Discussion Papers In Economics
And Business

Growth, Unemployment, and Fiscal Policy:
A Political Economy Analysis
Tetsuo Ono
Discussion Paper 14-30-Rev.2

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANGrowth, Unemployment, and Fiscal Policy:
A Political Economy Analysis
Tetsuo Ono
Discussion Paper 14-30-Rev.2

January 2017

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANGrowth, Unemployment, and Fiscal Policy: A Political
Economy Analysis∗
Tetsuo Ono†
Osaka University

Abstract
This study presents an overlapping-generations model featuring capital accumulation, collective wage-bargaining, and probabilistic voting over fiscal policy. The
study characterizes a Markov-perfect political equilibrium of the voting game within
and across generations and shows the following results. First, greater bargaining
power of unions lowers the growth rate of capital and creates a positive correlation between unemployment and public debt. Second, an increase in the politicalpower of the old lowers the growth rate and shifts government expenditure fromthe unemployed to the old. Third, prohibiting debt finance increases the growthrate and benefits future generations; however, it worsens the current employed andunemployed.
Keywords: Economic Growth; Fiscal Policy; Government Debt; Unemployment;
VotingJEL Classification: E24, E62, H60.

∗

The author would like to thank Yuki Uchida for his research assistance. This work was supported
in part by a Grant-in-Aid for Scientific Research (C) from the Ministry of Education, Culture, Sports,
Science and Technology of Japan No. 15K03509.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. Tel.: +81-6-6850-6111; Fax: +81-6-6850-5256. E-mail: tono@econ.osaka-u.ac.jp

1

Introduction

Public debt and economic growth have been major concerns for policymakers in mostadvanced countries in past decades; however, the debt-to-GDP ratio has increased inmany OECD (Organization for Economic Co-operation and Development) countries overthe last 20 years. The burden of debt repayment can crowd out private investment
and deteriorate economic performance in the long run. As illustrated in Figure 1, theevidence suggests a negative correlation between public debt and economic growth inadvanced economies (see, e.g., Reinhart, Reinhart, and Rogoﬀ, 2012; Kumar and Woo,2010; Checherita-Westphal and Rother, 2012).
[Figure 1 here.]
Prior studies suggest the negative growth eﬀect of public debt using a theoretical
approach based on neoclassical growth models (Diamond, 1965), and later, based on endogenous ones (Saint-Paul, 1992; Josten, 2000; Bräuninger, 2005). These studies assume
perfectly competitive labor markets with full employment. To consider the eﬀect in amore realistic environment, some recent studies extend these models by including unemployment and investigate the eﬀects of fiscal policy on economic growth in the presence of
unemployment (Kaas and von Thadden, 2004; Josten, 2006; Greiner and Flaschel, 2010;
Yakita, 2014).
However, the following issues remain unanswered in these studies. First, a high unemployment rate, possibly stemming from the power of trade unions, exerts political pressure
on the government to increase spending in favor of the unemployed. This pressure incentivizes the government to issue more public bonds to finance expanding expenditure,
resulting in crowding out of capital accumulation. That is, trade unions have a politicaleﬀect on fiscal policy and economic growth through unemployment. As depicted in Figure 2, the cross-country evidence shows a slight positive correlation between trade union
density and debt-GDP ratios, and a negative correlation between trade union density andper capita GDP growth rates. The aforementioned studies do not clarify the mechanisms
underlying these findings because they consider fiscal policy as exogenously given.
[Figure 2 here.]
Second, besides unemployment insurance, there is large public expenditure on intergenerational redistribution from the young to the old, such as public pensions and healthand nursing care systems for the aged in advanced countries. This implies that greater
political power of the old, possibly stemming from an increase in the old-age dependency
ratio, exerts pressure on the government to shift fiscal expenditure from the unemployed1young to the retired old. However, the cross-country evidence shows that the expenditureon the unemployed is positively correlated with the expenditure on the old (see Figure3). The evidence suggests that the political eﬀect of the old is outweighed by the politicaleﬀect of trade unions, and that increase in unemployment and old age benefits would be
financed by issuing bonds. Corneo and Marquardt (2000), Bräuninger (2005), and Ono(2007) analyze two types of expenditure in a unified framework; however, they assume no
public bonds and consider the tax rates for financing these expenditure as exogenously
given, and thus, set aside the political background behind the evidence.
[Figure 3 here.]
The argument, thus far, suggests that the two issues should be addressed together
because each component influences the other. In addition, the political conflict over the
allocation of public budgets should be addressed in analyzing two fiscal expenditures:
unemployment and old age benefits. To address these concerns, we employ a two-period
overlapping-generations model with AK technology (Romer, 1986) and collective wage
bargaining (see, e.g., Kaas and von Thadden, 2003, 2004; Coimbra, Lloyd-Braga, andModesto, 2005; Chang, Shaw, and Lai, 2007) to demonstrate capital accumulation andunemployment. Government spending is represented by unemployment-insurance benefits
for the unemployed and public services for the old.1 The spending is financed by taxing
the young and also by issuing public bonds.
The unemployment-insurance benefit creates a conflict of interest between the unemployed and the employed. Furthermore, spending on services for the old creates a conflictof interest between the young (both the employed and the unemployed) and the old.
To demonstrate this conflict, we assume probabilistic voting à la Lindbeck and Weibull(1987), where the government objective is to maximize the weighted sum of the utility of
the young employed, the young unemployed, and the retired old. In particular, we employ
a Markov strategy in which policy variables are conditioned on payoﬀ-relevant state variables (Krusell, Quadrini, and Rios-Rull, 1997). This strategy enables us to demonstrate
the forward-looking behavior of individuals who consider intertemporal interaction between current and future policies through capital accumulation (see, e.g., Gonzalez-Eirasand Niepelt, 2008, 2012; Song, 2011; Kunze, 2014; Lancia and Russo, 2016).
Within this framework, we show that increased power of unions results in higher unemployment and a higher debt-to-GDP ratio. Thus, a positive correlation holds between
unemployment and debt. This result fits the empirical evidence observed in advanced
countries, as illustrated in Figure 4. The result is also in line with previous studies by
1

In general, there are two types of public expenditures on the old: public pensions, which compensate
for the lack of income after retirement, and public services, which improve utility in old age. This study
focuses on the latter type of expenditure for a tractable analysis.

2Kaas and von Thadden (2004) who show a positive correlation under capital shortages
and Battaglini and Coate (2014) who show the pro-cyclical behavior of unemployment anddebt arising from time-varying productivity. This study presents an alternative approachfor explaining the positive correlation.
[Figure 4 here.]
Following this, we consider the second issue and show that an increase in the politicalpower of the old, possibly stemming from an increase in the old-age dependency ratio,
results in a higher ratio of spending to GDP for the old, a lower ratio of unemployment
insurance benefits to GDP, and a lower growth rate of capital. The result suggests that anincrease in political power of the old results in a shift of resources from the unemployedyoung to the old, and the old harm economic growth through redistributive politics.
However, this result should be considered together with the first result to obtain themodel prediction that fits the evidence in Figure 3.
Some OECD countries have introduced budget rules that control public bond issues
from the perspective of fiscal sustainability. To assess the impact of debt control, we
consider two alternative budget scenarios that limit public bond issues. The first is a
tax-finance rule where government spending is solely financed by tax. Then, we compare
the debt-finance and the tax-finance cases, and obtain the following result. When the government finances its spending by issuing public bonds (i.e., by borrowing in the capital
market), the introduction of a tax-finance requirement results in a higher growth rate, andthus, benefits future generations; however, it results in a lower unemployment-insurance
payment-to-GDP ratio and a higher tax rate in the initial period. Therefore, the introduction of the tax-finance rule is not Pareto-improving; it benefits future generations at
the expense of the current employed and unemployed young.
The second scenario is constrained debt finance in which the government self-imposesa public debt ceiling similar to the member countries of the European Union. Given thisconstraint, we show that the debt-capital ratio increases over time while the constraint
continues to be binding, and moves into a state where the debt constraint becomes nonbinding and the ratio remains constant through time. The result is qualitatively diﬀerent
form that in Azzimonti, Battgalini, and Coate (2016) who demonstrate that debt accumulates initially and then reaches its ceiling. The diﬀerence arises because there is no
capital accumulation and resources are limited across periods in the model of Azzimonti,
Battaglini, and Coate (2016), while there is capital accumulation that increases income,and thus, tax base over time in the present model.
This study contributes to the following strands of political economy literature. First,
the literature on the positive theory of fiscal policy (see, e.g., Battaglini and Coate,3

2008; Song, Storesletten, and Zilibotti, 2012; Barseghyan, Battaglini, and Coate, 2013;
Battaglini, 2014; Arai and Naito, 2014; Ono, 2015). In particular, the framework ofthis study is based on that of Arai and Naito (2014) and Ono (2015); the present study
introduces the managerial trade union as a source of unemployment into their model. The
study focuses on the same issue as Battaglini and Coate (2014), who analyze politicaldecisions on fiscal policy in the presence of unemployment. However, they assume (i)exogenous wage rigidity and time-varying productivity as a source of unemployment; and(ii) no saving behavior, and thus, no capital accumulation. In contrast, this study assumes
collective wage-bargaining to demonstrate the mechanism in which unemployment arises
as an equilibrium phenomenon, and the AK technology to demonstrate the eﬀect of fiscal
policy on capital accumulation in the presence of unemployment.
The second strand includes studies on intragenerational and intergenerational redistributive politics in models with physical and/or human capital accumulation (see, e.g.,
Poutvaara, 2006; Bassetto, 2008; Gonzalez-Eiras and Niepelt, 2008, 2012; Song, 2011;
Bernasconi and Profeta, 2012; Uchida, 2016). They assume competitive labor markets,and thus, no equilibrium unemployment. In contrast, this study presents equilibrium
unemployment and demonstrates an intragenerational conflict between the employed andthe unemployed and an intergenerational conflict between the young (both the employed
and the unemployed) and the old. Within this environment, we consider redistributive
politics over unemployment-insurance benefits and redistribution targeting the elderly.
We show the welfare eﬀects of intragenerational and intergenerational conflicts on fiscal
policy and capital accumulation in the presence of unemployment.
Section 2 of this paper presents the model and characterizes the economic equilibrium.
Section 3 characterizes a political equilibrium when government expenditure is financedby tax and issue of public bonds. Section 4 considers two alternative scenarios that limit
public bond issues and investigates their impact on fiscal policy, growth, and/or welfare.
Section 5 checks the robustness of the results under alternative assumptions. Section 6presents some caveats to the analysis along with our conclusions. The Appendix contains
the proofs.

2

Model and Economic Equilibrium

Consider a two-period-lived overlapping-generations model where the economy comprisesof perfectly competitive firms, ex ante identical individuals, a trade union, and a government. Time is discrete and denoted by t = 0, 1, 2, · · · . A new generation is born ineach period t = 0, 1, 2, · · · , and individuals in each generation live for two periods, youth
and old age. No population growth is assumed, and the population in each generation isnormalized to unity.
42.1

Preferences and Utility Maximization

An individual supplies one unit of labor inelastically in youth and retires in old age. The
lifetime utility of an individual born in period t is given byoi
Uti = ln cyi
t + β ln ct+1 + βη ln gt+1 ,oiwhere cyit is consumption in youth, ct+1 is consumption in old age, gt+1 is public services
for the old (e.g., medical care systems and nursing-care insurance systems), β ∈ (0, 1)

is a discount factor, and η(> 0) captures the preference weight for public services. The
subscript t denotes the period of consumption, and the superscript i denotes the statusof labor: i = e and i = u if an individual is employed and unemployed, respectively. The
status is assigned according to bargaining between the trade union and the firm (describedlater) at the beginning of each period. The specification of the logarithmic utility function
makes the aggregation of the savings functions tractable.
An individual chooses consumption and savings to maximize lifetime utility under thefollowing budget constraints:icyi
t + st ≤ xt (1 − τt )wt + (1 − xt )bt , xt ∈ {0, 1}icoi
t+1 ≤ Rt+1 st ,where xt = 1 and xt = 0 if an individual is employed and unemployed, respectively. wt
is wage, bt is the unemployment-insurance benefit, st is savings, Rt+1 is the gross interest
rate, and τt is the tax on labor income. Unemployment-insurance benefits are assumed
exempt from taxation.
By solving the utility-maximization problem, we obtain the savings function of a type-i
individual as follows:sit =β{xt (1 − τt )wt + (1 − xt )bt } .
1+βThe corresponding consumption functions are1{xt (1 − τt )wt + (1 − xt )bt } ,
1+β
βRt+1
coi
{xt (1 − τt )wt + (1 − xt )bt } .
t+1 =
1+β
cyi
t =

These functions state that a higher wage level or unemployment-insurance benefit
implies higher savings and consumption, whereas a higher tax rate implies lower savingsand consumption. Using these functions, the indirect utility functions of the employed,5the unemployed, and the old are given by
(

1
β
ln
+ β ln
1+β
1+β
)
(
1
β
+ β ln,Vtyu = (1 + β) ln bt + β ln Rt+1 + βη ln gt+1 + ln1+β1+βVto = η ln gt + ln Rt sit−1 , i = e, o,)Vtye = (1 + β) ln(1 − τt )wt + β ln Rt+1 + βη ln gt+1 +,respectively.

2.2

Technology and Profit Maximization

There is a continuum of identical firms that are perfectly competitive profit maximizers
producing the final product Yt with a constant returns-to-scale Cobb–Douglas production
function, Yt = At (Kt )α (Lt )1−α . Here, At is the productivity parameter, Kt is aggregate
capital, Lt is aggregate labor, and α ∈ (0, 1) is a constant parameter representing capitalshare. Capital is assumed to fully depreciate within a period.
In each period t, a firm chooses capital and labor in order to maximize its profit,
At (Kt )α (Lt )1−α − ρt Kt − wt Lt , where ρt is the rental price of capital and wt is the wage
rate. The firm takes these prices as given. The first-order conditions with respect to Kt
and Lt are given by
Kt : ρt = αAt (Kt )α−1 (Lt )1−α ,Lt : wt = (1 − α)At (Kt )α (Lt )−α .
The productivity parameter At is assumed to be proportional to the aggregate capitalin the overall economy: At = A (Kt )1−α . Thus, capital investment involves a technologicalexternality of the type often used in theories of endogenous growth (Romer, 1986). This
assumption, called the “AK” technology, results in a constant interest rate across periods,
as demonstrated below. This approach enables us to obtain an analytical solution for themodel, that is, our model becomes tractable.
Under this assumption, the first-order conditions are rewritten as follows:ρt = αA (lt )1−α = Rt ,(1)

wt = (1 − α)AKt (lt )−α ,(2)where lt is the employment rate in the economy, and lt = Lt holds because the number ofpeople in each generation is unity. The arbitrage condition ρt = Rt holds for all t, because
the market for capital is competitive and capital fully depreciates within a period.

6

2.3

Government Budget Constraint

Fiscal policy is determined through elections, and public bonds are traded in a domestic
capital market. Let Dt denote the aggregate inherited debt. A budget constraint in period
t is
Dt+1 + τt lt wt = gt + (1 − lt )bt + Rt Dt ,(3)
where Dt+1 is newly issued public bonds, τt lt wt is the labor income-tax revenue, gt is
expenditure for the old, (1 − lt )bt is unemployment-insurance payments, and Rt Dt is
debt repayment. We assume that the government in each period is committed to not
repudiating the debt.
Equation (3) indicates that the government can freely issue public bonds as long as
it satisfies the flow-budget constraint. We demonstrate the political equilibrium outcomeof this case in Section 3. In Section 4, we consider two alternative cases: the tax-financecase, in which the government is required to finance its spending solely by taxation, andthe constrained debt-finance case where the government self-imposes a public debt ceiling.

2.4

Right-to-manage Model

Following Pemberton (1988), we assume a managerial trade union whose objective is
to pursue two targets: a high real wage, wt , and a high rate of employment, lt . In
particular, the trade union’s objective function is specified using the following Cobb–
Douglas function:(wt − w̄t )δ · (lt )1−δ ,where w̄t is the reference wage of the trade union, and δ ∈ (0, 1) is a parameter capturing
the relative intensity of the two targets.
Following Corneo and Marquardt (2000), we assume that the reference wage is thecompetitive wage, which is calculated by setting lt = 1 in the first-order condition withrespect to labor, (2):w̄t = (1 − α)AKt .
Alternatively, we assume that the reference wage is set to the unemployment-insurance
benefits, w̄t = bt (see, e.g., Chang, Shaw, and Lai, 2007), which is discussed in Section 6.
The present study employs the right-to-manage model (see, e.g., Benassy, 2011, Chapter 15, and Heijdra, 2009, Chapter 7, for an overview of the model). The union andthe firm bargain over wages through a generalized Nash bargaining solution. Given thesolution, employment is determined to satisfy the labor demand function of the firm.
According to this solution, the wage chosen after bargaining maximizes the geometricallyweighted average of the gains to the union and the firm, subject to the firm’s demand for7

labor. Formally, the problem is as follows:
[
]θ [
]1−θ
max Ωt = (wt − w̄t )δ (lt )1−δ · At (Kt )α (lt )1−α − wt lt
wt

s.t. wt = (1 − α)At (Kt )α (lt )−αgiven w̄t ,where θ ∈ [0, 1] represents the relative strength of the union. The term (wt − w̄t )δ (lt )1−δis the gain to the union, whereas the term At (Kt )α (lt )1−α − wt lt is the gain to the firm.
To solve the problem, we impose the following assumption, which ensures the secondorder condition for an interior solution.
{}1−δθAssumption 1. α < min 1, (1−θ)+δθ.
Under Assumption 1, the wage determined through bargaining becomes
wt = ϕw̄t = ϕ(1 − α)AKt ,(4)

where the second equality comes from w̄t = (1 − α)AKt , and ϕ is defined byϕ≡(1 − δ)θ + (1 − α)(1 − θ)(> 1).
(1 − δ)θ + (1 − α)(1 − θ) − αδθThe derivation of (4) is provided in Appendix A.1.
We substitute (4) into the labor demand function wt = (1 − α)AKt (lt )−α to obtain
the employment rate determined through bargaining:
lt = l ≡ (1/ϕ)1/α .

(5)Given that ϕ is increasing in θ, we immediately understand that higher union power yields
lower employment, ∂l/∂θ < 0.
Equation (5) indicates that the employment rate (or unemployment rate) is independent of fiscal policy and the stock of capital. This implies that the present model demonstrates the eﬀect of (un)employment on fiscal policy and capital accumulation; however,it does not show the reverse eﬀect. This property, caused by the specification of collective wage-bargaining, shows some limitations of this study. However, the present model
enables us to consider the interaction between fiscal policy and capital accumulation in
the presence of unemployment.
Using (5), we can write the aggregate output and the gross interest rate in terms of
the employment rate:
Yt = AKt l(l)−α = AlϕKt ,(6)
Rt = αAl(l)−α = αAlϕ ≡ R.

(7)

We use the expressions in (5), (6), and (7) in the following analysis. Hereafter, we often
use R instead of Rt = αAlϕ to simplify the presentation.
8

2.5

Economic Equilibrium

A market-clearing condition for capital is Kt+1 + Dt+1 = st , which expresses the equalityof total savings by young agents in generation t, st ≡ lt set + (1 − lt )sut , to the sum of thestocks of aggregate physical capital and aggregate public debt, Kt+1 + Dt+1 :Dt+1 + Kt+1 =β{l(1 − τt )ϕ(1 − α)AKt + (1 − l)bt } .
1+β

(8)

We are now ready to formally define an economic equilibrium in the present model.
Definition 1. Given a sequence of policy parameters {τt , bt , gt , Dt+1 }∞t=0 , an economic{ yi oi}∞iequilibrium is a sequence of prices and allocations, ct , ct+1 , st , lt , Kt , wt , w̄t , ρt , Rt t=0 ,with initial conditions K0 and D0 such that the following conditions are satisfied:()ioi(i) given (wt , Rt+1 ) and a fiscal policy, cyit , ct+1 , st solves the utility-maximization
problem of a type-i agent; (ii) given (wt , ρt ), (lt , Kt ) solves the profit-maximization
problem of a firm; (iii) given (w̄t , Kt ), wt solves the Nash bargaining problem; (iv)the reference wage w̄t is computed by assuming full employment in the labor market;(v) given (lt , wt , Rt , Dt ), (τt , bt , gt , Dt+1 ) satisfies the government budget constraint;(vi) ρt = Rt holds; (vii) the capital market clears: Dt+1 + Kt+1 = lt set + (1 − lt )sut .
In each period, the timing of the events is as follows. First, the government representing the young and the old decides a fiscal policy to maximize its objective function(demonstrated later). Second, wage is determined by the bargaining process, taking asgiven that the agents understand how wage aﬀects labor demand. Then, the firm demands
capital and labor, and sets employment according to its labor demand curve. Given a
fiscal policy, wage, and an interest rate, each young agent sets savings and consumption
to maximize his/her utility. Finally, the capital market clears.

3

Political Equilibrium

To consider the behavior of the government, we need to determine its objective and theagents’ indirect utility functions. Recall that Vtye , Vtyu , and Vto denote the indirect utilityof a young-employed agent in period t, the indirect utility of a young-unemployed agent
in period t, and the indirect utility of an old agent in period t, respectively. These areexpressed as functions of government policy and/or the stock of capital as follows:
Vtye = (1 + β) ln(1 − τt )ϕ(1 − α)AKt + βη ln gt+1 + C;
Vtyu = (1 + β) ln bt + βη ln gt+1 + C;
Vto = ln R(Kt + Dt ) + η ln gt ,9where(C ≡ β ln R + ln

β1
+ β ln
1+β
1+β

)
.

The terms (1−τt )ϕ(1−α)AKt and bt correspond to the lifetime income (and thus, lifetime
consumption) for the employed and the unemployed, respectively.
This study assumes probabilistic voting à la Lindbeck and Weibull (1987) in thedemonstration of the political mechanism (see Acemoglu and Robinson, 2005, Appendix,
and Persson and Tabellini, 2000, pp. 54–58, for an overview of this voting mechanism).
In each period, the government in power chooses fiscal policy to maximize its politicalobjective. Formally, the political objective function in period t is given by
Pt = ωVto + (1 − ω) {l · Vtye + (1 − l) · Vtyu } ,orPt = ωη ln gt + (1 − ω)(1 + β) [l · ln(1 − τt )ϕ(1 − α)AKt + (1 − l) · ln bt ]+ (1 − ω)βη ln gt+1 ,where ω and 1−ω are the relative weights of old and young agents, respectively, and l and
1 − l are relative weights of the employed and the unemployed measured as a percentageof the young generation population, respectively.2 The terms unrelated to politics areomitted from the above expression.
The remainder of the analysis proceeds as follows. We consider the debt-finance case:the government may borrow or lend in the capital market to finance its spending or use
its surplus funds. In the next section, we consider two alternative budget rules that
constrain public bond issues. First, the tax-finance case in which the government is notallowed to issue public bonds by the constitution and finances its spending solely bytaxation. Second, the constrained debt-finance case in which the government self-imposesa public debt ceiling. Then, we compare the debt-finance case and the tax-finance case(or the constrained debt-finance case) in the size of government debt, economic growth,and welfare.

3.1

Debt-finance Political Equilibrium

Given Kt and Dt , the problem of the government in period t is to choose a set of fiscal policies, (τt , gt , bt , Dt+1 ) , to maximize Pt subject to the period-t government budget
constraint. The problem is dynamic in that the values of the next-period state variables,
Kt+1 and Dt+1 , passed from the current government to the next government, will aﬀect
the choice of gt+1 by the next government. This choice, in turn, has an eﬀect on the utilityof the current young, and thus, on the current government’s objective.
2

Appendix B provides the microfoundation of the political objective function.

10

To take account of the above feature, we restrict our attention to a Markov-perfectequilibrium. Markov-perfectness implies that the outcomes are history-dependent only on
the payoﬀ-relevant state variables, that is, capital K and public debt D. Therefore, theexpected level of public services in the next period, gt+1 , is given by a function of the next
period capital stock and public debt, gt+1 = G(Kt+1 , Dt+1 ). Using a recursive notation
with x′ denoting the next period x,, we can define a Markov-perfect political equilibrium
as follows.
⟨⟩
Definition 2. A Markov-perfect political equilibrium is a set of functions, T̃ , G̃, B̃, D̃ ,where T̃ : ℜ++ × ℜ → [0, 1] is a tax rule, τ = T̃ (K, D); G̃ : ℜ++ × ℜ → ℜ++ is a public
expenditure rule, g = G̃(K, D); B̃ : ℜ++ × ℜ → ℜ+ is an unemployment-insurance rule,b = B̃(K, D); and D̃ : ℜ++ × ℜ → ℜ is a debt rule, D′ = D̃(K, D), such that(i) the capital market clears:
D̃(K, D) + K ′ =(ii) given K and D,

)
]
β [ (
l 1 − T̃ (K, D) ϕ(1 − α)AK + (1 − l)B̃(K, D) ;
1+β

(9)

⟨
⟩
T̃ (K, D), G̃(K, D), B̃(K, D), D̃(K, D) = arg max P subject tog ′ = G̃(K ′ , D′ ), the capital market clearing condition in (9), and the government
budget constraint,G̃(K, D) + (1 − l)B̃(K, D) + RD = T̃ (K, D)lϕ(1 − α)AK + D̃(K, D),
where P is defined by
P (K, g, τ, b, g ′ ) = ωη ln g+(1−ω)(1+β) [l ln(1 − τ )ϕ(1 − α)AK + (1 − l) ln b]+(1−ω)βη ln g ′ .
.
The following proposition characterizes the debt-finance political equilibrium.
Proposition 1. Let denote η̃ ≡ (1−α)(1+β)/(1+αβ). Given K and D, a debt-financeMarkov-perfect political equilibrium, {τ, b, g, K ′ , D′ }, is characterized by1−ω(1 + β)2·· (lϕ(1 − α)AK − RD) ,(1 − ω) (1 + β (1 + η)) + ωη 1 + αβ(10)ωηg = G̃(K, D) ≡· (lϕ(1 − α)AK − RD) ,(11)(1 − ω) (1 + β (1 + η)) + ωη1−ωD′ = D̃(K, D) ≡· β · (η̃ − η) · (lϕ(1 − α)AK − RD) ,(1 − ω) (1 + β (1 + η)) + ωη(12)αD
τ = T̃ (K, D) ≡ Λ +· (1 − Λ) · ,
(13)
1−α
K
b = B̃(K, D) ≡11

wherel(1 + β)21−ω·,(1 − ω) (1 + β (1 + η)) + ωη 1 + αβ
and by the law of motion of capital,
{
}
K′
1−ω
β {α (1 + β) + (1 + αβ) η}D=·· lϕ(1 − α)A − R,
K(1 − ω) (1 + β (1 + η)) + ωη
1 + αβKΛ≡1−wherein
D
=
K

{

D0
K0
(1+αβ)·(η̃−η)α(1+β)+(1+αβ)η

for t = 0,
for t ≥ 1.

The tax rate is set within the rage (0, 1) for period t(≥ 0) if Λ > 0 and D0 /K0 <(1 − α)/α.
Proof. See Appendix A.2.
Proposition 1 implies that the economy has the following features. First, g and b are
linear functions of wage income lϕ(1 − α)AK minus debt repayment RD. The available
resources for the government are lϕ(1 − α)AK − RD, and it uses them for expenditureon public services for the old and on unemployment benefits for the young.
Second, the government borrows or lends in the capital market. The state of financialbalance depends on η, which captures the preference weight of public services for the old:D′ ≷ 0 if and only if η ≶ η̃ ≡ (1−α)(1+β)/(1+αβ). A higher η incentivizes young votersto lower public debt from the perspective of maintaining public services they will enjoy in
old age. This is the disciplined eﬀect produced by the young voters (Song, Storesletten,
and Zilibotti, 2012). In particular, when η is above η̃, the disciplinary eﬀect is so largethat the government does not need to issue bonds. Rather, the government lends in thecapital market to utilize its surplus funds.
Third, the growth rate is constant across periods except for the initial period. Thisis because the model exhibits a constant interest rate inherited from the AK technology.
However, the growth rate changes between the first two periods, that is, periods 0 and
1, because the government starts to borrow or lend in the capital market in period 0. In
particular, the growth rate decreases when the government borrows in the capital market.
The issue of public bonds by the period-0 government pushes the next-period government
to raise taxes to finance debt repayment. In addition, the issue of public bonds crowds
out capital accumulation, and thus, impedes economic growth. The opposite result holds
when the government lends in the capital market.

3.2

Political Powers of the Old and the Trade Union

Based on the result in Proposition 1, we now investigate how policies and the growthrate are aﬀected by an increase in power of the old and the trade unions. The following
proposition summarizes the result.
12

Proposition 2. In the debt-finance political equilibrium, increases in ω and θ cause thefollowing:(i) ∂ (D/Y ) /∂ω = 0, ∂τ /∂ω > 0, ∂ (g/Y ) /∂ω > 0, ∂ ((1 − l)b/Y ) /∂ω < 0, and ∂ (K ′ /K) /∂ω <0;(ii) ∂ (D/Y ) /∂θ > 0, ∂τ /∂θ > 0, ∂ (g/Y ) /∂θ = 0, ∂ ((1 − l)b/Y ) /∂θ > 0, and ∂ (K ′ /K) /∂θ <0.
Proof. See Appendix A.3.
To understand the result in Proposition 2, we first consider the D/Y ratio. This isrewritten as D/Y = (D/K)/A(l)1−α , where D/K is independent of ω and θ. The employment rate l is independent of the political power of the elderly, because it is determined tobalance the conflicting items between the young workers and the trade union. However,
the employment rate decreases as the political power of the union is strengthened. Thisin turn decreases the aggregate output, and thus, increases the D/Y ratio.
Second, the tax rate increases as the political power of the old is strengthened, because
the old owe no tax burden, and thus, want to increase public services at the expense offinancial burden on the young. The tax rate also increases as the bargaining power of theunion increases. Greater bargaining power of the union lowers the employment rate. In
response to this change, the government raises the tax rate on the employed to maintainthe tax revenue level.
Third, g/Y increases but (1 − l)b/Y decreases, as the political weight of the oldincreases. A larger weight on the old incentivizes the government to shift resources fromthe young (including the unemployed) to the old, resulting in a higher g/Y and a lower(1 − l)b/Y . The relative bargaining strength has no eﬀect on g/Y , because its eﬀect on
the policy function of g is oﬀset by its eﬀect on the aggregate output Y . However, therelative bargaining strength has a positive eﬀect on (1 − l)b/Y , because increased union
power decreases the number of employed, and thus, increases the aggregate spending on
unemployment-insurance payments.
Finally, the growth rate of capital decreases as the political weight of the old increases.
A larger weight on the old forces the government to focus on the old, and spend moreresources on public services, resulting in less resources for savings and capital accumulation. The bargaining power of the union has two opposing eﬀects on the growth rate: ahigher markup, ϕ, and a lower number of employed. The result suggests that the latter
negative eﬀect always outweighs the former positive eﬀect, resulting in a lower growthrate in response to an increase in the power of the union.
A noteworthy feature of the result in Proposition 2 is that a greater power of theunion leads to a higher unemployment rate and a higher debt-to-GDP ratio, suggesting13

a positive correlation between these two variables. This is in line with previous studies’
predictions, such as Kaas and von Thadden (2004) and Battaglini and Coate (2014).
However, Kaas and von Thadden (2004) assume fixed unemployment benefits and thetax rate, and thus, rule out the possibility of changes in these policy variables throughvoting. Battaglini and Coate (2014) overcome this issue by demonstrating voting on fiscal
policy; however, they subtract physical capital accumulation from their model. This study
instead demonstrates a politico-economic model with physical capital accumulation, andshows that the government responds to an increase in unemployment by raising the tax
rate as well as issuing more public bonds, and this, in turn, reduces the growth rate.
Another noteworthy feature is that an increase in the political power of the old results
in a higher ratio of spending for the old to GDP and a lower ratio of unemploymentinsurance benefits to GDP. This implies a shift of resources from the unemployed young
to the retired old. However, the evidence in Figure 3 shows that the two expenditures
are positively correlated. The evidence suggests that the political eﬀect of the old isoutweighed by the political eﬀect of the trade union. In addition, increased spending for
the unemployed and the old would be financed by issuing public bonds.

3.3

Utility Gap

The result in Proposition 2 suggests that the political powers of the old and the tradeunion may aﬀect the utility of the employed and the unemployed through changes inpublic services for the old, unemployment insurance benefits, and economic growth. To
investigate the welfare eﬀects of the powers on the employed and the unemployed, we
focus on the utility gap between them, Vtye − Vtyu = (1 + β) ln (1 − τt ) wt /bt . Substitutingof the policy functions τt and bt in Proposition 1 in this gap function enables us to obtain
Vtye − Vtyu = (1 + β) ln 1 = 0.
Thus, there is no income gap, that is, no utility gap, between the employed and theemployed. The political powers of the old and the trade union have no eﬀect on theutility gap.
To understand the mechanism behind this result, let us consider the marginal costs andbenefits of the unemployment insurance benefits. First, the marginal cost for each individual is (1 + β) / (1 − τ ) w. Given that the political weight on the employed is (1 − ω) l,
and that the ratio of the unemployed to the employed is (1 − l)/l, the marginal cost for
the government is(1 − ω) l ×

1−l1+β(1 − ω) (1 − l) (1 + β)×=.
l
(1 − τ ) w(1 − τ ) w
14

Second, the marginal benefit for each individual is (1 + β) /b. Given that the politicalweight on the unemployed is (1 − ω) (1 − l) , the marginal cost for the government is(1 − ω) (1 − l) (1 + β).
b
The government chooses b to balance its marginal costs and benefits to maximize its
objective. Therefore, b is set equal to (1 − τ ) w, implying that there is no utility gap
between the employed and the unemployed.
The above analysis suggests that the utility gap may arise if the government’s attention is biased toward the employed. For example, suppose that the government attaches
a weight (1 − ω) l (1 + ε) to the employed and a weight (1 − ω) (1 − l(1 + ε)) to the unemployed. A higher ε(> 0) implies that the government attaches more weight to theemployed. Such bias may arise if the turnover rate of the employed is higher than that of
the unemployed. In the presence of bias, the first-order condition with respect to b is
(1 − ω) (1 + ε) (1 − l) (1 + β)
(1 − ω) (1 − l (1 + ε)) (1 + β)=,
(1 − τ ) wb
where the left-hand side and the right-hand side show the marginal costs and benefits of
the unemployment insurance benefits, respectively.
The condition is reformulated asb
1 − l (1 + ε)=
< 1,
(1 − τ ) w(1 + ε) (1 − l)(14)

implying that the utility gap, represented by b/(1−τ )w, becomes wider as the bias toward
the employed increases. In addition, the condition in (14) suggests that the political powerof the trade union, represented by θ, could aﬀect the utility gap through a change in theemployment rate. As demonstrated in Section 2, greater power of the trade union lowers
the employment rate. This implies more (less) political weight on the unemployed (theemployed) in the political objective function, which in turn results in more benefits forthe unemployed, and thus, a lower utility gap. Although the bias has an eﬀect on theutility gap, the following analysis rules out its presence, and instead focuses on the eﬀects
of budget rules.

4

Budget Rules

In the previous section, we considered fiscal policy and economic growth when the government is able to issue public bonds to finance its expenditure. We assume no constraint
on public bond issues. However, in the real world, many countries have introduced fiscal
constitutions that govern the determination of fiscal policies. In particular, constitutional
15

balanced budget rules are in force in Austria, Germany, Italy, Slovenia, Switzerland,
Spain, and the United States (Azzimonti, Battaglini, and Coate, 2016).
To investigate the role and impact of budget rules, we consider two alternative cases:
the tax-finance case, in which the government is unable to issue public bonds and finances
its spending solely by taxation, and the constrained debt-finance case, in which the government self-imposes a public debt ceiling similar to European Union member countries.
In particular, we assume η < η̃ throughout this section. The assumption implies that in
the absence of a constraint, the government borrows in the capital market (see Proposition 1). In other words, the government wants to borrow in the capital market, but its
borrowing is restricted when a budget rule is introduced.
In Subsection 4.1, we compare the debt-finance and tax-finance cases in terms ofgovernment expenditure and economic growth, and investigate welfare consequences ofshifting from debt finance to tax finance. In Subsection 4.2, we consider a constrained
debt-finance requirement such as that of Azzimonti, Battaglini, and Coate (2016), andinvestigate its long-run consequence.

4.1

Tax FinanceFor comparison, we assume that capital K and debt D are given in the beginning ofeach period; however, the government is unable to issue new public bonds. Therefore,a tax-finance Markov-perfect political equilibrium is a set of functions, ⟨T, G, B⟩, where
T : ℜ++ × ℜ → [0, 1] is a tax rule, τ = T (K, D), G : ℜ++ × ℜ → ℜ++ is a public services
provision rule, g = G(K, D), and B : ℜ++ × ℜ → ℜ+ is an unemployment-insurance rule,b = B(K, D), such that(i) the capital market clears:K′ =β[l (1 − T (K, D)) ϕ(1 − α)AK + (1 − l)B(K, D)] ;
1+β

(15)

(ii) given K, ⟨T (K, D), G(K, D), B(K, D)⟩ = arg max P (K, D, τ, g, b, g ′ ) subject to g ′ =G(K ′ , D′ ), (15), and the government budget constraint,
G(K) + (1 − l)B(K) + RD = T (K)lϕ(1 − α)AK.
The following proposition provides a characterization of the tax-finance political equilibrium.
Proposition 3. Given K, a tax-finance Markov-perfect political equilibrium, {τ, b, g, K ′ },16

is characterized by the policy functions,1−ω(1 + β(1 + η)) (lϕ(1 − α)AK − RD) ,(1 − ω)(1 + β(1 + η)) + ωηωηg=(lϕ(1 − α)AK − RD) ,(1 − ω)(1 + β(1 + η)) + ωη()β(1 + αβ)(1 + αβ)(1 + β(1 + η))αDτ=(1 − Λ),· (η̃ − η) +· Λ+
(1 + β)2(1 + β)2
1−αKb=as well as by the law of motion of capital,
)
(
K′
1−ω
β(1 + β(1 + η))D,=·· lϕ(1 − α)A − R
K(1 − ω)(1 + β(1 + η)) + ωη
1+βK
wherein

Dt
=
Kt

{

D0 /K0 for t = 0,0
for t ≥ 1.

The tax rate is set within the range (0, 1) if Λ > 0 and D0 /K0 < (1 − α)/α.
Proof. See Appendix A.4.
The result in Proposition 3 indicates that the policy function of the expenditure forthe old, g, is identical between the debt-finance and the tax-finance cases, while theother policy functions and the law of motion of capital diﬀer between the two cases. To
investigate the diﬀerence in detail, we compare the two cases in terms of government
expenditure-GDP ratios, g/Y and (1 − l)b/Y, and economic growth, K ′ /K. We firstcompare those in period 0 and obtain the following result. Variables in the debt- and thetax-finance cases are denoted with subscripts “debt” and “tax”, respectively.
Proposition 4. Assume η < η̃. Given D0 /K0 < (1 − α)/α, K1 /K0 , g0 /Y0 , (1 − l)b0 /Y0 ,and τ0 in the debt- and the tax-finance cases are compared as follows:
K1
K1
g0
g0
(1 − l)b0(1 − l)b0<,=,>, and τ0 |debt < τ0 |tax .
K0 debt
K0 tax Y0 debt
Y0 tax
Y0
Y0
debt
tax
Proof. See Appendix A.5.
The result in Proposition 4 indicates that in the initial period, the growth rate in thedebt-finance case is lower than in the tax-finance case because public debt crowds out
private investment, and thus, capital formation. The result also indicates that the tax
rate in the debt-finance case is lower than in the tax-finance case. This is because thegovernment can utilize the issue of public bonds to finance the tax cut. Thus, prohibiting
public bond issue creates a tax-hike eﬀect.
To understand the eﬀects of fiscal stance on public services for the old and the unemployment insurance benefits, recall the political objective function in the debt-finance17

case:
P = ωη ln g + (1 − ω)(1 + β)l ln [(lϕ(1 − α)AK − RD) − g − (1 − l)b + D′ ]+ (1 − ω)(1 + β)(1 − l) ln b
[
(
) ]
β
lϕ(1 − α)A+ (1 − ω)βη ln lϕ(1 − α)A{(lϕ(1 − α)AK − RD) − g} −+ R D′ .
1+β
1+β
(16)
The objective function indicates that the fiscal stance on the taxpayers generates thefollowing two types of costs in terms of utility. First, the burden results in a decrease of
consumption of the employed. Second, the burden also results in a decrease of savingsof the employed, which in turn lowers the level of the next-period capital stock, andthus, future public services for the old. However, the unemployment-insurance benefit isirrelevant for the latter because it is an intragenerational transfer, and thus, has no eﬀect
on the aggregate saving.
With this diﬀerence in mind, let us consider the eﬀects of issuing public bonds on the
two types of costs. First, the issuing bonds enables the government to cut the tax rate,
thereby reducing the first cost as observed by the second term in Eq. (16). Second, issuingbonds crowds out capital formation, decreases the future provision of public services forthe old, and thus, increases the second cost as observed by the fourth term of the politicalobjective function. The provision of public services for the old is not aﬀected by theissue of public bonds because the first eﬀect is oﬀset by the second eﬀect. However, thebond issues do aﬀect the provision of unemployment insurance because the second eﬀect
is irrelevant. Therefore, the expenditure for the elderly is independent of fiscal stance,
whereas the level of unemployment insurance is higher in the debt-finance case than in
the tax-finance case.
Next, we compare the growth rate and the public-services-to-GDP ratio for the twocases in period t(≥ 1).
Proposition 5. Assume η < η̃. Given K0 and D0 , Kt+1 /Kt and gt /Yt in the debt- andthe tax-finance cases are compared as follows:
Kt+1
gt
gt
Kt+1
<
and
.
<
Kt debt
Kt taxYt debt
Yt tax
Proof. See Appendix A.5.
The eﬀect on economic growth in period t (≥ 1) is qualitatively similar to that in theinitial period. However, the eﬀect on public services for the elderly in period t(≥ 1) diﬀers
from that in the initial period. The public services-to-GDP ratio is independent of the
fiscal stance in the initial period, whereas it is critically aﬀected by the fiscal stance for
period t ≥ 1. To understand this diﬀerence, recall that for period t ≥ 1, the available
18

resources for the government in the debt-finance case are given by lϕ(1 − α)AK − RD,
which are smaller than those in the tax finance case, because the government in the debtfinance case must use part of its resources for debt repayment. Because of this diﬀerencein the available resources, the public services-to-GDP ratio in the debt-finance case islower than in the tax-finance case.
The tax rate and the unemployment insurance payments are directly aﬀected by thestate of financial balance in the initial period, but these eﬀects are not straightforward
for period t ≥ 1, as we demonstrate in the following.
Proposition 6. For period t ≥ 1, there is a critical value of η, denoted by η̂ ≡α(1 + β)/β(1 − α), such that
(1 − l)bt(1 − l)bt
β≥and τt |debt ≤ τt |tax if α <and η ∈ [η̂, η̃) ;Yt
Yt
1 + 2βdebt
tax(1 − l)bt(1 − l)bt<
and τt |debt > τt |tax
Yt
Yt
debt
tax
β
β
if either α ≥, or α <and η ∈ (0, η̂) .
1 + 2β
1 + 2βProof. See Appendix A.5.
The bond issue produces two opposing eﬀects on the tax rate. The first is a tax-cut
eﬀect as demonstrated in Proposition 4. The second is a tax-hike eﬀect: the bond issue
creates debt repayment costs from period 1 onward, and the government finances a partof the costs by raising the tax rate. The result in Proposition 6 suggests that if eitherα ≥ β/(1+2β), or α < β/(1+2β) and η ∈ (0, η̂), the tax-hike eﬀect outweighs the tax-cut
eﬀect; that is, prohibiting debt finance decreases the tax rate. Otherwise, the opposite istrue.
The bond issue also produces two opposing eﬀects on the unemployment insurance
benefits; they are observed in the second terms of the political objective function in thedebt-finance case in Eq. (16). The positive eﬀect is that the bond issue increases theavailable resources for the government, and thus, enables the government to increase theunemployment insurance payments. The negative eﬀect is the creation of debt repayment
costs from period 1 onward, which reduces the available resources for the government,
and hence, results in a decrease in the unemployment insurance payments.
The result in Proposition 6 suggests that if either α ≥ β/(1 + 2β), or α < β/(1 + 2β)
and η ∈ (0, η̂), the negative eﬀect on the unemployment insurance outweighs the positiveeﬀect on it. In other words, the shift from debt finance to tax finance may increase
the unemployment insurance payment-to-GDP ratio. With the result in Proposition 5,
we find that the shift increases the growth rate and spending for the old-to-GDP ratio,
and may also increase the unemployment-insurance payment-to-GDP ratio. This result
19suggests that the balanced-budget requirement, which has been or is being introduced insome countries or states for fiscal discipline, may benefit future generations at the expenseof the employed and unemployed young in the initial period. We investigate this welfare
implication further in the following.
Proposition 7. There is a positive integer, T (≥ 1), such that
Vtye |debt ≥ Vtye |tax and Vtyu |debt ≥ Vtyu |tax for t ≤ T,
Vtye |debt < Vtye |tax and Vtyu |debt < Vtyu |tax for t > T.
Proof. See Appendix A.6.
The result in Proposition 7 suggests that tax finance definitely benefits future generations at the expense of the current employed and unemployed. To understand this,
suppose that the tax-finance rule is introduced in period 0. As demonstrated in Proposition 4, the government in period 0 requires a higher tax rate and a lower unemployment
insurance benefit level than in debt finance because the government is unable to utilize
the issue of public bonds to finance its expenditure. This creates negative income eﬀects
on both the current employed and unemployed, and thus, realizes lower utility in the
tax-finance case than in the debt-finance case.
Proposition 7 shows that the result is reversed for future generations. As demonstrated
in Proposition 6, the future young benefit from tax finance if either α ≥ β/(1 + 2β), orα < β/(1+2β) and η ∈ (0, η̂). If this condition fails to hold, the future young suﬀer from anegative income eﬀect of tax finance. However, the current government can bequeath more
capital to future generations in the tax-finance case. This implies that future government
can utilize more resources for the provision of public services for the elderly. This positiveeﬀect outweighs the negative income eﬀect. Therefore, the future generations benefit fromthe tax finance, regardless of the parameter values.

4.2

Constrained Debt Finance

The analysis of tax finance and its comparison with debt finance enable us to oﬀer aninsight into the political economy of fiscal policy. However, the requirement for tax financeis somewhat extreme, because in reality the government is allowed to issue public bonds
as long as they are below the debt ceiling. To investigate the eﬀect of the debt ceiling,
we introduce the following debt constraint:D′ ≤ µ · (K + D) + AlϕD,
where µ ∈ ℜ. Appendix A.7 shows that we can obtain a Markov-perfect equilibrium inthe presence of the debt ceiling as long as the ceiling is given by the above condition.
20

In the following analysis, we set µ = 0 for the simplicity of analysis, and characterize
the debt-finance political equilibrium in the presence of the constraint, D′ ≤ AlϕD. If A
is normalized to satisfy Alϕ = 1, the constraint is reduced to D′ ≤ D. This corresponds
to the balanced-budget rule investigated in Azzimonti, Battaglini, and Coate (2016). The
constraint D′ ≤ D is equivalent to τ lw ≥ g + (1 − l)b + (R − 1)D, implying that tax
revenues are suﬃcient to cover spending, g + (1 − l)b, plus the costs of servicing the debt,(R − 1)D.
The problem of the government is now modified by adding the constraint D′ ≤ AlϕD
into the problem in Definition 2(ii). When the constraint is non-binding, the politicalequilibrium allocation matches that in Proposition 1. We consider a political equilibrium
when the constraint is binding, and obtain the following result.
Proposition 8. Consider a political equilibrium with the debt constraint, D′ ≤ AlϕD.
Let d denote a threshold ratio of D/K defined byd≡(1 − ω) (1 − α) β (η̃ − η).
{(1 − ω) (1 + β (1 + η)) + ωη} + (1 − ω) βα (η̃ − η)

If D/K > d, then the debt constraint is non-binding, and the political equilibrium ischaracterized as in Proposition 1. If D/K ≤ d, then the debt constraint is binding,and the political equilibrium is characterized by the following:(1 − ω) (1 + β (1 + η))· lϕ(1 − α)A (K + D) ,(1 − ω) (1 + β (1 + η)) + ωηωηg=
· lϕ(1 − α)A (K + D) ,(1 − ω) (1 + β (1 + η)) + ωη()
l (1 − ω) (1 + β (1 + η))
D
τ =1−
· 1+
,
(1 − ω) (1 + β (1 + η)) + ωηK
D′ = AlϕD,β(1 − ω) (1 + β (1 + η))K′ =·· lϕ(1 − α)A (K + D) − AlϕD,
1 + β (1 − ω) (1 + β (1 + η)) + ωη
b=

and

D′
=f
K′

where
χ≡

(

D
K

)

[
≡ χ·

(

)
]−1
1
+1 −1
,
D/K

β
(1 − ω) (1 + β (1 + η))·· (1 − α) ∈ (0, 1) .
1 + β (1 − ω) (1 + β (1 + η)) + ωηProof. See Appendix A.8.
The result in Propositions 1 and 8 indicate that for period t ≥ 1, the debt-capitalratio satisfies the following equation:
{
(D)
D
if K
≤ d,
f K
D′
=
(1+αβ)·(η̃−η)D′
if K > d,Kα(1+β)+(1+αβ)η
21

where f (·) is increasing and convex in D/K with f ′ (0) = 1/χ > 1. The propertiesof f (·) suggest that given D0 /K0 < d, the debt-capital ratio increases over time while
the constraint continues to be binding, and moves into a state where the debt constraint
becomes non-binding and the ratio remains constant through time, as illustrated in Figure
5.
[Figure 5 here.]
To understand the movement of debt-capital ratio, consider a situation where the initial condition D0 /K0 is less than the threshold value, d. Less public debt today implies
that the government can utilize the tax revenue for its expenditure. However, this incentivizes the government to issue more bonds because the government can aﬀord to repay
its debt using the tax revenue. Thus, the debt accumulates and reaches its ceiling, AlϕD.
When the debt constraint is binding, the households’ tax burden is less compensatedby public bond issue. This implies a negative income eﬀect on households, which in turnimplies negative eﬀects on savings and capital accumulation. Because of this negativeeﬀect on capital, the debt-capital ratio increases along the equilibrium path with D′ =AlϕD. On some future date, the ratio exceeds the threshold value d. Then, the ratio
continues to be below the threshold, and the debt and capital grow at the same constantrate. Therefore, the economy experiences a decrease in the growth rate followed by its
permanent increase.
The present result is qualitatively diﬀerent from that of Azzimonti, Battaglini, andCoate (2016). They demonstrate that the debt accumulates initially and then reaches
its ceiling, while the present study shows that debt is constrained by its ceiling initially,
and then moves into increasing debt accumulation. The diﬀerence arises because there
is no capital accumulation and resources are limited across periods in the model used by
Azzimonti, Battaglini, and Coate (2016), while there is capital accumulation that increases
income, and thus, tax base over times in the model used in this study. Therefore, theresult in this study demonstrates an alternative view of the role of debt constraint in thepolitical economy.

5

Extensions

The main analysis of this study assumed that the reservation wage is the competitive
wage. This section attempts to relax this assumption in two ways and briefly examines
how the analysis and results would change.

22

5.1

Unemployment Insurance Benefit as the Reservation WageAn alternative assumption here is that the reservation wage is given by the level of
unemployment-insurance benefits, bt (see, e.g., Bean and Pissarides, 1993; Chang, Shaw,and Lai, 2007). The union’s objective under this alternative assumption is ((1 − τt ) wt − bt )δ (lt )1−δ .
The objective function in the Nash bargaining problem is now modified as
[
]θ [
]1−θ
δ
1−δ
Ωt = ((1 − τt ) wt − bt ) (lt )· At (Kt )α (lt )1−α − wt lt,
and the solution to the problem becomes(1 − τt ) wt = ϕbt ,where ϕ(> 1) is defined in (4).
The indirect utility functions of the employed and the unemployed are now given by
Vtye = (1 + β) ln ϕbt + β ln Rt+1 + βη ln gt+1 ,Vtyu = (1 + β) ln bt + β ln Rt+1 + βη ln gt+1 ,respectively. An increase in b improves the utility of both the employed and the unemployed. That is, there is no conflict over the provision of unemployment-insurance benefits
between the employed and the unemployed, which is empirically implausible. Therefore,
this study uses the competitive wage as the reservation wage for the union.

5.2

After-tax Wage as a Union’s Target

In the main analysis, the unemployment rate is endogenous but independent of fiscal
policy. This enables us to solve the model in a tractable way; however, it fails to capturea relationship between unemployment and fiscal policy, as suggested by Battaglini and
Coate (2014). To overcome this limitation, we consider the following alternative union’s
objective function:((1 − τt ) wt − w̄t )δ · (lt )1−δ ,where one of the union’s targets is the after-tax wage, (1 − τt ) wt , rather than the beforetax wage, wt .
Solving the Nash bargaining problem with this modification leads to the following
employment rate:()1/α
1 − τtlt = l (τt ) ≡.
ϕ
The unemployment rate now depends on the tax rate. A higher tax rate implies a lessafter-tax wage income. This incentivizes the union to set a higher wage through bargaining. Therefore, the employment rate is lower when the union’s target is the after-tax wage
than when it is before-tax wage.
23

The political objective function, P , is now modified by replacing l with l (τt ). However,this modification makes it diﬃcult to obtain an analytical solution because the choice ofτt aﬀects the employment rate, and thus, the optimality condition with respect to τt . To
overcome this problem, we divide the decision process of fiscal policy into the following two
sub-stages: (1) given K and D, the government sets the tax rate to satisfy its constraint,with the expectation of b, g, and D′ determined through voting; and (2) given τt , thegovernment decides a set of (g, b, D′ ) to maximize its objective.
This modification enables us to take l (τt ) as given in choosing g, b, and D′ . Thisimplies that the political equilibrium policy and allocation, denoted by {g, τ, b, D′ , K ′ },are characterized by the equations demonstrated in Proposition 1 but ls in them are
replaced with l(τ ). Thus, g/Y and (1 − l(τ )) b/Y ratios and the growth rate become
(
)
g
ωη
D
=
· (1 − α) − α,Y(1 − ω) (1 + β (1 + η)) + ωηK()
(1 − l(τ )) b
(1 − l(τ )) (1 − ω)
(1 + β)2D=·· (1 − α) − α,Y(1 − ω) (1 + β (1 + η)) + ωη 1 + αβK()
K′
(1 − ω) β {α (1 + β) + (1 + αβ) η}D=· l(τ )ϕ(1 − α)A − αAl(τ )ϕ.
K{(1 − ω) (1 + β (1 + η)) + ωη} (1 + αβ)KThe equations suggest that changing the target from w to (1 − τ ) w creates the following three eﬀects. First, the ratio g/Y remains unchanged because the negative eﬀects
on the wage and GDP are oﬀset by the positive eﬀect of a reduction in the interest rate.
Second, the ratio (1 − l)b/Y increases because given b and Y, the employment decreases.
Third, the growth rate decreases because the negative eﬀect on the wage outweighs the
positive eﬀect of a reduction in the interest rate. Therefore, the conclusion is that themodel in the main analysis fails to capture the eﬀect of fiscal policy on employment;
however, the eﬀect could be easily included in the model.

6Concluding Remarks

This study shows a positive correlation between unemployment and the debt-to-GDP
ratio resulting from the strong political power of a trade union. The study also showsthat an increase in political power of the old results in a higher ratio of spending forthe old to GDP, a lower ratio of unemployment-insurance benefits to GDP, and a lower
growth rate of capital. In addition, the tax-finance requirement shifts resources from theyounger to the older generation via fiscal policy, and thus, benefits the present old at
the expense of the current young generation; however, it improves economic growth, andthus, the utility of future generations.
The key assumptions to demonstrate these results are (i) the additively separable,
logarithmic utility function; (ii) AK technology; and (iii) the union’s objective function24that targets the before-tax wage. The first assumption abstracts from the eﬀect of savingson the preferences of public services for the elderly. If we assume non-separable preferences for private consumption and public services, individuals could substitute private
consumption for public services, and thus, prefer less public services and issue of publicbonds.
The second assumption produces a constant interest rate that rules out the eﬀectof fiscal policy on the interest rate through capital accumulation. We can include theinterest rate eﬀect by assuming a Cobb-Douglas production function that abstracts fromthe capital externality. This assumption implies that the interest rate is decreasing in
capital, thereby incentivizing individuals to prefer fiscal policy that discourages capital
accumulation and economic growth.
The third assumption produces an unemployment rate that is independent of fiscal
policy. This result enables us to obtain the political-equilibrium solution analytically;
however it rules out the possibility of interaction between fiscal policy choice and unemployment. To overcome this limitation, Subsection 6.2 considers an alternative assumptionthat the union’s target is the after-tax wage. Under this assumption, we find that unemployment depends on fiscal policy, and that spending for the unemployment insurance islower but the growth rate is higher in comparison to the main analysis.
The mechanism behind the third assumption is briefly revealed by analyzing the political equilibrium under the alternative assumption. However, the roles of the first two
assumptions have not been analyzed fully. In particular, relaxing these assumptions would
require numerical computation. However, as our aim is to demonstrate definitive results,
this task is left for future studies.

25

A

ProofsA.1

Bargaining Solution

We substitute the constraint into the objective function to obtain the following unconstrained problem:
[
]θ [
]1−θ
max Ωt = {(1 − α)At }(1−δ)/α (Kt )1−δ · {(1 − α)At }1/α (1/(1 − α) − 1) Kt· Ω̃t ,
wt

where Ω̃t ≡ (wt − w̄t )δθ · (wt )−{(1−δ)θ+(1−α)(1−θ)}/α . Therefore, the problem is reduced tomax Ω̃t ≡ (wt − w̄t )δθ · (wt )−(1−δ)θ+(1−α)(1−θ)α

wtgiven w̄t .
The first derivative of Ω̃t with respect to wt is][(1−δ)θ+(1−α)(1−θ)
∂ Ω̃t
(1
−
δ)θ
+
(1
−
α)(1
−
θ)
δθ−1
−
−1
α
= (wt − w̄t )·(wt )(wt − w̄t ) ,· δθwt −∂wt
αindicating that ∂ Ω̃t /∂wt = 0 implies wt = ϕw̄t , where ϕ is defined in Section 2. The
second derivative of Ω̃t with respect to wt , evaluated at ∂ Ω̃t /∂wt = 0, is∂ 2 Ω̃t∂wt2= (wt − w̄t )

δθ−1

−

·(wt )(1−δ)θ+(1−α)(1−θ)−1α∂ Ω̃t /∂wt =0

where ∂ 2 Ω̃t /∂wt2()(1 − δ)θ + (1 − α)(1 − θ)· δθ −,α< 0 holds under Assumption 1.
∂ Ω̃t /∂wt =0■

A.2

Proof of Proposition 1

To find a set of policy functions, let us first recall the government budget constraint in
Definition 2(ii), which can be rewritten as follows:1−τ =lϕ(1 − α)AK − g − (1 − l)b − RD + D′.
lϕ(1 − α)AK

Using this constraint, the capital market-clearing condition can be rewritten as
D′ + K ′ =β[(lϕ(1 − α)AK − RD) − g + D′ ] ,
1+β

(17)

where D′ appearing on the left-hand side indicates borrowing (lending) by the government
if D′ > (<)0, whereas D′ appearing on the right-hand side implies the benefits (costs)
arising from the shift of fiscal resources from taxes to public bonds if D′ > (<)0.
26

To find the policy functions that maximize the political objective, P , we need toconjecture the future policy function g ′ = G̃(K ′ , D′ ). Here, we conjecture that g ′ =G0 · (lϕ(1 − α)AK ′ − RD′ ), where G0 (> 0) is constant. The term lϕ(1 − α)AK ′ − RD′in the conjecture shows the aggregate labor income minus debt repayment, and implies
the resources available to the government. Plugging the capital market-clearing condition(17) into this conjecture, we obtain
[
(
) ]
β
lϕ(1 − α)A′
g = G0 · lϕ(1 − α)A{(lϕ(1 − α)AK − RD) − g} −+ R D′ .
1+β1+βUsing this guessing function, we can reformulate the political objective function as follows:P = ωη ln g + (1 − ω)(1 + β)l ln [lϕ(1 − α)AK − g − (1 − l)b − RD + D′ ]+ (1 − ω)(1 + β)(1 − l) ln b
(
(
) )
β
lϕ(1 − α)A+ (1 − ω)βη ln lϕ(1 − α)A{(lϕ(1 − α)AK − RD) − g} −+ R D′ ,1+β1+βwhere the terms unrelated to political decisions are omitted from the expression.
The first-order conditions with respect to g, D′ , and b areωη(1 − ω)(1 + β)l (1 − ω)βηlϕ(1 − α)A 1+β
g:
=
+
,
˜
˜
g
I(K,
D, D′ )J(K,D, D′ )()
lϕ(1−α)A(1 − ω)βη+R
1+β(1 − ω)(1 + β)l
D′ :
=
,
˜
˜
I(K,
D, D′ )J(K,D, D′ )
(1 − ω)(1 + β)(1 − l)(1 − ω)(1 + β)l(1 − l)
=
,
b:
˜
b
I(K,
D, D′ )β˜˜where I(K,D, D′ ) and J(K,D, D′ ) are defined as follows:˜I(K,D, D′ ) ≡ (lϕ(1 − α)AK − RD) − g − (1 − l)b + D′ ,()β
lϕ(1 − α)A′˜
J(K, D, D ) ≡ lϕ(1 − α)A{(lϕ(1 − α)AK − RD) − g} −+ R D′ .
1+β1+βThe first-order conditions with respect to D′ is rewritten as[β′· {(lϕ(1 − α)AK − RD) − g}
D = (1 + β)l · lϕ(1 − α)A
1+β
(
)
]
lϕ(1 − α)A−βη+ R {(lϕ(1 − α)AK − RD) − g − (1 − l)b}
1+β
[
(
)]−1
lϕ(1 − α)A.
× {(1 + β)l + βη}
+R
1+β

27

(18)

˜
˜
Using (18), we can reformulate I(K,D, D′ ) and J(K,D, D′ ) as follows:[(1 − α) β(1 + β)l′˜·{(lϕ(1 − α)AK − RD) − g}I(K, D, D ) =(1 + β)l + βη
1 + αβ+ (lϕ(1 − α)AK − RD) − g − (1 − l)b] ,()βη
lϕ(1 − α)A′˜
J(K, D, D ) =·+R(1 + β)l + βη
1+β[](1 − α) β×{(lϕ(1 − α)AK − RD) − g} + (lϕ(1 − α)AK − RD) − g − (1 − l)b .
1 + αβWith the use of these expressions, we can rewrite the first-order conditions with respect
to g and b as (10) and (11), respectively. (10) and (11) lead to (12). The functions in(10), (11), and (12) constitute a Markov-perfect political equilibrium as long as G0 =
ωη · {(1 − ω) (1 + β (1 + η)) + ωη}−1 .
Using these policy functions, we compute the tax rate. Recall the government budget
constraint,g + (1 − l)b + RD − D′τ=.
lϕ(1 − α)AK
Plugging (10)–(12) into this expression and rearranging the terms, we obtain the following:τ =Λ+(1 − Λ) R D· ,lϕ(1 − α)A K

(19)

where
Λ≡1− (=1−(1 − ω)l(1 + β) (lϕ(1 − α)A + R))
lϕ(1−α)A+R{(1 − ω) (1 + β (1 + η)) + ωη}
1+β

1−ωl(1 + β)2·∈ (0, 1).
(1 − ω) (1 + β (1 + η)) + ωη 1 + αβ

To determine the tax rate, we need to compute the ratio D/K. This is completedusing the policy function D′ = D(K, D) and the capital market-clearing condition K ′ =K(K, D). We substitute the policy functions of D′ and g, given by (12) and (11), respectively, into the capital market-clearing condition (17), and rearrange the terms to
obtain1−ω
K′=K(1 − ω) (1 + β (1 + η)) + ωη()
β · [α(1 + β) + (1 + αβ)η]D
×· lϕ(1 − α)A − R.
1 + αβK
Using (12) and (20), we can compute the ratio D/K as follows:{
D0 /K0for t = 0;D=
(1+αβ)(η̃−η)
for t ≥ 1,Kα(1+β)+(1+αβ)η
28

(20)

(21)

where η̃ ≡ (1 − α)(1 + β)/ (1 + αβ) , and D0 /K0 is an initial condition and is taken as
given.
To find the conditions that ensure τt ∈ (0, 1) for all t ≥ 0, consider first the period-0
tax rate:αD0τ0 = Λ +(1 − Λ).
1−α
K0Given that Λ < 1, we obtain
α
D0
D0
1−α
(1 − Λ)
<1−Λ⇔
<
,
1−α
K0
K0
α
τ0 > 0 ⇔ Λ > 0.
τ0 < 1 ⇔Therefore, τ0 ∈ (0, 1) holds if Λ > 0 and D0 /K0 < (1 − α)/α.
Next, consider the period-t(≥ 1) tax rate. Using the ratio D/K in (21), we can write
τ for period t ≥ 1 as
τ =Λ+

(1 + αβ) (η̃ − η)α· (1 − Λ) ·,1−αα(1 + β) + (1 + αβ)ηwhere τ > 0 if Λ > 0. We also have[
τ < 1 ⇔ (1 − Λ) · 1 −]α(1 + αβ) (η̃ − η)·>0
1 − α α(1 + β) + (1 + αβ)ηα
(1 + αβ) (η̃ − η)⇔1>·; since Λ < 1
1 − α α(1 + β) + (1 + αβ)η⇔ (1 − α) (1 + αβ) η > −α (1 + αβ) η,

where the last expression always holds. Therefore, τ > 0 for t ≥ 1 if Λ > 0.
■

A.3

Proof of Proposition 2

A.3.1

Eﬀect of ω and θ on D/YGiven Yt = AlϕKt , Dt /Yt can be rewritten as
Dt
Dt
1
=
·
.
Yt
Kt AlϕThe ratio D/K can be rewritten as{
D0 /K0 for t = 0,
Dt
1+β−η
1+β−η
lϕ(1−α)A−ηR(1−α)−ηα
=
1+β
1+β
=
for t ≥ 1.
Ktηηlϕ(1−α)A+(1+η)R(1−α)+(1+η)α
1+β

1+β

The ratio Dt /Kt is independent of ω and θ for t ≥ 0, as observed in the previous expression:∂(Dt /Kt )/∂ω = 0 and ∂(Dt /Kt )/∂θ = 0 for t ≥ 0.
29

Next, consider the term 1/Alϕ appeared on the right-hand side of the expression,
D/Y = (D/K) · (1/Alϕ). The term can be rewritten as
1
1= (ϕ)1/α−1 ,AlϕAwhere the equality arises from l = (1/ϕ)1/α . Given ∂ϕ/∂ω = 0 and ∂ϕ/∂θ > 0, we obtain∂(D/Y )/∂ω = 0 and ∂(D/Y )/∂θ > 0 for t ≥ 0.
A.3.2

Eﬀects of ω and θ on τ

The tax rate in (13) is rewritten as follows:τ =1− (

(1−α)
+α
1+β

)

(1 − ω)l(1 + β){(1 − ω) (1 + β (1 + η)) + ωη}

(
· 1−

α
D
·
(1 − α) K).

Given that ∂(D/Y )/∂ω = 0 and ∂(D/Y )/∂θ > 0 as demonstrated in Section A.3.1, and∂l/∂ω = 0 and ∂l/∂θ < 0, we obtain ∂τ /∂ω > 0 and ∂τ /∂θ > 0 for t ≥ 0.
A.3.3

Eﬀects of ω and θ on g/Y

Using the policy function g = G̃(K, D) in (11), we can write g/Y as follows:
(
)
g
ωη
D
=
· (1 − α) − α.
Y(1 − ω) (1 + β (1 + η)) + ωηKGiven ∂(D/K)/∂ω = 0 and ∂(D/K)/∂θ = 0 (in Section A.3.1), we obtain ∂ (g/Y ) /∂ω >0 and ∂ (g/Y ) /∂θ = 0 for t ≥ 0.
A.3.4

Eﬀects of ω and θ on (1 − l)b/Y

Using the policy function b = B̃(K, D) in Eq. (10), we can write (1 − l)b/Y as follows:()(1 − l)bD(1 − l)(1 − ω)(1 + β))
· (1 − α) − α
=(
.
1−α
Y
K
+
α
{(1
−
ω)
(1
+
β
(1
+
η))
+
ωη}
1+β
Given that ∂l/∂ω = 0 and ∂l/∂θ < 0, and that ∂(D/K)/∂ω = 0 and ∂(D/K)/∂θ > 0 as
demonstrated in Section A.3.1, we obtain ∂ ((1 − l)b/Y ) /∂ω < 0 and ∂ ((1 − l)b/Y ) /∂θ >0 for t ≥ 0.
A.3.5

Eﬀect on K ′ /K

For t = 0, the growth rate demonstrated in Proposition 1 can be rewritten as1−ω
β {α (1 + β) + (1 + αβ)η}
K1
=
·
K0
(1 − ω) (1 + β (1 + η)) + ωη
1 + αβ{}D0× (ϕ)1−1/α A (1 − α) − α.
K030

Given ∂ϕ/∂ω = 0 and ∂ϕ/∂θ > 0, we obtain ∂ (K1 /K0 ) /∂ω < 0 and ∂ (K1 /K0 ) /∂θ < 0.
For t ≥ 1, the growth rate can be rewritten as
Kt+1(1 − ω)βη=(ϕ)1−1/α A,
Kt(1 − ω) (1 + β (1 + η)) + ωηindicating that ∂ (Kt+1 /Kt ) /∂ω < 0 and ∂ (Kt+1 /Kt ) /∂θ < 0.
■

A.4

Proof of Proposition 3

To find a set of policy functions, let us first recall the government budget constraint in(3), which can be rewritten as1−τ =lϕ(1 − α)AK − g − (1 − l)b − RD − D′,lϕ(1 − α)AK(22)

where D′ = 0 in the tax-finance case. Plugging (22) into the capital market-clearing
condition (15), we obtainK′ =β[(lϕ(1 − α)AK − RD) − g] .
1+βConjecture a linear policy function of public services in the next period as g ′ = G0 ·(lϕ(1 − α)AK ′ − RD′ ), org ′ = G0 · lϕ(1 − α)A ·β[(lϕ(1 − α)AK − RD) − g] ,1+βwhere G0 (> 0) is a constant parameter and D′ = 0. Given this conjecture and thegovernment budget constraint in (22), we can write the political objective function as
follows:
P = ωη ln g + (1 − ω)(1 + β)l ln [(lϕ(1 − α)AK − RD) − g − (1 − l)b]+ (1 − ω)(1 + β)(1 − l) ln b + (1 − ω)βη ln [(lϕ(1 − α)AK − RD) − g] ,where the terms unrelated to policy are omitted from the expression.
The first-order conditions with respect to g and b are summarized asωη(lϕ(1 − α)AK − RD) ,(1 − ω)(1 + β(1 + η)) + ωη
(1 − ω)(1 + β(1 + η))b = B(K) ≡(lϕ(1 − α)AK − RD) .
(1 − ω)(1 + β(1 + η)) + ωηg = G(K) ≡

(23)
(24)

These functions constitute a stationary Markov-perfect political equilibrium as long as
G0 = ωη/ {(1 − ω)(1 + β(1 + η)) + ωη} holds.
31

We substitute the policy functions (23) and (24) into the government budget constraintin (22) to obtain(1 − l)(1 − ω)(1 + β(1 + η)) + ωη
(1 − ω)(1 + β(1 + η)) + ωηαDl(1 − ω)(1 + β(1 + η))+·· ,(1 − ω)(1 + β(1 + η)) + ωη 1 − α K

τ=

or

(25)

(
τ = 1−)l(1 − ω)(1 + β(1 + η))(1 − ω)(1 + β(1 + η)) + ωηl(1 − ω)(1 + β(1 + η))
α
D
+
·
·
(1 − ω)(1 + β(1 + η)) + ωη 1 − α K)(
1 + αβ= 1−· (1 + β(1 + η)) · (1 − Λ)
(1 + β)2
1 + αβαD+· ,2 · (1 + β(1 + η)) · (1 − Λ) ·1−α K
(1 + β)where Λ is defined in Proposition 1 and the third equality comes froml (1 + β)2
1−ω
·
.
1−Λ=
(1 − ω)(1 + β(1 + η)) + ωη 1 + αβ
The above expression is reformulated as follows:()β (1 + αβ) (η̃ − η) (1 + αβ) (1 + β(1 + η))αDτ=+· Λ + (1 − Λ) ·
·
,
1−α K
(1 + β)2(1 + β)2where D/K = D0 /K0 for period 0 and D/K = 0 for period t ≥ 1.
To find the conditions that ensure τt ∈ (0, 1) for all t, consider first the period-0 tax
rate:
(
)
β (1 + αβ) (η̃ − η) (1 + αβ) (1 + β(1 + η))
α
D0
τ0 =
+
· Λ + (1 − Λ) ··,1 − α K0
(1 + β)2(1 + β)2where η̃ − η > 0 by the assumption and 1 − Λ > 0 by the definition of Λ. From thisexpression, we find that τ0 < 1 if

)
(
D0
α
(1 + αβ) (1 + β(1 + η))·· Λ + (1 − Λ) ·1 − α K0
(1 + β)2
β (1 + αβ) (η̃ − η)<1−
(1 + β)2(1 + αβ) (1 + β(1 + η))=(1 + β)2αD0⇔ Λ + (1 − Λ) ··<1
1 − α K0
D0
1−α
⇔
<
.
K0
α
32

Therefore, we have τ0 ∈ (0, 1) if Λ > 0 and D0 /K0 < (1 − α)/α.
Next, consider the period-t(≥ 1) tax rate:τ=β (1 + αβ) (η̃ − η) (1 + αβ) (1 + β(1 + η))+· Λ.
(1 + β)2(1 + β)2The expression suggests that τ > 0 if Λ > 0; and that τ < 1 if
β (1 + αβ) (η̃ − η) (1 + αβ) (1 + β(1 + η))+· Λ < 1,
(1 + β)2(1 + β)2or Λ < 1, which holds by the definition of Λ. Therefore, we have τt ∈ (0, 1) if Λ > 0.
We substitute (23) and (25) into the capital market-clearing condition to obtain thelaw of motion of capital:()
K′(1 − ω)β · {1 + β (1 + η)}D=·· lϕ(1 − α)A − R. (26)
K(1 − ω)(1 + β(1 + η)) + ωη
1+β
K
■

A.5

Proofs of Propositions 4, 5, and 6

First, consider the growth rate of capital. Recall the laws of motion of capital demonstrated in Propositions 1 and 3:()
Kt+1
β {α (1 + β) + (1 + αβ) η}Dt1−ω·· lϕ (1 − α) A − R=,
Kt debt (1 − ω)(1 + β(1 + η)) + ωη
1 + αβKt debt()
β {1 + β (1 + η)}
Dt
Kt+1
1−ω
·
· lϕ (1 − α) A − R=.
Kt tax (1 − ω)(1 + β(1 + η)) + ωη
1+βKt taxFor period 0, we find that, by direct comparison,
K1
K1(1 − α) (1 + β),<⇔ η < η̃ ≡K0 debt
K0 tax1 + αβwhich holds under the assumption of η < η̃.
For period t ≥ 1, we have
(1 + αβ) (η̃ − η)
Dt
Dt
=
= 0.
and
Kt debt (1 + αβ) η + α (1 + β)
Kt tax
We substitute this into the expression of Kt+1 /Kt and obtain
Kt+1
Kt+1
<
⇔ 0 < 1 + αβ,
Kt debt
Kt taxwhich holds for any α > 0 and β > 0.
33

Next, consider the spending for the public services-to-GDP ratio. From the results in
Propositions 1 and 3 and Yt = lϕAKt , g/Y |debt and g/Y |tax are computed as
(
)
ωη
D
g
=
(1 − α) − α,
Y unbalanced (1 − ω)(1 + β(1 + η)) + ωηK debt()
gωηD=(1 − α) − α,
Y unbalanced (1 − ω)(1 + β(1 + η)) + ωηK tax
respectively. Given K0 and D0 , we obtain g/Y |debt = g/Y |tax in period 0. For periodt ≥ 1, we obtain g/Y |debt < g/Y |tax because D/K|debt > 0 holds in the debt-finance casewhereas D/K|tax = 0 in the tax-finance case.
Third, consider the unemployment-insurance payments-to-GDP ratio. Using the resultin Propositions 1 and 3 and Yt = lϕAKt , g/Y |debt , we can compute (1 − l)b/Y |debt and(1 − l)b/Y |tax as()(1 − l)(1 − ω)
(1 + β)2D(1 − l)bt=,·· (1 − α) − αYt(1 − ω) (1 + β (1 + η)) + ωη 1 + αβ
K debtdebt()(1 − l)bt
D(1 − l)(1 − ω)
· (1 + β (1 + η)) · (1 − α) − α=,Yt(1 − ω) (1 + β (1 + η)) + ωη
K tax
tax

respectively.
For period 0, given K0 and D0 , we directly compare (1 − l)b0 /Y0 |debt and (1 − l)b0 /Y0 |tax
and obtain(1 − l)b0(1 − l)b0(1 + β)2≷ (1 + β (1 + η))
≷
⇔
Y0
Y0
1 + αβdebt
tax(1 − α) (1 + β)⇔ η ≶ η̃ ≡.
1 + αβGiven the assumption of η < η̃, we have (1 − l)b0 /Y0 |debt > (1 − l)b0 /Y0 |tax .
For period t ≥ 1, given Dt /Kt |tax = 0, we obtain()(1 − l)bt(1 − l)bt(1 + β)2D≷⇔· (1 − α) − α≷ (1 + β (1 + η)) · (1 − α)Yt
Yt
1 + αβK debtdebt
tax()(1 + αβ) (η̃ − η)
(1 + β)2· (1 − α) − α ·≷ (1 + β (1 + η)) · (1 − α)⇔
1 + αβ(1 + αβ) η + α (1 + β)
(1 + β) α⇔ η ≷ η̂ ≡,β (1 − α)where the second line comes from
(1 + αβ) (η̃ − η)Dt=.
Kt debt (1 + αβ) η + α (1 + β)
We compare η̃ and η̂, and obtainη̂ ≷ η̃ ⇔ α ≷34

β.
1 + 2βIf α ≥ β/(1 + 2β), then η̃ ≤ η̂, so η < η̂ holds under the assumption of η < η̃. We have
(1 − l)bt /Yt |debt < (1 − l)bt /Yt |tax for t ≥ 1 if α ≥ β/(1+2β). However, if α < β/(1+2β),then η̂ < η̃. We have
(1 − l)bt(1 − l)bt<
if η < η̂,
Yt
Yt
debt
tax
(1 − l)bt(1 − l)bt≥
if η̂ ≤ η < η̃.
Yt
Yt
debt
tax
Summarizing the results, we obtain(1 − l)bt
β(1 − l)bt
β<
if either α ≥, or α <and η < η̂,YtYt
1 + 2β
1 + 2βdebt
tax(1 − l)bt
β(1 − l)bt≥
if α <and η̂ ≤ η < η̃.
YtYt
1 + 2βdebt
tax
Finally, consider the tax rate. For period 0, τ0 |debt and τ0 |tax are compared as follows:
τ0 |debt ≷ τ0 |tax

(
)
α
D0
β(1 + αβ)(1 + αβ)(1 + β(1 + η))
α
D0
⇔Λ+
Λ
≷
· (η̃ − η) +· Λ+
(1 − Λ)1 − α K0
(1 + β)2(1 + β)2
1−αK0()
β(1 + αβ)αD0β(1 + αβ)⇔· (η̃ − η) · Λ +(1 − Λ)≷· (η̃ − η) .
2
(1 + β)1−αK0(1 + β)2

Dividing both sides by β(1 + αβ) (η̃ − η) /(1 + β)2 , we have
τ0 |debt ≷ τ0 |tax ⇔ Λ +⇔αD0(1 − Λ)
≷1
1−α
K0

1−α
D0
≷
.
K0
α

Under the assumption of D0 /K0 < (1 − α)/α, we obtain τ0 |debt < τ0 |tax .
For period t ≥ 1, τt |debt and τt |tax are compared as:
τt |debt ≷ τt |tax ⇔ Λ+αDt
(1 + αβ)(1 + β(1 + η))β(1 + αβ)(1 − Λ)≷·(η̃ − η)+
·Λ.
2
1−α
Kt debt(1 + β)(1 + β)2Plugging Dt /Kt |debt = (1 + αβ) (η̃ − η) / {(1 + αβ)η + α (1 + β)} into the above expression and rearranging the terms, we obtainτt |debt ≷ τt |tax ⇔ η ≶ η̂ ≡α(1 + β).
β(1 − α)Following the argument above, we can conclude as follows: for t ≥ 1,
τt |debt > τt |tax if either α ≥τt |debt ≤ τt |tax if α <ββ, or α <and η < η̂;
1 + 2β
1 + 2ββand η̂ ≤ η < η̃.
1 + 2β■35

A.6

Proof of Proposition 7

Recall that the indirect utility of the employed in generation t is given by
Vtye = (1 + β) ln(1 − τt )ϕ(1 − α)AKt + βη ln gt+1 + C,
where C ≡ β ln R + ln(1/(1 + β)) + β ln(β/(1 + β)) includes terms irrelevant to politicaldecisions. Substitution of the policy functions demonstrated in Proposition 1, we canwrite the indirect utility function in the debt-finance case as(1 − ω)ωηβη
22 · (lϕA) + C{(1 − ω) (1 + β (1 + η)) + ωη}+ (1 + β (1 + η)) ln ((1 − α) Kt |debt − α Dt |) .
(27)

Vtye |debt = (1 + β) ln (1 − Λ) ϕA + βη ln

The term (1 − α) Kt |debt − α Dt | in (27) is reformulated as follows:
[
]
Kt
Kt−1
K2
K1
α
D0
·
· ··· ··
· K0 · 1 −·(1 − α) Kt |debt − α Dt | = (1 − α) ·Kt−1 debt Kt−2 debt
K1 debt K0 debt
1 − α K0
( ′)tK
= (1 − α) ·· ((1 − α)K0 − αD0 ) ,(28)
K debtwhere the second equality comes from the fact that the growth rate is constant along theequilibrium path as demonstrated in Proposition 1.
Substitution of (28) into (27) leads to
)
( ′
K
ye
ye
,
(29)
Vt |debt = V0 |debt + t (1 + β (1 + η)) ln
K debtwhere
V0ye |debt = (1 + β) ln (1 − Λ) ϕA ((1 − α)K0 − αD0 )(1 − ω)ωηβη
2+ βη ln
2 · (lϕA) · ((1 − α)K0 − αD0 ) + C. (30){(1 − ω) (1 + β (1 + η)) + ωη}Following the same manner, Vtye |tax , Vtyu |debt , and Vtyu |tax are computed as follows:
( ′
)
K
ye
ye
Vt |tax = V0 |tax + t (1 + β (1 + η)) ln
,
(31)
K tax
( ′
)
K
yu
yu
Vt |debt = V0 |debt + t (1 + β (1 + η)) ln,(32)
K debt
)
( ′
K
yu
yu
Vt |tax = V0 |tax + t (1 + β (1 + η)) ln
,
(33)
K tax

36

where(1 − ω) (1 + β (1 + η))· lϕA · ((1 − α)K0 − αD0 )(1 − ω) (1 + β (1 + η)) + ωη
(1 − ω)ωη(1 − α)+ βη ln· (lϕA)2{(1 − ω) (1 + β (1 + η)) + ωη}2
β (1 + β (1 + η))×· ((1 − α)K0 − αD0 ) + C,
1+β
(1 − ω)(1 + β)2yu
V0 |debt = (1 + β) ln·· lϕA · ((1 − α)K0 − αD0 )(1 − ω) (1 + β (1 + η)) + ωη 1 + αβ
(1 − ω)ωηβη
2+ βη ln
2 · (lϕA) · ((1 − α)K0 − αD0 ) + C,{(1 − ω) (1 + β (1 + η)) + ωη}(1 − ω) (1 + β (1 + η))
V0yu |tax = (1 + β) ln· lϕA · ((1 − α)K0 − αD0 )(1 − ω) (1 + β (1 + η)) + ωη
(1 − ω)ωη(1 − α)2+ βη ln
2 · (lϕA){(1 − ω) (1 + β (1 + η)) + ωη}
β (1 + β (1 + η))×· ((1 − α)K0 − αD0 ) + C.
1+βV0ye |tax = (1 + β) ln

Let us first compare V0ye |debt and V0ye |tax . Direct calculation leads toV0ye |debt ≷ V0ye |tax ⇔ ϕ (η) ≡ (1+β) ln

1+β(1 + β)2+βη lnη−(1 + β (1 + η)) ln (1 + β (1 + η)) ≷ 0,
1 + αβ1−αwhere ϕ (·) has the following properties:
ϕ (0) = (1 + β) ln(1 + β)> 0,
1 + αβϕ (eη ) = 0,∂ϕ
η(1 + β)= β ln∂η(1 − α) (1 + β (1 + η))η̃(1 + β)< β ln; since η < η̃
(1 − α) (1 + β (1 + η))= 0.
Therefore, we obtain ϕ > 0∀η ∈ (0, η̃) , that is,V0ye |debt > V0ye |tax .
Next, compare Vtye |debt and Vtye |tax as follows:
Vtye |tax − Vtye |debt = ( V0ye |tax − V0ye |debt ) + t (1 + β (1 + η)) ln(

K ′ /K|tax
K ′ /K|debt).

The term ( V0ye |tax − V0ye |debt ) is negative and constant. The term t (1 + β (1 + η)) ln ( K ′ /K|tax / K ′ /K|deis positive and is increasing in t since K ′ /K|tax > K ′ /K|debt . Therefore, there is a positive
integer T (≥ 1) such that Vtye |tax ≤ Vtye |debt for t ≤ T and Vtye |tax > Vtye |debt for t > T .
37

Following the same procedure, we compare Vtyu |debt and Vtyu |tax , and obtain
V0yu |debt > V0yu |tax ⇔ ϕ > 0∀η ∈ (0, η̃) ,Vtyu |tax − Vtyu |debt = ( V0yu |tax − V0yu |debt ) + t (1 + β (1 + η)) ln(

K ′ /K|tax
K ′ /K|debt).

These imply that there is a positive integer T (≥ 1) such that Vtyu |tax ≤ Vtyu |debt for t ≤ T
and Vtyu |tax > Vtyu |debt for t > T .
■A.7

Supplement to Subsection 5.2

The objective here is to show the following proposition.
Proposition A.1. Consider a debt constraint, D′ ≤ Ψ(K, D). There is a Markovperfect political equilibrium with the debt constraint if Ψ(K, D) is specified by
Ψ(K, D) ≡ µ · (K + D) + lϕAD,
where µ ∈ ℜ is constant.
Consider the following debt constraint:D′ ≤ µK K + µD D,(34)

where µK , µD ∈ ℜ. Assume that this constraint is binding and guess the following linear
policy function of g ′ :g ′ = GK K ′ + GD D′ ,where GK , GD ∈ ℜ. Given this guess, we solve the maximization problem of the politicalobjective and obtain the policy function for g. Then we compute µK and µD that verify
the initial guess.
Suppose that the constraint is binding: D′ = µK K + µD D. The government budget
constraint, D′ = τ lϕ(1 − α)AK = g + (1 − l)b + RD, is reformulated as1−τ =lϕ(1 − α)AK − RD − g − (1 − l)b + µK K + µD D.
lϕ(1 − α)AK(35)

We substitute (35) into the capital-market-clearing condition and obtainK′ =β· [lϕ(1 − α)AK − RD − g + µK K + µD D] − D′ .
1+β

38

(36)

With (34) and (36), the guess of g ′ , g ′ = GK K ′ + GD D′ , is reformulated as follows:β· [lϕ(1 − α)AK − RD − g + µK K + µD D] − GK D′ + GD D′
1+β
β
= GK
· [lϕ(1 − α)AK − RD − g + µK K + µD D]
1+β+ (GD − GK ) · (µK K + µD D) .

g ′ = GK(37)

Thus, the problem is to maximize
P = ωη ln g + (1 − ω) (1 + β) l ln (1 − τ ) ϕ(1 − α)AK + (1 − ω) (1 + β) (1 − l) ln b
+ (1 − ω) βη ln g ′subject to (35) and (37).
The first-order conditions with respect to g and b areωη(1 − ω) (1 + β) l=glϕ(1 − α)AK − RD − g − (1 − l)b + µK K + µD D
β(1 − ω) βηGK 1+β,+β{lϕ(1 − α)AK − RD − g + µK K + µD D} + (GD − GK ) · (µK K + µD D)
GK 1+β

g:

b:

l
1
= .
lϕ(1 − α)AK − RD − (1 − l)b + µK K + µD Db

The first-order condition with respect to b leads tob = lϕ(1 − α)AK − RD + µK K + µD D.
Plugging this into the first-order condition with respect to g, we obtainωη(1 − ω) (1 + β)=glϕ(1 − α)AK − RD + µK K + µD D
β(1 − ω) βηGK 1+β+.
β
GK 1+β{lϕ(1 − α)AK − RD − g + µK K + µD D} + (GD − GK ) · (µK K + µD D)
If GD = GK , this condition leads to the following linear policy function of g:g=ωη· [lϕ(1 − α)AK − RD + µK K + µD D] .
(1 − ω) (1 + β (1 + η)) + ωη(38)

Eq. (38) suggests that the initial guess, g ′ = GK K ′ + GD D′ = GK · (K ′ + D′ ), iscorrect if
lϕA(1 − α) + µK = −R + µD ,that is, ifµD = lϕA + µK ,

(39)

ωη
GD = GK =· (lϕA(1 − α) + µK ) .
(1 − ω) (1 + β (1 + η)) + ωη
39

The condition (39) implies that we obtain a Markov-perfect political equilibrium if theconstraint is given byD′ ≤ µK K + (lϕA + µK ) D,or
D′ ≤ µ · (K + D) + lϕAD,
where µ ∈ ℜ.
■

A.8

Proof of Proposition 8

The problem of the government in the presence of the debt constraint is
max P = ωη ln g + (1 − ω)(1 + β)l ln (1 − τ ) ϕ(1 − α)AK+ (1 − ω)(1 + β)(1 − l) ln b + (1 − ω)βη ln g ′βs.t. K ′ + D′ =· [l (1 − τ ) ϕ(1 − α)AK + (1 − l)b] ,1+βg + (1 − l)b + RD = τ lϕ(1 − α)AK + D′ ,D′ ≤ AlϕD,
given K and D.
Suppose that the debt constraint is binding: D′ = AlϕD. Then, the government
budget constraint is g + (1 − l)b + RD = τ lϕ(1 − α)AK + AlϕD, or1−τ =lϕ(1 − α)A (K + D) − g − (1 − l)b.
lϕ(1 − α)AK(40)

Substitution of (40) into the capital-market-clearing condition leads toK′ =β· [lϕ(1 − α)A (K + D) − g] − D′ .
1+β

(41)

Here, we conjecture that g ′ = G · (K ′ + D′ ), where G(> 0) is constant. With (41), we
can reformulate this conjecture asg′ = G ·β· [lϕ(1 − α)A (K + D) − g] .
1+βPlugging this into P leads to
P = ωη ln g + (1 − ω)(1 + β)l ln (1 − τ ) ϕ(1 − α)AK+ (1 − ω)(1 + β)(1 − l) ln b + (1 − ω)βη ln [lϕ(1 − α)A (K + D) − g] ,where unrelated terms are omitted from the expression.
40

The first-order conditions with respect to g and b areωη(1 − ω) (1 + β) l
(1 − ω) βη=+,glϕ(1 − α)A (K + D) − g − (1 − l)b lϕ(1 − α)A (K + D) − g(1 − ω) (1 + β) l(1 − l)(1 − ω) (1 + β) (1 − l)b:=.
lϕ(1 − α)A (K + D) − g − (1 − l)b
b

g:

(42)
(43)

Condition (43) is reformulated asb = lϕ(1 − α)A (K + D) − g.

(44)

Using (42) and (44), we can obtain the policy functions of g and b as demonstrated in
Proposition 8.
We substitute the policy functions of g and b into the government budget constraintin (40) and obtain the tax rate as in Proposition 8. We also substitute them into (41) andobtain the capital-market-clearing condition as in Proposition 8. By using D′ = AlϕDand the capital-market-clearing condition, we can compute D′ K ′ as a function of D/K.
Taking account of the non-binding case, we can write the policy function of D′ as
}
{
1−ω
′
· β · (η̃ − η) · (lϕ(1 − α)AK − RD) .
D = min AlϕD,(1 − ω) (1 + β (1 + η)) + ωηThus, the debt constraint is binding if
AlϕD ≤1−ω· β · (η̃ − η) · (lϕ(1 − α)AK − RD) ,(1 − ω) (1 + β (1 + η)) + ωηthat is, if
D(1 − ω) (1 − α) β (η̃ − η)≤d≡.
K{(1 − ω) (1 + β (1 + η)) + ωη} + (1 − ω) βα (η̃ − η)41

BSupplementary Materials (Not for Publication)

This appendix explains the micro-foundation of the political objective function. The
following presentation focuses on the unbalanced-budget case. The same argument also
holds for the balanced-budget case.
Recall that the population of each generation has a unit measure, and that a young
generation consists of two groups of voters, the employed with a fraction of l and theunemployed with a fraction of 1 − l. The old generation also consists of two groups,
agents who were employed in youth and those who were unemployed in youth. However,these two types of old agents are included in a single group of voters, because they havethe same policy preferences over public services.
The electoral competition takes place between two oﬃce-seeking candidates, L (left)
and R (right). Each candidate announces a policy vector (τ, g, b) under the unbalancedbudget rule, subject to the government budget constraint. Elections are held in every
period, so the candidates today cannot make credible promises over future policies.
Let V o , V ye , and V yu denote the indirect utility functions of the old, the employedyoung, and the unemployed young, respectively. An old voter prefers candidate R over
L if given the inherited capital and debt, K and D, respectively, the following condition
holds:()()
V o pL ; K, D < V o pR ; K, D + σ io + δ,
where pL (pR ) is the policy vector proposed
L (R). Likewise, given K and⟨ by candidate⟩
D and the equilibrium policy functions T̃ , G̃, B̃, D̃ , an employed young voter prefers
candidate R over L if
(
)
(
)
( L)
( R)
ye
L
ye
R
V
p , G̃ p ; K, D < V
p , G̃ p ; K, D + σ ie + δ,and an unemployed young voter prefers candidate R over L if
(
)
(
)
( L)
( R)
yu
L
yu
R
V
p , G̃ p ; K, D < V
p , G̃ p ; K, D + σ iu + δ.
σ ij (where j ∈ {o, e, u}) is an individual-specific parameter drawn from a symmetricgroup-specific distribution that is assumed to be uniform in support [−1/2ϕj , 1/2ϕj ]. It
measures voter i’s individual ideological bias toward candidate R. Intuitively, a positive(negative) σ ij implies that voter i has a bias in favor of party R (L). The parameter δ
measures the relative popularity of candidate R in the population, which is assumed to
be uniform in support [−1/2ψ, 1/2ψ] with density ψ. Therefore, the sum of the terms σ ijand δ captures the relative appeal of candidate R.
To compute the vote share of each candidate, we identify the swing voter in groupj ∈ {o, e, u}, that is, a voter whose ideological bias, given the candidate’s platforms,
42

makes him indiﬀerent between the two parties:()()
σ o = V o pL ; K, D − V o pR ; K, D − δ,
(
)
(
)
( )
( )
σ j = V yj pL , G̃ pL ; K, D − V yj pR , G̃ pR ; K, D − δ, j = e, u.
All voters i in group j with σ ij > σ j prefer party R. Hence, given the distributional
assumptions, candidate R’s actual vote share, π R , is[{}( o( L
)
( R
)
)
1 o
1
o
R
− V p ; K, D − V p ; K, D − δ
π =
ϕ ·
2
2ϕo
{
(
(
)
(
)
)}
( L)
( R)
1
e
ye
L
ye
R
+ lϕ ·− V
p , G̃ p ; K, D − V
p , G̃ p ; K, D − δ
2ϕe
{
(
(
)
(
)
)}]
( L)
( R)
1
u
yu
L
yu
R
+(1 − l)ϕ ·− V
p , G̃ p ; K, D − V
p , G̃ p ; K, D − δ.
2ϕuCandidate R’s probability of winning the election is[]1
R
Prob π ≥
2
[
{
( (
)
(
))
1
= Prob δ ≥ δ̂ ≡ o· ϕo · V o pL ; K, D − V o pR ; K, Deu
ϕ + lϕ + (1 − l)ϕ
(
(
)
(
))
( )
( )
+ lϕe · V ye pL , G̃ pL ; K, D − V ye pR , G̃ pR ; K, D
(
(
)
(
))]
( )
( )
+(1 − l)ϕu · V yu pL , G̃ pL ; K, D − V yu pR , G̃ pR ; K, D
[ o ( o( L
)
( R
))
1
ψ
o
− o
·
ϕ
·
V
p
;
K,
D
−
V
p
;
K,
D
2 ϕ + lϕe + (1 − l)ϕu
(
(
)
(
))
( )
( )
+ lϕe · V ye pL , G̃ pL ; K, D − V ye pR , G̃ pR ; K, D
(
(
)
(
))]
( )
( )
+(1 − l)ϕu · V yu pL , G̃ pL ; K, D − V yu pR , G̃ pR ; K, D
( (
)
(
))
1
= − ω · V o pL ; K, D − V o pR ; K, D
2
[ ((
(
)
(
)))
( )
( )
− (1 − ω) · l · V ye pL , G̃ pL ; K, D − V ye pR , G̃ pR ; K, D
(
(
)
(
))]
( )
( )
−(1 − l) · V yu pL , G̃ pL ; K, D − V yu pR , G̃ pR ; K, D,=

where ϕe = ϕu is assumed, and ω is defined as
ω≡

ψϕo
.
ϕo + lϕe + (1 − l)ϕu

Because both candidates seek to maximize their probability of winning the election,

43

the Nash equilibrium is characterized by the following equation:( o( L
)
( R
))
o
ω
·
V
p
;
K,
D
−
V
p
;
K,
D



(
))) 
)
[ ((
(


( R)
( L)
( L∗ )
ye
R
ye
L
;
K,
D
−
V
p
,
G̃
p
;
K,
D
+
(1
−
ω)
·
l
·
V
p
,
G̃
p
,
p
= max
(
(
)
(
))]

( L)
( R)
pL 


yu
L
yu
R


+(1 − l) · V
p , G̃ p ; K, D − V
p , G̃ p ; K, D


( (
)
(
))
−ω · (V o pL ; K, D − V)o pR ; K,
D




[
((
(
)))


( L)
( R)
( R∗ )
ye
L
ye
R
D
p
= max − (1 − ω) · l (· V( p , G̃ p ; K,)D − V ( p , G̃ p ; K,))]
.

(
)
(
)
pR 




−(1 − l) · V yu pL , G̃ pL ; K, D − V yu pR , G̃ pR ; K, D
Therefore, the two candidates’ platforms converge in equilibrium to the same fiscal policythat maximizes the weighted-average utility of the old, the employed young, and theunemployed young, ωV o + (1 − ω) · (lV ye + (1 − l)V yu ) subject to the government budget
constraint. This is the political objective function given in the main body of the paper.
■

44

References[1] Acemoglu, D., and Robinson, J.A., 2005. Economic Origins of Dictatorship and Democracy,
Cambridge University Press.
[2] Arai, R., and Naito, K., 2014. A politico-economic approach on public debt in an endogenous
growth economy. MPRA Working Paper No.56213, Munich Personal RePEc Archive.
[3] Azzimonti, M., Battaglini, M., and Coate, S., 2016. The costs and benefits of balanced
budget rules: lessons from a political economy model of fiscal policy. Journal of Public
Economics 136, 45–61.
[4] Barseghyan, L., Battaglini, M., and Coate, S., 2013. Fiscal policy over the real businesscycle: A positive theory. Journal of Economic Theory 148, 2223–2265.
[5] Bassetto, M., 2008. Political economy of taxation in an overlapping-generations economy.
Review of Economic Dynamics 11, 18–43.
[6] Battaglini, M., 2014. A dynamic theory of electoral competition. Theoretical Economics 9,515–554.
[7] Battaglini, M., and Coate, S., 2008. A dynamic theory of public spending, taxation, anddebt. American Economic Review 98, 201–236.
[8] Battaglini, M., and Coate, S., 2014. A political economy theory of fiscal policy and unemployment. http://www.princeton.edu/˜mbattagl/fpumay31 S.pdf[9] Benassy, J-P., 2011. Macroeconomic Theory, Oxford University Press.
[10] Bernasconi, M., and Profeta, P., 2012. Public education and redistribution when talents
are mismatched. European Economic Review 56, 84-96.
[11] Bräuninger, M., 2005. Social security, unemployment, and growth. International Tax and
Public Finance 12, 423–434.
[12] Chang, J-J., Shaw, M-F., and Lai, C-C., 2007. A “managerial” trade union and economic
growth. European Economic Review 51, 365–384.
[13] Checherita-Westphal, C., and Rother, P., 2012. The impact of high government debt on
economic growth and its channels: An empirical investigation for the euro area. European
Economic Review 56, 1392–1405.
[14] Coimbra, R., Lloyd-Braga, T., and Modesto, L., 2005. Endogenous fluctuations in unionizedeconomies with productive externalities. Economic Theory 26, 629–649.
[15] Corneo, G., and Marquardt, M., 2000. Public pension, unemployment insurance, andgrowth. Journal of Public Economics 75, 293–311.
[16] Diamond, P.A., 1965. National debt in a neoclassical growth model. American Economic
Review 55, 1126–1150.
[17] Gonzalez-Eiras, M., and Niepelt, D., 2008. The future of social security. Journal of Monetary
Economics 55, 197–218.
[18] Gonzalez-Eiras, M., and Niepelt, D., 2012. Ageing, government budgets, retirement, andgrowth. European Economic Review 56, 97–115.
[19] Greiner, A., and Flaschel, P., 2010. Public debt and public investment in an endogenous
growth model with real wage rigidities. Scottish Journal of Political Economy 57, 68–84.
[20] Heijdra, B.J., 2009. Foundations of Modern Macroeconomics, Second Edition, Oxford University Press.

45[21] Josten, S.D., 2000. Public debt policy in an endogenous growth model of perpetual youth.
FinanzArchiv/Public Finance Analysis 57, 197–215.
[22] Josten, S.D., 2006. Dynamic fiscal policies and unemployment in a simple endogenous
growth model. International Tax and Public Finance 13, 701–716.
[23] Kaas, L., and von Thadden, L., 2003. Unemployment, factor substitution and capital formation. German Economic Review 4, 475-495.
[24] Kaas, L., and von Thadden, L., 2004. Budgetary policy and unemployment dynamics in an
OLG model with collective bargaining. Economic Journal 114, 867–889.
[25] Krusell, P., Quadrini, V., and Rios-Rull, J-V., 1997. Politico-economic equilibrium andeconomic growth. Journal of Economic Dynamics and Control 21, 243–272.
[26] Kumar, M., and Woo, J., 2010. Public debt and growth. IMF Working Paper 10/174, July
2010.
[27] Kunze, L., 2014. Life expectancy and economic growth. Journal of Macroeconomics 39,
54–65.
[28] Lancia F., and Russo, A., 2015. Public education and pensions in democracy: a political
economy theory. Journal of the European Economic Association 14, 1038–1073.
[29] Lindbeck, A., and Weibull, J., 1987. Balanced budget redistribution and the outcome ofpolitical competition. Public Choice 52, 273–297.
[30] Ono, T., 2007. Unemployment dynamics in an OLG Economy with public pensions. Economic Theory 33, 549–577.
[31] Ono, T., 2015. Intergenerational politics, government debt, and economic growth.
https://ideas.repec.org/p/osk/wpaper/1423r2.html (August 5, 2015).
[32] Pemberton, J., 1988. A “managerial” model of the trade union. Economic Journal 98,
755–771.
[33] Persson, T., and Tabellini, G., 2000. Political Economics: Explaining Economic Policy,
MIT Press.
[34] Poutvaara, P., 2006. On the political economy of social security and public education.
Journal of Population Economics 19, 345-365.
[35] Reinhart, C.M., Reinhart, V.R., and Rogoﬀ, K.S., 2012. Debt overhangs: past and present.
NBER Working Paper 18015.
[36] Romer, P.M., 1986. Increasing returns and long run growth. Journal of Political Economy
94, 1002–1037.
[37] Saint-Paul, G., 1992. Fiscal policy in an endogenous growth model. Quarterly Journal of
Economics 107, 1243–1259.
[38] Song, Z., 2011. The dynamics of inequality and social security in general equilibrium. Review
of Economic Dynamics 14, 613–635.
[39] 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.
[40] Uchida, Y., 2016. Education, social mobility, and talent mismatch. Economic Theory, forthcoming.
[41] Yakita, A., 2014. Involuntary unemployment and sustainability of bond-financed fiscal
deficit. Journal of Macroeconomics 41, 79–93.

46

5
Korea

Per capita GDP growth rates

4.5

Estonia

Poland

4
3.5

Czech
Republic
Slovenia
Hungary
Ireland
Chile
Iceland
New ZealandFinlandSwedenTurkeyAustralia
Spain
Netherlands
Austria
United Kingdom
CanadaSwitzerland
UnitedStatesNorwayMexico Germany
FranceDenmarkPortugal

3
2.5
2
1.5
1
0.5

GreeceIsraelBelgium

Japan

Italy

0
0

20

40

60

80

100

120

140

160

Debt-GDP ra!os

Figure 1: Debt-to-GDP ratio and per capita GDP growth rate for 32 countries, 1999-2009.
Source: OECD.Stat (December 17, 2016).

47

160
Japan

140

Dbt-GDP ra!os

120
Greece100

Italy
Israel80

Portugal Austria
Hungary
France Turkey
United Kingdom
Netherlands
PolandSpainStates
Germany
UnitedCanadaIrelandNew
Zealand
Switzerland
Korea
Slovenia
MexicoCzech Republic
Chile Australia
Estonia

60
40
20
0
0

20

40

Belgium

SwedenDenmark
Finland
Iceland
Norway60

80100

Trade union density

(a)
5
Korea
Estonia
PolandPer capita GDP growth rates

4.5
4
3.5

Czech Republic
Slovenia
Hungary
Chile
Greece Ireland

3
2.5
2

New ZealandTurkeySpainAustralia Israel
Netherlands
Austria
UnitedKingdom
CanadaUnitedSwitzerland
StatesGermany
FranceMexico
Portugal

1.5
1
0.5

Japan

0
0

20

IcelandFinlandSwedenBelgium
NorwayDenmark

Italy
40

60

80

100

Trade union density(b)

Figure 2: Trade union density and debt-GDP ratios (Panel (A)); trade union density andper capita GDP growth rates (Panel (B)) for 32 OECD countries, 1999-2009. Source:
OECD.Stat (December 17, 2016).

48

Old-age beneﬁts GDP ra"os

14.0
12.0

Italy
FranceAustria
GreeceSlovenia
PolandSweden Germany
Denmark Japan
Portugal
Finland
Hungary
Spain
Czech Republic
NorwayEstonia
Switzerland
United Kingdom
United States
Netherlands
Turkey IsraelAustralia
New
ZealandChile
CanadaIceland
Ireland

10.0
8.0
6.0
4.0
2.0

Belgium

Korea
Mexico

0.0
0

0.5

1

1.5

2

2.5

3

3.5

Unemployment beneﬁts GDP ra"os

Figure 3: Unemployment benefit-GDP ratios and Old-age benefit-GDP ratios for 32
OECD countries, 1999-2009. Source: OECD.Stat (December 17, 2016).

49

160
Japan

140

Debt GDP ra!os

120
Greece
Italy
Belgium
Israel100
80Austria Portugal
Hungary
France Turkey
UnitedSweden
Kingdom
Denmark
Netherlands
Spain
FinlandUnited States
Iceland
Canada Germany
Ireland
Switzerland
KoreaNew Zealand
Slovenia
MexicoCzech Republic
NorwayChile
Australia
Estonia

60
40
20
0
0

2

4

6

8

10

12

Poland14

16

Unemployment rates

Figure 4: Unemployment benefit-GDP ratios and debt-GDP ratios for 32 OECD countries,
1999-2009. Source: OECD.Stat (December 17, 2016).

50

䢲䢰䢻
䢲䢰䢺
䢲䢰䢹
䢲䢰䢸
䢲䢰䢷
䢲䢰䢶
䢲䢰䢵
䢲䢰䢴
䢲䢰䢳
䢲
䢲

䢲䢰䢲䢷

䢲䢰䢳

䢲䢰䢳䢷

䢲䢰䢴

䢲䢰䢴䢷

䢲䢰䢵

Figure 5: A numerical example of the D/K ratio. Parameters are set at β = (0.99)30 ,δ = 0.2, θ = 0.2, α = 0.3, η = 0.4, and ω = 0.3 to satisfy Assumption 1 and η < η̃.

51