Discussion Papers In Economics
And Business

Dynamic analysis of reductions in public debt in an
endogenous growth model with public capital
Noritaka Maebayashi, Takeo Hori and Koichi Futagami

Discussion Paper 12-08-Rev.3

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDynamic analysis of reductions in public debt in an
endogenous growth model with public capital
Noritaka Maebayashi, Takeo Hori and Koichi Futagami

Discussion Paper 12-08-Rev.3

August 2015

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANDynamic analysis of reductions in public debt in an
endogenous growth model with public capital∗
Noritaka Maebayashi†

Takeo Hori‡

Koichi Futagami§

Abstract
We construct an endogenous growth model that includes productive public capitaland government debt. We assume that the government debt-to-GDP ratio is gradually adjusted to a target level, reflecting the permanent commitment rules in the
Stability and Growth Pact or the Maastricht Treaty in the EU (i.e., the well-known
60% rule). These rules aﬀect government borrowing and public investment. Here,
we examine the welfare implications of the permanent commitment rules. We findthat fiscal consolidation based on the rules improves social welfare. Moreover, theimprovement in welfare accelerates as fiscal consolidation progresses more rapidly.
Lastly, we also discuss and derive the optimal long-run debt-to-GDP ratio.

JEL classification: E62, H54, H63
Keywords: Fiscal consolidation; Debt policy rule; Public capital; Welfare;
Endogenous growth∗

The authors would like to thank two anonymous referees for their helpful comments on an earlier
version of this paper. We are responsible for any remaining errors.
†
Faculty of Economics and Business Administration, The University of Kitakyushu, 4-2-1 Kitagata,
KokuraMinami-ku, Kitakyushu 802-8577, JAPAN; e-mail: non818mn@kitakyu-u.ac.jp‡
College of Economics, Aoyama Gakuin University, 4-4-25, Shibuya, Shibuya-ku, Tokyo, 150-8366,
JAPAN; e-mail: t-hori@aoyamagakuin.jp§
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
JAPAN; e-mail: futagami@econ.osaka-u.ac.jp1

1

Introduction

Government debt has increased in the EU countries since the onset of the 2008–2009
financial crisis owing to discretionary fiscal expansion. For example, in Greece, public
debt as a share of GDP equaled 166.1% in 2012, and the country has subsequently suﬀered
severe government financial failure. The debt-to-GDP ratios in Italy, Ireland, and Portugal
also exceeded 100% in 2012. Since then, many EU member countries have increased theireﬀorts to implement fiscal consolidation.
To reduce the risk of such government financial failure, the EU has implemented theMaastricht Treaty, which set two types of fiscal rules that work as permanent constraints
on fiscal policies. The Maastricht Treaty states that EU member states must keep theirgovernment deficit-to-GDP ratio below 3% and their debt-to-GDP ratio below 60%. The
debt reduction benchmark (rule) was introduced by the reform of the Stability and Growth
Pact (SGP), the so-called Six-Pack, in December 2011. The rule states that member states
whose current debt-to-GDP ratio is above the 60% threshold must reduce their ratios to
60% by an average rate of one-twentieth per year. We study these fiscal rules from atheoretical perspective, focusing in particular on the debt/GDP rules.
These fiscal rules on government borrowing are related to public investment policies,
because government borrowing is linked directly to public investment, which aﬀects economic growth. The IMF (2014) suggests that investment in productive public capital isimportant in countries with a high degree of public investment eﬃciency, because the positive eﬀects of such investments on potential growth are substantially stronger. Bom andLigthart (2014) estimate the average output elasticity of public capital, which suggeststhat public capital is undersupplied in OECD economies. Indeed, according to plans set
out in the Stability and Convergence Programs (SCPs) in the EU, which were submitted
to the Commission and Council in Spring 2012, the member countries are planning further
fiscal consolidation based on expenditure cuts, including reductions in public investment.
Thus, it is important that we study how to reduce government debt and create the fiscal
space for public investment.
Many recent studies consider permanent fiscal rules and public investment using endogenous growth models. For example, Greiner and Semmler (2000) investigate the longrun growth eﬀects of public investment policies under deficit/GDP rules. Ghosh andMourmouras (2004) extend the Greiner and Semmler framework into welfare analysis. In
contrast, Futagami, Iwaisako, and Ohdoi (2008) and Minea and Villieu (2013) focus on
debt/GDP rules, under which the government debt relative to the size of the economy isadjusted gradually to a target level in the long run. These authors show that permanent
fiscal rules may be important determinants of long-run economic growth and welfare, asemphasized by Barro (1990).1
Because we focus on the debt/GDP rule, the works of Futagami et al. (2008) and Minea
and Villieu (2013) are more relevant to our study. These authors successfully provide policy
implications of the debt/GDP rule. However, there is still room for discussion. First, they
focus on the current flow of public services rather than the stock of public capital or its

1

There is considerable literature on how fiscal debt policies aﬀect macroeconomic stability, growth, andwelfare in an endogenous growth model. See Bruce and Turnovsky (1999), Greiner (2007, 2010, 2012a,
2012b), Yakita (2008b), Arai (2011), Kamiguchi and Tamai (2012), and Teles and Mussolini (2014).

2

accumulation.2 Second, they do not derive and discuss an optimal debt-to-GDP ratio.
Third, they do not investigate the welfare eﬀects of debt reduction. Thus, the present
study addresses the following: (i) the eﬀects of the debt/GDP rule on the accumulation of
public capital; (ii) the optimal long-term debt-to-GDP ratio; and (iii) the welfare eﬀects
of debt reduction, including transition dynamics. Further, we pay attention to the pace ofdebt reduction (the timeline of debt reduction) because the timeline of a debt adjustmentaﬀects both government borrowing and public investment.
For our purpose, we construct an endogenous growth model that includes public debt
finance, in line with the aforementioned literature. In our model, the growth engine isproductive public capital, as in Futagami et al. (1993) and Turnovsky (1997). Public
capital accumulates through public investment financed by issuing bonds and taxes oncapital income, labor income, and household consumption. For a government to reduce
its debt to a target level, it must cut spending or increase tax rates. We focus mainlyon the former case, because fiscal consolidation based only on spending cuts is discussed
widely in the context of many European countries.3 However, we do also briefly examine
the latter case because fiscal consolidation in Spain, Austria, Cyprus, France, Malta, andRomania is relatively evenly balanced between these two methods.
First, we show that a steady state that is locally saddle stable exists. In contrast to
the models in Futagami et al. (2008) and Minea and Villieu (2013), our model includes
the possibilities that growth rates become negative and that the no-Ponzi game conditionof the government breaks because we consider the stock of public capital. Thus, we derive
the necessary and suﬃcient condition for strictly positive long-run growth.
Second, we derive the optimal long-term debt-to-GDP ratio. This ratio depends on
the tax rates on wage income and consumption, as well as the share of public investment
in total government spending. Furthermore, we find that the target debt ratio set by the
SGP and Maastricht Treaty, namely 60%, might be much higher than the optimal level.
Finally, we investigate the welfare eﬀects of debt reduction by considering the transition dynamics. For these analyses, we calibrate the model to the data of Greece as
an example of a country with a very high debt-to-GDP ratio. In the benchmark case,where debt reduction is based on expenditure cuts only, reducing the debt-to-GDP ratio
to 60% improves social welfare. This is because debt reduction not only releases resources
to private consumption in the short run, but also creates fiscal space to increase public
investment in the long run. In addition, we suggest two policy implications. First, theimprovement in welfare increases as the pace of debt reduction increases. Second, lowering
the target debt-to-GDP ratio from 60% to the optimal level increases the improvement inwelfare.
Then, we also examine the welfare eﬀects of debt reduction based on tax increases andexpenditure cuts. In this case, welfare does not always improve. In addition, even when
welfare does improve, the welfare gains are lower than those under expenditure cuts only.
2

Fisher and Turnovsky (1998) note that “... it is open to the criticism that insofar as productive
government expenditures are intended to represent public infrastructure, such as roads and education, itis the accumulated stock, rather than the current flow, that is relevant.”.
3
The European Commission proposes using spending cuts rather than tax increases for fiscal consolidation because past evidence indicates that expenditure-based consolidation (spending cuts) tends to havegreater success. According to Public finances in EMU – 2012, on average, the SCPs of both the Euro area
and the EU27 are based primarily on spending cuts. In addition, Greece, whose debt-to-GDP ratio is thehighest in the EU, announced deep spending cuts in their 2011 budget, under IMF and EU supervision.

3

This is because although the increases in taxes mitigate the decrease in public investmentin the short run, the negative welfare eﬀects caused by the tax increases become stronger,
mainly owing to their negative eﬀects on savings and consumption growth.
Some fiscal consolidation strategies are discussed in exogenous growth models, forexample, Coenen et al. (2008), Forni et al. (2010), Papageorgiou (2012), Bi et al. (2013),
Cogan et al. (2013), and Erceg and Lindé (2013). These authors focus mainly on the eﬀects
of fiscal consolidation on transitional dynamics. Then, less normative welfare analysesof the permanent debt policy rule have been conducted. Our study diﬀers from theseprior studies in the following ways. We consider the long-run growth eﬀects of public
investment that are linked to the permanent debt policy rule. This means that fiscal
consolidation is part of a long-run objective to balance government borrowing and tocreate the fiscal space for positive expenditure on long-run growth. More importantly, weinvestigate welfare analyses with the permanent debt policy rule, and derive and discuss
the long-run optimal debt-to-GDP ratio. Here, we focus on the pace at which consolidation
should be implemented.
The remainder of this paper is organized as follows. Section 2 presents the basic model.
Section 3 derives the equilibria. Section 4 examines the long-run optimal debt-to-GDP
ratio. Section 5 presents the welfare eﬀects of expenditure-based consolidation. Section 6
introduces an increase in taxes, along with expenditure cuts. Section 7 modifies the model
settings and checks the robustness of the results. Section 8 concludes the paper.

2

Model

Our model is based on those of Futagami et al. (2008) and Minea and Villieu (2013).
Our economy is populated by infinitely long-lived representative households who have an
infinite planning horizon and perfect foresight. Time is continuous and denoted as t ≥ 0.
We assume there is no population growth and that the population size is normalized toone, as in Futagami et al. (2008) and Minea and Villieu (2013).

2.1

Production Structure

There is a continuum of competitive firms whose size is normalized to one. Firm j producesαa single final good using the production technology given by Yj,t = AKj,t(ht Lj,t )1−α (0 <α < 1), where Yj,t , Kj,t , and Lj,t represent the output level, private capital, and labor
input of firm j, respectively. In addition, ht represents the labor productivity at timet. Through profit maximization, factor prices become equal to the marginal products:, where Rt and wt denote
and wt = (1 − α)A(Kj,t /Lj,t )α h1−α
Rt = αA(Kj,t /Lj,t )α−1 h1−αtt
the rental price of capital and wage rates, respectively.
Following Kalaitzidakis and Kalyvitis(2004) and Yakita (2008a), we assume that the∑aggregate private capital, Kt = j Kj,t , and public capital, Kg,t , have positive externalϵ(ϵ ∈ (0, 1)). Since Kj,t /Lj,t =eﬀects on labor productivity,and specify ht = Kt1−ϵ Kg,t∑
Kt /Lt and Lt = j Lj,t hold in equilibrium, the aggregate output and factor prices in4

period t are written, respectively, asβYt = Akg,t
Kt ,

(1a)

β
Rt = αAkg,t
,

(1b)

β
wt = (1 − α)Akg,t
Kt ,

(1c)

where β ≡ ϵ(1 − α) and kg,t (≡ Kg,t /Kt ) is the ratio of public capital to private capital.
Since we have Lt = 1 in equilibrium (as shown later), we omit Lt in the above equations.

2.2

Households

The utility function of a representative household is specified as∫ ∞ 1−σCt
U0 =e−ρt dt,
1−σ
0

(2)

where Ct is the household’s consumption at time t, and ρ(> 0) and 1/σ denote the subjective discount rate and intertemporal elasticity of substitution, respectively. Since mostempirical evidence suggests that the intertemporal elasticity of substitution is relatively
small, we assume that σ > 1. The household inelastically supplies one unit of labor. The
household’s budget constraint is given byẆt = (1 − τr,t )rt Wt + (1 − τw,t )wt − (1 + τc,t )Ct ,(3)

where Wt denotes assets, rt is the interest rate, τc,t (≥ 0) is the consumption tax rate, andτr,t (τw,t ) ∈ [0, 1) is the interest (labor) income tax rate. Taking rt , wt , τr,t , τw,t , and τc,t
as given, the household maximizes (2) subject to (3), which yields the Euler equation andthe transversality condition:
[
]
1
τ̇c,t
Ċt =
(1 − τr,t )rt − ρ −
Ct ,
(4a)
σ
1 + τc,t
lim Ct−σ Wt exp (−ρt) = 0.
(4b)
t→∞

2.3

Government

The government imposes taxes on income and consumption, and issues bonds, Bt , tofinance public expenditure, Gt . We restrict our attention to the case of Gt > 0. The
budget constraint of the government is
Ḃt = rt Bt + Gt − (τr,t rt Wt + τw,t wt + τc,t Ct ),(5)

where Bt represents the government’s outstanding debt and Ḃt denotes newly issued government bonds. ∫The government constraint must satisfy the no-Ponzi game condition,tlimt→∞ Bt exp[− 0 rv dv] = 0.4 A constant fraction, θ ∈ (0, 1), of Gt is used for public
capital investment, Ig,t , that is, θGt = Ig,t . The evolution of public capital is given byK̇g,t = Ig − δg Kg,t = θGt − δg Kg,t ,4(6)

It is well known that this no-Ponzi game condition of the government becomes equivalent to thehousehold’s transversality condition in the steady state.

5where δg ∈ (0, 1) is the depreciation of public capital. In Section 7, we briefly discuss thecase in which households derive their utility from public services, Cg,t ≡ (1 − θ)Gt , which
is not considered in the benchmark model.
Following Futagami et al. (2008) and Minea and Villieu (2013), we assume that the
government adjusts its bonds gradually to the target level. Let us define bt ≡ Bt /Yt andassume that the government adjusts bt according to the following rule:ḃt = −ϕ(bt − b̄),(7)

where b̄ and ϕ(> 0) denote the target level of government bonds and the adjustment
coeﬃcient of the rule, respectively. We consider the case of b̄ > 0. If bt is larger than b̄,
the government has to reduce its debt according to the diﬀerence between the current andtarget levels of b: bt − b̄. If the adjustment coeﬃcient, ϕ, takes a large (small) value, thegovernment adjusts bt to the target level at a fast (slow) pace. This policy rule indicatesthe long-run commitment of the SGP debt reduction benchmark, reformed in 2011, underwhich member states whose current debt-to-GDP ratio is above the 60% threshold must
reduce their ratio 60% (see the second paragraph of the introduction).5 Our investigation
focuses on the rule that the SGP expects.
The benchmark model considers fiscal consolidation based on expenditure cuts, andthe government sets constant tax rates, τr,t = τr , τw,t = τw , and τc,t = τc . Then, gt varies sothat it satisfies (5) and (7). This is because, according to Public finances in EMU – 2012,
the consolidations set out in the SCPs for both the Euro area and the EU27 are based
primarily on expenditure cuts. In contrast, in Spain, Austria, Cyprus, France, Malta, andRomania, fiscal consolidation is relatively evenly balanced between spending cuts and tax
increases, and is primarily tax-based in Belgium and Italy. Therefore, we examine fiscal
consolidation based on both expenditure cuts and tax increases in Section 6.

3

Equilibrium

3.1

Dynamic System

The labor market equilibrium condition is Lt = 1. The asset market clears as Wt = Kt +Bt .
Let us assume that private capital depreciates at the rate δk ∈ (0, 1), where the interest
rate is rt = Rt − δk .6 Substituting these into (5) and using (1a), (1b), and (1c), we obtainḂt = (1 − τr )(Rt − δk )Bt − (τ̃ Yt + τc C − Gt ) + τr δk Kt ,(8)

where τ̃ ≡ ατr + (1 − α)τw . The goods market equilibrium condition is given byK̇t = Yt − Ct − Gt − δk Kt .
5

(9)Such a policy rule would not expect uncertain fiscal policy behavior such as policy regime changes.
However, there may be some discussion on uncertain fiscal consolidation, as studied by Bi et al. (2013).
There may also be other discussions on the rule. For example, recent high-debt countries that are in theprocess of reducing their debt can have diﬃculty obeying the rule in the short run because of the adverse
eﬀects that contractionary fiscal policies and the increase in the interest payment have on GDP.
6
This is because, by the no-arbitrage argument in the asset market, government bonds have the samerate of return as capital.

6

Let us define ct ≡ Ct /Kt and gt ≡ Gt /Kt . The definition of bt implies ḃt = Ḃt /Yt −(Ẏt /Yt )bt . Substituting (1a), (1b), (7), and (8) into this expression, we obtain
]
[
} Ẏ
{
ct
gt
τr δk
t
β
bt − τ̃ − τc β ++. (10)ḃt = −ϕ(bt − b̄) = (1 − τr ) αAkg,t− δk −ββYtAkg,t Akg,t Akg,tUsing (1a), (6), and (9), we obtain
1+βk̇g,t = (θ + kg,t )gt − Akg,t+ ct kg,t + (δk − δg )kg,t .

(11)

βFrom Yt = Akg,tKt , (9), and (11), we derive the GDP growth rate as

Ẏt
k̇g,t
gt
β
β
=β
+ Akg,t− βδg .
− ct − gt − δk = (1 − β)(Akg,t− ct − δk ) − [(1 − β)kg,t − θβ]
Yt
kg,t
kg,t
(12)
Substituting (12) into (10), and solving for gt , we obtain

gt =

β 2
β
β
β
β
Γ(bt , ct , kg,t ) − αbt (Akg,t) − ϕ(bt − b̄)Akg,t+ (1 − β)(Akg,t− ct )Akg,tbt − β(δg − δk )Akg,tbt
β−1
1 + [(1 − β)kg,t − θβ] bt Akg,t≡ g(bt , ct , kg,t ),

(13)

β
β
where Γ(bt , ct , kg,t ) ≡ [τ̃ + τr (αAkg,t− δk )bt ]Akg,t+ τc ct − τr δk . The numerator on theright-hand side (RHS) of (13) represents the components of gt . The first term Γ(bt , ct , kg,t )is the tax revenue of the government. The second term represents the interest payment
on government debt. The third term shows that, given bt , if the government reduces b̄,public spending must fall in the short run. When ϕ is larger, the short-run decreases in gt
are also larger because the government must reduce its debt more rapidly, and hence its
budget becomes tight. We call this eﬀect of ϕ the timeline eﬀect. The fourth term implies
that a high growth of Yt has a positive eﬀect on public spending.7 The last is the term
related to depreciation.
Substituting τ̇c = 0 and (1b) into (4a), and using (1a), (9), and (13), we obtain
[
{
}
{
} ]
1
ρ
1 − τrβċt = ct + g(bt , ct , kg,t ) − 1 − (1 − τr )α Akg,t − + 1 −δk ct .
(14)σ
σ
σ

Using (1a), (6), (9), and (13), we then have1+βk̇g,t = (θ + kg,t )g(bt , ct , kg,t ) − Akg,t+ ct kg,t + (δk − δg )kg,t .

(15)

Equations (7), (14), and (15), together with the initial values, kg,0 and b0 , and the transversality condition, (4b), characterize the dynamics of the economy.
β
The goods market equilibrium condition, (9), shows that, given Gt , when Akg,t− ct becomes large,
the growth rate of private capital rises. When Kt grows at a high rate, Yt tends to grow more quickly. In
this case, the government requires relatively little eﬀort to reduce bt ≡ Bt /Yt to b̄. Then, a large increase
in Yt enables the government to increase gt .
7

7

3.2

Steady-State Equilibrium

This subsection derives the steady-state equilibrium in which ct , kg,t , and bt become constant over time. We omit the time index, t, from the variables that become constant overtime in the steady state. Setting ċt = 0, k̇g,t = 0, and bt = b̄ in (14) and (15), we obtain
]
[
]
[
1
ρ
1 − τr
∗ β
∗
∗
δk
(16)
c + g = 1 − (1 − τr )α A(kg ) + − 1 −
σ
σ
σ
(
)
θ
∗
∗ β
c = A(kg ) − 1 + ∗ g ∗ − (δk − δg ).
(17)kg
We use an asterisk to represent a steady-state variable. From (16) and (17), we obtaing∗ =(1 − τr )αA(kg∗ )1+β − [ρ + (1 − τr )δk − σδg ]kg∗
.
σθ

(18)

From (1b) and (4a), we derive the long-run growth rate, γ ∗ = Ċt /Ct = K̇t /Kt = K̇g,t /Kg,t :
γ∗ =

{
}
]
1[
(1 − τr ) αA(kg∗ )β − δk − ρ .
σ

(19)
1

Then, we have γ ∗ > 0 if and only if kg∗ > [{ρ + (1 − τr )δk }/{(1 − τr )αA}] β ≡ kg,γ .
Using (17), (19), γ ∗ = K̇t /Kt = K̇g,t /Kg,t , and bt = b̄, we can rearrange (10) as[()]θ
σ 1 + τc 1 + ∗g ∗ = ω(kg∗ )A(kg∗ )β − σ(δk − δg )τc − στr δk ,
kg

(20)

where ω(kg∗ ) ≡ (1 − σ)(1 − τr )αb̄A(kg∗ )β + σ(τ̃ + τc ) − [(1 − σ)(1 − τr )δk + ρ]b̄. Substituting(18) into (20), we obtain[][]θ Ω(kg )Akgβ + ζ = (1 + τc )kg (1 − τr )αAkgβ − {ρ + (1 − τr )δk } + σδg ,(21)

where Ω(kg ) ≡ ω(kg ) − τc (1 − τr )α and ζ ≡ τc {ρ + (1 − τr )δk } − σ(τc + τr )δk . This equation
determines kg∗ . Substituting kg∗ into (17), we obtain c∗ . Let us denote the RHS and theleft-hand side (LHS) of (21) as Π(kg ) and Λ(kg ), respectively. Appendices A and B prove
the next proposition.
Proposition 1
Suppose that Λ(kg† ) > Π(kg† ) and (A.5) or (A.6) are satisfied, where kg† , (A.5), and (A.6)
are defined in Appendix A. There exists a locally saddle-stable steady state where we haveγ ∗ > 0 and the no-Ponzi game condition of the government is satisfied.
In contrast to the models in Futagami et al. (2008) and Minea and Villieu (2013), ourmodel includes the possibilities that the growth rates become negative and the no-Ponzi
game condition of the government breaks in the long run. This is because we consider thestock of public capital rather than the flow of public services. A suﬃcient public capital8

accumulation is needed to sustain a large government debt. Conditions (A.5) or (A.6)
ensure such a public capital accumulation.8
Next, we examine the long-run eﬀects of reductions in b̄. Appendix C proves thefollowing proposition.
Proposition 2
A decrease in the target debt ratio, b̄, promotes economic growth and increases public
expenditure in the long run. That is, dkg∗ /db̄ < 0, dγ ∗ /db̄ < 0, and dg ∗ /db̄ < 0.
The intuition behind these policy eﬀects is simple. As b̄ decreases, outstanding public
debt reduces in the long run. The government’s interest payments also reduce, which
loosens its budget constraint and creates fiscal space to increase public investment. In
the long run, g increases and, hence, the long-run growth rate increases. Note that in
the short run, decreases in b̄ reduce g (see (13)). However, as bt decreases to b̄, thegovernment’s interest payment steadily decreases, which means it can gradually increase
public expenditure. The next subsection examines the transitional eﬀects of reductions in
b̄.

3.3

Transitional Dynamics

Here, we examine the eﬀects of b̄ on the transition path and show that these eﬀects depend
heavily on the timeline eﬀect of ϕ, which has no eﬀect in the long run.
We consider the following scenario: the economy is initially in the steady state withb̄ = b̄init , where b̄init denotes the initial level of b̄. Then, we have b0 = b̄init . At time 0, thegovernment reduces b̄ from b̄init to b̄new unexpectedly, where b̄new is the level of b̄ after thepolicy change. Then, the economy begins to move toward the new steady state.
Parameter Values
The complexity of the model does not allow us to obtain an analytical solution. Thus, we
conduct a numerical analysis. The debt-to-GDP ratio in the new steady state, b̄new , is set
to 0.6 following the target debt ratio in the SGP and the Maastricht Treaty. We use thefollowing five values of ϕ: ϕ = 0.01, 0.025, 0.05, 0.075, 0.1. The other parameter values arechosen based on the data of Greece (see Appendix D) as an example of a country witha very high debt-to-GDP ratio. Table 1 summarizes the parameter values. Under theseparameter values, we calculate the values of the consumption-to-GDP ratio (C0 /Y0 ), thetotal capital-to-GDP ratio ((K0 + Kg,0 )/Y0 ), and the ratio of tax revenue to output in theinitial steady state (see Appendix D). Table 2 compares these calculations with the data
averages in Greece for the period 2000–2008.9 Since the initial steady state of the model is
8

Dioikitopoulos and Kalyvitis (2008) also assess the possibility of a negative long-run growth rate inan endogenous growth model of productive public capital in Proposition 1 on pages 3766–3767. With thedepreciations of public and private capital, they show that the government should be cautious in using
scarce public expenditure to maintain the public capital that sustains the economic growth. However,rather than government borrowing, they consider a balanced budget. In our model, we would rather put
a caveat on huge government debt that hinges on sustainable economic growth.
9
The data average for the capital-to-GDP ratio is based on the AMECO database. The data averagefor tax revenue is based on the OECD Revenue Statistics.

9

in line with the data, it is a reasonable starting point for our policy experiments. Finally,note that the long-run growth rate in the initial steady state takes a positive value under∗= 0.035 (see Appendix D), and then (A.5)
the (parameter) values in Table 1, namely γinitor (A.6) in Proposition 1 is satisfied. Reductions in b̄ also satisfy either of them in thelong run because of Proposition 2.
[Tables 1 and 2]
Eﬀects of Reductions in b̄ on Transitional Dynamics
Here, we analyze the transition paths numerically using the relaxation algorithm.10 Panel(a) in Figure 1 shows that the debt-to-GDP ratio decreases monotonically toward its new
steady-state value. As ϕ increases, the debt-to-GDP ratio decreases at a higher rate.
[Figure 1]
Just after the policy change, the ratio of public investment to output, Ig,t /Yt , drops
sharply (see Panel (b)). To reduce bt , the government must initially reduce its expenditure,
as shown by the term ϕ(bt − b̄) in (13). However, as bt steadily declines, the interest
payments on government debt gradually decrease, which loosens the government’s budget
constraint and creates fiscal space to increase public investment. Then, Ig,t /Yt gradually
increases and eventually exceeds the initial level (see the last paragraph in subsection 3.2).
For larger values of ϕ, the initial drops in Ig,t /Yt are also larger because the government
must cut a larger amount of its expenditure to reduce its debt at a higher rate. Higher
values of ϕ also mean that it takes less time for Ig,t /Yt to recover its initial level.
Panel (c) shows that the growth rate of private capital, Kt , jumps just after the policychange because the decline in public expenditure (investment) releases resources to theprivate sector. Then, it gradually decreases to its new steady-state value.
The dynamics of public investment and private capital drive those of kg,t . During theearly stage of the transition, kg,t gradually decreases because of the reductions in Ig,t /Yt
and the increases in Kt . However, as Ig,t /Yt gradually increases, kg,t begins to increase,
eventually exceeding its initial level in the long run. Since the growth rate of consumption
is a function of kg,t (see (1b) and (4a)), it decreases during the early stage of the transition,
but then begins to increase in the latter stages, eventually exceeding its initial level (seePanel (d)). As ϕ increases, the decline in the growth rate of consumption in the early stageof the transition is large, whereas it also takes less time until the growth rate recovers its
initial level. The Euler equation (4a) shows that the interest rate exhibits the transition
path similar to that of the growth rate of consumption.11
Panel (e) provides the transitional paths of ct . From this panel, we know the eﬀects ofthe policy change on the initial consumption level, C0 , because C0 ≡ c0 K0 . For all values
10

Trimborn et al.
(2008) detail the relaxation algorithm.
They also provide MATLAB programs for the relaxation algorithm, freely downloadable at http://www.wiwi.unisiegen.de/vwli/forschung/relaxation/the relaxation method.html?lang=de.
11
The real interest rate becomes somewhat larger than the standard level. This comes from the simple
setting of the model where capital inflow is absent. If one carefully examines this situation in the realworld, huge models that incorporate the capital market in the open economy and the exchange rates andother fiscal and monetary policies in each of the countries might be necessary. However, this is beyond thescope of this study. Here, we investigate those eﬀects that are essential to social welfare and an optimal
target debt ratio, which have received less attention. Thus, our discussion restricts the movement of thereal interest rate.

10

of ϕ, C0 jumps just after the policy change because the decline in public expenditure(investment) releases resources to the private sector. As ϕ increases, these crowding-in
eﬀects become stronger and the initial increase in C0 becomes large.

4

Optimal Debt-to-GDP Ratio

In the SGP or the Maastricht Treaty, the target debt-to-GDP ratio is set at 60%. However,this level might be diﬀerent from the optimal level. This section examines the long-run
optimal debt-to-GDP ratio that can attain the first-best welfare maximizing allocation.
The long-run optimal debt-to-GDP ratio can be derived by considering the optimal fiscal
policy in the steady state. To this end, we solve a social planner’s problem.
Distortions from the following external eﬀects exist in this economy: the positive exαϵternal eﬀect of Kt and that of Kg,t (see Yj,t = AKj,t(ht Lj,t )1−α and ht = Kt1−ϵ Kg,tin
Subsection 2.1). Therefore, the decentralized economy cannot attain a first-best allocation in the long run. To attain the optimum allocation, the following fiscal policies are
required. First, the capital income tax rate τr is needed to remove the distortion from theexternal eﬀect of Kt . Second, the other policy instruments, τc , τw , θ, and b̄, are needed toremove the external eﬀects of Kg,t .
A social planner maximizes the household’s discounted lifetime utility, subject to theeconomy’s aggregate resource constraints, (6) and (9). Because θGt = Ig,t (see (6)) and(1a), the resource constraint (9) can be rewritten asβK̇t = AKg,tKt1−β − θ−1 Ig,t − Ct − δk Kt .

(22)

The social planner maximizes (2), subject to (6) and (22) with respect to Ct , Ig,t , Kt , andKg,t . From the first-order conditions, we obtain
]
Ċt
1[
β
(1 − β)Akg,t− δk − ρ ,=Ct
σ(1 − β)Akgβ − δk = θβAkgβ−1 − δg .

(23)
(24)

The condition (24) indicates that the marginal product of private capital is equal to thatof public capital. The LHS of (24) is monotonically increasing in kg , whereas the RHS ismonotonically decreasing in kg . Then, kgopt is uniquely determined. Substituting (24) into
(23), we obtain the following optimal long-run growth rate of consumption:
γ opt =]
1[
(1 − β)A(kgopt )β − δk − ρ .
σ(25)The government in the decentralized economy chooses the target debt ratio, b̄, to replicate
the first-best outcome in the long run.
Let us compare the corresponding relationships (21) and (19) with (24) and (25). The
steady-state equilibrium will replicate the first-best optimum: kg∗ = kgopt and γ ∗ = γ opt if
and only if(1 − β)A(kgopt )β − δkopt
1 − τr =,(26)
αA(kgopt )β − δk
11

and
[
]
[
]
θ Ω(kgopt )A(kgopt )β + ζ = (1 + τc )kgopt (1 − τropt )αA(kgopt )β − {ρ + (1 − τropt )δk } + σδg .
(27)
Recall that β = ϵ(1 − α). Condition (26) shows that τropt < 0 when 0 ≤ ϵ < 1, whereas
τropt = 0 when ϵ = 1. The former result corresponds to the usual policy implicationthat subsidies on capital are required to remove the external eﬀects of Kt (see Barro andSala-i-Martin (2004), pp 216-217). On the other hand, the latter case corresponds to amanifestation of the well-known result of Chamley (1986) that capital should be untaxed
when there are no externalities from Kt .
From (26) and (27), we obtain the optimal target debt ratio as follows:
]
[
−1
opt
opt
opt β
opt
)δ
}
+
σδ
)
−
{ρ
+
(1
−
τ
−ζ
θ
(1
+
τ
)k
(1
−
τ
)αA(k
k
g
c
g
r
g
r
}
b̄opt = {
opt
opt β
opt
opt β
(1 − σ)(1 − τr )αA(kg ) − [(1 − σ)(1 − τr )δk + ρ] A(kg )τc (1 − τropt )α − σ(τ̃ + τc )+.
(28)
(1 − σ)(1 − τropt )αA(kgopt )β − [(1 − σ)(1 − τropt )δk + ρ]
In the case of the parameter set in Table 1: (1 − α, A, β, σ, ρ, θ, δk , δg )=(0.5645, 0.4650,
0.0345, 2.5, 0.0412, 0.1590, 0.028, 0.031), we obtain (kgopt , τropt ) = (0.0056, −1.4567).
Figure 2 shows the relationship between b̄opt and the set of tax rates (τc , τw ). When(τc , τw )=(0.1293, 0.2818), as in Table 1, b̄opt is in the region of a negative value. Only
when both τc and τw are suﬃciently high do we have b̄opt > 0.
[Figure 2]
To simplify the discussion, we consider the case of no depreciation in both private
and public capital (δk = δg = 0). In this case, we obtain kgopt = θβ/(1 − β) and τropt =(α + β − 1)/α from (24) and (26). Therefore, (28) reduces to
)β
(
θβ
c +β)
+ ρ(τ1−β[στ̃ + (σ − 1)τc − β]A 1−β] (b̄opt = [
(29)
(
)β
)β ,
θβ
θβ
(σ − 1)(1 − β)A 1−β + ρ A 1−βwhere τ̃ ≡ (1 − α)τw + α + β − 1. Because σ > 1 is assumed, b̄opt is obviously increasing
in both τw and τc . After some calculation, we can easily show ∂ b̄opt /∂θ < 0 when b̄opt > 0.
Thus, we obtain the following proposition.
Proposition 3
If the depreciation of both private and public capital are zero, we have
opt
b̄opt
b̄opt
> 0 and ∂∂τ> 0, and (ii) ∂ b̄∂θ < 0, when b̄opt > 0.
(i) ∂∂τw
w
The intuitive explanation of this policy implication is simple. When tax revenue is suﬃciently large, the government’s debt finance can become positive as the optimal decision.
On the other hand, an increase in θ promotes public investment and leads to a rise in kg∗ ,which increases the interest rate of government debt. In order to reduce the cost of thisinterest payment, the planner decreases b̄opt in the long run.
12

5

Welfare Analysis: Expenditure Cuts Only

Focusing only on the steady state, we obtain the long-run optimal debt-to-GDP ratio in
the previous section. To conduct a complete welfare analysis of debt reduction, we haveto fully consider the transitional dynamics. We consider the same scenario as that in
Subsection 3.3.
∫tβ
Our welfare measure is (2). Using Ct = C0 exp[ 0 γC,v dv], where γC,t (≡ [(1−τr ){αAkg,t−δk } − ρ]/σ) is the growth rate of consumption, we rewrite (2) as
]
[
∫ t
∫
C01−σ ∞
γC,v dv − ρt dt.
exp (1 − σ)
U0 =
1−σ 0
0
This equation shows that decreases in b̄ aﬀect the welfare level through their eﬀects on
C0 = c0 K0 and the paths of γC,t . Because the eﬀects of the policy change on C0 and thetransitional paths of γC,t depend heavily on the timeline eﬀect of ϕ, as we observed in
Subsection 3.3, the welfare eﬀects of reductions in b̄ are also influenced by ϕ.
We examine the welfare eﬀects numerically using the relaxation algorithm. The initialsteady state is the same as that considered in Subsection 3.3. Again, we consider fivevalues of ϕ: 0.01, 0.025, 0.05, 0.075, and 0.1. The case in which ϕ = 0.05 matches the debt
reduction benchmark introduced by the SGP (see the introduction).
∗
Let us denote the welfare level without the policy change as U0,N≡ (c∗N K0 )1−σ /{(1 −∗∗σ)[ρ − γC,N
(1 − σ)]}, where c∗N and γC,N
are the initial steady-state values of c and γC ,∗respectively. To calculate U0,N , we set K0 = 1. The welfare level immediately after thepolicy change is denoted as U0∗∗ . Appendix E presents the calculation procedure. The∗∗
welfare gains (losses) of the policy change are measured by ∆U0 ≡ (U0∗∗ − U0,N)/|U0,N|.
Results
The second column in Table 3 provides the long-run growth rates of the new steady states.
Because the growth rate in the initial steady state is 0.035, debt reduction increases thelong-run growth rate, as shown in Proposition 2. However, the growth eﬀect is rather
modest. This reflects the small elasticity of output with respect to public capital, β =ϵ(1 − α) = 0.0345.12[Table 3]
The row labeled as ∆C0 represents percentage changes in the initial consumption levels.
As discussed in Subsection 3.3, as ϕ increases, the initial increases in C0 become large.
With regard to the welfare eﬀects, Table 3 reveals the following results: (i) for all
values of ϕ, a reduction of b̄ to 60% improves welfare; and (ii) as ϕ increases, welfare
improvements become large. This is because when ϕ is larger, more resources are released
to the private sector and, hence, the initial increases in C0 become large.
If the consumption growth rate remains constant at the initial rate, how much of anincrease in initial consumption is required to attain the same level of welfare improvements as that of ∆U0 ? The row labeled as ∆Cequiv shows the required increases in initial
12

Bom and Ligthart (2014) estimate that the average output elasticity of public capital amounts to
0.106 if we consider OECD countries. Such a large output elasticity of public capital can bring about thelarger long-run growth eﬀect of debt reductions.

13consumption. In all cases, ∆Cequiv is smaller than ∆C0 . This indicates that the decline in
the consumption growth rate during the early stage of the transition has a strong negative
welfare eﬀect. However, ∆C0 is large enough to counteract this negative eﬀect.
Furthermore, Table 3 shows that by setting the target level, b̄new , to zero, welfare
improves even further. This is because the long-run optimal target debt ratio is smaller
than 0.6 under the parameter sets of Table 1 (see the statements below Eq. (28) in Section
4).

6

Expenditure Cuts with Tax Increases

Thus far, we have discussed debt reductions based only on expenditure cuts because theconsolidations set out in the SCPs for both the Euro area and the EU27 are primarily
based on this method, according to the Public finances in EMU – 2012 (See footnote 3
in the introduction). The goal of this section is to compare the welfare eﬀects between
debt reductions based only on expenditure cuts and those based on both expenditure
cuts and tax increases. This is motivated by the following points. As noted earlier, incountries such as Spain, Austria, Cyprus, France, Malta, and Romania, consolidations are
relatively evenly spread between expenditure cuts and tax increases. If the government
could increase tax rates when attempting to decrease the debt-to-GDP ratio, the reductions
in public investment just after the policy change might be mitigated. Therefore, with tax
increases, the eﬀects of debt reductions might be diﬀerent from those examined thus far.
We assume that the government unexpectedly increases tax rates at the same time
as it decreases b̄. See Appendix F for more details. From then on, the tax rates remainunchanged because, in practice, frequent changes in tax rates might be diﬃcult to implement politically. We denote increases in the rates of interest income, labor income,and consumption taxes as ∆τr , ∆τw , and ∆τc , respectively. We assume that only one tax
rate is changed, while the other two rates remain unchanged. We choose values of ∆τr ,∆τw , and ∆τc so that the ratio of tax revenue to output in the new steady state becomeslarger than that in the initial steady state by x(= 0, 1, 2, 3)%. Naturally, x = 0 implies∆τr = ∆τw = ∆τc = 0.
We use the same parameter values as in the previous sections. Figure 3 shows thetransitional dynamics under b̄new = 0.6 and ϕ = 0.05. The solid lines represent debt
reductions without any tax increases (x = 0). The panels in the first column present theresults obtained when only τr is increased. The eﬀects of increases in τw and τc are shownin the second and last columns, respectively.
As x becomes larger, the initial decline in the ratio of public investment to output is
mitigated (see Panels (a)-1, (b)-1, and (c)-1). Panels (a)-2, 3, and 4 show that because
increases in τr have a negative distortionary eﬀect on households’ saving, the growth rates
of private capital and consumption decrease significantly and the initial increases in private
consumption become larger. Panels (b)-2 and (c)-2 show that an increase in either τw or
τc reduces the early increase in the growth rate of Kt . Panels (b)-4 and (c)-4 further showthat the initial increases in C0 are lower. This is because more resources are devoted to
public investment as a result of the tax increases. Panels (b)-3 and (c)-3 show that wheneither τw or τc increase, the decreases in the growth rate of consumption in the early stage
are mitigated because increased public investments have a positive eﬀect on the interest

14

rate.
Table 4 presents the welfare eﬀects. The results without a tax increase are presented
in the row labeled “Benchmark.” With a tax increase, the welfare gains from the debt
reduction are depressed and, in many cases, welfare deteriorates. In all cases, the debt
reduction based only on expenditure cuts generates the largest welfare gains. As shown inPanel (a)-3 in Figure 3, an increase in τr has a large and negative distortionary eﬀect on
the consumption growth rate, which has a significant negative welfare eﬀect. When either
τw or τc increase, there is less of an initial increase in C0 , which decreases the negative
welfare eﬀects.13[Figure 3 and Table 4]

7

Extension of the Model: Government Consumption
into Utility

Thus far, we have assumed that government consumption expenditure, Cg,t ≡ (1 − θ)Gt ,which accounts for a large part of government spending, is not valued by households.
Reductions in b̄ initially have a negative eﬀect on Cg,t . Hence, if households derive theirutility from Cg,t , the initial welfare loss becomes larger and a reduction in b̄ might havea negative welfare eﬀect. Incorporating government consumption into utility may be an
important modification.
However, if we assume non-separability between utilities from private and government
consumption, based on the empirical findings of Ni (1995), equilibrium indeterminacy
may arise.14 The multiplicity of equilibrium paths makes it diﬃcult to investigate thewelfare eﬀects. If the utility function takes a separable form with respect to Ct and Cg,t ,equilibrium indeterminacy never arises. Then, we consider the following utility function:)∫ ∞ ( 1−σCtU0 =+ ψ ln(Cg,t ) e−ρt dt,
(30)
1
−
σ
0
where ψ is a positive constant and Cg,t ≡ (1−θ)Gt denotes government consumption. This
modification has no influence on the dynamic system of the economy and the steady state.
The results thus far, except those regarding welfare eﬀects, hold perfectly, even under thismodification.
Using the same parameter values as in Subsections 3.3 and Section 5 and considering
the same scenario as in Section 5, we conduct a welfare analysis of reductions in b̄ for∗∗)/|U0,N|, where U0∗∗∗ denotes
several values of ψ.15 Table 5 presents ∆U0 ≡ (U0∗∗∗ − U0,N∗
the utility just after the policy change, and U0,N denotes the utility in the initial steadystate (as earlier).16 Table 5 also shows the results obtained under ψ = 0, which are
13

Note that τw and τc do not distort households’ behaviors. However, they aﬀect public investments
through the government budget constraint and the debt rule (see (13).).
14
Using a model with utility-generating government consumption, Guo and Harrison (2008) show thatequilibrium indeterminacy arises under a balanced budget.
15
Since this modification has no eﬀect on the dynamic system and the steady-state equilibrium, theinitial and new steady states are the same as in Subsection 3.3 and Section 5 under the same parametervalues.
16
See Appendix E for the calculation of U0∗∗∗ .

15

naturally the same as those in Table 3. Even under this modification, the reductions in b̄
improve welfare for all values of ϕ and b̄new , and these improvements in welfare increase
as ϕ increases and b̄new decreases.
If the government increases tax rates, as described in Section 6, the reductions in b̄
might have diﬀerent welfare eﬀects if ψ > 0, because tax increases mitigate the negativeeﬀects on Cg,t . Assuming ψ = 0.5, we conduct the same policy experiment as in Section 6
and present the results in Table 6. The table shows that, even when households place a highvalue on government consumption (ψ = 0.5), debt reductions based only on expenditure
cuts generate the largest positive welfare gains in all cases.
[Tables 5 and 6]

8

Conclusion

Following Futagami et al. (2008) and Minea and Villieu (2013), we considered a simple
debt policy rule in an endogenous growth model. Under the policy rule, the government
gradually reduces its debt to the target level. Departing from these two studies, weinvestigate (i) the eﬀects of the debt policy rule on the accumulation of public capital,(ii) the optimal target debt ratio in the long run, and (iii) the welfare analysis of the debt
reduction that fully considers the transition dynamics. We obtained the following results.
1. A steady state that is locally saddle stable exists. However, in contrast to themodels in Futagami et al. (2008) and Minea and Villieu (2013), our model includes
the possibilities that the growth rates become negative and that the no-Ponzi gamecondition of the government breaks in the long run. This result occurs because we
consider the stock of public capital rather than the flow of public services.
2. The optimal target debt ratio is uniquely determined. The ratio depends on the tax
rates on wage income and consumption, as well as the share of public investment in
total government spending. The target debt ratio set by the SGP and Maastricht
Treaty, namely 60%, might be higher than the optimal level.
3. Debt reduction based only on expenditure cuts improves welfare. As the pace ofdebt reduction increases, welfare improves further. Second, lowering the target debt
ratio of 60% to the optimal level increases the improvement in welfare.
4. Under fiscal consolidation based on expenditure cuts with a tax increase, welfare
does not always improve. Even when welfare does improve, the welfare gains arelower than those under expenditure cuts only.
5. Even when households derive their utility from public services, the above results
hold.
There remain areas for further study. First, we do not consider the heterogeneityof agents. Bastagli et al. (2012) discuss how some governments may increase the top
marginal income tax rates and avoid the decrease in redistributive transfers during fiscal
consolidation periods. Incorporating the heterogeneity of agents may be an interesting
extension. Second, we do not consider intergenerational conflicts. Expenditure cuts or tax
16increases may worsen the welfare of the current generation. However, if the consolidation
is not conducted appropriately, the burden on future generations may increase. Therefore,
incorporating intergenerational conflicts remains for future research. Third, some policy
shifts after a debt reduction would remain as an important policy issue. For example, thegovernment could use the policy space created by the debt reduction to decrease tax rates.

17

Appendix
A

Proof of Propositions 1-(a): Existence of the Steady StateThe first and second derivatives of Π(kg ) (the RHS of (21)) are[]Π′ (kg ) = (1 + τc ) (1 − τr )(1 + β)αAkgβ − {ρ + (1 − τr )δk } + σδg ,Π′′ (kg ) = αβ(1 + τc )(1 + β)(1 − τr )Akgβ−1 > 0.

(A.1)
(A.2)

Therefore, Π(kg ) is a convex and strictly increasing (decreasing) function of kg for kg ≥ (<
[
] β1
1
1
σδg
β
β
)[1/(1+β)] k g , where k g ≡ [1 − σδg /{ρ + (1 − τr )δk }] kg,γ . Note that k g ≡ 1 − ρ+(1−τr )δk kg,γ =[]1ρ+(1−τr )δk −σδg β
and k g < kg,γ . If 1 > σδg /{ρ + (1 − τr )δk }, Π(kg ) is equal to zero both(1−τr )αA1

when kg = 0 and kg = k g ≡ [1 − σδg /{ρ + (1 − τr )δk }] β kg,γ . On the other hand, if
1 ≤ σδg /{ρ + (1 − τr )δk }, Π(kg ) is equal to zero only when kg = 0. Figures 4 and 5 showthat Π(kg ) is a convex function of kg .
[Figures 4 and 5]
Next, we consider the properties of Λ(kg ) (the LHS of (21)). Seemingly, we haveΛ(0) = θζ. The first derivative of Λ(kg ) is[]Λ′ (kg ) = θβAkgβ−1 −2(σ − 1)(1 − τr )αb̄Akgβ + η ,(A.3)
where η ≡ σ(τ̃ + τc ) − [(1 − σ)(1 − τr )δk + ρ]b̄ − τc (1 − τr )α. If η > 0, Λ′ (kg ) < (=) > 0 if
and only if kg > (=) < k̃g . If η < 0, Λ′ (kg ) < 0 for kg ≥ 0. The second derivative of Λ(kg )is[]Λ′′ (kg ) = θβAkgβ−2 −2(2β − 1)(σ − 1)(1 − τr )αb̄Akgβ + (β − 1)η .
(A.4)
If η > 0, Λ′′ (kg ) has the following properties. When 12 ≤ β < 1, Λ′′ (kg ) < 0 holds for all
kg (≥ 0). When 0 < β < 12 , Λ′′ (kg ) ≤ 0 holds for 0 < kg < k̃g ≤ k̆g ≡ [(1−β)/(1−2β)]1/β k̃g ,and Λ′′ (kg ) > 0 for kg > k̆g . If η < 0, the following properties holds for Λ′′ (kg ). When1
≤ β < 1, Λ′′ (kg ) > 0 holds for 0 < kg ≤ k̆g , and Λ′′ (kg ) ≤ 0 for kg > k̆g .
2
In summary, Λ(kg ) has the following properties. When η > 0, Λ(kg ) is an increasing1and concave function for kg < k̃g ≡ [η/{2(σ − 1)(1 − τr )αb̄A}] β , while it decreases with kg
for kg > k̃g (see Figure 4). On the other hand, when η ≤ 0, Λ(kg ) becomes monotonicallydecreasing for kg > 0 (see Figure 5).
A steady state E ∗ always exists if and only if Λ(kg† ) > Π(kg† ), where kg† is the value ofkg (< kg∗ ) that satisfies Π′ (kg ) = Λ′ (kg ). Appendix B shows that this steady state is saddle
stable.17 Then, as shown in Figures 4 and 5, this steady state value kg∗ satisfies γ ∗ > 0 if
and only if Λ(kg,γ ) > Π(kg,γ ) or Λ′ (kg,γ ) > Π′ (kg,γ ) when Λ(kg,γ ) < Π(kg,γ ). That is,ρ + (1 − τr )δk< Θ,(1 − τr )αA(A.5)

It is possible that another steady state in which kg < kg∗ exists if ζ < 0 (see Figures 4 and 5). Let us
denote the value of kg in this steady state as kg,low . We find that this steady state is unstable in Appendix
B. Therefore, we find that the economy converges to the saddle stable economy kg∗ if kg,0 > kg,low .
Nevertheless, if kg,0 < kg,low , the economy can fail to develop, in which case the government in theeconomy with low public capital should avoid the fiscal policy of ζ < 0.
17

18

or
Θ<
where

ρ + (1 − τr )δk<Υ(1 − τr )αAwhen

ρ + (1 − τr )δk> Θ,(1 − τr )αA

(A.6)

{ [
] }β
θ (τ̃ + τc − ρb̄){ρ + (1 − τr )δk } − (1 − τr )(τc + τr )αδkΘ≡,(1 − τr )(1 + τc )αδg{Υ≡θβA[η − 2(σ − 1){ρ + (1 − τr )δk }b̄]
(1 + τc )[β{ρ + (1 − τr )δk } + σδg ]

β
} 1−β

.

Under (A.5) or (A.6), the conventional transversality condition (no-Ponzi game conditionof the government): (1 − σ)γ ∗ < ρ for σ > 1 is satisfied in the long run because of γ ∗ > 0.
Both Θ and Υ are decreasing in b̄. Then, the government can break the fiscal balance inthe long run if b̄ is large enough to break (A.5) or (A.6).

B

Proof of Proposition 1-(b): Stability of the Steady State

Approximating (7), (14), and (15) linearly around the steady states, we obtain
  


bt − b̄ḃ−ϕ00 ċ  =  Jcb Jcc Jckg   ct − c∗  .
kgt − kg∗Jkg b Jkg c Jkg kg
k˙g

(B.1)

Then, J = (Jij ) denotes the coeﬃcient matrix of the former system:
Jcb =g1 (b̄, c∗ , kg∗ )c∗ , Jcc = (1 + g2 (b̄, c∗ , kg∗ ))c∗ ,[][σ − (1 − τr )α]βA(kg∗ )β−1 ∗∗∗Jckg = g3 (b̄, c , kg ) −c,σJkg b =(θ + kg∗ )g1 (b̄, c∗ , kg∗ ), Jkg c = (θ + kg∗ )g2 (b̄, c∗ , kg∗ ) + kg∗ ,Jkg kg =g ∗ + (θ + kg∗ )g3 (b̄, c∗ , kg∗ ) − (1 + β)A(kg∗ )β + c∗ + δk − δg .

(B.2)

The structure of the first column of J means that one of the eigenvalues of J is −ϕ < 0. The¯ derived by deleting the first row
remaining two eigenvalues of J are those of the matrix J,
and column from J. To check for stability, we examine the sign of detJ¯ = Jcc Jkg kg −Jkg c Jckg .
Using (B.2) with (17), (18), and (21), we obtainc∗ Ψ(kg∗ )] ,}{detJ¯ = [1 + (1 − β)kg∗ − θβ b̄A(kg∗ )β−1 σ
where
[
]
Ψ(kg∗ ) ≡θβA(kg∗ )β−1 2(1 − σ)(1 − τr )αb̄A(kg∗ )β + η][− (1 + τc ) (1 − τr )α(1 + β)A(kg∗ )β − {ρ + (1 − τr )δk } + σδg .
If we use Λ′ (kg∗ ) and Π′ (kg∗ ), the above equation can be rewritten as][c∗] Λ′ (kg∗ ) − Π′ (kg∗ ) .
}{detJ¯ = [1 + (1 − β)kg∗ − θβ b̄A(kg∗ )β−1 σ19Figure 4 shows that in the steady state E ∗ , Λ′ (kg∗ ) < Π′ (kg∗ ) holds and, hence, detJ¯ < 0.
One of the eigenvalues of J¯ is positive and the other is negative. Then, the steady state islocally saddle-point stable. When ζ < 0, another steady state in which kg = kg,low can exist(see footnote 17). However, this steady state is unstable because Λ′ (kg,low ) > Π′ (kg,low )holds and, hence, detJ¯ > 0.

C

Proof of Proposition 2

From the LHS of (21), the magnitude of the shift in Λ(kg ) when b̄ increases is σθA(kg∗ )β [(1−σ)γ ∗ − ρ]. Because σ > 1, it takes a negative value for γ ∗ > 0. Thus, when b̄ falls, Λ(kg )shifts upward, as shown in Figures 4 and 5. As a result, kg∗ increases. Then, the long-run
growth rate, γ ∗ , increases (see (19)). Using (6) and the definitions of g, kg , and γ, we havethat θg ∗ = (γ ∗ + δg )kg∗ . Reductions in b̄ unambiguously increase g ∗ . Thus, Proposition 2
holds.

D

Parameter Values

We choose parameter values using the data of Greece. The labor income share is computedfrom the EU KLEMS database using data of labor compensation and gross value added,
which yield an average labor income share of 56.43% in the period 2000–2007.18 Thus, we
set 1 − α = 0.5643. According to the AMECO database, the average ratio of government
expenditure (including public investment) to GDP is 0.2170 in the period 2000–2008.19
During the same period, the average ratio of public investment to GDP is 0.0345. Since wehave θ = public investment/government expenditure, we set θ = 0.1590(= 0.0345/0.2170),
which means that investment by the government amounts to 15.9% of its total expenditure.
Following Baxter and King (1993), the elasticity of output with respect to public capitalis set to the ratio of public investment to GDP, namely β ≡ ϵ(1 − α) = 0.0345, whichyields ϵ = 0.0611. The intertemporal elasticity of substitution, 1/σ, is set to 0.4. Following
Kollintzas amd Vassilatos (2000), we set δk = 0.028 and δg = 0.031. These depreciation
rates are consistent with those of Papageorgiou (2012).
We calculate the tax rates based on a modified version of the methodology proposedby Mendoza et al. (1994), as described in Carey and Tchilinguirian (2000) in paragraphs
16–28, which yields average tax rates for 2000–2008 in Greece as τr = 0.2178, τw = 0.2818,
and τc = 0.1293. The results largely agree with those of Papageorgiou (2012). The only
diﬀerence is that he assumed self-employed earnings to be an imputed wage.
According to the AMECO database, the debt-to-GDP ratio of Greece in 2008 was
1.13, and the average annual compound growth rate of GDP for the period 2000–2008 was
about 3.5%. Then, we set binit = 1.13. The long-run growth rate in the initial steady state∗= 0.035. Using (19), (21), and G/Y = g/(Akgβ ), we choose the values of A
is set as γinit∗= 0.035 hold in the initialand ρ such that G0 /Y0 = 0.2170, B0 /Y0 (= b0 ) = 1.13, and γinit
steady state, which yields A = 0.465 and ρ = 0.0412.

18
19

The EU KLEMS database is freely accessible at http://www.euklems.net/.
The AMECO database is freely accessible at http://ec.europa.eu/economy finance/db indicators/ameco/index en.htm.

20

E

Welfare Eﬀects of b̄

the value of U0∗∗ , we calculate the dynamic path and initial value of Ut ≡∫To∞ calculatee−ρ(v−t) Cv1−σ /(1 − σ)dv using the relaxation algorithm. However, we cannot calculatetthe dynamic path and initial value of Ut directly because Ut does not remain constant in
the steady state. Let us define Xt ≡ Ut /Kt1−σ . Since Ct ≡ ct Kt , we haveẊt = ρXt −

c1−σ
β
t
− (1 − σ)(Akg,t− ct − gt − δk )Xt .
1−σ

In this case, Xt becomes constant over time in the high-growth steady state. Then, we
calculate the dynamic path and the initial value of Xt using the relaxation algorithm.
Since K0 is normalized to one, we have that U0∗∗ = X0 .
the utility function is given by (30), as in Section 7, we further define Xcg ,t ≡∫ ∞ If−ρteψ ln(Cg,t )dt − (ψ/ρ) ln Kt . Since Cg,t ≡ (1 − θ)gt Kt , we have0Ẋcg ,t = ρXcg ,t − ψ ln(1 − θ)gt −ψβ(Akg,t− ct − gt − δk ).
ρ

In this case, Xcg ,t becomes constant over time in the steady state, and we can calculate
the dynamic path and initial value of Xcg ,t . Since K0 = 1, we have that U0∗∗∗ = X0 + Xcg ,0under (30).

F

Debt Reduction based on Tax Increases and Expenditure Cuts

After the increase in the tax rates, the budget constraint of the government becomesḂt = rt Bt + Gt − [(τr + ∆τr )rt Wt + (τw + ∆τw )wt + (τc + ∆τc )Ct ] + Tt
= (1 − τr )rt Bt − (τ̃ Yt + τc Ct − Gt ) − {∆τr rt (Kt + Bt ) + ∆τw wt + ∆τc Ct − Tt },where the lump-sum transfer to the households is denoted as Tt , which we will specify
later. In addition, we have Ig,t = θGt .
In this setting, (13) is modified as
gt = g(bt , ct , kg,t ) +β∆t Akg,tβ−1
1 + [(1 − β)kg,t − θβ] bt Akg,t≡ g̃(bt , ct , kg,t , ∆t ),(F.1)

where g(bt , ct , kg,t ) is defined as in (13) and ∆t ≡ {∆τr rt (Kt +Bt )+∆τw wt +∆τc Ct −Tt }/Yt
represents the direct eﬀect of tax increases on public expenditure. If ∆t > 0, the tax
increases mitigate the negative eﬀect of the debt reduction on public expenditure. We
formulate Tt in such a way that ∆t > 0 gradually diminishes and eventually disappears
as the fiscal consolidation progresses (i.e., bt approaches b̄new ). Keeping in mind that the
β−1
β−1interest and wage rates can be rewritten as rt = αAkg,tkg,t −δk and wt = (1−α)Akg,t
Kg,t ,∗∗ β−1∗∗∗∗ β−1∗∗
Kg,t , where kg ∗∗kg,t − δk and wt = (1 − α)Akg
respectively, we define rt ≡ αAkg
denotes the new steady-state value of kg,t . Using these definitions, we specify Tt as
Tt = ∆τr r∗∗ (Kt + Bt ) + ∆τw wt∗∗ + ∆τc c∗∗ Kt ,

21

(F.2)

where c∗∗ is the new steady-state value of ct . Under (F.2), ∆t is expressed as[)1−β ][(]kct − c∗∗g,tβ∆t ≡ {α∆τr + (1 − α)∆τw + ∆τr αAkg,t bt } 1 −
+ ∆τc
.
kg∗∗
A(kg,t )βWhen ∆τr = ∆τw = ∆τc = 0, we have ∆t = 0. From the discussion in subsection 3.2, we
plausibly guess that, even with tax increases, debt reductions raise the steady-state valueof kg and, hence, kg∗∗ > kg,0 holds. The discussion in Subsection 3.3 reasonably suggeststhat ct increases initially in response to debt reductions. Then, under (F.2), we expectthat ∆t takes a positive value just after the policy change. As the fiscal consolidation
progresses, kg,t and ct converge to kg∗∗ and c∗∗ , respectively. Then, ∆t gradually diminishes
and eventually disappears.
Equations (14) and (15) are modified as
[
{
}
{
} ]
1
ρ
1 − τr − ∆τrβċt = ct + g̃(bt , ct , kg,t , ∆t ) − 1 − (1 − τr − ∆τr )α Akg,t − + 1 −δ k ct ,
σ
σ
σ
(F.3)
1+β
k̇g,t = (θ + kg,t )g̃(bt , ct , kg,t , ∆t ) − Akg,t+ ct kg,t + (δk − δg )kg,t .

(F.4)

The dynamic system of the economy is given by (7), (F.3), and (F.4). As in the benchmark
model, there exists a unique steady state. Note that in the steady state, ∆τw and ∆τcdisappear from the dynamic system. As long as ∆τr = 0, the steady state in this modified
model corresponds exactly to that of the benchmark model. When ∆τr > 0, this is not
the case, because ∆τr directly influences household behavior, and so is included in (F.3).

References[1] Arai, R., 2011. Productive government expenditure and fiscal sustainability. FinanzArchiv: Public Finance Analysis 67, 327–351.
[2] Barro, R.J., 1990. Government spending in a simple model of endogenous growth.
Journal of Political Economy 98, S103–S125.
[3] Barro, R.J., Sala-i-Martin, X., 2004. Economic Growth, 2nd edition. Cambridge, MA:
MIT Press.
[4] Bastagli, F., Coady, D., Gupta, S., 2012. Income inequality and fiscal policy. IMF Staﬀ
Discussion Note, No. SDN/12/08. Washington, DC: International Monetary Fund.
[5] Baxter, M., King, R., 1993. Fiscal policy in general equilibrium. American Economic
Review 83, 315–334.
[6] Bi, H., Leeper, M.E., Leith, C., 2013. Uncertain fiscal consolidations. The Economic
Journal 123, F31–F63.
[7] Bom, P.R.D., Ligthart, J.E., 2014. What have we learned from three decades ofresearch on the productivity of public capital? Journal of Economic Surveys 28,
889–916.
22

[8] Bruce, N., Turnovsky, S.J., 1999. Budget balance, welfare, and the growth rate: “Dynamic scoring” of the long-run government budget. Journal of Money, Credit, andBanking 31, 162–186.
[9] Carey, D., Tchilinguirian, H., 2000. Average eﬀective tax rates on capital, labour,and consumption. OECD Economics Department Working Papers, No. 258, OECD
Publishing. http://dx.doi.org/10.1787/247874530426.
[10] Chamley, C., 1986. Optimal taxation of capital in general equilibrium with finite lives.
Econometrica 54, 607–622.
[11] Coenen, G., Mohr, M., Straub, R., 2008. Fiscal consolidation in the Euro area: Longrun benefits and short-run costs. Economic Modeling 25, 912–932.
[12] Cogan, F.J., Taylor, B.J., Wieland, V., Wolters, H.M., 2013. Fiscal consolidation
strategy. Journal of Economic Dynamics and Control 37, 404–421.
[13] Dioikitopoulos, E., Kalyvitis, S., 2008. Public capital maintenance and congestion:
Long-run growth and fiscal policies. Journal of Economic Dynamics and Control 32,
3760–3779.
[14] Erceg, J.C., Lindé, J., 2013. Fiscal consolidation in a currency union: Spending cutsvs. tax hikes. Journal of Economic Dynamics and Control 37, 422–445.
[15] European Commission, 2012. Public finances in EMU – 2012, European Economy No.
4, 2012.
[16] Fisher, W.H., Turnovsky, S.J., 1998. Public investment, congestion, and private capital accumulation. The Economic Journal 108, 399–413.
[17] Forni, L., Gerali, A., Pisani, M., 2010. The macroeconomics of fiscal consolidations
in Euro area countries. Journal of Economic Dynamics and Control 34, 1791–1812.
[18] Futagami, K., Iwaisako, T., Ohdoi, R., 2008. Debt policy rule, productive government
spending, and multiple growth paths. Macroeconomic Dynamics 12, 445–462.
[19] Futagami, K., Morita, Y., Shibata, A., 1993. Dynamic analysis of an endogenous
growth model with public capital. Scandinavian Journal of Economics 95, 607–625.
[20] Ghosh, S., Mourmouras, I.A., 2004. Endogenous growth, welfare, and budgetary
regimes. Journal of Macroeconomics 26, 623–635.
[21] Greiner, A., 2007. An endogenous growth model with public capital and sustainable
government debt. Japanese Economic Review 58, 345–361.
[22] Greiner, A., 2010. Economic growth, public debt, and welfare: Comparing threebudgetary rules. German Economic Review 12, 205–222.
[23] Greiner, A., 2012a. Public debt in a basic endogenous growth model. Economic Modelling 29, 1344–1348.

23

[24] Greiner, A., 2012b. Public capital, sustainable debt, and endogenous growth. Researchin Economics 66, 230–238.
[25] Greiner, A., Semmler, W., 2000. Endogenous growth, government debt and budgetary
regimes. Journal of Macroeconomics 22, 363–384.
[26] Guo, J.-T., Harrison, S.G., 2008. Useful government spending and macroeconomic(in)stability under balanced-budget rules. Journal of Public Economic Theory 10,
383–397.
[27] IMF, 2014. World Economic Outlook: Legacies, Clouds, Uncertainties. October.
[28] Kalaitzidakis, P., Kalyvitis, S., 2004. On the macroeconomic implications of maintenance in public capital. Journal of Public Economics 88, 695–712.
[29] Kamiguchi, A., Tamai, T., 2012. Are fiscal sustainability and stable balanced growthequilibrium simultaneously attainable? Metroeconomica 63, 443–457.
[30] Kollintzas, T., Vassilatos, V., 2000. A small open economy model with transaction
costs in foreign capital. European Economic Review 44, 1515–1541.
[31] Mendoza, E., Razin, A., Tesar, L., 1994. Eﬀective tax rates in macroeconomics: Crosscountry estimates of tax rates on factor incomes and consumption. Journal of Monetary Economics 34, 297–323.
[32] Minea, A., Villieu, P., 2013. Debt policy rule, productive government spending, andmultiple growth paths: A note. Macroeconomic Dynamics 17, 947–954.
[33] Ni, S., 1995. An empirical analysis on the substitutability between private consumption and government purchases. Journal of Monetary Economics 36, 593–605.
[34] Papageorgiou, D., 2012. Fiscal policy reforms in general equilibrium: The case of
Greece. Journal of Macroeconomics 34, 504–522.
[35] Teles, V., Mussolini, C., 2014. Public debt and the limits of fiscal policy to increase
economic growth. European Economic Review 66, 1–15.
[36] Trimborn, T., Koch, K.J., Steger, T.M., 2008. Multidimensional transitional dynamics: A simple numerical procedure. Macroeconomic Dynamics 12, 301–319.
[37] Turnovsky, S.J., 1997. Fiscal policy in a growing economy with public capital. Macroeconomic Dynamics 1, 615–639.
[38] Yakita, A., 2008a. Aging and public capital accumulation. International Tax and
Public Finance 15, 582–598.
[39] Yakita, A., 2008b. Sustainability of public debt, public capital formation, and endogenous growth in an overlapping generations setting. Journal of Public Economics
92, 897–914.

24

25

Parameter or variable
1−α
A
ϵ
σ
ρ
K̇g,0 /Y0
G0 /Y0
θ
τr
τw
τc
b0
δk
δg
ϕ

Description
Labor income share
Total factor productivity
Public capital elasticity in labor productivity
Curvature parameter in the utility function
Time discount rate
Initial government investment-to-output ratio
Initial government expenditure-to-output ratio
Government investment-to-government expenditure ratio
Tax rate on capital income
Tax rate on labor income
Tax rate on consumption
Initial debt-to-GDP ratio
Depreciation rate for private capital
Depreciation rate for public capital
Adjustment parameter of the speed of debt reduction

Table 1: Benchmark Parameters
Value
0.56430.46500.06112.5
0.04120.03450.21700.15900.21780.28180.1293
1.13
0.028
0.031
0.01, 0.025, 0.05, 0.075, 0.1

SourceData average
Calibrated
Calibrated
Set
Calibrated
Data average
Data average
Data average
Data average
Data average
Data average
Data average
Set
Set
Set

Table 2: Data Averages and Solutions in the Initial Steady State
Variable
C0 /Y0(K0 + Kg,0 )/Y0
T R0 /Y0

Description
Consumption-to-output ratio
Total capital-to-output ratio
Tax revenue-to-output ratio

Data averages Solution
0.69850.6405
3.558
2.78500.32620.3634

Table 3: Welfare Eﬀects (θ = 0.1590)
b̄new
0.6

∗
γH,new
0.0353800.03575∆C0∆U0∆Cequiv∆C0∆U0∆Cequiv

ϕ =0.01 ϕ =0.025 ϕ =0.05 ϕ =0.075 ϕ =0.1
0.382%
0.9%
1.65%
2.28%
2.82%
0.559%
1.31%
2.38%
3.26%
4%
0.374%
0.883%
1.62%
2.23%
2.76%
(20.9)
(8.4)
(4.2)
(2.8)
(2.1)
0.819%
1.95%
3.62%
5.07%
6.34%
1.19%
2.8%
5.1%
7.01%
8.63%
0.803%
1.91%
3.55%
4.96%
6.21%
(9.3)
(3.7)
(1.9)
(1.2)
(0.9)

The figures in parentheses are the years taken for the debt-to-GDP ratio to decrease by
10% from 1.13 to 1.03.

26

Table 4: Welfare Eﬀects with Tax Increases
b̄new
0.6

Revenue ϕ =0.01 ϕ =0.025 ϕ =0.05 ϕ =0.075 ϕ =0.1
Benchmark0.559%
1.31%
2.38%
3.26%
4%
τr 1% ↑
-7.35%-6.59%-5.52%-4.63% -3.88%
2% ↑-10.3%-9.56%-8.47%-7.57% -6.81%
3% ↑-13.6%-12.8%-11.7%-10.8%-10%
τw 1% ↑
-0.549%0.174%
1.2%
2.04%
2.75%
2% ↑
-0.863%-0.148% 0.861%
1.69%
2.39%
3% ↑
-1.17%-0.467% 0.529%
1.35%
2.04%
τc 1% ↑
0.122%
0.874%
1.95%
2.84%
3.6%
2% ↑
0.00149% 0.753%
1.83%
2.72%
3.49%
3% ↑
-0.116%
0.635%
1.71%
2.61%
3.38%

Table 5: Welfare Eﬀects: Public Services (θ = 0.1590)
ψ ϕ =0.01 ϕ =0.025 ϕ =0.05 ϕ =0.075
00.559%
1.31%
2.38%
3.26%
0.1 0.692%
1.55%
2.7%
3.63%
0.3 0.904%
1.94%
3.23%
4.21%
0.5 1.06%
2.24%
3.63%
4.65%

ϕ =0.1
4%
4.39%
4.99%
5.46%

Table 6: Welfare Eﬀects: Public Services and Tax Increases
b̄new
0.6

Revenue ϕ =0.01 ϕ =0.025 ϕ =0.05 ϕ =0.075 ϕ =0.1
Benchmark
1.06%
2.24%
3.63%
4.65%
5.46%
τr 1% ↑
-6.1%
-4.95%-3.57%-2.56% -1.75%-8.59%-7.43%-6.05%-5.03% -4.23%
2% ↑
3% ↑
-11.3%-10.1%-8.73%-7.7%-6.89%
0.637%
1.76%
3.09%
4.07%
4.85%
τw 1% ↑
0.505%
1.61%
2.93%
3.9%
4.66%
2% ↑
3% ↑
0.371%
1.47%
2.77%
3.72%
4.48%
0.885%
2.02%
3.38%
4.4%
5.2%
τc 1% ↑
2% ↑0.835%
1.96%
3.32%
4.32%
5.13%
0.786%
1.9%
3.25%
4.25%
5.06%
3% ↑

27

(a)
1.1

B/Y

1
0.9
0.8
0.7
0

5

10

15

20

25

30

35

40

25

30

35

40

25

30

35

40

25

30

35

40

25

30

35

40

(b)
Public Investment /Y

0.04
0.035
0.03
0

5

10

15

20
(c)

growth rate of K

0.044
0.042
0.04
0.038
0.036
0

5

10

15

20
(d)

0.0352growth rate ofC
0.0351
0.035
0.0349
0

5

10

15

20
(e)

c = C/K

0.29
0.285
0.28
0.275
0.27
0.265
0

5

10

φ=0.0115

20

φ=0.025

φ=0.05φ=0.075

Figure 1: Transition Dynamics

28

φ=0.1

1
0.9

0

0.5
1

-0.
5

0.5

-0.

0.4

1

1

-1.
5

5

1.5

0.5

-2

0

0

1
-0.5

-1
-0.

0.1

0.5

0

0.2

0

1.5

2

1

τc

0.3

2

0.5

0

0.6

2.5

1.5

0.8
0.7

2.5

2

-1.
5
0.2

5

0

-1

0.5

-1.5

-1
0.4

τw

0.6

0.8

1

Figure 2: The Relationship between b̄opt and (τc , τw )

29

-2

bopt

τr

τw

(a)−1

(b)−1

Public Investment/Y

0.042

0.042

0.04

0.04

0.04

0.038

0.038

0.038

0.036

0.036

0.036

0.034

0.034

0.034

0.032

0.032

0.032

0

20

40

0

growth rate of K

20

40

0

(b)−2
0.039

0.039

0.036

0.038

0.038

0.037

0.037

0.036

0.036

0.034
0.032
0.03
0.028
20

40

0.035

0

(a)−3

20

40

0.035

0

(b)−3

0.034

20

40

(c)−2

0.038

0

growth rate of C

(c)−1

0.042

(a)−2

20

40

(c)−3

0.03520.03520.03510.0351

0.035

0.035

0.032
0.03
0.028
0

20

40

0.0349020

400.0349

0.29

0.28

0.28

0.285
0.28

0.275

0.275

0.27

0.27

0.27

0.265

0.265
20

40

40

(c)−4

0.285

0.275

20

0.29

0.285

0

0

(b)−4

(a)−4
0.29
c = C/K

τc

0.265
0

x=0

20

x=1%

40

0

x=2%

x=3%

Figure 3: Transition Dynamics with Tax Increases

30

20

40

σδg
(a) 1 > ρ+(1−τr )δkП(kg) Λ(kg)

П(kg)
E*
Λ(kg) upward shift
when b falls

Λ(kg); ζ ≥ 0

θζ
0

kg

~

kg kg,γ kg*kg

Λ(kg); ζ < 0
γ≤0

γ>0

σδg
(b) 1 ≤ ρ+(1−τr )δkП(kg) Λ(kg)П(kg)E*
Λ(kg) upward shift
when b falls

Λ(kg); ζ ≥ 0

θζ
0

kg

~

kg,γ kg* kg

Λ(kg); ζ < 0
γ≤0

γ>0

Figure 4: Steady State When η > 0
31

σδg(a) 1 > ρ+(1−τr )δkП(kg) Λ(kg)П(kg)θζΛ(kg)upward shiftE*when b falls

Λ(kg); ζ ≥ 0
0kg

kg kg,γ kg*

Λ(kg); ζ < 0
γ≤0

γ>0

σδg
(b) 1 ≤ ρ+(1−τr )δkП(kg) Λ(kg)
П(kg)

θζ

E*

Λ(kg)upward shift
when b falls

Λ(kg)

0

kg,γ
γ≤0

kg

kg*
γ>0

Figure 5: Steady State When η ≤ 032