Discussion Papers In Economics
And Business

Intergenerational Politics, Government Debt, andEconomic Growth
Tetsuo Ono
Discussion Paper 14-23-Rev.2

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANIntergenerational Politics, Government Debt, andEconomic Growth
Tetsuo Ono
Discussion Paper 14-23-Rev.2

June 2015

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANIntergenerational Politics, Government Debt, andEconomic Growth
Tetsuo Onoy
Osaka University

Abstract
This study presents a two-period overlapping-generations model featuring intergenerational con‡ict over …scal policy. In particular, we characterize a Markovperfect political equilibrium of the voting game between generations and show thefollowing three main results. First, population aging incentivizes the government
to invest more in capital for future public spending, positively a¤ecting economic
growth. Second, when the government …nances its spending by issuing bonds, theintroduction of the balanced budget rule results in a higher public spending-to-GDP
ratio and a higher growth rate. Third, to obtain a normative implication of the political equilibrium, we compare it with an allocation chosen by a benevolent planner
who takes care of all future generations. The planner’s allocation might feature lessgrowth and more borrowing than the political equilibrium if the planner attaches
low weights to future generations.
Keywords: Economic Growth; Government Debt; Overlapping Generations; Population Aging; Voting.
JEL Classi…cation: D72, D91, H63.

The author would like to thank seminar participants at Osaka City University, the University ofTokyo, and Waseda University for their comments and suggestions, and Yuki Uchida for his research
assistance. This work was supported in part by a Grant-in-Aid for Scienti…c Research (C) from the
Ministry of Education, Culture, Sports, Science, and Technology of Japan No. 15K03509.
yGraduate 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

Many OECD countries have experienced a declining fertility rate and increasing life expectancy over the past several decades (OECD, 2011). These demographic changes haveincreased the political power of the elderly in terms of voting, which has been expected
to increase government spending in their favor. An aging population is also expected
to increase the tax burden on the young as a by-product of this political pressure. Although these predictions are controversial (Razin, Sadka, and Swagel, 2002; Gradstein
and Kaganovich, 2004), changes in government spending and the tax burden deﬁnitely
aﬀect household savings, which, in turn, are expected to inﬂuence long-term economic
growth and welfare.
Several studies have attempted to investigate the political eﬀects of demographic
changes on government spending and economic growth. Examples include Gradstein andKaganovich (2004), Holtz-Eakin, Lovely, and Tosun (2004), Bassetto (2008), GonzalezEiras and Niepelt (2008, 2012), Tosun (2008), Kuehnel (2011), Iturbe-Ormaetxe andValera (2012), and Kaganovich and Meier (2012). These studies all assume a balanced
government budget. In other words, they ignore the possibility of government spending
being ﬁnanced by the issuance of government debt. However, each generation might havean incentive to shift the burden to future generations by issuing government debt. Thus,
from the practical viewpoint of ﬁscal policy, analyzing this scenario in the presence ofgovernment debt is necessary.
Several recent studies present politico-economic models of government debt, but they
abstract from economic growth and thus assume no capital accumulation (e.g., see Persson
and Svensson, 1989; Alesina and Tabellini, 1990; Tabellini, 1991; Battaglini and Coate,
2008; Azzimonti, Battaglini, and Coate, 2010; Caballero and Yared, 2010; Song, Storesletten, and Zilibotti, 2012). Two notable exceptions are Cukierman and Meltzer (1989) andArai and Naito (2014). Cukierman and Meltzer (1989) present a politico-economic modelof debt-ﬁnanced social security. An intergenerational conﬂict is inherent in their model,
but their focus is on an intragenerational conﬂict on ﬁscal policy. Therefore, little attention is given to the intergenerational conﬂict aﬀected by population aging or to its impact
on economic growth.
Arai and Naito (2014) independently develop a politico-economic model of government
debt and endogenous growth, which is similar to the model introduced in the present
study. However, they focus on the eﬀect of public spending preferences on ﬁscal policy andeconomic growth, whereas the present study examines the following three issues. First, we
focus on the aging eﬀect on ﬁscal policy and economic growth, which is a common policy
issue in advanced countries. Second, we compare the debt-ﬁnanced public spending casewith the balanced budget case and investigate the eﬀect of introducing a balanced budget1

rule into the former case on public spending and economic growth. Finally, we characterizea Ramsey allocation in which a benevolent planner with a commitment technology setsﬁscal policy over time to maximize the welfare of all generations and compare it with thepolitical equilibrium outcome to evaluate its normative implication.1
For the analysis, we use the two-period-lived overlapping-generations model of Diamond (1965). We employ AK technology, presented by Romer (1986), to demonstrate
capital accumulation. Public spending is shared by two successive generations, namely
the young and old, and is ﬁnanced by a tax on the young as well as by the issuance ofgovernment debt. In each period, tax, public spending, and a new debt issue are decided
by probabilistic voting, as in Lindbeck and Weibull (1987). In particular, we focus ona Markov-perfect political equilibrium in which the policy variables are conditioned by
payoﬀ-relevant state variables, namely the beginning-of-period government debt and capital in the present framework. This equilibrium concept enables us to demonstrate theforward-looking behavior of agents who consider this intertemporal eﬀect when voting(e.g., see Hassler et al., 2003, 2005; Hassler, Storesletten, and Zilibotti, 2007; Forni, 2005;
Gonzalez-Eiras, 2011; Song, 2011).
Based on the aforementioned framework, we ﬁrst characterize a political equilibrium
when the government is allowed to run an unbalanced budget. A natural prediction is that
population aging makes the government shift the ﬁscal burden to future generations by issuing more debt, which thus crowds out capital and decreases the growth rate. However,the present analysis shows that aging is beneﬁcial for economic growth. In particular,
greater longevity leads to a larger weight being placed on future public spending. This
incentivizes politicians to invest more in capital for future public spending and thus produces a positive eﬀect on the growth rate. The result indicates that the forward-looking
behavior of agents plays a role in determining the growth eﬀect of aging.2
Second, we consider a special case, called the balanced budget case. Here, the government is prohibited from borrowing or lending in the capital market and a balanced
budget is thus required by statute. To consider the role of government debt, we compare
the growth rate in the balanced budget case with that in the unbalanced budget case andshow that the introduction of a balanced budget rule results in a higher growth rate when
the government borrows in the capital market compared with the unbalanced budget case.
As a further analysis, we also consider a vote on ﬁscal rules and show that the balanced
1

A companion paper to the present study (Ono, 2014) introduces collective wage bargaining into thepresent framework and investigates the eﬀects of union power on capital accumulation and ﬁscal policy.
2
The present analysis of aging is also related to that proposed in Rangel (2003). He demonstrates athree-period overlapping-generations model where the young and old abstain from voting and only the
middle-aged segment of the population decides on the intergenerational reallocation of resources. Within
this framework, he considers how the aging of the electorate (i.e., the middle-aged population) inﬂuences
the intergenerational reallocation. On the contrary, the present study examines how the conﬂict of votinginterests between two successive generations aﬀects ﬁscal policy and capital accumulation.

2

budget rule is chosen in an election under plausible sets of parameters.3
Third, to consider the normative implication of the political equilibrium, we characterize the Ramsey allocation and compare it with the political equilibrium. The analysis
shows that the Ramsey allocation might feature less growth and more borrowing than thepolitical equilibrium if the planner attaches low weights to future generations. The opposite result holds when the planner attaches high weights to future generations. Therefore,
the eﬃciency of the political equilibrium in terms of growth and borrowing depends on
the planner’s weights on future generations.
The remainder of this paper is organized as follows. Section 2 presents the model andcharacterizes the economic equilibrium. Section 3 characterizes the political equilibrium
without a balanced budget rule. Section 4 characterizes the political equilibrium under thebalanced budget rule and compares the unbalanced and balanced budget cases in termsof public spending and economic growth. Section 5 demonstrates the Ramsey allocationand compares it with the political equilibrium outcomes. Section 6 provides a discussion
and extensions of the model. Section 7 concludes. Proofs are given in the appendix.

2

The Model and Economic Equilibrium

Consider an inﬁnite-horizon economy composed of identical agents, perfectly competitiveﬁrms, and perfect annuity markets. A new generation, called generation t, is born in eachperiod t = 0, 1, 2, .... Generation t is composed of a continuum of Nt > 0 units, which are
identical agents. We assume that Nt = (1 + n)Nt−1 : the net population growth rate isn > −1.

2.1

Preferences and Utility Maximization

Agents live a maximum of two periods, namely youth and old age. In youth, each agent
is endowed with one unit of labor, which is supplied inelastically to ﬁrms, and earns awage. An agent in generation t divides his or her wage wt between current consumption,
cyt , saving for consumption in old age, st , which is held as an annuity and invested inphysical capital and/or government debt, and the payment of tax, τt wt , which is quoted
as a proportion of the wage. Here, τt is the period-t tax rate on labor income. Thus, thebudget constraint for a period-t young agent is cyt + st ≤ (1 − τt )wt .4
3

The balanced budget rule in the present framework prohibits the issuance of government debt. Notably, this rule is stricter than that in Azzimonti, Battaglini, and Coate (2010), who propose a rulein which debt cannot increase across periods. We adopt this stricter rule to investigate how allowing
government debt aﬀects ﬁscal policymaking and economic growth.
4
In the present framework, wage income tax is equivalent to comprehensive income tax because labor
supply is assumed to be inelastic and therefore agents earn no imputed income. This property implies
that it does not matter if wage income tax is replaced by comprehensive income tax.

3

Agents are assumed to be faced with uncertain lifetimes. In particular, an agent diesat the end of youth with a probability of 1 − p ∈ (0, 1) and lives throughout old age witha probability of p. If an agent dies young, his or her annuitized wealth is transferredthrough the annuity markets to the agents who live throughout old age. If an agent isalive in old age, he or she consumes the return from savings. The budget constraint fora period-t + 1 old agent is given by cot+1 ≤ R̃t+1 st , where cot+1 is consumption in old age
and R̃t+1 is the return from the agent’s savings as an annuity.
Agents consume two goods: private goods, denoted by c, and public spending, i.e.,
particular types of publicly provided private goods, denoted by g. We assume additively
separable logarithmic preferences over these goods. Public spending must be consumed by
all agents in the same quantity; it cannot be provided by a competitive equilibrium market.
{}
The utility of a young agent in period t is written as ln cyt +θ ln gt +pβ · ln cot+1 + θ ln gt+1 ,where gt denotes per capita period-t public spending, θ(> 0) captures the preference
weight for public spending, and β ∈ (0, 1) is a discount factor.
The expected utility maximization problem of a period-t young agent can be written
as
maxo
y

{ct ,st ,ct+1 }{}ln cyt + θ ln gt + pβ · ln cot+1 + θ ln gt+1

s.t. cyt + st ≤ (1 − τt )wt ,cot+1 ≤ R̃t+1 st ,given τt , wt , and R̃t+1 .
Solving the problem leads to the following consumption and saving functions:cyt =1pβ R̃t+1pβ(1 − τt )wt , cot+1 =(1 − τt )wt , and st =(1 − τt )wt .
1 + pβ1 + pβ1 + pβIn period 0, there are both young agents in generation 0 and initial old agents ingeneration −1. Each agent in generation −1 is endowed with s−1 units of goods andearns a return of R̃0 s−1 , which is consumed. The measure of the initial old agents is
pN−1 . The utility of an agent in generation −1 is ln co0 + θ ln g0 .

2.2

Technology and Proﬁt Maximization

There is a continuum of identical ﬁrms. They are perfectly competitive proﬁt maximizers that produce output by using a constant-returns-to-scale Cobb–Douglas production
function, Yt = At (Kt )α (Nt )1−α , where Yt is aggregate output, At is the productivity parameter, Kt is aggregate capital, Nt is aggregate labor, and α ∈ (0, 1) is a constantparameter representing capital share. Capital is assumed to fully depreciate within aperiod.
4

The productivity parameter is assumed to be proportional to the aggregate capital perlabor unit in the overall economy: At = A(Kt /Nt )1−α . Thus, capital investment involves
a technological externality of the kind often used in theories of endogenous growth (e.g.,
Romer, 1986). This assumption, called AK technology, results in a constant interest rate
across periods as demonstrated below. This approach enables us to obtain an analyticalsolution for the model. Thus, we employ AK technology for its analytical tractability.
In each period t, a ﬁrm chooses capital and labor to maximize its proﬁts, Πt =At (Kt )α (Nt )1−α − Rt Kt − wt Nt , where Rt is the rental price of capital and wt is thewage rate. The ﬁrm takes these prices as given. The ﬁrst-order conditions for proﬁt
maximization are given by
Kt : Rt = αAt (Kt )α−1 (Nt )1−α ,Nt : wt = (1 − α)At (Kt )α (Nt )−α .

2.3

Government Budget Constraint

Fiscal policy is determined through elections. Government debt is traded in a domestic
capital market. Let Bt denote aggregate inherited debt and Gt denote aggregate public
spending. A dynamic budget constraint in period t is Bt+1 + Nt τt wt = Gt + Rt Bt , whereBt+1 is the newly issued debt, Nt τt wt is the aggregate tax revenue, and Rt Bt is the debt
repayment. We assume a one-period debt structure to simplify the voting strategy space
and to derive analytical solutions from the model.
Let bt ≡ Bt /Nt denote inherited debt per capita and gt ≡ Gt /(pNt−1 + Nt ) denote percapita period-t public spending. By dividing both sides of the above constraint by Nt , weobtain a per capita form of the government budget constraint:(1 + n)bt+1 + τt wt =p+1+n
gt + Rt bt ,1+nwhere τt > (<)0 holds when the government imposes a tax on (provides a subsidy to)individuals and bt+1 > (<)0 holds when the government borrows (lends) in the capital
market. The present analysis allows the government to oﬀer a subsidy and/or loans to
individuals.
Given bt , the elected government in period t chooses the labor income tax τt , percapita public spending gt , and newly issued debt bt+1 subject to the above constraint. We
assume that the government in each period is committed to not repudiating the debt.

2.4

Economic Equilibrium

The market clearing condition for capital is Kt+1 + Bt+1 = Nt st . This expresses theequality of total savings by young agents in generation t, namely Nt st , to the sum of the5stocks of aggregate physical capital and aggregate government debt. Dividing both sides
by Nt leads to(1 + n) · (kt+1 + bt+1 ) = st .
Since the market for capital is competitive, the following arbitrage condition holds under
perfect annuity:R̃t+1 = Rt+1 /p ∀t.
Formally, an economic equilibrium is deﬁned as follows.
{}∞
Deﬁnition 1. An economic equilibrium is a sequence of prices, wt , Rt , R̃t, a set=0∞∞
y oquence of allocations, {ct , ct , st }t=0 , a sequence of capital stock {kt }t=0 , and government debt {bt }∞t=0 with the initial conditions k0 > 0 and b0 , and a sequence ofpolicies {τt , gt }∞t=0 , such that the following conditions are met: (i) the conditionsof utility maximization with the budget constraints in youth and old age; (ii) the
conditions of proﬁt maximization; (iii) the government budget constraint; (iv) the
capital market clearing condition; and (v) the no arbitrage condition.
Under the assumption of a productive externality, At = A(Kt /Nt )1−α , the ﬁrst-orderconditions for proﬁt maximization are rewritten as
Rt = R ≡ αA and wt = (1 − α)Akt .
By using the saving function and ﬁrst-order conditions for proﬁt maximization, we canrewrite the capital market clearing condition as follows:(1 + n)(kt+1 + bt+1 ) =pβ· (1 − τt ) (1 − α)Akt .
1 + pβ(1)

In an economic equilibrium, the indirect utility of a young agent in period t, Vty , andthat of an old agent alive in period t, Vto , can be expressed as functions of government
policy, capital stock, and government debt as follows:
Vty = (1 + pβ) ln(1 − τt )(1 − α)Akt + θ ln gt + pβθ ln gt+1 ,Vto = ln(kt + bt ) + θ ln gt ,where some irrelevant terms are omitted from the expressions. The ﬁrst term of the young
agent’s indirect utility function corresponds to the utility of consumption in youth andold age. The second and third terms show the utility of ﬁrst- and second-period public
spending, respectively. The ﬁrst term of the old agent’s indirect utility corresponds tothe utility of consumption and the second shows the utility of public spending.

6

3

Political Equilibrium

This study assumes probabilistic voting to demonstrate the political mechanism. In eachperiod, the government in power maximizes a political objective function. Formally, thepolitical objective function in each period t is given byΩt = ωpVto + (1 + n)Vty ,where ωp and (1 + n) are the relative weights of old and young agents, respectively. In
particular, the parameter ω(> 0) represents the political power of old agents. An explicit
microfoundation for this objective function is explained in Persson and Tabellini (2000,
Chapter 3) and Acemoglu and Robinson (2006, Appendix). The government’s problem
in period t is to maximize Ωt subject to the government budget constraint, given the twostate variables kt and bt .
We restrict our attention to a Markov-perfect equilibrium. Markov perfection implies
that outcomes depend only on the payoﬀ-relevant state variables, namely capital (k)and government debt (b). Therefore, the expected level of public spending for the next
period, gt+1 , is given by a function of the next period stocks of capital and debt, gt+1 =G(kt+1 , bt+1 ). By using recursive notation, with x′ denoting next-period x, we can deﬁnea Markov-perfect political equilibrium in the present model as follows.
Deﬁnition 2. A Markov-perfect political equilibrium is a set of functions, ⟨T, G, B⟩,
where T : ℜ++ × ℜ → [0, 1] is a tax rule, with τ = T (k, b), G : ℜ++ × ℜ → ℜ++ isa government expenditure rule, with g = G(k, b), and B : ℜ++ × ℜ → ℜ is a debt
rule, with b′ = B(k, b), such that:(i) the capital market clears:(1 + n)(k ′ + B(k, b)) =pβ(1 − T (k, b)) · (1 − α)Ak,
1 + pβ(2)(ii) given k and b, ⟨T (k, b), G(k, b), B(k, b)⟩ = arg max Ω(k, b, g, b′ , g ′ ) subject to g ′ =G(k ′ , b′ ), (2), and the government budget constraint,(1 + n)B(k, b) + T (k, b)(1 − α)Ak =p+1+n
G(k, b) + Rb,
1+n

(3)

where Ω(k, b, g, b′ , g ′ ) is deﬁned byΩ(k, b, g, b′ , g ′ ) ≡ ωp {ln(k + b) + θ ln g} + (1 + n) {(1 + pβ) ln (1 − T (k, b)) (1 − α)Ak+θ ln g + pβθ ln g ′ } .

7

A new state variable, x, is introduced to solve the problem in a tractable way:x ≡ (1 − α)Ak − Rb,
where x represents the labor income less the government debt repayment. By using thisnew variable, we assume G(k ′ , b′ ) = G(x′ ) ≡ G((1 − α)Ak ′ − Rb′ ). Then, the problem in
Deﬁnition 2(ii) is reformulated as
{
(
)
p+1+n
⟨G(x), X(x)⟩ = arg max (1 + n)(1 + pβ) ln A · (x −G(x) − (1 + n) · X(x))
1+n+(ωp + 1 + n)θ ln G(x) + (1 + n)pβθ ln g ′ }(4)subject to g ′ = G(X(x)),
where X is a mapping from ℜ to ℜ. The proof of this reformulation is provided inAppendix A.1.
The reformulated problem implies that we can solve the government’s problem andthus ﬁnd policy functions in the following ways. First, we ﬁnd solutions to the reformulated problem, g = G(x) and x′ = X(x). Second, we use the solutions, the capital market
clearing condition, and the government budget constraint to ﬁnd the policy functionsb′ = B(k, b) and τ = T (b, k) and the law of the motion of capital, k ′ = K(b, k).
The analysis proceeds as follows. Section 3.1 characterizes a political equilibrium.
Section 3.2 investigates the eﬀects of population aging on the growth rate of capital.
Section 3.3 analyzes the political equilibrium further.

3.1

Characterization of the Political Equilibrium

To solve the aforementioned problem, we conjecture a linear function, g ′ = GDebt ·x′ , where
GDebt ∈ (0, ∞) is a constant parameter. Under this conjecture, we solve the problem andobtain the following policy functions:1+n(ωp + 1 + n)θ
·x,
p + 1 + n (1 + n) {1 + pβ(1 + θ)} + (ωp + 1 + n)θ
X(x) = XDebt · x,
G(x) =(5)(6)

where XDebt is a constant term deﬁned by
XDebt ≡

θpβA.
(1 + n) {1 + pβ(1 + θ)} + (ωp + 1 + n)θ

These functions constitute a Markov-perfect political equilibrium as long as GDebt ={(1 + n)/(p + 1 + n)} · (ωp + 1 + n)θ · [(1 + n) {1 + pβ(1 + θ)} + (ωp + 1 + n)θ]−1 .
Policy function (6) states that the wage income less the debt repayment in the next
period, (1 − α)Ak ′ − Rb′ , depends on that in the current period, (1 − α)Ak − Rb. By using
8the capital market clearing condition in (2) and government budget constraint in (3), we
ﬁnd that k ′ and b′ are determined as a function of (1 − α)Ak − Rb. Thus, the ratio b′ /k ′becomes constant across periods. The tax rate is determined to satisfy the government
budget constraint in each period. The following proposition formally states the ﬁndings
demonstrated so far.
Proposition 1. Consider an economy without a balanced budget rule. Given k0 > 0and b0 < (1 − α)k0 /α, a Markov-perfect political equilibrium is characterized by thefollowing policy functions:τ = T (k, b) ≡+pβ
(ωp + 1 + n)θ − (1 + n) 1+αpβ{(1 − θ + pβ) − α (1 + pβ(1 + θ))}(1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ
pβ(1 + n) (1 + pβ(1 + θ)) + (1 + n) 1+αpβ{(1 − θ + pβ) − α (1 + pβ(1 + θ))}

·

α
b
· ,
1−α k

(1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ
1+n(ωp + 1 + n)θ
g = G(k, b) ≡·· {(1 − α)Ak − Rb} ,p + 1 + n (1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ
(1 − θ + pβ) − α (1 + pβ(1 + θ))
pβ·· {(1 − α)Ak − Rb} ,b′ = B(k, b) ≡1 + αpβ (1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ
and the law of the motion of capital:k′ =pβ
θ + α (1 + pβ(1 + θ))·· {(1 − α)Ak − Rb} ,1 + αpβ (1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ

where

b
(1 − θ + pβ) − α (1 + pβ(1 + θ))=
Ak
A {θ + α (1 + pβ(1 + θ))}holds ∀t ≥ 1. The government borrows (lends) in the capital market, namely b′ >(<)0, if and only if α < (>)(1 − θ + pβ)/{1 + pβ(1 + θ)}.
Proof. See Appendix A.2.
Proposition 1 implies that the model economy has the following two features. First,
b0 < (1 − α)k0 /α must hold otherwise the debt repayment (Rb0 = αAb0 ) would outweighthe wage income ((1−α)Ak0 ), which implies that the government cannot provide a positive
level of public spending in period 0.
Second, the government borrows or lends in the capital market. Here, the state of
the ﬁnancial balance depends on the parameter α, representing the share of capital inproduction. To understand the mechanism behind this result, recall the policy function
B(k, b) in Proposition 1, which can be rearranged as follows:
z

(a.1)

}|
{
(a.2)
z }| {
pβ{}(1 + pβ(1 + θ)) (1 − α)A − pβθA
(1 − α)A
1 + pβ′+R b =·{(1 − α)Ak − Rb} . (7)1 + pβ(1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ
9The expression states that individuals devote a part of their available resources, their wage
income less their debt repayment, (1 − α)Ak − Rb, to saving, denoted by the term (a.1) inEquation (7). The government uses its ﬁscal policy to split the saving into an investment
for the next period stock of capital, denoted by the term (a.2) on the right-hand side, andbuying or selling government bonds, denoted by the term on the left-hand side.
Equation (7) implies that the government borrows (lends) in the capital market if thesaving is greater (less) than the investment in capital. Their relative strength depends
on the parameter α, which represents the capital share. If α is low and thus the labor
share, 1 − α, is high such that α < (1 − θ + pβ)/{1 + β(1 + θ)} holds, agents earn enoughof a wage income to be able to save. They can then aﬀord to lend in the capital market
and the government becomes a borrower. However, if the capital share is high such thatα > (1 − θ + pβ)/{1 + β(1 + θ)} holds, the opposite result is true: agents borrow in thecapital market and the government becomes a lender. Therefore, α plays a key role indetermining the government’s ﬁnancial balance.

3.2

Aging and GrowthBased on the result in Proposition 1, we derive the growth rate of per capita capital,k ′ /k, and investigate how this growth rate is aﬀected by population aging, namely alower population growth rate and greater longevity of agents. The following propositionsummarizes the result.
Proposition 2. Consider a political equilibrium in the unbalanced budget case.
(i) The growth rate of capital is(b.1)+(b.3)

z
}|
{

{
}


pβ
θ+α· (1 + pβ(1 + θ))b0
·· (1 − α) − α k0 A for t = 0pβ(1 + θ))+(ωp + 1 + n)θ
1 + αpβ (1 + n)(1 +


{z
} |
{z
}
| {z } |
kt+1(b.1)+(b.3)
(b.2)
(b.4)
=
(b.3)

kt

z}|{



pβθ

A
for t ≥ 1.


(1
+
n)(1
+
pβ(1
+ θ))+(ωp + 1 + n)θ


|
{z
} |
{z
}

(b.1)+(b.3)(b.2)(ii) The growth rate of capital is increased by a lower population growth rate and greater
longevity.
Proof. See Appendix A.3.
The growth rate of capital is constant after period 1 because the model exhibits aconstant interest rate inherited from AK technology. Here, we consider how the growth10rate is aﬀected by a lower population growth rate and greater longevity. To see the eﬀect,recall the political objective function Ω given by
(
)
p+1+n
Ω = (1 + n)(1 + pβ) ln A(x −G(x)) − (1 + n)X(x)
|
{z
}
1+n
(b.1)

+ (ωp + 1 + n)θ ln G(x) + (1 + n) pβθ ln G(X(x)).
|{z}
|
{z
}
(b.3)

(b.2)

The terms (b.1), (b.2), and (b.3) in the political objective function represent thepolitical weights on the utility of consumption, the utility of current public spending, andthe utility of future public spending, respectively. These terms correspond to those in
the equation for the growth rate demonstrated in Proposition 2(i). The equation statesthat the government allocates the wage income (1 − α)Ak0 in period 0 and the output
Ak in period t ≥ 1 to consumption, current public spending, and investment in capital,
which contributes to the formation of future public spending. In addition, the period-0
allocation is aﬀected by the saving rate, represented by the term (b.4).
The equation shows that the allocation is aﬀected by a decline in the population
growth rate and an increase in longevity. To see the eﬀect, consider ﬁrst a decline in
the population growth rate, which has the following implications for economic growth.
A lower population growth rate attaches lower weights to the utility of consumption andthat of current public spending, as observed in the terms (b.1) and (b.2), respectively.
These two terms indicate that a lower population growth rate incentivizes the government
to save more for future public spending and thus promotes economic growth. In addition,a lower population growth rate increases per capita equipment in the economy—this eﬀect
is observed in the term (1+n) in front of the term (b.3). This additional eﬀect also has
a positive impact on capital accumulation. Therefore, the growth rate increases as the
population growth rate decreases.5
Next, consider the eﬀect of greater longevity on the growth rate. In period t ≥ 1,
greater longevity implies larger weights on consumption and public spending for the old, as
shown by the terms (b.1) and (b.2) in the numerator. This incentivizes the government to
use the resources for current consumption and current public spending instead of investingin capital, thereby producing a negative eﬀect on the growth rate. On the contrary, greater
longevity leads to a larger weight on future public spending as shown by the term (b.3).
This gives the government an incentive to invest more in capital for future public spending,thereby producing a positive eﬀect on the growth rate. In period 0, the positive eﬀect is
5Note that the term (1 + n) in front of the term (b.3) in the political objective function has a diﬀerent
implication from that in the equation for the growth rate. The former represents a weight on the utilityof future public spending, whereas the latter represents the term (1 + n) in the capital market clearing
condition, (1 + n)(k + b) = s. The term (1 + n), representing the weight on the utility of future public
spending, is cancelled out through the calculation of the growth rate.

11

strengthened by an increase in the saving rate, as shown by the term (b.4).
The analysis shows that the negative eﬀect through the terms (b.1) and (b.2) is outweighed by the positive eﬀect through the term (b.3), thereby resulting in a higher growthrate. To understand the mechanism behind this result, recall the ﬁrst-order condition withrespect to x′ :
(b.3)

z}|{
z }| {
1+n(1 + n) · pβθ
(1 + n) · (1 + pβ) ·=,A · (x − g) − (1 + n) · x′
x′
(b.1)

where the left-hand side denotes the marginal cost of x′ and the right-hand side denotes
the marginal beneﬁt of it. As observed in the terms (b.1) and (b.3), given x′ and g, bothcosts and beneﬁts increase as longevity rises. However, there is an additional eﬀect on
the marginal cost through the term g. Given that g = (ωp + 1 + n)x′ /pβA holds at the
optimum, the government ﬁnds it optimal to reduce current public spending in responseto an increase in longevity. This works to increase the disposable income of the young
and to decrease the marginal cost of x′ . Because of this additional eﬀect of longevity on
the marginal cost, the eﬀect of longevity on the marginal beneﬁt outweighs the eﬀect on
the marginal cost.

3.3

Further Analysis of the Political Equilibrium

3.3.1

Economic and Political Eﬀects of Aging

Thus far, we have considered the aging eﬀect on the growth rate by focusing on the government’s ﬁscal policy decision. However, aging also aﬀects individual decisions on saving,which in turn inﬂuence the growth rate through the capital market. For instance, greater
longevity implies that individuals attach a higher weight to their old-age consumption.
This incentivizes them to save more for their future consumption.
To clarify the diﬀerence between the economic and political eﬀects of aging, we brieﬂy
present the eﬀect of aging on capital accumulation in the economic equilibrium. For thispurpose, recall the government budget constraint and capital market clearing condition
presented in Section 2. They lead to the following equation of the growth rate for a given
set of policies:















 pβ
 p



1
k′
1




=
·
(1 − τ ) + τ · (1 − α)Ak −+ 1 g + Rb 





k
+ n}  1 + pβ
k  |1 {z
+ n}



|1 {z
| {z }






(c.1)
(c.3)
(c.2)

This equation indicates three eﬀects of aging in the economic equilibrium. The ﬁrst
eﬀect, represented by the term (c.1), indicates that a lower population growth rate increases per capita equipment in the economy. The second eﬀect, represented by the term12

(c.2), shows the aging eﬀect on capital accumulation through individual saving decisions.
These two eﬀects have positive implications for economic growth. However, the ﬁnal effect, represented by the term (c.3), has a negative implication for the growth rate. A
lower population growth rate and/or greater longevity results in a larger burden on theyoung for the provision of public spending, which produces a negative income eﬀect on
saving.
In the political equilibrium, the negative eﬀect through the term (c.3) is oﬀset byan endogenous choice of spending. However, in the economic equilibrium, the negativeeﬀect remains active since the spending level (g) is taken as given. This suggests thatthe negative eﬀect through the term (c.3) might outweigh the positive eﬀects throughthe terms (c.1) and (c.2) in the economic equilibrium. In other words, the economic
equilibrium analysis might underestimate the positive eﬀect of aging on the growth rate.
3.3.2

Saving Rate and Economic Growth

The result in Proposition 2 also suggests that the growth rate is aﬀected by the savingrate. Given that a higher saving rate is associated with a higher discount factor, we here
consider how an increase in the discount factor aﬀects the growth rate through a change
in the saving rate in the short and the long run.
A higher discount factor implies that agents attach a larger weight to the utility of
old-age consumption. This incentivizes agents to increase after-tax income and thus toincrease the level of old-age consumption. For this purpose, they vote for a lower tax rate
as well as for higher government debt issue to compensate for the loss of tax revenue.
The capital market clearing condition indicates that a higher level of government debt
produces a crowding out eﬀect, thereby resulting in a lower level of capital. Therefore,
an increase in the saving rate, which is induced by a rise in the discount factor, createsa negative growth eﬀect in the short run. However, a higher discount factor gives all
successive agents an incentive to save more for old-age consumption. This promotes
capital accumulation and thus produces a positive growth eﬀect in the long run. In
particular, the positive eﬀect becomes perpetual since the present framework assumes theexistence of AK technology.

4

Balanced Budget Rule

So far, we have assumed that government expenditure can be ﬁnanced by issuing government debt. In a standard neoclassical growth model, the presence of government debt may
crowd out capital and lower economic growth and may also aﬀect the size of government
spending through the government budget constraint.
13

To understand the role of government debt in the present political economy model,
we focus on a special case in which a balanced budget is required by statute. Here, thegovernment is unable to issue government bonds and runs a balanced budget in eachperiod. We compute the government spending-to-GDP ratio and growth rate of thebalanced budget case. Then, in Subsection 4.1, we compare these ﬁgures with those in
the unbalanced budget case and investigate how the balanced budget rule aﬀects the
spending-to-GDP ratio and economic growth. In Subsection 4.2, we then consider a voteon ﬁscal rules.
Given the initial condition b0 , and assumption of the balanced budget rule, the government budget constraint becomes
p+1+n
g0 + Rb0 = τ0 w0 for t = 0,
1+n
p+1+n
gt = τt wt for t ≥ 1.
1+n
Government expenditure gt is ﬁnanced by labor income tax revenue from the young, τt wt .
The capital market clearing condition is Kt+1 = st Lt , expressing the equality of totalsavings by young agents to the stock of aggregate capital. We divide both sides by Nt
and substitute the saving function and government budget constraint into the clearing
condition to obtain the law of the motion of capital for a given level of government
expenditure as follows:{{}pβ(1 + n)k1 = 1+pβ
(1 − α)Ak0 − Rb0 − p+1+n
g0 for t = 0
1+n
{
}
pβ
(1 + n)kt+1 = 1+pβ
(1 − α)Akt − p+1+n
gt for t ≥ 1
1+n

(8)

The indirect utility functions of the old and young are now given by
Vto = θ ln gt for t ≥ 0,(){
(1 + pβ) ln( (1 − α)Ak0 − Rb0 − )p+1+n
g0 + θ ln g0 + pβθ ln g1 for t = 0,
y
1+n
Vt =
(1 + pβ) ln (1 − α)Akt − p+1+n
gt + θ ln gt + pβθ ln gt+1
for t ≥ 1,
1+nrespectively, where the terms unrelated to political decisions are omitted from the expressions. By using these functions, we can write the political objective function as
)
(
p+1+n
g0
Ω0 = (1 + n)(1 + pβ) ln (1 − α)Ak0 − Rb0 −
1+n+ (ωp + 1 + n)θ ln g0 + (1 + n)pβθ ln g1 ,
(
)
p+1+n
Ωt = (1 + n)(1 + pβ) ln (1 − α)Akt −gt
1+n+ (p + 1 + n)θ ln gt + (1 + n)pβθ ln gt+1 , for t ≥ 1.
The objective function indicates that capital is a payoﬀ-relevant state variable for periodt ≥ 1.
14

The government’s problem is to choose gt subject to constraint (8), given k0 and b0
in period 0 and given kt in period t ≥ 1. Solving the problem leads to the following
proposition.
Proposition 3. Consider an economy with a balanced budget rule. Given k0 (> 0)and b0 < (1 − α)k0 /α, a Markov-perfect political equilibrium is characterized by thefollowing policy functions:
τt =

Rbt
(ωp + 1 + n)θ + (1 + n)(1 + pβ(1 + θ)) (1−α)Akt,(1 + n) {1 + pβ(1 + θ)} + (ωp + 1 + n)θ
(1 + n)(ωp + 1 + n)θ
{(1 − α)Akt − Rbt } ,gt =(p + 1 + n) [(1 + n) {1 + pβ(1 + θ)} + (ωp + 1 + n)θ]and the law of the motion of capital:
}
{
pβ
kt+1
1 + pβ(1 + θ)bt=·
· (1 − α)A − R,kt
1 + pβ (1 + n) {1 + pβ(1 + θ)} + (ωp + 1 + n)θ
kt
where bt = 0 for t ≥ 1.
Proof. See Appendix A.4.
As in the unbalanced budget case demonstrated in Section 3.1, the tax rate and growthrate of capital are constant across periods except period 0. In addition, the solution in
the balanced budget case matches that in the unbalanced budget case if and only if
α = (1 − θ + pβ)/{1 + β(1 + θ)}. In other words, the solutions match if and only if there
happens to be no debt issue in the unbalanced budget case.

4.1

Comparing the Unbalanced and Balanced Budget Cases

To consider the role of government debt, we compare the spending-to-GDP ratio andgrowth rate in the balanced budget case with those in the unbalanced budget case andobtain the following result.
Proposition 4. Let x|Debt and x|Balanced denote the variable x in the unbalanced budget
case and that in the balanced budget case, respectively.
(i) For t = 0, g0 /Ak0 |Debt = g0 /Ak0 |Balanced holds. For t ≥ 1, gt /Akt |Debt Q gt /Akt |Balanced
holds if and only if α Q (1 − θ + pβ)/(1 + pβ(1 + θ)) for t ≥ 1.
(ii) For t ≥ 0, kt+1 /kt |Debt Q kt+1 /kt |Balanced holds if and only if α Q (1 − θ + pβ)/(1 +pβ(1 + θ)).

15

Proof. Direct calculation leads to the aforementioned result.
The ﬁrst result in Proposition 4 states that the spending-to-GDP ratios diﬀer between
the unbalanced and balanced budget cases. In the unbalanced budget case, the available
resources for the government are given by (1 − α)Ak − Rb, which are smaller (larger) thanthose in the balanced budget case when the government borrows (lends) in the capital
market in period t ≥ 1. As a result of this diﬀerence in the available resources, the ratio
in the unbalanced budget case becomes higher or lower than that in the balanced budget
case depending on the state of the ﬁnancial balance.
The growth rate also diﬀers between the two cases. Government debt crowds out
private investment and thus capital formation when α < (1 − θ + pβ)/(1 + pβ(1 + θ)) suchthat the government borrows in the capital market. However, when α > (1 − θ + pβ)/(1 +pβ(1+θ)) such that the government lends in the capital market, such state lending enables
households to save more, thereby enhancing capital formation. Therefore, the state of the
ﬁnancial balance is crucial when evaluating the performance of economic growth.
The case of government borrowing suggests that the unbalanced budget scenario attains lower economic growth than does the balanced budget scenario. This result implies anegative relation between government debt and economic growth. In other words, higher
government debt is associated with a lower economic growth rate. This model prediction
is consistent with the theoretical predictions in a competitive equilibrium (e.g., Saint-Paul,
1992; Josten, 2000; Bräuninger, 2005) and recent empirical evidence (e.g., Reinhart, Reinhart, and Rogoﬀ, 2012; Kumar and Woo, 2010; Checherita-Westphal and Rother, 2012).
The present study adds to the literature by providing a political economy perspective of
this negative relation, which has not been fully investigated in previous studies.

4.2

Vote on the Rule

The analysis has thus far assumed a given ﬁscal rule, namely either the rule where thegovernment is allowed to borrow or lend in the capital market (Section 3) or the rule where
the government is required to balance its budget in each period (Section 4.1). However,in the real world, the rule is also established through the political process: some countries
and states adopt the balanced budget rule or something similar, whereas others do not.
For example, the US federal government has no balanced budget requirement in the US
Constitution (Poterba, 1995), while some European countries have another form of thebalanced budget rule such as the Maastricht Treaty criteria (Corsetti and Roubini, 1996).
Therefore, a natural question is under what condition the government adopts the balanced
budget rule rather than allows accessing the ﬁnancial market.
To address this question, we consider a vote on the rule in the following way. In eachperiod, the government proposes the two ﬁscal rules (i.e., the unbalanced budget rule and16balanced budget rule), for a given set of state variables k and b. One of them is chosen
through voting from the viewpoint of maximizing the value of the political objectivefunction. Second, for a given rule, agents vote on ﬁscal policy (i.e., public spending anddebt issue). The model is solved by backward induction. We have already demonstrated
the vote on ﬁscal policy for a given set of state variables in Section 3 and Section 4.1.
Based on the result thus far, we can compare the value of the political objective function
under the unbalanced budget rule with that under the balanced budget rule for a given
set of k and b as follows:Ω|Debt ≷ Ω|Balanced

[

]
p+1+n
⇔ (1 + n)(1 + pβ) ln (1 − α)Ak − Rb −· g|Debt + (1 + n) · b|Debt
1+n
|
{z
}
(U.1)
′

+ (1 + n)pβθ ln G|Debt · {(1 − α)Ak − Rb′ }|Debt
|
{z
}
[

(U.2)

]
p+1+n
≷ (1 + n)(1 + pβ) ln (1 − α)Ak − Rb −· g|Balanced
1+n
{z
}
|
(B.1)

+ (1 + n)pβθ ln G|Balanced · {(1 − α)Ak ′ − Rb′ }|Balanced .
|
{z
}
(B.2)

The terms related to the utility of old-age consumption and utility of current public
spending cancel each other out because they are identical under the two rules. The
terms (U.1) and (B.1) denote the lifetime utility of consumption, while (U.2) and (B.2)
denote the utility of future public spending. We compare (U.1) with (B.1) and obtain(U.1) ≷ (B.1) ⇔ b′ |Debt ≷ 0. In other words, the government achieves a higher utilityof consumption by borrowing in the capital market. We also compare (U.2) with (B.2)
and obtain (U.2) ≷ (B.2) ⇔ b′ |Debt ≶ 0. This condition states that borrowing in thecapital market increases the cost of debt repayment in the next period and thus decreases
the next-period level of public spending. Therefore, allowing for an unbalanced budget
creates a trade-oﬀ between lifetime consumption and future public spending.
To ﬁnd the overall eﬀect of the interactions, we undertake the numerical analysis bysetting p = 0.8, θ = 0.9, and β = (0.99)30 .6 The result depicted in Figure 1 indicates thatfor most values of α, the political objective function is higher under the balanced budget
rule than under the unbalanced budget rule. In other words, under plausible values of α,
the balanced budget rule is chosen through voting regardless of the state of the ﬁnancial
6

The last assumption, β = (0.99)30 , implies that each generation lasts for 30 years. For example, theﬁrst and second periods correspond to ages 25–54 and 55–84, respectively. We also assume a single-period
discount factor of 0.99. Because agents under the current assumption plan over these 30-year generations,
we discount the future by (0.99)30 .

17

balance.7[Figure 1 about here.]
The numerical result in Figure 1 provides some insights into Pareto improvement. For
a given set of k and b, the old are indiﬀerent between the balanced and unbalanced budget
rules because the policy functions of public spending are identical between the two rules.
However, the young become better oﬀ by shifting from the unbalanced budget to thebalanced budget for most values of α. Hence, taking the balanced budget rule in eachperiod is desirable from the viewpoint of Pareto improvement.

5

Ramsey Allocation

This section characterizes a Ramsey allocation chosen by a benevolent planner. The planner has the ability to commit to all his or her future policy choices at the beginning of aperiod, subject to the competitive equilibrium constraints. These constraints include thecapital market clearing condition and government budget constraint. We compare theRamsey allocation with the unbalanced budget case of the political equilibrium demonstrated in Section 3. Then, we evaluate the normative aspect of the political equilibriumin terms of economic growth and the government’s ﬁnancial position.
The benevolent planner is assumed to value the welfare of all households. In particular, following Farhi and Werning (2007) and Song, Storesletten, and Zilibotti (2012),
we assume that the planner attaches geometrically decaying Pareto weights ρt to theutility of each generation t. In addition, the planner’s weight on generations is assumed
to reﬂect the cohort size. Therefore, the planner’s objective function is given by W =∑t−1 ypV0o + (1 + n) ∞Vt−1 , or
t=1 {(1 + n)ρ}
{ (
)
}
p+1+n
W = (1 + n)(1 + pβ) ln A · x0 −g0 − (1 + n)x1 + (p + 1 + n)θ ln g0
1+n
)
}
[
{ (
∞
∑
p+1+n
t
gt − (1 + n)xt+1+((1 + n)ρ) · (1 + n)(1 + pβ) ln A · xt −
1
+
n
t=1
)
]
(
pβ
+ 1 + n θ ln gt ,+ρwhere the terms unrelated to political decisions are omitted from the expression. We
assume (1 + n)ρ < 1. Given k0 (> 0) and b0 (< (1 − α)k0 /α), the planner’s problem is tochoose {gt , kt+1 , bt+1 }∞t=0 subject to the capital market clearing condition and government
budget constraint.
7

The aforementioned result seems to be inconsistent with the cross-country evidence. Many countries
run unbalanced budgets by borrowing in the capital market (OECD, 2014). However, ﬁnding the factor
that prevents the government from following the balanced budget rule is left for future research.

18

By using a method similar to that applied in Section 3, we can reformulate the aforementioned objective function in terms of xt ≡ (1 − α)Akt − Rbt and write down therecursive formulation of this problem as follows:
}
)
[
{ (
p+1+n
Ṽ (x0 ) = max (1 + n)(1 + pβ) ln A x0 −g0 − (1 + n)x1
{g0 ,x1 }1+n+(p + 1 + n)θ ln g0 + (1 + n)ρV (x1 )] ,for t = 0, and
[

{ (
)
}
p+1+n
′
V (x) = max(1 + n)(1 + pβ) ln A x −g − (1 + n)x
g,k′
1+n
(
)
]
pβ
′
+
+ 1 + n θ ln g + (1 + n)ρV (x ) ,ρ
for t ≥ 1. We solve the functional equations based on the guess-and-verify method andcompute the policy functions and growth rate (see Appendix A.5 for the derivation).
Here, we look at the growth rate and state of the ﬁnancial balance in the Ramsey
allocation to explore the normative implication of the political equilibrium. The growthrate in the Ramsey allocation changes from period 1 to period 2, but remains stable after
period 3 at the rate kt+1 /kt = ρA (t ≥ 2). The planner’s ﬁnancial position is determinedin the following way:
)]
[
(
pβ
1
− 1+pβpβ 1 − θ 1−α[()] .
b′ ≷ 0 ⇔ ρ ≶pβ
pβ
1(1 + n) 1−α+ θ 1−α− 1+pβHence, the Ramsey allocation might feature less growth and more borrowing than thepolitical equilibrium if the planner attaches low weights to future generations. Theselower weights imply that the planner has less incentive to save goods for future generations
through capital accumulation.

6

Discussion and Extensions

The aforementioned results depend on several assumptions. In this section, we brieﬂyconsider the role of each assumption and investigate how the results would change ifeither of them was relaxed or modiﬁed. In Section 6.1, we consider a more realistic casein which public spending beneﬁts diﬀer between generations. In Section 6.2, we compare
the political equilibrium outcome of the present model with that of the model with aneoclassical production function.

19

6.1

Age-dependent Public Spending Beneﬁts

We assumed that the young and old beneﬁt from public spending to the same degree.
However, in the real world, the young might beneﬁt more or less from public spending
than the old. For example, the young beneﬁt more from public educational services, while
the old beneﬁt more from medical services. To demonstrate such age-dependent cases, wefollow Song, Storesletten, and Zilibotti (2012) and assume the following utility function:}{
Ut = ln cyt + θ ln gt + β ln cot+1 + θλ ln gt+1 ,where λ(> 0) captures the preference weight of public spending on the old. When λ <(>)1, the young beneﬁt more (less) from public spending.
Under this assumption, the political objective function in Deﬁnition 2 is modiﬁed asfollows:

[

(
)
]
p+1+n
′
Ω(x, g, x , g ) = (1 + n)(1 + pβ) ln A · x −g − (1 + n)x
1+n+ (ωp + 1 + n) θλ ln g + (1 + n)pβθλ ln g ′ .
′′Solving the maximization problem of Ω leads to the following policy functions:(ωp + 1 + n)(1 + n)θλx,
(p + 1 + n) · [(1 + n) {1 + pβ(1 + θλ)} + θλ(ωp + 1 + n)]
pβθλA
x′ =
x,
(1 + n) {1 + pβ(1 + θλ)} + θλ(ωp + 1 + n)g=where ∂g/∂λ > 0 and ∂x′ /∂λ > 0 hold. Public spending and the growth rate rise as theweight λ increases.
A higher λ results in higher public spending since young individuals attach a higher
weight to public spending when old. In addition, a higher λ results in a higher growth rate.
The young attach a higher weight to the utility of old-age public spending. This ﬁnding
implies that the government, reﬂecting the preferences of the young, has an incentive
to increase future public spending and thus chooses ﬁscal policy to stimulate capital
accumulation. The result suggests that the introduction of diﬀerent preferences for public
spending quantitatively aﬀects ﬁscal policy. However, we should note that the main results
are qualitatively unchanged under this alternative setup.

6.2

A Neoclassical Production Function

We assumed AK technology throughout the analysis. This assumption results in a constant interest rate across periods, which enables us to obtain an analytical solution. However, AK technology ignores the possibility that the interest rate changes in response to20

capital accumulation. Given that current ﬁscal policy aﬀects saving and capital accumulation, the government might take into account the intertemporal mechanism between
ﬁscal policy and the interest rate through capital accumulation.
To demonstrate the aforementioned eﬀect, we here remove the assumption of AK technology and instead assume the neoclassical production function given by Yt = A(Kt )α (Lt )1−α .
The gross interest rate is now given by Rt+1 = R(kt+1 ) ≡ αA(kt+1 )α−1 , which is dependenton per capita capital. Following the same procedure as in Section 3, we can now write
the political objective function as follows:Ωt = (1 + n)(1 + pβ) ln(1 − τt )wt + pβ ln R(kt+1 ) + (ωp + 1 + n) θ ln gt + (1 + n)pβθ ln gt+1 ,where the second term on the right-hand side indicates that the government takes accountof the intertemporal eﬀect of its ﬁscal policy choice through capital accumulation. In thecase of AK technology, this term is taken as given.
The ﬁrst-order conditions with respect to gt and bt+1 are
gt : (1 + n)(1 + pβ)∂(1 − τt )wt∂R(kt+1 ) ∂kt+1
+ pβ
·
∂gt
∂kt+1
∂gt
{z
}
|
(R.1)

(ωp + 1 + n) θ
∂gt+1 ∂kt+1+ (1 + n)pβθ
·
= 0,
gt
∂kt+1 ∂gt
∂R(kt+1 ) ∂kt+1∂gt+1∂(1 − τt )wt+ pβ·+ (1 + n)pβθ
= 0,bt+1 : (1 + n)(1 + pβ)
∂bt+1
∂kt+1
∂bt+1
∂bt+1
|
{z
}
+

(R.2)

where the terms (R.1) and (R.2) indicate the eﬀects through the interest rate. The term(R.1) shows the marginal beneﬁt of public spending through the interest rate. A higher
level of public spending places a larger tax burden on the young, which creates a negative
income eﬀect on saving and capital accumulation. A decrease in capital increases themarginal productivity of capital, gross interest rate, and thus return from saving. Thisis an additional marginal beneﬁt of public spending. The term (R.2) shows the marginal
beneﬁt of government debt through the interest rate. A higher level of government debt
crowds out capital accumulation, which creates an additional marginal beneﬁt of government debt as a result of an increase in the interest rate. These two beneﬁts, peculiar
to the model with the neoclassical production function, are abstracted from the analysis
based on AK technology.

7

Concluding Remarks

How does intergenerational conﬂict on ﬁscal policy aﬀect public spending and economic
growth through voting? How does the issuance of government debt inﬂuence ﬁscal policy
21and economic growth? What is the normative implication of the political equilibriumoutcome? This study attempted to answer these questions by adopting an overlappinggenerations model in which public spending is ﬁnanced by tax and the issuance of government debt. In addition, ﬁscal policy is decided by probabilistic voting that capturesintergenerational conﬂict.
The ﬁndings of the present study are threefold. First, population aging incentivizes
the government to invest more in capital for future public spending, positively aﬀecting economic growth. Second, when the government borrows in the capital market, theintroduction of the balanced budget rule results in a higher growth rate. In addition,
under plausible conditions, voters prefer the balanced budget scenario to the unbalanced
budget scenario when the budget rule is also decided by voting. Third, we compare thepolitical equilibrium outcome with the Ramsey allocation in which an inﬁnitely lived planner commits to all his or her future policy choices and ﬁnd that the Ramsey allocation
might feature less growth and more borrowing than the political equilibrium if the planner
attaches low weights to future generations.
The main contribution of this study is that it demonstrates the growth and the normative implication of the issuance of government debt in the presence of intergenerational
conﬂict on ﬁscal policy. Such implications have not been fully examined in previous studies (Cukierman and Meltzer, 1989; Röhrs, 2010; Song, Storesletten, and Zilibotti, 2012;
Arai and Naito, 2014). To demonstrate the implications, this study relies on a logarithmic utility function, simple AK technology, and inelastic labor supply. These assumptions
thus enable us to solve the model in a tractable way.
In addition to the aforementioned points, the present study made the following assumptions. First, it assumed a closed economy with a constant interest rate stemmingfrom simple AK technology. Allowing for government spending on infrastructure as inBarro (1990) or on public education as in Glomm and Ravikumar (1992) would enable us
to demonstrate a more realistic scenario of ﬁscal policy and its eﬀect on economic growth.
Second, we assumed no altruism toward children. With some weight placed on children’s welfare, the old care about the future tax burden, which may have an additionaleﬀect on the choice of ﬁscal policy by the government, as demonstrated by Cukierman andMeltzer (1989) and discussed by Beauchemin (1998). Third, we assumed no pay-as-yougo (PAYG) pension provision for the old. Although pension and debt are economicallyequivalent (Barro, 1974), the introduction of a PAYG pension as an alternative to government debt might be expected to result in a diﬀerent equilibrium allocation. However,these eﬀects are omitted from the analysis here and are left to future work.

22

A

ProofsRecall that R = αA holds in the equilibrium. In the following series of proofs, we often
use R instead of αA to remind readers of the importance of the term Rb in the equations.

A.1

Reformulation of the Problem

First, we substitute the government budget constraint (1 + n)b′ + τ (1 − α)Ak = (p +1 + n)g/(1 + n) + Rb into the capital market clearing condition (1 + n)(k ′ + b′ ) =(pβ/(1 + pβ)) (1 − τ )(1 − α)Ak to replace τ by k, b and b′
{
}
p+1+n
pβ
′
′
′
(1 − α)Ak −(1 + n)(k + b ) =g − Rb + (1 + n)b .
1 + pβ1+nThis expression is reformulated as follows:
{
}
p+1+n
1+n
pβ
′
(1 + n)b =(1 − α)Ak −g − Rb −{(1 − α)Ak ′ − Rb′ }1 + pβ1+n(1 − α)A
{
}
1+n
pβ ′
−
Rb′ − (1 − α)Ab .
(1 − α)A
1 + pβWe then move the third term on the right-hand side to the left-hand side and rearrange
the terms to obtain
{
}
]
[
pβ
p+1+n
1+n
′
′
′
((1 − α)Ak − Rb) −g −{(1 − α)Ak − Rb }(1 + n)b =1 + pβ1+n(1 − α)A
(
)−1
R
1
×
+
.
(9)
(1 − α)A 1 + pβ
Next, we rewrite the indirect utility function of the young, V y = (1 + pβ) ln(1 − τ )(1 −α)Ak + θ ln g + pβθ ln g ′ , as follows:
{
}
p+1+n
y
′
g + (1 + n)b + θ ln g + pβθ ln g ′V = (1 + pβ) ln ((1 − α)Ak − Rb) −
1+n
[
{
{
}
p+1+n
pβ
p+1+n
= (1 + pβ) ln ((1 − α)Ak − Rb) −g+((1 − α)Ak − Rb) −g1+n
1 + pβ
1+n
} (
)−1 ]
1+n
R
1
−
{(1 − α)Ak ′ − Rb′ } ×++ θ ln g + pβθ ln g ′ ,(1 − α)A(1 − α)A 1 + pβ
where the ﬁrst equality comes from substituting in the government budget constraint andthe second equality comes from substituting in (9). The above expression is rewritten as
[(
){
}
R
p+1+n
y
V = (1 + pβ) ln+1((1 − α)Ak − Rb) −g(1 − α)A
1+n
]
1+n
′
′
−
{(1 − α)Ak − Rb } + θ ln g + pβθ ln g ′ .
(1 − α)A23

By using R = αA, the term R/(1 − α)A + 1 is rewritten as 1/(1 − α). Therefore, theabove expression is reduced to
[ {
}
]
p+1+n
y
′
′
V = (1 + pβ) ln A ((1 − α)Ak − Rb) −g − (1 + n) ((1 − α)Ak − Rb )1+n
+ θ ln g + pβθ ln g ′ ,(10)where constant terms are omitted from the expression.
By using (10) and x ≡ (1 − α)Ak − Rb, the political objective function is now given
by
{

p+1+n
Ω (x, g, x , g ) = (1 + n)(1 + pβ) ln A(x −g) − (1 + n)x′
1+n
+ (ωp + 1 + n)θ ln g + (1 + n)pβθ ln g ′ ,′}′where the unrelated terms are omitted from the expression. Since the capital market
clearing condition and government budget constraint are included in Ω (x, g, x′ , g ′ ), theproblem is now to maximize Ω (x, g, x′ , g ′ ), subject to g ′ = G(k, b) and given x, k and b.
Therefore, the problem in Deﬁnition 2(ii) is reformulated as in the statement in (4), if we
assume G(k, b) = G(x) ≡ G ((1 − α)Ak − Rb).


A.2

Proof of Proposition 1

Consider the reformulated problem demonstrated in (4). Given the guess of g ′ = GDebt ·x′ ,we obtain the following ﬁrst-order conditions with respect to x′ and g :1+n(1 + n)pβθ
,
x′

(11)

A

(ωp + 1 + n)θ
.
g

(12)

x′ : (1 + n)(1 + pβ)=g) − (1 + n)x′A(x − p+1+n
1+ng : (1 + n)(1 + pβ)=′
A(x − p+1+n
g)
−
(1
+
n)x
1+n

Conditions (11) and (12) lead to the following relation between g and x′g=(ωp + 1 + n)(1 + n) ′x.
pβA(p + 1 + n)

(13)

Substituting (13) into (11) leads to the following optimality condition for x′
x′ =

θpβA
x.
(1 + n) {1 + pβ(1 + θ)} + (ωp + 1 + n)θ

(14)

With (13) and (14), the optimality condition for g becomesg=1+n(ωp + 1 + n)θ
·x.
p + 1 + n (1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ
24

(15)

Therefore, the function g ′ = GDebt · x′ constitutes a stationary Markov-perfect politicalequilibrium as long as the following holds:
GDebt =

1+n
(ωp + 1 + n)θ
·.
p + 1 + n (1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ

To ﬁnd the policy functions B(k, b) and T (k, b), recall the capital market clearing
condition and government budget constraint given bypβ(1 − τ )(1 − α)Ak,
1 + pβp+1+ng + Rb,(1 + n)b′ + τ (1 − α)Ak =1+n(1 + n)(k ′ + b′ ) =

(16)
(17)

respectively. Given k and b, the four variables, g, k ′ , b′ and τ , are determined by (14),(15), (16), and (17).
Substituting (15) and (17) into (16) leads topβ{1 + pβ(1 + θ)} (1 − α)A(1 − α)A ′··{(1 − α)Ak − Rb}−b.
1 + pβ (1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ
1 + pβ(18)
We substitute (18) into (14) and rearrange the terms to obtain the policy function B(k, b)(1−α)Ak ′ =pβ(1 − θ + pβ) − α (1 + pβ(1 + θ))·· {(1 − α)Ak − Rb} .
1 + αpβ (1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ
(19)
By using (18) and (19), we obtain the law of the motion of capital as follows:b′ = B(k, b) ≡

k′ =

pβ
θ + α (1 + pβ(1 + θ))·· {(1 − α)Ak − Rb} .
1 + αpβ (1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ

(20)

Here, (19) and (20) imply that b′ /Ak ′ is constant across periods after period 1:b′(1 − θ + pβ) − α (1 + pβ(1 + θ))=∀t ≥ 1.
′
Ak
A {θ + α (1 + pβ(1 + θ))}Given k ′ > 0, this equation states that b′ ≷ 0 holds if and only if α ≶ (1 − θ + pβ)/{1 +pβ(1 + θ)}.
To determine the policy function T (k, b), recall the government budget constraint (17),which is rewritten as
p+1+n
g + Rb − (1 + n)b′1+n(ωp + 1 + n)θ
=· {(1 − α)Ak − Rb} + Rb
(1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ
(1 − θ + pβ) − α (1 + pβ(1 + θ))
pβ·· {(1 − α)Ak − Rb} ,− (1 + n)
1 + αpβ (1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ

τ (1 − α)Ak =25

where the second equality is derived from (15) and (19).
By dividing both sides by (1 − α)Ak and rearranging the terms, we obtainτ = T (k, b) ≡+pβ
(ωp + 1 + n)θ − (1 + n) 1+αpβ{(1 − θ + pβ) − α (1 + pβ(1 + θ))}(1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ
pβ(1 + n) (1 + pβ(1 + θ)) + (1 + n) 1+αpβ{(1 − θ + pβ) − α (1 + pβ(1 + θ))}(1 + n) (1 + pβ(1 + θ)) + (ωp + 1 + n)θ

where
b
=
k

{

·

α
b
· ,
1−α k
(21)

b0
for t = 0,k0
(1−θ+pβ)−α(1+pβ(1+θ))for t ≥ 1.
θ+α(1+pβ(1+θ))(22)

The remaining task is to show that g > 0 and τ < 1 hold ∀t ≥ 0. In period 0, given
k0 (> 0), b0 must satisfy (1 − α)Ak0 − Rb0 > 0 and T (k0 , b0 ) < 1. Both conditions are
reduced to b0 < (1 − α)k0 /α.
Next, consider g in period t ≥ 1. Equation (15) implies that g > 0 holds if and only
if (1 − α) − αb/k > 0 holds. Given (22), the necessary and suﬃcient condition for g > 0
in period t ≥ 1 becomes θ(1 + αpβ) > 0, which holds for any set of parameters.
Finally, consider τ in period t ≥ 1. We substitute (22) into (21) and rearrange theterms to obtain




pβ


τ < 1 ⇔ (1 + n) · (1 + pβ(1 + θ)) +{(1 − θ + pβ) − α (1 + pβ(1 + θ))}
1 + αpβ


|
{z
}
(A.1)






 α
(1 − θ + pβ) − α (1 + pβ(1 + θ))×·− 1 < 0,θ + α (1 + pβ(1 + θ))
1 − α

{z
}
|
(A.2)

(23)
where the sign of the term (A.1) is positive and the sign of the term (A.2) is negative.
Therefore, the condition (23) holds for any set of parameters.


A.3

Proof of Proposition 2(i) Recall the law of the motion of capital demonstrated in Proposition 1. Given the initialcondition b0 (< (1 − α)k0 /α), the growth rate of capital in period 0, k1 /k0 , is immediately
computed as demonstrated in Proposition 2(i).
26Next, recall the law of the motion of capital in period t ≥ 1. Dividing both sides ofthe equation by kt leads to
{
}
kt+1
pβ
θ + α (1 + pβ(1 + θ))bt=·· (1 − α) − αA.
kt
1 + αpβ (1 + n)(1 + pβ(1 + θ)) + (ωp + 1 + n)θ
kt
The substitution of the ratio bt /kt shown in Proposition 1 into the above expression leads
to
kt+1
xt+1
pβθ
=
=
A for t ≥ 1.
kt
xt
(1 + n)(1 + pβ(1 + θ)) + (ωp + 1 + n)θ
(ii) The eﬀect of a lower population growth rate is immediate from the expression
k1 /k0 and kt+1 /kt (t ≥ 1) in Proposition 2(i).
(iii) To show the eﬀect of greater longevity, recall the growth rate of capital for t ≥ 1,
which is reformulated as
βθ
kt+1
(
) (
) A.
=
kt
(1 + n) 1 + β(1 + θ) + ω + 1+n θppThis indicates that ∂ (kt+1 /kt ) /∂p > 0 holds for t ≥ 1.
The diﬀerentiation of k1 /k0 with respect to p yields
[{

} ]−1
b0
∂ (k1 /k0 )(1 + αpβ) {(1 + n)(1 + pβ(1 + θ)) + (ωp + 1 + n)θ} ·(1 − α) − α
A
·
k0
∂p
= β {θ + α (1 + pβ(1 + θ))} (1 + αpβ) {(1 + n)(1 + pβ(1 + θ)) + (ωp + 1 + n)θ}
|
{z
}
2

2

(B1)

+ pβαβ(1 + θ)(1 + αpβ) {(1 + n)(1 + pβ(1 + θ)) + (ωp + 1 + n)θ}
{z
}
|
(B2)

− pβ {θ + α(1 + pβ(1 + θ))} αβ {(1 + n)(1 + pβ(1 + θ)) + (ωp + 1 + n)θ}
|
{z
}
(B3)

− pβ {θ + α (1 + pβ(1 + θ))} (1 + αpβ) {(1 + n)β(1 + θ) + ωθ}.
|
{z
}
(B4)

Here,
(B2) − (B3) = pβαp(1 − α) {(1 + n)(1 + pβ(1 + θ)) + (ωp + 1 + n)θ} > 0,
(B1) − (B4) = β {θ + α(1 + pβ(1 + θ))} (1 + αpβ)(1 + n)(1 + θ) > 0.
Therefore, we obtain ∂ (k1 /k0 ) /∂p > 0.


27

A.4

Proof of Proposition 3

To solve the problem, we conjecture the following linear policy function:
gt+1 = GBalanced · {(1 − α)Akt+1 − Rbt+1 } ,where GBalanced ∈ (0, ∞) is a constant parameter. Under this conjecture and the capital
market clearing condition (8), we can reformulate the problem as
)
(
p+1+n
max(1 + n) {1 + pβ(1 + θ)} ln (1 − α)Akt − Rbt −gt + (ωp + 1 + n)θ ln gt ,
{gt }
1+n
where bt = 0 for t ≥ 1.
Solving this problem leads to the following policy function:
gt =

1+n
(ωp + 1 + n)θ
·{(1 − α)Akt − Rbt } .
p + 1 + n (1 + n) {1 + pβ(1 + θ)} + (ωp + 1 + n)θ

This function constitutes a Markov-perfect political equilibrium as long as
GBalanced =1+n(ωp + 1 + n)θ
·= GDebt .
p + 1 + n (1 + n) {1 + pβ(1 + θ)} + (ωp + 1 + n)θ

By using the policy function of gt and government budget constraint, we can compute thetax rate as follows:
τt =
=

p+1+n
gt + Rbt
1+n(1 − α)Akt
Rbt(ωp + 1 + n)θ + (1 + n) {1 + pβ(1 + θ)} (1−α)Akt(1 + n) {1 + pβ(1 + θ)} + (ωp + 1 + n)θ

,where bt = 0 for t ≥ 1. Finally, we substitute the policy function of gt into the constraint(8) to obtain the law of the motion of capital as presented in Proposition 3.


A.5

Ramsey Allocation

We ﬁrst solve the functional equation for t ≥ 1. We make the guess V (x′ ) = v0 + v1 ln x′ ,where v0 and v1 are undetermined coeﬃcients. For this guess, the recursive formulationof the problem is
{
{ (
)
}
p+1+n
′
v(x) = max(1 + n)(1 + pβ) ln A x −g − (1 + n)x{g,x′ }
1+n
(
)
}
pβ
′
+
+ 1 + n θ ln g + (1 + n)ρ · (v0 + v1 ln x ) .
ρ28

Solving this functional equation leads to
1+n
p+1+n

g=

(
(

(1 + n)(1 + pβ) +x′ =((1 + n)(1 + pβ) +

pβ
+1+n
ρ

pβ
+1+n
ρ

ρv1 A
pβ
+1+n
ρ

)

)

θ
x,
θ + (1 + n)ρv1)x.
θ + (1 + n)ρv1

Substituting these into the recursive formulation gives
{
(
)
}
pβ
v(x) = (const) + (1 + n)(1 + pβ) ++ 1 + n θ + ρ(1 + n)v1 ln x,
ρwhere (const) is the term including the constant terms. The guess is veriﬁed if v1 =(1 + n)(1 + pβ) + (pβ/ρ + 1 + n)θ + ρ(1 + n)v1 and v0 = (const). Therefore, v1 is given by
(
)
+
1
+
n
θ
(1 + n)(1 + pβ) + pβρv1 =,1 − (1 + n)ρ
and the corresponding policy functions of x′ and g are given as follows:x′ = ρAx,

(

)
pβ
1+n
{1 − (1 + n)ρ} p+1+n ρ + 1 + n θ
(
) x.
g=
pβ
(1 + n)(1 + pβ) + ρ + 1 + n θ

(24)
(25)

Next, we consider the problem in period 0. By using the aforementioned result, thefunctional equation in period 0 is written as
[
{ (
)
}
p+1+n
Ṽ (x0 ) = max (1 + n)(1 + pβ) ln A x0 −g0 − (1 + n)x1
{g0 ,x1 }1+n+(p + 1 + n)θ ln g0 + (1 + n)ρv0 + (1 + n)ρv1 ln x1 ] .
Solving this functional equation leads to the following policy functions in period 0:(1 − (1 + n)ρ) (1 + n)θ
()
x0 ,(1 + n)(1 + pβ) + (1 + n)ρ pβ
+
1
+
n
θ
+
(p
+
1
+
n)θ
(1
−
(1
+
n)ρ)
ρ
{
(
) }
(1 + n)(1 + pβ) + pβ
+
1
+
n
θ ρA
ρ
(
)
x1 =
x0 .
(1 + n)(1 + pβ) + (1 + n)ρ pβ
+
1
+
n
θ
+
(p
+
1
+
n)θ
(1
−
(1
+
n)ρ)
ρ
g0 =

Policy Functions of k ′ and b′
To ﬁnd the policy functions of b′ and k ′ in period t(≥ 1), recall the capital market
clearing condition and government budget constraint. Together, they are put into thefollowing condition:
]
[
pβ
p+1+n
′
′
′
(1 + n)(k + b ) =g − Rb + (1 + n)b .
(1 − α)Ak −1 + pβ1+n29

By substituting the policy function of g in (25) into the above expression, we obtain
(
)
pβ
(1
+
n)(1
+
pβ)
+
(1
+
n)ρ
+
1
+
n
θ
ρ
1+n ′
pβ
(
)
b =
·
·{(1 − α)Ak − Rb}−(1+n)b′ .
pβ
1 + pβ1 + pβ(1 + n)(1 + pβ) ++1+n θ
ρ

From this and (24), we obtain the policy functions of k ′ and b′ in period t(≥ 1) as follows:[{()}−1 ]
ρ
α
1
α
k′ =
+
(1 + n)+· Φ · {(1 − α)Ak − Rb} , (26)1−α 1−α1 + pβ 1 − α
{
(
)}−1
α
1
′
b = (1 + n)+
· Φ · {(1 − α)Ak − Rb} ,(27)
1 + pβ 1 − α
where

(
)
pβ
(1
+
n)(1
+
pβ)
+
(1
+
n)ρ
+
1
+
n
θ
ρ
pβ
ρ
(
)
Φ≡
·
− (1 + n).
1 + pβ1−α(1 + n)(1 + pβ) + pβ
+
1
+
n
θ
ρ

Following the same manner, we obtain the policy functions of k1 and b1 as follows:
[
]
{
(
)}−1
1
α
α
k1 = Ψ 0 +(1 + n)+· Φ0 · {(1 − α)Ak0 − Rb0 } , (28)1−α1 + pβ 1 − α
{
(
)}−1
1
α
b1 = (1 + n)+
· Φ0 · {(1 − α)Ak0 − Rb0 } ,(29)
1 + pβ 1 − α
where

[
(
) ]
[
(
) ]
pβ
1+n
(1 + n)(1 + pβ) + (1 + n)ρ pβ
+
1
+
n
θ
−
ρ
(1
+
n)(1
+
pβ)
+
+
1
+
n
θ
ρ
1−α
ρ
(
)
Φ0 ≡
,
(1 + n)(1 + pβ) + (1 + n)ρ pβ
+
1
+
n
θ
+
(p
+
1
+
n)θ(1
−
(1
+
n)ρ)
ρ
[
(
) ]
ρ
pβ
(1 + n)(1 + pβ) + ρ + 1 + n θ
1−α
(
)
Ψ0 ≡
.
(1 + n)(1 + pβ) + (1 + n)ρ pβ
+
1
+
n
θ
+
(p
+
1
+
n)θ(1
−
(1
+
n)ρ)
ρ
pβ
·
1+pβ

Growth Rates
Given k0 and b0 , the growth rate in period 1, k1 /k0 , is immediately obtained from
(28). To compute the growth rate in period t ≥ 2, recall that the following holds from
(26):
[
}
{
(
)}−1 ] {
kt+1
α
α
ρ
1
bt
=
+
+
· Φ · (1 − α)A − R.
(1 + n)kt
1−α 1−α1 + pβ 1 − αkt
We can compute the growth rate in period 2, k2 /k1 , by substituting (28) and (29) into
the above expression, and the growth rate in period t (≥ 3) by substituting (26) and (27)30

into the above expression. In particular, the growth rate in period t (≥ 3) is given by
kt+1 /kt = ρA.
State of the Financial Balance
Recall (29) that presents the policy function of b1 . By using (29), we can determine
the state of the ﬁnancial balance in period 0 as b1 ≷ 0 ⇔ Φ0 ≷ 0. For t ≥ 1, the stateof the ﬁnancial balance is determined by using (27): b′ ≷ 0 ⇔ Φ ≷ 0. The condition for
period t ≥ 1 is reformulated as follows:
[
(
)]
pβ
1
pβ 1 − θ 1−α − 1+pβ([)] .
b′ ≷ 0 ⇔ Φ ≷ 0 ⇔ ρ ≶pβpβ1+ θ 1−α− 1+pβ(1 + n) 1−α31References[1] Acemoglu, D., and Robinson, J.A., 2006. Economic Origins of Dictatorship and Democracy,
Cambridge University Press.
[2] Alesina, A., and Tabellini, G., 1990. A positive theory of ﬁscal deﬁcits and government
debt. Review of Economic Studies 57, 403–414.
[3] Arai, R., and Naito, K., 2014. A politico-economic approach on public debt in an endogenous growth economy. Available at http://EconPapers.repec.org/RePEc:pra:mprapa:56213(December 26, 2014).
[4] Azzimonti, M., Battaglini, M., and Coate, S., 2010. On the casea balanced budget amendment to the U.S. Constitution. Available
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1

Unbalanced
Balanced

0.5

Values of the political objective function

0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5

0

0.2

0.4

Alpha

0.6

0.8

1

Figure 1: The horizontal axis is α and the vertical axis is Ω. The solid curve shows thevalue of Ω for the unbalanced budget case and the dotted curve shows the value of Ω forthe balanced budget case. The vertical line is the critical value of α that distinguishes
the state of the ﬁnancial balance.

35