GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 267

Dynamic analysis of reductions in public debt in an
endogenous growth model with public capital

Noritaka Maebayashi, Takeo Hori and Koichi Futagami

June 2012GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, 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 with productive public capital andgovernment debt when government debt is adjusted to the target level. We examine
how reducing public debt in an economy with a large public debt aﬀects the transitionof the economy and welfare. We ﬁnd that the government faces the following trade oﬀ
when reducing its debt. In the short run, public investment is reduced to decrease thedebt and this has a negative eﬀect on welfare. However, as the interest payments on
the debt decrease, public investment begins to increase. Eventually, the government
can increase the amount of public investment by more than before. This has apositive eﬀect on welfare, implying that reducing the debt is welfare improving.
Furthermore, we ﬁnd that the adjustment speed of reductions in debt aﬀects welfare
crucially. The relationships between the welfare gains and the adjustment speed are
U-shaped in many cases. However, they are decreasing monotonically when the tax
rate is low and the initial debt–GDP ratio is suﬃciently large.

JEL classiﬁcation: E62, H54, H63
Keywords: Reductions in public debt, Debt policy rule, Public capital, Endogenous Growth1

Introduction

Recently, many developed countries have been suﬀering from large levels of government
debt. In Europe, Greece has experienced severe government ﬁnancial failure. Public debt
as a share of GDP in Greece equaled 142.8 percent in 2010. The average level of public
debt in the European Union (EU) was close to 80 percent of GDP in 2010.1 In the UnitedStates (USA), the ratio was around 62 percent of GDP in 2010. In Japan, it is 225.8∗

Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
JAPAN; e-mail: gge010mn@mail2.econ.osaka-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.jp
1
The source of these data is Eurostat, which is owned by the European Commission. The debt–
GDP ratios of the main countries of the EU are as follows: Austria, 72.3%; Belgium, 96.8%; Denmark,43.6%; Finland, 48.4%; France, 81.7%; Germany, 83.2%; Hungary, 80.2%; Ireland, 96.2%; Italy, 119%;
Netherlands, 62.7%; Poland, 55%; Portugal, 93%; Spain, 60.1%; and United Kingdom, 80%.

1

percent. If no adjustments are made and public debt keeps increasing, these countries
may face a situation similar to that of Greece.
In the EU, there is a constraint on the size of government debt to reduce the chanceof government ﬁnancial failure. The Maastricht Treaty states that EU Member States are
required to ensure that their government debt is below 60 percent of GDP. However, many
countries in the EU have violated this criterion. In response, the European Commission has
given the Member States projections for ﬁscal consolidation. In Public ﬁnances in EMU– 2011, published in 2011, the European Commission states that the public deﬁcits must
decrease faster and also that public debts should hit their inﬂection points in 2012, andthen begin to fall as a share of GDP from then on. In September 2011, President Barack
Obama sent a blueprint to Congress on how the USA can reduce its public deﬁcit and pay
down public debt. He presented a plan that aims to realize more than three trillion in net
deﬁcit reduction over the next 10 years. In his statement, the debt under this plan would
be declining as a share of GDP over the next decade, falling from a high of 77 percent of
GDP in 2013 to 73 percent of GDP in 2021. When considering the restructure of public
ﬁnances in many developed countries, it is important to investigate how a reduction in
public debt aﬀects economic growth and welfare.
There are two main ways to reduce the debt levels. One is to increase tax rates, while
the other is to cut spending. Some countries insist that spending cuts should be chosen,
and others insist on a mix of spending cuts and tax increases. In the USA, the president
plans to use both. On the other hand, in the EU, the European Commission proposes
to use not tax increases but spending cuts for ﬁscal consolidation because evidence fromthe past indicates that expenditure-based consolidations (spending cuts) tend to havegreater success. In November 2011, Germany, the Netherlands, Romania and the United
Kingdom implemented spending cuts whereas France, Italy, Ireland, Greece, Portugal andSpain unveiled tax increases in addition to spending cuts.
This paper examines how reducing public debt aﬀects economic growth and welfare
when the government implements spending cuts, which are being discussed widely in USA
and many European countries. When a government reduces debt levels by decreasing public spending, it faces the following trade oﬀ. In the short run, a decline in public spending
worsens welfare levels because it implies a reduction in the amount of public services orinfrastructure necessary for economic activities. However, a decline in public debt reduces
interest payments and hence enables the government to increase public spending in thelong run, thus improving welfare levels. Therefore, it is important to investigate whether
reducing public debt by cutting public spending is really welfare improving. If it is welfare
improving, should the government reduce its debt at a slower pace or at a more rapid pace?
If the pace of the reduction is slow, the initial decline in public spending may be small.
However, because the government takes a longer time to reduce its debt to the target level,
the decline in public spending may be prolonged. On the other hand, a rapid reduction in
debt may result in large initial declines in public spending. However, these large declines
may be short lived because the government will take less time to reduce its debt level.
Elucidation of this relationship between welfare and the speed of debt reduction would be
helpful to the policy makers of countries with large public debt.
Furthermore, although there may be many types of government spending that canbe cut, we discuss cuts in public investment spending in this paper. This is because
in some countries, public investment is cut to reduce debt. In a paper published by
2

European Union, Rubianes (2010) writes, “In the event that the necessary adjustment ofpublic ﬁnance is made at the expense of public investment, it could even fall below thecurrent level and compromise competitiveness and future gains of productivity over the
medium and long run.” In addition, when the government holds too much debt, private
investment in the economy is crowded out severely. In this situation, public investment
may not contribute adequately to growth if there is complementarity in goods production
between private and public capital.2 . Then, ineﬃcient public investment ﬁnanced bypublic debts may only increase the interest payments on the debt, resulting in an increasein outstanding debt. Therefore, cuts in public investment to reduce the debts may benecessary to improve the eﬃciency of public investment.
We construct an endogenous growth model with public debt ﬁnance, where the growthengine is productive public capital as in Futagami et al. (1993) and Turnovsky (1997).
In order to investigate the welfare eﬀects of reducing debt, we introduce the debt policy
rule introduced by Futagami, Iwaisako and Ohdoi (2008). Under this rule, the government
sets a target level of the debt-to-private-capital ratio and reduces its debt gradually to thetarget level. Public debt is reduced by spending cuts in public investment.
In this paper, we obtain the following three main results from the numerical analyses.
First, we obtain the transitional behavior of the economy when the government reduces
public debt. In the short run, reducing public debts decreases the growth rate of public
capital and this enables more resources to be allocated to private investment. Therefore,
the growth rate of private capital increases initially. As a result, the return to private
capital (the interest rate) decreases and then the growth rate of consumption declines inthe early stages of the transition. However, as the level of public debt becomes small, public
investment begins to increase and this, in turn, crowds out private investment. Then, thegrowth rate of private capital begins to decline. This decline of private investment increases
the interest rate and then the growth rate of consumption begins to recover. In the longrun, the economy can attain higher growth rates of public capital, private capital andconsumption than those in the state before the government reduced debt. The reason
for this is as follows. Because government debt is reduced, the interest payments become
smaller than in the initial state. Hence, public investment becomes larger than before thegovernment began to reduce its debt, which makes it possible to attain higher rates of
economic growth in the long run.
Second, we ﬁnd that reducing debt at the expense of public investment improves welfare. In the short run, households change their consumption level immediately after thepolicy change. Whether this short-run eﬀect becomes positive or negative depends on how
expected future income changes because of the policy change. Then, when the government reduces its debt, the growth rate in consumption decreases in the early stage of thetransition, as we stated above. This has a negative welfare eﬀect during the transition.
However, in the long run, the growth rate increases above the initial level, which has apositive welfare eﬀect. Moreover, countries whose initial debt–GDP ratio is larger canachieve larger welfare gains. This is mainly because when the initial debt–GDP ratio islarge, the growth rate of consumption in the new steady state is much higher than that in
the initial steady state. In addition, although governments must make larger reductions
in public debt and the decline of public investment is larger, more resources are released
2

Futagami, Morita and Shibata (1993) and Turnovsky (1997) apply complementarity between private
and public capital in goods production in their analysis.

3

to the private sector and households can consume more.
Third, the adjustment speed of the reduction in debt is an important determinant ofwelfare. The extent of welfare gains from a decrease in government debt as the government
increases the speed of debt reduction depends on the tax rate and on how much the debt–
GDP ratio exceeds the target level initially. In many cases, (i) the relationships between
the welfare gains and adjustment speed are U-shaped. However, (ii) the welfare gains
decrease monotonically with respect to the adjustment speed when the tax rate is low andthe initial debt–GDP ratio is suﬃciently large. If the government increases the speed of
public debt reduction, it faces the following trade-oﬀ. On the one hand, the initial declineof public investment increases. On the other hand, the growth of consumption recovers
sooner. When the tax rate is low and the initial debt–GDP ratio is large, the initialdecline of public investment is too large when the government increases the adjustment
speed. Then, the welfare gains decrease monotonically with respect to the adjustment
speed. In contrast, when the tax rate is high or the initial debt–GDP ratio is small, theinitial decline of public investment is small. Then, when the adjustment speed is somewhat
high, the positive eﬀect of the earlier recovery of the growth of consumption dominates thenegative eﬀect of the initial decline of public investment. Thus, the relationship between
the welfare gain and the adjustment speed is U-shaped.
Besides these main results, there is also a possibility that the economy will fail to develop sustainably. When the initial level of public capital is suﬃciently small relative to
private capital, if the ﬁnancing of public spending relies largely on issuing debt, the economy may fall into a development trap. In this trap, there exists no path the representative
households with rational expectation and perfect foresight can follow. This occurs because
tax revenue is too low to pay interest payments on debt because of scarce public capital.
We show that the economy can avoid this situation by selecting tax ﬁnancing rather thandebt ﬁnancing in its development process.
In the theoretical literature, many studies examine the ﬁscal policy of productive public
spending with debt ﬁnancing; for example, see Greiner and Semmler (2000), Ghosh andMourmouras (2004) and Yakita (2008).3 However, these studies deal not with the debt size
criterion, but rather with the deﬁcit constraint in the Maastricht treaty.4 In contrast tothese studies, Futagami et al. (2008) introduce a target rule for public debt size as we statedabove, whereby public debt is issued to ﬁnance productive public ﬂow services such as inBarro (1990). They show that there exist two steady states, and the transition paths areindeterminate.5 Because the reduction of public debt generates transitional dynamics, theoccurrence of indeterminacy of transition paths makes welfare analysis diﬃcult. Therefore,
3

Greiner and Semmler (2000) study how debt-ﬁnanced ﬁscal policy aﬀects growth in the long run.
They introduce the golden rule of public ﬁnance (GRPF). Under this rule, public debt is issued only for
public investment and government consumption, and the repayment of the principle and interest payments
must be ﬁnanced by tax revenue while the deﬁcits are assumed to be a constant percentage of GDP. Theyshow that a less strict policy rule does not necessarily promote growth. Ghosh and Mourmouras (2004)
show the GRPF can avoid over-accumulation of public capital and be more eﬃcient than the standard
constraint of government. Yakita (2008) calculates the threshold that determines whether government
debt is sustainable or not in an overlapping generations model.
4
Under the Maastricht treaty, EU Member States are required to ensure their public deﬁcits do not
exceed three percent of GDP.
5
Minea and Villieu (2012) ﬁnd that if the government does not set the debt target, deﬁned as the ratioof bonds to GDP, equal to the value of private capital as in Futagami et al. (2008), there exists a uniquesteady state and the indeterminacy is removed.

4they do not investigate the welfare eﬀects of reductions in public debt. We solve thisproblem by replacing the ﬂow of public services with the stock of public capital. In thispaper, the economically meaningful steady state is unique and thus the transition path isdetermined uniquely. Therefore, we obtain interesting results regarding the welfare eﬀects
and transitional responses of debt reduction.
The remainder of this paper is organized as follows. Section 2 presents the basicmodel. Section 3 derives the equilibria and transition dynamics of the economy. Section 4
conducts the comparative statics analyses of the equilibria and analyzes the policy eﬀects
on economic growth and public investment. Section 5 examines the transitional responses
when the government lowers the target level of debt. Section 6 presents the policy eﬀects
on welfare. Section 7 concludes the paper.

2

The Model

Our model is based on the model of Futagami et al. (2008). The only diﬀerence between
their model and ours is that productive government inputs used in the production of
goods are public capital not public ﬂow services. We consider an economy populatedby an inﬁnitely long-lived representative household with an inﬁnite planning horizon andperfect foresight. Time is denoted as t ≥ 0. Without any loss of generality, we assume
that there is no population growth and the population size is normalized to unity.

2.1

Production structure

A single ﬁnal good, Yt , is produced with private capital, Kt , public capital, Kg,t , and labor,
Lt . The production function takes the Cobb-Douglas form as follows:
Yt = AKtα (Kg,t Lt )1−α ,α ∈ (0, 1).

(1)

The ﬁrst-order conditions for proﬁt maximization are given by:1−αrt = αAkg,t
,

(2)

1−α
wt = (1 − α)Akg,tK,(3)

where rt and wt denote the interest rate and the wage rate, and kg,t (≡ Kg,t /Kt ) is theratio of public capital to private capital.

2.2

Households

The utility function of the representative household is speciﬁed as:∫ ∞U0 =(ln Ct )e−ρt dt,(4)0

where Ct and ρ(> 0) denote consumption and the subjective discount rate, respectively.
The household’s budget constraint is given by:Ẇt = (1 − τ )(rt Wt + wt ) − Ct ,5(5)where Wt denotes assets and τ ∈ [0, 1) is the income tax rate, which is assumed to be timeinvariant, as in Futagami et al. (2008). Taking the interest rate, wage rate and tax rate asgiven, the household chooses its consumption path so as to maximize (4) subject to (5).
Intertemporal maximization yields the following condition:Ċt= (1 − τ )rt − ρ.
Ct

(6)In addition, the following transversality condition must hold:lim Ct−1 Wt exp (−ρt) = 0.

t→∞

2.3

(7)

Government

The government in this economy imposes income taxation, and issues bonds, Bt , to ﬁnancepublic capital investment, Gt . For simplicity, we assume that the depreciation rate of public
capital is zero. The evolution of public capital is given by:K̇g,t = Gt .

(8)

The budget constraint of the government is:Ḃt = rt Bt + Gt − τ (rt Wt + wt ).

(9)

Following Futagami et al. (2008), we assume that the government adjusts its bonds
gradually to a target level. In this paper, we gauge the size of the economy by the levelof private capital, Kt , as in Futagami et al. (2008). We assume the government adjusts
bt ≡ Bt /Kt according to the following rule:b˙t = −ϕ(bt − b̄),

(10)

where b̄ and ϕ(> 0) denote the target level of government bonds and the adjustment
coeﬃcient of the rule, respectively.6 We consider the case of b̄ > 0. The government must
adjust the level of government spending Gt at every point in time, given this bond issuancerule and the tax revenue. When the government reduces the target, b̄, it has to reduce
debt according to the diﬀerence between the current and the target level of b: bt − b̄. If theadjustment coeﬃcient, ϕ, takes a large value, the government adjusts bt to the target level
rapidly. When ϕ is relatively small, the government adjusts bt slowly to the target level.
Under this debt policy rule, the reduction of b̄ accompanies the cut in public investment,
G, in the short run in order to satisfy (9). We see this clearly in the next section. Then,the expenditure-based reduction of public debt is formulated as following the discussionof ﬁscal consolidation in Public ﬁnances in EMU – 2011 and Rubianes (2010).

6

We can set the ratio of debt to GDP as the target. By using b̄, we can calculate the ratio of debt toGDP because B/Y = (B/K)(K/Y ) = b̄/(A(kg )1−α ).

63Equilibrium

3.1

Dynamic system

In this section, we ﬁrst derive the dynamic system of the economy, and then examine theexistence and the stability of the steady-state equilibria in Subsections 3.2 and 3.3.
We derive the equilibrium of this economy by using market equilibrium conditions.
The labor market equilibrium condition is Lt = 1 because the population size is unityand each household supplies one unit of labor inelastically. The asset market clears as
Wt = Kt + Bt . Substituting these into (9) and using (1), (2) and (3), we obtain:Ḃt = (1 − τ )rt Bt − (τ Yt − Gt ).

(11)

We assume that the depreciation rate of private capital is zero for simplicity. Then, thegoods market equilibrium condition is given by:K̇t = Yt − Ct − Gt .

(12)

Let us deﬁne ct ≡ Ct /Kt and gt ≡ Gt /Kt . By substituting (2) into (6), and using (1) and(12), we obtain:
1−αc˙t = [ct + gt − {1 − (1 − τ )α} Akg,t− ρ]ct .
(13)
By using (1), (8) and (12) we have:2−αk̇g,t = (1 + kg,t )gt − Akg,t+ ct kg,t .

By using (11) with (1), (2) and the deﬁnition of bt , we can rewrite (10) as:
}
{
K̇
t
1−α
1−α
bt − τ Akg,t+ gt .
b˙t = −ϕ(bt − b̄) = (1 − τ )αAkg,t
−
Kt

(14)

(15)

Substituting (12) into (15) and solving for gt , we obtain the ratio of public investment to
private capital, gt ≡ Gt /Kt , as follows:
gt =

}
1 {
1−α
1−α
1−α
τ (1 + αbt )Akg,t− αbt Akg,t− ϕ(bt − b̄) + (Akg,t− ct )bt .
1 + bt(16)

The ﬁrst term of the right-hand side (RHS) denotes the tax revenue of the government andshows that, other things being equal, gt increases as τ rises. The second term represents
the interest payments on the government debt. Other things being equal, gt increases as
bt decreases. The third and the last terms are given by the debt policy rule (10). The
third term shows that given bt , if the government reduces the target level of its debt b̄,the volume of public investment falls in the short run. When ϕ takes a larger value,decreases in public investment in the short run are also larger because the government
must reduce its debt more rapidly and hence its budget becomes tight. However, as bt
decreases steadily because of a decrease in b̄ under the rule given by (10), the interest
payments on government debt decrease gradually. This enables the government to spend
more of its revenue on public investment. Finally, the last term implies that a higher ratioof public to private capital, kg , because of a high growth rate of Kt , has a positive eﬀect
7

on public investment.7 When Kt grows at a high rate, the government requires little eﬀort
to reduce bt ≡ Bt /Kt to b̄. Then, high growth of Kt enables the government to increase
public investment.
Substituting (16) into (13) and (14), respectively, we obtain the following dynamic
system with respect to ct and kg,t .
1[ct − ζ(kg,t , τ, bt ) − ϕ(bt − b̄)]ct ,1 + bt}1 {(kg,t − bt )ct − η(kg,t , τ, bt ) − ϕ(bt − b̄)(1 + kg,t ) ,k̇g,t =
1 + bt
c˙t =

(17)
(18)

where
1−α
+ ρ(1 + bt ),ζ(kg,t , τ, bt ) ≡ (1 − τ )(1 − α)Akg,tη(kg,t , τ, bt ) ≡ [(1 − τ )(1 + αbt )kg,t − {[1 − (1 − τ )α]bt + τ }]Akg1−α .
Equations (10), (17) and (18), together with the initial values kg,0 and b0 , and the transversality condition (7), characterize the dynamics of the economy.

3.2

Steady states

Now we derive the steady state of the economy where ct , kg,t and bt become constant overtime. Setting c˙t = 0 and k̇g,t = 0 and bt = b̄ in (17) and (18), respectively, results in:c = ζ(kg , τ, b̄), and (kg − b̄)c = η(kg , τ, b̄).

(19)

We omit the time index t in the above because c and kg become constant over time in thesteady state. By eliminating c from the two equations of (19), we have:[(1 − τ )αkg − τ ]Akg1−α = ρ(kg − b̄).

(20)This equation determines the steady-state value kg∗ . Substituting kg∗ into the ﬁrst equationof (19), we obtain the steady-state value c∗ . Let us denote the left-hand side (LHS) of(20) as Λ(kg ). As shown in Figure 1, Λ(kg ) is a convex function of kg that equals zeroτ
when kg = 0 and kg = (1−τ(see Appendix A). We denote the RHS of (20) as Π(kg ).
)αApparently, Π(kg ) is a straight line whose slope is ρ, and equals zero when kg = b̄ holds.
Let us deﬁne k̃g by Λ′ (k̃g ) = ρ. As shown in Appendix A, there exists a unique k̃g > 0.
Note that at kg = k̃g , both sides of (20) have the same slope.
[Figure 1]
Figure 1 shows that if Λ(0) > Π(0) and Λ(k̃g ) < Π(k̃g ) hold, there exist two steadystates. Because Λ(0) ≡ 0, Π(0) ≡ −ρb̄ and b̄ > 0, the ﬁrst inequality is satisﬁed. The
second inequality is equivalent to:1−α[(1 − τ )αk˜g − τ ]Ak˜g˜.
b̄ < kg −
ρ

(21)

1−α
Eq.(12) shows that given Gt , when the diﬀerence between Yt /Kt = Akg,tand Ct /Kt = ct becomeslarge, the growth rate of private capital rises.
78

If b̄ is larger than the RHS of the above inequality, there exist no paths that the rational
representative households can choose.8 Therefore, we henceforth assume (21). We denote∗∗∗∗and kg,L(kg,H> k̃g > kg,L
> 0).
the steady-state value of kg in each steady state as kg,H
From (2) and (6), the long-run growth rate, γ = Ċt /Ct = K̇t /Kt = K̇g /Kg , is given by:
γ = (1 − τ )αA(kg∗ )1−α − ρ,

(22)

∗
∗
where kg∗ is equal to kg,Hor kg,L. We can conﬁrm immediately that the long-run growth∗rate increases with kg . This is because the return on private capital increases with kg∗ and∗the high-growth
the households save more. From now on, we call the steady state with kg,H∗
steady state, whereas we call the steady state with kg,L the low-growth steady state. From
the discussion so far, we obtain the following proposition.

Proposition 1
There exist two steady states, the high-growth and the low-growth steady states, if (21)holds.
As in Futagami et al. (2008), there exist two steady states in our model. In their model,
both of the steady states are meaningful economically. In our model, only the high-growthsteady state is meaningful economically, as we will show in the next subsection.

3.3

Stability

We next examine the stability of the steady states. The dynamic adjustment of bt isdetermined autonomously by (10) and is always stable, whereas (ct , kg,t ) evolves over time
according to (17) and (18). When ḃt = 0 (bt = b̄) holds, from (17) and (18), the dynamics
of ct and kg,t are obtained as:
1[ct − ζ(kg,t , τ, b̄)]ct ,1 + b̄
}1 {k̇g,t =
(kg,t − b̄)ct − η(kg,t , τ, b̄)] .
1 + b̄
c˙t =

(23)
(24)

Using these, we can depict the movement of (ct , kg,t ) in (c, kg ) space as shown in Figure
2. Depending on the value of b̄, the k̇g = 0 locus takes diﬀerent shapes. As a result, thereτare ﬁve diﬀerent phase diagrams.9 When b̄ is relatively small, b̄ < (1−τ, and we obtain)ατthe phase diagram in Figure 2-(b). When b̄ equals (1−τ )α , we have two types of diagrams
represented in Figure 2-(a). Figure 2-(a)-(i) is the case when kg = b̄ is the high-growth1{ ρ(1+ τ ) } 1−α(1−τ )αsteady state, while x ≡ A[1−(1−τ )(1−α)]is the low-growth steady state. Figure 2-(a)(ii), in contrast, is the case where kg = b̄ is the low-growth steady state, while x is theτ, and two types
high-growth steady state. Finally, when b̄ is relatively large, b̄ > (1−τ)α
of diagrams emerge. Figure 2-(c)-(i) is the case where both of the steady-state values kg
8Transversality condition (7) excludes all paths if there exist no steady states. We see this in the next
subsection.
9
Please see Appendix B regarding the shapes and positions of the ċ = 0 and k̇g,t = 0 loci in the phase
diagrams.

9

become smaller than b̄, whereas Figure 2-(c)-(ii) is the case where both of them becomelarger than b̄. In any case, we have the following common features regarding the dynamic
behaviors. The low-growth steady state is unstable, whereas there exists a saddle pathconverging to the high-growth steady state. The stable arm is always upward sloping. If(21) is violated, there are no steady states. In this case, there are no paths that satisfy
transversality condition (7). The paths that go to kg = 0 or c = 0 do not satisfy (7)obviously. The paths in which both kg and c continue to increase are excluded because
private capital, Kt , keeps decreasing and eventually becomes zero on those paths.10[Figure 2]
Next, we check the local stability of the system including the case of bt ̸= b̄. We obtainthe following proposition on the stability of the dynamic system (10), (17) and (18).
Proposition 2
The high-growth steady state is locally saddle-point stable, whereas the low-growth steadystate is unstable.
Poof. See Appendix C.
Proposition 2 and the phase diagrams state the following three facts. First, although
there are two steady states, only the high-growth steady state is meaningful economically.
Second, it is determinate. Third, given b0 ≃ b̄, the economy falls into the development∗
trap if the initial ratio of public capital to private capital, kg,0 , is smaller than kg,L. In
the trap, there exists no path that the representative household with perfect foresight andrational expectations can follow, and hence the economy cannot develop.11
The intuition for the development trap is as follows: when kg is small, production
relative to private capital is small.12 The tax revenue of the government becomes small
relative to private capital, which tightens the budget constraint of the government. Then,public investment becomes small relative to private capital. This can be veriﬁed using
bt = b̄, (15) and (22), so that we can derive:g = τ Akg 1−α − ρb̄.

(25)

This equation shows that in the steady state, g is small when kg is small. A small g implies
that the government cannot accumulate enough public capital. Then, the economy falls
into the development trap when kg,0 is suﬃciently small.
Our results are totally diﬀerent from those of Futagami et al. (2008), where a productive input of the government is a ﬂow variable. The results in Futagami et al. (2008) can
1−αThe reason for this is as follows. From (12), if private capital keeps decreasing, Akg,t− ct − gt <1−α0 holds. By using (16), this condition can be rewritten as ct > (1 − τ )(1 + αb̄)Akg,t . Next, from{})αb̄−τ
(24), the k̇g,t = 0 locus is rewritten as c = (1 − τ )(1 + αb̄) + (1−τ
(1 + b̄) Akg1−α . Because thekg −b̄
10

paths in which both kg and c continue
to increase pass through the k̇g,t = 0 locus, they satisfy ct >{}(1−τ )αb̄−τ
1−ατ
(1 − τ )(1 + αb̄) + kg,t −b̄ (1 + b̄) Akg,t . Because b̄ > (1−τ)α when (21) is violated, these paths
1−αsatisfy ct > (1 − τ )(1 + αb̄)Akg,tand then Kt keeps decreasing on the paths.
11
The transversality condition (7) excludes any paths that go to the states in which kg becomes zero.
12
Production relative to private capital is given by Yt /Kt = Akg,t 1−α .

10be summarized as follows. First, the two steady states are both stable and meaningful
economically. Second, the high-growth steady state can become locally indeterminate.
Finally, because the low-growth steady state is stable and meaningful economically, theeconomy can be stuck in a low-growth trap.13 The low-growth trap in Futagami et al.
(2008) occurs when the household has pessimistic expectations that the low-growth steadystate will be realized. The development trap in this paper occurs when kg,0 is smaller than∗kg,L. Therefore, the low-growth trap in Futagami et al. (2008) has a feature diﬀerent fromthe development trap in this paper.
In Futagami et al. (2008) where public services enter the production function, public
productive expenditure comes into operation in ﬁnal goods production immediately. If thegovernment increases public productive expenditure, ﬁnal goods production and also thetax revenue of the government increase immediately, which allows the government to spend
more on productive public expenditure. There is a complementarity between productive
public expenditure and ﬁnal goods production. As a result, the development trap does
not occur and equilibrium indeterminacy can arise in Futagami et al. (2008).

4

Characteristics of the Steady States

This section analyzes the eﬀects of changes in the income tax rate and in the target levelof government bonds in the steady states. Because the economically meaningful steadystate is unique (the high-growth steady state), we should examine the policy eﬀect in the∗high-growth steady state. Because the eﬀects of policy changes on kg,Lare crucial for the∗development trap, we also pay attention to the policy eﬀects on kg,L .
We ﬁrst examine the eﬀects of changes in b̄. As shown in Figure 3-(a), when b̄ rises,
only the straight line Π(kg ), which represents the RHS of (20), shifts downward. As a∗∗result, kg,Lrises whereas kg,Hfalls. We next examine the eﬀects of changes in τ . As shownin Figure 3-(b), when τ rises, only the U-shaped curve Λ(kg ), which represents the LHS of∗∗(20), shifts downward. Then, kg,Lfalls whereas kg,Hrises. We obtain the following lemma.
[Figure 3]
Lemma
The eﬀects of changes in policy variables b̄ and τ are as follows:
∗
∗
∂kg,L
∂kg,H
(i)
> 0,
< 0.
∂ b̄
∂ b̄

∗
∗
∂kg,L
∂kg,H
(ii)
< 0,
> 0.
∂τ
∂τ

This lemma reveals the following. In the stable high-growth steady state, increases in b̄∗(τ ) have negative (positive) eﬀects on the ratio of public capital to private capital, kg,H.
∗In the unstable low-growth steady state, the ratio of public capital to private capital, kg,L ,increases (decreases) when b̄ (τ ) increases. Hence, when public capital is scarce relative to
private capital, the economy is more (less) likely to fall into the development trap.
13

In Futagami et al. (2008), the welfare level in the low-growth steady state is lower than that in thehigh-growth steady state.

11

When b̄ increases, the interest payments on government debt increase in the steadystate, as (25) shows. As a result, public investment becomes small and the accumulationof public capital is depressed. Consequently, when public capital is scarce relative to
private capital, the economy is more likely to fall into the development trap. Besides, thedepressed accumulation of public capital decreases the ratio of public capital to private∗
capital, kg,H, in the high-growth steady state. When τ increases, tax revenue increases.
Public investment relative to private capital, g, increases, as (25) shows. Then, an increasein τ has eﬀects opposite to those for an increase in b̄.
In Futagami et al. (2008), where public services enter the production function, theeconomy falls into a low-growth trap when the household has pessimistic expectations.
Because it is not easy for the government to control households’ expectations, the government has diﬃculty in helping the economy escape from the low-growth trap. In contrast,
in our case, the stock of public capital enters the production function, the government canrelatively easily help the economy to escape from the development trap by decreasing b̄,or increasing τ .
We next investigate the relationship between the long-run growth rate and the main
policy variable, b̄.14 From now on, we focus only on the high-growth steady state thatis stable. As the growth rate in the high-growth steady state is given by γH = (1 −∗τ )αA(kg,H)1−α − ρ, we can prove the next proposition by using Lemma-(i).
Proposition 3
A decrease in the target level, b̄, enhances the growth rate in the high-growth steady state.

In the rest of this section, we investigate the eﬀect of b̄ on the ratio of public investment∗
to private capital in the high growth steady state, gH. From (25) and Lemma-(i), we obtain
immediately:
∗
∗
∂kg,H
∂gH
∗
−α
= τ (1 − α)A(kg,H )− ρ < 0.
(26)
∂ b̄
∂ b̄
This states the following proposition.
Proposition 4
A decrease in the target level, b̄, boosts the ratio of public investment to private capital in
the high-growth steady state.
∗
Proposition 4 states that if the government reduces b̄, gH
always increases in the longrun. The reason is simple. As b̄ decreases, outstanding public debt is reduced in the longrun. Therefore, the interest payments are reduced and then the budget constraint of thegovernment becomes loose. Please remember that in the short run, a decrease in b̄ reduces
public investment. However, this steadily decreases outstanding debt and the interest
payments and can increase public investment gradually. In the long run, Proposition 4states that the volume of public investment can become larger than that in the original
14

How the tax rate inﬂuences the growth rate in the long run may be an important issue. We obtain a
tax rate that maximizes the growth rate in the high-growth steady state in this model. Furthermore, we
ﬁnd this tax is increasing in b̄.

12state.

5Transitional Dynamics

This section examines the transitional dynamics numerically. Our scenario is as follows:
the economy is in a high-growth steady state initially with b̄ = b̄init , where b̄init denotes theinitial level of b̄. At time 0, the government reduces b̄ from b̄init to b̄new unexpectedly, whereb̄new is the level of b̄ after the policy change. Then, the economy begins to move toward
the new high-growth steady state along the saddle path and the transitional dynamics are
generated.
In Section 4, we examined the long-run eﬀects of changes in b̄: decreases in b̄ havepositive eﬀects on both the growth rate and on public investment in the long run. Thissection shows the policy eﬀects of b̄ in the short run and on the transition path. We also
show that the transitional dynamics and the strength of policy eﬀects in the short run andon the transition path depend heavily on the value of ϕ, which has no eﬀect in the longrun. Remember that ϕ determines the adjustment speed of bt (see (10)).
We analyze the transition paths numerically using the relaxation algorithm.15 As abenchmark, we choose the following parameter values. The subjective time discount rate,ρ, is set to 0.05, which is an often-used value in the growth literature.16 The elasticity of
production with respect to public capital, 1 − α, is set to 0.25. This value is often used instudies such as Barro (1990), Greiner (2007) and Gómez (2004). We use τ = 0.1 following
Greiner (2007), who studies how public investment ﬁnanced by public debts aﬀects thetransitional dynamics of the economy. The value of A is chosen so that the long-run
growth rate of the new steady state is equal to 0.02, which results in A = 0.1313. The
values of b̄init and b̄new are chosen so that the debt–GDP ratio in the initial and new steadystates is equal to 80 percent and 50 percent, respectively. This yields b̄init = 0.0802 andb̄new = 0.0519. The reason for setting the initial debt–GDP ratio to 80 percent is thatthe average debt–GDP ratio in the EU27 was almost 80 percent in 2010, as we mentioned
in Section 1. Because of the recent public ﬁnance crises in the EU, using the EU in thedetermination of the parameter values is appropriate. Because the member countries of
the EU must maintain a debt–GDP ratio of less than 60 percent under the Maastricht
criterion, we assume that the debt–GDP ratio in the new steady state is less than 60
percent. It is also noted that the debt–GDP ratio in the high-growth steady state is given∗
by B/Y = b̄/(A(kg,H)1−α ) and hence it increases with b̄ in the high-growth steady state∗(see Lemma-(i)). Under these parameter values, kg,H(≡ Kg /K) is around 0.34 and 0.39 in
the initial and new steady states, respectively.
The value of ϕ has no eﬀect on the steady state. However, as for transitional dynamics,
ϕ has an important roles because it governs the dynamics of bt , and hence the time pathsof other endogenous variables are aﬀected strongly by the value of ϕ. We use ﬁve values:
ϕ = 0.05, 0.1, 0.2, 0.3 and 0.4. In the case of ϕ = 0.05, it takes about 25.5 years until thedebt–GDP ratio reduces to around 60 percent. As ϕ increases, it takes fewer years. When
15

Trimborn et al.
(2008) detail the relaxation algorithm.
They also provide MATLAB programs for the relaxation algorithm,
freely downloadable at http://www.unisiegen.de/fb5/vwli/forschung/relaxation/the relaxation method.html?lang=de.
16
Greiner (2008) also sets ρ = 0.05.

13ϕ = 0.1, 0.2, 0.3 and 0.4, the debt–GDP ratio reduces to around 60 percent in around 13,
6.5, 4.5 and 3.5 years.17[Figure 4]
Under these parameter values, we examine numerically the transitional dynamics generated by the reduction of b̄ from b̄init to b̄new , using the relaxation algorithm. Figure 4
presents the results. Panels (a) and (b) show that bt and the debt–GDP ratio decrease
monotonically and converge to their new-steady state values. As ϕ increases, the speedsof convergence of bt and the debt–GDP ratio increase.
Panel (c) shows that just after the policy change, gt (≡ G/K) drops sharply below theinitial level and before rising to the new steady-state level, which is higher than the initiallevel. To reduce bt to b̄new , the government must reduce its expenditure initially, which
results in a short-run decrease in gt , as is shown by the third term in parentheses in (16).
However, as bt declines steadily, the interest payments on the government debt decrease
gradually and its budget constraint becomes gradually loose. Then, gt increases graduallyand eventually exceeds the initial level (see (25) and Proposition 4). When the values of ϕ
are larger, the initial declines in gt also become larger. A large ϕ means that the government
must reduce its debt at a higher speed and hence its budget becomes tight immediately
after the policy change. As a result, the government must reduce public investment by alarge amount. When ϕ is as large as 0.3 or 0.4, gt becomes negative immediately after thepolicy change, which implies that the government must sell its capital in order to meet its
budget. Although the initial declines in gt increase with ϕ, it takes shorter periods of time
until gt recovers its initial level when the value of ϕ is large: when ϕ is equal to 0.05, 0.1,
0.2, 0.3 and 0.4, it takes about 14.5, 11.5, 8.5, 6.5 and 6 years, respectively, until gt returns
back to its initial level. Because public investment is the source of the accumulation of
public capital Kg,t , the growth rates of public capital follow dynamical paths similar to
those of gt , as shown in Panel (d).
In contrast to gt , the growth rate of private capital Kt increases sharply above its initiallevel immediately after the policy change, and then decreases (see Panel (e)). After thepolicy change, public investment decreases as we have just discussed, which has crowdingin eﬀects on private investment. Then, the growth rate of Kt increases initially. However,as gt increases gradually, the crowding-out eﬀects on private investment begin to prevail,which leads to gradual declines in the growth rate of Kt . When ϕ is as large as 0.3 or 0.4,
the crowding-out eﬀects of public investment become so strong that the growth rate of Kt
follows a nonmonotonic transitional path.
The dynamics of the growth rates of Kg,t and Kt give rise to the nonmonotonic transitional paths of kg,t ≡ Kg,t /Kt , as shown in Panel (f). In the early stages of transition,
kg,t decreases gradually because of decreased public investment and increased private investment. The larger ϕ is, the larger the declines in kg are. However, because of gradualincreases in public investment and gradual decreases in private investment, kg,t starts toincrease to its new steady-state level several years after the policy change. When ϕ becomes larger, it takes less time until kg,t begins to increase: when ϕ is equal to 0.05, 0.1,
0.2, 0.3 and 0.4, kg,t begins to increase in about 13, 10, 7, 6 and 5 years, respectively.
When ϕ = 0.05, 0.1, 0.2, 0.3 and 0.4, bt − b̄ reduces to 50 percent of its initial size in around 14, 7,
3.5, 2.5 and 1.5 years, respectively.
17

14

From (2) and (6), the growth rate of private consumption becomes a function of kg . Then,as shown in Panel (g), the growth rates of private consumption follow dynamical paths
similar to those of kg,t .
Panel (h) provides the transitional paths of ct . From the panel, we know the eﬀects
of the policy change on the initial consumption levels, C0 , because C0 ≡ c0 K0 holds.
The eﬀects on C0 depend apparently on the value of ϕ. When ϕ is small (ϕ = 0.05), C0
decreases in reaction to the policy change. As ϕ becomes larger, C0 increases to a higher
level. For suﬃciently large values of ϕ(= 0.4), C0 increases to a higher level than the initiallevel as a result of the policy change. As the value of ϕ increases, the initial declines inpublic investment also become large, and hence more resources are released to the private
sector (see (12)). Consequently, the household can consume more as ϕ increases. To see
why, let us integrate (5) and use (6).
C0 = ρ(W0 + w̃0 ).

(27)

C0 increasesthe )initial{( with} asset, W0 , and the present value of wage income, w̃0 =∫∞∫t
wt exp − 0 rω dω t dt. wt depends on kg,t and Kt (see (3)). After the policy
0change, the growth rate of private capital Kt increases beyond the initial level (see Panel(e)), which enhances wt and exerts positive eﬀects on w̃0 . In contrast, kg decreases inthe early stages of transition as shown in Panel (f), which lowers wt and has negativeeﬀects on w̃0 . As ϕ increases, the positive eﬀects tend to dominate the negative eﬀects.
Therefore, C0 tends to be larger (smaller) than the initial level when ϕ is large (small) andC0 increases with ϕ.

6

WelfareAs in Section 5, we consider the following scenario. The economy is initially in a highgrowth steady state with b̄ = b̄init . At time 0, the government reduces b̄ from b̄init to b̄new
unexpectedly. Then, the transitional dynamics are generated. The purpose of this section
is to investigate the welfare eﬀects of this policy change.
Our welfare measure is (4). Because the ∫growth rate of consumption is given by γC,t ≡t(1 − τ )αAkg,t 1−α − ρ and we have Ct = C0 e 0 γC,ν dν , we can rewrite (4) as:)∫ ∞ (∫ t
ln C0
U0 =
+
γC,ν dν e−ρt dt.
ρ00
The above equation shows that decreases in b̄ aﬀect the welfare level through its eﬀects on
C0 and the paths of γC,t . As we observed in Section 5, the eﬀects of the policy change on
C0 and the transitional paths of γC,t depend heavily on the value of ϕ. Consequently, we
will observe that the welfare eﬀects of the policy change are inﬂuenced by the value of ϕ.
We examine the welfare eﬀects numerically by considering the case where the government reduces its debt-GDP ratio from B0 /Y0 to 0.5, where B0 /Y0 =0.6, 0.7, 0.8, 0.9 and 1.
As in Section 5, we assume that the debt–GDP ratio in the new steady state is less than
60 percent because the member countries of the EU must maintain a debt–GDP ratio ofless than 60 percent under the Maastricht criterion. In the benchmark case, the values ofA, ρ and α are the same as those employed in Section 5. The values of ϕ we consider are15

ϕ = 0.01, 0.025, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.75 and 1. When ϕ is smaller (larger) than
0.05 (0.4), it takes considerably more (fewer) years until the debt–GDP ratio reduces to 60
percent. However, to clarify the welfare eﬀects of the policy change, we include extremely
small and large values of ϕ. We consider three values of income tax rates: τ =0.1, 0.2
and 0.3. The values of b̄init and b̄SS are chosen for each τ and B0 /Y0 so that the initialdebt–GDP ratio equals B0 /Y0 and the debt–GDP ratio in the new steady state equals 0.5.
∗∗
Let us denote the welfare level without the policy change as U0,N
. We have U0,N
=
∗
∗
2
∗
∗
ln(cH,N K0 )/ρ + γC,N /ρ , where cH,N and γC,N are the initial steady-state values of c and∗γC . To calculate U0,N
, we need the initial value of private capital, K0 . Here, we set K0 = 1.
The welfare level immediately after the policy change is denoted as U0∗∗ . We calculate U0∗∗by setting K0 = 1 and using the relaxation algorithm (see Appendix D for more detail).
[Table 1]∗
In Table 1, we present the diﬀerence between U0∗∗ and U0,N
. The table shows thatwhen τ = 0.1, B0 /Y0 = 1 and ϕ = 0.05 hold, and social welfare can be improved by∗
0.04048(= U0∗∗ − U0,N
) if the government reduces its debt–GDP ratio. In addition, thetable shows that it takes about 37 years until the debt–GDP ratio reduces to about 60
percent.18
The ﬁrst notable result in Table 1 is that in all the cases we consider, reductions inb̄ have positive welfare eﬀects. As shown in Section 5, when the government reduces b̄,whether C0 decreases or increases in the short run depends on the values of ϕ. Furthermore,
the growth rate of consumption,γC , decreases in the early stages of transition, which has
negative eﬀects on welfare. However, in the long run, the growth rate becomes higher
than the initial level, which has positive welfare eﬀects. Our results show that in all cases,
the positive eﬀects dominate the negative eﬀects, and hence social welfare improves if thegovernment reduces b̄ and decreases its debt–GDP ratio.
Table 1 also shows that given τ and ϕ, when the initial debt–GDP ratio is larger, thewelfare gains are also larger. For example, when τ = 0.1 and ϕ = 0.05, the welfare gains
decrease from 0.03379 to 0.02627 as B0 /Y0 decreases from 0.9 to 0.8. A large B0 /Y0 meansthat there are large gaps between the initial and target debt–GDP ratios. To ﬁll this large
gap, the government must reduce gt initially by a large amount, as shown in Table 2.19
As discussed in Section 5, the initial declines in gt , leading to decreases in kg,t at the earlystages of transition, are the source of the negative welfare eﬀect (the declines in γC,t ).
Then, the negative welfare eﬀect is large when B0 /Y0 is large. However, the large initialdeclines in gt suggest that more resources are released to the private sector and hence thehousehold can consume more compared with the case where B0 /Y0 is small. Then, theshort-run eﬀect (changes in C0 ) tends to be weakly negative or even positive when B0 /Y0
is large. In addition, a large B0 /Y0 indicates a large diﬀerence between the values of kg in
the initial and new steady states. Then, the growth rate of consumption in the new steadystate is much higher than that in the initial steady state, which suggests that the positive
long-run eﬀect is large when B0 /Y0 is large. Because the dominance of the positive eﬀectover the negative eﬀect increases as B0 /Y0 increases, the welfare gains from the policychange are large when B0 /Y0 is large.
18

When B0 /Y0 is equal to 0.6, we do not calculate the years that it takes until the debt–GDP ratio
reduces to about 60 percent.
19
The initial decline of gt is calculated by (g0 − gint )/gint where gint is the initial steady-state value andg0 is the value immediately after the policy change.

16

[Table 2]
As expected, the values of ϕ aﬀect the welfare eﬀects. Table 1 shows that, as for the
relationship between the welfare gains and ϕ, the following two patterns are observed.
In many cases, (i) the welfare gains decrease in ϕ when ϕ is small, but increase in ϕ as
ϕ becomes relatively large. Thus, the relationships between the welfare gains and ϕ are
U-shaped in many cases. However, (ii) the welfare gains decrease monotonically in ϕ when
τ is low and B0 /Y0 is large: τ = 0.1 and B0 /Y0 ≥ 0.9.
To understand the intuition for these results, the following points should be noted.
Table 2 illustrates that the size of the initial declines of gt under ϕ = 0.05 depend cruciallyon τ and B0 /Y0 . When τ is lower (higher), and B0 /Y0 is larger (smaller), the initial declines
in gt become larger (smaller). This is because when the tax rate is lower (higher), thegovernment can rely less (more) on tax revenue to reduce its debt–GDP ratio. Furthermore,when B0 /Y0 is larger (smaller), the initial declines of gt also become larger (smaller) to ﬁlllarger gaps between the initial and target debt–GDP ratios. Next, the increase in ϕ has
the following two opposing eﬀects on welfare. It accelerates the decline of gt , as we see in
Figure 4-(c). However, it leads to a recovery in the growth rate of consumption sooner, as
we see in Figure 4-(g). Which eﬀect is dominant depends on the size of the initial declineof gt . When the tax rate is low and the initial debt–GDP ratio is large, such as τ = 0.1
and B0 /Y0 ≥ 0.9, the initial decline of gt is too large when the government increases ϕ.
Then, the welfare gains decrease monotonically in ϕ. In contrast, when the tax rate ishigh or B0 /Y0 is small, the initial decline of gt becomes small. Then, when ϕ is somewhat
large, the positive eﬀect of the earlier recovery of the growth of consumption dominates
the negative eﬀect of the initial decline of gt . Thus, the relationships between the welfare
gains and ϕ are U-shaped.
The results we have obtained may be crucially dependent on the subjective discountrate. When ρ is small (large), it is likely that the welfare eﬀects in the long run (in the
short run and on the transition path) dominate those in the short run and on the transition
path (in the long run). Hence, the total welfare eﬀects may change if we choose diﬀerent
values of ρ. We conduct the same exercise under diﬀerent values of the discount rates (seeTable 3). The values of A, α and ϕ are the same as those employed in Table 1. The valuesof b̄init and b̄new are chosen for each ρ, τ and B0 /Y0 so that the initial debt–GDP ratio
equals B0 /Y0 and the debt–GDP ratio in the new steady state equals 0.5. Table 2 shows
again that: (i) in all cases we consider, reductions in b̄ have positive welfare eﬀects; and(ii) as the initial debt–GDP ratio is larger, the welfare gains are also larger. With respect
to the inﬂuences of ϕ, the ﬁndings when ρ is 0.03 are identical to those in Table 1. When
τ is low and B0 /Y0 is large (τ = 0.1 and B0 /Y0 ≥ 0.9), they are decreasing monotonically.
The relationships between the welfare gains and ϕ are U-shaped in many cases. In thecase of ρ = 0.07, the results are also the same when B0 /Y0 is small. Therefore, the results
are robust.
However, in the case of ρ = 0.07, some results that are diﬀerent from those in Table 1
are obtained when B0 /Y0 is large. When τ ≥ 0.2 holds, the welfare gains increase monotonically with ϕ if B0 /Y0 is large enough. As discussed above, when B0 /Y0 is large, theshort-run eﬀect (changes in C0 ) tends to be weakly negative or even positive. In addition,
C0 increases with ϕ as shown in Panel (h) of Figure 4. Because a large ρ indicates that thehousehold cares more about the short-run eﬀect, the welfare gains increase monotonicallywith ϕ when B0 /Y0 is large and τ ≥ 0.2 holds. In the case of τ = 0.1 and B0 /Y0 = 0.9, the17welfare gains increase with ϕ for large values of ϕ, in contrast to the results obtained inTable 1. These increases in the welfare gains also reﬂect the short-run eﬀect. Furthermore,when τ = 0.1 and B0 /Y0 = 1 hold, the short-run eﬀect (changes in C0 ) tends to be weakly
negative or even positive and C0 increases with ϕ. However, the negative eﬀect comingfrom the decline of gt is so strong that the welfare gains decrease with ϕ even when the
discount rate is large (ρ = 0.07).
[Table 3]
In the benchmark case, when τ is equal to 0.2 or 0.3, the long-run growth rates aremuch larger than 0.02. When τ is equal to 0.2 and 0.3, the long-run growth rate is about
0.026 and 0.027, respectively, in the benchmark case. A small diﬀerence by 0.006 or 0.007
in the long-run growth rate may lower the reliability of the result. Therefore, in the casethat τ is 0.2 and 0.3, we conduct the same numerical exercise as when we change the valueof total factor productivity, A, so that the long-run growth rate equals 0.02. However,Table 4 shows that the results are qualitatively the same as those obtained in Table 1.
When τ = 0.2, the welfare gains decrease at ﬁrst and then begin to increase as the valueof ϕ increases. On the other hand, the welfare gains increase with ϕ when τ is as large as
0.3.
[Table 4]

7

Conclusion

We investigated the policy eﬀects of a reduction in the government debt–GDP ratio tobelow the 60 percent criterion. For an expenditure-based reduction of public debt, thegovernment decreases public investment initially to reduce its debt. This policy change
causes the following response during the transition and in the steady state, and causes
opposite eﬀects regarding welfare changes between the transition path and in the longrun. In the short run, the investment in private capital rises initially because of the fallin public investment, and subsequently the return from private capital decreases. Next,
the growth rate of consumption begins to decline after the policy change. This exerts
negative eﬀects on welfare on the transition path. However, as the size of public debt
decreases, the interest payments on the debt decline and the government can spend moreof its tax revenue on public investment. The recovery of public investment, in turn, hinges
on private investment, and subsequently the return from private capital decreases. As aresult, in the long run, the growth rates of consumption, private capital and public capital
exceed those in the initial state. This exerts positive eﬀects on welfare in the long run.
We ﬁnd the positive eﬀects dominate the negative eﬀects, meaning that the policy change
improves welfare.
The adjustment speed of debt reduction has important eﬀects on welfare. How the
welfare gains are inﬂuenced as the government changes the speed of debt reduction depends on the tax rate and on how much the debt–GDP ratio initially exceeds the target
level. The relationships between the welfare gains and the adjustment speed are U-shapedin many cases. However, they decrease monotonically when the tax rate is low and initial debt-GDP ratio is suﬃciently large. When the tax rate is low, the government must18

reduce the gap between the initial debt–GDP ratio and the target without adequate tax
revenue. Then the initial decline of public investment becomes large. When the initialdebt–GDP ratio is large, the government must reduce the large gap between the initial
and the target level of debt. Then, the initial decline of public investment also becomeslarge. The increase in the speed of reduction of public debt leads to the following tradeoﬀ. On the one hand, the initial decline of public investment increases. On the other
hand, the growth of consumption recovers sooner. When the tax rate is low and the initialdebt–GDP ratio is large, the initial decline of public investment is too large when the government increases the adjustment speed. Then, the welfare gains decrease monotonicallyin the adjustment speed. In contrast, when the tax rate is high or the initial debt–GDP
ratio is small, the initial decline of public investment becomes small. Next, when the adjustment speed is somewhat high, the positive eﬀect of the earlier recovery of the growthof consumption dominates the negative eﬀect of the initial decline of public investment.
Then, the relationship between the welfare gains and the adjustment speed are U-shaped.
Besides these main results, this paper shows the possibility that the economy fails todevelop sustainably when the initial level of public capital is suﬃciently small relative
to private capital. In the development process, the government should rely more on tax
ﬁnance rather than debt to avoid falling into the development trap.
Finally, we provided directions for future research. For countries facing government
ﬁnancial crises, there may be two ways to improve government ﬁnancial conditions. The
ﬁrst is to reduce government spending. The second is to increase the tax rate. We
investigate the ﬁrst way in this paper. However, countries such as France, Italy, Ireland,
Greece, Portugal, Spain and the USA have attempted to implement both spending cuts andtax increases. Investigating debt reduction by including tax increases is very important.
An additional possibility is that under some conditions, reductions in tax rates rather
than increases in tax rates may be a better policy for reducing debt. Bruce and Turnovsky(1999) show that both expenditure cuts and cuts in the tax rate generate higher economic
growth, which leads to higher tax revenue in the future. In turn, these increase the present
discounted value of all future tax revenues. However, they do not investigate the policy
eﬀects on the transition dynamics but rather those in the steady state. Investigating which
is better, increasing or decreasing tax rates, for welfare including transition paths is also
important. Furthermore, we assumed public infrastructure only inﬂuences the productionof goods as in Futagami et al. (1993) and Turnovsky (1997) and so on. However, it may
also aﬀect households’ utility functions if public capital includes public health capital, as
in Agénor (2008). How this extension changes the policy eﬀects on the economy may be
worth further research. These are left for future research.

Acknowledgments
We would like to thank Akira Momota, Akira Yakita and Kazuki Hiraga for helpful comments and suggestions. We are responsible for any remaining errors.

19Appendix
A

Properties of Λ(kg )We can show that Λ(kg ) has the following properties:
Λ′ (kg ) = Akg−α {(1 − τ )α(2 − α)kg − τ (1 − α)} < (=)(>)0 if and only if kg < (=)(>)k˘g ,Λ′′ (kg ) = (1 − τ )α(1 − α)(2 − α)Akg−α + τ α(1 − α)Akg−α−1 > 0,
)
(
τ
′
′
= 0,
lim Λ (kg ) = −∞, lim Λ (kg ) = ∞, and Λ(0) = Λ
kg →0
kg →∞
(1 − τ )α(1−α)τ
where k˘g ≡ (1−τ. Apparently, Λ(kg ) is a convex function of kg as shown in Figure)(2−α)α

1. The above properties ensure the existence and the uniqueness of k̃g (> 0) that satisﬁes
Λ′ (k̃g ) = ρ.

B

Phase diagram of (kg , c)From (23), the ċ = 0 locus is given by c = ζ(kg , τ, b̄). It is easy to show ζ(0, τ, b̄) > 0,∂ζ(kg , τ, b̄)/∂kg > 0 and ∂ 2 ζ(kg , τ, b̄)/∂kg2 < 0. Eq. (23) shows that ċt ≥ (<)0 holds whenct ≥ (<)ζ(kg,t , τ, b̄). Summarizing these results, Figures 2 (a)–2(c) show the shape of theċ = 0 locus and the motion of ct .
Next, we move on to the k˙g = 0 locus. Depending on the value of b̄, the k̇g = 0 locus
takes diﬀerent shapes. We consider the following three cases: (i) b̄ = τ /{(1 − τ )α}, (ii)b̄ < τ /{(1 − τ )α} and (iii) b̄ > τ /{(1 − τ )α}.
(i) We consider the case of b̄ = τ /{(1 − τ )α}. Because η(kg,t , τ, b̄) = (kg,t − b̄)Akg1−α holds,
we know from (24) that the k˙g = 0 locus is given by:
kg = b̄ =τ
(1 − τ )αandc = Akg1−α .

(B.1)

The ﬁrst equation of (B.1) indicates that one of the steady-state values of kg is given bykg∗ = b̄. Solving the ċ = 0 locus and the second equation of (B.1) for kg , we know that theother steady-state value of kg is given by:
{
x≡

τ
)
ρ(1 + (1−τ)α

A[1 − (1 − τ )(1 − α)]

1
} 1−α

.

The upper panel of Figure 2-(a) presents the k̇g = 0 locus in the case when ρ is smallenough to satisfy b̄ > x. When ρ is large enough to satisfy b̄ < x, the k̇g = 0 locus isshown in the lower panel of Figure 2-(a).
We know from (24) that:k̇g,t ≥ 0, (a) if ct ≤ Akg1−α and kg ≤ b̄, or (b) if ct ≥ Akg1−α and kg ≥ b̄,k̇g,t < 0, (c) if ct < Akg1−α and kg > b̄, or (d) if ct > Akg1−α and kg < b̄.
Based on the discussion so far, the phase diagrams are represented in Figure 2-(a) whenτb̄ = (1−τholds.
)α20τWe next move on to cases (ii) and (iii) where b̄ ̸= (1−τ
. From (24) and the deﬁnition)αof η(kg,t , τ, b̄), the k̇g = 0 locus is given by:{}[(1 − τ )(1 + αb̄)kg − [1 − (1 − τ )α]b̄ + τ ]Akg1−αc = Γ(kg , τ, b̄) ≡.
kg − b̄(B.2))α]b̄+τ
Apparently, Γ(kg , τ, b̄) equals zero when kg = 0 and kg = k̂g ≡ [1−(1−τ. Then, the(1−τ )(1+αb̄)k̇g = 0 locus passes through (0, 0) and (k̂g , 0) as shown in Figures 2-(b) and (c). Γ(kg , τ, b̄)is continuous with respect to kg except at kg = b̄.
τ(ii) We consider the case where b̄ < (1−τholds. In this case, b̄ < k̂g holds because we)αb̄)[τ −(1−τ )αb̄]
have kˆg − b̄ = (1+(1−τ. We rewrite (B.2) as:)(1+αb̄)}{(1 − τ )αb̄ − τ(1 + b̄) Akg1−α .
(B.3)
Γ(kg , τ, b̄) = (1 − τ )(1 + αb̄) +kg − b̄τ
Because b̄ < (1−τholds, ∂Γ(kg , τ, b̄)/∂kg has a positive sign if kg ̸= b̄. We also have)αlimkg →b̄−0 Γ(kg , τ, b̄) = +∞. Because we have Γ(0, τ, b̄) = 0 < ζ(0, τ, b̄), the continuityof Γ(kg , τ, b̄) and ζ(kg , τ, b̄) indicates that the k̇g = 0 locus has at least one intersection
with the ċ = 0 locus in the region of kg < b̄ as shown in Figure 2-(b). We can show
limkg →∞ Γ(kg , τ, b̄)/ζ(kg , τ, b̄) = (1 + αb̄)/(1 − α) > 1, which means that for large kg , thek̇g = 0 locus is located above the ċt = 0 locus. Because we have Γ(k̂g , τ, b̄) = 0 < ζ(k̂g , τ, b̄),the continuity indicates that the k̇g = 0 locus intersects with the ċ = 0 locus at least oncein the region of kg > b̄. Because there exist two steady states, we now know that one of
the two intersection of the ċ = 0 and k̇g = 0 loci is in the region of kg < b̄ and the otherintersection is in the region of kg > b̄. From the discussion so far, we depict the k̇g = 0
locus as shown in Figure 2-(b). From (24), we obtain:k̇g,t ≥ 0, (a) if ct ≤ Γ(kg,t , τ, b̄) and kg ≤ b̄, or (b) if ct ≥ Γ(kg,t , τ, b̄) and kg ≥ b̄, (B.4)
k̇g,t < 0, (c) if ct < Γ(kg,t , τ, b̄) and kg > b̄, or (d) if ct > Γ(kg,t , τ, b̄) and kg < b̄.

(B.5)

Thus, we can draw the phase diagram as shown in Figure 2-(b).
τ(iii) We ﬁnally turn to the case of b̄ > (1−τ. The second term in parentheses of the RHS)αof (B.3) is negative (positive) when kg < (>)b̄ holds. In the region where kg < (>)b̄ holds,
the graph of c = Γ(kg , τ, b̄) is located below (above) the graph of c = (1 − τ )(1 + αb̄)Akg1−α ,as is shown in Figure 2-(c). Eq (B.3) also reveals limkg →b̄+0 Γ(kg , τ, b̄) = +∞. In addition,
the graph of c = Γ(kg , τ, b̄) asymptotically becomes close to the graph of c = (1 − τ )(1 +αb̄)Akg1−α as kg increases to +∞. Therefore, in the region of kg > b̄, the k̇g = 0 locus takes
the shape shown in Figure 2-(c).
By using (B.2), we next show that Γ(kg , τ, b̄) > 0 holds in the region of kg < k̂g . From
(B.2), we know that the denominator of Γ(kg , τ, b̄) becomes negative when kg < b̄ and theτnumerator becomes negative when kg < k̂g . Because we have k̂g < b̄ when b̄ > (1−τ)α
holds, both the denominator and numerator of Γ(kg , τ, b̄) are negative in the region wherekg < k̂g holds. Therefore, Γ(kg , τ, b̄) > 0 holds in the region of kg < k̂g .
From the discussion so far, together with (B.4) and (B.5), we can draw the phase
diagram as shown in Figure 2-(c). The upper (lower) panel shows the case where both∗∗kg,Land kg,H
are smaller (larger) than b̄.

21

C

Proof of Proposition 2

Approximating (10), (17) and (18) linearly in the neighborhood of the steady states, we
obtain:  
bt − b̄ḃ−ϕ00 ċ  =  Jcb Jcc Jckg   ct − c∗  .
(C.1)
∗
˙
k gt − k g
Jkg b Jkg c Jkg kgkg
J = (Jij ) denotes the coeﬃcient matrix of the former system:
ρ+ϕ ∗c∗(1 − α)2 (1 − τ )c∗c , Jcc =, Jckg = −A(kg∗ )−α ,1 + b̄
1 + b̄
1 + b̄
kg∗ − b̄ρ+ϕJkg b = − (1 + kg∗ ), J kg c =,1 + b̄
1 + b̄
(1 − α)η(kg∗ , τ, b̄)Jkg kg = −− [(1 − τ )αA(kg∗ )1−α − ρ].
∗(1 + b̄)kgJcb = −

(C.2)∗∗
or kg,H, and c∗ = ζ(kg∗ , τ, b̄).
where kg∗ = kg,LLet us denote the eigenvalues of the coeﬃcient matrix J as νi (i = 1, 2 and 3). The
structure of the ﬁrst column of J entails that −ϕ is an eigenvalue ν1 . The remaining¯ derived by deleting the ﬁrst row
eigenvalues, ν2 and ν3 , of J are those of the matrix, J,
and column from J. The eigenvalues, ν2 and ν3 , are the solution of the characteristic
equation, ν 2 − (Jcc + Jkg kg )ν + Jcc Jkg kg − Jkgc Jckg = 0. Because we will use ν2 and ν3 inAppendix E, we derive these by solving this equation:√
Jcc + Jkg kg + (Jcc + Jkg kg )2 − 4(Jcc Jkg kg − Jkg c Jckg )
ν2 =
,
(C.3)
2
√
Jcc + Jkg kg − (Jcc + Jkg kg )2 − 4(Jcc Jkg kg − Jkg c Jckg )
.
(C.4)
ν3 =
2

To check the stability, we examine the sign of detJ¯ = Jcc Jkg kg − Jkg c Jckg . Using (C.2),
we obtain:{}c∗
detJ¯ = −(2 − α)(1 − τ )αA(kg∗ )1−α − (1 − α)τ A(kg∗ )−α − ρ .
(C.5)
2(1 + b̄)To simplify the above equation, we subtract Π′ (kg∗ ) from Λ′ (kg∗ ).
Λ′ (kg∗ ) − Π′ (kg∗ ) = (2 − α)(1 − τ )αA(kg∗ )1−α − (1 − α)τ A(kg∗ )−α − ρ.
Thus, (C.5) can be rewritten as:
detJ¯ = −

{ ′ ∗
}
c∗
′ ∗
Λ
(k
)
−
Π
(k
)
.
g
g
(1 + b̄)2∗∗Figure 1 shows that in the high-growth steady state, Λ′ (kg,H) > Π′ (kg,H) holds and¯¯hence detJ < 0. One of the eigenvalues of J has a positive real part and the other hasa negative real part. Moreover, because the inequality detJ¯ < 0 implies (Jcc + Jkg kg )2 −4(Jcc Jkg kg − Jkg c Jckg ) > 0, both ν2 and ν3 are real numbers. We then have ν2 > 0 andν3 < 0. The high-growth steady state is saddle-point stable.

22

∗
∗
Figure 1 shows that in the low-growth steady state, Λ′ (kg,L) < Π′ (kg,L) holds and then¯
detJ > 0 holds. This implies that the real parts of ν2 and ν3 have the same signs. To
determine the sign of the real parts, we check the sign of T rJ¯ = Jcc + Jkg kg . As we havejust shown, we have detJ¯ = Jcc Jkg kg − Jkg c Jckg > 0 in the low-growth steady state. Thisinequality can be written as:(1 − α)2 (1 − τ )(kg∗ − b̄) c∗
Jcc Jkg kg > Jkg c Jckg = −A(kg∗ )−α .
1 + b̄
1 + b̄
We divide both sides of the above inequality by Jcc = c∗ /(1 + b̄)(> 0).
Jkg kg > −(1 − α)2 (1 − τ )(kg∗ − b̄)A(kg∗ )−α .
1 + b̄

Adding Jcc to both sides of the above inequality and using c∗ = ζ(kg∗ , τ, b̄), we obtain:}
1 { ∗c − (1 − α)2 (1 − τ )(kg∗ − b̄)A(kg∗ )−α ,1 + b̄
}1 {=(1 − τ )(1 − α)[α + (1 − α)b̄]A(kg∗ )1−α + ρ(1 + b̄) ,1 + b̄
> 0.

Jcc + Jkg kg >In the low-growth steady state, T rJ¯ = Jcc + Jkg kg > 0 holds. The real parts of ν2 and ν3
are positive. The low-growth steady state is unstable.

D

Welfare eﬀects of b̄

To calculate
the value of U0∗∗ , we calculate the dynamic path and the initial value of∫∞Ut ≡ t (ln Cv )e−ρ(v−t) dv by using the relaxation algorithm. However, we cannot calculate
the dynamic path of and the initial value of Ut directly because Ut does not remain constantin the high-growth steady state. Let us deﬁne Xt ≡ Ut − ln Kt /ρ. Because Ct ≡ ct Kt , wehave:1Ẋt = ρXt − ln ct − (Akg,t 1−α − ct − ρ).
ρ
Xt becomes constant over time in the high-growth steady state. Then, we calculate thedynamic path of and the initial value of Xt using the relaxation algorithm. As K0 is
normalized to one, we have U0∗∗ = X0 .

References[1] Agénor, P.R., 2008. Health and infrastructure in a model of endogenous growth.
Journal of Macroeconomics 30, 1407–1422.
[2] Barro, R.J., 1990. Government spending in a simple model of endogenous growth.
Journal of Political Economy 98, S103–S125.

23

[3] 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.
[4] European Commission, 2011. Public ﬁnances in EMU – 2011, European Economy No.
3, 2011.
[5] Futagami, K., Iwaisako, T., Ohdoi, R., 2008. Debt policy rule, productive government
spending, and multiple growth paths. Macroeconomic Dynamics 12, 445–462.
[6] Futagami, K., Morita, Y., Shibata, A., 1993. Dynamic analysis of an endogenous
growth model with public capital. Scandinavian Journal of Economics 95, 607–625.
[7] Ghosh, S., Mourmouras, I.A., 2004. Endogenous growth, welfare and budgetary
regimes. Journal of Macroeconomics 26, 623–635.
[8] Gómez, M.A., 2004. Optimal ﬁscal policy in a growing economy with public capital.
Macroeconomic Dynamics 8, 419–435.
[9] Greiner, A., Semmler, W., 2000. Endogenous growth, government debt and budgetary
regimes. Journal of Macroeconomics 22, 363–384.
[10] Greiner, A., 2007. An endogenous growth model with public capital and sustainable
government debt. Japanese Economic Review 58, 345–361.
[11] Greiner, A., 2008. Human capital formation, public debt and economic growth. Journal of Macroeconomics 30, 415–427.
[12] Minea, A., Villieu, P., 2012. Debt policy rule, productive government
spending, and multiple growth paths: A note. Macroeconomic Dynamics,
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[13] Rubianes, Á.C., 2010. Patterns and trends of public investment in the new member
states of the European Union. Working Paper No. 3, 2010.
[14] Trimborn, T., Koch, K.J., Steger, T.M., 2008. Multidimensional transitional dynamics: a simple numerical procedure. Macroeconomic Dynamics 12, 301–319.
[15] Turnovsky, S.J., 1997. Fiscal policy in a growing economy with public capital. Macroeconomic Dynamics 1,615–639.
[16] Yakita, A., 2008. Sustainability of public debt, public capital formation, and endogenous growth in an overlapping generations setting. Journal of Public Economics 92,
897–914.

24

Λ(kg ) : [(1 − τ )αkg − τ ]Akg1−αΠ(kg ) : ρ(kg − b̄)

∗
kg,L

∗
kg,H

0
b̄

k˜g

τ
(1−τ )αρ

Figure 11kg

(i) b̄ > xk̇g = 0(kg = b̄)ck̇g = 0(c = Akg1−α )ċ = 0ρ(1 + b̄)

0

∗
kg,L

∗
kg,H

kg

(ii) b̄ < xk̇g = 0(kg = b̄)ck̇g = 0(c = Akg1−α )ċ = 0ρ(1 + b̄)

0

∗
kg,H

∗
kg,L

τ
Figure 2-(a) b̄ = (1−τ
)α

2

kg

k̇g = 0ck̇g = 0ċ = 0

0

∗
kg,L

k̂g

b̄

τ
Figure 2-(b) b̄ < (1−τ
)α

3

∗
kg,H

kg

∗
∗
(i) kg,L< kg,H< b̄ck̇g = 0c = (1 − τ )(1 + αb̄)Akg1−αċ = 0k̇g = 0

0

∗
kg,H
k̂g

∗
kg,L

kg

b̄

∗
∗
(ii) b̄ < kg,L< kg,Hc = (1 − τ )(1 + αb̄)Akg1−αk̇g = 0cċ = 0k̇g = 00k̂g b̄

∗
kg,L

∗
kg,H

τ
Figure 2-(c) b̄ > (1−τ
)α

4

kg

(i) Policy eﬀect with respect to b̄(b̄ ↑)Λ(kg ) : [(1 − τ )αkg − τ ]Akg1−αΠ(kg ) : ρ(kg − b̄)

∗
kg,L

∗
kg,H

0

kg

(ii) Policy eﬀect with respect to τ (τ ↑)Λ(kg ) : [(1 − τ )αkg − τ ]Akg1−αΠ(kg ) : ρ(kg − b̄)

0

∗
kg,L

∗
kg,H

Figure 3

5

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0.02481(9y)
0.02191(6.5y)
0.01653(4y)
0.00911
0.05815(9.5y)
0.04785(8y)0.03688(6y)
0.02524(4y)
0.012780.06557(9y)0.05356(7.5y)
0.04090(6y)
0.02793(4y)0.014270.3
0.02261(7y)0.02412(6y)
0.02199(4.5y)
0.01692(3y)0.009460.06041(6.5y)
0.04988(5y)0.03855(4y)
0.02646(2.5y)
0.013430.06888(6y)
0.05630(5y)0.04303(4y)
0.02941(2.5y)
0.015030.4
0.02163(5.5y)
0.02389(4.5y)
0.02217(3.5y)
0.01725(2y)0.00971
0.06201(5y)
0.05129(4y)0.03971(3y)0.02729(2y)0.01388
0.07110(4.5y)
0.05815(4y)
0.04446(3y)0.03040(2y)0.015540.5
0.02104(4.5y)
0.02380(4y)0.02234(3y)
0.01750(1.5y)
0.00989
0.06314(4y)0.05229(3y)
0.04053(2.5y)
0.02788(1.5y)
0.014190.07264(4y)0.05943(3y)0.04546(2.5y)
0.03109(1.5y)
0.015900.75
0.02023(3y)0.02371(2.5y)
0.02263(2y)0.01789(1y)0.01018
0.06484(2.5y)
0.05380(2y)
0.04177(1.6y)
0.02877(1y)0.01467
0.07493(2.5y)
0.06135(2y)
0.04697(1.6y)
0.03213(1y)0.0164510.01976(2y)0.02364(2y)0.02278(1.5y)
0.01811(0.8y)
0.01034
0.06570(2y)0.05459(1.5y)
0.04243(1.3y)
0.02926(0.8y)
0.01494
0.07606(2y)
0.06232(1.5y)
0.04776(1.3y)
0.03270(0.8y)
0.01675

Table 1. Welfare eﬀects when the debt–GDP ratio falls to 50 percent.
The numbers in parentheses denote the time required to reduce the debt–GDP ratio to approximately 60 percent.

B0 /Y0
1
0.9
0.8
0.7
0.6

0.1
41%
30%
21%
13%
6%

τ
0.2
0.3
14%
8%
11% 6.5%
8%
5%
5%
3%
2.5% 1.5%

Table 2. Initial declines of gt under ϕ = 0.05.

τ B0 /Y0
0.1
1
0.9
0.8
0.7
0.6
0.2

1
0.9
0.8
0.7
0.6

0.3

1
0.9
0.8
0.7
0.6

ρ = 0.03
ρ = 0.07
ρ = 0.03
ρ = 0.07
ρ = 0.03
ρ = 0.07
ρ = 0.03
ρ = 0.07
ρ = 0.03
ρ = 0.07
ρ = 0.03
ρ = 0.07
ρ = 0.03
ρ = 0.07
ρ = 0.03
ρ = 0.07
ρ = 0.03
ρ = 0.07
ρ = 0.03
ρ = 0.07
ρ = 0.03
ρ = 0.07
ρ = 0.03
ρ = 0.07
ρ = 0.03
ρ = 0.07
ρ = 0.03
ρ = 0.07
ρ = 0.03
ρ = 0.07

0.01
0.20037
0.01863
0.15852
0.01501
0.11770.01155
0.07776
0.00801
0.03857
0.004210.23350.01576
0.18760.014310.14130.012120.09472
0.00909
0.047530.00505
0.24580.009220.19930.010150.15160.00987
0.10260.00815
0.05206
0.00491

0.025
0.144410.017290.11767
0.013900.089690.010610.06074
0.007260.03074
0.00376
0.19600.017280.15780.014860.11910.011990.079880.008610.04001
0.004610.21410.013240.17310.012310.13120.01066
0.088440.00809
0.04473
0.00456

0.05
0.10650.01516
0.09197
0.01238
0.07356
0.009540.051810.00655
0.027170.003390.18110.01779
0.14660.01487
0.11130.01168
0.075020.008170.037830.00426
0.20550.01560
0.16620.01353
0.12600.011030.08492
0.007960.04295
0.004300.1
0.08091
0.012700.076970.01084
0.06600
0.008640.04899
0.00607
0.02677
0.003190.18380.018050.15000.01495
0.11470.011630.07786
0.00805
0.03951
0.004150.21590.017190.17490.01446
0.13290.011470.08974
0.00806
0.045460.00425ϕ
0.2
0.3
0.06595 0.06034
0.01097 0.01043
0.07021 0.06826
0.00999 0.009847
0.06440 0.06461
0.00833 0.00841
0.05000 0.05112
0.00606 0.00622
0.02821 0.02921
0.00327 0.00339
0.19610.20360.01887 0.01962
0.16110.16760.01568 0.01635
0.12380.12920.01221 0.01277
0.08453 0.08838
0.00846 0.00887
0.04311 0.04517
0.00436 0.00458
0.23550.24650.01879 0.01983
0.19120.20030.01569 0.01656
0.14550.15250.01235 0.01302
0.09841 0.1032
0.00862 0.00908
0.04992 0.05240
0.00451 0.00475

Table 3. Welfare eﬀects under ρ = 0.03 and 0.07.

0.4
0.057310.010200.06734
0.009846
0.064900.00853
0.051900.006360.02985
0.00349
0.20820.020190.17170.01686
0.13250.013190.09075
0.00917
0.046430.00474
0.25320.020560.20580.017170.15680.01350
0.10620.009410.053910.00492

0.5
0.055370.01001
0.06679
0.00988
0.065140.08627
0.052440.006480.030290.00367
0.21130.020610.17440.017230.13470.01350
0.09233
0.009390.047280.004860.25770.021070.02095
0.0176
0.15960.01384
0.10810.00966
0.054910.005050.75
0.052530.00994
0.06600
0.00994
0.06552
0.00880
0.05325
0.00666
0.03094
0.003700.21580.021230.17830.017800.13790.01397
0.094640.00974
0.04851
0.00506
0.2641
0.02180.21480.018240.16370.014360.11090.01003
0.05637
0.005261
0.050910.009810.06552
0.00996
0.06569
0.00888
0.05368
0.00677
0.031290.00377
0.021800.02150
0.18040.018070.13960.014220.09586
0.009930.049170.005160.26730.022090.21750.01853
0.16590.014620.11240.010230.057160.00537

τ B0 /Y0
0.2
1
0.9
0.8
0.7
0.6
0.3
1
0.9
0.8
0.7
0.6

A = 0.1161
A = 0.1161
A = 0.1161
A = 0.1161
A = 0.1161
A = 0.1146
A = 0.1146
A = 0.1146
A = 0.1146
A = 0.1146

0.01
0.04687
0.038760.030110.020820.010810.042100.03599
0.028810.02049
0.01091

ϕ
0.025
0.05
0.1
0.2
0.3
0.4
0.5
0.75
1
0.04316 0.04046 0.04005 0.04220 0.04397 0.04518 0.04604 0.04731 0.04792
0.03528 0.03296 0.03274 0.03469 0.03623 0.03729 0.03803 0.03913 0.03969
0.02707 0.02519 0.02509 0.02671 0.02796 0.02881 0.02941 0.03031 0.03078
0.01849 0.01713 0.01709 0.01828 0.01917 0.01978 0.0202 0.02084 0.02119
0.00947 0.00873 0.00872 0.00936 0.00984 0.01016 0.01039 0.01073 0.01092
0.04116 0.04045 0.04165 0.04502 0.04733 0.04886 0.04991 0.05145 0.05218
0.03428 0.03325 0.03409 0.03685 0.03877 0.04003 0.04091 0.04220 0.04283
0.02678 0.02564 0.02617 0.02830 0.02977 0.03076 0.03144 0.03245 0.03297
0.0186 0.01759 0.01787 0.01932 0.02033 0.02101 0.02148 0.02219 0.02256
0.00969 0.00905 0.00915 0.00989 0.01042 0.01077 0.01101 0.01138 0.01158

Table 4. Welfare eﬀects when the long-run growth rate is equal to 0.02.