reduces total savings. In addition, a higher level of public debt crowds
out the investment in private capital. As a result, the long-run growth rate declines.

6

Redistributive public policy

We have considered the relationship between the sustainability of public debt and wealthinequality. In many developed countries, policies aiming for a reduction in wealth inequality are implemented.13 Thus, we then wonder how a public redistributive policy aﬀects thesustainability of public debt, wealth inequality, and economic growth. In order to tackle
this problem, we introduce a public redistributive policy in the following simple way.
The government taxes bequests of the rich at rate τ b and redistributes its revenue
lump-sum to the poor in youth. This is the reduced form of the redistributive policy
considered in Bossmann et al. (2007).14 The budget constraint for the rich is given by
b R
R
2R
R
R
c1R
t = (1 − τt )wt − st + bt and ct+1 = [1 + (1 − τt+1 )rt+1 ] st − (1 + τ )bt+1 and that for the
P
P
2P
P
P
poor is given by c1P
t = (1 − τt )wt − st + bt + Tt and ct+1 = [1 + (1 − τt+1 )rt+1 ] st − bt+1 ,where Tt is the uniform lump-sum transfer by the redistributive policy. The government
redistributive policy is represented as δN τ b bR
t = (1 − δ)N Tt .
13

For instance, France, Germany, Japan, the United States, and the United Kingdom have inheritanceor estate taxes.
14
In Bossmann et al. (2007), the government taxes bequests of both the rich and the poor and redistributes the revenue lump-sum to the young generation independently of whether the young belong to therich or the poor.

17

Appendix F shows that the growth of aggregate savings and that of private capital are
given by()[]ᾱ(1 − γ)µ1
γµ1
αR − αPG (xt , φt−1 ; τ ) =+β 1+
φt−1 + αP ,(1 + γxt )(1 + xt )1 + γxt
1 + τbAb(25)

GK (xt , φt−1 ; τ b ) = (1 + xt )GA (xt , φt−1 ; τ b ) − (xt + λΓ).

(26)

and

Furthermore, Appendix F shows that φt = φt−1 and xt+1 = xt loci are rewritten asε(xt ) =1 + τbbαP
1 − (αR −ατP )(1−φt−1 )η(φt−1 ) ≡ η̃(φt−1 ),

(27)

and
φt−1 =

1 + τbζ(xt ),
αR − αP(28)

respectively. As shown in Appendix G, the introduction of redistributive policy does not
aﬀect the main properties of φt = φt−1 locus that we observe in Section 4 if we assume
αR > (1 + τ b )αP . Then, we easily recognize that (i) φt = φt−1 locus: (27) is an upwardsloping curve and xt+1 = xt locus: (28) is a U-shaped curve on the (xt , φt−1 ) plane, and(ii) two steady states corresponding to S and U exists as depicted in Figure 3.
[Figure 3]
We next investigate the eﬀects of a rise in the bequest tax rate. Taking the total diﬀerentials of (27) yields
η̃(φt−1 ) [αR (1 − φt−1 ) + αP φt−1 ]dφt−1[] < 0.
=−bbdτ φt =φt−1(1 + τ ) η̃(φt−1 )(αR − αP ) + (1 + τ b )(1 − γ)δαR φ−2
t−1
Thus, when the government increases τ b , φt = φt−1 locus shifts downward. We thenexamine the eﬀect of changes in τ b on xt+1 = xt locus. Diﬀerentiating (28) with respectto τ b , we obtain
ζ(xt )
dφt−1
=
> 0.
b
dτ xt+1 =xt
αR − αP18

Therefore, when the government increases τ b , xt+1 = xt locus shifts upward. These results
imply that the eﬀects of an increase in the bequest tax rate are qualitatively similar to
those of a rise in the public debt finance ratio. We can state the following proposition insummary.
Proposition 3. Taxing bequests of the rich and redistributing it to the poor (or a rise in
the bequest tax) reduces the range of sustainable initial public debt and makes fiscal policy
less sustainable.
An intuitive explanation is as follows. The redistributive policy increases income andsavings of the poor whereas it decreases income and savings of the rich. We find thatthe latter dominates the former and then the growth of aggregate savings declines by anincrease in τ b as follows:()
αR − αPdGA (xt , φt−1 ; τ b )γµ1= −β 1 +φt−1 < 0.
bdτ
1 + γxt (1 + τ b )2(29)

This is because the rich are more patient than the poor. If the initial public debt/private
capital ratio, x, is suﬃciently large, the growth of private capital declines suﬃciently to be
below that of public debt. Thus, the redistributive policy makes the government budget
deficit policy less sustainable. If the economy falls into a region where the public debt isunsustainable, then wealth inequality increases during the bankruptcy path. Hence, it isnoticeable that the redistributive policy can result in widening wealth inequality.
We investigate the eﬀects of changes in the redistributive policy at the steady state
S. When the government raises τ b , the steady state S shifts rightward. The public
debt/private capital ratio at the steady state S increases (i.e., dx∗S /dτ b > 0). From (22), it
is obvious that an increase in τ b reduces the growth rate at the steady state S. The resultof the declining growth rate is brought about by the reduction in the growth of aggregate
savings, as in (29). On the other hand, the eﬀect of τ b on wealth inequality at the steadystate S is ambiguous because it is not certain whether the steady state S shifts upward or
downward. To clarify this ambiguity, we conduct a numerical analysis with the following
parameters: γ = 0.2, Γ = 12, g = 0.2, δ = 0.5, β = 0.3, αR = 0.45, and αP = 0.25. Let∗∗∗∗
us denote the steady state values after the policy change as x∗∗
k , φk , and Ŷk . The eﬀects
∗
∗
∗
∗∗
∗
∗
of the policy change are measured by ∆x∗k ≡ (x∗∗k − xk ) /xk , ∆φk ≡ (φk − φk ) /φk , and

19

(
)
∆Ŷk∗ ≡ Ŷk∗∗ − Ŷk∗ /Ŷk∗ (k = {S, U }).
[Table 2]
Table 2 represents percentage changes in the steady state values when the government
increases the bequest tax rate, τ b , from 0 to 0.3. This numerical analysis shows thefollowing. An increase in τ b reduces wealth inequality. However, the eﬀect of decreasing
wealth inequality is relatively small (only about 2% or 3% changes in φ∗S even by a 30%
increase in τb ). This is attributed to the adverse eﬀect of τ b on wealth inequality. As
mentioned in the paragraph below Proposition 3, a rise in τ b not only redistributes income
from the rich to the poor but also decreases the long-run growth rate and the investmentin private capital. The latter leads to a decrease in the wage income (from (7a)), and thebequest income plays a more important role in savings. Because the rich receive morebequest, wealth inequality increases. This mitigates the eﬀect of τ b on wealth inequality.
We summarize the redistributive policy eﬀect in the steady state S.
Result 2. An increase in τ b reduces wealth inequality and the growth rate at the steadystate S. The eﬀect of decreasing wealth inequality is relatively small.

Our investigation throughout this section leads to the following policy implication. A
redistributive policy aimed at reducing wealth inequality faces a trade-oﬀ between equality and growth in the steady state S. Moreover, in the economy with a large public
debt/private capital ratio, implementing such a redistributive policy might be very risky
because it can take the economy on an unstable path from which wealth inequality andpublic debt continue to increase.

7

Conclusion

This study constructs an endogenous growth model with heterogeneous agents to examine
the correlation between the sustainability of the public debt and wealth inequality. We
show there is a threshold of public debt for each level of wealth inequality in order forgovernment to sustain the fiscal policy and the threshold of public debt is increasing in
wealth inequality. In addition, we investigate the eﬀects of budget deficit and redistributive
20

policies on the sustainability of the public debt. We show that an increase in the deficit
ratio or the redistributive tax reduces the range of sustainable public debt. That is, inthe economy with large public debt, such a policy change makes the economy fall into theunsustainable region in which both public debt and wealth inequality continue to increase.
There are several interesting directions for future research. First, this study used an
OLG model with the AK production structure and assumed that the government expenditure is public consumption. However, it is well known that productive government
spending is one of the important factors of economic growth. In future research, it would beinteresting to consider Barro’s (1990) model. Second, we do not consider the heterogeneity of wage income. Incorporating human capital accumulation generates an endogenous
disparity of wage income and could allow us to study how the growth–income inequality
relationship aﬀects the sustainability of public debt.

AppendixA

Proof of Lemma 1

As shown in Figure 4, the intersection of ε(xt ) and η(φt−1 ) determines the value of xt ,which satisfies (19) for a given φt−1 . When φt−1 increases, the function η(φt−1 ) increases,
as depicted by the broken line in Figure 4. Thus, xt rises correspondingly. As a result,φt = φt−1 locus can be depicted as an upward-sloping curve on the (xt , φt−1 ) plane in Figure
1. Note that the definition of φt implies φt−1 ∈ (0, 1). Let us define φ̃ as φ̃ ≡ η −1 (ε(0)).
Since limφt−1 →1 η(φt−1 ) = +∞ and limφt−1 →0 η(φt−1 ) = −∞, φt = φt−1 locus has an
asymptote φt−1 = 1 when xt → ∞ and φt−1 has a lower limit φ̃ when xt = 0.15
[Figure 4]

15

Taking the total diﬀerentials of (19) yields[]
β (αR − αP ) (1 − φt−1 ) µ−1
dφt−1
1 (1 + γ + 2γxt ) + γ[ −1] > 0.
=
dxt φt =φt−1(1 − γ)δαR φ−2
t−1 + β (αR − αP ) (1 + xt ) µ1 (1 + γxt ) + γ

Therefore, we can observe that φt = φt−1 locus is an upward-sloping curve and has an asymptote φt−1 = 1
on the (xt , φt−1 ) plane.

21

B

Proof of Lemma 2

We derive xt+1 = xt locus. Diﬀerentiating ζ(xt ) with respect to xt yields[
ζ ′ (xt ) =(γ(1 + xt )−1 + λΓ
xt)(+ (1 + γxt )1 + λΓ
xt)− (1 + γxt )(1 + xt ) λΓx2t]β(1 + xt )[1 + γ(xt + µ1 )][()](1 + γxt )(1 + xt ) 1 + λΓ
−
ᾱ(1
−
γ)µ
1 [1 + γ(1 + 2xt + µ1 )]xt
β(1 + xt )2 [1 + γ(xt + µ1 )]2

.

(B.1)

We rearrange (B.1) as follows:ζ ′ (xt ) =

]
[
(
)
λΓ
+ ᾱ(1 − γ)µ1 [1 + γ(1 + 2xt + µ1 )](1 + xt )2 γ 2 µ1 1 + λΓ
−
(1
+
γx
)(1
+
γx
+
γµ
)
2
t
t
1
xt
x
t

β(1 + xt )2 [1 + γ(xt + µ1 )]2

Γ(1 + xt )2 [γ 2 (1 − g)x2t − λ{1 + γ(2xt + µ1 )} + ᾱ(1 − γ)µ1 x2t [1 + γ(1 + 2xt + µ1 )]],βx2t (1 + xt )2 [1 + γ(xt + µ1 )]2
σ(xt ).
=
2βxt (1 + xt )2 [1 + γ(xt + µ1 )]2
=

where
[
]
σ(xt ) ≡ Γ(1+xt )2 γ 2 (1 − g)x2t − λ{1 + γ(2xt + µ1 )} + ᾱ(1−γ)µ1 x2t [1 + γ(1 + 2xt + µ1 )] .
We then diﬀerentiate σ(xt ) with respect to xt as follows:σ ′ (xt ) = µ2 x3t + µ3 x2t + µ4 xt + µ5 ,where
µ2 ≡ 4Γγ 2 (1 − g) + 2γ ᾱ(1 − γ)µ1 > 0,µ3 ≡ 6Γγ{γ(1 − g) − λ} + 2ᾱ(1 − γ)µ1 (1 + 2γ + γµ1 ),µ4 ≡ 2Γγ 2 (1 − g) + 2γ ᾱ(1 − γ)µ1 − 8λΓγ − 2λΓ(1 + γµ1 ),µ5 ≡ −2λΓ(1 + γ + γµ1 ) < 0.
Assuming that γ(1 − g) > λ, we obtain µ4 > 0. As a result, there is a unique x̄¯ > 0 thatsatisfies σ ′ (x̄¯) = 0. σ ′ (xt ) < 0 holds when 0 < xt < x̄¯ and σ ′ (xt ) > 0 holds when xt > x̄¯.
22,Moreover, since σ(0) = −λΓ(1 + γµ1 ) < 0 and limxt →∞ σ(xt ) = ∞, there is a unique x̄ > 0
that satisfies σ(x̄) = 0. σ(xt ) < 0 holds when 0 < xt < x̄ and σ(xt ) > 0 holds when xt > x̄.
That is, we obtain
ζ ′ (xt ) < 0

if

0 < xt < x̄,

ζ ′ (xt ) > 0

if

xt > x̄.

In addition, we obtain the following results:
lim ζ(xt ) = ∞,

xt →0

(

lim ζ(xt ) = lim 

xt →∞

xt →∞

1
+γ
xt

)(

)

ᾱ(1−γ)µ1
1 + λΓ
−
2
xt
xt
1
(
)(
)
− αP  = − αP .
β1
β x1t + 1 1+γµ
+γ
xt
1
+1
xt

)(

By using these results, we obtain a curve, xt+1 = xt locus, which is U-shaped and has1(β −1 − αP ) > 1 when xt → ∞ and φt−1 → ∞ when xt → 0.
asymptotes φt−1 = αR −αP

xt+1 = xt locus is depicted in Figure 1.

C

Properties of Kt+1 /Kt = 0 locus

Diﬀerentiating Λ(xt ) with respect to xt , we obtain[
Λ′ (xt ) =

(
)
(
)
]
λΓ
λΓ
(1 + γxt ) 1 + λΓ
+
γx
1
+
−
(1
+
γx
)
t
t
xt
xt
xt

−

β(1 + xt )[1 + γ(xt + µ1 )]
)
]
[
(
−
ᾱ(1
−
γ)µ
xt (1 + γxt ) 1 + λΓ
1 [1 + γ(1 + 2xt + µ1 )]xt,
β(1 + xt )2 [1 + γ(xt + µ1 )]2(1 + 2γxt + γλΓ)(1 + xt )[1 + γ(xt + µ1 )] − (1 + γxt )(xt + λΓ)[1 + γ(1 + 2xt + µ1 )]=β(1 + xt )2 [1 + γ(xt + µ1 )]2ᾱ(1 − γ)µ1 [1 + γ(1 + 2xt + µ1 )]+,β(1 + xt )2 [1 + γ(xt + µ1 )]2(1 − λΓ)(1 + γxt )2 + γµ1 (γx2t + 2γxt + 1 − λΓ + γλΓ) + ᾱ(1 − γ)µ1 [1 + γ(1 + 2xt + µ1 )].
=β(1 + xt )2 [1 + γ(xt + µ1 )]2
(C.1)

23

Using µ1 ≡ Γ(1 + λ − g), we can rewrite the numerator of (C.1) as
ν1 x2t + ν2 xt + ν3 ,where
ν1 ≡ γ 2 [1 + (1 − g)Γ],ν2 ≡ 2γ{1 + [γλ + (1 − g)γ − λ]Γ + (1 − γ)ᾱµ1 },ν3 ≡ (1 + γµ1 ){1 + [ᾱ(1 − γ)(1 + λ − g) − λ]Γ} + γ 2 µ1 λΓ + ᾱ(1 − γ)γµ1 .
Assuming that ᾱ(1 − γ)(1 + λ − g) > λ and using the assumption γ(1 − g) > λ in Lemma
2, ν1 > 0, ν2 > 0, and ν3 > 0 hold, and hence, Λ′ (xt ) > 0 holds for all xt ≥ 0. In addition,
we obtain the following property for Λ(xt ):(lim Λ(xt ) = lim 

xt →∞

xt →∞

)

ᾱ(1−γ)µ1
1 + λΓ
−
xt
x2t
1
(
)(
)
− αP  = − αP .
β1
β x1t + 1 1+γµ
+γ
xt

1
+γ
xt

)(

Thus, Kt+1 /Kt = 0 locus is upward sloping if ᾱ(1−γ)(1+λ−g) > λ and has an asymptote1(β −1 − αP ) > 1 when xt → ∞ on the (xt , φt−1 ) plane.
φt−1 = αR −αP

DPhase diagram

First, we examine whether φt > φt−1 or φt < φt−1 at each point of the (xt , φt−1 ) plane.
By using (9), we obtainφt ⋛ φt−1 ⇔ ε(xt ) ⋛ η(φt−1 ).
Suppose that (x, φ) is a combination that satisfies (19); that is, ε(x) = η(φ) holds. Moreover, let us define x by x > x. Since ε(xt ) is increasing in xt , ε(x) < η(φ) holds. As aresult, we obtain φt < φt−1 on the left of φt = φt−1 locus. Similarly, we obtain φt > φt−1
on the right of φt = φt−1 locus.
Next, we investigate whether xt+1 > xt or xt+1 < xt at each point of the (xt , φt−1 )24plane. From (6), we obtain
xt+1 ⋛ xt ⇔ φt−1 ⋚ ζ(xt ).
Therefore, we obtain xt+1 > xt below xt+1 = xt locus and xt+1 < xt above xt+1 = xt locus.
By using these results, we can depict a phase diagram, as in Figure 1.

E

Local stability around the steady states

Approximating (17) and (18) linearly around the steady state k (k ∈ {S, U }), we obtain

φ − φ∗k t
xt+1 − x∗k




k
k
∗
Jφφ Jφx
φ − φk  t−1
,
=
k
k
Jxx
Jxφ
xt − x∗k

(E.1)

where
∗
∗
R ∗
∗
∗ A ∗
∗
φ∗k GRφ (xk , φk ) + G (xk , φk ) − φk Gφ (xk , φk ),GA (x∗k , φ∗k )φ∗ GR (x∗ , φ∗ ) − φ∗ GA (x∗ , φ∗ )kJφx
= k x k Ak ∗ k∗ x k k ,G (xk , φk )
A ∗
∗
∗
∗
G (xk , φk )(1 − λΓ) − (1 + x∗k )(x∗k + λΓ)GAx (xk , φk )
i
Jxx
=
,
[(1 + x∗k )GA (x∗k , φ∗k ) − (x∗k + λΓ)]2∗∗(x∗k + λΓ)(1 + x∗k )GAφ (xk , φk )
k
,
Jxφ
=−
[(1 + x∗k )GA (x∗k , φ∗k ) − (x∗k + λΓ)]2

k
Jφφ
=

R

A

∂G (xt ,φt−1 )∂G (xt ,φt−1 )∗∗∗∗where GRand GAz (xk , φk ) ≡z (xk , φk ) ≡∂z∂z(xt ,φt−1 )=(x∗ ,φ∗ )(xt ,φt−1 )=(x∗ ,φ∗ )
k

k

k

k

for z ∈ {xt , φt−1 }. Note that GA (x∗k , φ∗k ) = GR (x∗k , φ∗k ) holds. Let us denote the two eigenvalues of the Jacobian matrix of the linearized system as ek1 and ek2 . These eigenvalues are
k
k
k
k
k
k
).
Jxφ
− Jφx
Jφφ
)e + (Jxx+ Jxx
the roots of the characteristic polynomial: P (e) = e2 − (JφφTo consider this, we conduct a numerical analysis. We adopt the following benchmark
parameters: γ = 0.2, Γ = 12, g = 0.2, δ = 0.5, β = 0.3, αR = 0.45, and αP = 0.25. Table
3 shows the two eigenvalues for each steady state.
[Table 3]
From Table 3, we find that both eS1 and eS2 take real positive values and satisfy 0 <eS1 < eS2 < 1, and then the steady state S is a sink. We further find that both eU1 and eU225

take real positive values and satisfy 0 < eU1 < 1 < eU2 , and then the steady state U is asaddle point.

F

Introduction of a redistributive public policy

Under the redistributive public policy, we rewrite the following optimal conditions of individuals:
β [1 + (1 − τt+1 )rt+1 ] sR
t
,
b
1+τ
[
]
P
c1P
t = (1 − αP ) (1 − τt )wt + bt + Tt ,[]sPt = αP (1 − τt )wt + bPt + Tt ,

bR
t+1 =

(F.1)
(F.2)
(F.3)

Other optimal conditions are the same as (4a)-(4d). In addition, (5) becomes[sRt = αR]β{1 + (1 − τt )rt } R
st−1 ,(1 − τt )wt +1 + τb[]sPt = αP (1 − τt )wt + Tt + β{1 + (1 − τt )rt }sPt−1 .

(F.4)
(F.5)

Substituting (F.1) into the government redistributive policy δN τ b bR
t = (1 − δ)N Tt , weobtain
(
Tt =

δ
1−δ

)(

τb
1 + τb)β[1 + (1 − τt )rt ]sR
t−1 .

(F.6)

The asset market-clearing condition (6), equations determined in the production sector((7a), (7b) and (7c)), and those in the public sector ((8), (9) and (10)) remain unchanged.
Furthermore, derivations of the other equations are conducted in the same manner as
Section 3. Some algebra rewrite GA (xt , φt−1 ) (in (13), (16), (17), and (18)), GK (xt , φt−1 )(in (16) and (18)), and GR (xt , φt−1 ) (in (14) and (17)) into()[]ᾱ(1 − γ)µ1
γµ1
αR − αPG (xt , φt−1 ; τ ) =+β 1+
φt−1 + αP ,(1 + γxt )(1 + xt )1 + γxt
1 + τbAb(F.7)

GK (xt , φt−1 ; τ b ) = (1 + xt )GA (xt , φt−1 ; τ b ) − (xt + λΓ),

(F.8)

26

and
]
[
(
)
αR
δ(1 − γ)µ1
βγµ1
G (xt , φt−1 ; τ ) =1+
φt−1 ,+
φt−1 (1 + γxt )(1 + xt ) 1 + τ b
1 + γxt
R

b

(F.9)

respectively. Translating these into (17), and (18) transforms the dynamic systems as
GR (xt , φt−1 ; τ b )
φt
= A
φt−1
G (xt , φt−1 ; τ b )1 + λΓ/xt
xt+1
=
.
A
xt
(1 + xt )G (xt , φt−1 ; τ b ) − (xt + λΓ)(F.10)

Setting φt = φt−1 and xt = xt+1 in (F.10) yields (27) and (28).

G

φt = φt−1 locus under the redistributive public policy

We rearrange (27) as follows:(1 + τ b )(1 − γ)η̃(φt−1 ) =[αR − (1 + τ b )αP ] − (αR − αP )φt−1

(
)
δαR
ᾱ −
.
φt−1

Here, we assume αR > (1 + τ b )αP . Under this assumption, when φt−1 increases, thefunction η̃(φt−1 ) increases. Similarly to the analysis in Appendix A, φt = φt−1 locus canbe depicted as an upward-sloping curve on the (xt , φt−1 ) plane. Moreover, we define φ̂
(
)
b
αR
−
α
< 1 and φ̃(τ b ) ≡ η̃ −1 (ε(0)), respectively. Sinceand φ̃(τ b ) as φ̂ ≡ α1+τ
P
1+τ b
R −αP
limφt−1 →φ̂ η̃(φt−1 ) = +∞ and limφt−1 →0 η̃(φt−1 ) = −∞, φt = φt−1 locus has an asymptote
φt−1 = φ̂ when xt → ∞ and φt−1 has a lower limit φ̃(τ b ) when xt = 0.16

16

Taking the total diﬀerentials of (27) yields
)
][
[(
]
αP τ b
αR
−
α
(1
−
φ
)
−
φ
µ−1
β
P
t−1
1 (1 + γ + 2γxt ) + γ
1+τ b
1+τ b t−1
dφt−1
(
)
=
[
].
b
dxt φt =φt−1(1 − γ)δα φ−2 + β αR − α + αP τ (1 + x ) µ−1 (1 + γx ) + γ
R

t−1

1+τ b

P

1+τ b

t

1

t

From the definition of φ̂, dφdxt−1> 0 holds when 0 < φt−1 < φ̂. Therefore, we can also observe
φt =φt−1tthat φt = φt−1 locus is an upward-sloping curve and has an asymptote φt−1 = φ̂ on the (xt , φt−1 ) plane.

27References[1] Abel, A. B. and M. Warshawsky (1988) Specification of the joy of giving: Insightsfrom altruism. Review of Economics and Statistics 70, 145–149.
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29

φt –1

xt＋1 = xt locus

1H

φt = φt–1 locus

U

S
Q

2

Thr
e

sho

ld

φ˜

0.5

Q1

Kt+1
Kt = 0

H

0

–x

xH

Figure 1: Phase Diagram on the (xt , φt−1 ) plane.

30xt

φt –1

xt＋1 = xt locus
φt = φt–1 locus

U
¸↑
U’
S

S’

¸↑
Thresholdxt

0

Figure 2: Eﬀects of an increase in λ.

31

φt –1

xt＋1 = xt locus

φt = φt–1 locus

U
τ b↑
U’

S
S’

τb↑
Thresholdxt

0

Figure 3: Eﬀects of an increase in τ b .

32

ε(xt)η(φt –1)β( μ1-1 + γ)

xt

0Figure 4: .Derivation of xt+1 = xt locus.

33

λ

x∗S

φ∗S

ŶS∗

x∗U

φ∗U

ŶU∗

0.01

0.06460.65852.8582

1.61210.68351.07440.02

0.14310.65962.67701.44700.6806

1.16590.03

0.2454

0.66112.46691.25880.6773

1.28600.04

0.40330.66352.19011.0156

0.6732

1.4726

Table 1: Deficit ratio and steady state values.

λ

∆x∗S

∆φ∗S

∆ŶS∗

∆x∗U

∆φ∗U

∆ŶU∗

0.01

1.76%

-2.46%-1.12%-4.28%-6.17%

0.31%

0.02

2.27%

-2.64%-1.39%-4.75%-5.75%

0.71%

0.03

3.25%

-2.87%-1.87%-5.65%-5.27%

1.33%

0.04

6.61%

-3.23%-3.37%-8.62%-4.67%

3.03%

Table 2: Eﬀects of an increase in bequest tax rate.
Note: The changes in the steady state values are expressed in percentage points.

λ(eS1 , eS2 )(eU1 , eU2 )0.01(0.098, 0.378)(0.219, 2.566)

0.02(0.104, 0.439)(0.207, 2.227)

0.03(0.113, 0.527)(0.193, 1.874)

0.04(0.126, 0.678)(0.174, 1.467)

Table 3: Eigenvalues at the steady states S and U .

34