Discussion Papers In Economics
And Business

Debt Policy Rules in an Open Economy
Keiichi Morimoto, Takeo Hori, Noritaka Maebayashi and Koichi Futagami

Discussion Paper 13-07-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDebt Policy Rules in an Open Economy
Keiichi Morimoto, Takeo Hori, Noritaka Maebayashi and Koichi Futagami

Discussion Paper 13-07-Rev.

August 2015

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDebt Policy Rules in an Open Economy∗
Keiichi Morimoto†Takeo Hori‡

Noritaka Maebayashi§

Koichi Futagami¶

Abstract
In a small open economy model of endogenous growth with public capital accumulation, we examine the eﬀects of a debt policy rule under which the government
must reduce its debt–GDP ratio if it exceeds the criterion level. To sustain public
debt at a ﬁnite level, the government should adjust public spending rather than theincome tax rate. The long run debt–GDP ratio should be kept suﬃciently low to
avoid equilibrium indeterminacy. Under sustainability and determinacy, a tighter(looser) debt rule brings welfare gains when the world interest rate is relatively high(low).

JEL classiﬁcation: E62; H54; H63
Keywords: Fiscal policy, Public debt, Welfare, Small open economy, Indeterminacy, Limit
cycles∗

We would like to thank Luca Agnello, Tomohiro Hirano, Nobuhiro Kiyotaki, Ryuichiro Murota,
Yoshiyasu Ono, Josef Schroth, Takayuki Tsuruga, Stephan Turnovsky, Tuan Khai Vu, Akira Yakita,
Cheng-Chen Yang, and the participants at the 2013 Asian Meeting of Econometric Society, the 2013 European Meeting of Econometric Society, and other conferences for their helpful comments and suggestions.
We are responsible for any remaining errors.
†
Corresponding author: Department of Economics, Meisei University, 2-1-1 Hodokubo, Hino, Tokyo
191-8506, JAPAN; e-mail: keiichi.morimoto@meisei-u.ac.jp; Tel: +81-42-591-6964‡
College of Economics, Aoyama Gakuin University, 4-4-25 Shibuya, Shibuya-ku, Tokyo 150-8366,
JAPAN; e-mail: t-hori@aoyamagakuin.jp§
Faculty of Economics and Business Administration, The University of Kitakyushu, 4-2-1 Kitagata,
Kokura Minami-ku, Kitakyushu, Fukuoka 802-8577, JAPAN; e-mail: non818mn@kitakyu-u.ac.jp¶
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
JAPAN; e-mail: futagami@econ.osaka-u.ac.jp

1

Introduction

Discretionary ﬁscal policies during the 2008–2009 world crisis resulted in serious increases
in government debt in the Euro area. In 2011, the average debt–GDP ratio in the Euro
area reached 88 percent of GDP, some 20 percentage points higher than at the start ofthe crisis in 2007. Public debt as a share of GDP in Greece equaled 166.1 percent in 2012.
Debt–GDP ratios in Italy, Ireland, and Portugal also exceeded 100 percent. These weakﬁscal conditions raised doubts about these countries’ abilities to ﬁnance their increased
debt. As a response to the crisis, the EU has introduced strong ﬁscal consolidations under
the surveillance of the European Commission. Overall public deﬁcits were reduced thanks
to expenditure cuts, especially lower public investments, as stated in Public ﬁnances in
EMU (2012).1 According to the Stability and Convergence Programmes submitted to
the Commission and Council in Spring 2012, EU member states plan to base further
ﬁscal consolidation on expenditure cuts that include reductions in public investment.
According to the debt reduction benchmark introduced by the reform of the Stability
and Growth Pact (SGP), the so-called Six-Pack in December 2011, member states whose
current debt-to-GDP ratio is above the 60% threshold have to reduce the distance to 60%
by an average rate of one-twentieth per year.2 It is important to investigate the eﬀects ofthe debt-reduction rule proposed by the SGP under its requirements.
Some authors have examined the eﬀects of such a debt-reduction rule. In an endogenous growth model whose growth engine is the ﬂow of public service as in Barro (1990),
Futagami et al. (2008) investigate the eﬀects of a government bond-issuance rule that
requires the government to reduce its debt at a steady pace if its debt is beyond thecriterion level. Maebayashi et al. (2013) uses an endogenous growth model whose engineof growth is public capital accumulation to study the same issue. These authors provide
interesting results, but their investigations are conﬁned to closed economies; accordingly,
1

In the Euro area, the average general government deﬁcit fell from 6.2 percent of GDP in 2010 to 4.1
percent of GDP in 2011.
2
The Maastrichit Treaty asks EU countries to keep their deﬁcit and debt levels below 3 and 60 percent,respectively, to ensure compliance with budgetary discipline.

1transactions in foreign capital markets are removed. In reality, both the government andprivate sector can borrow and lend their assets in the foreign capital market. Countries
holding large levels of debt such as Greece, Italy, Ireland, and Portugal hold large external
debt as well. This shows the signiﬁcance of studying the debt policy rule described here
in a model of an open economy.
For our purpose, we consider an endogenously growing small open economy where thegovernment adopts a debt-reduction rule. As in Futagami et al. (1993) and Turnovsky(1997), public capital accumulated through public investment has positive eﬀects on private goods production. The government ﬁnances its spending on public investment by
imposing a tax on income and by issuing bonds. Public bond-issuance is under the restriction of the same debt policy rule as that in Futagami et al. (2008). We consider two
types of public ﬁnance budget regimes. In budgetary regime (I), if the debt–GDP ratio
exceeds the criterion level, the government adjusts its expenditure with a ﬁxed tax rate
to reduce this ratio. In budgetary regime (II), if the debt–GDP ratio exceeds the criterion
level, the government controls the tax rate to reduce its debt with a ﬁxed expenditure
ratio. In both regimes, the debt–GDP ratio tends to the criterion level in the long run.
The criterion level can be considered as the long-run debt–GDP ratio.
In budgetary regime (I), there exists a unique steady-state equilibrium. The long-run
debt–GDP ratio is a crucial determinant of the steady-state stability and equilibrium(in)determinacy. When the long-run debt–GDP ratio is suﬃciently low, the steady stateis saddle stable and hence exhibits equilibrium determinacy. However, if the government
sets a high criterion debt–GDP ratio, equilibrium indeterminacy arises because the steadystate is a sink or there exists a limit cycle around the steady state.3 The data in countries
in the Euro area show that the 60% criterion level of the debt–GDP ratio proposed by
the SGP may not be suﬃciently low to ensure equilibrium determinacy.
3Some authors study the relation between ﬁscal policy and indeterminacy. Focusing on balanced
budget rules, Schmitt-Grohé and Uribe (1997) and Guo and Harrison (2004, 2008) discuss the eﬀects ofﬁscal policies on equilibrium indeterminacy in closed economies. Farmer (1986) and Greiner (2007) showthat in closed economies, limit cycles emerge and equilibrium indeterminacy arises when the government
controls the value of its deﬁcit rather than the value of its debt.

2

We also examine the welfare eﬀects of debt reduction under budgetary regime (I),assuming that the long-run debt–GDP ratio is suﬃciently low to ensure determinacy.
The welfare eﬀects of debt reduction depend on the rates of returns from private savingsand public investment. In our small open economy, the rate of return from private savings
is equal to the world interest rate. When the world interest rate is higher (lower) than therate of return from public investment, reductions in government debt improve (deteriorate)welfare. Furthermore, the pace of debt reduction is an important determinant of themagnitude of welfare gains (losses). When the world interest rate is higher than the rateof return from public savings, the government can further improve welfare by reducing
the debt at a faster pace. In contrast, if the world interest rate is relatively low, thegovernment can mitigate welfare losses by reducing the debt at a slower pace.
In budgetary regime (II), there exists a unique steady-state equilibrium. However,the steady state is always unstable under budgetary regime (II). Because the economy
cannot reach the unstable steady state, it makes little sense to study the characteristics of the unstable steady state, and hence, we do not examine the welfare eﬀects ofdebt reduction in regime (II). Nevertheless, our model provides the following important
implication. Suppose that the initial private domestic savings cannot aﬀord to absorb
the initial outstanding government debt and the government then borrows from foreign
investors. In such a situation, under regime (II), households eventually become overextended with foreign debt, and there exists no equilibrium such that the government canfollow the debt-reduction rule. Then, regime (II) is unsuitable for sustaining public debt.
There exist studies on public debt ﬁnance in endogenous growth models where government services or public capital are inputs for private goods production (see, e.g., Bruce
and Turnovsky (1999), Greiner and Semmler (2000), Ghosh and Mourmouras (2004),
Greiner (2007, 2012), and Yakita (2008)). These studies explore the policy implicationsof public debt ﬁnance for equilibrium dynamics, long-run growth, and welfare. However,few studies investigate the debt-reduction rule found in the Maastricht Treaty and the
SGP, except Futagami et al. (2008), Minea and Villieu (2013), and Maebayashi et al.
3(2013). The present study diﬀers from these studies on the debt-reduction rule in thefollowing three points.
First, although these studies focus only on closed economies, we consider an open
economy and show that the openness of the economy provides important implications
for equilibrium (in)determinacy. Assuming that the public debt-to-private-capital ratio
is constrained by the debt-reduction rule and focusing only on regime (I), Futagami etal. (2008) show that the debt-reduction rule may be a source of indeterminacy. However,
Minea and Villieu (2013) indicate that this result crucially depends on how to construct
the dynamic system, showing that indeterminacy never arises if the debt–GDP ratio isused as the policy target as in the present model. Maebayashi et al. (2013) show thatindeterminacy does not arise in a closed economy version of our model. In contrast,
we show that the debt-reduction rule found in the SGP may be a source of equilibrium
indeterminacy in a small open economy.
Second, we provide sharp insights on the welfare eﬀects of debt-reduction rules. Although Futagami et al. (2008) and Minea and Villieu (2013) compare multiple balanced
growth paths in terms of growth rate, they do not conduct welfare analysis explicitly. Maebayashi et al. (2013) conduct welfare analysis numerically in a closed economy model,
but they do not show the exact conditions under which debt reduction improves welfare. We derive the analytical expression of the welfare eﬀects and provide an intuitive
interpretation of them.
Finally, we shed light on how diﬀerent budgetary regimes generate diﬀerent macroeconomic consequences in a small open economy. In contrast to our study, Futagami etal. (2008) and Minea and Villieu (2013) focus only on regime (I). In a closed version of
our model, Maebayashi et al. (2013) show that under both regimes (I) and (II), a saddle stable steady state exists. In our small open model, the stability of the steady state
under regime (I) crucially depends on the long-run debt–GDP ratio, whereas the steadystate is always unstable under regime (II), which implies that regime (II) is unsuitable
for sustaining public debt.
4

Furthermore, our study is related to the literature on the relationship between ﬁscal
policy and sustainability of economies. It can be compared to the existing results onﬁscal sustainability. Assuming that the government adjusts the income tax rate in closed
economy models, Bräuninger (2005) and Yakita (2008) show that when the initial debt is
too large, the debt–GDP ratio grows unboundedly; hence, public debt is not sustainable.4
These papers explore the conditions under which public debt can be sustained. In contrast,
we investigate which budgetary regime the government should adopt to sustain its debt.
Under regime (I), the government can sustain its debt on the condition that the criterion
level of the debt–GDP ratio is suﬃciently low. However, if the government adopts regime(II) when it borrows from foreign investors, households eventually become overextended
with foreign debt, and the economy loses the ability to pay back its debt.
The remainder of this paper is organized as follows. Section 2 presents the model.
Section 3 investigates the local stability and social welfare under budgetary regime (I).
Section 4 examines budgetary regime (II). Section 5 concludes.

2

Model

We consider a small open economy. A single ﬁnal good is produced using labor, private
capital, and public capital (infrastructure). It is assumed that the ﬁnal good and private
capital are freely traded beyond the country’s borders. However, individuals cannot migrate, and public capital (social infrastructure) cannot cross borders. To construct social
infrastructure, the government must make public investments. 5

4

Diamond (1965) and Chalk (2000) show that permanent budget deﬁcits cannot be sustainable unlessthe interest rate is less than the growth rate in closed economies.
5
The benchmark model assumes that the agents are faced with the constant world interest rate.
Following Chatterjee et al. (2003), Appendix G extends the benchmark model assuming that the interest
rate varies depending on the ﬁscal condition of the economy. This extension does not aﬀect our main
results.

5

2.1

Households

We consider a representative household. The size of the population is normalized to one.
Let Ct be consumption at time t. The utility of the representative household is given by
∫ +∞
U0 =

[ln Ct ]e−ρt dt,(1)0

where ρ > 0 is the subjective discount rate. The household inelastically supplies one
unit of labor at each moment of time. The government imposes a tax on the household’s
income, It ≡ rAt +wt , where r is the (constant and exogenously given) world interest rate,
wt is the wage rate at time t, and At is the asset holdings of the household at time t. In
this small open economy, there are three types of assets, private capital (Kt ), government
bonds (Bt ), and foreign assets (F At ). Hence, we have At = Kt + Bt + F At . If F At < 0,some fractions of private capital or government bonds are owned by foreign agents. We
assume that the tax takes the residence base form. Then, residents’ income is taxed ata uniform rate regardless of its source country, while non-residents’ income is not taxed.
Thus, the ﬂow budget constraint of the representative household isȦt = (1 − τt )It − Ct ,(2)

where τt denotes the income tax rate at time t. In contrast to Futagami et al. (2008) who
assume a constant tax rate, we allow τt to vary over time, as we discuss later. In an open
economy, the household can borrow from foreign countries, and hence, At can be negative.
∫T

However, At must satisfy the no-Ponzi game (NPG) condition, limT →∞ AT e− t (1−τv )rdv ≥0. The household maximizes (1) subject to (2), which yields
Ċt = {(1 − τt )r − ρ}Ct ,∫T

and the transversality condition (TVC), limT →∞ AT e− t (1−τv )rdv = 0.

6

(3)

2.2

Firms

As in Futagami et al. (1993), the production function of the representative ﬁrm is given
by Yt = F (Kt , Kg,t Lt ), where Yt , Kt , Kg,t , and Lt are output, private capital, stockof infrastructure, and labor input at time t, respectively.6 The presence of Kg,t in the
production function reﬂects the external eﬀects. Infrastructure stock, Kg,t , accumulates
through government investments. The production function satisﬁes the standard neoclassical characteristics, especially the constant returns to scale with respect to Kt and Kg,t Lt .
Accordingly, we can transform this into the following intensive form:

Yt = F (xt , 1)Kg,t Lt = f (xt )Kg,t Lt ,(4)

where xt ≡ Kt /(Kg,t Lt ) and f (xt ) ≡ F (xt , 1). Given perfect competition and proﬁt
maximization, we obtainr = f ′ (xt ),(5a)

wt = {f (xt ) − f ′ (xt )xt }Kg,t ≡ ωt Kg,t .

(5b)

The world interest rate, r, is constant because of the assumption of a small country. Thus,
xt and ωt become constant over time. The following discussion omits time index t fromxt and ωt . In equilibrium, we have Lt = 1. Then, Yt grows at the same rate as Kg,t (see(4)). For later use, we deﬁne Kg,t /Yt = 1/f (x) ≡ kg .

2.3

Government

To construct infrastructure, the government makes public investments. The amount ofpublic investments in time t is denoted by Gt .7 Then, stock of infrastructure accumulates
according to K̇g,t = Gt . For later use, we deﬁne gt ≡ Gt /Yt . The government ﬁnances its
6

In contrast to us, Futagami et al. (2008) assume that the ﬂow of public service enters the production
function as an input.
7
As in Futagami et al. (1993), we ignore the depreciation of public capital to simplify the analysis.

7expenditure in two ways. One is by levying an income tax, and the other is by issuingbonds. Thus, the government’s budget constraint is

rBt + Gt = Ḃt + τt (rAt + wt ),(6)

where Bt stands for outstanding government debt, and Ḃt denotes newly-issued government bonds. Using At = Kt + Bt + F At , we rewrite (6) as Ḃt = (1 − τt )rBt + Gt −
τt {r(Kt + F At ) + wt }. If the government debt increases at a rate higher than (1 − τt )r,no agents are willing to hold government bonds. Therefore, the government must satisfy∫T

the NPG condition, limT →∞ BT e− t (1−τv )rdv ≤ 0.
Similar to the reform of the SGP in 2011 for EU countries, we assume that the government must reduce its debt–GDP ratio at a steady pace if its level is beyond the criterion
level. To simply formulate this rule, we follow Futagami et al. (2008) and assume thatthe government adjusts its debt–GDP ratio, bt ≡ Bt /Yt , according to the following rule:b˙t = −ϕ(bt − b̄), ϕ > 0,(7)where b̄ > 0 and ϕ > 0 represent the criterion level of the government’s debt–GDP ratio
and the adjustment coeﬃcient of the rule, respectively. If bt > b̄ holds, the government
reduces its debt by 100ϕ percent of the diﬀerence between the current and target levelsof b. Then, if ϕ = 0.05 and b̄ = 0.6, the debt policy rule, (7), is well suited to the debt
reduction benchmark introduced by the SGP (i.e., member states whose current debt–
GDP ratio is above 60 percent must reduce their debt–GDP ratio distance to 60 percent
by an average rate of one twentieth per year). We assume b0 ≥ b̄, because the averagedebt–GDP ratio in the Euro area in 2011 reached 88 percent of GDP (which is higher
than the criterion level of 60 percent). If ϕ takes a large (small) value, the government
adjusts bt to the criterion level at a fast (slow) pace.
Given this adjustment rule, the government chooses either τt or gt to satisfy the budget8

constraint, (6).8 The present study considers two types of budgetary regimes: (I) the
government sets a constant τ and adjusts gt to satisfy (6) and (7), or (II) the government
chooses a constant g, and τt is then endogenously determined to satisfy (6) and (7).

3Adjustments in Public Investments: Regime (I)We ﬁrst consider the economy under budgetary regime (I), where the government sets a
constant τ ∈ (0, 1) and adjusts public investments to satisfy (6) and (7). We begin withthe derivation of the dynamic system under regime (I). We deﬁne ct ≡ Ct /Yt , at ≡ At /Yt ,and γt ≡ Ẏt /Yt . Here, Yt is given by (4), x remains constant, Kg,t is a state variable, and
Lt = 1 holds in equilibrium. We then have to treat Yt as a predetermined variable. Then,at and bt are state variables and ct is a jump variable. From (4) and K̇g,t = Gt , we have
gt = kg γt . From (2), (3), (5b), (6), and the deﬁnitions of It and γt , we obtainḃt = (r − γt )bt + gt − τt (rat + ωkg ),(8a)ȧt = {(1 − τt )r − γt }at − ct + (1 − τt )ωkg ,(8b)ċt = {(1 − τt )r − ρ − γt }ct .

(8c)Eliminating ḃt from (7) and (8a) and solving for γt using gt = kg γt , we obtain9()
τ (rat + ωkg ) − rbt − ϕ(bt − b̄)gt=≡ γ(at , bt ).
γt =kgkg − bt(9)This equation shows that if the government reduces b̄, γt decreases (increases) in the
short run if bt < (>)kg . The government can reduce its debt–GDP ratio in two ways.
One is reducing its debt, Bt , and the other is enhancing the output growth through public
investments. When bt is suﬃciently small to satisfy bt < kg , the government can reduce
Dividing both sides of (6) by Yt , we obtain ḃt = (r − γt )bt + gt − τt (rat + ωkg ). The ﬂow budget
constraint of the government is represented by τt and gt . This is Equation (8a) we derive later.
9
Because gt = kg γt , the choice of gt is equivalent to that of γt . When we discuss regime (I) in thefollowing, we mainly focus on γt rather than gt .
89

bt easier by reducing Bt rather than by enhancing output growth. In contrast, whenbt > kg holds, enhancing output growth is the easier way to reduce bt . Hence, when thegovernment reduces b̄, γt decreases (increases) in the short run if bt < (>)kg . Note that (i)the eﬀects of reductions in b̄ on γt (= gt /kg ) become stronger as ϕ becomes larger, because
the government must reduce bt at a faster pace and (ii) eﬀects on γt (= gt /kg ) graduallydisappear as bt becomes close to b̄.
Substituting (9) into (8b) and (8c) yields
ȧt = {(1 − τ )r − γ(at , bt )}at − ct + (1 − τ )ωkg ,(10a)ċt = {(1 − τ )r − ρ − γ(at , bt )}ct .

(10b)

The dynamic system is then given by (7), (10a), and (10b) together with the initial values,b0 and a0 .

3.1

Steady State and Stability

We derive the steady-state equilibrium where ḃt = ȧt = ċt = 0 holds. We set ḃt =ȧt = ċt = 0 in (7), (10a), and (10b), and we solve for bt , at , ct , and γt using (9). The
steady-state values of bt , at , ct , and γt are, respectively, given by b∗I = b̄(> 0),(γI∗ − τ ω)kg + (ρ + τ r)b̄,τrγ ∗ (ρ + τ ω)kg + ρ(ρ + τ r)b̄c∗I = I
> 0,
τr

a∗I =

γI∗ = (1 − τ )r − ρ.

(11a)
(11b)
(11c)

To ensure positive growth, we assume that γI∗ = (1 − τ )r − ρ > 0. From (11a), householdincome at the steady state is given by (ra∗I + ωkg )Yt =(τ r+ρ)b̄+γI∗ kg
Yt that is apparentlyτ

positive. Because τ ∈ (0, 1) and ρ > 0, we have γI∗ < (1 − τ )r; this ensures TVC, becauseȦt /At = γI∗ holds at the steady state. The NPG condition of the government is alsosatisﬁed because Ḃt /Bt = γI∗ < (1 − τ )r in the steady state. The policy parameters, b̄ and10

ϕ, have no eﬀects on the long-run growth rate, γI∗ . We obtain the following proposition.
Proposition 1 Consider budgetary regime (I). Suppose that γI∗ = (1−τ )r −ρ > 0. Thereexists a unique steady-state equilibrium where b∗I = b̄ and (11a)–(11c) hold. b̄ and ϕ haveno eﬀects on the long-run growth rate, γI∗ .
We next examine the stability of the steady state characterized by b∗I = b̄ and (11a)–(11c). We linearize the dynamic system around the steady state characterized by b∗I = b̄and (11a)–(11c), and then we obtain










ḃ
−ϕ
0
0
b − b̄
 t   t∗  ∂γ(a∗I ,b̄) ∗∗ I ,b̄) ∗ ȧt  =  − ∂b aI ρ − ∂γ(a


,
a
−1
a
−
a
I
I 
∂at
t

 
 t
∂γ(a∗ ,b̄)∂γ(a∗ ,b̄)ċt
− ∂bIt c∗I− ∂aIt c∗I0ct − c∗I

(12)

where
∂γ(at , bt )∂γ(a∗I , b̄)
ϕ + ρ + τr
≡
,
=−
∂bt
∂bt
kg − b̄
(at ,bt )=(a∗ ,b̄)I∂γ(a∗I , b̄)∂at≡∂γ(at , bt )
τr
=
.
∂at
kg − b̄
(at ,bt )=(a∗ ,b̄)

(13)

I

We denote the Jacobian matrix of (12) by M . Let us denote the (i, j) element of M bymij (i, j = 1, 2, 3). One of the eigenvalues of M is equal to −ϕ < 0. The other two, λ1
and λ2 , are the solutions of the characteristic equation Ω(z) ≡ z 2 − m22 z + m32 = 0, thus
satisfying
λ1 + λ2 = m22 = ρ −∂γ(a∗I , b̄) ∗aI∂atand λ1 · λ2 = m32 = −∂γ(a∗I , b̄) ∗
cI .
∂at

(14)

The signs of m22 and m32 indicate the stability of the steady-state equilibrium and canprove the next proposition.
Proposition 2 Consider budgetary regime (I). Suppose that γI∗ = (1 − τ )r − ρ > 0.
1. If b̄ < kg , the steady state is locally saddle stable and exhibits local determinacy.
11

2. If b̄ > kg , the following hold:ω−(1−τ )r(a) Let b̃ ≡ 2ρ+τ2ρ+τ
kg . When τ > r/ω,r

i. if kg < b̄ < b̃, the steady state is locally stable and exhibits local indeterminacy.
ii. if b̄ > b̃, the steady state is locally unstable.10(b) When τ < r/ω, the steady state is locally unstable.
(Proof) See Appendix A.
Proposition 2 shows that when b̄ is large (b̄ > kg ), the steady state becomes unstable under some conditions. However, this does not mean that there is no equilibrium.
Proposition 3 shows the possibility of an equilibrium that exhibits a limit cycle.
Proposition 3 Consider budgetary regime (I). Suppose that γI∗ = (1 − τ )r − ρ > 0 andb̄ > kg . In addition, suppose that r/ω < 1 and b̂− < b̃ < b̂+ and that b̄ is suﬃciently close
to b̃, where b̂− and b̂+ are as deﬁned in Appendix B. Then, there exists at least a limit
cycle around the steady state, and the steady state exhibits equilibrium indeterminacy.
(Proof) See Appendix B.
The properties of the equilibrium path heavily depend on b̄. When b̄ is suﬃciently small
to satisfy b̄ < kg , the equilibrium path is uniquely determined. However, when b̄ takes amoderate value (kg < b̄ < b̃) and τ > r/ω holds, the steady state is locally stable andexhibits indeterminacy. In other cases, the steady state is unstable. However, as provenin Proposition 3, there exists a limit cycle around the steady state. In Figures 1 and 2, we
illustrate a numerical example of a stable limit cycle.11 In this example, indeterminacyarises because there are multiple equilibrium paths that converge to a stable limit cycle.
10Note that b̃ is larger than kg if τ > r/ω holds. See Appendix A.
In this example, we assume the Cobb–Douglas production function: Yt = Kt α (Kg,t Lt )1−α . We use
the following parameters: α = 0.36, r = 0.03, ρ = 0.01, ϕ = 0.05, and τ = 0.3. Under these parameters,
we have ω ≈ 2.59, kg ≈ 0.25, b̂− ≈ 2.01, b̂+ ≈ 238.61, and b̃ ≈ 6.612. We then assume that b0 ≈ 6.613.
These values satisfy 1 > τ > r/ω, b̄ > kg , and b̂− < b̃ < b̂+ .
11

12

7.4
7.3
7.2

b

t

7.1
7
6.9
6.8
6.7
6.6
0.6
0.55
−2

0.5
0.45

−4
−6

0.4
0.35
c

t

−8
0.3
0.25

−10
−12

at

Figure 1: An Example of Limit Cycle

The intuition behind indeterminacy is as follows: Suppose that households expect
future increases in Gt . Because increases in Gt have positive eﬀects on output growthand labor wage (see (4) and (5b)), this expectation implies that households also expect
future increases in output growth and labor income. Households then have a lesser incentive to save, and at thus decreases in the future. As a result, the government’s tax
revenue decreases, which tightens its budget constraint. As discussed just below (9), thegovernment—which is constrained with tight budgets—has two ways to reduce bt ≡ Bt /Ytaccording to (7): one is to reduce the debt, Bt , which requires reductions in public investments, Gt , and the other is to increase public investments, Gt , in order to stimulate
output growth. When bt > kg holds, enhancing output growth is an easier way to reduce
bt than paying back Bt . Then, if b̄ > kg , households’ expectations are self-fulﬁlling andequilibrium indeterminacy arises. In the numerical example of the limit cycle in Figure
2, government spending and private asset holdings move in opposite directions, which is
13

−3

x 10
5

g

t

4
3
2
1
0

200

400

600
time

800

1000

1200

0

200

400

600
time

800

1000

1200

0

200

400

600
time

800

1000

1200

−2

−6

a

t

−4

−8
−10
−12

0.7

0.5

c

t

0.6

0.4
0.3
0.2

Figure 2: Fluctuations of (gt , at , ct )consistent with this interpretation for indeterminacy. When bt < kg , the government can
reduce bt easier by paying back Bt rather than by enhancing output growth. Then, ifb̄ < kg , expectations of future increases in Gt are not self-fulﬁlling. Hence, the steadystate exhibits equilibrium determinacy.
Proposition 2 also shows that the equilibrium determinacy condition is independent of
ϕ. Intuitively, equilibrium determinacy depends only on the directions of the simultaneous
movement of (bt , at , ct ) as explained above. Because adjustment speed ϕ does not change
the directions, ϕ does not aﬀect equilibrium determinacy.
Maebayashi et al. (2013) show that in a closed economy where the government follows(7) and adopts regime (I), equilibrium indeterminacy does not arise. Consider a closed
economy where public capital is productive. As in our open economy, increases in public
investments have positive growth eﬀects. Concurrently, increases in public investments

14

year
Italy
1990
49.0
1991
49.7
1992
50.6
1993
52.1
1994
51.6
1995
50.4
1996
50.1
1997
49.4
1998
48.9
1999
48.6
2000
47.9
2001
47.5
average 49.7

Netherlands
68.9
67.8
66.8
66.9
65.2
64.3
62.9
61.2
59.2
57.6
56.4
56.7
62.8

Norway Portugal Spain
52.5
32.0
40.9
52.4
32.2
43.1
52.5
33.4
45.9
52.7
36.4
48.8
51.3
38.5
49.9
50.6
38.9
50.7
49.4
39.5
51.2
48.4
40.5
50.3
48.9
41.4
49.3
50.1
42.3
48.7
50.5
43.3
48.0
50.9
44.4
47.8
50.9
38.6
47.9

Sweden Switzerland UK
40.2
48.4
48.5
41.5
50.2
49.6
43.1
51.7
49.8
45.0
53.4
49.1
44.2
54.4
47.5
44.1
55.5
46.8
44.9
56.4
46.0
45.0
56.3
44.4
44.0
55.7
42.9
42.8
55.6
41.8
42.0
54.7
40.3
42.0
54.7
39.3
43.2
53.9
45.5

Table 1: Real government net capital stocks as a percentage of real GDP (data source:
Database on Capital Stocks in OECD Countries of Kiel Institute for the World Economy)

crowd out private investments, which counteracts the positive growth eﬀects. Then, thegrowth eﬀects of increases in public investments become weaker compared to those in
our small open economy. The government—constrained with tight budgets—has only
one choice: to cut public investments to pay back its debt. Households’ expectations ofincreases in Gt are not self-fulﬁlling, and thus indeterminacy does not arise in a closed
economy. In contrast, because an open economy can import capital from abroad andbecause economic activities in a small economy do not aﬀect the world resource constraint,
increases in public investments do not necessarily crowd out private investments in a small
open economy and can thus have stronger growth eﬀects. As such, in our small open
economy, equilibrium indeterminacy can arise under some conditions.
Finally, we refer to an empirical implication of the determinacy condition: b̄ < kg .
Table 1 shows the real government net capital stocks as a percentage of real GDP
from 1990 to 2001 in several countries in the Euro area.12 On average, except in the12Constructing new data is not the scope of this paper. Therefore, we use existing data provided
by the Database on Capital Stocks in OECD Countries of Kiel Institute for the World Economy. It
is downloadable at http://www.ifw-kiel.de/forschung/Daten/netcap. Because only data up to 2001 areavailable for our purpose, we focus on data from 1990 to 2001. For details on data construction, see
Kamps (2006).

15

Netherlands, the 60% criterion level of the debt–GDP ratio proposed by the SGP may
not be suﬃciently low to ensure equilibrium determinacy. Besides, this result holds stably
in that term.

3.2

Welfare Analysis

To examine the welfare eﬀects of reductions in b̄ under budgetary regime (I), we consider
the following scenario. The economy is initially in the steady-state equilibrium characterized by b∗I = b̄ and (11a)–(11c). At time 0, the government unexpectedly reduces b̄
marginally. After marginal reductions in b̄, if the new steady state is stable, the economy
begins to move toward the new steady state. If the steady state exhibits local indeterminacy, the transitional path is not unique and welfare eﬀects of this policy change cannot
be examined. Hence, here, we assume b̄ < kg to ensure determinacy. However, we analyze
the case of indeterminacy in an extended model where the interest rate depends on theﬁscal conditions of the economy. See Appendix G.
Our welfare measure is (1). Because Ct = C0 e{(1−τ )r−ρ}t = c0 f (x)Kg,0 e{(1−τ )r−ρ}t , we
can rewrite (1) as
1
1
U0 = ln c0 +ρρ{}
(1 − τ )r − ρ
ln f (x)Kg,0 +
.
ρ

(15)

When the government unexpectedly reduces b̄ at time 0, c0 jumps to its new value justat time 0; however, other variables in (15) do not change. Then, marginal reductions inb̄ aﬀect household welfare only through c0 . Because b̄ < kg holds and the steady stateis saddle-stable, we can consider λ1 < 0 and λ2 > 0 as solutions of the characteristic
equation Ω(z) = 0, as shown in Subsection 3.1. Appendix C shows that using λ2 > 0, c0
can be expressed as
c0 = c∗I + (b0 − b̄)v3 + {a0 − a∗I − (b0 − b̄)v2 }λ2 ,

16

(16)

where v2 and v3 are deﬁned in (C.3a) and (C.3b).
We now explain our scenario in more detail. Initially, b̄ is set equal to b̄init . We denotethe initial steady-state value of at as ainit
I . Because the economy is initially in the steadystate equilibrium, we have b0 = b̄init and a0 = ainitin (16). Denote the criterion level ofI
bt and the steady-state value of at after the policy change as b̄new and anew
, respectively.
I
Because we consider marginal changes, b̄new and anew
are approximately equal to b̄init andIinit≈ b̄new and a0 = ainit≈ anew. After the policy
ainit
I , respectively. This generates b0 = b̄IIchange, we have b̄ = b̄new and a∗I = anewin (16), which implies that b0 − b̄ = b̄init − b̄new ≈ 0Iand a0 − a∗I = ainit− anew≈ 0. Then, the eﬀect of marginal changes in b̄ can be expressed
I
I
as
∂c0
∂c∗ ∂a∗= I − I λ 2 + v2 λ 2 − v3 ,∂ b̄ (b0 ,a0 )≈(b̄new ,anew )
∂ b̄
∂ b̄
I
{
}
λ2 − ρ λ2 (ϕ + ρ + τ r)=
− (ρ + τ r) ≡ Ψ(ϕ).
τr
ϕ + λ2(17)

The calculation procedure from the ﬁrst to the second lines in (17) is presented in Appendix D. We can easily show that λ2 − ρ > 0 and Ψ(0) = 0.13 Diﬀerentiating Ψ(ϕ), we
obtainΨ′ (ϕ) =λ2 (λ2 − ρ)
{λ2 − (ρ + τ r)}.
τ r(ϕ + λ2 )We can show that if r < (>)ω holds, we have ρ + τ r < (>)λ2 .14 Then, if r < (>)ω,Ψ(ϕ) > (<)0 and Ψ′ (ϕ) > (<)0 for ϕ > 0. From (15), we obtain ∂U0 /∂ b̄ (b0 ,a0 )≈(b̄new ,anew ) =I

Ψ(ϕ)/ρc∗I . Because ϕ has no eﬀect on c∗I , the discussion so far yields the following proposition.
Proposition 4 Consider budgetary regime (I). Suppose that γI∗ = (1 − τ )r − ρ > 0 andb̄ < kg and that the economy is initially in the steady-state equilibrium.
13

λ2 is a solution of Ω(z) = 0. 0 < z < λ2 (z > λ2 ) if Ω(z) < (>)0 holds for z > 0. BecauseΩ(ρ) = ρ2 − m22 ρ + m32 = −(1 − τ )ωkg τ r/(kg − b̄) < 0, we have λ2 − ρ > 0.
14
As in footnote 13, 0 < z < λ2 (z > λ2 ) holds if Ω(z) < (>)0 holds for z > 0. We have Ω(ρ + τ r) =τ rkg (r − ω)/(kg − b̄). Because b̄ < kg is assumed, ρ + τ r < (>)λ2 if r < (>)ω.

17

(a) If r < ω holds, marginal reductions in b̄ reduce households’ welfare. As ϕ > 0 becomes
smaller, reductions in household welfare also become smaller.
(b) If r > ω holds, marginal reductions in b̄ raise households’ welfare. As ϕ > 0 increases,
increases in household welfare also become larger.
The welfare eﬀects of reductions in b̄ depend on the relationship between r and ω.
When r > ω, reductions in b̄ improve welfare. Furthermore, the value of ϕ provides
an important implication. When r > ω, the government can further improve welfare by
reducing bt at a faster pace. In contrast, when r < ω, the government can mitigate welfare
losses by reducing bt at a slower pace.
To interpret Proposition 4 intuitively, we rewrite households’ income, It = rAt + wt , as
It = rAt +ωKg,t by using (5b). Households’ income depends on their assets, At , and public
capital, Kg,t . At accumulates through savings of the private sector (households), whereas
Kg,t accumulates through public investments by the government. We can consider public
investments as savings of the public sector. Consequently, r and ω can be considered therates of return on private and public savings, respectively. If r < ω holds, accumulating
assets through public savings is socially more eﬃcient. Because reductions in b̄ havenegative eﬀects on public investments (savings) when b̄ < kg holds as discussed below (9),reductions in b̄ reduce households’ welfare. When ϕ is small, the initial decline in γt isalso small, which mitigates the negative welfare eﬀects. When r > ω holds, accumulating
assets through public savings is not socially eﬃcient. Because reductions in b̄ depress
public investments (savings), welfare improves. When ϕ is large, the initial decline in
γt is also large. Then, public investments are further depressed and households’ welfare
further improves.

4

Tax Adjustments: Regime(II)We now move onto budgetary regime (II), where the government sets a constant γ(= g/kg )and adjusts τt to satisfy (7) and (8a). Under regime (II), it will be shown that there exists
18

a unique unstable steady state. This is a new result by considering a small open economy
because Maebayashi et al. (2013) have shown that the steady state is stable under regime(II) in the closed economy.
We ﬁrst derive the dynamic system. Eliminating ḃt from (7) and (8a) and solving forτt , we obtainτt =(r − γ)bt + kg γ + ϕ(bt − b̄)≡ τ (at , bt ).
rat + ωkg

(18)

When the government reduces b̄, τt increases in the short run because the government
must earn a larger primary surplus to reduce bt . The eﬀect on τt becomes stronger as ϕincreases and gradually disappears as bt approaches b̄. The following two points shouldbe mentioned. First, when at = −ωkg /r, τt cannot be deﬁned; thus, the optimization
problem of the household is not well-deﬁned. Then, the transitional paths along which atmoves across −ωkg /r must not be equilibrium. Second, when at is so small that householdincome (rat + ωkg )Yt is negative, τt tends to be negative; thus, household tax paymentsτt (rat + ωkg )Yt tend to be positive despite that household income is negative. Then,the representative household must borrow from abroad to make tax payments beyond its
income. In other words, the government forces the household to borrow from abroad inorder to meet the budget constraint of the government. Although the case makes little
sense in practice, it is theoretically possible and provides important implications as welater show. Substituting (18) into (8b) and (8c) yields
ȧt = (r − γ)(at − bt ) − ϕ(bt − b̄) − ct + (ω − γ)kg ,(19a)ċt = {(1 − τ (at , bt ))r − ρ − γ}ct .

(19b)

The dynamic system is given by (7), (19a), and (19b) along with b0 and a0 .
We next derive the steady-state equilibrium, where ḃt = ȧt = ċt = 0 holds. We setḃt = ȧt = ċt = 0 in (7), (18), (19a), and (19b), and we solve for bt , at , ct , and τt . The

19

steady-state values of bt , at , ct , and τt are respectively given by b∗II = b̄(> 0),(r − γ)b̄ + γkg ωkg
−
,
r−ρ−γ
r
(r − γ)b̄ + γkgγωkgc∗II =
ρ+,r−ρ−γ
rr − (ρ + γ)
τII∗ =.
ra∗II =

(20a)
(20b)
(20c)

b̄+γkgFrom (20a), households’ income is given by (ra∗II + ωkg )Yt = (r−γ)
rYt at the steadyr−ρ−γ

state. If γ < r − ρ holds, the steady-state income level becomes strictly positive. Thisinequality ensures that τII∗ ∈ (0, 1) and c∗II > 0. Accordingly, the remainder of this section
assumes that γ < r − ρ. Because Ȧt /At = Ḃt /Bt = γ holds in the steady state and (20c)∗
implies (1 − τII)r − γ = ρ > 0, the TVC of the household and the NPG condition of thegovernment are both satisﬁed.
As shown in Appendix E, the Jacobian matrix of the linearized dynamic system has
only one stable root, −ϕ. Because there are two state variables, bt and at , the economy
cannot approach the steady-state equilibrium unless b0 and a0 are respectively equal to b̄
and a∗II by chance. We thus obtain the next proposition.
Proposition 5 Consider budgetary regime (II). Suppose that γ < r − ρ. There exists a
unique steady-state equilibrium where b∗II = b̄ and (20a)–(20c) hold. Unless b0 and a0 are
equal to b̄ and a∗II , respectively, the economy cannot approach the steady-state equilibrium.
The intuition of Proposition 5 is as follows. We focus on the marginal rate of intertemporal substitution in the Euler equation of the household. Under regime (II), the income
tax rate is inversely related to asset holding of the household. When at is lower (higher)
than that of the steady state, the income tax rate is raised (lowered) to ﬁnance the constant government spending. This lowers (raises) the rate of return of savings. Then, thehousehold decreases (increases) asset holdings and at gets away from the steady state.
Therefore, the steady state is unstable in regime (II).
Because the economy cannot reach the unstable steady-state equilibrium, it makes
20

little sense to continue studying the characteristics of the steady state. However, ourmodel provides other important implications. Using (7), we solve (19a) and then rearrange
the solution using inequality ct > 0:15
{

}
ω−γ
ω−γ
at < bt + (a0 − b0 ) +kg e(r−γ)t −
kg .
r−γ
r−γ

(21)

Note that the debt rule, (7), ensures limt→+∞ bt = b̄ and r − γ > 0 is assumed. Supposethat a0 − b0 is suﬃciently small so that a0 − b0 < −(ω − γ)kg /(r − γ). Then, (21) implies
limt→∞ at = −∞. If a0 > −ωkg /r holds, at necessarily moves across −ωkg /r. As discussed
below (18), such transitional paths cannot be an equilibrium, because τt is not deﬁned at
at = −ωkg /r and the households’ optimization problem is not well-deﬁned.
It is also shown that even if a0 < −ωkg /r holds and at does not move across −ωkg /ralong the transition, there exists no equilibrium. As shown in (18), τt tends to zero as at
tends to −∞. Then, for a large t, the discounted sum of households’ labor income can be
written as∫ +∞∫− tv (1−τs )rds

(1 − τv )wv e

∫ +∞
dv =

t

ωkg Yv e−r(v−t) dv =t

ωkg
Yt > 0.
r−γ(22)

In the ﬁrst equality, we use τt = 0 for a large t, (5b), and the deﬁnition of kg . The second
equality uses Ẏt /Yt = γ and r > γ. The assumption γ < r − ρ(< r) ensures the lastinequality. Because at = −∞ for a large t, the inequality −at > ωkg /(r − γ) holds for alarge t. From (22) and the deﬁnition of at , the following inequality is obtained for a large
t:
∫ +∞
−At >

∫v

(1 − τv )wv e− t (1−τs )rds dv.

tThis inequality implies that for a large t, households’ borrowing eventually exceeds thediscounted sum of their labor income, and households would thus be unable to pay oﬀ
15

Appendix G provides the derivation of (21).

21their borrowing. Therefore, rational expectations and perfect foresight ensure that foreign
agents do not lend to such households. The next remark summarizes the discussion sofar.
RemarkConsider budgetary regime (II). Suppose that γ < r − ρ and a0 − b0 < −(ω − γ)kg /(r − γ).
Then, there exists no equilibrium such that the government can follow the debt-reduction
rule, (7).
Inequality a0 − b0 < −(ω − γ)kg /(r − γ) indicates that households’ initial asset holdings tend to be smaller than the government’s initial outstanding debt, which means thatprivate domestic savings cannot absorb the outstanding government debt; thus, the government borrows from foreign investors. Our result implies that in such a case, underregime (II), households eventually become overextended with foreign debt. Therefore,
there exists no equilibrium where the government can follow the debt-reduction rule, (7).
Maebayashi et al. (2013) show that in a closed economy where the government follows(7) and adopts budgetary regime (II), the unique steady state is always saddle stable andthe economy converges to the steady state where the debt–GDP ratio remains constantover time. Then, there always exists an equilibrium where the government can follow thedebt-reduction rule, (7), if it is a closed economy. To reduce its debt–GDP ratio, thegovernment imposes a high tax rate on the interest income under regime (II) (see (18)),
which discourages household savings and negatively aﬀects at . However, households ina closed economy cannot borrow from abroad, and hence, at cannot be smaller than bt .
Therefore, households in a closed economy do not lose the ability to pay back debt. In
contrast, because households in an open economy can borrow from abroad and becausehouseholds in a small economy do not aﬀect the world asset market, households in a small
open economy are at risk of becoming overextended with foreign debt (although this does
not happen in equilibrium).

22

5Concluding Remarks

We examine how the debt-reduction rule found in the SGP aﬀects the dynamics of theeconomy and welfare in a small open economy model where the government can borrowfrom abroad to ﬁnance its debt. Public capital accumulated by public investments has
positive externalities on goods production.
In budgetary regime (I), where the government controls public spending to follow adebt policy rule, there exists a unique steady state equilibrium. When the long-run debt–
GDP ratio is suﬃciently low (high), the steady state exhibits (in)determinacy. Focusing
on the case where the steady state exhibits determinacy, we obtain the following welfare
implications. If the rate of return from private savings is larger (smaller) than thatfrom public savings, reductions in government debt improve (reduce) welfare and thegovernment can realize (may suﬀer) larger welfare gains (losses) by reducing its debt at
a faster pace. In budgetary regime (II), where the government controls the income tax
rate to follow a debt policy rule, the steady state is always unstable. If the initial asset
holdings of households are smaller than the initial outstanding government debt, there
exists no equilibrium where the government can follow the debt-reduction rule, (7).

AppendixA

Proof of Proposition 2

1. If b̄ < kg holds, we have m32 < 0 because c∗I > 0 and ∂γ(a∗I , b̄)/∂at = τ r/(kg − b̄) > 0(see (13)). From the second equation of (14), one of the eigenvalues of the Jacobian matrixin (12) is negative while the other is positive. Then, the steady state is locally saddle
stable and exhibits local determinacy.
2. If b̄ > kg holds, m32 > 0 (see (14) and (13)). Using (11a) and (13), we rewrite m22 asm22 ={(1 − τ )r − 2ρ − τ ω}kg + (2ρ + τ r)b̄.
b̄ − kg23(A.1)

ω−(1−τ )r(a) If τ > r/ω holds, (1 − τ )r − 2ρ − τ ω takes a negative value. Let b̃ ≡ 2ρ+τ2ρ+τ
kg .
r
ω−(1−τ )rThen, if kg < b̄ < b̃ holds, m22 < 0. Inequality τ > r/ω ensures that 2ρ+τ2ρ+τ
=
r
τ ω−r
1 + 2ρ+τ> 1 and thus ensures the existence of a b̄ that satisﬁes kg < b̄ < b̃. Fromr

m22 < 0, m32 > 0, and (14), we know that the real parts of both λ1 and λ2 are negative.
The steady state is locally stable and exhibits local indeterminacy.
ω−(1−τ )r
If b̄ > b̃ holds, b̄ > kg because inequality τ > r/ω ensures that 2ρ+τ2ρ+τ
= 1+
r
τ ω−r
2ρ+τ r> 1. Then, m22 > 0 and m32 > 0, which implies that the real parts of the two

eigenvalues are positive.
(b) We now examine the signs of m22 and m32 in the case of τ < r/ω and b̄ > kg . m32 > 0
holds. Because b̄ > kg , we havem22 >{(1 − τ )r − 2ρ − τ ω}kg + (2ρ + τ r)kg
r − τω
=kgb̄ − kgb̄ − kg> 0.

Then, m22 > 0 and m32 > 0, which implies that the real parts of the two eigenvalues are
positive.

B

Proof of Proposition 3

We show the existence of limit cycles by applying the Hopf bifurcation theorem.16 Because
Jacobian matrix M (deﬁned in (12)) has one negative eigenvalue, −ϕ, the dynamic system
can exhibit Hopf bifurcation if its submatrixM′ ≡∂γ(a∗ ,b̄)ρ − ∂aIt a∗I−∂γ(a∗I ,b̄)
∂at

16

c∗I

−1




0

To see the Hopf bifurcation theorem in detail, refer to Theorem 3.4.2 (pp. 151) of Guckenheimer andHolmes (1983).

24

has a couple of complex eigenvalues. A necessary and suﬃcient condition for complex
eigenvalues is (trM ′ )2 − 4detM ′ < 0, that is,17
(
ρ−

τr ∗
a
kg − b̄ I

)2
< −4

τr ∗
c.
kg − b̄ I(B.1)

Substituting (11a) and (11b) into (B.1), we obtain[(γ ∗ − τ ω)kg + (ρ + τ r)b̄ρ− I
kg − b̄]2γ ∗ (ρ + τ ω)kg + ρ(ρ + τ r)b̄.
< −4 I
kg − b̄Multiplying both sides by (kg − b̄)2 and arranging it as a polynomial of b̄, we have[][]2(τ r)2 b̄2 − 2τ r (1 − τ )(ω − r) + ω kg b̄ + (1 − τ )r + τ ω kg2 < 0.

(B.2)

The critical values of quadratic inequality (B.2) are√(1 − τ )(ω − r) + ω − 2 (1 − τ )(ω − r)ω
b̂− ≡
,
τr √
(1 − τ )(ω − r) + ω + 2 (1 − τ )(ω − r)ω
b̂+ ≡
,
τr
where these real numbers are well-deﬁned under the assumption that r/ω < 1. Thus, anecessary and suﬃcient condition for complex eigenvalues is b̂− < b̄ < b̂+ . M ′ has complex′

eigenvalues because b̂− < b̃ < b̂+ . The real part of these eigenvalues, detM
= m222 , equals
2
zero only when b̄ = b̃; it is also diﬀerentiable and increasing in b̄. Thus, b̃ is the uniquebifurcation value of parameter b̄, and submatrix M ′ has a couple of complex eigenvalues
in any suﬃciently small neighborhood of b̃. Therefore, by the Hopf bifurcation theorem,
there exists a suﬃciently small ε > 0 such that either of the following cases is true: (i)there exists a stable limit cycle for every b̄ such that b̃ < b̄ < b̃ + ε, or (ii) there exists an
unstable limit cycle for every b̄ such that b̃ − ε < b̄ < b̃. Because any solution emerging

This inequality requires b̄ > kg , and holds even in case 2.(a).ii of Proposition 2 where τ > ωr andb̄ > b̃, as mentioned previously.
17

25

from the neighborhoods of the cycle winds around it in case (i) and remains in the cycle(asymptotically on the (at , ct ) plane) in case (ii), there are innumerable initial values of ct
consistent with TVC in each case. This implies the multiplicity of equilibria. The proofof Proposition 3 is thus complete.

C

Derivation of c0

We derive the saddle path that converges to the steady state we consider. Because b̄ < kgholds, we can assume λ1 < 0 and λ2 > 0. Because m32 < 0 holds in the case of b̄ < kg√(see Appendix A), m22 2 − 4m32 > |m22 |. Then, λ1 and λ2 are given by

λ1 =

m22 −

√
√
m22 2 − 4m32
m22 + m22 2 − 4m32
< 0 and λ2 =
> 0.
2
2

(C.1)

Given b0 and a0 , we solve (12) and obtain



bt



b∗I






1




0


 






  ∗  −ϕt at  =  aI  + θ1  v2  e + θ2  1  eλ1 t .

 





∗
ct
cI
v3
λ2

(C.2)

In (C.2), θ1 ≡ b0 − b̄ and θ2 ≡ a0 − a∗I − (b0 − b̄)v2 are determined by the initial condition.
Vectors (1 v2 v3 )T and (0 1 λ2 )T are the eigenvectors associated with eigenvalues −ϕ and
λ1 , respectively.18 v2 and v3 are expressed asv2 =m11 m21 − m31∂γ(a∗I , b̄)
ϕa∗I + c∗I∂γ(a∗I , b̄)
ϕa∗I + c∗I==,
m11 (m11 − m22 ) + m32∂bt ϕ2 + m22 ϕ + m32∂bt (ϕ + λ1 )(ϕ + λ2 )(C.3a)v3 =∂γ(a∗I , b̄)(ρ + ϕ)c∗I∂γ(a∗I , b̄)(ρ + ϕ)c∗I(m11 − m22 )m31 + m21 m32==.
m11 (m11 − m22 ) + m32∂bt ϕ2 + m22 ϕ + m32∂bt (ϕ + λ1 )(ϕ + λ2 )(C.3b)

To derive the eigenvector associated with λ1 , we use (14). Let us denote the eigenvector as v̂ T ≡(v̂1 1 v̂3 )T . From M v̂ = λv̂, we obtain v̂1 = 0 and v̂3 = m22 − λ1 = m32 /λ1 . Then, the two equations of(14) imply that v̂3 = λ2 .
1826

In (C.3a) and (C.3b), we use (14), which implies that ϕ2 + m22 ϕ + m32 = (ϕ + λ1 )(ϕ + λ2 ).
From (C.2), c0 is given by (16).

D

Derivation of (17)From (11a) and (11b), we have ∂a∗I /∂ b̄ = (ρ + τ r)/(τ r) and ∂c∗I /∂ b̄ = ρ∂a∗I /∂ b̄. Using thedeﬁnitions of v2 and v3 , we rearrange v2 λ − v3 as
v2 λ2 − v3 =ϕ + ρ + τr{(ρ − λ2 )c∗I + ϕ(c∗I − λ2 a∗I )}.
(kg − b̄)(ϕ + λ1 )(ϕ + λ2 )(D.1)

Because λ2 is a solution of Ω(z) = 0, Ω(λ2 ) = λ2 2 − m22 λ2 + m32 = 0, which implies that(c∗I − λ2 a∗I =∂γ(a∗I , b̄)∂at)−1λ2 (λ2 − ρ),

where ∂γ(a∗I , b̄)/∂at is given by (13). In deriving this equation, we use the deﬁnitions of
m22 and m32 . Substituting this equation into (D.1) yields()∂γ(a∗I , b̄) ∗(λ2 − ρ)(ϕ + ρ + τ r)
ϕλ2 −
cI ,
v2 λ 2 − v3 =τ r(ϕ + λ1 )(ϕ + λ2 )∂at
(λ2 − ρ)(ϕ + ρ + τ r)=
(ϕλ2 + m32 ) ,τ r(ϕ + λ1 )(ϕ + λ2 )(λ2 − ρ)λ2 (ϕ + ρ + τ r)=.
τ r(ϕ + λ2 )On the second line, we use the deﬁnition of m32 . The second equation of (14) is used on thelast line. Substituting the above equation, ∂a∗I /∂ b̄ = (ρ + τ r)/(τ r) and ∂c∗I /∂ b̄ = ρ∂a∗I /∂ b̄
into the ﬁrst line of (17) yields the second line of (17).

27

E

Stability of the Steady State: Regime (II)To examine the stability of the steady-state equilibrium, we linearize the dynamic system
around the steady state characterized by b∗II = b̄ and (20a)–(20c):






ḃt



−ϕ


 

 
 ȧt  =  −(r − γ + ϕ)

 
∗
ċt
− rar−γ+ϕ∗ +ωk rcII
g
II

0
r−γ
r−ρ−γ
rc∗II
ra∗II +ωkg

0

bt − b̄







∗
−1   at − aII  .
∗0
ct − cII

One of the eigenvalues of the Jacobian matrix is given by −ϕ < 0. The other two, µ1∗
and µ2 , satisfy µ1 + µ2 = r − γ > 0 and µ1 · µ2 = rar−ρ−γ∗ +ωk rcII > 0. These inequalitiesgIIhold because γ < r − ρ is assumed. The real parts of both µ1 and µ2 are positive. There
are two state variables, bt and at . Unless b0 and a0 are respectively equal to b̄ and a∗II bychance, the economy cannot achieve the steady-state equilibrium.

F

Derivation of (21)

We deﬁne zt ≡ at − bt . Using (7) and the deﬁnition of zt , we rewrite (19a) as żt =(r − γ)zt − ct + (ω − γ)kg . Given z0 ≡ a0 − b0 , the solution for this is
)
(
∫ t
ω−γ
ω−γ
(r−γ)tkg e
kg −cv e(r−γ)(t−v) dv.
−
zt = z0 +r−γr−γ0Because ct > 0 holds for all t ≥ 0, this equation implies the following inequality:
)
(
ω−γ
ω−γ
kg e(r−γ)t −kg .
zt < z0 +r−γ
r−γ
Substituting zt = at − bt into this equation yields (21).

G

An Extention: Variable Interest Rate

We extend the model in the main text to include an interest rate that is aﬀected by an
endogenous variable representative of the ﬁscal conditions of the economy. Intuitively,28

even in a small open economy, an interest rate for the agents in the economy can vary
according to ﬁscal conditions because the risk premium for debt depends on them.
Some works such as Chatterjee et al. (2003) incorporate it in the following way. Lettnt = BtY−A. This is net foreign debt per GDP. We think of this as an indicator of ﬁscaltconditions and interest rate rt depends on nt :rt = r(nt ) > 0,

where r(·) is a nondecreasing function. Following Chatterjee et al. (2003), we adopt thefollowing speciﬁcation of r(·) when we conduct numerical analyses:
rt = r∗ + exp(σnt ) − 1 ≡ r(nt ),(G.1)

where r∗ is a constant and parameter σ > 0 is response strength of the interest rate for theﬁscal conditions. When nt > 0, the economy borrows from abroad and then the interest
rate for the economy rises from natural level r∗ , which reﬂects a risk premium.
G.1

Model

We incorporate a positive rate of depreciation of private capital, denoted by δk , and adopt
a Cobb–Douglas type production function. The proﬁt of the representative ﬁrm isθKtα (Kg,t Lt )1−α − (rt + δk )Kt − wt Lt .

(G.2)

By (G.2), the ﬁrst-order conditions of the ﬁrm’s problem are− δk ,
rt = αθxα−1
t

(G.3)

wt = (1 − α)θxαt Kg,t ,

(G.4)

29

Kt
where xt = Kg,t
. From (G.3) and (G.4), we have
Lt

[

αθ
xt =
r(nt ) + δk

1
] 1−α

≡ x(nt ).

Besides, public capital-to-GDP ratio is

kg,t =

Kg,t1=≡ kg (nt ).
Yt
θx(nt )αNote that by the speciﬁcation of r(·), the price system and other key variables depend on
nt and vary with time.
The government adopts the same debt policy rule:ḃt = −ϕ(bt − b̄),and faces the ﬂow-budget constraint

rt Bt + Gt = Ḃt + τ (rt At + wt ).

For numerical analyses, we assume a positive rate of depreciation for public capital, denoted by δg . The dynamic equation of public capital accumulation isK̇g,t = Gt − δg Kg,t .
G.2

Regime (I)We ﬁrst consider regime (I) where the government sets a constant τ ∈ (0, 1) and adjusts
public investment. As in the basic model, using the budget constraint of the representative30

household, we can obtain government expenditure per GDP as a function of (bt , nt , ct ):gt = g(bt , nt , ct ) =[[kg (nt )()τ (ω − r(nt )nt ) − (1 − τ )r(nt ) + δg btt ) (bt −nt )kg (nt )kg (nt ) − bt + αε(n1−αnt]] αε(n ) []ct − ωtτ (ω + r(nt )(bt − nt ) +bt − ϕ(bt − b̄) ,− ϕ(bt − b̄) −
1−α
nt

′

r (nt )nt
where ε(nt ) = r(n.
t )+δgTogether with the Euler equation, the model is reduced to a dynamic system of bt , nt ,and ct . Through some long manipulations, we obtain the following diﬀerential equations:ḃt = −ϕ(bt − b̄),ṅt = ∆(nt )Γ(bt , nt , ct )nt ,[]ċt = (1 − τ )r(nt ) − ρ + (∆(nt ) − 1)Γ(bt , nt , ct ) − g(bt , nt , ct )kg (nt )−1 + δg ct ,[]−1αt ,ct )
t ,ct )
where ∆(nt ) = 1 − 1−αε(nt )and Γ(bt , nt , ct ) = r(nt ) + g(bt ,n− g(bktg,n+ ctn−ω+ δg .
nt
(nt )
t
G.3

Local Determinacy

Let (b∗ , n∗ , c∗ ) be the stationary point. It satisﬁesb∗ = b̄,Γ(b∗ , n∗ , c∗ ) = 0,
(1 − τ )r(n∗ ) − ρ − g(b∗ , n∗ , c∗ )kg (n∗ )−1 + δg = 0.

Because this nonlinear system has no closed-form solution, we hereafter conduct numericalanalyses. We linearize the dynamical system around the stationary point and check thesigns of the eigenvalues of the Jacobian matrix. If the number of eigenvalues with anegative real part is three, the stationary point is a sink. Equilibrium indeterminacyarises because the system includes just one jumpable variable, ct .
We adopt a benchmark parameter value as follows. On the household’s preference31and the production function, we choose a popular value: ρ = 0.05, θ = 1, α = 0.36. Asa standard income tax rate and interest rate, we set τ = 0.15 and r∗ = 0.05. Following
Chatterjee et al. (2003), we set δk = 0.05, δg = 0.04. We analyze the model undervarious values of (ϕ, b̄, σ) because these are key parameters. As a benchmark value, we
adopt (ϕ, b̄, σ) = (0.05, 1, 0.1). We follow Chatterjee et al. (2003) and choose σ = 0.1
in (G.1). According to the SGP, we set ϕ = 0.05. Although the long-run target of bt is
0.6 in the SGP, we adopt b̄ = 1. This is because the actual debt–GDP ratios in the EU
countries are distributed around 1 (with slightly large variance) and such a value ensuressome goodness of ﬁt for this model. With this parameter set, consumption–output ratio
is 0.584, capital–output ratio is 2.915, and growth rate is 0.020 in the steady state.

Figure 3: ϕ and equilibrium determinacyFirst, we explore whether adjustment speed ϕ aﬀects equilibrium determinacy under
an endogenously varying interest rate. Figure 3 provides the answer: ϕ does not aﬀect
equilibrium determinacy. As in the original model of the main text, only the directions of
the simultaneous movement of (at , bt , gt ) are relevant to equilibrium determinacy. Therefore, adjustment speed ϕ does not aﬀect equilibrium determinacy.
Second, we investigate the relationship between equilibrium determinacy and the re-

32

1

0.9

Indeterminacy

bbar

0.8

0.7

Determinacy

0.6

0.5

0.4
0

0.05

0.1

0.15

0.2

0.25

sigma

Figure 4: σ and equilibrium determinacysponse strength of the interest rate to ﬁscal conditions, σ. Figure 4 illustrates the same.
As σ increases, the unique equilibrium area becomes larger. This result is very intuitive.
Since one of the sources of indeterminacy in a small open economy is ﬁxed interest rates,
the existence of a variable component in the interest rate weakens this indeterminacy.
However, more importantly, we should note that a variable interest rate does not eliminate equilibrium indeterminacy. Besides, for plausible values of σ, indeterminacy canarises under a not so high b̄ in our model. The mechanism underlying indeterminacy isgiven in the main text, and is important when creating debt-policy rules.
G.4

WelfareThe lifetime utility of the representative household is∫ ∞U=

e−ρt log Ct dt.

(G.5)

0
∫t

From the Euler equation, we have Ct = C0 e 0 γc,u du , where γc,u = (1 − τ )r(nu ) − ρ. By∫t

C0 = c0 Y0 and Y0 = θxα0 Kg,0 , we have Ct = c0 θx(n0 )α Kg,0 e 0 γc,u du . Substituting this into33

(G.5) and diﬀerentiating it with respcet to b̄, we obtain19
1 ∂c0
∂U
=
+
ρc0 ∂ b̄
∂ b̄

∫ ∞
e

−ρt

∫ t

0

(1 − τ )r′ (nu )

0

∂nu
dudt.
∂ b̄

(G.6)

We analyze E( ∂U) for both cases of unique equilibrium and multiple equilibria.20∂ b̄
Based on the welfare analysis in the main text, we consider the following scenario.
At the initial time, the economy stays at the stationary point: b0 = b̄init , n0 = ninit .
Then, the government unexpectedly marginally reduces b̄ to b̄new . Since this is a marginalchange, we may take the following approximation:b0 = b̄init ≈ b̄new = b̄,(G.7)

n0 = n̄init ≈ n̄new = n∗ .

(G.8)

In the case of a unique equilibrium, the economy jumps into a new saddle path and monotonically converges to the new stationary point. In the case of equilibrium indeterminacy,
for simplicity, we assume that a sunspot shock hits the economy only at the same time thegovernment reduces b̄. Wherever it jumps, the economy converges as in the determinatecase because the stationary point of the local dynamical system is a sink.
Case of a Unique Equilibrium
In the case of a unique equilibrium, using (G.7) and (G.8), we ﬁnd the equilibrium pathof the linearized model and substitute it into (G.6). Table 2 provides the results.21 See
the rows labeled as “D”. The main ﬁndings are as follows.
• For plausible parameter values, a marginal reduction in b̄ worsens social welfare.
This is consistent with the result in the original model (see Proposition 4). In the0
Note that ∂n= 0 because nt is a predetermined variable.
∂ b̄
In the case of multiple equilibria, we focus on solutions with sunspot shocks. Therefore, the partial
derivative should be evaluated by the expected value. In the case of a unique equilibrium, the model does
not contain any stochastic component.
21
Because we should treat both cases of unique and multiple equilibria and the threshold (with respect
to b̄) depends strongly on σ, we properly change the sets of alternative values of b̄ according to the valuesof σ as described in Table 1.

19
20

34

0.075

0.1

0.2

σ

b̄

D or I

0.01

0

0.45
0.5

D
I/N
I/U
I/N
I/U

-212.15 -525.36
510.21 1279.55
509.86 1279.57
96.67
247.1196.88
247.04-1034.43
2575.532576.57
514.38513.95-1527.95
3894.83
3893.95
802.92804.05-2006.63
5225.155228.69
1118.721119.22-3785.51
10750.92
10755.31
2709.082709.03

0.55

0.025

ϕ
0.05

0.075

0.55
0.6
0.65

D
D
I/N
I/U

-1.51
-1.86
-17.49-17.92-3.62-4.50-45.24-45.32-6.95-8.67-93.99-94.59-10.17-12.96-149.50-149.87-13.33-17.14-210.91-211.67-25.44-33.77-546.70-546.47

0.1

0.55
0.6
0.65

D
D
I/N
I/U

-1.02
-1.17
-37.13-37.47-2.38-2.77-94.53-94.82-4.47-5.25-200.37-200.73

-6.46
-7.65
-320.06-319.97

-8.39
-9.99
-455.25-455.54-15.65-19.05-1256.82-1257.18

0.125

0.65
0.7
0.75

D
D
I/N
I/U

-0.95
-1.11
-4.10
-3.98

-2.21
-2.61
-9.75
-9.96

-4.13
-4.94
-20.06-20.30-5.97-7.21-31.30-31.20-7.77-9.46-42.95-43.31-14.71-18.38-104.79-104.97

1.5

0.75
0.8

D
I/N
I/U
I/N
I/U

-0.90
-3.17
-3.26
-2.05
-2.61

-2.07
-7.86
-7.69
-5.83
-5.91

-3.85
-16.00-15.89-12.02-11.42-5.55-25.06-24.59-18.34-18.32-7.23-34.39-34.40-24.76-24.86-13.82-83.65-83.65-56.51-57.100.85

Table 2: E( ∂U) given (σ, b̄, ϕ). D (I) means equilibrium is determinate (indeterminate)∂ b̄under (σ, b̄). N (U) means sunspot shocks follow a normal distribution (uniform distribution). Note that ϕ does not aﬀect equilibrium determinacy.

35

original model, a reduction in b̄ has a detrimental eﬀect, if r < ω(= 1 − α), which
widely holds in usual cases. Inequality r < ω(= 1 − α) also holds in this numericalanalysis, and accordingly, a marginal reduction in b̄ worsens social welfare.
• The role of adjustment speed too is similar to that in the case of ﬁxed interest rates.
We can conclude that at least with regard to the qualitative aspect, the welfare
eﬀect of a reduction in b̄ and the mechanism underlying it do not change basicallyeven under an endogenous interest rate.
• The welfare implication of a change in (b̄, ϕ) is similar to the basic model in which
the interest rate is perfectly ﬁxed.
• The existence of a variable component in the interest rate (i.e., σ > 0) yields aquantitatively crucial eﬀect on social welfare.22 This is because the mechanism
underlying the welfare implication in the basic model depends on the assumptionthat the interest rate is perfectly ﬁxed. Thus, it is natural that the absolute valueof the welfare eﬀect is decreasing in the degree of variability of the interest rate.
Case of Multiple Equilibria
In the case of multiple equilibria, we consider sunspot solutions.23 Denote the threeeigenvalues of the Jacobian matrix by λ1 , λ2 , and λ3 and the associated eigenvectors by
v1 , v2 , and v3 , respectively. The general solution of the linearized model is
zt = z ∗ + κ1 exp(λ1 t)v1 + κ2 exp(λ2 t)v2 + κ3 exp(λ3 t)v3 ,(G.9)

where zt = (bt , nt , ct )′ , z ∗ = (b∗ , n∗ , c∗ )′ , and κ1 , κ2 , and κ3 are coeﬃcients. Because one of
the eigenvalues is −ϕ < 0, we may set λ1 = −ϕ. In the case of indeterminacy, Re(λ2 ) < 0
and Re(λ3 ) < 0, where Re(λi ) is the real part of λi . Suppose that coeﬃcient κ2 in (G.9) isa random variable that generates a sunspot shock that hits the economy only at the initial
22
23

Note that this model coincides with the original model in the main text if σ = 0.
We use the method of Lubik and Schorfheide (2003) to compose sunspot equilibria.

36

time at which the government sets a new long-run target for the debt–GDP ratio.24 We
consider the cases where κ2 is (i) normally distributed such a way that c0 jumps into therange [0.95c∗ , 1.05c∗ ] with probability greater than 0.999 and (ii) is uniformly distributed
over [0.97c∗ , 1.03c∗ ]. Since c0 is now a random variable, we should consider an expectedwelfare eﬀect for a marginal reduction in b̄: E( ∂U). In Table 2, the rows labeled “I”∂ b̄
present the results for this analysis. Labels, “N” and “U” indicate that sunspot shocks
follow a normal and uniform distribution, respectively. The main ﬁndings are as follows.
• Both the distributions yield qualitatively and quantitatively similar results. Although the welfare eﬀect in general varies for each realization of κ2 , the expectedvalue is stable due to the symmetry of the distributions. We conjecture that thisis from sunspot shocks following any symmetric distributions with suﬃciently small
variances.
• Except for the case of σ = 0, the relationships between adjustment speed ϕ and) are similar to that in the case of equilibrium determinacy andwelfare eﬀect E( ∂U∂ b̄hence the original model. In the case of σ = 0, the sign of E( ∂U) reverses. Its∂ b̄absolute value increases in ϕ even in this case.25
G.5

Regime (II)We now consider regime (II) where τ is endogenous and g = G/Y is exogenous. The
dynamics system is given byḃt = −ϕ(bt − b̄),ṅt = ∆(nt )Ψ(bt , nt , ct )nt ,[]ċt = (1 − τ (bt , nt , ct ))r(nt ) − ρ + (∆(nt ) − 1)Ψ(bt , nt , ct ) − gkg (nt )−1 + δg ct ,According to the dynamical equations, κ1 = b0 − b̄. Given κ2 , the value of κ3 is determined so thatthe initial value satisﬁes (G.9) .
25
We obtain similar results for suﬃciently small values of σ > 0, for example, σ = 10−5 . Thus, thewelfare eﬀect seems to have continuity at σ = 0 with respect to σ. Besides, similar results hold when
there are no sunspot shocks. The phase diagram seems to exhibit some complicated changes between thecases of a saddle and sink nearby σ = 0.
24

37

[
]−1
g
α
where ∆(nt ) = 1 − 1−αε(nt )and Ψ(bt , nt , ct ) = r(nt ) + ngt − kg (n+ ctn−ω+ δg . Thet)ttax rate is a function of bt , nt , and ct :τ (bt , nt , ct ) =α(r(nt ) + 1−αε(nt ) ṅntt − gkg (nt )−1 + δg )bt + g − ϕ(bt − b̄)r(nt )(bt − nt ) + ω.

We derive the steady state where ḃt = ṅt = ċt = 0. From ṅt = Ψ(bt , nt , ct ) = 0 andċt = 0, we obtain
1
τ =1−
r(n)

{

}
g
ρ+
− δg .
kg (n)(G.10)

If r is ﬁxed, the right-hand side (RHS) of (G.10) is constant. If r′ (n) > 0, the RHS is
an increasing function of n as long as the long-run growth rate, γ = kgg(n) − δg , is strictly
positive. From the deﬁnition of nt = bt − at , we have ȧt = 0 in the steady state, which
implies c = ρ(b̄ − n) + (1 − τ )ω. Using this equation and ṅt = Ψ(bt , nt , ct ) = 0, we obtainτ=ρb̄ + ρ.
ω − r(n)n(G.11)

Irrespective of whether r′ (n) > 0 or r′ (n) = 0, there exists a unique n̄ such that ω = r(n̄)n̄
and the denominator of the RHS is strictly positive if n < n̄. The RHS of (G.11) increases
with n. Equations (G.10) and (G.11) determine the steady-state value of n.
As shown in Proposition 5 of the main text, if r is ﬁxed, there exists a unique steadystate. Point A in panel (a) of Figure 5 corresponds to this steady state. The curve of(G.11) intersects that of (G.10) from below, at the steady state. The main text showsthat this steady state is unstable. Even when r increases with n, there exists a steadystate where the curve of (G.11) intersects that of (G.10) from below (see Point B in panel(b) of Figure 5).
To examine the stability of this steady state, we use the speciﬁcation of (G.1). As
in Subsection G.2, we assume ρ = 0.05, α = 0.36, δk = 0.05, δg = 0.04, ϕ = 0.05, andr∗ = 0.05. To ensure positive growth, we use θ = 1.3. We set g = 0.038. The values of b̄38

(a) r ’ (n)=0t(G.11)

1

A

n*II(G.10)nn(b) r ’ (n)>0t(G.11)

1

B

n*II(G.10)

nn

Figure 5: Steady-state equilibrium under regime (II)range from 0.1 to 3. We linearly approximate the dynamic system around this steady state
and then calculate the eigenvalues for σ = 0.1 and σ = 0.2. Our calculation shows thatfor all values of b̄, this steady state is unstable. For example, when σ = 0.1 and b̄ = 0.6,
the Jacobian of the linearized system has the following eigenvalues: 0.866, 0.0757, -0.05.
This result shows that even when r is endogenously determined, the steady state may beunstable for a wide range of parameters.

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