Discussion Papers In Economics
And Business

Inequality and Public Debt: A Positive Analysis
Ryo Arawatari and Tetsuo Ono
Discussion Paper 15-01-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANInequality and Public Debt: A Positive Analysis
Ryo Arawatari and Tetsuo Ono
Discussion Paper 15-01-Rev.

September 2015

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANInequality and Public Debt: A Positive Analysis
Ryo Arawatariy

Tetsuo Onoz

Nagoya University

Osaka University

Abstract
This study extends the multi-country, politico-economic model of …scal policy
developed by Song, Storesletten, and Zilibotti (2012) to incorporate wage inequality within each country. In this extended framework, we present con‡ict over …scal
policy within and across generations and show that a low-inequality country realizes tight …scal policy with low public debt accumulation, whereas a high-inequality
country experiences loose …scal policy with high public debt. This model prediction is consistent with empirical evidence from OECD countries for the past three
decades.
Keywords: …scal policy; inequality; probabilistic voting; public debt; small open
economies.
JEL Classi…cation: D72; E62; F34; H41;H60

Ryo Arawatari is supported by a grant from the Japan Society for the Promotion of Science (JSPS)
through a Grant-in-Aid for Young Scientists (B) (No. 23730189). Tetsuo Ono is supported by Grantin-Aid for Scienti…c Research (C) from the Ministry of Education, Science and Culture of Japan No.
15K03509.
yCorresponding author: Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku,
Nagoya, Aichi 464-8601, Japan. E-mail: arawatari@soec.nagoya-u.ac.jpzGraduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. Email: tono@econ.osaka-u.ac.jp

1

Introduction

Conventional economic theory suggests that higher income inequality is associated witha higher level of income redistribution (see, e.g., Romer, 1975; Roberts, 1977; Meltzer
and Richard, 1981). Given government budget constraints, higher inequality should increase pressure on politicians to shift the fiscal burden from the present generation to
future generations. This pressure incentivizes politicians to finance a part of government
expenditure by issuing public debt, which may result in a higher debt-to-gross domestic
product (GDP) ratio in the long run.
The purpose of this study is to develop a simple model that examines the aforementioned argument from a theoretical point of view. For this purpose, we use Song,Storesletten, and Zilibotti’s (2012) multi-country politico-economic model of public debt.
They present a two-period overlapping-generations model with many small open countries that diﬀer in their public goods preferences. Each country decides its public goods
provision financed by taxes and public debt through probabilistic voting, reflecting theconflicting preferences of two successive generations. In this model, they show that public
goods preferences shape cross-country diﬀerences in fiscal policy.
This study modifies their framework by assuming away diﬀerences in preferencesamong countries and instead introduces wage inequality within each country. In thisalternative framework, we present conflict over fiscal policy within and across generations
and show that when the intertemporal elasticity of substitution (EIS) of agents is lessthan one, a low-inequality country realizes tight fiscal policy with low public debt accumulation, whereas a high-inequality country experiences loose fiscal policy with highpublic debt. The reverse occurs when EIS is greater than one: a higher inequality level isassociated with a lower level of public debt.
To evaluate the empirical plausibility of the two conflicting model predictions, wefirst look at the evidence of inequality and public debt from Organisation for Economic
Co-operation and Development (OECD) countries for the past three decades. We find apositive and highly significant correlation between inequality and public debt. We alsofind that the assumption of EIS below one is in line with many empirical estimates (see,e.g., Kydland and Prescott, 1982; Hall, 1988; Campbell and Mankiw, 1989; Browning,
Hansen, and Heckman, 1999; Campbell, 1999) based on a literature review. These twosources of evidence suggest that EIS below one is an empirically valid prediction.
The present study contributes to the literature on the political economy of public
debt. While many studies consider how politics determines the size of public debt, mostabstract away the role of inequality among voters. Previous studies instead focus on
the roles of common pool problems (Tabellini, 1986; Velasco, 1999), political instability(Persson and Svensson, 1989; Aghion and Bolton, 1990; Alesina and Tabellini, 1990;
1

Tabellini and Alesina, 1990; Natvik, 2013), altruistic and selfish agents (de Walque andGevers, 2001), tax smoothing (Battaglini and Coate, 2008), and intergenerational conflict(Song, Storesletten, and Zilibotti, 2012).
However, studies by Cukierman and Meltzer (1989) and Azzimonti, de Francisco, andQuadrini (2014) are exceptions. Cukierman and Meltzer (1989) develop an overlappinggenerations model with bequest motives and income inequality, wherein high-income individuals are bequest unconstrained and indiﬀerent to the intergenerational reallocationof taxes. By contrast, low-income individuals are bequest constrained, and thus benefit
from substituting future taxes for current taxes by means of debt issued to finance government expenditure. In particular, when low-income individuals are decisive in voting,they prefer a higher level of debt over a reduction of their wages.
Cukierman and Meltzer (1989) suggest a positive relationship between inequality andpublic debt. However, their analysis is confined to a closed economy, ignoring crosscountry diﬀerences in fiscal policy. Azzimonti, de Francisco, and Quadrini (2014) overcome
this problem by developing a multi-country model with income risk. They show thathigher risk in a home country results in more public debt issues in home and foreigncountries, and argue that public debt responds positively to income inequality, provided
that rising income inequality is associated with an increase in individual income risk.
Our study diﬀers from that of Azzimonti, de Francisco, and Quadrini (2014) in that
we focus directly on income inequality, particularly wage inequality, and show that when
EIS is below one, rising inequality in a home country may result in an increase in its
public debt and a decrease in public debt in foreign countries. These results based on adiﬀerent mechanism from that in Azzimonti, de Francisco, and Quadrini (2014), could beviewed as a component of an alternative testable hypothesis for the relationship between
inequality and public debt from a political economy perspective.
The remainder of this paper is organized as follows. Section 2 presents our model.
Section 3 provides a characterization of political equilibrium. Section 4 demonstrates
numerical examples and discusses the implications of the results. Proofs are given in the
appendices.

2

Model

We base our model on that developed by Song, Storesletten, and Zilibotti (2012) andconsider a discrete-time economy where time is denoted by t = 0, 1, 2, · · · . We assume
that the world economy consists of a unit mass of small open countries. Each country is
populated with overlapping generations of agents who live for two periods: they work in
the first and retire in the second. Each generation has a unit mass.
2Within each generation, agents belong to either of two income classes based on theirfirst-period wages: the poor or the rich. Let wjp and wjr denote the exogenous wages ofthe poor and the rich, respectively, in country j, and let π ∈ (0, 1) denote the fraction of
the poor within a generation. The average income level, assumed to be identical acrosscountries, is denoted by w̄ = w̄j ≡ πwjp + (1 − π)wjr . We introduce this assumption tocompare two groups of countries, high-inequality countries and low inequality countries,
which have similar average income levels but diﬀerent levels of wage inequality.
Countries are classified into two types according to wage inequality, high and low(j = H, L), the proportions of which are ν and 1 − ν, respectively. The wage profilepr}, and the wage profile of L-type countries isof H-type countries is WH ≡ {wH
, wH
p
p
p
r
r
r
WL ≡ {wL , wL }, where wH /wH > wL /wL holds. The ratio of high-to-low income is higher
in high-inequality countries than in low-inequality countries.

2.1

Utility Maximization

Each agent receives utility from private consumption and publicly provided goods. The
utility of a i-type young agent in country j = {L, H}, born in period t, is
1−σ
(cyi
−1
(gj,t )1−σ − 1j,t )yiUj,t =
+θ·
+β·
1−σ
1−σ

{

1−σ
1−σ
(gj,t+1)−1(coi−1j,t+1 )+ θλ ·
1−σ
1−σ

}
,

oi
where cyi
j,t is consumption during youth, cj,t+1 is consumption during old age, gj,t is public
goods provision in period t, β ∈ (0, 1] is the discount factor, and θ(> 0) and θλ(> 0)
capture the preference weights for public goods for the young and the elderly agents,
respectively. In particular, the parameter λ(> 0) captures the relative strength of theelderly agents’ preference for public goods.
The parameter σ(> 0) measures the elasticity of the marginal utility of its argument.
An inverse of σ, 1/σ, measures the elasticity of intertemporal substitution (henceforth,
EIS). The case σ = 1 leads to a logarithmic utility function of the form Ujtyi = ln cyi
jt +
( oi
)
θ ln gjt + β · ln cjt+1 + θ ln gjt+1 . We abstract this case away in the following analysis
because the eﬀect of wage inequality on fiscal policy disappears under this specification,
an empirically implausible scenario.
The individual budget constraints of i-type agents during youth and old age are given
by
i
i
cyi
j,t + sj,t ≤ (1 − τj,t ) · wj ,icoi
j,t+1 ≤ Rsj,t ,where sij,t is savings, τj,t is the income tax rate in period t, and R is the (endogenous) world3

interest rate. Following Song, Storesletten, and Zilibotti (2012), we focus on stationary
equilibria and thus characterize the allocation of each country as functions of a constantR.
We solve the utility maximization problem and obtain the following consumption andsavings functions:
cyi
j,t =

1
1−σ

1 + β(βR) σ· (1 − τj,t ) · wji ,

1−σ

coi
j,t+1 =

β(βR) σ1 + β(βR)

1−σσ· R · (1 − τj,t ) · wji ,1−σsijt =

β(βR) σ1 + β(βR)

1−σσ· (1 − τj,t ) · wji .

Ignoring irrelevant terms, we can express the indirect utility function of an i-type
young agent as:
yi
Vj,t
=

}σ {
[{
]
}
1−σ
1
i 1−σ
1−σ
1−σ
σ
· 1 + β(βR)· (1 − τj,t ) · wj
+ θ · (gj,t )+ βλθ · (gj,t+1 ).
1−σ

The first term on the right-hand side denotes the utility of consumption during youth andold age, the second term denotes the utility of public goods during youth, and the third
term denotes the utility of public goods during old age.
For elderly agents in period t, their indirect utility functions are expressed as:
op
or
o
Vj,t
= Vj,t= Vj,t=1· λθ · (gj,t )1−σ ,1−σwhere the irrelevant terms are omitted from the expression. All elderly agents have thesame indirect utility function regardless of their type because they have predeterminedsavings during youth and we assume an individual’s utility function is additively separable.

2.2

Government Budget Constraints

Government bonds are traded in international asset markets. Given inherited debt peryoung agent, bj,t , the government of country j chooses the income tax rate, τj,t ; public
goods expenditure, gj,t ; and to new issue bonds, bj,t+1 ; subject to the following dynamic
budget constraint:
bj,t+1 = gj,t + Rbj,t − τj,t w̄.
(1)
Governments are committed to not repudiating debt. Thus, sovereign debt cannot
exceed the present value of the maximum feasible tax revenue (i.e., the natural debt
limit). In an environment with a constant interest rate and exogenous wages, the tax
4revenue in period t is maximized at τj,t = 1. Therefore, the natural debt limit, denotedby b̄, is given by:w̄
bj,t+1 ≤ b̄ ≡.
R−1
In the following analysis, we also assume a lower bound of b, denoted as b, and an upperbound of g, denoted as ḡ, for technical reasons.

3

Political Equilibrium

We adopt Lindbeck and Weibull’s (1987) probabilistic voting model in the operationof the political mechanism. In this model, the equilibrium fiscal policy maximizes aweighted sum of young and elderly voters’ indirect utilities. The weights capture therelative political clout of each group, reflecting both relative sizes and (exogenous) voting
turnouts. Formally, the political objective function is given by:
( yp
)
yr
o
Ω(τj,t , gj,t , gj,t+1 ; wjp , wjr , R) = ωVj,t+ (1 − ω) · πVj,t+ (1 − π) · Vj,t,where ω ∈ (0, 1) is the elderly agents’ relative weight.1
Each country determines fiscal policy by the dynamic games between successive generations of voters. An international asset market clearing condition determines the worldinterest rate. Given the assumption of small open economies, voters take the equilibrium
interest rate sequence as given.
We restrict this study to stationary Markov perfect equilibria. Voters base their strategies only on the payoﬀ-relevant state variable. In our model, private wealth does not aﬀect
elderly voters’ political preferences. Therefore, the public debt, denoted by b, is the only
payoﬀ-relevant state variable. Hereafter, we omit time index t and use recursive notion
with x′ denoting next-period x.
Definition 1. A stationary Markov-perfect political equilibrium is an interest rate, R,a stationary debt distribution {bj }j∈{H,L} , and a three-tuple ⟨B, G, T ⟩ , where B :
[ ]
[ ]
[ ]
b, b̄ × Wj × ℜ+ → b, b̄ is a debt rule, b′ = B(b; wjp , wjr , R); G : b, b̄ × Wj × ℜ+ →[ ][0, ḡ] is a government expenditure rule, g = G(b; wjp , wjr , R); and T : b, b̄ × Wj ×ℜ+ → [0, 1] is a tax rule, τ = T (b; wjp , wjr , R), such that the following conditionshold:(i) ⟨B(bj ; wjp , wjr , R), G(bj ; wjp , wjr , R), T (bj ; wjp , wjr , R)⟩ = arg max{b′ ,g,τ }Ω(τj , gj , gj′ ; wjp , wjr , R), subject to (1), where g ′ = G(bj ; wjp , wjr , R), and the1

Persson and Tabellini (2000, Chapter 3) and Acemoglu and Robinson (2006, Appendix B) provide
an explicit micro-foundation for this model. Song, Storesletten, and Zilibotti (2012, Appendix B) outline
the process to derive the political objective function used in our study.

5

government budget constraint is satisfied as follows:
B(bj ; wjp , wjr , R) = G(bj ; wjp , wjr , R) + Rbj − T (bj ; wjp , wjr , R)w̄.
(ii) The international asset market clearing condition isν {πspH + (1 − π)srH } + (1 − ν) {πspL + (1 − π)srL } = νb′H + (1 − ν)b′L ,where b′j = B(b; wjp , wjr , R) and sijt =

1−σ
σ
1−σ
1+β(βR) σβ(βR)· (1 − τj,t ) · wji .

yioi
We substitute Vj,tand Vj,t
into the political objective function Ω and obtain

[
{
}σ
1−σ
1
· (1 − ω) · 1 + β(βR) σ· (1 − τj )1−σ · w
ej
1−σ]+ {(1 − ω) + ωλ} · θ · (gj )1−σ + (1 − ω)βλθ · (gj′ )1−σ ,Ω(τj , gj , gj′ ; wjp , wjr , R) =

(2)

where
w
ej ≡ π(wjp )1−σ + (1 − π)(wjr )1−σ .
The term w
ej measures the impact of wage inequality on the political objective function.
Rising inequality in country j increases w
ej if σ > 1 (i.e., if 1/σ < 1), while it decreasesw
ej if σ < 1 (i.e., if 1/σ > 1). The impact of inequality on w
ej is crucial in the following
analysis.
The small open economies assumption implies that voters take the equilibrium interest
rate as given. Given R, the first-order conditions with respect to gj and τj are:( ′ )σ∂G(b′j ; wjp , wjr , ω, R)
gj(1 − ω)βλ= (−1) ··,gj(1 − ω) + ωλ
∂b′j
(
)1
1−σ
1
1 + β(βR) σθ{(1 − ω) + ωλ} σ
1
1/σ
· (w
ej ) =· (w̄) σ · ,1 − τj
1−ω
gj

(3)
(4)

where gj = G(bj ; wjp , wjr , R), gj′ = G(bj ; wjp , wjr , R), τj = T (bj ; wjp , wjr , R), and b′j = gj +Rbj − τj w̄ ≡ B(bj ; wjp , wjr , R). Appendix A.1 provides the derivations for these conditions.
Condition (3) is a generalized Euler equation for public goods provision. The term∂G(b′j ; wjp , wjr , ω, R)/∂b′j on the right-hand side captures young voters’ disciplining eﬀect,
which is qualitatively similar to that demonstrated in Song, Storesletten, and Zilibotti(2012). The young agents’ concern for future public goods provision controls current
fiscal policy and prevents the government from running up too much public debt. The
term (1−ω)βλ/ {(1 − ω) + ωλ} suggests that the disciplining eﬀect strengthens as elderly
agents’ preference weight for public goods, λ, increases.
6Condition (4) states that the government chooses a tax rate to equate the marginal
cost on the left-hand side to the marginal benefit on the right-hand side. In particular, themarginal cost depends on wage inequality within a generation captured by the term w
ej .
Wage inequality aﬀects public goods provision and public debt through the determinationof the tax rate. This point will be further investigated in the following analysis.
To find policy functions that satisfy (3) and (4), we presume and verify linear equilibrium policy functions and obtain the following result:Lemma 1. Given R and bj , country j’s policy functions in a stationary Markov perfectpolitical equilibrium are given byG(bj ; wjp , wjr , R) = γj∗ · (b̄ − bj ),
{
} (
1−σ
p
r
T (bj ; wj , wj , R) = 1 − 1 + β(βR) σ ·(1 − ω)θ{(1 − ω) + ωλ}() σ1(1 − ω)βλB(bj ; wjp , wjr , R) = b̄ −· γ∗· (b̄ − bj ),(1 − ω) + ωλ j

) σ1 ( ) σ1wej·
· γj∗ · (b̄ − bj ),w̄

where γj∗ (> 0) satisfies the following condition:
[

} (
{
1−σ
R − γj · 1 + w̄ · 1 + β(βR) σ ·1σ(= (γj ) ·(1 − ω) βλ(1 − ω) + ωλ

) σ1

1−ω
θ{(1 − ω) + ωλ}) σ1 ( ) σ1 ]
w
ej
·
w̄

.

Proof. See Appendix A.2.
Hereafter, we assume that a natural stability condition holds and focus on a steadystate equilibrium where the relevant terms are constant across periods. Our task is todetermine the world interest rate and the associated steady-state debt distribution. We
thus rewrite the law of motion of debt in Lemma 1 as:)(b̄ − b′j = ϕ∗j · b̄ − bj ,whereϕ∗j = ϕ∗(wjp , wjr , R)(≡(1 − ω) βλ· γ∗(1 − ω) + ωλ j

) σ1.

We define ϕ̄ as:pr, R)} .
, wH
ϕ̄ ≡ max {ϕ∗ (wLp , wLr , R) , ϕ∗ (wH

Recall that there are two groups of countries, high- and low-inequality countries, denoted by j = H and L, respectively. Following Song, Storesletten, and Zilibotti (2012),7

we can use the following to compute the steady-state world interest rate. First, there is()
no country with ϕ∗ wjp , wjr , R > 1. Otherwise, either the high- or low-inequality countries would accumulate ever-increasing surpluses, while the other country groups could
accumulate a maximum debt of b̄. This condition prevents the international asset market
from clearing. Second, there is no R such that ϕ̄ < 1. Otherwise, both groups of countries
would accumulate a maximum public debt of b̄, which also prevents the international asset
market from clearing.
Given these two conditions, we can conclude that the steady-state equilibrium interest
rate is determined by ϕ̄ = 1. Because ϕ∗j is increasing in γj∗ , the countries with a higher γj∗are characterized by ϕ∗j = 1 and accumulate public debt below b̄, whereas countries witha lower γj∗ are characterized by ϕ∗j < 1 and accumulate debt up to b̄, which lowers theminto poverty. The following proposition shows that EIS determines the financial balanceof low and high inequality countries.
Proposition 1. Consider a stationary Markov-perfect political equilibrium with the setof policy functions in Lemma 1. The world interest rate R∗ and the steady-state
distribution (bH , bL ) are as follows: (i) the world interest rate R∗ satisfies ϕ∗L = 1∗
if( σ > 1, and) ϕH = 1 if σ (< 1; (ii) the) steady-state debt distribution, (bH , bL ), is∗∗
are defined byb̄, ξL∗ · b̄ if σ > 1, and ξH· b̄, b̄ if σ < 1, where ξL∗ and ξH]−1) σ1 ( ) σ1
1
−
ω
w
e
L
·
·
· γL∗
,
ξL∗ ≡ 1 − (1 − ν) · 1 + w̄β (βR∗ )θ{(1 − ω) + ωλ}
w̄
[
]−1
(
) σ1 (
) σ1
1−σ
1
−
ω
w
e
H
∗
∗
ξH
≡ 1 − v + (1 − v)w̄β (βR∗ ) σ ··· γH.
θ{(1 − ω) + ωλ}
w̄
[

−1

1−σ
σ

(

Proof. See Appendix A.3.
Proposition 1 states that when σ > 1, a high-inequality country has a high tax rate,
no public goods provision, and a high public debt level such that τH = 1, gH = 0, andbH = b̄; while a low-inequality country has low tax, high levels of public goods, and alow public debt level such that τL ∈ (0, 1), gL > 0, and bL = ξL∗ · b̄ < b̄. The fiscal policy
direction reverses if σ < 1. The result in Proposition 1 suggests that wage inequality and
EIS play key roles in explaining the diﬀerences in fiscal policies between the two groupsof countries.
The result in Proposition 1 is somewhat extreme in the sense that either a low or ahigh-inequality country is distinguished by a corner solution: 100% taxation, no provisionof public goods, and a debt level approaching the natural debt limit. This property stems
from the assumption of inelastic labor supply. If we alternatively assume an elastic labor
supply, either country group is distinguished by an interior solution (Song, Storesletten,
8and Zilibotti, 2012), which is more plausible from an empirical viewpoint. However,
our study adopts an inelastic labor supply assumption to simply obtain a solution anddemonstrate the correlation between inequality and fiscal policy.
To understand the role of wage inequality and EIS in shaping fiscal policy, recall thefirst-order conditions for τj in (4). We can reformulate the expression in (4) as follows:[ p yp −σ]∂gj′r yr −σ′ −σw̄,πwj (cj ) + (1 − π)wj (cj )= (−1)βλθ(gj )
∂b′j

(5)

where the left-hand side is the marginal cost of taxation and the right-hand side is themarginal benefit of taxation. Appendix A.4 provides the derivation of (5).
Given the assumption that average income levels are identical across countries, weconsider a mean-preserving spread of wage distribution to compare the two groups ofcountries with similar average income levels but diﬀerent levels of wage inequality. To
evaluate the impact of the spread, we focus on the i-type young agents’ marginal cost of−σ
taxation, wji (cyi
, which is rewritten as:j ){−σwji (cyi= wji
j )wji (1 − τj )1 + β(βR)

1−σ
σ

}−σ

(

)
i 1−σ

= wj

{

(1 − τj )1−σ

1 + β(βR) σ}−σ, i = p, r,(6)

where the first equality is derived using the consumption function in Section 2.
The expression above suggests that an increase in the wages of the rich, wjr , has−σtwo opposing eﬀects on the marginal cost of taxation, wji (cyi
. The first is an inj )crease in the tax burden expressed by wjr , which increases the marginal cost of taxation−σin response to an increase in the wage. The second eﬀect is expressed by (cyi
=
j )
}]
[
{
−σ
1−σ
, representing the marginal utility of consumption. A
wji (1 − τj ) / 1 + β(βR) σ
one-unit increase in consumption caused by a rise in wages leads to a decrease in themarginal utility of consumption. This term works to decrease the marginal cost of taxation.
The net eﬀect depends on the magnitude of the elasticity of the marginal utility of consumption, represented by σ. A higher σ implies that the marginal utility of consumption
decreases further in response to an increase in wages. To examine σ’s role more precisely,
consider first the case where σ > 1. Eq. (6) indicates that the second eﬀect outweighs
the first. That is, a rise in the wages of the rich results in a decrease in their marginal
cost of taxation and thus an improvement in their utility. This result implies that therich tend to prefer a higher tax rate and a higher level of public goods provision as theirwages increase. However, the poor tend to prefer a lower tax rate and fewer public goods
because the mean-preserving spread of wage distribution is associated with a decline in
the wages of the poor.
9

Having established the eﬀect of wage inequality on the choice of tax rate and thelevel of public goods provision, we keep the σ > 1 assumption and consider the eﬀect
of wage inequality on the choice of public bond issues. The mean-preserving spread ofwage distribution results in a widening gap in utility between the rich and the poor
when σ > 1. The gap in utility incentivizes the government to smooth utility between
them by choosing a lower level of public goods provision in response to the spread ofwage distribution. However, the government can partially avoid decreasing public goods
provision by accumulating public debt. Therefore, higher inequality is associated with ahigher level of public debt when σ > 1. When σ < 1, the mean-preserving spread of wage
distribution leads to a shrinking, rather than widening, gap in utility between the richand the poor. Therefore, higher inequality is associated with a lower level of public debt
when σ < 1.
To evaluate the empirical plausibility of the two conflicting model predictions, we
review the empirical evidence from OECD countries for the past three decades. Table
1 presents the empirical investigation into the relationship between the Gini index andpublic debt using cross-country panel data. The first column shows the relationshipwithout control variables, while the second and third columns show the estimation results
for cases that include some control variables. The result indicates that the coeﬃcient on
the Gini index is positive and highly significant. Azzimonti, de Francisco, and Quadrini(2014) report a similar finding. Therefore, assuming that σ > 1 (i.e., EIS is below one)is an empirically valid perspective of inequality and public debt. This assumption is also
in line with many empirical EIS estimates (see, e.g., Kydland and Prescott, 1982; Hall,
1988; Campbell and Mankiw, 1989; Browning et al., 1999; Campbell, 1999).
[Table 1 here.]4

Numerical Examples and Discussion

We provide numerical simulations to further evaluate the eﬀects of the spread of wage
distribution on public debt, focusing on the case where σ > 1. Figure 1 plots the tax rate
(Panel (a)), the level of public goods provision (Panel (b)), the debt-to-GDP ratio (Panel(c)) in low-inequality countries, and the equilibrium interest rate (Panel (d)). The figure
shows how the mean-preserving spread of wage distribution in low-inequality countries
aﬀects these variables. The results demonstrated in Panels (a), (b), and (c) support theargument in the previous section: a rise in inequality in a country leads to decreases inthe tax rate and level of public goods provision as well as an increase in the debt-to-GDP
ratio.
[Figure 1 here.]10

To understand the eﬀects on the interest rate and public debt in high-inequality countries, Panel (c) indicates that a rise in inequality in low-inequality countries increases the
public debt in those countries, implying an increase in the supply of public debt in theinternational asset market that results in a decline in the price of public debt, that is,
an increase in the equilibrium interest rate as demonstrated in Panel (d).2 Because thepublic debt level in high-inequality countries equals the natural debt limit, b̄ = w̄/(R − 1),
debt in high-inequality countries decreases in response to an increase in the interest rate.
This finding is relevant in terms of Azzimonti, de Francisco, and Quadrini’s (2014)
investigation into the eﬀect of rising income risk in a home country on the choice of
public debt in home and foreign countries. They present a model with two types of
agents: entrepreneurs who encounter investment risk and workers who face no risk. In
this setting, they show that higher risk strengthens entrepreneurs’ demand for safe assets(that is, public debt), thus reducing the interest rate. They also show that a lower
interest rate makes public debt more attractive for foreign countries. Therefore, theirresults suggest that higher income risk in a home country results in more public debt
issues in home and foreign countries.
Azzimonti, de Francisco, and Quadrini (2014) argue that public debt responds positively to income inequality, provided that rising income inequality is associated with anincrease in individual income risk. Our study diﬀers that we focus directly on income
inequality within a generation and show that the conflict between the rich and the poor
does aﬀect the choice of public debt through voting. In particular, we demonstrate thatrising inequality within a home country may result in an increase in its public debt anda decrease in public debt in foreign countries, which is diﬀerent from Azzimonti, de Francisco, and Quadrini’s (2014) model prediction. Our study can be viewed as a componentof an alternative testable hypothesis for the relationship between inequality and public
debt from a political economy perspective.

2

Appendix A.5 provides a formal proof of this statement.

11

A

ProofsA.1

Derivation of (3) and (4)We diﬀerentiate (2) with respect to gj and τj and obtain{(1 − ω) + ωλ}θ(gj )−σ + (1 − ω)βλθ(gj′ )−σ∂gj′ ∂b′j= 0,∂b′j ∂gj{}σ∂gj′ ∂b′j1−σ(−1)(1 − τj )−σ w
ej + (1 − ω)βλθ(gj′ )−σ ′= 0.
(1 − ω) 1 + β(βR) σ∂bj ∂τj
Using ∂b′j /∂gj = 1 derived from the government budget constraint in (1), we can write
the first expression as in (3). The second expression is rewritten as:{}σ
1−σ
1 + β(βR) σ(1 − τj )σ

∂gj′
w̄
∂b′j
1
(1 − ω) + ωλ
w̄
,
=θ
1−ω
(gj )σ·w
ej = (−1)βλθ(gj′ )−σwhere we derive the first equality using ∂b′j /∂τj = −w̄, and derive the second using (3).
By rearranging the terms, we obtain (4).
■A.2

Proof of Lemma 1

Recall the first-order conditions with respect to gj and τj , given by (3) and (4), respectively. We substitute (4) into the government budget constraint and obtain:[b′j = gj + Rbj − w̄ ·

{
} (
1−σ
σ
1 − 1 + β(βR)·

1−ω
θ{(1 − ω) + ωλ}]) σ1 ( ) σ1wej·
· gj ,w̄orb′j = gj + Rbj − (R − 1)b̄ + w̄ ·

{
} (
1−σ
1 + β(βR) σ ·1−ωθ{(1 − ω) + ωλ}) σ1 ( ) σ1wej·
· gj , (7)w̄where we derive the second expression using w̄ = (R − 1)b̄.
To find the solution satisfying (3), (4), and (7), we presume)(
gj′ = γj · b̄ − b′j ,

12

(8)

where γj is an undetermined coeﬃcient. We substitute (8) into (3) and obtain
(

) )σ
(
γj · b̄ − b′j(1 − ω)βλ= (−1) ·· (−γj ) ,gj(1 − ω) + ωλor(b′j = b̄ − (γj )1/σ−1 ·(1 − ω) βλ(1 − ω) + ωλ) σ1

· gj .

Plugging this expression into (7) and rearranging the terms, we obtain()
R · b̄ − bj
gj =
} (
{
) σ1 ( ) σ1(
) σ1 .
1
1−σ
w
ej
−1
(1−ω)βλ1−ωσ1 + w̄ · 1 + β(βR) σ · θ{(1−ω)+ωλ}· w̄
+ (γj )· (1−ω)+ωλ
Therefore, our estimate is verified if, for a given R, γj satisfies the following condition:
γj =

R
}
(
{
) σ1 ( ) σ1(
) σ1 ,
1
1−σ
w
ej
−1
(1−ω)βλ1−ωσ1 + w̄ · 1 + β(βR) σ · θ{(1−ω)+ωλ}· w̄
+ (γj )· (1−ω)+ωλ

or
[

{
} (
1−σ
R − γj · 1 + w̄ · 1 + β(βR) σ ·1σ= (γj ) ·((1 − ω) βλ(1 − ω) + ωλ

) σ1
.

1−ω
θ{(1 − ω) + ωλ}) σ1 ( ) σ1 ]
w
ej
·
w̄
(9)

The left-hand side of (9), denoted by LHS, is decreasing in γj with LHS|γj =0 = R andlimγj →∞ LHS = −∞. The right-hand side of (9), denoted by RHS, is increasing in γjwith RHS|γj =0 = 0 and limγj →∞ RHS = ∞. Therefore, there exists a unique γj (> 0)that satisfies (9). We can obtain the corresponding tax and debt policy functions by()
substituting gj = γj · b̄ − bj into (4) and (7).
■A.3

Proof of Proposition 1

Given the argument in the text, each country’s steady-state debt level, j(= H, L), bj ,satisfies{∗
bH < b̄ and bL = b̄ if γH> γL∗ ,
(10)∗bH = b̄ and bL < b̄ if γH< γL∗ ,

13

where γj∗ (j = H, L) satisfies (9). The equation (9) implies that∗γH
≶ γL∗ ⇔ w
eH ≷ weL .

(11)

To compare w
eH and w
eL , we consider a mean-preserving spread of wage distribution;that is increasing wages for the rich coupled with decreasing wages for the poor, keeping w̄
unchanged. Given the definition of w̄ = πwjp + (1 − π)wjr , the spread of wage distribution
results in
1−π r
dwj .
(12)
dw̄ = 0 ⇔ dwjp = (−1)π
Using (12), we can compute a change in w
ej with:( )−σ p( )−σ rdw
ej = π (1 − σ) wjp
dwj + (1 − π) (1 − σ) wjr
dwj[]( r )−σ r( p )−σ1−π r= (1 − σ) π wj(−1)
dwj + (1 − π) wj
dwj ,π
where we derive the second equality from (12).
The above expression is reformulated as(dw
ej = (1 − σ) (1 − π)11( r )σ − ( p )σwj
wj)
dwjr ,

(13)

( )σ
( )σ
where 1 − π > 0 and 1/ wjr − 1/ wjp < 0. This expression indicates that dw
ej > 0 if
σ > 1, and dw
ej < 0 otherwise. That is, the mean-preserving spread of wage distribution
leads to{w
eH > weL if σ > 1,w
eH < weL if σ < 1.
With (10) and (11), we obtain{

bH = b̄ and bL < b̄ if σ > 1,bH < b̄ and bL = b̄ if σ < 1.

The remaining task is to determine R∗ (> 0) and the public debt level of low-inequality(high-inequality) countries for either case.
Case where σ > 1.
p∗r
Based on the argument above, we have w
eH > weL ⇔ γH< γL∗ ⇔ ϕ∗ (wH, wH, R) <p∗r
ϕ (wL , wL , R). The results demonstrated in the text suggest that the equilibrium worldinterest rate, R∗ , satisfies(
ϕ∗(wLp , wLr , R) =(1 − ω) βλ· γ∗(1 − ω) + ωλ L
14

)1/σ
= 1.

The second equality determines γL∗ as follows:γL∗ =(1 − ω) + ωλ.
(1 − ω) βλWe substitute this into (9) with j = L to obtain[) σ1 ( ) σ1 ]
{
} (
1−σ
1
−
ω
w
eL
(1
−
ω)
+
ωλ
R∗ −
· 1 + w̄ · 1 + β(βR∗ ) σ ··(1 − ω) βλθ{(1 − ω) + ωλ}
w̄
(
)1 (
)1
(1 − ω) + ωλ σ(1 − ω) βλ σ=·,(1 − ω) βλ(1 − ω) + ωλor,[
) σ1 ( ) σ1 ]
{
} (
1−σ
(1
−
ω)
+
ωλ
1
−
ω
w
eL
R∗ = 1 +· 1 + w̄ · 1 + β(βR∗ ) σ ··,(1 − ω) βλθ{(1 − ω) + ωλ}
w̄
(14)
∗
where R satisfies this condition.
To illustrate the fiscal policy for the case where σ > 1, recall the policy functions
established in Lemma 1. Given that bH = b̄H , we can find that fiscal policy in highinequality countries is distinguished by gH = 0 and τH = 1. Given a 100 percent tax rate,
there is no saving in high-inequality countries: spH = srH = 0.
Fiscal policy in low-inequality countries is distinguished by)(
gL = γL∗ · b̄ − bL ,
} (
{
∗ 1−σ
τL = 1 − 1 + β(βR ) σ ·1−ωθ{(1 − ω) + ωλ}) σ1 ( ) σ1
(
)
w
eL
·
· γL∗ · b̄ − bL .
w̄We substitute τL into the saving function and obtain() σ1 ( ) σ1
(
)
w
eL
·
· γL∗ · b̄ − bL ,w̄() σ1 ( ) σ1
)
(
1−σ
1−ω
w
eL
·
· γL∗ · b̄ − bL .
srL = β(βR∗ ) σ · wLr ·θ{(1 − ω) + ωλ}w̄
1−σspL = β(βR∗ ) σ · wLp ·1−ω
θ{(1 − ω) + ωλ}Therefore, the aggregate saving in the world is(1−σ
ν · (πspL + (1 − π)srL ) = ν · w̄ · β(βR∗ ) σ ·1−ωθ{(1 − ω) + ωλ}) σ1 ( ) σ1
)
(
w
eL
·
· γL∗ · b̄ − bL ,w̄where we use w̄ ≡ (πwLp + (1 − π)wLr ) to derive the expression.
The international asset market clearing condition is ν · (πspL + (1 − π)srL ) = νbH + (1 −15ν)bL , or,(∗ν · w̄ · β(βR )

1−σ
σ

·

1−ω
θ{(1 − ω) + ωλ}) σ1 ( ) σ1
(
)
w
eL
·
· γL∗ · b̄ − bL = ν b̄ + (1 − ν)bL ,w̄

where R∗ is determined in (14). We compute the public debt level bL by solving the above
market clearing condition for bL .
Case where σ < 1.
pr∗
> γL∗ ⇔ ϕ∗ (wH, wH, R) > ϕ∗ (wLp , wLr , R). The
In this case, we have w
eH < weL ⇔ γHequilibrium world interest rate R∗ satisfies
(
ϕ

∗

p
r
(wH
, wH
, R) =(1 − ω) βλ· γ∗(1 − ω) + ωλ H

)1/σ
= 1.

∗
The second equality determines γH
as
∗
γH
=

(1 − ω) + ωλ.
(1 − ω) βλWe substitute this equation into (9) with j = H to obtain[) σ1 () σ1 ]
{
} (
1−σ
1
−
ω
w
e
(1
−
ω)
+
ωλ
H
R∗ = 1 +· 1 + w̄ · 1 + β(βR∗ ) σ ··,(1 − ω) βλθ{(1 − ω) + ωλ}w̄
(15)
where R∗ satisfies this condition.
Given bL = b̄, fiscal policy in low-inequality countries is τL = 1 and gL = 0. Given a
100 percent tax rate, there is no saving in low-inequality countries, and high-inequality
countries provide savings for the international asset market. Following the same procedurein the σ > 1 case, we can determine the international asset market clearing condition as
follows:(∗(1 − ν) · w̄ · β(βR )

1−σ
σ

·

1−ω
θ{(1 − ω) + ωλ}) σ1 ()1)
w
eH σ ∗ (·· γH · b̄ − bH = νbH + (1 − ν)b̄,w̄

where R∗ satisfies (15). We compute the public debt level, bH by solving the above market
clearing condition for bH .
■

16

A.4

Derivation of Eq. (5)We first reformulate Eq. (4) as(

1−σ

1 + β(βR) σ
1 − τj)σ·wej =θ{(1 − ω) + ωλ}1· w̄ ·.
1−ω(gj )σThe left-hand side, denoted by LHS, becomes
)σ
1−σ
[
]
1 + β(βR) σ
LHS =· π(wjp )1−σ + (1 − π)(wjr )1−σ
1 − τj( r )σ( p )σwj
wj
p 1−σ· π(wj )+ yr· (1 − π)(wjr )1−σ
=
yp
cj
cj
( )−σ
( )−σ
= πwjp · cyp+ (1 − π)wjr · cyr,jj(where the first equality comes from the definition of w
ej and the second equality comesfrom the consumption function.
The right-hand side, denoted by RHS, becomesθ{(1 − ω) + ωλ}(1 − ω) βλ ( ′ )−σ ∂gj′
RHS =· w̄ · (−1) ·
g
· ′
1−ω
(1 − ω) + ωλ j
∂bj
′
∂gj
= −θβλ(gj′ )−σ ′ w̄,∂bj
where the first equality comes from Eq. (3).
■A.5

Eﬀect of Inequality on R∗ when σ > 1

Recall Eq. (14), which determines the equilibrium interest rate when σ > 1. Eq. (14)has the following properties. The left-hand side, denoted by LHS, is increasing in R∗
with LHS|R∗ =1 = 0 and limR∗ →+∞ LHS = +∞. The right-hand side, denoted by RHS,
is decreasing in R∗ with RHS|R∗ =1 > 1 and limR∗ →+∞ RHS ∈ (1, +∞). Given that theterm weL on the right-hand side of Eq. (14) increases in response to the spread in wage
distribution, we can verify that mean-preserving spread of wage distribution increases R∗
when σ > 1.
■

17

B

Estimation Procedure and Data Description

The estimated regression of the form is
ln (Gross Debtj,t ) = α0 + βj + α1 · ln (Gini Indexj,t ) + αX · Xj,t + uj,t ,where j denotes country, t denotes year, βj is the fixed country-specific eﬀect, Gross Debt
is the ratio of general government gross debt to GDP, Gini Index is the estimated Gini
index for household gross (pre-tax, pre-transfer) income, X is a vector of control variables,
and u is the residual containing country- and year-fixed eﬀects.
The first column in Table 1 reports the estimated results without control variables. The
second column reports the results when we include Government Expenditure measured
by the ratio of general government total expenditure to GDP as a control variable. The
third column reports the results when we further include Age Dependency Ratio, whichis the ratio of dependents older than 64 years of age to the working-age population aged
from 15 to 64 years.
We obtain the data for the ratio of general government gross debt to GDP and theratio of general government total expenditure to GDP from the World Economic Outlook
database (IMF, 2012). The Gini index (Solt, 2009) is the estimated index of inequality
in household gross (pre-tax, pre-transfer) income. We obtain the age dependency ratio
from World Development Indicators (World Bank). The sample period is 1980–2010 andthe sample countries are Australia, Austria, Belgium, Canada, Chile, the Czech Republic,
Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Israel,
Italy, Japan, Luxembourg, Mexico, the Netherlands, New Zealand, Norway, Poland, Portugal, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey, the United Kingdom, andthe United States.

18

References[1] Acemoglu, D., and Robinson, J., 2006. Economic Origins of Dictatorship and Democracy.
Cambridge University Press, Cambridge, UK.
[2] Aghion, P., and Bolton, P., 1990. Government domestic debt and the risk of default: A
political-economic model of the strategic role of debt. In: Dornbusch, R., and Draghi,M., (eds.), Public Debt Management: Theory and History, 315–344. Cambridge University
Press, Cambridge, UK.
[3] Alesina, A., and Tabellini, G., 1990. A positive theory of fiscal deficits and government
debt. Review of Economic Studies 57, 403-414.
[4] Azzimonti, M., de Francisco, E., and Quadrini, V., 2014. Financial globalization, inequality,
and the rising public debt. American Economic Review 104, 2267-2302.
[5] Battaglini, M., and Coate, S., 2008. A dynamic theory of public spending, taxation, anddebt. American Economic Review 98, 201-236.
[6] Browning, M., Hansen, L.P., and Heckman, J., 1999. Micro data and general equilibrium
models. In: Taylor, J., and Woodford, M., (eds.), Handbook of Macroeconomics, Volume
1A. North-Holland, Amsterdam.
[7] Campbell, J., 1999. Asset prices, consumption and the business cycle. In: Taylor, J., andWoodford, M., (eds.), Handbook of Macroeconomics, Volume 1C, North-Holland, Amsterdam.
[8] Campbell, J.Y., and Mankiw, N.G., 1989. Consumption, income, and interest rate: Reinterpreting the time series evidence. NBER Macroeconomics Annual. MIT Press, Cambridge,
MA.
[9] Cukierman, A., and Meltzer, A.H., 1989. A political theory of government debt and deficits
in a neo-Ricardian framework. American Economic Review 79, 713-732.
[10] de Walque, G., and Gevers, L., 2001. Heterogeneous dynasties and the political economyof public debt. Scandinavian Journal of Economics 103, 369-389.
[11] Hall, R., 1988. Intertemporal substitution in consumption. Journal of Political Economy
96, 339-357.
[12] IMF, 2012. World Economic Outlook database, April.
⟨http://www.imf.org/external/pubs/ft/weo/2012/01/weodata/weoselgr.aspx⟩ (January 6,2015).
[13] Kydland, F., and Prescott, E., 1982. Time to build and aggregate fluctuations. Econometrica 50, 1345-1370.
[14] Lindbeck, A., and Weibull, J., 1987. Balanced-budget redistribution as the outcome ofpolitical competition. Public Choice 52, 273-297.
[15] Meltzer, A., and Richard, S., 1981. A rational theory of the size of government. Journal of
Political Economy 89, 914-927.
[16] Natvik, G.J., 2013. The political economy of fiscal deficits and government production.
European Economic Review 58, 81-94.
[17] Persson, T., and Svensson, L.E.O., 1989. Why a stubborn conservative would run a deficit:
policy with time-inconsistent preferences. Quarterly Journal of Economics 104, 325-345.
[18] Persson, T., and Tabellini, G., 2000. Political Economics: Explaining Economic Policy,
MIT Press, Cambridge, MA.

19

[19] Roberts, K.W.S., 1977. Voting over income tax schedules. Journal of Public Economics 8,329-340.
[20] Romer, T., 1975. Individual welfare, majority voting, and the properties of a linear income
Tax. Journal of Public Economics 4, 163-185.
[21] Solt, F., 2009. Standardizing the World Income Inequality Database. Social Science Quarterly 90, 231-242.
[22] Song, Z., Storesletten, K., and Zilibotti, F., 2012. Rotten parents and disciplined children:a politico-economic theory of public expenditure and debt. Econometrica 80, 2785-2803.
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20

ln(Gini Index)
0.000
ln(Government Expenditure)
0.000
ln(Age Dependency Ratio)
0.000
Constant
0.000
R-squared
Number of observations
Number of countries

(1)

(2)

0.9921***(0.1436)

1.0878***(0.1169)
2.1334***(0.1121)0.0783(0.5396)
0.006
749
34

−8.2150***(0.6187)0.1748742
34(3)0.6986***(0.1239)
1.8885***(0.1127)
0.7880***(0.1051)−8.2031***(0.5958)
0.1714742
34

Table 1: Country fixed-eﬀect regression results. The dependent variable is the logarithmof general government gross debt. The estimation procedure and data descriptions are
given in Appendix B.

21

Figure 1: The figure depicts the eﬀects of the mean-preserving spread of wage distribution
in low-inequality countries on the tax rate (Panel (a)), the level of public goods provision(Panel (b)), the debt-to-GDP ratio (Panel (c)) in low-inequality countries, and the equilibrium interest rate R∗ (Panel (d)). The horizontal axis takes wLr /wLp in all panels. The parameters are set as σ = 1/0.66, ω = 0.25, β = (0.973)30 , λ = 2.2, π = 0.8, θ = 2.0, v = 0.2,
and w̄ = 1. The value of σ is taken from Kydland and Prescott (1982) and the values for
ω, β, and λ are taken from Song, Storesletten, and Zilibotti (2012).

22