GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 261

Redistributive Politics and Government Debt in a
Borrowing-constrained Economy

Ryo Arawatari and Tetsuo Ono

May 2012

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Redistributive Politics and Government Debt in a
Borrowing-constrained Economy
Ryo Arawatari∗
Nagoya University

Tetsuo Ono†
Osaka University

May 8, 2012

Abstract
We develop a two-period, three-class of income model where low-income agentsare borrowing constrained because of capital market imperfections, and where redistributive expenditure is ﬁnanced by tax and government debt. When the degreeof capital market imperfection is high, there is an ends-against-the-middle equilibrium where the constrained low-income and the unconstrained high-income agentsfavor low levels of government debt and redistributive expenditure; these agents
form a coalition against the middle. In this equilibrium, the levels of government
debt and expenditure might be below the eﬃcient levels, and the spread of income
distribution results in a lower debt-to-GDP ratio.
Keywords: Government debt; Borrowing constraints; Voting; Structure-induced
equilibrium; Income inequality.
JEL Classification: D72, H52, H60.

∗

Corresponding author. Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku,
Nagoya, Aichi, 464-8601 Japan. Email: arawatari@soec.nagoya-u.ac.jp†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. Email: tono@econ.osaka-u.ac.jp

1

Introduction

The conventional wisdom is that a higher inequality of income results in a larger redistributive public expenditure and that it results in a greater issue of government debt thatﬁnances redistribution. This theoretical prediction builds on a median-voter framework,in which a higher level of inequality translates into a poorer decisive agent in the politicalarena, who will then demand more redistribution (Romer, 1975; Roberts, 1977; Meltzer
and Richard, 1981; Krusell and Rios-Rull, 1999) and, thus, more government debt issue
for ﬁnancing redistribution (Cukierman and Meltzer, 1989).
The empirical evidence, however, does not necessarily support the above-mentioned
theoretical predictions. OECD cross-country data show that the volume of redistribution
is negatively correlated with income inequality (for example, Gottschalk and Smeeding,1997; Chen and Song, 2009). The theoretical prediction of inequality and government
debt is also controversial. For instance, Belgium, France and Germany, located in theContinental Europe, show low Gini coeﬃcients and high debt-to-GDP ratios, whereas theUnited Kingdom and the United States, included in the Anglo-Saxon group, show highGini coeﬃcients and low debt-to-GDP ratios.1 A negative correlation between inequalityand debt-to-GDP ratio is observed for some OECD countries. This indicates that therelationship between inequality and the size of government is not as simple as the standard
theory might expect.
Several theories have been provided to make sense of the above-mentioned puzzles.
Examples include political bias toward the rich (Benabou, 2000), the prospect of upward
mobility by low-income agents (Quadrini, 1999; Benabou and Ok, 2001; Alesina and La
Ferrara, 2005; Arawatari and Ono, 2009), lobbying and campaign contributions by therich (Rodriguez, 2004; Campante, 2010), voters’ preferences for redistribution (Creedy
and Moslehi, 2009; Creedy, Li and Moslehi, 2010), and borrowing constraints that hit
low-income agents (Casamatta, Cremer and Pestieau, 2000; Bellettini and Berti Ceroni,
2007; Cremer et al., 2007). These studies, however, assume that redistributive expenditureis ﬁnanced only via income tax. In other words, they abstract away government debt as anadditional option for ﬁnancing redistributive expenditure even though government debt
is one of the major sources of government revenue in OECD countries.
The purpose of this paper is to consider the relationship between inequality and thesize of government debt when voting results in a negative correlation between inequality
and redistributive expenditure. In particular, we focus on the role of borrowing constraints
1

Gini coeﬃcients in Belgium, France and Germany in the mid-2000s were 0.271, 0.270 and 0.298,
respectively; debt-to-GDP ratios in those countries in 2005 were 0.957, 0.760 and 0.711, respectively. In
contrast, Gini coeﬃcients in the United Kingdom and the United States in the mid-2000s were 0.335 and
0.381, respectively; debt-to-GDP ratios in these countries in 2005 were 0.461 and 0.623, respectively. The
source of the Gini coeﬃcients is OECD (2008), and the source of the debt-to-GDP ratio is OECD (2009).

1

(or, equivalently, capital market imperfection) as a source of the negative correlation, andexamine how politically determined government debt and expenditure are aﬀected by the
degree of capital market imperfection. In addition, we consider the spread of income
distribution and examine its impact on the debt-to-GDP ratio in relation to the degreeof capital market imperfection.
For the purpose of analysis, we utilize the two-period, three-class of income model ofBellettini and Berti Ceroni (2007). In their framework, redistributive expenditures, suchas publicly provided education and investment in infrastructure such as schools, libraries
and research institutes are ﬁnanced through the ﬁrst-period income tax; the expenditures
improve the productivity of all agents in the second period. We introduce government
debt as an additional policy option for ﬁnancing redistributive expenditure into theirframework. That is, the redistributive expenditure is ﬁnanced by the ﬁrst-period income
tax as well as by government debt issue. The debt repayment is ﬁnanced by the secondperiod income tax.
Under this extended framework, voters cast a ballot over the ﬁrst-period income tax,
and also over government debt issue. Under this type of voting game, the existence of a
Condorcet winner of the majority voting game is not necessarily guaranteed because of
the multidimensionality of the issue space (see, for example, Persson and Tabellini, 2000,
Chapter 2). To deal with this problem, we utilize the concept of a structure-induced
equilibrium (Shepsle, 1979). We determine the decisive voter over one issue given theother issue and derive his/her reaction function for each policy issue. We then ﬁnd thepoint where the two reaction functions cross; this point corresponds to the structureinduced outcome of the majority voting game.
Our model demonstrates that voting over policy produces two opposing eﬀects on
agents via the government expenditure: a negative eﬀect that results in a greater tax
burden for ﬁnancing expenditure and, thus, the utility loss today; and a positive eﬀect
that results in an improvement of labor productivity in the second period and, thus, theutility gain in the future. When agents are borrowing unconstrained, they prefer a higher
level of government expenditure and, thus, a higher ﬁrst-period tax and government debt
as their ﬁrst-period income becomes lower. Because of the borrowing, unconstrained
agents can reallocate resources freely from the second period to the ﬁrst period, and they
can compensate the utility loss today by the utility gain tomorrow. Therefore, they want
to increase government expenditure in order to get the beneﬁt of the second-period laborproductivity improvement.
However, the opposite result holds when agents are borrowing constrained. Borrowingconstrained agents prefer lower government debt and expenditure as their ﬁrst-period
income becomes lower. Borrowing-constrained agents are unable to reallocate resources2

freely from the second period to the ﬁrst period. Because of this constraint, the utility
gain from government expenditure in the future is valued less than the utility loss fromgovernment expenditure today for the borrowing-constrained agents. Therefore, they
prefer less government expenditure and, thus, lower ﬁrst-period income tax and debt as
their ﬁrst-period income becomes lower.
Given the above-mentioned feature of the model, we obtain the following three results.
First, the type of decisive voter depends on the degree of capital market imperfection. The
decisive voter is the middle-income, borrowing-unconstrained agent when the degree ofcapital market imperfection is low. However, the decisive voter becomes the low-income,borrowing-constrained agent when the degree is high. That is, the economy displays theends-against-the-middle equilibrium, as in Epple and Romano (1996), where the highand low-income agents, who favor low government expenditure and debt, form a coalition
against the middle who favor high government expenditure and debt.
The second result is that the political equilibrium generally fails to attain the eﬃcient
allocation. That is, ﬁrst-period tax, government debt and expenditure in the politicalequilibrium are higher or lower than the eﬃcient levels depending on either the income
level of a decisive voter or the degree of capital market imperfection. In particular, undera certain condition, there exists a critical level of capital market imperfection such thatthe political equilibrium levels of ﬁrst-period tax, government debt and expenditure arelower than the eﬃcient levels when the degree of capital market imperfection is above thecritical level. This result implies that countries with less access to capital markets are
more likely to attain lower levels of ﬁrst-period tax, government debt and expenditurethan the eﬃcient levels.
The third result is that the eﬀect of income distribution on the debt-to-GDP ratio
depends on the degree of capital market imperfection. In particular, there exists a criticallevel of capital market imperfection, which is diﬀerent from that described in the lastparagraph, such that the income distribution results in a higher debt-to-GDP ratio when
the degree of capital market imperfection is below the critical level, and the standard
result a la Cukierman and Meltzer (1989) holds. However, when the degree is above thecritical level, the opposite result holds because the decisive voter, who is a borrowingconstrained low-income agent, wants to choose lower government expenditure and debt.
That is, there is a negative correlation between inequality and debt-to-GDP ratio when
the degree of capital market imperfection is high.
Our analysis and results contribute to the following three strands of literature. The
ﬁrst strand is the literature on inequality and redistribution in the presence of borrowing constraints. Examples are Casamatta, Cremer and Pestieau (2000), Bellettini andBerti Ceroni (2007), Cremer et al. (2007) and Arawatari and Ono (2011). These stud-3

ies demonstrate that the decisive voter prefers a lower income tax as his/her income isdecreased when he/she is borrowing constrained. Thus, they clarify the role of borrowing constraint in presenting the negative correlation between inequality and the preferred
tax for redistribution. However, government debt is abstracted away in these studies.
The current paper contributes to the literature by demonstrating how the politically determined size of government debt is aﬀected by income distribution in the presence ofborrowing constraints.
The second strand is the literature on tax smoothing and government debt (for example, Barro, 1979; Lucas and Stokey, 1983; Aiyagari et al., 2002). Our model demonstrates
that, in an eﬃcient allocation, the tax rates should be equal between two periods, andthe size of government debt is adjusted to smooth tax rates across periods. However,
the model demonstrates that, in political equilibrium, the ﬁrst-period tax rate becomes
lower or higher than the eﬃcient rate depending on either the income level of a decisive
voter or the degree of capital market imperfection. The result suggests that the presenceof borrowing constraint might prevent the realization of tax smoothing in the political
economy.
The third strand is the literature on the politics of government debt. Although there
are many studies that consider how the size of government debt is determined via politics,
most of them abstract away the role of income inequality among voters in the determination of government debt. Previous studies instead have focused on the roles of common
pool problems (for example, Tabellini, 1986; Velasco, 1999), political instability (for example, Persson and Svensson, 1989; Aghion and Bolton, 1990; Alesina and Tabellini,
1990; Tabellini and Alesina, 1990), altruistic and selﬁsh agents (de Walque and Gevers,
2001) and intergenerational conﬂict (for example, Song, Storesletten and Zilibotti, 2012).
An exception is the study of Cukierman and Meltzer (1989), which demonstrates the
positive correlation between inequality and government debt: a higher level of inequality
results in a greater issue of government debt. However, as mentioned above, empiricalevidence suggests that the relationship is not so straightforward: some countries arefeatured by low inequality and high debt-to-GDP ratio, whereas others are characterizedby high inequality and low debt-to-GDP ratio. The current paper demonstrates that
there is a negative correlation between inequality and debt-to-GDP ratio when the degreeof capital market imperfection is high. The result could provide one possible explanation
for the empirical evidence among some OECD countries.
The organization of this paper is as follows. Section 2 introduces the model. Section
3 characterizes political equilibrium. Section 4 compares the political equilibrium withthe eﬃcient allocation. Section 5 examines the eﬀect of income inequality on the debt-toGDP ratio. Section 6 provides concluding remarks. Proofs of Propositions are given in4the Appendix.

2

The Model

We consider a small open economy model that is based on Bellettini and Berti Ceroni(2007). Agents live in two periods; they are indexed by their ﬁrst-period labor productivity
ei1 , which is also equal to their income. They belong to three income classes (low, middle
and high classes) in terms of their ﬁrst-period labor productivity, denoted by ei1 (i =hiil, m, h) : el1 < em
1 < e1 . The fraction of people in each class is given by ϕ with ϕ ∈ (0, 0.5)∑∑
and i∈{l,m,h} ϕi = 1. The average ﬁrst-period income, E1 ≡ i∈{l,m,h} ϕi ei1 , which ishequal to aggregate labor income, is assumed to satisfy em
1 < E1 < e1 ; the distribution of
the ﬁrst-period income is right-skewed.
In their ﬁrst period of life, agents allocate their labor income between consumption
and saving. Because of the assumption of a small open economy, the aggregate return on
saving is exogenous and equal to R(> 1). Following De Gregorio (1996) and Bellettini
and Berti Ceroni (2007), we assume that, in the ﬁrst period, agents cannot borrow morethan ψ − 1 times their after-tax income to ﬁnance current consumption. When ψ = 1,
agents cannot borrow at all; when ψ → +∞, agents can borrow as much as they want.
Therefore, the index ψ ∈ [1, +∞) represents the degree of capital market imperfection.
A lower ψ implies a higher degree of capital market imperfection.
Preferences are speciﬁed by the following intertemporal utility function:
U i = log ci1 + β log ci2 ,where ci1 and ci2 represent consumption of a type- i agent in periods 1 and 2, respectively,and β ∈ (0, 1) denotes the discount factor. We employ the above-mentioned speciﬁcation
for the tractability of analysis. The role of this assumption will be discussed later.
Labor income in the second period is equal to his/her labor productivity in that period.
Following Bellettini and Berti Ceroni (2007), we assume that a type-i’s productivity inthe second period, deﬁned by ei2 , depends on the public expenditure in the ﬁrst period. In
particular, we assume ei2 = A(ei1 )µ G, where A(> 0) is an exogenous parameter, µ ∈ (0, 1)
is a coeﬃcient of labor productivity depreciation and G is public expenditure in theﬁrst period. The expenditure G increases the productivity of labor of all agents in thesecond period. Examples of the public expenditure are publicly ﬁnanced education andpublic investment in infrastructure such as schools, universities, research institutions andlibraries.
The public expenditure in the ﬁrst period is ﬁnanced through linear income taxationand government debt issue. We assume convex costs of collecting taxes in order to avoid
corner solutions for the endogenous tax rate. In particular, if τt is the tax rate in the tth5period, the actual tax revenue in that period is τt (1 − τt )Et , where Et ≡∑i ii∈{l,m,h} ϕ et isthe aggregate labor income (i.e., GDP) in the tth period.2
The government budget constraints in the ﬁrst and the second periods, respectively,
are given by:
G = τ1 (1 − τ1 )E1 + B,
RB = τ2 (1 − τ2 )E2 ,

(1)
(2)

where B denotes government debt issue. In the ﬁrst period, the revenue from tax anddebt issue is used for public expenditure G. In the second period, government debt is
paid oﬀ by the second-period tax revenue. We assume that the government is not allowed
to default.

3

The Political Equilibrium

At the beginning of period 1, each agent votes over τ1 and B to maximize his/her indirect utility subject to the government budget constraints. The corresponding level ofgovernment expenditure is determined via the ﬁrst-period government budget constraint(1). After that, given policies, each agent chooses consumption ci1 and ci2 to maximize
his/her utility subject to individual budget constraints. At the beginning of period 2, thegovernment imposes the tax τ2 to ﬁnance the repayment of government debt.
The tax revenue in the second period is solely used to ﬁnance debt repayment. Thisimplies that setting B is equivalent to setting τ2 . Given this property, we hereafter focus on
the political determination of τ1 and τ2 , rather than τ1 and B, and consider the following
two-stage maximization problem. In the ﬁrst stage, agents vote over policies to maximize
their indirect utility subject to the government budget constraints (1) and (2). In thesecond stage, given policies, agents choose consumption to maximize their utility subject
to individual budget constraints.
In what follows, we induce the political equilibrium by backward induction. First, we
solve the utility maximization problems of agents (Subsection 3.1). Then, we deﬁne thepolitical institution and describe policy preferences of agents (Subsection 3.2). Finally,
we characterize political equilibrium of the voting game (Subsection 3.3).

2

The assumption of convex costs could be viewed as the reduced form of distortion in the labor market
produced by labor–leisure choice (see, for example, Casamatta, Cremer and Pestieau, 2000; Bellettini andBerti Ceroni, 2007; Cremer et al., 2007). The assumption of distortionary taxation is solely to ensure aninterior solution to preferred tax rates and otherwise plays no role.

6

3.1

Economic Decisions

The utility maximization problem of a type-i agent is as follows:max
i i

U i = log ci1 + β log ci2 ,s.t.

ci1 = (1 − τ1 )ei1 − si ,c1 ,c2ci2 = (1 − τ2 )ei2 + Rsi ,si ≥ (1 − ψ)(1 − τ1 )ei1 ,(3)where si denotes the saving of a type-i agent. The ﬁrst and the second constraints areindividual i’s budget constraints in the ﬁrst and the second periods, respectively. The
third constraint is the borrowing constraint.
In order to solve the problem, suppose ﬁrst that the borrowing constraint is not binding. The solution to the utility maximization problem yields:[](1 − τ2 )ei2
1iiu
· (1 − τ1 )e1 +
,
(4)
c1 =
1+β
R
[
]
βR
(1 − τ2 )ei2
iu
i
c2 =
· (1 − τ1 )e1 +
,
(5)
1+β
R
1
β
(1 − τ1 )ei1 −(1 − τ2 )ei2 ,
(6)
siu =
1+β
(1 + β)R
where the superscript u denotes “unconstrained”.
We substitute the saving function into the borrowing constraint (3) to obtain thecondition where a type-i agent is actually unconstrained in terms of the tax rates τ1 andτ2 :(1 − τ2 )ei2
R {(1 + β)ψ − 1} >.
(7)
(1 − τ1 )ei1
This condition states that the borrowing constraint of a type-i agent does not bind when
his/her after-tax income is high in the ﬁrst period and is low in the second period.
Alternatively, suppose that the borrowing constraint is binding; that is, (7) fails to
hold. The solution to the utility maximization problem yields:icic
1 = ψ(1 − τ1 )e1 ,{}(1 − τ2 )ei2iii
ic
c2 = R(1 − ψ)(1 − τ1 )e1 + (1 − τ2 )e2 = R · (1 − ψ)(1 − τ1 )e1 +,R
sic = (1 − ψ)(1 − τ1 )ei1 ,where the superscript c denotes “constrained”.

7

(8)
(9)
(10)

3.2

Policy Preferences, the Political Institution and Voting

In the voting stage, agents vote over policies to maximize their indirect utility. In order
to set up this maximization problem, we ﬁrst derive the indirect utility functions in termsof tax rates. For this purpose, we substitute the consumption functions derived in theprevious subsection into the utility function U i and obtain the following:

)
(
1
i
i


·
[(1
−
τ
)e
+
(1
−
τ
)e
/R]
log
1
2
2

1+β

( 1
)


(1−τ )ei
βR
i
i
+
β
·
log
·
[(1
−
τ
)e
+
(1
−
τ
)e
/R]
if R {(1 + β)ψ − 1} > (1−τ21 )ei2 ,
i
1 1
2 2
1+β
U =
1
i


log
(ψ(1
−
τ
)e
)
1
1



(1−τ )ei+β · log [(1 − τ2 )ei2 + R(1 − ψ)(1 − τ1 )ei1 ]if R {(1 + β)ψ − 1} ≤ (1−τ21 )e2i .
1

The productivity in the second stage, ei2 , depends on the government expenditure G
and, thus, on the tax rates τ1 and τ2 . In order to write ei2 as a function of τ1 and τ2 , wedeﬁne:
e1 ≡ AE∑

ϕi (ei1 )µi

e1 :and rewrite E2 as a function of E
E2 ≡

∑

e1 .
ϕi ei2 = GEie1 shows the marginal eﬀect of public expenditure G on the second-period
The variable E
GDP, denoted by E2 .
e1 ≡ A ∑ ϕi (ei )µ and E2 = GE
e1 to rewrite two government budget
We utilize E
1iconstraints (1) and (2) as follows:τ1 (1 − τ1 )E1,
1 − γ 2 τ2 (1 − τ2 )
τ1 (1 − τ1 )E1
B = γ 2 τ2 (1 − τ2 ) ·,1 − γ 2 τ2 (1 − τ2 )
G=

where

(11)
(12)

√
e1 /R.
γ≡ E

We impose the following assumption:
Assumption 1: γ ∈ (1, 2).
The inequality condition of 1 < γ ensures that the tax rate in the second period is
set within the range (0, 1/2) in equilibrium. Given τ ∈ (0, 1/2), the term appeared in(11) and (12) is positive as long as γ < 2. Therefore, the condition of γ < 2 ensures thatB > 0 and G > 0 hold in equilibrium; otherwise the economy experiences (a) international8

lending of assets rather than borrowing and (b) no government expenditure, both of which
are not considered in this paper. The inequality condition of γ < 2 also guarantees the
single-peakedness of preferences over τ1 and τ2 . These roles of Assumption 1 are found in
the following analysis.
By the use of (11), we can present the lifetime income of a type-i agent as a functionof tax rates:(1 − τ2 )ei2
(1 − τ2 )i(1 − τ1 )e1 += (1 − τ1 )ei1 +· A(ei1 )µ ·RR

τ1 (1 − τ1 )E1.
1 − γ 2 τ2 (1 − τ2 )With the above-mentioned lifetime income, the indirect utility function of a type-i agent
becomes: iu
V ≡ β · log (βR) −[ (1 + β) · log(1 + β)]
(1−τ2 )
τ1 (1−τ1 )E1

i
i µ

+
(1
+
β)
·
log
(1
−
τ
)e
+
·
A(e
)
·
1
2

1
1
R
1−γ ·τ2 (1−τ2 )
(1−τ )ei
if R {(1 + β)ψ − 1} > (1−τ21 )e2i ,i1
V (τ1 , τ2 ) =V ic ≡ log (ψei1[) + log(1 − τ1 ) + β · log(R)


]


(1−τ2 )
τ1 (1−τ1 )E1
i
i µ

+
β
·
log
(1
−
ψ)(1
−
τ
)e
·
A(e
+
)
·

1
2
1
1

R
1−γ ·τ2 (1−τ2 )(1−τ )ei
if R {(1 + β)ψ − 1} ≤ (1−τ21 )e2i .
1(13)
The tax rates τ1 and τ2 are determined by individuals through a political process ofmajoritarian voting. Because the issue space is bidimensional, the Nash equilibrium of amajoritarian voting game may fail to exist. To deal with this feature, we use the conceptof issue-by-issue voting, or structure-induced equilibrium, as formalized by Shepsle (1979).
In particular, if preferences are single peaked for each policy issue, a suﬃcient condition
for (τ1∗ , τ2∗ ) to be an equilibrium of the voting game is that τ1∗ represents the outcomeof majority voting over τ1 when the other dimension is ﬁxed at τ2∗ , and vice versa. In
Appendix A.1, it is shown that preferences are indeed single peaked along every dimensionof the issue space.
We can now solve the problem in the voting stage. A type-i agent chooses τ1 to
maximize his/her indirect utility given τ2 ; and he/she chooses τ2 to maximize his/herindirect utility given τ1 . The threshold level of capital market imperfection for a type-i
agent, denoted by ψ i , is derived by substituting his/her preferred pair of tax rates into
the condition (7). The following proposition states policy preferences of agents.
Proposition 1. The most preferred policy by a type-i agent satisfies:{
τ1iu ≡ 12 − Rγ(2−γ)· (ei1 )1−µ < 21
if ψ > ψ i ,2AE1(τ1i )∗ =β
τ1ic ≡ 1+2β
+ (1+β)R(ψ−1)γ(2−γ)· (ei1 )1−µ if ψ ≤ ψ i ,(1+2β)AE1
(
)
1
1
i ∗
(τ2 ) = 1 − ∈ 0,∀ψ ≥ 1, ∀i ∈ {l, m, h},
γ
2
9


[
]
(
)2

Rγ(2−γ)Ei2(1−µ)iu11−· (e1 )
if ψ > ψ i ,G ≡ 4(2−γ)

AE1



[
i ∗
E1
(G ) =
Gic ≡ (2−γ)(1+2β)
β(1 + β) + (1+β)R(ψ−1)γ(2−γ)· (ei1 )1−µ
2
AE1

]

(
)2


(1+β)R(ψ−1)γ(2−γ)
i 2(1−µ)−· (e1 )
if ψ ≤ ψ i ,

AE1
{
i ∗

(B ) =

B iu = (γ − 1) · Giu if ψ > ψ i ,
B ic = (γ − 1) · Gic if ψ ≤ ψ i ,(14)

where ψ i is given by:
ψi ≡

AE1
1
+
· (ei1 )µ−1 .
2(1 + β) 2R(1 + β)γ(2 − γ)

Proof. See Appendix A.2.
For each type of i, there is a threshold level of capital market imperfection, denoted by
ψ . When the degree of imperfection is higher, such that ψ ≤ ψ i , any agent belonging toiclass i is borrowing constrained. The threshold level ψ i becomes higher as the productivity
in the ﬁrst period becomes lower: ψ l > ψ m > ψ h . That is, an agent belonging to a lower
class is more likely to be borrowing constrained.
The degree of capital market imperfection critically aﬀects the preferences over policy.
In order to understand the role of capital market imperfection, we ﬁrst consider its impact
on the preference over the government expenditure ﬁnanced by the ﬁrst-period tax anddebt. On the one hand, raising the government expenditure decreases the utility today
through an increase in the ﬁrst-period tax burden. On the other hand, raising the government expenditure, ﬁnanced by an increase in the ﬁrst-period tax and debt issue, increases
the utility in the future through an improvement in labor productivity. Therefore, themost preferred ﬁrst-period tax and debt are determined to equate negative and positiveeﬀects at a margin.
When a type-i agent is borrowing unconstrained, he/she can choose τ1i and B i thatattain the maximum of utility. His/her preferences over τ1i and B i follow the standard
result in the literature of the political economy of redistribution (Romer, 1975; Roberts,
1977; Meltzer and Richard, 1981; Cukierman and Meltzer, 1989): a richer agent prefers
lower tax and debt issue. However, when he/she is borrowing constrained, he/she cannot
reallocate income freely from the second period to the ﬁrst period. Given this limitation,
the utility gain in the future is less valued than the utility loss today. Therefore, aconstrained agent prefers a lower level of government expenditure and, thus, lower ﬁrstperiod tax and debt as he/she becomes poorer.
Voters’ preferences over the second-period tax rate are unaﬀected by types and capital market imperfection. This property depends on the speciﬁcation of the second-period
10

productivity, ei2 = A(ei1 )µ G. The second-period productivity ei2 is linearly related to government expenditure G. This implies that maximizing V i with respect to τ2i is equivalent
to maximizing G with respect to τ2i . Because G is unaﬀected by types and capital market
imperfection, as shown in (11), the choice of τ2i is independent of them. This result also
holds true as long as preferences are characterized by a constant intertemporal elasticity
of substitution (see Appendix A.3).
The most preferred levels of government expenditure and public debt by a type-i agent
are derived by substituting (τ1i )∗ and (τ2i )∗ into the government budget constraints (11)and (12). The preferred levels satisfy the following properties. First, (Gi )∗ and (B i )∗ areaﬀected by the degree of capital market imperfection when a type-i agent is borrowing
constrained. Second, (B i )∗ is linearly related to (Gi )∗ because (B i )∗ = (γ)2 τ2 (1 − τ2 )(Gi )∗holds from (11) and (12), and (τ2i )∗ is independent of the type of an agent and is simply
given by 1 − 1/γ.
The following corollary states how policy preferences of agents are aﬀected by capital
market imperfection when agents are borrowing constrained.
Corollary 1(i) ∂τ1ic /∂ψ > 0, ∂Gic /∂ψ > 0, and ∂B ic /∂ψ > 0.
(ii) τ1iu = τ1ic , Giu = Gic , and B iu = B ic at ψ = ψ i .
Proof. See Appendix A.4.

[Figure 1 about here.]
Figure 1 illustrates how voters’ preferences over policies (τ1 , G, B) are aﬀected by
the degree of capital market imperfection, denoted by ψ. When they are borrowing
unconstrained, their choice of τ1 , G and B is independent of capital market imperfection.
However, when they are borrowing constrained, their choice depends on the degree ofcapital market imperfection. A lower ψ implies that they are more constrained. In orderto relax the constraint, they prefer a lower ﬁrst-period tax rate and, thus, prefer a lower
level of government expenditure.

3.3

Political Equilibrium

Based on the characterization of voting behavior of each type of an agent, we now consider
the determination of structure-induced equilibrium policies via majority voting. The
second-period tax rate is given by τ2 = 1 − 1/γ because all types of agents prefer thisrate. Given τ2 = 1 − 1/γ, the equilibrium ﬁrst-period tax rate is given by the most11

preferred τ1 by the median voter. The equilibrium levels of government expenditure anddebt are then given by the preferred levels by the median voter over τ1 .
In order to determine the median voter over τ1 , we introduce the critical level of theﬁrst-period income, ee1 , deﬁned by:1

ee1 ≡ [AE1 /((1 + 2β)Rγ(2 − γ))] 1−µ .
Direct calculation leads to:
ei1 R ee1 ⇔ ψ i Q 1.
When the ﬁrst-period productivity of a type-i agent is high such that ei1 > ee1 , ψ i < 1
always holds for a type-i agent. In other words, he/she is borrowing unconstrained forany degree of capital market imperfection of ψ ∈ [1, +∞). He/she has enough income inthe ﬁrst period that is beyond the critical level ee1 and, thus, can choose a combination of
consumption and saving that does not hit the borrowing constraint from the viewpointof utility maximization. In contrast, an agent might be borrowing constrained for a lowψ if ei1 ≤ ee1 holds. In order to reduce a set of possible political equilibria, we impose thefollowing assumption with respect to ee1 .
Assumption 2. el1 < eme1 < eh1 .
1 < eAssumption 2 ensures that type-h agents are never borrowing constrained. Therefore,
given the properties of preferred ﬁrst-period tax rate, demonstrated in Proposition 1 andCorollary 1, the decisive voter will be a type-l or a type-m agent. The following proposition
determines the decisive voter, contingent on the degree of capital market imperfection.
Proposition 2 The decisive voter over τ1 is a borrowing-constrained type-l agent if
b and a borrowing-unconstrained type-m agent if ψ ∈ (ψ,b ∞) where:ψ ∈ [1, ψ],
[
]
l 1−µ
m 1−µ
AE
+
Rγ(2
−
γ)
·
2(1
+
β)
·
(e
)
−
(1
+
2β)
·
(e
)
1
1
1
∈ (ψ m , ψ l ).
ψb ≡2(1 + β)Rγ(2 − γ) · (el1 )1−µPolitical equilibrium policies, {τ1∗ , τ2∗ , G∗ , B ∗ }, are given by:{b
τ1lc if ψ ∈ [1, ψ]τ1∗ =b ∞) ,
τ1mu if ψ ∈ (ψ,1τ2∗ = 1 −∀ψ ≥ 1,γ{b
Glc if ψ ∈ [1, ψ]G∗ =b ∞) ,
Gmu if ψ ∈ (ψ,{b
B lc if ψ ∈ [1, ψ]B∗ =b ∞) .
B mu if ψ ∈ (ψ,12Proof. See Appendix A.5.
Figure 2 illustrates the political equilibrium policies τ1∗ , G∗ and B ∗ . As demonstrated
in the ﬁgure, the identity of the median voter depends on the degree of capital market
imperfection, denoted by ψ: a higher ψ means a lower degree of capital market imperb thefection. When an agent is less likely to be borrowing constrained, such that ψ > ψ,
decisive voter becomes the type-m agent who is borrowing unconstrained. Their preferred
policies lie between those by type-l and type-h agents. However, when an agent is moreb there is an ends-against-the-middle
likely to be borrowing constrained, such that ψ < ψ,
equilibrium: the decisive voter is the borrowing-constrained, type-l agent. He/she prefers
a lower tax rate and, thus, lower levels of government expenditure and debt than thetype-m agent because of the strict ﬁnancial constraint. The policies preferred by theborrowing-constrained type-l agent lie between those by type-m and type-h agents.
[Figure 2 about here.]4

Welfare Implication

In this section, we evaluate the political equilibrium in terms of eﬃciency. Following
Bellettini and Berti Ceroni (2007), we focus on the set of policy variables that maximize
the present discounted value (PDV) of aggregate disposable income. We compare itwith the political equilibrium policies and investigate under what condition the politicalequilibrium results in an excess burden of tax and debt.
The PDV of aggregate disposable income, denoted by I, is given by:I≡∑

ϕi (1 − τ1 )ei1 +i∈{l,m,h}that is,I = (1 − τ1 )E1 +

1R∑

ϕi (1 − τ2 )ei2 ,i∈{l,m,h}(1 − τ2 )Aτ1 (1 − τ1 )E1 e· E1 .
R {1 − (γ)2 τ2 (1 − τ2 )}The ﬁrst term on the right-hand side is the aggregate disposable income in the ﬁrst period,
and the second term is the present value of the aggregate disposable income in the second
period.
The ﬁrst-period tax rate τ1 has two competing eﬀects on PDV of aggregate disposable
income. An increase in τ1 creates a negative eﬀect via a decrease in the ﬁrst-period aftertax income, and a positive eﬀect via an increase in the government expenditure devoted
to education. A social planner chooses τ1 to equate these two competing eﬀects at amargin. The second-period tax τ2 also has two competing eﬀects via the second-period
13after-tax income and government expenditure. The planner chooses τ2 to equate thesetwo competing eﬀects at a margin.
Let {τ1e , τ2e } denote the eﬃcient tax rates that maximize the PDV of aggregate disposable income, and {Ge , B e } denote the corresponding levels of government expenditureand debt, respectively. The eﬃcient tax rates τ1e and τ2e are derived by setting ∂I/∂τ1 = 0
and ∂I/∂τ2 = 0, respectively:1τ1e = τ2e = 1 − .
γ
The planner chooses the same tax rates between the two periods in order to smooth tax
burdens over periods. The corresponding levels of government expenditure and debt arecalculated by substituting τ1e and τ2e into the government budget constraints:
Ge =(γ − 1) · E1
(γ − 1)2 · E1e;B=
= (γ − 1) · Ge .
γ 2 (2 − γ)
γ 2 (2 − γ)

A noteworthy feature is that the second-period tax rate in the political equilibrium
is eﬃcient: τ2e = τ2∗ . In order to understand the mechanism behind this result, we focus
on the term related to τ2 . In the political equilibrium, the decisive voter j chooses τ2 tomaximize his/her second-period after-tax income, given by:(1 − τ2 )A(ei1 )µ G = (1 − τ2 )A(ei1 )µ .

τ1 (1 − τ1 )E1.
1 − γ 2 τ2 (1 − τ2 )In contrast, in the eﬃcient allocation, the planner chooses τ2 to maximize the aggregate
second-period after-tax income given by:∑∑

ϕi (1 − τ2 )A(ei1 )µ G =i∈{l,m,h}ϕi (1 − τ2 )A(ei1 )µ .

i∈{l,m,h}τ1 (1 − τ1 )E1
1 − γ 2 τ2 (1 − τ2 )The objective functions are diﬀerent between the two problems. However, the terms
related to τ2 are equivalent between the two problems; both are given by (1 − τ2 )/{1 −γ 2 τ2 (1 − τ2 )}. This is because the government expenditure G, given in (1), enters into
these two objective functions in a linear fashion. Therefore, we can obtain the eﬃcientsecond-period tax rate in the political equilibrium.
The eﬃciency generally fails to hold as regards τ1 , G and B. The following proposition
demonstrates the conditions for which the political equilibrium levels of τ1 , G and B are
higher or lower than the eﬃcient levels.
Proposition 3.
(i) Suppose that em
1 >

(

AE1
Rγ 2

1
) 1−µ

τ1e > τ1∗ ,holds. Then,
Ge > G ∗ ,14Be > B∗,∀ψ ≥ 1.

(ii) Suppose that γ < 1+2β
holds. Then,1+βτ1e ≤ τ1∗ ,(iii) Suppose that em
1 ≤{where:

(

AE1
Rγ 2

1
) 1−µ

τ1e > τ1∗ ,τ1e ≤ τ1∗ ,G e ≤ G∗ ,Be ≤ B∗,∀ψ ≥ 1.

hold. Then,and γ ≥ 1+2β
1+β
Ge > G ∗ ,Ge ≤ G ∗ ,B e > B ∗ , if ψ ∈ [1, ψ ∗ ),B e ≤ B ∗ , if ψ ∈ [ψ ∗ , ∞),][(1 + 2β)AE1
11+βb· (el1 )µ−1 ∈ [1, ψ].
ψ ≡1+
·
−
(1 + β)Rγ(2 − γ) 1 + 2β γ∗

Proof. See Appendix A.6.
Figure 3 illustrates the result in Proposition 3. In order to provide the interpretation of
the result in Proposition 3, we focus on the ﬁrst-period tax rate because the properties ofgovernment expenditure and debt are qualitatively similar to those of the ﬁrst-period tax
rate. After describing the eﬃciency of the ﬁrst-period tax rate, we discuss the eﬃciency
of debt issue in relation to the previous studies.
[Figure 3 about here.]In the current framework, eﬃciency requires the same tax rates between the two periods in order to smooth the tax burden across periods. However, the political equilibrium
realizes a lower or a higher ﬁrst-period tax rate than the eﬃcient one depending on either
the income level of a decisive voter or the degree of capital market imperfection.
To conﬁrm this argument, consider ﬁrst the case of a low degree of capital market imperfection such that ψ > ψ̂: an unconstrained middle-income agent becomes the decisive
voter. He/she prefers a lower ﬁrst-period tax rate than the eﬃcient one when his/her
12 1−µincome is high such that em; otherwise, he/she prefers a higher tax
1 > (AE1 /Rγ )rate. Therefore, the decisive voter’s income level aﬀects the eﬃciency of the politicalequilibrium ﬁrst-period tax rate when he/she is borrowing unconstrained.
Next, consider the case of a high degree of capital market imperfection such that
ψ < ψ̂: the decisive voter is the constrained low-income agent. As demonstrated in
Subsection 3.2, a constrained agent prefers a lower tax rate as the degree of imperfection
is increased. This implies that a further imperfection improves tax smoothing when thepreferred tax rate by the decisive voter is initially higher than the eﬃcient one, while itprevents tax smoothing when the preferred tax rate is initially lower than the eﬃcient level.
Therefore, the degree of capital market imperfection determines the degree of diﬀerence
from the eﬃcient rate when the decisive voter is borrowing constrained.
15

Given the discussion so far, we can now state that the political equilibrium does
not necessarily result in an overissue of public debt in the current framework. Previous
studies, in general, show that the politics results in an overissue of public debt because ofcommon-pool problems (Tabellini, 1986; Velasco, 1999) and political instability (Persson
and Svensson, 1989; Aghion and Bolton, 1990; Alesina and Tabellini, 1990). In the realworld, however, some countries, such as Australia, Korea and New Zealand, do control
successfully the issue of public debt (OECD, 2009). These countries could be viewed as
issuing public debt below the eﬃcient level. The result of underissue of public debt in thecurrent framework may provide one possible explanation for these countries.

5

Income Inequality and Borrowing Constraint

The analysis so far has demonstrated the political equilibrium policy given the distributionof income. This section takes a step toward changing a situation. In particular, weconsider an increase in the initial income of the high type, eh1 , coupled with a decreasee1 ≡ A ∑ ϕi (ei )µ ,in the initial income of the low type, el1 .3 We focus on the term E
1iwhich represents the marginal impact of government expenditure on the second-period
e1 unchanged. Under
GDP. Then, we consider the spread of income distribution, keeping E
e1 -preserving spread of income distribution, the second-period tax rate is unchanged
the E√
e1 /R. Thus, the speciﬁcation enables us to concentratebecause it is given by τ = 1−1/ E
2on the decisive voter’s choice over the ﬁrst-period tax τ1 and the government debt B.
Proposition 4. Suppose that the first-period income of the high type, eh1 , increases ande1 unchanged. Then,the first-period income of the low type, el1 , decreases, leaving E
b such that the equilibrium debtthere exists a critical level of ψ, denoted by ψel (> ψ),
to-GDP ratio decreases if ψ ≤ ψel ; the equilibrium debt-to-GDP ratio increases if
ψ > ψel .

Proof. See Appendix A.7.
The result in Proposition 4 states that the eﬀect of income inequality on the debt-toGDP ratio depends heavily on the degree of capital market imperfection, denoted by ψ.
The spread of income distribution results in a lower debt-to-GDP ratio when the degreeof capital market imperfection is high such that ψ ∈ [1, ψel ]. However, it results in a higher
debt-to-GDP ratio when the degree is low such that ψ ∈ (ψel , ∞). In order to understandthe mechanism behind this result, we present three key results about agents’ preferences
3lAssuming a decrease in em
1 instead of a decrease in e1 does not qualitatively aﬀect the result shown
below.

16

being aﬀected by the spread of income distribution. Based on the results, we demonstrate
illustratively the eﬀect of income distribution on the debt-to-GDP ratio.
First, recall the threshold level of capital market imperfection for a type-j agent (j =l, m), denoted by ψ j (Proposition 1):
ψj ≡

1
AE1
+
(ej1 )µ−1 .
2(1 + β) 2R(1 + β)γ(2 − γ)

The threshold level is increased by the spread of income distribution in the following two
ways. Given ψ, the spread of income distribution decreases type-l′ s labor earnings and,thus, decreases his/her after-tax income. In addition, a decrease in the type-l′ s income
induces him/her to choose a higher ﬁrst-period tax rate, which results in a lower aftertax income for every agent as long as the type-l agent is a decisive voter. Because of
these two negative eﬀects on the after-tax income, the type-j (j = l, m) agent, who isinitially borrowing unconstrained, could become borrowing constrained. This implies thatthe range of ψ that makes the type-j agent (j = l, m) free from borrowing constraint,(ψ j , ∞), becomes narrower as the income inequality becomes higher.
Second, the spread of income distribution produces positive and negative eﬀects on
e1 -preserving spread of income
the preferred debt-to-GDP ratio by each agent. The E
distribution increases the ﬁrst-period GDP, E1 , implying an expansion of the tax basein the ﬁrst period. This gives the decisive voter an incentive to increase the government
spending and, thus, to issue more public debt. In addition, the type-l agent chooses higher
government spending and, thus, a higher level of public debt when he/she is a decisive
voter because his/her income is decreased by the spread of income distribution. Theseare positive eﬀects on the debt-to-GDP ratio. In contrast, an increase in the ﬁrst-period
GDP, E1 , makes the debt-to-GDP ratio lower for a given B; this is a negative eﬀect.
When the type-j (= l or m) agent is borrowing constrained, the positive eﬀect becomes
smaller than the negative eﬀect because he/she is unable to freely reallocate resources
between the two periods. His/her policy choice results in a lower debt-to-GDP ratio in
response to the spread of income distribution. In contrast, the positive eﬀect overcomesthe negative eﬀect when he/she is borrowing unconstrained.
Third, under Assumption 2, a type-h agent is always borrowing unconstrained and,thus, prefers a higher B/E1 in response to the spread of income distribution. However,the preferred level of B/E1 by a type-h agent is always lower than that by the other two
types of agents under Assumption 2. Therefore, the decisive voter is a type-l or type-m
agent who prefers a lower B/E1 between them (Proposition 2).
Based on the above-mentioned three results, we now illustrate changes in preferences
over B/E1 for each type of agent as in Figure 4. The dotted and solid curves in panel (a)depict the type-j’s choices over B/E1 before and after the spread of income distribution,17

respectively. The spread of income distribution increases the threshold level of ψ that
distinguishes the type of decisive voter from ψb to ψb′ . The bold dotted and solid curvesin panel (b) depict the decisive voter’s choices over B/E1 before and after the spread ofb ψb′ )
income distribution, respectively. There exists a unique level of ψ, denoted by ψel ∈ (ψ,such that the bold dotted and solid curves cross at this level. Therefore, the range of ψb (ψ,b ψel ], (ψel , ψb′ ] andis divided into four subranges of ψ, as illustrated in Figure 4: [1, ψ],(ψb′ , +∞).

[Figure 4 about here.]b the initial decisive voter is a borrowing-constrained, type-l agent, andFor ψ ∈ [1, ψ],
he/she still remains the decisive voter after the spread of income distribution. His/her
choice over policies results in a lower level of B/E1 in response to the decrease in his/herb ψel ], the initial decisive voter is a borrowing-unconstrained
income. In contrast, for ψ ∈ (ψ,type-m agent. Because of the spread of income distribution, the decisive voter changes
from a borrowing-unconstrained type-m agent to a borrowing-constrained type-l agent.
The policy choice by the latter results in a lower level of B/E1 than that by the former,
as demonstrated in Figure 4.
For ψ ∈ (ψel , ψb′ ], the initial decisive voter is a borrowing-unconstrained type-m agent.
The decisive voter changes from a borrowing-unconstrained type-m agent to a borrowingconstrained type-l agent. The policy choice by the latter results in a higher level of B/E1
than that by the former as demonstrated in Figure 4. Finally, for ψ ∈ (ψb′ , +∞), theinitial decisive voter is a borrowing-unconstrained type-m agent, and he/she still remains
the decisive voter after the spread of income distribution. His/her choice over policies
results in a higher level of B/E1 in response to the spread of income distribution.

6

Conclusion

This paper developed the two-period, three-class of income model with borrowing constraints (i.e., capital market imperfection), and studied the politics of redistributive
expenditure ﬁnanced by tax and government debt. In our framework, a borrowingunconstrained agent prefers higher government expenditure and debt as his/her ﬁrstperiod income becomes lower because he/she wants to compensate his/her income loss
today by redistributive expenditure in the future. However, an opposite result holds for aborrowing-constrained agent: he/she prefers lower government expenditure and debt because his/her income becomes lower because he/she is unable to make such compensation
under borrowing constraints.
18

We showed the following three results. First, the type of a decisive voter depends
on the degree of capital market imperfection. In particular, the decisive voter becomesthe low-income, borrowing-constrained agent, rather than the middle, when the degree ofcapital market imperfection is high. The economy displays the ends-against-the-middle
equilibrium. Second, the political equilibrium generally fails to attain the eﬃcient allocation. In particular, under a certain condition, the political equilibrium attains anunderissue of government debt when the degree of capital market imperfection is high.
Third, the spread of income distribution results in a lower debt-to-GDP ratio when the
degree of capital market imperfection is high. This is because the decisive voter, who is aborrowing-constrained, low-income agent, wants to choose lower government expenditureand debt. The result implies that there is a negative correlation between inequality anddebt-to-GDP ratio when the degree of capital market imperfection is low.
The standard result in the literature of politics and government debt is that (i) thepolitical economy displays an overissue of government debt, and (ii) the spread of income distribution results in a greater issue of government debt. Our analysis and results,
therefore, show that the standard result does not hold when the borrowing constraint
is considered. More importantly, the result of the negative correlation between inequality and debt-to-GDP ratio could provide one possible explanation for the cross-country
evidence among OECD countries.
Several research directions for future research are highlighted by our results in thispaper. First, we characterized the structure-induced equilibrium of the voting game where
agents simultaneously vote over tax and government debt. An interesting question iswhether the current result is still true when an alternative concept of equilibrium is
adopted. For example, one may study sequential voting or probabilistic voting. Second,
we assumed a small open economy where the interest rate is exogenous. This simpliﬁes
the analysis and guarantees the analytical treatment of our political game but at the costof abstracting general equilibrium eﬀects. Third, the default of government debt was notconsidered in our analysis. Relaxing these assumptions are interesting areas for future
research.

19

A

Appendix

A.1

Single-peakedness of Preferences

A.1.1

Single-peakedness of preferences over τ1 given τ2

Given τ2 , let τ̂1 (τ2 , ei1 ) denote the ﬁrst-period tax rate satisfying
R {(1 + β)ψ − 1} =(1 − τ2 )ei2.
(1 − τ1 )ei1

(15)Then, the preferences of a type-i agent over τ1 are:{ iu
V if τ1 < τ̂1 (τ2 , ei1 ),iV (τ1 , τ2 ) =V ic if τ1 ≥ τ̂1 (τ2 , ei1 ),where V iu and V ic are deﬁned in (13). In what follows, we ﬁrst show that both V iu andV ic are singled peaked over τ1 . Then, we demonstrate that V iu = V ic and ∂V iu /∂τ1 =∂V ic /∂τ1 hold at τ1 = τ̂1 (τ2 , ei1 ), implying that V i has a unique local maximum over thewhole range of τ1 and, thus, V i is single peaked.
In order to show that V iu and V ic are single peaked over τ1 , we take the ﬁrst derivativesof V iu and V ic with respect to τ1 , and obtain:(1−τ2 )· A(ei1 )µ · 1−γ 2 ·τE21(1−τ2 )∂V iu (τ1 , τ2 )1R
R 0 ⇔ −(1 + β) ·+ (1 + β) ·
τ1 E1
∂τ1
1 − τ1ei1 + (1−τ2 ) · A(ei1 )µ ·2RR01−γ ·τ2 (1−τ2 )(1 − τ2 )(1 − 2τ1 )E1⇔· A(ei1 )µ ·R ei1R1 − γ 2 · τ2 (1 − τ2 )R · {1 − γ 2 · τ2 (1 − τ2 )}⇔ 1 − 2τ1 R· (ei1 )1−µAE1 (1 − τ2 )1 R · {1 − γ 2 · τ2 (1 − τ2 )}⇔ τ1 Q τ1iu (τ2 ; ei1 ) ≡ −· (ei1 )1−µ ,2
2AE1 (1 − τ2 )

(16)

and
(1 − τ2 )A(ei1 )µ · 1−γ 2 τE21(1−τ2 )1∂V ic (τ1 , τ2 )R 0 ⇔ −(1 + β) ·
+β·
R0
1
∂τ1
1 − τ1(1 − τ2 )A(ei1 )µ · 1−γ 2ττ12E(1−τ+ R(1 − ψ)ei1
2){β − (1 + 2β)τ1 } E1
R (1 + β)R(1 − ψ)ei12
1 − γ τ2 (1 − τ2 )β
(1 + β)R(ψ − 1) · {1 − γ 2 · τ2 (1 − τ2 )}⇔ τ1 Q τ1ic (τ2 ; ei1 ) ≡+· (ei1 )1−µ ,1 + 2β(1 + 2β)AE1 (1 − τ2 )(17)

⇔ (1 − τ2 )A(ei1 )µ ·where the second lines for both calculations come from the fact that 1−γ 2 ·τ2 (1−τ2 ) > 0.4
These results state that V i u and V i c have a unique local maximum at τ1 = τ1i u(τ2 ; ei1 )and τ1 = τ1i c(τ2 ; ei1 ), respectively, and, thus, are single-peaked.
The condition 1 − γ 2 · τ2 (1 − τ2 ) > 0 always holds because γ is set within the range (1, 2) under
Assumption 1, and τ2 is set within the range (0, 1/2) under Assumption 1 and the convex costs ofcollecting taxes.
420

Next, we show that V iu = V ic and ∂V iu /∂τ1 = ∂V ic /∂τ1 hold at τ1 = τ̂1 (τ2 , ei1 ). From(11) and (15), we have:(1 − τ̂1 ) =τ̂1 (1 − τ̂1 )E1(1 − τ2 )· A(ei1 )µ ·i
e1 R {(1 + β)ψ − 1}
1 − γ 2 τ2 (1 − τ2 )R {(1 + β)ψ − 1} · {1 − γ 2 τ2 (1 − τ2 )} · (ei1 )1−µ.
AE1 (1 − τ2 )⇔ τ̂1 (τ2 , ei1 ) =(18)

By direct calculation, we have:
V iu τ1 =τ̂1 (τ2 ,ei ) = V ic τ1 =τ̂1 (τ2 ,ei ) = β log(βR) + (1 + β) · log(ψei1 ) + (1 + β) · log(1 − τ̂1 (τ2 , ei1 )),
1

1

∂V iu
∂V ic
=
∂τ1 τ1 =τ̂1 (τ2 ,ei )∂τ1 τ1 =τ̂1 (τ2 ,ei )11= −(1 + β) ·1
1 − τ2
E1+· A(ei1 )µ−1 ·.
i
2
1 − τ̂1 (τ2 , e1 )ψR
1 − γ · τ2 (1 − τ2 )In words, V i is continuous at τ1 = τ̂1 (τ2 , ei1 ), and V iu and V ic have the same slope atτ1 = τ̂1 (τ2 , ei1 ). Given these results with the single peakedness of V iu and V ic , it canbe concluded that V i has a unique local Maximum, as demonstrated in Figure 5. In
particular, V i is maximized at τ1 = τ1iu (τ2 ; ei1 ) if τ1iu (τ2 ; ei1 ) < τ̂1 (τ2 , ei1 ); it is maximized at
τ1 = τ1ic (τ2 ; ei1 ) otherwise.
[Figure 5 about here.]
A.1.2

Single-peakedness of preferences over τ2 given τ1

Given τ1 , let τ̂2 (τ1 ; ei1 ) denote the second-period tax rate satisfying
R {(1 + β)ψ − 1} =(1 − τ2 )ei2.
(1 − τ1 )ei1

From (13), the preferences of a type-i agent over τ1 are:{ iu
V if τ2 > τ̂2 (τ1 ; ei1 ),iV (τ1 , τ2 ) =V ic if τ2 ≤ τ̂2 (τ1 ; ei1 ).
We take the ﬁrst derivatives of V iu and V ic with respect to τ2 and obtain:{}∂V iu (τ1 , τ2 )R 0 ⇔ − 1 − γ 2 τ2 (1 − τ2 ) + (1 − τ2 )γ 2 (1 − 2τ2 ) R 0∂τ2
1⇔ τ2 Q 1 − ,
γ
21

(19)

and
{
}
∂V ic (τ1 , τ2 )R 0 ⇔ − 1 − γ 2 τ2 (1 − τ2 ) + (1 − τ2 )γ 2 (1 − 2τ2 ) R 0∂τ2
1⇔ τ2 Q 1 − .
γ(20)V iu and V ic are maximized at the same tax rate τ2 = 1 − 1/γ. Thus, we can conclude
that V i has a unique local maximum at τ2 = 1 − 1/γ; that is, V i is single peaked over τ2 .

A.2

Proof of Proposition 1

From (19) and (20), the most preferred second-period tax rate by a type-i agent is givenby:1(τ2i )∗ = 1 − ,(21)γ
where (τ2i )∗ ∈ (0, 1/2) holds under Assumption 1. We substitute τ2 = (τ2i )∗ into (16) and(17) and obtain the following most preferred ﬁrst-period tax rates by a type-i agent when
he/she is unconstrained and constrained, respectively:
1 Rγ(2 − γ)−
· (ei1 )1−µ ,2
2AE1(1 + β)R(ψ − 1)γ(2 − γ)β+
· (ei1 )1−µ .
τ1ic ≡
1 + 2β(1 + 2β)AE1

τ1iu ≡(22)(23)

Next, we derive (Gi )∗ and (B i )∗ . We substitute (21) and (22) into (11) and (12) toobtain Giu and B iu :[] []1 Rγ(2 − γ)
1 Rγ(2 − γ)
E1
iu
i 1−µ
i 1−µ
G =
·
−
· (e1 )·+· (e1 )
2−γ
2
2AE1
2
2AE1
[
]
(
)2
E1
Rγ(2 − γ)i 2(1−µ)=1−· (e1 ),
4(2 − γ)AE1
B iu = γ(τ2i )∗ (1 − (τ2i )∗ ) · Giu= (γ − 1) · Giu

]
[
(
)2
Rγ(2 − γ)(γ − 1) · E1
i 2(1−µ)· (e1 )
1−;=4(2 − γ)AE1

22and we substitute (21) and (23) into (11) and (12) to obtain Gic and B ic :][E1
β(1 + β)R(ψ − 1)γ(2 − γ)
ic
i 1−µ
G =
+
· (e1 )
2 − γ 1 + 2β
(1 + 2β)AE1[]
1+β(1 + β)R(ψ − 1)γ(2 − γ)
i 1−µ
×
−
· (e1 )
1 + 2β(1 + 2β)AE1[(1 + β)R(ψ − 1)γ(2 − γ)E1β(1 + β) +=· (ei1 )1−µ2
(2 − γ)(1 + 2β)
AE1
]
(
)2
(1 + β)R(ψ − 1)γ(2 − γ)−
· (ei1 )2(1−µ) ,AE1
B ic = γ(τ2i )∗ (1 − (τ2i )∗ ) · Gic= (γ − 1) · Gic[(γ − 1) · E1
(1 + β)R(ψ − 1)γ(2 − γ)=β(1 + β) +· (ei1 )1−µ2
(2 − γ)(1 + 2β)
AE1
]
(
)2
(1 + β)R(ψ − 1)γ(2 − γ)−
· (ei1 )2(1−µ) .
AE1
Finally, we substitute τ1iu , (τ2i )∗ and Giu into the condition (7) to ﬁnd the thresholdlevel of capital market imperfection for a type-i agent:
ψi ≡

AE1
1
+
· (ei1 )µ−1 .
2(1 + β) 2R(1 + β) · γ(2 − γ)A.3

Voting Over τ2 When Preferences Are Characterized by aConstant Intertemporal Elasticity of Substitution

Proposition 1 demonstrates that the second-period tax rate is independent of types andcapital market imperfections under a logarithmic utility function. Here, we show thatthe result also holds true as long as the preferences are characterized by a constant intertemporal elasticity of substitution (CIES). For the purpose of analysis, we consider thefollowing utility function:
Ui =(ci1 )1−σ(ci )1−σ+β· 2, σ ̸= 1, σ > 0,1−σ1−σwhere σ represents the inverse of the intertemporal elasticity of substitution.
In order to solve the problem, suppose that the borrowing constraint is not binding.

23

The solution to the utility maximization problem yields:[]
1(1 − τ2 )ei2
iu
i
c1 =
,
1 · (1 − τ1 )e1 +R
R + (βR) σ[]
1
R(βR) σ(1 − τ2 )ei2
iu
i
c2 =
,
1 · (1 − τ1 )e1 +R
R + (βR) σ
[
]
1
i
σ
R
(βR)
(1
−
τ
)e
2
2
siu =
· (1 − τ1 )ei1 −
.
1 ·
R
R
R + (βR) σ

(24)
(25)
(26)

We substitute (26) into the borrowing constraint (3) to obtain the condition where a
type-i agent is actually unconstrained in terms of the tax rates τ1 and τ2 :[{}
] (1 − τ )ei
1
2 2
σ
R + (βR)·ψ−R >.
(27)(1 − τ1 )ei1
This condition states that the borrowing constraint of a type-i agent does not bind when
his/her after-tax income is high in the ﬁrst period and is low in the second period. Alternatively, suppose that the borrowing constraint is binding; that is, (27) fails to hold.
The solution to the utility maximization problem is given by (8), (9) and (10).
We substitute the consumption functions and the ﬁrst-period government budget constraint (11) into the utility function U i , and obtain the following indirect utility functionof a type-i agent:()1−σ {
}

1−σ
R
1

iu
σ

V
≡
·
1
+
β(βR)
· 1−σ
1


R+(βR) σ


[
]1−σ


(1−τ2 )
τ1 (1−τ1 )E1ii µ× (1 − τ1 )e1 + R · A(e1 ) · 1−γ 2 ·τ2 (1−τ2 )+ (const)



}
]
[{

1
(1−τ )ei
σ
i
·
ψ
−
R
> (1−τ21 )e2i ,
if
R
+
(βR)
V (τ1 , τ2 ) =
1

1
ic
i 1−σ


V
≡
{ψ(1
−
τ
)e
}
1

1
1−σ

[
]1−σ


(1−τ2 )
τ1 (1−τ1 )E1
R
i
i µ

+β
·
·
(1
−
ψ)(1
−
τ
)e
+
·
A(e
)
·

1
2
1

1−σ
R
1−γ ·τ2 (1−τ2 )

[{ 1
}
]

1

(1−τ )eiif
R + (βR) σ · ψ − R ≤ (1−τ21 )e2i .
1(28)
We take the ﬁrst derivative of V iu and V ic with respect to τ2 and obtain:[]∂(1 − τ2 )∂V iu (τ1 , τ2 )
R⇔
0
R0
∂τ2
∂τ2 1 − γ 2 · τ2 (1 − τ2 )1
⇔ τ2 Q 1 − ,
γ
and
[
]
∂V ic (τ1 , τ2 )∂(1 − τ2 )
R⇔
0
R0
∂τ2
∂τ2 1 − γ 2 · τ2 (1 − τ2 )1
⇔ τ2 Q 1 − .
γ24The result implies that voter’s preferences over τ2 are unaﬀected by types and capital
market imperfections as long as the utility function is characterized by CIES.
The intuition behind the result is as follows. An agent chooses τ2 to maximize his/hersecond-period after-tax income given by:(1 − τ2 )ei2 = (1 − τ2 ) · A(ei1 )µ · G= (1 − τ2 ) · A(ei1 )µ ·

τ1 (1 − τ1 )E1,
1 − γ 2 · τ2 (1 − τ2 )regardless of whether he/she is borrowing constrained or not; and whether his/her utility
function is characterized by a logarithmic function or CIES. The ﬁrst line comes from theassumption of ei2 = A(ei1 )µ · G; and the second line comes from the government budget
constraints (1) and (2). Thus, maximizing (1 − τ2 )ei2 with respect to τ2 is equivalent tomaximizing (1 − τ2 )/ {1 − γ 2 · τ2 (1 − τ2 )} that is independent of types and capital market
imperfections.

A.4

Proof of Corollary 1(i) The diﬀerentiation of τ1ic with respect to ψ leads to:∂τ ic(1 + β)Rγ(2 − γ)
=
> 0.
∂ψ
(1 + 2β)AE1
Next, we diﬀerentiate Gic with respect to ψ and derive the condition for which∂Gic /∂ψ > 0 holds:∂Gic(1 + β)Rγ(2 − γ)
>0 ⇔· (ei1 )1−µ∂ψAE1
(1 + β)R(ψ − 1)γ(2 − γ) (1 + β)Rγ(2 − γ)−2··
· (ei1 )2(1−µ) > 0
AE1
AE1(1 + β)R(ψ − 1)γ(2 − γ)
⇔ 1−2·· (ei1 )1−µ > 0
AE1
AE1⇔ ψ <1+(29)· (ei1 )µ−1 .
2R(1 + β)γ(2 − γ)
Based on the result in Proposition 1, the following condition holds when a type-i agent
is borrowing constrained:ψ < ψi ≡1
AE1+· (ei1 )µ−1 .
2(1 + β) 2R(1 + β)γ(2 − γ)

Because the right-hand side of (29) is greater than ψ i , we obtain:ψ < ψi < 1 +AE1· (ei1 )µ−1 ,2R(1 + β)γ(2 − γ)
25

implying that ∂Gic /∂ψ > 0 actually holds. Finally, given B ic = (γ − 1) · Gic , ∂B ic /∂ψ > 0
also holds in equilibrium.
(ii) From the result in Proposition 1, we obtain:][β(1 + β)Rγ(2 − γ)
1 + 2βAE1
ic
i µ−1
τ1 ψ=ψi =+· −+· (e1 )· (ei1 )1−µ
1 + 2β(1 + 2β)AE12(1 + β) 2R(1 + β)γ(2 − γ)βRγ(2 − γ)1=−· (ei1 )1−µ +1 + 2β2AE12(1 + 2β)
1 Rγ(2 − γ)= −· (ei1 )1−µ
2
2AE1
(30)
= τ1iu .
From (11), we also obtain:
τ1iu (1 − τ1iu )E1
τ ic (1 − τ1ic )E1
= 1
2−γ
2−γ
iu
ic
⇔ τ1 = τ1 ,

Giu = Gic ⇔which impliesψ = ψi.
Finally, we obtain:
B ic ψ=ψi = (γ − 1) · Gic ψ=ψi
= (γ − 1) · Giu= B iu .


A.5

Proof of Proposition 2

Consider ﬁrst the political determination of the second-period tax rate. As demonstrated
in Proposition 1, all agents choose the same second-period tax rate, given by τ2∗ = 1−1/γ.
This is the equilibrium tax rate in majority voting.
Next, consider the political determination of the ﬁrst-period tax rate. As demonstrated
in Proposition 1, the preferred tax rate by a type-i agent, (τ1i )∗ , is:{ ic
τ1 for ψ ≤ ψ i ,i ∗(τ1 ) =
τ1iu for ψ > ψ i .
Under Assumption 2, the critical degrees of the three types of agents, ψ l , ψ m , and ψ h areordered as:
ψh < 1 < ψm < ψl .
26

Given this order, a type-h agent is borrowing unconstrained and chooses (τ1h )∗ = τ1hu ∀ψ ∈[1, ∞). In contrast, a type-i (i = l, m) agent is borrowing constrained and chooses (τ1i )∗ =
τ1ic for ψ ≤ ψ i ; he/she is borrowing unconstrained and chooses (τ1i )∗ = τ1iu for ψ > ψ i .
Preferred ﬁrst-period tax rates by the three types of agents are illustrated in Panel(a) of Figure 2. Because (τ1l )∗ < (τ1m )∗ holds for ψ ∈ [1, ψ l ] and (τ1l )∗ > (τ1m )∗ holds forψ ∈ [ψ m , ∞), there exists a critical degree of ψ, denoted by ψ̂, such that (τ1l )∗ = (τ1m )∗holds. We solve (τ1l )∗ = (τ1m )∗ for ψ and obtain the following critical degree that changes
the order of preferred ﬁrst-period tax rates by types l and m agents:[]
1−µAE1 + Rγ(2 − γ) · 2(1 + β) · (el1 )1−µ − (1 + 2β) · (em1 )bψ=∈ (ψ m , ψ l ).
2(1 + β)Rγ(2 − γ) · (el1 )1−µb we now divide the range of ψ into the following four
Given the deﬁnition of ψ i and ψ,b [ψ,b ψ l ), and [ψ l , ∞). As illustrated in Panel (a) of Figure 2,
subranges: [1, ψ m ), [ψ m , ψ),preferred tax rates by the three types of agents and their order are:(τ1h )∗ = τ1hu < (τ1l )∗ = τ1lc < (τ1m )∗ = τ1mc for ψ ∈ [1, ψ m ]b(τ1h )∗ = τ1hu < (τ1l )∗ = τ1lc < (τ1m )∗ = τ1mu for ψ ∈ (ψ m , ψ]b ψl ]
 (τ1h )∗ = τ1hu < (τ1m )∗ = τ1mu < (τ1l )∗ = τ1lc for ψ ∈ (ψ,


h ∗
hu
m ∗
mu
l ∗
lu
(τ1 ) = τ1 < (τ1 ) = τ1
< (τ1 ) = τ1for ψ ∈ (ψ l , ∞).
Under the assumption of distribution of three types of agents, a type ranked in the
middle of preferred tax rates becomes the decisive voter. That is, the decisive voter is ab it is a borrowing-unconstrained type-mborrowing-constrained type-l agent for ψ ∈ [1, ψ);b ∞). Thus, the equilibrium tax rate determined in majority voting is:agent for ψ ∈ [ψ,{bτ1lc for ψ ∈ [1, ψ](τ1 )∗ =b ∞).
τ1mu for ψ ∈ (ψ,The corresponding levels of G and B are:{b
Glc for ψ ∈ [1, ψ]
G∗ =
b ∞)
Gmu for ψ ∈ (ψ,{b
B lc for ψ ∈ [1, ψ]
B∗ =
b ∞).
B mu for ψ ∈ (ψ,These are calculated based on the result in Proposition 1.


A.6

Proof of Proposition 3

As demonstrated in Proposition 2, τ1∗ is equal to τ1lc and is increasing in ψ for ψ ∈ [1, ψ̂);
τ1∗ is equal to τ1mu and is constant for ψ ∈ [ψ̂, ∞). That is, τ1∗ attains the lowest value27

β/(1 + 2β) at ψ = 1 and the highest value at ψ ∈ [ψ̂, ∞). Thus, τ1e > τ1∗ holds ∀ψ ≥ 1 if
τ1e > τ1mu ; and τ1e < τ1∗ holds ∀ψ ≥ 1 if τ1e < β/(1 + 2β). That is,
(

AE1
Rγ 2
1 + 2βτ1e < τ1∗ ∀ψ ≥ 1 if γ <.
1+β
τ1e > τ1∗ ∀ψ ≥ 1 if

em
1 >

1
) 1−µ

,

1

2 1−µ
Suppose that the above-mentioned two conditions fail to hold; that is, em
1 ≤ (AE1 /Rγ )and γ ≥ (1 + 2β)/(1 + β) hold. Because of the continuity of τ1lc with respect to ψ, there
exists a critical level of ψ, denoted by ψ ∗ , such that τ1e = τ1lc holds. Solving τ1e = τ1lc forψ leads to:[](1 + 2β)AE1
1+β1· (el1 )µ−1 .
ψ =ψ ≡1+−(1 + β)Rγ(2 − γ) 1 + 2β γ∗Thus, we obtain τ1e R τ1∗ if ψ Q ψ ∗ .
The same argument holds for G∗ and B ∗ because they have the same property as τ1∗with respect to ψ, as demonstrated in Proposition 2.


A.7

Proof of Proposition 4

Consider an increase in the initial income of the high type, eh1 , coupled with a decrease
e1 unchanged. Given the deﬁnition of E
e1 ≡in the initial income of the type-l, keeping E∑ i i µe1 -preserving spread of income distribution results in:ϕ (e ) , the EAi1e1 = 0 ⇔ deh1 = (−1)
dE

ϕl (eh1 )1−µ·
· del1 .
ϕh (el1 )1−µ

Under this spread of income distribution, the ﬁrst-period GDP, given by E1 =

(31)
∑

i i
i ϕ e1 ,changes as follows:
dE1 = ϕl · del1 + ϕh · deh1[]
(eh1 )1−µ= 1 − l 1−µ · ϕl · del1
(e1 )
> 0.

(32)

where the equality in the second line comes from (31).
By the use of the result in (32), we hereafter investigate how the type-j ′ s (j = l, m, h)
e1 -preserving spread of income dispreference over B/E1 changes in response to the Etribution. We ﬁrst consider borrowing-unconstrained and borrowing-constrained cases
separately. Then, we unify the two cases to determine the decisive voter for each level of28

e1 -preserving spread of income distribution, and demonstrate theψ before and after the Eeﬀect of this distribution spread on the decisive voter’s choice over B/E1 .
Under Assumption 2, a type-h agent is always borrowing unconstrained. The preferredlevel of B/E1 by a type-h agent is always lower than that by the other two types of agents,
under Assumption 2. The decisive voter is a type-l or type-m agent who prefers a lower
B/E1 (Proposition 1) between them. Therefore, we hereafter focus on types-l and magents.
A.7.1

An unconstrained type-j’s (j = l, m) preference over B/E1

e1 We consider how the type-j’s (j = l, m) preference over B/E1 is aﬀected by the E
preserving spread of income distribution when he/she is borrowing unconstrained. From
(14) in Proposition 1, the preferred debt-to-GDP ratio by an unconstrained type-j’s individual is given by:
]
[
(
)2
B ju
γ−1
Rγ(2 − γ)j 2(1−µ),=· 1−· (e1 )
E14(2 − γ)AE1

j = l, m.

Diﬀerentiation of the above equation with respect to B ju /E1 , E1 and ej1 leads to:4(2 − γ)
·d
γ−1

(

B ju
E1

)

(

)2 ( )31=2·· (ej1 )2(1−µ) · dE1
E1()2
Rγ(2 − γ)− 2(1 − µ) ·· (ej1 )1−2µ · dej1
AE1
{
}
(
)2
j
Rγ(2 − γ)e1=2· (ej1 )1−2µ ·· dE1 − (1 − µ) · dej1 .
AE1
E1
Rγ(2 − γ)
A

(33)

(
)
(33) indicates that d B lu /E1 > 0 always holds because dE1 > 0 as shown in (32) anddej1 ≤ 0 (j = l, m), under the current assumption.
A.7.2

A constrained type-j’s (j = l, m) preference over B/E1

Next, we consider how the type-j’s (j = l, m) preference over B/E1 is aﬀected by thee1 -preserving spread of income distribution when he/she is borrowing constrained. FromE(14) in Proposition 1, the preferred debt-to-GDP ratio by a constrained type-j’s individualis given by:
[
{
}]
B jc
γ−1
(ej1 )1−µ(ej1 )2(1−µ)=β(1 + β) + Γ ·−Γ·, j = l, m; (34)E1(2 − γ)(1 + 2β)2E1(E1 )2whereΓ≡(1 + β)R(ψ − 1)γ(2 − γ).
A29

Diﬀerentiation of the above equation with respect to B jc /E1 , E1 and ej1 leads to:( jc ){}(2 − γ)(1 + 2β)2B
1 − µ j −µ2(1 − µ)j 1−2µ=Γ·(e1 ) − Γ ·· (e1 )· dej1·dγ−1E1E1(E1 )2{}(ej1 )1−µ(ej1 )2(1−µ)+Γ· −+ 2Γ ·· dE1(E1 )2(E1 )3{}2Γ(ej1 )1−µ
(1 − µ)Γ(ej1 )−µ=· 1−· dej1E1
E1{}
Γ(ej1 )1−µ2Γ(ej1 )1−µ
−
· 1−
· dE1
(E1 )2E1]{} [Γ(ej1 )−µ2Γ(ej1 )1−µ
ej1
j
=
· 1−
· (1 − µ) · de1 −· dE1 ,
E1
E1
E1
(35)
{
}
where the sign of the term 1 − 2Γ(ej1 )1−µ /E1 in the last line is positive under the as()
sumption that the type-j agent is borrowing constrained.5 (35) indicates that d B lc /E1 >0 always holds because dE1 > 0 as shown in (32) and dej1 ≤ 0 (j = l, m), under the current
assumption.
A.7.3

Decisive voter’s choice over B/E1

Based on the results established so far, we can illustrate changes in preferences over B/E1
by three types of agents, as in Figure 4.
The bold dotted and solid curves depict the decisive voter’s choices over B/E1 before
and after the spread of income distribution, respectively. The spread of income distribution increases the threshold level of ψ that distinguishes the type of a decisive voter,
The parameter ψ included in Γ satisfy ψ ≤ ψ j because we assume that the type-j agent (j = l, m) is
now borrowing constrained (Proposition 1). By the use of the deﬁnition of ψ j in Proposition 1, we canwrite:
51−

2Γ(ej1 )1−µ2(1 + β)R(ψ − 1)γ(2 − γ) · (ej1 )1−µ
=1−
E1
AE1
j
2(1 + β)R(ψ − 1)γ(2 − γ) · (ej1 )1−µ≥1−AE12(1 + β)Rγ(2 − γ) · (ej1 )1−µ AE1 − (1 + 2β)Rγ(2 − γ) · (ej1 )1−µ
=1−
·
AE1
2(1 + β)Rγ(2 − γ) · (ej1 )1−µ
AE1 − (1 + 2β)Rγ(2 − γ) · (ej1 )1−µAE1
(1 + 2β)Rγ(2 − γ) · (ej1 )1−µ
=
AE1
> 0,
=1−

where the inequality in the second line comes from ψ ≤ ψ j , and the inequality in the last line comes from
Assumption 1.

30

b ψb′ )denoted by ψ̂, from ψb to ψb′ . There exists a unique level of ψ, denoted by ψel ∈ (ψ,such that the bold dotted and solid curves cross at this level. Therefore, there are fourb (ψ,b ψel ], (ψel , ψb′ ] and (ψb′ , +∞).
subranges of ψ, as illustrated in Figure 4: [1, ψ],b the initial decisive voter is a borrowing-constrained, type-l agent, andFor ψ ∈ [1, ψ],
he/she still remains the decisive voter after the spread of income distribution. His/her
choice over policies results in a lower level of B/E1 in response to the spread of income
e1 -preserving spread of
distribution, as demonstrated in Subsection A.7.2. Thus, the E
income distribution leads to a lower level of B/E1 .
b ψel ], the initial decisive voter is a borrowing-unconstrained type-m agent.
For ψ ∈ (ψ,Because of the spread of income distribution, the decisive voter changes from a borrowingunconstrained type-m agent to a borrowing-constrained type-l agent. The policy choiceby the latter results in a lower level of B/E1 than that by the former, as demonstrated
in Figure 4.
For ψ ∈ (ψel , ψb′ ], the initial decisive voter is a borrowing-unconstrained type-m agent.
The decisive voter changes from a borrowing-unconstrained type-m agent to a borrowingunconstrained type-l agent. The policy choice by the latter results in a higher level of
B/E1 than that by the former, as demonstrated in Figure 4.
Finally, for ψ ∈ (ψb′ , +∞), the initial decisive voter is a borrowing-unconstrained typem agent, and he/she still remains the decisive voter after the spread of income distribution.
His/her choice over policies results in a higher level of B/E1 in response to the spreade1 -preserving spread ofof income distribution, as shown in Subsection A.7.1. Thus, the E
income distribution leads to a higher level of B/E1 .


31

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Figure 1: The most preferred policies for a type-i agent.

35

Figure 2: The bold curves show equilibrium policies.

36Figure 3: Panel (a) is the case of em
1 >panel (c) is the case of em
1 ≤

(

AE1
Rγ 2

1
) 1−µ

(

AE1
Rγ 2

1
) 1−µ

; and
; panel (b) is the case of γ < 1+2β
1+βand γ ≥ 1+2β
.
1+β

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Figure 4: In panel (a), the dotted and solid curves depict the choice of a type-j(= l, m)agent over B/E1 before and after the spread of income distribution, respectively. In panel(b), the bold dotted and solid curves depict the choice of the decisive voter over B/E1
before and after the spread of income distribution, respectively.

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Figure 5: The bold curve illustrates the type-i’s indirect utility as a function of τ1 givenτ2 . The ﬁgure is the case of τ1iu (τ2 , ei1 ) < τb1 (τ2 , ei1 ).

39